{"id": "2512.00318v1", "paper_link": "http://arxiv.org/abs/2512.00318v1", "theorems_cnt": 7, "theorem": {"env_name": "theorem", "content": "\\label{thm:main1}\nLet $\\mathcal {A}$ be an admissible affine arrangement in $\\mathbb R^n$ which is invariant under the action of a discrete translation subgroup $\\mathbb Z^n$ of $\\mathbb R^n$ (this does not have to be the usual embedding of $\\mathbb Z^n$). Suppose $n\\le 4$. Then $\\mathcal {A}$ is a $K(\\pi,1)$ arrangement. More generally, modulo a group theoretical conjecture on the spherical Artin group of type $D_n$ (Conjecture~\\ref{conj:dn}), $\\mathcal {A}$ is a $K(\\pi,1)$ arrangement for any $n$.", "start_pos": 28627, "end_pos": 29134, "label": "thm:main1"}, "ref_dict": {"def:admissible1": "\\begin{definition}\n \\label{def:admissible1}\nWe say an affine hyperplane arrangement $\\ca$ in $\\mathbb R^n$ is \\emph{admissible}, if at each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of the following four types:\n\\begin{enumerate}\n    \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n    \\item (type $D_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$;\n    \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n    \\item or a product of the previous types.\n\\end{enumerate}\n\\end{definition}", "thm:main1": "\\begin{theorem}\n\\label{thm:main1}\nLet $\\ca$ be an admissible affine arrangement in $\\mathbb R^n$ which is invariant under the action of a discrete translation subgroup $\\mathbb Z^n$ of $\\mathbb R^n$ (this does not have to be the usual embedding of $\\mathbb Z^n$). Suppose $n\\le 4$. Then $\\ca$ is a $K(\\pi,1)$ arrangement. More generally, modulo a group theoretical conjecture on the spherical Artin group of type $D_n$ (Conjecture~\\ref{conj:dn}), $\\ca$ is a $K(\\pi,1)$ arrangement for any $n$.\n\\end{theorem}", "subsec:deligne complex": "\\label{subsec:deligne complex}\n\n\\subsection{Falk complexes}\n\nIn \\cite{falk1995k}, for each affine arrangement in $\\mathbb C^2$ which is the complexification of a real arrangement, Falk described a loc", "conj:dn": "\\begin{conj}(Haettel)\n\\label{conj:dn}\nSuppose $\\Lambda$ is of type $D_n$ for $n\\geq 3$. Then $((\\Delta'_\\Lambda)^0,<)$ is a poset that is bowtie free and upward flag.\n\\end{conj}", "cor:AB": "\\begin{cor}\n\\label{cor:AB}\nSuppose $\\ca$ is a complete finite shape affine arrangement in $\\mathbb R^n$ such that for each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of one of the following three types:\n\\begin{enumerate}\n    \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n    \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n    \\item or a product of the previous types.\n\\end{enumerate}\nThen $(\\falk_\\ca,d_\\infty)$ is an injective metric space and $\\ca$ is a $K(\\pi,1)$ arrangement.\n\\end{cor}", "thm:main": "\\begin{thm}\n\\label{thm:main}\nSuppose $\\ca$ is a complete admissible affine arrangement in $\\mathbb R^n$ with finite shape. Suppose Conjecture~\\ref{conj:dn} holds. Then $(\\falk_\\ca,d_\\infty)$ is an injective metric space and $\\ca$ is a $K(\\pi,1)$ arrangement.\n\\end{thm}", "prop:key": "\\begin{prop}\n\\label{prop:key}\nThe poset $((\\Delta'_{\\ca(3)})^0,<)$ is bowtie free and upward flag.\n\\end{prop}", "thm:examples": "\\begin{thm}\n\\label{thm:examples}\nSuppose Conjecture~\\ref{conj:dn} holds in dimension $n$. Then $(\\falk_{\\ch_{k,n}},d_\\infty)$ and $(\\falk_{\\ck_{k,n}},d_\\infty)$ are injective metric spaces, and \n$\\ch_{k,n}$ and $\\ck_{k,n}$ are $K(\\pi,1)$ arrangements for any $k\\ge 1$. \n\nThus by Theorem~\\ref{thm:dn dim 3 and 4}, for $n=2,3,4$ and any $k\\ge 1$, the arrangements $\\ch_{k,n}$ and $\\ck_{k,n}$ are $K(\\pi,1)$ arrangements. \n\\end{thm}", "rmk:contrast": "\\begin{remark}\n\\label{rmk:contrast}\n    There is an interesting contrast between this proposition and Theorem~\\ref{thm:bowtie free An}, as the poset in Theorem~\\ref{thm:bowtie free An} is not upward flag.\n\\end{remark}"}, "pre_theorem_intro_text_len": 3961, "pre_theorem_intro_text": "Let $\\mathcal {A}$ be an \\emph{affine hyperplane arrangement} in $\\mathbb R^n$, i.e., a locally finite collection of affine hyperplanes in $\\mathbb R^n$. We consider the complex manifold which is the complement of the following collection of hyperplanes in $\\mathbb C^n$:\n$$M(\\mathcal {A})=\\mathbb C^n-\\bigcup_{H\\in \\mathcal {A}}(H\\otimes \\mathbb C).$$\nIt is an important question to understand the topology of $M(\\mathcal {A})$, see e.g.~\\cite{falk1986homotopy,falk1998homotopy}. We will be specifically interested in the asphericity of $M(\\mathcal {A})$. \nIf the manifold $M(\\mathcal {A})$ is aspherical, we call $\\mathcal {A}$ a \\emph{$K(\\pi,1)$ arrangement}. \n\nUnlike the situation of knot complements in $\\mathbb S^3$, asphericity of $M(\\mathcal {A})$ is a relatively rare phenomenon. However, there are some specific classes of $\\mathcal {A}$ where asphericity is known, for example:\n\\begin{enumerate}\n    \\item $\\mathcal {A}$ is central and simplicial by Deligne \\cite{deligne};\n    \\item $\\mathcal {A}$ is supersolvable by Terao \\cite{terao1986modular};\n    \\item $\\mathcal {A}$ is certain type of line arrangement in $\\mathbb R^2$ by Falk \\cite{falk1995k};\n    \\item $\\mathcal {A}$ is the collection of reflection hyperplanes associated with an affine Coxeter group by Paolini and Salvetti \\cite{paolini2021proof}.\n\\end{enumerate}\nThese results are obtained through different means: (1) and (4) rely heavily on Garside theory; (2) is obtained through a fibration argument; (3) uses a form of conformal non-positive curvature for 2-dimensional complexes, allowing one to compute the second homotopy group directly. \nGiven that there are relatively few methods and examples of aspherical arrangements when $n\\ge 3$, it is desirable to extend Falk's method over dimension 2, which is the goal of this article. \nIn higher dimensions, we must use a different notion of non-positive curvature in place of the conformal non-positive curvature in \\cite{falk1995k} which can only be used in dimension 2.\n\nGiven an affine arrangement $\\mathcal {A}$, an \\emph{$\\mathcal {A}$-vertex} is a point in $\\mathbb R^n$ which can be realized as intersection of elements of $\\mathcal {A}$. The \\emph{local arrangement} at an $\\mathcal {A}$-vertex $x$ is the collection of all hyperplanes in $\\mathcal {A}$ that contain $x$. An interesting feature of Falk's result, is the local-to-global phenonmenon that for certain classes of arrangements $\\mathcal {A}$, one can detect the asphericity of $M(\\mathcal {A})$ by looking at the combinatorial features of its local arrangements. Motivated by this, we consider the following class of arrangements characterized by their local arrangements.\n\\begin{definition}\n \\label{def:admissible1}\nWe say an affine hyperplane arrangement $\\mathcal {A}$ in $\\mathbb R^n$ is \\emph{admissible}, if at each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of the following four types:\n\\begin{enumerate}\n    \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n    \\item (type $D_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$;\n    \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n    \\item or a product of the previous types.\n\\end{enumerate}\n\\end{definition}\n\nNote that any affine Coxeter arrangement associated with a non-exceptional affine Coxeter group (i.e., types $\\widetilde A_n,\\widetilde B_n,\\widetilde C_n,\\widetilde D_n$) is an admissible arrangement\\footnote{We use a different description of the $\\widetilde A_n$ arrangement, where the hyperplanes are $x_i\\in \\mathbb Z$ for $1\\le i\\le n$ and $x_i-x_j\\in \\mathbb Z$ for $1\\le i\\neq j\\le n$. This does not affect the topology of $M(\\mathcal {A})$.}. Although most of the arrangements in Definition~\\ref{def:admissible1} are not Coxeter arrangements.", "context": "Let $\\mathcal {A}$ be an \\emph{affine hyperplane arrangement} in $\\mathbb R^n$, i.e., a locally finite collection of affine hyperplanes in $\\mathbb R^n$. We consider the complex manifold which is the complement of the following collection of hyperplanes in $\\mathbb C^n$:\n$$M(\\mathcal {A})=\\mathbb C^n-\\bigcup_{H\\in \\mathcal {A}}(H\\otimes \\mathbb C).$$\nIt is an important question to understand the topology of $M(\\mathcal {A})$, see e.g.~\\cite{falk1986homotopy,falk1998homotopy}. We will be specifically interested in the asphericity of $M(\\mathcal {A})$. \nIf the manifold $M(\\mathcal {A})$ is aspherical, we call $\\mathcal {A}$ a \\emph{$K(\\pi,1)$ arrangement}.\n\nUnlike the situation of knot complements in $\\mathbb S^3$, asphericity of $M(\\mathcal {A})$ is a relatively rare phenomenon. However, there are some specific classes of $\\mathcal {A}$ where asphericity is known, for example:\n\\begin{enumerate}\n \\item $\\mathcal {A}$ is central and simplicial by Deligne \\cite{deligne};\n \\item $\\mathcal {A}$ is supersolvable by Terao \\cite{terao1986modular};\n \\item $\\mathcal {A}$ is certain type of line arrangement in $\\mathbb R^2$ by Falk \\cite{falk1995k};\n \\item $\\mathcal {A}$ is the collection of reflection hyperplanes associated with an affine Coxeter group by Paolini and Salvetti \\cite{paolini2021proof}.\n\\end{enumerate}\nThese results are obtained through different means: (1) and (4) rely heavily on Garside theory; (2) is obtained through a fibration argument; (3) uses a form of conformal non-positive curvature for 2-dimensional complexes, allowing one to compute the second homotopy group directly. \nGiven that there are relatively few methods and examples of aspherical arrangements when $n\\ge 3$, it is desirable to extend Falk's method over dimension 2, which is the goal of this article. \nIn higher dimensions, we must use a different notion of non-positive curvature in place of the conformal non-positive curvature in \\cite{falk1995k} which can only be used in dimension 2.\n\nGiven an affine arrangement $\\mathcal {A}$, an \\emph{$\\mathcal {A}$-vertex} is a point in $\\mathbb R^n$ which can be realized as intersection of elements of $\\mathcal {A}$. The \\emph{local arrangement} at an $\\mathcal {A}$-vertex $x$ is the collection of all hyperplanes in $\\mathcal {A}$ that contain $x$. An interesting feature of Falk's result, is the local-to-global phenonmenon that for certain classes of arrangements $\\mathcal {A}$, one can detect the asphericity of $M(\\mathcal {A})$ by looking at the combinatorial features of its local arrangements. Motivated by this, we consider the following class of arrangements characterized by their local arrangements.\n\\begin{definition}\n \\label{def:admissible1}\nWe say an affine hyperplane arrangement $\\mathcal {A}$ in $\\mathbb R^n$ is \\emph{admissible}, if at each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of the following four types:\n\\begin{enumerate}\n \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n \\item (type $D_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$;\n \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n \\item or a product of the previous types.\n\\end{enumerate}\n\\end{definition}\n\nNote that any affine Coxeter arrangement associated with a non-exceptional affine Coxeter group (i.e., types $\\widetilde A_n,\\widetilde B_n,\\widetilde C_n,\\widetilde D_n$) is an admissible arrangement\\footnote{We use a different description of the $\\widetilde A_n$ arrangement, where the hyperplanes are $x_i\\in \\mathbb Z$ for $1\\le i\\le n$ and $x_i-x_j\\in \\mathbb Z$ for $1\\le i\\neq j\\le n$. This does not affect the topology of $M(\\mathcal {A})$.}. Although most of the arrangements in Definition~\\ref{def:admissible1} are not Coxeter arrangements.\n\n\\begin{conj}(Haettel)\n\\label{conj:dn}\nSuppose $\\Lambda$ is of type $D_n$ for $n\\geq 3$. Then $((\\Delta'_\\Lambda)^0,<)$ is a poset that is bowtie free and upward flag.\n\\end{conj}\n\n\\begin{definition}\n \\label{def:admissible1}\nWe say an affine hyperplane arrangement $\\ca$ in $\\mathbb R^n$ is \\emph{admissible}, if at each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of the following four types:\n\\begin{enumerate}\n    \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n    \\item (type $D_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$;\n    \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n    \\item or a product of the previous types.\n\\end{enumerate}\n\\end{definition}", "full_context": "Let $\\mathcal {A}$ be an \\emph{affine hyperplane arrangement} in $\\mathbb R^n$, i.e., a locally finite collection of affine hyperplanes in $\\mathbb R^n$. We consider the complex manifold which is the complement of the following collection of hyperplanes in $\\mathbb C^n$:\n$$M(\\mathcal {A})=\\mathbb C^n-\\bigcup_{H\\in \\mathcal {A}}(H\\otimes \\mathbb C).$$\nIt is an important question to understand the topology of $M(\\mathcal {A})$, see e.g.~\\cite{falk1986homotopy,falk1998homotopy}. We will be specifically interested in the asphericity of $M(\\mathcal {A})$. \nIf the manifold $M(\\mathcal {A})$ is aspherical, we call $\\mathcal {A}$ a \\emph{$K(\\pi,1)$ arrangement}.\n\nUnlike the situation of knot complements in $\\mathbb S^3$, asphericity of $M(\\mathcal {A})$ is a relatively rare phenomenon. However, there are some specific classes of $\\mathcal {A}$ where asphericity is known, for example:\n\\begin{enumerate}\n \\item $\\mathcal {A}$ is central and simplicial by Deligne \\cite{deligne};\n \\item $\\mathcal {A}$ is supersolvable by Terao \\cite{terao1986modular};\n \\item $\\mathcal {A}$ is certain type of line arrangement in $\\mathbb R^2$ by Falk \\cite{falk1995k};\n \\item $\\mathcal {A}$ is the collection of reflection hyperplanes associated with an affine Coxeter group by Paolini and Salvetti \\cite{paolini2021proof}.\n\\end{enumerate}\nThese results are obtained through different means: (1) and (4) rely heavily on Garside theory; (2) is obtained through a fibration argument; (3) uses a form of conformal non-positive curvature for 2-dimensional complexes, allowing one to compute the second homotopy group directly. \nGiven that there are relatively few methods and examples of aspherical arrangements when $n\\ge 3$, it is desirable to extend Falk's method over dimension 2, which is the goal of this article. \nIn higher dimensions, we must use a different notion of non-positive curvature in place of the conformal non-positive curvature in \\cite{falk1995k} which can only be used in dimension 2.\n\nGiven an affine arrangement $\\mathcal {A}$, an \\emph{$\\mathcal {A}$-vertex} is a point in $\\mathbb R^n$ which can be realized as intersection of elements of $\\mathcal {A}$. The \\emph{local arrangement} at an $\\mathcal {A}$-vertex $x$ is the collection of all hyperplanes in $\\mathcal {A}$ that contain $x$. An interesting feature of Falk's result, is the local-to-global phenonmenon that for certain classes of arrangements $\\mathcal {A}$, one can detect the asphericity of $M(\\mathcal {A})$ by looking at the combinatorial features of its local arrangements. Motivated by this, we consider the following class of arrangements characterized by their local arrangements.\n\\begin{definition}\n \\label{def:admissible1}\nWe say an affine hyperplane arrangement $\\mathcal {A}$ in $\\mathbb R^n$ is \\emph{admissible}, if at each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of the following four types:\n\\begin{enumerate}\n \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n \\item (type $D_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$;\n \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n \\item or a product of the previous types.\n\\end{enumerate}\n\\end{definition}\n\nNote that any affine Coxeter arrangement associated with a non-exceptional affine Coxeter group (i.e., types $\\widetilde A_n,\\widetilde B_n,\\widetilde C_n,\\widetilde D_n$) is an admissible arrangement\\footnote{We use a different description of the $\\widetilde A_n$ arrangement, where the hyperplanes are $x_i\\in \\mathbb Z$ for $1\\le i\\le n$ and $x_i-x_j\\in \\mathbb Z$ for $1\\le i\\neq j\\le n$. This does not affect the topology of $M(\\mathcal {A})$.}. Although most of the arrangements in Definition~\\ref{def:admissible1} are not Coxeter arrangements.\n\n\\begin{conj}(Haettel)\n\\label{conj:dn}\nSuppose $\\Lambda$ is of type $D_n$ for $n\\geq 3$. Then $((\\Delta'_\\Lambda)^0,<)$ is a poset that is bowtie free and upward flag.\n\\end{conj}\n\n\\begin{definition}\n \\label{def:admissible1}\nWe say an affine hyperplane arrangement $\\ca$ in $\\mathbb R^n$ is \\emph{admissible}, if at each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of the following four types:\n\\begin{enumerate}\n    \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n    \\item (type $D_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$;\n    \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n    \\item or a product of the previous types.\n\\end{enumerate}\n\\end{definition}\n\nGiven an affine arrangement $\\ca$, an \\emph{$\\ca$-vertex} is a point in $\\mathbb R^n$ which can be realized as intersection of elements of $\\ca$. The \\emph{local arrangement} at an $\\ca$-vertex $x$ is the collection of all hyperplanes in $\\ca$ that contain $x$. An interesting feature of Falk's result, is the local-to-global phenonmenon that for certain classes of arrangements $\\ca$, one can detect the asphericity of $M(\\ca)$ by looking at the combinatorial features of its local arrangements. Motivated by this, we consider the following class of arrangements characterized by their local arrangements.\n\\begin{definition}\n \\label{def:admissible1}\nWe say an affine hyperplane arrangement $\\ca$ in $\\mathbb R^n$ is \\emph{admissible}, if at each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of the following four types:\n\\begin{enumerate}\n \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n \\item (type $D_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$;\n \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n \\item or a product of the previous types.\n\\end{enumerate}\n\\end{definition}\n\nNote that any affine Coxeter arrangement associated with a non-exceptional affine Coxeter group (i.e., types $\\widetilde A_n,\\widetilde B_n,\\widetilde C_n,\\widetilde D_n$) is an admissible arrangement\\footnote{We use a different description of the $\\widetilde A_n$ arrangement, where the hyperplanes are $x_i\\in \\mathbb Z$ for $1\\le i\\le n$ and $x_i-x_j\\in \\mathbb Z$ for $1\\le i\\neq j\\le n$. This does not affect the topology of $M(\\ca)$.}. Although most of the arrangements in Definition~\\ref{def:admissible1} are not Coxeter arrangements.\n\nIn the situation of the above theorem, we have a free action of $\\mathbb Z^n$ on $M(\\ca)$. Then the fundamental group of $M(\\ca)/\\mathbb Z^n$ can be viewed as a generalization of the affine Artin groups (when $\\ca$ is an affine Coxeter arrangement, this gives a finite index subgroup of the corresponding affine Artin group).\n\n\\begin{thm}(=Theorem~\\ref{thm:examples})\nFor $n\\le 4$ and any $k\\ge 1$, the arrangements $\\ch_{k,n}$ and $\\ck_{k,n}$ are $K(\\pi,1)$ arrangements. More generally, modulo a group theoretical conjecture on the spherical Artin group of type $D_n$ (Conjecture~\\ref{conj:dn}), $\\ch_{k,n}$ and $\\ck_{k,n}$ are $K(\\pi,1)$ arrangements for any $n,k$.\n\\end{thm}\n\n\\begin{cor} \\textup{(= \\Cref{cor:AB})}\n\\label{cor:AB intro}\nSuppose $\\ca$ is a complete, finite shape, affine arrangement in $\\mathbb R^n$ such that for each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of one of the following three types:\n\\begin{enumerate}\n \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n \\item or a product of the previous types.\n\\end{enumerate}\nThen $(\\falk_\\ca,d_\\infty)$ is an injective metric space and $\\ca$ is a $K(\\pi,1)$ arrangement.\n\\end{cor}\n\nThe Coxeter diagram $\\Lambda$ of type $D_n$ (for $n\\ge 3$) is shown in Figure~\\ref{fig:dn}, where all edges are labeled by $3$. The associated reflection arrangement is $\\{x_i=\\pm x_j\\}_{1\\le i < j\\le n}$ in $\\mathbb R^n$.\nWe subdivide each edge of $\\Delta_\\Lambda$ connecting a vertex of type $\\hat s_n$ and a vertex of type $\\hat s_{n-1}$, and declare the middle point of such edge is of type $m$. Cut each top dimensional simplex in $\\Delta_\\Lambda$ into two simplices along the codimension 1 simplex spanned by vertices of type $m$ and $\\{\\hat s_i\\}_{i=1}^{n-2}$. This gives a new simplicial complex, denoted by $\\Delta'_\\Lambda$. Define a map $t$ from $(\\Delta'_\\Lambda)^0$ to $\\{1,2,\\ldots,n\\}$ by sending vertices of type $\\hat s_i$ to $i$ for $1\\le i\\le n-2$, vertices of type $m$ to $n-1$, and vertices of type $\\hat s_n$ and $\\hat s_{n-1}$ to $n$. We define a relation $<$ on $(\\Delta'_\\Lambda)^0$ as follows. For two vertices $x,y$ of $\\Delta'_\\Lambda$, $x<y$ if $x$ and $y$ are adjacent in $\\Delta'_\\Lambda$ and $t(x)<t(y)$. The simplicial complex $\\Delta'_{\\Lambda}$, together with this relation, is called the \\emph{$(s_n,s_{n-1})$-subdivision of $\\Delta_\\Lambda$}. Similarly, we can define $(s_n,s_{n-1})$-subdivision $\\bC'_\\Lambda$ of $\\bC_\\Lambda$. Note that $\\bC'_\\Lambda$ is isomorphic to the Coxeter complex of type $B_n$ via a type-preserving isomorphism. While $\\Delta'_\\Lambda$ is not isomophic to the Artin complex of type $B_n$, the following conjecture, due to Haettel, predicts that these two complexes share the following property -- his original motivation for this conjecture is that it leads an alternative proof that Artin group of type $\\widetilde D_n$ satisfies the $K(\\pi,1)$-conjecture.\n\n\\subsection{Falk complexes for admissible arrangements}\n\\label{subsec:admissible}\nFor $i=1,2$, let $\\ca_i$ be an affine hyperplane arrangement in $\\mathbb R^{m_i}$. Then the \\emph{product} of $\\ca_1$ and $\\ca_2$ is an arrangement $\\ca$ in $\\mathbb R^{m_1+m_2}$ whose hyperplanes are of form $H\\times \\mathbb R^{m_2}$ for $H\\in \\ca_1$ or $\\mathbb R^{m_1}\\times H$ for $H\\in \\ca_2$. \n\\begin{definition}\n\\label{def:admissible}\nWe say an affine hyperplane arrangement $\\ca$ in $\\mathbb R^n$ is \\emph{admissible}, if at each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of one of the following four types:\n\\begin{enumerate}\n \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n \\item (type $D_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$;\n \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n \\item or a product of the previous types.\n\\end{enumerate}\n\\end{definition}\n\nThe next result can be proved in the same way as Theorem~\\ref{thm:main}, using Proposition~\\ref{prop:key} and Theorem~\\ref{thm:Bnflag}.\n\\begin{cor}\n\\label{cor:AB}\nSuppose $\\ca$ is a complete finite shape affine arrangement in $\\mathbb R^n$ such that for each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of one of the following three types:\n\\begin{enumerate}\n \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n \\item or a product of the previous types.\n\\end{enumerate}\nThen $(\\falk_\\ca,d_\\infty)$ is an injective metric space and $\\ca$ is a $K(\\pi,1)$ arrangement.\n\\end{cor}", "post_theorem_intro_text_len": 5858, "post_theorem_intro_text": "In the situation of the above theorem, we have a free action of $\\mathbb Z^n$ on $M(\\mathcal {A})$. Then the fundamental group of $M(\\mathcal {A})/\\mathbb Z^n$ can be viewed as a generalization of the affine Artin groups (when $\\mathcal {A}$ is an affine Coxeter arrangement, this gives a finite index subgroup of the corresponding affine Artin group). \n\n\\begin{cor}\nUnder the assumption of Theorem~\\ref{thm:main1}, the manifold $M(\\mathcal {A})/\\mathbb Z^n$ is homotopy equivalent to a finite aspherical cell complex, which is the quotient of the Salvetti complex of $\\mathcal {A}$ (\\cite{s87}) by a free action of $\\mathbb Z^n$. In particular, the fundamental group of $M(\\mathcal {A})/\\mathbb Z^n$ is of type $F$.\n\\end{cor}\n\n\\Cref{thm:main1} is a special case of a more general statement which does not require $\\mathcal {A}$ to be $\\mathbb Z^n$-invariant, see \\Cref{thm:main}.\n\nNow we give more concrete examples of arrangements where Theorem~\\ref{thm:main1} (or more specifically, \\Cref{thm:main}) applies. For each dimension $n$, we will construct an infinite family of finite affine arrangements $\\ch_{k,n}$, and an infinite family of $\\mathbb Z^n$-invariant affine arrangements $\\ck_{k,n}$. These arrangements are not Coxeter arrangements, and their complexified complement do not admit iterated fibration structure in an obvious way. The finite arrangements are not simplicial. So the $K(\\pi,1)$ results for these arrangements are new. \n\n\\begin{definition}\nFor $k\\ge 1$, let $\\ch_{k,n}$ be the affine hyperplane arrangement in $\\mathbb R^n$ given by $x_i \\in \\{-2k-1, -2k+1, \\dots,-3, -1, 1, 3, \\dots, 2k - 1, 2k + 1\\}$ for $1 \\le i \\le n$ and $x_i \\pm x_j = 0$ for $1\\le i\\neq j\\le n$. This family $\\ch_{k,n}$ generalizes Falk's \\cite[Example 3.13]{falk1995k} which is neither supersolvable or simpicial.\n\nFor $k\\ge 1$, let $\\ck_{k,n}$ be the affine hyperplane arrangement in $\\mathbb R^n$ given by $x_i\\in \\mathbb Z$ for $1\\le i\\le n$, and $x_i+x_j \\in 2k\\mathbb Z+1$, $x_i-x_j \\in 2k\\mathbb Z$ for $1\\le i\\neq j\\le n$.\n\\end{definition}\n\n\\begin{thm}(=Theorem~\\ref{thm:examples})\nFor $n\\le 4$ and any $k\\ge 1$, the arrangements $\\ch_{k,n}$ and $\\ck_{k,n}$ are $K(\\pi,1)$ arrangements. More generally, modulo a group theoretical conjecture on the spherical Artin group of type $D_n$ (Conjecture~\\ref{conj:dn}), $\\ch_{k,n}$ and $\\ck_{k,n}$ are $K(\\pi,1)$ arrangements for any $n,k$.\n\\end{thm}\n\nFor $n=2$, Falk \\cite{falk1995k} constructed a (locally infinite) 2-dimensional complex $\\falk_\\mathcal {A}$ that is homotopy equivalent to the universal cover of $M(\\mathcal {A})$. This construction of Falk can be generalized to higher dimensions without too much difficulty \\cite{huang2024cycles}. Later, we show $\\falk_\\mathcal {A}$ is homotopy equivalent to the universal cover of $M(\\mathcal {A})$ whenever all the local arrangements are $K(\\pi,1)$. Thus all the above results rely on showing the complex $\\falk_\\mathcal {A}$ is contractible. \n\nRecall that a geodesic metric space $X$ is \\emph{injective} if any pairwise intersecting closed metric balls in $X$ have non-empty common intersection. For example, $\\mathbb R^n$ equipped with the $\\ell^\\infty$ metric is injective. Injective metric spaces are contractible, and they are connected to the above theorems in the following way.\n\n\\begin{thm}\n\\textup{(= \\Cref{thm:main} and \\Cref{thm:examples})}\nUnder the assumptions of any of the previous theorems, the Falk complex $\\falk_\\mathcal {A}$ admits a metric which makes it an injective metric space.\n\\end{thm}\n\nThe reason for us to consider skewed type $A_n$ arrangements as local arrangements, rather than the standard $A_n$-arrangements, is for the compatibility of arranging injective metrics on $\\falk_\\mathcal {A}$. A substantial part of the article (Section~\\ref{sec:check links}) is devoted to checking such compatibility (Proposition~\\ref{prop:key} and Remark~\\ref{rmk:contrast}). \nIt is a topic of independent interest to produce natural examples of injective metric spaces arising from group theory, and the above theorem gives many such examples. As the above theorem becomes conditional for $n\\ge 5$, we also have the following variation that is unconditional for all dimensions, where we allow the local arrangements to be a mixture of $A_n$-type and $B_n$-type.\nThe following corollary is more interesting in terms of providing new examples of injective metric spaces, rather than $K(\\pi,1)$ results, as one can prove the arrangements in the following corollary are $K(\\pi,1)$ via an iterated fibration argument.\n\n\\begin{cor} \\textup{(= \\Cref{cor:AB})}\n\\label{cor:AB intro}\nSuppose $\\mathcal {A}$ is a complete, finite shape, affine arrangement in $\\mathbb R^n$ such that for each $\\mathcal A$-vertex $x$, the local arrangement at $x$ is a translate of one of the following three types:\n\\begin{enumerate}\n    \\item (type $B_n$) $x_i\\pm x_j=0$ for $1\\le i\\neq j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n    \\item (skewed type $A_n$) $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\neq j\\le n$, or any image of this this arrangement under the $(\\bb{Z}/2\\bb{Z})^n$ action on $\\bb[n]{R}$ by reflections about the coordinate hyperplanes;\n    \\item or a product of the previous types.\n\\end{enumerate}\nThen $(\\falk_\\mathcal {A},d_\\infty)$ is an injective metric space and $\\mathcal {A}$ is a $K(\\pi,1)$ arrangement.\n\\end{cor}\n\n\\subsection*{Structure of the article} In Section~\\ref{sec:prelim} we collect some background material. In Section~\\ref{subsec:deligne complex} we define Falk complexes of affine hyperplane arrangements and prove some of them admit injective metrics, modulo a key proposition (Proposition~\\ref{prop:key}) about skewed $A_n$ arrangements. Section~\\ref{sec:check links} is devoted to this key proposition. Section~\\ref{sec:example} contains some new, concrete examples of affine arrangements where our results apply.", "sketch": "The introduction explains that the results (including Theorem~\\ref{thm:main1}) “rely on showing the complex $\\falk_\\mathcal{A}$ is contractible.” For $n=2$, Falk constructs a complex $\\falk_\\mathcal{A}$ “that is homotopy equivalent to the universal cover of $M(\\mathcal{A})$,” and this construction “can be generalized to higher dimensions.” Later they “show $\\falk_\\mathcal{A}$ is homotopy equivalent to the universal cover of $M(\\mathcal{A})$ whenever all the local arrangements are $K(\\pi,1)$,” so contractibility of $\\falk_\\mathcal{A}$ implies $M(\\mathcal{A})$ is aspherical.\n\nThey then connect contractibility to injective metric spaces: “Injective metric spaces are contractible,” and, under the assumptions of the main theorems, “the Falk complex $\\falk_\\mathcal{A}$ admits a metric which makes it an injective metric space.” Achieving this requires “compatibility of arranging injective metrics on $\\falk_\\mathcal{A}$,” motivating the use of “skewed type $A_n$ arrangements as local arrangements,” and a substantial part of the article checks this compatibility (via “Proposition~\\ref{prop:key}”).", "expanded_sketch": "The introduction explains that the results (including the main theorem) “rely on showing the complex $\\falk_\\mathcal{A}$ is contractible.” For $n=2$, Falk constructs a complex $\\falk_\\mathcal{A}$ “that is homotopy equivalent to the universal cover of $M(\\mathcal{A})$,” and this construction “can be generalized to higher dimensions.” Later they “show $\\falk_\\mathcal{A}$ is homotopy equivalent to the universal cover of $M(\\mathcal{A})$ whenever all the local arrangements are $K(\\pi,1)$,” so contractibility of $\\falk_\\mathcal{A}$ implies $M(\\mathcal{A})$ is aspherical.\n\nThey then connect contractibility to injective metric spaces: “Injective metric spaces are contractible,” and, in establishing the main theorem, “the Falk complex $\\falk_\\mathcal{A}$ admits a metric which makes it an injective metric space.” Achieving this requires “compatibility of arranging injective metrics on $\\falk_\\mathcal{A}$,” motivating the use of “skewed type $A_n$ arrangements as local arrangements,” and a substantial part of the article checks this compatibility via the following proposition.\n\n\\begin{prop}\n\\label{prop:key}\nThe poset $((\\Delta'_{\\ca(3)})^0,<)$ is bowtie free and upward flag.\n\\end{prop}", "expanded_theorem": "\\label{thm:main1}\nLet $\\mathcal {A}$ be an admissible affine arrangement in $\\mathbb R^n$ which is invariant under the action of a discrete translation subgroup $\\mathbb Z^n$ of $\\mathbb R^n$ (this does not have to be the usual embedding of $\\mathbb Z^n$). Suppose $n\\le 4$. Then $\\mathcal {A}$ is a $K(\\pi,1)$ arrangement. More generally, modulo the following group theoretical conjecture on the spherical Artin group of type $D_n$, $\\mathcal {A}$ is a $K(\\pi,1)$ arrangement for any $n$. \\begin{conj}(Haettel)\n\\label{conj:dn}\nSuppose $\\Lambda$ is of type $D_n$ for $n\\geq 3$. Then $((\\Delta'_\\Lambda)^0,<)$ is a poset that is bowtie free and upward flag.\n\\end{conj}", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let $\\mathcal A$ be an affine hyperplane arrangement in $\\mathbb R^n$, meaning a locally finite collection of affine hyperplanes. Its complexified complement is\n$$M(\\mathcal A)=\\mathbb C^n\\setminus \\bigcup_{H\\in \\mathcal A}(H\\otimes \\mathbb C).$$\nCall $\\mathcal A$ a $K(\\pi,1)$ arrangement if $M(\\mathcal A)$ is aspherical. Assume that $\\mathcal A$ is invariant under the action by translations of a discrete subgroup of $\\mathbb R^n$ isomorphic to $\\mathbb Z^n$. Also assume that $\\mathcal A$ is admissible in the following sense: for every $\\mathcal A$-vertex $x$ (a point that is an intersection of hyperplanes of $\\mathcal A$), the local arrangement at $x$ (the set of hyperplanes of $\\mathcal A$ containing $x$) is an affine translate of one of these types: \n1. type $B_n$: $x_i\\pm x_j=0$ for $1\\le i\\ne j\\le n$ and $x_i=0$ for $1\\le i\\le n$;\n2. type $D_n$: $x_i\\pm x_j=0$ for $1\\le i\\ne j\\le n$;\n3. skewed type $A_n$: $x_i=0$ for $1\\le i\\le n$ and $x_i=x_j$ for $1\\le i\\ne j\\le n$, or any image of this arrangement under coordinate-sign changes by $(\\mathbb Z/2\\mathbb Z)^n$;\n4. a product of arrangements of the previous types.\n\nUnder these hypotheses, which conclusion is valid about $\\mathcal A$ and $M(\\mathcal A)$?", "correct_choice": {"label": "A", "text": "If $n\\le 4$, then $M(\\mathcal A)$ is aspherical, so $\\mathcal A$ is a $K(\\pi,1)$ arrangement. More generally, for arbitrary $n$, the same conclusion holds provided Haettel's conjecture is assumed: for every type $D_n$ object $\\Lambda$ with $n\\ge 3$, the poset $((\\Delta'_\\Lambda)^0,<)$ is bowtie free and upward flag."}, "choices": [{"label": "B", "text": "If $n<4$, then $M(\\mathcal A)$ is aspherical, so $\\mathcal A$ is a $K(\\pi,1)$ arrangement. More generally, for arbitrary $n$, the same conclusion holds provided Haettel's conjecture is assumed only for every type $D_n$ object $\\Lambda$ with $n\\ge 4$, namely that the poset $((\\Delta'_\\Lambda)^0,<)$ is bowtie free and upward flag."}, {"label": "C", "text": "If $n\\le 4$, then $M(\\mathcal A)$ is aspherical, so $\\mathcal A$ is a $K(\\pi,1)$ arrangement."}, {"label": "D", "text": "If $n\\le 4$, then $M(\\mathcal A)$ is aspherical, so $\\mathcal A$ is a $K(\\pi,1)$ arrangement. More generally, for arbitrary $n$, for each such arrangement $\\mathcal A$ there exists a type $D_n$ object $\\Lambda$ for which, if the poset $((\\Delta'_\\Lambda)^0,<)$ is bowtie free and upward flag, then $M(\\mathcal A)$ is aspherical."}, {"label": "E", "text": "If $n\\le 4$, then the Falk complex associated with $\\mathcal A$ is contractible, so $\\mathcal A$ is a $K(\\pi,1)$ arrangement. More generally, for arbitrary $n$, the same conclusion holds provided that for every type $D_n$ object $\\Lambda$ with $n\\ge 3$, the poset $((\\Delta'_\\Lambda)^0,<)$ is bowtie free, upward flag, and itself contractible."}], "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": "dimension threshold and conjecture range", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped arbitrary-$n$ conditional extension via Haettel conjecture", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "universal conjecture requirement replaced by arrangement-dependent existence", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "bowtie free and upward flag strengthened by unnecessary contractibility of the poset", "template_used": "wildcard"}]}}
{"id": "2512.00445v1", "paper_link": "http://arxiv.org/abs/2512.00445v1", "theorems_cnt": 1, "theorem": {"env_name": "teo", "content": "\\label{maintheorem0}\nLet $T>0$. Suppose that hypothesis \\eqref{H_0} and \\eqref{H2} are satisfied. Then there exists $\\delta = \\delta(T)>0$ such that, for any initial data $y_0, z_0 \\in L^2(\\Omega)$ satisfying \n\\begin{equation}\n\\|y_0\\|_{L^2(\\Omega)} + \\|z_0\\|_{L^2(\\Omega)} < \\delta,\n\\end{equation}\nand for any $F(r,s), G(r,s) : \\mathbb{R} \\times \\mathbb{R} \\to \\mathbb{R}$, $C^1$-functions fulfilling\n\\begin{equation}\\label{BoundM}\n\\max \\left\\{ \\left|\\frac{\\partial F}{\\partial r} (r,s)\\right|, \\left|\\frac{\\partial F}{\\partial s} (r,s)\\right|,  \\left|\\frac{\\partial G}{\\partial r} (r,s)\\right|, \\left|\\frac{\\partial G}{\\partial s} (r,s)\\right|\n\\right\\} \\leq M,\n\\end{equation}\nfor all $(r,s) \\in \\mathbb{R}^2$, with $\\frac{\\partial G}{\\partial r} (r,s) \\neq 0$ in \\eqref{sistema0}, $F(0,0)=0$, $G(0,0)=0$, then there exists a control $\\nu \\in L^2(\\omega \\times (0,T))$ such that the solution of the system \\eqref{sistema0} verifies $y(x,T)=0$ and $z(x,T)=0$.", "start_pos": 9781, "end_pos": 10728, "label": "maintheorem0"}, "ref_dict": {"H_0": "\\begin{equation} \\label{H_0}\n\\overline{K}=\\sup_{t \\in  [0,T]} exp\\left(\\frac{2\\sigma^-}{t(T-t)} \\right)\\lambda^2_i(t)\\int_{-1}^1 \\vert J_i(z) \\vert^2 dz < + \\infty \n\\end{equation}", "maintheorem0": "\\begin{teo}\\label{maintheorem0}\nLet $T>0$. Suppose that hypothesis \\eqref{H_0} and \\eqref{H2} are satisfied. Then there exists $\\delta = \\delta(T)>0$ such that, for any initial data $y_0, z_0 \\in L^2(\\Omega)$ satisfying \n\\begin{equation}\n\\|y_0\\|_{L^2(\\Omega)} + \\|z_0\\|_{L^2(\\Omega)} < \\delta,\n\\end{equation}\nand for any $F(r,s), G(r,s) : \\R \\times \\R \\to \\R$, $C^1$-functions fulfilling\n\\begin{equation}\\label{BoundM}\n\\max \\left\\{ \\left|\\frac{\\partial F}{\\partial r} (r,s)\\right|, \\left|\\frac{\\partial F}{\\partial s} (r,s)\\right|,  \\left|\\frac{\\partial G}{\\partial r} (r,s)\\right|, \\left|\\frac{\\partial G}{\\partial s} (r,s)\\right|\n\\right\\} \\leq M,\n\\end{equation}\nfor all $(r,s) \\in \\R^2$, with $\\frac{\\partial G}{\\partial r} (r,s) \\neq 0$ in \\eqref{sistema0}, $F(0,0)=0$, $G(0,0)=0$, then there exists a control $\\nu \\in L^2(\\omega \\times (0,T))$ such that the solution of the system \\eqref{sistema0} verifies $y(x,T)=0$ and $z(x,T)=0$.\n\\end{teo}", "sistema0": "\\begin{equation} \\label{sistema0}\n\\begin{cases}\ny_t=a_1  y_{xx}+b_1 y_x + c_1 y +\\lambda_1(t)\\int_0^1 J_1(\\zeta-x)y(\\zeta,t) d\\zeta + F(y,z) + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nz_t=a_2  z_{xx}+b_2 z_x + c_2 z +\\lambda_2(t)\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)d\\zeta+ G(y,z), \\ \\ \\mbox{ in } Q,\\\\\ny(0,t) = y(1,t) = z(0,t) = z(1,t)  = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\ny(x,0) = y_0(x), \\  z(x,0) = z_0(x), \\mbox{ in }  \\Omega.\n\\end{cases}\n\\end{equation}", "sistema_anterior": "\\begin{equation} \\label{sistema_anterior}\n\\begin{cases}\nu_t=a_1  u_{xx}+b_1 u_x + c_1 u +\\lambda_1\\int_0^1 J_1(z-x)u(z,t) dz+\\\\\n\\qquad \\qquad \\qquad  \\qquad \\qquad \\qquad \\quad  \\qquad \\qquad \\qquad  + q_{11}u+q_{12}v + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nv_t=a_2  v_{xx}+b_2v_x +c_2v +\\lambda_2\\int_0^1J_2(z-x)v(z,t)dz+q_{21}u+q_{22}v, \\ \\ \\mbox{ in } Q,\\\\\nu(0,t) = u(1,t)=v(0,t) = v(1,t)  = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\nu(x,0) = u_0(x), \\  v(x,0) = v_0(x), \\mbox{ in }  \\Omega,\n\\end{cases}\n\\end{equation}", "H2": "\\begin{equation} \\label{H2}\n\\sup_{(x,t) \\in [0,1]\\times [0,T]} exp\\left(2s \\alpha^{-}(t) \\right)\\lambda_1^2(t)\\int_{0}^1 \\vert J_1(x-z) \\vert^2 dz < \\overline{\\delta},\n\\end{equation}"}, "pre_theorem_intro_text_len": 5900, "pre_theorem_intro_text": "\\label{sec:intro}\n\nFor any $T>0$, let us consider the spacial domain $\\Omega = (0,1)$ and $Q = (0,1) \\times (0,T)$. We study the null controllability of the following systems of semilinearly coupled equations with kernel terms,\n\\begin{equation} \\label{sistema0}\n\\begin{cases}\ny_t=a_1  y_{xx}+b_1 y_x + c_1 y +\\lambda_1(t)\\int_0^1 J_1(\\zeta-x)y(\\zeta,t) d\\zeta + F(y,z) + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nz_t=a_2  z_{xx}+b_2 z_x + c_2 z +\\lambda_2(t)\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)d\\zeta+ G(y,z), \\ \\ \\mbox{ in } Q,\\\\\ny(0,t) = y(1,t) = z(0,t) = z(1,t)  = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\ny(x,0) = y_0(x), \\  z(x,0) = z_0(x), \\mbox{ in }  \\Omega.\n\\end{cases}\n\\end{equation}\nwhere $\\nu$ is a control acting on the open set $\\omega \\times (0,T)$ with $ \\omega \\subset (0,1)$, $y$ and $z$ are state functions, and, for $i=1,2$, $\\lambda_i(t)$ are time dependent functions. In \\eqref{sistema0} $a_i$, $b_i$ and $c_i$ are real constants such that $a_i >0$, for $i = 1,2$. The precise hypothesis about the functions $F$ and $G$ will be stated in our main theorem.\n\nThe present system \\eqref{sistema0} aims to analyze the null controllability within a broader class of equations that extends the model below considered in the previous article of the authors \\cite{Limaco-Lobosco-Yapu_24}: \n\n\\begin{equation} \\label{sistema_anterior}\n\\begin{cases}\nu_t=a_1  u_{xx}+b_1 u_x + c_1 u +\\lambda_1\\int_0^1 J_1(z-x)u(z,t) dz+\\\\\n\\qquad \\qquad \\qquad  \\qquad \\qquad \\qquad \\quad  \\qquad \\qquad \\qquad  + q_{11}u+q_{12}v + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nv_t=a_2  v_{xx}+b_2v_x +c_2v +\\lambda_2\\int_0^1J_2(z-x)v(z,t)dz+q_{21}u+q_{22}v, \\ \\ \\mbox{ in } Q,\\\\\nu(0,t) = u(1,t)=v(0,t) = v(1,t)  = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\nu(x,0) = u_0(x), \\  v(x,0) = v_0(x), \\mbox{ in }  \\Omega,\n\\end{cases}\n\\end{equation}\nwhere $\\nu$ is a control acting on the open set $\\omega \\times (0,T)$, $u$ and $v$ are state functions, $\\lambda_i(t)$ are time dependent functions with $i=1,2$ and $q_{ij}$, $a_i$, $b_i$ and $c_i$, with $i,j = 1,2$, are real constants such that $a_i >0$;\n$q_{ij}$ are transition rates; $q_{ij}\\geq 0$ if $i \\neq j$; and  $q_{i1}+q_{i2}=0$. \n\nProblems of the type \\eqref{sistema_anterior} were studied, for instance, when the control $\\nu$ is not considered and \n$$J_i(x) = \\lambda_i C_i e^{-\\frac{(x-d_i)^2}{2k_i^2}},$$\nwith $\\lambda_i$, $C_i$, $d_i$ and $k_i$ are constants. In that case, we have the model proposed by \\cite{acoplado} for pricing of European, American and Butterfly options whose asset price dynamics follow the regime switching jump diffusion process. This model generalized the classical work \\cite{BS} where the Black-Scholes equation was established for derivative pricing of European options. Due to the complexity of financial markets, that generalization became necessary in order to have the ability of efficiently interpret the economic cycles and the changes in the financial time series data due to the regime shifts, as was analysed in \\cite{acoplado}. For other financial options pricing models, see the articles \\cite{BS2, BS7, BS14, FM1}. \n\nConcerning partial differential equations with nonlocal terms (kernels) there are controllability results for one partial differential equation. In fact, for the linear equation\n\\begin{equation*} \n\\begin{cases}\nu_t= u_{xx} + \\int_0^1 K(x,z,t)u(z,t) dz + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nu(0,t) = u(1,t)= 0 ,\\ \\ \\mbox{ for } 0<t<T,\\\\\nu(x,0) = u_0(x), \\mbox{ in }  \\Omega,\n\\end{cases}\n\\end{equation*}\nit is known that the equation is null controllable when the nonlocal term is independent of time and analytic \\cite{fernandez2016null}, see also \\cite{lissy2018internal} for a coupled system. For a kernel independent of time and with separable variables, the controllability was established in \\cite{micu2018local}. For one equation with kernel depending on time,  Biccari--Hernandez-Santamaría \\cite{biccari2019} showed the null controllability supposing an exponential decay at the ends of the interval $[0,T]$ such as the condition  \\eqref{H_0} below.  \nWith the same decay condition, and the smallness condition \\eqref{H2}, the authors showed in \\cite{Limaco-Lobosco-Yapu_24} the controllability of a coupled linear system.\nOn the other hand, Nina-Huaman and the first author showed in \\cite{Limaco-Dany-25} a Carleman inequality and null controllability of a parabolic PDE whose linearization about a trajectory has a non-local integral term, but with kernel $K$ identically 1. In this case no decay condition is needed.  \n\nIn the present article, we investigate whether the controllability properties established for the original system remain valid under the introduction of additional terms $F$ and $G$, thereby providing a natural generalization of the earlier results. In fact, the conditions imposed in the funcions $F$ and $G$ for the system \\eqref{sistema0} permit to include the terms of system \\eqref{sistema_anterior} that do not appear explicitly in system \\eqref{sistema0} and to introduce nonlinearities. \n\nWe consider the case where the functions $J_i\\in L^{\\infty}(-1,1)$ and $\\lambda_i\\in L^{\\infty}(0,T)$ satisfy the following condition of exponential decay at the extremity points of the interval $[0,T]$:\n\\begin{equation} \\label{H_0}\n\\overline{K}=\\sup_{t \\in  [0,T]} exp\\left(\\frac{2\\sigma^-}{t(T-t)} \\right)\\lambda^2_i(t)\\int_{-1}^1 \\vert J_i(z) \\vert^2 dz < + \\infty \n\\end{equation}\nfor $i=1,2$, where $\\sigma^-$ is a positive constant that will be defined later.   \n\nIn order to get null controllability with only one control, we need the following additional hypothesis on the decay of the first kernel:\n\\begin{equation} \\label{H2}\n\\sup_{(x,t) \\in [0,1]\\times [0,T]} exp\\left(2s \\alpha^{-}(t) \\right)\\lambda_1^2(t)\\int_{0}^1 \\vert J_1(x-z) \\vert^2 dz < \\overline{\\delta},\n\\end{equation}\nfor $\\overline{\\delta}>0$ which will be chosen small enough and $\\alpha^{-}$ will be defined later.\n\nOur main result is:", "context": "For any $T>0$, let us consider the spacial domain $\\Omega = (0,1)$ and $Q = (0,1) \\times (0,T)$. We study the null controllability of the following systems of semilinearly coupled equations with kernel terms,\n\\begin{equation} \\label{sistema0}\n\\begin{cases}\ny_t=a_1 y_{xx}+b_1 y_x + c_1 y +\\lambda_1(t)\\int_0^1 J_1(\\zeta-x)y(\\zeta,t) d\\zeta + F(y,z) + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nz_t=a_2 z_{xx}+b_2 z_x + c_2 z +\\lambda_2(t)\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)d\\zeta+ G(y,z), \\ \\ \\mbox{ in } Q,\\\\\ny(0,t) = y(1,t) = z(0,t) = z(1,t) = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\ny(x,0) = y_0(x), \\ z(x,0) = z_0(x), \\mbox{ in } \\Omega.\n\\end{cases}\n\\end{equation}\nwhere $\\nu$ is a control acting on the open set $\\omega \\times (0,T)$ with $ \\omega \\subset (0,1)$, $y$ and $z$ are state functions, and, for $i=1,2$, $\\lambda_i(t)$ are time dependent functions. In \\eqref{sistema0} $a_i$, $b_i$ and $c_i$ are real constants such that $a_i >0$, for $i = 1,2$. The precise hypothesis about the functions $F$ and $G$ will be stated in our main theorem.\n\n\\begin{equation} \\label{sistema_anterior}\n\\begin{cases}\nu_t=a_1 u_{xx}+b_1 u_x + c_1 u +\\lambda_1\\int_0^1 J_1(z-x)u(z,t) dz+\\\\\n\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\quad \\qquad \\qquad \\qquad + q_{11}u+q_{12}v + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nv_t=a_2 v_{xx}+b_2v_x +c_2v +\\lambda_2\\int_0^1J_2(z-x)v(z,t)dz+q_{21}u+q_{22}v, \\ \\ \\mbox{ in } Q,\\\\\nu(0,t) = u(1,t)=v(0,t) = v(1,t) = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\nu(x,0) = u_0(x), \\ v(x,0) = v_0(x), \\mbox{ in } \\Omega,\n\\end{cases}\n\\end{equation}\nwhere $\\nu$ is a control acting on the open set $\\omega \\times (0,T)$, $u$ and $v$ are state functions, $\\lambda_i(t)$ are time dependent functions with $i=1,2$ and $q_{ij}$, $a_i$, $b_i$ and $c_i$, with $i,j = 1,2$, are real constants such that $a_i >0$;\n$q_{ij}$ are transition rates; $q_{ij}\\geq 0$ if $i \\neq j$; and $q_{i1}+q_{i2}=0$.\n\nConcerning partial differential equations with nonlocal terms (kernels) there are controllability results for one partial differential equation. In fact, for the linear equation\n\\begin{equation*} \n\\begin{cases}\nu_t= u_{xx} + \\int_0^1 K(x,z,t)u(z,t) dz + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nu(0,t) = u(1,t)= 0 ,\\ \\ \\mbox{ for } 0<t<T,\\\\\nu(x,0) = u_0(x), \\mbox{ in } \\Omega,\n\\end{cases}\n\\end{equation*}\nit is known that the equation is null controllable when the nonlocal term is independent of time and analytic \\cite{fernandez2016null}, see also \\cite{lissy2018internal} for a coupled system. For a kernel independent of time and with separable variables, the controllability was established in \\cite{micu2018local}. For one equation with kernel depending on time, Biccari--Hernandez-Santamaría \\cite{biccari2019} showed the null controllability supposing an exponential decay at the ends of the interval $[0,T]$ such as the condition \\eqref{H_0} below. \nWith the same decay condition, and the smallness condition \\eqref{H2}, the authors showed in \\cite{Limaco-Lobosco-Yapu_24} the controllability of a coupled linear system.\nOn the other hand, Nina-Huaman and the first author showed in \\cite{Limaco-Dany-25} a Carleman inequality and null controllability of a parabolic PDE whose linearization about a trajectory has a non-local integral term, but with kernel $K$ identically 1. In this case no decay condition is needed.\n\nWe consider the case where the functions $J_i\\in L^{\\infty}(-1,1)$ and $\\lambda_i\\in L^{\\infty}(0,T)$ satisfy the following condition of exponential decay at the extremity points of the interval $[0,T]$:\n\\begin{equation} \\label{H_0}\n\\overline{K}=\\sup_{t \\in [0,T]} exp\\left(\\frac{2\\sigma^-}{t(T-t)} \\right)\\lambda^2_i(t)\\int_{-1}^1 \\vert J_i(z) \\vert^2 dz < + \\infty \n\\end{equation}\nfor $i=1,2$, where $\\sigma^-$ is a positive constant that will be defined later.\n\nIn order to get null controllability with only one control, we need the following additional hypothesis on the decay of the first kernel:\n\\begin{equation} \\label{H2}\n\\sup_{(x,t) \\in [0,1]\\times [0,T]} exp\\left(2s \\alpha^{-}(t) \\right)\\lambda_1^2(t)\\int_{0}^1 \\vert J_1(x-z) \\vert^2 dz < \\overline{\\delta},\n\\end{equation}\nfor $\\overline{\\delta}>0$ which will be chosen small enough and $\\alpha^{-}$ will be defined later.\n\nOur main result is:\n\n\\begin{equation} \\label{H_0}\n\\overline{K}=\\sup_{t \\in  [0,T]} exp\\left(\\frac{2\\sigma^-}{t(T-t)} \\right)\\lambda^2_i(t)\\int_{-1}^1 \\vert J_i(z) \\vert^2 dz < + \\infty \n\\end{equation}\n\n\\begin{equation} \\label{sistema0}\n\\begin{cases}\ny_t=a_1  y_{xx}+b_1 y_x + c_1 y +\\lambda_1(t)\\int_0^1 J_1(\\zeta-x)y(\\zeta,t) d\\zeta + F(y,z) + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nz_t=a_2  z_{xx}+b_2 z_x + c_2 z +\\lambda_2(t)\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)d\\zeta+ G(y,z), \\ \\ \\mbox{ in } Q,\\\\\ny(0,t) = y(1,t) = z(0,t) = z(1,t)  = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\ny(x,0) = y_0(x), \\  z(x,0) = z_0(x), \\mbox{ in }  \\Omega.\n\\end{cases}\n\\end{equation}", "full_context": "For any $T>0$, let us consider the spacial domain $\\Omega = (0,1)$ and $Q = (0,1) \\times (0,T)$. We study the null controllability of the following systems of semilinearly coupled equations with kernel terms,\n\\begin{equation} \\label{sistema0}\n\\begin{cases}\ny_t=a_1 y_{xx}+b_1 y_x + c_1 y +\\lambda_1(t)\\int_0^1 J_1(\\zeta-x)y(\\zeta,t) d\\zeta + F(y,z) + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nz_t=a_2 z_{xx}+b_2 z_x + c_2 z +\\lambda_2(t)\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)d\\zeta+ G(y,z), \\ \\ \\mbox{ in } Q,\\\\\ny(0,t) = y(1,t) = z(0,t) = z(1,t) = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\ny(x,0) = y_0(x), \\ z(x,0) = z_0(x), \\mbox{ in } \\Omega.\n\\end{cases}\n\\end{equation}\nwhere $\\nu$ is a control acting on the open set $\\omega \\times (0,T)$ with $ \\omega \\subset (0,1)$, $y$ and $z$ are state functions, and, for $i=1,2$, $\\lambda_i(t)$ are time dependent functions. In \\eqref{sistema0} $a_i$, $b_i$ and $c_i$ are real constants such that $a_i >0$, for $i = 1,2$. The precise hypothesis about the functions $F$ and $G$ will be stated in our main theorem.\n\n\\begin{equation} \\label{sistema_anterior}\n\\begin{cases}\nu_t=a_1 u_{xx}+b_1 u_x + c_1 u +\\lambda_1\\int_0^1 J_1(z-x)u(z,t) dz+\\\\\n\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\quad \\qquad \\qquad \\qquad + q_{11}u+q_{12}v + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nv_t=a_2 v_{xx}+b_2v_x +c_2v +\\lambda_2\\int_0^1J_2(z-x)v(z,t)dz+q_{21}u+q_{22}v, \\ \\ \\mbox{ in } Q,\\\\\nu(0,t) = u(1,t)=v(0,t) = v(1,t) = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\nu(x,0) = u_0(x), \\ v(x,0) = v_0(x), \\mbox{ in } \\Omega,\n\\end{cases}\n\\end{equation}\nwhere $\\nu$ is a control acting on the open set $\\omega \\times (0,T)$, $u$ and $v$ are state functions, $\\lambda_i(t)$ are time dependent functions with $i=1,2$ and $q_{ij}$, $a_i$, $b_i$ and $c_i$, with $i,j = 1,2$, are real constants such that $a_i >0$;\n$q_{ij}$ are transition rates; $q_{ij}\\geq 0$ if $i \\neq j$; and $q_{i1}+q_{i2}=0$.\n\nConcerning partial differential equations with nonlocal terms (kernels) there are controllability results for one partial differential equation. In fact, for the linear equation\n\\begin{equation*} \n\\begin{cases}\nu_t= u_{xx} + \\int_0^1 K(x,z,t)u(z,t) dz + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nu(0,t) = u(1,t)= 0 ,\\ \\ \\mbox{ for } 0<t<T,\\\\\nu(x,0) = u_0(x), \\mbox{ in } \\Omega,\n\\end{cases}\n\\end{equation*}\nit is known that the equation is null controllable when the nonlocal term is independent of time and analytic \\cite{fernandez2016null}, see also \\cite{lissy2018internal} for a coupled system. For a kernel independent of time and with separable variables, the controllability was established in \\cite{micu2018local}. For one equation with kernel depending on time, Biccari--Hernandez-Santamaría \\cite{biccari2019} showed the null controllability supposing an exponential decay at the ends of the interval $[0,T]$ such as the condition \\eqref{H_0} below. \nWith the same decay condition, and the smallness condition \\eqref{H2}, the authors showed in \\cite{Limaco-Lobosco-Yapu_24} the controllability of a coupled linear system.\nOn the other hand, Nina-Huaman and the first author showed in \\cite{Limaco-Dany-25} a Carleman inequality and null controllability of a parabolic PDE whose linearization about a trajectory has a non-local integral term, but with kernel $K$ identically 1. In this case no decay condition is needed.\n\nWe consider the case where the functions $J_i\\in L^{\\infty}(-1,1)$ and $\\lambda_i\\in L^{\\infty}(0,T)$ satisfy the following condition of exponential decay at the extremity points of the interval $[0,T]$:\n\\begin{equation} \\label{H_0}\n\\overline{K}=\\sup_{t \\in [0,T]} exp\\left(\\frac{2\\sigma^-}{t(T-t)} \\right)\\lambda^2_i(t)\\int_{-1}^1 \\vert J_i(z) \\vert^2 dz < + \\infty \n\\end{equation}\nfor $i=1,2$, where $\\sigma^-$ is a positive constant that will be defined later.\n\nIn order to get null controllability with only one control, we need the following additional hypothesis on the decay of the first kernel:\n\\begin{equation} \\label{H2}\n\\sup_{(x,t) \\in [0,1]\\times [0,T]} exp\\left(2s \\alpha^{-}(t) \\right)\\lambda_1^2(t)\\int_{0}^1 \\vert J_1(x-z) \\vert^2 dz < \\overline{\\delta},\n\\end{equation}\nfor $\\overline{\\delta}>0$ which will be chosen small enough and $\\alpha^{-}$ will be defined later.\n\nOur main result is:\n\n\\begin{equation} \\label{H_0}\n\\overline{K}=\\sup_{t \\in  [0,T]} exp\\left(\\frac{2\\sigma^-}{t(T-t)} \\right)\\lambda^2_i(t)\\int_{-1}^1 \\vert J_i(z) \\vert^2 dz < + \\infty \n\\end{equation}\n\n\\begin{equation} \\label{sistema0}\n\\begin{cases}\ny_t=a_1  y_{xx}+b_1 y_x + c_1 y +\\lambda_1(t)\\int_0^1 J_1(\\zeta-x)y(\\zeta,t) d\\zeta + F(y,z) + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nz_t=a_2  z_{xx}+b_2 z_x + c_2 z +\\lambda_2(t)\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)d\\zeta+ G(y,z), \\ \\ \\mbox{ in } Q,\\\\\ny(0,t) = y(1,t) = z(0,t) = z(1,t)  = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\ny(x,0) = y_0(x), \\  z(x,0) = z_0(x), \\mbox{ in }  \\Omega.\n\\end{cases}\n\\end{equation}\n\nFor any $T>0$, let us consider the spacial domain $\\Omega = (0,1)$ and $Q = (0,1) \\times (0,T)$. We study the null controllability of the following systems of semilinearly coupled equations with kernel terms,\n\\begin{equation} \\label{sistema0}\n\\begin{cases}\ny_t=a_1 y_{xx}+b_1 y_x + c_1 y +\\lambda_1(t)\\int_0^1 J_1(\\zeta-x)y(\\zeta,t) d\\zeta + F(y,z) + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nz_t=a_2 z_{xx}+b_2 z_x + c_2 z +\\lambda_2(t)\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)d\\zeta+ G(y,z), \\ \\ \\mbox{ in } Q,\\\\\ny(0,t) = y(1,t) = z(0,t) = z(1,t) = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\ny(x,0) = y_0(x), \\ z(x,0) = z_0(x), \\mbox{ in } \\Omega.\n\\end{cases}\n\\end{equation}\nwhere $\\nu$ is a control acting on the open set $\\omega \\times (0,T)$ with $ \\omega \\subset (0,1)$, $y$ and $z$ are state functions, and, for $i=1,2$, $\\lambda_i(t)$ are time dependent functions. In \\eqref{sistema0} $a_i$, $b_i$ and $c_i$ are real constants such that $a_i >0$, for $i = 1,2$. The precise hypothesis about the functions $F$ and $G$ will be stated in our main theorem.\n\nIn order to get null controllability with only one control, we need the following additional hypothesis on the decay of the first kernel:\n\\begin{equation} \\label{H2}\n\\sup_{(x,t) \\in [0,1]\\times [0,T]} exp\\left(2s \\alpha^{-}(t) \\right)\\lambda_1^2(t)\\int_{0}^1 \\vert J_1(x-z) \\vert^2 dz < \\overline{\\delta},\n\\end{equation}\nfor $\\overline{\\delta}>0$ which will be chosen small enough and $\\alpha^{-}$ will be defined later.\n\nThis paper is organized as follows. In Section~\\ref{sec:linear_case0} we formulate the linear problem associated with system~\\eqref{sistema0}. This system differs from that in~\\cite{Limaco-Lobosco-Yapu_24} as its coefficients are variable rather than constant, and the arguments from the earlier article are adapted here in detail. In Section~\\ref{sec:proof_nonlinear0} we prove the main result, Theorem~\\ref{maintheorem0}, for our system~\\eqref{sistema0}, via Kakutani’s fixed-point theorem. In Section~\\ref{sec:add_coments} we present related problems and additional remarks.\n\n\\begin{teo}\\label{theorem_linear} \nIf the hypothesis \\eqref{H_0} and \\eqref{H2} are satisfied, $\\tilde c \\in W^{2,\\infty}(Q)$ with $\\tilde c(x,t) \\neq 0$ in $\\bar \\omega \\times (0,T)$, then there exists a control $\\nu \\in L^2(\\omega \\times (0,T))$ such that the solution of the system \\eqref{eq:linearized_fixed_point} verifies $y(x,T)=z(x,T)=0$ and\n for a constant $C=C(M,\\omega,T)$ we have an estimate for the control of the form\n\\begin{equation}\\label{ControlEstimate}\n\\|\\nu\\|_{L^2(\\omega\\times(0,T))} \\leq C (\\|y_0\\|_{L^2(\\Omega)} + \\|z_0\\|_{L^2(\\Omega)}). \n\\end{equation}\n\\end{teo}\n\n\\begin{prop} \\label{carleman_um_controle0}\nConsider the adjoint system \\eqref{sistema_adj0} with $\\tilde c \\in W^{2,\\infty}(Q)$, $\\tilde c(x,t) \\neq 0$ in $\\bar \\omega \\times (0,T)$ and $\\phi_T \\in L^2(\\Omega)$, $\\psi_T \\in L^2(\\Omega)$. Moreover assume that the kernels $\\lambda_i(t) J_i(x)$, $i=1,2$, satisfy \\eqref{H_0} and suppose that the kernel $\\lambda_1(t) J_1(x)$ satisfies \\eqref{H2}. Then, there exist constants $C=C(\\omega,a_1,a_2,b_1,c_1,T,\\tilde c, \\tilde a)$, $\\kappa_1=C(\\omega, M, b_1,b_2,c_1,c_2)$ and $s_1=C(\\omega)(aT+ (aT)^2+\\overline{K}^\\frac{2}{3}T^2)$, where $\\overline{K}$ was defined in \\eqref{H_0} and $a = \\max(a_1,a_2)$, such that the solution $\\phi$, $\\psi$ of the system \\eqref{sistema_adj0} satisfy\n\\begin{equation}\\label{Carleman2_coeff33}\nI(\\phi) + I(\\psi) \\leq Cs^7 \\kappa^8\\left(\\iint_{\\omega \\times (0,T)} e^{-2s\\alpha}\\xi^7 \\vert \\phi \\vert^2dxdt \\right),\n\\end{equation}\nfor any constants $\\kappa>\\kappa_1$ and $s>s_1$.\n\n\\begin{prop}[\\cite{GlobalCarleman}]\\label{prop2.2}\nThere exist positive constants $C=C(\\Omega,\\omega,a)$, $s_1 = C(\\Omega,\\omega)(aT+ (aT)^2)$ and $\\kappa_1=C(\\Omega,\\omega)\n$ such that, for any $s>s_1$, $\\kappa > \\kappa_1$, $F \\in L^2(Q)$, $z_T\\in L^2(\\Omega)$ and $a>0$ being a positive constant, the solution of the equation \n\\begin{equation} \\label{sistema4}\n\\begin{cases}\n\\displaystyle z_t+a\\Delta z = F, \\ \\ &\\mbox{ in } \\ \\ 0 < x < 1,\\text{ } 0<t<T, \\\\\nz(0,t) = z(1,t) = 0 ,\\ \\ &\\mbox{ in } 0<t<T,\\\\\n\\displaystyle z(x,T) = z_T(x), &\\mbox{ in } 0 < x < 1,\n\\end{cases}\n\\end{equation}\nsatisfy\n\\begin{equation}\\label{Carleman11}\nI(z)\\leq C \\left[ s^3 \\kappa^4 \\iint_{\\omega\\times(0,T)}e^{-2s\\alpha}\\xi^3 \\vert z \\vert^2 dxdt + \\int_0^T \\int_0^1 e^{-2s\\alpha} \\vert F \\vert^2 dx dt \\right].\n\\end{equation}\n\\end{prop}\n\\begin{proof}\n\n\\begin{prop}\\label{Carleman1}\nLet $\\phi_T, \\psi_T \\in L^2(\\Omega)$ be given and assume that the kernels $\\lambda_i(t) J_i(x)$, $i=1,2$, satisfy \\eqref{H_0}. Then, there exist constants $C=C(\\Omega,\\omega,a_1,a_2)$, $\\kappa_1=C(\\Omega,\\omega, M, b_1,b_2,c_1,c_2)$ and $s_1=C(\\Omega,\\omega)(aT+ (aT)^2+\\overline{K}^\\frac{2}{3}T^2)$, where $\\overline{K}$ was defined in \\eqref{H_0} and $a = \\max(a_1,a_2)$, such that a solution $\\phi$, $\\psi$ of the system \\eqref{sistema_adj0} with final conditions $\\phi_T$ and $\\psi_T$, satisfy\n\\begin{equation}\n\\label{eq:carleman_dois_controles}\nI(\\phi) + I(\\psi) \\leq Cs^3 \\kappa^4\\left(\\iint_{\\omega \\times (0,T)} \\! \\! \\! \\! \\! e^{-2s\\alpha}\\xi^3 \\vert \\phi \\vert^2dxdt+\\iint_{\\omega \\times (0,T)} \\! \\! \\! \\! e^{-2s\\alpha}\\xi^3 \\vert \\psi \\vert^2dxdt\\right)\n\\end{equation}\nfor any constants $\\kappa > \\kappa_1$ and $s>s_1$.\n\\end{prop}\n\nAs a consequence of the Carleman estimate, we prove now the observability inequality.\n\\begin{prop}\\label{prop_observabilidade}\nConsider the adjoint system \\eqref{sistema_adj0} with $\\tilde c \\in W^{2,\\infty}(Q)$, $\\tilde c(x,t) \\neq 0$ in $\\bar \\omega \\times (0,T)$ and $\\phi_T \\in L^2(\\Omega)$, $\\psi_T \\in L^2(\\Omega)$. Moreover, assume that the kernels $\\lambda_i(t) J_i(x)$, $i=1,2$, satisfy \\eqref{H_0} and suppose that the kernel $\\lambda_1(t) J_1(x)$ satisfies \\eqref{H2}. Then, there exist constants $C=C(M,\\Omega,\\omega,a_1,a_2,b_1,b_2,c_1,c_2,T,\\tilde c, \\overline{K})$, where $\\overline{K}$ was defined in \\eqref{H_0}, such that the solution $\\phi$, $\\psi$ of the system \\eqref{sistema_adj0} satisfy \\begin{equation}\\label{obsin}\n \\parallel\\phi(0)\\parallel^2_{L^2(\\Omega)} + \\parallel\\psi(0)\\parallel^2_{L^2(\\Omega)} \\leq C \\iint_{\\omega \\times (0,T)} \\vert \\phi \\vert^2 dxdt.\n \\end{equation}\n\\end{prop}\n\n\\begin{equation} \\label{H_0}\n\\overline{K}=\\sup_{t \\in  [0,T]} exp\\left(\\frac{2\\sigma^-}{t(T-t)} \\right)\\lambda^2_i(t)\\int_{-1}^1 \\vert J_i(z) \\vert^2 dz < + \\infty \n\\end{equation}\n\n\\begin{teo}\\label{maintheorem0}\nLet $T>0$. Suppose that hypothesis \\eqref{H_0} and \\eqref{H2} are satisfied. Then there exists $\\delta = \\delta(T)>0$ such that, for any initial data $y_0, z_0 \\in L^2(\\Omega)$ satisfying \n\\begin{equation}\n\\|y_0\\|_{L^2(\\Omega)} + \\|z_0\\|_{L^2(\\Omega)} < \\delta,\n\\end{equation}\nand for any $F(r,s), G(r,s) : \\R \\times \\R \\to \\R$, $C^1$-functions fulfilling\n\\begin{equation}\\label{BoundM}\n\\max \\left\\{ \\left|\\frac{\\partial F}{\\partial r} (r,s)\\right|, \\left|\\frac{\\partial F}{\\partial s} (r,s)\\right|,  \\left|\\frac{\\partial G}{\\partial r} (r,s)\\right|, \\left|\\frac{\\partial G}{\\partial s} (r,s)\\right|\n\\right\\} \\leq M,\n\\end{equation}\nfor all $(r,s) \\in \\R^2$, with $\\frac{\\partial G}{\\partial r} (r,s) \\neq 0$ in \\eqref{sistema0}, $F(0,0)=0$, $G(0,0)=0$, then there exists a control $\\nu \\in L^2(\\omega \\times (0,T))$ such that the solution of the system \\eqref{sistema0} verifies $y(x,T)=0$ and $z(x,T)=0$.\n\\end{teo}\n\n\\begin{equation} \\label{sistema0}\n\\begin{cases}\ny_t=a_1  y_{xx}+b_1 y_x + c_1 y +\\lambda_1(t)\\int_0^1 J_1(\\zeta-x)y(\\zeta,t) d\\zeta + F(y,z) + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nz_t=a_2  z_{xx}+b_2 z_x + c_2 z +\\lambda_2(t)\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)d\\zeta+ G(y,z), \\ \\ \\mbox{ in } Q,\\\\\ny(0,t) = y(1,t) = z(0,t) = z(1,t)  = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\ny(x,0) = y_0(x), \\  z(x,0) = z_0(x), \\mbox{ in }  \\Omega.\n\\end{cases}\n\\end{equation}", "post_theorem_intro_text_len": 903, "post_theorem_intro_text": "This paper is organized as follows. In Section~\\ref{sec:linear_case0} we formulate the linear problem associated with system~\\eqref{sistema0}. This system differs from that in~\\cite{Limaco-Lobosco-Yapu_24} as its coefficients are variable rather than constant, and the arguments from the earlier article are adapted here in detail. In Section~\\ref{sec:proof_nonlinear0} we prove the main result, Theorem~\\ref{maintheorem0}, for our system~\\eqref{sistema0}, via Kakutani’s fixed-point theorem. In Section~\\ref{sec:add_coments} we present related problems and additional remarks.\n\nThroughout the paper, we use \\(C\\) and \\(C_i\\) (\\(i=1,2,\\ldots\\)) to denote generic positive constants whose value may change from line to line. When it is necessary to emphasize parameter dependence, we indicate it explicitly; for example, we write \\(C=C(\\omega,T)\\) to mean that \\(C\\) depends only on \\(\\omega\\) and \\(T\\).", "sketch": "In Section~\\ref{sec:proof_nonlinear0} we prove the main result, Theorem~\\ref{maintheorem0}, for our system~\\eqref{sistema0}, via Kakutani’s fixed-point theorem.", "expanded_sketch": "Next we prove the main result, To prove the main theorem, for our system\n\\begin{equation} \\label{sistema0}\n\\begin{cases}\ny_t=a_1  y_{xx}+b_1 y_x + c_1 y +\\lambda_1(t)\\int_0^1 J_1(\\zeta-x)y(\\zeta,t) d\\zeta + F(y,z) + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nz_t=a_2  z_{xx}+b_2 z_x + c_2 z +\\lambda_2(t)\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)d\\zeta+ G(y,z), \\ \\ \\mbox{ in } Q,\\\\\ny(0,t) = y(1,t) = z(0,t) = z(1,t)  = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\ny(x,0) = y_0(x), \\  z(x,0) = z_0(x), \\mbox{ in }  \\Omega.\n\\end{cases}\n\\end{equation}\nvia Kakutani’s fixed-point theorem.", "expanded_theorem": "\\label{maintheorem0}\nLet $T>0$. Suppose that hypothesis \n\\begin{equation} \\label{H_0}\n\\overline{K}=\\sup_{t \\in  [0,T]} exp\\left(\\frac{2\\sigma^-}{t(T-t)} \\right)\\lambda^2_i(t)\\int_{-1}^1 \\vert J_i(z) \\vert^2 dz < + \\infty \n\\end{equation}\nand\n\\begin{equation} \\label{H2}\n\\sup_{(x,t) \\in [0,1]\\times [0,T]} exp\\left(2s \\alpha^{-}(t) \\right)\\lambda_1^2(t)\\int_{0}^1 \\vert J_1(x-z) \\vert^2 dz < \\overline{\\delta},\n\\end{equation}\nare satisfied. Then there exists $\\delta = \\delta(T)>0$ such that, for any initial data $y_0, z_0 \\in L^2(\\Omega)$ satisfying \n\\begin{equation}\n\\|y_0\\|_{L^2(\\Omega)} + \\|z_0\\|_{L^2(\\Omega)} < \\delta,\n\\end{equation}\nand for any $F(r,s), G(r,s) : \\mathbb{R} \\times \\mathbb{R} \\to \\mathbb{R}$, $C^1$-functions fulfilling\n\\begin{equation}\\label{BoundM}\n\\max \\left\\{ \\left|\\frac{\\partial F}{\\partial r} (r,s)\\right|, \\left|\\frac{\\partial F}{\\partial s} (r,s)\\right|,  \\left|\\frac{\\partial G}{\\partial r} (r,s)\\right|, \\left|\\frac{\\partial G}{\\partial s} (r,s)\\right|\n\\right\\} \\leq M,\n\\end{equation}\nfor all $(r,s) \\in \\mathbb{R}^2$, with $\\frac{\\partial G}{\\partial r} (r,s) \\neq 0$ in\n\\begin{equation} \\label{sistema0}\n\\begin{cases}\ny_t=a_1  y_{xx}+b_1 y_x + c_1 y +\\\\lambda_1(t)\\int_0^1 J_1(\\zeta-x)y(\\zeta,t) d\\zeta + F(y,z) + \\nu 1_\\omega, \\ \\ \\mbox{ in } Q,\\\\\nz_t=a_2  z_{xx}+b_2 z_x + c_2 z +\\\\lambda_2(t)\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)d\\zeta+ G(y,z), \\ \\ \\mbox{ in } Q,\\\\\ny(0,t) = y(1,t) = z(0,t) = z(1,t)  = 0 ,\\ \\ \n\\mbox{ for } 0<t<T,\\\\\ny(x,0) = y_0(x), \\\\  z(x,0) = z_0(x), \\\\mbox{ in }  \\Omega.\n\\end{cases}\n\\end{equation}\n$F(0,0)=0$, $G(0,0)=0$, then there exists a control $\\nu \\in L^2(\\omega \\times (0,T))$ such that the solution of the system above verifies $y(x,T)=0$ and $z(x,T)=0$.", "theorem_type": ["Existential–Universal", "Implication"], "mcq": {"question": "Let T>0, Ω=(0,1), Q=Ω×(0,T), and let ω be an open subset of (0,1). Consider the semilinear coupled system\n\\[\n\\begin{cases}\ny_t=a_1 y_{xx}+b_1 y_x+c_1 y+\\lambda_1(t)\\displaystyle\\int_0^1 J_1(\\zeta-x)y(\\zeta,t)\\,d\\zeta+F(y,z)+\\nu 1_\\omega, & \\text{in }Q,\\\\\nz_t=a_2 z_{xx}+b_2 z_x+c_2 z+\\lambda_2(t)\\displaystyle\\int_0^1 J_2(\\zeta-x)z(\\zeta,t)\\,d\\zeta+G(y,z), & \\text{in }Q,\\\\\ny(0,t)=y(1,t)=z(0,t)=z(1,t)=0, & 0<t<T,\\\\\ny(x,0)=y_0(x),\\quad z(x,0)=z_0(x), & x\\in\\Omega,\n\\end{cases}\n\\]\nwhere a_i>0 and b_i,c_i are real constants. Assume the kernel hypotheses\n\\[\n\\sup_{t\\in[0,T]}\\exp\\!\\left(\\frac{2\\sigma^-}{t(T-t)}\\right)\\lambda_i(t)^2\\int_{-1}^1 |J_i(z)|^2\\,dz<\\infty\\quad (i=1,2),\n\\]\nand\n\\[\n\\sup_{(x,t)\\in[0,1]\\times[0,T]}\\exp\\!\\bigl(2s\\alpha^-(t)\\bigr)\\lambda_1(t)^2\\int_0^1 |J_1(x-z)|^2\\,dz<\\overline\\delta\n\\]\nare satisfied. Which statement holds for this system?", "correct_choice": {"label": "A", "text": "There exists \\(\\delta=\\delta(T)>0\\) such that for every initial datum \\((y_0,z_0)\\in L^2(\\Omega)\\times L^2(\\Omega)\\) with \\(\\|y_0\\|_{L^2(\\Omega)}+\\|z_0\\|_{L^2(\\Omega)}<\\delta\\), and for every pair of \\(C^1\\) functions \\(F,G:\\mathbb{R}\\times\\mathbb{R}\\to\\mathbb{R}\\) satisfying\n\\[\n\\max\\left\\{\\left|\\frac{\\partial F}{\\partial r}(r,s)\\right|,\\left|\\frac{\\partial F}{\\partial s}(r,s)\\right|,\\left|\\frac{\\partial G}{\\partial r}(r,s)\\right|,\\left|\\frac{\\partial G}{\\partial s}(r,s)\\right|\\right\\}\\le M\n\\quad\\text{for all }(r,s)\\in\\mathbb{R}^2,\n\\]\nwith \\(F(0,0)=0\\), \\(G(0,0)=0\\), and \\(\\frac{\\partial G}{\\partial r}(r,s)\\neq 0\\) for all \\((r,s)\\in\\mathbb{R}^2\\), there exists a control \\(\\nu\\in L^2(\\omega\\times(0,T))\\) such that the corresponding solution satisfies \\(y(x,T)=0\\) and \\(z(x,T)=0\\) in \\(\\Omega\\)."}, "choices": [{"label": "B", "text": "There exists \\(\\delta=\\delta(T)>0\\) such that for every initial datum \\((y_0,z_0)\\in L^2(\\Omega)\\times L^2(\\Omega)\\) with \\(\\|y_0\\|_{L^2(\\Omega)}+\\|z_0\\|_{L^2(\\Omega)}<\\delta\\), and for every pair of \\(C^1\\) functions \\(F,G:\\mathbb{R}\\times\\mathbb{R}\\to\\mathbb{R}\\) satisfying\n\\[\n\\max\\left\\{\\left|\\frac{\\partial F}{\\partial r}(r,s)\\right|,\\left|\\frac{\\partial F}{\\partial s}(r,s)\\right|,\\left|\\frac{\\partial G}{\\partial r}(r,s)\\right|,\\left|\\frac{\\partial G}{\\partial s}(r,s)\\right|\\right\\}\\le M\n\\quad\\text{for all }(r,s)\\in\\mathbb{R}^2,\n\\]\nwith \\(F(0,0)=0\\), \\(G(0,0)=0\\), there exists a control \\(\\nu\\in L^2(\\omega\\times(0,T))\\) such that the corresponding solution satisfies \\(y(x,T)=0\\) and \\(z(x,T)=0\\) in \\(\\Omega\\)."}, {"label": "C", "text": "There exists \\(\\delta=\\delta(T)>0\\) such that for every initial datum \\((y_0,z_0)\\in L^2(\\Omega)\\times L^2(\\Omega)\\) with \\(\\|y_0\\|_{L^2(\\Omega)}+\\|z_0\\|_{L^2(\\Omega)}<\\delta\\), and for every pair of \\(C^1\\) functions \\(F,G:\\mathbb{R}\\times\\mathbb{R}\\to\\mathbb{R}\\) satisfying the same derivative bound as in option A together with \\(F(0,0)=0\\), \\(G(0,0)=0\\), and \\(\\frac{\\partial G}{\\partial r}(r,s)\\neq 0\\) for all \\((r,s)\\in\\mathbb{R}^2\\), there exists a control \\(\\nu\\in L^2(\\omega\\times(0,T))\\) such that the corresponding solution satisfies \\(y(x,T)=0\\) in \\(\\Omega\\)."}, {"label": "D", "text": "For every initial datum \\((y_0,z_0)\\in L^2(\\Omega)\\times L^2(\\Omega)\\), and for every pair of \\(C^1\\) functions \\(F,G:\\mathbb{R}\\times\\mathbb{R}\\to\\mathbb{R}\\) satisfying\n\\[\n\\max\\left\\{\\left|\\frac{\\partial F}{\\partial r}(r,s)\\right|,\\left|\\frac{\\partial F}{\\partial s}(r,s)\\right|,\\left|\\frac{\\partial G}{\\partial r}(r,s)\\right|,\\left|\\frac{\\partial G}{\\partial s}(r,s)\\right|\\right\\}\\le M\n\\quad\\text{for all }(r,s)\\in\\mathbb{R}^2,\n\\]\nwith \\(F(0,0)=0\\), \\(G(0,0)=0\\), and \\(\\frac{\\partial G}{\\partial r}(r,s)\\neq 0\\) for all \\((r,s)\\in\\mathbb{R}^2\\), there exists a control \\(\\nu\\in L^2(\\omega\\times(0,T))\\) such that the corresponding solution satisfies \\(y(x,T)=0\\) and \\(z(x,T)=0\\) in \\(\\Omega\\)."}, {"label": "E", "text": "There exists \\(\\delta=\\delta(T)>0\\), depending only on \\(T\\), such that for every initial datum \\((y_0,z_0)\\in L^2(\\Omega)\\times L^2(\\Omega)\\) with \\(\\|y_0\\|_{L^2(\\Omega)}+\\|z_0\\|_{L^2(\\Omega)}<\\delta\\), and for every pair of \\(C^1\\) functions \\(F,G:\\mathbb{R}\\times\\mathbb{R}\\to\\mathbb{R}\\) satisfying\n\\[\n\\max\\left\\{\\left|\\frac{\\partial F}{\\partial r}(r,s)\\right|,\\left|\\frac{\\partial F}{\\partial s}(r,s)\\right|,\\left|\\frac{\\partial G}{\\partial r}(r,s)\\right|,\\left|\\frac{\\partial G}{\\partial s}(r,s)\\right|\\right\\}\\le M\n\\quad\\text{for all }(r,s)\\in\\mathbb{R}^2,\n\\]\nwith \\(F(0,0)=0\\), \\(G(0,0)=0\\), and \\(\\frac{\\partial G}{\\partial r}(r,s)\\neq 0\\) for all \\((r,s)\\in\\mathbb{R}^2\\), there exists a control \\(\\nu\\in L^\\infty(\\omega\\times(0,T))\\) such that the corresponding solution satisfies \\(y(x,T)=0\\) and \\(z(x,T)=0\\) in \\(\\Omega\\)."}], "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": "nondegeneracy condition on \\(\\partial G/\\partial r\\)", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped null controllability requirement for the second component at time \\(T\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "smallness of the initial data", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "control-space conclusion \\(L^2\\to L^\\infty\\)", "template_used": "stronger_trap"}]}}
{"id": "2512.00568v1", "paper_link": "http://arxiv.org/abs/2512.00568v1", "theorems_cnt": 2, "theorem": {"env_name": "lettertheorem", "content": "\\label{thm:main-theorem-intro}\n  Let $p>3$ be prime. Let $K$ be a $p$-adic field that is totally ramified over $\\Q_p$. Let $E_1$ and $E_2$ be elliptic curves defined over $K$ with full $K$-rational 2-torsion and supersingular reduction. Then for any $n>0$, there exists a WR relation $\\{P_1,P_2\\}_{K/K}=0$ in $K(K;E_1,E_2)$ of signature $(n,n)$.", "start_pos": 28636, "end_pos": 29022, "label": "thm:main-theorem-intro"}, "ref_dict": {"thm:main-theorem-intro": "\\begin{lettertheorem}\\label{thm:main-theorem-intro}\n  Let $p>3$ be prime. Let $K$ be a $p$-adic field that is totally ramified over $\\Q_p$. Let $E_1$ and $E_2$ be elliptic curves defined over $K$ with full $K$-rational 2-torsion and supersingular reduction. Then for any $n>0$, there exists a WR relation $\\{P_1,P_2\\}_{K/K}=0$ in $K(K;E_1,E_2)$ of signature $(n,n)$.\n\\end{lettertheorem}", "def:signature-of-formal-point-mod-p": "\\begin{definition}\\label{def:signature-of-formal-point-mod-p}\n  Let $K$ be a $p$-adic field with $e<p-1$. Let $E$ be an elliptic curve over $K$. The \\emph{signature} of a class $[\\a]\\in\\E(\\m)/p$ is defined as\n  \\[\n    \\sS([\\a]):=\n    \\begin{cases}\n      \\nu(\\a) &\\text{if}\\; \\a\\not\\equiv 0\\spmod{p}\\\\\n      +\\infty &\\text{otherwise}.\n    \\end{cases}\n  \\]\n\\end{definition}"}, "pre_theorem_intro_text_len": 6158, "pre_theorem_intro_text": "Let $X$ be a smooth projective variety defined over a field $K$. A fundamental invariant of $X$ is the Chow group $\\CH_0(X)$ of 0-cycles modulo rational equivalence. It is a generalization of the Picard group of curves and similarly comes equipped with an Abel-Jacobi map\n\\[\n    \\mathrm{alb_X}:F^1(X)\\rightarrow\\mathrm{Alb}_X(K),\n\\]\ncalled the \\emph{Albanese} map, where the domain $F^1(X)\\subseteq\\CH_0(X)$ is the subgroup of 0-cycles of degree zero and $\\mathrm{Alb}_X$ is an abelian variety with the same universal property the Jacobian of a curve has, namely, when $X$ possesses a $k$-rational point, there is a map $X\\rightarrow\\mathrm{Alb}_X$ through which every map from $X$ into an abelian variety factors uniquely. \n\nAs opposed to the case of curves, the Albanese map need not be injective and its kernel, denoted by $F^2(X)$, is known to vary wildly. For instance it is finite when $K$ is finite \\cite{KatoSaito1983}, but can be uncountable, when $K=\\mathbb{C}$ \\cite{Mumford1969}. When $K$ is a number field the problem of determining $F^2(X)$ remains wide open. In this direction, there are far-reaching conjectures by Beilinson and Bloch \\cite{Beilinson1985,Bloch1984} that suggest that $F^2(X)$ is finite.\n\nA historically fruitful testing ground for such conjectures is the case when $K$ is a $p$-adic field, i.e. a finite extension of $\\mathbb{Q}_p$. In this case, we have the following conjecture of Colliot-Th\\'el\\`ene, later refined by Raskind and Spiess.\n\n\\begin{letterconjecture}\\label{conj:CT}\n  (\\cite[Conjecture 1.4]{Colliot-Thelene1995}, \\cite[Conjecture 3.5.4]{RaskindSpiess2000}) Let $X$ be a smooth projective geometrically connected variety defined over a $p$-adic field $K$. Then $F^2(X)\\cong D\\oplus F$, where $D$ is a divisible group and $F$ is a finite group.\n\\end{letterconjecture}\n\nAn important advancement in this direction is the work of Saito and Sato \\cite{SaitoSato2010} where they show, under some mild technical conditions, that the larger group $F^1(X)$ is the direct sum of a finite group and a group that is divisible by every integer coprime to $p$. Thus the key unresolved aspect of Colliot-Th\\'el\\`ene's conjecture is proving that the quotients $F^2(X)/p^n$ are finite for all $n\\geq1$, along with determining the asymptotic behavior of their sizes as $n\\rightarrow\\infty$.\n\nThe full conjecture has been established only in very limited cases and for specific classes of varieties. In 2000, Raskind and Spiess proved the conjecture for varieties that are products of curves whose jacobians have a mixture of good ordinary reduction and split multiplicative reduction. Later, Gazaki and Leal were able to prove the conjecture for $X=E_1\\times E_2$ where at most one curve was allowed to have supersingular reduction \\cite{GazakiLeal2022}. The case when both curves have supersingular reduction is still unknown and is the case of study of this paper.\n\nThe starting point of the above results is to show that $F^2(E_1\\times E_2)$ is isomorphic to a certain $K$-group $K(K;E_1,E_2)$ called the \\emph{Somekawa} $K$-group \\cite{Somekawa1990}. It is known that $F^2(E_1\\times E_2)$ is generated by the 0-cycles on $E_1\\times E_2$ that come from intersecting the pullbacks of 0-cycles on $E_1$ and $E_2$. The bilinear nature of these generators yields a surjection\n\\[\n    \\bigoplus_{L/K\\,\\text{finite}}E_1(L)\\otimes_\\mathbb{Z} E_2(L) \\twoheadrightarrow F^2(E_1\\times E_2).\n\\]\nRaskind and Spiess were able to compute the kernel and show that the quotient is precisely the Somekawa $K$-group $K(K;E_1,E_2)$. This quotient is visibly generated by equivalence classes of simple tensors of the form $P_1\\otimes P_2$ with $P_i\\in E_i(L)$ as $L$ ranges over all finite extensions of $K$; such a class is called an $L/K$-\\emph{symbol} and is denoted by $\\{P_1,P_2\\}_{L/K}$.\n\nFrom this description, one can see why determining the size of $F^2(X)/p^n$ is difficult. Even showing that $F^2(X)/p$ is finite is still very hard. Indeed, each summand $E_1(L)/p\\otimes E_2(L)/p$ of $K(K;E_1,E_2)/p$ can be quite large, requiring many relations to offset the shear number of generators. Furthermore, the structure of each summand depends sensitively on both the ramification of $K$ and on the reduction type of the elliptic curves.\n\nWhen $E_1$ and $E_2$ have good reduction and the absolute ramification index $e$ of $K$ is small, i.e. $e<p-1$, it is expected that\n\\[\n    K(k;E_1,E_2)/p=0.\n\\]\nThis expectation comes from the vanishing of a certain cycle map on $K(K;E_1,E_2)/p$ \\cite[Corollary 8.1]{Gazaki2018}. So in order to prove a statement of this type, one must be able to produce enough relations in $K(K;E_1,E_2)$ to cancel the generators of the Somekawa $K$-group modulo $p$. In this paper, we focus on the ``first level'' of $K(K;E_1,E_2)/p$, that is the subgroup of $K/K$-symbols.\n\nThere are two types of relations in $K(K;E_1,E_2)$ one can draw from: the first, called the \\emph{Projection Formula} (PF), and the second, called \\emph{Weil Reciprocity} (WR). The former type of relation encodes how points on $E_1\\times E_2$, when viewed over different fields extensions, interact with each other via norm and restriction maps. In fact, these relations were a key component of the previously mentioned methods of Raskind-Spiess and Gazaki-Leal. However, when both elliptic curves have supersingular reduction, PF relations fail to produce enough cancellations.\n\nIn this paper we produce infinitely many distinct WR relations in $K(K;E_1,E_2)$. For these to be useful, we need a way to classify them. One can show that the supersingularity assumption implies that every $\\{P_1,P_2\\}_{K/K}$ will be congruent modulo $p$ to a $K/K$-symbol $\\{\\widehat{P}_1,\\widehat{P}_2\\}_{K/K}$ where $\\widehat{P}_i$ lies in the formal group $\\widehat{E}_i(\\mathfrak{m})$ (see Section \\ref{sec:ssing}). We can hence distinguish WR relations with the \\emph{signature} of $\\{P_1,P_2\\}_{K/K}$, defined to be the pair of integers $(n_1,n_2)$ where $n_i$ is the valuation of $\\widehat{ P }_i$ viewed as an element of the formal group (cf. Definition \\ref{def:signature-of-formal-point-mod-p}). The first main result of this paper is the following.", "context": "Let $X$ be a smooth projective variety defined over a field $K$. A fundamental invariant of $X$ is the Chow group $\\CH_0(X)$ of 0-cycles modulo rational equivalence. It is a generalization of the Picard group of curves and similarly comes equipped with an Abel-Jacobi map\n\\[\n \\mathrm{alb_X}:F^1(X)\\rightarrow\\mathrm{Alb}_X(K),\n\\]\ncalled the \\emph{Albanese} map, where the domain $F^1(X)\\subseteq\\CH_0(X)$ is the subgroup of 0-cycles of degree zero and $\\mathrm{Alb}_X$ is an abelian variety with the same universal property the Jacobian of a curve has, namely, when $X$ possesses a $k$-rational point, there is a map $X\\rightarrow\\mathrm{Alb}_X$ through which every map from $X$ into an abelian variety factors uniquely.\n\n\\begin{letterconjecture}\\label{conj:CT}\n (\\cite[Conjecture 1.4]{Colliot-Thelene1995}, \\cite[Conjecture 3.5.4]{RaskindSpiess2000}) Let $X$ be a smooth projective geometrically connected variety defined over a $p$-adic field $K$. Then $F^2(X)\\cong D\\oplus F$, where $D$ is a divisible group and $F$ is a finite group.\n\\end{letterconjecture}\n\nThe full conjecture has been established only in very limited cases and for specific classes of varieties. In 2000, Raskind and Spiess proved the conjecture for varieties that are products of curves whose jacobians have a mixture of good ordinary reduction and split multiplicative reduction. Later, Gazaki and Leal were able to prove the conjecture for $X=E_1\\times E_2$ where at most one curve was allowed to have supersingular reduction \\cite{GazakiLeal2022}. The case when both curves have supersingular reduction is still unknown and is the case of study of this paper.\n\nThe starting point of the above results is to show that $F^2(E_1\\times E_2)$ is isomorphic to a certain $K$-group $K(K;E_1,E_2)$ called the \\emph{Somekawa} $K$-group \\cite{Somekawa1990}. It is known that $F^2(E_1\\times E_2)$ is generated by the 0-cycles on $E_1\\times E_2$ that come from intersecting the pullbacks of 0-cycles on $E_1$ and $E_2$. The bilinear nature of these generators yields a surjection\n\\[\n \\bigoplus_{L/K\\,\\text{finite}}E_1(L)\\otimes_\\mathbb{Z} E_2(L) \\twoheadrightarrow F^2(E_1\\times E_2).\n\\]\nRaskind and Spiess were able to compute the kernel and show that the quotient is precisely the Somekawa $K$-group $K(K;E_1,E_2)$. This quotient is visibly generated by equivalence classes of simple tensors of the form $P_1\\otimes P_2$ with $P_i\\in E_i(L)$ as $L$ ranges over all finite extensions of $K$; such a class is called an $L/K$-\\emph{symbol} and is denoted by $\\{P_1,P_2\\}_{L/K}$.\n\nThere are two types of relations in $K(K;E_1,E_2)$ one can draw from: the first, called the \\emph{Projection Formula} (PF), and the second, called \\emph{Weil Reciprocity} (WR). The former type of relation encodes how points on $E_1\\times E_2$, when viewed over different fields extensions, interact with each other via norm and restriction maps. In fact, these relations were a key component of the previously mentioned methods of Raskind-Spiess and Gazaki-Leal. However, when both elliptic curves have supersingular reduction, PF relations fail to produce enough cancellations.\n\nIn this paper we produce infinitely many distinct WR relations in $K(K;E_1,E_2)$. For these to be useful, we need a way to classify them. One can show that the supersingularity assumption implies that every $\\{P_1,P_2\\}_{K/K}$ will be congruent modulo $p$ to a $K/K$-symbol $\\{\\widehat{P}_1,\\widehat{P}_2\\}_{K/K}$ where $\\widehat{P}_i$ lies in the formal group $\\widehat{E}_i(\\mathfrak{m})$ (see Section \\ref{sec:ssing}). We can hence distinguish WR relations with the \\emph{signature} of $\\{P_1,P_2\\}_{K/K}$, defined to be the pair of integers $(n_1,n_2)$ where $n_i$ is the valuation of $\\widehat{ P }_i$ viewed as an element of the formal group (cf. Definition \\ref{def:signature-of-formal-point-mod-p}). The first main result of this paper is the following.\n\n\\begin{definition}\\label{def:signature-of-formal-point-mod-p}\n  Let $K$ be a $p$-adic field with $e<p-1$. Let $E$ be an elliptic curve over $K$. The \\emph{signature} of a class $[\\a]\\in\\E(\\m)/p$ is defined as\n  \\[\n    \\sS([\\a]):=\n    \\begin{cases}\n      \\nu(\\a) &\\text{if}\\; \\a\\not\\equiv 0\\spmod{p}\\\\\n      +\\infty &\\text{otherwise}.\n    \\end{cases}\n  \\]\n\\end{definition}", "full_context": "Let $X$ be a smooth projective variety defined over a field $K$. A fundamental invariant of $X$ is the Chow group $\\CH_0(X)$ of 0-cycles modulo rational equivalence. It is a generalization of the Picard group of curves and similarly comes equipped with an Abel-Jacobi map\n\\[\n \\mathrm{alb_X}:F^1(X)\\rightarrow\\mathrm{Alb}_X(K),\n\\]\ncalled the \\emph{Albanese} map, where the domain $F^1(X)\\subseteq\\CH_0(X)$ is the subgroup of 0-cycles of degree zero and $\\mathrm{Alb}_X$ is an abelian variety with the same universal property the Jacobian of a curve has, namely, when $X$ possesses a $k$-rational point, there is a map $X\\rightarrow\\mathrm{Alb}_X$ through which every map from $X$ into an abelian variety factors uniquely.\n\n\\begin{letterconjecture}\\label{conj:CT}\n (\\cite[Conjecture 1.4]{Colliot-Thelene1995}, \\cite[Conjecture 3.5.4]{RaskindSpiess2000}) Let $X$ be a smooth projective geometrically connected variety defined over a $p$-adic field $K$. Then $F^2(X)\\cong D\\oplus F$, where $D$ is a divisible group and $F$ is a finite group.\n\\end{letterconjecture}\n\nThe full conjecture has been established only in very limited cases and for specific classes of varieties. In 2000, Raskind and Spiess proved the conjecture for varieties that are products of curves whose jacobians have a mixture of good ordinary reduction and split multiplicative reduction. Later, Gazaki and Leal were able to prove the conjecture for $X=E_1\\times E_2$ where at most one curve was allowed to have supersingular reduction \\cite{GazakiLeal2022}. The case when both curves have supersingular reduction is still unknown and is the case of study of this paper.\n\nThe starting point of the above results is to show that $F^2(E_1\\times E_2)$ is isomorphic to a certain $K$-group $K(K;E_1,E_2)$ called the \\emph{Somekawa} $K$-group \\cite{Somekawa1990}. It is known that $F^2(E_1\\times E_2)$ is generated by the 0-cycles on $E_1\\times E_2$ that come from intersecting the pullbacks of 0-cycles on $E_1$ and $E_2$. The bilinear nature of these generators yields a surjection\n\\[\n \\bigoplus_{L/K\\,\\text{finite}}E_1(L)\\otimes_\\mathbb{Z} E_2(L) \\twoheadrightarrow F^2(E_1\\times E_2).\n\\]\nRaskind and Spiess were able to compute the kernel and show that the quotient is precisely the Somekawa $K$-group $K(K;E_1,E_2)$. This quotient is visibly generated by equivalence classes of simple tensors of the form $P_1\\otimes P_2$ with $P_i\\in E_i(L)$ as $L$ ranges over all finite extensions of $K$; such a class is called an $L/K$-\\emph{symbol} and is denoted by $\\{P_1,P_2\\}_{L/K}$.\n\nThere are two types of relations in $K(K;E_1,E_2)$ one can draw from: the first, called the \\emph{Projection Formula} (PF), and the second, called \\emph{Weil Reciprocity} (WR). The former type of relation encodes how points on $E_1\\times E_2$, when viewed over different fields extensions, interact with each other via norm and restriction maps. In fact, these relations were a key component of the previously mentioned methods of Raskind-Spiess and Gazaki-Leal. However, when both elliptic curves have supersingular reduction, PF relations fail to produce enough cancellations.\n\nIn this paper we produce infinitely many distinct WR relations in $K(K;E_1,E_2)$. For these to be useful, we need a way to classify them. One can show that the supersingularity assumption implies that every $\\{P_1,P_2\\}_{K/K}$ will be congruent modulo $p$ to a $K/K$-symbol $\\{\\widehat{P}_1,\\widehat{P}_2\\}_{K/K}$ where $\\widehat{P}_i$ lies in the formal group $\\widehat{E}_i(\\mathfrak{m})$ (see Section \\ref{sec:ssing}). We can hence distinguish WR relations with the \\emph{signature} of $\\{P_1,P_2\\}_{K/K}$, defined to be the pair of integers $(n_1,n_2)$ where $n_i$ is the valuation of $\\widehat{ P }_i$ viewed as an element of the formal group (cf. Definition \\ref{def:signature-of-formal-point-mod-p}). The first main result of this paper is the following.\n\n\\begin{definition}\\label{def:signature-of-formal-point-mod-p}\n  Let $K$ be a $p$-adic field with $e<p-1$. Let $E$ be an elliptic curve over $K$. The \\emph{signature} of a class $[\\a]\\in\\E(\\m)/p$ is defined as\n  \\[\n    \\sS([\\a]):=\n    \\begin{cases}\n      \\nu(\\a) &\\text{if}\\; \\a\\not\\equiv 0\\spmod{p}\\\\\n      +\\infty &\\text{otherwise}.\n    \\end{cases}\n  \\]\n\\end{definition}\n\nIn this paper we produce infinitely many distinct WR relations in $K(K;E_1,E_2)$. For these to be useful, we need a way to classify them. One can show that the supersingularity assumption implies that every $\\{P_1,P_2\\}_{K/K}$ will be congruent modulo $p$ to a $K/K$-symbol $\\{\\widehat{P}_1,\\widehat{P}_2\\}_{K/K}$ where $\\widehat{P}_i$ lies in the formal group $\\widehat{E}_i(\\mathfrak{m})$ (see Section \\ref{sec:ssing}). We can hence distinguish WR relations with the \\emph{signature} of $\\{P_1,P_2\\}_{K/K}$, defined to be the pair of integers $(n_1,n_2)$ where $n_i$ is the valuation of $\\wh{P}_i$ viewed as an element of the formal group (cf. Definition \\ref{def:signature-of-formal-point-mod-p}). The first main result of this paper is the following.\n\nThe WR relations produced in Theorem \\ref{thm:main-theorem-intro} come from genus 2 covers of $E_1$ and $E_2$. More precisely, there is a WR relation in $K(K;E_1,E_2)$ for every curve $C$ over $K$, whose Jacobian can be mapped into $E_1\\times E_2$, and a choice of rational function $f\\in K(C)^\\times$. In this paper, we use a construction due to J. Scholten of a genus 2 curve $C$ and explicit nonconstant maps $\\f_1:C\\ra E_1$ and $\\f_2:C\\ra E_2$. Since $C$ is hyperelliptic it has particularly simple principal divisors, so the choice of rational function can be made flexibly. Armed with a suitable choice of $C,\\f_1,\\f_2$ and $f$, we can then $p$-adically approximate the resulting WR relation to determine its signature.\n\n\\begin{definition}\n Let $K$ be a $p$-adic field and let $E_1,E_2$ be elliptic curves over $K$ with supersingular reduction. Let $\\{P_1,P_2\\}_{K/K}$ be a symbol in $K(K;E_1,E_2)$, where $P_\\ell\\in E_\\ell(L)$ for $\\ell=1,2$. The \\emph{signature} of $\\{P_1,P_2\\}_{K/K}$ is defined as the pair of integers\n \\[\n \\sS(\\{P_1,P_2\\}_{K/K})=\\left(\\nu(\\wh{P}_1),\\nu(\\wh{P}_2)\\right)\n \\]\n where $\\wh{P}_\\ell\\in\\E_\\ell(\\m)$ is the associated formal point of $P_\\ell$ (see Definition \\ref{def:associated-formal-point}).\n\\end{definition}\n\n\\begin{proposition}\\label{prop:necessary-conditions-for-PhiL-0}\n Let $K$ be a $p$-adic field with absolute ramification index $e<p-1$. Let $E_1,E_2$ be elliptic curves over $K$ with supersingular reduction. Suppose we have a set of $K/K$-symbols\n \\[\n \\left\\{\\{P_{ij},Q_{i'j'}\\}_{K/K}\\in K(K;E_1,E_2)\\mid 1\\leq i,i'\\leq e,1\\leq j,j'\\leq f\\right\\}\n \\]\n such that\n \\begin{enumerate}[label=(\\roman*)]\n \\item\\label{item:1-Phi_L} $\\{P_{ij},Q_{i'j'}\\}_{K/K}\\equiv 0\\pmod{p}$ for all $i,i',j,'$.\n \\item\\label{item:2-Phi_L} $\\sS(\\{P_{ij},Q_{i'j'}\\}_{K/K})=(i,i')$,\n \\item\\label{item:3-Phi_L} Both $\\{\\wh{P}_{i1}+\\E_1(\\m^{i+1}),\\ldots,\\wh{P}_{if}+\\E_1(\\m^{i+1})\\}$ and $\\{\\wh{Q}_{i'1}+\\E_2(\\m^{i+1}),\\ldots,\\wh{Q}_{i'f}+\\E_1(\\m^{i+1})\\}$ are $\\F_p$-basis of $k\\cong\\E_\\ell(\\m^i)/\\E_\\ell(\\m^{i+1})$ ($\\ell=1,2$) for all $i$ and $i'$.\n \\end{enumerate}\n Then $\\Symb_{K/K}=0$.\n\\end{proposition}\n\n\\begin{lemma}\\label{lemma:p-3mod4-and-legendre-form}\n Suppose $K$ is a totally ramified extension of $\\Q_p$ with $p>3$. Let $E$ be an elliptic curve will full 2-torsion and supersingular reduction. Then\n \\begin{enumerate}[label=(\\roman*)]\n \\item $p\\equiv3\\pmod{4}$, in particular $-1$ is not a square in $K^\\times$.\n \\item There exists $\\l\\in\\O$ such that $\\l\\not\\equiv 0,1\\spmod{\\m}$ and such that $E$ is isomorphic over $K$ to a Legendre form elliptic curve\n \\[\n E_\\l:y^2=x(x-1)(x-\\l).\n \\]\n \\item The $j$-invariant $j(E)$ satisfies $j(E)\\not\\equiv 0\\spmod{\\m}$.\n \\end{enumerate}\n\\end{lemma}\n\\begin{proof}$\\;$\n\nThe above lemma tells us that if we have elliptic curves $E_1$ and $E_2$ over a totally ramified field $K$ with full 2-torsion and supersingular reduction, then we may assume that $E_1=E_\\l$ and $E_2=E_\\mu$ for some $\\l,\\mu\\in\\O$. The Legendre form of an elliptic curve is not unique up to isomorphism, so choosing a different form will produce a different genus 2 curve.\n\n\\begin{lemma}\\label{lemma:existence-of-good-span}\n Let $K/\\Q_p$ be a totally ramified extension with $p>3$. Let $E_1$ and $E_2$ be elliptic curves over $K$ with full 2-torsion and supersingular reduction. Then there exists a smooth genus 2 curve $C$ that spans $E_1$ and $E_2$. The curve $C$ has defining equation of the form\n \\[\n C:Y^2=u(X^2-r)(X^2-s)(X^2-t),\n \\]\n where $u,r,s,t\\in\\O^\\times$, and the spanning maps $\\s_1:C\\ra E_1$ and $\\s_2:C\\ra E_2$ have the following form\n \\[\n \\s_1(X,Y)=(u(X^2-v),uY),\\quad \\s_2(X,Y)=(ust(X^2-r)X^{-2},-urst X^{-3}Y),\n \\]\n where $v$ is a unit. Furthermore, $C$ has a $K$-rational Weierstrass point $W=(w,0)$ where $w\\in \\O^\\times$.\n\\end{lemma}\n\n\\begin{theorem}\\label{thm:main-theorem}\n Let $K$ be a totally ramified extension of $\\Q_p$, with $p>3$. Let $E_1$ and $E_2$ be elliptic curves with full 2-torsion and supersingular reduction. Then for every positive integer $N$, there exist points $P_N\\in \\E_1(K)$ and $Q_N\\in \\E_2(K)$ such that the $K/K$-symbol $\\{P_N,Q_N\\}_{K/K}\\in K(K;E_1,E_2)$ has signature $(N,N)$ and such that\n \\[\n \\{P_N,Q_N\\}_{K/K}\\equiv 0\\pmod{p}.\n \\]\n\\end{theorem}\n\n\\begin{definition}\\label{def:signature-of-formal-point-mod-p}\n  Let $K$ be a $p$-adic field with $e<p-1$. Let $E$ be an elliptic curve over $K$. The \\emph{signature} of a class $[\\a]\\in\\E(\\m)/p$ is defined as\n  \\[\n    \\sS([\\a]):=\n    \\begin{cases}\n      \\nu(\\a) &\\text{if}\\; \\a\\not\\equiv 0\\spmod{p}\\\\\n      +\\infty &\\text{otherwise}.\n    \\end{cases}\n  \\]\n\\end{definition}\n\n\\begin{lettertheorem}\\label{thm:main-theorem-intro}\n  Let $p>3$ be prime. Let $K$ be a $p$-adic field that is totally ramified over $\\Q_p$. Let $E_1$ and $E_2$ be elliptic curves defined over $K$ with full $K$-rational 2-torsion and supersingular reduction. Then for any $n>0$, there exists a WR relation $\\{P_1,P_2\\}_{K/K}=0$ in $K(K;E_1,E_2)$ of signature $(n,n)$.\n\\end{lettertheorem}", "post_theorem_intro_text_len": 1659, "post_theorem_intro_text": "The WR relations produced in Theorem \\ref{thm:main-theorem-intro} come from genus 2 covers of $E_1$ and $E_2$. More precisely, there is a WR relation in $K(K;E_1,E_2)$ for every curve $C$ over $K$, whose Jacobian can be mapped into $E_1\\times E_2$, and a choice of rational function $f\\in K(C)^\\times$. In this paper, we use a construction due to J. Scholten of a genus 2 curve $C$ and explicit nonconstant maps $\\f_1:C\\rightarrow E_1$ and $\\f_2:C\\rightarrow E_2$. Since $C$ is hyperelliptic it has particularly simple principal divisors, so the choice of rational function can be made flexibly. Armed with a suitable choice of $C,\\f_1,\\f_2$ and $f$, we can then $p$-adically approximate the resulting WR relation to determine its signature.\n\nThis paper is organized into three sections. In section \\ref{sec:prelim}, we review all the basic theory we will need about formal groups of elliptic curves, supersingular reduction, and the Somekawa $K$-group for products of elliptic curves. Along the way we define the signature of a $K/K$-symbol and determine the type and quantity of WR relations one needs to show the vanishing of all $K/K$-symbols modulo $p$. In section \\ref{sec:scholten_curves} we review the construction of the hyperelliptic curves that map to $E_1\\times E_2$, compute the WR relations they produce, and use them to prove Theorem \\ref{thm:main-theorem-intro}. Lastly, in section \\ref{sec:computations}, we focus on the case of quadratic ramified extensions $K$ and present computational evidence that the genus 2 spans of section \\ref{sec:scholten_curves} produce the necessary WR relations to show that all $K/K$-symbols vanish modulo $p$.", "sketch": "The WR relations in Theorem \\ref{thm:main-theorem-intro} are obtained from genus 2 covers of $E_1$ and $E_2$: there is a WR relation in $K(K;E_1,E_2)$ for every curve $C/K$ whose Jacobian maps into $E_1\\times E_2$, together with a choice of $f\\in K(C)^\\times$. The paper uses J. Scholten's construction of a genus 2 (hyperelliptic) curve $C$ and explicit nonconstant maps $\\f_1:C\\to E_1$ and $\\f_2:C\\to E_2$. Because $C$ is hyperelliptic, it has \"particularly simple principal divisors\", so $f$ can be chosen flexibly. With a suitable choice of $C,\\f_1,\\f_2$ and $f$, the resulting WR relation is then $p$-adically approximated \"to determine its signature\" (yielding signature $(n,n)$).", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:main-theorem-intro}\n  Let $p>3$ be prime. Let $K$ be a $p$-adic field that is totally ramified over $\\Q_p$. Let $E_1$ and $E_2$ be elliptic curves defined over $K$ with full $K$-rational 2-torsion and supersingular reduction. Then for any $n>0$, there exists a WR relation $\\{P_1,P_2\\}_{K/K}=0$ in $K(K;E_1,E_2)$ of signature $(n,n)$.", "theorem_type": ["Existence", "Universal"], "mcq": {"question": "Let p>3 be a prime, let K be a p-adic field totally ramified over \\(\\mathbb{Q}_p\\), and let \\(E_1\\) and \\(E_2\\) be elliptic curves over K with full K-rational 2-torsion and supersingular reduction. Let \\(K(K;E_1,E_2)\\) denote the Somekawa K-group, whose generators include K/K-symbols \\(\\{P_1,P_2\\}_{K/K}\\) with \\(P_i\\in E_i(K)\\). A WR relation means a vanishing relation in this group coming from Weil reciprocity. For such a K/K-symbol, its signature is the pair \\((n_1,n_2)\\) obtained from the valuations of the corresponding formal-group representatives modulo p; thus saying the symbol has signature \\((n,n)\\) means both valuation entries equal n. Which statement holds for every such choice of p, K, \\(E_1\\), \\(E_2\\), and every positive integer n?", "correct_choice": {"label": "A", "text": "There exist points \\(P_1\\in E_1(K)\\) and \\(P_2\\in E_2(K)\\) such that \\(\\{P_1,P_2\\}_{K/K}=0\\) is a WR relation in \\(K(K;E_1,E_2)\\) and the symbol \\(\\{P_1,P_2\\}_{K/K}\\) has signature \\((n,n)\\)."}, "choices": [{"label": "B", "text": "There exist points \\(P_1\\in E_1(K)\\) and \\(P_2\\in E_2(K)\\) such that \\(\\{P_1,P_2\\}_{K/K}=0\\) is a WR relation in \\(K(K;E_1,E_2)\\) and the symbol \\(\\{P_1,P_2\\}_{K/K}\\) has signature \\((n,m)\\) for every pair of positive integers \\(n,m\\)."}, {"label": "C", "text": "There exist points \\(P_1\\in E_1(K)\\) and \\(P_2\\in E_2(K)\\) such that \\(\\{P_1,P_2\\}_{K/K}=0\\) is a WR relation in \\(K(K;E_1,E_2)\\) and the symbol \\(\\{P_1,P_2\\}_{K/K}\\) has signature \\((n_1,n_2)\\) for some positive integers \\(n_1,n_2\\)."}, {"label": "D", "text": "There exists a single pair of points \\(P_1\\in E_1(K)\\) and \\(P_2\\in E_2(K)\\), independent of \\(n\\), such that for every positive integer \\(n\\) the relation \\(\\{P_1,P_2\\}_{K/K}=0\\) is a WR relation in \\(K(K;E_1,E_2)\\) and the symbol \\(\\{P_1,P_2\\}_{K/K}\\) has signature \\((n,n)\\)."}, {"label": "E", "text": "For every positive integer \\(n\\), there exist points \\(P_1\\in E_1(K)\\) and \\(P_2\\in E_2(K)\\) such that the \\(K/K\\)-symbol \\(\\{P_1,P_2\\}_{K/K}\\in K(K;E_1,E_2)\\) has signature \\((n,n)\\) and satisfies \\(\\{P_1,P_2\\}_{K/K}\\equiv 0\\pmod p\\)."}], "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": "diagonal-signature-only", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "exact prescribed signature (n,n)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "existential dependence on n", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "WR-vanishing replaced by congruence mod p", "template_used": "wildcard"}]}}
{"id": "2512.00787v1", "paper_link": "http://arxiv.org/abs/2512.00787v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{theorem: intrinsic torsion}\n  Let $E$ be an elliptic curve over $\\mathbb{Q}$.\nThen there is a pair $(A, B)$ of an abelian group $A$\nand its subgroup $B$ in the following table,\ntogether with \nan isomorphism $\\alpha : E(\\mathbb{Q})_\\Tor \\cong A$, \nsuch that $\\alpha(E(\\mathbb{Q})_\\Tor^\\is)=B$.\n  $$ \\extrarowheight3pt\n  \\begin{array}{ll} \\hline\\hline\n    A & \\text{order of}\\ B \\\\ \\hline\n    0 & 1 \\\\\n    \\mathbb{Z}/2\\mathbb{Z} & 1,2\\\\\n    \\mathbb{Z}/3\\mathbb{Z} & 1,3\\\\\n    \\mathbb{Z}/4\\mathbb{Z} & 1,2,4 \\\\\n    \\mathbb{Z}/5\\mathbb{Z} & 1,5 \\\\\n    \\mathbb{Z}/6\\mathbb{Z} & 1,2,3 \\\\\n    \\mathbb{Z}/7\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/8\\mathbb{Z} & 1,2 \\\\\n    \\mathbb{Z}/9\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/10\\mathbb{Z} & 1\\\\\n    \\mathbb{Z}/12\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/2\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1,2\\\\\n    \\mathbb{Z}/4\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1,2 \\ \\\\\n    \\mathbb{Z}/6\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/8\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1 \\\\ \\hline\\hline\n  \\end{array}\n  $$\n(For $A=\\mathbb{Z}/4\\mathbb{Z} \\times \\mathbb{Z}/2\\mathbb{Z}$,\nthere are two $\\operatorname{Aut}(A)$-conjugacy classes of order two subgroups,\nboth of which can be taken as $B$.)\n  Moreover, for each possible pair $(A, B)$ as above,\nthere are infinitely many (mutually non-isomorphic) elliptic curves $E$ over $\\mathbb{Q}$ \nfor which there is an isomorphism $\\alpha : E(\\mathbb{Q})_\\Tor \\cong A$\nsuch that $\\alpha(E(\\mathbb{Q})_\\Tor^\\is)=B$.", "start_pos": 11662, "end_pos": 12798, "label": "theorem: intrinsic torsion"}, "ref_dict": {"prop:two-def-same": "\\begin{proposition}\\label{prop:two-def-same}\nLet $v$ be a discrete valuation on $k$, and $O$ its valuation ring.\nLet $\\sX \\to \\Spec O$ be a smooth proper morphism with geometrically connected fibers,\nand denote by $X$ the generic fiber.\nThen we have\n\\[ \\Im(\\langle \\cdot, \\cdot \\rangle : \n\\CH^1(X)_\\Tor \\times A_0(X) \\to k^\\times \\otimes \\Q/\\Z)\n\\subset\n\\Im(O^\\times \\otimes \\Q/\\Z \\to k^\\times \\otimes \\Q/\\Z).\n\\]\n\\end{proposition}", "prop:TateCv": "\\begin{proposition}\\label{prop:TateCv}\n\\begin{enumerate}\n\\item \nLet $a, b \\in k^\\times$ and $m>0$.\nIf $[a],  [b] \\in E_q(k)[m]$, we have\n\\[ \\langle [a], [b] \\rangle \n= a^{-s_m([b])} \\otimes \\frac{1}{m}\n= b^{-s_m([a])} \\otimes \\frac{1}{m}\n= q^{-s_m([a])s_m([b])} \\otimes \\frac{1}{m^2}.\n\\]\n\\item\nWe have $E_q(k)_\\Tor^\\is = \\{ [\\zeta] \\mid \\zeta \\in \\mu(k) \\}$.\n\\end{enumerate}\n\\end{proposition}", "cor:kodairatype": "\\begin{corollary}\\label{cor:kodairatype}\nLet $E$ be an elliptic curve over a number field $k$,\nand $P \\in E(k)_\\Tor^\\is$ an element of the intrinsic subgroup of order $m>0$.\nLet $v$ be a finite place of $k$\nsuch that the completion $k_v$ of $k$ at $v$ satisfies $\\mu_m(k_v)= \\{ 1 \\}$.\nIf $E$ has split multiplicative reduction of Kodaira type $I_n$ at $v$,\nthen $n$ is divisible by $m^2$.\n\\end{corollary}", "eq:pairing-intro": "\\begin{equation}\\label{eq:pairing-intro}\n\\langle \\cdot , \\cdot \\rangle :\n\\Pic(X)_\\Tor \\times \\Pic(X)_\\Tor \\to k^\\times \\otimes \\Q/\\Z,\n\\end{equation}", "theorem: intrinsic torsion": "\\begin{theorem} \\label{theorem: intrinsic torsion}\n  Let $E$ be an elliptic curve over $\\Q$.\nThen there is a pair $(A, B)$ of an abelian group $A$\nand its subgroup $B$ in the following table,\ntogether with \nan isomorphism $\\alpha : E(\\Q)_\\Tor \\cong A$, \nsuch that $\\alpha(E(\\Q)_\\Tor^\\is)=B$.\n  $$ \\extrarowheight3pt\n  \\begin{array}{ll} \\hline\\hline\n    A & \\text{order of}\\ B \\\\ \\hline\n    0 & 1 \\\\\n    \\Z/2\\Z & 1,2\\\\\n    \\Z/3\\Z & 1,3\\\\\n    \\Z/4\\Z & 1,2,4 \\\\\n    \\Z/5\\Z & 1,5 \\\\\n    \\Z/6\\Z & 1,2,3 \\\\\n    \\Z/7\\Z & 1 \\\\\n    \\Z/8\\Z & 1,2 \\\\\n    \\Z/9\\Z & 1 \\\\\n    \\Z/10\\Z & 1\\\\\n    \\Z/12\\Z & 1 \\\\\n    \\Z/2\\Z\\times\\Z/2\\Z & 1,2\\\\\n    \\Z/4\\Z\\times\\Z/2\\Z & 1,2 \\ \\\\\n    \\Z/6\\Z\\times\\Z/2\\Z & 1 \\\\\n    \\Z/8\\Z\\times\\Z/2\\Z & 1 \\\\ \\hline\\hline\n  \\end{array}\n  $$\n(For $A=\\Z/4\\Z \\times \\Z/2\\Z$,\nthere are two $\\Aut(A)$-conjugacy classes of order two subgroups,\nboth of which can be taken as $B$.)\n  Moreover, for each possible pair $(A, B)$ as above,\nthere are infinitely many (mutually non-isomorphic) elliptic curves $E$ over $\\Q$ \nfor which there is an isomorphism $\\alpha : E(\\Q)_\\Tor \\cong A$\nsuch that $\\alpha(E(\\Q)_\\Tor^\\is)=B$.\n\\end{theorem}", "rem:FreyRuck": "\\begin{remark}\\label{rem:FreyRuck}\nAs is seen from the proof,\nthere is an analogous biadditive pairing\n\\[\n\\langle \\cdot, \\cdot \\rangle_n : \n\\Pic(X)[n] \\times \\Pic^0(X)/n \\to k^\\times/(k^\\times)^n\n\\]\nfor each $n>0$,\nwhich satisfies\n$\\langle d, e \\rangle_n \n=\n\\WP_n(d, e)\n\\langle e, d \\rangle_n\n$for any $d, e \\in \\Pic(X)[n]$.\nThis pairing has been constructed by Frey and R\\\"uck in \\cite{FreyRuck}\nwhen $k$ is a finite field\nand shown to be perfect if $|k| \\equiv 1 \\bmod n$.\n(This is called the \nLichtenbaum-Tate pairing in some literature,\nsee e.g. \\cite[XI.9]{sil2}.)\n\\end{remark}", "proposition: pairing 1-mod": "\\begin{theorem}\\label{proposition: pairing 1-mod}\n  Assume that $N\\in\\{4,5,6,7,8,9,10,12\\}$ and\n  let $M$ be a positive divisor of $N$.\n  Denote by $X_1(MN,N)$ the modular curve over $\\Q$ \n  associated to the congruence subgroup\n    $$\n    \\Gamma_1(MN,N):= \\Gamma_0(MN) \\cap \\Gamma_1(N)\n    =\\left\\{\\M abcd\\in\\SL(2,\\Z) : \\ \n      a,d\\equiv1\\bmod N, \\ MN|c \\right\\},\n    $$\n  on which the cusp $0$ is $\\Q$-rational.\n  Then $X_1(N)_M^+$ is isomorphic to $X_1(MN,N)$ over $\\Q$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2339, "pre_theorem_intro_text": "Let $X$ be a geometrically irreducible smooth projective curve\nover a field $k$. In \\S \\ref{sect:pairing},\nwe shall construct a biadditive symmetric pairing\n\\begin{equation}\\label{eq:pairing-intro}\n\\langle \\cdot , \\cdot \\rangle :\n\\operatorname{Pic}(X)_\\Tor \\times \\operatorname{Pic}(X)_\\Tor \\to k^\\times \\otimes \\mathbb{Q}/\\mathbb{Z},\n\\end{equation}\nwhere $\\operatorname{Pic}(X)_\\Tor$ is the \ntorsion part of the Picard group $\\operatorname{Pic}(X)$ of $X$.\nWe then define the \\emph{intrinsic subgroup}\nof $\\operatorname{Pic}(X)_\\Tor$ by\n\\[\n\\operatorname{Pic}(X)_\\Tor^\\is \n:= \\{ a \\in \\operatorname{Pic}(X)_\\Tor \\mid \\langle a, b \\rangle = 0\n\\text{ for all } b \\in \\operatorname{Pic}(X)_\\Tor \\}.\n\\]\n\nWe remark that \\eqref{eq:pairing-intro} is not entirely new.\nIndeed, its finite coefficient analogue \n(with values in $k^\\times/(k^\\times)^m$)  is considered by Frey and R\\\"uck in \\cite{FreyRuck}\n(see Remark \\ref{rem:FreyRuck}).\nThe first and second authors encountered with\na disguised version of \\eqref{eq:pairing-intro} in \\cite{YY}\n(see Remark \\ref{sect:genJac}).\nObviously, \\eqref{eq:pairing-intro} is trivial if $k^\\times \\otimes \\mathbb{Q}/\\mathbb{Z}=0$,\ne.g. when $k$ is finite, algebraically closed, or $k=\\mathbb{R}$.\nOn the other hand,\n$\\operatorname{Pic}(X)_\\Tor^\\is$ appears to be a non-trivial new invariant\nif $k$ is a global field or a $p$-adic field.\n\nWe will mainly consider the case where $X=E$ is an elliptic curve\nso that we may identify $E(k)_\\Tor= \\operatorname{Pic}(E)_\\Tor$\nby $P \\mapsto [P-O]$, where $O \\in E(k)$ is the identity element.\nThus we write \n$\\langle P,Q\\rangle:=\\langle [P-O], [Q-O]\\rangle$ for $P, Q \\in E(k)_\\Tor$\nand \n$E(k)_\\Tor^\\is:=\\operatorname{Pic}(E)_\\Tor^\\is$.\nOur main result is a classification of possible structures of\n$(E(\\mathbb{Q})_\\Tor,E(\\mathbb{Q})_\\Tor^\\is)$ for an elliptic curve $E$ over $\\mathbb{Q}$.\nA celebrated theorem of Mazur \\cite[Theorem (8)]{Mazur} states\nthat $E(\\mathbb{Q})_\\Tor$ is either cyclic of order\n$1, \\ldots,10$, or $12$, or is isomorphic to $(\\mathbb{Z}/2N\\mathbb{Z}) \\times (\\mathbb{Z}/2\\mathbb{Z})$,\n$N=1, 2,3,4$,\nand all cases are realized by infinitely many (mutually non-isomorphic) elliptic curves.\nThe classification of\n$E(\\mathbb{Q})_\\Tor^\\is$ is given in the following theorem,\nwhose proof will occupy \\S \\ref{sect:ref-mazur}--\\ref{sect:ref-mazur2}.", "context": "Let $X$ be a geometrically irreducible smooth projective curve\nover a field $k$. In \\S \\ref{sect:pairing},\nwe shall construct a biadditive symmetric pairing\n\\begin{equation}\\label{eq:pairing-intro}\n\\langle \\cdot , \\cdot \\rangle :\n\\operatorname{Pic}(X)_\\Tor \\times \\operatorname{Pic}(X)_\\Tor \\to k^\\times \\otimes \\mathbb{Q}/\\mathbb{Z},\n\\end{equation}\nwhere $\\operatorname{Pic}(X)_\\Tor$ is the \ntorsion part of the Picard group $\\operatorname{Pic}(X)$ of $X$.\nWe then define the \\emph{intrinsic subgroup}\nof $\\operatorname{Pic}(X)_\\Tor$ by\n\\[\n\\operatorname{Pic}(X)_\\Tor^\\is \n:= \\{ a \\in \\operatorname{Pic}(X)_\\Tor \\mid \\langle a, b \\rangle = 0\n\\text{ for all } b \\in \\operatorname{Pic}(X)_\\Tor \\}.\n\\]\n\nWe remark that \\eqref{eq:pairing-intro} is not entirely new.\nIndeed, its finite coefficient analogue \n(with values in $k^\\times/(k^\\times)^m$) is considered by Frey and R\\\"uck in \\cite{FreyRuck}\n(see Remark \\ref{rem:FreyRuck}).\nThe first and second authors encountered with\na disguised version of \\eqref{eq:pairing-intro} in \\cite{YY}\n(see Remark \\ref{sect:genJac}).\nObviously, \\eqref{eq:pairing-intro} is trivial if $k^\\times \\otimes \\mathbb{Q}/\\mathbb{Z}=0$,\ne.g. when $k$ is finite, algebraically closed, or $k=\\mathbb{R}$.\nOn the other hand,\n$\\operatorname{Pic}(X)_\\Tor^\\is$ appears to be a non-trivial new invariant\nif $k$ is a global field or a $p$-adic field.\n\nWe will mainly consider the case where $X=E$ is an elliptic curve\nso that we may identify $E(k)_\\Tor= \\operatorname{Pic}(E)_\\Tor$\nby $P \\mapsto [P-O]$, where $O \\in E(k)$ is the identity element.\nThus we write \n$\\langle P,Q\\rangle:=\\langle [P-O], [Q-O]\\rangle$ for $P, Q \\in E(k)_\\Tor$\nand \n$E(k)_\\Tor^\\is:=\\operatorname{Pic}(E)_\\Tor^\\is$.\nOur main result is a classification of possible structures of\n$(E(\\mathbb{Q})_\\Tor,E(\\mathbb{Q})_\\Tor^\\is)$ for an elliptic curve $E$ over $\\mathbb{Q}$.\nA celebrated theorem of Mazur \\cite[Theorem (8)]{Mazur} states\nthat $E(\\mathbb{Q})_\\Tor$ is either cyclic of order\n$1, \\ldots,10$, or $12$, or is isomorphic to $(\\mathbb{Z}/2N\\mathbb{Z}) \\times (\\mathbb{Z}/2\\mathbb{Z})$,\n$N=1, 2,3,4$,\nand all cases are realized by infinitely many (mutually non-isomorphic) elliptic curves.\nThe classification of\n$E(\\mathbb{Q})_\\Tor^\\is$ is given in the following theorem,\nwhose proof will occupy \\S \\ref{sect:ref-mazur}--\\ref{sect:ref-mazur2}.\n\n\\begin{equation}\\label{eq:pairing-intro}\n\\langle \\cdot , \\cdot \\rangle :\n\\Pic(X)_\\Tor \\times \\Pic(X)_\\Tor \\to k^\\times \\otimes \\Q/\\Z,\n\\end{equation}\n\n\\begin{remark}\\label{rem:FreyRuck}\nAs is seen from the proof,\nthere is an analogous biadditive pairing\n\\[\n\\langle \\cdot, \\cdot \\rangle_n : \n\\Pic(X)[n] \\times \\Pic^0(X)/n \\to k^\\times/(k^\\times)^n\n\\]\nfor each $n>0$,\nwhich satisfies\n$\\langle d, e \\rangle_n \n=\n\\WP_n(d, e)\n\\langle e, d \\rangle_n\n$for any $d, e \\in \\Pic(X)[n]$.\nThis pairing has been constructed by Frey and R\\\"uck in \\cite{FreyRuck}\nwhen $k$ is a finite field\nand shown to be perfect if $|k| \\equiv 1 \\bmod n$.\n(This is called the \nLichtenbaum-Tate pairing in some literature,\nsee e.g. \\cite[XI.9]{sil2}.)\n\\end{remark}", "full_context": "Let $X$ be a geometrically irreducible smooth projective curve\nover a field $k$. In \\S \\ref{sect:pairing},\nwe shall construct a biadditive symmetric pairing\n\\begin{equation}\\label{eq:pairing-intro}\n\\langle \\cdot , \\cdot \\rangle :\n\\operatorname{Pic}(X)_\\Tor \\times \\operatorname{Pic}(X)_\\Tor \\to k^\\times \\otimes \\mathbb{Q}/\\mathbb{Z},\n\\end{equation}\nwhere $\\operatorname{Pic}(X)_\\Tor$ is the \ntorsion part of the Picard group $\\operatorname{Pic}(X)$ of $X$.\nWe then define the \\emph{intrinsic subgroup}\nof $\\operatorname{Pic}(X)_\\Tor$ by\n\\[\n\\operatorname{Pic}(X)_\\Tor^\\is \n:= \\{ a \\in \\operatorname{Pic}(X)_\\Tor \\mid \\langle a, b \\rangle = 0\n\\text{ for all } b \\in \\operatorname{Pic}(X)_\\Tor \\}.\n\\]\n\nWe remark that \\eqref{eq:pairing-intro} is not entirely new.\nIndeed, its finite coefficient analogue \n(with values in $k^\\times/(k^\\times)^m$) is considered by Frey and R\\\"uck in \\cite{FreyRuck}\n(see Remark \\ref{rem:FreyRuck}).\nThe first and second authors encountered with\na disguised version of \\eqref{eq:pairing-intro} in \\cite{YY}\n(see Remark \\ref{sect:genJac}).\nObviously, \\eqref{eq:pairing-intro} is trivial if $k^\\times \\otimes \\mathbb{Q}/\\mathbb{Z}=0$,\ne.g. when $k$ is finite, algebraically closed, or $k=\\mathbb{R}$.\nOn the other hand,\n$\\operatorname{Pic}(X)_\\Tor^\\is$ appears to be a non-trivial new invariant\nif $k$ is a global field or a $p$-adic field.\n\nWe will mainly consider the case where $X=E$ is an elliptic curve\nso that we may identify $E(k)_\\Tor= \\operatorname{Pic}(E)_\\Tor$\nby $P \\mapsto [P-O]$, where $O \\in E(k)$ is the identity element.\nThus we write \n$\\langle P,Q\\rangle:=\\langle [P-O], [Q-O]\\rangle$ for $P, Q \\in E(k)_\\Tor$\nand \n$E(k)_\\Tor^\\is:=\\operatorname{Pic}(E)_\\Tor^\\is$.\nOur main result is a classification of possible structures of\n$(E(\\mathbb{Q})_\\Tor,E(\\mathbb{Q})_\\Tor^\\is)$ for an elliptic curve $E$ over $\\mathbb{Q}$.\nA celebrated theorem of Mazur \\cite[Theorem (8)]{Mazur} states\nthat $E(\\mathbb{Q})_\\Tor$ is either cyclic of order\n$1, \\ldots,10$, or $12$, or is isomorphic to $(\\mathbb{Z}/2N\\mathbb{Z}) \\times (\\mathbb{Z}/2\\mathbb{Z})$,\n$N=1, 2,3,4$,\nand all cases are realized by infinitely many (mutually non-isomorphic) elliptic curves.\nThe classification of\n$E(\\mathbb{Q})_\\Tor^\\is$ is given in the following theorem,\nwhose proof will occupy \\S \\ref{sect:ref-mazur}--\\ref{sect:ref-mazur2}.\n\n\\begin{equation}\\label{eq:pairing-intro}\n\\langle \\cdot , \\cdot \\rangle :\n\\Pic(X)_\\Tor \\times \\Pic(X)_\\Tor \\to k^\\times \\otimes \\Q/\\Z,\n\\end{equation}\n\n\\begin{remark}\\label{rem:FreyRuck}\nAs is seen from the proof,\nthere is an analogous biadditive pairing\n\\[\n\\langle \\cdot, \\cdot \\rangle_n : \n\\Pic(X)[n] \\times \\Pic^0(X)/n \\to k^\\times/(k^\\times)^n\n\\]\nfor each $n>0$,\nwhich satisfies\n$\\langle d, e \\rangle_n \n=\n\\WP_n(d, e)\n\\langle e, d \\rangle_n\n$for any $d, e \\in \\Pic(X)[n]$.\nThis pairing has been constructed by Frey and R\\\"uck in \\cite{FreyRuck}\nwhen $k$ is a finite field\nand shown to be perfect if $|k| \\equiv 1 \\bmod n$.\n(This is called the \nLichtenbaum-Tate pairing in some literature,\nsee e.g. \\cite[XI.9]{sil2}.)\n\\end{remark}\n\nWe will mainly consider the case where $X=E$ is an elliptic curve\nso that we may identify $E(k)_\\Tor= \\Pic(E)_\\Tor$\nby $P \\mapsto [P-O]$, where $O \\in E(k)$ is the identity element.\nThus we write \n$\\gen{P,Q}:=\\gen{[P-O], [Q-O]}$ for $P, Q \\in E(k)_\\Tor$\nand \n$E(k)_\\Tor^\\is:=\\Pic(E)_\\Tor^\\is$.\nOur main result is a classification of possible structures of\n$(E(\\Q)_\\Tor,E(\\Q)_\\Tor^\\is)$ for an elliptic curve $E$ over $\\Q$.\nA celebrated theorem of Mazur \\cite[Theorem (8)]{Mazur} states\nthat $E(\\Q)_\\Tor$ is either cyclic of order\n$1, \\ldots,10$, or $12$, or is isomorphic to $(\\Z/2N\\Z) \\times (\\Z/2\\Z)$,\n$N=1, 2,3,4$,\nand all cases are realized by infinitely many (mutually non-isomorphic) elliptic curves.\nThe classification of\n$E(\\Q)_\\Tor^\\is$ is given in the following theorem,\nwhose proof will occupy \\S \\ref{sect:ref-mazur}--\\ref{sect:ref-mazur2}.\n\nWe briefly explain the outline of the proof.\nOur approach is completely explicit. \nAs usual, we let $X_1(N)$ denote the modular curve associated to the\ncongruence subgroup $\\Gamma_1(N)$. It possesses a model over $\\Q$ on\nwhich the cusp $0$ is $\\Q$-rational. For a subfield $k$ of $\\C$, \nthe non-cuspidal points in $X_1(N)(k)$ parameterize isomorphism\nclasses of pairs $(E, P)$ of an elliptic curve $E$ over $k$ \nand a $k$-rational point $P$ of order $N$. \nMore precisely, if the coordinates of a point \n$\\tau\\in\\H$ on this model of $X_1(N)$ is $k$-rational, \nthen the isomorphism class of $(\\C/(\\Z\\tau+\\Z),1/N)$ contains a pair $(E,P)$\nof an elliptic curve $E$ over $k$ and a $k$-rational $N$-torsion point $P$.\nA famous theorem of Mazur \\cite[Theorem (7)]{Mazur} says that \n$X_1(N)(\\Q)$ has a non-cuspidal point precisely\nwhen $N=1,\\ldots,10$, or $N=12$.\n\nSimilarly, for a positive even\ninteger $N$, we let $X_1^0(N,2)$ denote the modular curve associated\nto the congruence subgroup\n\\begin{equation}\\label{eq:G10-N2}\n\\Gamma_1^0(N,2):=\\Gamma_1(N)\\cap\\Gamma^0(2)\n=\\left\\{\\M abcd\\in\\SL(2,\\Z):a,d\\equiv 1\\bmod N,~~2|b,~N|c\\right\\}.\n\\end{equation}\nThe modular curve $X_1^0(N,2)$ parameterizes isomorphism classes of\ntriples $(E, P, Q)$ of an elliptic curve $E$\nand $k$-rational points $P, Q$ \nsuch that \n$P$ and $Q$ are of order $N$ and $2$ respectively,\nand $Q\\notin\\gen P$.\nFor $N \\in \\{4, 6, 8 \\}$\nand $\\ul{M}=(M_1, M_2, M_3)$ with $M_1|N, M_2, M_3 \\in \\{ 1, 2\\}$,\nwe will construct a smooth projective curve \n$X_1^0(N, 2)_{\\ul{M}}^\\pm$ over $\\Q$\nequipped with a finite morphism to $X_1^0(N, 2)$,\nby which one can interpret the conditions\n\\[ \\langle P, \\frac{N}{M_1}P \\rangle \n= \\langle Q, \\frac{2}{M_2}Q \\rangle \n= \\langle P, \\frac{2}{M_3}Q \\rangle = 0\n\\]\nfor a triple $(E, P, Q)$ as above corresponding to a point of $X_1^0(N, 2)(\\Q)$.\nTherefore Theorem \\ref{theorem: intrinsic torsion} is reduced to a study of the $\\Q$-rational points of \n$X_1(N)_M^\\pm$ and $X_1^0(N, 2)_{\\ul{M}}^\\pm$.\nAs our construction of these curves are explicit,\nthis can be done by a (more or less) direct computation.\n\n\\begin{lemma} \\label{lemma: ENt}\n \\begin{enumerate}\n \\item Assume that the characteristic of $k$ is not $2$.\n If $E$ is an elliptic curve over $k$ with a $k$-rational point\n $P$ of order two, then $(E,P)$ is isomorphic to $(E_{2,t,a},(0,0))$\n for some $a,t\\in k^\\times$, where\n \\begin{equation} \\label{eq: E2ta}\n E_{2,t,a}:\n \\begin{cases}\n \\displaystyle y^2=x\\left(x^2+ax+\\frac{a^2t}{4(t+1)}\\right),\n &\\text{if }t\\neq-1, \\\\\n y^2=x(x^2+a), &\\text{if }t=-1.\n \\end{cases}\n \\end{equation}\n Moreover, two elliptic curves $(E_{2,t,a},(0,0))$ and\n $(E_{2,t',a'},(0,0))$ are isomorphic over $k$ if and only\n if $t=t'\\neq-1$ and $a/a'$ is a square in $k^\\times$\n or $t=t'=-1$ and $a/a'\\in(k^\\times)^4$ . \n \\item Assume that the characteristic of $k$ is not $3$.\n If $E$ is an elliptic curve over $k$ with a $k$-rational\n point $P$ of order $3$, then $(E,P)$ is isomorphic to $(E_{3,t},(0,0))$\n for some $t\\neq-1$ in $k^\\times$, where\n \\begin{equation} \\label{eq: E3t}\n E_{3,t}:y^2+xy+\\frac t{27(t+1)}y=x^3\n \\end{equation}\n or isomorphic to $(E_{3,-1,a},(0,0))$ for some $a\\in k^\\times$,\n where\n \\begin{equation} \\label{eq: E3a}\n E_{3,-1,a}:y^2+ay=x^3.\n \\end{equation}\n Two elliptic curves $(E_{3,t},(0,0))$ and\n $(E_{3,t'},(0,0))$ with $t,t'\\neq-1$ are isomorphic if and only\n if $t=t'$. Also, $(E_{3,-1,a},(0,0))\\simeq(E_{3,-1,a'},(0,0))$\n if and only if $a/a'\\in(k^\\times)^3$. \n \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n Consider first the case $N=2$. \nWe start with the equation \\eqref{eq: E2}\nand introduce a new parameter\n $t=4b/(a^2-4b)$ (so $b=a^2t/4(t+1)$). When $a\\neq0$, we have\n $t\\neq-1$ and the equation \\eqref{eq: E2} in the new parameters is\n $y^2=x(x^2+ax+a^2t/4(t+1))$. When $a=0$, we change the notation $b$\n in \\eqref{eq: E2} to $-a$ and get $y^2=x(x^2-a)$. Now it is easy to\n see that an isomorphism\n $\\phi:(E_{2,t,a},(0,0))\\to(E_{2,t',a'},(0,0))$ must be of the form\n $\\phi(x,y)=(u^2x,u^3y)$ (c.f. \\cite[Proposition 3.1]{sil}). From\n this, we immediately get the condition for $E_{2,t,a}$ and\n $E_{2,t',a'}$ to be isomorphic.\n\nThis leaves us with the cases\n$(N, M) \\in \\{ (6,6), (8, 4), (9,3), (10, 2), (12, 2) \\}$ with $\\epsilon=-$.\n(For $(N, M)=(9,3)$, the value of $\\epsilon$ does not matter\nbecause of Remark \\ref{rem:plus-minus} (1).)\nIn all cases, there exists \nan isomorphism $\\alpha : E \\to X_1(N)_M^\\epsilon$,\nwhere\n$E$ is an elliptic curve over $\\Q$\nwith finitely many $\\Q$-rational points,\nas exhibited in the following table,\nwhere we set\n$h_1(x)=(1-x)(1+x)^4(1+x-x^2)^2$ and\n$h_2(x)=(1+x)(1+x+x^2)(1+x^2)^2(1-x)^3$.\n $$ \\extrarowheight3pt\n \\begin{array}{c|c|c|c} \\hline\\hline\n (N, M) & E & E(\\Q) \\setminus \\{ O \\} & \n(s, t)=\\alpha(x,y) \\\\ \\hline\n(6,6) & y^2=x^3-1 & (1,0) &\n\\left(-\\frac{xy}{1-x^3},\\frac1{1-x^3}\\right)\n\\\\\n(8,4) & y^2=x^3+x & (0,0) &\n\\left(\\frac y{x^2}(1-\\frac{1}{x^2}),-\\frac1{x^2}\\right)\n\\\\\n(9,3) & y^2+y=x^3 & (0,0), (0,-1) &\n\\left(xy(y^2+y+1),y+1\\right)\n\\\\\n(10,2) & y^2=x^3-x^2-x & (0,0) &\n\\left(yh_1(x),x\\right)\n\\\\\n(12,2) & y^2=x^3+x^2+x & (0,0) &\n\\left(yh_2(x),x\\right)\n\\\\\n \\hline\\hline\n \\end{array}\n $$\nSince all points of $\\alpha(E(\\Q))$ are cusps \nby Lemma \\ref{lemma: uniformizers},\nwe get $Y_1(\\Q)^\\epsilon_M=\\emptyset$ in these cases as well.\nWe are done.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:kodairatype}\nLet $E$ be an elliptic curve over a number field $k$,\nand $P \\in E(k)_\\Tor^\\is$ an element of the intrinsic subgroup of order $m>0$.\nLet $v$ be a finite place of $k$\nsuch that the completion $k_v$ of $k$ at $v$ satisfies $\\mu_m(k_v)= \\{ 1 \\}$.\nIf $E$ has split multiplicative reduction of Kodaira type $I_n$ at $v$,\nthen $n$ is divisible by $m^2$.\n\\end{corollary}\n\\begin{proof}\nThe assumption implies that\nthe base change of $E$ to $k_v$ is \nisomorphic to the Tate curve $E_q=\\G_m/q^\\Z$\nfor some $q \\in k_v$ such that $v(q)=n$\n(\\cite[Chapter V, Theorem 5.3]{sil})\nLet us write the image of $P$ in $E_q(k_v)$ as $[a]$ with $a \\in k_v$.\nSince $\\mu_m(k_v)= \\{ 1 \\}$, \nwe find from \\eqref{eq:ex-TateCv} that\n$E_q(k_v)[m]$ is a cyclic group of order $m$ generated by $[a]$.\nIn particular, $s:= s_m([a])$ is invertible in $\\Z/m\\Z$.\nWe have $\\langle [a], [a] \\rangle = 0$ since $P \\in E(k)_\\Tor^\\is$.\nOn the other hand,\nProposition \\ref{prop:TateCv} (1) shows\n$\\langle [a], [a] \\rangle = q^{-s^2} \\otimes (1/m^2)$.\nBy looking at its image under the map\n$v \\otimes \\id_{\\Q/\\Z} : k_v^\\times \\otimes \\Q/\\Z \\to \\Q/\\Z$,\nwe conclude that $-s^2 n/m^2 = 0$ holds in $\\Q/\\Z$,\nthat is, $m^2$ divides $n$.\n\\end{proof}", "post_theorem_intro_text_len": 4805, "post_theorem_intro_text": "We briefly explain the outline of the proof.\nOur approach is completely explicit. \nAs usual, we let $X_1(N)$ denote the modular curve associated to the\ncongruence subgroup $\\Gamma_1(N)$. It possesses a model over $\\mathbb{Q}$ on\nwhich the cusp $0$ is $\\mathbb{Q}$-rational. For a subfield $k$ of $\\mathbb{C}$, \nthe non-cuspidal points in $X_1(N)(k)$ parameterize isomorphism\nclasses of pairs $(E, P)$ of an elliptic curve $E$ over $k$ \nand a $k$-rational point $P$ of order $N$. \nMore precisely, if the coordinates of a point \n$\\tau\\in\\mathbb H$ on this model of $X_1(N)$ is $k$-rational, \nthen the isomorphism class of $(\\mathbb{C}/(\\mathbb{Z}\\tau+\\mathbb{Z}),1/N)$ contains a pair $(E,P)$\nof an elliptic curve $E$ over $k$ and a $k$-rational $N$-torsion point $P$.\nA famous theorem of Mazur \\cite[Theorem (7)]{Mazur} says that \n$X_1(N)(\\mathbb{Q})$ has a non-cuspidal point precisely\nwhen $N=1,\\ldots,10$, or $N=12$. \n\nFor a pair $(N, M)$ of $N \\in \\{4, \\dots, 10, 12\\}$ and a positive divisor $M$ of $N$,\nwe will construct a smooth projective curve $X_1 (N)^\\pm_M$ over $\\mathbb{Q}$ equipped with\na finite morphism $X_1(N)_M^\\pm \\to X_1(N)$\nthat enjoys the following property:\nwhen $t \\in X_1(N)(\\mathbb{Q})$ corresponds to a pair $(E, P)$ as above (with $k=\\mathbb{Q}$),\none has \n\\[\n\\langle P, (N/M)P \\rangle = 0 \\Leftrightarrow\nt \\in \\operatorname{Im}(X_1(N)^\\pm_M(\\mathbb{Q}) \\to X_1(N)(\\mathbb{Q})).\n\\]\nIt is worth  mentioning that\n$X_1(N)^+_M$ is isomorphic to the modular curve\n$X_1(MN, N)$ associated to $\\Gamma_1(MN, N):=\\Gamma_0(MN) \\cap \\Gamma_1(N)$\n(Theorem \\ref{proposition: pairing 1-mod})\nand $X_1(N)^-_M$ is its twist.\n\nSimilarly, for a positive even\ninteger $N$, we let $X_1^0(N,2)$ denote the modular curve associated\nto the congruence subgroup\n\\begin{equation}\\label{eq:G10-N2}\n\\Gamma_1^0(N,2):=\\Gamma_1(N)\\cap\\Gamma^0(2)\n=\\left\\{\\M abcd\\in\\SL(2,\\mathbb{Z}):a,d\\equiv 1\\bmod N,~~2|b,~N|c\\right\\}.\n\\end{equation}\nThe modular curve $X_1^0(N,2)$ parameterizes isomorphism classes of\ntriples $(E, P, Q)$ of an elliptic curve $E$\nand $k$-rational points $P, Q$ \nsuch that \n$P$ and $Q$ are of order $N$ and $2$ respectively,\nand $Q\\notin\\gen P$.\nFor $N \\in \\{4, 6, 8 \\}$\nand $\\underline{M}=(M_1, M_2, M_3)$ with $M_1|N, M_2, M_3 \\in \\{ 1, 2\\}$,\nwe will construct a smooth projective curve  \n$X_1^0(N, 2)_{\\underline{M}}^\\pm$ over $\\mathbb{Q}$\nequipped with a finite morphism to $X_1^0(N, 2)$,\nby which one can interpret the conditions\n\\[ \\langle P, \\frac{N}{M_1}P \\rangle \n= \\langle Q, \\frac{2}{M_2}Q \\rangle \n= \\langle P, \\frac{2}{M_3}Q \\rangle = 0\n\\]\nfor a triple $(E, P, Q)$ as above corresponding to a point of $X_1^0(N, 2)(\\mathbb{Q})$.\nTherefore Theorem \\ref{theorem: intrinsic torsion} is reduced to a study of the $\\mathbb{Q}$-rational points of \n$X_1(N)_M^\\pm$ and $X_1^0(N, 2)_{\\underline{M}}^\\pm$.\nAs our construction of these curves are explicit,\nthis can be done by a (more or less) direct computation.\n\nIn the last section \\S \\ref{sect:high-dim},\nwe generalize the pairing \\eqref{eq:pairing-intro}\nand the intrinsic subgroup \nto higher dimensional varieties.\nThis new construction enables us to prove that,\nif $X$ has good reduction with respect to a discrete valuation of $k$,\nthere is a strong restriction on the values of the pairing \n(see Proposition \\ref{prop:two-def-same}).\nFinally, as a sample for the case of bad reduction,\nwe compute the intrinsic subgroup of \nTate elliptic curves (see Proposition \\ref{prop:TateCv}).\nAs an application,\nwe show that the intrinsic subgroup of an elliptic curve over a number field\nimposes some constraint on its reduction type\n(see Corollary \\ref{cor:kodairatype}).\n\n\\subsection*{Acknowledgement}\nIt is Kenneth Ribet who formulated \\eqref{eq:pairing-intro} as a pairing \nand asked if it is symmetric.\nThe authors express deep gratitude to him for asking this question,\nwhich was a starting point of our work.\n\nThe first author (T. Y.) is supported by JSPS KAKENHI Grant (25K06961). \nThe second author (Y. Y.) is supported by \nGrant 113-2115-M-002-003-MY3 of the National Science and Technology\n  Council of the Republic of China (Taiwan). \nThe third author (H. Y.) is supported by National Research Foundation of Korea(NRF) grant \nfunded by the Korea government(MSIT) (No. RS-2023-00239918  and No. 2020R1A5A1016126).\nThe fourth author (M. Y.) is supported by the National Research Foundation of Korea (NRF) grant \nfunded by the\nKorea government (MSIT) (RS-2025-23525445).\n\n\\subsection*{Notation and convention}\nFor an abelian group $A$, \nwe write $A[n]$ and $A/n$ for the kernel and cokernel\nof $A \\to A, \\ a \\mapsto na$ for each $n \\in \\mathbb{Z}$,\nand put $A_\\Tor := \\cup_{n > 0} A[n]$.\nFor a field $k$,\nwe set $\\mu_n(k):=(k^\\times)[n]$ and $\\mu(k):=(k^\\times)_\\Tor$.\n\nThe identity element of an elliptic curve is denoted by $O$.", "sketch": "The authors say: “We briefly explain the outline of the proof. Our approach is completely explicit.” They use modular curves $X_1(N)$: non-cuspidal points in $X_1(N)(k)$ “parameterize isomorphism classes of pairs $(E,P)$ of an elliptic curve $E$ over $k$ and a $k$-rational point $P$ of order $N$.” Using Mazur’s theorem that $X_1(N)(\\mathbb{Q})$ has a non-cuspidal point precisely for $N=1,\\ldots,10$ or $N=12$, they restrict to these torsion orders.\n\nFor each $(N,M)$ with $N\\in\\{4,\\dots,10,12\\}$ and $M\\mid N$, they “construct a smooth projective curve $X_1(N)^\\pm_M$ over $\\mathbb{Q}$ equipped with a finite morphism $X_1(N)_M^\\pm\\to X_1(N)$” such that for $t\\in X_1(N)(\\mathbb{Q})$ corresponding to $(E,P)$ one has\n\\[\n\\langle P,(N/M)P\\rangle=0\\Longleftrightarrow t\\in\\operatorname{Im}\\bigl(X_1(N)^\\pm_M(\\mathbb{Q})\\to X_1(N)(\\mathbb{Q})\\bigr).\n\\]\nThey note $X_1(N)^+_M$ is isomorphic to a modular curve $X_1(MN,N)$ and $X_1(N)^-_M$ is its twist.\n\nFor even $N$, they similarly use the modular curve $X_1^0(N,2)$, which “parameterizes isomorphism classes of triples $(E,P,Q)$” with $P$ of order $N$, $Q$ of order $2$, and $Q\\notin\\langle P\\rangle$. For $N\\in\\{4,6,8\\}$ and $\\underline{M}=(M_1,M_2,M_3)$ with $M_1\\mid N$ and $M_2,M_3\\in\\{1,2\\}$, they construct curves $X_1^0(N,2)^{\\pm}_{\\underline{M}}$ with finite morphisms to $X_1^0(N,2)$ “by which one can interpret the conditions”\n\\[\n\\langle P,\\tfrac{N}{M_1}P\\rangle=\\langle Q,\\tfrac{2}{M_2}Q\\rangle=\\langle P,\\tfrac{2}{M_3}Q\\rangle=0.\n\\]\nThey conclude that “Therefore Theorem~\\ref{theorem: intrinsic torsion} is reduced to a study of the $\\mathbb{Q}$-rational points of $X_1(N)_M^\\pm$ and $X_1^0(N,2)_{\\underline{M}}^\\pm$,” and since “our construction of these curves are explicit, this can be done by a (more or less) direct computation.”", "expanded_sketch": "The authors say: “We briefly explain the outline of the proof. Our approach is completely explicit.” They use modular curves $X_1(N)$: non-cuspidal points in $X_1(N)(k)$ “parameterize isomorphism classes of pairs $(E,P)$ of an elliptic curve $E$ over $k$ and a $k$-rational point $P$ of order $N$.” Using Mazur’s theorem that $X_1(N)(\\mathbb{Q})$ has a non-cuspidal point precisely for $N=1,\\ldots,10$ or $N=12$, they restrict to these torsion orders.\n\nFor each $(N,M)$ with $N\\in\\{4,\\dots,10,12\\}$ and $M\\mid N$, they “construct a smooth projective curve $X_1(N)^\\pm_M$ over $\\mathbb{Q}$ equipped with a finite morphism $X_1(N)_M^\\pm\\to X_1(N)$” such that for $t\\in X_1(N)(\\mathbb{Q})$ corresponding to $(E,P)$ one has\n\\[\n\\langle P,(N/M)P\\rangle=0\\Longleftrightarrow t\\in\\operatorname{Im}\\bigl(X_1(N)^\\pm_M(\\mathbb{Q})\\to X_1(N)(\\mathbb{Q})\\bigr).\n\\]\nThey note $X_1(N)^+_M$ is isomorphic to a modular curve $X_1(MN,N)$ and $X_1(N)^-_M$ is its twist.\n\nFor even $N$, they similarly use the modular curve $X_1^0(N,2)$, which “parameterizes isomorphism classes of triples $(E,P,Q)$” with $P$ of order $N$, $Q$ of order $2$, and $Q\\notin\\langle P\\rangle$. For $N\\in\\{4,6,8\\}$ and $\\underline{M}=(M_1,M_2,M_3)$ with $M_1\\mid N$ and $M_2,M_3\\in\\{1,2\\}$, they construct curves $X_1^0(N,2)^{\\pm}_{\\underline{M}}$ with finite morphisms to $X_1^0(N,2)$ “by which one can interpret the conditions”\n\\[\n\\langle P,\\tfrac{N}{M_1}P\\rangle=\\langle Q,\\tfrac{2}{M_2}Q\\rangle=\\langle P,\\tfrac{2}{M_3}Q\\rangle=0.\n\\]\nThey conclude that, in establishing the main theorem, everything is reduced to a study of the $\\mathbb{Q}$-rational points of $X_1(N)_M^\\pm$ and $X_1^0(N,2)_{\\underline{M}}^\\pm$, and since “our construction of these curves are explicit, this can be done by a (more or less) direct computation.”", "expanded_theorem": "\\label{theorem: intrinsic torsion}\n  Let $E$ be an elliptic curve over $\\mathbb{Q}$.\nThen there is a pair $(A, B)$ of an abelian group $A$\nand its subgroup $B$ in the following table,\ntogether with \nan isomorphism $\\alpha : E(\\mathbb{Q})_\\Tor \\cong A$, \nsuch that $\\alpha(E(\\mathbb{Q})_\\Tor^\\is)=B$.\n  $$ \\extrarowheight3pt\n  \\begin{array}{ll} \\hline\\hline\n    A & \\text{order of}\\ B \\\\ \\hline\n    0 & 1 \\\\\n    \\mathbb{Z}/2\\mathbb{Z} & 1,2\\\\\n    \\mathbb{Z}/3\\mathbb{Z} & 1,3\\\\\n    \\mathbb{Z}/4\\mathbb{Z} & 1,2,4 \\\\\n    \\mathbb{Z}/5\\mathbb{Z} & 1,5 \\\\\n    \\mathbb{Z}/6\\mathbb{Z} & 1,2,3 \\\\\n    \\mathbb{Z}/7\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/8\\mathbb{Z} & 1,2 \\\\\n    \\mathbb{Z}/9\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/10\\mathbb{Z} & 1\\\\\n    \\mathbb{Z}/12\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/2\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1,2\\\\\n    \\mathbb{Z}/4\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1,2 \\ \\\\\n    \\mathbb{Z}/6\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1 \\\\\n    \\mathbb{Z}/8\\mathbb{Z}\\times\\mathbb{Z}/2\\mathbb{Z} & 1 \\\\ \\hline\\hline\n  \\end{array}\n  $$\n(For $A=\\mathbb{Z}/4\\mathbb{Z} \\times \\mathbb{Z}/2\\mathbb{Z}$,\nthere are two $\\operatorname{Aut}(A)$-conjugacy classes of order two subgroups,\nboth of which can be taken as $B$.)\n  Moreover, for each possible pair $(A, B)$ as above,\nthere are infinitely many (mutually non-isomorphic) elliptic curves $E$ over $\\mathbb{Q}$ \nfor which there is an isomorphism $\\alpha : E(\\mathbb{Q})_\\Tor \\cong A$\nsuch that $\\alpha(E(\\mathbb{Q})_\\Tor^\\is)=B$.,", "theorem_type": ["Classification or Bijection", "Existence"], "mcq": {"question": "Let $E$ be an elliptic curve over $\\mathbb{Q}$ with identity element $O$. Via the identification $P \\mapsto [P-O]$, identify $E(\\mathbb{Q})_{\\mathrm{Tor}}$ with $\\operatorname{Pic}(E)_{\\mathrm{Tor}}$. Let\n\\[\n\\langle\\cdot,\\cdot\\rangle: E(\\mathbb{Q})_{\\mathrm{Tor}}\\times E(\\mathbb{Q})_{\\mathrm{Tor}}\\to \\mathbb{Q}^\\times\\otimes \\mathbb{Q}/\\mathbb{Z}\n\\]\nbe the induced biadditive symmetric pairing, and define the intrinsic subgroup by\n\\[\nE(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}}:=\\{P\\in E(\\mathbb{Q})_{\\mathrm{Tor}}\\mid \\langle P,Q\\rangle=0\\text{ for all }Q\\in E(\\mathbb{Q})_{\\mathrm{Tor}}\\}.\n\\]\nWhich existence statement holds for the pair consisting of the torsion subgroup $E(\\mathbb{Q})_{\\mathrm{Tor}}$ and its intrinsic subgroup $E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}}$?", "correct_choice": {"label": "A", "text": "For every elliptic curve $E/\\mathbb{Q}$, there exist an abelian group $A$ and a subgroup $B\\le A$, together with an isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim} A$, such that $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$, where $A$ and the possible orders of $B$ are exactly as follows: $A=0$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/3\\mathbb{Z}$ with $|B|=1$ or $3$; $A=\\mathbb{Z}/4\\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\\mathbb{Z}/5\\mathbb{Z}$ with $|B|=1$ or $5$; $A=\\mathbb{Z}/6\\mathbb{Z}$ with $|B|=1,2,$ or $3$; $A=\\mathbb{Z}/7\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/8\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/9\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/10\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/12\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/6\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$; and $A=\\mathbb{Z}/8\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$. For $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$, there are two $\\operatorname{Aut}(A)$-conjugacy classes of subgroups of order $2$, and either class can occur as $B$. Moreover, for every such possible pair $(A,B)$, there exist infinitely many mutually non-isomorphic elliptic curves $E/\\mathbb{Q}$ for which some isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim}A$ satisfies $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$."}, "choices": [{"label": "B", "text": "For every elliptic curve $E/\\mathbb{Q}$, there exist an abelian group $A$ and a subgroup $B\\le A$, together with an isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim} A$, such that $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$, where $A$ and the possible orders of $B$ are exactly as follows: $A=0$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/3\\mathbb{Z}$ with $|B|=1$ or $3$; $A=\\mathbb{Z}/4\\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\\mathbb{Z}/5\\mathbb{Z}$ with $|B|=1$ or $5$; $A=\\mathbb{Z}/6\\mathbb{Z}$ with $|B|=1,2,$ or $3$; $A=\\mathbb{Z}/7\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/8\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/9\\mathbb{Z}$ with $|B|=1$ or $3$; $A=\\mathbb{Z}/10\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/12\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/6\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$; and $A=\\mathbb{Z}/8\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$. For $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$, there are two $\\operatorname{Aut}(A)$-conjugacy classes of subgroups of order $2$, and either class can occur as $B$. Moreover, for every such possible pair $(A,B)$, there exists at least one elliptic curve $E/\\mathbb{Q}$ for which some isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim}A$ satisfies $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$."}, {"label": "C", "text": "For every elliptic curve $E/\\mathbb{Q}$, there exist an abelian group $A$ and a subgroup $B\\le A$, together with an isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim} A$, such that $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$, where $A$ and the possible orders of $B$ are among the following: $A=0$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/3\\mathbb{Z}$ with $|B|=1$ or $3$; $A=\\mathbb{Z}/4\\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\\mathbb{Z}/5\\mathbb{Z}$ with $|B|=1$ or $5$; $A=\\mathbb{Z}/6\\mathbb{Z}$ with $|B|=1,2,$ or $3$; $A=\\mathbb{Z}/7\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/8\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/9\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/10\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/12\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/6\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$; and $A=\\mathbb{Z}/8\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$."}, {"label": "D", "text": "For every elliptic curve $E/\\mathbb{Q}$, there exist an abelian group $A$ and a subgroup $B\\le A$, together with an isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim} A$, such that $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$, where $A$ and the possible orders of $B$ are exactly as follows: $A=0$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/3\\mathbb{Z}$ with $|B|=1$ or $3$; $A=\\mathbb{Z}/4\\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\\mathbb{Z}/5\\mathbb{Z}$ with $|B|=1$ or $5$; $A=\\mathbb{Z}/6\\mathbb{Z}$ with $|B|=1,2,$ or $3$; $A=\\mathbb{Z}/7\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/8\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/9\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/10\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/12\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\\mathbb{Z}/6\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$; and $A=\\mathbb{Z}/8\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$. For $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$, there are two $\\operatorname{Aut}(A)$-conjugacy classes of subgroups of order $2$, and either class can occur as $B$. Moreover, for every such possible pair $(A,B)$, there exist infinitely many mutually non-isomorphic elliptic curves $E/\\mathbb{Q}$ for which some isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim}A$ satisfies $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$."}, {"label": "E", "text": "For every elliptic curve $E/\\mathbb{Q}$, there exist an abelian group $A$ and a subgroup $B\\le A$, together with an isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim} A$, such that $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$, where $A$ and the possible orders of $B$ are exactly as follows: $A=0$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/3\\mathbb{Z}$ with $|B|=1$ or $3$; $A=\\mathbb{Z}/4\\mathbb{Z}$ with $|B|=1,2,$ or $4$; $A=\\mathbb{Z}/5\\mathbb{Z}$ with $|B|=1$ or $5$; $A=\\mathbb{Z}/6\\mathbb{Z}$ with $|B|=1,2,$ or $3$; $A=\\mathbb{Z}/7\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/8\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/9\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/10\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/12\\mathbb{Z}$ with $|B|=1$; $A=\\mathbb{Z}/2\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$ or $2$; $A=\\mathbb{Z}/6\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$; and $A=\\mathbb{Z}/8\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ with $|B|=1$. For $A=\\mathbb{Z}/4\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$, there are two $\\operatorname{Aut}(A)$-conjugacy classes of subgroups of order $2$, and both classes occur simultaneously for every elliptic curve $E/\\mathbb{Q}$ with $E(\\mathbb{Q})_{\\mathrm{Tor}}\\cong A$. Moreover, for every such possible pair $(A,B)$, there exist infinitely many mutually non-isomorphic elliptic curves $E/\\mathbb{Q}$ for which some isomorphism $\\alpha:E(\\mathbb{Q})_{\\mathrm{Tor}}\\xrightarrow{\\sim}A$ satisfies $\\alpha(E(\\mathbb{Q})_{\\mathrm{Tor}}^{\\mathrm{is}})=B$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "infinitely_many_realizations", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped_exactness_and_infinite_realization_clause", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "computational_check", "tampered_component": "order_4_subgroup_for_Z4xZ2", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "outer_automorphism", "tampered_component": "existential_occurrence_of_two_Aut_conjugacy_classes", "template_used": "wildcard"}]}}
{"id": "2512.00802v1", "paper_link": "http://arxiv.org/abs/2512.00802v1", "theorems_cnt": 8, "theorem": {"env_name": "thm", "content": "[Arakelian \\cite{Ar}]\\label{thm:Ar}\nA closed set $F\\subset \\mathbb{C}$ is a set of uniform approximation if and only if $F$ is an Arakelian set.", "start_pos": 5733, "end_pos": 5899, "label": "thm:Ar"}, "ref_dict": {"thm:Ar": "\\begin{thm}[Arakelian \\cite{Ar}]\\label{thm:Ar}\nA closed set $F\\subset \\mathbb{C}$ is a set of uniform approximation if and only if $F$ is an Arakelian set. \n\\end{thm}", "thm:F": "\\begin{thm}\\label{thm:F}\nA closed set $F\\subset \\mathbb{C}$ is an Arakelian set if and only if it possesses a neighborhood basis of simply connected open sets. \n\\end{thm}", "thm:main": "\\begin{thm}\\label{thm:main}\nA closed set $F\\subset \\mathbb{C}$ is an Arakelian set, if and only if \nfor  every $f\\in A(F)$ with no zeros in $F$, there exists $g\\in A(F)$ such that $e^g=f$ in $F$.\n\\end{thm}", "def:Arak": "\\begin{dfn}[Arakelian set]\\label{def:Arak}\nLet $F\\subset \\mathbb{C}$ be a closed set. We say that $F$ is an Arakelian set, if and only if the following points hold:\\\\[5pt]\n1. $F$ is without holes;\\\\[5pt]\n2. The union of all holes of $F\\cup K$ is bounded, for any compact subset $K\\subset \\mathbb{C}$. \n\\end{dfn}"}, "pre_theorem_intro_text_len": 2314, "pre_theorem_intro_text": "Approximation theorems (Runge \\cite{Run}, Mergelyan \\cite{Mer}, Arakelian \\cite{Ar}) play a central role in the field of complex approximation, since they give necessary and sufficient conditions for functions to be uniformly approximated on a given set by model functions, such as polynomials, rational functions or more generally entire/meromorphic functions. Our goal in the present paper is to give a new characterization of closed sets $F\\subset \\mathbb{C}$ in the complex plane, for which uniform approximation by entire functions is possible. \n\nTo this end, define the classes\n\\begin{align*}\nH(F)=&\\,\\lbrace  f:U\\to\\mathbb{C} \\text{ holomorphic for some open set $U\\supset F$}\\rbrace,\\\\\n A(F)=&\\,\\lbrace  f:F\\to\\mathbb{C} \\text{ holomorphic in }F^\\circ\\text{ and continuous in F}\\rbrace,\n\\end{align*}\nwhere $F^\\circ$ is the interior of $F$.\n\\begin{dfn}\\label{def:hole}\nAny bounded connected component $B$ of $\\mathbb{C}\\setminus F$ is called a hole of $F$.\n\\end{dfn}\nArakelian's theorem \\cite{Ar} gives necessary and sufficient conditions such that every function $f\\in A(F)$ can be uniformly approximated on $F$ by entire functions..\n\\begin{dfn}[Arakelian set]\\label{def:Arak}\nLet $F\\subset \\mathbb{C}$ be a closed set. We say that $F$ is an Arakelian set, if and only if the following points hold:\\\\[5pt]\n1. $F$ is without holes;\\\\[5pt]\n2. The union of all holes of $F\\cup K$ is bounded, for any compact subset $K\\subset \\mathbb{C}$. \n\\end{dfn}\n\\begin{rmk}\\label{rem:K}\nPoint 2. in Definition \\ref{def:Arak} is equivalent to the one where $K$ is replaced by $\\overline{D}_0(n)=\\{z\\in\\mathbb{C}: |z|\\leq n\\}$, for any $n\\in\\mathbb{N}$. \n\\end{rmk}\n\\begin{rmk}\nIn fact, Definition \\ref{def:Arak} appears in a later work \\cite{RR}, where a new proof of Arakelian's theorem was given (see also \\cite{Got}). The original conditions in \\cite{Ar} are:\\\\[5pt]\n1. $\\mathbb{C}\\setminus F$ is connected;\\\\[5pt]\n2. $\\mathbb{C}\\setminus F$ is locally connected at infinity.\\\\[5pt]\nNevertheless, they are easily seen to be equivalent.\n\\end{rmk}\n\n\\begin{dfn}[Uniform approximation set]\nLet $F\\subset \\mathbb{C}$ be a closed set. We say that $F$ is a set of uniform approximation, if for every $\\epsilon>0$ and $f\\in A(F)$, there exists an entire function $g$ such that $|f(z)-g(z)|<\\epsilon$, for all $z\\in F$. \n\\end{dfn}", "context": "Approximation theorems (Runge \\cite{Run}, Mergelyan \\cite{Mer}, Arakelian \\cite{Ar}) play a central role in the field of complex approximation, since they give necessary and sufficient conditions for functions to be uniformly approximated on a given set by model functions, such as polynomials, rational functions or more generally entire/meromorphic functions. Our goal in the present paper is to give a new characterization of closed sets $F\\subset \\mathbb{C}$ in the complex plane, for which uniform approximation by entire functions is possible.\n\nTo this end, define the classes\n\\begin{align*}\nH(F)=&\\,\\lbrace f:U\\to\\mathbb{C} \\text{ holomorphic for some open set $U\\supset F$}\\rbrace,\\\\\n A(F)=&\\,\\lbrace f:F\\to\\mathbb{C} \\text{ holomorphic in }F^\\circ\\text{ and continuous in F}\\rbrace,\n\\end{align*}\nwhere $F^\\circ$ is the interior of $F$.\n\\begin{dfn}\\label{def:hole}\nAny bounded connected component $B$ of $\\mathbb{C}\\setminus F$ is called a hole of $F$.\n\\end{dfn}\nArakelian's theorem \\cite{Ar} gives necessary and sufficient conditions such that every function $f\\in A(F)$ can be uniformly approximated on $F$ by entire functions..\n\\begin{dfn}[Arakelian set]\\label{def:Arak}\nLet $F\\subset \\mathbb{C}$ be a closed set. We say that $F$ is an Arakelian set, if and only if the following points hold:\\\\[5pt]\n1. $F$ is without holes;\\\\[5pt]\n2. The union of all holes of $F\\cup K$ is bounded, for any compact subset $K\\subset \\mathbb{C}$. \n\\end{dfn}\n\\begin{rmk}\\label{rem:K}\nPoint 2. in Definition \\ref{def:Arak} is equivalent to the one where $K$ is replaced by $\\overline{D}_0(n)=\\{z\\in\\mathbb{C}: |z|\\leq n\\}$, for any $n\\in\\mathbb{N}$. \n\\end{rmk}\n\\begin{rmk}\nIn fact, Definition \\ref{def:Arak} appears in a later work \\cite{RR}, where a new proof of Arakelian's theorem was given (see also \\cite{Got}). The original conditions in \\cite{Ar} are:\\\\[5pt]\n1. $\\mathbb{C}\\setminus F$ is connected;\\\\[5pt]\n2. $\\mathbb{C}\\setminus F$ is locally connected at infinity.\\\\[5pt]\nNevertheless, they are easily seen to be equivalent.\n\\end{rmk}\n\n\\begin{dfn}[Uniform approximation set]\nLet $F\\subset \\mathbb{C}$ be a closed set. We say that $F$ is a set of uniform approximation, if for every $\\epsilon>0$ and $f\\in A(F)$, there exists an entire function $g$ such that $|f(z)-g(z)|<\\epsilon$, for all $z\\in F$. \n\\end{dfn}", "full_context": "Approximation theorems (Runge \\cite{Run}, Mergelyan \\cite{Mer}, Arakelian \\cite{Ar}) play a central role in the field of complex approximation, since they give necessary and sufficient conditions for functions to be uniformly approximated on a given set by model functions, such as polynomials, rational functions or more generally entire/meromorphic functions. Our goal in the present paper is to give a new characterization of closed sets $F\\subset \\mathbb{C}$ in the complex plane, for which uniform approximation by entire functions is possible.\n\nTo this end, define the classes\n\\begin{align*}\nH(F)=&\\,\\lbrace f:U\\to\\mathbb{C} \\text{ holomorphic for some open set $U\\supset F$}\\rbrace,\\\\\n A(F)=&\\,\\lbrace f:F\\to\\mathbb{C} \\text{ holomorphic in }F^\\circ\\text{ and continuous in F}\\rbrace,\n\\end{align*}\nwhere $F^\\circ$ is the interior of $F$.\n\\begin{dfn}\\label{def:hole}\nAny bounded connected component $B$ of $\\mathbb{C}\\setminus F$ is called a hole of $F$.\n\\end{dfn}\nArakelian's theorem \\cite{Ar} gives necessary and sufficient conditions such that every function $f\\in A(F)$ can be uniformly approximated on $F$ by entire functions..\n\\begin{dfn}[Arakelian set]\\label{def:Arak}\nLet $F\\subset \\mathbb{C}$ be a closed set. We say that $F$ is an Arakelian set, if and only if the following points hold:\\\\[5pt]\n1. $F$ is without holes;\\\\[5pt]\n2. The union of all holes of $F\\cup K$ is bounded, for any compact subset $K\\subset \\mathbb{C}$. \n\\end{dfn}\n\\begin{rmk}\\label{rem:K}\nPoint 2. in Definition \\ref{def:Arak} is equivalent to the one where $K$ is replaced by $\\overline{D}_0(n)=\\{z\\in\\mathbb{C}: |z|\\leq n\\}$, for any $n\\in\\mathbb{N}$. \n\\end{rmk}\n\\begin{rmk}\nIn fact, Definition \\ref{def:Arak} appears in a later work \\cite{RR}, where a new proof of Arakelian's theorem was given (see also \\cite{Got}). The original conditions in \\cite{Ar} are:\\\\[5pt]\n1. $\\mathbb{C}\\setminus F$ is connected;\\\\[5pt]\n2. $\\mathbb{C}\\setminus F$ is locally connected at infinity.\\\\[5pt]\nNevertheless, they are easily seen to be equivalent.\n\\end{rmk}\n\n\\begin{dfn}[Uniform approximation set]\nLet $F\\subset \\mathbb{C}$ be a closed set. We say that $F$ is a set of uniform approximation, if for every $\\epsilon>0$ and $f\\in A(F)$, there exists an entire function $g$ such that $|f(z)-g(z)|<\\epsilon$, for all $z\\in F$. \n\\end{dfn}\n\n\\begin{abstract}\nArakelian's classical approximation theorem \\cite{Ar} gives necessary and sufficient conditions such that functions can be uniformly approximated in (unbounded) closed sets $F\\subset \\mathbb{C}$ by entire functions. The conditions are purely topological and concern the connectedness of the complement of $F$. We give a new characterization of Arakelian sets in terms of logarithmic branches of functions $f\\in A(F)$, which are continuous in $F$ and holomorphic in its interior $F^\\circ$. Our proof is based on a contradiction argument and the counterexample function that we use is furnished by the Weierstrass factorization theorem. \n\\end{abstract}\n\nApproximation theorems (Runge \\cite{Run}, Mergelyan \\cite{Mer}, Arakelian \\cite{Ar}) play a central role in the field of complex approximation, since they give necessary and sufficient conditions for functions to be uniformly approximated on a given set by model functions, such as polynomials, rational functions or more generally entire/meromorphic functions. Our goal in the present paper is to give a new characterization of closed sets $F\\subset \\mathbb{C}$ in the complex plane, for which uniform approximation by entire functions is possible.\n\nTo this end, define the classes\n\\begin{align*}\nH(F)=&\\,\\lbrace f:U\\to\\mathbb{C} \\text{ holomorphic for some open set $U\\supset F$}\\rbrace,\\\\\n A(F)=&\\,\\lbrace f:F\\to\\mathbb{C} \\text{ holomorphic in }F^\\circ\\text{ and continuous in F}\\rbrace,\n\\end{align*}\nwhere $F^\\circ$ is the interior of $F$.\n\\begin{dfn}\\label{def:hole}\nAny bounded connected component $B$ of $\\mathbb{C}\\setminus F$ is called a hole of $F$.\n\\end{dfn}\nArakelian's theorem \\cite{Ar} gives necessary and sufficient conditions such that every function $f\\in A(F)$ can be uniformly approximated on $F$ by entire functions..\n\\begin{dfn}[Arakelian set]\\label{def:Arak}\nLet $F\\subset \\mathbb{C}$ be a closed set. We say that $F$ is an Arakelian set, if and only if the following points hold:\\\\[5pt]\n1. $F$ is without holes;\\\\[5pt]\n2. The union of all holes of $F\\cup K$ is bounded, for any compact subset $K\\subset \\mathbb{C}$. \n\\end{dfn}\n\\begin{rmk}\\label{rem:K}\nPoint 2. in Definition \\ref{def:Arak} is equivalent to the one where $K$ is replaced by $\\overline{D}_0(n)=\\{z\\in\\mathbb{C}: |z|\\leq n\\}$, for any $n\\in\\mathbb{N}$. \n\\end{rmk}\n\\begin{rmk}\nIn fact, Definition \\ref{def:Arak} appears in a later work \\cite{RR}, where a new proof of Arakelian's theorem was given (see also \\cite{Got}). The original conditions in \\cite{Ar} are:\\\\[5pt]\n1. $\\mathbb{C}\\setminus F$ is connected;\\\\[5pt]\n2. $\\mathbb{C}\\setminus F$ is locally connected at infinity.\\\\[5pt]\nNevertheless, they are easily seen to be equivalent.\n\\end{rmk}\n\n\\begin{dfn}[Uniform approximation set]\nLet $F\\subset \\mathbb{C}$ be a closed set. We say that $F$ is a set of uniform approximation, if for every $\\epsilon>0$ and $f\\in A(F)$, there exists an entire function $g$ such that $|f(z)-g(z)|<\\epsilon$, for all $z\\in F$. \n\\end{dfn}\n\n\\subsection{Main result}\n\nWe can now state the main result:\n\\begin{thm}\\label{thm:main}\nA closed set $F\\subset \\mathbb{C}$ is an Arakelian set, if and only if \nfor every $f\\in A(F)$ with no zeros in $F$, there exists $g\\in A(F)$ such that $e^g=f$ in $F$.\n\\end{thm}\nAt first glance, the existence of logarithmic branches of functions $f\\in A(F)$ might seem to have little to do with the conditions in Definition \\ref{def:Arak}. Intuitively, it is easier to make the connection by first stating yet another characterization of Arakelian sets, proved in \\cite{F}, in terms of simply connected neighborhoods of $F$, that is to say, open sets $V\\supset F$ whose connected components are simply connected.\n\\begin{thm}\\label{thm:F}\nA closed set $F\\subset \\mathbb{C}$ is an Arakelian set if and only if it possesses a neighborhood basis of simply connected open sets. \n\\end{thm}\nAt least one of the directions in Theorem \\ref{thm:main} is clearer now. If a function $f\\in A(F)$ could be extended to a neighborhood of $F$, then the existence of an intermediate simply connected neighborhood would allow for a logarithmic branch of $f$ to be well-defined in the latter.\n\nTheorem \\ref{thm:Ar} has been successfully extended to relatively closed subsets of planar domains $G\\subset \\mathbb{C}$ \\cite{Ar2} and non-planar Riemann surfaces of finite genus \\cite{GH,Sch}. One of the outstanding open problems in complex approximation is to characterize sets of uniform approximation in Riemann surfaces of infinite genus, where Arakelian sets (Definition \\ref{def:Arak}) are not always sets of uniform approximation, see \\cite{BG} for a counterexample. Moreover, it was shown by Scheinberg \\cite{Sch} that such a characterization, if it exists, cannot be purely topological, but it has to take into account the complex analytic structure of the given Riemann surface.\n\nThe one direction of the characterization in Theorem \\ref{thm:main} is contained in \\cite{GP}. We include the proof here for the sake of completeness.\n\\begin{prp}[Gauthier-Pouryayevali \\cite{GP}]\\label{prop:GP}\nSuppose that the closed set $F\\subset \\mathbb{C}$ is an Arakelian set. For every function $f\\in A(F)$ with no zeros in $F$, there exists a function $g\\in A(F)$ such that $e^g=f$ in $F$.\n\\end{prp}\n\\begin{proof}\nBy Tietze's extension theorem, there exists a continuous extension of $f$ in $\\mathbb{C}$, which we\ndenote by $\\widetilde{f}:\\mathbb{C}\\to\\C$. Our assumption implies that the open set $U=\\mathbb{C}\\setminus \\widetilde{f}^{-1}(\\{0\\})$ contains $F$.\nThus, by Theorem \\ref{thm:F} there exists a simply connected open set $V$ with $F\\subset V\\subset U$.\nIf we consider the covering map $\\exp: \\mathbb{C}\\to\\mathbb{C}\\setminus\\{0\\}$, then the latter implies that\n$\\widetilde{f}_{\\big|V}:V\\to\\mathbb{C}\\setminus\\{0\\}$ can be lifted\nto a continuous function $\\widetilde{g}:V\\to\\mathbb{C}$, such that $\\widetilde{f}_{\\big|V}=e^{\\widetilde{g}}$.\nThe function $g=\\widetilde{g}_{\\big|F}$ is obviously continuous and $e^g=f$ in $F$.\nSince $f_{\\big|F^0}$ is holomorphic, by Lemma \\ref{lem:Car}, $g$ is also holomorphic in $F^0$.\nThus, $g\\in A(F)$ and the proof of the proposition is complete.\n\\end{proof}\nThe reverse direction is contained in the following proposition that, combined with Proposition \\ref{prop:GP}, completes the proof of Theorem \\ref{thm:main}. \n\\begin{prp}\\label{prop:FNP}\nLet $F\\subset\\mathbb{C}$ be a closed set, such that for every $f\\in A(F)$ with no zeros in $F$, there exists a $g\\in A(F)$ satisfying $e^g=f$ in $F$. Then $F$ is an Arakelian set.\n\\end{prp}\n\\begin{proof}\n{\\it Step 1. $F$ has no holes.} Arguing by contradiction, suppose that $F$ has a hole $B$. Let $\\zeta\\in B$ and let $f(z)=z-\\zeta$. Since $f$ has no zeros in $F$, there exists $g\\in A(F)$, such that $e^g=f$ in $F$.\n\n\\begin{thm}[Arakelian \\cite{Ar}]\\label{thm:Ar}\nA closed set $F\\subset \\mathbb{C}$ is a set of uniform approximation if and only if $F$ is an Arakelian set. \n\\end{thm}\n\n\\begin{thm}\\label{thm:main}\nA closed set $F\\subset \\mathbb{C}$ is an Arakelian set, if and only if \nfor  every $f\\in A(F)$ with no zeros in $F$, there exists $g\\in A(F)$ such that $e^g=f$ in $F$.\n\\end{thm}", "post_theorem_intro_text_len": 3824, "post_theorem_intro_text": "\\subsection{Main result}\n\nWe can now state the main result:\n\\begin{thm}\\label{thm:main}\nA closed set $F\\subset \\mathbb{C}$ is an Arakelian set, if and only if \nfor  every $f\\in A(F)$ with no zeros in $F$, there exists $g\\in A(F)$ such that $e^g=f$ in $F$.\n\\end{thm}\nAt first glance, the existence of logarithmic branches of functions $f\\in A(F)$ might seem to have little to do with the conditions in Definition \\ref{def:Arak}. Intuitively, it is easier to make the connection by first stating yet another characterization of Arakelian sets, proved in \\cite{F}, in terms of simply connected neighborhoods of $F$, that is to say, open sets $V\\supset F$ whose connected components are simply connected.\n\\begin{thm}\\label{thm:F}\nA closed set $F\\subset \\mathbb{C}$ is an Arakelian set if and only if it possesses a neighborhood basis of simply connected open sets. \n\\end{thm}\nAt least one of the directions in Theorem \\ref{thm:main} is clearer now. If a function $f\\in A(F)$ could be extended to a neighborhood of $F$, then the existence of an intermediate simply connected neighborhood would allow for a logarithmic branch of $f$ to be well-defined in the latter.\n\nWe should give credit here to the influential work of Gauthier-Pouryayevali \\cite{GP}, which inspired us to pursue the above characterizations. In fact, the directions in both Theorems \\ref{thm:main}, \\ref{thm:F} where we assume that $F$ is an Arakelian set, are contained in \\cite{GP}.\n\nOur contribution in the present paper is to prove the reverse direction stated in Theorem \\ref{thm:main}. We argue by contradiction, assuming $F$ is not an Arakelian set. Then a special function $f\\in A(F)$ is constructed, which is finally proven not to satisfy the initial assumption. Interestingly, the function $f$ that we use is entire and it is produced by invoking the Weierstrass factorization theorem, unlike the meromorphic functions that are usually employed in such proofs \\cite[Chapter IV, \\S2.$C_2$]{Gaier}. \n\n\\subsection{Outlook}\n\nTheorem \\ref{thm:Ar} has been successfully extended to relatively closed subsets of planar domains $G\\subset \\mathbb{C}$ \\cite{Ar2} and non-planar Riemann surfaces of finite genus \\cite{GH,Sch}. One of the outstanding open problems in complex approximation is to characterize sets of uniform approximation in Riemann surfaces of infinite genus, where Arakelian sets (Definition \\ref{def:Arak}) are not always sets of uniform approximation, see \\cite{BG} for a counterexample. Moreover, it was shown by Scheinberg \\cite{Sch} that such a characterization, if it exists, cannot be purely topological, but it has to take into account the complex analytic structure of the given Riemann surface. \n\nIt would be interesting to seek potential extensions of Theorem \\ref{thm:main} to Arakelian sets of planar domains $G\\subset \\mathbb{C}$, as in \\cite{Ar2}. The present logarithmic characterization and our method of proof could be naturally extended to Arakelian sets of simply connected domains $G\\subset\\mathbb{C}$, see \\cite[Corollary 2.15]{F} for one of the directions. However, we do not know how it could be generalized to planar domains $G$ with holes, even when $G$ is an annulus. Finally, coming up with a generalization of our characterization that would take into account the analytic structure of a given domain, could hopefully be of use for characterizing uniform approximation sets in Riemann surfaces.\n\n\\subsection{Acknowledgments}\n\nWe would like to thank Paul M. Gauthier for his interest in this work and insightful comments.\nG.F. and S.P. would like to acknowledge the support of the ERC starting grant 101078061 SINGinGR, under the European Union's Horizon Europe program for research and innovation, and the H.F.R.I. grant 7126, under the 3rd call for H.F.R.I. research projects to support post-doctoral researchers.", "sketch": "For the direction assuming $F$ is Arakelian, the text explains that using Theorem~\\ref{thm:F} (Arakelian $\\Leftrightarrow$ a neighborhood basis of simply connected open sets), “at least one of the directions in Theorem \\ref{thm:main} is clearer now”: if $f\\in A(F)$ “could be extended to a neighborhood of $F$,” then “the existence of an intermediate simply connected neighborhood would allow for a logarithmic branch of $f$ to be well-defined in the latter,” yielding $g$ with $e^g=f$ on $F$.\n\nFor the reverse direction in Theorem~\\ref{thm:main}, the authors state: “We argue by contradiction, assuming $F$ is not an Arakelian set. Then a special function $f\\in A(F)$ is constructed, which is finally proven not to satisfy the initial assumption.” They add that this $f$ “is entire and it is produced by invoking the Weierstrass factorization theorem,” in contrast to “the meromorphic functions that are usually employed in such proofs.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "[Arakelian \\cite{Ar}]\\label{thm:Ar}\nA closed set $F\\subset \\mathbb{C}$ is a set of uniform approximation if and only if $F$ is an Arakelian set.", "theorem_type": ["Biconditional or Equivalence"], "mcq": {"question": "Let $F\\subset \\mathbb{C}$ be a closed set, and let\n\\[\nA(F)=\\{f:F\\to\\mathbb{C}: f \\text{ is holomorphic on } F^\\circ \\text{ and continuous on } F\\}.\n\\]\nSay that $F$ is a set of uniform approximation if for every $\\varepsilon>0$ and every $f\\in A(F)$, there exists an entire function $g$ such that $|f(z)-g(z)|<\\varepsilon$ for all $z\\in F$. Which of the following statements is equivalent to $F$ being a set of uniform approximation?", "correct_choice": {"label": "A", "text": "$F$ is an Arakelian set; that is, $F$ has no holes, and for every compact set $K\\subset \\mathbb{C}$, the union of all holes of $F\\cup K$ is bounded, where a hole of a closed set means a bounded connected component of its complement in $\\mathbb{C}$."}, "choices": [{"label": "B", "text": "$F$ is an Arakelian set; that is, $F$ has no holes, and there exists a compact set $K\\subset \\mathbb{C}$ such that the union of all holes of $F\\cup K$ is bounded, where a hole of a closed set means a bounded connected component of its complement in $\\mathbb{C}$."}, {"label": "C", "text": "$F$ has no holes; equivalently, $\\mathbb{C}\\setminus F$ has no bounded connected components."}, {"label": "D", "text": "$F$ is an Arakelian set; that is, $\\mathbb{C}\\setminus F$ is connected and, for every compact set $K\\subset \\mathbb{C}$, the complement $\\mathbb{C}\\setminus (F\\cup K)$ is connected."}, {"label": "E", "text": "$F$ is an Arakelian set; that is, $F$ has no holes, and for every compact set $K\\subset \\mathbb{C}$, the union of all connected components of $\\mathbb{C}\\setminus (F\\cup K)$ is bounded, where a hole of a closed set means a bounded connected component of its complement in $\\mathbb{C}$."}], "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 quantifier over compact sets", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the condition on holes of $F\\cup K$ for all compact $K$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "replaced boundedness of holes of $F\\cup K$ by connectedness of full complement after adjoining $K$", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "boundedness applies only to holes, not to all components of the complement", "template_used": "wildcard"}]}}
{"id": "2512.03188v1", "paper_link": "http://arxiv.org/abs/2512.03188v1", "theorems_cnt": 2, "theorem": {"env_name": "prop", "content": "\\label{prop:bound_a}\n    If $a!b!=c!$ is a solution of class $k$, i.e., $c-b = k$, then $k < a < k + 2 \\lceil \\log_2 c \\rceil$.", "start_pos": 5961, "end_pos": 6112, "label": "prop:bound_a"}, "ref_dict": {"cor:SerreSet": "\\begin{cor} \\label{cor:SerreSet}\n    Fix $k > 1$.  For at least a $1/k$ natural density subset of primes $p$, the equation\n    $$\n    (p-1)! b! = (b+k)!\n    $$\n    has no solutions for integers $b$.\n\\end{cor}"}, "pre_theorem_intro_text_len": 3235, "pre_theorem_intro_text": "Erd\\H{o}s asks the following problem in \\cite{ErdosGraham1980}: for which integers $a_1, \\ldots, a_t$ with $t \\geq 2$ and $a_i \\geq 2$ for each $i$ does there exist an integer $c$ so that $\\prod_{i=1}^t a_i! = c!$?  This question, which is open even for $t=2$, has been the inspiration for dozens of papers in elementary number theory since at least the 1970s.  It is widely believed that there are only finitely many ``nontrivial'' solutions, i.e., those for which $c - \\max_i a_i > 1$.  In 1991, Erd\\H{o}s \\cite{Erdos91} showed that for sufficiently large $c$, $k \\leq 5 \\log\\log c$; the constant was subsequently lowered, most recently by \\cite{BkhatRamachandra10}, to $(1+o(1))/\\log 2 \\approx 1.44$. In 2007, Luca \\cite{Luca07} showed that there are only finitely many nontrivial solutions if the notorious $abc$ Conjecture holds, and Nair-Shorey \\cite{NAIR2016307} refined this to a complete list of solutions conditional on an explicit form of the $abc$ Conjecture due to Baker.  In the same paper, Luca showed that the set of $c$'s that arise in solutions is of asymptotic density zero.  Here, we focus on the $t=2$ case, i.e., $a!b!=c!$, showing that the possible values of $a$ are also rather sparse.\n\nThe following definition will play a key role throughout.\n\n\\begin{defn}\nA solution to $a! b! = c!$ with $a<b$ in the integers is said to be ``class k'' if $c - b = k$.\n\\end{defn}\n\nNote that class $0$ solutions must have $a=0$ or $a=1$ and $b=c$; it is easy to see that the solution is class $1$ iff $b=a!-1$ and $c=a!$.  Thus, a solution to $a!b!=c!$ is said to be ``trivial'' if $k \\leq 1$. It is conjectured that the only non-trivial solution is the triple $(6,7,10)$.  Already the conjecture that there are no class $2$ solutions is open (see \\cite{BerendHarmse06}).  In the definition above, and throughout the sequel, we assume that $a < b$.  It is straightforward to see that $a \\neq b$ for nontrivial solutions.  Indeed, if $a=b$, then $c!=a!^2$.  However, writing $\\nu_p(n!)$ for the $p$-adic order of $n$, this implies that $\\nu_p(c!)=2\\nu_p(a!)$ for every prime $p$, contradicting that $\\nu_p(c!) = 1$ for every prime in $(c/2,c]$, which exists by Bertrand's Postulate.\n\nIn Section \\ref{sec:primes}, we show that (Corollary \\ref{cor:SerreSet}) for any fixed $k>1$, there exists $1/k$ natural density fraction of primes $p$ such that if $a = p-1$, there are no solutions to $a!b!=c!$.  The proof makes use of some interesting elementary number theory: old results of Westland-Fl\\\"uge on irreducible polynomials which are sparse in the falling-factorial basis, Wilson's Theorem, and Dirichlet's Theorem.  Then, in Section \\ref{sec:equidistribution}, by analyzing the asymptotics of falling factorials, we show that $c$ falls into a small interval defined by $a$ and $k$, implying that if $\\sqrt[k]{a!} \\pmod 1$ is sufficiently equidistributed as $a,k \\rightarrow \\infty$ -- a much stronger version of which we conjecture is true -- then the set of $a$’s that appear in a solution is also of asymptotic density zero.  Below we write $x^{\\underline{k}}$ for the $k$-th falling factorial of $x$, i.e., $x(x-1)\\cdots (x-k+1)$.\n\nHere we collect a few useful bounds on possible solutions, especially, that $k < a < b < c$.", "context": "Erd\\H{o}s asks the following problem in \\cite{ErdosGraham1980}: for which integers $a_1, \\ldots, a_t$ with $t \\geq 2$ and $a_i \\geq 2$ for each $i$ does there exist an integer $c$ so that $\\prod_{i=1}^t a_i! = c!$? This question, which is open even for $t=2$, has been the inspiration for dozens of papers in elementary number theory since at least the 1970s. It is widely believed that there are only finitely many ``nontrivial'' solutions, i.e., those for which $c - \\max_i a_i > 1$. In 1991, Erd\\H{o}s \\cite{Erdos91} showed that for sufficiently large $c$, $k \\leq 5 \\log\\log c$; the constant was subsequently lowered, most recently by \\cite{BkhatRamachandra10}, to $(1+o(1))/\\log 2 \\approx 1.44$. In 2007, Luca \\cite{Luca07} showed that there are only finitely many nontrivial solutions if the notorious $abc$ Conjecture holds, and Nair-Shorey \\cite{NAIR2016307} refined this to a complete list of solutions conditional on an explicit form of the $abc$ Conjecture due to Baker. In the same paper, Luca showed that the set of $c$'s that arise in solutions is of asymptotic density zero. Here, we focus on the $t=2$ case, i.e., $a!b!=c!$, showing that the possible values of $a$ are also rather sparse.\n\nThe following definition will play a key role throughout.\n\n\\begin{defn}\nA solution to $a! b! = c!$ with $a<b$ in the integers is said to be ``class k'' if $c - b = k$.\n\\end{defn}\n\nNote that class $0$ solutions must have $a=0$ or $a=1$ and $b=c$; it is easy to see that the solution is class $1$ iff $b=a!-1$ and $c=a!$. Thus, a solution to $a!b!=c!$ is said to be ``trivial'' if $k \\leq 1$. It is conjectured that the only non-trivial solution is the triple $(6,7,10)$. Already the conjecture that there are no class $2$ solutions is open (see \\cite{BerendHarmse06}). In the definition above, and throughout the sequel, we assume that $a < b$. It is straightforward to see that $a \\neq b$ for nontrivial solutions. Indeed, if $a=b$, then $c!=a!^2$. However, writing $\\nu_p(n!)$ for the $p$-adic order of $n$, this implies that $\\nu_p(c!)=2\\nu_p(a!)$ for every prime $p$, contradicting that $\\nu_p(c!) = 1$ for every prime in $(c/2,c]$, which exists by Bertrand's Postulate.\n\nIn Section \\ref{sec:primes}, we show that (Corollary \\ref{cor:SerreSet}) for any fixed $k>1$, there exists $1/k$ natural density fraction of primes $p$ such that if $a = p-1$, there are no solutions to $a!b!=c!$. The proof makes use of some interesting elementary number theory: old results of Westland-Fl\\\"uge on irreducible polynomials which are sparse in the falling-factorial basis, Wilson's Theorem, and Dirichlet's Theorem. Then, in Section \\ref{sec:equidistribution}, by analyzing the asymptotics of falling factorials, we show that $c$ falls into a small interval defined by $a$ and $k$, implying that if $\\sqrt[k]{a!} \\pmod 1$ is sufficiently equidistributed as $a,k \\rightarrow \\infty$ -- a much stronger version of which we conjecture is true -- then the set of $a$’s that appear in a solution is also of asymptotic density zero. Below we write $x^{\\underline{k}}$ for the $k$-th falling factorial of $x$, i.e., $x(x-1)\\cdots (x-k+1)$.\n\nHere we collect a few useful bounds on possible solutions, especially, that $k < a < b < c$.\n\n\\begin{cor} \\label{cor:SerreSet}\n    Fix $k > 1$.  For at least a $1/k$ natural density subset of primes $p$, the equation\n    $$\n    (p-1)! b! = (b+k)!\n    $$\n    has no solutions for integers $b$.\n\\end{cor}", "full_context": "Erd\\H{o}s asks the following problem in \\cite{ErdosGraham1980}: for which integers $a_1, \\ldots, a_t$ with $t \\geq 2$ and $a_i \\geq 2$ for each $i$ does there exist an integer $c$ so that $\\prod_{i=1}^t a_i! = c!$? This question, which is open even for $t=2$, has been the inspiration for dozens of papers in elementary number theory since at least the 1970s. It is widely believed that there are only finitely many ``nontrivial'' solutions, i.e., those for which $c - \\max_i a_i > 1$. In 1991, Erd\\H{o}s \\cite{Erdos91} showed that for sufficiently large $c$, $k \\leq 5 \\log\\log c$; the constant was subsequently lowered, most recently by \\cite{BkhatRamachandra10}, to $(1+o(1))/\\log 2 \\approx 1.44$. In 2007, Luca \\cite{Luca07} showed that there are only finitely many nontrivial solutions if the notorious $abc$ Conjecture holds, and Nair-Shorey \\cite{NAIR2016307} refined this to a complete list of solutions conditional on an explicit form of the $abc$ Conjecture due to Baker. In the same paper, Luca showed that the set of $c$'s that arise in solutions is of asymptotic density zero. Here, we focus on the $t=2$ case, i.e., $a!b!=c!$, showing that the possible values of $a$ are also rather sparse.\n\nThe following definition will play a key role throughout.\n\n\\begin{defn}\nA solution to $a! b! = c!$ with $a<b$ in the integers is said to be ``class k'' if $c - b = k$.\n\\end{defn}\n\nNote that class $0$ solutions must have $a=0$ or $a=1$ and $b=c$; it is easy to see that the solution is class $1$ iff $b=a!-1$ and $c=a!$. Thus, a solution to $a!b!=c!$ is said to be ``trivial'' if $k \\leq 1$. It is conjectured that the only non-trivial solution is the triple $(6,7,10)$. Already the conjecture that there are no class $2$ solutions is open (see \\cite{BerendHarmse06}). In the definition above, and throughout the sequel, we assume that $a < b$. It is straightforward to see that $a \\neq b$ for nontrivial solutions. Indeed, if $a=b$, then $c!=a!^2$. However, writing $\\nu_p(n!)$ for the $p$-adic order of $n$, this implies that $\\nu_p(c!)=2\\nu_p(a!)$ for every prime $p$, contradicting that $\\nu_p(c!) = 1$ for every prime in $(c/2,c]$, which exists by Bertrand's Postulate.\n\nIn Section \\ref{sec:primes}, we show that (Corollary \\ref{cor:SerreSet}) for any fixed $k>1$, there exists $1/k$ natural density fraction of primes $p$ such that if $a = p-1$, there are no solutions to $a!b!=c!$. The proof makes use of some interesting elementary number theory: old results of Westland-Fl\\\"uge on irreducible polynomials which are sparse in the falling-factorial basis, Wilson's Theorem, and Dirichlet's Theorem. Then, in Section \\ref{sec:equidistribution}, by analyzing the asymptotics of falling factorials, we show that $c$ falls into a small interval defined by $a$ and $k$, implying that if $\\sqrt[k]{a!} \\pmod 1$ is sufficiently equidistributed as $a,k \\rightarrow \\infty$ -- a much stronger version of which we conjecture is true -- then the set of $a$’s that appear in a solution is also of asymptotic density zero. Below we write $x^{\\underline{k}}$ for the $k$-th falling factorial of $x$, i.e., $x(x-1)\\cdots (x-k+1)$.\n\nHere we collect a few useful bounds on possible solutions, especially, that $k < a < b < c$.\n\n\\begin{cor} \\label{cor:SerreSet}\n    Fix $k > 1$.  For at least a $1/k$ natural density subset of primes $p$, the equation\n    $$\n    (p-1)! b! = (b+k)!\n    $$\n    has no solutions for integers $b$.\n\\end{cor}\n\nHere we collect a few useful bounds on possible solutions, especially, that $k < a < b < c$.\n\n\\section{Solutions Modulo a Prime} \\label{sec:primes}\n\n\\begin{prop} There are no class $2$ solutions to $a!b!=c!$ in positive integers if $a+1$ is a prime congruent to $5 \\pmod{6}$.\n\\end{prop}\n\\begin{proof} \nNote that $c-b=2$ and $a!b!=c!$ implies $a! = c(c-1)$. Suppose $a!=c(c-1)$ where $a+1=p$ is a prime congruent to $5 \\pmod{6}$. Then\n$$\nc^2 - c - a! = 0\n$$ \nso we have $c = \\frac{1\\pm \\sqrt{1-4(-a!)}}{2}$. Thus, $1+4a! = 1 + 4(p-1)!$ is a perfect square. By Wilson's Theorem, $1+4(p-1)! \\equiv 1+4(-1) = -3 \\pmod{p}$, so $-3$ is a quadratic residue modulo $p$. However, this contradicts Lemma \\ref{lem:-3}.\n\\end{proof}\n\n\\begin{prop} \\label{thm:pIs4} There are no class $4$ solutions to $a!b!=c!$ in positive integers if $a+1$ is a prime congruent to $2$ or $3 \\pmod{5}$.\n\\end{prop}\n\\begin{proof}\nSuppose $a! = c(c-1)(c-2)(c-3)$, where $a=p-1$ for some prime $p$. Note that $b>0$ implies $c=b+4>4$, so $c!/b! \\geq 24 > 3!$, so we may assume $a>3$ and so $p>4$; furthermore, if $b>0$ then $4! = c(c-1)(c-2)(c-3)$ only has the solution $c=4$, but then $b = c-4 = 0$, a contradiction, so $p \\neq 5$. Let $r \\equiv c/2 \\pmod{p}$, and apply Wilson's Theorem to conclude that\n$$\n2r(2r-1)(2r-2)(2r-3) \\equiv -1 \\pmod{p}.\n$$\nNow, \n$$\n2r(2r-1)(2r-2)(2r-3) = (4r^2-6r+1)^2-1,\n$$\nso \n$$\n(4r^2-6r+1)^2 \\equiv 0 \\pmod{p},\n$$\ni.e., $p | 4r^2 - 6r + 1$. However, $(2r-3/2)^2 - 5/4 = 4r^2 - 6r + 1$, so $5/4$ is a quadratic residue mod $p$, which happens iff $5$ is a quadratic residue mod $p$. It is known this occurs only for primes $p \\equiv 1,4 \\pmod{5}$.\n\\end{proof}\n\n\\begin{theorem} \\label{thm:PositiveDensityIfIrreducible}Suppose $F_k(x) = x(x-1)\\cdots (x-k+1)+1$ is irreducible over $\\mathbb{Z}$. Let $P_0$ be the set of primes so that, if $a=p-1$ for some $p \\in P_0$, then $a!b!=c!$ has no class $k$ solutions in positive integers. Then the natural density of $P_0$ in the primes is at least $1/k$.\n\\end{theorem}\n\\begin{proof}\n Suppose $a!b!=c!$ is class $k$ and $a = p-1$ for some prime $p$. Then $c = b+k$ and $a! = c!/b! = F_k(c)-1$. Then, reducing this equation modulo $p$ yields $c^{\\underline{k}} \\equiv a! \\pmod{p}$, and by Wilson's Theorem, this is equivalent to $F_k(x) \\equiv 0 \\pmod{p}$. However, by Theorem 1 and 2 of \\cite{Serre03}, the set $P_0$ of primes $p$ for which $F_k(x)$ does not have solution modulo $p$ has positive density, and the density is at least $1/k$.\n\\end{proof}\n\n\\begin{prop}\\label{thm:WestlundFluge}\n For $k \\geq 2$, $F_k(x)$ is reducible over $\\mathbb{Z}$ iff $k=4$. \n\\end{prop}\n\\begin{proof}\n This is an immediate consequence of a result by Westlund and Fl\\\"{u}ge, answering a question of Schur, as discussed in \\cite{DorwartOre33}.\n\\end{proof}\n\n\\begin{cor} \\label{cor:SerreSet}\n Fix $k > 1$. For at least a $1/k$ natural density subset of primes $p$, the equation\n $$\n (p-1)! b! = (b+k)!\n $$\n has no solutions for integers $b$.\n\\end{cor}\n\\begin{proof}\n By Theorem \\ref{thm:WestlundFluge}, $F_k$ is irreducible if $k \\neq 4$, so Theorem \\ref{thm:PositiveDensityIfIrreducible} implies that $a!b!=c!$ has no solutions in the integers when $c = b+k$ and $a=p-1$, for an at least $1/k$ density subset of the primes. Proposition \\ref{thm:pIs4} yields the remaining case, by Dirichlet's Theorem on primes in arithmetic progressions.\n\\end{proof}\n\n\\begin{prop}\n For $p$ prime, the equation\n $$\n (p-1)!b! = (b+k)!\n $$\n has at most $nk/p + 1$ solutions in the integers with $b \\leq n$.\n\\end{prop}\n\\begin{proof}\n Suppose $(p-1)! b! = (b+k)!$. Then, $(p-1)! = F_k(b)-1$. Taking this equation modulo $p$ and applying Wilson's Theorem gives\n $$\n -1 \\equiv F_k(b) - 1 \\pmod{p},\n $$\n i.e., $F_k(b) \\equiv 0 \\pmod{p}$. However, this equation has at most $k$ solutions $A_k \\subseteq \\mathbb{Z}/p\\mathbb{Z}$. If $b \\pmod{p} \\not \\in A_k$, then $(p-1)!b! \\neq (b+k)!$. Since the set of $b \\in [1,n]$ so that $b \\pmod{p} \\in A_k$ has cardinality $\\lceil nk/p \\rceil \\leq nk/p + 1$, the conclusion follows.\n\\end{proof}", "post_theorem_intro_text_len": 874, "post_theorem_intro_text": "\\begin{proof}\n    Write $s_q(n)$ for the sum of the base-$q$ digits of $n$. Clearly, $\\nu_p(a!)+\\nu_p(b!)=\\nu_p(c!)$.  Applying Legendre's Formula, we obtain\n    $$\n    \\frac{a-s_p(a)}{p-1} + \\frac{b-s_p(b)}{p-1} = \\frac{c-s_p(c)}{p-1},\n    $$\n    from which it follows that $a-(c-b) = s_p(a)+s_p(b)-s_p(c)$ for every prime $p$.\n    Since $a < b < c$, in particular\n    $$\n    | s_2(a)+s_2(b)-s_2(c) | < 2 \\lceil \\log_2 c \\rceil .\n    $$\n    Thus, $|a-k| < 2 \\lfloor \\log_2 c \\rfloor$.  Now, we argue that $a > k$.  First,\n    \\begin{align*}\n    a! & = \\frac{c!}{b!} = c(c-1)\\cdots(c-k+1) \\\\\n    & > (c-k+1)^k = (b+1)^k \\geq (a+2)^k.\n    \\end{align*}\n    Then taking the log of both sides above and applying Stirling's approximation (that $\\log a! < a \\log a$) gives\n    $$\n    a \\log a > \\log a! > k \\log (a+2)\n    $$\n    so that $k < a \\log a / \\log(a+2) < a$.\n\\end{proof}", "sketch": "Using Legendre's formula with digit sums $s_p(n)$, from $\\nu_p(a!)+\\nu_p(b!)=\\nu_p(c!)$ one gets\n\\[\n\\frac{a-s_p(a)}{p-1}+\\frac{b-s_p(b)}{p-1}=\\frac{c-s_p(c)}{p-1},\n\\]\nso for every prime $p$,\n\\[\na-(c-b)=s_p(a)+s_p(b)-s_p(c).\n\\]\nWith $k=c-b$ and $a<b<c$, this yields the base-$2$ bound\n\\[\n|s_2(a)+s_2(b)-s_2(c)|<2\\lceil\\log_2 c\\rceil,\n\\]\nhence $|a-k|<2\\lfloor\\log_2 c\\rfloor$, giving the upper bound $a<k+2\\lceil\\log_2 c\\rceil$.\n\nTo show $a>k$, rewrite\n\\[\na!=\\frac{c!}{b!}=c(c-1)\\cdots(c-k+1)>(c-k+1)^k=(b+1)^k\\ge (a+2)^k.\n\\]\nTaking logs and using Stirling's approximation (namely $\\log a!<a\\log a$) gives\n\\[\na\\log a>\\log a!>k\\log(a+2),\n\\]\nso $k<a\\log a/\\log(a+2)<a$, i.e. $k<a$.", "expanded_sketch": "Using Legendre's formula with digit sums $s_p(n)$, from $\\nu_p(a!)+\\nu_p(b!)=\\nu_p(c!)$ one gets\n\\[\n\\frac{a-s_p(a)}{p-1}+\\frac{b-s_p(b)}{p-1}=\\frac{c-s_p(c)}{p-1},\n\\]\nso for every prime $p$,\n\\[\na-(c-b)=s_p(a)+s_p(b)-s_p(c).\n\\]\nWith $k=c-b$ and $a<b<c$, this yields the base-$2$ bound\n\\[\n|s_2(a)+s_2(b)-s_2(c)|<2\\lceil\\log_2 c\\rceil,\n\\]\nhence $|a-k|<2\\lfloor\\log_2 c\\rfloor$, giving the upper bound $a<k+2\\lceil\\log_2 c\\rceil$.\n\nTo show $a>k$, rewrite\n\\[\na!=\\frac{c!}{b!}=c(c-1)\\cdots(c-k+1)>(c-k+1)^k=(b+1)^k\\ge (a+2)^k.\n\\]\nTaking logs and using Stirling's approximation (namely $\\log a!<a\\log a$) gives\n\\[\na\\log a>\\log a!>k\\log(a+2),\n\\]\nso $k<a\\log a/\\log(a+2)<a$, i.e. $k<a$.", "expanded_theorem": "\\label{prop:bound_a}\nIf $a!b!=c!$ is a solution of class $k$, i.e., $c-b = k$, then $k < a < k + 2 \\lceil \\log_2 c \\rceil$.", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let $a,b,c$ be nonnegative integers with $a<b$ such that $a!\\,b!=c!$. A solution is said to be of class $k$ when $c-b=k$. Under these assumptions, which statement about $a$ must hold? (Here $\\lceil \\cdot \\rceil$ denotes the ceiling function and $\\log_2$ is the base-$2$ logarithm.)", "correct_choice": {"label": "A", "text": "If $a!\\,b!=c!$ and $c-b=k$, then $k<a<k+2\\lceil \\log_2 c\\rceil$."}, "choices": [{"label": "B", "text": "If $a!\\,b!=c!$ and $c-b=k$, then $k\\le a<k+2\\lceil \\log_2 c\\rceil$."}, {"label": "C", "text": "If $a!\\,b!=c!$ and $c-b=k$, then $a<k+2\\lceil \\log_2 c\\rceil$."}, {"label": "D", "text": "If $a!\\,b!=c!$ and $c-b=k$, then $k<a<k+2\\lceil \\log_2 a\\rceil$."}, {"label": "E", "text": "If $a!\\,b!=c!$ and $c-b=k$, then for every prime $p$ one has $k<a<k+2\\lceil \\log_p c\\rceil$."}], "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_lower_bound_k_lt_a", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped_lower_bound_k_lt_a", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "logarithm_depends_on_c_not_a", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "special_role_of_base_2_digit_sum_bound", "template_used": "wildcard"}]}}
{"id": "2512.03527v1", "paper_link": "http://arxiv.org/abs/2512.03527v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{main_theorem} Let $X$ be a Gorenstein del Pezzo\nsurface of Picard rank $1$ over an algebraically closed field of\ncharacteric zero which is not isomorphic to the surface $S'(E_8)$. Then\n$X$ admits an int-amplified endomorphism (i.e. an endomorphism of degree\n$>1$) if and only if $X$ is a quotient of a toric variety by a finite\ngroup acting freely in codimension one that preserves the open torus.", "start_pos": 18365, "end_pos": 18798, "label": "main_theorem"}, "ref_dict": {"main_theorem": "\\begin{theorem}\\label{main_theorem} Let $X$ be a Gorenstein del Pezzo\nsurface of Picard rank $1$ over an algebraically closed field of\ncharacteric zero which is not isomorphic to the surface $S'(E_8)$. Then\n$X$ admits an int-amplified endomorphism (i.e. an endomorphism of degree\n$>1$) if and only if $X$ is a quotient of a toric variety by a finite\ngroup acting freely in codimension one that preserves the open torus.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 3666, "pre_theorem_intro_text": "A surjective endomorphism of a projective algebraic variety $f\\colon X\n\\to X$ is said to be \\textit{int-amplified} if there exists an ample\nCartier divisor $A$ on $X$ such that $f^*A - A$ is ample. Equivalently,\n$f$ is int-amplified if all eigenvalues of the operator $f^* \\colon\nN^1(X)_{\\mathbb{C}} \\to N^1(X)_{\\mathbb{C}}$ have magnitude greater than $1$\n\\cite[Theorem 1.1]{Meng-Building-Blocks}, where $N^1(X)$ is the group\nof divisors on $X$ up to numerical equivalence. Roughly speaking, an\nendomorphism is int-amplified if it expands the algebraic variety ``in\nall directions.''\n\nWhether a given variety admits any int-amplified endomorphism is an\ninteresting question of algebraic dynamics and seems to be a rare\nproperty. Abelian varieties admit int-amplified endomorphisms (e.g. the\nmultiplication-by-$m$ endomorphism) and normal toric varieties admit\nint-amplified endomorphisms (e.g. the $m$-th power Frobenius morphism).\nMeng and Zhong conjecture \\cite[Question 1.2]{Meng-Rigidity} that a\nsmooth projective rationally connected complex variety admits an\nint-amplified endomorphism if and only if it is a toric variety (see\nalso \\cite[Question 4.4]{Fakhruddin}). This has been proved for smooth\nsurfaces \\cite[Theorem 3]{Nakayama02} and smooth Fano threefolds\n\\cite[Theorem 1.4]{Meng-Zhang-Zhong}, \\cite[Theorem\n6.1]{Totaro-Log-Bott}. \n\nWhen we consider singular Fano varieties, we find examples that admit\nint-amplified endomorphisms which are not toric, but are finite\nquotients of toric varieties.\n\\begin{example}\\label{quaternion-example}\nConsider the two-dimensional representation of the quaternion group\ngiven by\n\\[ \n   i\\mapsto \\begin{pmatrix} i&0\\\\0&-i\\end{pmatrix},\\quad\n   j\\mapsto \\begin{pmatrix}0&1\\\\-1&0\\end{pmatrix},\\quad\n   k=ij\\mapsto \\begin{pmatrix}0&i\\\\i&0\\end{pmatrix}.\n\\]\nExtending this to a projective representation in $\\PGL(3)$, we can\nconsider it as an action of $Q_8$ on $\\mathbb{P}^2$. Let $X$ be the quotient of\n$\\mathbb{P}^2$ by this action. Then the image of $[0:0:1]$ is a $D_4$\nsingularity: $D_4$ singularities are not toric so $X$ cannot be a toric\nvariety. Nevertheless we can easily see that the ``Frobenius\nendomorphism'' $[x:y:z] \\mapsto [x^5:y^5:z^5]$ of $\\mathbb{P}^2$ commutes with\nthe action of $Q_8$ and thus descends to an int-amplified endomorphism\nof $X$. \n\\end{example}\n\nIn view of such examples, it is a folklore conjecture that a klt Fano\nvariety admits an int-amplified endomorphism if and only if it a finite\nquotient of a toric variety. Nakayama proved \\cite[Lemma\n2.6]{NakayamaStruct} that a quotient of a toric surface by a finite\ngroup which acts freely in codimension one and preserves the open torus\ninherits an int-amplified endomorphism from the toric variety. Nakayama\nalso proved that for a klt del Pezzo surface $X$ of Picard rank $>1$,\n$X$ admits an int-amplified endomorphism iff $X$ is a quotient of a\ntoric variety by a finite group acting freely in codimension $1$ such\nthat the action preserves the open torus \\cite[Theorem 1.3]{Nakayama}. \n\nIn this article, I study the case of Picard rank $1$ and consider del\nPezzo surfaces with Gorenstein singularities. Note that an\nendomorphism of a projective variety of Picard rank $1$ is int-amplified\nif and only if it has degree greater than one, since in this case\n$N^1(X)_{\\mathbb{C}} \\cong \\mathbb{C}$ and $f^*: N^1(X)_{\\mathbb{C}} \\to N^1(X)_{\\mathbb{C}}$ is just\nmultiplication by $(\\deg f)^{\\dim X}$. Define $S'(E_8)$ as the weighted\nsextic surface $\\{ X^5Y + X^4Z + Z^3+ W^2 = 0 \\} \\subset \\mathbb{P}(1, 1, 2, 3)$\n\\cite{Gurjar}: it is a Gorenstein del Pezzo surface of Picard rank 1.\nThen the main theorem of this paper is the following.", "context": "A surjective endomorphism of a projective algebraic variety $f\\colon X\n\\to X$ is said to be \\textit{int-amplified} if there exists an ample\nCartier divisor $A$ on $X$ such that $f^*A - A$ is ample. Equivalently,\n$f$ is int-amplified if all eigenvalues of the operator $f^* \\colon\nN^1(X)_{\\mathbb{C}} \\to N^1(X)_{\\mathbb{C}}$ have magnitude greater than $1$\n\\cite[Theorem 1.1]{Meng-Building-Blocks}, where $N^1(X)$ is the group\nof divisors on $X$ up to numerical equivalence. Roughly speaking, an\nendomorphism is int-amplified if it expands the algebraic variety ``in\nall directions.''\n\nWhether a given variety admits any int-amplified endomorphism is an\ninteresting question of algebraic dynamics and seems to be a rare\nproperty. Abelian varieties admit int-amplified endomorphisms (e.g. the\nmultiplication-by-$m$ endomorphism) and normal toric varieties admit\nint-amplified endomorphisms (e.g. the $m$-th power Frobenius morphism).\nMeng and Zhong conjecture \\cite[Question 1.2]{Meng-Rigidity} that a\nsmooth projective rationally connected complex variety admits an\nint-amplified endomorphism if and only if it is a toric variety (see\nalso \\cite[Question 4.4]{Fakhruddin}). This has been proved for smooth\nsurfaces \\cite[Theorem 3]{Nakayama02} and smooth Fano threefolds\n\\cite[Theorem 1.4]{Meng-Zhang-Zhong}, \\cite[Theorem\n6.1]{Totaro-Log-Bott}.\n\nWhen we consider singular Fano varieties, we find examples that admit\nint-amplified endomorphisms which are not toric, but are finite\nquotients of toric varieties.\n\\begin{example}\\label{quaternion-example}\nConsider the two-dimensional representation of the quaternion group\ngiven by\n\\[ \n i\\mapsto \\begin{pmatrix} i&0\\\\0&-i\\end{pmatrix},\\quad\n j\\mapsto \\begin{pmatrix}0&1\\\\-1&0\\end{pmatrix},\\quad\n k=ij\\mapsto \\begin{pmatrix}0&i\\\\i&0\\end{pmatrix}.\n\\]\nExtending this to a projective representation in $\\PGL(3)$, we can\nconsider it as an action of $Q_8$ on $\\mathbb{P}^2$. Let $X$ be the quotient of\n$\\mathbb{P}^2$ by this action. Then the image of $[0:0:1]$ is a $D_4$\nsingularity: $D_4$ singularities are not toric so $X$ cannot be a toric\nvariety. Nevertheless we can easily see that the ``Frobenius\nendomorphism'' $[x:y:z] \\mapsto [x^5:y^5:z^5]$ of $\\mathbb{P}^2$ commutes with\nthe action of $Q_8$ and thus descends to an int-amplified endomorphism\nof $X$. \n\\end{example}\n\nIn view of such examples, it is a folklore conjecture that a klt Fano\nvariety admits an int-amplified endomorphism if and only if it a finite\nquotient of a toric variety. Nakayama proved \\cite[Lemma\n2.6]{NakayamaStruct} that a quotient of a toric surface by a finite\ngroup which acts freely in codimension one and preserves the open torus\ninherits an int-amplified endomorphism from the toric variety. Nakayama\nalso proved that for a klt del Pezzo surface $X$ of Picard rank $>1$,\n$X$ admits an int-amplified endomorphism iff $X$ is a quotient of a\ntoric variety by a finite group acting freely in codimension $1$ such\nthat the action preserves the open torus \\cite[Theorem 1.3]{Nakayama}.\n\nIn this article, I study the case of Picard rank $1$ and consider del\nPezzo surfaces with Gorenstein singularities. Note that an\nendomorphism of a projective variety of Picard rank $1$ is int-amplified\nif and only if it has degree greater than one, since in this case\n$N^1(X)_{\\mathbb{C}} \\cong \\mathbb{C}$ and $f^*: N^1(X)_{\\mathbb{C}} \\to N^1(X)_{\\mathbb{C}}$ is just\nmultiplication by $(\\deg f)^{\\dim X}$. Define $S'(E_8)$ as the weighted\nsextic surface $\\{ X^5Y + X^4Z + Z^3+ W^2 = 0 \\} \\subset \\mathbb{P}(1, 1, 2, 3)$\n\\cite{Gurjar}: it is a Gorenstein del Pezzo surface of Picard rank 1.\nThen the main theorem of this paper is the following.", "full_context": "A surjective endomorphism of a projective algebraic variety $f\\colon X\n\\to X$ is said to be \\textit{int-amplified} if there exists an ample\nCartier divisor $A$ on $X$ such that $f^*A - A$ is ample. Equivalently,\n$f$ is int-amplified if all eigenvalues of the operator $f^* \\colon\nN^1(X)_{\\mathbb{C}} \\to N^1(X)_{\\mathbb{C}}$ have magnitude greater than $1$\n\\cite[Theorem 1.1]{Meng-Building-Blocks}, where $N^1(X)$ is the group\nof divisors on $X$ up to numerical equivalence. Roughly speaking, an\nendomorphism is int-amplified if it expands the algebraic variety ``in\nall directions.''\n\nWhether a given variety admits any int-amplified endomorphism is an\ninteresting question of algebraic dynamics and seems to be a rare\nproperty. Abelian varieties admit int-amplified endomorphisms (e.g. the\nmultiplication-by-$m$ endomorphism) and normal toric varieties admit\nint-amplified endomorphisms (e.g. the $m$-th power Frobenius morphism).\nMeng and Zhong conjecture \\cite[Question 1.2]{Meng-Rigidity} that a\nsmooth projective rationally connected complex variety admits an\nint-amplified endomorphism if and only if it is a toric variety (see\nalso \\cite[Question 4.4]{Fakhruddin}). This has been proved for smooth\nsurfaces \\cite[Theorem 3]{Nakayama02} and smooth Fano threefolds\n\\cite[Theorem 1.4]{Meng-Zhang-Zhong}, \\cite[Theorem\n6.1]{Totaro-Log-Bott}.\n\nWhen we consider singular Fano varieties, we find examples that admit\nint-amplified endomorphisms which are not toric, but are finite\nquotients of toric varieties.\n\\begin{example}\\label{quaternion-example}\nConsider the two-dimensional representation of the quaternion group\ngiven by\n\\[ \n i\\mapsto \\begin{pmatrix} i&0\\\\0&-i\\end{pmatrix},\\quad\n j\\mapsto \\begin{pmatrix}0&1\\\\-1&0\\end{pmatrix},\\quad\n k=ij\\mapsto \\begin{pmatrix}0&i\\\\i&0\\end{pmatrix}.\n\\]\nExtending this to a projective representation in $\\PGL(3)$, we can\nconsider it as an action of $Q_8$ on $\\mathbb{P}^2$. Let $X$ be the quotient of\n$\\mathbb{P}^2$ by this action. Then the image of $[0:0:1]$ is a $D_4$\nsingularity: $D_4$ singularities are not toric so $X$ cannot be a toric\nvariety. Nevertheless we can easily see that the ``Frobenius\nendomorphism'' $[x:y:z] \\mapsto [x^5:y^5:z^5]$ of $\\mathbb{P}^2$ commutes with\nthe action of $Q_8$ and thus descends to an int-amplified endomorphism\nof $X$. \n\\end{example}\n\nIn view of such examples, it is a folklore conjecture that a klt Fano\nvariety admits an int-amplified endomorphism if and only if it a finite\nquotient of a toric variety. Nakayama proved \\cite[Lemma\n2.6]{NakayamaStruct} that a quotient of a toric surface by a finite\ngroup which acts freely in codimension one and preserves the open torus\ninherits an int-amplified endomorphism from the toric variety. Nakayama\nalso proved that for a klt del Pezzo surface $X$ of Picard rank $>1$,\n$X$ admits an int-amplified endomorphism iff $X$ is a quotient of a\ntoric variety by a finite group acting freely in codimension $1$ such\nthat the action preserves the open torus \\cite[Theorem 1.3]{Nakayama}.\n\nIn this article, I study the case of Picard rank $1$ and consider del\nPezzo surfaces with Gorenstein singularities. Note that an\nendomorphism of a projective variety of Picard rank $1$ is int-amplified\nif and only if it has degree greater than one, since in this case\n$N^1(X)_{\\mathbb{C}} \\cong \\mathbb{C}$ and $f^*: N^1(X)_{\\mathbb{C}} \\to N^1(X)_{\\mathbb{C}}$ is just\nmultiplication by $(\\deg f)^{\\dim X}$. Define $S'(E_8)$ as the weighted\nsextic surface $\\{ X^5Y + X^4Z + Z^3+ W^2 = 0 \\} \\subset \\mathbb{P}(1, 1, 2, 3)$\n\\cite{Gurjar}: it is a Gorenstein del Pezzo surface of Picard rank 1.\nThen the main theorem of this paper is the following.\n\n\\begin{abstract}\nWe prove that, in all except one case, a Gorenstein del Pezzo surface of\nPicard rank $1$ admits an int-amplified endomorphism if and only if it\nis a quotient of a toric variety by a finite group which acts freely in\ncodimension one and preserves the open torus. We classify all such\nquotients.\n\\end{abstract}\n\\maketitle\n\nWhen we consider singular Fano varieties, we find examples that admit\nint-amplified endomorphisms which are not toric, but are finite\nquotients of toric varieties.\n\\begin{example}\\label{quaternion-example}\nConsider the two-dimensional representation of the quaternion group\ngiven by\n\\[ \n i\\mapsto \\begin{pmatrix} i&0\\\\0&-i\\end{pmatrix},\\quad\n j\\mapsto \\begin{pmatrix}0&1\\\\-1&0\\end{pmatrix},\\quad\n k=ij\\mapsto \\begin{pmatrix}0&i\\\\i&0\\end{pmatrix}.\n\\]\nExtending this to a projective representation in $\\PGL(3)$, we can\nconsider it as an action of $Q_8$ on $\\P^2$. Let $X$ be the quotient of\n$\\P^2$ by this action. Then the image of $[0:0:1]$ is a $D_4$\nsingularity: $D_4$ singularities are not toric so $X$ cannot be a toric\nvariety. Nevertheless we can easily see that the ``Frobenius\nendomorphism'' $[x:y:z] \\mapsto [x^5:y^5:z^5]$ of $\\P^2$ commutes with\nthe action of $Q_8$ and thus descends to an int-amplified endomorphism\nof $X$. \n\\end{example}\n\nIn view of such examples, it is a folklore conjecture that a klt Fano\nvariety admits an int-amplified endomorphism if and only if it a finite\nquotient of a toric variety. Nakayama proved \\cite[Lemma\n2.6]{NakayamaStruct} that a quotient of a toric surface by a finite\ngroup which acts freely in codimension one and preserves the open torus\ninherits an int-amplified endomorphism from the toric variety. Nakayama\nalso proved that for a klt del Pezzo surface $X$ of Picard rank $>1$,\n$X$ admits an int-amplified endomorphism iff $X$ is a quotient of a\ntoric variety by a finite group acting freely in codimension $1$ such\nthat the action preserves the open torus \\cite[Theorem 1.3]{Nakayama}.\n\nIn this article, I study the case of Picard rank $1$ and consider del\nPezzo surfaces with Gorenstein singularities. Note that an\nendomorphism of a projective variety of Picard rank $1$ is int-amplified\nif and only if it has degree greater than one, since in this case\n$N^1(X)_{\\C} \\isom \\C$ and $f^*: N^1(X)_{\\C} \\to N^1(X)_{\\C}$ is just\nmultiplication by $(\\deg f)^{\\dim X}$. Define $S'(E_8)$ as the weighted\nsextic surface $\\{ X^5Y + X^4Z + Z^3+ W^2 = 0 \\} \\subset \\P(1, 1, 2, 3)$\n\\cite{Gurjar}: it is a Gorenstein del Pezzo surface of Picard rank 1.\nThen the main theorem of this paper is the following.\n\nWe give an overview of rank $1$ Gorenstein del Pezzo surfaces below and\ntheir classification by singularity type. To prove Theorem\n\\ref{main_theorem}, we first identify which rank $1$ Gorenstein del\nPezzo surfaces are quotients of toric surfaces. The ``easy direction''\n(that is, the statement that if $X$ is such a quotient of a toric\nvariety, it admits an int-amplified endomorphism) follows from\n\\cite[Lemma 2.6]{NakayamaStruct}. For the rest, following Kawakami and\nTotaro \\cite{Kawakami-Totaro}, we use the tool of Bott vanishing,\ncombined with lifting to universal covers, to deduce that they do not\nadmit int-amplified endomorphisms. To find failures of Bott vanishing,\nwe use a version of the Riemann-Roch formula for normal surfaces. This\nformula is automated in a Python program which is in the accompanying\ncode repository \\cite{CodeRepo}.\n\nWe work in characteristic zero, but everything in this paper should\nhold over an algebraically closed field of characteristic $\\neq 2, 3$,\nas long as we everywhere replace ``int-amplified endomorphism'' with\n``int-amplified endomorphism of degree invertible in the base field''.\nThe classification of rank $1$ Gorenstein del Pezzo surfaces is the same\nover algebraically closed fields of characteristic $\\neq 2, 3$\n\\cite[Theorem B.6]{Lacini}, and all the endomorphisms described in this\npaper have degree coprime to $2$ and $3$.\n\n\\begin{proposition}\\label{toric_quotient} The five surfaces\n$S(3A_1+D_4)$, $S(A_1+2A_3)$, $S(A_1+A_2+A_5)$, $S(4A_2)$ and\n$S(2A_1+2A_3)$ are quotients of toric surfaces by a finite group that\nacts freely in codimension one and preserves the open torus.\n\\end{proposition}\n\\begin{proof}\nWe describe each of these surfaces as a quotient of a toric variety, and\nfind an int-amplified endomorphism of the toric variety that descends to\nthe quotient (we can also apply \\cite[Lemma 2.6]{NakayamaStruct} to show\nthat the quotients admit int-amplified endomorphisms).\n\\begin{itemize}\n\\item $S(3A_1+D_4)$ is the surface constructed in Example\n\\ref{quaternion-example}.\n\nThe quasi-universal covers of $S(A_2+E_6)$ and $S'(A_2+E_6)$ are\nGorenstein del Pezzo surfaces of rank $3$ with a single $D_4$\nsingularity. The quasi-universal covers of $S(A_1+E_7)$ and\n$S'(A_1+E_7)$ are Gorenstein del Pezzo surfaces of rank $2$ with a\nsingle $E_6$ singularity. Let $Y$ be one of the four surfaces\n$S(A_2+E_6)$, $S'(A_2+E_6)$, $S(A_1+E_7)$ or $S'(A_1+E_7)$ and let $\\pi:\nX \\to Y$ be the quasi-universal cover. Since $D_4$ and $E_6$\nsingularities are not toric, $X$ is not a toric variety. Furthermore, we\nknow $f^{-1}(Y^0)$ is simply-connected and is $X^0$ minus a finite set\nof points: since $X^0$ is smooth, $X^0$ is also simply-connected. Thus\n$X$ cannot be a quotient of a toric variety by a finite group acting\nfreely in codimension one. However, by \\cite[Theorem 1.3]{Nakayama}, if\na klt del Pezzo surface of Picard rank $>1$ admits a endomorphism of\ndegree $>1$, it is a quotient of a toric variety by a finite group\nacting freely in codimension one. If $Y$ admitted an int-amplified\nendomorphism, it would lift to endomorphism of $X$ of degree $>1$, which\nis impossible. Thus the surfaces $S(A_2+E_6)$, $S'(A_2+E_6)$,\n$S(A_1+E_7)$ and $S'(A_1+E_7)$ do not admit int-amplified endomorphisms.\n\n\\begin{theorem}\\label{main_theorem} Let $X$ be a Gorenstein del Pezzo\nsurface of Picard rank $1$ over an algebraically closed field of\ncharacteric zero which is not isomorphic to the surface $S'(E_8)$. Then\n$X$ admits an int-amplified endomorphism (i.e. an endomorphism of degree\n$>1$) if and only if $X$ is a quotient of a toric variety by a finite\ngroup acting freely in codimension one that preserves the open torus.\n\\end{theorem}", "post_theorem_intro_text_len": 1560, "post_theorem_intro_text": "We give an overview of rank $1$ Gorenstein del Pezzo surfaces below and\ntheir classification by singularity type. To prove Theorem\n\\ref{main_theorem}, we first identify which rank $1$ Gorenstein del\nPezzo surfaces are quotients of toric surfaces. The ``easy direction''\n(that is, the statement that if $X$ is such a quotient of a toric\nvariety, it admits an int-amplified endomorphism) follows from\n\\cite[Lemma 2.6]{NakayamaStruct}. For the rest, following Kawakami and\nTotaro \\cite{Kawakami-Totaro}, we use the tool of Bott vanishing,\ncombined with lifting to universal covers, to deduce that they do not\nadmit int-amplified endomorphisms. To find failures of Bott vanishing,\nwe use a version of the Riemann-Roch formula for normal surfaces. This\nformula is automated in a Python program which is in the accompanying\ncode repository \\cite{CodeRepo}.\n\nWe work in characteristic zero, but everything in this paper should\nhold over an algebraically closed field of characteristic $\\neq 2, 3$,\nas long as we everywhere replace ``int-amplified endomorphism'' with\n``int-amplified endomorphism of degree invertible in the base field''.\nThe classification of rank $1$ Gorenstein del Pezzo surfaces is the same\nover algebraically closed fields of characteristic $\\neq 2, 3$\n\\cite[Theorem B.6]{Lacini}, and all the endomorphisms described in this\npaper have degree coprime to $2$ and $3$.\n\n\\subsection*{Acknowledgments} I would like to thank Burt Totaro for many\nhelpful conversations. The author is supported by NSF Graduate Research\nFellowship Grant No. DGE-2034835.", "sketch": "To prove Theorem~\\ref{main_theorem}, we first identify which rank $1$ Gorenstein del Pezzo surfaces are quotients of toric surfaces. The ``easy direction'' (if $X$ is such a quotient of a toric variety, then it admits an int-amplified endomorphism) follows from \\cite[Lemma 2.6]{NakayamaStruct}. For the remaining direction, following Kawakami and Totaro \\cite{Kawakami-Totaro}, we use Bott vanishing, combined with lifting to universal covers, to deduce that the other surfaces do not admit int-amplified endomorphisms. To find failures of Bott vanishing, we use a version of the Riemann--Roch formula for normal surfaces (implemented in a Python program in \\cite{CodeRepo}).", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{main_theorem} Let $X$ be a Gorenstein del Pezzo\nsurface of Picard rank $1$ over an algebraically closed field of\ncharacteric zero which is not isomorphic to the surface $S'(E_8)$. Then\n$X$ admits an int-amplified endomorphism (i.e. an endomorphism of degree\n$>1$) if and only if $X$ is a quotient of a toric variety by a finite\ngroup acting freely in codimension one that preserves the open torus.", "theorem_type": ["Biconditional or Equivalence", "Implication"], "mcq": {"question": "Let $X$ be a Gorenstein del Pezzo surface of Picard rank $1$ over an algebraically closed field of characteristic zero, and assume that $X$ is not isomorphic to the surface $S'(E_8)$. An int-amplified endomorphism here means an endomorphism of degree $>1$. Which of the following statements is equivalent to the existence of such an endomorphism on $X$?", "correct_choice": {"label": "A", "text": "$X$ is a quotient of a toric variety by a finite group acting freely in codimension one and preserving the open torus (that is, the dense torus of the toric variety)."}, "choices": [{"label": "B", "text": "$X$ is a quotient of a toric variety by a finite group preserving the open torus."}, {"label": "C", "text": "$X$ is a quotient of a toric variety by a finite group acting freely in codimension one."}, {"label": "D", "text": "$X$ is a quotient of a toric variety by a finite group acting freely in codimension one."}, {"label": "E", "text": "$X$ is itself a toric variety."}], "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": "free_in_codimension_one", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "preserves_open_torus", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "preserves_open_torus", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "finite_quotient_vs_toric", "template_used": "wildcard"}]}}
{"id": "2512.03670v1", "paper_link": "http://arxiv.org/abs/2512.03670v1", "theorems_cnt": 5, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\n    For any sequence of binary alphabet $\\A_1,\\A_2,\\dots$, there exists an infinite cube-free word in $\\A_1\\times\\A_2\\times\\dots$.", "start_pos": 12428, "end_pos": 12606, "label": "thm:main"}, "ref_dict": {"thm:growth": "\\begin{theorem} \\label{thm:growth}  \nFor any sequence of binary alphabet $\\A_1,\\A_2,\\dots$, and for every $n$, there exist at least $1.35^n$ cube-free words in $\\A_1\\times\\A_2\\times\\dots\\times\\A_n$. \n\\end{theorem}", "thm:computable": "\\begin{theorem} \\label{thm:computable}\nLet $\\A_1,\\A_2,\\dots$ be a computable sequence of binary alphabet. Then there exists a computable infinite cube-free word $w=w_1w_2w_3\\ldots$ in $\\A_1\\times\\A_2\\times\\dots$. Moreover, there is an algorithm that given $w_1\\ldots w_n$ and\\ $\\A_{n+1}$ computes $w_{n+1}$ in time polynomial in $n$.\n\\end{theorem}", "conj:ThueListe": "\\begin{conjecture}[{\\cite[\\ldots]{rosenfeldThueList, CzerwinskiThueChoiceNumber, GagolThueChoiceNumber, GrytczukSurveyThueChoiceNumber, GrytczukThueChoiceNumber, WoodThueChoiceNumber,ZhaoThueChoiceNumber,Grytczuk2011Jan,Fiorenzi2011Oct,Skrabulakova2015Aug}}]\\label{conj:ThueListe}\n  The Thue list number of the infinite path is $3$.\n\\end{conjecture}", "thm:main": "\\begin{theorem} \\label{thm:main}\n    For any sequence of binary alphabet $\\A_1,\\A_2,\\dots$, there exists an infinite cube-free word in $\\A_1\\times\\A_2\\times\\dots$. \n\\end{theorem}", "lem:eigenvector": "\\begin{lemma}\\label{lem:eigenvector}\n    Let $p=12$. There exist nonnegative weights $(\\lambda_s)_{s\\in \\minsuff}$ with $\\lambda_{\\varepsilon}>0$ such that for every word $w$ over the alphabet $\\mathbb{N}$\n    and every $\\A\\in \\binom{\\mathbb{N}}{2}$,\n    \\begin{equation}\\label{ineq:eigen}\n        \\alpha \\lambda_{\\minsuff(w)} \\le \\sum_{\\substack{ c\\in\\A\\\\wc\\in \\widetilde{\\mathcal{C}}^{(p)}}} \\lambda_{\\minsuff(wc)},\n    \\end{equation}\n    where $\\alpha = 1.457$.\n\\end{lemma}"}, "pre_theorem_intro_text_len": 3447, "pre_theorem_intro_text": "A \\emph{square} (\\textit{resp.} a \\emph{cube}) is a word $uu$ (\\textit{resp.} $uuu$) for some nonempty word $u$. A \\emph{cube-free word} (resp. \\emph{square-free word}) is a word that contains no cube (resp. square) as a factor.\nThue proved that there exists an infinite cube-free word over the binary alphabet and an infinite square-free word over the ternary alphabet \\cite{Thue1}. These results are regarded as the first results in combinatorics on words, and many generalizations of these problems have been considered.\n\nOne such generalization is the notion of nonrepetitive coloring of graphs introduced by Alon et al. \\cite{Alon2002}. A graph coloring (of the edges or of the vertices) is said to be \\emph{nonrepetitive} if the sequence of colors along any path avoids squares (see \\cite{WoodThueChoiceNumber} for a recent survey on this topic). This notion naturally led to the notion of nonrepetitive list-coloring, where instead of having one fixed set of colors every vertex has to choose a color from a list of colors specific to this vertex. The \\emph{Thue-list number} of a graph is the smallest integer $k$ such that if all the lists have size at least $k$ then the graph can be nonrepetitively colored in such a way that each vertex receives a color from its list. In this article, we will consider a variant of the following challenging conjecture.\n\\begin{conjecture}[{\\cite[\\ldots]{rosenfeldThueList, CzerwinskiThueChoiceNumber, GagolThueChoiceNumber, GrytczukSurveyThueChoiceNumber, GrytczukThueChoiceNumber, WoodThueChoiceNumber,ZhaoThueChoiceNumber,Grytczuk2011Jan,Fiorenzi2011Oct,Skrabulakova2015Aug}}]\\label{conj:ThueListe}\n  The Thue list number of the infinite path is $3$.\n\\end{conjecture}\n\nWe provide a formulation of this conjecture in terms of combinatorics on words. By an \\emph{alphabet}, we mean any finite set. Given a sequence $(\\A_i)_{i\\ge1}$ of alphabets, we say that a word $w=w_1\\ldots w_n$ \\emph{respects} $(\\A_i)_{i\\ge 1}$, if for all $i$, $w_i\\in\\A_i$ (the definition naturally extends to infinite words). Conjecture \\ref{conj:ThueListe} can be rephrased as follows.\n\\begin{conjecture}\nGiven a sequence of alphabets $(\\A_i)_{i\\ge1}$ with  $|A_i|\\ge3$ for all $i$, there exists an infinite square-free word that respects $(\\A_i)_{i\\ge 1}$.\n\\end{conjecture}\n\nWithout loss of generality, we can assume that all the $\\A_i$ are sets of integers, so we adopt this convention for the sake of notation in the rest of the article. It was first proven by Grytczuk, Przyby{\\l}o, and Zhu that the conjecture is true if the condition $|\\A_i|\\ge3$ is replaced by $|\\A_i|\\ge4$ \\cite{Grytczuk2011Jan}.\nThe second author recently proved that if $|\\bigcup_{i\\ge1} \\A_i|=4$ and $|\\A_i|\\ge3$ for every $i$ then there exists an infinite square-free word that respects $(\\A_i)_{i\\ge 1}$ \\cite{rosenfeld2022avoiding}. This result is far from proving the conjecture, but the author argues that if it were not for the computational limitations, the technique used in this could certainly prove the full conjecture. Indeed, this technique requires one to verify by a computer-assisted proof some growth property of a very large automaton associated with square-free words.\n\nIn the current article, we consider a natural variant of the question. Since cubes are avoidable over the binary alphabet, is it possible to avoid cubes if all of the lists are of size $2$? We give a positive answer to this question in the following theorem.", "context": "A \\emph{square} (\\textit{resp.} a \\emph{cube}) is a word $uu$ (\\textit{resp.} $uuu$) for some nonempty word $u$. A \\emph{cube-free word} (resp. \\emph{square-free word}) is a word that contains no cube (resp. square) as a factor.\nThue proved that there exists an infinite cube-free word over the binary alphabet and an infinite square-free word over the ternary alphabet \\cite{Thue1}. These results are regarded as the first results in combinatorics on words, and many generalizations of these problems have been considered.\n\nOne such generalization is the notion of nonrepetitive coloring of graphs introduced by Alon et al. \\cite{Alon2002}. A graph coloring (of the edges or of the vertices) is said to be \\emph{nonrepetitive} if the sequence of colors along any path avoids squares (see \\cite{WoodThueChoiceNumber} for a recent survey on this topic). This notion naturally led to the notion of nonrepetitive list-coloring, where instead of having one fixed set of colors every vertex has to choose a color from a list of colors specific to this vertex. The \\emph{Thue-list number} of a graph is the smallest integer $k$ such that if all the lists have size at least $k$ then the graph can be nonrepetitively colored in such a way that each vertex receives a color from its list. In this article, we will consider a variant of the following challenging conjecture.\n\\begin{conjecture}[{\\cite[\\ldots]{rosenfeldThueList, CzerwinskiThueChoiceNumber, GagolThueChoiceNumber, GrytczukSurveyThueChoiceNumber, GrytczukThueChoiceNumber, WoodThueChoiceNumber,ZhaoThueChoiceNumber,Grytczuk2011Jan,Fiorenzi2011Oct,Skrabulakova2015Aug}}]\\label{conj:ThueListe}\n The Thue list number of the infinite path is $3$.\n\\end{conjecture}\n\nWe provide a formulation of this conjecture in terms of combinatorics on words. By an \\emph{alphabet}, we mean any finite set. Given a sequence $(\\A_i)_{i\\ge1}$ of alphabets, we say that a word $w=w_1\\ldots w_n$ \\emph{respects} $(\\A_i)_{i\\ge 1}$, if for all $i$, $w_i\\in\\A_i$ (the definition naturally extends to infinite words). Conjecture \\ref{conj:ThueListe} can be rephrased as follows.\n\\begin{conjecture}\nGiven a sequence of alphabets $(\\A_i)_{i\\ge1}$ with $|A_i|\\ge3$ for all $i$, there exists an infinite square-free word that respects $(\\A_i)_{i\\ge 1}$.\n\\end{conjecture}\n\nWithout loss of generality, we can assume that all the $\\A_i$ are sets of integers, so we adopt this convention for the sake of notation in the rest of the article. It was first proven by Grytczuk, Przyby{\\l}o, and Zhu that the conjecture is true if the condition $|\\A_i|\\ge3$ is replaced by $|\\A_i|\\ge4$ \\cite{Grytczuk2011Jan}.\nThe second author recently proved that if $|\\bigcup_{i\\ge1} \\A_i|=4$ and $|\\A_i|\\ge3$ for every $i$ then there exists an infinite square-free word that respects $(\\A_i)_{i\\ge 1}$ \\cite{rosenfeld2022avoiding}. This result is far from proving the conjecture, but the author argues that if it were not for the computational limitations, the technique used in this could certainly prove the full conjecture. Indeed, this technique requires one to verify by a computer-assisted proof some growth property of a very large automaton associated with square-free words.\n\nIn the current article, we consider a natural variant of the question. Since cubes are avoidable over the binary alphabet, is it possible to avoid cubes if all of the lists are of size $2$? We give a positive answer to this question in the following theorem.\n\n\\begin{conjecture}[{\\cite[\\ldots]{rosenfeldThueList, CzerwinskiThueChoiceNumber, GagolThueChoiceNumber, GrytczukSurveyThueChoiceNumber, GrytczukThueChoiceNumber, WoodThueChoiceNumber,ZhaoThueChoiceNumber,Grytczuk2011Jan,Fiorenzi2011Oct,Skrabulakova2015Aug}}]\\label{conj:ThueListe}\n  The Thue list number of the infinite path is $3$.\n\\end{conjecture}", "full_context": "A \\emph{square} (\\textit{resp.} a \\emph{cube}) is a word $uu$ (\\textit{resp.} $uuu$) for some nonempty word $u$. A \\emph{cube-free word} (resp. \\emph{square-free word}) is a word that contains no cube (resp. square) as a factor.\nThue proved that there exists an infinite cube-free word over the binary alphabet and an infinite square-free word over the ternary alphabet \\cite{Thue1}. These results are regarded as the first results in combinatorics on words, and many generalizations of these problems have been considered.\n\nOne such generalization is the notion of nonrepetitive coloring of graphs introduced by Alon et al. \\cite{Alon2002}. A graph coloring (of the edges or of the vertices) is said to be \\emph{nonrepetitive} if the sequence of colors along any path avoids squares (see \\cite{WoodThueChoiceNumber} for a recent survey on this topic). This notion naturally led to the notion of nonrepetitive list-coloring, where instead of having one fixed set of colors every vertex has to choose a color from a list of colors specific to this vertex. The \\emph{Thue-list number} of a graph is the smallest integer $k$ such that if all the lists have size at least $k$ then the graph can be nonrepetitively colored in such a way that each vertex receives a color from its list. In this article, we will consider a variant of the following challenging conjecture.\n\\begin{conjecture}[{\\cite[\\ldots]{rosenfeldThueList, CzerwinskiThueChoiceNumber, GagolThueChoiceNumber, GrytczukSurveyThueChoiceNumber, GrytczukThueChoiceNumber, WoodThueChoiceNumber,ZhaoThueChoiceNumber,Grytczuk2011Jan,Fiorenzi2011Oct,Skrabulakova2015Aug}}]\\label{conj:ThueListe}\n The Thue list number of the infinite path is $3$.\n\\end{conjecture}\n\nWe provide a formulation of this conjecture in terms of combinatorics on words. By an \\emph{alphabet}, we mean any finite set. Given a sequence $(\\A_i)_{i\\ge1}$ of alphabets, we say that a word $w=w_1\\ldots w_n$ \\emph{respects} $(\\A_i)_{i\\ge 1}$, if for all $i$, $w_i\\in\\A_i$ (the definition naturally extends to infinite words). Conjecture \\ref{conj:ThueListe} can be rephrased as follows.\n\\begin{conjecture}\nGiven a sequence of alphabets $(\\A_i)_{i\\ge1}$ with $|A_i|\\ge3$ for all $i$, there exists an infinite square-free word that respects $(\\A_i)_{i\\ge 1}$.\n\\end{conjecture}\n\nWithout loss of generality, we can assume that all the $\\A_i$ are sets of integers, so we adopt this convention for the sake of notation in the rest of the article. It was first proven by Grytczuk, Przyby{\\l}o, and Zhu that the conjecture is true if the condition $|\\A_i|\\ge3$ is replaced by $|\\A_i|\\ge4$ \\cite{Grytczuk2011Jan}.\nThe second author recently proved that if $|\\bigcup_{i\\ge1} \\A_i|=4$ and $|\\A_i|\\ge3$ for every $i$ then there exists an infinite square-free word that respects $(\\A_i)_{i\\ge 1}$ \\cite{rosenfeld2022avoiding}. This result is far from proving the conjecture, but the author argues that if it were not for the computational limitations, the technique used in this could certainly prove the full conjecture. Indeed, this technique requires one to verify by a computer-assisted proof some growth property of a very large automaton associated with square-free words.\n\nIn the current article, we consider a natural variant of the question. Since cubes are avoidable over the binary alphabet, is it possible to avoid cubes if all of the lists are of size $2$? We give a positive answer to this question in the following theorem.\n\n\\begin{conjecture}[{\\cite[\\ldots]{rosenfeldThueList, CzerwinskiThueChoiceNumber, GagolThueChoiceNumber, GrytczukSurveyThueChoiceNumber, GrytczukThueChoiceNumber, WoodThueChoiceNumber,ZhaoThueChoiceNumber,Grytczuk2011Jan,Fiorenzi2011Oct,Skrabulakova2015Aug}}]\\label{conj:ThueListe}\n  The Thue list number of the infinite path is $3$.\n\\end{conjecture}\n\n\\section{Introduction}\nA \\emph{square} (\\textit{resp.} a \\emph{cube}) is a word $uu$ (\\textit{resp.} $uuu$) for some nonempty word $u$. A \\emph{cube-free word} (resp. \\emph{square-free word}) is a word that contains no cube (resp. square) as a factor.\nThue proved that there exists an infinite cube-free word over the binary alphabet and an infinite square-free word over the ternary alphabet \\cite{Thue1}. These results are regarded as the first results in combinatorics on words, and many generalizations of these problems have been considered.\n\nWe provide a formulation of this conjecture in terms of combinatorics on words. By an \\emph{alphabet}, we mean any finite set. Given a sequence $(\\A_i)_{i\\ge1}$ of alphabets, we say that a word $w=w_1\\ldots w_n$ \\emph{respects} $(\\A_i)_{i\\ge 1}$, if for all $i$, $w_i\\in\\A_i$ (the definition naturally extends to infinite words). Conjecture \\ref{conj:ThueListe} can be rephrased as follows.\n\\begin{conjecture}\nGiven a sequence of alphabets $(\\A_i)_{i\\ge1}$ with $|A_i|\\ge3$ for all $i$, there exists an infinite square-free word that respects $(\\A_i)_{i\\ge 1}$.\n\\end{conjecture}\n\nIn the current article, we consider a natural variant of the question. Since cubes are avoidable over the binary alphabet, is it possible to avoid cubes if all of the lists are of size $2$? We give a positive answer to this question in the following theorem.\n\nThe idea behind the first proof is to show that the growth rate of the language is not far from the growth rate of the regular approximation of the language which yields the following result.\n\\begin{theorem} \\label{thm:growth} \nFor any sequence of binary alphabet $\\A_1,\\A_2,\\dots$, and for every $n$, there exist at least $1.35^n$ cube-free words in $\\A_1\\times\\A_2\\times\\dots\\times\\A_n$. \n\\end{theorem}\nThe core idea allowing us to deduce bounds on the growth of a power-free language $\\mathcal L$ from its approximation $\\mathcal L$ was first used by Kolpakov to estimate the growth rate of power-free languages \\cite{kopakov2007efficient,Kolpakov2007Dec}. Shur later improved upon this technique \\cite{shur2010Jul,Shur2012Nov}. This technique was later adapted by the second author to provide a partial answer to Conjecture \\ref{conj:ThueListe} in \\cite{rosenfeld2022avoiding}.\nThe proof of this result applies the technique developed in \\cite{rosenfeld2022avoiding} for squares, but provides a complete answer in the case of cubes.\nThe second proof of our main result does not imply anything about the number of solutions, but it contains algorithmic implications.\n\\begin{theorem} \\label{thm:computable}\nLet $\\A_1,\\A_2,\\dots$ be a computable sequence of binary alphabet. Then there exists a computable infinite cube-free word $w=w_1w_2w_3\\ldots$ in $\\A_1\\times\\A_2\\times\\dots$. Moreover, there is an algorithm that given $w_1\\ldots w_n$ and\\ $\\A_{n+1}$ computes $w_{n+1}$ in time polynomial in $n$.\n\\end{theorem}\nThis proof also relies on computing properties of the same automaton, but then we use a different argument for adding the large cubes back.\nIntuitively, the idea is to introduce a weight function that measures how difficult it is to extend a word, and to prove that under some condition if the weight is not too large then at least one of the extensions also has a small weight (which we can then use inductively to construct a word). This idea was originally due to Miller \\cite{miller2012two}, and was recently generalized in \\cite{Rosenfeld2025Mar}. However, the weight functions in \\cite{miller2012two} and \\cite{Rosenfeld2025Mar} are much simpler and in our article the precise definition of the function depends on the automaton. Our result is in fact even stronger, as everything still holds if for all $i$ an opponent is allowed to choose the alphabet $\\A_{i+1}$ after we choose $w_i$.\n\n\\section{Approximation by a regular language}\\label{sec:notation}\nOur approach is to bound the difference between the number of cube-free words and the number of words that avoids all cubes up to a given length $p$. \nWe let $\\mathcal{C}$ be the set of all cube-free words. \nFor a given integer $p$, we let $\\widetilde{\\mathcal{C}}^{(p)}$ be the set of words that avoid cubes of period at most $p$.\nGiven a sequence of alphabets $(\\A_i)_{i\\ge1}$, we let $\\mathcal{C}[(\\A_i)_{i\\ge1}]$ (resp. $\\widetilde{\\mathcal{C}}^{(p)}[(\\A_i)_{i\\ge1}]$) be the elements of $\\mathcal{C}$ (resp. $\\widetilde{\\mathcal{C}}^{(p)}$) that respect $(\\A_i)_{i\\ge1}$.\n\nThis implies \n\\begin{align*}\n \\sum_{\\substack{c\\in\\A\\\\uc\\in \\widetilde{\\mathcal{C}}^{(p)}}} \\omega(uc)\n &\\le \\beta\\omega(u)+\\sum_{i>p} \\lambda_{\\minsuff(u)}\\beta^{1-i}\\\\\n &<\\lambda_{\\minsuff(u)}\\left(\\beta+\\frac{\\beta^{1-p}}{\\beta-1}\\right)\\\\\n &\\le\\lambda_{\\minsuff(u)}\\alpha\\\\\n &\\le \\sum_{\\substack{c\\in\\A\\\\uc\\in \\widetilde{\\mathcal{C}}^{(p)}}} \\lambda_{\\minsuff(uc)}\\,,\n\\end{align*}\nwhere the third inequality is by our theorem hypothesis, and the last inequality was proven earlier in this proof.\nThis implies that there exists $c\\in \\A$ such that $uc\\in \\widetilde{\\mathcal{C}}^{(p)}$ and $\\omega(uc)<\\lambda_{\\minsuff(uc)}$ as desired.\n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{thm:computable}]\n In particular, with $p=12$ and $\\alpha=1.457$ we can set $\\beta=1.35$ to satisfy the condition of Lemma \\ref{lem:betacond}.\n So we inductively define $u_0=\\varepsilon$, and for all $n$, $u_{n+1} = u_n a_{n+1}$ where $a_{n+1}\\in\\A_{n+1}$ is chosen such that $\\omega(u_{n+1})<\\lambda_{\\minsuff(u_{n+1})}$. By Lemma \\ref{lem:smallweighisfine}, each of the $u_i$ is cube-free, and so is the infinite word $a_1a_2a_3\\ldots$ and by construction it belongs to $\\A_1\\times\\A_2\\times\\A_3\\ldots$ as desired.\n\n\\begin{lemma}\\label{lem:counting_argument}\nFor all $n\\in\\mathbb{N}$ and $t\\in\\minsuff$, we have the following inequality\n \\begin{equation}\\label{eq:exclusion}\n |C_{n+1}^t| \\ge \\sum_{v\\in C_n}\\sum_{\\substack{c\\in\\A_{n+1}\\\\ vc\\in\\widetilde{\\mathcal{C}}^{(p)}\\\\\\minsuff(vc)=t}} 1 - \\sum_{i=p+1}^{n+1} |C_{n+1-i}^t|.\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\n We let $G^t$ be the set of words of length $n+1$ in $A_1\\times\\ldots\\times A_{n+1}$ so that\n \\begin{itemize}\n \\item the prefix of size $n$ is cube-free,\n \\item the word does not contain any cubes of period at most $p$,\n \\item the word ends in state $t$.\n \\end{itemize} \n In other words,\n \\[\n G^t= \\left\\{va\\in \\widetilde{\\mathcal{C}}^{(p)}: v\\in C_{n}, a\\in A_{n+1}, \\minsuff(va)=t\\right\\}\\,.\n \\]\n Hence,\n \\[\n |G^t|= \\sum_{v\\in C_n}\\sum_{\\substack{c\\in\\A_{n+1}\\\\ vc\\in\\widetilde{\\mathcal{C}}^{(p)}\\\\\\minsuff(vc)=t}} 1\\,.\n \\]\n Let $F_i$ be the set of words from $G_t$ that contain a cube of period $i$ as a suffix. By definition, a word from $G_t$ can only contain a cube as a suffix and this cube must be of period more than $p$ (and at most $n+1$).\\footnote{We could easily replace this $n+1$ by $\\lfloor(n+1)/3\\rfloor$, but it does not make any difference in the latter part of the proof.} We get\n \\[\n C_{n+1}^t= G^t \\setminus \\bigcup_{i=p+1}^{n+1} F_i\\,,\n \\]\n which implies\n \\[\n |C_{n+1}^t|\\ge |G^t| - \\sum_{i=p+1}^{n+1} |F_i|\\,.\n \\]\n It remains to prove that $|F_i|\\le |C_{n+1-i}^t|$ for every $i$ to conclude the proof.\n Let $va\\in F_i$, then by definition $va=uyyy$ with $|y|=i>p$. Since $v\\in C_n$ and $va\\in \\widetilde{\\mathcal{C}}^{(p)}$, we can apply Corollary \\ref{cor:easingOnePeriodPreservesState} to deduce that $\\minsuff(uyy)=\\minsuff(vc)=t$, that is, $uyy\\in C_{n+1-i}^t$. Given the word $uyy$ and the period $i=|y|$, we can uniquely reconstruct $vc=uyyy$. This implies $|F_i|\\le |C_{n+1-i}^t|$, which concludes our proof.\n\\end{proof}", "post_theorem_intro_text_len": 4027, "post_theorem_intro_text": "In fact, we do not prove this result directly, but we prove two different stronger results, each implying Theorem \\ref{thm:main}. \nThese two proofs share a key idea: we can use an automaton to study the variant of the problem where we only forbid cubes of length at most $p$, for some arbitrary choice of $p$. Intuitively, short cubes are more difficult to avoid than long cubes, so by taking $p$ large enough, we should obtain a good approximation of the problem. Adapting standard tools from combinatorics on words and automata theory, studying this approximation is quite simple to achieve.\nThe delicate part is then to prove that if $p$ is large enough this approximation is good enough to deduce something about the original problem where all cubes are forbidden. \n\nThe idea behind the first proof is to show that the growth rate of the language is not far from the growth rate of the regular approximation of the language which yields the following result.\n\\begin{theorem} \\label{thm:growth}  \nFor any sequence of binary alphabet $\\A_1,\\A_2,\\dots$, and for every $n$, there exist at least $1.35^n$ cube-free words in $\\A_1\\times\\A_2\\times\\dots\\times\\A_n$. \n\\end{theorem}\nThe core idea allowing us to deduce bounds on the growth of a power-free language $\\mathcal L$ from its approximation $\\mathcal L$ was first used by Kolpakov to estimate the growth rate of power-free languages \\cite{kopakov2007efficient,Kolpakov2007Dec}. Shur later improved upon this technique \\cite{shur2010Jul,Shur2012Nov}. This technique was later adapted by the second author to provide a partial answer to Conjecture \\ref{conj:ThueListe} in  \\cite{rosenfeld2022avoiding}.\nThe proof of this result applies the technique developed in \\cite{rosenfeld2022avoiding} for squares, but provides a complete answer in the case of cubes.\nThe second proof of our main result does not imply anything about the number of solutions, but it contains algorithmic implications.\n\\begin{theorem} \\label{thm:computable}\nLet $\\A_1,\\A_2,\\dots$ be a computable sequence of binary alphabet. Then there exists a computable infinite cube-free word $w=w_1w_2w_3\\ldots$ in $\\A_1\\times\\A_2\\times\\dots$. Moreover, there is an algorithm that given $w_1\\ldots w_n$ and\\ $\\A_{n+1}$ computes $w_{n+1}$ in time polynomial in $n$.\n\\end{theorem}\nThis proof also relies on computing properties of the same automaton, but then we use a different argument for adding the large cubes back.\nIntuitively, the idea is to introduce a weight function that measures how difficult it is to extend a word, and to prove that under some condition if the weight is not too large then at least one of the extensions also has a small weight (which we can then use inductively to construct a word). This idea was originally due to Miller \\cite{miller2012two}, and was recently generalized in \\cite{Rosenfeld2025Mar}. However, the weight functions in \\cite{miller2012two} and \\cite{Rosenfeld2025Mar} are much simpler and in our article the precise definition of the function depends on the automaton. Our result is in fact even stronger, as everything still holds if for all $i$ an opponent is allowed to choose the alphabet $\\A_{i+1}$ after we choose $w_i$.\n\nNote that Theorem \\ref{thm:growth} and Theorem \\ref{thm:computable} both independently imply Theorem \\ref{thm:main}, but none of them seem to imply the other. We discuss the possible connection between this result in Section \\ref{sec:discussion}.\n\nThe article is organized as follows. In Section \\ref{sec:notation}, we introduce some common notations. In particular, we define the language $\\widetilde{\\mathcal{C}}^{(p)}$ that approximate the language of cube-free words, and we give the statement of Lemma \\ref{lem:eigenvector} that is later crucial to our two proofs. In Section \\ref{sec:potential}, we prove Theorem \\ref{thm:computable}. In Section \\ref{sec:growth}, we prove Theorem \\ref{thm:growth}. Finally, we provide in Section \\ref{sec:computations} a short description of the computation used to verify Lemma \\ref{lem:eigenvector}.", "sketch": "We \"do not prove [Theorem \\ref{thm:main}] directly\", but instead prove \"two different stronger results, each implying Theorem \\ref{thm:main}\" (Theorem~\\ref{thm:growth} and Theorem~\\ref{thm:computable}). Both proofs share a key idea: \"use an automaton\" to study an approximation where one \"only forbid[s] cubes of length at most $p$\". The motivation is that \"short cubes are more difficult to avoid than long cubes\", so with $p$ large this gives \"a good approximation\"; the \"delicate part\" is to show that for $p$ large enough, this approximation lets one conclude results for the original language where \"all cubes are forbidden\".\n\nFor the route via Theorem~\\ref{thm:growth}, the idea is \"to show that the growth rate of the language is not far from the growth rate of the regular approximation of the language\" and then apply the technique (Kolpakov; improved by Shur; adapted in \\cite{rosenfeld2022avoiding}) to \"deduce bounds on the growth of a power-free language\" from its approximation.\n\nFor the route via Theorem~\\ref{thm:computable}, one again uses \"computing properties of the same automaton\", but with \"a different argument for adding the large cubes back\": \"introduce a weight function that measures how difficult it is to extend a word\" and prove that \"under some condition if the weight is not too large then at least one of the extensions also has a small weight\", enabling an inductive construction of an infinite cube-free word; the weight function here \"depends on the automaton\" (in contrast to simpler weight functions in \\cite{miller2012two,Rosenfeld2025Mar}).", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:main}\nFor any sequence of binary alphabet $\\A_1,\\A_2,\\dots$, there exists an infinite cube-free word in $\\A_1\\times\\A_2\\times\\dots$.", "theorem_type": ["Universal–Existential", "Existence"], "mcq": {"question": "Let \\((\\mathcal A_i)_{i\\ge 1}\\) be any sequence of alphabets, where each \\(\\mathcal A_i\\) is binary (that is, \\(|\\mathcal A_i|=2\\)). A cube is a word of the form \\(uuu\\) for some nonempty word \\(u\\), and an infinite word is cube-free if none of its finite factors is a cube. Which statement holds about infinite words \\(w=a_1a_2a_3\\cdots\\) with \\(a_i\\in \\mathcal A_i\\) for every \\(i\\)?", "correct_choice": {"label": "A", "text": "There exists an infinite cube-free word \\(w=a_1a_2a_3\\cdots\\) such that \\(a_i\\in \\mathcal A_i\\) for every \\(i\\); equivalently, for any sequence of binary alphabets \\(\\mathcal A_1,\\mathcal A_2,\\dots\\), there exists an infinite cube-free word in \\(\\mathcal A_1\\times \\mathcal A_2\\times \\cdots\\)."}, "choices": [{"label": "B", "text": "There exists an infinite word \\(w=a_1a_2a_3\\cdots\\) such that \\(a_i\\in \\mathcal A_i\\) for every \\(i\\), and \\(w\\) is square-free; equivalently, for any sequence of binary alphabets \\(\\mathcal A_1,\\mathcal A_2,\\dots\\), there exists an infinite square-free word in \\(\\mathcal A_1\\times \\mathcal A_2\\times \\cdots\\)."}, {"label": "C", "text": "There exists an infinite word \\(w=a_1a_2a_3\\cdots\\) such that \\(a_i\\in \\mathcal A_i\\) for every \\(i\\), and every finite prefix of \\(w\\) is cube-free; in particular, \\(w\\) has arbitrarily long cube-free finite prefixes respecting \\((\\mathcal A_i)_{i\\ge 1}\\)."}, {"label": "D", "text": "For every sequence of binary alphabets \\(\\mathcal A_1,\\mathcal A_2,\\dots\\), there exists an integer \\(N\\) such that for every \\(n\\ge N\\) one can choose letters \\(a_1\\in\\mathcal A_1,\\dots,a_n\\in\\mathcal A_n\\) so that \\(a_1\\cdots a_n\\) is cube-free; equivalently, every such sequence admits cube-free words of all sufficiently large lengths."}, {"label": "E", "text": "There exists a single infinite cube-free binary word \\(w=b_1b_2b_3\\cdots\\) with the following universal property: for every sequence of binary alphabets \\(\\mathcal A_1,\\mathcal A_2,\\dots\\), one has \\(b_i\\in \\mathcal A_i\\) for every \\(i\\); equivalently, the same infinite cube-free word belongs to \\(\\mathcal A_1\\times\\mathcal A_2\\times\\cdots\\) for all choices of binary alphabets."}], "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": "cube_free_vs_square_free_target", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "drop_global_factor_condition_to_prefixwise_cube_free", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "replace_single_infinite_word_with_nonuniform_per_length_existence", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "dependence_of_word_on_alphabet_sequence", "template_used": "quantifier_dependence"}]}}
{"id": "2512.04053v1", "paper_link": "http://arxiv.org/abs/2512.04053v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{thm:main}\nLet $\\beta(n)\\colonequals \\max_{w\\in S_n}|\\mathrm{supp}(\\mathfrak S_w)|$. Then\n\\[\n\\lim_{n\\to\\infty}\\frac{\\ln(\\beta(n))}{n\\ln(n)} = 1.\n\\]\nMore precisely,\n\\[\n-\\ln(4)-1 \\leq \\liminf_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n \\leq \\limsup_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n \\leq -1.\n\\]", "start_pos": 9311, "end_pos": 9633, "label": "thm:main"}, "ref_dict": {"thm:main": "\\begin{thm}\n\\label{thm:main}\nLet $\\beta(n)\\colonequals \\max_{w\\in S_n}|\\supp(\\mathfrak S_w)|$. Then\n\\[\n\\lim_{n\\to\\infty}\\frac{\\ln(\\beta(n))}{n\\ln(n)} = 1.\n\\]\nMore precisely,\n\\[\n-\\ln(4)-1 \\leq \\liminf_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n \\leq \\limsup_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n \\leq -1.\n\\]\n\\end{thm}"}, "pre_theorem_intro_text_len": 1066, "pre_theorem_intro_text": "Schubert polynomials $\\mathfrak S_w(x_1, \\dots, x_n)$, indexed by permutations $w\\in S_n$, are lifts of Schubert cycles in the cohomology of the flag variety \\cite{ls82}. The specialization $\\mathfrak S_w(1,\\dots,1)$ is equal to the number of reduced pipe dreams of $w$, and has a geometric interpretation as the degree of the matrix Schubert variety of $w$. Writing $u(n)\\colonequals \\max_{w\\in S_n}\\mathfrak S_w(1, \\dots,1)$, Stanley \\cite{stanley17} observed that\n\\[\n\\frac14 \\leq \\liminf_{n\\to\\infty} \\frac{\\log_2(u(n))}{n^2} \\leq \\limsup_{n\\to\\infty}\\frac{\\log_2(u(n))}{n^2} \\leq \\frac12\n\\]\nand asked whether $\\lim_{n\\to\\infty}\\frac{\\log_2(u(n))}{n^2}$ exists, and, if so, what the value of this limit is. His question remains open, but see \\cite{mpp19,gao21,gl24,mppy25,zhang25} for recent progress on this problem and its variants.\n\nWe study the growth rate of the maximal sizes of \\emph{supports} of Schubert polynomials. Write $\\mathrm{supp}(\\mathfrak S_w)\\colonequals \\{\\alpha \\in \\ZZ_{\\geq 0}^n\\colon \\mathbf x^\\alpha\\textup{ appears in } \\mathfrak S_w\\}$.", "context": "Schubert polynomials $\\mathfrak S_w(x_1, \\dots, x_n)$, indexed by permutations $w\\in S_n$, are lifts of Schubert cycles in the cohomology of the flag variety \\cite{ls82}. The specialization $\\mathfrak S_w(1,\\dots,1)$ is equal to the number of reduced pipe dreams of $w$, and has a geometric interpretation as the degree of the matrix Schubert variety of $w$. Writing $u(n)\\colonequals \\max_{w\\in S_n}\\mathfrak S_w(1, \\dots,1)$, Stanley \\cite{stanley17} observed that\n\\[\n\\frac14 \\leq \\liminf_{n\\to\\infty} \\frac{\\log_2(u(n))}{n^2} \\leq \\limsup_{n\\to\\infty}\\frac{\\log_2(u(n))}{n^2} \\leq \\frac12\n\\]\nand asked whether $\\lim_{n\\to\\infty}\\frac{\\log_2(u(n))}{n^2}$ exists, and, if so, what the value of this limit is. His question remains open, but see \\cite{mpp19,gao21,gl24,mppy25,zhang25} for recent progress on this problem and its variants.\n\nWe study the growth rate of the maximal sizes of \\emph{supports} of Schubert polynomials. Write $\\mathrm{supp}(\\mathfrak S_w)\\colonequals \\{\\alpha \\in \\ZZ_{\\geq 0}^n\\colon \\mathbf x^\\alpha\\textup{ appears in } \\mathfrak S_w\\}$.", "full_context": "Schubert polynomials $\\mathfrak S_w(x_1, \\dots, x_n)$, indexed by permutations $w\\in S_n$, are lifts of Schubert cycles in the cohomology of the flag variety \\cite{ls82}. The specialization $\\mathfrak S_w(1,\\dots,1)$ is equal to the number of reduced pipe dreams of $w$, and has a geometric interpretation as the degree of the matrix Schubert variety of $w$. Writing $u(n)\\colonequals \\max_{w\\in S_n}\\mathfrak S_w(1, \\dots,1)$, Stanley \\cite{stanley17} observed that\n\\[\n\\frac14 \\leq \\liminf_{n\\to\\infty} \\frac{\\log_2(u(n))}{n^2} \\leq \\limsup_{n\\to\\infty}\\frac{\\log_2(u(n))}{n^2} \\leq \\frac12\n\\]\nand asked whether $\\lim_{n\\to\\infty}\\frac{\\log_2(u(n))}{n^2}$ exists, and, if so, what the value of this limit is. His question remains open, but see \\cite{mpp19,gao21,gl24,mppy25,zhang25} for recent progress on this problem and its variants.\n\nWe study the growth rate of the maximal sizes of \\emph{supports} of Schubert polynomials. Write $\\mathrm{supp}(\\mathfrak S_w)\\colonequals \\{\\alpha \\in \\ZZ_{\\geq 0}^n\\colon \\mathbf x^\\alpha\\textup{ appears in } \\mathfrak S_w\\}$.\n\nThe key ingredient in our proof of Theorem~\\ref{thm:main} is the observation that a certain \\emph{layered} permutation has support of size asymptotically at least $n!/4^n$. Layered permutations have previously appeared in Merzon--Smirnov's conjecture \\cite{ms16} that $\\mathfrak S_w(1,\\dots,1)$ is maximized at a layered permutation. Guo--Lin conjectured \\cite[Prob 5.3]{gl24} that there exist permutations simultaneously maximizing $\\mathfrak S_w(1, \\dots,1)$ and $|\\supp(\\mathfrak S_w)|$.\n\nWe will use the following (crude) estimates for the factorial function, whose proofs we include for completeness.\n\\begin{lem}\n\\label{lem:factorial-estimate}\nThe inequalities\n\\[\ne\\left(\\frac ne\\right)^n \\leq n! \\leq e\\left(\\frac{n+1}e\\right)^{n+1}\n\\]\nhold for all positive integers $n$.\n\\end{lem}\n\\begin{proof}\nEstimate $\\ln(n!) = \\sum_{k=1}^n\\ln(k)$ by the integrals\n\\[\n\\int_1^n \\ln(x)\\,dx \\leq \\sum_{k=1}^n \\ln(k) \\leq \\int_1^{n+1}\\ln(x)\\,dx.\n\\]\nThe definite integrals evaluate to $n\\ln(n) - n + 1$ and $(n+1)\\ln(n+1) - (n+1) + 1$. The claim follows.\n\\end{proof}\n\\begin{lem}\n\\label{lem:floor-fact-est}\nThe inequality\n\\[\n\\lfloor k\\rfloor! \\geq \\frac1{2k}\\left(\\frac ke\\right)^k\n\\]\nholds for all real numbers $k>1$.\n\\end{lem}\n\\begin{proof}\nUsing Lemma~\\ref{lem:factorial-estimate}, compute that\n\\[\n\\lfloor k\\rfloor! \\geq \\frac1{k+1} \\lceil k\\rceil! \\geq \\frac1{k+1} \\cdot \\frac{\\lceil k\\rceil^{\\lceil k\\rceil}}{e^{\\lceil k\\rceil-1}} \\geq \\frac1{k+1}\\cdot \\frac{k^k}{e^k}.\n\\]\nAs $k > 1$, the claim follows.\n\\end{proof}\n\\section{A layered permutation with large Schubitope}\nOur main technical result is as follows.\n\\begin{thm}\n\\label{thm:tech-main}\nLet $w = w(b_1, \\dots, b_m)$ be a layered permutation in $S_n$. If $b_m \\leq n/2$, then\n\\[\n|\\supp(\\mathfrak S_w)| \\geq b_m!\\cdot |\\supp(\\mathfrak S_{w'})|, \\qquad \\textup{ where } w'\\colonequals w(b_1, \\dots, b_{m-1})\\in S_{n-b_m}.\n\\]\n\\end{thm}\nOur proof of Theorem~\\ref{thm:tech-main} will use the following lemma.\n\\begin{lem}\n\\label{lem:last-block}\nLet $w\\in S_n$ be a layered permutation of the form $w = w(1, \\dots, 1, b_m)$ for $b_m \\leq n/2$. For any subset $S\\subseteq D(w)$, there exists a diagram $C$ satisfying $C\\leq D(w)$ and $C\\cap D(w) = S$.\n\\end{lem}\n\\begin{proof}\nEach column $D_j$ of $D(w)$ is of the form $\\{n-b_m + 1, \\dots, n-c\\}$ for some $c \\geq 1$. As $|D_j| \\leq b_m-1 < n-b_m$, it follows that for any subset $S_j$ of $D_j$ there exists a set $C_j\\leq D_j$ with $C_j \\cap D_j = S_j$. Applying this column by column to $D(w)$, the claim follows.\n\\end{proof}\n\\begin{proof}[Proof of Theorem~\\ref{thm:tech-main}]\nFor each of the $b_m!$ many vectors $\\alpha\\in\\ZZ_{\\geq0}^{b_m}$ satisfying $(0,\\dots,0) \\leq \\alpha \\leq (b_m - 1, b_m - 2, \\dots, 1, 0)$, we construct an embedding $i_\\alpha\\colon \\mathcal S_{D(w')} \\hookrightarrow \\mathcal S_{D(w)}$ so that $\\img(i_\\alpha)\\cap\\img(i_{\\alpha'}) = \\emptyset$ for all $\\alpha\\neq\\alpha'$.\n\nFor each $\\alpha$ satisfying $(0,\\dots,0)\\leq\\alpha\\leq (b_m - 1, b_m - 2, \\dots, 1)$, use Lemma~\\ref{lem:last-block} to choose a diagram $C_\\alpha \\leq D(w_{\\mathrm{LB}})$ whose weight satisfies $\\wt(C_\\alpha)_{n-b_i + j} = \\alpha_j$. Then define the map\n\\begin{align*}\ni_\\alpha\\colon \\mathcal S_{D(w')}&\\to \\mathcal S_{D(w)}\\\\\n\\wt(C')&\\mapsto\\wt((C', C_\\alpha));\n\\end{align*}\nthe map $i_\\alpha$ is well-defined and injective because it is in fact the translation map $\\gamma \\mapsto \\gamma +\\wt(C_\\alpha)$. Furthermore, any $\\gamma' \\in \\mathcal S_{D(w')}$ satisfies $\\gamma'_{n-b_m+j} = 0$ for all $j\\geq 0$, so any vector $\\gamma\\in\\img(i_\\alpha)$ satisfies $\\gamma_{n-b_m+j} = \\alpha_j$. In particular, the $\\img(i_\\alpha)$ are disjoint subsets of $\\mathcal S_{D(w)}$.\n\\end{proof}\n\\begin{cor}\n\\label{cor:upshot}\nFix an integer $n\\geq 3$ and let $c\\colonequals \\max\\{k \\colon \\lfloor n/2^k\\rfloor \\geq 1\\}$ and $d \\colonequals n - \\sum_{k=1}^c \\lfloor n/2^k\\rfloor$. Let $w$ be the layered permutation with blocks of size\n\\[\n\\underbrace{1,\\dots,1}_{d\\textup{ many}}, \\left\\lfloor \\frac n{2^c}\\right\\rfloor, \\left\\lfloor \\frac n{2^{c-1}}\\right\\rfloor, \\dots, \\left\\lfloor\\frac n4\\right\\rfloor,\\left\\lfloor \\frac n2\\right\\rfloor.\n\\]\nThen \n\\[\n|\\supp(\\mathfrak S_w)| \\geq \\prod_{k=1}^c\\left\\lfloor \\frac n{2^k}\\right\\rfloor!.\n\\]\nIn particular, \n\\[\n\\ln(|\\supp(\\mathfrak S_w)|) \\geq \\left(n - \\frac1{\\ln(2)}\\ln(n) - 2\\right)\\ln(n) - n\\ln(4) - (n-2).\n\\]\n\\end{cor}\n\\begin{proof}\nFrom the inequality\n\\[\n\\left\\lfloor \\frac n{2^k}\\right\\rfloor \\leq \\frac n{2^k} = \\frac12\\left(n - \\sum_{i=1}^{k-1} \\frac n{2^i}\\right) \\leq \\frac12\\left(n - \\sum_{i=1}^{k-1} \\left\\lfloor\\frac n{2^i}\\right\\rfloor\\right),\n\\]\nTheorem~\\ref{thm:tech-main} may be repeatedly applied to the layered permutation $w$ to obtain\n\\[\n|\\supp(\\mathfrak S_w)| \\geq \\prod_{k=1}^c\\left\\lfloor \\frac n{2^k}\\right\\rfloor!.\n\\]\nLemma~\\ref{lem:floor-fact-est} implies\n\\begin{align*}\n\\prod_{k=1}^c\\left \\lfloor \\frac n{2^k}\\right\\rfloor! &\\geq \\prod_{k=1}^c\\left(\\frac{2^{k-1}}n \\left(\\frac n{2^k e}\\right)^{\\frac n{2^k}}\\right) \\\\&\\geq \\frac 1{n^c} \\cdot \\underbrace{\\prod_{k=1}^c \\left(\\frac1{2^k}\\right)^{\\frac n{2^k}}}_{\\geq 2^{-2n}}\\cdot \\underbrace{\\prod_{k=1}^c \\left(\\frac ne\\right)^{\\frac n{2^k}}}_{\\geq \\left(\\frac ne\\right)^{n-2}}\\label{eqn:estimate}\\tag{$*$}\\\\\n&\\geq \\frac{n^{n-c-2}}{4^n\\cdot e^{n-2}}\\\\&\\geq \\frac{n^{n - \\log_2(n) - 2}}{4^n \\cdot e^{n-2}}\n\\end{align*}\nwhere the estimates in~\\eqref{eqn:estimate} follow from\n\\[\n\\sum_{k=1}^c k\\cdot \\frac n{2^k} \\leq \\sum_{k=1}^\\infty k \\cdot \\frac n{2^k} = 2n\n\\]\nand\n\\[\n\\frac n2 + \\dots + \\frac n{2^c} = n - \\frac n{2^c} \\geq n-2.\n\\]\n\\end{proof}\n\n\\newtheorem*{thm:main}{Theorem~\\ref{thm:main}}\n\\begin{thm:main}\nLet $\\beta(n)\\colonequals \\max_{w\\in S_n}|\\supp(\\mathfrak S_w)|$. Then\n\\[\n\\lim_{n\\to\\infty}\\frac{\\ln(\\beta(n))}{n\\ln(n)} = 1.\n\\]\nMore precisely,\n\\[\n-\\ln(4)-1 \\leq \\liminf_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n \\leq \\limsup_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n \\leq -1.\n\\]\n\\end{thm:main}\n\\begin{proof}[Proof of Theorem~\\ref{thm:main}]\nBy Lemma~\\ref{lem:upper-bound}, the inequality \n\\[\n\\ln(|\\supp(\\mathfrak S_w)|) \\leq \\ln(n!) \\leq (n+1)\\ln(n+1) - n\n\\]\nholds for all $w\\in S_n$. By Corollary~\\ref{cor:upshot}, there exists $w\\in S_n$ with \n\\[\n\\ln(|\\supp(\\mathfrak S_w)|) \\geq \\left(n-\\frac1{\\ln(2)}\\ln(n) - 2\\right)\\ln(n) - n\\ln(4) - (n-2).\n\\]\nThe inequalities\n\\[\n-\\ln(4)-1 \\leq \\liminf_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n \\leq \\limsup_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n \\leq -1\n\\]\nfollow.\n\\end{proof}\n\n\\begin{prop}\n\\label{prop:groth-precise}\nFix an integer $n$, let $k$ be the unique integer so that $\\binom{k+1}2\\leq n<\\binom{k+2}2$, and let $b \\colonequals n - \\sum_{i=1}^k i$. Let $w$ be the layered permutation with blocks of size $1, 2, \\dots, k, b$. Then\n\\[\n|\\supp(\\mathfrak G_w)| \\geq \\frac{n!}{n^{k+1}}.\n\\]\nIn particular,\n\\[\n\\ln(|\\supp(\\mathfrak G_w)|) \\geq n\\ln(n) - n - (\\sqrt{2n}+1)\\ln(n)\n\\]\n\\end{prop}\n\\begin{proof}\nLet $\\mathbf c = \\mathrm{wt}(D(w))$ denote the Lehmer code of $w$, and let $\\mathbf d = \\mathrm{wt}(\\overline{D(w)})$. Proposition~\\ref{prop:fireworks-grothendieck} implies that $[\\mathbf c, \\mathbf d] \\subseteq \\supp(\\mathfrak G_w)$, so\n\\begin{equation}\n\\label{eqn:crude-grothendieck}\n|\\supp(\\mathfrak G_w)| \\geq \\prod_{i=1}^n (d_i - c_i + 1).\n\\end{equation}\n\n\\newtheorem*{thm:groth}{Theorem~\\ref{thm:groth}}\n\\begin{thm:groth}\nLet $\\beta^{\\mathfrak G}(n)\\colonequals\\max_{w\\in S_n}|\\supp(\\mathfrak G_w)|$. Then\n\\[\n\\lim_{n\\to\\infty}\\frac{\\ln(\\beta^{\\mathfrak G}(n)) - n\\ln(n)}n = -1.\n\\]\n\\end{thm:groth}\n\\begin{proof}[Proof of Theorem~\\ref{thm:groth}]\nBy Lemma~\\ref{lem:upper-bound}, the inequality\n\\[\n\\ln(|\\supp(\\mathfrak G_w)|)\\leq \\ln(n!) \\leq (n+1)\\ln(n+1)-n\n\\]\nholds for all $w\\in S_n$. By Proposition~\\ref{prop:groth-precise}, there exists $w\\in S_n$ with\n\\[\n\\ln(|\\supp(\\mathfrak G_w)|) \\geq n\\ln(n) - n - (\\sqrt{2n}+1)\\ln(n).\n\\]\nThe claim follows.\n\\end{proof}", "post_theorem_intro_text_len": 1386, "post_theorem_intro_text": "Theorem~\\ref{thm:main} answers a problem \\cite[Prob 5.5]{gl24} posed by Guo--Lin. We do not know if the limit $\\lim_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n$ exists, nor do we have a conjecture for its value.\n\nThe key ingredient in our proof of Theorem~\\ref{thm:main} is the observation that a certain \\emph{layered} permutation has support of size asymptotically at least $n!/4^n$. Layered permutations have previously appeared in Merzon--Smirnov's conjecture \\cite{ms16} that $\\mathfrak S_w(1,\\dots,1)$ is maximized at a layered permutation. Guo--Lin conjectured \\cite[Prob 5.3]{gl24} that there exist permutations simultaneously maximizing $\\mathfrak S_w(1, \\dots,1)$ and $|\\mathrm{supp}(\\mathfrak S_w)|$.\n\nGrothendieck polynomials $\\mathfrak G_w$ are inhomogeneous deformations of Schubert polynomials; they are generating functions for (possibly nonreduced) pipe dreams of $w$. We are able to produce a layered permutation with support of size asymptotically at least $n!/e^{\\sqrt{2n} \\cdot \\ln(n)}$, i.e., of size $n!$ up to a subexponential factor. In particular, the maximal sizes of supports of Grothendieck polynomials satisfy the following more precise asymptotics.\n\\begin{thm}\n\\label{thm:groth}\nLet $\\beta^{\\mathfrak G}(n)\\colonequals\\max_{w\\in S_n}|\\mathrm{supp}(\\mathfrak G_w)|$. Then\n\\[\n\\lim_{n\\to\\infty}\\frac{\\ln(\\beta^{\\mathfrak G}(n)) - n\\ln(n)}n = -1.\n\\]\n\\end{thm}", "sketch": "The post-theorem introduction indicates that “the key ingredient in our proof of Theorem~\\ref{thm:main} is the observation that a certain \\emph{layered} permutation has support of size asymptotically at least $n!/4^n$.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:main}\nLet $\\beta(n)\\colonequals \\max_{w\\in S_n}|\\mathrm{supp}(\\mathfrak S_w)|$. Then\n\\[\n\\lim_{n\\to\\infty}\\frac{\\ln(\\beta(n))}{n\\ln(n)} = 1.\n\\]\nMore precisely,\n\\[\n-\\ln(4)-1 \\leq \\liminf_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n \\leq \\limsup_{n\\to\\infty}\\frac{\\ln(\\beta(n)) - n\\ln(n)}n \\leq -1.\n\\],", "theorem_type": ["Asymptotic or Limit", "Inequality or Bound"], "mcq": {"question": "For each positive integer \\(n\\), let \\(S_n\\) be the symmetric group on \\(n\\) letters, and for \\(w\\in S_n\\) let \\(\\mathfrak S_w(x_1,\\dots,x_n)\\) denote the corresponding Schubert polynomial. Define its support by\n\\[\n\\mathrm{supp}(\\mathfrak S_w)\\colonequals \\{\\alpha\\in \\mathbb Z_{\\ge 0}^n : \\mathbf x^\\alpha \\text{ appears in } \\mathfrak S_w\\},\n\\]\nand set\n\\[\n\\beta(n)\\colonequals \\max_{w\\in S_n} |\\mathrm{supp}(\\mathfrak S_w)|.\n\\]\nUsing \\(\\ln\\) for the natural logarithm, as \\(n\\to\\infty\\), which asymptotic statement holds for \\(\\beta(n)\\)?", "correct_choice": {"label": "A", "text": "The quantity \\(\\beta(n)\\) satisfies\n\\[\n\\lim_{n\\to\\infty}\\frac{\\ln(\\beta(n))}{n\\ln(n)}=1,\n\\]\nand more precisely,\n\\[\n-\\ln(4)-1\\le \\liminf_{n\\to\\infty}\\frac{\\ln(\\beta(n))-n\\ln(n)}{n}\n\\le \\limsup_{n\\to\\infty}\\frac{\\ln(\\beta(n))-n\\ln(n)}{n}\n\\le -1.\n\\]"}, "choices": [{"label": "B", "text": "The quantity \\(\\beta(n)\\) satisfies\n\\[\n\\lim_{n\\to\\infty}\\frac{\\ln(\\beta(n))}{n\\ln(n)}=1,\n\\]\nand more precisely,\n\\[\n-\\ln(4)-1\\le \\liminf_{n\\to\\infty}\\frac{\\ln(\\beta(n))-n\\ln(n)}{n}\n\\le \\limsup_{n\\to\\infty}\\frac{\\ln(\\beta(n))-n\\ln(n)}{n}\n\\le 0.\n\\]"}, {"label": "C", "text": "The quantity \\(\\beta(n)\\) satisfies\n\\[\n\\lim_{n\\to\\infty}\\frac{\\ln(\\beta(n))}{n\\ln(n)}=1.\n\\]"}, {"label": "D", "text": "The quantity \\(\\beta(n)\\) satisfies\n\\[\n\\lim_{n\\to\\infty}\\frac{\\ln(\\beta(n))}{n\\ln(n)}=1,\n\\]\nand more precisely,\n\\[\n-\\ln(4)-1\\le \\liminf_{n\\to\\infty}\\frac{\\ln(\\beta(n))-n\\ln(n)}{n}\n\\le \\limsup_{n\\to\\infty}\\frac{\\ln(\\beta(n))-n\\ln(n)}{n}\n\\le -\\ln(4)-1.\n\\]"}, {"label": "E", "text": "The quantity \\(\\beta(n)\\) satisfies\n\\[\n\\limsup_{n\\to\\infty}\\frac{\\ln(\\beta(n))}{n\\ln(n)}=1,\n\\]\nand more precisely,\n\\[\n-\\ln(4)-1\\le \\liminf_{n\\to\\infty}\\frac{\\ln(\\beta(n))-n\\ln(n)}{n}\n\\le \\limsup_{n\\to\\infty}\\frac{\\ln(\\beta(n))-n\\ln(n)}{n}\n\\le -1.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "upper_bound_constant_in_second_order_term", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped_refined_liminf_limsup_bounds", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "collapsed_liminf_and_limsup_to_same_value", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "replaced_limit_with_limsup_only", "template_used": "quantifier_dependence"}]}}
{"id": "2512.04410v2", "paper_link": "http://arxiv.org/abs/2512.04410v2", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:opt-iid-homo}\nAssume {\\rm(A1), (A2), (A3)}, and $d\\ge 3$. Let $\\bar a,\\bar\\psi$ be as in Theorem~\\ref{thm:opt-quant-homo}. Suppose that $f\\in C^{4,\\alpha}(\\bar\\B_1)$ and $g\\in C^{6,\\alpha}(\\partial\\B_1)$ for some $\\alpha\\in (0,1)$. Then, there exists a $C^6(\\bar\\B_1)$-extension of $g$ such that the solutions  $u, \\bar u$ of \\eqref{eq:elliptic-dirich}, \\eqref{eq:effective-ellip} satisfy\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar u(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n  \\begin{array}{rl}\n  \tR^{-3/2}\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\tx &\\text{ when } d=3,\\\\\n  \tR^{-2}(\\log R)\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\tx&\\text{ when }d\\ge 4.\n  \\end{array}\n\\right.\n\\]\nAn example of such an extension is $g=\\bar u$.", "start_pos": 19166, "end_pos": 19887, "label": "thm:opt-iid-homo"}, "ref_dict": {"thm:opt-iid-homo": "\\begin{theorem}\n\\label{thm:opt-iid-homo}\nAssume {\\rm(A1), (A2), (A3)}, and $d\\ge 3$. Let $\\bar a,\\bar\\psi$ be as in Theorem~\\ref{thm:opt-quant-homo}. Suppose that $f\\in C^{4,\\alpha}(\\bar\\B_1)$ and $g\\in C^{6,\\alpha}(\\partial\\B_1)$ for some $\\alpha\\in (0,1)$. Then, there exists a $C^6(\\bar\\B_1)$-extension of $g$ such that the solutions  $u, \\bar u$ of \\eqref{eq:elliptic-dirich}, \\eqref{eq:effective-ellip} satisfy\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar u(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n  \\begin{array}{rl}\n  \tR^{-3/2}\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\tx &\\text{ when } d=3,\\\\\n  \tR^{-2}(\\log R)\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\tx&\\text{ when }d\\ge 4.\n  \\end{array}\n\\right.\n\\]\nAn example of such an extension is $g=\\bar u$.\n\\end{theorem}", "rmk:optimality": "\\begin{remark}\\label{rmk:optimality}\nThis proposition identifies $O(R^{-1})$ as the optimal homogenization rate unless $\\bar z \\equiv 0$, and so no improvement beyond $O(R^{-1})$ should be expected in general. \n         We further expect that Proposition~\\ref{lem:two-scale-exp} remains valid for environments with finite range of dependence, for the same reasons discussed in the comments following Theorem~\\ref{thm:opt-quant-homo}.\n\\end{remark}", "eq:effective-ellip": "\\begin{equation}\\label{eq:effective-ellip}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr(\\bar a\\, D^2\\bar{u}(x))\n=f(x)\\,\\bar\\psi &x\\in\\B_1,\\\\ \n\\bar u(x)=g(x) &x\\in\\partial \\B_1.\n\\end{array}\n\\right.\n\\end{equation}", "thm:c11": "\\begin{thmx}\n\\label{thm:c11}\nAssume {\\rm(A1), (A2)},  and that $\\psi$ is a {\\bf local} function. Let $R\\ge 1$.  \nThere exists $\\alpha=\\alpha(d,\\kappa)\\in(0,\\tfrac13)$ such that,\nfor any $u$ with $L_\\omega u(x)=\\psi(\\theta_x\\omega)+f(x)$ on $B_R$, $j\\in\\{1,2\\}$, $\\tx\\le r<R$, we have that \n\\begin{equation}\n\\label{eq:oscillation-estimates}\n\\frac{1}{r^{j}}\\inf_{p\\in\\fct_{j-1}}\\osc_{B_r}\\,(u-p)\n\\lesssim\n\\frac{1}{R^{j}}\n\\inf_{p\\in\\fct_{j-1}}\\osc_{B_R}\\,(u-p)+A_{j},\n\\end{equation}\nwhere the terms $A_1=A_1(R)$ and $A_2=A_2(R,r)$ obey the bounds\n\\begin{align*}\n&A_1\\le R^{1-\\alpha}\\norm{\\psi}_\\infty+R\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_1\\le R^{1-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^{1+\\sigma}[f]_{\\sigma;B_R},\n\\\\\n&A_2\\le r^{-\\alpha}\\norm{\\psi}_\\infty+\\log(\\tfrac{R}{r})\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_2\\le r^{-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^\\sigma[f]_{\\sigma;B_R}\n\\end{align*}\nfor any $\\sigma\\in(0,1]$. In particular, for any $R>1$ and $j\\in\\{1,2\\}$, there holds\n\\begin{equation}\\label{eq:c2}\n|\\nabla^ju(0)|\\lesssim \\left(\\frac{\\tx}{R}\\right)^j \n\\left(\n\\norm{u-u(0)}_{1;B_R}+R^2\\norm{\\psi+f(0)}_\\infty+R^{2+\\sigma}[f]_{\\sigma;B_{R}}\n\\right).\n\\end{equation}\n\\end{thmx}", "eq:def-lambda": "\\begin{align}\n&\\lambda^k_j(x)=\\lambda^k_j(x;\\omega) := a_j(x)\\left[v^k(x+e_j)-v^k(x-e_j)\\right],\\;\\;\n\\bar{\\lambda}^k_j(x) :=E_{\\mb P}\\left[\\rho(x)\\lambda^k_j(x)\\right],\\label{eq:def-lambda}\\\\\n&\\eta_j(x)=\\eta_j(x;\\omega)\\; :=a_j(x)\\left[\\xi(x+e_j)-\\xi(x-e_j)\\right],\n\\quad\\;\\;\\,\n\\bar{\\eta}_j(x) :=E_{\\mb P}\\left[\\rho(x)\\eta_j(x)\\right],\\label{eq:def-eta}\n\\end{align}", "eq:elliptic-dirich": "\\begin{equation}\\label{eq:elliptic-dirich}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr\\big(\\omega(x)\\nabla^2u(x)\\big)=\\frac{1}{R^2}f\\left(\\tfrac{x}{R}\\right)\\zeta(\\theta_x\\omega) & x\\in \\inB R,\\\\[5 pt]\nu(x)=g\\left(\\tfrac{x}{R}\\right) & x\\in \\partial \\inB R,\n\\end{array}\n\\right.\n\\end{equation}", "thm:opt-quant-homo": "\\begin{thmx}\n\\label{thm:opt-quant-homo}\nAssume {\\rm(A1)}, {\\rm(A2)}, and {\\rm(A3)}. Let $\\bar a :=E_{\\Q}[\\frac{\\omega(0)}{\\tr\\omega(0)}]>0$ and $\\bar\\psi :=E_{\\Q}[\\frac{\\zeta}{\\tr\\omega(0)}]$. Suppose $f,g$ are both in $C^4(\\R^d)$. Then, for any $\\error\\in(0,1)$ and $R\\ge 2$, there exists a random variable $\\ms Y=\\ms Y(R,\\error, \\omega)>1$ with $\\mb E[\\exp(\\ms Y^{{d}/{(2d+2)}-\\error})]<C$ such that the solution $u$ of \\eqref{eq:elliptic-dirich} satisfies\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar{u}(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n\\begin{array}{rl}\n(\\log R)^{2+\\error}R^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d=2,\\\\\nR^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d\\ge 3,\n\\end{array}\n\\right.\n\\]\nwhere $\\bar u$ is the solution of the Dirichlet problem \\eqref{eq:effective-ellip}.\n\\end{thmx}", "thm:global_krt": "\\begin{thmx}\n\\label{thm:global_krt}\nLet $\\psi$ be an $L^\\infty(\\mb P)$-bounded function of $\\omega(0)$ with $\\norm{\\psi}_\\infty=1$.\nFor each $d\\ge 2$ and $\\mb P$-a.e. $\\omega$, there exists a function $\\krt=\\krt_\\omega:\\Z^d\\to\\R$ that solves \n\\[\nL_\\omega\\krt(x)=\\psi(\\theta_x\\omega)-\\bar\\psi \\quad\\text{ for }x\\in\\Z^d\n\\]\nwith the following properties:\n\\begin{enumerate}[(i)]\n\\item\\label{item:gkrt-2} When $d\\ge 5$,\\quad $\\mb E[\\exp(c|\\krt(x)/\\mu(|x|)|^p)]<C_p$ for any $p\\in(0,\\tfrac{2d}{3d+2}),x\\in\\Z^d$;\n\nWhen $d=3,4$,\\; $\\mb E[\\exp(c|\\krt(x)/\\mu(|x|)|^p)]<C_p$ for any $p\\in(0,\\tfrac{2d}{3d+4}),x\\in\\Z^d$;\n\nWhen $d=2$,\\quad\n$\\mb E\\left[\n\\exp\\big(\nc\\abs{\\tfrac{\\krt(x)}{|x|\\delta(|x|)}}^p\n\\big)\n\\right]\\le C_p$ for any $p\\in(0,\\tfrac13), x\\in\\Z^d$;\n\n\\item\\label{item:gkrt-3} $\\mb E[\\exp(c|\\nabla\\krt(x)/\\delta(|x|)|^q)]\\le C_q$ for any $q\\in(0,\\tfrac{d}{2d+2})$ and $x\\in\\Z^d$;\n\\item\\label{item:gkrt-2nd-der}\n $\\mb E[\\exp(c|\\nabla^2\\krt(x)|^r)]\\le C_r$ for any $r\\in(0,\\tfrac{d}{2})$ and $x\\in\\Z^d$;\n\\item\\label{item:gkrt-4}(Stationarity properties)\n\\begin{itemize}\n\\item When $d\\ge 5$,  the field $\\{\\krt_\\omega(x):x\\in\\Z^d\\}$ is stationary (under $\\mb P$);\n\\item When $d\\ge 3$, we have $\\nabla\\krt_\\omega(x)=\\nabla\\krt_{\\theta_x\\omega}(0)$ for all $x\\in\\Z^d$, and the field $\\{\\nabla\\krt_\\omega(x):x\\in\\Z^d\\}$  is stationary;\n\\item When $d=2$, we have $\\nabla^2\\krt_\\omega(x)=\\nabla^2\\krt_{\\theta_x\\omega}(0)$ for all $x\\in\\Z^d$, and the field $\\{\\nabla^2\\krt_\\omega(x):x\\in\\Z^d\\}$  is stationary.\n\\end{itemize}\n\\end{enumerate}\nMoreover,  such a corrector $\\krt$ is unique up to an additive constant when $d\\ge 3$,  and is unique up to an affine transformation when $d=2$.\n\\end{thmx}", "lem:two-scale-exp": "\\begin{proposition}\n\\label{lem:two-scale-exp}\nSuppose that $d\\geq 3$, $f\\in C^{4,\\alpha}(\\bar\\B_1)$, and $g\\in C^{6,\\alpha}(\\partial\\B_1)$. Let $u,\\bar u$ be the solutions of \\eqref{eq:elliptic-dirich}, \\eqref{eq:effective-ellip}, respectively. Recall $a,\\bar a$ from \\eqref{eq:def-a}, \\eqref{eq:def-abar}, and let $v^k,\\xi,\\lambda^k_j, \\bar\\lambda^k_j,\\eta_j,\\bar\\eta_j$ be as above.  We extend $g$ so that $g=\\bar{u}$ on $\\bar\\B_1$. Let $\\bar z:\\bar\\B_1\\to\\R$ be the solution of \n\\[\n\\left\\{\n\\begin{array}{lr}\n\\frac{1}{2}\\tr(\\bar{a}D^2 \\bar{z}(x))=\\tfrac{1}{2}\\bar{\\lambda}^k_i\\partial_{kki}\\bar{u}(x)-\\tfrac{1}{2}\\bar\\eta_j\\partial_j f(x) & x\\in\\B_1,\\\\\n\\bar z(x)=0 &x\\in\\partial\\B_1.\n\\end{array}\n\\right.\n\\]\nFor $\\omega\\in\\Omega$, $R>1$, and $j,k\\in\\{1,\\ldots,d\\}$, let $p^k_j, s_j:\\Z^d \\rightarrow \\R$ be solutions of \n\\begin{align}\nL_\\omega p^k_j(x)&=\\lambda^k_j(x)-\\bar\\lambda^k_j \\quad\\text{ for }x\\in B_R,\\label{eq:def-p}\\\\\nL_\\omega s_j(x)&=\\eta_j(x)-\\bar\\eta_j \\quad\\;\\text{ for }x\\in B_R\\label{eq:def-sj},\n\\end{align}\nrespectively. Then, for $\\omega\\in\\Omega$, $R>1$, we have\n\\begin{align}\\label{eq:two-scale-further}\n&\\max_{x\\in B_R}\\left|u(x)-\\bar{u}(\\tfrac xR)-\\tfrac 1R\\bar z(\\tfrac xR)\\right|\\nn\\\\\n&\\qquad\\lesssim\n\\tfrac 1{R^2}\\norm{\\bar u}_{C^6(\\bar{\\B}_1)}\\max_{i,j,k=1}^d\n\\bigg[1+\\left\\| |\\nabla p^k_j|+|\\nabla s_j|+(|\\bar\\lambda_j^i|+|\\bar\\eta_j|)|\\nabla v^k|\\right\\|_{d;B_R}\n\\nn\\\\\n&\\qquad\n\\qquad\\qquad\\qquad\\qquad\\quad\\;\\;\\,+\\max_{\\bar B_R}\\left(|\\xi|+|v^k|+\\tfrac{1}{R}(|p_i^k|+|s_i|)\\right)\\bigg].\n\\end{align}\n\\end{proposition}", "thm:QCLT": "\\begin{thmx}\\label{thm:QCLT}\nAssume \\rm{(A2)} and that law $\\mb P$ of the environment is ergodic under spatial shifts $\\{\\theta_x:x\\in\\Z^d\\}$. Then, the following assertions hold. \n\\begin{enumerate}[(i)]\n\\item\\label{item:ergodic} There exists a probability measure $\\mb Q\\approx\\mb P$ such that $(\\evp{i})_{i\\ge 0}$ is an ergodic (with respect to time shifts) sequence under law $\\mb Q\\times P_\\omega$.\n\\item For $\\mb P$-almost every $\\omega$, the rescaled path $X_{n^2t}/n$ converges weakly (under law $P_\\omega$) to a Brownian motion with covariance matrix \n\\begin{equation}\n\\label{eq:def-abar}\n\\bar a=\\diag[\\bar a_1,\\ldots,\\bar a_d]:=E_\\Q[a]=E_\\Q[\\tfrac{\\omega(0)}{\\tr\\omega(0)}]>0.\n\\end{equation}\n\\end{enumerate}\n\\end{thmx}", "eq:def-eta": "\\begin{align}\n&\\lambda^k_j(x)=\\lambda^k_j(x;\\omega) := a_j(x)\\left[v^k(x+e_j)-v^k(x-e_j)\\right],\\;\\;\n\\bar{\\lambda}^k_j(x) :=E_{\\mb P}\\left[\\rho(x)\\lambda^k_j(x)\\right],\\label{eq:def-lambda}\\\\\n&\\eta_j(x)=\\eta_j(x;\\omega)\\; :=a_j(x)\\left[\\xi(x+e_j)-\\xi(x-e_j)\\right],\n\\quad\\;\\;\\,\n\\bar{\\eta}_j(x) :=E_{\\mb P}\\left[\\rho(x)\\eta_j(x)\\right],\\label{eq:def-eta}\n\\end{align}", "eq:two-scale-further": "\\begin{align}\\label{eq:two-scale-further}\n&\\max_{x\\in B_R}\\left|u(x)-\\bar{u}(\\tfrac xR)-\\tfrac 1R\\bar z(\\tfrac xR)\\right|\\nn\\\\\n&\\qquad\\lesssim\n\\tfrac 1{R^2}\\norm{\\bar u}_{C^6(\\bar{\\B}_1)}\\max_{i,j,k=1}^d\n\\bigg[1+\\left\\| |\\nabla p^k_j|+|\\nabla s_j|+(|\\bar\\lambda_j^i|+|\\bar\\eta_j|)|\\nabla v^k|\\right\\|_{d;B_R}\n\\nn\\\\\n&\\qquad\n\\qquad\\qquad\\qquad\\qquad\\quad\\;\\;\\,+\\max_{\\bar B_R}\\left(|\\xi|+|v^k|+\\tfrac{1}{R}(|p_i^k|+|s_i|)\\right)\\bigg].\n\\end{align}"}, "pre_theorem_intro_text_len": 17605, "pre_theorem_intro_text": "\\label{sec:intro}\nIn this paper, we consider the homogenization convergence rates of purely second-order non-divergence form difference operators in an independent and identically distributed (i.i.d.) random environment on $\\Z^d$ for dimensions $d\\ge 3$.\n\nLarge-scale behavior of random walks in i.i.d.\\ environments has been intensively studied; see \\cite{BoSz-02, OZ-04, MB-11, DreRam-14, Kumagai-14} and the references therein. Since the dynamics of random walks in random environments (RWRE) can be described by difference equations, stochastic homogenization of difference equations in random environments is crucial for understanding the large-scale behavior of RWRE.\nFor environments in which spectral-gap or Efron--Stein type inequalities are available, the strong concentration properties of the medium can be exploited to obtain quantitative homogenization results; see, for example, \\cite{GNO-15, AL-17, GT-23}. In this paper, we show that, for non-divergence form difference operators in the i.i.d.\\ setting, the reflection symmetry of the law of the environment can in fact be used to extract additional cancellations. Somewhat unexpectedly, this yields homogenization rates for the Dirichlet problem that are strictly better than the standard optimal rate $O(R^{-1})$ obtained in \\cite{GT-23}. To achieve this, we require a more delicate analysis of the fluctuations of the homogenization error, relying in particular on a deeper understanding of higher-order correctors. \n\n\\subsection{Settings}\nLet $\\mb S_{d\\times d}$ denote the set of real $d\\times d$ positive definite diagonal matrices. A map \n\\[\n\\omega:\\Z^d\\to\\mb S_{d\\times d}\n\\] is called an {\\it environment}, and the set of all such environments is denoted by $\\Omega$. \nLet $\\mb P$ be a probability measure on $\\Omega$ such that \n\\[\n\\left\\{\\omega(x)=\\mathrm{diag}[\\omega_1(x),\\ldots, \\omega_d(x)], \\;x\\in\\Z^d\\right\\}\n\\] \nare i.i.d. under $\\mb P$. We denote the expectation with respect to $\\mb P$ by $\\mb E$ (or $E_{\\mb P}$). \n\n\\begin{definition}\\label{def:differences}\nLet $\\{e_1,\\ldots,e_d\\}$ be the canonical basis for $\\R^d$, and let \n\\[\nU:=\\left\\{e\\in\\Z^d:|e|=1\\right\\}=\\left\\{\\pm e_1,\\ldots,\\pm e_d\\right\\}\n\\]\nbe the set of unit vectors in $\\Z^d$. We define the difference operators $\\nabla=(\\nabla_e)_{e\\in U}$  and \n$\\nabla^2= \\mathrm{diag}[\\nabla_1^2,\\ldots,\\nabla_d^2]$ by\n\\begin{equation}\\label{eq:def-nabla}\n\\nabla_e u(x) := u(x+e)-u(x), \\qquad\n\\nabla_i^2u(x) := u(x+e_i)+u(x-e_i)-2u(x)  \n\\end{equation}\nfor $e\\in U$ and $i\\in\\{1,\\ldots,d\\}$. \nClearly, $\\nabla$ and $\\nabla^2$ are linear operators.  \n\\end{definition}\n\nFor $r>0$ and $y\\in\\R^d$, we write\n\\[\n\\B_r(y) := \\left\\{x\\in\\R^d: |x-y|<r\\right\\}, \\qquad\nB_r(y):=\\B_r(y)\\cap\\Z^d\n\\]\nto denote the continuous and discrete ball with center $y$ and radius $r$, respectively.\nWhen $y=0$, we write $\\B_r := \\B_r(0)$ and $B_r := B_r(0)$ for simplicity. \nFor any $B\\subset\\Z^d$, its {\\it discrete boundary} $\\partial B$ and {\\it discrete closure} $\\bar B$ are defined as\n\\[\n\\partial B:=\\left\\{z\\in\\Z^d\\setminus B: |z-x|=1 \\text{ for some }x\\in B\\right\\},\\qquad \\bar B := B\\cup\\partial B.\n\\]\nBy abuse of notation, we also use $\\partial A$ and $\\bar A$ to denote the usual continuous boundary and closure of $A\\subset\\R^d$, respectively, if there is no confusion.  \nThe interior of $B_R$ is denoted by\n\\[\n\\inB{R} := \\{x\\in B_R\\,:\\,  |x-y| \\ge 1 \\text{ for all }y\\in\\partial\\B_R\\}.\n\\]\nNote that for any $R\\ge 1$, there holds\n\\[\n\\bar{\\B}_{R-1}\\cap\\Z^d\\subset\\inB \nR\\subset B_R\\subset\\binB{R}\\subset\\bar\\B_R\\cap\\Z^d\\subset\\bar B_R.\n\\]\n\n For $x\\in\\Z^d$, a {\\it spatial shift} is defined by \n \\[\n \\theta_x:\\Omega\\to\\Omega,\\qquad \\theta_x\\omega := \\omega(x+\\cdot).\n \\] \nIn a random environment $\\omega\\in\\Omega$ in dimensions $d\\ge 2$,  we consider the discrete {\\it non-divergence form}   elliptic Dirichlet problem\n\\begin{equation}\\label{eq:elliptic-dirich}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr\\big(\\omega(x)\\nabla^2u(x)\\big)=\\frac{1}{R^2}f\\left(\\tfrac{x}{R}\\right)\\zeta(\\theta_x\\omega) & x\\in \\inB R,\\\\[5 pt]\nu(x)=g\\left(\\tfrac{x}{R}\\right) & x\\in \\partial \\inB R,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $f,g\\in\\R^{\\bar\\B_1}$ are sufficiently regular, and $\\zeta\\in \\R^\\Omega$ satisfies suitable integrability conditions.\nAs $R\\to\\infty$, it is expected that $\\vert u - \\bar u(\\frac{\\cdot}{R})\\rvert$ converges to $0$, where $\\bar u$ solves \n\\begin{equation}\\label{eq:effective-ellip}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr(\\bar a\\, D^2\\bar{u}(x))\n=f(x)\\,\\bar\\psi &x\\in\\B_1,\\\\ \n\\bar u(x)=g(x) &x\\in\\partial \\B_1.\n\\end{array}\n\\right.\n\\end{equation}\nHere, $D^2 \\bar{u}$ denotes the Hessian matrix of $\\bar{u}$, and $\\bar a=\\bar a(\\mb P)\\in\\mb S_{d\\times d}$ and $\\bar\\psi=\\bar\\psi(\\mb P,\\zeta)\\in\\R$ are {\\it deterministic} and do not depend on the realization of the random environment; see Theorem~\\ref{thm:QCLT} and Theorem~\\ref{thm:opt-quant-homo} for formulas of $\\bar{a}$ and $\\bar{\\psi}$.\n\n\\medskip\n\nWe remark that in the continuous PDE setting, one only needs a function $g\\in C^{2,\\alpha}(\\partial\\B_1)$ on the boundary $\\partial\\B_1$ to consider Dirichlet problems of types \\eqref{eq:elliptic-dirich} and \\eqref{eq:effective-ellip}. However, in the discrete setting, \nthe discrete and continuous boundaries do not agree, and therefore the function $g$ in $\\eqref{eq:elliptic-dirich}$ should be extended to $\\frac{1}{R}\\partial\\inB R\\not\\subset\\partial\\B_1$.  This discrepancy typically induces an error of order $\\tfrac{1}{R}$ which automatically becomes the ceiling of the optimal homogenization error in the discrete setting,  cf.  \\cite{GT-23}. \nIndeed, this $O(R^{-1})$ rate is the optimal rate obtained in \\cite{GT-23} for $d\\geq 3$.\nWhereas, when such a discretization issue is not present, e.g., in the continuous periodic setting \\cite{GTY-19, ST-21, Spr-24},  one needs to go deeper into higher order correctors to justify the optimality of the rate $O(R^{-1})$ for the homogenization of the Dirichlet problem. We refer to \\cite{CSS-20, FGP-24, SSZ-25} for recent developments in numerical homogenization of PDEs in non-divergence form.\n\n\\medskip\n\nA goal of this article is to isolate and examine the homogenization rate that does {\\bf not} originate from boundary discretization.\nTo this end, we assume that the boundary data $g$ is properly extended to be a function on $\\frac{1}{R}\\partial\\inB R$ in the Dirichlet problem \\eqref{eq:elliptic-dirich}.\n\n\\medskip\n\nThe non-divergence form difference equation \\eqref{eq:elliptic-dirich} is used to describe random walks in a random environment (RWRE) in $\\Z^d$. To be specific, for $x,y\\in \\Z^d$, we set\n\\begin{equation}\\label{def:omegaxei}\n\\omega(x,x\\pm e_i):=\\frac{\\omega_i(x)}{2\\,\\tr\\omega(x)} \\quad  \\text{ for } i=1,\\ldots, d,\n\\end{equation}\n and \n \\[\n \\omega(x,y) := 0 \\quad \\text{ if $|x-y|\\neq 1$},\n \\]\n that is, we normalize $\\omega$ to get a transition probability.  \n We note that the configuration of $\\{\\omega(x,y):x,y\\in\\Z^d\\}$ is also called a {\\it balanced environment} in the literature \\cite{L-82,GZ-12,BD-14,DGR-15, BCDG-18,DG-19}.\n\\begin{definition}\\label{def:rwre-discrete}\nFor each fixed $\\omega\\in\\Omega$, the random walk  $(X_n)_{n\\ge 0}$ in the environment $\\omega$ with $X_0=x$ is a Markov chain  in $\\Z^d$ with transition probability $P_\\omega^x$ specified by\n\\begin{equation}\\label{eq:def-RW}\nP_\\omega^x\\left(X_{n+1}=z|X_n=y\\right) := \\omega(y,z).\n\\end{equation}\nThe {\\it continuous-time} RWRE $(Y_t)_{t\\ge 0}$ in $\\omega$ is defined to be the Markov process on $\\Z^d$ with jump rates $\\omega(x,y)$ from $x$ to $y$. With abuse of notations, we still denote by $P_\\omega^x$  the quenched law of $(Y_t)$ if there is no ambiguity from the context.\n\\end{definition} \nThe expectation with respect to $P_\\omega^x$ is denoted by $E_\\omega^x$. When the starting point of the random walk is $0$, we sometimes omit the superscript and write $P_\\omega^0$ and $E_\\omega^0$ as $P_\\omega$ and $E_\\omega$, respectively. Note that for random walks $(X_n)$ in an environment $\\omega$, we have that\n\\begin{equation}\\label{def:omegabar}\n\\bar\\omega^i := \\theta_{X_i}\\omega\\in\\Omega, \\qquad i\\ge 0,\n\\end{equation}\nis also a Markov chain, called the {\\it environment viewed from the particle} process. By abuse of notation, we enlarge our probability space so that $P_\\omega$ still denotes the joint law of the random walks and $(\\bar\\omega^i)_{i\\ge 0}$. \n\nFor $\\omega\\in\\Omega$ and $i\\in \\{1,\\ldots,d\\}$, we set $a_i(x)=a_i^\\omega(x) := \\frac{\\omega_i(x)}{\\tr\\omega(x)} = 2\\omega(x,x+e_i)$ and\n\\begin{equation}\n\\label{eq:def-a}\na(x)=a^\\omega(x):=\\diag[a_1(x),\\ldots, a_d(x)]=\\frac{\\omega(x)}{\\tr\\omega(x)}, \\quad x\\in\\Z^d.\n\\end{equation}\nFinally, for $\\omega\\in \\Omega$, we introduce the difference operator $L_{\\omega}$ given by\n\\begin{align*}\n  L_\\omega u(x) := \\sum_{y\\in \\Z^d}\\omega(x,y)[u(y)-u(x)]=\\tfrac{1}{2}\\tr(a(x)\\nabla^2 u(x)).  \n\\end{align*}\nThat is, $L_\\omega$ is the generator of the process $(Y_t)_{t\\ge 0}$.\n\n\\subsection{Main assumptions and some notations}\nThroughout the paper, the following assumptions are always in force:\n\\begin{enumerate}[(\\text{A}1)]\n\\item $\\left\\{\\omega(x),\\, x\\in \\Z^d \\right\\}$ are i.i.d. under the probability measure $\\mb P$.\n\\item\\label{item:ellip-cond} $\\frac{\\omega}{\\tr\\omega}\\ge 2\\kappa{\\rm I}$ for $\\mb P$-almost every $\\omega$ and some constant $\\kappa\\in(0,\\tfrac{1}{2d}]$.\n\\item $\\psi(\\omega):=\\tfrac{\\zeta(\\omega)}{\\tr\\omega(0)}$ is a $L^\\infty(\\mb P)$-bounded measurable function of $\\omega(0)$.\n\\end{enumerate} \nThroughout this paper, we use $c, C$ to denote positive constants which may change from line to line but only depend on the dimension $d$ and the ellipticity constant $\\kappa$ unless stated otherwise. We write $A\n\\lesssim B$ if $A\\le CB$, and $A\\asymp B$ if $A\\lesssim B$ and $A\\gtrsim B$. \nWe also use the notations $A\\lesssim_j B$ and $A\\asymp_j B$ to indicate that the multiplicative constant depends on the variable $j$ other than $(d,\\kappa)$.\nWe will use the convention of summation over repeated indices.  \nFor example, $a_kb_k$ represents $\\sum_{k=1}^d a_kb_k$.\nThis convention is not used in definitions \\eqref{eq:def-lambda} and \\eqref{eq:def-eta}. \nFor simplicity, we use $\\tx=\\tx(\\omega)$ to denote a generic {\\it stretched-exponentially integrable} random variable (i.e., $\\mb E[\\exp(\\tx^c)]<C$) whose value may differ from line to line. \nWe write $\\tx_x(\\omega):=\\tx(\\theta_x\\omega)$. \nWe note that $\\tx$ may depend on other parameters as well, although the constants $c, C$ in the stretched-exponential moment bound only depend on $(d,\\kappa)$.\n\n\\subsection{Earlier results in the literature}\n\nWe first recall the following quenched central limit theorem (QCLT) proved in \\cite{L-82}, which is a discrete version of \\cite{PV-82}.\n\\begin{thmx}\\label{thm:QCLT}\nAssume \\rm{(A2)} and that law $\\mb P$ of the environment is ergodic under spatial shifts $\\{\\theta_x:x\\in\\Z^d\\}$. Then, the following assertions hold. \n\\begin{enumerate}[(i)]\n\\item\\label{item:ergodic} There exists a probability measure $\\mb Q\\approx\\mb P$ such that $(\\evp{i})_{i\\ge 0}$ is an ergodic (with respect to time shifts) sequence under law $\\mb Q\\times P_\\omega$.\n\\item For $\\mb P$-almost every $\\omega$, the rescaled path $X_{n^2t}/n$ converges weakly (under law $P_\\omega$) to a Brownian motion with covariance matrix \n\\begin{equation}\n\\label{eq:def-abar}\n\\bar a=\\diag[\\bar a_1,\\ldots,\\bar a_d]:=E_\\Q[a]=E_\\Q[\\tfrac{\\omega(0)}{\\tr\\omega(0)}]>0.\n\\end{equation}\n\\end{enumerate}\n\\end{thmx}\nWe denote the Radon--Nikodym derivative of $\\Q$ with respect to $\\mb P$ by\n\\begin{equation}\\label{eq:def-rho}\n\\rho(\\omega)=\\dd\\mb Q/\\dd\\mb P.\n\\end{equation}\nIf the environment is i.i.d., it was proved in \\cite[Theorem 1.5]{GT-22} that \n\\begin{equation}\n    \\label{eq:rho-bounds}\n    E[\\exp(\\rho^{-c})]+E[\\exp(\\rho^{c})]<\\infty\n\\end{equation}\nfor some constant $c=c(\\kappa,d)>0$.\nFor any $x\\in\\Z^d$, we write \n\\[\n\\rho_\\omega(x):=\\rho(\\theta_x\\omega).\n\\]\nFor any $\\alpha\\in(0,1]$ and any function $f$ on a set $A$, we define\n\\[\n\\osc_A f:=\\sup_{x,y\\in A}|f(x)-f(y)|,\n\\qquad\n[f]_{\\alpha;A} :=\\sup_{x,y\\in A,\\; x\\neq y}\\frac{|f(x)-f(y)|}{|x-y|^\\alpha},\n\\]\nand, if $A$ is a finite set, for $p\\in(0,\\infty)$, we define \n\\[\n\\norm{f}_{p;A}:=\\left(\\frac{1}{\\#A}\\sum_{x\\in A}|f(x)|^p\\right)^{1/p}, \\quad\n\\norm{f}_{\\infty;A} :=\\max_{x\\in A}|f(x)|.\n\\]\n\nFor any integer $j\\ge 0$, let $\\fct_j$ denote the set of $j$-th order polynomials, with $\\fct_0=\\R$. In fact, in our paper, we will only use the cases $j=0,1$. \n\nThe following large-scale regularity result can be found in \\cite{GT-23}, which is a discrete version of the $C^{1,1}$ regularity estimate in \\cite[Theorem 3.1]{AL-17}.\nThis was first done in the periodic setting in \\cite{AL-87,AL-89}.\n\n\\begin{thmx}\n\\label{thm:c11}\nAssume {\\rm(A1), (A2)},  and that $\\psi$ is a {\\bf local} function. Let $R\\ge 1$.  \nThere exists $\\alpha=\\alpha(d,\\kappa)\\in(0,\\tfrac13)$ such that,\nfor any $u$ with $L_\\omega u(x)=\\psi(\\theta_x\\omega)+f(x)$ on $B_R$, $j\\in\\{1,2\\}$, $\\tx\\le r<R$, we have that \n\\begin{equation}\n\\label{eq:oscillation-estimates}\n\\frac{1}{r^{j}}\\inf_{p\\in\\fct_{j-1}}\\osc_{B_r}\\,(u-p)\n\\lesssim\n\\frac{1}{R^{j}}\n\\inf_{p\\in\\fct_{j-1}}\\osc_{B_R}\\,(u-p)+A_{j},\n\\end{equation}\nwhere the terms $A_1=A_1(R)$ and $A_2=A_2(R,r)$ obey the bounds\n\\begin{align*}\n&A_1\\le R^{1-\\alpha}\\norm{\\psi}_\\infty+R\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_1\\le R^{1-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^{1+\\sigma}[f]_{\\sigma;B_R},\n\\\\\n&A_2\\le r^{-\\alpha}\\norm{\\psi}_\\infty+\\log(\\tfrac{R}{r})\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_2\\le r^{-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^\\sigma[f]_{\\sigma;B_R}\n\\end{align*}\nfor any $\\sigma\\in(0,1]$. In particular, for any $R>1$ and $j\\in\\{1,2\\}$, there holds\n\\begin{equation}\\label{eq:c2}\n|\\nabla^ju(0)|\\lesssim \\left(\\frac{\\tx}{R}\\right)^j \n\\left(\n\\norm{u-u(0)}_{1;B_R}+R^2\\norm{\\psi+f(0)}_\\infty+R^{2+\\sigma}[f]_{\\sigma;B_{R}}\n\\right).\n\\end{equation}\n\\end{thmx}\n\nIntroducing the quantities\n\\begin{equation}\\label{eq:def-mu}\n\\mu(R):=\n\\left\\{\n\\begin{array}{lr}\nR & d=2,\\\\\nR^{1/2} &d=3,\\\\\n(1\\vee \\log R)^{1/2} &d=4,\\\\\n1 &d\\ge 5,\n\\end{array}\n\\right.\n\\quad\n\\delta(R):=\\left\\{\n\\begin{array}{lr}\n(\\log (R\\vee 2))^{3/2} &d=2,\\\\\n1 &d\\ge 3,\n\\end{array}\n\\right.\n\\end{equation}\nthe following theorems on properties of the (global) correctors and quantitative homogenization of the Dirichlet problem were established in \\cite{GT-23}.\n\n\\begin{thmx}\n\\label{thm:global_krt}\nLet $\\psi$ be an $L^\\infty(\\mb P)$-bounded function of $\\omega(0)$ with $\\norm{\\psi}_\\infty=1$.\nFor each $d\\ge 2$ and $\\mb P$-a.e. $\\omega$, there exists a function $\\krt=\\krt_\\omega:\\Z^d\\to\\R$ that solves \n\\[\nL_\\omega\\krt(x)=\\psi(\\theta_x\\omega)-\\bar\\psi \\quad\\text{ for }x\\in\\Z^d\n\\]\nwith the following properties:\n\\begin{enumerate}[(i)]\n\\item\\label{item:gkrt-2} When $d\\ge 5$,\\quad $\\mb E[\\exp(c|\\krt(x)/\\mu(|x|)|^p)]<C_p$ for any $p\\in(0,\\tfrac{2d}{3d+2}),x\\in\\Z^d$;\n\nWhen $d=3,4$,\\; $\\mb E[\\exp(c|\\krt(x)/\\mu(|x|)|^p)]<C_p$ for any $p\\in(0,\\tfrac{2d}{3d+4}),x\\in\\Z^d$;\n\nWhen $d=2$,\\quad\n$\\mb E\\left[\n\\exp\\big(\nc\\abs{\\tfrac{\\krt(x)}{|x|\\delta(|x|)}}^p\n\\big)\n\\right]\\le C_p$ for any $p\\in(0,\\tfrac13), x\\in\\Z^d$;\n\n\\item\\label{item:gkrt-3} $\\mb E[\\exp(c|\\nabla\\krt(x)/\\delta(|x|)|^q)]\\le C_q$ for any $q\\in(0,\\tfrac{d}{2d+2})$ and $x\\in\\Z^d$;\n\\item\\label{item:gkrt-2nd-der}\n $\\mb E[\\exp(c|\\nabla^2\\krt(x)|^r)]\\le C_r$ for any $r\\in(0,\\tfrac{d}{2})$ and $x\\in\\Z^d$;\n\\item\\label{item:gkrt-4}(Stationarity properties)\n\\begin{itemize}\n\\item When $d\\ge 5$,  the field $\\{\\krt_\\omega(x):x\\in\\Z^d\\}$ is stationary (under $\\mb P$);\n\\item When $d\\ge 3$, we have $\\nabla\\krt_\\omega(x)=\\nabla\\krt_{\\theta_x\\omega}(0)$ for all $x\\in\\Z^d$, and the field $\\{\\nabla\\krt_\\omega(x):x\\in\\Z^d\\}$  is stationary;\n\\item When $d=2$, we have $\\nabla^2\\krt_\\omega(x)=\\nabla^2\\krt_{\\theta_x\\omega}(0)$ for all $x\\in\\Z^d$, and the field $\\{\\nabla^2\\krt_\\omega(x):x\\in\\Z^d\\}$  is stationary.\n\\end{itemize}\n\\end{enumerate}\nMoreover,  such a corrector $\\krt$ is unique up to an additive constant when $d\\ge 3$,  and is unique up to an affine transformation when $d=2$.\n\\end{thmx}\n\n\\begin{thmx}\n\\label{thm:opt-quant-homo}\nAssume {\\rm(A1)}, {\\rm(A2)}, and {\\rm(A3)}. Let $\\bar a :=E_{\\Q}[\\frac{\\omega(0)}{\\tr\\omega(0)}]>0$ and $\\bar\\psi :=E_{\\Q}[\\frac{\\zeta}{\\tr\\omega(0)}]$. Suppose $f,g$ are both in $C^4(\\R^d)$. Then, for any $\\error\\in(0,1)$ and $R\\ge 2$, there exists a random variable $\\ms Y=\\ms Y(R,\\error, \\omega)>1$ with $\\mb E[\\exp(\\ms Y^{{d}/{(2d+2)}-\\error})]<C$ such that the solution $u$ of \\eqref{eq:elliptic-dirich} satisfies\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar{u}(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n\\begin{array}{rl}\n(\\log R)^{2+\\error}R^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d=2,\\\\\nR^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d\\ge 3,\n\\end{array}\n\\right.\n\\]\nwhere $\\bar u$ is the solution of the Dirichlet problem \\eqref{eq:effective-ellip}.\n\\end{thmx}\n\nWe point out that Theorem \\ref{thm:opt-quant-homo} also remains valid for certain finite range dependent environments that are not i.i.d.\\  (for example, the ``checkerboard\" environment considered in \\cite{AL-17}), provided that an Efron--Stein-type concentration inequality holds. Indeed, to obtain the heat-kernel bounds with exponential integrability \\cite{GT-22} and the optimal convergence rates \\cite{GT-23}, the only requirements are an Efron--Stein-type concentration inequality and a finite range of dependence.\nWe refer the reader to \\cite{CaSu10, AS-14, AL-17, BCDG-18, GPT-19, GT-22, AFL-22, JZ-23,GT-23} for recent developments of quantitative stochastic homogenization of non-divergence form elliptic PDE in both discrete and continuous settings.\n\n\\subsection{Main results}", "context": "For $x\\in\\Z^d$, a {\\it spatial shift} is defined by \n \\[\n \\theta_x:\\Omega\\to\\Omega,\\qquad \\theta_x\\omega := \\omega(x+\\cdot).\n \\] \nIn a random environment $\\omega\\in\\Omega$ in dimensions $d\\ge 2$, we consider the discrete {\\it non-divergence form} elliptic Dirichlet problem\n\\begin{equation}\\label{eq:elliptic-dirich}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr\\big(\\omega(x)\\nabla^2u(x)\\big)=\\frac{1}{R^2}f\\left(\\tfrac{x}{R}\\right)\\zeta(\\theta_x\\omega) & x\\in \\inB R,\\\\[5 pt]\nu(x)=g\\left(\\tfrac{x}{R}\\right) & x\\in \\partial \\inB R,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $f,g\\in\\R^{\\bar\\B_1}$ are sufficiently regular, and $\\zeta\\in \\R^\\Omega$ satisfies suitable integrability conditions.\nAs $R\\to\\infty$, it is expected that $\\vert u - \\bar u(\\frac{\\cdot}{R})\\rvert$ converges to $0$, where $\\bar u$ solves \n\\begin{equation}\\label{eq:effective-ellip}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr(\\bar a\\, D^2\\bar{u}(x))\n=f(x)\\,\\bar\\psi &x\\in\\B_1,\\\\ \n\\bar u(x)=g(x) &x\\in\\partial \\B_1.\n\\end{array}\n\\right.\n\\end{equation}\nHere, $D^2 \\bar{u}$ denotes the Hessian matrix of $\\bar{u}$, and $\\bar a=\\bar a(\\mb P)\\in\\mb S_{d\\times d}$ and $\\bar\\psi=\\bar\\psi(\\mb P,\\zeta)\\in\\R$ are {\\it deterministic} and do not depend on the realization of the random environment; see Theorem~\\ref{thm:QCLT} and Theorem~\\ref{thm:opt-quant-homo} for formulas of $\\bar{a}$ and $\\bar{\\psi}$.\n\nWe first recall the following quenched central limit theorem (QCLT) proved in \\cite{L-82}, which is a discrete version of \\cite{PV-82}.\n\\begin{thmx}\\label{thm:QCLT}\nAssume \\rm{(A2)} and that law $\\mb P$ of the environment is ergodic under spatial shifts $\\{\\theta_x:x\\in\\Z^d\\}$. Then, the following assertions hold. \n\\begin{enumerate}[(i)]\n\\item\\label{item:ergodic} There exists a probability measure $\\mb Q\\approx\\mb P$ such that $(\\evp{i})_{i\\ge 0}$ is an ergodic (with respect to time shifts) sequence under law $\\mb Q\\times P_\\omega$.\n\\item For $\\mb P$-almost every $\\omega$, the rescaled path $X_{n^2t}/n$ converges weakly (under law $P_\\omega$) to a Brownian motion with covariance matrix \n\\begin{equation}\n\\label{eq:def-abar}\n\\bar a=\\diag[\\bar a_1,\\ldots,\\bar a_d]:=E_\\Q[a]=E_\\Q[\\tfrac{\\omega(0)}{\\tr\\omega(0)}]>0.\n\\end{equation}\n\\end{enumerate}\n\\end{thmx}\nWe denote the Radon--Nikodym derivative of $\\Q$ with respect to $\\mb P$ by\n\\begin{equation}\\label{eq:def-rho}\n\\rho(\\omega)=\\dd\\mb Q/\\dd\\mb P.\n\\end{equation}\nIf the environment is i.i.d., it was proved in \\cite[Theorem 1.5]{GT-22} that \n\\begin{equation}\n \\label{eq:rho-bounds}\n E[\\exp(\\rho^{-c})]+E[\\exp(\\rho^{c})]<\\infty\n\\end{equation}\nfor some constant $c=c(\\kappa,d)>0$.\nFor any $x\\in\\Z^d$, we write \n\\[\n\\rho_\\omega(x):=\\rho(\\theta_x\\omega).\n\\]\nFor any $\\alpha\\in(0,1]$ and any function $f$ on a set $A$, we define\n\\[\n\\osc_A f:=\\sup_{x,y\\in A}|f(x)-f(y)|,\n\\qquad\n[f]_{\\alpha;A} :=\\sup_{x,y\\in A,\\; x\\neq y}\\frac{|f(x)-f(y)|}{|x-y|^\\alpha},\n\\]\nand, if $A$ is a finite set, for $p\\in(0,\\infty)$, we define \n\\[\n\\norm{f}_{p;A}:=\\left(\\frac{1}{\\#A}\\sum_{x\\in A}|f(x)|^p\\right)^{1/p}, \\quad\n\\norm{f}_{\\infty;A} :=\\max_{x\\in A}|f(x)|.\n\\]\n\n\\begin{thmx}\n\\label{thm:c11}\nAssume {\\rm(A1), (A2)}, and that $\\psi$ is a {\\bf local} function. Let $R\\ge 1$. \nThere exists $\\alpha=\\alpha(d,\\kappa)\\in(0,\\tfrac13)$ such that,\nfor any $u$ with $L_\\omega u(x)=\\psi(\\theta_x\\omega)+f(x)$ on $B_R$, $j\\in\\{1,2\\}$, $\\tx\\le r<R$, we have that \n\\begin{equation}\n\\label{eq:oscillation-estimates}\n\\frac{1}{r^{j}}\\inf_{p\\in\\fct_{j-1}}\\osc_{B_r}\\,(u-p)\n\\lesssim\n\\frac{1}{R^{j}}\n\\inf_{p\\in\\fct_{j-1}}\\osc_{B_R}\\,(u-p)+A_{j},\n\\end{equation}\nwhere the terms $A_1=A_1(R)$ and $A_2=A_2(R,r)$ obey the bounds\n\\begin{align*}\n&A_1\\le R^{1-\\alpha}\\norm{\\psi}_\\infty+R\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_1\\le R^{1-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^{1+\\sigma}[f]_{\\sigma;B_R},\n\\\\\n&A_2\\le r^{-\\alpha}\\norm{\\psi}_\\infty+\\log(\\tfrac{R}{r})\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_2\\le r^{-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^\\sigma[f]_{\\sigma;B_R}\n\\end{align*}\nfor any $\\sigma\\in(0,1]$. In particular, for any $R>1$ and $j\\in\\{1,2\\}$, there holds\n\\begin{equation}\\label{eq:c2}\n|\\nabla^ju(0)|\\lesssim \\left(\\frac{\\tx}{R}\\right)^j \n\\left(\n\\norm{u-u(0)}_{1;B_R}+R^2\\norm{\\psi+f(0)}_\\infty+R^{2+\\sigma}[f]_{\\sigma;B_{R}}\n\\right).\n\\end{equation}\n\\end{thmx}\n\n\\begin{thmx}\n\\label{thm:opt-quant-homo}\nAssume {\\rm(A1)}, {\\rm(A2)}, and {\\rm(A3)}. Let $\\bar a :=E_{\\Q}[\\frac{\\omega(0)}{\\tr\\omega(0)}]>0$ and $\\bar\\psi :=E_{\\Q}[\\frac{\\zeta}{\\tr\\omega(0)}]$. Suppose $f,g$ are both in $C^4(\\R^d)$. Then, for any $\\error\\in(0,1)$ and $R\\ge 2$, there exists a random variable $\\ms Y=\\ms Y(R,\\error, \\omega)>1$ with $\\mb E[\\exp(\\ms Y^{{d}/{(2d+2)}-\\error})]<C$ such that the solution $u$ of \\eqref{eq:elliptic-dirich} satisfies\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar{u}(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n\\begin{array}{rl}\n(\\log R)^{2+\\error}R^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d=2,\\\\\nR^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d\\ge 3,\n\\end{array}\n\\right.\n\\]\nwhere $\\bar u$ is the solution of the Dirichlet problem \\eqref{eq:effective-ellip}.\n\\end{thmx}\n\nWe point out that Theorem \\ref{thm:opt-quant-homo} also remains valid for certain finite range dependent environments that are not i.i.d.\\ (for example, the ``checkerboard\" environment considered in \\cite{AL-17}), provided that an Efron--Stein-type concentration inequality holds. Indeed, to obtain the heat-kernel bounds with exponential integrability \\cite{GT-22} and the optimal convergence rates \\cite{GT-23}, the only requirements are an Efron--Stein-type concentration inequality and a finite range of dependence.\nWe refer the reader to \\cite{CaSu10, AS-14, AL-17, BCDG-18, GPT-19, GT-22, AFL-22, JZ-23,GT-23} for recent developments of quantitative stochastic homogenization of non-divergence form elliptic PDE in both discrete and continuous settings.\n\n\\subsection{Main results}", "full_context": "For $x\\in\\Z^d$, a {\\it spatial shift} is defined by \n \\[\n \\theta_x:\\Omega\\to\\Omega,\\qquad \\theta_x\\omega := \\omega(x+\\cdot).\n \\] \nIn a random environment $\\omega\\in\\Omega$ in dimensions $d\\ge 2$, we consider the discrete {\\it non-divergence form} elliptic Dirichlet problem\n\\begin{equation}\\label{eq:elliptic-dirich}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr\\big(\\omega(x)\\nabla^2u(x)\\big)=\\frac{1}{R^2}f\\left(\\tfrac{x}{R}\\right)\\zeta(\\theta_x\\omega) & x\\in \\inB R,\\\\[5 pt]\nu(x)=g\\left(\\tfrac{x}{R}\\right) & x\\in \\partial \\inB R,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $f,g\\in\\R^{\\bar\\B_1}$ are sufficiently regular, and $\\zeta\\in \\R^\\Omega$ satisfies suitable integrability conditions.\nAs $R\\to\\infty$, it is expected that $\\vert u - \\bar u(\\frac{\\cdot}{R})\\rvert$ converges to $0$, where $\\bar u$ solves \n\\begin{equation}\\label{eq:effective-ellip}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr(\\bar a\\, D^2\\bar{u}(x))\n=f(x)\\,\\bar\\psi &x\\in\\B_1,\\\\ \n\\bar u(x)=g(x) &x\\in\\partial \\B_1.\n\\end{array}\n\\right.\n\\end{equation}\nHere, $D^2 \\bar{u}$ denotes the Hessian matrix of $\\bar{u}$, and $\\bar a=\\bar a(\\mb P)\\in\\mb S_{d\\times d}$ and $\\bar\\psi=\\bar\\psi(\\mb P,\\zeta)\\in\\R$ are {\\it deterministic} and do not depend on the realization of the random environment; see Theorem~\\ref{thm:QCLT} and Theorem~\\ref{thm:opt-quant-homo} for formulas of $\\bar{a}$ and $\\bar{\\psi}$.\n\nWe first recall the following quenched central limit theorem (QCLT) proved in \\cite{L-82}, which is a discrete version of \\cite{PV-82}.\n\\begin{thmx}\\label{thm:QCLT}\nAssume \\rm{(A2)} and that law $\\mb P$ of the environment is ergodic under spatial shifts $\\{\\theta_x:x\\in\\Z^d\\}$. Then, the following assertions hold. \n\\begin{enumerate}[(i)]\n\\item\\label{item:ergodic} There exists a probability measure $\\mb Q\\approx\\mb P$ such that $(\\evp{i})_{i\\ge 0}$ is an ergodic (with respect to time shifts) sequence under law $\\mb Q\\times P_\\omega$.\n\\item For $\\mb P$-almost every $\\omega$, the rescaled path $X_{n^2t}/n$ converges weakly (under law $P_\\omega$) to a Brownian motion with covariance matrix \n\\begin{equation}\n\\label{eq:def-abar}\n\\bar a=\\diag[\\bar a_1,\\ldots,\\bar a_d]:=E_\\Q[a]=E_\\Q[\\tfrac{\\omega(0)}{\\tr\\omega(0)}]>0.\n\\end{equation}\n\\end{enumerate}\n\\end{thmx}\nWe denote the Radon--Nikodym derivative of $\\Q$ with respect to $\\mb P$ by\n\\begin{equation}\\label{eq:def-rho}\n\\rho(\\omega)=\\dd\\mb Q/\\dd\\mb P.\n\\end{equation}\nIf the environment is i.i.d., it was proved in \\cite[Theorem 1.5]{GT-22} that \n\\begin{equation}\n \\label{eq:rho-bounds}\n E[\\exp(\\rho^{-c})]+E[\\exp(\\rho^{c})]<\\infty\n\\end{equation}\nfor some constant $c=c(\\kappa,d)>0$.\nFor any $x\\in\\Z^d$, we write \n\\[\n\\rho_\\omega(x):=\\rho(\\theta_x\\omega).\n\\]\nFor any $\\alpha\\in(0,1]$ and any function $f$ on a set $A$, we define\n\\[\n\\osc_A f:=\\sup_{x,y\\in A}|f(x)-f(y)|,\n\\qquad\n[f]_{\\alpha;A} :=\\sup_{x,y\\in A,\\; x\\neq y}\\frac{|f(x)-f(y)|}{|x-y|^\\alpha},\n\\]\nand, if $A$ is a finite set, for $p\\in(0,\\infty)$, we define \n\\[\n\\norm{f}_{p;A}:=\\left(\\frac{1}{\\#A}\\sum_{x\\in A}|f(x)|^p\\right)^{1/p}, \\quad\n\\norm{f}_{\\infty;A} :=\\max_{x\\in A}|f(x)|.\n\\]\n\n\\begin{thmx}\n\\label{thm:c11}\nAssume {\\rm(A1), (A2)}, and that $\\psi$ is a {\\bf local} function. Let $R\\ge 1$. \nThere exists $\\alpha=\\alpha(d,\\kappa)\\in(0,\\tfrac13)$ such that,\nfor any $u$ with $L_\\omega u(x)=\\psi(\\theta_x\\omega)+f(x)$ on $B_R$, $j\\in\\{1,2\\}$, $\\tx\\le r<R$, we have that \n\\begin{equation}\n\\label{eq:oscillation-estimates}\n\\frac{1}{r^{j}}\\inf_{p\\in\\fct_{j-1}}\\osc_{B_r}\\,(u-p)\n\\lesssim\n\\frac{1}{R^{j}}\n\\inf_{p\\in\\fct_{j-1}}\\osc_{B_R}\\,(u-p)+A_{j},\n\\end{equation}\nwhere the terms $A_1=A_1(R)$ and $A_2=A_2(R,r)$ obey the bounds\n\\begin{align*}\n&A_1\\le R^{1-\\alpha}\\norm{\\psi}_\\infty+R\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_1\\le R^{1-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^{1+\\sigma}[f]_{\\sigma;B_R},\n\\\\\n&A_2\\le r^{-\\alpha}\\norm{\\psi}_\\infty+\\log(\\tfrac{R}{r})\\norm{f}_\\infty\n\\quad\\text{ and }\\quad\nA_2\\le r^{-\\alpha}\\norm{\\psi+f(0)}_\\infty+R^\\sigma[f]_{\\sigma;B_R}\n\\end{align*}\nfor any $\\sigma\\in(0,1]$. In particular, for any $R>1$ and $j\\in\\{1,2\\}$, there holds\n\\begin{equation}\\label{eq:c2}\n|\\nabla^ju(0)|\\lesssim \\left(\\frac{\\tx}{R}\\right)^j \n\\left(\n\\norm{u-u(0)}_{1;B_R}+R^2\\norm{\\psi+f(0)}_\\infty+R^{2+\\sigma}[f]_{\\sigma;B_{R}}\n\\right).\n\\end{equation}\n\\end{thmx}\n\n\\begin{thmx}\n\\label{thm:opt-quant-homo}\nAssume {\\rm(A1)}, {\\rm(A2)}, and {\\rm(A3)}. Let $\\bar a :=E_{\\Q}[\\frac{\\omega(0)}{\\tr\\omega(0)}]>0$ and $\\bar\\psi :=E_{\\Q}[\\frac{\\zeta}{\\tr\\omega(0)}]$. Suppose $f,g$ are both in $C^4(\\R^d)$. Then, for any $\\error\\in(0,1)$ and $R\\ge 2$, there exists a random variable $\\ms Y=\\ms Y(R,\\error, \\omega)>1$ with $\\mb E[\\exp(\\ms Y^{{d}/{(2d+2)}-\\error})]<C$ such that the solution $u$ of \\eqref{eq:elliptic-dirich} satisfies\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar{u}(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n\\begin{array}{rl}\n(\\log R)^{2+\\error}R^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d=2,\\\\\nR^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d\\ge 3,\n\\end{array}\n\\right.\n\\]\nwhere $\\bar u$ is the solution of the Dirichlet problem \\eqref{eq:effective-ellip}.\n\\end{thmx}\n\nWe point out that Theorem \\ref{thm:opt-quant-homo} also remains valid for certain finite range dependent environments that are not i.i.d.\\ (for example, the ``checkerboard\" environment considered in \\cite{AL-17}), provided that an Efron--Stein-type concentration inequality holds. Indeed, to obtain the heat-kernel bounds with exponential integrability \\cite{GT-22} and the optimal convergence rates \\cite{GT-23}, the only requirements are an Efron--Stein-type concentration inequality and a finite range of dependence.\nWe refer the reader to \\cite{CaSu10, AS-14, AL-17, BCDG-18, GPT-19, GT-22, AFL-22, JZ-23,GT-23} for recent developments of quantitative stochastic homogenization of non-divergence form elliptic PDE in both discrete and continuous settings.\n\n\\subsection{Main results}\n\n\\subsection{Main results}\n\n\\begin{proposition}\n\\label{lem:two-scale-exp}\nSuppose that $d\\geq 3$, $f\\in C^{4,\\alpha}(\\bar\\B_1)$, and $g\\in C^{6,\\alpha}(\\partial\\B_1)$. Let $u,\\bar u$ be the solutions of \\eqref{eq:elliptic-dirich}, \\eqref{eq:effective-ellip}, respectively. Recall $a,\\bar a$ from \\eqref{eq:def-a}, \\eqref{eq:def-abar}, and let $v^k,\\xi,\\lambda^k_j, \\bar\\lambda^k_j,\\eta_j,\\bar\\eta_j$ be as above. We extend $g$ so that $g=\\bar{u}$ on $\\bar\\B_1$. Let $\\bar z:\\bar\\B_1\\to\\R$ be the solution of \n\\[\n\\left\\{\n\\begin{array}{lr}\n\\frac{1}{2}\\tr(\\bar{a}D^2 \\bar{z}(x))=\\tfrac{1}{2}\\bar{\\lambda}^k_i\\partial_{kki}\\bar{u}(x)-\\tfrac{1}{2}\\bar\\eta_j\\partial_j f(x) & x\\in\\B_1,\\\\\n\\bar z(x)=0 &x\\in\\partial\\B_1.\n\\end{array}\n\\right.\n\\]\nFor $\\omega\\in\\Omega$, $R>1$, and $j,k\\in\\{1,\\ldots,d\\}$, let $p^k_j, s_j:\\Z^d \\rightarrow \\R$ be solutions of \n\\begin{align}\nL_\\omega p^k_j(x)&=\\lambda^k_j(x)-\\bar\\lambda^k_j \\quad\\text{ for }x\\in B_R,\\label{eq:def-p}\\\\\nL_\\omega s_j(x)&=\\eta_j(x)-\\bar\\eta_j \\quad\\;\\text{ for }x\\in B_R\\label{eq:def-sj},\n\\end{align}\nrespectively. Then, for $\\omega\\in\\Omega$, $R>1$, we have\n\\begin{align}\\label{eq:two-scale-further}\n&\\max_{x\\in B_R}\\left|u(x)-\\bar{u}(\\tfrac xR)-\\tfrac 1R\\bar z(\\tfrac xR)\\right|\\nn\\\\\n&\\qquad\\lesssim\n\\tfrac 1{R^2}\\norm{\\bar u}_{C^6(\\bar{\\B}_1)}\\max_{i,j,k=1}^d\n\\bigg[1+\\left\\| |\\nabla p^k_j|+|\\nabla s_j|+(|\\bar\\lambda_j^i|+|\\bar\\eta_j|)|\\nabla v^k|\\right\\|_{d;B_R}\n\\nn\\\\\n&\\qquad\n\\qquad\\qquad\\qquad\\qquad\\quad\\;\\;\\,+\\max_{\\bar B_R}\\left(|\\xi|+|v^k|+\\tfrac{1}{R}(|p_i^k|+|s_i|)\\right)\\bigg].\n\\end{align}\n\\end{proposition}\n\n\\begin{proof}\nLet $z:\\binB R\\to\\R$ be a solution of the Dirichlet problem\n\\[\n\\left\\{\n\\begin{array}{lr}\nL_\\omega z(x)=\\tfrac{1}{R^2}\\left[ \\tfrac{1}{2}\\bar{\\lambda}^k_i\\partial_{kki}\\bar{u}(\\tfrac xR)-\\tfrac{1}{2}\\bar\\eta_i\\partial_i f(\\tfrac xR)\\right] & x\\in\\inB R,\\\\\nz(x)=0 &x\\in\\partial\\inB R.\n\\end{array}\n\\right.\n\\]\nConsider the function\n\\begin{align}\\label{eq:def-w}\n&w(x)=u(x)-\\bar u(\\tfrac{x}{R})-\\tfrac 1R z(x)\\nn\\\\\n&\n+\\tfrac{1}{R^2}\\left[v^k(x)\\partial_{kk}\\bar u(\\tfrac{x}{R})\n-f(\\tfrac{x}R)\\xi(x)\\right]- {\\tfrac{1}{2}} \\tfrac{1}{R^3}\\left[\\partial_{kki}\\bar{u}(\\tfrac{x}{R})p_i^k(x)-\\partial_i f(\\tfrac{x}{R})s_i(x)\\right]\n\\end{align}\nfor $x\\in\\binB R\\subset\\bar\\B_R\\cap\\Z^d$. Then, applying the formula\n\\begin{align*}\nL_\\omega(uv)(x)&=u(x)L_\\omega v(x)+v(x) L_\\omega u(x)+\n\\sum_{y:y\\sim x}\\omega(x,y)[u(y)-u(x)][v(y)-v(x)]\\\\\n&=u(x)L_\\omega v(x)+v(x) L_\\omega u(x) \\\\ &\\qquad+ \\tfrac{1}{2}\na_i(x)\\left(\\nabla_{e_i}u(x)[v(x+e_i)-v(x-e_i)]\n+\\nabla_i^2u(x)\\nabla_{-e_i}v(x)\n\\right)\n\\end{align*}\nto the products within \\eqref{eq:def-w}, we obtain in $\\inB R$ that\n\\begin{align}\n&L_\\omega[v^k\\,\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})]\\nn\\\\\n&=\nL_\\omega v^k \\,\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})\n+v^k \\, L_\\omega[\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})]+\\tfrac{1}{2}\\lambda^k_i\\nabla_{e_i}[\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})]+\\tfrac{1}{2}a_i \\,\\nabla_i^2[\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})]\\,\\nabla_{-e_i}v^k \n\\nn\\\\\n&=\\tfrac1{2}(a_k -\\bar a_k)\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})+\\tfrac{1}{2}\\tfrac{1}{R}\\lambda^k_i\\partial_{kki}\\bar u(\\tfrac{\\cdot}{R})+O(\\tfrac 1{R^2})\\norm{\\bar u}_{C^5(\\bar\\B_1)}\\max_{\\binB R}\\sum_{k = 1}^d|v^k|,\\label{eq:term1}\n\\end{align}\nwhere we used the fact that $|\\lambda^k_i|\\lesssim|\\nabla v^k|\\lesssim\\max_{\\binB R}|v^k|$ in $\\inB R$. Similarly,\n\\begin{align}\nL_\\omega[f(\\tfrac{\\cdot}R)\\xi ]\n&=(\\psi-\\bar\\psi)f(\\tfrac{\\cdot}R)+{\\tfrac{1}{2}}\\tfrac{1}{R}\\eta_i\\partial_i f(\\tfrac{\\cdot}{R})+O(\\tfrac 1{R^2})\\norm{\\bar u}_{C^5(\\bar\\B_1)}\\max_{\\inB R}|\\xi|,\\label{eq:term2}\\\\\nL_\\omega[\\partial_{kki}\\bar{u}(\\tfrac{\\cdot}{R})p_i^k ]\n&=(\\lambda^k_i-\\bar\\lambda^k_i)\\partial_{kki}\\bar{u}(\\tfrac{\\cdot}{R})\n+{\\tfrac{1}{2}}\\tfrac{1}{R}\\partial_{kkij}\\bar{u}(\\tfrac{\\cdot}{R})a_j[p^k_i(\\cdot+e_j)-p^k_i(\\cdot-e_j)]\\nn\\\\\n&+O(\\tfrac 1{R^2})\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\max_{\\binB R}\\sum_{i,k = 1}^d|p^k_i|,\\label{eq:term3}\\\\\nL_\\omega[\\partial_i f(\\tfrac{\\cdot}{R})s_i ]\n&=(\\eta_i-\\bar\\eta_i)\\partial_i f(\\tfrac{\\cdot}{R})\n+{\\tfrac{1}{2}}\\tfrac{1}{R}\\partial_{ij}f(\\tfrac{\\cdot}{R})a_j [s_i(\\cdot+e_j)-s_i(\\cdot-e_j)]\\nn\\\\\n&+O(\\tfrac 1{R^2})\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\max_{\\binB R}\\sum_{i=1}^d|s_i|.\\label{eq:term4}\n\\end{align}\nNote that \n$\\abs{L_\\omega[\\bar u(\\tfrac{\\cdot}{R})]-\\tfrac1{2R^2}\\tr[a D^2\\bar{u}(\\tfrac{\\cdot}{R})]}\\lesssim\\tfrac1{R^4}\\norm{\\bar u}_{C^4(\\bar\\B_1)}$, and \n\\begin{align*}\nL_\\omega u \n=\\tfrac1{R^2}f(\\tfrac{\\cdot}{R})(\\psi -\\bar\\psi)+\\tfrac1{R^2}f(\\tfrac{\\cdot}{R})\\bar\\psi =\\tfrac1{R^2}\\big[f(\\tfrac{\\cdot}{R})L_\\omega\\xi +\n\\tfrac1{2}\\bar{a}_k\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})\\big].\n\\end{align*}\nThen, recalling the definition of $w$ in \\eqref{eq:def-w} and writing \n\\[\nA:=\\max_{\\binB R}\\sum_{i,k=1}^d\\left[|\\xi|+|v^k|+\\tfrac{1}{R}(|p_i^k|+|s_i|)\\right],\n\\]\nby \\eqref{eq:term1}, \\eqref{eq:term2},\\eqref{eq:term3}, \\eqref{eq:term4},\nwe have in $\\inB R$ that\n\\begin{align*}\n&\\abs{L_\\omega w } \\\\\n&=\n\\tfrac1{R^2}\\Abs{\nf(\\tfrac{\\cdot}{R})L_\\omega\\xi +\\tfrac1{2}\\bar{a}_k\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})-R^2L_\\omega[\\bar{u}(\\tfrac{\\cdot}{R})]-{\\tfrac{1}{2}}\\tfrac{1}{R} \\bar{\\lambda}^k_i\\partial_{kki}\\bar{u}(\\tfrac{\\cdot}{R})+{\\tfrac{1}{2}}\\tfrac 1R\\bar\\eta_i\\partial_i f(\\tfrac{\\cdot}{R})\n\\\\\n&\\qquad+\\tfrac1{2}(a_k-\\bar a_k)\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})+{\\tfrac{1}{2}}\\tfrac{1}{R}\\lambda^k_i\\partial_{kki}\\bar u(\\tfrac{\\cdot}{R})\n\\nn\\\\\n&\\qquad-(\\psi-\\bar\\psi)f(\\tfrac{\\cdot}{R})-{\\tfrac{1}{2}}\\tfrac{1}{R}\\eta_i\\partial_i f(\\tfrac{\\cdot}{R})+O(\\tfrac{1}{R^2})\\norm{\\bar u}_{C^6(\\bar\\B_1)}A\\\\\n&\\qquad-{\\tfrac{1}{2}}\\tfrac{1}{R}(\\lambda^k_i-\\bar\\lambda^k_i)\\partial_{kki}\\bar{u}(\\tfrac{\\cdot}{R})\n-{\\tfrac{1}{4}}\\tfrac{1}{R^2}\\partial_{kkij}\\bar{u}(\\tfrac{\\cdot}{R})a_j[p^k_i(\\cdot+e_j)-p^k_i(\\cdot-e_j)]\\\\\n&\\qquad+{\\tfrac{1}{2}}\\tfrac{1}{R}(\\eta_i-\\bar\\eta_i)\\partial_i f(\\tfrac{\\cdot}{R})\n+{\\tfrac{1}{4}}\\tfrac{1}{R^2}\\partial_{ij}f(\\tfrac{\\cdot}{R})a_j[s_i(\\cdot+e_j)-s_i(\\cdot-e_j)]\n}\n\\nn\\\\\n&\\lesssim\n\\tfrac1{R^4}\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\sum_{i,k=1}^d\\left[1+\n|\\nabla p_i^k|+|\\nabla s_i|\n+\\max_{\\binB R}\\left(|\\xi|+|v^k|+\\tfrac{1}{R}(|p_i^k|+ |s_i|)\\right)\n\\right].\n\\end{align*} \nThus, by the above inequality, the definition of $w$ from \\eqref{eq:def-w}, and the ABP maximum principle, we obtain that\n\\begin{align}\n&\\max_{\\binB R}|u -\\bar u(\\tfrac{\\cdot}{R})-\\tfrac 1R z |\\label{eq:ineq1}\\\\\n&\\le \\max_{\\binB R}|w|+\n\\max_{\\binB R}\\Abs{\\tfrac{1}{R^2}[v^k\\partial_{kk}\\bar u(\\tfrac{\\cdot}{R})\n-f(\\tfrac{\\cdot}{R})\\xi ]\n-{\\tfrac{1}{2}}\\tfrac{1}{R^3}[\\partial_{kki}\\bar{u}(\\tfrac{\\cdot}{R})p_i^k -\\partial_i f(\\tfrac{\\cdot}{R})s_i ]}\\nn\\\\\n&\\lesssim\nR^2\\norm{L_\\omega w}_{d;\\inB R}\n+\\tfrac1{R^2}\\norm{\\bar u}_{C^3}\\max_{\\binB R}\\left(|\\xi|+|v^k|+\\tfrac{1}{R}(|p_i^k|+ |s_i|)\\right)\\nn\n\\\\&\n\\lesssim\n\\tfrac1{R^2}\\norm{\\bar u}_{C^6}\\sum_{i,k=1}^d\n\\big[1+\n\\left\\||\\nabla p_i^k|+|\\nabla s_i|\\right\\|_{d;B_R}+\\max_{\\binB R}\\left(|\\xi|+|v^k|+\\tfrac{1}{R}(|p_i^k|+|s_i|)\\right)\n\\big].\\nn\n\\end{align}\nFurthermore, by \\cite[Lemma~30]{GT-23}, \n\\[\n\\max_{\\binB R}\\abs{z-\\bar z(\\tfrac{\\cdot}{R})}\n\\lesssim\n\\tfrac1{R^2}\\max_{j,\\ell=1}^d(|\\bar\\lambda_j^\\ell|+|\\bar\\eta_j|)\\norm{\\bar u}_{C^5(\\bar\\B_1)}\n\\sum_{k=1}^d(1+R \\norm{\\nabla v^k}_{d;\\inB R}+\\osc_{\\binB R}v^k).\n\\]\nThis inequality, together with \\eqref{eq:ineq1}, yields the claimed bound \\eqref{eq:two-scale-further}.\n\\end{proof}", "post_theorem_intro_text_len": 3841, "post_theorem_intro_text": "In the i.i.d. setting with well-prepared boundary conditions, Theorem \\ref{thm:opt-iid-homo} yields improved convergence rates compared to the rate $O(R^{-1})$ in Theorem \\ref{thm:opt-quant-homo} when $d\\geq 3$.\nOur approaches and contributions are as follows.\nWe employ the two-scale asymptotic expansion to obtain the estimate \\eqref{eq:two-scale-further} in Proposition \\ref{lem:two-scale-exp}.\nDue to the i.i.d. structure, the environment enjoys a reflection symmetry; that is, the random fields $\\omega(\\cdot)$ and $\\omega(-\\,\\cdot)$ are identically distributed. \n(Of course, this symmetry fails for environments with only a finite range of dependence.)\nThis reflection symmetry implies that the third-order homogenized tensor vanishes, which in turn suggests that a homogenization error bound sharper than\n $O(R^{-1})$ may be attainable.\n\nWe remark that such improved rates should not be expected in environments with correlations even slightly weaker than in the point-wise i.i.d.\\ case, since the distribution's reflective symmetry is no longer present.\n\nIn our i.i.d.\\ setting, without the presence of the third-order tensor, the homogenization rate is completely determined by the $C^{0,1}$-bounds of higher-order correctors. \nRoughly speaking, the (first-order) correctors “correct” local coefficients, while the higher-order correctors account for the derivatives of the lower-order correctors. \nHence, unlike the correctors themselves, the source terms in the equations for higher-order correctors are highly nonlocal, and thus the large-scale regularity estimates in Theorem~\\ref{thm:c11}, which apply only to equations with {\\it local} source terms, no longer hold.\n\nOur improved convergence rates in Theorem \\ref{thm:opt-iid-homo} are inspired by earlier results in the periodic setting \\cite{GST-22}, where it was proved that if the environment has ``one degree of freedom\" $a(x)$, i.e., the coefficient matrix is of the form $\\omega(x)=C+a(x)M$ for some constant symmetric matrices $C$ and $M$, then the third-order homogenized tensor vanishes, leading to an improved rate of $O(R^{-2})$.\nThe periodic setting, however, is quite rigid.\nWe provided examples in \\cite{GST-22} that show that if the environment has two degrees of freedom, then the third-order homogenized tensor does not vanish in general, and the optimal convergence rate in such cases is only $O(R^{-1})$. \n\nIn contrast, in our random setting, $\\omega(x)=\\mathrm{diag}[\\omega_1(x),\\ldots, \\omega_d(x)]$ has $d$ degrees of freedom, which is fundamentally different from the periodic case considered in \\cite{GST-22}.\nMoreover, in the periodic setting, the $C^{0,1}$-boundedness of higher-order correctors is automatically guaranteed by periodicity.\n\nOne of the main contributions of our work is to establish $C^{0,1}$ bounds for higher-order correctors in the random setting. Our approach is based on comparing the true higher-order correctors with certain {\\it localized} higher-order correctors that possess good regularity properties and {\\it approximate} higher-order correctors that are stationary.\n\n\\begin{remark}\n\tWhen $d=2$, the corrector $\\upsilon^k$ grows super-linearly (Theorem~\\ref{thm:global_krt}). Consequently, via the expansion \\eqref{eq:two-scale-further}, it produces an error term of order $R^{-2}|v^k|$ that dominates the desired $O(R^{-1})$ scale. Therefore, in two dimensions, we should not expect an improvement beyond the $O(R^{-1})$ rate in i.i.d.\\ environments.\n\n    All in all, this demonstrates that, in the absence of additional structural assumptions on the random environment, the boundary conditions, or the dimension, the best convergence rate that one can expect is $O(R^{-1})$, as established in \\cite{GT-23}. We refer to Remark~\\ref{rmk:optimality} for a discussion of the optimality of the $O(R^{-1})$ rate.\n\\end{remark}", "sketch": "To prove Theorem~\\ref{thm:opt-iid-homo}, the authors describe the following strategy and key ingredients.\n\n- They “employ the two-scale asymptotic expansion to obtain the estimate \\eqref{eq:two-scale-further} in Proposition~\\ref{lem:two-scale-exp}.”\n\n- They use a special feature of the point-wise i.i.d. setting: “Due to the i.i.d. structure, the environment enjoys a reflection symmetry; that is, the random fields $\\omega(\\cdot)$ and $\\omega(-\\,\\cdot)$ are identically distributed.” This “reflection symmetry implies that the third-order homogenized tensor vanishes,” which “suggests that a homogenization error bound sharper than $O(R^{-1})$ may be attainable.”\n\n- With “the absence of the third-order tensor,” they explain that “the homogenization rate is completely determined by the $C^{0,1}$-bounds of higher-order correctors.” They motivate the difficulty: while first-order correctors “correct local coefficients,” “the higher-order correctors account for the derivatives of the lower-order correctors,” and “the source terms in the equations for higher-order correctors are highly nonlocal,” so “the large-scale regularity estimates in Theorem~\\ref{thm:c11}, which apply only to equations with {\\it local} source terms, no longer hold.”\n\n- A main technical step is then to “establish $C^{0,1}$ bounds for higher-order correctors in the random setting,” by “comparing the true higher-order correctors with certain {\\it localized} higher-order correctors that possess good regularity properties and {\\it approximate} higher-order correctors that are stationary.”\n\n- They also note limitations: the improved rates rely on i.i.d.-driven reflection symmetry (“such improved rates should not be expected in environments with correlations… since the distribution's reflective symmetry is no longer present”), and in $d=2$ the corrector “grows super-linearly,” so the expansion yields an error term that “dominates the desired $O(R^{-1})$ scale,” hence “we should not expect an improvement beyond the $O(R^{-1})$ rate” in two dimensions.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:opt-iid-homo}\nAssume {\\rm(A1), (A2), (A3)}, and $d\\ge 3$. Let $\\bar a,\\bar\\psi$ be as in the following theorem.\n\n\\begin{thmx}\n\\label{thm:opt-quant-homo}\nAssume {\\rm(A1)}, {\\rm(A2)}, and {\\rm(A3)}. Let $\\bar a :=E_{\\Q}[\\frac{\\omega(0)}{\\tr\\omega(0)}]>0$ and $\\bar\\psi :=E_{\\Q}[\\frac{\\zeta}{\\tr\\omega(0)}]$. Suppose $f,g$ are both in $C^4(\\R^d)$. Then, for any $\\error\\in(0,1)$ and $R\\ge 2$, there exists a random variable $\\ms Y=\\ms Y(R,\\error, \\omega)>1$ with $\\mb E[\\exp(\\ms Y^{{d}/{(2d+2)}-\\error})]<C$ such that the solution $u$ of\n\\begin{equation}\\label{eq:elliptic-dirich}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr\\big(\\omega(x)\\nabla^2u(x)\\big)=\\frac{1}{R^2}f\\left(\\tfrac{x}{R}\\right)\\zeta(\\theta_x\\omega) & x\\in \\inB R,\\\\[5 pt]\n\\nu(x)=g\\left(\\tfrac{x}{R}\\right) & x\\in \\partial \\inB R,\n\\end{array}\n\\right.\n\\end{equation}\nsatisfies\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar{u}(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n\\begin{array}{rl}\n(\\log R)^{2+\\error}R^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d=2,\\\\\nR^{-1}\\norm{\\bar u}_{C^4(\\bar\\B_1)}\\ms Y &\\text{ when }d\\ge 3,\n\\end{array}\n\\right.\n\\]\nwhere $\\bar u$ is the solution of the Dirichlet problem\n\\begin{equation}\\label{eq:effective-ellip}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr(\\bar a\\, D^2\\bar{u}(x))\n=f(x)\\,\\bar\\psi &x\\in\\B_1,\\\\ \n\\bar u(x)=g(x) &x\\in\\partial \\B_1.\n\\end{array}\n\\right.\n\\end{equation}\n\\end{thmx}\n\nSuppose that $f\\in C^{4,\\alpha}(\\bar\\B_1)$ and $g\\in C^{6,\\alpha}(\\partial\\B_1)$ for some $\\alpha\\in (0,1)$. Then, there exists a $C^6(\\bar\\B_1)$-extension of $g$ such that the solutions $u, \\bar u$ of\n\\begin{equation}\\label{eq:elliptic-dirich}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr\\big(\\omega(x)\\nabla^2u(x)\\big)=\\frac{1}{R^2}f\\left(\\tfrac{x}{R}\\right)\\zeta(\\theta_x\\omega) & x\\in \\inB R,\\\\[5 pt]\n\\nu(x)=g\\left(\\tfrac{x}{R}\\right) & x\\in \\partial \\inB R,\n\\end{array}\n\\right.\n\\end{equation}\nand\n\\begin{equation}\\label{eq:effective-ellip}\n\\left\\{\n\\begin{array}{lr}\n\\tfrac 12\\tr(\\bar a\\, D^2\\bar{u}(x))\n=f(x)\\,\\bar\\psi &x\\in\\B_1,\\\\ \n\\bar u(x)=g(x) &x\\in\\partial \\B_1.\n\\end{array}\n\\right.\n\\end{equation}\nsatisfy\n\\[\n\\max_{x\\in B_R}|u(x)-\\bar u(\\tfrac{x}{R})|\n\\lesssim\n\\left\\{\n  \\begin{array}{rl}\n  \tR^{-3/2}\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\tx &\\text{ when } d=3,\\\\\n  \tR^{-2}(\\log R)\\norm{\\bar u}_{C^6(\\bar\\B_1)}\\tx&\\text{ when }d\\ge 4.\n  \\end{array}\n\\right.\n\\]\nAn example of such an extension is $g=\\bar u$.", "theorem_type": ["Existence", "Inequality or Bound"], "mcq": {"question": "Assume \\((A1)\\), \\((A2)\\), and \\((A3)\\), and let \\(d\\ge 3\\). For a random environment \\(\\omega\\), define the spatial shift by \\(\\theta_x\\omega=\\omega(x+\\cdot)\\). Let \\(\\bar a:=E_{\\mathbb Q}[\\omega(0)/\\operatorname{tr}\\omega(0)]>0\\) and \\(\\bar\\psi:=E_{\\mathbb Q}[\\zeta/\\operatorname{tr}\\omega(0)]\\). Suppose \\(f\\in C^{4,\\alpha}(\\overline{\\mathbb B}_1)\\) and \\(g\\in C^{6,\\alpha}(\\partial\\mathbb B_1)\\) for some \\(\\alpha\\in(0,1)\\). Let \\(u\\) solve the discrete Dirichlet problem on the lattice ball \\(B_R\\subset\\mathbb Z^d\\),\n\\[\n\\tfrac12\\operatorname{tr}(\\omega(x)\\nabla^2u(x))=\\frac1{R^2}f\\!\\left(\\tfrac{x}{R}\\right)\\zeta(\\theta_x\\omega)\\quad \\text{for }x\\in B_R,\\qquad\nu(x)=g\\!\\left(\\tfrac{x}{R}\\right)\\quad \\text{for }x\\in \\partial B_R,\n\\]\nand let \\(\\bar u\\) solve on the unit ball \\(\\mathbb B_1\\subset\\mathbb R^d\\),\n\\[\n\\tfrac12\\operatorname{tr}(\\bar a\\,D^2\\bar u(x))=f(x)\\bar\\psi\\quad \\text{for }x\\in\\mathbb B_1,\\qquad\n\\bar u(x)=g(x)\\quad \\text{for }x\\in\\partial\\mathbb B_1.\n\\]\nWhich quantitative estimate holds for the homogenization error \\(\\max_{x\\in B_R}|u(x)-\\bar u(x/R)|\\)?", "correct_choice": {"label": "A", "text": "There exists a \\(C^6(\\overline{\\mathbb B}_1)\\)-extension of \\(g\\) such that the corresponding solutions satisfy\n\\[\n\\max_{x\\in B_R}\\left|u(x)-\\bar u\\!\\left(\\tfrac{x}{R}\\right)\\right|\n\\lesssim\n\\begin{cases}\nR^{-3/2}\\,\\|\\bar u\\|_{C^6(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d=3,\\\\[4pt]\nR^{-2}(\\log R)\\,\\|\\bar u\\|_{C^6(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d\\ge 4.\n\\end{cases}\n\\]"}, "choices": [{"label": "B", "text": "There exists a \\(C^6(\\overline{\\mathbb B}_1)\\)-extension of \\(g\\) such that the corresponding solutions satisfy\n\\[\n\\max_{x\\in B_R}\\left|u(x)-\\bar u\\!\\left(\\tfrac{x}{R}\\right)\\right|\n\\lesssim\n\\begin{cases}\nR^{-2}\\,\\|\\bar u\\|_{C^6(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d=3,\\\\[4pt]\nR^{-2}\\,\\|\\bar u\\|_{C^6(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d\\ge 4.\n\\end{cases}\n\\]"}, {"label": "C", "text": "There exists a \\(C^6(\\overline{\\mathbb B}_1)\\)-extension of \\(g\\) such that the corresponding solutions satisfy\n\\[\n\\max_{x\\in B_R}\\left|u(x)-\\bar u\\!\\left(\\tfrac{x}{R}\\right)\\right|\n\\lesssim R^{-1}\\,\\|\\bar u\\|_{C^6(\\overline{\\mathbb B}_1)}\\,\\tilde X,\n\\qquad d\\ge 3.\n\\]"}, {"label": "D", "text": "For every \\(\\varepsilon\\in(0,1)\\), there exists a random variable \\(\\tilde X=\\tilde X(R,\\varepsilon,\\omega)>1\\) with stretched-exponential moments such that the corresponding solutions satisfy\n\\[\n\\max_{x\\in B_R}\\left|u(x)-\\bar u\\!\\left(\\tfrac{x}{R}\\right)\\right|\n\\lesssim\n\\begin{cases}\nR^{-3/2+\\varepsilon}\\,\\|\\bar u\\|_{C^6(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d=3,\\\\[4pt]\nR^{-2+\\varepsilon}\\,\\|\\bar u\\|_{C^6(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d\\ge 4.\n\\end{cases}\n\\]"}, {"label": "E", "text": "There exists a \\(C^4(\\overline{\\mathbb B}_1)\\)-extension of \\(g\\) such that the corresponding solutions satisfy\n\\[\n\\max_{x\\in B_R}\\left|u(x)-\\bar u\\!\\left(\\tfrac{x}{R}\\right)\\right|\n\\lesssim\n\\begin{cases}\nR^{-3/2}(\\log R)\\,\\|\\bar u\\|_{C^4(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d=3,\\\\[4pt]\nR^{-2}\\,\\|\\bar u\\|_{C^4(\\overline{\\mathbb B}_1)}\\,\\tilde X, & d\\ge 4.\n\\end{cases}\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": "dimension-dependent rates and logarithmic factor", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "sharp improved rates dropped in favor of the coarser \\(R^{-1}\\) decay", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "exact rate without epsilon-loss", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "required \\(C^6\\)-extension/norm and placement of logarithmic correction across dimensions", "template_used": "wildcard"}]}}
{"id": "2512.04422v1", "paper_link": "http://arxiv.org/abs/2512.04422v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{thm:thm1} With notation as above, the Dirichlet heat trace $H^D_{\\Omega}(t)$ has an expansion as $t\\to 0$ given by\n\\begin{multline}\\label{eq:heattrace}\nH^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t} - \\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}} + \\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\, ds + \\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)\\\\\n+ \\sqrt t\\Big(\\frac{1}{256\\sqrt{\\pi}}\\int_{\\partial\\Omega}\\kappa^2\\, ds + \\sum_{j=1}^n\\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\Big) +O(t\\log t),\n\\end{multline}\nwhere $\\mathcal C_{1/2}$ is a function only of the angle $\\alpha$ and the limit curvatures $\\kappa_{\\pm}$.", "start_pos": 10880, "end_pos": 11538, "label": "thm:thm1"}, "ref_dict": {"eq:general": "\\begin{equation}\\label{eq:general} \\begin{aligned}\n        c_{1/2}(\\alpha)\n        &= -\\,\\fp \n        \\iint_{(0,\\infty)\\times(0,\\alpha)}\n        \\int_{0}^{1} (1-\\sigma)^{-1}\\sigma^{-1/2}\n        \\iint_{(0,\\infty)\\times(0,\\alpha)} \\\\\n        &\\quad a\\left(\\frac{R}{\\sqrt{1-\\sigma}},\\theta,\n        \\frac{\\sqrt{\\sigma}\\,\\rho}{\\sqrt{1-\\sigma}},\\phi\\right)\\,\n        b\\left(\\rho,\\phi,\\frac{R}{\\sqrt{\\sigma}},\\theta\\right)\\,\n        \\dmup\\,\\dmu.\n        \\end{aligned}\n\\end{equation}", "eq:heattrace": "\\begin{multline}\\label{eq:heattrace}\nH^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t} - \\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}} + \\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\, ds + \\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)\\\\\n+ \\sqrt t\\Big(\\frac{1}{256\\sqrt{\\pi}}\\int_{\\partial\\Omega}\\kappa^2\\, ds + \\sum_{j=1}^n\\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\Big) +O(t\\log t),\n\\end{multline}", "eq:form": "\\begin{equation}\\label{eq:form} \\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-}) = c_{1/2}(\\alpha)r_0(\\alpha,\\kappa_+,\\kappa_-)\n\\end{equation}", "thm:thm1": "\\begin{theorem}\\label{thm:thm1} With notation as above, the Dirichlet heat trace $H^D_{\\Omega}(t)$ has an expansion as $t\\to 0$ given by\n\\begin{multline}\\label{eq:heattrace}\nH^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t} - \\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}} + \\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\, ds + \\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)\\\\\n+ \\sqrt t\\Big(\\frac{1}{256\\sqrt{\\pi}}\\int_{\\partial\\Omega}\\kappa^2\\, ds + \\sum_{j=1}^n\\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\Big) +O(t\\log t),\n\\end{multline}\nwhere $\\mathcal C_{1/2}$ is a function only of the angle $\\alpha$ and the limit curvatures $\\kappa_{\\pm}$.\n\\end{theorem}", "thm:moreid": "\\begin{theorem}\\label{thm:moreid} For $\\alpha\\ne\\pi$, the function $\\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-})$ has the form \n\\begin{equation}\\label{eq:form} \\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-}) = c_{1/2}(\\alpha)r_0(\\alpha,\\kappa_+,\\kappa_-)\n\\end{equation}\nwhere $c_{1/2}(\\alpha)$ is a function depending only on $\\alpha$, whose general expression is given by a Hadamard renormalized trace on an exact sector:\n\\begin{equation}\\label{eq:general} \\begin{aligned}\n        c_{1/2}(\\alpha)\n        &= -\\,\\fp \n        \\iint_{(0,\\infty)\\times(0,\\alpha)}\n        \\int_{0}^{1} (1-\\sigma)^{-1}\\sigma^{-1/2}\n        \\iint_{(0,\\infty)\\times(0,\\alpha)} \\\\\n        &\\quad a\\left(\\frac{R}{\\sqrt{1-\\sigma}},\\theta,\n        \\frac{\\sqrt{\\sigma}\\,\\rho}{\\sqrt{1-\\sigma}},\\phi\\right)\\,\n        b\\left(\\rho,\\phi,\\frac{R}{\\sqrt{\\sigma}},\\theta\\right)\\,\n        \\dmup\\,\\dmu.\n        \\end{aligned}\n\\end{equation}\nHere $a$ and $b$ are functions related to the exact heat kernel on a sector of angle $\\alpha$, explicitly given by \\eqref{eq:a-def} and \\eqref{eq:bexpression-clean}, and $\\fp$ denotes the Hadamard finite part at $R=\\infty$ of the improper integral in $R$. \n\nMoreover, $c_{1/2}(\\alpha)$ has the special value\n\\begin{equation}\\label{eq:specialpi2} c_{1/2}(\\frac{\\pi}{2}) = \\frac{1}{16\\sqrt\\pi}.\n\\end{equation}\n\\end{theorem}", "eq:a-def": "\\begin{equation}\\label{eq:a-def}\n    a(R,\\theta,R',\\theta')\n    =\\frac12\\,\\exp \\Big[-\\frac{R^2+(R')^2}{4}\\Big]\n    \\sum_{j=1}^\\infty I_{\\mu_j} \\Big(\\frac{RR'}{2}\\Big)\\,\n    \\varphi_j(\\theta)\\,\\varphi_j(\\theta').\n\\end{equation}", "eq:bexpression-clean": "\\begin{equation}\\label{eq:bexpression-clean}\n    \\begin{aligned}\n    b(R,\\theta,R',\\theta')&=\n    \\Big(\\tfrac12 D^2S\\Big)\\,a(R,\\theta,R',\\theta')\\\\\n    &\\quad+\\,4 R\\cos(\\theta+\\theta_0)\\,\\Delta_{R,\\theta}a(R,\\theta,R',\\theta')\\\\\n    &\\quad+\\,\\tfrac{1}{2}\\,\\Delta_{R,\\theta}\\big(D^2S\\cdot a(R,\\theta,R',\\theta')\\big)\n    -\\tfrac{1}{2}\\,(D^2S)\\,\\Delta_{R,\\theta}a(R,\\theta,R',\\theta').\n    \\end{aligned}\n    \\end{equation}", "eq:specialpi2": "\\begin{equation}\\label{eq:specialpi2} c_{1/2}(\\frac{\\pi}{2}) = \\frac{1}{16\\sqrt\\pi}.\n\\end{equation}"}, "pre_theorem_intro_text_len": 2402, "pre_theorem_intro_text": "A classical problem in spectral geometry is to determine what geometric information is encoded in the spectrum of the Laplacian on a Riemannian manifold \\cite{kac1966can}. This is frequently approached using heat trace methods. Specifically, one examines the short-time asymptotic expansion of the trace of the heat kernel. This trace is spectrally determined, and therefore the coefficients in the asymptotic expansion are \\emph{spectral invariants} -- geometric quantities determined by the spectrum. In 1967, McKean and Singer computed the first three heat invariants for a smoothly bounded domain in $\\mathbb R^2$ \\cite{mckean1967curvature}. Using these, they showed that the area, perimeter, and Euler characteristic of such a domain are all spectral invariants. It is natural to ask whether one can deduce similar information for a less smooth domain, and indeed this has been studied in depth \\cite{Cheeger1983,vanDenBergSrisat88,NRS}.\n\nWe now explain the setting of this paper. Let $\\Omega\\subseteq\\mathbb R^2$ be a curvilinear polygon, with piecewise smooth boundary, smooth except for a finite number $n$ of corners $P_j$ of interior angles $\\alpha_j>0$, $j=1,\\ldots,n$. Let $\\kappa$ be the inward-pointing curvature of the boundary, and let $\\gamma(s)$ be an arc-length parametrization of the boundary, oriented counterclockwise. For each $j$, let $\\kappa_{j,\\pm}$ be the limits of $\\kappa(\\gamma(s))$ as $s$ approaches $\\gamma^{-1}(P_j)$ from above and below respectively.\n\nIt is well known that the trace of the Dirichlet heat kernel on $\\Omega$, $H^D_{\\Omega}(t)$, has an asymptotic expansion as $t\\to 0$, beginning with a $t^{-1}$ term. The $t^{-1}$, $t^{-1/2}$, and $t^0$ terms of this expansion have been computed, first in the exact polygonal setting in unpublished work of Ray, as cited by \\cite{Cheeger1983} and \\cite{vdBS}, and then extended to the general curvilinear setting \\cite{NRS}. Our interest here is in the next term, which we expect at order $t^{1/2}$. In \\cite{EGS_2017}, based on results in the literature such as \\cite{branson1990asymptotics}, the authors give the coefficient of this $t^{1/2}$ term in the ``straight corners\" case, where a neighborhood of each corner $P_j$ is isometric to an exact sector. Without this assumption, the curvature of the sides may interact with the angle at the corner. As we show, this interaction produces a term at order $t^{1/2}$.", "context": "A classical problem in spectral geometry is to determine what geometric information is encoded in the spectrum of the Laplacian on a Riemannian manifold \\cite{kac1966can}. This is frequently approached using heat trace methods. Specifically, one examines the short-time asymptotic expansion of the trace of the heat kernel. This trace is spectrally determined, and therefore the coefficients in the asymptotic expansion are \\emph{spectral invariants} -- geometric quantities determined by the spectrum. In 1967, McKean and Singer computed the first three heat invariants for a smoothly bounded domain in $\\mathbb R^2$ \\cite{mckean1967curvature}. Using these, they showed that the area, perimeter, and Euler characteristic of such a domain are all spectral invariants. It is natural to ask whether one can deduce similar information for a less smooth domain, and indeed this has been studied in depth \\cite{Cheeger1983,vanDenBergSrisat88,NRS}.\n\nWe now explain the setting of this paper. Let $\\Omega\\subseteq\\mathbb R^2$ be a curvilinear polygon, with piecewise smooth boundary, smooth except for a finite number $n$ of corners $P_j$ of interior angles $\\alpha_j>0$, $j=1,\\ldots,n$. Let $\\kappa$ be the inward-pointing curvature of the boundary, and let $\\gamma(s)$ be an arc-length parametrization of the boundary, oriented counterclockwise. For each $j$, let $\\kappa_{j,\\pm}$ be the limits of $\\kappa(\\gamma(s))$ as $s$ approaches $\\gamma^{-1}(P_j)$ from above and below respectively.\n\nIt is well known that the trace of the Dirichlet heat kernel on $\\Omega$, $H^D_{\\Omega}(t)$, has an asymptotic expansion as $t\\to 0$, beginning with a $t^{-1}$ term. The $t^{-1}$, $t^{-1/2}$, and $t^0$ terms of this expansion have been computed, first in the exact polygonal setting in unpublished work of Ray, as cited by \\cite{Cheeger1983} and \\cite{vdBS}, and then extended to the general curvilinear setting \\cite{NRS}. Our interest here is in the next term, which we expect at order $t^{1/2}$. In \\cite{EGS_2017}, based on results in the literature such as \\cite{branson1990asymptotics}, the authors give the coefficient of this $t^{1/2}$ term in the ``straight corners\" case, where a neighborhood of each corner $P_j$ is isometric to an exact sector. Without this assumption, the curvature of the sides may interact with the angle at the corner. As we show, this interaction produces a term at order $t^{1/2}$.", "full_context": "A classical problem in spectral geometry is to determine what geometric information is encoded in the spectrum of the Laplacian on a Riemannian manifold \\cite{kac1966can}. This is frequently approached using heat trace methods. Specifically, one examines the short-time asymptotic expansion of the trace of the heat kernel. This trace is spectrally determined, and therefore the coefficients in the asymptotic expansion are \\emph{spectral invariants} -- geometric quantities determined by the spectrum. In 1967, McKean and Singer computed the first three heat invariants for a smoothly bounded domain in $\\mathbb R^2$ \\cite{mckean1967curvature}. Using these, they showed that the area, perimeter, and Euler characteristic of such a domain are all spectral invariants. It is natural to ask whether one can deduce similar information for a less smooth domain, and indeed this has been studied in depth \\cite{Cheeger1983,vanDenBergSrisat88,NRS}.\n\nWe now explain the setting of this paper. Let $\\Omega\\subseteq\\mathbb R^2$ be a curvilinear polygon, with piecewise smooth boundary, smooth except for a finite number $n$ of corners $P_j$ of interior angles $\\alpha_j>0$, $j=1,\\ldots,n$. Let $\\kappa$ be the inward-pointing curvature of the boundary, and let $\\gamma(s)$ be an arc-length parametrization of the boundary, oriented counterclockwise. For each $j$, let $\\kappa_{j,\\pm}$ be the limits of $\\kappa(\\gamma(s))$ as $s$ approaches $\\gamma^{-1}(P_j)$ from above and below respectively.\n\nIt is well known that the trace of the Dirichlet heat kernel on $\\Omega$, $H^D_{\\Omega}(t)$, has an asymptotic expansion as $t\\to 0$, beginning with a $t^{-1}$ term. The $t^{-1}$, $t^{-1/2}$, and $t^0$ terms of this expansion have been computed, first in the exact polygonal setting in unpublished work of Ray, as cited by \\cite{Cheeger1983} and \\cite{vdBS}, and then extended to the general curvilinear setting \\cite{NRS}. Our interest here is in the next term, which we expect at order $t^{1/2}$. In \\cite{EGS_2017}, based on results in the literature such as \\cite{branson1990asymptotics}, the authors give the coefficient of this $t^{1/2}$ term in the ``straight corners\" case, where a neighborhood of each corner $P_j$ is isometric to an exact sector. Without this assumption, the curvature of the sides may interact with the angle at the corner. As we show, this interaction produces a term at order $t^{1/2}$.\n\nIt is well known that the trace of the Dirichlet heat kernel on $\\Omega$, $H^D_{\\Omega}(t)$, has an asymptotic expansion as $t\\to 0$, beginning with a $t^{-1}$ term. The $t^{-1}$, $t^{-1/2}$, and $t^0$ terms of this expansion have been computed, first in the exact polygonal setting in unpublished work of Ray, as cited by \\cite{Cheeger1983} and \\cite{vdBS}, and then extended to the general curvilinear setting \\cite{NRS}. Our interest here is in the next term, which we expect at order $t^{1/2}$. In \\cite{EGS_2017}, based on results in the literature such as \\cite{branson1990asymptotics}, the authors give the coefficient of this $t^{1/2}$ term in the ``straight corners\" case, where a neighborhood of each corner $P_j$ is isometric to an exact sector. Without this assumption, the curvature of the sides may interact with the angle at the corner. As we show, this interaction produces a term at order $t^{1/2}$.\n\nMoreover, $c_{1/2}(\\alpha)$ has the special value\n\\begin{equation}\\label{eq:specialpi2} c_{1/2}(\\frac{\\pi}{2}) = \\frac{1}{16\\sqrt\\pi}.\n\\end{equation}\n\\end{theorem}\n\\begin{remark} A few comments:\n\\begin{itemize}\n\\item The $O(t\\log t)$ error in \\eqref{eq:heattrace} is not necessarily optimal, and in fact we expect that it should be $O(t)$.\n\\item Each of $a$ and $b$ is a function only of $\\alpha$ and the variables of integration, see \\eqref{eq:a-def}, \\eqref{eq:bexpression-clean}. Thus \\eqref{eq:general} is indeed independent of $\\kappa_{\\pm}$.\n\\item For inverse spectral results, it is crucial that \\eqref{eq:specialpi2} is nonzero. We can only verify this explicitly for $\\alpha=\\pi/2$. It would be of significant interest to find out for which $\\alpha$ we have $c_{1/2}(\\alpha)\\ne 0$. \n\\item Theorem \\ref{thm:moreid} does not work for $\\alpha=\\pi$ because the conformal model that we use breaks down in that case, as can be seen from the fact that $r_0$ is undefined when $\\alpha=\\pi$. So the case $\\alpha=\\pi$, where the boundary has an abrupt change in second derivative, remains open. \n\\end{itemize}\n\\end{remark}\n\n\\begin{proposition}\n Suppose that an admissible curvilinear polygon $\\Omega$ is Dirichlet isospectral to a polygon. Then $\\Omega$ is a polygon.\n\\end{proposition}\n\\begin{proof}\nFor a polygon, the $t^{1/2}$ term in the heat trace vanishes. So it must vanish for $\\Omega$. But under our assumptions, by Theorems \\ref{thm:thm1} and \\ref{thm:moreid}, the $t^{1/2}$ coefficient in $H^D_{\\Omega}(t)$ is given by \n \\begin{equation}\n \\frac{1}{256\\sqrt\\pi}\\int_{\\partial\\Omega}\\kappa^2\\, ds + \\sum_{j=1}^n c_{1/2}(\\alpha_j)\\sqrt{\\kappa_{+,j}^2+\\kappa_{-,j}^2}.\n \\end{equation}\n Since this coefficient is non-negative and only zero for an exact polygon, $\\Omega$ is an exact polygon.\n\\end{proof}\n\\begin{remark}\nObserve from \\eqref{eq:specialpi2} that any polygon where all non-straight corners are right angles is admissible.\n\\end{remark}\n\n\\subsection{Plan of the paper}\nIn section \\ref{sec:gm} we prove Theorem \\ref{thm:thm1}. This is done by examining the geometric microlocal description of the heat kernel given in \\cite{NRS}. We show that there is a well-defined corner contribution $\\mathcal C_{1/2}\\sqrt t$, and then prove that any two corners with the same angle $\\alpha$ and limit curvatures $\\kappa_{\\pm}$ yield the same contribution, regardless of the rest of $\\Omega$. To take advantage of this uniformity, in section \\ref{sec:conformal} we introduce a simple conformal model for a curvilinear corner and compute the contribution $\\mathcal C_{1/2}\\sqrt t$ in that model. This computation proceeds via a parametrix construction as in \\cite{NRS} and allows us to obtain both \\eqref{eq:form} and \\eqref{eq:general}. \nFinally, in section \\ref{sec:specialpi2}, we prove \\eqref{eq:specialpi2} by computing a sufficient number of terms in the Dirichlet heat trace expansion for a semicircle and comparing with \\eqref{eq:heattrace}.\n\n\\begin{lemma}\\label{lem:local-corner}\nLet $\\Omega$ and $\\Omega'$ be domains with a single corner having the same opening angle\n$\\alpha$ and the same one‑sided boundary curvatures $\\kappa_\\pm$ at that corner.\nThen $H^D_\\Omega-H^D_{\\Omega'}$ has order $\\le T^{0}$ (at worst $T^0 = t^0$) at ff.\nEquivalently, their ff coefficients of orders $T^{-2} = t^{-1}$ and $T^{-1} = t^{-1/2}$ agree.\n\\end{lemma}\n\nWe can now prove Theorem \\ref{thm:thm1}.\n\\begin{proof}[Proof of Theorem \\ref{thm:thm1}] As in \\cite{NRS}, the heat trace is obtained by restricting $H^{D}_{\\Omega}$ to the spatial diagonal and then integrating over $z\\in\\Omega$. The technical tool used is Melrose's pushforward theorem. We will not repeat all the details here, but the upshot is that each face td, sf, and ff gives a separate contribution to the expansion \\eqref{eq:heattrace}. A priori, the expansions can interact to give logarithmic terms, but for the same reasons as in \\cite[p. 49]{NRS}, keeping in mind that we now have two orders at ff rather than one, there are no logarithmic terms until at least $O(T^2\\log T)=O(t\\log t)$ (and potentially much later, if at all). The contribution from td is \n\\[\\frac{|\\Omega|}{4\\pi t} + O(t^{\\infty}).\\]\nThe contributions from sf are\n\\[-\\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}} + \\frac{1}{12\\pi}\\int_{\\partial\\Omega}\\kappa\\, ds +\\frac{\\sqrt t}{256\\sqrt\\pi}\\int_{\\partial\\Omega}\\kappa^2\\, ds + O(t).\\]\nAnd the contributions from ff, since the first is already known and the second depends only on $\\alpha$ and $\\kappa_{\\pm}$, are\n\\[\\sum_{j=1}^{n}\\frac{\\pi^2-\\alpha_j^2}{24\\pi\\alpha_j} + \\sqrt t\\sum_{j=1}^{n}\\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-}) + O(t),\\]\nfor some unknown function $\\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-})$. Summing the three faces gives Theorem \\ref{thm:thm1}.\n\\end{proof}\n\nFor $H(t)$, we use the asymptotics of Bessel function zeroes due to McMahon \\cite[(10.21.19)]{NIST:DLMF}:\n\\begin{equation}\nj_{0,k}= (k-\\frac 14)\\pi + \\frac{1}{8(k-\\frac 14)\\pi} + O(k^{-3}),\n\\end{equation}\nwhich imply\n\\begin{equation}\nj_{0,k}^{2} = (k-\\frac 14)^2\\pi^2 + \\frac 14 + O(k^{-2}).\n\\end{equation}\nBased on this we define the function\n\\begin{equation}\n\\tilde H(t) = \\sum_{k=1}^{\\infty} \\exp[-t((k-\\frac 14)^2\\pi^2 + \\frac 14)]\n\\end{equation}\nand estimate its difference with $H(t)$.\n\\begin{lemma}\\label{lem:difference} With all notation as above, as $t\\to 0$,\n\\begin{equation}\nH(t) - \\tilde H(t) = O(t).\n\\end{equation}\n\\end{lemma}\nThis is helpful because we also have the following lemma.\n\\begin{lemma}\\label{lem:compute} As $t\\to 0$,\n\\begin{equation}\n\\tilde H(t) = \\frac{1}{2\\sqrt\\pi}t^{-1/2} + c - \\frac{1}{8\\sqrt\\pi}t^{1/2}+ O(t).\n\\end{equation}\nHere $c$ is a constant which is irrelevant to our purposes.\n\\end{lemma}\nFrom these two Lemmas and \\eqref{eq:relationship} we immediately deduce this proposition.\n\\begin{proposition} As $t\\to 0$,\n\\begin{equation}\n\\Tr e^{-t\\Delta_{\\Omega}} = \\frac 18 t^{-1} - (\\frac{\\sqrt\\pi}{8}+\\frac{1}{4\\sqrt\\pi})t^{-1/2}-\\frac 12c + (\\frac{\\sqrt\\pi}{256}+\\frac{1}{16\\sqrt\\pi})t^{1/2}+ O(t).\n\\end{equation}\n\\end{proposition}\n\\begin{remark} We could reverse engineer $c$ from the known expansion from \\cite{NRS} but there is no need.\n\\end{remark}", "post_theorem_intro_text_len": 4928, "post_theorem_intro_text": "We can say substantially more about $\\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-})$ whenever $\\alpha\\ne\\pi$. Define\n\\begin{equation}\\label{eq:defofr0}\n    r_0=r_0(\\alpha,\\kappa_{+},\\kappa_-)=\\frac 12\\csc\\alpha\\sqrt{\\kappa_+^2+\\kappa_-^2-2\\kappa_+\\kappa_-\\cos\\alpha}.\n\\end{equation}\n\\begin{theorem}\\label{thm:moreid} For $\\alpha\\ne\\pi$, the function $\\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-})$ has the form \n\\begin{equation}\\label{eq:form} \\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-}) = c_{1/2}(\\alpha)r_0(\\alpha,\\kappa_+,\\kappa_-)\n\\end{equation}\nwhere $c_{1/2}(\\alpha)$ is a function depending only on $\\alpha$, whose general expression is given by a Hadamard renormalized trace on an exact sector:\n\\begin{equation}\\label{eq:general} \\begin{aligned}\n        c_{1/2}(\\alpha)\n        &= -\\,\\operatorname*{fp} \n        \\iint_{(0,\\infty)\\times(0,\\alpha)}\n        \\int_{0}^{1} (1-\\sigma)^{-1}\\sigma^{-1/2}\n        \\iint_{(0,\\infty)\\times(0,\\alpha)} \\\\\n        &\\quad a\\left(\\frac{R}{\\sqrt{1-\\sigma}},\\theta,\n        \\frac{\\sqrt{\\sigma}\\,\\rho}{\\sqrt{1-\\sigma}},\\phi\\right)\\,\n        b\\left(\\rho,\\phi,\\frac{R}{\\sqrt{\\sigma}},\\theta\\right)\\,\n        \\,\\rho\\,d\\rho\\,d\\phi\\,\\,R\\,dR\\,d\\theta.\n        \\end{aligned}\n\\end{equation}\nHere $a$ and $b$ are functions related to the exact heat kernel on a sector of angle $\\alpha$, explicitly given by \\eqref{eq:a-def} and \\eqref{eq:bexpression-clean}, and $\\operatorname*{fp}$ denotes the Hadamard finite part at $R=\\infty$ of the improper integral in $R$. \n\nMoreover, $c_{1/2}(\\alpha)$ has the special value\n\\begin{equation}\\label{eq:specialpi2} c_{1/2}(\\frac{\\pi}{2}) = \\frac{1}{16\\sqrt\\pi}.\n\\end{equation}\n\\end{theorem}\n\\begin{remark} A few comments:\n\\begin{itemize}\n\\item The $O(t\\log t)$ error in \\eqref{eq:heattrace} is not necessarily optimal, and in fact we expect that it should be $O(t)$.\n\\item Each of $a$ and $b$ is a function only of $\\alpha$ and the variables of integration, see \\eqref{eq:a-def}, \\eqref{eq:bexpression-clean}. Thus \\eqref{eq:general} is indeed independent of $\\kappa_{\\pm}$.\n\\item For inverse spectral results, it is crucial that \\eqref{eq:specialpi2} is nonzero. We can only verify this explicitly for $\\alpha=\\pi/2$. It would be of significant interest to find out for which $\\alpha$ we have $c_{1/2}(\\alpha)\\ne 0$. \n\\item Theorem \\ref{thm:moreid} does not work for $\\alpha=\\pi$  because the conformal model that we use breaks down in that case, as can be seen from the fact that $r_0$ is undefined when $\\alpha=\\pi$. So the case $\\alpha=\\pi$, where the boundary has an abrupt change in second derivative, remains open. \n\\end{itemize}\n\\end{remark}\n\n\\subsection{Inverse spectral applications}\n\nIn \\cite{EGS_2017}, the authors prove that any curvilinear polygon with straight corners is in fact isospectral to a polygon. They require straight corners because of a lack of a formula in the curved corner case. We can strengthen their result.\n\\begin{definition}\n    A curvilinear polygon $\\Omega$ is \\emph{admissible} if, for each $j$, either $c_{1/2}(\\alpha_j)>0$  or $\\Omega$ is straight near $P_j$.\n\\end{definition}\n\n\\begin{proposition}\n    Suppose that an admissible curvilinear polygon $\\Omega$ is Dirichlet isospectral to a polygon. Then $\\Omega$ is a polygon.\n\\end{proposition}\n\\begin{proof}\nFor a polygon, the $t^{1/2}$ term in the heat trace vanishes. So it must vanish for $\\Omega$. But under our assumptions, by Theorems \\ref{thm:thm1} and \\ref{thm:moreid}, the $t^{1/2}$ coefficient in $H^D_{\\Omega}(t)$ is given by \n    \\begin{equation}\n    \\frac{1}{256\\sqrt\\pi}\\int_{\\partial\\Omega}\\kappa^2\\, ds + \\sum_{j=1}^n c_{1/2}(\\alpha_j)\\sqrt{\\kappa_{+,j}^2+\\kappa_{-,j}^2}.\n    \\end{equation}\n Since this coefficient is non-negative and only zero for an exact polygon, $\\Omega$ is an exact polygon.\n\\end{proof}\n\\begin{remark}\nObserve from \\eqref{eq:specialpi2} that any polygon where all non-straight corners are right angles is admissible.\n\\end{remark}\n\n\\subsection{Plan of the paper}\nIn section \\ref{sec:gm} we prove Theorem \\ref{thm:thm1}. This is done by examining the geometric microlocal description of the heat kernel given in \\cite{NRS}. We show that there is a well-defined corner contribution $\\mathcal C_{1/2}\\sqrt t$, and then prove that any two corners with the same angle $\\alpha$ and limit curvatures $\\kappa_{\\pm}$ yield the same contribution, regardless of the rest of $\\Omega$. To take advantage of this uniformity, in section \\ref{sec:conformal} we introduce a simple conformal model for a curvilinear corner and compute the contribution $\\mathcal C_{1/2}\\sqrt t$ in that model. This computation proceeds via a parametrix construction as in \\cite{NRS} and allows us to obtain both \\eqref{eq:form} and \\eqref{eq:general}. \nFinally, in section \\ref{sec:specialpi2}, we prove \\eqref{eq:specialpi2} by computing a sufficient number of terms in the Dirichlet heat trace expansion for a semicircle and comparing with \\eqref{eq:heattrace}.", "sketch": "In the “Plan of the paper” the authors outline the proof of Theorem~\\ref{thm:thm1}: in Section~\\ref{sec:gm} they “examin[e] the geometric microlocal description of the heat kernel given in \\cite{NRS},” “show that there is a well-defined corner contribution $\\mathcal C_{1/2}\\sqrt t$,” and “prove that any two corners with the same angle $\\alpha$ and limit curvatures $\\kappa_{\\pm}$ yield the same contribution, regardless of the rest of $\\Omega$.” Then, “to take advantage of this uniformity,” in Section~\\ref{sec:conformal} they “introduce a simple conformal model for a curvilinear corner and compute the contribution $\\mathcal C_{1/2}\\sqrt t$ in that model,” via “a parametrix construction as in \\cite{NRS},” which “allows [them] to obtain both \\eqref{eq:form} and \\eqref{eq:general}.” Finally, in Section~\\ref{sec:specialpi2} they prove \\eqref{eq:specialpi2} “by computing a sufficient number of terms in the Dirichlet heat trace expansion for a semicircle and comparing with \\eqref{eq:heattrace}.”", "expanded_sketch": "In the “Plan of the paper” the authors outline the proof of the main theorem: next they “examin[e] the geometric microlocal description of the heat kernel given in NRS,” “show that there is a well-defined corner contribution $\\mathcal C_{1/2}\\sqrt t$,” and “prove that any two corners with the same angle $\\alpha$ and limit curvatures $\\kappa_{\\pm}$ yield the same contribution, regardless of the rest of $\\Omega$.” Then, “to take advantage of this uniformity,” they “introduce a simple conformal model for a curvilinear corner and compute the contribution $\\mathcal C_{1/2}\\sqrt t$ in that model,” via “a parametrix construction as in NRS,” which “allows [them] to obtain both\n\\begin{equation}\\label{eq:form} \\mathcal C_{1/2}(\\alpha,\\kappa_{+},\\kappa_{-}) = c_{1/2}(\\alpha)r_0(\\alpha,\\kappa_+,\\kappa_-)\n\\end{equation}\nand\n\\begin{equation}\\label{eq:general} \\begin{aligned}\n        c_{1/2}(\\alpha)\n        &= -\\,\\fp \n        \\iint_{(0,\\infty)\\times(0,\\alpha)}\n        \\int_{0}^{1} (1-\\sigma)^{-1}\\sigma^{-1/2}\n        \\iint_{(0,\\infty)\\times(0,\\alpha)} \\\\\n        &\\quad a\\left(\\frac{R}{\\sqrt{1-\\sigma}},\\theta,\n        \\frac{\\sqrt{\\sigma}\\,\\rho}{\\sqrt{1-\\sigma}},\\phi\\right)\\,\\\n        b\\left(\\rho,\\phi,\\frac{R}{\\sqrt{\\sigma}},\\theta\\right)\\,\\\n        \\dmup\\,\\dmu.\n        \\end{aligned}\n\\end{equation}.\nFinally, they prove\n\\begin{equation}\\label{eq:specialpi2} c_{1/2}(\\frac{\\pi}{2}) = \\frac{1}{16\\sqrt\\pi}.\n\\end{equation}\n“by computing a sufficient number of terms in the Dirichlet heat trace expansion for a semicircle and comparing with\n\\begin{multline}\\label{eq:heattrace}\nH^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t} - \\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}} + \\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\, ds + \\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)\\\\\n+ \\sqrt t\\Big(\\frac{1}{256\\sqrt{\\pi}}\\int_{\\partial\\Omega}\\kappa^2\\, ds + \\sum_{j=1}^n\\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\Big) +O(t\\log t),\n\\end{multline}.”,", "expanded_theorem": "\\label{thm:thm1} With notation as above, the Dirichlet heat trace $H^D_{\\Omega}(t)$ has an expansion as $t\\to 0$ given by\n\\begin{multline}\\label{eq:heattrace}\nH^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t} - \\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}} + \\frac{1}{12\\pi}\\Big(\\int_{\\partial\\Omega}\\kappa\\, ds + \\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\Big)\\\\\n+ \\sqrt t\\Big(\\frac{1}{256\\sqrt{\\pi}}\\int_{\\partial\\Omega}\\kappa^2\\, ds + \\sum_{j=1}^n\\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\Big) +O(t\\log t),\n\\end{multline}\nwhere $\\mathcal C_{1/2}$ is a function only of the angle $\\alpha$ and the limit curvatures $\\kappa_{\\pm}..", "theorem_type": ["Asymptotic or Limit", "Inequality or Bound"], "mcq": {"question": "Let \\(\\Omega\\subseteq\\mathbb R^2\\) be a curvilinear polygon with piecewise smooth boundary, smooth except at finitely many corners \\(P_j\\) with interior angles \\(\\alpha_j>0\\), \\(j=1,\\dots,n\\). Let \\(\\kappa\\) denote the inward-pointing curvature of \\(\\partial\\Omega\\), and for each corner \\(P_j\\), let \\(\\kappa_{j,+}\\) and \\(\\kappa_{j,-}\\) be the one-sided limits of \\(\\kappa\\) along the two boundary arcs meeting at \\(P_j\\). If \\(H^D_{\\Omega}(t)\\) denotes the trace of the Dirichlet heat kernel on \\(\\Omega\\), then as \\(t\\to 0\\), which asymptotic expansion holds?", "correct_choice": {"label": "A", "text": "\\[H^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t}-\\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}}+\\frac{1}{12\\pi}\\left(\\int_{\\partial\\Omega}\\kappa\\,ds+\\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\right)+\\sqrt t\\left(\\frac{1}{256\\sqrt\\pi}\\int_{\\partial\\Omega}\\kappa^2\\,ds+\\sum_{j=1}^n \\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\right)+O(t\\log t),\\]\\nwhere \\(\\mathcal C_{1/2}(\\alpha,\\kappa_+,\\kappa_-)\\) depends only on the corner angle \\(\\alpha\\) and the one-sided curvature limits \\(\\kappa_\\pm\\)."}, "choices": [{"label": "B", "text": "\\[H^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t}-\\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}}+\\frac{1}{12\\pi}\\left(\\int_{\\partial\\Omega}\\kappa\\,ds+\\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\right)+\\sqrt t\\left(\\frac{1}{256\\sqrt\\pi}\\int_{\\partial\\Omega}\\kappa^2\\,ds+\\sum_{j=1}^n \\mathcal C_{1/2}(\\alpha_j)\\right)+O(t\\log t),\\]\\nwhere \\(\\mathcal C_{1/2}(\\alpha)\\) depends only on the corner angle."}, {"label": "C", "text": "\\[H^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t}-\\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}}+\\frac{1}{12\\pi}\\left(\\int_{\\partial\\Omega}\\kappa\\,ds+\\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\right)+O(\\sqrt t).\\]"}, {"label": "D", "text": "\\[H^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t}-\\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}}+\\frac{1}{12\\pi}\\left(\\int_{\\partial\\Omega}\\kappa\\,ds+\\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\right)+\\sqrt t\\left(\\frac{1}{256\\sqrt\\pi}\\int_{\\partial\\Omega}\\kappa^2\\,ds+\\sum_{j=1}^n \\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+},\\kappa_{j,-})\\right)+O(t),\\]\\nwhere \\(\\mathcal C_{1/2}(\\alpha,\\kappa_+,\\kappa_-)\\) depends only on the corner angle \\(\\alpha\\) and the one-sided curvature limits \\(\\kappa_\\pm\\)."}, {"label": "E", "text": "\\[H^D_{\\Omega}(t)=\\frac{|\\Omega|}{4\\pi t}-\\frac{|\\partial\\Omega|}{8\\sqrt{\\pi t}}+\\frac{1}{12\\pi}\\left(\\int_{\\partial\\Omega}\\kappa\\,ds+\\sum_{j=1}^n\\frac{\\pi^2-\\alpha_j^2}{2\\alpha_j}\\right)+\\sqrt t\\left(\\frac{1}{256\\sqrt\\pi}\\int_{\\partial\\Omega}\\kappa^2\\,ds+\\sum_{j=1}^n \\mathcal C_{1/2}(\\alpha_j,\\kappa_{j,+}+\\kappa_{j,-})\\right)+O(t\\log t),\\]\\nwhere \\(\\mathcal C_{1/2}(\\alpha,\\kappa_++\\kappa_-)\\) depends only on the corner angle \\(\\alpha\\) and the sum of the one-sided curvature limits."}], "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": "corner dependence on both angle and one-sided curvatures", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "explicit identification of the \\sqrt t coefficient and sharper remainder", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "first possible logarithmic remainder order", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "separate dependence on \\kappa_+ and \\kappa_- rather than only their sum", "template_used": "wildcard"}]}}
{"id": "2512.04640v2", "paper_link": "http://arxiv.org/abs/2512.04640v2", "theorems_cnt": 1, "theorem": {"env_name": "teo", "content": "\\label{thm:main}\n\tLet $\\Omega\\subset\\mathbb{G}$ be an open and bounded set\n\twith smooth enough boundary $\\partial \\Omega$, and let $p\\in (0,1)$. Then, there exists $\\Lambda > 0$\n\tsuch that\n\t\\begin{itemize}\n\t\t\\item[A)]problem \\eqref{EqProblem}$_{\\lambda}$ does not admit weak solutions\n\t\tfor every $\\lambda>\\Lambda$;\n\t\t\\item[B)] problem \\eqref{EqProblem}$_{\\lambda}$ admits at least one weak solution for every $\\lambda \\in (0,\\Lambda]$;\n\t\t\\item[C)] problem \\eqref{EqProblem}$_{\\lambda}$ admits at least two weak solutions for every $0<\\lambda<\\Lambda$.\n\t\\end{itemize}", "start_pos": 9823, "end_pos": 10412, "label": "thm:main"}, "ref_dict": {"thm:Existence_First": "\\begin{teo}\\label{thm:Existence_First}\n\tProblem \\eqref{EqProblem}$_\\lambda$ admits at least\n\tone weak solution $u_\\lambda\\in S^{1}_{0}(\\Omega)$ for every $\\lambda \\in (0,\\Lambda]$.\n\\end{teo}", "def:weak_sub_super_sol": "\\begin{defin}\\label{def:weak_sub_super_sol}\n\tLet $\\Omega\\subseteq\\mathbb{G}$ be an open, bounded and connected set.\nWe say that a function $u \\in S^{1}_{0}(\\Omega)$ is a weak subsolution (resp. supersolution) of \\eqref{EqProblem}$_\\lambda$ if it satisfies the following properties:\n\t\\begin{itemize}\n\t\t\\item[(i)] $u>0$ in $\\Omega$.\n\t\t\\item[(ii)] For every $0\\leq \\varphi \\in C^{\\infty}_{0}(\\Omega)$, it holds that\n\t\t\\begin{equation}\\label{eq:weak_sub_super_sol}\n\t\t\t\\int_{\\Omega}\\langle \\nabla_{\\mathbb{G}}u, \\nabla_{\\mathbb{G}}\\varphi \\rangle_{\\mathfrak{g}_{1}} \\leq (\\textrm{resp. } \\geq)  \\int_{\\Omega}\\left(\\lambda u^{q}  + \\ u^{2^{\\star}_{Q}}\\right)\\varphi.\n\t\t\\end{equation}\n\t\\end{itemize}\n\tFinally, we say that $u \\in S^{1}_{0}(\\Omega)$ is a weak solution of \\eqref{EqProblem}$_\\lambda$ if it is both a weak subsolution and a weak supersolution of \\eqref{EqProblem}$_\\lambda$ without the non-negativity condition on $\\varphi$.\n\\end{defin}", "EqProblem": "\\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n\t\\left\\{\\begin{array}{rll}\n\t\t-\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\tu&=0 & \\textrm{ on } \\partial \\Omega.\n\t\\end{array}\\right.\n\\end{equation}", "eq:Sublinear_Problem": "\\begin{equation}\\label{eq:Sublinear_Problem}\n\t\\left\\{\\begin{array}{rll}\n\t\t\t\t-\\Delta_{\\mathbb{G}}u &= \\lambda\\, u^{q} & \\textrm{ in } \\Omega,\\\\\n\t\t\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\t\t\tu&=0 & \\textrm{ on } \\partial \\Omega,\n\t\\end{array}\\right.\n\\end{equation}", "lem:locmin": "\\begin{lemma}\\label{lem:locmin}\n\tFor $\\lambda\\in (0,\\Lambda)$, if $u_\\lambda$ is the solution\n\tpresented in Theorem \\ref{thm:Existence_First}, then\n\t$u_\\lambda$ is a local minimum for $I_\\lambda$ in the $S^1_0(\\Omega)$-topology,\n\tmeaning that there exists $r_0>0$ such that for any $u\\in S^1_0(\\Omega)$,\n\t\\[\n\tI_\\lambda(u_\\lambda)\\leq I_\\lambda(u) \\quad \\textrm{ for all } u\\in S^{1}_{0}(\\Omega) \\textrm{ with } \\|u-u_\\lambda\\|_{S_{0}^{1}(\\Omega)}< r_0 .\n\t\\]\n\\end{lemma}", "thm:main": "\\begin{teo} \\label{thm:main}\n\tLet $\\Omega\\subset\\mathbb{G}$ be an open and bounded set\n\twith smooth enough boundary $\\partial \\Omega$, and let $p\\in (0,1)$. Then, there exists $\\Lambda > 0$\n\tsuch that\n\t\\begin{itemize}\n\t\t\\item[A)]problem \\eqref{EqProblem}$_{\\lambda}$ does not admit weak solutions\n\t\tfor every $\\lambda>\\Lambda$;\n\t\t\\item[B)] problem \\eqref{EqProblem}$_{\\lambda}$ admits at least one weak solution for every $\\lambda \\in (0,\\Lambda]$;\n\t\t\\item[C)] problem \\eqref{EqProblem}$_{\\lambda}$ admits at least two weak solutions for every $0<\\lambda<\\Lambda$.\n\t\\end{itemize}\n\\end{teo}"}, "pre_theorem_intro_text_len": 1436, "pre_theorem_intro_text": "Let $\\mathbb{G}$ be a Carnot group and let $\\Omega \\subset \\mathbb{G}$ be an bounded and connected\nopen set with smooth enough boundary $\\partial \\Omega$. Let $q \\in (0,1)$, let $2^{\\star}_{Q}:=\\tfrac{2Q}{Q-2}$ be the critical Sobolev exponent related to the Sobolev inequality in $\\mathbb{G}$, and let $\\lambda >0$. We consider the following Dirichlet boundary value problem\n\\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n\t\\left\\{\\begin{array}{rll}\n\t\t-\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\tu&=0 & \\textrm{ on } \\partial \\Omega.\n\t\\end{array}\\right.\n\\end{equation}\n\nWe stress that $-\\Delta_{\\mathbb{G}}$ denotes here the sub-laplacian on $\\mathbb{G}$ which is a second-order differential operator with non-negative characteristic form that can be explicitly expressed as a sum of squares of vector fields satisfying the H\\\"{o}rmander condition, see e.g. \\cite{Hormander}. We refer to Section~\\ref{sec:Prel} for more details, including the Folland-Stein Sobolev spaces we will work with.\\\\\n\nAlong the paper it will sometimes be useful to denote the above problem as \\eqref{EqProblem}$_\\lambda$ \nto make it clear the choice of the parameter.\nWe immediately state the main result of this paper. In what follows,\nwe refer to Definition \\ref{def:weak_sub_super_sol} for the precise definition of \\emph{weak solution}\nof \\eqref{EqProblem}$_{\\lambda}$.", "context": "Let $\\mathbb{G}$ be a Carnot group and let $\\Omega \\subset \\mathbb{G}$ be an bounded and connected\nopen set with smooth enough boundary $\\partial \\Omega$. Let $q \\in (0,1)$, let $2^{\\star}_{Q}:=\\tfrac{2Q}{Q-2}$ be the critical Sobolev exponent related to the Sobolev inequality in $\\mathbb{G}$, and let $\\lambda >0$. We consider the following Dirichlet boundary value problem\n\\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n \\left\\{\\begin{array}{rll}\n -\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n u&>0 & \\textrm{ in } \\Omega,\\\\\n u&=0 & \\textrm{ on } \\partial \\Omega.\n \\end{array}\\right.\n\\end{equation}\n\nWe stress that $-\\Delta_{\\mathbb{G}}$ denotes here the sub-laplacian on $\\mathbb{G}$ which is a second-order differential operator with non-negative characteristic form that can be explicitly expressed as a sum of squares of vector fields satisfying the H\\\"{o}rmander condition, see e.g. \\cite{Hormander}. We refer to Section~\\ref{sec:Prel} for more details, including the Folland-Stein Sobolev spaces we will work with.\\\\\n\nAlong the paper it will sometimes be useful to denote the above problem as \\eqref{EqProblem}$_\\lambda$ \nto make it clear the choice of the parameter.\nWe immediately state the main result of this paper. In what follows,\nwe refer to Definition \\ref{def:weak_sub_super_sol} for the precise definition of \\emph{weak solution}\nof \\eqref{EqProblem}$_{\\lambda}$.\n\n\\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n\t\\left\\{\\begin{array}{rll}\n\t\t-\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\tu&=0 & \\textrm{ on } \\partial \\Omega.\n\t\\end{array}\\right.\n\\end{equation}\n\n\\begin{defin}\\label{def:weak_sub_super_sol}\n\tLet $\\Omega\\subseteq\\mathbb{G}$ be an open, bounded and connected set.\nWe say that a function $u \\in S^{1}_{0}(\\Omega)$ is a weak subsolution (resp. supersolution) of \\eqref{EqProblem}$_\\lambda$ if it satisfies the following properties:\n\t\\begin{itemize}\n\t\t\\item[(i)] $u>0$ in $\\Omega$.\n\t\t\\item[(ii)] For every $0\\leq \\varphi \\in C^{\\infty}_{0}(\\Omega)$, it holds that\n\t\t\\begin{equation}\\label{eq:weak_sub_super_sol}\n\t\t\t\\int_{\\Omega}\\langle \\nabla_{\\mathbb{G}}u, \\nabla_{\\mathbb{G}}\\varphi \\rangle_{\\mathfrak{g}_{1}} \\leq (\\textrm{resp. } \\geq)  \\int_{\\Omega}\\left(\\lambda u^{q}  + \\ u^{2^{\\star}_{Q}}\\right)\\varphi.\n\t\t\\end{equation}\n\t\\end{itemize}\n\tFinally, we say that $u \\in S^{1}_{0}(\\Omega)$ is a weak solution of \\eqref{EqProblem}$_\\lambda$ if it is both a weak subsolution and a weak supersolution of \\eqref{EqProblem}$_\\lambda$ without the non-negativity condition on $\\varphi$.\n\\end{defin}", "full_context": "Let $\\mathbb{G}$ be a Carnot group and let $\\Omega \\subset \\mathbb{G}$ be an bounded and connected\nopen set with smooth enough boundary $\\partial \\Omega$. Let $q \\in (0,1)$, let $2^{\\star}_{Q}:=\\tfrac{2Q}{Q-2}$ be the critical Sobolev exponent related to the Sobolev inequality in $\\mathbb{G}$, and let $\\lambda >0$. We consider the following Dirichlet boundary value problem\n\\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n \\left\\{\\begin{array}{rll}\n -\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n u&>0 & \\textrm{ in } \\Omega,\\\\\n u&=0 & \\textrm{ on } \\partial \\Omega.\n \\end{array}\\right.\n\\end{equation}\n\nWe stress that $-\\Delta_{\\mathbb{G}}$ denotes here the sub-laplacian on $\\mathbb{G}$ which is a second-order differential operator with non-negative characteristic form that can be explicitly expressed as a sum of squares of vector fields satisfying the H\\\"{o}rmander condition, see e.g. \\cite{Hormander}. We refer to Section~\\ref{sec:Prel} for more details, including the Folland-Stein Sobolev spaces we will work with.\\\\\n\nAlong the paper it will sometimes be useful to denote the above problem as \\eqref{EqProblem}$_\\lambda$ \nto make it clear the choice of the parameter.\nWe immediately state the main result of this paper. In what follows,\nwe refer to Definition \\ref{def:weak_sub_super_sol} for the precise definition of \\emph{weak solution}\nof \\eqref{EqProblem}$_{\\lambda}$.\n\n\\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n\t\\left\\{\\begin{array}{rll}\n\t\t-\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\tu&=0 & \\textrm{ on } \\partial \\Omega.\n\t\\end{array}\\right.\n\\end{equation}\n\n\\begin{defin}\\label{def:weak_sub_super_sol}\n\tLet $\\Omega\\subseteq\\mathbb{G}$ be an open, bounded and connected set.\nWe say that a function $u \\in S^{1}_{0}(\\Omega)$ is a weak subsolution (resp. supersolution) of \\eqref{EqProblem}$_\\lambda$ if it satisfies the following properties:\n\t\\begin{itemize}\n\t\t\\item[(i)] $u>0$ in $\\Omega$.\n\t\t\\item[(ii)] For every $0\\leq \\varphi \\in C^{\\infty}_{0}(\\Omega)$, it holds that\n\t\t\\begin{equation}\\label{eq:weak_sub_super_sol}\n\t\t\t\\int_{\\Omega}\\langle \\nabla_{\\mathbb{G}}u, \\nabla_{\\mathbb{G}}\\varphi \\rangle_{\\mathfrak{g}_{1}} \\leq (\\textrm{resp. } \\geq)  \\int_{\\Omega}\\left(\\lambda u^{q}  + \\ u^{2^{\\star}_{Q}}\\right)\\varphi.\n\t\t\\end{equation}\n\t\\end{itemize}\n\tFinally, we say that $u \\in S^{1}_{0}(\\Omega)$ is a weak solution of \\eqref{EqProblem}$_\\lambda$ if it is both a weak subsolution and a weak supersolution of \\eqref{EqProblem}$_\\lambda$ without the non-negativity condition on $\\varphi$.\n\\end{defin}\n\nThe above theorem is the natural generalization to Carnot groups of classical results of \\cite{ABC}. We refer e.g. to \\cite{BESS,CCP,CoPe,BV} for further generalizations.\n\nWe are now ready to properly set the definition of weak sub/supersolution of \\eqref{EqProblem}$_\\lambda$.\n\\begin{defin}\\label{def:weak_sub_super_sol}\n Let $\\Omega\\subseteq\\mathbb{G}$ be an open, bounded and connected set.\nWe say that a function $u \\in S^{1}_{0}(\\Omega)$ is a weak subsolution (resp. supersolution) of \\eqref{EqProblem}$_\\lambda$ if it satisfies the following properties:\n \\begin{itemize}\n \\item[(i)] $u>0$ in $\\Omega$.\n \\item[(ii)] For every $0\\leq \\varphi \\in C^{\\infty}_{0}(\\Omega)$, it holds that\n \\begin{equation}\\label{eq:weak_sub_super_sol}\n \\int_{\\Omega}\\langle \\nabla_{\\mathbb{G}}u, \\nabla_{\\mathbb{G}}\\varphi \\rangle_{\\mathfrak{g}_{1}} \\leq (\\textrm{resp. } \\geq) \\int_{\\Omega}\\left(\\lambda u^{q} + \\ u^{2^{\\star}_{Q}}\\right)\\varphi.\n \\end{equation}\n \\end{itemize}\n Finally, we say that $u \\in S^{1}_{0}(\\Omega)$ is a weak solution of \\eqref{EqProblem}$_\\lambda$ if it is both a weak subsolution and a weak supersolution of \\eqref{EqProblem}$_\\lambda$ without the non-negativity condition on $\\varphi$.\n\\end{defin}\n\n\\begin{lemma}\\label{lem:Perron}\n Let $\\underline{u}, \\overline{u}\\in S^{1}_{0}(\\Omega)$ be a weak subsolution and a weak supersolution, respectively, of problem \\eqref{EqProblem}$_\\lambda$. \n We assume that \n \\begin{itemize}\n \\item[a)] $\\underline{u}(g) \\leq \\overline{u}(g)$ for a.e.\\,$g\\in \\Omega$;\n \\item[b)] for every open set $\\Oo\\Subset\\Omega$ there exists\n $C = C(\\Oo,\\underline{u}) > 0$ such that\n $$\\text{$\\underline{u}\\geq C$ a.e.\\,in $\\Oo$}.$$\n \\end{itemize}\n Then, there exists a weak solution $u \\in S^{1}_{0}(\\Omega)$ of \\eqref{EqProblem}$_{\\lambda}$ such that \n $$\\text{$\\underline{u}(g) \\leq u(g) \\leq \\overline{u}(g)$ for a.e. $g \\in \\Omega$}.$$\n\\end{lemma}\n\n\\noindent We split the proof of I) in two lemmas. \n\\begin{lemma}\\label{lem:lambda0}\n Let $\\Lambda$ as defined in \\eqref{eq:DefinitionLambda}. Then $\\Lambda >0$.\n \\begin{proof}\n We will show that there exists a sufficiently small $\\lambda >0$ such that \n \\eqref{EqProblem}$_{\\lambda}$ has a solution. To this aim, we will use Lemma \\ref{lem:Perron} exhibiting both a super and a subsolution.\n Looking for a supersolution, we consider the following auxiliary torsion problem\n \\begin{equation}\n \\left\\{\\begin{array}{rl}\n -\\Delta_{\\mathbb{G}} V = 1 & \\textrm{in } \\Omega,\\\\\n V = 0 & \\textrm{on } \\partial \\Omega,\n \\end{array}\\right.\n \\end{equation}\n \\noindent whose unique solution is provided by Lax-Milgram Theorem. Moreover, by a classical Stampacchia iteration method, it holds that $V\\in L^{\\infty}(\\Omega)$. Observe further that for any positive constant $C$,\n the function $C\\cdot x^p-x$, with $p>1$, has negative value for $x>0$ sufficiently small.\n Therefore, for every $C,C'\\in \\R^+$\n there exists $\\lambda^*>0$ such that \n for every $\\lambda<\\lambda^*$, \n \\[\n \\exists \\, m_\\lambda\\in \\R^+:\\quad \\lambda\\cdot C'\\cdot m_\\lambda^q+C\\cdot m_\\lambda^p-m_\\lambda\\leq0.\n \\]\n We fix $\\lambda<\\lambda^*$ and set $C=\\|V\\|_{L^{\\infty}(\\Omega)}^p$, $C'=\\|V\\|_{L^{\\infty}(\\Omega)}^q$. \n We define $\\overline u_1:=m_\\lambda V$, which weakly verifies\n \\[\n \\left\\{\\begin{array}{rll}\n -\\Delta_{\\G}\\overline u_1&=m_\\lambda \\geq \\lambda\\overline u_1^q+\\overline u_1^p &\\ \\m{in }\\Omega\\\\\n \\overline u_1 &>0 &\\ \\m{in }\\Omega\\\\\n \\overline u_1 &=0 &\\ \\m{on }\\de\\Omega,\n \\end{array}\\right.\n \\]\n therefore it is a weak supersolution of \\eqref{EqProblem}$_\\lambda$.\\\\\n Regarding the weak subsolution to \\eqref{EqProblem}, we choose the unique solution $\\underline u_\\lambda$ to \\eqref{eq:Sublinear_Problem}. \n We can now conclude the proof by appealing Lemma \\ref{lem:Perron}. Indeed, by Lemma \\ref{lem:Weak_Comparison_Model} with $w=\\overline{u}_1$ and $v=\\underline{u}_{\\lambda}$, we get that \n $\\overline{u}_1 \\geq \\underline{u}_{\\lambda}$, which is condition a) of Lemma \\ref{lem:Perron}. Regarding b) of Lemma \\ref{lem:Perron} it is enough to recall \\cite[Corollary 2.3]{BiGaVe}. This closes the proof. \n \\end{proof}\n\\end{lemma}\n\nWe now set $\\overline{u}= u_{\\lambda'}$ and $\\underline{u} = w_{\\lambda}$, \n and we apply Lemma \\ref{lem:Perron}: this immediately yields that problem \\eqref{EqProblem}$_{\\lambda}$ admits a weak solution $u_{\\lambda}$ for every $\\lambda \\in (0,\\Lambda)$. Moreover, recalling the definition of $I_{\\lambda}$ in \\eqref{eq:Def_Ilambda}, such a solution satisfies that\n $$I_{\\lambda}(u_\\lambda) = \\min\\{u\\in S_0^1(\\Omega):\\,w_{\\lambda}\\leq u\\leq u_{\\lambda'}\\}\n \\leq I_{\\lambda}(w_{\\lambda}).$$\n In particular,\n by Theorem \\ref{thm:Sublinear_Problem} we have\n \\begin{equation} \\label{eq:Ilambdaulambdaneg}\n I_{\\lambda}(u_{\\lambda}) \\leq I_{\\lambda}(w_{\\lambda}) \\leq J_{\\lambda}(w_{\\lambda}) <0.\n \\end{equation}\n It remains to consider the case $\\lambda = \\Lambda$. The proof\n is rather standard and pretty similar to that of \\cite[Lemma 3.5]{BiGaVe}. We report it here for the sake of completeness.\n To begin with, we choose a monotone increasing sequence $\\{\\lambda_k\\}_k\\subseteq(0,\\Lambda)$ such that \n $\\lambda_{k} \\to \\Lambda$ as $k\\to+\\infty$. Now, for each $k \\in \\mathbb{N}$, we set\n $$u_k := u_{\\lambda_k}\\in S^{1}_{0}(\\Omega),$$\n \\noindent where $u_{\\lambda_k}$ is the weak solution of problem \\eqref{EqProblem}$_{\\lambda_k}$ constructed\n as above by means of Lemma \\ref{lem:Perron}. Thanks to \n \\eqref{eq:Ilambdaulambdaneg}, for every $k\\geq 1$ we have\n \\begin{equation} \\label{eq:Ilambdakneg}\n I_{\\lambda_k,}(u_{k}) \n = \\dfrac{1}{2} \\int_{\\Omega}|\\nabla_{\\mathbb{G}}u_k|^2 - \\dfrac{\\lambda_k}{q+1}\\int_{\\Omega}|u_k|^{q+1} - \\dfrac{1}{2^{\\star}_{Q}}\\int_{\\Omega}|u_k|^{2^{\\star}_{Q}} <0.\n \\end{equation}\n Moreover, by using $\\varphi = u_k$ in \\eqref{eq:weak_sub_super_sol}, and recalling that $u_k$\n solves \\eqref{EqProblem}$_{\\lambda_k}$, we get\n \\begin{equation} \\label{eq:testwithukzero}\n \\int_{\\Omega}|\\nabla_{\\mathbb{G}}u_k|^2 -\\lambda_k\\int_\\Omega u_k^{q+1}\n -\\int_\\Omega u_k^{2^{\\star}_{Q}} = 0.\n \\end{equation}\n Combining \\eqref{eq:Ilambdakneg} with \\eqref{eq:testwithukzero},\n we notice that the sequence $\\{u_k\\}_k$ is bounded in~$S^{1}_{0}(\\Omega)$.\n Therefore, we can find a function\n $$u_{\\Lambda}\\in S^{1}_{0}(\\Omega)$$ \n such that\n (up to a subsequence and as $k\\to+\\infty$)\n \\begin{itemize}\n \\item[a)] $u_k\\to u_{\\Lambda}$ weakly in $S^{1}_{0}(\\Omega)$ and strongly\n in $L^p(\\Omega)$ for $1\\leq p <2^{\\star}_{Q}$;\n \\item[b)] $u_k\\to u_{\\Lambda}$ a.e.\\,in $\\Omega$.\n \\end{itemize}\n We now observe that, being $\\{\\lambda_k\\}_k$ increasing, it follows that $\\lambda_k\\geq \\lambda_1$ for every $k\\geq 1$.\n Moreover, arguing as above yields that\n $u_{\\lambda_k}\\geq w_{\\lambda_1}$, and thus\n $$u_{\\Lambda} > 0\\quad\\text{a.e.\\,in $\\Omega$}.$$\n Moreover, since $u_k$ solves problem \\eqref{EqProblem}$_{\\lambda_k}$,\n we have\n $$\\int_{\\Omega}\\langle \\nabla_{\\mathbb{G}}u_k,\\nabla_{\\mathbb{G}}\\varphi\\rangle_{\\mathfrak{g}_{1}} -\\lambda_k\\int_\\Omega u_k^{q}\\varphi \n -\\int_\\Omega u_k^{2^{\\star}_{Q}-1}\\varphi = 0\\quad\\text{for every $\\varphi\\in S^{1}_{0}(\\Omega)$}.$$\nTherefore, passing to the limit as $k\\to+\\infty$ in the above identity, and by dominated convergence, we get that $u_{\\Lambda}$ satisfies\n $$\\int_{\\Omega}\\langle \\nabla_{\\mathbb{G}}u_{\\Lambda},\\nabla_{\\mathbb{G}}\\varphi\\rangle_{\\mathfrak{g}_{1}} -\\Lambda\\int_\\Omega u_{\\Lambda}^{q}\\varphi\n -\\int_\\Omega u_{\\Lambda}^{2^{\\star}_{Q}-1}\\varphi = 0\\quad\\text{for every $\\varphi\\in S^{1}_{0}(\\Omega)$},$$\n \\noindent which shows that $u_{\\Lambda}$ is actually a weak solution of\n problem \\eqref{EqProblem}$_{\\Lambda}$. \n This closes the proof.\n\\end{proof}", "post_theorem_intro_text_len": 5030, "post_theorem_intro_text": "The above theorem is the natural generalization to Carnot groups of classical results of \\cite{ABC}. We refer e.g. to \\cite{BESS,CCP,CoPe,BV} for further generalizations. \n\nThe interest in studying existence of positive solutions to critical problems in the Carnot group setting,\n is in the geometric significance of the purely critical problem in the model case of the Heisenberg group.\n Indeed, when $\\lambda =0$ and $\\Omega = \\mathbb{H}^n$, the problem \\eqref{EqProblem} becomes the famous CR-Yamabe problem studied by Jerison and Lee \\cite{JerisonLee,JerisonLee2,JerisonLee3}. The problem  we are interested in is settled on bounded domains, where tipycally one can prove non-existence of positive solutions, at least in star-shaped domains, by appealing suitable versions of the Pohozaev identity. Because of this, the seminal paper by Brezis and Nirenberg \\cite{BN} showed that adding a perturbative term, linear in \\cite{BN}, but subsequently extended to much more general perturbations, may allow to prove the existence of one or more positive solutions. A crucial tool in the argument performed in \\cite{BN} is provided by the use of the Aubin-Talenti functions, whose analogue in $\\mathbb{H}^n$ made its appearance in \\cite{JerisonLee2}. This was a key ingredient which gave rise to a prolific study of critical problems in $\\mathbb{H}^n$, see e.g. \\cite{Citti, GaLa, Ugu1, CitUg, MaUg, FelliUgu, MaMaPi, PaPiTe}.\n\nAs long as one needs explicit knowledge of proper replacements of the Aubin-Talenti functions, the only other sub-Riemannian structure where they are known is that of groups of Iwasawa type, see \\cite{GaroVa2, GaroVa}. As far as we know, there are no other structures, nor Sobolev inequalities with $p\\neq 2$, for which the minimizers are explicitly known. \nOn the other hand, since the best constant in the Sobolev inequality is achieved in all Carnot groups (see \\cite{GaroVa2}), it has been proved to be enough to know  the asymptotic behaviour at infinity of the minimizers. This is now known for $p\\neq 2$ as well, see \\cite{Loiudice3}, and it paved the way for a series of \nexistence, multiplicity or non-existence of positive solutions for critical problems  à la Br\\'{e}zis-Nirenberg in $\\mathbb{G}$: we refer e.g. to \\cite{BoUg, Loiudice1, Loiudice2, Loiudice4, BiGaVe}.\n\n\\medskip\n\nLet us now briefly describe the proof of Theorem \\ref{thm:main}: \n\\begin{itemize}\n\t\\item in Theorem \\ref{thm:Existence_First} we prove the existence of a first solution by means of a variational Perron method which transfers the approach of Struwe \\cite{Struwe} to the Carnot group setting. In particular, setting\n\t\\begin{equation*}\n\t\\Lambda := \\sup \\{ \\lambda >0: \\eqref{EqProblem}_\\lambda \\textrm{ admits a weak solution}\\},\n\\end{equation*}\n\t\\noindent we show first that $0<\\Lambda<+\\infty$, and this immediately provides a threshold for the non-existence of weak solutions. Once this is done, we use the unique solution of the purely sublinear problem \\eqref{eq:Sublinear_Problem} as a weak subsolution and we construct a weak supersolution for fixed $\\lambda$ by using the weak solution for a bigger $\\lambda'$; \n\t\\item we show that for $\\lambda \\in (0,\\Lambda)$ the first solution obtained as described before is a local minimizer in the natural topology associated with problem \\eqref{EqProblem}, see Lemma \\ref{lem:locmin}. We stress here that in \\cite{ABC} the authors made use of a famous result by Brezis and Nirenberg \\cite{BNH1C1} which does not have an analog in the Carnot group setting. This is due to the fact that $C^{1,\\alpha}$ regularity up to the boundary is still a delicate issue at the so called characteristic points: the first obstructions have been observed by Jerison \\cite{Jerison,Jerison2}, but this is still an active field of research, see e.g. \\cite{BaCiCu,BaGaMu,AbTr}. For this reason we follow here a more variational approach based on a paper by Alama \\cite{Alama}, already used in a different setting in \\cite{AbDiVa};\n\t\\item we prove the existence of a second solution following an argument originally due to Tarantello \\cite{Tarantello}: this combines the Ekeland variational principle \\cite{Ekeland} with the fine asymptotic expansions proved in \\cite{Loiudice1}.\n\\end{itemize} \n\n\\medskip\n\nWe stress that the multiplicity result obtained in Theorem \\ref{thm:main} can be easily extended to cover the convex-case of a Sobolev sub-critical nonlinearity.\n\n\\medskip\n\nThe paper is organized as follows: in Section \\ref{sec:Prel} we recall the basic facts on Carnot groups and we set the variational functional setting necessary for the study of \\eqref{EqProblem} We also recall the basic result regarding the purely sublinear problems, like existence and uniqueness of a positive solution and a comparison principle resembling the classical one. In Section \\ref{sec:First_Solution} we prove the existence of a first solution as described before, while the existence of a second solution (for $\\lambda \\in (0,\\Lambda)$) is postponed to the final Section \\ref{sec:Second_Solution}.", "sketch": "Let us now briefly describe the proof of Theorem \\ref{thm:main}:\n\\begin{itemize}\n\\item In Theorem \\ref{thm:Existence_First} the existence of a first solution is proved “by means of a variational Perron method which transfers the approach of Struwe \\cite{Struwe} to the Carnot group setting.” Defining\n\\[\n\\Lambda := \\sup\\{\\lambda>0:\\ \\eqref{EqProblem}_\\lambda\\ \\textrm{admits a weak solution}\\},\n\\]\none “show[s] first that $0<\\Lambda<+\\infty$, and this immediately provides a threshold for the non-existence of weak solutions.” Then one uses “the unique solution of the purely sublinear problem \\eqref{eq:Sublinear_Problem} as a weak subsolution” and constructs “a weak supersolution for fixed $\\lambda$ by using the weak solution for a bigger $\\lambda'$.”\n\\item For $\\lambda\\in(0,\\Lambda)$, one shows the first solution “is a local minimizer in the natural topology associated with problem \\eqref{EqProblem}, see Lemma \\ref{lem:locmin}.” Since the Brezis–Nirenberg boundary regularity tool used in \\cite{ABC} “does not have an analog in the Carnot group setting” (due to delicate boundary regularity at characteristic points), the paper “follow[s] here a more variational approach based on a paper by Alama \\cite{Alama}.”\n\\item The second solution is obtained by “an argument originally due to Tarantello \\cite{Tarantello},” combining “the Ekeland variational principle \\cite{Ekeland} with the fine asymptotic expansions proved in \\cite{Loiudice1}.”\n\\end{itemize}", "expanded_sketch": "Let us now briefly describe the proof of Theorem \\ref{thm:main}:\n\\begin{itemize}\n\\item We first prove the following theorem.\n\\begin{teo}\\label{thm:Existence_First}\n\tProblem \\eqref{EqProblem}$_\\lambda$ admits at least\n\tone weak solution $u_\\lambda\\in S^{1}_{0}(\\Omega)$ for every $\\lambda \\in (0,\\Lambda]$.\n\\end{teo}\nIn this step, the existence of a first solution is proved “by means of a variational Perron method which transfers the approach of Struwe \\cite{Struwe} to the Carnot group setting.” Defining\n\\[\n\\Lambda := \\sup\\{\\lambda>0:\\ \\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n\t\\left\\{\\begin{array}{rll}\n\t\t-\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\tu&=0 & \\textrm{ on } \\partial \\Omega.\n\t\\end{array}\\right.\n\\end{equation}_\\lambda\\ \\textrm{admits a weak solution}\\},\n\\]\none “show[s] first that $0<\\Lambda<+\\infty$, and this immediately provides a threshold for the non-existence of weak solutions.” Then one uses “the unique solution of the purely sublinear problem\n\\begin{equation}\\label{eq:Sublinear_Problem}\n\t\\left\\{\\begin{array}{rll}\n\t\t\t\t-\\Delta_{\\mathbb{G}}u &= \\lambda\\, u^{q} & \\textrm{ in } \\Omega,\\\\\n\t\t\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\t\t\tu&=0 & \\textrm{ on } \\partial \\Omega,\n\t\\end{array}\\right.\n\\end{equation}\nas a weak subsolution” and constructs “a weak supersolution for fixed $\\lambda$ by using the weak solution for a bigger $\\lambda'$.”\n\\item For $\\lambda\\in(0,\\Lambda)$, one shows the first solution “is a local minimizer in the natural topology associated with the equation above.” We now record the corresponding statement.\n\\begin{lemma}\\label{lem:locmin}\n\tFor $\\lambda\\in (0,\\Lambda)$, if $u_\\lambda$ is the solution\n\tpresented in Theorem \\ref{thm:Existence_First}, then\n\t$u_\\lambda$ is a local minimum for $I_\\lambda$ in the $S^1_0(\\Omega)$-topology,\n\tmeaning that there exists $r_0>0$ such that for any $u\\in S^1_0(\\Omega)$,\n\t\\[\n\tI_\\lambda(u_\\lambda)\\leq I_\\lambda(u) \\quad \\textrm{ for all } u\\in S^{1}_{0}(\\Omega) \\textrm{ with } \\|u-u_\\lambda\\|_{S_{0}^{1}(\\Omega)}< r_0 .\n\t\\]\n\\end{lemma}\nSince the Brezis–Nirenberg boundary regularity tool used in \\cite{ABC} “does not have an analog in the Carnot group setting” (due to delicate boundary regularity at characteristic points), the paper “follow[s] here a more variational approach based on a paper by Alama \\cite{Alama}.”\n\\item The second solution is obtained by “an argument originally due to Tarantello \\cite{Tarantello},” combining “the Ekeland variational principle \\cite{Ekeland} with the fine asymptotic expansions proved in \\cite{Loiudice1}.”\n\\end{itemize}", "expanded_theorem": "\\label{thm:main}\n\tLet $\\Omega\\subset\\mathbb{G}$ be an open and bounded set\n\twith smooth enough boundary $\\partial \\Omega$, and let $p\\in (0,1)$. Then, there exists $\\Lambda > 0$\n\tsuch that\n\t\\begin{itemize}\n\t\t\\item[A)]problem \\begin{equation}\\tag{{$\\mathrm{P}$}}\\label{EqProblem}\n\t\\left\\{\\begin{array}{rll}\n\t\t-\\Delta_{\\mathbb{G}}u& = \\lambda\\, u^{q} + u^{2^{\\star}_{Q}-1} & \\textrm{ in } \\Omega,\\\\\n\t\tu&>0 & \\textrm{ in } \\Omega,\\\\\n\t\tu&=0 & \\textrm{ on } \\partial \\Omega.\n\t\\end{array}\\right.\n\\end{equation}$_{\\lambda}$ does not admit weak solutions\n\t\tfor every $\\lambda>\\Lambda$;\n\t\t\\item[B)] problem the equation above$_{\\lambda}$ admits at least one weak solution for every $\\lambda \\in (0,\\Lambda]$;\n\t\t\\item[C)] problem the equation above$_{\\lambda}$ admits at least two weak solutions for every $0<\\lambda<\\Lambda$.\n\t\\end{itemize}", "theorem_type": ["Existential–Universal", "Existence"], "mcq": {"question": "Let \\(\\mathbb G\\) be a Carnot group, let \\(\\Omega\\subset \\mathbb G\\) be an open bounded set with sufficiently smooth boundary \\(\\partial\\Omega\\), let \\(q\\in(0,1)\\), and let \\(2_Q^*:=\\frac{2Q}{Q-2}\\) be the critical Sobolev exponent associated with \\(\\mathbb G\\). For \\(\\lambda>0\\), consider the Dirichlet problem\n\\[\n\\begin{cases}\n-\\Delta_{\\mathbb G}u=\\lambda u^q+u^{2_Q^*-1} & \\text{in }\\Omega,\\\\\nu>0 & \\text{in }\\Omega,\\\\\nu=0 & \\text{on }\\partial\\Omega.\n\\end{cases}\n\\]\nHere \\(-\\Delta_{\\mathbb G}\\) is the sub-Laplacian on \\(\\mathbb G\\), and a weak solution means a function \\(u\\in S_0^1(\\Omega)\\) with \\(u>0\\) in \\(\\Omega\\) such that\n\\[\n\\int_\\Omega \\langle \\nabla_{\\mathbb G}u,\\nabla_{\\mathbb G}\\varphi\\rangle\\,=\\int_\\Omega \\bigl(\\lambda u^q+u^{2_Q^*-1}\\bigr)\\varphi\n\\quad\\text{for every }\\varphi\\in S_0^1(\\Omega).\n\\]\nWhich existence statement holds for this problem?", "correct_choice": {"label": "A", "text": "There exists \\(\\Lambda>0\\) such that: (i) for every \\(\\lambda>\\Lambda\\), the problem admits no weak solution; (ii) for every \\(\\lambda\\in(0,\\Lambda]\\), the problem admits at least one weak solution; and (iii) for every \\(0<\\lambda<\\Lambda\\), the problem admits at least two weak solutions."}, "choices": [{"label": "B", "text": "There exists \\(\\Lambda>0\\) such that: (i) for every \\(\\lambda\\ge \\Lambda\\), the problem admits no weak solution; (ii) for every \\(\\lambda\\in(0,\\Lambda)\\), the problem admits at least one weak solution; and (iii) for every \\(0<\\lambda<\\Lambda\\), the problem admits at least two weak solutions."}, {"label": "C", "text": "There exists \\(\\Lambda>0\\) such that: (i) for every \\(\\lambda>\\Lambda\\), the problem admits no weak solution; and (ii) for every \\(\\lambda\\in(0,\\Lambda]\\), the problem admits at least one weak solution."}, {"label": "D", "text": "There exists \\(\\Lambda>0\\) such that: (i) for every \\(\\lambda>\\Lambda\\), the problem admits no weak solution; (ii) for every \\(\\lambda\\in(0,\\Lambda]\\), the problem admits a unique weak solution; and (iii) for every \\(0<\\lambda<\\Lambda\\), the problem admits at least two weak solutions."}, {"label": "E", "text": "There exists \\(\\Lambda>0\\) such that: (i) for every \\(\\lambda>\\Lambda\\), the problem admits no weak solution; (ii) for every \\(\\lambda\\in(0,\\Lambda]\\), the problem admits at least one weak solution; and (iii) for every \\(0<\\lambda\\le \\Lambda\\), the problem admits at least two weak solutions."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "endpoint solvability at \\(\\lambda=\\Lambda\\)", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "multiplicity conclusion for \\(0<\\lambda<\\Lambda\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "construction of a second solution below \\(\\Lambda\\) does not imply uniqueness of the first/existing solutions", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "strict interior range for the second-solution result excludes the endpoint \\(\\lambda=\\Lambda\\)", "template_used": "boundary_range"}]}}
{"id": "2512.04933v1", "paper_link": "http://arxiv.org/abs/2512.04933v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\nLet $\\beta\\in \\{1,2,4\\}$. If $p\\ge \\frac{3}{2}$, then, as $n\\to\\infty$, \n\\begin{align}\\label{eq:main-1}\n\\ln \\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)\n=&-\\frac{\\beta}{2}\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n^2\\ln n\\notag\\\\\n&+\\frac{\\beta}{2}\\Big(\\frac{1}{2}\\ln\\frac{4\\pi}{\\beta}+\\frac{3}{4}+\\ln A(p)\\Big)n^2\\notag\\\\\n&-\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n\\ln n\\notag\\\\\n&+\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac{1}{2}\\ln\\frac{4}{\\beta\\pi}+\\frac{1}{2}+\\frac{1}{2p}+\\ln A(p)+\\mathrm{Ent}(\\mu_p)\\Big)n +o(n).\n\\end{align}\nMoreover, if $\\beta=2$ and $p\\ge 1$, then, for some constant $M_p\\in \\mathbb{R}$,\n\\begin{equation} \\label{eq:main-2}\n\\ln \\mathrm{vol}(\\IB_{p,2}^n)\n=-\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n^2\\ln n\n+\\Big(\\frac{1}{2}\\ln 2\\pi +\\frac{3}{4}+\\ln A(p)\\Big)n^2\n-\\ln n+M_p+o(1).\n\\end{equation}", "start_pos": 19899, "end_pos": 20703, "label": "thm:main"}, "ref_dict": {"eq:main-2": "\\begin{equation} \\label{eq:main-2}\n\\ln \\vol(\\IB_{p,2}^n)\n=-\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n^2\\ln n\n+\\Big(\\frac{1}{2}\\ln 2\\pi +\\frac{3}{4}+\\ln A(p)\\Big)n^2\n-\\ln n+M_p+o(1).\n\\end{equation}", "eq:delta": "\\begin{equation} \\label{eq:delta}\nA(p)\n=\\frac{1}{2}\\bigg(\\frac{p\\sqrt{\\pi}\\Gamma(\\frac{p}{2})}{\\sqrt{e}\\Gamma(\\frac{p+1}{2})}\\bigg)^{1/p}\n\\end{equation}", "thm:LS": "\\begin{theorem} \\label{thm:LS}\nLet $\\beta>0$ and $p\\ge \\frac{3}{2}$. As $n\\to\\infty$,\n\\[\n\\ln Z_{n,p,\\beta}\n=-\\frac{\\beta}{2}n^2 I_p(\\mu_p)+\\frac{\\beta}{2}n\\ln n- C(\\beta)n+\\Big(1-\\frac{\\beta}{2}\\Big)\\Ent(\\mu_p)n +o(n),\n\\]\nwhere \n\\begin{equation} \\label{eq:c-beta}\nC(\\beta)\n=\\Big(1-\\frac{\\beta}{2}\\Big)\\ln \\frac{\\beta}{2}-\\frac{\\beta}{2}\\ln 2\\pi+\\frac{\\beta}{2}+\\ln \\Gamma\\Big(\\frac{\\beta}{2}\\Big).\n\\end{equation}\n\\end{theorem}", "eq:Z-first": "\\begin{equation} \\label{eq:Z-first}\n\\lim_{n\\to\\infty}\\frac{2}{\\beta n^2}\\ln Z_{n,p,\\beta}\n= I_p(\\mu_p)\n=\\ln 2+\\frac{3}{2p}.\n\\end{equation}", "eq:main-1": "\\begin{align}\\label{eq:main-1}\n\\ln \\vol(\\bpb)\n=&-\\frac{\\beta}{2}\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n^2\\ln n\\notag\\\\\n&+\\frac{\\beta}{2}\\Big(\\frac{1}{2}\\ln\\frac{4\\pi}{\\beta}+\\frac{3}{4}+\\ln A(p)\\Big)n^2\\notag\\\\\n&-\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n\\ln n\\notag\\\\\n&+\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac{1}{2}\\ln\\frac{4}{\\beta\\pi}+\\frac{1}{2}+\\frac{1}{2p}+\\ln A(p)+\\Ent(\\mu_p)\\Big)n +o(n).\n\\end{align}", "eq:sim-sym": "\\begin{equation} \\label{eq:sim-sym}\n\t\\lim_{n\\to \\infty}n^{1/2+1/p}\\vol(\\bpb)^{1/d_n}\n\t=\\sqrt{\\frac{4\\pi}{\\beta}}e^{3/4}A(p).\n\\end{equation}", "eq:sim-asym": "\\begin{equation} \\label{eq:sim-asym}\n\\lim_{n\\to \\infty}n^{1/2+1/p}\\vol(B_{p,\\beta}^n)^{1/\\beta n^2}\n=\\sqrt{\\frac{2\\pi}{\\beta}}e^{3/4}A(p)\n\\end{equation}", "lem:vol-partition": "\\begin{lemma} \\label{lem:vol-partition}\nLet $\\beta\\in\\{1,2,4\\}$ and $1\\le p<\\infty$. Then\n\\begin{align*}\n\\vol(\\bpb)\n&=c_n\\int_{\\IB_p^n} \\prod_{1\\le i< j\\le n}|x_i-x_j|^{\\beta}\\dd x_1 \\cdots \\dd x_n\\\\\n&=\\frac{c_{n}}{\\Gamma(1+\\frac{d_n}{p})}\\Big(\\frac{n\\beta v_p}{2}\\Big)^{d_n/p}Z_{n,p,\\beta}\n\\end{align*}\nwhere $Z_{n,p,\\beta}$ and $v_p$ are as in \\eqref{eq:Z} and\n\\[\nc_{n}\n=\\frac{1}{n!}\\bigg(\\frac{\\Gamma(\\frac{\\beta}{2})}{(2\\pi)^{\\beta/2}}\\bigg)^{n}\\prod_{k=1}^n\\frac{(2\\pi)^{\\beta k/2}}{\\Gamma(\\frac{\\beta k}{2})}.\n\\]\n\\end{lemma}", "thm:main": "\\begin{theorem} \\label{thm:main}\nLet $\\beta\\in \\{1,2,4\\}$. If $p\\ge \\frac{3}{2}$, then, as $n\\to\\infty$, \n\\begin{align}\\label{eq:main-1}\n\\ln \\vol(\\bpb)\n=&-\\frac{\\beta}{2}\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n^2\\ln n\\notag\\\\\n&+\\frac{\\beta}{2}\\Big(\\frac{1}{2}\\ln\\frac{4\\pi}{\\beta}+\\frac{3}{4}+\\ln A(p)\\Big)n^2\\notag\\\\\n&-\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n\\ln n\\notag\\\\\n&+\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac{1}{2}\\ln\\frac{4}{\\beta\\pi}+\\frac{1}{2}+\\frac{1}{2p}+\\ln A(p)+\\Ent(\\mu_p)\\Big)n +o(n).\n\\end{align}\nMoreover, if $\\beta=2$ and $p\\ge 1$, then, for some constant $M_p\\in \\IR$,\n\\begin{equation} \\label{eq:main-2}\n\\ln \\vol(\\IB_{p,2}^n)\n=-\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n^2\\ln n\n+\\Big(\\frac{1}{2}\\ln 2\\pi +\\frac{3}{4}+\\ln A(p)\\Big)n^2\n-\\ln n+M_p+o(1).\n\\end{equation}\n\\end{theorem}", "lem:cn": "\\begin{lemma} \\label{lem:cn}\nLet $\\beta\\in \\{1,2,4\\}$. Then, as $n\\to\\infty$,\n\\begin{align*}\n\\ln c_{n}\n=&-\\frac{\\beta}{4}n^2\\ln n + \\frac{\\beta}{2}\\Big(\\frac{1}{2}\\ln \\frac{4\\pi}{\\beta}+\\frac{3}{4}\\Big)n^2-\\frac{1}{2}\\Big(1+\\frac{\\beta}{2}\\Big)n\\ln n\\\\\n&+\\Big(\\frac{1}{2}\\Big(1-\\frac{\\beta}{2}\\Big)\\ln \\beta -\\frac{1}{2}\\Big(1+\\frac{\\beta}{2}\\Big)(\\ln \\pi-1)-\\ln 2+\\ln \\Gamma\\Big(\\frac{\\beta}{2}\\Big)\\Big)n\\\\\n&-\\frac{3+\\frac{\\beta}{2}+\\frac{2}{\\beta}}{12}\\ln n\n+a_{\\beta}+o(1),\n\\end{align*}\nwhere $a_{\\beta}\\in \\IR$ is some constant.\n\\end{lemma}", "thm:CKM": "\\begin{theorem} \\label{thm:CKM}\nLet $p\\ge 1$. There exists $M_p'\\in \\IR$ such that, as $n\\to\\infty$,\n\\[\n\\ln Z_{n,p,2}\n=-n^2 I_p(\\mu_p)+n\\ln n+(\\ln 2\\pi -1) n +\\frac{5}{12}\\ln n +M_p'+o(1).\n\\]\n\\end{theorem}"}, "pre_theorem_intro_text_len": 6252, "pre_theorem_intro_text": "The Schatten or Schatten-von Neumann class consists of all compact operators between two Hilbert spaces whose sequence of singular values belongs to the Lebesgue sequence space $\\ell_p$. It can be seen a non-commutative version of $\\ell_p$ and is named after Schatten and von Neumann \\cite{Sch46,Sch50,SVN46}. We refer to \\cite{Pie07,PX03} for an overview of their role in functional analysis and Banach space geometry.\n\nIn this article, we study the geometry of finite-dimensional Schatten classes and in particular their unit balls. These have been investigated as test cases for conjectures in asymptotic geometric analysis \\cite{DFG+23,GMP25,GP07,KMP98,RV20}, in the context of low-rank matrix recovery and information-based complexity \\cite{CR12,CK15,HPV17,HPV21,PS22} or in quantum information theory \\cite{ASW11,Wil13}. Let $\\mathbb{F}_1=\\mathbb{R}$, $\\mathbb{F}_2=\\mathbb{C}$ and $\\mathbb{F}_4=\\mathbb{H}$ and denote the singular values of a matrix $A\\in \\mathbb{F}_{\\beta}^{n\\times n}$, where $\\beta\\in\\{1,2,4\\}$, by \n\\[\ns_1(A)\\ge \\cdots \\ge s_n(A)\\ge 0.\n\\] \nFor $1\\le p\\le \\infty$, the $p$-Schatten norm of $A$ is \n\\[\n\\|A\\|_p\n=\n\\begin{cases}\n\t\\Big(\\sum_{j=1}^{n}s_j(A)^p\\Big)^{1/p}&\\colon p<\\infty\\\\\n\ts_1(A) &\\colon p=\\infty.\n\\end{cases}\n\\]\n\nThe associated unit ball and its intersection with the subspace of self-adjoint matrices $\\cH_n(\\IF_{\\beta})=\\{A\\in \\IF_{\\beta}^{n\\times n}\\colon A^{*}=A\\}$  are denoted by\n\\[\nB_{p,\\beta}^n\n=\\{A\\in \\IF_{\\beta}^{n\\times n}\\colon \\|A\\|_p\\le 1\\}\\quad \\text{and}\\quad\n\\mathbb{B}_{p,\\beta}^n\n= B_{p,\\beta}^n \\cap \\cH_n(\\IF_{\\beta}).\n\\]\nFor self-adjoint matrices the Schatten $p$-norm $\\|A\\|_p$ becomes a $p$-norm of eigenvalues. In case of non-self adjoint matrices one can also study the rectangular case, see e.g. \\cite{JKP24}.\n\nWe shall identify $\\IF_{\\beta}^{n\\times n}$ with $\\mathbb{R}^{\\beta n^2}$ and $\\cH_n(\\IF_{\\beta})$ with $\\mathbb{R}^{d_n}$, where \n\\begin{align}\\label{eq:dn}\nd_n:=\\beta\\frac{n(n-1)}{2}+n.\n\\end{align}\nThen $B_{p,\\beta}^n$ and $\\mathbb{B}_{p,\\beta}^n$ are convex bodies (i.e. compact convex sets with nonempty interior) in $\\mathbb{R}^{\\beta n^2}$ and $\\mathbb{R}^{d_n}$, respectively. In the following, we study their volumes $\\mathrm{vol}(B_{p,\\beta}^n)$ and $\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)$. Our aim is to give more precise asymptotics of the volume of $\\mathbb{B}_{p,\\beta}^n$ as $n\\to \\infty$.\n\nThe exact volume of $B_{p,\\beta}^n$ and $\\mathbb{B}_{p,\\beta}^n$ is only known in the cases $p=2$ and $p=\\infty$. For $p=2$, the Schatten $2$-balls are Euclidean unit balls of corresponding dimension and thus\n\\begin{equation*} \n\\mathrm{vol}(B_{2,\\beta}^n)\n=\\frac{\\pi^{\\beta n^2/2}}{\\Gamma(1+\\frac{\\beta n^2}{2})}\n\\qquad\\text{and}\\qquad\n\\mathrm{vol}(\\IB_{2,\\beta}^n)\n=\\frac{\\pi^{d_n/2}}{\\Gamma(1+\\frac{d_n}{2})},\n\\end{equation*} \nwhere $\\Gamma(x)=\\int_0^{\\infty}t^{x-1}e^{-t}\\mathrm{d} t$ denotes the Gamma function. For $p=\\infty$ it follows from Saint Raymond's work \\cite{SR84} and Selberg's integral formula \\cite{And91,Sel44} that\n\\begin{equation} \\label{eq:inf-exact}\n\\mathrm{vol}(B_{\\infty,\\beta}^n)\n=\\frac{\\prod_{j=0}^{n-1}\\Gamma( 1+j\\frac{\\beta}{2} )}{\\prod_{j=n}^{2n-1}\\Gamma( 1+j\\frac{\\beta}{2})}\\pi^{\\beta n^2/2},\n\\end{equation}\nand in the self-adjoint case it holds that\n\\begin{equation} \\label{eq:inf-sa-exact}\n\\mathrm{vol}(\\IB_{\\infty,\\beta}^n)\n=2^{d_n}(2\\pi)^{\\beta n(n-1)/4}\\prod_{j=0}^{n-1}\\frac{\\Gamma( 1+j\\frac{\\beta}{2})^2\\Gamma( (j+1)\\frac{\\beta}{2} )}{\\Gamma( 2+(n+j-1)\\frac{\\beta}{2} )\\Gamma( j\\frac{\\beta}{2} )}.\n\\end{equation}\nFor general $1\\le p\\le \\infty$, exact volumes are unknown and instead their asymptotic behavior has been studied. Saint Raymond \\cite{SR84} derived $A(p)>0$ such that, for $\\beta\\in\\{1,2\\}$, \n\\begin{equation} \\label{eq:sim-asym}\n\\lim_{n\\to \\infty}n^{1/2+1/p}\\mathrm{vol}(B_{p,\\beta}^n)^{1/\\beta n^2}\n=\\sqrt{\\frac{2\\pi}{\\beta}}e^{3/4}A(p)\n\\end{equation}\nand determined $A(2)=e^{-1/4}$ and $A(\\infty)=\\frac{1}{2}$. Gu\\'{e}don and Paouris \\cite{GP07} showed that $\\mathrm{vol}(B_{p,4}^n)^{1/4 n^2}$ and $\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)^{1/d_n}, \\beta\\in\\{1,2,4\\}$ are of the same order $n^{-1/2-1/p}$. Kabluchko, Prochno and Thäle \\cite{KPT20a} determined\n\\begin{equation} \\label{eq:delta}\nA(p)\n=\\frac{1}{2}\\bigg(\\frac{p\\sqrt{\\pi}\\Gamma(\\frac{p}{2})}{\\sqrt{e}\\Gamma(\\frac{p+1}{2})}\\bigg)^{1/p}\n\\end{equation}\nand moreover showed in \\cite{KPT20b} that, for $\\beta\\in \\{1,2,4\\}$,\n\\begin{equation} \\label{eq:sim-sym}\n\t\\lim_{n\\to \\infty}n^{1/2+1/p}\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)^{1/d_n}\n\t=\\sqrt{\\frac{4\\pi}{\\beta}}e^{3/4}A(p).\n\\end{equation}\nWe remark that one way to see that $A(p)$ is decreasing in $p$ is to observe that the same is true for the normalized volume $\\mathrm{vol}(n^{1/p}\\mathbb{B}_{p,\\beta}^n)$. Recently, Dadoun, Fradelizi, Gu\\'{e}don and Zitt \\cite[Remark 4.8]{DFG+23} showed \\eqref{eq:sim-asym} for $\\beta=4$. Thus, the asymptotics in \\eqref{eq:sim-asym} and \\eqref{eq:sim-sym} hold in all cases $\\beta\\in \\{1,2,4\\}$ and $1\\le p\\le \\infty$. \n\nIn case of $p=\\infty$, the quantity $A(\\infty)=\\frac{1}{2}$ can be expressed in terms of the minimal logarithmic energy of probability measures supported in $[-1,1]$ attained by the arcsine distribution, see \\cite{ST97}. We refer to \\cite{BDF+24} and \\cite{Bra24} for related asymptotics for discrete minimal logarithmic energy. \n\nIn case of $p<\\infty$, the quantity $A(p)$ in \\eqref{eq:delta} can be expressed as \n\\[\nA(p)\n=\\frac{1}{2}\\Big(\\frac{p\\sqrt{\\pi}\\Gamma(\\frac{p}{2})}{\\sqrt{e}\\Gamma(\\frac{p+1}{2})}\\Big)^{1/p}\n=\\frac{1}{2}\\Big(\\frac{1}{\\sqrt{e}\\alpha_p}\\Big)^{1/p},\n\\]\nwhere $\\alpha_p=\\int_{\\mathbb{R}} |x|^p\\mathrm{d}\\mu_p(x)$ is the $p$-th absolute moment of the Ullman distribution $\\mu_p$ which has density\n\\begin{equation} \\label{eq:ullman-1}\nf_p(x)\n= \\frac{p}{\\pi}\\int_{|x|}^1 \\frac{t^{p-1}}{\\sqrt{t^2-x^2}}\\mathrm{d} t, \\qquad x\\in [-1,1].\n\\end{equation}\n\nOur main contribution is the following asymptotic expansion for the logarithmic volume $\\ln \\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)$. Recall that the differential entropy of a probability measure $\\mu$ on $\\mathbb{R}$ is\n\\[\n\\mathrm{Ent}(\\mu)\n= -\\int_{\\mathbb{R}}\\ln(f(x))f(x)\\mathrm{d} x, \n\\]\nwhenever $\\mu$ admits a Lebesgue density $f$ and $\\mathrm{Ent}(\\mu)=-\\infty$ otherwise.", "context": "In this article, we study the geometry of finite-dimensional Schatten classes and in particular their unit balls. These have been investigated as test cases for conjectures in asymptotic geometric analysis \\cite{DFG+23,GMP25,GP07,KMP98,RV20}, in the context of low-rank matrix recovery and information-based complexity \\cite{CR12,CK15,HPV17,HPV21,PS22} or in quantum information theory \\cite{ASW11,Wil13}. Let $\\mathbb{F}_1=\\mathbb{R}$, $\\mathbb{F}_2=\\mathbb{C}$ and $\\mathbb{F}_4=\\mathbb{H}$ and denote the singular values of a matrix $A\\in \\mathbb{F}_{\\beta}^{n\\times n}$, where $\\beta\\in\\{1,2,4\\}$, by \n\\[\ns_1(A)\\ge \\cdots \\ge s_n(A)\\ge 0.\n\\] \nFor $1\\le p\\le \\infty$, the $p$-Schatten norm of $A$ is \n\\[\n\\|A\\|_p\n=\n\\begin{cases}\n \\Big(\\sum_{j=1}^{n}s_j(A)^p\\Big)^{1/p}&\\colon p<\\infty\\\\\n s_1(A) &\\colon p=\\infty.\n\\end{cases}\n\\]\n\nThe associated unit ball and its intersection with the subspace of self-adjoint matrices $\\cH_n(\\IF_{\\beta})=\\{A\\in \\IF_{\\beta}^{n\\times n}\\colon A^{*}=A\\}$ are denoted by\n\\[\nB_{p,\\beta}^n\n=\\{A\\in \\IF_{\\beta}^{n\\times n}\\colon \\|A\\|_p\\le 1\\}\\quad \\text{and}\\quad\n\\mathbb{B}_{p,\\beta}^n\n= B_{p,\\beta}^n \\cap \\cH_n(\\IF_{\\beta}).\n\\]\nFor self-adjoint matrices the Schatten $p$-norm $\\|A\\|_p$ becomes a $p$-norm of eigenvalues. In case of non-self adjoint matrices one can also study the rectangular case, see e.g. \\cite{JKP24}.\n\nWe shall identify $\\IF_{\\beta}^{n\\times n}$ with $\\mathbb{R}^{\\beta n^2}$ and $\\cH_n(\\IF_{\\beta})$ with $\\mathbb{R}^{d_n}$, where \n\\begin{align}\\label{eq:dn}\nd_n:=\\beta\\frac{n(n-1)}{2}+n.\n\\end{align}\nThen $B_{p,\\beta}^n$ and $\\mathbb{B}_{p,\\beta}^n$ are convex bodies (i.e. compact convex sets with nonempty interior) in $\\mathbb{R}^{\\beta n^2}$ and $\\mathbb{R}^{d_n}$, respectively. In the following, we study their volumes $\\mathrm{vol}(B_{p,\\beta}^n)$ and $\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)$. Our aim is to give more precise asymptotics of the volume of $\\mathbb{B}_{p,\\beta}^n$ as $n\\to \\infty$.\n\nThe exact volume of $B_{p,\\beta}^n$ and $\\mathbb{B}_{p,\\beta}^n$ is only known in the cases $p=2$ and $p=\\infty$. For $p=2$, the Schatten $2$-balls are Euclidean unit balls of corresponding dimension and thus\n\\begin{equation*} \n\\mathrm{vol}(B_{2,\\beta}^n)\n=\\frac{\\pi^{\\beta n^2/2}}{\\Gamma(1+\\frac{\\beta n^2}{2})}\n\\qquad\\text{and}\\qquad\n\\mathrm{vol}(\\IB_{2,\\beta}^n)\n=\\frac{\\pi^{d_n/2}}{\\Gamma(1+\\frac{d_n}{2})},\n\\end{equation*} \nwhere $\\Gamma(x)=\\int_0^{\\infty}t^{x-1}e^{-t}\\mathrm{d} t$ denotes the Gamma function. For $p=\\infty$ it follows from Saint Raymond's work \\cite{SR84} and Selberg's integral formula \\cite{And91,Sel44} that\n\\begin{equation} \\label{eq:inf-exact}\n\\mathrm{vol}(B_{\\infty,\\beta}^n)\n=\\frac{\\prod_{j=0}^{n-1}\\Gamma( 1+j\\frac{\\beta}{2} )}{\\prod_{j=n}^{2n-1}\\Gamma( 1+j\\frac{\\beta}{2})}\\pi^{\\beta n^2/2},\n\\end{equation}\nand in the self-adjoint case it holds that\n\\begin{equation} \\label{eq:inf-sa-exact}\n\\mathrm{vol}(\\IB_{\\infty,\\beta}^n)\n=2^{d_n}(2\\pi)^{\\beta n(n-1)/4}\\prod_{j=0}^{n-1}\\frac{\\Gamma( 1+j\\frac{\\beta}{2})^2\\Gamma( (j+1)\\frac{\\beta}{2} )}{\\Gamma( 2+(n+j-1)\\frac{\\beta}{2} )\\Gamma( j\\frac{\\beta}{2} )}.\n\\end{equation}\nFor general $1\\le p\\le \\infty$, exact volumes are unknown and instead their asymptotic behavior has been studied. Saint Raymond \\cite{SR84} derived $A(p)>0$ such that, for $\\beta\\in\\{1,2\\}$, \n\\begin{equation} \\label{eq:sim-asym}\n\\lim_{n\\to \\infty}n^{1/2+1/p}\\mathrm{vol}(B_{p,\\beta}^n)^{1/\\beta n^2}\n=\\sqrt{\\frac{2\\pi}{\\beta}}e^{3/4}A(p)\n\\end{equation}\nand determined $A(2)=e^{-1/4}$ and $A(\\infty)=\\frac{1}{2}$. Gu\\'{e}don and Paouris \\cite{GP07} showed that $\\mathrm{vol}(B_{p,4}^n)^{1/4 n^2}$ and $\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)^{1/d_n}, \\beta\\in\\{1,2,4\\}$ are of the same order $n^{-1/2-1/p}$. Kabluchko, Prochno and Thäle \\cite{KPT20a} determined\n\\begin{equation} \\label{eq:delta}\nA(p)\n=\\frac{1}{2}\\bigg(\\frac{p\\sqrt{\\pi}\\Gamma(\\frac{p}{2})}{\\sqrt{e}\\Gamma(\\frac{p+1}{2})}\\bigg)^{1/p}\n\\end{equation}\nand moreover showed in \\cite{KPT20b} that, for $\\beta\\in \\{1,2,4\\}$,\n\\begin{equation} \\label{eq:sim-sym}\n \\lim_{n\\to \\infty}n^{1/2+1/p}\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)^{1/d_n}\n =\\sqrt{\\frac{4\\pi}{\\beta}}e^{3/4}A(p).\n\\end{equation}\nWe remark that one way to see that $A(p)$ is decreasing in $p$ is to observe that the same is true for the normalized volume $\\mathrm{vol}(n^{1/p}\\mathbb{B}_{p,\\beta}^n)$. Recently, Dadoun, Fradelizi, Gu\\'{e}don and Zitt \\cite[Remark 4.8]{DFG+23} showed \\eqref{eq:sim-asym} for $\\beta=4$. Thus, the asymptotics in \\eqref{eq:sim-asym} and \\eqref{eq:sim-sym} hold in all cases $\\beta\\in \\{1,2,4\\}$ and $1\\le p\\le \\infty$.\n\nIn case of $p<\\infty$, the quantity $A(p)$ in \\eqref{eq:delta} can be expressed as \n\\[\nA(p)\n=\\frac{1}{2}\\Big(\\frac{p\\sqrt{\\pi}\\Gamma(\\frac{p}{2})}{\\sqrt{e}\\Gamma(\\frac{p+1}{2})}\\Big)^{1/p}\n=\\frac{1}{2}\\Big(\\frac{1}{\\sqrt{e}\\alpha_p}\\Big)^{1/p},\n\\]\nwhere $\\alpha_p=\\int_{\\mathbb{R}} |x|^p\\mathrm{d}\\mu_p(x)$ is the $p$-th absolute moment of the Ullman distribution $\\mu_p$ which has density\n\\begin{equation} \\label{eq:ullman-1}\nf_p(x)\n= \\frac{p}{\\pi}\\int_{|x|}^1 \\frac{t^{p-1}}{\\sqrt{t^2-x^2}}\\mathrm{d} t, \\qquad x\\in [-1,1].\n\\end{equation}\n\nOur main contribution is the following asymptotic expansion for the logarithmic volume $\\ln \\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)$. Recall that the differential entropy of a probability measure $\\mu$ on $\\mathbb{R}$ is\n\\[\n\\mathrm{Ent}(\\mu)\n= -\\int_{\\mathbb{R}}\\ln(f(x))f(x)\\mathrm{d} x, \n\\]\nwhenever $\\mu$ admits a Lebesgue density $f$ and $\\mathrm{Ent}(\\mu)=-\\infty$ otherwise.\n\n\\begin{equation} \\label{eq:sim-asym}\n\\lim_{n\\to \\infty}n^{1/2+1/p}\\vol(B_{p,\\beta}^n)^{1/\\beta n^2}\n=\\sqrt{\\frac{2\\pi}{\\beta}}e^{3/4}A(p)\n\\end{equation}\n\n\\begin{equation} \\label{eq:sim-sym}\n\t\\lim_{n\\to \\infty}n^{1/2+1/p}\\vol(\\bpb)^{1/d_n}\n\t=\\sqrt{\\frac{4\\pi}{\\beta}}e^{3/4}A(p).\n\\end{equation}", "full_context": "In this article, we study the geometry of finite-dimensional Schatten classes and in particular their unit balls. These have been investigated as test cases for conjectures in asymptotic geometric analysis \\cite{DFG+23,GMP25,GP07,KMP98,RV20}, in the context of low-rank matrix recovery and information-based complexity \\cite{CR12,CK15,HPV17,HPV21,PS22} or in quantum information theory \\cite{ASW11,Wil13}. Let $\\mathbb{F}_1=\\mathbb{R}$, $\\mathbb{F}_2=\\mathbb{C}$ and $\\mathbb{F}_4=\\mathbb{H}$ and denote the singular values of a matrix $A\\in \\mathbb{F}_{\\beta}^{n\\times n}$, where $\\beta\\in\\{1,2,4\\}$, by \n\\[\ns_1(A)\\ge \\cdots \\ge s_n(A)\\ge 0.\n\\] \nFor $1\\le p\\le \\infty$, the $p$-Schatten norm of $A$ is \n\\[\n\\|A\\|_p\n=\n\\begin{cases}\n \\Big(\\sum_{j=1}^{n}s_j(A)^p\\Big)^{1/p}&\\colon p<\\infty\\\\\n s_1(A) &\\colon p=\\infty.\n\\end{cases}\n\\]\n\nThe associated unit ball and its intersection with the subspace of self-adjoint matrices $\\cH_n(\\IF_{\\beta})=\\{A\\in \\IF_{\\beta}^{n\\times n}\\colon A^{*}=A\\}$ are denoted by\n\\[\nB_{p,\\beta}^n\n=\\{A\\in \\IF_{\\beta}^{n\\times n}\\colon \\|A\\|_p\\le 1\\}\\quad \\text{and}\\quad\n\\mathbb{B}_{p,\\beta}^n\n= B_{p,\\beta}^n \\cap \\cH_n(\\IF_{\\beta}).\n\\]\nFor self-adjoint matrices the Schatten $p$-norm $\\|A\\|_p$ becomes a $p$-norm of eigenvalues. In case of non-self adjoint matrices one can also study the rectangular case, see e.g. \\cite{JKP24}.\n\nWe shall identify $\\IF_{\\beta}^{n\\times n}$ with $\\mathbb{R}^{\\beta n^2}$ and $\\cH_n(\\IF_{\\beta})$ with $\\mathbb{R}^{d_n}$, where \n\\begin{align}\\label{eq:dn}\nd_n:=\\beta\\frac{n(n-1)}{2}+n.\n\\end{align}\nThen $B_{p,\\beta}^n$ and $\\mathbb{B}_{p,\\beta}^n$ are convex bodies (i.e. compact convex sets with nonempty interior) in $\\mathbb{R}^{\\beta n^2}$ and $\\mathbb{R}^{d_n}$, respectively. In the following, we study their volumes $\\mathrm{vol}(B_{p,\\beta}^n)$ and $\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)$. Our aim is to give more precise asymptotics of the volume of $\\mathbb{B}_{p,\\beta}^n$ as $n\\to \\infty$.\n\nThe exact volume of $B_{p,\\beta}^n$ and $\\mathbb{B}_{p,\\beta}^n$ is only known in the cases $p=2$ and $p=\\infty$. For $p=2$, the Schatten $2$-balls are Euclidean unit balls of corresponding dimension and thus\n\\begin{equation*} \n\\mathrm{vol}(B_{2,\\beta}^n)\n=\\frac{\\pi^{\\beta n^2/2}}{\\Gamma(1+\\frac{\\beta n^2}{2})}\n\\qquad\\text{and}\\qquad\n\\mathrm{vol}(\\IB_{2,\\beta}^n)\n=\\frac{\\pi^{d_n/2}}{\\Gamma(1+\\frac{d_n}{2})},\n\\end{equation*} \nwhere $\\Gamma(x)=\\int_0^{\\infty}t^{x-1}e^{-t}\\mathrm{d} t$ denotes the Gamma function. For $p=\\infty$ it follows from Saint Raymond's work \\cite{SR84} and Selberg's integral formula \\cite{And91,Sel44} that\n\\begin{equation} \\label{eq:inf-exact}\n\\mathrm{vol}(B_{\\infty,\\beta}^n)\n=\\frac{\\prod_{j=0}^{n-1}\\Gamma( 1+j\\frac{\\beta}{2} )}{\\prod_{j=n}^{2n-1}\\Gamma( 1+j\\frac{\\beta}{2})}\\pi^{\\beta n^2/2},\n\\end{equation}\nand in the self-adjoint case it holds that\n\\begin{equation} \\label{eq:inf-sa-exact}\n\\mathrm{vol}(\\IB_{\\infty,\\beta}^n)\n=2^{d_n}(2\\pi)^{\\beta n(n-1)/4}\\prod_{j=0}^{n-1}\\frac{\\Gamma( 1+j\\frac{\\beta}{2})^2\\Gamma( (j+1)\\frac{\\beta}{2} )}{\\Gamma( 2+(n+j-1)\\frac{\\beta}{2} )\\Gamma( j\\frac{\\beta}{2} )}.\n\\end{equation}\nFor general $1\\le p\\le \\infty$, exact volumes are unknown and instead their asymptotic behavior has been studied. Saint Raymond \\cite{SR84} derived $A(p)>0$ such that, for $\\beta\\in\\{1,2\\}$, \n\\begin{equation} \\label{eq:sim-asym}\n\\lim_{n\\to \\infty}n^{1/2+1/p}\\mathrm{vol}(B_{p,\\beta}^n)^{1/\\beta n^2}\n=\\sqrt{\\frac{2\\pi}{\\beta}}e^{3/4}A(p)\n\\end{equation}\nand determined $A(2)=e^{-1/4}$ and $A(\\infty)=\\frac{1}{2}$. Gu\\'{e}don and Paouris \\cite{GP07} showed that $\\mathrm{vol}(B_{p,4}^n)^{1/4 n^2}$ and $\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)^{1/d_n}, \\beta\\in\\{1,2,4\\}$ are of the same order $n^{-1/2-1/p}$. Kabluchko, Prochno and Thäle \\cite{KPT20a} determined\n\\begin{equation} \\label{eq:delta}\nA(p)\n=\\frac{1}{2}\\bigg(\\frac{p\\sqrt{\\pi}\\Gamma(\\frac{p}{2})}{\\sqrt{e}\\Gamma(\\frac{p+1}{2})}\\bigg)^{1/p}\n\\end{equation}\nand moreover showed in \\cite{KPT20b} that, for $\\beta\\in \\{1,2,4\\}$,\n\\begin{equation} \\label{eq:sim-sym}\n \\lim_{n\\to \\infty}n^{1/2+1/p}\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)^{1/d_n}\n =\\sqrt{\\frac{4\\pi}{\\beta}}e^{3/4}A(p).\n\\end{equation}\nWe remark that one way to see that $A(p)$ is decreasing in $p$ is to observe that the same is true for the normalized volume $\\mathrm{vol}(n^{1/p}\\mathbb{B}_{p,\\beta}^n)$. Recently, Dadoun, Fradelizi, Gu\\'{e}don and Zitt \\cite[Remark 4.8]{DFG+23} showed \\eqref{eq:sim-asym} for $\\beta=4$. Thus, the asymptotics in \\eqref{eq:sim-asym} and \\eqref{eq:sim-sym} hold in all cases $\\beta\\in \\{1,2,4\\}$ and $1\\le p\\le \\infty$.\n\nIn case of $p<\\infty$, the quantity $A(p)$ in \\eqref{eq:delta} can be expressed as \n\\[\nA(p)\n=\\frac{1}{2}\\Big(\\frac{p\\sqrt{\\pi}\\Gamma(\\frac{p}{2})}{\\sqrt{e}\\Gamma(\\frac{p+1}{2})}\\Big)^{1/p}\n=\\frac{1}{2}\\Big(\\frac{1}{\\sqrt{e}\\alpha_p}\\Big)^{1/p},\n\\]\nwhere $\\alpha_p=\\int_{\\mathbb{R}} |x|^p\\mathrm{d}\\mu_p(x)$ is the $p$-th absolute moment of the Ullman distribution $\\mu_p$ which has density\n\\begin{equation} \\label{eq:ullman-1}\nf_p(x)\n= \\frac{p}{\\pi}\\int_{|x|}^1 \\frac{t^{p-1}}{\\sqrt{t^2-x^2}}\\mathrm{d} t, \\qquad x\\in [-1,1].\n\\end{equation}\n\nOur main contribution is the following asymptotic expansion for the logarithmic volume $\\ln \\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)$. Recall that the differential entropy of a probability measure $\\mu$ on $\\mathbb{R}$ is\n\\[\n\\mathrm{Ent}(\\mu)\n= -\\int_{\\mathbb{R}}\\ln(f(x))f(x)\\mathrm{d} x, \n\\]\nwhenever $\\mu$ admits a Lebesgue density $f$ and $\\mathrm{Ent}(\\mu)=-\\infty$ otherwise.\n\n\\begin{equation} \\label{eq:sim-asym}\n\\lim_{n\\to \\infty}n^{1/2+1/p}\\vol(B_{p,\\beta}^n)^{1/\\beta n^2}\n=\\sqrt{\\frac{2\\pi}{\\beta}}e^{3/4}A(p)\n\\end{equation}\n\n\\begin{equation} \\label{eq:sim-sym}\n\t\\lim_{n\\to \\infty}n^{1/2+1/p}\\vol(\\bpb)^{1/d_n}\n\t=\\sqrt{\\frac{4\\pi}{\\beta}}e^{3/4}A(p).\n\\end{equation}\n\nOur main contribution is the following asymptotic expansion for the logarithmic volume $\\ln \\vol(\\bpb)$. Recall that the differential entropy of a probability measure $\\mu$ on $\\IR$ is\n\\[\n\\Ent(\\mu)\n= -\\int_{\\IR}\\ln(f(x))f(x)\\dd x, \n\\]\nwhenever $\\mu$ admits a Lebesgue density $f$ and $\\Ent(\\mu)=-\\infty$ otherwise.\n\nIn case of $p=2$ it is known that $\\Ent(\\mu_2)=\\ln \\pi -\\frac{1}{2}$ where $\\mu_2$ is the semi-circle law. Otherwise the entropy $\\Ent(\\mu_p)$ of the Ullman distribution appears to be unknown. Independently, the expansion in \\eqref{eq:main-1} was recently obtained by Dworaczek Guera, Memin and Pain \\cite{GMP25} in case of $p\\ge 2$.\n\nIn this context the Ullman distribution arises as the corresponding equilibrium measure minimizing the functional\n\\[\n\\mu\\mapsto I_p(\\mu)\n=-\\int_{\\IR}\\int_{\\IR} \\ln|x-y|\\dd\\mu(x)\\dd\\mu(y)+v_p\\int_{\\IR} |x|^p\\dd\\mu(x)\n\\]\nover $\\mu\\in \\cP(\\IR)$, where $\\cP(\\IR)$ denotes the space of probability measures on $\\IR$. More precisely, if $X=(X_1,\\dots,X_n)$ is distributed according to the density $f_{n,p,\\beta}$, then the random empirical measure $ \\mu_{n,p} =\\frac{1}{n}\\sum_{i=1}^{n}\\delta_{X_i} $ tends to $\\mu_p$ as $n\\to\\infty$. Moreover, large deviation principles due to \\cite{BG97} and \\cite{HP00} yield \n\\begin{equation} \\label{eq:Z-first}\n\\lim_{n\\to\\infty}\\frac{2}{\\beta n^2}\\ln Z_{n,p,\\beta}\n= I_p(\\mu_p)\n=\\ln 2+\\frac{3}{2p}.\n\\end{equation}\nEarlier work \\cite{DPS95,Joh98} also identified the limiting constant but with different techniques. In \\cite{DFG+23}, the limit \\eqref{eq:Z-first} is used to prove \\eqref{eq:sim-sym}. The large deviation principle for tagged empirical fields due to Lebl\\'e and Serfaty \\cite{LS17} allows to refine \\eqref{eq:Z-first}. The following result forms the core of the proof of \\eqref{eq:main-1} in Theorem~\\ref{thm:main}.\n\n\\begin{theorem} \\label{thm:LS}\nLet $\\beta>0$ and $p\\ge \\frac{3}{2}$. As $n\\to\\infty$,\n\\[\n\\ln Z_{n,p,\\beta}\n=-\\frac{\\beta}{2}n^2 I_p(\\mu_p)+\\frac{\\beta}{2}n\\ln n- C(\\beta)n+\\Big(1-\\frac{\\beta}{2}\\Big)\\Ent(\\mu_p)n +o(n),\n\\]\nwhere \n\\begin{equation} \\label{eq:c-beta}\nC(\\beta)\n=\\Big(1-\\frac{\\beta}{2}\\Big)\\ln \\frac{\\beta}{2}-\\frac{\\beta}{2}\\ln 2\\pi+\\frac{\\beta}{2}+\\ln \\Gamma\\Big(\\frac{\\beta}{2}\\Big).\n\\end{equation}\n\\end{theorem}\n\nIt remains to check Hölder continuity at zero. Let $h\\in (0,1)$ and write\n\\begin{align*}\n\\frac{\\pi}{p}\\big(f_p(h)-f_p(0)\\big)\n&=\\int_h^1 \\frac{t^{p-1}}{\\sqrt{t^2-h^2}}\\dd t - \\int_0^1 t^{p-2}\\dd t\\\\\n&=\\int_h^1 t^{p-1}\\Big(\\frac{1}{\\sqrt{t^2-h^2}}-\\frac{1}{t}\\Big)\\dd t -\\frac{h^{p-1}}{p-1}.\n\\end{align*}\nThe latter integral can be estimated by\n\\begin{align*}\n\\int_h^1 t^{p-1}\\Big(\\frac{1}{\\sqrt{t^2-h^2}}-\\frac{1}{t}\\Big)\\dd t\n&\\le h^2\\int_h^1 \\frac{t^{p-3}}{\\sqrt{t^2-h^2}}\\dd t\\\\\n&= h^{p-1}\\int_1^{1/h} \\frac{u^{p-3}}{\\sqrt{u^2-1}}\\dd u.\n\\end{align*}\nIn particular, if $1< p<2$, then\n\\begin{equation} \\label{eq:hoelder}\n0\\le \\frac{\\pi}{p}\\frac{f_p(h)-f_p(0)}{h^{p-1}}\n\\le \\int_1^{\\infty} \\frac{u^{p-3}}{\\sqrt{u^2-1}}\\dd u\n<\\infty.\n\\end{equation}\nThus, in this case we obtain that $f_p$ is Hölder continuous of order $p-1$ at $0$. For $p\\ge 2$, a similar argument shows that $f_p$ is continuously differentiable at $0$ with $f_p'(0)=0$. In conclusion, if $p\\ge\\frac{3}{2}$, the function $f_p$ is Hölder continuous of order $\\frac{1}{2}$ on $[-1,1]$. This completes the proof.\n\\end{proof}\n\n\\begin{lemma} \\label{lem:cn}\nLet $\\beta\\in \\{1,2,4\\}$. Then, as $n\\to\\infty$,\n\\begin{align*}\n\\ln c_{n}\n=&-\\frac{\\beta}{4}n^2\\ln n + \\frac{\\beta}{2}\\Big(\\frac{1}{2}\\ln \\frac{4\\pi}{\\beta}+\\frac{3}{4}\\Big)n^2-\\frac{1}{2}\\Big(1+\\frac{\\beta}{2}\\Big)n\\ln n\\\\\n&+\\Big(\\frac{1}{2}\\Big(1-\\frac{\\beta}{2}\\Big)\\ln \\beta -\\frac{1}{2}\\Big(1+\\frac{\\beta}{2}\\Big)(\\ln \\pi-1)-\\ln 2+\\ln \\Gamma\\Big(\\frac{\\beta}{2}\\Big)\\Big)n\\\\\n&-\\frac{3+\\frac{\\beta}{2}+\\frac{2}{\\beta}}{12}\\ln n\n+a_{\\beta}+o(1),\n\\end{align*}\nwhere $a_{\\beta}\\in \\IR$ is some constant.\n\\end{lemma}\n\\begin{proof}\nIt follows from the definition of $c_{n}$ that \n\\[\nc_{n}\n=\\frac{1}{n!}\\Gamma_2\\Big(n+1;\\frac{2}{\\beta},1\\Big)\\bigg(\\frac{\\Gamma(\\frac{\\beta}{2})}{(\\frac{\\beta}{2})^{\\frac{\\beta}{2}}}\\bigg)^n\\Big(\\frac{4\\pi}{\\beta}\\Big)^{\\frac{\\beta}{4}n^2-(\\frac{1}{2}+\\frac{\\beta}{4})n},\n\\]\nwhere $\\Gamma_2(x;\\frac{2}{\\beta},1)$ is a particular instance of the Barnes double Gamma function, see \\cite{BG24,Spr09}. Adapting \\cite[eq. (7.17)]{BG24}, the following asymptotic expansion holds\n\\begin{align*}\n\\ln \\Gamma_2\\Big(n+1;\\frac{2}{\\beta},1\\Big) \n=&-\\frac{\\beta}{4}n^2\\ln n+\\frac{3\\beta}{8}n^2+\\frac{1}{2}\\Big(1-\\frac{\\beta}{2}\\Big)n\\ln n-\\frac{1}{2}\\Big(1-\\frac{\\beta}{2}\\Big)n\\\\\n&+\\bigg(\\frac{1}{2} -\\frac{3+\\frac{\\beta}{2}+\\frac{2}{\\beta}}{12}\\bigg)\\ln n +a_{\\beta}'+o(1),\n\\end{align*}\nwhere $a_{\\beta}'\\in \\IR$ is some constant. Combined with Stirling's approximation $\\ln n!=n\\ln n-n+\\frac{1}{2}\\ln(2\\pi n)+o(1)$ we deduce the claimed expansion.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:main}]\nLet $\\beta\\in \\{1,2,4\\}$ and $p\\ge \\frac{3}{2}$. Recall that $d_n=\\frac{\\beta}{2}n^2+(1-\\frac{\\beta}{2})n$. By Lemma~\\ref{lem:vol-partition} it holds that\n\\begin{equation} \\label{eq:vol-asymp-parts}\n\\ln \\vol(\\bpb)\n=\\ln c_{n}+\\frac{d_n}{p}\\ln \\Big(\\frac{\\beta nv_p}{2}\\Big)-\\ln \\Gamma\\Big(1+\\frac{d_n}{p}\\Big)+\\ln Z_{n,p,\\beta}.\n\\end{equation}\nBy Stirling's approximation, as $n\\to\\infty$,\n\\begin{align}\\label{eq:p-term-1}\n\\ln \\Gamma\\Big(1+\\frac{d_n}{p}\\Big)\n&=\\frac{d_n}{p}\\Big(\\ln \\frac{d_n}{p}-1\\Big)+o(n)\\notag\\\\\n&=\\frac{d_n}{p}\\Big(2\\ln n+\\ln \\frac{\\beta}{2p}-1+\\Big(1-\\frac{\\beta}{2}\\Big)\\frac{2}{\\beta n}\\Big)+o(n).\n\\end{align}\nTherefore, \n\\begin{align}\\label{eq:p-term-2}\n\\frac{d_n}{p}\\ln\\Big(\\frac{\\beta n v_p}{2}\\Big)\n-\\ln \\Gamma\\Big(1+\\frac{d_n}{p}\\Big)\n=&-\\frac{d_n}{p}\\Big(\\ln n -\\ln(p v_p)-1+\\Big(1-\\frac{\\beta}{2}\\Big)\\frac{2}{\\beta n}\\Big)\\notag\\\\\n=&-\\frac{\\beta}{2p}n^2\\ln n+\\frac{\\beta}{2p}\\big(\\ln(pv_p)+1\\big)n^2\\notag\\\\\n&-\\frac{1}{p}\\Big(1-\\frac{\\beta}{2}\\Big)n\\ln n+\\frac{\\ln(pv_p)}{p}\\Big(1-\\frac{\\beta}{2}\\Big)n +o(n).\n\\end{align}", "post_theorem_intro_text_len": 7939, "post_theorem_intro_text": "In case of $p=2$ it is known that $\\mathrm{Ent}(\\mu_2)=\\ln \\pi -\\frac{1}{2}$ where $\\mu_2$ is the semi-circle law. Otherwise the entropy $\\mathrm{Ent}(\\mu_p)$ of the Ullman distribution appears to be unknown. Independently, the expansion in \\eqref{eq:main-1} was recently obtained by Dworaczek Guera, Memin and Pain \\cite{GMP25} in case of $p\\ge 2$. \n\nThe proof of Theorem~\\ref{thm:main} relies on the fact that the volume of $\\mathbb{B}_{p,\\beta}^n$ can be expressed in terms of the integral\n\\begin{equation} \\label{eq:Z}\nZ_{n,p,\\beta}\n=\\int_{\\mathbb{R}^n}\\prod_{1\\le i<j\\le n}|x_i-x_j|^{\\beta} e^{-\\frac{\\beta}{2}n\\, v_p\\sum_{i=1}^{n}|x_i|^p}\\mathrm{d} x,\n\\end{equation}\nwhere\n\\[\nv_p =\\frac{\\sqrt{\\pi}\\Gamma(\\frac{p}{2})}{\\Gamma(\\frac{p+1}{2})}=\\frac{1}{p\\alpha_p}.\n\\]\nThe quantity $Z_{n,p,\\beta}$ is the partition function of a one-dimensional $\\beta$-ensemble with potential $V(x)=v_p |x|^p$, where eigenvalues/particles are distributed with joint density \n\\begin{equation} \\label{eq:density}\nf_{n,p,\\beta}(x)\n=\\frac{1}{Z_{n,p,\\beta}}\\prod_{1\\le i<j\\le n}|x_i-x_j|^{\\beta} e^{-\\frac{\\beta}{2}n\\, v_p\\sum_{i=1}^{n}|x_i|^p}\\mathrm{d} x, \\qquad x\\in \\mathbb{R}^n.\n\\end{equation}\nThe parameter $\\beta$ stands for the inverse temperature and may take any positive value, while for the matrix representation we restrict to $\\beta\\in \\{1,2,4\\}$. We refer to \\cite{AGZ10} and \\cite{For10} for more information on $\\beta$-ensembles. \n\nIn this context the Ullman distribution arises as the corresponding equilibrium measure minimizing the functional\n\\[\n\\mu\\mapsto I_p(\\mu)\n=-\\int_{\\mathbb{R}}\\int_{\\mathbb{R}} \\ln|x-y|\\mathrm{d}\\mu(x)\\mathrm{d}\\mu(y)+v_p\\int_{\\mathbb{R}} |x|^p\\mathrm{d}\\mu(x)\n\\]\nover $\\mu\\in \\mathcal{P}(\\mathbb{R})$, where $\\mathcal{P}(\\mathbb{R})$ denotes the space of probability measures on $\\mathbb{R}$.  More precisely, if $X=(X_1,\\dots,X_n)$ is distributed according to the density $f_{n,p,\\beta}$, then the random empirical measure $ \\mu_{n,p} =\\frac{1}{n}\\sum_{i=1}^{n}\\delta_{X_i} $  tends to $\\mu_p$ as $n\\to\\infty$. Moreover, large deviation principles due to \\cite{BG97} and \\cite{HP00} yield \n\\begin{equation} \\label{eq:Z-first}\n\\lim_{n\\to\\infty}\\frac{2}{\\beta n^2}\\ln Z_{n,p,\\beta}\n= I_p(\\mu_p)\n=\\ln 2+\\frac{3}{2p}.\n\\end{equation}\nEarlier work \\cite{DPS95,Joh98} also identified the limiting constant but with different techniques. In \\cite{DFG+23}, the limit \\eqref{eq:Z-first} is used to prove \\eqref{eq:sim-sym}. The large deviation principle for tagged empirical fields due to Lebl\\'e and Serfaty \\cite{LS17} allows to refine \\eqref{eq:Z-first}. The following result forms the core of the proof of \\eqref{eq:main-1} in Theorem~\\ref{thm:main}. \n\n\\begin{theorem} \\label{thm:LS}\nLet $\\beta>0$ and $p\\ge \\frac{3}{2}$. As $n\\to\\infty$,\n\\[\n\\ln Z_{n,p,\\beta}\n=-\\frac{\\beta}{2}n^2 I_p(\\mu_p)+\\frac{\\beta}{2}n\\ln n- C(\\beta)n+\\Big(1-\\frac{\\beta}{2}\\Big)\\mathrm{Ent}(\\mu_p)n +o(n),\n\\]\nwhere \n\\begin{equation} \\label{eq:c-beta}\nC(\\beta)\n=\\Big(1-\\frac{\\beta}{2}\\Big)\\ln \\frac{\\beta}{2}-\\frac{\\beta}{2}\\ln 2\\pi+\\frac{\\beta}{2}+\\ln \\Gamma\\Big(\\frac{\\beta}{2}\\Big).\n\\end{equation}\n\\end{theorem}\n\nIndependently, in the recent work \\cite{GMP25} the authors deduced the statement of Theorem~\\ref{thm:LS} in case of $p\\ge 2$ from a central limit theorem for linear eigenvalue statistics. The restriction to $p\\ge 2$ results from regularity requirements of the underlying approach, see also \\cite{DFG+23}. The slightly milder restriction on $p$ in Theorem~\\ref{thm:LS} is due to the fact that the Ullman density $f_p$ is Hölder continuous of order $\\frac{1}{2}$ if and only if $p\\ge\\frac{3}{2}$. This is required to apply the results in \\cite{LS17}. \n\nThe term $\\mathrm{Ent}(\\mu_p)$ in Theorem~\\ref{thm:LS} arises via a rescaling property of the specific entropy which is part of the rate function in the large deviation principle in \\cite{LS17}. In case of $\\beta=2$, the contribution of $\\mathrm{Ent}(\\mu_p)$ vanishes and more precise asymptotics hold. The following is a consequence of \\cite[Prop.~1.1]{CKM23} which is based on \\cite{KM99}.\n\n\\begin{theorem} \\label{thm:CKM}\nLet $p\\ge 1$. There exists $M_p'\\in \\mathbb{R}$ such that, as $n\\to\\infty$,\n\\[\n\\ln Z_{n,p,2}\n=-n^2 I_p(\\mu_p)+n\\ln n+(\\ln 2\\pi -1) n +\\frac{5}{12}\\ln n +M_p'+o(1).\n\\]\n\\end{theorem}\n\nTheorem~\\ref{thm:CKM} combined with Lemma~\\ref{lem:vol-partition} and Lemma~\\ref{lem:cn} below yields \\eqref{eq:main-2} in Theorem~\\ref{thm:main}. In case of $p$ an even integer, the potential $V(x)=v_p |x|^p$ is an analytic function and results of \\cite{BG13} yield a full asymptotic expansion of $\\ln Z_{n,p,\\beta}$, and consequently of $\\ln \\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)$. In case of $p=2$ the integral $Z_{n,2,\\beta}$ is known via a Mehta-Selberg integral. For $p\\neq 2$ the coefficients in the full asymptotic expansion are defined in terms of integrals of the Stieltjes transform of $\\mu_p$, see also \\cite[Section 7]{BG24} and \\cite{Shc13,Shc14}.\n\nIn the case of non self-adjoint matrices, the volume of the unit ball $B_{p,\\beta}^n$ may be written in terms of \n\\[\n\\tilde{Z}_{n,p,\\beta}\n=\\int_{(\\IR_+)^n}\\prod_{1\\le i<j\\le n}|x_i-x_j|^{\\beta}\\prod_{i=1}^n x_i^{\\beta/2-1} e^{-\\beta n\\, v_p\\sum_{i=1}^{n}x_i^{p/2}}\\mathrm{d} x,\n\\]\nsee e.g.~\\cite[Sec.~4.2]{DFG+23}. This corresponds to the partition function of a Laguerre $\\beta$-ensemble with potential $V(x)=v_p|x|^{p/2}$. The leading order asymptotics for $\\tilde{Z}_{n,p,\\beta}$ can again be obtained via large deviations, leading to \\eqref{eq:sim-asym}. However, the results in \\cite{LS17} do not apply and it goes beyond the scope of the present work to derive next-order asymptotics for $\\tilde{Z}_{n,p,\\beta}$ and $B_{p,\\beta}^n$. \n\n\\begin{remark}\n\tThe unit balls $B_{p,\\beta}^n$ and $\\mathbb{B}_{p,\\beta}^n$ can also be studied for $0<p<1$. Then \\eqref{eq:sim-asym} and \\eqref{eq:sim-sym} as well as Theorem~\\ref{thm:CKM} continue to hold. However, even neglecting regularity assumptions, the proof of Theorem~\\ref{thm:main} and the underlying large deviation principle in \\cite{LS17} are limited to $p\\ge 1$. This is because, while the microscopic limit point process in the bulk is the universal sine$_2$-process for $p\\ge 1$, it becomes dependent on $p$ for $p<1$, see \\cite{BEY14,CKM23}. Since the minimizer of the rate function is the free energy which is uniquely given by the sine$_{\\beta}$-process, see \\cite{EHL21,HM25}, the large deviation principle in \\cite{LS17} cannot hold in its present form if $p<1$. \n\\end{remark}\n\n\\begin{remark}\n\tHeuristically, the unit balls $B_{p,\\beta}^n$ and $\\IB_{p,\\beta}^n$ are non-commutative versions of $\\IB_p^n=\\{x\\in \\mathbb{R}^n\\colon \\sum_{i=1}^{n}|x_i|^p\\le 1\\}$. To compare, note that\n\\[\n\\mathrm{vol}(n^{1/p}\\IB_{p}^n)\n=n^{n/p}\\frac{2^n\\Gamma(1+\\frac{1}{p})^n}{\\Gamma(1+\\frac{n}{p})}\n=\\frac{1+o(1)}{\\sqrt{2\\pi n/p}} e^{\\mathrm{Ent}(\\nu_p)n},\n\\]\nwhere $\\nu_p$ has density proportional to $e^{-|x|^p/p}$ and maximizes entropy over $\\mu\\in \\mathcal{P}(\\mathbb{R})$ with $\\int |x|^p\\mathrm{d}\\mu(x) \\le 1$, see also \\cite{KP21}. Similarly, the Ullman distribution $\\mu_p$ maximizes free entropy over $\\mu\\in \\mathcal{P}(\\mathbb{R})$ with $\\int |x|^p\\mathrm{d}\\mu(x)\\le \\alpha_p$, see \\cite[Prop.~5.34]{HP00}. In case of $p=\\infty$, one can interpret $\\nu_{\\infty}$ as the uniform distribution on $[-1,1]$, which maximizes entropy, and $\\mu_{\\infty}$ as the arcsine distribution, which maximizes free entropy. Formally, see Lemma~\\ref{lem:vol-partition} below, the case $\\beta=0$ corresponds to the commutative case where a Poisson point process appears in the microscopic limit, see also \\cite{AD14,BP15,Leb16}. \n\\end{remark}\n\n\\begin{remark}\nTheorem~\\ref{thm:main} can be combined with the stochastic representation in [Cor.~4.3]{KPT20b} of a random matrix drawn uniformly from a Schatten ball $\\mathbb{B}_{p,\\beta}^n$ to obtain information on critical intersections of Schatten $p$-balls in the spirit of \\cite{KPT20b} and \\cite{ST25}. \n\\end{remark}", "sketch": "The proof of Theorem~\\ref{thm:main} is based on rewriting $\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)$ in terms of the partition function\n\\[\nZ_{n,p,\\beta}=\\int_{\\mathbb{R}^n}\\prod_{1\\le i<j\\le n}|x_i-x_j|^{\\beta}\\, e^{-\\frac{\\beta}{2}n\\, v_p\\sum_{i=1}^{n}|x_i|^p}\\,\\mathrm{d} x,\n\\]\nwhich is the partition function of a one-dimensional $\\beta$-ensemble with potential $V(x)=v_p|x|^p$ and joint density $f_{n,p,\\beta}$.\n\nIn this setting, the Ullman distribution $\\mu_p$ appears as the equilibrium measure minimizing\n\\[\nI_p(\\mu)=-\\iint\\ln|x-y|\\,\\mathrm{d}\\mu(x)\\mathrm{d}\\mu(y)+v_p\\int |x|^p\\,\\mathrm{d}\\mu(x),\n\\]\nand the empirical measure $\\mu_{n,p}=\\frac1n\\sum_{i=1}^n\\delta_{X_i}$ (for $X\\sim f_{n,p,\\beta}$) converges to $\\mu_p$. Large deviations yield the leading-order limit\n\\[\n\\lim_{n\\to\\infty}\\frac{2}{\\beta n^2}\\ln Z_{n,p,\\beta}=I_p(\\mu_p)=\\ln 2+\\frac{3}{2p}.\n\\]\n\nTo obtain the next-order terms needed for \\eqref{eq:main-1}, the argument uses the large deviation principle for tagged empirical fields of Lebl\\'e--Serfaty \\cite{LS17} to refine this to the expansion stated in Theorem~\\ref{thm:LS}:\n\\[\n\\ln Z_{n,p,\\beta}=-\\frac{\\beta}{2}n^2 I_p(\\mu_p)+\\frac{\\beta}{2}n\\ln n- C(\\beta)n+\\Big(1-\\frac{\\beta}{2}\\Big)\\mathrm{Ent}(\\mu_p)n+o(n).\n\\]\nThe restriction $p\\ge\\tfrac32$ is tied to the requirement that the Ullman density is H\\\"older $1/2$, which is needed to apply \\cite{LS17}. The appearance of $\\mathrm{Ent}(\\mu_p)$ is attributed to a rescaling property of the specific entropy in the rate function of \\cite{LS17}.\n\nFor the $\\beta=2$ refinement \\eqref{eq:main-2}, the text explains that the $\\mathrm{Ent}(\\mu_p)$ contribution vanishes and one invokes Theorem~\\ref{thm:CKM} (a consequence of \\cite[Prop.~1.1]{CKM23}) giving sharper asymptotics for $\\ln Z_{n,p,2}$; then “Theorem~\\ref{thm:CKM} combined with Lemma~\\ref{lem:vol-partition} and Lemma~\\ref{lem:cn} below yields \\eqref{eq:main-2} in Theorem~\\ref{thm:main}.”", "expanded_sketch": "The proof of the main theorem is based on rewriting $\\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)$ in terms of the partition function\n\\[\nZ_{n,p,\\beta}=\\int_{\\mathbb{R}^n}\\prod_{1\\le i<j\\le n}|x_i-x_j|^{\\beta}\\, e^{-\\frac{\\beta}{2}n\\, v_p\\sum_{i=1}^{n}|x_i|^p}\\,\\mathrm{d} x,\n\\]\nwhich is the partition function of a one-dimensional $\\beta$-ensemble with potential $V(x)=v_p|x|^p$ and joint density $f_{n,p,\\beta}$.\n\nIn this setting, the Ullman distribution $\\mu_p$ appears as the equilibrium measure minimizing\n\\[\nI_p(\\mu)=-\\iint\\ln|x-y|\\,\\mathrm{d}\\mu(x)\\mathrm{d}\\mu(y)+v_p\\int |x|^p\\,\\mathrm{d}\\mu(x),\n\\]\nand the empirical measure $\\mu_{n,p}=\\frac1n\\sum_{i=1}^n\\delta_{X_i}$ (for $X\\sim f_{n,p,\\beta}$) converges to $\\mu_p$. Large deviations yield the leading-order limit\n\\[\n\\lim_{n\\to\\infty}\\frac{2}{\\beta n^2}\\ln Z_{n,p,\\beta}=I_p(\\mu_p)=\\ln 2+\\frac{3}{2p}.\n\\]\n\nTo obtain the next-order terms needed for\n\\begin{align}\\label{eq:main-1}\n\\ln \\vol(\\bpb)\n=&-\\frac{\\beta}{2}\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n^2\\ln n\\notag\\\\\n&+\\frac{\\beta}{2}\\Big(\\frac{1}{2}\\ln\\frac{4\\pi}{\\beta}+\\frac{3}{4}+\\ln A(p)\\Big)n^2\\notag\\\\\n&-\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n\\ln n\\notag\\\\\n&+\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac{1}{2}\\ln\\frac{4}{\\beta\\pi}+\\frac{1}{2}+\\frac{1}{2p}+\\ln A(p)+\\Ent(\\mu_p)\\Big)n +o(n).\n\\end{align}\nthe argument uses the large deviation principle for tagged empirical fields of Lebl\\'e--Serfaty (LS17) to refine this to the following result.\n\n\\begin{theorem} \\label{thm:LS}\nLet $\\beta>0$ and $p\\ge \\frac{3}{2}$. As $n\\to\\infty$,\n\\[\n\\ln Z_{n,p,\\beta}\n=-\\frac{\\beta}{2}n^2 I_p(\\mu_p)+\\frac{\\beta}{2}n\\ln n- C(\\beta)n+\\Big(1-\\frac{\\beta}{2}\\Big)\\Ent(\\mu_p)n +o(n),\n\\]\nwhere \n\\begin{equation} \\label{eq:c-beta}\nC(\\beta)\n=\\Big(1-\\frac{\\beta}{2}\\Big)\\ln \\frac{\\beta}{2}-\\frac{\\beta}{2}\\ln 2\\pi+\\frac{\\beta}{2}+\\ln \\Gamma\\Big(\\frac{\\beta}{2}\\Big).\n\\end{equation}\n\\end{theorem}\n\nThe restriction $p\\ge\\tfrac32$ is tied to the requirement that the Ullman density is H\\\"older $1/2$, which is needed to apply LS17. The appearance of $\\mathrm{Ent}(\\mu_p)$ is attributed to a rescaling property of the specific entropy in the rate function of LS17.\n\nFor the $\\beta=2$ refinement\n\\begin{equation} \\label{eq:main-2}\n\\ln \\vol(\\IB_{p,2}^n)\n=-\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n^2\\ln n\n+\\Big(\\frac{1}{2}\\ln 2\\pi +\\frac{3}{4}+\\ln A(p)\\Big)n^2\n-\\ln n+M_p+o(1).\n\\end{equation}\none notes that the $\\mathrm{Ent}(\\mu_p)$ contribution vanishes and one invokes the following sharper asymptotics.\n\n\\begin{theorem} \\label{thm:CKM}\nLet $p\\ge 1$. There exists $M_p'\\in \\IR$ such that, as $n\\to\\infty$,\n\\[\n\\ln Z_{n,p,2}\n=-n^2 I_p(\\mu_p)+n\\ln n+(\\ln 2\\pi -1) n +\\frac{5}{12}\\ln n +M_p'+o(1).\n\\]\n\\end{theorem}\n\nThen the theorem above, combined with the following two lemmas,\n\n\\begin{lemma} \\label{lem:vol-partition}\nLet $\\beta\\in\\{1,2,4\\}$ and $1\\le p<\\infty$. Then\n\\begin{align*}\n\\vol(\\bpb)\n&=c_n\\int_{\\IB_p^n} \\prod_{1\\le i< j\\le n}|x_i-x_j|^{\\beta}\\dd x_1 \\cdots \\dd x_n\\\\\n&=\\frac{c_{n}}{\\Gamma(1+\\frac{d_n}{p})}\\Big(\\frac{n\\beta v_p}{2}\\Big)^{d_n/p}Z_{n,p,\\beta}\n\\end{align*}\nwhere $Z_{n,p,\\beta}$ and $v_p$ are as in \\eqref{eq:Z} and\n\\[\nc_{n}\n=\\frac{1}{n!}\\bigg(\\frac{\\Gamma(\\frac{\\beta}{2})}{(2\\pi)^{\\beta/2}}\\bigg)^{n}\\prod_{k=1}^n\\frac{(2\\pi)^{\\beta k/2}}{\\Gamma(\\frac{\\beta k}{2})}.\n\\]\n\\end{lemma}\n\n\\begin{lemma} \\label{lem:cn}\nLet $\\beta\\in \\{1,2,4\\}$. Then, as $n\\to\\infty$,\n\\begin{align*}\n\\ln c_{n}\n=&-\\frac{\\beta}{4}n^2\\ln n + \\frac{\\beta}{2}\\Big(\\frac{1}{2}\\ln \\frac{4\\pi}{\\beta}+\\frac{3}{4}\\Big)n^2-\\frac{1}{2}\\Big(1+\\frac{\\beta}{2}\\Big)n\\ln n\\\\\n&+\\Big(\\frac{1}{2}\\Big(1-\\frac{\\beta}{2}\\Big)\\ln \\beta -\\frac{1}{2}\\Big(1+\\frac{\\beta}{2}\\Big)(\\ln \\pi-1)-\\ln 2+\\ln \\Gamma\\Big(\\frac{\\beta}{2}\\Big)\\Big)n\\\\\n&-\\frac{3+\\frac{\\beta}{2}+\\frac{2}{\\beta}}{12}\\ln n\n+a_{\\beta}+o(1),\n\\end{align*}\nwhere $a_{\\beta}\\in \\IR$ is some constant.\n\\end{lemma}\n\nyields the stated $\\beta=2$ volume asymptotics, completing that part of the main theorem.", "expanded_theorem": "\\label{thm:main}\nLet $\\beta\\in \\{1,2,4\\}$. If $p\\ge \\frac{3}{2}$, then, as $n\\to\\infty$, \n\\begin{align}\\label{eq:main-1}\n\\ln \\mathrm{vol}(\\mathbb{B}_{p,\\beta}^n)\n=&-\\frac{\\beta}{2}\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n^2\\ln n\\notag\\\\\n&+\\frac{\\beta}{2}\\Big(\\frac{1}{2}\\ln\\frac{4\\pi}{\\beta}+\\frac{3}{4}+\\ln A(p)\\Big)n^2\\notag\\\\\n&-\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n\\ln n\\notag\\\\\n&+\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac{1}{2}\\ln\\frac{4}{\\beta\\pi}+\\frac{1}{2}+\\frac{1}{2p}+\\ln A(p)+\\mathrm{Ent}(\\mu_p)\\Big)n +o(n).\n\\end{align}\nMoreover, if $\\beta=2$ and $p\\ge 1$, then, for some constant $M_p\\in \\mathbb{R}$,\n\\begin{equation} \\label{eq:main-2}\n\\ln \\mathrm{vol}(\\IB_{p,2}^n)\n=-\\Big(\\frac{1}{2}+\\frac{1}{p}\\Big)n^2\\ln n\n+\\Big(\\frac{1}{2}\\ln 2\\pi +\\frac{3}{4}+\\ln A(p)\\Big)n^2\n-\\ln n+M_p+o(1).\n\\end{equation},", "theorem_type": ["Asymptotic or Limit", "Implication"], "mcq": {"question": "Let \\(\\mathbb F_1=\\mathbb R\\), \\(\\mathbb F_2=\\mathbb C\\), and \\(\\mathbb F_4=\\mathbb H\\). For \\(\\beta\\in\\{1,2,4\\}\\) and \\(A\\in \\mathbb F_\\beta^{n\\times n}\\), let \\(s_1(A)\\ge\\cdots\\ge s_n(A)\\ge0\\) be the singular values and define the Schatten \\(p\\)-norm by \\(\\|A\\|_p=(\\sum_{j=1}^n s_j(A)^p)^{1/p}\\) for \\(1\\le p<\\infty\\) and \\(\\|A\\|_\\infty=s_1(A)\\). Let \\(\\mathcal H_n(\\mathbb F_\\beta)=\\{A\\in\\mathbb F_\\beta^{n\\times n}:A^*=A\\}\\), and let \\(\\mathbb B_{p,\\beta}^n=\\{A\\in\\mathcal H_n(\\mathbb F_\\beta):\\|A\\|_p\\le1\\}\\), whose volume is taken in the corresponding real vector space of self-adjoint matrices. Also let \\(A(p)=\\frac12\\Big(\\frac{p\\sqrt\\pi\\,\\Gamma(p/2)}{\\sqrt e\\,\\Gamma((p+1)/2)}\\Big)^{1/p}\\) for finite \\(p\\) (and \\(A(\\infty)=\\tfrac12\\)), and let \\(\\mathrm{Ent}(\\mu_p)\\) denote the differential entropy of the Ullman distribution \\(\\mu_p\\). Which asymptotic statement for \\(\\ln \\mathrm{vol}(\\mathbb B_{p,\\beta}^n)\\) as \\(n\\to\\infty\\) holds for \\(p\\ge \\tfrac32\\), and what sharper asymptotic statement holds in the special case \\(\\beta=2\\) and \\(p\\ge1\\)?", "correct_choice": {"label": "A", "text": "As \\(n\\to\\infty\\), for every \\(\\beta\\in\\{1,2,4\\}\\) and every \\(p\\ge \\tfrac32\\),\n\\[\n\\ln \\mathrm{vol}(\\mathbb B_{p,\\beta}^n)\n=-\\frac{\\beta}{2}\\Big(\\frac12+\\frac1p\\Big)n^2\\ln n\n+\\frac{\\beta}{2}\\Big(\\frac12\\ln\\frac{4\\pi}{\\beta}+\\frac34+\\ln A(p)\\Big)n^2\n-\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac12+\\frac1p\\Big)n\\ln n\n+\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac12\\ln\\frac{4}{\\beta\\pi}+\\frac12+\\frac{1}{2p}+\\ln A(p)+\\mathrm{Ent}(\\mu_p)\\Big)n\n+o(n).\n\\]\nMoreover, in the complex self-adjoint case \\((\\beta=2)\\), for every \\(p\\ge1\\) there exists a constant \\(M_p\\in\\mathbb R\\) such that\n\\[\n\\ln \\mathrm{vol}(\\mathbb B_{p,2}^n)\n=-\\Big(\\frac12+\\frac1p\\Big)n^2\\ln n\n+\\Big(\\frac12\\ln 2\\pi+\\frac34+\\ln A(p)\\Big)n^2\n-\\ln n+M_p+o(1).\n\\]"}, "choices": [{"label": "B", "text": "As \\(n\\to\\infty\\), for every \\(\\beta\\in\\{1,2,4\\}\\) and every \\(p\\ge 1\\),\n\\[\n\\ln \\mathrm{vol}(\\mathbb B_{p,\\beta}^n)\n=-\\frac{\\beta}{2}\\Big(\\frac12+\\frac1p\\Big)n^2\\ln n\n+\\frac{\\beta}{2}\\Big(\\frac12\\ln\\frac{4\\pi}{\\beta}+\\frac34+\\ln A(p)\\Big)n^2\n-\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac12+\\frac1p\\Big)n\\ln n\n+\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac12\\ln\\frac{4}{\\beta\\pi}+\\frac12+\\frac{1}{2p}+\\ln A(p)+\\mathrm{Ent}(\\mu_p)\\Big)n\n+o(n).\n\\]\nMoreover, in the complex self-adjoint case \\((\\beta=2)\\), for every \\(p\\ge1\\) there exists a constant \\(M_p\\in\\mathbb R\\) such that\n\\[\n\\ln \\mathrm{vol}(\\mathbb B_{p,2}^n)\n=-\\Big(\\frac12+\\frac1p\\Big)n^2\\ln n\n+\\Big(\\frac12\\ln 2\\pi+\\frac34+\\ln A(p)\\Big)n^2\n-\\ln n+M_p+o(1).\n\\]"}, {"label": "C", "text": "As \\(n\\to\\infty\\), for every \\(\\beta\\in\\{1,2,4\\}\\) and every \\(p\\ge \\tfrac32\\),\n\\[\n\\ln \\mathrm{vol}(\\mathbb B_{p,\\beta}^n)\n=-\\frac{\\beta}{2}\\Big(\\frac12+\\frac1p\\Big)n^2\\ln n\n+\\frac{\\beta}{2}\\Big(\\frac12\\ln\\frac{4\\pi}{\\beta}+\\frac34+\\ln A(p)\\Big)n^2\n+o(n^2).\n\\]\nMoreover, in the complex self-adjoint case \\((\\beta=2)\\), for every \\(p\\ge1\\) there exists a constant \\(M_p\\in\\mathbb R\\) such that\n\\[\n\\ln \\mathrm{vol}(\\mathbb B_{p,2}^n)\n=-\\Big(\\frac12+\\frac1p\\Big)n^2\\ln n\n+\\Big(\\frac12\\ln 2\\pi+\\frac34+\\ln A(p)\\Big)n^2\n+o(n^2).\n\\]"}, {"label": "D", "text": "As \\(n\\to\\infty\\), for every \\(\\beta\\in\\{1,2,4\\}\\) and every \\(p\\ge \\tfrac32\\),\n\\[\n\\ln \\mathrm{vol}(\\mathbb B_{p,\\beta}^n)\n=-\\frac{\\beta}{2}\\Big(\\frac12+\\frac1p\\Big)n^2\\ln n\n+\\frac{\\beta}{2}\\Big(\\frac12\\ln\\frac{4\\pi}{\\beta}+\\frac34+\\ln A(p)\\Big)n^2\n-\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac12+\\frac1p\\Big)n\\ln n\n+\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac12\\ln\\frac{4}{\\beta\\pi}+\\frac12+\\frac{1}{2p}+\\ln A(p)\\Big)n\n+o(n).\n\\]\nMoreover, in the complex self-adjoint case \\((\\beta=2)\\), for every \\(p\\ge1\\) there exists a constant \\(M_p\\in\\mathbb R\\) such that\n\\[\n\\ln \\mathrm{vol}(\\mathbb B_{p,2}^n)\n=-\\Big(\\frac12+\\frac1p\\Big)n^2\\ln n\n+\\Big(\\frac12\\ln 2\\pi+\\frac34+\\ln A(p)\\Big)n^2\n-\\ln n+M_p+o(1).\n\\]"}, {"label": "E", "text": "As \\(n\\to\\infty\\), for every \\(\\beta\\in\\{1,2,4\\}\\) and every \\(p\\ge \\tfrac32\\),\n\\[\n\\ln \\mathrm{vol}(\\mathbb B_{p,\\beta}^n)\n=-\\frac{\\beta}{2}\\Big(\\frac12+\\frac1p\\Big)n^2\\ln n\n+\\frac{\\beta}{2}\\Big(\\frac12\\ln\\frac{4\\pi}{\\beta}+\\frac34+\\ln A(p)\\Big)n^2\n-\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac12+\\frac1p\\Big)n\\ln n\n+\\Big(1-\\frac{\\beta}{2}\\Big)\\Big(\\frac12\\ln\\frac{4}{\\beta\\pi}+\\frac12+\\frac{1}{p}+\\ln A(p)+\\mathrm{Ent}(\\mu_p)\\Big)n\n+o(n).\n\\]\nMoreover, in the complex self-adjoint case \\((\\beta=2)\\), for every \\(p\\ge1\\) there exists a constant \\(M_p\\in\\mathbb R\\) such that\n\\[\n\\ln \\mathrm{vol}(\\mathbb B_{p,2}^n)\n=-\\Big(\\frac12+\\frac1p\\Big)n^2\\ln n\n+\\Big(\\frac12\\ln 2\\pi+\\frac34+\\ln A(p)\\Big)n^2\n-\\ln n+M_p+o(1).\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": "threshold_p_ge_3_2", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped_nlogn_and_linear_terms", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "entropy_linear_term_from_tagged_field_LDP", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "coefficient_of_linear_1_over_2p_term", "template_used": "wildcard"}]}}
{"id": "2512.05068v1", "paper_link": "http://arxiv.org/abs/2512.05068v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm_sepsys}\nWith high probability\\footnote{that is, with probability tending to $1$ as $n \\to \\infty$, abbreviated later as whp.}, the separating systole of $\\mathbf{T}_{2n,g_n}$ is of logarithmic order, i.e. there exists two constants $0<A<B$ (depending on $\\theta$) such that whp\n\\[A\\log(n)\\leq \\text{sepsys}(\\bT)\\leq B\\log(n)\\]", "start_pos": 52799, "end_pos": 53128, "label": "thm_sepsys"}, "ref_dict": {"eq_psi": "\\begin{equation}\\label{eq_psi}\n\\psi(\\theta)=\\frac{1}{36\\lambda(\\theta)^4}\\left(1-12\\lambda(\\theta)-24\\lambda(\\theta)^2H(\\lambda(\\theta))\\right).\n\\end{equation}", "thm_sepsys": "\\begin{theorem}\\label{thm_sepsys}\nWith high probability\\footnote{that is, with probability tending to $1$ as $\\nto$, abbreviated later as whp.}, the separating systole of $\\bT$ is of logarithmic order, i.e. there exists two constants $0<A<B$ (depending on $\\theta$) such that whp\n\\[A\\log(n)\\leq \\sep\\leq B\\log(n)\\] \n\\end{theorem}"}, "pre_theorem_intro_text_len": 2484, "pre_theorem_intro_text": "In this paper we study random maps on surfaces, in a regime in which both their size $n$ and their genus $g$ go to infinity. Recall that a \\emph{map} is a discrete surface made by gluing polygons along their edges, and a \\emph{triangulation} is a map built entirely from triangles.\n\nIn the planar case ($g=0$, $n \\rightarrow \\infty$), the geometry of random maps is now well understood. On a local scale, their structure is described by their local limit \\cite{AS03,Kri05, CD06, Bud15}, while on a global scale, results cover their diameter and scaling limit~\\cite{CS04,LG11, Mie11, Mar16}. These results rely heavily on enumerative formulas~\\cite{Tut62,Tut63} and explicit bijections~\\cite{CV81,Sch98these,BDG04}. Similar results have also been established for maps of fixed genus $g>0$ \\cite{Chapuy:profile, BM22}.\n\nWhen both $n$ and $g$ grow large, exact asymptotics are not available. However, for \\emph{unicellular maps} (maps with a single face), a bijection~\\cite{CFF13} provides a way to understand their geometry~\\cite{ACCR13,Ray13a, Louf-expander, JansonLouf1, JansonLouf2}.\n\nBeyond the unicellular case, a series of recent results study more general classes of maps, such as triangulations or maps with prescribed face degrees. Their local behavior has been uncovered~\\cite{BL19,BL20}, confirming a conjecture of Benjamini and Curien (see ~\\cite[Chapter 6]{B13book} and ~\\cite{CurPSHIT}). Global properties have also been investigated: both the \\emph{planarity radius}~\\cite{Louf} (a discrete analogue of the injectivity radius) and the \\emph{diameter}~\\cite{BCL} grow logarithmically with the size of the map.\n\nResults for high-genus maps often resemble those for another model of random surfaces: large-genus Weil--Petersson hyperbolic surfaces (including a strong conjectural link with unicellular maps~\\cite{JansonLouf2, MP19}). Thanks again to precise asymptotics, many geometric properties have been established in that model (see e.g. \\cite{Mir13,GPY11,NWX,MP19,PWX,AM,HMT}).\n\nIn this paper, we focus on a different global observable for random high-genus triangulations: the \\emph{separating systole}, which is the length of the shortest separating but non-contractible cycle. Once again, it is analogous to existing results for hyperbolic surfaces~\\cite{NWX,PWX}.\n\nIn all that follows, we fix $\\theta\\in(0,1/2)$ and a sequence $(g_n)$ such that $\\frac{g_n}{n}\\rightarrow \\theta$. Let $\\mathbf{T}_{2n,g_n}$ be a uniform triangulation of genus $g_n$ with $2n$ faces.", "context": "In this paper we study random maps on surfaces, in a regime in which both their size $n$ and their genus $g$ go to infinity. Recall that a \\emph{map} is a discrete surface made by gluing polygons along their edges, and a \\emph{triangulation} is a map built entirely from triangles.\n\nIn the planar case ($g=0$, $n \\rightarrow \\infty$), the geometry of random maps is now well understood. On a local scale, their structure is described by their local limit \\cite{AS03,Kri05, CD06, Bud15}, while on a global scale, results cover their diameter and scaling limit~\\cite{CS04,LG11, Mie11, Mar16}. These results rely heavily on enumerative formulas~\\cite{Tut62,Tut63} and explicit bijections~\\cite{CV81,Sch98these,BDG04}. Similar results have also been established for maps of fixed genus $g>0$ \\cite{Chapuy:profile, BM22}.\n\nBeyond the unicellular case, a series of recent results study more general classes of maps, such as triangulations or maps with prescribed face degrees. Their local behavior has been uncovered~\\cite{BL19,BL20}, confirming a conjecture of Benjamini and Curien (see ~\\cite[Chapter 6]{B13book} and ~\\cite{CurPSHIT}). Global properties have also been investigated: both the \\emph{planarity radius}~\\cite{Louf} (a discrete analogue of the injectivity radius) and the \\emph{diameter}~\\cite{BCL} grow logarithmically with the size of the map.\n\nResults for high-genus maps often resemble those for another model of random surfaces: large-genus Weil--Petersson hyperbolic surfaces (including a strong conjectural link with unicellular maps~\\cite{JansonLouf2, MP19}). Thanks again to precise asymptotics, many geometric properties have been established in that model (see e.g. \\cite{Mir13,GPY11,NWX,MP19,PWX,AM,HMT}).\n\nIn this paper, we focus on a different global observable for random high-genus triangulations: the \\emph{separating systole}, which is the length of the shortest separating but non-contractible cycle. Once again, it is analogous to existing results for hyperbolic surfaces~\\cite{NWX,PWX}.\n\nIn all that follows, we fix $\\theta\\in(0,1/2)$ and a sequence $(g_n)$ such that $\\frac{g_n}{n}\\rightarrow \\theta$. Let $\\mathbf{T}_{2n,g_n}$ be a uniform triangulation of genus $g_n$ with $2n$ faces.", "full_context": "In this paper we study random maps on surfaces, in a regime in which both their size $n$ and their genus $g$ go to infinity. Recall that a \\emph{map} is a discrete surface made by gluing polygons along their edges, and a \\emph{triangulation} is a map built entirely from triangles.\n\nIn the planar case ($g=0$, $n \\rightarrow \\infty$), the geometry of random maps is now well understood. On a local scale, their structure is described by their local limit \\cite{AS03,Kri05, CD06, Bud15}, while on a global scale, results cover their diameter and scaling limit~\\cite{CS04,LG11, Mie11, Mar16}. These results rely heavily on enumerative formulas~\\cite{Tut62,Tut63} and explicit bijections~\\cite{CV81,Sch98these,BDG04}. Similar results have also been established for maps of fixed genus $g>0$ \\cite{Chapuy:profile, BM22}.\n\nBeyond the unicellular case, a series of recent results study more general classes of maps, such as triangulations or maps with prescribed face degrees. Their local behavior has been uncovered~\\cite{BL19,BL20}, confirming a conjecture of Benjamini and Curien (see ~\\cite[Chapter 6]{B13book} and ~\\cite{CurPSHIT}). Global properties have also been investigated: both the \\emph{planarity radius}~\\cite{Louf} (a discrete analogue of the injectivity radius) and the \\emph{diameter}~\\cite{BCL} grow logarithmically with the size of the map.\n\nResults for high-genus maps often resemble those for another model of random surfaces: large-genus Weil--Petersson hyperbolic surfaces (including a strong conjectural link with unicellular maps~\\cite{JansonLouf2, MP19}). Thanks again to precise asymptotics, many geometric properties have been established in that model (see e.g. \\cite{Mir13,GPY11,NWX,MP19,PWX,AM,HMT}).\n\nIn this paper, we focus on a different global observable for random high-genus triangulations: the \\emph{separating systole}, which is the length of the shortest separating but non-contractible cycle. Once again, it is analogous to existing results for hyperbolic surfaces~\\cite{NWX,PWX}.\n\nIn all that follows, we fix $\\theta\\in(0,1/2)$ and a sequence $(g_n)$ such that $\\frac{g_n}{n}\\rightarrow \\theta$. Let $\\mathbf{T}_{2n,g_n}$ be a uniform triangulation of genus $g_n$ with $2n$ faces.\n\nIn this paper, we focus on a different global observable for random high-genus triangulations: the \\emph{separating systole}, which is the length of the shortest separating but non-contractible cycle. Once again, it is analogous to existing results for hyperbolic surfaces~\\cite{NWX,PWX}.\n\nNote that in this theorem, we do not require the shortest separating cycle to be simple. In fact, even the existence of a simple separating non-contractible cycle in triangulations is a longstanding conjecture \\cite{MT01}[Chapter 5]. However, we do conjecture that \\whp{}, it does exist in $\\bT$, and what's more that it is again of logarithmic length. We will discuss this, and other conjectures such as the convergence of $\\frac{\\sep}{\\log(n)}$ in probability, in more details in~\\cref{sec_conj}.\n\nIn the process of proving the lower bound, we can adapt some of the techniques to establish a new enumerative result on high genus triangulation that we call the convergence of the \\emph{genus ratio}. Let us give more context. Let $\\tau(n,g)$ be the number of triangulations of genus $g$ with $2n$ faces, we established in~\\cite{BL19}, together with Budzinski, the convergence of the \"\\emph{size ratio}\"\n\\begin{equation}\\label{eq_ratio_size}\n\\frac{\\tau(n-1,g_n)}{\\tau(n,g_n)}\\to \\lambda(\\theta),\n\\end{equation}\nas $\\nto$, where $\\lambda$ is an explicit function.\n\n\\begin{proposition}\\label{prop_asympto_ratio}\nFor all $\\theta \\in \\left( 0,\\frac{1}{2} \\right)$, there is a constant $a_{\\theta} \\in (0,1)$ such that the following holds. Let $(g_n)$ be a sequence such that $0 \\leq g_n \\leq \\frac{n+1}{2}$ for every $n$ and $\\frac{g_n}{n} \\to \\theta$. For all integers $m$ and $h$ satisfying $0\\leq m\\leq \\frac{n}{2}$ and $0\\leq h\\leq \\min \\left( \\frac{g_n}{2}, \\frac{m+1}{2} \\right)$, we have\n\\[\n\\frac{\\tau(n,g_n)}{\\tau(n-m,g_n-h)}\\geq a_{\\theta}^{h}\\frac{n^{2g_n}}{(n-m)^{2(g_n-h)}}\\exp \\left( f\\left(\\frac{g_n}{n}\\right) n -f\\left(\\frac{g_n-h}{n-m}\\right)(n-m)+ o(m) \\right),\n\\]\nwhere the $o(m)$ is uniform in $(m,h)$ as $n \\to +\\infty$ (that is, it is bounded by $m\\eps(n)$ with $\\eps(n) \\to 0$ as $n \\to +\\infty$).\n\\end{proposition}\n\n\\paragraph{\\textbf{Case 4:}} To show~\\cref{eq_ineq_smaller_n2} (and thus finish the proof of the lemma), it remains to show that \n\\[\\sum_{n_1+n_2=n\\atop n_2\\leq (5+b)\\log n}n_1^a\\tau(n_1,g_n-1)n_2^b\\tau(n_2,1)\\leq n^{a-2+o(1)}\\tau(n,g_n)).\\]\nBut, on the one hand, by \\cref{lem_asympto_genus_ratio}, uniformly for all $n_2\\leq (5+b)\\log n$, we have\n\\[\\frac{\\tau(n-n_2,g_n-1)}{\\tau(n,g_n-1)}\\leq n^{-2+o(1)}\\lambda(\\theta)^{n_2}.\\]\nAnd on the other hand, it is well-known~(see for instance~\\cite{Kri07}) that the generating series\n\\[H_b(x)=\\sum{n\\geq 0} n^b\\tau(n,1)x^n\\]\nhas radius of convergence $\\lambda(0)>\\lambda(\\theta)$, therefore $H_b(\\lambda(\\theta))<\\infty$ and\n\\[\\sum_{n_1+n_2=n\\atop n_2\\leq (5+b)\\log n}n_1^a\\tau(n_1,g_n-1)n_2^b\\tau(n_2,1)\\leq n^{a-2+o(1)}\\sum_{k\\geq 0}\\lambda(\\theta)^kk^b\\tau(k,1)=n^{a-2+o(1)}H_b(\\lambda(\\theta)).\\]\nThis finishes the proof.\n\\end{proof}\n\n\\begin{proposition}\\label{prop_simple}\nLet $A=\\frac{1}{2\\log\\lambda(\\theta)}$, we have\n\\[\\sum_{\\ell\\leq A\\log n} \\frac{|\\Ts(2n,g_n,\\ell)|}{\\tau(n,g_n)}\\to 0\\]\nas $\\nto$.\n\\end{proposition}\nLet us start with an inequality that holds for all $n,g,\\ell$.\n\n\\subsection{Counting \\te{s}}\nIn this section, we prove that $\\bT$ does not contain small \\te{s}, that is:\n\\begin{proposition}\\label{prop_thin}\nLet $A=\\frac{1}{2\\log\\lambda(\\theta)}$, we have\n\\[\\sum_{\\ell\\leq A\\log n} \\frac{\\Tt(2n,g_n,\\ell)}{\\tau(n,g_n)}\\to 0\\]\nas $\\nto$.\n\\end{proposition}\n\n\\begin{conjecture}\\label{conj_constant}\nLet $\\sep^*$ be the simple separating systole of $\\bT$.\nFor every $\\theta\\in(0,1/2)$, there exists two constants $0<S(\\theta)<S^*(\\theta)$ such that\n\\[\\frac{\\sep}{\\log(n)}\\to S(\\theta)\\quad \\text{and}\\quad \\frac{\\sep^*}{\\log(n)}\\to S^*(\\theta)\\]\nin probability as $\\nto$.\n\\end{conjecture}", "post_theorem_intro_text_len": 2390, "post_theorem_intro_text": "Note that in this theorem, we do not require the shortest separating cycle to be simple. In fact, even the existence of a simple separating non-contractible cycle in triangulations is a longstanding conjecture \\cite{MT01}[Chapter 5]. However, we do conjecture that with high probability{}, it does exist in $\\mathbf{T}_{2n,g_n}$, and what's more that it is again of logarithmic length. We will discuss this, and other conjectures such as the convergence of $\\frac{\\text{sepsys}(\\bT)}{\\log(n)}$ in probability, in more details in~\\cref{sec_conj}.\n\nThe upper bound in~\\cref{thm_sepsys} follows directly from a theorem of~\\cite{CVHM14}. To establish the upper bound, we will resort to a similar approach as in \\cite{Louf,BCL}: cut along cycles and estimate the number of triangulations with boundaries, and then  use a first moment method. However, contrary to previous works, we are dealing this time with non simple cycles, with can become quite intricate, especially as topology is involved. This is one of the main challenges of this work, and it is largely dealt with in~\\cref{sec_geom}.\n\nIn the process of proving the lower bound, we can adapt some of the techniques to establish a new enumerative result on high genus triangulation that we call the convergence of the \\emph{genus ratio}. Let us give more context. Let $\\tau(n,g)$ be the number of triangulations of genus $g$ with $2n$ faces, we established in~\\cite{BL19}, together with Budzinski, the convergence of the \"\\emph{size ratio}\"\n\\begin{equation}\\label{eq_ratio_size}\n\\frac{\\tau(n-1,g_n)}{\\tau(n,g_n)}\\rightarrow \\lambda(\\theta),\n\\end{equation}\nas $n \\to \\infty$, where $\\lambda$ is an explicit function.\n\nIt is a natural question to ask for a similar result when the genus varies and the size is fixed. We answer it with the following theorem:\n\n\\begin{theorem}\\label{thm_ratio}\nThere exists an explicit function $\\psi$ (see~\\cref{eq_psi}) such that\n\\begin{equation} \n\\frac{n^2\\tau(n,g_n-1)}{\\tau(n,g_n)}\\rightarrow \\psi(\\theta),\n\\end{equation}\nas $n \\to \\infty$.\n\\end{theorem}\n\n\\paragraph{Acknowledgments:} Thanks to Thomas Budzinski, Guillaume Chapuy, Andrew Elvey-Price, Wenjie Fang and Michael Wallner for numerous joint works and enriching discussions about high genus geometry and enumeration. Thanks also to Arnaud de Mesmay and Hugo Parlier for interesting exchanges around \\cref{thm_upper,conj_Barnette,conj_decomp}.", "sketch": "The upper bound in~\\cref{thm_sepsys} \"follows directly from a theorem of~\\cite{CVHM14}.\" For the lower bound, the authors \"resort to a similar approach as in \\cite{Louf,BCL}: cut along cycles and estimate the number of triangulations with boundaries, and then use a first moment method.\" They emphasize that, \"contrary to previous works,\" they must handle \"non simple cycles\" which \"can become quite intricate, especially as topology is involved,\" and that this challenge is \"largely dealt with in~\\cref{sec_geom}.\" In proving the lower bound they also \"adapt some of the techniques to establish\" an enumerative input, the \"convergence of the genus ratio\" (Theorem~\\ref{thm_ratio}).", "expanded_sketch": "The upper bound in establishing the main theorem follows directly from a theorem of~\\cite{CVHM14}. For the lower bound, the authors \"resort to a similar approach as in \\cite{Louf,BCL}: cut along cycles and estimate the number of triangulations with boundaries, and then use a first moment method.\" They emphasize that, \"contrary to previous works,\" they must handle \"non simple cycles\" which \"can become quite intricate, especially as topology is involved,\" and that this challenge is \"largely dealt with\" later. In proving the lower bound they also \"adapt some of the techniques to establish\" an enumerative input, the \"convergence of the genus ratio\" (Theorem~\\ref{thm_ratio}).,", "expanded_theorem": "\\label{thm_sepsys}\nWith high probability\\footnote{that is, with probability tending to $1$ as $n \\to \\infty$, abbreviated later as whp.}, the separating systole of $\\mathbf{T}_{2n,g_n}$ is of logarithmic order, i.e. there exists two constants $0<A<B$ (depending on $\\theta$) such that whp\n\\[A\\log(n)\\leq \\text{sepsys}(\\bT)\\leq B\\log(n)\\],", "theorem_type": "unknown", "mcq": {"question": "Fix \\(\\theta\\in(0,1/2)\\) and let \\((g_n)\\) be a sequence with \\(g_n/n\\to\\theta\\). For each \\(n\\), let \\(\\mathbf{T}_{2n,g_n}\\) be a uniform triangulation of genus \\(g_n\\) with \\(2n\\) faces. Define the separating systole \\(\\operatorname{sepsys}(\\mathbf{T}_{2n,g_n})\\) to be the length of the shortest cycle in \\(\\mathbf{T}_{2n,g_n}\\) that is separating and non-contractible (the cycle is not required to be simple). Which statement holds for \\(\\operatorname{sepsys}(\\mathbf{T}_{2n,g_n})\\) as \\(n\\to\\infty\\)?", "correct_choice": {"label": "A", "text": "There exist constants \\(0<A<B\\), depending on \\(\\theta\\), such that \\[\\mathbb{P}\\!\\left(A\\log n\\le \\operatorname{sepsys}(\\mathbf{T}_{2n,g_n})\\le B\\log n\\right)\\to 1\\quad\\text{as }n\\to\\infty.\\] Equivalently, with high probability, the separating systole of \\(\\mathbf{T}_{2n,g_n}\\) is of logarithmic order."}, "choices": [{"label": "B", "text": "There exist constants \\(0<A<B\\), depending only on \\(\\theta\\), such that \\[\\mathbb{P}\\!\\left(A\\log n\\le \\operatorname{sepsys}(\\mathbf{T}_{2n,g_n})\\le B\\right)\\to 1\\quad\\text{as }n\\to\\infty.\\] In particular, with high probability the separating systole is bounded above by an absolute constant while remaining bounded below by a logarithmic term."}, {"label": "C", "text": "There exists a constant \\(A>0\\), depending on \\(\\theta\\), such that \\[\\mathbb{P}\\!\\left(\\operatorname{sepsys}(\\mathbf{T}_{2n,g_n})\\ge A\\log n\\right)\\to 1\\quad\\text{as }n\\to\\infty.\\] Thus, with high probability, the separating systole is at least of logarithmic order."}, {"label": "D", "text": "For every sequence \\((g_n)\\) with \\(g_n/n\\to\\theta\\), there exist universal constants \\(0<A<B\\), independent of \\(\\theta\\), such that \\[\\mathbb{P}\\!\\left(A\\log n\\le \\operatorname{sepsys}(\\mathbf{T}_{2n,g_n})\\le B\\log n\\right)\\to 1\\quad\\text{as }n\\to\\infty.\\] Equivalently, the logarithmic bounds can be chosen uniformly over all \\(\\theta\\in(0,1/2)\\)."}, {"label": "E", "text": "There exist constants \\(0<A<B\\), depending on \\(\\theta\\), such that \\[\\mathbb{P}\\!\\left(A\\log n\\le \\operatorname{sepsys}(\\mathbf{T}_{2n,g_n})\\le B\\log n\\right)=1-o(n^{-1})\\quad\\text{as }n\\to\\infty.\\] In other words, the separating systole is of logarithmic order with an explicit polynomially fast error probability."}], "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": "upper-bound scale from logarithmic to constant", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "dropped the upper logarithmic bound", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "dependence of constants on theta", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "non-effective whp strengthened to polynomial-rate error bound", "template_used": "uniformity_effectivity"}]}}
{"id": "2512.05340v1", "paper_link": "http://arxiv.org/abs/2512.05340v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "[\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016,FangKoike2022}]\n  \\label{Theorem: high temp O(N)}\n  Fix $N \\geq 2$ and let $W_{n} := \\sqrt{N-\\beta}\\,S_{n}/\\sqrt{n}$ and $Z \\sim N\\left(0, \\mathbf{I}\\right)$, where $\\mathbf{I}\\in\\mathbb{R}^{d\\times d}$ is the\n  identity matrix. For any $\\beta < N$ there exists $c(N,\\beta)<\\infty$ such that\n    \\begin{equation}\n         d(\\mathcal{L}(W_n),\\mathcal{L}(Z)) \\leq \\frac{c(N,\\beta)}{\\sqrt{n}}.\n    \\end{equation}", "start_pos": 10002, "end_pos": 10457, "label": "Theorem: high temp O(N)"}, "ref_dict": {"Theorem: main result": "\\begin{theorem}\\label{Theorem: main result}\n  Suppose $V$ satisfies Assumption \\ref{assumption: assumptions on V}, and let $\\mu$ have Lebesgue density proportional to\n  $e^{-V}$.\n  Let $W,W'$ be identically distributed $\\RR^d$-valued random variables, defined on the same probability space, such that if $\\delta:=W'-W$ then\n  $W$ and $\\delta\\delta^\\transpose$ are both integrable. \n  Let $\\sF\\supseteq\\sigma(W)$ be a $\\sigma$-algebra, let $\\lambda>0$, and define $R_1$ and $R_2$ so that\n    \\begin{equation}\n      \\mathbb{E}[\\delta|\\mathcal{F}] = \\lambda(-\\grad V(W) + R_{1}),\n      \\label{eq: Regression Property}\n    \\end{equation}\n    and\n    \\begin{equation}\n      \\mathbb{E}\\left[\\delta\\,\\delta^{\\transpose}\\big|\\mathcal{F}\\right] = 2\\lambda(\\id + R_{2}).\n      \\label{eq: Correlation Property}\n    \\end{equation}\n    Then there exists a constant $C\\in(0,\\infty)$ such that\n    \\begin{equation}\n      d(\\sL(W),\\mu)\n      \\leq C\\left\\lbrace \\frac{1}{\\lambda}\\mathbb{E} [|\\delta|^{3}(|\\log|\\delta|| \\vee 1)] + \\mathbb{E}|R_{1}|\n      + \\sqrt{d}\\,\\mathbb{E}\\,\\|R_{2}\\| \\right\\rbrace. \n    \\end{equation}\n\\end{theorem}", "Theorem: Critical O(N)": "\\begin{theorem}\n  \\label{Theorem: Critical O(N)}\n  Fix $N \\geq 2$ and let $W_{n} := n^{-3/4}\\,S_{n}$. If $\\beta=N$, then there exists $c(N)<\\infty$ such that\n    \\begin{equation}\n        \\wass(\\sL(W_n),\\mu) \\leq \\frac{c(N)}{\\sqrt{n}},\n    \\end{equation}\n  where $\\mu$ has Lebesgue density proportional to $\\exp\\left(-a_N |x|^{4}\\right)$ and $a_N:=N^2/(4N+8)$.\n\\end{theorem}", "eq:mean-field O(N) measure": "\\begin{equation}\n  \\exp\\left(\\frac{\\beta}{2n}\\sum_{i,j=1}^n \\sigma_i\\cdot\\sigma_j\\right), \\qquad \\sigma\\in (\\sphere)^n.\n  \\label{eq:mean-field O(N) measure}\n\\end{equation}"}, "pre_theorem_intro_text_len": 5192, "pre_theorem_intro_text": "The $O(N)$ model was introduced in~\\cite{Stanley1968} as a natural generalization of the Ising, $XY$ and Heisenberg\nmodels. For integer $N,n\\ge1$, let $\\SS^{N-1}:=\\{x\\in\\mathbb{R}^N:|x|=1\\}$ denote the set of Euclidean unit vectors in $\\mathbb{R}^N$, and let $P_{N,n}$\ndenote the $n$-fold product of uniform measure on $\\SS^{N-1}$. The mean-field $O(N)$ model is then defined for $\\beta\\ge0$ by the probability\nmeasure $\\PP_{N,n,\\beta}$ whose density with respect to $P_{N,n}$ is proportional to\n\\begin{equation}\n  \\exp\\left(\\frac{\\beta}{2n}\\sum_{i,j=1}^n \\sigma_i\\cdot\\sigma_j\\right), \\qquad \\sigma\\in (\\SS^{N-1})^n.\n  \\label{eq:mean-field O(N) measure}\n\\end{equation}\n\nIt is known that, with respect to the mean-field measure~\\eqref{eq:mean-field O(N) measure}, the magnetization\n\\begin{equation}\n  \\label{eq:magnetization definition}\nS_n(\\sigma):=\\sum_{i=1}^n\\sigma_i\n\\end{equation}\nobeys a law of large numbers iff $\\beta\\le N$; see~\\cite{EllisNewman1978,Ellis1985} for the $N=1$ case, \\cite{KirkpatrickMeckes2013} for\nthe $N=3$ case, and~\\cite{KirkpatrickNawaz2016} for the general case of $N\\ge2$.\nIn the subcritical phase, $\\beta<N$, it is also known~\\cite{EllisNewman1978,Ellis1985,KirkpatrickMeckes2013,KirkpatrickNawaz2016} that\n$S_n/\\sqrt{n}$ obeys a Gaussian central limit theorem. By contrast, at the critical point, $\\beta=N$,\nit has long been known that $n^{-3/4}\\,S_n$ converges weakly to a random vector in $\\mathbb{R}^N$ with Lebesgue density\nproportional to $\\exp(-a_N |x|^4)$, where $a_N:=N^2/(4N+8)$. This was established for $N=1$ in~\\cite{SimonGriffiths1973},\nand then generalised to $N\\ge2$ in~\\cite{DunlopNewman1975}.\nAnalogous limit theorems have been subsequently established for other mean-field spin models, including the Curie-Weiss-Potts\nmodel~\\cite{EllisWang1990}, and Blume-Emery-Griffiths model~\\cite{EllisOttoTouchette2005}.\n\nA natural question associated with a given limit theorem is to determine its rate of convergence. It was shown\nin~\\cite{EichelsbacherLowe2010} via Stein's method, that for the $N=1$ model with $\\beta<1$, the Kolmogorov distance\nbetween the distribution of $\\sqrt{1-\\beta}\\, S_n/\\sqrt{n}$ and a standard univariate normal is bounded above by\n$c(\\beta)/\\sqrt{n}$. Utilising a multivariate generalisation of Stein's method established\nin~\\cite{ChatterjeeMeckes2008,ReinertRollin2009,Meckes2009}, analogous results~\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were then\nshown to hold in the disordered phase, $\\beta<N$, of the $O(N)$ model for any $N\\ge2$.\n\nIn the critical case, it was shown in~\\cite{EichelsbacherLowe2010,ChatterjeeShao2011} using nonnormal extensions of Stein's method,\nthat for the $N=1$ model with $\\beta=1$ the Kolmogorov distance between the distribution of $n^{-3/4}\\, S_n$ and the probability\nmeasure with Lebesgue density proportional to $\\exp(-x^4/12)$ is again bounded above by $c/\\sqrt{n}$. We establish here a natural\nmultivariate generalisation of this result to all $N\\ge 2$.\n\nOur method relies on a general multivariate nonnormal Stein's method framework established\nin~\\cite{FangShaoXu2019,FangShaoXu2019correction}, which in turn can be viewed as a multivariate generalization\nof~\\cite{ChatterjeeShao2011}. \nAs explicitly noted in~\\cite{FangShaoXu2019}, however, their underlying assumptions\ndo not hold for densities of the form $\\exp(-a|x|^m)$ with $m>2$, and so their results cannot be applied to the critical mean-field $O(N)$\nmodel. Nevertheless, as shown here, a generalization of~\\cite[Theorem 2.5]{FangShaoXu2019} can be established which does indeed cover the\ncase of the critical $O(N)$ model. We present this result in Theorem~\\ref{Theorem: main result} of Section~\\ref{sec: Stein's method}.\n\nThe distributional bounds we present are in terms of the Wasserstein distance. The Wasserstein distance between the laws $\\mathcal{L}(X)$, $\\mathcal{L}(Y)$\nof two random vectors $X,Y\\in\\mathbb{R}^d$ is \n\\begin{equation}\n  d(\\mathcal{L}(X),\\mathcal{L}(Y)):= \\inf_{(X,Y)}\\mathbb{E}|X-Y| = \\sup_{h \\in \\mathrm{Lip}}\\left|\\mathbb{E}\\, h(X) - \\mathbb{E}\\, h(Y)\\right|\n  \\label{eq: Wasserstein definition}\n\\end{equation}\nwhere the infimum is taken over all couplings of $\\mathcal{L}(X)$ and $\\mathcal{L}(Y)$, and $\\mathrm{Lip}$ denotes the set of all functions $h:\\mathbb{R}^d\\to\\mathbb{R}$ such\nthat $|h(x)-h(y)|\\le |x-y|$ for all $x,y\\in\\mathbb{R}^d$. We note that convergence in Wasserstein distance implies weak convergence; see\ne.g.~\\cite{GibbsSu2002}.  \n\nIn the disordered phase, the bounds originally presented in~\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were not in terms of the\nWasserstein distance, but instead in terms of an integral probability metric defined by a smaller, smoother, class of test functions. These\nbounds were proved by applying a general multivariate normal approximation theorem presented in~\\cite{Meckes2009}. However, recent results\ngiving multivariate normal approximation in the Wasserstein distance allow the results from~\\cite{KirkpatrickNawaz2016} to be immediately\nsharpened to the Wasserstein distance. Indeed, substituting results given in~\\cite[Lemmas 1 and 2]{KirkpatrickNawaz2016} into~\\cite[Theorem\n  1.1]{FangKoike2022} yields the following.", "context": "The $O(N)$ model was introduced in~\\cite{Stanley1968} as a natural generalization of the Ising, $XY$ and Heisenberg\nmodels. For integer $N,n\\ge1$, let $\\SS^{N-1}:=\\{x\\in\\mathbb{R}^N:|x|=1\\}$ denote the set of Euclidean unit vectors in $\\mathbb{R}^N$, and let $P_{N,n}$\ndenote the $n$-fold product of uniform measure on $\\SS^{N-1}$. The mean-field $O(N)$ model is then defined for $\\beta\\ge0$ by the probability\nmeasure $\\PP_{N,n,\\beta}$ whose density with respect to $P_{N,n}$ is proportional to\n\\begin{equation}\n \\exp\\left(\\frac{\\beta}{2n}\\sum_{i,j=1}^n \\sigma_i\\cdot\\sigma_j\\right), \\qquad \\sigma\\in (\\SS^{N-1})^n.\n \\label{eq:mean-field O(N) measure}\n\\end{equation}\n\nIt is known that, with respect to the mean-field measure~\\eqref{eq:mean-field O(N) measure}, the magnetization\n\\begin{equation}\n \\label{eq:magnetization definition}\nS_n(\\sigma):=\\sum_{i=1}^n\\sigma_i\n\\end{equation}\nobeys a law of large numbers iff $\\beta\\le N$; see~\\cite{EllisNewman1978,Ellis1985} for the $N=1$ case, \\cite{KirkpatrickMeckes2013} for\nthe $N=3$ case, and~\\cite{KirkpatrickNawaz2016} for the general case of $N\\ge2$.\nIn the subcritical phase, $\\beta<N$, it is also known~\\cite{EllisNewman1978,Ellis1985,KirkpatrickMeckes2013,KirkpatrickNawaz2016} that\n$S_n/\\sqrt{n}$ obeys a Gaussian central limit theorem. By contrast, at the critical point, $\\beta=N$,\nit has long been known that $n^{-3/4}\\,S_n$ converges weakly to a random vector in $\\mathbb{R}^N$ with Lebesgue density\nproportional to $\\exp(-a_N |x|^4)$, where $a_N:=N^2/(4N+8)$. This was established for $N=1$ in~\\cite{SimonGriffiths1973},\nand then generalised to $N\\ge2$ in~\\cite{DunlopNewman1975}.\nAnalogous limit theorems have been subsequently established for other mean-field spin models, including the Curie-Weiss-Potts\nmodel~\\cite{EllisWang1990}, and Blume-Emery-Griffiths model~\\cite{EllisOttoTouchette2005}.\n\nA natural question associated with a given limit theorem is to determine its rate of convergence. It was shown\nin~\\cite{EichelsbacherLowe2010} via Stein's method, that for the $N=1$ model with $\\beta<1$, the Kolmogorov distance\nbetween the distribution of $\\sqrt{1-\\beta}\\, S_n/\\sqrt{n}$ and a standard univariate normal is bounded above by\n$c(\\beta)/\\sqrt{n}$. Utilising a multivariate generalisation of Stein's method established\nin~\\cite{ChatterjeeMeckes2008,ReinertRollin2009,Meckes2009}, analogous results~\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were then\nshown to hold in the disordered phase, $\\beta<N$, of the $O(N)$ model for any $N\\ge2$.\n\nIn the critical case, it was shown in~\\cite{EichelsbacherLowe2010,ChatterjeeShao2011} using nonnormal extensions of Stein's method,\nthat for the $N=1$ model with $\\beta=1$ the Kolmogorov distance between the distribution of $n^{-3/4}\\, S_n$ and the probability\nmeasure with Lebesgue density proportional to $\\exp(-x^4/12)$ is again bounded above by $c/\\sqrt{n}$. We establish here a natural\nmultivariate generalisation of this result to all $N\\ge 2$.\n\nThe distributional bounds we present are in terms of the Wasserstein distance. The Wasserstein distance between the laws $\\mathcal{L}(X)$, $\\mathcal{L}(Y)$\nof two random vectors $X,Y\\in\\mathbb{R}^d$ is \n\\begin{equation}\n d(\\mathcal{L}(X),\\mathcal{L}(Y)):= \\inf_{(X,Y)}\\mathbb{E}|X-Y| = \\sup_{h \\in \\mathrm{Lip}}\\left|\\mathbb{E}\\, h(X) - \\mathbb{E}\\, h(Y)\\right|\n \\label{eq: Wasserstein definition}\n\\end{equation}\nwhere the infimum is taken over all couplings of $\\mathcal{L}(X)$ and $\\mathcal{L}(Y)$, and $\\mathrm{Lip}$ denotes the set of all functions $h:\\mathbb{R}^d\\to\\mathbb{R}$ such\nthat $|h(x)-h(y)|\\le |x-y|$ for all $x,y\\in\\mathbb{R}^d$. We note that convergence in Wasserstein distance implies weak convergence; see\ne.g.~\\cite{GibbsSu2002}.\n\nIn the disordered phase, the bounds originally presented in~\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were not in terms of the\nWasserstein distance, but instead in terms of an integral probability metric defined by a smaller, smoother, class of test functions. These\nbounds were proved by applying a general multivariate normal approximation theorem presented in~\\cite{Meckes2009}. However, recent results\ngiving multivariate normal approximation in the Wasserstein distance allow the results from~\\cite{KirkpatrickNawaz2016} to be immediately\nsharpened to the Wasserstein distance. Indeed, substituting results given in~\\cite[Lemmas 1 and 2]{KirkpatrickNawaz2016} into~\\cite[Theorem\n 1.1]{FangKoike2022} yields the following.\n\n\\begin{equation}\n  \\exp\\left(\\frac{\\beta}{2n}\\sum_{i,j=1}^n \\sigma_i\\cdot\\sigma_j\\right), \\qquad \\sigma\\in (\\sphere)^n.\n  \\label{eq:mean-field O(N) measure}\n\\end{equation}", "full_context": "The $O(N)$ model was introduced in~\\cite{Stanley1968} as a natural generalization of the Ising, $XY$ and Heisenberg\nmodels. For integer $N,n\\ge1$, let $\\SS^{N-1}:=\\{x\\in\\mathbb{R}^N:|x|=1\\}$ denote the set of Euclidean unit vectors in $\\mathbb{R}^N$, and let $P_{N,n}$\ndenote the $n$-fold product of uniform measure on $\\SS^{N-1}$. The mean-field $O(N)$ model is then defined for $\\beta\\ge0$ by the probability\nmeasure $\\PP_{N,n,\\beta}$ whose density with respect to $P_{N,n}$ is proportional to\n\\begin{equation}\n \\exp\\left(\\frac{\\beta}{2n}\\sum_{i,j=1}^n \\sigma_i\\cdot\\sigma_j\\right), \\qquad \\sigma\\in (\\SS^{N-1})^n.\n \\label{eq:mean-field O(N) measure}\n\\end{equation}\n\nIt is known that, with respect to the mean-field measure~\\eqref{eq:mean-field O(N) measure}, the magnetization\n\\begin{equation}\n \\label{eq:magnetization definition}\nS_n(\\sigma):=\\sum_{i=1}^n\\sigma_i\n\\end{equation}\nobeys a law of large numbers iff $\\beta\\le N$; see~\\cite{EllisNewman1978,Ellis1985} for the $N=1$ case, \\cite{KirkpatrickMeckes2013} for\nthe $N=3$ case, and~\\cite{KirkpatrickNawaz2016} for the general case of $N\\ge2$.\nIn the subcritical phase, $\\beta<N$, it is also known~\\cite{EllisNewman1978,Ellis1985,KirkpatrickMeckes2013,KirkpatrickNawaz2016} that\n$S_n/\\sqrt{n}$ obeys a Gaussian central limit theorem. By contrast, at the critical point, $\\beta=N$,\nit has long been known that $n^{-3/4}\\,S_n$ converges weakly to a random vector in $\\mathbb{R}^N$ with Lebesgue density\nproportional to $\\exp(-a_N |x|^4)$, where $a_N:=N^2/(4N+8)$. This was established for $N=1$ in~\\cite{SimonGriffiths1973},\nand then generalised to $N\\ge2$ in~\\cite{DunlopNewman1975}.\nAnalogous limit theorems have been subsequently established for other mean-field spin models, including the Curie-Weiss-Potts\nmodel~\\cite{EllisWang1990}, and Blume-Emery-Griffiths model~\\cite{EllisOttoTouchette2005}.\n\nA natural question associated with a given limit theorem is to determine its rate of convergence. It was shown\nin~\\cite{EichelsbacherLowe2010} via Stein's method, that for the $N=1$ model with $\\beta<1$, the Kolmogorov distance\nbetween the distribution of $\\sqrt{1-\\beta}\\, S_n/\\sqrt{n}$ and a standard univariate normal is bounded above by\n$c(\\beta)/\\sqrt{n}$. Utilising a multivariate generalisation of Stein's method established\nin~\\cite{ChatterjeeMeckes2008,ReinertRollin2009,Meckes2009}, analogous results~\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were then\nshown to hold in the disordered phase, $\\beta<N$, of the $O(N)$ model for any $N\\ge2$.\n\nIn the critical case, it was shown in~\\cite{EichelsbacherLowe2010,ChatterjeeShao2011} using nonnormal extensions of Stein's method,\nthat for the $N=1$ model with $\\beta=1$ the Kolmogorov distance between the distribution of $n^{-3/4}\\, S_n$ and the probability\nmeasure with Lebesgue density proportional to $\\exp(-x^4/12)$ is again bounded above by $c/\\sqrt{n}$. We establish here a natural\nmultivariate generalisation of this result to all $N\\ge 2$.\n\nThe distributional bounds we present are in terms of the Wasserstein distance. The Wasserstein distance between the laws $\\mathcal{L}(X)$, $\\mathcal{L}(Y)$\nof two random vectors $X,Y\\in\\mathbb{R}^d$ is \n\\begin{equation}\n d(\\mathcal{L}(X),\\mathcal{L}(Y)):= \\inf_{(X,Y)}\\mathbb{E}|X-Y| = \\sup_{h \\in \\mathrm{Lip}}\\left|\\mathbb{E}\\, h(X) - \\mathbb{E}\\, h(Y)\\right|\n \\label{eq: Wasserstein definition}\n\\end{equation}\nwhere the infimum is taken over all couplings of $\\mathcal{L}(X)$ and $\\mathcal{L}(Y)$, and $\\mathrm{Lip}$ denotes the set of all functions $h:\\mathbb{R}^d\\to\\mathbb{R}$ such\nthat $|h(x)-h(y)|\\le |x-y|$ for all $x,y\\in\\mathbb{R}^d$. We note that convergence in Wasserstein distance implies weak convergence; see\ne.g.~\\cite{GibbsSu2002}.\n\nIn the disordered phase, the bounds originally presented in~\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were not in terms of the\nWasserstein distance, but instead in terms of an integral probability metric defined by a smaller, smoother, class of test functions. These\nbounds were proved by applying a general multivariate normal approximation theorem presented in~\\cite{Meckes2009}. However, recent results\ngiving multivariate normal approximation in the Wasserstein distance allow the results from~\\cite{KirkpatrickNawaz2016} to be immediately\nsharpened to the Wasserstein distance. Indeed, substituting results given in~\\cite[Lemmas 1 and 2]{KirkpatrickNawaz2016} into~\\cite[Theorem\n 1.1]{FangKoike2022} yields the following.\n\n\\begin{equation}\n  \\exp\\left(\\frac{\\beta}{2n}\\sum_{i,j=1}^n \\sigma_i\\cdot\\sigma_j\\right), \\qquad \\sigma\\in (\\sphere)^n.\n  \\label{eq:mean-field O(N) measure}\n\\end{equation}\n\n\\begin{lemma}\\label{lemma: Critical remainders}\n Let $(W, W')$ be as described above. Then\n \\begin{equation}\\label{eq: W' - W conditioned on sigma}\n \\EE[W'-W|\\sigma] = \\frac{1}{Nn^{3/2}}\\left(-\\frac{N^{2}}{N+2}|W|^{2}W + R_{1}\\right),\n \\end{equation}\n where\n \\begin{equation}\\label{eq: remainder 1}\n \\begin{split}\n R_{1} &= \\frac{N}{n^{1/4}}\\sum_{i=1}^{n}\\left[m_{(i)}f_{N}(|m_{(i)}|) - mf_{N}(|m|)\\right] \\\\\n & \\quad + Nn^{3/4}\\left(f_{N}(|m|) - 1 + \\frac{N|m|^{2}}{(N+2)}\\right)m.\n \\end{split}\n \\end{equation}\n and \n \\begin{equation}\\label{eq: W' - W W' - W T conditioned on sigma}\n \\begin{split}\n \\EE[(W'-W)(W'-W)^{\\transpose}|\\sigma] &= 2\\frac{1}{Nn^{3/2}}\\left(\\id + R_{2}\\right),\n \\end{split}\n \\end{equation}\n where \n \\begin{equation}\\label{eq: remainder 2}\n \\begin{split}\n R_{2} &= \\left[\\frac{N}{2n}\\sum_{i=1}^{n}\\sigma_{i}\\sigma_{i}^{\\transpose} - \\frac{1}{2}\\id\\right] -\n \\frac{N}{\\sqrt{n}}WW^{\\transpose} + \\frac{N}{n^{2}}\\sum_{i=1}^{n}\\sigma_{i}\\sigma_{i}^{\\transpose}\\\\ \n &\\quad + \\frac{N}{2n}\\sum_{i=1}^{n}\\,g_{N}(|m_{(i)}|)\\,|m_{(i)}|^2\\,\n \\left[\\sigma_{i}m_{(i)}^{\\transpose}+ m_{(i)} \\sigma_{i}^{\\transpose} - \\frac{\\id}{N}\\right]\n + \\frac{N}{2n}\\sum_{i=1}^{n}g_{N}(|m_{(i)}|)m_{(i)}m_{(i)}^\\transpose. \n \\end{split}\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\n We follow a similar strategy to the proof~\\cite[Lemma 1]{KirkpatrickNawaz2016} (see also \\cite[Lemma 11]{KirkpatrickMeckes2013}).\n The key ingredients are the small $x$ asymptotics for $f_N(x)$ given in~\\eqref{eq: fN asymptotics}, combined with the fact\n that, as shown in~\\cite[Theorem 3]{KirkpatrickNawaz2016}, the quantity $m$ concentrates at 0 as $n\\to \\infty$ when $\\beta = N$.\n\nFinally, to establish~\\eqref{eq: Bound hilbert schmidt norm 2}, we fix $\\epsilon>0$ and $y\\in\\RR^d$ with $|y| \\leq 1$ and apply a\n similar decomposition to that used in~\\cite{FangShaoXu2019}. Specifically, defining\n \\begin{equation}\\label{eq: Phi definition}\n \\Phi(t) :=\\left[D^{2}P_{t}h(x + \\epsilon y)[u, v] - D^{2}P_{t}h(x)[u, v]\\right],\n \\end{equation}\n it follows by~\\eqref{eq: second derivative of f} that\n \\begin{equation}\\label{eq: rewriting f(x + epsilon u) - f(x)}\n D_{v}D_{u}f(x + \\epsilon y) - D_{v}D_{u}f(x) = -\\int_{0}^{\\epsilon^{2}\\wedge \\epsilon}\\Phi(t)\\,\\dt -\n \\int_{\\epsilon^{2}\\wedge \\epsilon}^{\\infty}\\Phi(t)\\,\\dt.\n \\end{equation}\n Equation~\\eqref{eq: Ptphi derivative 2} implies\n \\begin{equation}\\label{eq: bound int 0 to epsilon squared}\n \\begin{split}\n \\left|\\int_{0}^{\\epsilon^{2}\\wedge \\epsilon}\\Phi(t) \\, \\dt\\right| &\\leq 2|u||v|\\int_{0}^{\\epsilon^{2}\\wedge \\epsilon} S(t)\\, \\dt,\\\\\n &\\leq 2\\,|u| |v|\\,(\\epsilon\\wedge \\sqrt{\\epsilon})\\, \\left(s_0( \\epsilon\\wedge \\sqrt{\\epsilon}) +2s_{-1/2}\\right).\n \\end{split}\n \\end{equation}\n If $\\epsilon\\le 1$ then $\\epsilon\\wedge \\sqrt{\\epsilon}=\\epsilon$, in which case\n \\begin{equation}\n (\\epsilon\\wedge \\sqrt{\\epsilon})\\, \\left(s_0( \\epsilon\\wedge \\sqrt{\\epsilon}) +2s_{-1/2}\\right) = \n \\epsilon\\, (s_0\\epsilon +2s_{-1/2})\\,\\leq \\epsilon (s_0 + 2s_{-1/2}).\n \\end{equation}\n While if $\\epsilon>1$ then $\\epsilon\\wedge\\sqrt{\\epsilon}=\\sqrt{\\epsilon}$, and so \n \\begin{equation}\n (\\epsilon\\wedge \\sqrt{\\epsilon})\\, \\left(s_0( \\epsilon\\wedge \\sqrt{\\epsilon}) +2s_{-1/2}\\right) =\n \\sqrt{\\epsilon}\\, \\left(s_0\\sqrt{\\epsilon}+2s_{-1/2}\\right)\\leq\n \\epsilon\\left(s_{0} +2s_{-1/2}\\right).\n \\end{equation}\n We therefore conclude from~\\eqref{eq: bound int 0 to epsilon squared} that for all $\\epsilon>0$\n \\begin{equation}\\label{eq: int bound 0 to min}\n \\left|\\int_{0}^{\\epsilon^{2}\\wedge \\epsilon}\\Phi(t) \\, \\dt\\right| \\leq 2\\epsilon (s_0 + 2s_{-1/2})|u| |v|.\n \\end{equation}\n Now consider the remaining term in~\\eqref{eq: rewriting f(x + epsilon u) - f(x)}. It follows from~\\eqref{eq: Ptphi derivative 3} that\n $D^2P_t h(x+\\cdot)[u,v]$ is differentiable with bounded derivative, and so the fundamental theorem of calculus implies \n \\begin{equation}\n \\int_{\\epsilon^{2}\\wedge \\epsilon}^{\\infty}\\Phi(t)\\,\\dt\n = \\epsilon\\int_{\\epsilon^{2}\\wedge \\epsilon}^{\\infty}\\int_{0}^{1} D^{3}P_{t}h(x + \\epsilon r y)[u, v, y]\\, \\dr\\, \\dt. \n \\end{equation}\n Therefore, using $|y|\\le1$, and again applying~\\eqref{eq: Ptphi derivative 3}, yields\n \\begin{equation}\n \\label{eq: phi tail bound}\n \\begin{split}\n \\left|\\int_{\\epsilon^{2}\\wedge \\epsilon}^{\\infty}\\Phi\\,\\dt\\right| \n &\\leq \\epsilon |u||v| \\int_{\\epsilon\\wedge\\epsilon^{2}}^{\\infty}\\, Q(t) \\,e^{-\\theta t/4} \\dt,\\\\\n &\\leq \\epsilon |u||v|\n \\left[\\frac{128q_2}{\\theta^3}+\\frac{16q_1}{\\theta^2}+\\frac{4q_0}{\\theta}+\\frac{2\\sqrt{\\pi}q_{-1/2}}{\\sqrt{\\theta}}\n + q_{-1} E_1\\left(\\frac{\\theta(\\epsilon^{2}\\wedge \\epsilon)}{4}\\right)\\right]\n \\end{split}\n \\end{equation}\n where $E_{1}(\\cdot)$ denotes the exponential integral~\\cite[Equation 5.1.1]{AbramowitzStegun1972}. Applying~\\cite[Equation 5.1.20]{AbramowitzStegun1972} \n yields\n \\begin{equation}\n \\label{eq: bound on exponential integral}\n \\begin{split}\n E_1\\left(\\frac{\\theta(\\epsilon^{2}\\wedge \\epsilon)}{4}\\right) &\\le 2\\left(1\\vee \\left|\\log \\frac{\\theta(\\epsilon^{2}\\wedge \\epsilon)}{4} \\right|\\right)\\\\\n &\\le 4\\left(1+\\left|\\log\\frac{\\theta}{4}\\right|\\right)\\left(1\\vee |\\log\\epsilon|\\right).\n \\end{split}\n \\end{equation}\n Applying~\\eqref{eq: bound on exponential integral} to~\\eqref{eq: phi tail bound}, and combining the result with~\\eqref{eq: int bound 0\n to min} and~\\eqref{eq: rewriting f(x + epsilon u) - f(x)} then yields \n \\begin{equation}\n \\sup_{x,y\\in\\RR^d : |y|\\le 1} |D_{v}D_{u}f(x + \\epsilon y) - D_{v}D_{u}f(x)|\n \\le\n K_3\\,|u|\\, |v|\\, \\epsilon\\, (1\\vee|\\log\\epsilon|)\n \\label{eq: third main bound}\n \\end{equation}\n with\n \\begin{equation}\n K_3 :=\n 2(s_0 +2s_{-1/2}) + \n \\frac{128q_2}{\\theta^3}+\\frac{16q_1}{\\theta^2}+\\frac{4q_0}{\\theta}+\\frac{2\\sqrt{\\pi}q_{-1/2}}{\\sqrt{\\theta}}\n + 4\\left(1+\\left|\\log\\frac{\\theta}{4}\\right|\\right) q_{-1}.\n \\end{equation}\n\nThe analogue of Proposition~\\ref{prop: mean-square bounds} in~\\cite{FangShaoXu2019} is Lemma 5.2. While the proof of the latter is\nessentially immediate under their assumption of strict convexity of $V$,\nthe proof of Proposition~\\ref{prop: mean-square bounds} under Assumption~\\ref{assumption: assumptions on V} is somewhat more involved. A key\ningredient in its proof is the following lemma, whose proof is deferred to the end of this section. \n\\begin{lemma}\\label{lemma: E(t) bound}\n If $V$ satisfies Assumption~\\ref{assumption: assumptions on V}, then for all $x\\in\\RR^d$ and $0\\le s\\le t<\\infty$ \n $$\n \\EE \\exp\\left(-2\\int_s^t \\rho(X_r^x) \\dr \\right) \\le C_2^2 e^{-2\\theta (t-s)}.\n $$\n\\end{lemma}\nIn addition, we will also require the following Gr\\\"{o}nwall-type lemma.\n\\begin{lemma}\\label{lemma: Gronwall type}\n Let $u: \\RR \\to \\RR^{d}$ be such that $|u(t)|^{2} \\in C^{1}(\\RR, \\RR)$. Suppose it satisfies the following differential inequality\n \\begin{equation}\n \\frac{\\diff}{\\dt}|u(t)|^{2} \\leq 2a(t)|u(t)|^{2} + 2b(t)|u(t)|, \\qquad \\forall t \\in (0, T),\n \\end{equation}\n where $T \\in (0, \\infty)$, $a, b \\in L^{1}([0, T])$ and $b \\geq 0$. If $|u(0)| = 0$ then\n \\begin{equation}\n |u(t)| \\leq \\int_{0}^{t}b(s)\\exp\\left(\\int_{s}^{t}a(r)\\, \\dr\\right)\\, \\ds, \\qquad \\forall t \\in [0, T).\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\n Fix $t \\in (0, T)$ and let $\\tau := \\sup\\{0 \\leq s \\leq t: |u(s)| = 0\\}$. If $\\tau = t$ then the statement holds trivially since $b\n \\geq 0$. If $\\tau < t$ then $|u(t)|$ is differentiable on $(\\tau, t]$ and gives \n \\begin{equation}\n \\frac{\\diff}{\\ds}|u(s)| \\leq a(s)|u(s)| + b(s), \\qquad s \\in (\\tau, t].\n \\end{equation}\n Since $|u(\\tau)| = 0$, by the Generalised Jones Inequality \\cite[Theorem 1.2.2]{Qin2016} we have\n \\begin{equation}\n |u(t)| \\leq \\int_{\\tau}^{t}b(s)\\exp\\left(\\int_{s}^{t}a(r)\\, \\dr\\right)\\, \\ds \\leq \\int_{0}^{t}b(s)\\exp\\left(\\int_{s}^{t}a(r)\\, \\dr\\right)\\, \\ds,\n \\end{equation}\n using $b \\geq 0$. This completes the proof.\n\\end{proof}", "post_theorem_intro_text_len": 4700, "post_theorem_intro_text": "Our main result in the current work is the analogous multivariate nonnormal approximation holding at criticality.\n\\begin{theorem}\n  \\label{Theorem: Critical O(N)}\n  Fix $N \\geq 2$ and let $W_{n} := n^{-3/4}\\,S_{n}$. If $\\beta=N$, then there exists $c(N)<\\infty$ such that\n    \\begin{equation}\n        d(\\mathcal{L}(W_n),\\mu) \\leq \\frac{c(N)}{\\sqrt{n}},\n    \\end{equation}\n  where $\\mu$ has Lebesgue density proportional to $\\exp\\left(-a_N |x|^{4}\\right)$ and $a_N:=N^2/(4N+8)$.\n\\end{theorem}\n\n\\begin{remark}\nAs a corollary of Theorem~\\ref{Theorem: Critical O(N)} it follows that for any $N\\ge2$, the Wasserstein distance between the distribution of\n$n^{-3/4}\\,|S_n|$ and the probability measure with Lebesgue density proportional to $r^{N-1}\\exp(-a_N r^4)$, is again bounded above by\n$c(N)/\\sqrt{n}$. This is analogous to the univariate nonnormal approximation presented in~\\cite[Theorem 6]{KirkpatrickNawaz2016} for\n$|S_n|^2$, however the bound given there is not in terms of Wasserstein distance, but in terms of an integral probability metric defined by\na smaller class of test functions, and contains an additional logarithmic factor.\n\\end{remark}\n\nThe remainder of this paper is organised as follows. Section~\\ref{sec: Stein's method} introduces the relevant background on Stein's method, and\nprovides the statement of our main result on Wasserstein approximation, Theorem~\\ref{Theorem: main result}. In Section \\ref{sec: O(N) model}\nwe apply Theorem~\\ref{Theorem: main result} to prove Theorem~\\ref{Theorem: Critical O(N)}. \nSection~\\ref{sec: sdes and semigroups} provides preliminary results related to a class of stochastic differential equations, and their\nrelated stochastic semigroups, which provide the framework for the proof of Theorem \\ref{Theorem: main result} given in Section \\ref{sec: proof of main\n  result}. The key result used in this proof is a bound on the derivatives of the relevant stochastic semigroups.\nThese bounds are proved in Section~\\ref{sec: semigroup bounds}, using Elworthy-Li formulae and\nbounds on the spatial derivatives of the solutions to the corresponding stochastic differential equation\nproved in Section~\\ref{sec: variation process bounds}.\nFinally, the appendix contains the proof of a proposition stated in Section~\\ref{sec: sdes and semigroups}.\n\n\\subsection{Notation}\n\\label{subsec:notation}\nWe will denote the set of positive integers by $\\ZZ_+$, and the set of natural numbers by $\\mathbb{N}:=\\ZZ_+\\cup\\{0\\}$.\nFor $d\\in\\ZZ_+$ and $x,y\\in \\mathbb{R}^{d}$, we let $\\<x,y\\>$ denote the Euclidean inner product, and set $|x| = \\sqrt{\\<x,x\\>}$. For two\nmatrices $A, B \\in \\mathbb{R}^{d \\times d}$ we let $\\<A,B\\>_{{}} = \\sum_{i, j = 1}^{d}A_{i, j}B_{i, j}$ denote the Hilbert-Schmidt inner product,\nand denote the corresponding norm by $\\|A\\|_{{}} = \\sqrt{\\< A, A \\>_{{}}} = \\sqrt{\\text{Tr}(A^{\\mathrm{T}}A)}$. We will denote the operator\nnorm of $A\\in\\mathbb{R}^{d\\times d}$ by $\\|A\\|_{\\mathrm{op}}:=\\sup_{x\\in\\mathbb{S}^{d-1}}|Ax|$. \n\nFor open sets $U_1\\subseteq \\mathbb{R}^\\ell$ and $U_2\\subseteq\\mathbb{R}^m$ we let $C^{k}(U_1;U_2)$ denote the set of $k$-times continuously differentiable\n$f:U_1\\to U_2$. If $U_2=\\mathbb{R}$ we abbreviate $C^k(U_1):=C^k(U_1,U_2)$, and if additionally $U_1=\\mathbb{R}^d$ we write simply $C^k$.\nWe let $B_b$ denote the\nspace of $f:\\mathbb{R}^d\\to\\mathbb{R}$ which are bounded and Borel measurable, and let $C_b$ denote the subspace of $B_b$ of uniformly continuous such functions. We will then make\nuse of the convenient abbreviation $\\mathrm{Lip}^{\\infty}_b:=\\mathrm{Lip}\\cap C^{\\infty}\\cap C_b$.\n\nFor $f\\in C^1(\\mathbb{R}^d;\\mathbb{R})$, we let $D_u f(x)$ denote the directional derivative of $f$ in the direction $u\\in\\mathbb{R}^d$, and we extend the\ndefinition to $f\\in C^1(\\mathbb{R}^d;\\mathbb{R}^m)$, by then defining $D_u f(x)$ entrywise. For given $f\\in C^3(\\mathbb{R}^d;\\mathbb{R}^m)$ and $u,v,w,x\\in\\mathbb{R}^d$ we\ndefine\n\\begin{equation}\n  \\label{eq: D notation}\n  \\begin{split}\n    Df(x)[u]&:= D_u f(x)\\\\\n    D^2f(x)[u,v]&:=D_v D_u f(x)\\\\\n    D^3f(x)[u,v,w]&:=D_w D_v D_u f(x)\n  \\end{split}\n\\end{equation}\nWe note that for each given $x\\in\\mathbb{R}^d$ and $i=1,2,3$, the map $D^i g(x) : (\\mathbb{R}^d)^i\\to\\mathbb{R}^d$ is multilinear and symmetric.\n\nFor $f \\in C^{1}(\\mathbb{R}^d;\\mathbb{R})$, we let $\\nabla f$ denote the gradient of $f$, so that\n\\begin{equation}\n  Df(x)[u]=\\<\\nabla f(x),u\\>,\n\\end{equation}\nand set $\\|\\nabla f\\|_\\infty:=\\sup_{x \\in \\mathbb{R}^{d}}|\\nabla f(x)|$. \nFinally, for $f \\in C^{2}(\\mathbb{R}^{d};\\mathbb{R})$, we let $\\nabla^2 f$ denote the Hessian of $f$,\nso that\n\\begin{equation}\nD^2 f(x)[u,v] =\\<u,\\nabla^2 f(x) v\\>,\n\\end{equation}\nand let $\\Delta$ denote the Laplacian.", "sketch": "The introduction does not give a step-by-step proof of Theorem~\\ref{Theorem: high temp O(N)}, but it does indicate how the analogous critical result is proved: the paper \"introduces the relevant background on Stein's method\" and states a main Wasserstein-approximation theorem (Theorem~\\ref{Theorem: main result}); then \"we apply Theorem~\\ref{Theorem: main result} to prove Theorem~\\ref{Theorem: Critical O(N)}.\" The proof framework for Theorem~\\ref{Theorem: main result} uses \"a class of stochastic differential equations, and their related stochastic semigroups\"; \"the key result used in this proof is a bound on the derivatives of the relevant stochastic semigroups.\" These derivative bounds are proved \"using Elworthy-Li formulae\" and \"bounds on the spatial derivatives of the solutions to the corresponding stochastic differential equation\" (via \"variation process bounds\"). No direct proof sketch for Theorem~\\ref{Theorem: high temp O(N)} itself appears in the post-theorem introduction.", "expanded_sketch": "The introduction does not give a step-by-step proof of the main theorem, but it does indicate how the analogous critical result is proved: the paper introduces the relevant background on Stein's method and states a main Wasserstein-approximation theorem.\n\n\\begin{theorem}\\label{Theorem: main result}\n  Suppose $V$ satisfies Assumption \\ref{assumption: assumptions on V}, and let $\\mu$ have Lebesgue density proportional to\n  $e^{-V}$.\n  Let $W,W'$ be identically distributed $\\RR^d$-valued random variables, defined on the same probability space, such that if $\\delta:=W'-W$ then\n  $W$ and $\\delta\\delta^\\transpose$ are both integrable. \n  Let $\\sF\\supseteq\\sigma(W)$ be a $\\sigma$-algebra, let $\\lambda>0$, and define $R_1$ and $R_2$ so that\n    \\begin{equation}\n      \\mathbb{E}[\\delta|\\mathcal{F}] = \\lambda(-\\grad V(W) + R_{1}),\n      \\label{eq: Regression Property}\n    \\end{equation}\n    and\n    \\begin{equation}\n      \\mathbb{E}\\left[\\delta\\,\\delta^{\\transpose}\\big|\\mathcal{F}\\right] = 2\\lambda(\\id + R_{2}).\n      \\label{eq: Correlation Property}\n    \\end{equation}\n    Then there exists a constant $C\\in(0,\\infty)$ such that\n    \\begin{equation}\n      d(\\sL(W),\\mu)\n      \\leq C\\left\\lbrace \\frac{1}{\\lambda}\\mathbb{E} [|\\delta|^{3}(|\\log|\\delta|| \\vee 1)] + \\mathbb{E}|R_{1}|\n      + \\sqrt{d}\\,\\mathbb{E}\\,\\|R_{2}\\| \\right\\rbrace. \n    \\end{equation}\n\\end{theorem}\n\nThen this theorem is applied to prove the following critical result.\n\n\\begin{theorem}\n  \\label{Theorem: Critical O(N)}\n  Fix $N \\geq 2$ and let $W_{n} := n^{-3/4}\\,S_{n}$. If $\\beta=N$, then there exists $c(N)<\\infty$ such that\n    \\begin{equation}\n        \\wass(\\sL(W_n),\\mu) \\leq \\frac{c(N)}{\\sqrt{n}},\n    \\end{equation}\n  where $\\mu$ has Lebesgue density proportional to $\\exp\\left(-a_N |x|^{4}\\right)$ and $a_N:=N^2/(4N+8)$.\n\\end{theorem}\n\nThe proof framework for the Wasserstein-approximation theorem above uses a class of stochastic differential equations and their related stochastic semigroups; the key result used in this proof is a bound on the derivatives of the relevant stochastic semigroups. These derivative bounds are proved using Elworthy--Li formulae and bounds on the spatial derivatives of the solutions to the corresponding stochastic differential equation (via variation process bounds). No direct proof sketch for the main theorem itself appears in the post-theorem introduction.", "expanded_theorem": "[\\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016,FangKoike2022}]\n  \\label{Theorem: high temp O(N)}\n  Fix $N \\geq 2$ and let $W_{n} := \\sqrt{N-\\beta}\\,S_{n}/\\sqrt{n}$ and $Z \\sim N\\left(0, \\mathbf{I}\\right)$, where $\\mathbf{I}\\in\\mathbb{R}^{d\\times d}$ is the\n  identity matrix. For any $\\beta < N$ there exists $c(N,\\beta)<\\infty$ such that\n    \\begin{equation}\n         d(\\mathcal{L}(W_n),\\mathcal{L}(Z)) \\leq \\frac{c(N,\\beta)}{\\sqrt{n}}.\n    \\end{equation}", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Let $N\\ge 2$. In the mean-field $O(N)$ model, the spin configuration $\\sigma=(\\sigma_1,\\dots,\\sigma_n)\\in (\\mathbb S^{N-1})^n$ has law with density proportional to\n\\[\n\\exp\\!\\left(\\frac{\\beta}{2n}\\sum_{i,j=1}^n \\sigma_i\\cdot \\sigma_j\\right)\n\\]\nwith respect to the $n$-fold product of the uniform measure on the unit sphere $\\mathbb S^{N-1}\\subset\\mathbb R^N$. Let the magnetization be\n\\[\nS_n=\\sum_{i=1}^n \\sigma_i,\n\\]\nand define\n\\[\nW_n:=\\sqrt{N-\\beta}\\,\\frac{S_n}{\\sqrt n}.\n\\]\nLet $Z\\sim N(0,\\mathbf I)$ be the standard Gaussian vector in $\\mathbb R^N$, where $\\mathbf I$ is the identity matrix, and let the Wasserstein distance between laws be\n\\[\nd(\\mathcal L(X),\\mathcal L(Y)):=\\inf_{(X,Y)}\\mathbb E|X-Y|=\\sup_{h\\in \\mathrm{Lip}}\\big|\\mathbb Eh(X)-\\mathbb Eh(Y)\\big|,\n\\]\nwhere the supremum is over all $1$-Lipschitz functions $h$. For a fixed inverse temperature $\\beta<N$, which quantitative estimate holds?", "correct_choice": {"label": "A", "text": "There exists a finite constant $c(N,\\beta)$, depending only on $N$ and $\\beta$, such that for every $n\\ge 1$,\n\\[\nd\\big(\\mathcal L(W_n),\\mathcal L(Z)\\big)\\le \\frac{c(N,\\beta)}{\\sqrt n}.\n\\]"}, "choices": [{"label": "B", "text": "There exists a finite constant $c(N)$, depending only on $N$, such that for every $\\beta<N$ and every $n\\ge 1$,\n\\[\nd\\big(\\mathcal L(W_n),\\mathcal L(Z)\\big)\\le \\frac{c(N)}{\\sqrt n}.\n\\]"}, {"label": "C", "text": "For each fixed $\\beta<N$, one has\n\\[\nd\\big(\\mathcal L(W_n),\\mathcal L(Z)\\big)\\longrightarrow 0\\qquad\\text{as }n\\to\\infty.\n\\]"}, {"label": "D", "text": "There exists a finite constant $c(N,\\beta)$, depending only on $N$ and $\\beta$, such that for every $n\\ge 1$,\n\\[\nd\\big(\\mathcal L(W_n),\\mathcal L(Z)\\big)\\le \\frac{c(N,\\beta)}{n}.\n\\]"}, {"label": "E", "text": "There exists a finite constant $c(N,\\beta)<\\infty$, depending only on $N$ and $\\beta$, such that for every $n\\ge 1$,\n\\[\nd\\big(\\mathcal L\\!\\left(n^{-3/4}S_n\\right),\\mathcal L(Z)\\big)\\le \\frac{c(N,\\beta)}{\\sqrt n}.\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": "dependence of the constant on $\\beta$", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "explicit $n^{-1/2}$ Wasserstein rate and quantitative constant", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "sharp order of convergence $n^{-1/2}$", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "subcritical scaling/normal limit replaced by critical scaling", "template_used": "wildcard"}]}}
{"id": "2512.05828v1", "paper_link": "http://arxiv.org/abs/2512.05828v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{theorem: bound rango}\nFor every $k\\geq 2$ and every $d_1,\\dots,d_k\\geq 3$ we have\n$$\n\\rk_{\\mathbf{d}}(W_{d_1}\\otimes \\cdots \\otimes W_{d_k})\\leq  2^{k-1}\\left(d_1+\\cdots+d_k-2k+2\\right).\n$$", "start_pos": 11405, "end_pos": 11624, "label": "theorem: bound rango"}, "ref_dict": {"theorem: bound rango": "\\begin{theorem}\\label{theorem: bound rango}\nFor every $k\\geq 2$ and every $d_1,\\dots,d_k\\geq 3$ we have\n$$\n\\rk_{\\bfd}(W_{d_1}\\otimes \\cdots \\otimes W_{d_k})\\leq  2^{k-1}\\left(d_1+\\cdots+d_k-2k+2\\right).\n$$\n\\end{theorem}", "subsection: commenti sul bound": "\\label{subsection: commenti sul bound}\n\nAs already emphasized, the general idea of bounding the rank of tensor product of $W$-states by looking at the containment of $W_{d_1}\\otimes \\cdots \\otimes W_{", "remark: commento algo": "\\begin{remark}\\label{remark: commento algo}\nLet $\\bfd\\in\\NN^k_{\\geq 3}$. For $s\\in[0,d]\\setminus\\{d-k\\}$ we choose one $\\bf i \\in \\A_{\\bf d,s }$ and set $\\varepsilon^J_{\\bf i}=1$ for any $J\\subseteq[2,k]$, and for $s=d-k$ and $\\bfi=(d_1-1,\\dots,d_k-1)$ we set $\\varepsilon_\\bfi^J=1$ for any $J\\subseteq[2,k]$. In order to find a decomposition of $W_{d_1}\\otimes\\cdots\\otimes W_{d_k}$ following our procedure, we have to first find $T^J\\in\\spann(\\mathcal C_{\\bfd}^J)$ such that \n$$\\prod_{j=1}^kd_jW_{d_1}\\otimes\\cdots\\otimes W_{d_k}=\\sum_{J\\subseteq[2,k]}T^J.$$\nThe coordinates of the $T^J$'s are of the form\n$$T^J=(\\varepsilon_{\\bfi}^J\\alpha_s^J)_{\\substack{0\\leq s\\leq d\\\\ \\bfi\\in\\A_{\\bfd,s}}}$$\nand, by \\Cref{prop: rango TJ}, it is enough to take $\\alpha_{k-2}^J\\neq0$ and $\\alpha_{d-k}^J\\neq 0$ for every $J$ to minimize the length of the decomposition. The easiest possible choice is given by \\Cref{prop: mortale}: we take $\\alpha_s=0$ for $s\\in[0,d]\\setminus\\{k-2,d-k\\}$, $\\alpha_{k-2}=(-1)^{|J|}$ and $\\alpha_{d-k}=2^{k-1}$. Hence, when we compute $\\varphi^J(T^J)$ we only have two possibilities:\n$$\\varphi^J(T^J)=\\begin{cases}\n     \\binom{d}{k-2}u^{d-k+2}v^{k-2}+2^{k-1}\\binom{d}{d-k}u^kv^{d-k},&\\text{ if $|J|$ is even}\\\\\n     -\\binom{d}{k-2}u^{d-k+2}v^{k-2}+2^{k-1}\\binom{d}{d-k}u^kv^{d-k},&\\text{ otherwise.}\n\\end{cases}$$\nAs a consequence, it is enough to apply the Sylvester's algorithm to only two binary forms. Once a minimal decomposition of the two possible $\\varphi^J(T^J)$'s is found it is enough to apply $(\\varphi^J)^{-1}$ for all the $J\\subseteq[2,k]$ to find a partially symmetric decomposition of the product of $W$-states. Note that this drastically reduce the computational cost. Indeed, the decompositions of the two binary forms and the application of $(\\varphi^J)^{-1}$ are computationally cheap. The core of the advantage of this method is that the combinatorics of the $\\varepsilon_\\bfi^J$'s allows to avoid many computations. We summarized the procedure in \\Cref{algo}.\n\\end{remark}", "algo": "\\label{algo}\n\\end{algorithm}\n\\subsection{Further remarks on \\Cref{theorem: bound rango} }\\label{subsection: commenti sul bound}\n\nAs already emphasized, the general idea of bounding the rank of tensor", "theorem: bound rango bordo": "\\begin{theorem}\\label{theorem: bound rango bordo}\nLet $k\\geq 2$, let $\\bfd\\in \\NN_{\\geq 3}^k$ and let $d=d_1+\\cdots+d_k$.  Then\n$$\n\\underline{\\rk}_\\bfd(W_{d_1}\\otimes\\cdots\\otimes W_{d_k})= 2^{k}.\n$$\n\\end{theorem}", "prop: rango TJ": "\\begin{proposition}\\label{prop: rango TJ}\nLet $k\\geq 2$, let $\\bfd\\in \\NN_{\\geq 3}^k$ and let $d=d_1+\\cdots +d_k$. Let $\\mathcal{F}\\subseteq \\CC[u,v]_d$ be the family of degree $d$-homogenous polynomial defined as \n$$\n\\mathcal{F}= \\left\\{  F=\\sum_{i=0}^{k-2}a_i\\binom{d}{i}u^{d-i}v^i+\\binom{d}{d-k}a_{d-k}u^{k}v^{d-k}+\\sum_{i=d-k+2}^d a_i \\binom{d}{i}u^{d-i}v^i, \\text{ for }a_i\\in \\CC \\text{ with } a_{d-k}\\neq 0 \\right\\}.\n$$\nThen \n$$\n\\min_{F\\in\\mathcal F}\\rk F=d-2k+2.\n$$\nIn particular, if  $2k\\leq\\left\\lfloor \\frac d2\\right\\rfloor +1$ the minimum is achieved by any $F\\in\\mathcal F$ with $a_{k-2}\\neq 0$. Otherwise, the minimum is achieved by any $F\\in \\mathcal{F}$ with $a_{k-2}\\neq 0$ and $a_i=0$ for $i\\in \\{0,\\dots,k-3\\} \\cup \\{d-k+2,\\dots,d \\}$.\n\\end{proposition}"}, "pre_theorem_intro_text_len": 5580, "pre_theorem_intro_text": "Computing the rank of a tensor is generally NP-hard, see \\cite{Has90, rankNPhard}. For this reason, one often focuses on special families of tensors that serve as meaningful benchmarks either because they originate from applications, or because they exhibit particularly interesting behavior. One such family is the tensor product of generalized $W$-state: $$W_{d_1}\\otimes \\cdots \\otimes W_{d_k}\\in \\underbrace{\\mathbb{C}^2\\otimes \\cdots \\otimes \\mathbb{C}^2}_{d_1+\\cdots +d_k-\\text{times}}\n$$ \nwhere each $W$-state $W_d$ is defined as\n$$\nW_d=\\sum_{i=1}^d e^{(1)}_{1}\\otimes \\cdots \\otimes e^{(i-1)}_1\\otimes e^{(i)}_2 \\otimes e^{(i+1)}_1\\otimes \\cdots \\otimes e^{(d)}_1 \\in (\\mathbb{C}^2)^{\\otimes d}, \\quad \\spann\\{e^{(i)}_1,e^{(i)}_2 \\}\\cong \\mathbb{C}^2.\n$$ \nFocusing for a moment on a single copy, the generalized $W$-state itself is a highly interesting tensor. It is a symmetric tensor and one of the simplest examples illustrating the non-semicontinuity of tensor rank. Moreover, the expression of the general element of the tangential variety $\\tau(\\nu((\\mathbb{P}^1)^{\\times d}))$ of the Segre image $\\nu((\\mathbb{P}^1)^{\\times d})$ is $W_d$. This is because actually $W_d\\in \\langle \\nu(Z)\\rangle$, where $Z\\subset (\\mathbb{P}^1)^{\\times d}$ is a zero-dimensional scheme of length 2 supported at $\\otimes_{i=1}^d e^{(i)}_{1} $, called a 2-jet \\cite{alexander1997interpolation}, so actually $W_d$ lies on a line that is tangent to $\\nu((\\mathbb{P}^1)^{\\times d})$ at $\\otimes_{i=1}^d e^{(i)}_{1} $. The $W$-state is a rank-$d$ tensor and it is an example of tensor for which tensor rank and symmetric tensor rank coincide \\cite{rankTangentialBB}. It has infinitely many decompositions computing its rank and in particular, one can choose any elementary tensor except for $\\otimes_{i=1}^d e^{(i)}_{1}$ to be part of one of its minimal (symmetric) rank decomposition \\cite{carlini2017waring, BOS}. \n\n From the point of view of complexity theory, $W$-states are related to the so-called Coppersmith-Winogard tensors \\cite{coppersmith1987matrix}, which are tensors of interest in the study of the matrix multiplication tensor. In particular, $W$-states are the outer structure of many tensors, including Strassen tensor and truncated polynomial multiplication, used in the so-called \\emph{laser method}, a strategy introduced by V. Strassen in \\cite{Str86} for computing the complexity of matrix multiplication.\n\nIn the three-factors case $W_3$ plays a historical central role in quantum information theory, being one of the two fundamental classes of genuine tripartite entanglement \\cite{dur2000three}, and this is where the name \\emph{$W$-state} comes from. This makes the $W$-state a structurally very rich object at the intersection of geometry, algebraic complexity, and quantum information.\n\n Given how special $W_d$ is, it is natural to expect that the tensor product of several copies, $W_{d_1}\\otimes \\cdots \\otimes W_{d_k} $, is also quite special. This expectation is reinforced for instance by the role that the tensor product of two copies of $W$-states had in the study of the multiplicativity of tensor rank under tensor product: indeed, \\cite{CJZ} proved that the rank of $W_3\\otimes W_3$ is strictly less than the naive multiplicative guess $3\\cdot 3$. This was the first explicit example showing that the rank of the tensor product of two tensors is not the product of the two ranks. Shortly after, \\cite{CF18} proved that the rank of  $W_3\\otimes W_3$ is 8. A systematic approach to the strict submulticativity property can be found in \\cite{BBGOV}, while, motivated by applications in complexity theory and quantum information theory, the submultiplicativity of the Kronecker tensor product of many copies of $W$-states has been studied for instance in \\cite{chen2010tensor, zuiddam2017note}. Some Kronecker products of $W$-states show also other interesting connections. For instance, $W_3\\boxtimes W_3$ is a symmetric tensor corresponding to the cubic hypersurface of $\\mathbb{P}^3$ made of a quadric and a tangent hyperplane and, by \\cite{Seg42}, this is the only quaternary cubic having maximal rank 7. As a second example, a particular projection of $W_3\\boxtimes W_3\\boxtimes W_3$ is the so-called Bjorklund-Kaski tensor, which appears in \\cite{BK24} and is used for proving that the asymptotic rank conjecture \\cite{Str94} and the set cover conjecture \\cite{CFKLMMPS15, KT19} cannot be both true.\n\n Hence, determining the rank of $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$ becomes a particularly meaningful problem. \\cite{BBCG} gave the first bounds on the rank of the tensor product of an arbitrary number of $W$-states \n and these remain the best known results to date, with the notable exception of the case $k=2$ where the bound was later improved in \\cite[Theorem 1.8(i)]{gal}. \n\n  In this paper, we make progress along this direction by providing a new sharp upper bound on the rank of tensor products of generalized $W$-states $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}\\in \\bigotimes_{i=1}^k(\\mathbb{C}^{2})^{\\otimes d_i}$. We remark that $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$ is a partially symmetric tensor and for a partially symmetric tensor $T\\in \\mathop{\\rm Sym}\\nolimits^{d_1}\\mathbb{C}^{n_1}\\otimes \\cdots \\otimes \\mathop{\\rm Sym}\\nolimits^{d_k}\\mathbb{C}^{n_k}$ the partially symmetric rank of $T$, denoted as $\\rk_{\\mathbf{d}}(T)$, is the minimum integer $r$ such that $T=\\sum_{i=1}^r (v^{(1)}_i)^{\\otimes d_1}\\otimes \\cdots \\otimes (v^{(k)}_i)^{\\otimes d_k}$, where $v^{(j)}_i\\in \\mathbb{C}^{n_j}$. Our main result is the following.", "context": "Computing the rank of a tensor is generally NP-hard, see \\cite{Has90, rankNPhard}. For this reason, one often focuses on special families of tensors that serve as meaningful benchmarks either because they originate from applications, or because they exhibit particularly interesting behavior. One such family is the tensor product of generalized $W$-state: $$W_{d_1}\\otimes \\cdots \\otimes W_{d_k}\\in \\underbrace{\\mathbb{C}^2\\otimes \\cdots \\otimes \\mathbb{C}^2}_{d_1+\\cdots +d_k-\\text{times}}\n$$ \nwhere each $W$-state $W_d$ is defined as\n$$\nW_d=\\sum_{i=1}^d e^{(1)}_{1}\\otimes \\cdots \\otimes e^{(i-1)}_1\\otimes e^{(i)}_2 \\otimes e^{(i+1)}_1\\otimes \\cdots \\otimes e^{(d)}_1 \\in (\\mathbb{C}^2)^{\\otimes d}, \\quad \\spann\\{e^{(i)}_1,e^{(i)}_2 \\}\\cong \\mathbb{C}^2.\n$$ \nFocusing for a moment on a single copy, the generalized $W$-state itself is a highly interesting tensor. It is a symmetric tensor and one of the simplest examples illustrating the non-semicontinuity of tensor rank. Moreover, the expression of the general element of the tangential variety $\\tau(\\nu((\\mathbb{P}^1)^{\\times d}))$ of the Segre image $\\nu((\\mathbb{P}^1)^{\\times d})$ is $W_d$. This is because actually $W_d\\in \\langle \\nu(Z)\\rangle$, where $Z\\subset (\\mathbb{P}^1)^{\\times d}$ is a zero-dimensional scheme of length 2 supported at $\\otimes_{i=1}^d e^{(i)}_{1} $, called a 2-jet \\cite{alexander1997interpolation}, so actually $W_d$ lies on a line that is tangent to $\\nu((\\mathbb{P}^1)^{\\times d})$ at $\\otimes_{i=1}^d e^{(i)}_{1} $. The $W$-state is a rank-$d$ tensor and it is an example of tensor for which tensor rank and symmetric tensor rank coincide \\cite{rankTangentialBB}. It has infinitely many decompositions computing its rank and in particular, one can choose any elementary tensor except for $\\otimes_{i=1}^d e^{(i)}_{1}$ to be part of one of its minimal (symmetric) rank decomposition \\cite{carlini2017waring, BOS}.\n\nFrom the point of view of complexity theory, $W$-states are related to the so-called Coppersmith-Winogard tensors \\cite{coppersmith1987matrix}, which are tensors of interest in the study of the matrix multiplication tensor. In particular, $W$-states are the outer structure of many tensors, including Strassen tensor and truncated polynomial multiplication, used in the so-called \\emph{laser method}, a strategy introduced by V. Strassen in \\cite{Str86} for computing the complexity of matrix multiplication.\n\nIn the three-factors case $W_3$ plays a historical central role in quantum information theory, being one of the two fundamental classes of genuine tripartite entanglement \\cite{dur2000three}, and this is where the name \\emph{$W$-state} comes from. This makes the $W$-state a structurally very rich object at the intersection of geometry, algebraic complexity, and quantum information.\n\nGiven how special $W_d$ is, it is natural to expect that the tensor product of several copies, $W_{d_1}\\otimes \\cdots \\otimes W_{d_k} $, is also quite special. This expectation is reinforced for instance by the role that the tensor product of two copies of $W$-states had in the study of the multiplicativity of tensor rank under tensor product: indeed, \\cite{CJZ} proved that the rank of $W_3\\otimes W_3$ is strictly less than the naive multiplicative guess $3\\cdot 3$. This was the first explicit example showing that the rank of the tensor product of two tensors is not the product of the two ranks. Shortly after, \\cite{CF18} proved that the rank of $W_3\\otimes W_3$ is 8. A systematic approach to the strict submulticativity property can be found in \\cite{BBGOV}, while, motivated by applications in complexity theory and quantum information theory, the submultiplicativity of the Kronecker tensor product of many copies of $W$-states has been studied for instance in \\cite{chen2010tensor, zuiddam2017note}. Some Kronecker products of $W$-states show also other interesting connections. For instance, $W_3\\boxtimes W_3$ is a symmetric tensor corresponding to the cubic hypersurface of $\\mathbb{P}^3$ made of a quadric and a tangent hyperplane and, by \\cite{Seg42}, this is the only quaternary cubic having maximal rank 7. As a second example, a particular projection of $W_3\\boxtimes W_3\\boxtimes W_3$ is the so-called Bjorklund-Kaski tensor, which appears in \\cite{BK24} and is used for proving that the asymptotic rank conjecture \\cite{Str94} and the set cover conjecture \\cite{CFKLMMPS15, KT19} cannot be both true.\n\nHence, determining the rank of $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$ becomes a particularly meaningful problem. \\cite{BBCG} gave the first bounds on the rank of the tensor product of an arbitrary number of $W$-states \n and these remain the best known results to date, with the notable exception of the case $k=2$ where the bound was later improved in \\cite[Theorem 1.8(i)]{gal}.\n\nIn this paper, we make progress along this direction by providing a new sharp upper bound on the rank of tensor products of generalized $W$-states $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}\\in \\bigotimes_{i=1}^k(\\mathbb{C}^{2})^{\\otimes d_i}$. We remark that $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$ is a partially symmetric tensor and for a partially symmetric tensor $T\\in \\mathop{\\rm Sym}\\nolimits^{d_1}\\mathbb{C}^{n_1}\\otimes \\cdots \\otimes \\mathop{\\rm Sym}\\nolimits^{d_k}\\mathbb{C}^{n_k}$ the partially symmetric rank of $T$, denoted as $\\rk_{\\mathbf{d}}(T)$, is the minimum integer $r$ such that $T=\\sum_{i=1}^r (v^{(1)}_i)^{\\otimes d_1}\\otimes \\cdots \\otimes (v^{(k)}_i)^{\\otimes d_k}$, where $v^{(j)}_i\\in \\mathbb{C}^{n_j}$. Our main result is the following.", "full_context": "Computing the rank of a tensor is generally NP-hard, see \\cite{Has90, rankNPhard}. For this reason, one often focuses on special families of tensors that serve as meaningful benchmarks either because they originate from applications, or because they exhibit particularly interesting behavior. One such family is the tensor product of generalized $W$-state: $$W_{d_1}\\otimes \\cdots \\otimes W_{d_k}\\in \\underbrace{\\mathbb{C}^2\\otimes \\cdots \\otimes \\mathbb{C}^2}_{d_1+\\cdots +d_k-\\text{times}}\n$$ \nwhere each $W$-state $W_d$ is defined as\n$$\nW_d=\\sum_{i=1}^d e^{(1)}_{1}\\otimes \\cdots \\otimes e^{(i-1)}_1\\otimes e^{(i)}_2 \\otimes e^{(i+1)}_1\\otimes \\cdots \\otimes e^{(d)}_1 \\in (\\mathbb{C}^2)^{\\otimes d}, \\quad \\spann\\{e^{(i)}_1,e^{(i)}_2 \\}\\cong \\mathbb{C}^2.\n$$ \nFocusing for a moment on a single copy, the generalized $W$-state itself is a highly interesting tensor. It is a symmetric tensor and one of the simplest examples illustrating the non-semicontinuity of tensor rank. Moreover, the expression of the general element of the tangential variety $\\tau(\\nu((\\mathbb{P}^1)^{\\times d}))$ of the Segre image $\\nu((\\mathbb{P}^1)^{\\times d})$ is $W_d$. This is because actually $W_d\\in \\langle \\nu(Z)\\rangle$, where $Z\\subset (\\mathbb{P}^1)^{\\times d}$ is a zero-dimensional scheme of length 2 supported at $\\otimes_{i=1}^d e^{(i)}_{1} $, called a 2-jet \\cite{alexander1997interpolation}, so actually $W_d$ lies on a line that is tangent to $\\nu((\\mathbb{P}^1)^{\\times d})$ at $\\otimes_{i=1}^d e^{(i)}_{1} $. The $W$-state is a rank-$d$ tensor and it is an example of tensor for which tensor rank and symmetric tensor rank coincide \\cite{rankTangentialBB}. It has infinitely many decompositions computing its rank and in particular, one can choose any elementary tensor except for $\\otimes_{i=1}^d e^{(i)}_{1}$ to be part of one of its minimal (symmetric) rank decomposition \\cite{carlini2017waring, BOS}.\n\nFrom the point of view of complexity theory, $W$-states are related to the so-called Coppersmith-Winogard tensors \\cite{coppersmith1987matrix}, which are tensors of interest in the study of the matrix multiplication tensor. In particular, $W$-states are the outer structure of many tensors, including Strassen tensor and truncated polynomial multiplication, used in the so-called \\emph{laser method}, a strategy introduced by V. Strassen in \\cite{Str86} for computing the complexity of matrix multiplication.\n\nIn the three-factors case $W_3$ plays a historical central role in quantum information theory, being one of the two fundamental classes of genuine tripartite entanglement \\cite{dur2000three}, and this is where the name \\emph{$W$-state} comes from. This makes the $W$-state a structurally very rich object at the intersection of geometry, algebraic complexity, and quantum information.\n\nGiven how special $W_d$ is, it is natural to expect that the tensor product of several copies, $W_{d_1}\\otimes \\cdots \\otimes W_{d_k} $, is also quite special. This expectation is reinforced for instance by the role that the tensor product of two copies of $W$-states had in the study of the multiplicativity of tensor rank under tensor product: indeed, \\cite{CJZ} proved that the rank of $W_3\\otimes W_3$ is strictly less than the naive multiplicative guess $3\\cdot 3$. This was the first explicit example showing that the rank of the tensor product of two tensors is not the product of the two ranks. Shortly after, \\cite{CF18} proved that the rank of $W_3\\otimes W_3$ is 8. A systematic approach to the strict submulticativity property can be found in \\cite{BBGOV}, while, motivated by applications in complexity theory and quantum information theory, the submultiplicativity of the Kronecker tensor product of many copies of $W$-states has been studied for instance in \\cite{chen2010tensor, zuiddam2017note}. Some Kronecker products of $W$-states show also other interesting connections. For instance, $W_3\\boxtimes W_3$ is a symmetric tensor corresponding to the cubic hypersurface of $\\mathbb{P}^3$ made of a quadric and a tangent hyperplane and, by \\cite{Seg42}, this is the only quaternary cubic having maximal rank 7. As a second example, a particular projection of $W_3\\boxtimes W_3\\boxtimes W_3$ is the so-called Bjorklund-Kaski tensor, which appears in \\cite{BK24} and is used for proving that the asymptotic rank conjecture \\cite{Str94} and the set cover conjecture \\cite{CFKLMMPS15, KT19} cannot be both true.\n\nHence, determining the rank of $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$ becomes a particularly meaningful problem. \\cite{BBCG} gave the first bounds on the rank of the tensor product of an arbitrary number of $W$-states \n and these remain the best known results to date, with the notable exception of the case $k=2$ where the bound was later improved in \\cite[Theorem 1.8(i)]{gal}.\n\nIn this paper, we make progress along this direction by providing a new sharp upper bound on the rank of tensor products of generalized $W$-states $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}\\in \\bigotimes_{i=1}^k(\\mathbb{C}^{2})^{\\otimes d_i}$. We remark that $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$ is a partially symmetric tensor and for a partially symmetric tensor $T\\in \\mathop{\\rm Sym}\\nolimits^{d_1}\\mathbb{C}^{n_1}\\otimes \\cdots \\otimes \\mathop{\\rm Sym}\\nolimits^{d_k}\\mathbb{C}^{n_k}$ the partially symmetric rank of $T$, denoted as $\\rk_{\\mathbf{d}}(T)$, is the minimum integer $r$ such that $T=\\sum_{i=1}^r (v^{(1)}_i)^{\\otimes d_1}\\otimes \\cdots \\otimes (v^{(k)}_i)^{\\otimes d_k}$, where $v^{(j)}_i\\in \\mathbb{C}^{n_j}$. Our main result is the following.\n\nIn this paper, we make progress along this direction by providing a new sharp upper bound on the rank of tensor products of generalized $W$-states $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}\\in \\bigotimes_{i=1}^k(\\CC^{2})^{\\otimes d_i}$. We remark that $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$ is a partially symmetric tensor and for a partially symmetric tensor $T\\in \\sym^{d_1}\\CC^{n_1}\\otimes \\cdots \\otimes \\sym^{d_k}\\CC^{n_k}$ the partially symmetric rank of $T$, denoted as $\\rk_{\\bfd}(T)$, is the minimum integer $r$ such that $T=\\sum_{i=1}^r (v^{(1)}_i)^{\\otimes d_1}\\otimes \\cdots \\otimes (v^{(k)}_i)^{\\otimes d_k}$, where $v^{(j)}_i\\in \\CC^{n_j}$. Our main result is the following.\n\nIn addition to providing a theoretical upper bound on the tensor rank, \\Cref{theorem: bound rango} is constructive. Indeed, the proof yields an explicit decomposition of length equal to the bound that can be achieved by computing the rank decomposition of essentially one binary form (see \\Cref{algo}). This decreases a lot the computational cost for computing a partially symmetric decomposition of $W_{d_1}\\otimes\\cdots\\otimes W_{d_k}$; see the forthcoming \\Cref{remark: commento algo}.\n\nAs an immediate consequence the same bound holds also for the rank of tensor product of $W$-states. \\begin{corollary}\\label{corollary: bound rango vero}\nLet $k\\geq 2$, fix positive integers $d_i\\geq 3$ for $i=1,\\dots,k$. \nThe rank of $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}\\in \\bigotimes_{i=1}^k(\\CC^{2})^{\\otimes d_i}$ is at most $2^{k-1}(d_1+\\dots+d_k-2k+2)$.\n\\end{corollary}\n\nWe have now all the tools to prove our main \\Cref{theorem: bound rango}, which we restate below for convenience.\n\\begin{theorem*}(\\Cref{theorem: bound rango}) Let $k\\geq 2$, $\\bfd\\in \\NN_{\\geq 3}^k$ and $d=d_1+\\cdots +d_k$. Then\n$$\n\\rk_{\\bfd}(W_{d_1}\\otimes \\cdots \\otimes W_{d_k})\\leq 2^{k-1}(d-2k+2).\n$$\n\\end{theorem*}\n\\begin{proof}\nBy \\Cref{remark: W-state sta nel quadrello} $W_{d_1}\\otimes\\cdots \\otimes W_{d_k}$ belongs to $\\spann\\left\\{ \\cup_{J\\subseteq [2,k]} \\mathcal{C}_{\\bfd}^J\\right\\} $, so we have\n$$\n\\prod_{j=1}^kd_j W_{d_1}\\otimes\\cdots \\otimes W_{d_k}=\\sum_{J\\subseteq [2,k]}T^J, \\,\\hbox{ for some }T^J\\in\\spann\\mathcal{C}_{\\bfd}^J. \n$$\nWriting $T^J$ with respect to the basis of \\Cref{remark: multi hom binary in forma binaria emplice}, we have\n$$T^J=\\sum_{s=0}^{d}\\alpha_s^J\\left(\\sum_{\\bfi\\in\\mathcal A_{\\bfd,s}}\\varepsilon_\\bfi^Jx_{1,0}^{i_1}x_{1,1}^{d_1-i_1}\\otimes\\cdots\\otimes x_{k,0}^{i_k}x_{k,1}^{d_k-i_k}\\right),$$\nwhere the $\\alpha^J_s$'s are a solution of the linear system $\\mathcal S$ of \\Cref{prop: mortale}.\nUsing the isomorphism $\\varphi^J$ defined in \\Cref{remark: multi hom binary in forma binaria emplice}, each $T^J$ can be seen as a binary form of degree $d$ in $\\spann\\mathcal C_\\bfd^J$. More precisely, we have\n$$\\varphi^J(T^J)=\\sum_{s=0}^d\\alpha_s^J\\binom{d}{s}u^{d-s}v^s.$$\nIt follows that $\\rk_\\bfd T^J\\leq\\rk\\varphi^J(T^J)$, where $\\rk\\varphi^J(T^J)$ is the Waring rank of the binary form $\\varphi^J(T^J)$ and, by \\Cref{remark: multi hom binary in forma binaria emplice}, it is equal to the $\\mathcal C_\\bfd^J$-rank of $T^J$.\nBy \\Cref{prop: mortale}, we have $\\alpha_s^J=0$ for any $s\\in[k-1,d-k-1]\\cup\\left\\{d-k+1\\right\\}$ and $\\alpha^J_{d-k}\\neq 0$. Moreover, whether $2k\\leq\\left\\lfloor\\frac{d}{2}\\right\\rfloor+1$ or $2k>\\left\\lfloor\\frac{d}{2}\\right\\rfloor+1$, by \\Cref{prop: mortale} we can choose the $\\alpha_s^J$'s such that $\\varphi^J(T^J)$ is a minimizer as in \\Cref{prop: rango TJ}, so that $\\rk (\\varphi^J(T^J))=d-2k+2$. Hence, we have\n\\begin{equation*}\n\\rk_{\\bfd}(W_{d_1}\\otimes \\cdots \\otimes W_{d_k})\\leq\\sum_{J\\subseteq[2,k]}\\rk_\\bfd T^J\\leq \\sum_{J\\subseteq[2,k]}\\rk \\varphi^J(T^J)= 2^{k-1}(d-2k+2).\\qedhere\n\\end{equation*}\n\\end{proof}\nSince for a tensor $T\\in \\sym^{d_1}\\CC^2\\otimes \\cdots \\otimes \\sym^{d_k}\\CC^2$ we have $\\rk(T)\\leq \\rk_{\\bf d}(T)$, the previous result gives a bound also on the tensor rank of $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$.\n\n\\Cref{example: step4}, together with \\Cref{example: step1}, \\Cref{example: step2} and \\Cref{example: step3}, concretely shows that, following our procedure, it is possible to construct an explicit decomposition of $W_{d_1}\\otimes\\cdots\\otimes W_{d_k}$ having length as in the bound of \\Cref{theorem: bound rango} as we now explain in detail.\n\\begin{remark}\\label{remark: commento algo}\nLet $\\bfd\\in\\NN^k_{\\geq 3}$. For $s\\in[0,d]\\setminus\\{d-k\\}$ we choose one $\\bf i \\in \\A_{\\bf d,s }$ and set $\\varepsilon^J_{\\bf i}=1$ for any $J\\subseteq[2,k]$, and for $s=d-k$ and $\\bfi=(d_1-1,\\dots,d_k-1)$ we set $\\varepsilon_\\bfi^J=1$ for any $J\\subseteq[2,k]$. In order to find a decomposition of $W_{d_1}\\otimes\\cdots\\otimes W_{d_k}$ following our procedure, we have to first find $T^J\\in\\spann(\\mathcal C_{\\bfd}^J)$ such that \n$$\\prod_{j=1}^kd_jW_{d_1}\\otimes\\cdots\\otimes W_{d_k}=\\sum_{J\\subseteq[2,k]}T^J.$$\nThe coordinates of the $T^J$'s are of the form\n$$T^J=(\\varepsilon_{\\bfi}^J\\alpha_s^J)_{\\substack{0\\leq s\\leq d\\\\ \\bfi\\in\\A_{\\bfd,s}}}$$\nand, by \\Cref{prop: rango TJ}, it is enough to take $\\alpha_{k-2}^J\\neq0$ and $\\alpha_{d-k}^J\\neq 0$ for every $J$ to minimize the length of the decomposition. The easiest possible choice is given by \\Cref{prop: mortale}: we take $\\alpha_s=0$ for $s\\in[0,d]\\setminus\\{k-2,d-k\\}$, $\\alpha_{k-2}=(-1)^{|J|}$ and $\\alpha_{d-k}=2^{k-1}$. Hence, when we compute $\\varphi^J(T^J)$ we only have two possibilities:\n$$\\varphi^J(T^J)=\\begin{cases}\n \\binom{d}{k-2}u^{d-k+2}v^{k-2}+2^{k-1}\\binom{d}{d-k}u^kv^{d-k},&\\text{ if $|J|$ is even}\\\\\n -\\binom{d}{k-2}u^{d-k+2}v^{k-2}+2^{k-1}\\binom{d}{d-k}u^kv^{d-k},&\\text{ otherwise.}\n\\end{cases}$$\nAs a consequence, it is enough to apply the Sylvester's algorithm to only two binary forms. Once a minimal decomposition of the two possible $\\varphi^J(T^J)$'s is found it is enough to apply $(\\varphi^J)^{-1}$ for all the $J\\subseteq[2,k]$ to find a partially symmetric decomposition of the product of $W$-states. Note that this drastically reduce the computational cost. Indeed, the decompositions of the two binary forms and the application of $(\\varphi^J)^{-1}$ are computationally cheap. The core of the advantage of this method is that the combinatorics of the $\\varepsilon_\\bfi^J$'s allows to avoid many computations. We summarized the procedure in \\Cref{algo}.\n\\end{remark}\n\n\\begin{theorem}\\label{theorem: bound rango bordo}\nLet $k\\geq 2$, let $\\bfd\\in \\NN_{\\geq 3}^k$ and let $d=d_1+\\cdots+d_k$. Then\n$$\n\\underline{\\rk}_\\bfd(W_{d_1}\\otimes\\cdots\\otimes W_{d_k})= 2^{k}.\n$$\n\\end{theorem}\n\\begin{proof}\nWe use the catalecticant bound for border rank of \\cite[Corollary 5.5]{gal}. In order to do that we have to produce a partially symmetric flattening with rank at least $2^k$. Let us consider the following flattening of $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$:\n\\[\nF : \\sym^2(\\mathbb{C}^2) \\otimes \\dots \\otimes \\sym^2(\\mathbb{C}^2) \\rightarrow \n\\sym^{d_1-2}(\\mathbb{C}^2)^* \\otimes \\dots \\otimes \\sym^{d_k-2}(\\mathbb{C}^2)^*.\n\\]\nLet $\\mathbb{C}[\\alpha_{1,0},\\alpha_{1,1},\\dots,\\alpha_{k,0},\\alpha_{k,1}]$ be the ring of multigraded derivations on $\\CC[x_{1,0},x_{1,1},\\dots,x_{k,0},x_{k,1}]$ with $\\alpha_{i,j}$ defined as the dual of $x_{i,j}$. By apolarity theory, the rank of the linear map $F$ is exactly the dimension of the degree $(2,\\dots,2)$ component of the quotient of the multigraded ring $\\mathbb{C}[\\alpha_{1,0},\\alpha_{1,1},\\dots,\\alpha_{k,0},\\alpha_{k,1}]$ by the ideal $I_{d}$ apolar to the W-state, i.e. $I_{d}=(\\alpha_{1,0}^2,\\dots,\\alpha_{k,0}^2,\\alpha_{1,1}^{d_1},\\dots,\\alpha_{k,1}^{d_k})$. For $\\varepsilon_i \\in \\{0,1\\}$ the $2^k$ forms\n$$F_{\\varepsilon_1,\\dots,\\varepsilon_{k}}=\\alpha_{1,0}^{\\varepsilon_1}\\alpha_{1,1}^{2-\\varepsilon_1} \\dots \\alpha_{k,0}^{\\varepsilon_k}\\alpha_{k,1}^{2-\\varepsilon_k}$$ are linear independent of multidegree $(2,\\dots, 2)$, with $F_{\\varepsilon_1,\\dots,\\varepsilon_{k}} \\notin I_d$. This concludes the proof. \n\\end{proof}", "post_theorem_intro_text_len": 2935, "post_theorem_intro_text": "\\Cref{theorem: bound rango} improves \\cite[Theorem 3.6]{BBCG} by $2^k(k-1)$. For $k=2$ it coincides with the bound provided in \\cite[Theorem 1.8(i)]{gal} and when all $d_i=3$ it matches the bound of \\cite[Theorem 3.3]{BBCG}. The bound of \\Cref{theorem: bound rango} is sharp as for $k=2$ and $d_1=d_2=3$ it gives rank 8.\n\n In addition to providing a theoretical upper bound on the tensor rank, \\Cref{theorem: bound rango} is constructive. Indeed, the proof yields an explicit decomposition of length equal to the bound that can be achieved by computing the rank decomposition of essentially one binary form (see \\Cref{algo}). This decreases a lot the computational cost for computing a partially symmetric decomposition of $W_{d_1}\\otimes\\cdots\\otimes W_{d_k}$; see the forthcoming \\Cref{remark: commento algo}.\n\n As an immediate consequence the same bound holds also for the rank of tensor product of $W$-states. \\begin{corollary}\\label{corollary: bound rango vero}\nLet $k\\geq 2$, fix positive integers $d_i\\geq 3$ for $i=1,\\dots,k$. \nThe rank of $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}\\in \\bigotimes_{i=1}^k(\\mathbb{C}^{2})^{\\otimes d_i}$ is at most $2^{k-1}(d_1+\\dots+d_k-2k+2)$.\n\\end{corollary}\n\nOur methods to achieve this result are geometric and, as in \\cite[Theorem 3.6]{BBCG}, they rely on finding a curve in $\\mathcal{C}\\subset (\\mathbb{P}^1)^{\\times k}$ such that the span of the image of $\\mathcal{C}$ under the Segre-Veronese embedding contains $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$. \n\n\\subsection*{Outline of the paper}  We set up our notation and give standard preliminaries in \\Cref{section: preliminaries}, while \\Cref{section: solving the system} contains the main technical results to get our bounds. We start \\Cref{section: computing the bounds} by briefly reviewing the theory of catalecticants needed to compute the rank of a family of binary homogeneous polynomials (\\Cref{prop: rango TJ}). Then, we prove our main \\Cref{theorem: bound rango} and in \\Cref{algo} we give a recipe to explicitly compute a partially symmetric decomposition of length given by our bound. \\Cref{theorem: bound rango bordo} computes the border partially symmetric rank of $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$. We conclude with a detailed comment on the method in \\Cref{subsection: commenti sul bound}.\n\n\\subsection*{Acknowledgements}\nThe authors are grateful to Fulvio Gesmundo, Alessandro Gimigliano and Joachim Jelisiejew for interesting conversations on the topic.\n\nS. Canino has been funded by the Italian Ministry of\nUniversity and Research in the framework of the Call for Proposals for\nscrolling of final rankings of the PRIN 2022 call - Protocol no.\n2022NBN7TL.\nA. Casarotti has been funded by the European Union under the project NextGenerationEU. PRIN 2022, CUP: F53D23002600006.\nP. Santarsiero was supported by the European Union under NextGenerationEU. PRIN 2022, Prot. 2022E2Z4AK and PRIN 2022 SC-CUP: I53C24002240006.", "sketch": "The post-theorem text does not give a step-by-step proof sketch for \\Cref{theorem: bound rango}, but it indicates the proof’s nature and main idea: the result is obtained by “geometric” methods and “rely on finding a curve in $\\mathcal{C}\\subset (\\mathbb{P}^1)^{\\times k}$ such that the span of the image of $\\mathcal{C}$ under the Segre-Veronese embedding contains $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$.” It also states the proof is “constructive,” yielding “an explicit decomposition of length equal to the bound” that “can be achieved by computing the rank decomposition of essentially one binary form (see \\Cref{algo}).”", "expanded_sketch": "The post-theorem text does not give a step-by-step proof sketch for the main theorem, but it indicates the proof’s nature and main idea: the result is obtained by “geometric” methods and “rely on finding a curve in $\\mathcal{C}\\subset (\\mathbb{P}^1)^{\\times k}$ such that the span of the image of $\\mathcal{C}$ under the Segre-Veronese embedding contains $W_{d_1}\\otimes \\cdots \\otimes W_{d_k}$.” It also states the proof is “constructive,” yielding “an explicit decomposition of length equal to the bound” that “can be achieved by computing the rank decomposition of essentially one binary form (see\n\\label{algo}\n\\end{algorithm}\n\\subsection{Further remarks on \\Cref{theorem: bound rango} }\\label{subsection: commenti sul bound}\n\nAs already emphasized, the general idea of bounding the rank of tensor).”", "expanded_theorem": "\\label{theorem: bound rango}\nFor every $k\\geq 2$ and every $d_1,\\dots,d_k\\geq 3$ we have\n$$\n\\rk_{\\mathbf{d}}(W_{d_1}\\otimes \\cdots \\otimes W_{d_k})\\leq  2^{k-1}\\left(d_1+\\cdots+d_k-2k+2\\right).\n$$", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Let \\(k\\ge 2\\) and let \\(d_1,\\dots,d_k\\ge 3\\) be integers, and write \\(\\mathbf d=(d_1,\\dots,d_k)\\). For each \\(d\\), define the generalized \\(W\\)-state\n\\[\nW_d=\\sum_{i=1}^d e^{(1)}_1\\otimes\\cdots\\otimes e^{(i-1)}_1\\otimes e^{(i)}_2\\otimes e^{(i+1)}_1\\otimes\\cdots\\otimes e^{(d)}_1\\in (\\mathbb C^2)^{\\otimes d},\n\\]\nwhere each \\(\\operatorname{span}\\{e^{(i)}_1,e^{(i)}_2\\}\\cong \\mathbb C^2\\). View\n\\(W_{d_1}\\otimes\\cdots\\otimes W_{d_k}\\) as an element of\n\\(\\operatorname{Sym}^{d_1}(\\mathbb C^2)\\otimes\\cdots\\otimes \\operatorname{Sym}^{d_k}(\\mathbb C^2)\\), and let \\(\\operatorname{rk}_{\\mathbf d}\\) denote its partially symmetric rank, i.e. the minimum number of summands of the form \\(\\ell_1^{d_1}\\otimes\\cdots\\otimes \\ell_k^{d_k}\\) needed to express it. Which statement holds for every such choice of \\(k,d_1,\\dots,d_k\\)?", "correct_choice": {"label": "A", "text": "\\[\\operatorname{rk}_{\\mathbf d}(W_{d_1}\\otimes\\cdots\\otimes W_{d_k})\\le 2^{k-1}\\bigl(d_1+\\cdots+d_k-2k+2\\bigr).\\]"}, "choices": [{"label": "B", "text": "\\[\\operatorname{rk}_{\\mathbf d}(W_{d_1}\\otimes\\cdots\\otimes W_{d_k})\\le 2^{k}\\bigl(d_1+\\cdots+d_k-2k+2\\bigr).\\]"}, {"label": "C", "text": "\\[\\operatorname{rk}_{\\mathbf d}(W_{d_1}\\otimes\\cdots\\otimes W_{d_k})\\le 2^{k-1}(d_1+\\cdots+d_k).\\]"}, {"label": "D", "text": "\\[\\operatorname{rk}_{\\mathbf d}(W_{d_1}\\otimes\\cdots\\otimes W_{d_k})\\le 2^{k-1}\\bigl(d_1\\cdots d_k-2k+2\\bigr).\\]"}, {"label": "E", "text": "\\[\\operatorname{rk}_{\\mathbf d}(W_{d_1}\\otimes\\cdots\\otimes W_{d_k})= 2^{k-1}\\bigl(d_1+\\cdots+d_k-2k+2\\bigr).\\]"}], "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": "sharp_constant_2^{k-1}", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "geometric_construction", "tampered_component": "dropped_negative_correction_-2k+2", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "sum_of_degrees_replaced_by_product", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "upper_bound_promoted_to_exact_formula", "template_used": "stronger_trap"}]}}
{"id": "2512.05945v1", "paper_link": "http://arxiv.org/abs/2512.05945v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\nLet $M,N\\in\\mathbb{N}$, with $\\lcm(M,N)$ not\ndivisible by $2^4$ or the square of an odd prime.\nLet $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ and $g\\in S_\\ell^{\\rm new}(\\Gamma_0(N))$\nbe non-zero cusp forms with real Fourier coefficients\n$a_f(n),a_g(n)$. Then the following are equivalent:\n\\begin{enumerate}\n\\item $a_f(n)a_g(n)\\ge0$ for all sufficiently large $n\\in\\mathbb{N}$;\n\\item $(M,k)=(N,\\ell)$ and $f/g$ is a positive constant.\n\\end{enumerate}\nThus if $f$ and $g$ are not proportional then\n$\\{n\\in\\mathbb{N}:\\epsilon a_f(n)a_g(n)>0\\}$\nis infinite for each $\\epsilon\\in\\{\\pm1\\}$.", "start_pos": 11842, "end_pos": 12439, "label": "thm:main"}, "ref_dict": {"thm:main": "\\begin{theorem}\\label{thm:main}\nLet $M,N\\in\\N$, with $\\lcm(M,N)$ not\ndivisible by $2^4$ or the square of an odd prime.\nLet $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ and $g\\in S_\\ell^{\\rm new}(\\Gamma_0(N))$\nbe non-zero cusp forms with real Fourier coefficients\n$a_f(n),a_g(n)$. Then the following are equivalent:\n\\begin{enumerate}\n\\item $a_f(n)a_g(n)\\ge0$ for all sufficiently large $n\\in\\N$;\n\\item $(M,k)=(N,\\ell)$ and $f/g$ is a positive constant.\n\\end{enumerate}\nThus if $f$ and $g$ are not proportional then\n$\\{n\\in\\N:\\epsilon a_f(n)a_g(n)>0\\}$\nis infinite for each $\\epsilon\\in\\{\\pm1\\}$.\n\\end{theorem}", "prop:dense": "\\begin{proposition}\\label{prop:dense}\nLet $p_1,\\ldots,p_d\\in S$ be distinct primes. Then\n\\[\n\\bigl\\{\n\\bigl[a_1\\bigl(p_1^{k_1}\\cdots p_d^{k_d}\\bigr):\\cdots:a_d\\bigl(p_1^{k_1}\\cdots p_d^{k_d}\\bigr)\\bigl]\\in\\RP^{d-1}:\nk_1,\\ldots,k_d\\ge0\n\\bigl\\}\n\\]\nis dense in $\\RP^{d-1}$.\n\\end{proposition}", "lem:linindep": "\\begin{lemma}\\label{lem:linindep}\nThere is a set $S\\subseteq\\{p\\text{ prime}:p\\nmid M\\}$ of natural density $1$\nsuch that, for any $p\\in S$ and any pair $i\\ne j$, the numbers\n$1$, $\\theta_i(p)$, and $\\theta_j(p)$ are linearly independent\nover $\\Q$.\n\\end{lemma}"}, "pre_theorem_intro_text_len": 1431, "pre_theorem_intro_text": "Let $f\\in S_k(\\Gamma_0(M))$ and $g\\in S_\\ell(\\Gamma_0(N))$ be modular\nforms with real Fourier coefficients $a_f(n)$ and $a_g(n)$. Various\nauthors have investigated the extent to which $f$ and $g$ are distinguished\nby the signs of $a_f(n)$ and $a_g(n)$; see for instance\n\\cite{KLSW,Matomaki,GKR} and the references therein.\nFor Hecke eigenforms,\nit follows from the recent proof of joint Sato--Tate by Wong \\cite{Wong}\nthat if $f$ and $g$ are distinct, non-CM, normalized newforms\nthen each of the sets\n$\\{p\\text{ prime}:\\epsilon a_f(p)a_g(p)>0\\}$\nfor $\\epsilon\\in\\{\\pm1\\}$\nhas natural density $\\frac12$.\nIn another direction, Gun, Kohnen, and Rath \\cite{GKR} considered\nforms $f$ and $g$ of distinct weights that are not necessarily\nHecke eigenforms; it follows from their result that if $k\\ne\\ell$ and\n$a_f(n)a_g(n)$ is not identically zero\\footnote{One can\nfind examples of non-zero $f$ and $g$ of distinct weights such that\n$a_f(n)a_g(n)=0$ identically, so this hypothesis cannot be removed.}\nthen each of the sets $\\{n\\in\\mathbb{N}:\\epsilon a_f(n)a_g(n)>0\\}$ is infinite.\n\nIn both of these results the forms in question are orthogonal,\nso it is natural to expect sign changes to occur.\nWithout orthogonality the situation is much less clear, but\none might guess that the signs of $a_f(n)$ determine $f$ up to\nscalar multiplication. That is not always the case (see the remarks below),\nbut we show that it is true in many cases:", "context": "Let $f\\in S_k(\\Gamma_0(M))$ and $g\\in S_\\ell(\\Gamma_0(N))$ be modular\nforms with real Fourier coefficients $a_f(n)$ and $a_g(n)$. Various\nauthors have investigated the extent to which $f$ and $g$ are distinguished\nby the signs of $a_f(n)$ and $a_g(n)$; see for instance\n\\cite{KLSW,Matomaki,GKR} and the references therein.\nFor Hecke eigenforms,\nit follows from the recent proof of joint Sato--Tate by Wong \\cite{Wong}\nthat if $f$ and $g$ are distinct, non-CM, normalized newforms\nthen each of the sets\n$\\{p\\text{ prime}:\\epsilon a_f(p)a_g(p)>0\\}$\nfor $\\epsilon\\in\\{\\pm1\\}$\nhas natural density $\\frac12$.\nIn another direction, Gun, Kohnen, and Rath \\cite{GKR} considered\nforms $f$ and $g$ of distinct weights that are not necessarily\nHecke eigenforms; it follows from their result that if $k\\ne\\ell$ and\n$a_f(n)a_g(n)$ is not identically zero\\footnote{One can\nfind examples of non-zero $f$ and $g$ of distinct weights such that\n$a_f(n)a_g(n)=0$ identically, so this hypothesis cannot be removed.}\nthen each of the sets $\\{n\\in\\mathbb{N}:\\epsilon a_f(n)a_g(n)>0\\}$ is infinite.\n\nIn both of these results the forms in question are orthogonal,\nso it is natural to expect sign changes to occur.\nWithout orthogonality the situation is much less clear, but\none might guess that the signs of $a_f(n)$ determine $f$ up to\nscalar multiplication. That is not always the case (see the remarks below),\nbut we show that it is true in many cases:", "full_context": "Let $f\\in S_k(\\Gamma_0(M))$ and $g\\in S_\\ell(\\Gamma_0(N))$ be modular\nforms with real Fourier coefficients $a_f(n)$ and $a_g(n)$. Various\nauthors have investigated the extent to which $f$ and $g$ are distinguished\nby the signs of $a_f(n)$ and $a_g(n)$; see for instance\n\\cite{KLSW,Matomaki,GKR} and the references therein.\nFor Hecke eigenforms,\nit follows from the recent proof of joint Sato--Tate by Wong \\cite{Wong}\nthat if $f$ and $g$ are distinct, non-CM, normalized newforms\nthen each of the sets\n$\\{p\\text{ prime}:\\epsilon a_f(p)a_g(p)>0\\}$\nfor $\\epsilon\\in\\{\\pm1\\}$\nhas natural density $\\frac12$.\nIn another direction, Gun, Kohnen, and Rath \\cite{GKR} considered\nforms $f$ and $g$ of distinct weights that are not necessarily\nHecke eigenforms; it follows from their result that if $k\\ne\\ell$ and\n$a_f(n)a_g(n)$ is not identically zero\\footnote{One can\nfind examples of non-zero $f$ and $g$ of distinct weights such that\n$a_f(n)a_g(n)=0$ identically, so this hypothesis cannot be removed.}\nthen each of the sets $\\{n\\in\\mathbb{N}:\\epsilon a_f(n)a_g(n)>0\\}$ is infinite.\n\nIn both of these results the forms in question are orthogonal,\nso it is natural to expect sign changes to occur.\nWithout orthogonality the situation is much less clear, but\none might guess that the signs of $a_f(n)$ determine $f$ up to\nscalar multiplication. That is not always the case (see the remarks below),\nbut we show that it is true in many cases:\n\nOur proof makes use of the joint Sato--Tate equidistribution\nof any two non-CM, twist-inequivalent newforms, proven by\nWong \\cite{Wong}, based on the spectacular results\nof Barnet-Lamb et al.\\ \\cite{BGG}. (See also Newton and Thorne\n\\cite{Newton-Thorne} for the recent strengthening\nto full automorphy of symmetric powers for Hilbert modular forms\nof regular weight, and Thorner \\cite{Thorner} for results with an\neffective rate of convergence.)\nPrecisely, let $\\varphi\\in S_k^{\\rm new}(\\Gamma_0(M))$,\n$\\psi\\in S_\\ell^{\\rm new}(\\Gamma_0(N))$\nbe distinct normalized Hecke eigenforms of conductors $M,N$ as in the theorem,\nand write their Fourier coefficients at primes $p$ in the form\n\\[\na_\\varphi(p)=2p^{\\frac{k-1}{2}}\\cos(2\\pi\\theta_\\varphi(p)),\n\\quad\na_\\psi(p)=2p^{\\frac{\\ell-1}{2}}\\cos(2\\pi\\theta_\\psi(p)),\n\\quad\\text{where }\n(\\theta_\\varphi(p),\\theta_\\psi(p))\\in\\bigl[0,\\tfrac12\\bigr]^2.\n\\]\nThen by \\cite[Theorem~1.1]{Wong}, for any box\n$B=[\\alpha,\\beta]\\times[\\gamma,\\delta]\\subseteq[0,\\frac12]^2$, we have\n\\[\n\\lim_{x\\to\\infty}\n\\frac{\\#\\{p\\le x:(\\theta_\\varphi(p),\\theta_\\psi(p))\\in B\\}}{\\pi(x)}\n=\\int_B\\bigl(4\\sin(2\\pi u)\\sin(2\\pi v)\\bigr)^2\\,du\\,dv.\n\\]\nAs we will see in Lemma~\\ref{lem:linindep}, it follows that\n$1$, $\\theta_\\varphi(p)$, and $\\theta_\\psi(p)$ are linearly independent\nover $\\Q$ for all $p$ in a set of density $1$.\n\nWe note that mutual independence of the angles $\\theta_\\varphi$ for\nmore than two newforms is not known, though it would follow from\nthe functoriality of arbitrary products of symmetric powers.\nThis presents an obstacle to proving Theorem~\\ref{thm:main} using\nonly information on $a_f(p)a_g(p)$ at primes $p$.\n(We will however give such a proof when $f$ and $g$ have different\nweights or levels.) Instead we take an approach that is possibly of independent\ninterest, showing that for almost any tuple of\ndistinct primes $p_1,\\ldots,p_d$, where $d=\\dim S_k^{\\rm new}(\\Gamma_0(M))$,\nthe non-zero $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ with real coefficients\nare determined modulo scalars by the signs of $a_f(n)$\nfor $n\\in\\{p_1^{k_1}\\cdots p_d^{k_d}:k_1,\\ldots,k_d\\ge0\\}$;\nsee Proposition~\\ref{prop:dense} for\nthe precise statement, and \\cite{Amri,GMP} for some related results.\n\nLet $\\{\\varphi_1,\\ldots,\\varphi_d\\}$\nand $\\{\\psi_1,\\ldots,\\psi_{d'}\\}$ be normalized Hecke eigenbases\nfor $S_k^{\\rm new}(\\Gamma_0(M))$ and $S_\\ell^{\\rm new}(\\Gamma_0(N))$, respectively.\nWrite\n\\[\n\\varphi_i(z)=\\sum_{n=1}^\\infty\n\\lambda_{\\varphi_i}(n)n^{\\frac{k-1}{2}}e(nz),\n\\quad\n\\psi_j(z)=\\sum_{n=1}^\\infty\n\\lambda_{\\psi_j}(n)n^{\\frac{\\ell-1}{2}}e(nz),\n\\]\nand\n\\[\n\\lambda_f(n)\\coloneq\\frac{a_f(n)}{n^{\\frac{k-1}{2}}}\n=\\sum_{i=1}^d u_i\\lambda_{\\varphi_i}(n),\n\\quad\n\\lambda_g(n)\\coloneq\\frac{a_g(n)}{n^{\\frac{\\ell-1}{2}}}\n=\\sum_{j=1}^{d'} v_j\\lambda_{\\psi_j}(n).\n\\]\nOur hypothesis on $\\lcm(M,N)$ implies that\n$\\varphi_1,\\ldots,\\varphi_d,\\psi_1,\\ldots,\\psi_{d'}$ are all\ntwist minimal without CM, and no two are twist equivalent.\nBy \\cite[\\S3, Theorem~M]{Ramakrishnan}, for any pair of indices $(i,j)$,\nthere is a cuspidal automorphic representation $\\pi_{ij}$ of $\\GL(4)$\nwith $p$th Dirichlet coefficient $\\lambda_{\\pi_{ij}}(p)$ satisfying\n$\\lambda_{\\pi_{ij}}(p)=\\lambda_{\\varphi_i}(p)\\lambda_{\\psi_j}(p)$\nfor primes $p\\nmid MN$.\nFurthermore, following the proof of\n\\cite[Lemma~4.5.8]{Ramakrishnan}, one can see that\n$\\pi_{ij}\\cong\\pi_{i'j'}$ if and only if $(i,j)=(i',j')$.\n\n\\begin{lemma}\\label{lem:dense}\nFor any $p\\in S$ and any index $i_1$, there are distinct\nnon-zero integers $n_1,\\ldots,n_d$ such that\n\\begin{enumerate}\n\\item $n_{i_1}\\nmid n_i$ for $i\\ne i_1$;\n\\item for any $t\\in\\R$ and any $\\varepsilon>0$, there exists\n$n\\in\\N$ such that\n\\[\n\\min\\{|n\\theta_i(p)-n_it-m|:m\\in\\Z\\}<\\varepsilon\n\\quad\\text{for }i=1,\\ldots,d.\n\\]\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nFix $p\\in S$ and an index $i_1$, and consider the vector space\n\\[\nV=\\Q+\\Q\\theta_1(p)+\\cdots+\\Q\\theta_d(p).\n\\]\nWrite $\\dim V=m+1\\ge3$.\nBy permuting the indices if necessary, we may assume that\n$\\{1,\\theta_1(p),\\ldots,\\theta_m(p)\\}$ is a basis for $V$, and we may choose\nthe permutation to map $i_1$ to $1$.\nFor $i=1,\\ldots,m$ we write\n\\[\n\\theta_i(p)=\\sum_{j=1}^m\\alpha_{ij}\\theta_j(p) + \\beta_i\n\\]\nfor some $\\alpha_{ij},\\beta_i\\in\\Q$.\nBy our construction, no two of the vectors\n$(\\alpha_{ij})_{j=1,\\ldots,d}$ are colinear.\n\n\\begin{proposition}\\label{prop:dense}\nLet $p_1,\\ldots,p_d\\in S$ be distinct primes. Then\n\\[\n\\bigl\\{\n\\bigl[a_1\\bigl(p_1^{k_1}\\cdots p_d^{k_d}\\bigr):\\cdots:a_d\\bigl(p_1^{k_1}\\cdots p_d^{k_d}\\bigr)\\bigl]\\in\\RP^{d-1}:\nk_1,\\ldots,k_d\\ge0\n\\bigl\\}\n\\]\nis dense in $\\RP^{d-1}$.\n\\end{proposition}\n\\begin{proof}\nLet $\\theta_{ij}=\\theta_j(p_i)$. For each $i$ we\napply Lemma~\\ref{lem:dense} with $p=p_i$ and $i_1=i$,\nand we denote the resulting integers $n_{i1},\\ldots,n_{id}$.\nLet $t=(t_1,\\ldots,t_d)$, and\nconsider the vectors $F(t)=(F_j(t))_{j=1,\\ldots,d}$, where\n\\[\nF_j(t)=\\prod_{i=1}^d\\frac{\\sin(2\\pi n_{ij}t_i)}{\\sin(2\\pi\\theta_{ij})}\n=c_j\\prod_{i=1}^d\\sin(2\\pi n_{ij}t_i),\n\\]\nfor some $c_j\\in\\R_{>0}$.\nThen\n\\[\n\\frac{\\partial F_j}{\\partial t_i}\n=2\\pi c_jn_{ij}\\cos(2\\pi n_{ij}t_i)\n\\prod_{\\substack{1\\le i'\\le d\\\\i'\\ne i}}\n\\sin(2\\pi n_{i'j}t_{i'}).\n\\]\nBy construction we have $n_{ii}\\nmid n_{ij}$ for $j\\ne i$,\nand it follows that $F$ vanishes at the point\n$t=(\\frac1{2n_{ii}})_{i=1,\\ldots,d}$, and\nthe Jacobian matrix there is diagonal with non-zero determinant.\nBy the inverse function theorem, the image of $F$ contains\nan open neighborhood of the origin.\n\nTo complete the proof of Theorem~\\ref{thm:main}, write\n$f=\\sum_{i=1}^d u_i\\varphi_i$ and\n$g=\\sum_{i=1}^d v_i\\varphi_i$\nfor some non-zero vectors $u=(u_1,\\ldots,u_d),v=(v_1,\\ldots,v_d)\\in\\R^d$.\nIf $f$ and $g$ are not proportional\nthen we may choose a vector $w$ such that\n$w\\cdot u=0$ and $w\\cdot v\\ne0$.\nThen for any $\\delta\\in\\R$, we have\n\\[\n(w+\\delta u)\\cdot u=\\delta(u\\cdot u)\n\\quad\\text{and}\\quad\n(w+\\delta u)\\cdot v=(w\\cdot v)+\\delta(u\\cdot v).\n\\]\nChoosing $\\delta$ such that $\\delta(w\\cdot v)<0$ and\n$\\delta|u\\cdot v|<|w\\cdot v|$, we see that\n$(w+\\delta u)\\cdot u$ and $(w+\\delta u)\\cdot v$ have different signs.\nApplying Proposition~\\ref{prop:dense} we can find\nan arbitrarily large $n$ such that $[a_1(n):\\cdots:a_d(n)]$ approximates\nthe image of $w+\\delta u$ in $\\RP^{d-1}$\narbitrarily closely, and it follows that\n$a_f(n)$ and $a_g(n)$ have different signs.\nThis is a contradiction, so $f$ and $g$ must be proportional,\nmeaning that $c=f/g$ is constant.\nSince $f$ and $g$ are non-zero and $a_f(n)a_g(n)\\ge0$\nfor sufficiently large $n$, it follows that $c>0$.", "post_theorem_intro_text_len": 3906, "post_theorem_intro_text": "\\begin{remarks}\\\n\\begin{enumerate}\n\\item The restriction on $\\lcm(M,N)$\nis necessary when $(M,k)=(N,\\ell)$ and $Mk$ is sufficiently large,\nsince otherwise one can\nchoose a fundamental discriminant $\\Delta\\ne1$ with $\\Delta^2\\mid M$\nand a twist-minimal, non-CM newform $f$ of conductor $M$.\nSetting $g=2f+f\\times\\left(\\frac{\\Delta}{\\cdot}\\right)$, we have\n$a_f(n)a_g(n)=a_f(n)^2(2+\\left(\\frac{\\Delta}{n}\\right))\\ge0$.\n\\item The restriction to the new subspaces of\n$S_k(\\Gamma_0(M))$ and $S_\\ell(\\Gamma_0(N))$ is also\nnecessary in some cases, even under the assumption that $M=N$.\nFor example, let $f$ be the newform associated to an\nelliptic curve $E$ of squarefree conductor $N_E$, and set $g(z)=f(z)+f(pz)$,\nwhere $p>3$ is a supersingular prime for $E$. Then\n$f,g\\in S_2(\\Gamma_0(pN_E))$, and $a_f(n)a_g(n)=a_f(n)^2\\ge0$.\n\\item It will be clear from the proof that condition (1) of the theorem\ncan be substantially weakened. For instance, it suffices to have\n$\\liminf_{\\substack{n\\to\\infty\\\\\\gcd(n,q)=1}}\\frac{a_f(n)a_g(n)}{n^{\\frac{k+\\ell}{2}-1}}\\ge0$\nfor some fixed modulus $q$.\n\\item\nHowever, infinitely many $n$ are required when $(M,k)=(N,\\ell)$\nand $\\dim S_k^{\\rm new}(\\Gamma_0(M))>1$, i.e.\\ the first sign\nchange of $a_f(n)a_g(n)$ cannot be effectively bounded.\nTo see this, let $\\varphi,\\psi\\in S_k^{\\rm new}(\\Gamma_0(M))$ be distinct\nnormalized newforms, and set $f=\\varphi+\\pi\\psi$,\n$g_\\varepsilon=f+\\varepsilon\\varphi$ for some $\\varepsilon\\ne0$.\nThen $\\min\\{n\\in\\mathbb{N}:a_f(n)a_{g_\\varepsilon}(n)<0\\}\\to\\infty$ as $\\varepsilon\\to0$.\n\\end{enumerate}\n\\end{remarks}\n\nOur proof makes use of the joint Sato--Tate equidistribution\nof any two non-CM, twist-inequivalent newforms, proven by\nWong \\cite{Wong}, based on the spectacular results\nof Barnet-Lamb et al.\\ \\cite{BGG}. (See also Newton and Thorne\n\\cite{Newton-Thorne} for the recent strengthening\nto full automorphy of symmetric powers for Hilbert modular forms\nof regular weight, and Thorner \\cite{Thorner} for results with an\neffective rate of convergence.)\nPrecisely, let $\\varphi\\in S_k^{\\rm new}(\\Gamma_0(M))$,\n$\\psi\\in S_\\ell^{\\rm new}(\\Gamma_0(N))$\nbe distinct normalized Hecke eigenforms of conductors $M,N$ as in the theorem,\nand write their Fourier coefficients at primes $p$ in the form\n\\[\na_\\varphi(p)=2p^{\\frac{k-1}{2}}\\cos(2\\pi\\theta_\\varphi(p)),\n\\quad\na_\\psi(p)=2p^{\\frac{\\ell-1}{2}}\\cos(2\\pi\\theta_\\psi(p)),\n\\quad\\text{where }\n(\\theta_\\varphi(p),\\theta_\\psi(p))\\in\\bigl[0,\\tfrac12\\bigr]^2.\n\\]\nThen by \\cite[Theorem~1.1]{Wong}, for any box\n$B=[\\alpha,\\beta]\\times[\\gamma,\\delta]\\subseteq[0,\\frac12]^2$, we have\n\\[\n\\lim_{x\\to\\infty}\n\\frac{\\#\\{p\\le x:(\\theta_\\varphi(p),\\theta_\\psi(p))\\in B\\}}{\\pi(x)}\n=\\int_B\\bigl(4\\sin(2\\pi u)\\sin(2\\pi v)\\bigr)^2\\,du\\,dv.\n\\]\nAs we will see in Lemma~\\ref{lem:linindep}, it follows that\n$1$, $\\theta_\\varphi(p)$, and $\\theta_\\psi(p)$ are linearly independent\nover $\\mathbb{Q}$ for all $p$ in a set of density $1$.\n\nWe note that mutual independence of the angles $\\theta_\\varphi$ for\nmore than two newforms is not known, though it would follow from\nthe functoriality of arbitrary products of symmetric powers.\nThis presents an obstacle to proving Theorem~\\ref{thm:main} using\nonly information on $a_f(p)a_g(p)$ at primes $p$.\n(We will however give such a proof when $f$ and $g$ have different\nweights or levels.) Instead we take an approach that is possibly of independent\ninterest, showing that for almost any tuple of\ndistinct primes $p_1,\\ldots,p_d$, where $d=\\dim S_k^{\\rm new}(\\Gamma_0(M))$,\nthe non-zero $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ with real coefficients\nare determined modulo scalars by the signs of $a_f(n)$\nfor $n\\in\\{p_1^{k_1}\\cdots p_d^{k_d}:k_1,\\ldots,k_d\\ge0\\}$;\nsee Proposition~\\ref{prop:dense} for\nthe precise statement, and \\cite{Amri,GMP} for some related results.\n\n\\subsection*{Acknowledgements}\nI thank Jonathan Bober and Oleksiy Klurman for\nmany helpful conversations.", "sketch": "Our proof uses the joint Sato--Tate equidistribution of any two non-CM, twist-inequivalent newforms (Wong), based on Barnet-Lamb et al.  Writing for primes $p$\n\\[\na_\\varphi(p)=2p^{\\frac{k-1}{2}}\\cos(2\\pi\\theta_\\varphi(p)),\\qquad a_\\psi(p)=2p^{\\frac{\\ell-1}{2}}\\cos(2\\pi\\theta_\\psi(p)),\\qquad (\\theta_\\varphi(p),\\theta_\\psi(p))\\in[0,\\tfrac12]^2,\n\\]\nWong’s theorem gives equidistribution of $(\\theta_\\varphi(p),\\theta_\\psi(p))$ in $[0,\\tfrac12]^2$ with respect to density $(4\\sin(2\\pi u)\\sin(2\\pi v))^2\\,du\\,dv$.  As noted, “it follows that $1$, $\\theta_\\varphi(p)$, and $\\theta_\\psi(p)$ are linearly independent over $\\mathbb{Q}$ for all $p$ in a set of density $1$” (via Lemma~\\ref{lem:linindep}).\n\nBecause “mutual independence of the angles $\\theta_\\varphi$ for more than two newforms is not known,” this “presents an obstacle to proving Theorem~\\ref{thm:main} using only information on $a_f(p)a_g(p)$ at primes $p$” (except that “we will however give such a proof when $f$ and $g$ have different weights or levels”).  Instead, the approach is to show that “for almost any tuple of distinct primes $p_1,\\ldots,p_d$, where $d=\\dim S_k^{\\rm new}(\\Gamma_0(M))$, the non-zero $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ with real coefficients are determined modulo scalars by the signs of $a_f(n)$ for $n\\in\\{p_1^{k_1}\\cdots p_d^{k_d}:k_1,\\ldots,k_d\\ge0\\}$,” referring to Proposition~\\ref{prop:dense} for the precise statement.", "expanded_sketch": "Our proof uses the joint Sato--Tate equidistribution of any two non-CM, twist-inequivalent newforms (Wong), based on Barnet-Lamb et al.  Writing for primes $p$\n\\[\na_\\varphi(p)=2p^{\\frac{k-1}{2}}\\cos(2\\pi\\theta_\\varphi(p)),\\qquad a_\\psi(p)=2p^{\\frac{\\ell-1}{2}}\\cos(2\\pi\\theta_\\psi(p)),\\qquad (\\theta_\\varphi(p),\\theta_\\psi(p))\\in[0,\\tfrac12]^2,\n\\]\nWong’s theorem gives equidistribution of $(\\theta_\\varphi(p),\\theta_\\psi(p))$ in $[0,\\tfrac12]^2$ with respect to density $(4\\sin(2\\pi u)\\sin(2\\pi v))^2\\,du\\,dv$.  As noted, “it follows that $1$, $\\theta_\\varphi(p)$, and $\\theta_\\psi(p)$ are linearly independent over $\\mathbb{Q}$ for all $p$ in a set of density $1$”, i.e.\n\\begin{lemma}\\label{lem:linindep}\nThere is a set $S\\subseteq\\{p\\text{ prime}:p\\nmid M\\}$ of natural density $1$\nsuch that, for any $p\\in S$ and any pair $i\\ne j$, the numbers\n$1$, $\\theta_i(p)$, and $\\theta_j(p)$ are linearly independent\nover $\\Q$.\n\\end{lemma}\n\nBecause “mutual independence of the angles $\\theta_\\varphi$ for more than two newforms is not known,” this “presents an obstacle to proving the main theorem using only information on $a_f(p)a_g(p)$ at primes $p$” (except that “we will however give such a proof when $f$ and $g$ have different weights or levels”).  Instead, the approach is to show that “for almost any tuple of distinct primes $p_1,\\ldots,p_d$, where $d=\\dim S_k^{\\rm new}(\\Gamma_0(M))$, the non-zero $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ with real coefficients are determined modulo scalars by the signs of $a_f(n)$ for $n\\in\\{p_1^{k_1}\\cdots p_d^{k_d}:k_1,\\ldots,k_d\\ge0\\}$,” and for the precise statement we use the following proposition.\n\\begin{proposition}\\label{prop:dense}\nLet $p_1,\\ldots,p_d\\in S$ be distinct primes. Then\n\\[\n\\bigl\\{\n\\bigl[a_1\\bigl(p_1^{k_1}\\cdots p_d^{k_d}\\bigr):\\cdots:a_d\\bigl(p_1^{k_1}\\cdots p_d^{k_d}\\bigr)\\bigl]\\in\\RP^{d-1}:\nk_1,\\ldots,k_d\\ge0\n\\bigl\\}\n\\]\nis dense in $\\RP^{d-1}$.\n\\end{proposition}\n", "expanded_theorem": "\\label{thm:main}\nLet $M,N\\in\\mathbb{N}$, with $\\lcm(M,N)$ not\ndivisible by $2^4$ or the square of an odd prime.\nLet $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ and $g\\in S_\\ell^{\\rm new}(\\Gamma_0(N))$\nbe non-zero cusp forms with real Fourier coefficients\n$a_f(n),a_g(n)$. Then the following are equivalent:\n\\begin{enumerate}\n\\item $a_f(n)a_g(n)\\ge0$ for all sufficiently large $n\\in\\mathbb{N}$;\n\\item $(M,k)=(N,\\ell)$ and $f/g$ is a positive constant.\n\\end{enumerate}\nThus if $f$ and $g$ are not proportional then\n$\\{n\\in\\mathbb{N}:\\epsilon a_f(n)a_g(n)>0\\}$\nis infinite for each $\\epsilon\\in\\{\\pm1\\}$.,", "theorem_type": ["Biconditional or Equivalence", "Implication"], "mcq": {"question": "Let $M,N\\in\\mathbb N$ be such that $\\operatorname{lcm}(M,N)$ is divisible neither by $2^4$ nor by the square of any odd prime. Let $f\\in S_k^{\\rm new}(\\Gamma_0(M))$ and $g\\in S_\\ell^{\\rm new}(\\Gamma_0(N))$ be nonzero new cusp forms of weights $k,\\ell$ and levels $M,N$, with real Fourier coefficients $a_f(n)$ and $a_g(n)$. Which statement about the signs of the products $a_f(n)a_g(n)$ is valid under these hypotheses?", "correct_choice": {"label": "A", "text": "The condition $a_f(n)a_g(n)\\ge 0$ for all sufficiently large $n\\in\\mathbb N$ is equivalent to saying that $(M,k)=(N,\\ell)$ and $f/g$ is a positive constant, i.e. $f=cg$ for some $c>0$. Consequently, if $f$ and $g$ are not proportional, then for each $\\epsilon\\in\\{\\pm1\\}$ the set $\\{n\\in\\mathbb N: \\epsilon\\, a_f(n)a_g(n)>0\\}$ is infinite."}, "choices": [{"label": "B", "text": "The condition $a_f(n)a_g(n)\\ge 0$ for all sufficiently large $n\\in\\mathbb N$ is equivalent to saying that $(M,k)=(N,\\ell)$ and $f/g$ is a constant, possibly negative. Consequently, if $f$ and $g$ are not proportional, then there is at least one $\\epsilon\\in\\{\\pm1\\}$ for which the set $\\{n\\in\\mathbb N: \\epsilon\\, a_f(n)a_g(n)>0\\}$ is infinite."}, {"label": "C", "text": "If $(M,k)=(N,\\ell)$ and $f/g$ is a positive constant, then $a_f(n)a_g(n)\\ge 0$ for all sufficiently large $n\\in\\mathbb N$. In particular, this condition is sufficient for eventual nonnegativity of the products $a_f(n)a_g(n)$."}, {"label": "D", "text": "The condition $a_f(p)a_g(p)\\ge 0$ for all sufficiently large primes $p$ is equivalent to saying that $(M,k)=(N,\\ell)$ and $f/g$ is a positive constant, i.e. $f=cg$ for some $c>0$. Consequently, if $f$ and $g$ are not proportional, then for each $\\epsilon\\in\\{\\pm1\\}$ the set $\\{p\\text{ prime}: \\epsilon\\, a_f(p)a_g(p)>0\\}$ is infinite."}, {"label": "E", "text": "If $a_f(n)a_g(n)\\ge 0$ for infinitely many $n\\in\\mathbb N$, then necessarily $(M,k)=(N,\\ell)$ and $f/g$ is a positive constant, i.e. $f=cg$ for some $c>0$. Consequently, if $f$ and $g$ are not proportional, then $a_f(n)a_g(n)<0$ for all but finitely many $n$."}], "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": "positivity_of_scalar", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_equivalence_and_converse", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "all_n_vs_primes_only", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "sufficiently_large_replaced_by_infinitely_many", "template_used": "wildcard"}]}}
{"id": "2512.06696v1", "paper_link": "http://arxiv.org/abs/2512.06696v1", "theorems_cnt": 5, "theorem": {"env_name": "theorem", "content": "\\label{theorem:1.1}\nLet $\\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\\in \\mathbb{Z}\\cong \\pi_{4}(BG_2)$. The following holds.\n\\begin{itemize}\n\\item[{\\rm (1)}] \nIf $\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}$, then $(k, 84)=(k', 84)$ and \n\\item[{\\rm (2)}]  $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$ if $(k, 84)=(k', 84)$.\n\\end{itemize}", "start_pos": 4739, "end_pos": 5172, "label": "theorem:1.1"}, "ref_dict": {"theorem:1.1": "\\begin{theorem}\\label{theorem:1.1}\nLet $\\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\\in \\mathbb{Z}\\cong \\pi_{4}(BG_2)$. The following holds.\n\\begin{itemize}\n\\item[{\\rm (1)}] \nIf $\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}$, then $(k, 84)=(k', 84)$ and \n\\item[{\\rm (2)}]  $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$ if $(k, 84)=(k', 84)$.\n\\end{itemize}\n\\end{theorem}", "proposition:1.5": "\\begin{proposition} \\label{proposition:1.5}\nWe have  $4(\\psi\\circ s'')\\simeq 0$  in $\\pi_{14}(G_2)_{(2)}$. \\end{proposition}", "proposition:1.2": "\\begin{proposition}\\label{proposition:1.2}\nLet $\\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\\in \\mathbb{Z}\\cong \\pi_{4}(BG_2)$.\nThen, we have \n\\begin{itemize}\n\\item[{\\rm (1)}] $\\mathcal{G}_k\\simeq_{(2)} \\mathcal{G}_{k'}$  if $(k, 4)=(k', 4)$, \n\\item[{\\rm (2)}] \n$\\mathcal{G}_k\\simeq_{(3)} \\mathcal{G}_{k'}$  if $(k, 3)=(k', 3)$, \n\\item[{\\rm (3)}] $\\mathcal{G}_k\\simeq_{(7)} \\mathcal{G}_{k'}$  if $(k, 7)=(k', 7)$\nand \n\\item[{\\rm (4)}] $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'} \\simeq _{(p)} G_2 \\times \\Omega_0^4 G_2$ for $p\\not=2,3,7$.\n\\end{itemize}\n\\end{proposition}", "proposition:1.3": "\\begin{proposition}\\label{proposition:1.3}  The order of the Samelson product $\\langle i_3, 1\\rangle$ in $[\\Sigma^3 G_2, G_2]_{(2)}$ is at most $4$.\n\\end{proposition}"}, "pre_theorem_intro_text_len": 2019, "pre_theorem_intro_text": "\\label{sec1}\n\nLet $G$ be a compact Lie group, and $P$ a principal $G$-bundle over the $4$-dimensional sphere $S^4$. The $G$-gauge group over the base space $S^4$ is the topological group of $G$-bundle automorphisms of $P$. Suppose that $G$ is simply-connected and simple. Its classifying space $BG$ is $3$-connected and $\\pi_4(BG)\\cong \\mathbb{Z}$. Let us denote the homotopy class of a map $f$ by the same symbol $f$. Then, the principal $G$-bundle $P$ is classified by a map $k\\colon S^4 \\to BG$ in $\\pi_4(BG)\\cong \\mathbb{Z}$ and there are infinitely many isomorphism classes of principal $G$-bundles over $S^4$. In \\cite{kono-1991}, Kono classified the homotopy types of $SU(2)$-gauge groups over $S^4$ and showed that there are six homotopy types of these groups.\n\nSince then, many classification results on the homotopy types of gauge groups of low-rank simple Lie groups have been obtained. Among compact Lie groups, $SU(2)$ is the rank $1$ simply-connected simple Lie group. There are three simply-connected simple compact Lie groups of rank $2$: $SU(3)$, $Sp(2)$, and $G_2$. For $G=SU(3)$, Hamanaka and Kono \\cite{hamanaka-kono-2006} proved that $\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}$ if and only if $(k, 24)=(k', 24)$, where $(m,n)$ is the greatest common divisor of $m$ and $n$. For $G=Sp(2)$, Theriault \\cite{theriault-2010} showed $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$ if and only if $(k, 40)=(k', 40)$, where $\\simeq_{(p)}$ means $p$-local homotopy equivalence. Then, for $G=G_2$, Kishimoto, Theriault and Tsutaya \\cite{ktt-2017} showed that $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$ if $(k, 168)=(k', 168)$ and if $\\mathcal{G}_k \\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$, then $(k,84)=(k', 84)$. The discrepancy between $(k, 168)=(k',168)$ and $(k, 84)=(k', 84)$ has remained as an open problem. \n\nIn this paper, we complete the classification for rank $2$ simple Lie groups up to $p$-local homotopy equivalence at any prime $p$ by proving the following.", "context": "\\label{sec1}\n\nLet $G$ be a compact Lie group, and $P$ a principal $G$-bundle over the $4$-dimensional sphere $S^4$. The $G$-gauge group over the base space $S^4$ is the topological group of $G$-bundle automorphisms of $P$. Suppose that $G$ is simply-connected and simple. Its classifying space $BG$ is $3$-connected and $\\pi_4(BG)\\cong \\mathbb{Z}$. Let us denote the homotopy class of a map $f$ by the same symbol $f$. Then, the principal $G$-bundle $P$ is classified by a map $k\\colon S^4 \\to BG$ in $\\pi_4(BG)\\cong \\mathbb{Z}$ and there are infinitely many isomorphism classes of principal $G$-bundles over $S^4$. In \\cite{kono-1991}, Kono classified the homotopy types of $SU(2)$-gauge groups over $S^4$ and showed that there are six homotopy types of these groups.\n\nSince then, many classification results on the homotopy types of gauge groups of low-rank simple Lie groups have been obtained. Among compact Lie groups, $SU(2)$ is the rank $1$ simply-connected simple Lie group. There are three simply-connected simple compact Lie groups of rank $2$: $SU(3)$, $Sp(2)$, and $G_2$. For $G=SU(3)$, Hamanaka and Kono \\cite{hamanaka-kono-2006} proved that $\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}$ if and only if $(k, 24)=(k', 24)$, where $(m,n)$ is the greatest common divisor of $m$ and $n$. For $G=Sp(2)$, Theriault \\cite{theriault-2010} showed $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$ if and only if $(k, 40)=(k', 40)$, where $\\simeq_{(p)}$ means $p$-local homotopy equivalence. Then, for $G=G_2$, Kishimoto, Theriault and Tsutaya \\cite{ktt-2017} showed that $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$ if $(k, 168)=(k', 168)$ and if $\\mathcal{G}_k \\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$, then $(k,84)=(k', 84)$. The discrepancy between $(k, 168)=(k',168)$ and $(k, 84)=(k', 84)$ has remained as an open problem.\n\nIn this paper, we complete the classification for rank $2$ simple Lie groups up to $p$-local homotopy equivalence at any prime $p$ by proving the following.", "full_context": "\\label{sec1}\n\nLet $G$ be a compact Lie group, and $P$ a principal $G$-bundle over the $4$-dimensional sphere $S^4$. The $G$-gauge group over the base space $S^4$ is the topological group of $G$-bundle automorphisms of $P$. Suppose that $G$ is simply-connected and simple. Its classifying space $BG$ is $3$-connected and $\\pi_4(BG)\\cong \\mathbb{Z}$. Let us denote the homotopy class of a map $f$ by the same symbol $f$. Then, the principal $G$-bundle $P$ is classified by a map $k\\colon S^4 \\to BG$ in $\\pi_4(BG)\\cong \\mathbb{Z}$ and there are infinitely many isomorphism classes of principal $G$-bundles over $S^4$. In \\cite{kono-1991}, Kono classified the homotopy types of $SU(2)$-gauge groups over $S^4$ and showed that there are six homotopy types of these groups.\n\nSince then, many classification results on the homotopy types of gauge groups of low-rank simple Lie groups have been obtained. Among compact Lie groups, $SU(2)$ is the rank $1$ simply-connected simple Lie group. There are three simply-connected simple compact Lie groups of rank $2$: $SU(3)$, $Sp(2)$, and $G_2$. For $G=SU(3)$, Hamanaka and Kono \\cite{hamanaka-kono-2006} proved that $\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}$ if and only if $(k, 24)=(k', 24)$, where $(m,n)$ is the greatest common divisor of $m$ and $n$. For $G=Sp(2)$, Theriault \\cite{theriault-2010} showed $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$ if and only if $(k, 40)=(k', 40)$, where $\\simeq_{(p)}$ means $p$-local homotopy equivalence. Then, for $G=G_2$, Kishimoto, Theriault and Tsutaya \\cite{ktt-2017} showed that $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$ if $(k, 168)=(k', 168)$ and if $\\mathcal{G}_k \\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$, then $(k,84)=(k', 84)$. The discrepancy between $(k, 168)=(k',168)$ and $(k, 84)=(k', 84)$ has remained as an open problem.\n\nIn this paper, we complete the classification for rank $2$ simple Lie groups up to $p$-local homotopy equivalence at any prime $p$ by proving the following.\n\nIn this paper, we complete the classification for rank $2$ simple Lie groups up to $p$-local homotopy equivalence at any prime $p$ by proving the following.\n\nTheorem~\\ref{theorem:1.1} (2) is equivalent to the following local form. \\begin{proposition}\\label{proposition:1.2}\nLet $\\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\\in \\mathbb{Z}\\cong \\pi_{4}(BG_2)$.\nThen, we have \n\\begin{itemize}\n\\item[{\\rm (1)}] $\\mathcal{G}_k\\simeq_{(2)} \\mathcal{G}_{k'}$ if $(k, 4)=(k', 4)$, \n\\item[{\\rm (2)}] \n$\\mathcal{G}_k\\simeq_{(3)} \\mathcal{G}_{k'}$ if $(k, 3)=(k', 3)$, \n\\item[{\\rm (3)}] $\\mathcal{G}_k\\simeq_{(7)} \\mathcal{G}_{k'}$ if $(k, 7)=(k', 7)$\nand \n\\item[{\\rm (4)}] $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'} \\simeq _{(p)} G_2 \\times \\Omega_0^4 G_2$ for $p\\not=2,3,7$.\n\\end{itemize}\n\\end{proposition}\n\nFirst, we recall the basic method for determining the homotopy types of $G$-gauge groups over $S^4$, which relates their homotopy types to Samelson products. We begin with the following homotopy fiber sequence of mapping spaces.\n\\[\n\\Omega {\\mathrm{Map}} (S^4, BG)_k \\to G \\to {\\mathrm{Map}}_* (S^4, BG)_k \\to {\\mathrm{Map}} (S^4, BG)_k \\stackrel{\\mathrm{ev}}{\\longrightarrow} BG,\n\\]\nwhere $\\mathrm{Map} (S^4, BG)_k$ is the connected component of the space of continuous maps containing the map $k\\colon S^4\\to BG$, $\\mathrm{Map}_* (S^4, BG)_k$ is its subspace consisting of base point preserving maps, and $\\mathrm{ev}$ is the evaluation map.\nThere is a homotopy equivalence $\\Omega_0^3 G\\simeq \\mathrm{Map}_{*}(S^4, BG)_0 \\simeq \\mathrm{Map}_{*}(S^4, BG)_k$ and Gottlieb \\cite{gottlieb-1972} showed that the classifying space of the gauge group $B\\mathcal{G}_k$ is homotopy equivalent to the mapping space $ \\mathop{\\mathrm{Map}} (S^4, BG)_k$. Therefore, we have the following fiber sequence.\n\\[\n\\mathcal{G}_k \\to G \\stackrel{\\partial_k}{\\longrightarrow} \\Omega_0^3 G \\to B\\mathcal{G}_k \\to BG.\n\\]\nThus, the $G$-gauge group $\\mathcal{G}_k$ is homotopy equivalent to the homotopy fiber of the map $\\partial_k$. Let $i_3\\colon S^3 \\to G$ be the inclusion map of the bottom cell. Then, Lang \\cite{lang-1973} proved that the map $\\partial_k$ is the triple adjoint of the Samelson product $\\langle k\\cdot i_3, 1\\rangle$ where $1$ is the identity map of $G$. By the linearity of the Samelson product, we have $\\langle k\\cdot i_3, 1\\rangle\\simeq k \\cdot \\langle i_3, 1\\rangle$. Furthermore, in \\cite{theriault-2010}, Theriault showed that $\\mathcal{G}_k$ is $p$-locally homotopy equivalent to $\\mathcal{G}_{k'}$ at all prime $p$ if $(k, m)=(k',m)$ where $m$ is the order of the Samelson product $\\langle i_3, 1\\rangle$.\n\n\\begin{proposition}\\label{proposition:2.1}\nThe following holds.\n\\begin{itemize}\n\\item[{\\rm (1)}] $\\pi_{8}(S^{6})=\\mathbb{Z}/2\\{ \\eta_{6}^{2}\\}$, $\\pi_{11}(S^{6})=\\mathbb{Z}\\{ \\Delta(\\iota_{13})\\}$, $\\pi_{14}(S^{6})_{{(2)}}=\\mathbb{Z}/8\\{\\bar{\\nu}_6 \\} \\oplus \\mathbb{Z}/2\\{ \\varepsilon_6 \\}$, \n\\item[{\\rm (2)}] $\\pi_{14}(S^{8})=\\mathbb{Z}/2\\{ \\nu_{8}^{2} \\}$, $\\pi_{14}(S^{9})=\\{0\\}$, $\\pi_{14}(S^{10})=\\{0\\}$.\n\\item[{\\rm (3)}] $\\eta_{n}\\nu_{n+1}\\simeq 0$ for $n\\geq 5$.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proposition}[Mimura] \\label{proposition:2.2}\nLet $i\\colon SU(3)\\to G_2$ be the inclusion map and \n$p\\colon G_2\\to S^6=G_2/SU(3)$ the projection map.\nBy $\\langle \\alpha \\rangle$, we denote an element in $\\pi_i(G_2)$ such that $p_*(\\langle \\alpha\\rangle)\\simeq \\alpha$. \nThe following holds.\n\\begin{itemize}\n\\item[{\\rm (1)}] $\\pi_{11}(G_{2})=\\mathbb{Z}\\{ \\langle 2 \\Delta(\\iota_{13})\\rangle\\} \\oplus \\mathbb{Z}/2\\{ i_{*}([\\nu_{5}^{2}])\\}$. \n\\item[{\\rm (2)}] $\\pi_{14}(G_{2})_{(2)}=\\mathbb{Z}/8\\{ \\langle \\bar{\\nu}_{6}+\\varepsilon_{6}\\rangle\\} \\oplus \\mathbb{Z}/2\\{ i_{*}([\\nu_{5}^{2}])\\circ \\nu_{11}\\}$\n\\end{itemize}\n\\end{proposition}\n\n\\begin{theorem}[{\\oshima} \\cite{oshima-2005}*{Theorem 2.1}] \\label{theorem:3.1}\nThere is a map $\\gamma \\colon S^{11}\\to G_2$ such that \n\\begin{itemize}\n\\item[{\\rm (1)}] \n$p\\circ \\gamma\\simeq 2\\Delta(\\iota_{13})$, \n\\item[{\\rm (2)}] $\\langle i_3, \\gamma\\rangle\\in \\pi_{14}(G_2)$ has order $21=3\\cdot 7$.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proposition}\\label{proposition:3.2}\nLet $i_{8}\\colon S^8\\to P^9(2)$ be the inclusion map of the bottom cell. Then, the following hold.\n\\begin{itemize}\n\\item[{\\rm (1)}] ${i_8}_{*}\\colon \\pi_{14}(S^8)\\to \\pi_{14}(P^9(2))$ is an isomorphism.\n\\item[{\\rm (2)}] For maps $f\\colon S^{14}\\to S^8$, and $g\\colon S^{8}\\to S^{6}$, \n$g \\circ f \\simeq 0$ in $\\pi_{14}(S^6)$. \n\\item[{\\rm (3)}] If a map $f\\colon S^{14}\\to S^{6}$ factors through $P^9(2)$, then $f\\simeq 0$ in $\\pi_{14}(S^{6})$.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proposition}\\label{proposition:3.3}\nThe following holds.\n\\begin{itemize}\n\\item[{\\rm (1)}] $\\pi_{11}(C)\\simeq \\mathbb{Z}\\{ \\bar{i}_{6}\\circ \\Delta(\\iota_{13})\\} \\oplus \\mathbb{Z}\\{ \\bar{s}\\}$, \n\\item[{\\rm (2)}] $\\mathop{\\mathrm{Ker}} \\bar{p}_{6\\; *} \\colon \\pi_{11}(C)\\to \\pi_{11}(S^6)$ is generated by $w\\simeq a \\bar{i}_{6}\\circ \\Delta(\\iota_{13})+b \\bar{s}$ where $a\\in \\mathbb{Z}$ and $b=1$ or $a$ is odd and $b=2$.\n\\item[{\\rm (3)}] $j\\circ q\\circ \\gamma\\simeq cw$ where $c\\in \\{ \\pm 1, \\pm 2\\}$.\n\\item[{\\rm (4)}] $p_{11}\\circ q\\circ \\gamma \\simeq bc (\\iota_{11})$ where $bc \\in \\{ \\pm 1, \\pm 2, \\pm 4\\}$.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{theorem}\\label{theorem:1.1}\nLet $\\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\\in \\mathbb{Z}\\cong \\pi_{4}(BG_2)$. The following holds.\n\\begin{itemize}\n\\item[{\\rm (1)}] \nIf $\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}$, then $(k, 84)=(k', 84)$ and \n\\item[{\\rm (2)}]  $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$ if $(k, 84)=(k', 84)$.\n\\end{itemize}\n\\end{theorem}", "post_theorem_intro_text_len": 6277, "post_theorem_intro_text": "Theorem~\\ref{theorem:1.1} (2) is equivalent to the following local form. \\begin{proposition}\\label{proposition:1.2}\nLet $\\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\\in \\mathbb{Z}\\cong \\pi_{4}(BG_2)$.\nThen, we have \n\\begin{itemize}\n\\item[{\\rm (1)}] $\\mathcal{G}_k\\simeq_{(2)} \\mathcal{G}_{k'}$  if $(k, 4)=(k', 4)$, \n\\item[{\\rm (2)}] \n$\\mathcal{G}_k\\simeq_{(3)} \\mathcal{G}_{k'}$  if $(k, 3)=(k', 3)$, \n\\item[{\\rm (3)}] $\\mathcal{G}_k\\simeq_{(7)} \\mathcal{G}_{k'}$  if $(k, 7)=(k', 7)$\nand \n\\item[{\\rm (4)}] $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'} \\simeq _{(p)} G_2 \\times \\Omega_0^4 G_2$ for $p\\not=2,3,7$.\n\\end{itemize}\n\\end{proposition}\n\nTheorem~\\ref{theorem:1.1} (1) was established in \\cite{ktt-2017}*{Theorem 1.2 (a)} and Proposition~\\ref{proposition:1.2} (2), (3), (4) were established in \\cite{ktt-2017}*{Propositions 4.6 and 3.2}. In this paper, we prove Proposition~\\ref{proposition:1.2} (1) and complete the proof of Theorem~\\ref{theorem:1.1} assuming the results of \\cite{ktt-2017}.\n\nFirst, we recall the basic method for determining the homotopy types of $G$-gauge groups over $S^4$, which relates their homotopy types to Samelson products. We begin with the following homotopy fiber sequence of mapping spaces.\n\\[\n\\Omega {\\mathrm{Map}} (S^4, BG)_k  \\to G \\to  {\\mathrm{Map}}_* (S^4, BG)_k \\to {\\mathrm{Map}} (S^4, BG)_k \\stackrel{\\mathrm{ev}}{\\longrightarrow} BG,\n\\]\nwhere $\\mathrm{Map} (S^4, BG)_k$ is the connected component of the space of continuous maps containing the map $k\\colon S^4\\to BG$, $\\mathrm{Map}_* (S^4, BG)_k$ is its subspace consisting of base point preserving maps,  and $\\mathrm{ev}$ is the evaluation map.\nThere is a homotopy equivalence  $\\Omega_0^3 G\\simeq \\mathrm{Map}_{*}(S^4, BG)_0 \\simeq \\mathrm{Map}_{*}(S^4, BG)_k$ and Gottlieb \\cite{gottlieb-1972} showed that the classifying space of the gauge group $B\\mathcal{G}_k$ is homotopy equivalent to the mapping space $ \\mathop{\\mathrm{Map}} (S^4, BG)_k$. Therefore, we have the following fiber sequence.\n\\[\n\\mathcal{G}_k  \\to G \\stackrel{\\partial_k}{\\longrightarrow} \\Omega_0^3 G \\to B\\mathcal{G}_k \\to BG.\n\\]\nThus, the $G$-gauge group $\\mathcal{G}_k$ is homotopy equivalent to the homotopy fiber of the map $\\partial_k$. Let $i_3\\colon S^3 \\to G$ be the inclusion map of the bottom cell. Then, Lang \\cite{lang-1973} proved that the map $\\partial_k$ is the triple adjoint of the Samelson product $\\langle k\\cdot i_3, 1\\rangle$ where $1$ is the identity map of $G$. By the linearity of the Samelson product, we have $\\langle k\\cdot i_3, 1\\rangle\\simeq k \\cdot \\langle i_3, 1\\rangle$. Furthermore, in \\cite{theriault-2010}, Theriault showed that $\\mathcal{G}_k$ is $p$-locally homotopy equivalent to $\\mathcal{G}_{k'}$ at all prime $p$ if $(k, m)=(k',m)$ where $m$ is the order of the Samelson product $\\langle i_3, 1\\rangle$. \n\nFor an abelian group $A$, we denote by $A_{(p)}$ its localization at the prime $p$. If $A$ is finite, then $A_{(p)}$ is the $p$-primary subgroup of $A$. By \\cite{ktt-2017}*{Corollary 6.6}, the order of $\\langle i_3, 1\\rangle$ in $[\\Sigma^3 G_2, G_2]_{(2)}$ is at least $4$. Therefore, to show Proposition~\\ref{proposition:1.2} (1), it suffices to prove the following proposition.\n\n\\begin{proposition}\\label{proposition:1.3}  The order of the Samelson product $\\langle i_3, 1\\rangle$ in $[\\Sigma^3 G_2, G_2]_{(2)}$ is at most $4$.\n\\end{proposition}\n\nAs for the order of the Samelson product $\\langle i_3, 1\\rangle$ in $[\\Sigma^3 G_2, G_2]$, together with \\cite{ktt-2017}*{Lemma 3.1, Proposition 4.5 and Corollary 6.6}, by Proposition~\\ref{proposition:1.3}, we have the following theorem.\n\n\\begin{theorem}\\label{theorem:1.4} \nThe order of the Samelson product $\\langle i_3, 1\\rangle$ in $[\\Sigma^3 G_2, G_2]$ is $84$.\n\\end{theorem}\n\nNext, we recall some properties of a space $V:=G_2/SU(2)$, which played an important role in \\cite{ktt-2017} and is crucial in this paper. Let $q\\colon G_2\\to V$ be the obvious projection map. By \\cite{ktt-2017}*{Lemmas 7.1 and 7.2}, the Samelson product $3 \\cdot \\langle i_3, 1\\rangle\\colon \\Sigma^3 G_2\\to G_2$ factors through $\\Sigma^3 q \\colon \\Sigma^3 G_2 \\to \\Sigma^3 V$. \nLet us write this factorization as follows.\n \\[\n3\\cdot \\langle i_3, 1\\rangle\\simeq \\psi\\circ \\Sigma^3 q.\n\\]\nLet $P^{n+1}(2)$ be the mapping cone of the degree $2$ map $\\times 2 \\colon S^n\\to S^n$.\nThe $6$-skeleton of $V=S^5\\cup e^6\\cup e^{11}$ is $P^6(2)$ and, by \\cite{ktt-2017}{Lemma 7.4}, there is a homotopy equivalence $\\Sigma^3 V\\simeq P^9(2) \\vee S^{14}$.  Let \\[\np_{11}\\colon V \\to S^{11}\n\\]\n be the pinch map to the top cell, collapsing the $6$-skeleton to the base point. We denote by \n \\[\n i_6\\colon P^6(2) \\to V\n \\]\n  the inclusion map. Then, the above homotopy equivalence provides maps $s'\\colon \\Sigma^3 V \\to P^9(2)$ and $s''\\colon S^{14}\\to \\Sigma^3V$ such that the identity map of $\\Sigma^3 V$ is homotopic to \n  \\[\n  \\Sigma^3 i_6 \\circ s' + s'' \\circ \\Sigma^3 p_{11}.\n  \\] The following proposition is what we prove in the rest of this paper.\n\n\\begin{proposition} \\label{proposition:1.5}\nWe have  $4(\\psi\\circ s'')\\simeq 0$  in $\\pi_{14}(G_2)_{(2)}$. \\end{proposition}\n\n\\begin{proof}[Proof of Proposition~\\ref{proposition:1.3}] We consider the following decomposition in $[\\Sigma^3 G_2, G_2]$. \\[ 3 \\cdot \\langle i_3, 1\\rangle\\simeq \\psi \\circ \\Sigma^3 i_6\\circ s' \\circ \\Sigma^3 q +\\psi \\circ s'' \\circ \\Sigma^3 (p_{11}\\circ q). \\]  \nBy \\cite{ktt-2017}*{Lemma 5.2 (a) and (b)}, the order of the identity map of $P^9(2)$ is $4$. Hence, we have\n$4 (\\Sigma^3 i_6\\circ s' )\\simeq 0$. Therefore, by Proposition~\\ref{proposition:1.5}, the order of $ \\langle i_3, 1\\rangle$ in $[\\Sigma^3 G_2, G_2]_{(2)}$ is at most $4$.  \\end{proof}\n\nIn what follows, we use the symbol $p$ to express the map $p\\colon G_2\\to S^6\\simeq G_2/SU(3)$ only. When we need to indicate the generator of a cyclic group, we write $\\mathbb{Z}/m\\{ a\\}$ for the cyclic group of order $m$ generated by $a$.  \n\nThis paper is organized as follows. In Section \\ref{sec2}, we collect some facts on homotopy groups of $G_2$ and $S^6$ and reduce the problem on the $2$-primary homotopy group $\\pi_{14}(G_2)_{(2)}$ to that of $\\pi_{14}(S^6)_{(2)}$. In Section \\ref{sec3}, we prove Proposition~\\ref{proposition:1.5}.", "sketch": "To complete Theorem~\\ref{theorem:1.1} (2) it suffices to prove the local statement Proposition~\\ref{proposition:1.2}. The paper reduces this to proving Proposition~\\ref{proposition:1.2} (1) (the $2$-local case), since (2),(3),(4) are cited from \\cite{ktt-2017}.\n\nThe method recalled is: using Gottlieb’s identification $B\\mathcal{G}_k\\simeq \\mathrm{Map}(S^4,BG)_k$ and the resulting fiber sequence\n\\[\n\\mathcal{G}_k\\to G\\xrightarrow{\\partial_k} \\Omega_0^3 G\\to B\\mathcal{G}_k\\to BG,\n\\]\nso $\\mathcal{G}_k$ is the homotopy fiber of $\\partial_k$. By Lang, $\\partial_k$ is the triple adjoint of the Samelson product $\\langle k\\cdot i_3,1\\rangle$, hence by linearity $\\langle k\\cdot i_3,1\\rangle\\simeq k\\cdot\\langle i_3,1\\rangle$. Theriault’s result is then invoked: $\\mathcal{G}_k\\simeq_{(p)}\\mathcal{G}_{k'}$ for all primes $p$ if $(k,m)=(k',m)$ where $m$ is the order of $\\langle i_3,1\\rangle$.\n\nFor $G=G_2$ at $p=2$, \\cite{ktt-2017} gives that the order of $\\langle i_3,1\\rangle$ in $[\\Sigma^3G_2,G_2]_{(2)}$ is at least $4$, so “to show Proposition~\\ref{proposition:1.2} (1), it suffices” to prove Proposition~\\ref{proposition:1.3}: the order is at most $4$.\n\nThe proof sketch for Proposition~\\ref{proposition:1.3} is given: using that $3\\cdot\\langle i_3,1\\rangle$ factors through $\\Sigma^3q\\colon \\Sigma^3G_2\\to \\Sigma^3V$ (with $V=G_2/SU(2)$), write\n\\[\n3\\cdot\\langle i_3,1\\rangle\\simeq \\psi\\circ \\Sigma^3q.\n\\]\nWith the homotopy decomposition $\\Sigma^3V\\simeq P^9(2)\\vee S^{14}$, choose maps $s'\\colon \\Sigma^3V\\to P^9(2)$ and $s''\\colon S^{14}\\to \\Sigma^3V$ so that\n\\[\n\\mathrm{id}_{\\Sigma^3V}\\simeq \\Sigma^3i_6\\circ s'\\; +\\; s''\\circ \\Sigma^3p_{11}.\n\\]\nThen\n\\[\n3\\cdot\\langle i_3,1\\rangle\\simeq \\psi\\circ \\Sigma^3i_6\\circ s'\\circ \\Sigma^3q\\; +\\; \\psi\\circ s''\\circ \\Sigma^3(p_{11}\\circ q).\n\\]\nUsing \\cite{ktt-2017} (Lemma 5.2(a),(b)) that “the order of the identity map of $P^9(2)$ is $4$,” one gets $4(\\Sigma^3i_6\\circ s')\\simeq 0$. The remaining summand is controlled by Proposition~\\ref{proposition:1.5}, namely $4(\\psi\\circ s'')\\simeq 0$ in $\\pi_{14}(G_2)_{(2)}$. Combining these shows “the order of $\\langle i_3,1\\rangle$ in $[\\Sigma^3G_2,G_2]_{(2)}$ is at most $4$,” hence Proposition~\\ref{proposition:1.2}(1), and therefore Theorem~\\ref{theorem:1.1}(2) via the Samelson-product/Thériault criterion.\n\n(They additionally note that with Proposition~\\ref{proposition:1.3} and results of \\cite{ktt-2017}, the global order is computed as $84$ (Theorem~\\ref{theorem:1.4}).)", "expanded_sketch": "To establish the main theorem (2) it suffices to prove the local statement\n\n\\begin{proposition}\\label{proposition:1.2}\nLet $\\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\\in \\mathbb{Z}\\cong \\pi_{4}(BG_2)$.\nThen, we have \n\\begin{itemize}\n\\item[{\\rm (1)}] $\\mathcal{G}_k\\simeq_{(2)} \\mathcal{G}_{k'}$  if $(k, 4)=(k', 4)$, \n\\item[{\\rm (2)}] \n$\\mathcal{G}_k\\simeq_{(3)} \\mathcal{G}_{k'}$  if $(k, 3)=(k', 3)$, \n\\item[{\\rm (3)}] $\\mathcal{G}_k\\simeq_{(7)} \\mathcal{G}_{k'}$  if $(k, 7)=(k', 7)$\nand \n\\item[{\\rm (4)}] $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'} \\simeq _{(p)} G_2 \\times \\Omega_0^4 G_2$ for $p\\not=2,3,7$.\n\\end{itemize}\n\\end{proposition}\n\nThe paper reduces this to proving the preceding proposition (1) (the $2$-local case), since (2),(3),(4) are cited from \\cite{ktt-2017}.\n\nThe method recalled is: using Gottlieb’s identification $B\\mathcal{G}_k\\simeq \\mathrm{Map}(S^4,BG)_k$ and the resulting fiber sequence\n\\[\n\\mathcal{G}_k\\to G\\xrightarrow{\\partial_k} \\Omega_0^3 G\\to B\\mathcal{G}_k\\to BG,\n\\]\nso $\\mathcal{G}_k$ is the homotopy fiber of $\\partial_k$. By Lang, $\\partial_k$ is the triple adjoint of the Samelson product $\\langle k\\cdot i_3,1\\rangle$, hence by linearity $\\langle k\\cdot i_3,1\\rangle\\simeq k\\cdot\\langle i_3,1\\rangle$. Theriault’s result is then invoked: $\\mathcal{G}_k\\simeq_{(p)}\\mathcal{G}_{k'}$ for all primes $p$ if $(k,m)=(k',m)$ where $m$ is the order of $\\langle i_3,1\\rangle$.\n\nFor $G=G_2$ at $p=2$, \\cite{ktt-2017} gives that the order of $\\langle i_3,1\\rangle$ in $[\\Sigma^3G_2,G_2]_{(2)}$ is at least $4$, so to show the preceding proposition (1), it suffices to prove the following.\n\n\\begin{proposition}\\label{proposition:1.3}  The order of the Samelson product $\\langle i_3, 1\\rangle$ in $[\\Sigma^3 G_2, G_2]_{(2)}$ is at most $4$.\n\\end{proposition}\n\nThe proof sketch for the preceding proposition is given: using that $3\\cdot\\langle i_3,1\\rangle$ factors through $\\Sigma^3q\\colon \\Sigma^3G_2\\to \\Sigma^3V$ (with $V=G_2/SU(2)$), write\n\\[\n3\\cdot\\langle i_3,1\\rangle\\simeq \\psi\\circ \\Sigma^3q.\n\\]\nWith the homotopy decomposition $\\Sigma^3V\\simeq P^9(2)\\vee S^{14}$, choose maps $s'\\colon \\Sigma^3V\\to P^9(2)$ and $s''\\colon S^{14}\\to \\Sigma^3V$ so that\n\\[\n\\mathrm{id}_{\\Sigma^3V}\\simeq \\Sigma^3i_6\\circ s'\\; +\\; s''\\circ \\Sigma^3p_{11}.\n\\]\nThen\n\\[\n3\\cdot\\langle i_3,1\\rangle\\simeq \\psi\\circ \\Sigma^3i_6\\circ s'\\circ \\Sigma^3q\\; +\\; \\psi\\circ s''\\circ \\Sigma^3(p_{11}\\circ q).\n\\]\nUsing \\cite{ktt-2017} (Lemma 5.2(a),(b)) that “the order of the identity map of $P^9(2)$ is $4$,” one gets $4(\\Sigma^3i_6\\circ s')\\simeq 0$. The remaining summand is controlled by\n\n\\begin{proposition} \\label{proposition:1.5}\nWe have  $4(\\psi\\circ s'')\\simeq 0$  in $\\pi_{14}(G_2)_{(2)}$. \\end{proposition}\n\nCombining these shows that the order of $\\langle i_3,1\\rangle$ in $[\\Sigma^3G_2,G_2]_{(2)}$ is at most $4$, hence the preceding proposition (1), and therefore the main theorem (2) via the Samelson-product/Thériault criterion.\n\n(They additionally note that with the proposition above and results of \\cite{ktt-2017}, the global order is computed as $84$ (Theorem~\\ref{theorem:1.4}).)", "expanded_theorem": "\\label{theorem:1.1}\nLet $\\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\\in \\mathbb{Z}\\cong \\pi_{4}(BG_2)$. The following holds.\n\\begin{itemize}\n\\item[{\\rm (1)}] \nIf $\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}$, then $(k, 84)=(k', 84)$ and \n\\item[{\\rm (2)}]  $\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}$ at any prime $p$ if $(k, 84)=(k', 84)$.\n\\end{itemize}", "theorem_type": "unknown", "mcq": {"question": "Let \\(\\mathcal{G}_k\\) denote the gauge group of a principal \\(G_2\\)-bundle over \\(S^4\\) whose classifying map represents \\(k\\in \\pi_4(BG_2)\\cong \\mathbb Z\\). Write \\((m,n)\\) for the greatest common divisor of integers \\(m\\) and \\(n\\), write \\(\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}\\) for homotopy equivalence, and write \\(\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}\\) for \\(p\\)-local homotopy equivalence. Which statement holds for integers \\(k\\) and \\(k'\\)?", "correct_choice": {"label": "A", "text": "If \\(\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}\\), then \\((k,84)=(k',84)\\). Moreover, if \\((k,84)=(k',84)\\), then \\(\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}\\) for every prime \\(p\\)."}, "choices": [{"label": "B", "text": "If \\(\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}\\), then \\((k,168)=(k',168)\\). Moreover, if \\((k,168)=(k',168)\\), then \\(\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}\\) for every prime \\(p\\)."}, {"label": "C", "text": "If \\((k,84)=(k',84)\\), then \\(\\mathcal{G}_k\\simeq_{(2)} \\mathcal{G}_{k'}\\), \\(\\mathcal{G}_k\\simeq_{(3)} \\mathcal{G}_{k'}\\), and \\(\\mathcal{G}_k\\simeq_{(7)} \\mathcal{G}_{k'}\\)."}, {"label": "D", "text": "If \\(\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}\\), then \\((k,84)=(k',84)\\). Moreover, if \\((k,84)=(k',84)\\), then \\(\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}\\)."}, {"label": "E", "text": "If \\(\\mathcal{G}_k\\simeq \\mathcal{G}_{k'}\\), then \\((k,84)=(k',84)\\). Moreover, if \\((k,4)=(k',4)\\), then \\(\\mathcal{G}_k\\simeq_{(p)} \\mathcal{G}_{k'}\\) for every prime \\(p\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "counting_estimate", "tampered_component": "global_order_84_replaced_by_168", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "dropped_global_all-primes_conclusion_and_global_homotopy_necessity", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "p-local_equivalence_strengthened_to_integral_homotopy_equivalence", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "local_primewise_conditions_collapsed_to_2-local_gcd_condition", "template_used": "wildcard"}]}}
{"id": "2512.06908v1", "paper_link": "http://arxiv.org/abs/2512.06908v1", "theorems_cnt": 1, "theorem": {"env_name": "maintheorem", "content": "[Structure of $f(R)$ gravity in spherical symmetry]\n\\label{main-theo}\nConsider the field equations in $f(R)$ gravity \\eqref{Eq1-14}, coupled to a \npossibly massive, real-valued scalar field $\\phi$. Suppose that the defining function $f= f(R)$ and the matter potential $U=U(\\phi)$ satisfy the following conditions: \n\\bel{hypo-Einstein}\n\\aligned \n& (1) \n\\, &  \\phi \\, U'(\\phi) &\\geq&& 0, \n\\\\\n& (2) \n\\, &  U(\\phi) &\\geq&& 0,   \n\\\\\n& (3) \n\\, & f'(R) &>&&0, \n\\\\\n& (4)  \n\\, &  \nf(R) &\\leq&& R \\, f'(R).  \n\\endaligned\n\\ee \nThen, the field equations of $f(R)$ gravity for spherically symmetric spacetimes can be reduced to an \\emph{integro-differential system} consisting of two first-order coupled, nonlinear hyperbolic equations, stated in Proposition~\\ref{main-propo-first-order}, below. \nIndeed, this system is equivalent to the full Einstein equations in the class of $C^1$ solutions that are suitably regular at the center in the sense of Definition~\\ref{definition}, below. Furthermore, the Hawking mass is future non-decreasing along radial directions and non-increasing along null directions. In the limit ${f(R) \\to R}$ and \n$U(\\phi)\\to 0$, one recovers Christodoulou's formulation of the Einstein-massless scalar field system.", "start_pos": 16280, "end_pos": 17547, "label": "main-theo"}, "ref_dict": {"main-propo-first-order": "\\begin{proposition}[A first-order formulation of $f(R)$ gravity]\n\\label{main-propo-first-order}\n\\bse\\label{Bondi system int augmented3} \nThe equations of $f(R)$ gravity take the form of a first-order system, in which $h$ and $l$ are the main unknowns: \n\\be\n\\aligned\n& Dh = \\frac{1}{2r}\\big(g- \\gb\\big)(h- \\hb) +\n\\frac{\\kappa}{2}\\big(q(h- \\hb) + p(l- \\lb)\\big)-{r \\over 2} e^{\\nu+\\lambda- \\kappa\\lb}U'(\\hb),\n\\\\\n& Dl = \\frac{1}{2r}\\big(g- \\gb\\big)(l- \\lb)\n- \\frac{8\\pi}{3\\kappa}e^{- \\kappa\\lb}(h- \\hb) p-{r \\over \\kappa} e^{\\nu+\\lambda}\\Big(\\Wstar(\\lb)- \\frac{16\\pi}{3}e^{-2\\kappa\\lb}U(\\hb)\\Big),\n\\endaligned\n\\ee\nin which the metric coefficient $\\nu + \\lambda$ is computed by \n\\be\n\\aligned \n& \\nu + \\lambda = - \\frac{3\\kappa^{2}}{4}\\int_r^{+ \\infty}\\frac{|l- \\lb|^2}{s} \\, ds - 4\\pi\\int_r^{+ \\infty}\\frac{|h- \\hb|^2}{se^{\\kappa\\lb}} \\, ds,\n\\endaligned\n\\ee\nand the auxiliary variables $p$ and $q$ are defined explicitly by \n\\bel{p,q}\n\\aligned \n&p = \\frac{\\kappa}{2r}\\int_0^r\\Big((l- \\lb)p + (h- \\hb)q\\Big) \\, ds +\\frac{1}{2r}\\int_0^r\\frac{e^{\\nu- \\lambda}(h- \\hb)}{s} \\, ds\n- \\frac{1}{2r}\\int_0^rse^{\\nu+\\lambda- \\kappa\\lb}U'(\\hb) \\, ds, \n\\\\\n& \\kappa q \n= - \\frac{8\\pi}{3 r}\\int_0^r\\frac{(h- \\hb)p}{e^{\\kappa\\lb}} \\, ds+ \\frac{\\kappa}{2r}\\int_0^r{\\frac{e^{\\nu- \\lambda}(l- \\lb)}{s} \\, ds}\n - \\frac{1}{r}\\int_0^r se^{\\nu+\\lambda}\\Big (\\Wstar(\\lb)-\n\\frac{16\\pi}{3} e^{-2\\kappa\\lb}U(\\hb)\\Big) \\, ds. \n\\endaligned\n\\ee\n\\ese\n\\end{proposition}", "definition": "\\begin{definition}  \n\\label{definition}\n\\emph{The \\emph{first regularity condition at the center\\footnote{In view of \\eqref{eq fR components} and \\eqref{eq Bondi components T}, this condition can be expressed in terms of first-order derivatives of the unknowns.}}} is defined as \n\\be\n\\label{regularity} \n\\lim_{r\\to0} r^2e^{-\\kappa\\rho}\\big(E - 8\\pi T\\big)(D,D) = 0.\n\\ee\nThe \\emph{second regularity condition at the center} is defined as \n\\be\n\\label{regularity2}\n\\del_r(r\\phi) \\text{ and }\\del_r(r\\rho) \\text{ are } C^1\\text{ on } [0,u_0]\\times [0,+ \\infty).\n\\ee\n\\end{definition}", "Eq1-01": "\\label{Eq1-01}\nG := \\Ric - {R_g \\over 2} \\, g = 8 \\pi \\, T[\\phi,g]\n\\ee\nin which the right-hand side \n\\bel{Eq:12}\nT_{ab}[\\phi,g] := -2 \\, {\\delta L \\over \\delta g^{ab}} [\\phi,g]  + g_{ab}\\, L[\\phi,g]\n\\"}, "pre_theorem_intro_text_len": 8262, "pre_theorem_intro_text": "\\label{section---1}\n\n\\subsection{Purpose of this paper}\n\n\\paragraph{Theories of modified gravity}\n\nIn recent years, new observational data suggest that extensions of Einstein's field equations may be relevant for explaining the accelerated expansion of the Universe and certain galactic-scale instabilities. Many of the proposed physical theories of modified gravity exhibit undesirable features, including lack of strong hyperbolicity in commonly used gauges, or non-uniqueness of solutions, or changes in the character of the evolution equations that may alter the system's causal structure. The development of robust numerical simulations in nonlinear, physically relevant regimes is hampered by the lack of mathematically rigorous formulations. \nIndeed, only a few studies address the formulation and well-posedness of these theories \\cite{Avalos,Lehner2,Lehner,Cocke,Hilditch1,Felice,Figueras,LeFlochMa17a,Reall,Reall2,Pretorius,Salgado}. \nMore recently, alternative, well-posed theories have been proposed; cf.~\\cite{Brady-Figueras,Figueras} (and the references therein).\n\nAmong the various extensions of general relativity, $f(R)$ gravity is widely regarded as a natural and physically viable alternative to Einstein's theory \\cite{Felice,Ferreira}. The $f(R)$ field equations are considerably more involved than the Einstein equations: in addition to the second-order Ricci curvature terms, they contain \\emph{fourth-order} metric derivatives (in particular, second-order derivatives of the scalar curvature). This partly explains why, despite their physical importance, rigorous mathematical results to date encompass only local well-posedness~\\cite{Felice,LeFlochMa17a} and the global nonlinear stability in the near-Minkowski regime \\cite{LeFlochMa23}.\n\n\\paragraph{Evolution in spherical symmetry.}\n\nIn this paper, we initiate the study of the global evolution of a (massive) scalar field in $f(R)$ gravity in spherical symmetry. Christodoulou \\cite{Chr,Chr2,Chr3} developed, within general relativity, a framework to analyze a massless scalar field in spherical symmetry using Bondi-type coordinates. This framework was also employed to study black-hole formation via numerical methods; cf.~\\cite{Goldwirth-Piran} and the references therein. These developments inspired further investigations of gravitational collapse~\\cite{Brady-etal,Hilditch2,Zhang-Lu, Michel-Moss}, as well as additional mathematical results, cf.~\\cite{Chae,Costa-Mena,Costa-Duarte-Mena} (classical solutions) and \\cite{LeFloch-Mena} (generalized solutions). Building on Christodoulou's subsequent work using a double-null foliation~\\cite{Chr-bv,Chr-naked}, many studies have also addressed the gravitational collapse toward black holes in spherical symmetry. Next, with the growing interest in modified gravity, subsequent works include \\cite{Quo-Joshi, Zhang-etal-2016, Chow}, which mostly consider quadratic curvature corrections to the Einstein--Hilbert action. Despite these advances, rigorous mathematical results on the \\emph{global geometry} of Cauchy developments in $f(R)$ gravity are still lacking.\n\n\\paragraph{Aim of this paper.}\n\nOur aim here is to advance the mathematical analysis of $f(R)$ gravity by building on the work of Christodoulou  \\cite{Chr,Chr2,Chr3}. \nWe are also relying on the formulation proposed in LeFloch and Ma \\cite{LeFlochMa17a,LeFlochMa23} to deal with the equations of $f(R)$ gravity in the near-Minkowski regime. In the present paper, we focus on spacetimes containing a (possibly massive) scalar field evolving in spherical symmetry and, drawing on Christodoulou's  insights, we provide a framework suited to addressing the global causal structure of such spacetimes.\nOur companion paper \\cite{LeFloch-Mena} considered the evolution of a self-gravitating massive scalar field and provided a first step toward understanding the role of a massive scalar field on the global spacetime geometry. While in \\cite{LeFloch-Mena} we considered generalized solutions, in the present paper we consider solutions of class $C^1$ that satisfy regularity conditions at the center and are asymptotically Euclidean. Following Christodoulou, we use a generalization of the Bondi--Sachs coordinates and formulate the characteristic initial value problem with data imposed on a future light cone.\n\nThe new system is significantly more involved than the Einstein system, and a remarkable mathematical structure is uncovered here by introducing  an augmented formulation in which the metric and its scalar curvature are regarded as independent unknowns. We thus obtain a new integro-differential system of two coupled, first-order, nonlinear hyperbolic equations whose unknowns are the spacetime scalar curvature and, in this case, the scalar field. We then prove several consistency and regularity properties for the new system and its solutions as well as monotonicity properties of the Hawking mass in the $f(R)$ setting. In the process we also recover the general relativity limit (when ${f(R) \\to R}$), as well as the special case of a massless scalar field. \n\n\\subsection{A first-order formulation of $f(R)$ gravity}\n\n\\paragraph{Action of $f(R)$ modified gravity.}\n\nRecall first that Einstein's theory is based on the Einstein--Hilbert action\n\\be\n\\Acal_{\\text{EH}}[\\phi, g] := \\int_M \\Big( {R_g  \\over 16 \\pi} + L[\\phi,g] \\Big) \\, dV_g,\n\\ee\nassociated with a $(1+3)$--dimensional spacetime $({\\Mcal},g)$ with signature $(-, +,+,+)$.\nHence, the functional $\\Acal_{\\text{EH}}[\\phi, g] $ is determined from the scalar curvature $R_g$ of the metric $g$ representing the geometry of the spacetime and\n a Lagrangian $L[\\phi,g]$ describing the matter content and represented by certain fields $\\phi$ defined on $\\Mcal$. Here, $dV_g$ denotes the canonical volume form associated with $g$.\nAs is well-known, the action $\\Acal_{\\text{EH}}[\\phi, g]$ is (formally) critical at metrics $g$\nsatisfying Einstein's field equations\n\\be\n\\label{Eq1-01}\nG := \\Ric - {R_g \\over 2} \\, g = 8 \\pi \\, T[\\phi,g]\n\\ee\nin which the right-hand side \n\\bel{Eq:12}\nT_{ab}[\\phi,g] := -2 \\, {\\delta L \\over \\delta g^{ab}} [\\phi,g]  + g_{ab}\\, L[\\phi,g]\n\\ee\nis the stress-energy tensor of the matter system (Latin indices $a,b,..= 0, \\ldots, 3$ denoting spacetime components).\nOur emphasis in this paper is on massive scalar fields $\\phi$ described by \n\\be\n\\label{stress-energy}\nT_{ab} = \\nabla_a \\phi \\nabla_b \\phi- \\big( {1 \\over 2} \\nabla^c \\phi \\nabla_c \\phi + U(\\phi) \\Big) g_{ab}, \n\\ee\nin which the potential function is $U=U(\\phi)$, so that the field $\\phi$ satisfies the Klein-Gordon equation \n\\be\n\\Box_g \\phi = U'(\\phi).\n\\end{equation}\nWe study an extension of Einstein's theory, defined as follows. A function $f: \\RR \\to \\RR$ being given, we consider the modified gravity action for the $f(R)$ theory\n$$\n\\Acal_{\\text{MG}}[\\phi,g] =: \\int_M \\Big( {f(R) \\over 16 \\pi} + L[\\phi, g]\\Big) \\, dV,\n$$\nwhose critical points satisfy the following \\emph{field equations of modified gravity}\n\\bel{Eq1-14}\nE_{ab} :=\nf'(R) \\, G_{ab} - \\frac{1}{2} \\Big( f(R) - R f'(R) \\Big) g_{ab}\n+  \\big( g_{ab} \\, \\Box_g   - \\nabla_a\\nabla_b \\big)  \\big( f'(R) \\big) = 8 \\pi \\, T_{ab}[\\phi,g]. \n\\ee\nThe right-hand side is still given by \\eqref{Eq:12}. The modified gravity tensor $E_{ab}$ replaces\nthe Einstein tensor $G_{ab}$ and satisfies $\\nabla^a E_{ab} = 0$, \n so that $T_{ab}$ is also divergence-free. \n\n\\paragraph{Main result.}\n\nIn local coordinates, \\eqref{Eq1-14} consists of a nonlinear system of fourth-order partial differential equations, while Einstein theory in \\eqref{Eq1-01} leads to second-order equations. We investigate\nhow to include the effect of these fourth-order terms in techniques developed earlier for the Einstein equations.\nThe condition $f'(R) >0$ in \\eqref{hypo-Einstein}, below, is fundamental throughout the $f(R)$ theory and, for instance, we use it to be able to introduce the \\emph{conformal metric} $\\gt$ and the \\emph{augmented variable} $\\rho$ (in \\eqref{equa-233}, below):  \n\\be\n\\gt_{ab} := e^{\\kappa \\rho} g_{ab}, \n\\qquad \n\\rho := {1 \\over \\kappa} \\log f'(R). \n\\ee\nHere, $\\kappa>0$ is a parameter  for the definition of $\\rho$, so that the general relativity limit corresponds to $f(R)\\to R$ when ${\\kappa \\to 0}$. We summarize our results as follows. \n\n\\vskip.15cm", "context": "Our aim here is to advance the mathematical analysis of $f(R)$ gravity by building on the work of Christodoulou \\cite{Chr,Chr2,Chr3}. \nWe are also relying on the formulation proposed in LeFloch and Ma \\cite{LeFlochMa17a,LeFlochMa23} to deal with the equations of $f(R)$ gravity in the near-Minkowski regime. In the present paper, we focus on spacetimes containing a (possibly massive) scalar field evolving in spherical symmetry and, drawing on Christodoulou's insights, we provide a framework suited to addressing the global causal structure of such spacetimes.\nOur companion paper \\cite{LeFloch-Mena} considered the evolution of a self-gravitating massive scalar field and provided a first step toward understanding the role of a massive scalar field on the global spacetime geometry. While in \\cite{LeFloch-Mena} we considered generalized solutions, in the present paper we consider solutions of class $C^1$ that satisfy regularity conditions at the center and are asymptotically Euclidean. Following Christodoulou, we use a generalization of the Bondi--Sachs coordinates and formulate the characteristic initial value problem with data imposed on a future light cone.\n\nThe new system is significantly more involved than the Einstein system, and a remarkable mathematical structure is uncovered here by introducing an augmented formulation in which the metric and its scalar curvature are regarded as independent unknowns. We thus obtain a new integro-differential system of two coupled, first-order, nonlinear hyperbolic equations whose unknowns are the spacetime scalar curvature and, in this case, the scalar field. We then prove several consistency and regularity properties for the new system and its solutions as well as monotonicity properties of the Hawking mass in the $f(R)$ setting. In the process we also recover the general relativity limit (when ${f(R) \\to R}$), as well as the special case of a massless scalar field.\n\nRecall first that Einstein's theory is based on the Einstein--Hilbert action\n\\be\n\\Acal_{\\text{EH}}[\\phi, g] := \\int_M \\Big( {R_g \\over 16 \\pi} + L[\\phi,g] \\Big) \\, dV_g,\n\\ee\nassociated with a $(1+3)$--dimensional spacetime $({\\Mcal},g)$ with signature $(-, +,+,+)$.\nHence, the functional $\\Acal_{\\text{EH}}[\\phi, g] $ is determined from the scalar curvature $R_g$ of the metric $g$ representing the geometry of the spacetime and\n a Lagrangian $L[\\phi,g]$ describing the matter content and represented by certain fields $\\phi$ defined on $\\Mcal$. Here, $dV_g$ denotes the canonical volume form associated with $g$.\nAs is well-known, the action $\\Acal_{\\text{EH}}[\\phi, g]$ is (formally) critical at metrics $g$\nsatisfying Einstein's field equations\n\\be\n\\label{Eq1-01}\nG := \\Ric - {R_g \\over 2} \\, g = 8 \\pi \\, T[\\phi,g]\n\\ee\nin which the right-hand side \n\\bel{Eq:12}\nT_{ab}[\\phi,g] := -2 \\, {\\delta L \\over \\delta g^{ab}} [\\phi,g] + g_{ab}\\, L[\\phi,g]\n\\ee\nis the stress-energy tensor of the matter system (Latin indices $a,b,..= 0, \\ldots, 3$ denoting spacetime components).\nOur emphasis in this paper is on massive scalar fields $\\phi$ described by \n\\be\n\\label{stress-energy}\nT_{ab} = \\nabla_a \\phi \\nabla_b \\phi- \\big( {1 \\over 2} \\nabla^c \\phi \\nabla_c \\phi + U(\\phi) \\Big) g_{ab}, \n\\ee\nin which the potential function is $U=U(\\phi)$, so that the field $\\phi$ satisfies the Klein-Gordon equation \n\\be\n\\Box_g \\phi = U'(\\phi).\n\\end{equation}\nWe study an extension of Einstein's theory, defined as follows. A function $f: \\RR \\to \\RR$ being given, we consider the modified gravity action for the $f(R)$ theory\n$$\n\\Acal_{\\text{MG}}[\\phi,g] =: \\int_M \\Big( {f(R) \\over 16 \\pi} + L[\\phi, g]\\Big) \\, dV,\n$$\nwhose critical points satisfy the following \\emph{field equations of modified gravity}\n\\bel{Eq1-14}\nE_{ab} :=\nf'(R) \\, G_{ab} - \\frac{1}{2} \\Big( f(R) - R f'(R) \\Big) g_{ab}\n+ \\big( g_{ab} \\, \\Box_g - \\nabla_a\\nabla_b \\big) \\big( f'(R) \\big) = 8 \\pi \\, T_{ab}[\\phi,g]. \n\\ee\nThe right-hand side is still given by \\eqref{Eq:12}. The modified gravity tensor $E_{ab}$ replaces\nthe Einstein tensor $G_{ab}$ and satisfies $\\nabla^a E_{ab} = 0$, \n so that $T_{ab}$ is also divergence-free.\n\nIn local coordinates, \\eqref{Eq1-14} consists of a nonlinear system of fourth-order partial differential equations, while Einstein theory in \\eqref{Eq1-01} leads to second-order equations. We investigate\nhow to include the effect of these fourth-order terms in techniques developed earlier for the Einstein equations.\nThe condition $f'(R) >0$ in \\eqref{hypo-Einstein}, below, is fundamental throughout the $f(R)$ theory and, for instance, we use it to be able to introduce the \\emph{conformal metric} $\\gt$ and the \\emph{augmented variable} $\\rho$ (in \\eqref{equa-233}, below): \n\\be\n\\gt_{ab} := e^{\\kappa \\rho} g_{ab}, \n\\qquad \n\\rho := {1 \\over \\kappa} \\log f'(R). \n\\ee\nHere, $\\kappa>0$ is a parameter for the definition of $\\rho$, so that the general relativity limit corresponds to $f(R)\\to R$ when ${\\kappa \\to 0}$. We summarize our results as follows.\n\n\\vskip.15cm", "full_context": "Our aim here is to advance the mathematical analysis of $f(R)$ gravity by building on the work of Christodoulou \\cite{Chr,Chr2,Chr3}. \nWe are also relying on the formulation proposed in LeFloch and Ma \\cite{LeFlochMa17a,LeFlochMa23} to deal with the equations of $f(R)$ gravity in the near-Minkowski regime. In the present paper, we focus on spacetimes containing a (possibly massive) scalar field evolving in spherical symmetry and, drawing on Christodoulou's insights, we provide a framework suited to addressing the global causal structure of such spacetimes.\nOur companion paper \\cite{LeFloch-Mena} considered the evolution of a self-gravitating massive scalar field and provided a first step toward understanding the role of a massive scalar field on the global spacetime geometry. While in \\cite{LeFloch-Mena} we considered generalized solutions, in the present paper we consider solutions of class $C^1$ that satisfy regularity conditions at the center and are asymptotically Euclidean. Following Christodoulou, we use a generalization of the Bondi--Sachs coordinates and formulate the characteristic initial value problem with data imposed on a future light cone.\n\nThe new system is significantly more involved than the Einstein system, and a remarkable mathematical structure is uncovered here by introducing an augmented formulation in which the metric and its scalar curvature are regarded as independent unknowns. We thus obtain a new integro-differential system of two coupled, first-order, nonlinear hyperbolic equations whose unknowns are the spacetime scalar curvature and, in this case, the scalar field. We then prove several consistency and regularity properties for the new system and its solutions as well as monotonicity properties of the Hawking mass in the $f(R)$ setting. In the process we also recover the general relativity limit (when ${f(R) \\to R}$), as well as the special case of a massless scalar field.\n\nRecall first that Einstein's theory is based on the Einstein--Hilbert action\n\\be\n\\Acal_{\\text{EH}}[\\phi, g] := \\int_M \\Big( {R_g \\over 16 \\pi} + L[\\phi,g] \\Big) \\, dV_g,\n\\ee\nassociated with a $(1+3)$--dimensional spacetime $({\\Mcal},g)$ with signature $(-, +,+,+)$.\nHence, the functional $\\Acal_{\\text{EH}}[\\phi, g] $ is determined from the scalar curvature $R_g$ of the metric $g$ representing the geometry of the spacetime and\n a Lagrangian $L[\\phi,g]$ describing the matter content and represented by certain fields $\\phi$ defined on $\\Mcal$. Here, $dV_g$ denotes the canonical volume form associated with $g$.\nAs is well-known, the action $\\Acal_{\\text{EH}}[\\phi, g]$ is (formally) critical at metrics $g$\nsatisfying Einstein's field equations\n\\be\n\\label{Eq1-01}\nG := \\Ric - {R_g \\over 2} \\, g = 8 \\pi \\, T[\\phi,g]\n\\ee\nin which the right-hand side \n\\bel{Eq:12}\nT_{ab}[\\phi,g] := -2 \\, {\\delta L \\over \\delta g^{ab}} [\\phi,g] + g_{ab}\\, L[\\phi,g]\n\\ee\nis the stress-energy tensor of the matter system (Latin indices $a,b,..= 0, \\ldots, 3$ denoting spacetime components).\nOur emphasis in this paper is on massive scalar fields $\\phi$ described by \n\\be\n\\label{stress-energy}\nT_{ab} = \\nabla_a \\phi \\nabla_b \\phi- \\big( {1 \\over 2} \\nabla^c \\phi \\nabla_c \\phi + U(\\phi) \\Big) g_{ab}, \n\\ee\nin which the potential function is $U=U(\\phi)$, so that the field $\\phi$ satisfies the Klein-Gordon equation \n\\be\n\\Box_g \\phi = U'(\\phi).\n\\end{equation}\nWe study an extension of Einstein's theory, defined as follows. A function $f: \\RR \\to \\RR$ being given, we consider the modified gravity action for the $f(R)$ theory\n$$\n\\Acal_{\\text{MG}}[\\phi,g] =: \\int_M \\Big( {f(R) \\over 16 \\pi} + L[\\phi, g]\\Big) \\, dV,\n$$\nwhose critical points satisfy the following \\emph{field equations of modified gravity}\n\\bel{Eq1-14}\nE_{ab} :=\nf'(R) \\, G_{ab} - \\frac{1}{2} \\Big( f(R) - R f'(R) \\Big) g_{ab}\n+ \\big( g_{ab} \\, \\Box_g - \\nabla_a\\nabla_b \\big) \\big( f'(R) \\big) = 8 \\pi \\, T_{ab}[\\phi,g]. \n\\ee\nThe right-hand side is still given by \\eqref{Eq:12}. The modified gravity tensor $E_{ab}$ replaces\nthe Einstein tensor $G_{ab}$ and satisfies $\\nabla^a E_{ab} = 0$, \n so that $T_{ab}$ is also divergence-free.\n\nIn local coordinates, \\eqref{Eq1-14} consists of a nonlinear system of fourth-order partial differential equations, while Einstein theory in \\eqref{Eq1-01} leads to second-order equations. We investigate\nhow to include the effect of these fourth-order terms in techniques developed earlier for the Einstein equations.\nThe condition $f'(R) >0$ in \\eqref{hypo-Einstein}, below, is fundamental throughout the $f(R)$ theory and, for instance, we use it to be able to introduce the \\emph{conformal metric} $\\gt$ and the \\emph{augmented variable} $\\rho$ (in \\eqref{equa-233}, below): \n\\be\n\\gt_{ab} := e^{\\kappa \\rho} g_{ab}, \n\\qquad \n\\rho := {1 \\over \\kappa} \\log f'(R). \n\\ee\nHere, $\\kappa>0$ is a parameter for the definition of $\\rho$, so that the general relativity limit corresponds to $f(R)\\to R$ when ${\\kappa \\to 0}$. We summarize our results as follows.\n\n\\vskip.15cm\n\nOur aim here is to advance the mathematical analysis of $f(R)$ gravity by building on the work of Christodoulou \\cite{Chr,Chr2,Chr3}. \nWe are also relying on the formulation proposed in LeFloch and Ma \\cite{LeFlochMa17a,LeFlochMa23} to deal with the equations of $f(R)$ gravity in the near-Minkowski regime. In the present paper, we focus on spacetimes containing a (possibly massive) scalar field evolving in spherical symmetry and, drawing on Christodoulou's insights, we provide a framework suited to addressing the global causal structure of such spacetimes.\nOur companion paper \\cite{LeFloch-Mena} considered the evolution of a self-gravitating massive scalar field and provided a first step toward understanding the role of a massive scalar field on the global spacetime geometry. While in \\cite{LeFloch-Mena} we considered generalized solutions, in the present paper we consider solutions of class $C^1$ that satisfy regularity conditions at the center and are asymptotically Euclidean. Following Christodoulou, we use a generalization of the Bondi--Sachs coordinates and formulate the characteristic initial value problem with data imposed on a future light cone.\n\n\\vskip.15cm\n\n\\vskip.15cm\n\nIn the modified field equations, there are terms that contain derivatives of the scalar\ncurvature, that is, third order derivatives of the metric functions $\\nu$ and $\\lambda$. This leads to an essential difficulty in dealing with the equations.\nFollowing LeFloch and Ma~\\cite{LeFlochMa17a}, we introduce an \\emph{augmented system}\n where the relation\n\\be\n\\rho = \\frac{1}{\\kappa}\\ln f'(R) \\quad \\text{(conformal formulation)} \n\\ee\nis no longer imposed but $\\rho$ \n is regarded as an independent variable.\nTo clarify the notation, we introduce a new independent variable denoted by \n\\be\n\\rhoh \\quad \\text{(augmented conformal formulation)} \n\\ee\nwhich plays the role of $\\rho$ and will coincides with $\\rho$ once the constraint below is enforced. \nIn view of Proposition~\\ref{prop essential f(R) Bondi}, this leads us to the following system \n\\bel{Bondi system diff augmented1.5}\n\\aligned\n\\del_r(\\nu+\\lambda)\n& = \\frac{3}{4}\\kappa^{2}r|\\del_r\\rhoh|^2 + 4\\pi r e^{- \\kappa\\rhoh} \\, |\\del_r\\phi|^2,\n\\\\\n\\del_r\\big(re^{\\nu- \\lambda}\\big)\n & = \\big(1 - r^{2}e^{-2\\kappa\\rhoh} \\big( \\Vstar(\\rhoh) + 8\\pi U(\\phi) \\big) \\big)e^{\\nu+\\lambda},\n \\\\\nD(\\del_r(r\\phi)) + {r \\over 2} e^{\\nu+\\lambda- \\kappa\\rhoh}U'(\\phi)\n& = \\frac{1}{2}e^{\\nu+\\lambda}\\Big(1 - e^{-2\\lambda} - r^2e^{-2\\kappa\\rhoh}\\big(\\Vstar(\\rhoh)+8\\pi U(\\phi)\\big)\\Big)\\del_r\\phi \\\\\n& \\quad + \\frac{\\kappa r}{2}\\big(D\\rhoh \\del_r\\phi + D\\phi \\del_r\\rhoh\\big), \n\\\\\nD\\big(\\del_r(r\\rhoh)\\big) + {r \\over \\kappa} e^{\\nu+\\lambda} \\Wstar(\\rhoh)\n& = \\frac{1}{2}e^{\\nu+\\lambda}\\Big(1 - e^{-2\\lambda} - r^2e^{-2\\kappa\\rhoh}\\big(\\Vstar(\\rhoh)+8\\pi U(\\phi)\\big)\\Big)\\del_r\\rhoh\n\\\\\n& - \\frac{8\\pi r}{3\\kappa} e^{- \\kappa\\rhoh}\\del_r\\phi D\\phi\n + {r \\over \\kappa} e^{\\nu+\\lambda} \\frac{16\\pi}{3}e^{-2\\kappa\\rhoh}U(\\phi).\n\\endaligned\n\\ee\nObserve that, as before, the last equation of \\eqref{Bondi system diff augmented1.5} is equivalent to\n\\be\n\\label{box-tilde-rho}\n\\Box_{\\gt}\\rhoh - \\frac{2}{\\kappa} \\Wstar(\\rhoh)\n= - {8\\pi \\over 3\\kappa}e^{-2\\kappa\\rhoh}\\big(\\sigma+4 \\, U(\\phi)\\big).\n\\ee\nNow that we have defined the augmented system, the next question is whether a solution of \\eqref{Bondi system diff augmented1.5} (with a certain regularity) is also a solution of the original system. We emphasize that to be a solution of the full set of field equations, a solution of \\eqref{equa-three-equa}, \\eqref{Bondi scalar}, and \\eqref{eq:599} \nmust additionally satisfy the condition\n\\bel{Bondi system diff augmented1.5 constraint}\nf'(R) = e^{\\kappa\\rhoh},\n\\ee\nwhich we regard as a nonlinear differential constraint on the solutions.\n Conversely, any classical solution of the original $f(R)$ system with $\\rho=\\frac{1}{\\kappa}\\ln f'(R)$ satisfies \\eqref{Bondi system diff augmented1.5} upon setting $\\rhoh=\\rho$; thus the two formulations are equivalent in the admissible class once \\eqref{Bondi system diff augmented1.5 constraint} holds on the initial cone (and is then propagated by the evolution)\n\nUnder the regularity conditions at the center and the asymptotic flatness condition, we can reduce the field equations (cf.~Appendix~\\ref{section---A2})) to the the following integro-differential system with the main unknowns $\\phi$ and $\\rho$: \n\\bel{notresysteme}\n\\aligned\nD(\\del_r(r\\phi))\n& = \\frac{1}{2}e^{\\nu+\\lambda}\\Big(1 - e^{-2\\lambda} - r^2e^{-2\\kappa\\rho}\\big(\\Vstar(\\rho)+8\\pi U(\\phi)\\big)\\Big)\\del_r\\phi\n\\\\\n & \\quad + \\frac{\\kappa r}{2}\\big(D\\rho \\del_r\\phi + D\\phi \\del_r\\rho\\big)-{r \\over 2} e^{\\nu+\\lambda- \\kappa\\rho}U'(\\phi),\n\\\\\nD\\big(\\del_r(r\\rho)\\big)\n& = \\frac{1}{2}e^{\\nu+\\lambda}\\Big(1 - e^{-2\\lambda} - r^2e^{-2\\kappa\\rho}\\big(\\Vstar(\\rho)+8\\pi U(\\phi)\\big)\\Big)\\del_r\\rho\n\\\\\n& \\quad - \\frac{8\\pi r}{3\\kappa} e^{- \\kappa\\rho}\\del_r\\phi D\\phi- {r \\over \\kappa} e^{\\nu+\\lambda}\\Big(\\Wstar(\\rho) - \\frac{16\\pi}{3}e^{-2\\kappa\\rho}U(\\phi)\\Big),\n\\endaligned\n\\ee\nin which the metric coefficients $\\nu, \\lambda$ are given by \n\\bel{eq:403}\n\\aligned\n\\nu + \\lambda\n& = - \\frac{3}{4}\\kappa^{2} \\int_r^{+ \\infty}s|\\del_r\\rho|^2ds - 4\\pi\\int_r^{+ \\infty} e^{- \\kappa\\rho} \\, s|\\del_r\\phi|^2 \\, ds,\n\\\\\ne^{\\nu- \\lambda}\n& = {1 \\over r} \\int_0^r\\Big(1 - s^{2}e^{-2\\kappa\\rho} \\big( \\Vstar(\\rho) + 8\\pi U(\\phi) \\big)\\Big)e^{\\nu+\\lambda} \\, ds. \n\\endaligned\n\\ee\nWe refer to \\eqref{notresysteme} as the \\emph{augmented conformal system} of modified gravity.\n\nSuppose now that $h$ and $l$ are solutions to the system \\eqref{Bondi system int augmented3} defined on $[0,u_0]\\times [0,+ \\infty)$. If \n\\bel{center.condition.2}\nh,l\\in C^1\\big([0,u_0]\\times [0,+ \\infty)\\big),\n\\ee\nthen the first and second regularity conditions at the center hold. In fact, it follows from the system \\eqref{Bondi system int augmented3} that\n\\be\n\\label{more-conditions-center}\n\\hb,\\,\\lb,\\,\\nu,\\,\\lambda\\in C^1\\big([0,u_0]\\times [0,+ \\infty)\\big).\n\\ee\nIn view of \\eqref{eq fR components} and \\eqref{eq Bondi components T}, we get\n\\be\n\\aligned\n& r^2{e^{-\\kappa\\lb}}\\big(E-8\\pi T\\big)(D,D) \n\\\\\n& = r{e^{\\nu- \\lambda}}\\Big(\\frac{e^{\\nu- \\lambda}}{2}\\del_r(\\nu+\\lambda) - 2\\del_u\\lambda\\Big)\n- \\frac{3r^2\\kappa^{2}{ } }{2}(D\\lb)^2-8\\pi r^2{ }(D\\hb)^2.\n\\endaligned\n\\ee\nHence, from the regularity conditions, and taking into account \\eqref{more-conditions-center}, we deduce that\n\\be\n\\lim_{r\\to0}{\\Big(r^2e^{-\\kappa\\rho}\\big(E-8\\pi T\\big)(D,D)\\Big)}= 0, \n\\ee\nsince $\\nu,\\lambda=O(r)$ and $D\\hb,D\\lb$ are bounded near $r=0$. For the second regularity condition at the center, we compute \n\\be\nr\\del_r\\lb = l- \\lb, \\qquad r\\del_r\\hb = h- \\hb.\n\\ee \n This shows that $h=\\partial_r(r\\phi)$ and $l=\\partial_r(r\\rho)$ are $C^1$ on $[0,u_0]\\times[0,+\\infty)$, as claimed. Conversely, if the two center regularity conditions hold and $(h,l)\\in C^0$, the right-hand sides of \\eqref{Bondi system int augmented3} are continuous; integrating along characteristics then yields $(h,l)\\in C^1$, so the two formulations are equivalent in the admissible class.", "post_theorem_intro_text_len": 1131, "post_theorem_intro_text": "\\vskip.15cm\n\nWe point out that our assumptions are quite natural since they ensure positivity and monotonicity properties that also arise in the massless case. The condition $\\phi \\, U'(\\phi) \\geq 0$ is imposed since it guarantees that the Klein-Gordon energy is \\emph{defocusing}, so that the forward evolution will not be limited by the matter model. \nThe conditions ${U(\\phi) \\geq 0}$ and ${f(R) \\leq R \\, f'(R)}$ are required to prove that the Hawking-mass is non-negative. \n\n\\subsection{Outline of this paper.}\n\nIn Section~\\ref{section---2}, we introduce Bondi coordinates and express the field equations of $f(R)$ gravity for a scalar field. We then identify the essential equations, which imply the full set of $f(R)$ field equations, provided a mild regularity condition is assumed at the center. In Section~\\ref{section---A2}, we analyze the regularity at the center. In Section~\\ref{section---3}, we introduce a first-order formulation and establish that it is equivalent to the full set of $f(R)$ equations. Finally, Section~\\ref{section--- 4} is devoted to the study of the Hawking mass and its monotonicity properties.", "sketch": "In the discussion after Theorem~\\ref{main-theo}, the paper indicates the following argument structure. First, introduce Bondi coordinates and write the spherically symmetric $f(R)$-scalar field equations; then “identify the essential equations, which imply the full set of $f(R)$ field equations, provided a mild regularity condition is assumed at the center.” Next, “analyze the regularity at the center.” Then “introduce a first-order formulation and establish that it is equivalent to the full set of $f(R)$ equations,” yielding the claimed integro-differential system. Finally, “study of the Hawking mass and its monotonicity properties,” using the assumptions (notably $U(\\phi)\\ge 0$ and $f(R)\\le R f'(R)$) which are said to be “required to prove that the Hawking-mass is non-negative,” and $\\phi U'(\\phi)\\ge 0$ to ensure the Klein--Gordon energy is “defocusing.”", "expanded_sketch": "In the discussion after the main theorem, the paper indicates the following argument structure. First, introduce Bondi coordinates and write the spherically symmetric $f(R)$-scalar field equations; then “identify the essential equations, which imply the full set of $f(R)$ field equations, provided a mild regularity condition is assumed at the center.” Next, “analyze the regularity at the center.” Then “introduce a first-order formulation and establish that it is equivalent to the full set of $f(R)$ equations,” yielding the claimed integro-differential system. Finally, “study of the Hawking mass and its monotonicity properties,” using the assumptions (notably $U(\\phi)\\ge 0$ and $f(R)\\le R f'(R)$) which are said to be “required to prove that the Hawking-mass is non-negative,” and $\\phi U'(\\phi)\\ge 0$ to ensure the Klein--Gordon energy is “defocusing.”", "expanded_theorem": "[Structure of $f(R)$ gravity in spherical symmetry]\n\\label{main-theo}\nConsider the field equations in $f(R)$ gravity \\eqref{Eq1-14}, coupled to a \npossibly massive, real-valued scalar field $\\phi$. Suppose that the defining function $f= f(R)$ and the matter potential $U=U(\\phi)$ satisfy the following conditions: \n\\bel{hypo-Einstein}\n\\aligned \n& (1) \n\\, &  \\phi \\, U'(\\phi) &\\geq&& 0, \n\\\\\n& (2) \n\\, &  U(\\phi) &\\geq&& 0,   \n\\\\\n& (3) \n\\, & f'(R) &>&&0, \n\\\\\n& (4)  \n\\, &  \nf(R) &\\leq&& R \\, f'(R).  \n\\endaligned\n\\ee \nThen, the field equations of $f(R)$ gravity for spherically symmetric spacetimes can be reduced to an \\emph{integro-differential system} consisting of two first-order coupled, nonlinear hyperbolic equations. We first state the following proposition.\n\n\\begin{proposition}[A first-order formulation of $f(R)$ gravity]\n\\label{main-propo-first-order}\n\\bse\\label{Bondi system int augmented3} \nThe equations of $f(R)$ gravity take the form of a first-order system, in which $h$ and $l$ are the main unknowns: \n\\be\n\\aligned\n& Dh = \\frac{1}{2r}\\big(g- \\gb\\big)(h- \\hb) +\n\\frac{\\kappa}{2}\\big(q(h- \\hb) + p(l- \\lb)\\big)-{r \\over 2} e^{\\nu+\\lambda- \\kappa\\lb}U'(\\hb),\n\\\\\n& Dl = \\frac{1}{2r}\\big(g- \\gb\\big)(l- \\lb)\n- \\frac{8\\pi}{3\\kappa}e^{- \\kappa\\lb}(h- \\hb) p-{r \\over \\kappa} e^{\\nu+\\lambda}\\Big(\\Wstar(\\lb)- \\frac{16\\pi}{3}e^{-2\\kappa\\lb}U(\\hb)\\Big),\n\\endaligned\n\\ee\nin which the metric coefficient $\\nu + \\lambda$ is computed by \n\\be\n\\aligned \n& \\nu + \\lambda = - \\frac{3\\kappa^{2}}{4}\\int_r^{+ \\infty}\\frac{|l- \\lb|^2}{s} \\, ds - 4\\pi\\int_r^{+ \\infty}\\frac{|h- \\hb|^2}{se^{\\kappa\\lb}} \\, ds,\n\\endaligned\n\\ee\nand the auxiliary variables $p$ and $q$ are defined explicitly by \n\\bel{p,q}\n\\aligned \n&p = \\frac{\\kappa}{2r}\\int_0^r\\Big((l- \\lb)p + (h- \\hb)q\\Big) \\, ds +\\frac{1}{2r}\\int_0^r\\frac{e^{\\nu- \\lambda}(h- \\hb)}{s} \\, ds\n- \\frac{1}{2r}\\int_0^rse^{\\nu+\\lambda- \\kappa\\lb}U'(\\hb) \\, ds, \n\\\\\n& \\kappa q \n= - \\frac{8\\pi}{3 r}\\int_0^r\\frac{(h- \\hb)p}{e^{\\kappa\\lb}} \\, ds+ \\frac{\\kappa}{2r}\\int_0^r{\\frac{e^{\\nu- \\lambda}(l- \\lb)}{s} \\, ds}\n - \\frac{1}{r}\\int_0^r se^{\\nu+\\lambda}\\Big (\\Wstar(\\lb)-\n\\frac{16\\pi}{3} e^{-2\\kappa\\lb}U(\\hb)\\Big) \\, ds. \n\\endaligned\n\\ee\n\\ese\n\\end{proposition}\n\nIndeed, this system is equivalent to the full Einstein equations in the class of $C^1$ solutions that are suitably regular at the center in the following sense.\n\n\\begin{definition}  \n\\label{definition}\n\\emph{The \\emph{first regularity condition at the center\\footnote{In view of \\eqref{eq fR components} and \\eqref{eq Bondi components T}, this condition can be expressed in terms of first-order derivatives of the unknowns.}}} is defined as \n\\be\n\\label{regularity} \n\\lim_{r\\to0} r^2e^{-\\kappa\\rho}\\big(E - 8\\pi T\\big)(D,D) = 0.\n\\ee\nThe \\emph{second regularity condition at the center} is defined as \n\\be\n\\label{regularity2}\n\\del_r(r\\phi) \\text{ and }\\del_r(r\\rho) \\text{ are } C^1\\text{ on } [0,u_0]\\times [0,+ \\infty).\n\\ee\n\\end{definition}\n\nFurthermore, the Hawking mass is future non-decreasing along radial directions and non-increasing along null directions. In the limit ${f(R) \\to R}$ and \n$U(\\phi)\\to 0$, one recovers Christodoulou's formulation of the Einstein-massless scalar field system.,", "theorem_type": ["Implication", "Biconditional or Equivalence"], "mcq": {"question": "Consider the modified-gravity field equations E_{ab}=8π T_{ab} on a spherically symmetric spacetime, where E_{ab}:=f'(R)G_{ab}−(1/2)(f(R)−R f'(R))g_{ab}+(g_{ab}□_g−∇_a∇_b)(f'(R)) and T_{ab}=∇_aφ∇_bφ−( (1/2)∇^cφ∇_cφ+U(φ) )g_{ab} for a real-valued scalar field φ. Assume that φ U'(φ)≥0, U(φ)≥0, f'(R)>0, and f(R)≤R f'(R). Introduce ρ:=(1/κ)log f'(R) with κ>0, and in Bondi-type coordinates let h:=∂_r(rφ) and l:=∂_r(rρ). Which conclusion about the resulting spherically symmetric f(R) system holds under these assumptions?", "correct_choice": {"label": "A", "text": "For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. In the class of C^1 solutions satisfying the two center regularity conditions lim_{r→0} r^2 e^{−κρ}(E−8πT)(D,D)=0 and ∂_r(rφ), ∂_r(rρ) ∈ C^1([0,u_0]×[0,∞))—where D is the characteristic Bondi derivative—this first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations. Moreover, the Hawking mass of the symmetry spheres is future non-decreasing along radial directions and non-increasing along null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system."}, "choices": [{"label": "B", "text": "For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. This first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations for every C^1 solution, without any additional regularity assumptions at the center. Moreover, the Hawking mass of the symmetry spheres is future non-decreasing along radial directions and non-increasing along null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system."}, {"label": "C", "text": "For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. Moreover, in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system."}, {"label": "D", "text": "For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. In the class of C^1 solutions satisfying the two center regularity conditions lim_{r→0} r^2 e^{−κρ}(E−8πT)(D,D)=0 and ∂_r(rφ), ∂_r(rρ) ∈ C^1([0,u_0]×[0,∞))—where D is the characteristic Bondi derivative—this first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations. Moreover, the Hawking mass of the symmetry spheres is future non-decreasing both along radial directions and along null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system."}, {"label": "E", "text": "For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. In the class of C^1 solutions satisfying the two center regularity conditions lim_{r→0} r^2 e^{−κρ}(E−8πT)(D,D)=0 and ∂_r(rφ), ∂_r(rρ) ∈ C^1([0,u_0]×[0,∞))—where D is the characteristic Bondi derivative—this first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations. Moreover, the Hawking mass of the symmetry spheres is non-negative and therefore constant along radial directions and null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system."}], "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": "center regularity needed for equivalence", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "equivalence and Hawking-mass monotonicity clauses removed", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "direction/sign of Hawking-mass monotonicity along null directions", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "nonnegative Hawking mass incorrectly upgraded to constancy", "template_used": "wildcard"}]}}
{"id": "2512.07362v1", "paper_link": "http://arxiv.org/abs/2512.07362v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{th:main}\nLet $a$, $b$ and $d$ be given positive constants such that $a\\ge 4$ and $d<b$. Then there exists a positive solution $(\\phi,\\psi)$ of \\eqref{TWS}-\\eqref{BC} for each $s > s^{*}$. This existence also holds for $s=s^*$, if we further assume that $J_2$ has a compact support. Moreover, there exist no positive solutions of \\eqref{TWS}-\\eqref{BC} for $s < s^{*}$.", "start_pos": 8781, "end_pos": 9185, "label": "th:main"}, "ref_dict": {"th:main": "\\begin{theorem}\\label{th:main}\nLet $a$, $b$ and $d$ be given positive constants such that $a\\ge 4$ and $d<b$. Then there exists a positive solution $(\\phi,\\psi)$ of \\eqref{TWS}-\\eqref{BC} for each $s > s^{*}$. This existence also holds for $s=s^*$, if we further assume that $J_2$ has a compact support. Moreover, there exist no positive solutions of \\eqref{TWS}-\\eqref{BC} for $s < s^{*}$.\n\\end{theorem}", "TWS": "\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}", "pp": "\\begin{equation}\\label{pp}\n\\begin{cases}\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-V](x,t), \\; x\\in\\bR,\\, t>0,\\\\\nV_t(x,t)=d \\bN_2[V(\\cdot,t)](x)+[bV(1-V/U)](x,t), \\; x\\in\\bR,\\, t>0,\n\\end{cases}\n\\end{equation}", "BC": "\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}"}, "pre_theorem_intro_text_len": 4485, "pre_theorem_intro_text": "\\setcounter{equation}{0}\n\nWe consider the following diffusive prey-predator model with nonlocal dispersal\n\\begin{equation}\\label{pp}\n\\begin{cases}\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-V](x,t), \\; x\\in\\bR,\\, t>0,\\\\\nV_t(x,t)=d \\bN_2[V(\\cdot,t)](x)+[bV(1-V/U)](x,t), \\; x\\in\\bR,\\, t>0,\n\\end{cases}\n\\end{equation}\nwhere the unknown functions $U,V$ stand for the population densities of prey and predator species at position $x$ and time $t$, respectively, $d,a,b$ are positive constants such that $1,d$ are diffusion coefficients and $a,b$ are intrinsic growth rates of species $U,V$, respectively. The functional response of predator to prey is normalized to be $1$. The prey obeys the logistic growth and its carrying capacity is normalized to be 1. However, the density of predator follows a logistic dynamics with a varying carrying capacity proportional to the density of prey. Moreover, for $i=1,2$, $\\bN_i$ formulates the spatial nonlocal dispersal of individuals and is defined by\n\\beaa\n\\bN_i[u(\\cdot,t)](x):=\\int_{\\mathbb{R}}J_i(y)u(x-y,t)dy-u(x,t),\\; u=U,V,\n\\eeaa\nwhere $J_i$ is a probability density function satisfying the following conditions: \n\\begin{enumerate}\n\\item[(H1)]\\,  $J_i$ is a nonnegative continuous function defined in $\\bR$;\n\\item[(H2)]\\, $\\int_{\\mathbb{R}}J_i(y)dy=1$ and $J_i(y)=J_i(-y)$ for all $y\\in \\mathbb{R}$;\n\\item[(H3)] there exists $\\hat{\\lambda}_i\\in(0,\\infty]$ such that  $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy < \\infty$ for any $\\lambda \\in (0, \\hat{\\lambda_i})$ and $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy\\to\\infty$ as $\\lambda \\uparrow \\hat{\\lambda}_i$.\n\\end{enumerate}\nNote that there is always the predator-free state $(1,0)$, and there is the unique constant co-existence state $(a^*,a^*)$, $a^*:=1-1/a\\in(0,1)$, when $a>1$.\n\nWhen the nonlocal dispersal in system \\eqref{pp} is replaced by the classical diffusion, the dynamical behaviors of the corresponding system was investigated in the survey paper \\cite{CGS21}. In fact, system \\eqref{pp} without diffusion (the ODE system) arises in the control of introduced rabbits to protect native birds from introduced cat predation in an island (cf. \\cite{CLS}), when we consider the case without rabbits and control. For the detailed biological background of the full ODE system including the rabbits and control, we refer the reader to \\cite{CLS,CS}. However, it is reasonable and more realistic to take into account the influence of spatial movements of birds and cats, namely, the effect of diffusion. On the other hand, to model the long range movements and nonadjacent interactions of individuals it is more realistic to consider the nonlocal dispersal instead of the random movement with classical diffusion. This motivates us to study system \\eqref{pp}.\n\nThe main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \\eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \\eqref{pp} is called a traveling wave solution of \\eqref{pp}, if there exist a constant $s\\in\\R$ ({\\it the wave speed}) and a function $(\\phi,\\psi)$ ({\\it the wave profile}) of class $C^1(\\R)$ such that\n$$(U,V)(x,t)=(\\phi,\\psi)( z),\\ \\ z:=x-st.$$\nWe are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\\{s,\\phi,\\psi\\}$ satisfies the following system of equations:\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\ntogether with the following asymptotic boundary conditions\n\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}\n\nFor convenience, for $i=1,2$, we set\n\\beaa\nI_i(\\lambda):=\\int_\\bR J_i(y)e^{\\lambda y}dy,\\ \\ \\lambda\\in\\R,\n\\eeaa\nwith $I_i(\\lambda)\\in(0,\\infty)$ if $0\\le\\lambda<\\hat{\\lambda}_i$ and $I_i(\\lambda)=\\infty$ if $\\lambda\\ge\\hat{\\lambda}_i$. Note that, due to the symmetry of $J_i$, the function $I_i$ is even, and it is also strictly convex in $(-\\hat{\\lambda}_i,\\hat{\\lambda}_i)$. Also, we introduce the quantity\n\\beaa\ns^*:=\\inf_{\\lambda\\in(0,\\hat{\\lambda}_2)}\\frac{d[I_2(\\lambda)-1]+b}{\\lambda}.\n\\eeaa\nNote that $s^*$ is well-defined, the infimum is reached, and $s^*>0$.\n\nWe now state our main theorem as follows.", "context": "We consider the following diffusive prey-predator model with nonlocal dispersal\n\\begin{equation}\\label{pp}\n\\begin{cases}\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-V](x,t), \\; x\\in\\bR,\\, t>0,\\\\\nV_t(x,t)=d \\bN_2[V(\\cdot,t)](x)+[bV(1-V/U)](x,t), \\; x\\in\\bR,\\, t>0,\n\\end{cases}\n\\end{equation}\nwhere the unknown functions $U,V$ stand for the population densities of prey and predator species at position $x$ and time $t$, respectively, $d,a,b$ are positive constants such that $1,d$ are diffusion coefficients and $a,b$ are intrinsic growth rates of species $U,V$, respectively. The functional response of predator to prey is normalized to be $1$. The prey obeys the logistic growth and its carrying capacity is normalized to be 1. However, the density of predator follows a logistic dynamics with a varying carrying capacity proportional to the density of prey. Moreover, for $i=1,2$, $\\bN_i$ formulates the spatial nonlocal dispersal of individuals and is defined by\n\\beaa\n\\bN_i[u(\\cdot,t)](x):=\\int_{\\mathbb{R}}J_i(y)u(x-y,t)dy-u(x,t),\\; u=U,V,\n\\eeaa\nwhere $J_i$ is a probability density function satisfying the following conditions: \n\\begin{enumerate}\n\\item[(H1)]\\, $J_i$ is a nonnegative continuous function defined in $\\bR$;\n\\item[(H2)]\\, $\\int_{\\mathbb{R}}J_i(y)dy=1$ and $J_i(y)=J_i(-y)$ for all $y\\in \\mathbb{R}$;\n\\item[(H3)] there exists $\\hat{\\lambda}_i\\in(0,\\infty]$ such that $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy < \\infty$ for any $\\lambda \\in (0, \\hat{\\lambda_i})$ and $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy\\to\\infty$ as $\\lambda \\uparrow \\hat{\\lambda}_i$.\n\\end{enumerate}\nNote that there is always the predator-free state $(1,0)$, and there is the unique constant co-existence state $(a^*,a^*)$, $a^*:=1-1/a\\in(0,1)$, when $a>1$.\n\nWhen the nonlocal dispersal in system \\eqref{pp} is replaced by the classical diffusion, the dynamical behaviors of the corresponding system was investigated in the survey paper \\cite{CGS21}. In fact, system \\eqref{pp} without diffusion (the ODE system) arises in the control of introduced rabbits to protect native birds from introduced cat predation in an island (cf. \\cite{CLS}), when we consider the case without rabbits and control. For the detailed biological background of the full ODE system including the rabbits and control, we refer the reader to \\cite{CLS,CS}. However, it is reasonable and more realistic to take into account the influence of spatial movements of birds and cats, namely, the effect of diffusion. On the other hand, to model the long range movements and nonadjacent interactions of individuals it is more realistic to consider the nonlocal dispersal instead of the random movement with classical diffusion. This motivates us to study system \\eqref{pp}.\n\nThe main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \\eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \\eqref{pp} is called a traveling wave solution of \\eqref{pp}, if there exist a constant $s\\in\\R$ ({\\it the wave speed}) and a function $(\\phi,\\psi)$ ({\\it the wave profile}) of class $C^1(\\R)$ such that\n$$(U,V)(x,t)=(\\phi,\\psi)( z),\\ \\ z:=x-st.$$\nWe are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\\{s,\\phi,\\psi\\}$ satisfies the following system of equations:\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n& \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z) =0, &\\; z\\in \\mathbb{R},\\\\\n& d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\ntogether with the following asymptotic boundary conditions\n\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}\n\nFor convenience, for $i=1,2$, we set\n\\beaa\nI_i(\\lambda):=\\int_\\bR J_i(y)e^{\\lambda y}dy,\\ \\ \\lambda\\in\\R,\n\\eeaa\nwith $I_i(\\lambda)\\in(0,\\infty)$ if $0\\le\\lambda<\\hat{\\lambda}_i$ and $I_i(\\lambda)=\\infty$ if $\\lambda\\ge\\hat{\\lambda}_i$. Note that, due to the symmetry of $J_i$, the function $I_i$ is even, and it is also strictly convex in $(-\\hat{\\lambda}_i,\\hat{\\lambda}_i)$. Also, we introduce the quantity\n\\beaa\ns^*:=\\inf_{\\lambda\\in(0,\\hat{\\lambda}_2)}\\frac{d[I_2(\\lambda)-1]+b}{\\lambda}.\n\\eeaa\nNote that $s^*$ is well-defined, the infimum is reached, and $s^*>0$.\n\nWe now state our main theorem as follows.\n\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}", "full_context": "We consider the following diffusive prey-predator model with nonlocal dispersal\n\\begin{equation}\\label{pp}\n\\begin{cases}\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-V](x,t), \\; x\\in\\bR,\\, t>0,\\\\\nV_t(x,t)=d \\bN_2[V(\\cdot,t)](x)+[bV(1-V/U)](x,t), \\; x\\in\\bR,\\, t>0,\n\\end{cases}\n\\end{equation}\nwhere the unknown functions $U,V$ stand for the population densities of prey and predator species at position $x$ and time $t$, respectively, $d,a,b$ are positive constants such that $1,d$ are diffusion coefficients and $a,b$ are intrinsic growth rates of species $U,V$, respectively. The functional response of predator to prey is normalized to be $1$. The prey obeys the logistic growth and its carrying capacity is normalized to be 1. However, the density of predator follows a logistic dynamics with a varying carrying capacity proportional to the density of prey. Moreover, for $i=1,2$, $\\bN_i$ formulates the spatial nonlocal dispersal of individuals and is defined by\n\\beaa\n\\bN_i[u(\\cdot,t)](x):=\\int_{\\mathbb{R}}J_i(y)u(x-y,t)dy-u(x,t),\\; u=U,V,\n\\eeaa\nwhere $J_i$ is a probability density function satisfying the following conditions: \n\\begin{enumerate}\n\\item[(H1)]\\, $J_i$ is a nonnegative continuous function defined in $\\bR$;\n\\item[(H2)]\\, $\\int_{\\mathbb{R}}J_i(y)dy=1$ and $J_i(y)=J_i(-y)$ for all $y\\in \\mathbb{R}$;\n\\item[(H3)] there exists $\\hat{\\lambda}_i\\in(0,\\infty]$ such that $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy < \\infty$ for any $\\lambda \\in (0, \\hat{\\lambda_i})$ and $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy\\to\\infty$ as $\\lambda \\uparrow \\hat{\\lambda}_i$.\n\\end{enumerate}\nNote that there is always the predator-free state $(1,0)$, and there is the unique constant co-existence state $(a^*,a^*)$, $a^*:=1-1/a\\in(0,1)$, when $a>1$.\n\nWhen the nonlocal dispersal in system \\eqref{pp} is replaced by the classical diffusion, the dynamical behaviors of the corresponding system was investigated in the survey paper \\cite{CGS21}. In fact, system \\eqref{pp} without diffusion (the ODE system) arises in the control of introduced rabbits to protect native birds from introduced cat predation in an island (cf. \\cite{CLS}), when we consider the case without rabbits and control. For the detailed biological background of the full ODE system including the rabbits and control, we refer the reader to \\cite{CLS,CS}. However, it is reasonable and more realistic to take into account the influence of spatial movements of birds and cats, namely, the effect of diffusion. On the other hand, to model the long range movements and nonadjacent interactions of individuals it is more realistic to consider the nonlocal dispersal instead of the random movement with classical diffusion. This motivates us to study system \\eqref{pp}.\n\nThe main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \\eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \\eqref{pp} is called a traveling wave solution of \\eqref{pp}, if there exist a constant $s\\in\\R$ ({\\it the wave speed}) and a function $(\\phi,\\psi)$ ({\\it the wave profile}) of class $C^1(\\R)$ such that\n$$(U,V)(x,t)=(\\phi,\\psi)( z),\\ \\ z:=x-st.$$\nWe are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\\{s,\\phi,\\psi\\}$ satisfies the following system of equations:\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n& \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z) =0, &\\; z\\in \\mathbb{R},\\\\\n& d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\ntogether with the following asymptotic boundary conditions\n\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}\n\nFor convenience, for $i=1,2$, we set\n\\beaa\nI_i(\\lambda):=\\int_\\bR J_i(y)e^{\\lambda y}dy,\\ \\ \\lambda\\in\\R,\n\\eeaa\nwith $I_i(\\lambda)\\in(0,\\infty)$ if $0\\le\\lambda<\\hat{\\lambda}_i$ and $I_i(\\lambda)=\\infty$ if $\\lambda\\ge\\hat{\\lambda}_i$. Note that, due to the symmetry of $J_i$, the function $I_i$ is even, and it is also strictly convex in $(-\\hat{\\lambda}_i,\\hat{\\lambda}_i)$. Also, we introduce the quantity\n\\beaa\ns^*:=\\inf_{\\lambda\\in(0,\\hat{\\lambda}_2)}\\frac{d[I_2(\\lambda)-1]+b}{\\lambda}.\n\\eeaa\nNote that $s^*$ is well-defined, the infimum is reached, and $s^*>0$.\n\nWe now state our main theorem as follows.\n\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\n\nThe main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \\eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \\eqref{pp} is called a traveling wave solution of \\eqref{pp}, if there exist a constant $s\\in\\R$ ({\\it the wave speed}) and a function $(\\phi,\\psi)$ ({\\it the wave profile}) of class $C^1(\\R)$ such that\n$$(U,V)(x,t)=(\\phi,\\psi)( z),\\ \\ z:=x-st.$$\nWe are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\\{s,\\phi,\\psi\\}$ satisfies the following system of equations:\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n& \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z) =0, &\\; z\\in \\mathbb{R},\\\\\n& d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\ntogether with the following asymptotic boundary conditions\n\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}\n\nWe now state our main theorem as follows.\n\nHereafter, a function $(\\phi,\\psi)$ is positive if $\\phi,\\psi>0$ in $\\bR$. A corresponding theorem to Theorem~\\ref{th:main}, under the condition $a\\ge 4$, on the traveling waves in the classical diffusion case was derived in \\cite{CGS21}. Theorem~\\ref{th:main} characterizes the minimal speed $s^*$ in the nonlocal dispersal case, but it needs the extra condition $d<b$ in the construction of lower and upper solutions to~\\eqref{TWS}.\n\nThe existence of wave profiles $(\\phi,\\psi)$ to \\eqref{TWS}-\\eqref{BC} for all speeds $s>s^*$ is based on the construction of lower and upper solutions, which is new for this singular nonlocal system. The case $s=s^*$ is more involved, it requires a special care and uses the boundedness of the support of $J_2$. The characterization of the limiting state behind the front, that is, $(\\phi,\\psi)(-\\infty)=(a^*,a^*)$ is carried out with a squeezing method and an argument by contradiction. The proof of the nonexistence of wave profiles for all speeds $s<s^*$ is also done by contradiction. Section~\\ref{sec2} is devoted to the existence part in Theorem~\\ref{th:main}, while the nonexistence is shown in Section~\\ref{sec3}.\n\n\\begin{proposition}\\label{le6}\nLet $s>0$ be given. Let $(\\overline{\\phi },\\overline{\\psi })$ and $(\\underline{\\phi},\\underline{\\psi})$ be a pair of bounded upper and lower solutions of \\eqref {TWS}. Then system \\eqref {TWS} admits a positive solution $(\\phi,\\psi)$ of class $C^1(\\R)$ such that\n\\begin{equation*}\n\\underline{\\phi }( z )\\leq \\phi(z)\\leq \\overline{\\phi}( z ),\\; \\upsi(z)\\leq \\psi(z)\\leq\\overline{\\psi}( z ),\\; z \\in \\mathbb{R}.\n\\end{equation*}\n\\end{proposition}\n\nFor a given $s>s^*$, we consider the quantity\n\\beaa\nA(\\lambda)=A(\\lambda;s):=d[I_2(\\lambda)-1]-s\\lambda+b,\\; \\lambda>0.\n\\eeaa\nIt follows from the definition of $s^*$ and the strict convexity of $A(\\lambda)$ that there are two positive constants $\\lambda_1<\\lambda_2<\\hat{\\ld}_2$ such that\n$$A(\\lambda_1)=A(\\ld_2)=0$$\nand $A(\\lambda)<0$ for all $\\lambda\\in(\\lambda_1,\\lambda_2)$. Also, set \n$$B(\\ld)=B(\\ld;s):=[I_1(\\ld)-1]-s\\ld.$$ \nSince $B(0)=0$ and $B'(0)=-s<0$, we can choose a constant $\\ld_0\\in(0,\\min\\{\\ld_1,\\hat{\\ld}_1\\})$ small enough such that\n$$B(\\ld_0)<0.$$\nNow, for a fixed constant $\\mu\\in(1,\\min{\\{ \\lambda_2/ \\lambda_1,2\\}})$, we choose \n\\be\\label{q0}\nq>\\max\\Big\\{1,\\frac{2b}{-A(\\mu\\ld_1)}\\Big\\}.\n\\ee\nThen, the function $f(z) := e^{ -\\lambda_1z}-qe^{-\\mu \\lambda_1z}$ has exactly one zero $z_0>0$ and exactly one maximum point $z_M\\in(z_0,\\infty)$, and there holds $f(z_M)>0$. Thus, using $a\\ge4>1$ and $d<b$, we can choose $\\delta$ such that\n$$0<\\delta<\\min\\Big\\{f(z_M),\\underbrace{1-\\frac{1}{a}}_{=a^*},\\frac12\\Big(1-\\frac{d}{b}\\Big)\\Big\\},$$\nand a point $z_1\\in(z_0,z_M)$ such that $f(z_1)=\\delta$. Note that\n\\be\\label{delta}\nd<b(1-2\\delta).\n\\ee\nFurthermore, the inequalities $z_1>z_0>0$, $\\ld_1>0$ and $\\mu>1$ imply\n\\be\\label{q}\n{e^{(\\mu-1)\\ld_1z_1}>e^{(\\mu-1)\\ld_1z_0}=q.}\n\\ee\nWith these choices of $\\mu,q,\\delta$, we finally choose $\\e$ such that\n\\be\\label{ep1} \n0<\\e<\\min{\\left\\{\\frac{\\delta}{1+s\\ld_1+a},\\frac{e^{(\\mu-1)\\lambda_1z_1}-q}{(1+s\\lambda_1+a)e^{(\\mu-1)\\lambda_1z_1}} \\right\\}}.\n\\ee\nNote that the constant $\\e$ is admissible, due to \\eqref{q}, and that $0<\\e<\\delta<1/2$.\n\n\\begin{lemma}\\label{la:upper-lower2}\nAssume that $a\\ge 4$, $d<b$, and that $J_2$ has a compact support such that its support is contained in $[-S,S]$ for some $S\\in(0,\\infty)$. For $s=s^{*}$, the functions $(\\overline{\\phi},\\overline{\\psi})$ and~$(\\underline{\\phi},\\underline{\\psi})$ defined in \\eqref{uplosol2} are a pair of upper and lower solutions of \\eqref{TWS}.\n\\end{lemma}\n\nFirst, for any $s\\ge s^*$, the existence of a $C^1(\\R)$ solution $(\\phi,\\psi)$ to~\\eqref{TWS} such that $1/2\\le\\underline{\\phi}\\le\\phi\\le\\overline{\\phi}<1$ and $0<\\underline{\\psi}\\le\\psi\\le\\overline{\\psi}\\le1$ in $\\R$ follows from Lemmas~\\ref{la:upper-lower1}-\\ref{la:upper-lower2} and Proposition~\\ref{le6}. Moreover, by the construction of upper and lower solutions, it is clear that $(\\phi,\\psi)(\\infty)=(1,0)$.\n\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\n\n\\begin{equation}\\label{pp}\n\\begin{cases}\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-V](x,t), \\; x\\in\\bR,\\, t>0,\\\\\nV_t(x,t)=d \\bN_2[V(\\cdot,t)](x)+[bV(1-V/U)](x,t), \\; x\\in\\bR,\\, t>0,\n\\end{cases}\n\\end{equation}\n\n\\begin{theorem}\\label{th:main}\nLet $a$, $b$ and $d$ be given positive constants such that $a\\ge 4$ and $d<b$. Then there exists a positive solution $(\\phi,\\psi)$ of \\eqref{TWS}-\\eqref{BC} for each $s > s^{*}$. This existence also holds for $s=s^*$, if we further assume that $J_2$ has a compact support. Moreover, there exist no positive solutions of \\eqref{TWS}-\\eqref{BC} for $s < s^{*}$.\n\\end{theorem}", "post_theorem_intro_text_len": 1964, "post_theorem_intro_text": "Hereafter, a function $(\\phi,\\psi)$ is positive if $\\phi,\\psi>0$ in $\\bR$. A corresponding theorem to Theorem~\\ref{th:main}, under the condition $a\\ge 4$, on the traveling waves in the classical diffusion case was derived in \\cite{CGS21}. Theorem~\\ref{th:main} characterizes the minimal speed $s^*$ in the nonlocal dispersal case, but it needs the extra condition $d<b$ in the construction of lower and upper solutions to~\\eqref{TWS}.\n\nIt is worth to note that there are some difficulties in the study of system \\eqref{pp}. It is a non-monotone system which is lack of comparison principle and a singularity may occur in the $V$-equation when $U$ reaches zero in a finite time which has been confirmed numerically in \\cite{CLS}. Thus system \\eqref{pp} is named as a {\\it singular} prey-predator model due to the latter fact. This is in contrast to the system with $U$-equation in \\eqref{pp} replaced by\n\\beaa\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-kUV](x,t), \\; x\\in\\bR,\\, t>0,\n\\eeaa\nin which the functional response of predation is a linear function of prey. In this case, the strong maximum principle for the scalar equation gives that $U>0$ for all $t>0$. Hence no singularity can occur for the nonlinear term $V/U$ in the $V$-equation.\n\nThe existence of wave profiles $(\\phi,\\psi)$ to \\eqref{TWS}-\\eqref{BC} for all speeds $s>s^*$ is based on the construction of lower and upper solutions, which is new for this singular nonlocal system. The case $s=s^*$ is more involved, it requires a special care and uses the boundedness of the support of $J_2$. The characterization of the limiting state behind the front, that is, $(\\phi,\\psi)(-\\infty)=(a^*,a^*)$ is carried out with a squeezing method and an argument by contradiction. The proof of the nonexistence of wave profiles for all speeds $s<s^*$ is also done by contradiction. Section~\\ref{sec2} is devoted to the existence part in Theorem~\\ref{th:main}, while the nonexistence is shown in Section~\\ref{sec3}.", "sketch": "The post-theorem introduction outlines the proof of Theorem~\\ref{th:main} as follows:\n\n- **Existence for $s>s^*$:** “based on the construction of lower and upper solutions,” which the authors note “is new for this singular nonlocal system,” and the extra condition $d<b$ is needed “in the construction of lower and upper solutions to~\\eqref{TWS}.”\n\n- **Existence at $s=s^*$:** “more involved,” requiring “special care” and using “the boundedness of the support of $J_2$” (i.e., compact support).\n\n- **Limiting state behind the front:** the claim “$(\\phi,\\psi)(-\\infty)=(a^*,a^*)$” is proved “with a squeezing method and an argument by contradiction.”\n\n- **Nonexistence for $s<s^*$:** proved “by contradiction.”\n\n- **Organization:** “Section~\\ref{sec2} is devoted to the existence part in Theorem~\\ref{th:main}, while the nonexistence is shown in Section~\\ref{sec3}.”", "expanded_sketch": "The post-theorem introduction outlines the proof strategy used to establish the main theorem as follows:\n\n- **Existence for $s>s^*$:** “based on the construction of lower and upper solutions,” which the authors note “is new for this singular nonlocal system,” and the extra condition $d<b$ is needed “in the construction of lower and upper solutions to\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}.”\n\n- **Existence at $s=s^*$:** “more involved,” requiring “special care” and using “the boundedness of the support of $J_2$” (i.e., compact support).\n\n- **Limiting state behind the front:** the claim “$(\\phi,\\psi)(-\\infty)=(a^*,a^*)$” is proved “with a squeezing method and an argument by contradiction.”\n\n- **Nonexistence for $s<s^*$:** proved “by contradiction.”\n\n- **Organization:** Next they prove the existence part in the main theorem, while the nonexistence is shown later.", "expanded_theorem": "\\label{th:main}\nLet $a$, $b$ and $d$ be given positive constants such that $a\\ge 4$ and $d<b$. Then there exists a positive solution $(\\phi,\\psi)$ of\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\nwith boundary conditions\n\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}\nfor each $s > s^{*}$. This existence also holds for $s=s^*$, if we further assume that $J_2$ has a compact support. Moreover, there exist no positive solutions of the system \\eqref{TWS} with boundary conditions \\eqref{BC} for $s < s^{*}$.", "theorem_type": ["Existence", "Nonexistence"], "mcq": {"question": "Let \\(a,b,d>0\\) with \\(a\\ge 4\\) and \\(d<b\\), and set \\(a^*=1-\\frac1a\\). For \\(i=1,2\\), define the nonlocal dispersal operator\n\\[\n\\mathcal N_i[u](z)=\\int_{\\mathbb R}J_i(y)u(z-y)\\,dy-u(z),\n\\]\nwhere each kernel \\(J_i\\) is a nonnegative continuous symmetric probability density on \\(\\mathbb R\\) satisfying the exponential-moment condition: there exists \\(\\hat\\lambda_i\\in(0,\\infty]\\) such that \\(\\int_{\\mathbb R}J_i(y)e^{\\lambda y}\\,dy<\\infty\\) for \\(0<\\lambda<\\hat\\lambda_i\\), and this integral tends to \\(\\infty\\) as \\(\\lambda\\uparrow \\hat\\lambda_i\\). Let\n\\[\nI_2(\\lambda)=\\int_{\\mathbb R}J_2(y)e^{\\lambda y}\\,dy,\n\\qquad\ns^*=\\inf_{\\lambda\\in(0,\\hat\\lambda_2)}\\frac{d\\,[I_2(\\lambda)-1]+b}{\\lambda}.\n\\]\nConsider the traveling-wave profile system\n\\[\n\\begin{cases}\n\\mathcal N_1[\\phi](z)+s\\phi'(z)+a\\phi(z)(1-\\phi(z))-\\psi(z)=0, & z\\in\\mathbb R,\\\\\n d\\,\\mathcal N_2[\\psi](z)+s\\psi'(z)+b\\psi(z)\\left(1-\\dfrac{\\psi(z)}{\\phi(z)}\\right)=0, & z\\in\\mathbb R,\n\\end{cases}\n\\]\nwith boundary conditions\n\\[\n(\\phi,\\psi)(+\\infty)=(1,0),\\qquad (\\phi,\\psi)(-\\infty)=(a^*,a^*).\n\\]\nWhich statement holds for positive solutions \\((\\phi,\\psi)\\) of this boundary-value problem?", "correct_choice": {"label": "A", "text": "A positive solution exists for every wave speed \\(s>s^*\\). It also exists for the critical speed \\(s=s^*\\) provided that \\(J_2\\) has compact support. For every \\(s<s^*\\), no positive solution exists."}, "choices": [{"label": "B", "text": "A positive solution exists for every wave speed \\(s\\ge s^*\\). Moreover, for every \\(s<s^*\\), no positive solution exists."}, {"label": "C", "text": "For every \\(s>s^*\\), there exists a positive solution. In addition, for every \\(s<s^*\\), no positive solution exists."}, {"label": "D", "text": "A positive solution exists for every wave speed \\(s>s^*\\), and also for \\(s=s^*\\) without any additional assumption on \\(J_2\\). For every \\(s<s^*\\), no positive solution exists."}, {"label": "E", "text": "A positive solution exists for every wave speed \\(s>s^*\\). It also exists for the critical speed \\(s=s^*\\) provided that both \\(J_1\\) and \\(J_2\\) have compact support. For every \\(s<s^*\\), no positive solution exists."}], "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": "critical-speed compact-support hypothesis at s=s^*", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped the critical-speed existence clause", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "dependence of critical-speed existence on compact support of J_2", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "which kernel requires compact support at the critical speed", "template_used": "wildcard"}]}}
{"id": "2512.07362v1", "paper_link": "http://arxiv.org/abs/2512.07362v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{th:main}\nLet $a$, $b$ and $d$ be given positive constants such that $a\\ge 4$ and $d<b$. Then there exists a positive solution $(\\phi,\\psi)$ of \\eqref{TWS}-\\eqref{BC} for each $s > s^{*}$. This existence also holds for $s=s^*$, if we further assume that $J_2$ has a compact support. Moreover, there exist no positive solutions of \\eqref{TWS}-\\eqref{BC} for $s < s^{*}$.", "start_pos": 8781, "end_pos": 9185, "label": "th:main"}, "ref_dict": {"th:main": "\\begin{theorem}\\label{th:main}\nLet $a$, $b$ and $d$ be given positive constants such that $a\\ge 4$ and $d<b$. Then there exists a positive solution $(\\phi,\\psi)$ of \\eqref{TWS}-\\eqref{BC} for each $s > s^{*}$. This existence also holds for $s=s^*$, if we further assume that $J_2$ has a compact support. Moreover, there exist no positive solutions of \\eqref{TWS}-\\eqref{BC} for $s < s^{*}$.\n\\end{theorem}", "TWS": "\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}", "pp": "\\begin{equation}\\label{pp}\n\\begin{cases}\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-V](x,t), \\; x\\in\\bR,\\, t>0,\\\\\nV_t(x,t)=d \\bN_2[V(\\cdot,t)](x)+[bV(1-V/U)](x,t), \\; x\\in\\bR,\\, t>0,\n\\end{cases}\n\\end{equation}", "BC": "\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}"}, "pre_theorem_intro_text_len": 4485, "pre_theorem_intro_text": "\\setcounter{equation}{0}\n\nWe consider the following diffusive prey-predator model with nonlocal dispersal\n\\begin{equation}\\label{pp}\n\\begin{cases}\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-V](x,t), \\; x\\in\\bR,\\, t>0,\\\\\nV_t(x,t)=d \\bN_2[V(\\cdot,t)](x)+[bV(1-V/U)](x,t), \\; x\\in\\bR,\\, t>0,\n\\end{cases}\n\\end{equation}\nwhere the unknown functions $U,V$ stand for the population densities of prey and predator species at position $x$ and time $t$, respectively, $d,a,b$ are positive constants such that $1,d$ are diffusion coefficients and $a,b$ are intrinsic growth rates of species $U,V$, respectively. The functional response of predator to prey is normalized to be $1$. The prey obeys the logistic growth and its carrying capacity is normalized to be 1. However, the density of predator follows a logistic dynamics with a varying carrying capacity proportional to the density of prey. Moreover, for $i=1,2$, $\\bN_i$ formulates the spatial nonlocal dispersal of individuals and is defined by\n\\beaa\n\\bN_i[u(\\cdot,t)](x):=\\int_{\\mathbb{R}}J_i(y)u(x-y,t)dy-u(x,t),\\; u=U,V,\n\\eeaa\nwhere $J_i$ is a probability density function satisfying the following conditions: \n\\begin{enumerate}\n\\item[(H1)]\\,  $J_i$ is a nonnegative continuous function defined in $\\bR$;\n\\item[(H2)]\\, $\\int_{\\mathbb{R}}J_i(y)dy=1$ and $J_i(y)=J_i(-y)$ for all $y\\in \\mathbb{R}$;\n\\item[(H3)] there exists $\\hat{\\lambda}_i\\in(0,\\infty]$ such that  $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy < \\infty$ for any $\\lambda \\in (0, \\hat{\\lambda_i})$ and $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy\\to\\infty$ as $\\lambda \\uparrow \\hat{\\lambda}_i$.\n\\end{enumerate}\nNote that there is always the predator-free state $(1,0)$, and there is the unique constant co-existence state $(a^*,a^*)$, $a^*:=1-1/a\\in(0,1)$, when $a>1$.\n\nWhen the nonlocal dispersal in system \\eqref{pp} is replaced by the classical diffusion, the dynamical behaviors of the corresponding system was investigated in the survey paper \\cite{CGS21}. In fact, system \\eqref{pp} without diffusion (the ODE system) arises in the control of introduced rabbits to protect native birds from introduced cat predation in an island (cf. \\cite{CLS}), when we consider the case without rabbits and control. For the detailed biological background of the full ODE system including the rabbits and control, we refer the reader to \\cite{CLS,CS}. However, it is reasonable and more realistic to take into account the influence of spatial movements of birds and cats, namely, the effect of diffusion. On the other hand, to model the long range movements and nonadjacent interactions of individuals it is more realistic to consider the nonlocal dispersal instead of the random movement with classical diffusion. This motivates us to study system \\eqref{pp}.\n\nThe main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \\eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \\eqref{pp} is called a traveling wave solution of \\eqref{pp}, if there exist a constant $s\\in\\R$ ({\\it the wave speed}) and a function $(\\phi,\\psi)$ ({\\it the wave profile}) of class $C^1(\\R)$ such that\n$$(U,V)(x,t)=(\\phi,\\psi)( z),\\ \\ z:=x-st.$$\nWe are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\\{s,\\phi,\\psi\\}$ satisfies the following system of equations:\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\ntogether with the following asymptotic boundary conditions\n\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}\n\nFor convenience, for $i=1,2$, we set\n\\beaa\nI_i(\\lambda):=\\int_\\bR J_i(y)e^{\\lambda y}dy,\\ \\ \\lambda\\in\\R,\n\\eeaa\nwith $I_i(\\lambda)\\in(0,\\infty)$ if $0\\le\\lambda<\\hat{\\lambda}_i$ and $I_i(\\lambda)=\\infty$ if $\\lambda\\ge\\hat{\\lambda}_i$. Note that, due to the symmetry of $J_i$, the function $I_i$ is even, and it is also strictly convex in $(-\\hat{\\lambda}_i,\\hat{\\lambda}_i)$. Also, we introduce the quantity\n\\beaa\ns^*:=\\inf_{\\lambda\\in(0,\\hat{\\lambda}_2)}\\frac{d[I_2(\\lambda)-1]+b}{\\lambda}.\n\\eeaa\nNote that $s^*$ is well-defined, the infimum is reached, and $s^*>0$.\n\nWe now state our main theorem as follows.", "context": "We consider the following diffusive prey-predator model with nonlocal dispersal\n\\begin{equation}\\label{pp}\n\\begin{cases}\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-V](x,t), \\; x\\in\\bR,\\, t>0,\\\\\nV_t(x,t)=d \\bN_2[V(\\cdot,t)](x)+[bV(1-V/U)](x,t), \\; x\\in\\bR,\\, t>0,\n\\end{cases}\n\\end{equation}\nwhere the unknown functions $U,V$ stand for the population densities of prey and predator species at position $x$ and time $t$, respectively, $d,a,b$ are positive constants such that $1,d$ are diffusion coefficients and $a,b$ are intrinsic growth rates of species $U,V$, respectively. The functional response of predator to prey is normalized to be $1$. The prey obeys the logistic growth and its carrying capacity is normalized to be 1. However, the density of predator follows a logistic dynamics with a varying carrying capacity proportional to the density of prey. Moreover, for $i=1,2$, $\\bN_i$ formulates the spatial nonlocal dispersal of individuals and is defined by\n\\beaa\n\\bN_i[u(\\cdot,t)](x):=\\int_{\\mathbb{R}}J_i(y)u(x-y,t)dy-u(x,t),\\; u=U,V,\n\\eeaa\nwhere $J_i$ is a probability density function satisfying the following conditions: \n\\begin{enumerate}\n\\item[(H1)]\\, $J_i$ is a nonnegative continuous function defined in $\\bR$;\n\\item[(H2)]\\, $\\int_{\\mathbb{R}}J_i(y)dy=1$ and $J_i(y)=J_i(-y)$ for all $y\\in \\mathbb{R}$;\n\\item[(H3)] there exists $\\hat{\\lambda}_i\\in(0,\\infty]$ such that $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy < \\infty$ for any $\\lambda \\in (0, \\hat{\\lambda_i})$ and $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy\\to\\infty$ as $\\lambda \\uparrow \\hat{\\lambda}_i$.\n\\end{enumerate}\nNote that there is always the predator-free state $(1,0)$, and there is the unique constant co-existence state $(a^*,a^*)$, $a^*:=1-1/a\\in(0,1)$, when $a>1$.\n\nWhen the nonlocal dispersal in system \\eqref{pp} is replaced by the classical diffusion, the dynamical behaviors of the corresponding system was investigated in the survey paper \\cite{CGS21}. In fact, system \\eqref{pp} without diffusion (the ODE system) arises in the control of introduced rabbits to protect native birds from introduced cat predation in an island (cf. \\cite{CLS}), when we consider the case without rabbits and control. For the detailed biological background of the full ODE system including the rabbits and control, we refer the reader to \\cite{CLS,CS}. However, it is reasonable and more realistic to take into account the influence of spatial movements of birds and cats, namely, the effect of diffusion. On the other hand, to model the long range movements and nonadjacent interactions of individuals it is more realistic to consider the nonlocal dispersal instead of the random movement with classical diffusion. This motivates us to study system \\eqref{pp}.\n\nThe main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \\eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \\eqref{pp} is called a traveling wave solution of \\eqref{pp}, if there exist a constant $s\\in\\R$ ({\\it the wave speed}) and a function $(\\phi,\\psi)$ ({\\it the wave profile}) of class $C^1(\\R)$ such that\n$$(U,V)(x,t)=(\\phi,\\psi)( z),\\ \\ z:=x-st.$$\nWe are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\\{s,\\phi,\\psi\\}$ satisfies the following system of equations:\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n& \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z) =0, &\\; z\\in \\mathbb{R},\\\\\n& d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\ntogether with the following asymptotic boundary conditions\n\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}\n\nFor convenience, for $i=1,2$, we set\n\\beaa\nI_i(\\lambda):=\\int_\\bR J_i(y)e^{\\lambda y}dy,\\ \\ \\lambda\\in\\R,\n\\eeaa\nwith $I_i(\\lambda)\\in(0,\\infty)$ if $0\\le\\lambda<\\hat{\\lambda}_i$ and $I_i(\\lambda)=\\infty$ if $\\lambda\\ge\\hat{\\lambda}_i$. Note that, due to the symmetry of $J_i$, the function $I_i$ is even, and it is also strictly convex in $(-\\hat{\\lambda}_i,\\hat{\\lambda}_i)$. Also, we introduce the quantity\n\\beaa\ns^*:=\\inf_{\\lambda\\in(0,\\hat{\\lambda}_2)}\\frac{d[I_2(\\lambda)-1]+b}{\\lambda}.\n\\eeaa\nNote that $s^*$ is well-defined, the infimum is reached, and $s^*>0$.\n\nWe now state our main theorem as follows.\n\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}", "full_context": "We consider the following diffusive prey-predator model with nonlocal dispersal\n\\begin{equation}\\label{pp}\n\\begin{cases}\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-V](x,t), \\; x\\in\\bR,\\, t>0,\\\\\nV_t(x,t)=d \\bN_2[V(\\cdot,t)](x)+[bV(1-V/U)](x,t), \\; x\\in\\bR,\\, t>0,\n\\end{cases}\n\\end{equation}\nwhere the unknown functions $U,V$ stand for the population densities of prey and predator species at position $x$ and time $t$, respectively, $d,a,b$ are positive constants such that $1,d$ are diffusion coefficients and $a,b$ are intrinsic growth rates of species $U,V$, respectively. The functional response of predator to prey is normalized to be $1$. The prey obeys the logistic growth and its carrying capacity is normalized to be 1. However, the density of predator follows a logistic dynamics with a varying carrying capacity proportional to the density of prey. Moreover, for $i=1,2$, $\\bN_i$ formulates the spatial nonlocal dispersal of individuals and is defined by\n\\beaa\n\\bN_i[u(\\cdot,t)](x):=\\int_{\\mathbb{R}}J_i(y)u(x-y,t)dy-u(x,t),\\; u=U,V,\n\\eeaa\nwhere $J_i$ is a probability density function satisfying the following conditions: \n\\begin{enumerate}\n\\item[(H1)]\\, $J_i$ is a nonnegative continuous function defined in $\\bR$;\n\\item[(H2)]\\, $\\int_{\\mathbb{R}}J_i(y)dy=1$ and $J_i(y)=J_i(-y)$ for all $y\\in \\mathbb{R}$;\n\\item[(H3)] there exists $\\hat{\\lambda}_i\\in(0,\\infty]$ such that $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy < \\infty$ for any $\\lambda \\in (0, \\hat{\\lambda_i})$ and $\\int_{\\mathbb{R}}J_i(y)e^{\\lambda y} dy\\to\\infty$ as $\\lambda \\uparrow \\hat{\\lambda}_i$.\n\\end{enumerate}\nNote that there is always the predator-free state $(1,0)$, and there is the unique constant co-existence state $(a^*,a^*)$, $a^*:=1-1/a\\in(0,1)$, when $a>1$.\n\nWhen the nonlocal dispersal in system \\eqref{pp} is replaced by the classical diffusion, the dynamical behaviors of the corresponding system was investigated in the survey paper \\cite{CGS21}. In fact, system \\eqref{pp} without diffusion (the ODE system) arises in the control of introduced rabbits to protect native birds from introduced cat predation in an island (cf. \\cite{CLS}), when we consider the case without rabbits and control. For the detailed biological background of the full ODE system including the rabbits and control, we refer the reader to \\cite{CLS,CS}. However, it is reasonable and more realistic to take into account the influence of spatial movements of birds and cats, namely, the effect of diffusion. On the other hand, to model the long range movements and nonadjacent interactions of individuals it is more realistic to consider the nonlocal dispersal instead of the random movement with classical diffusion. This motivates us to study system \\eqref{pp}.\n\nThe main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \\eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \\eqref{pp} is called a traveling wave solution of \\eqref{pp}, if there exist a constant $s\\in\\R$ ({\\it the wave speed}) and a function $(\\phi,\\psi)$ ({\\it the wave profile}) of class $C^1(\\R)$ such that\n$$(U,V)(x,t)=(\\phi,\\psi)( z),\\ \\ z:=x-st.$$\nWe are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\\{s,\\phi,\\psi\\}$ satisfies the following system of equations:\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n& \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z) =0, &\\; z\\in \\mathbb{R},\\\\\n& d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\ntogether with the following asymptotic boundary conditions\n\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}\n\nFor convenience, for $i=1,2$, we set\n\\beaa\nI_i(\\lambda):=\\int_\\bR J_i(y)e^{\\lambda y}dy,\\ \\ \\lambda\\in\\R,\n\\eeaa\nwith $I_i(\\lambda)\\in(0,\\infty)$ if $0\\le\\lambda<\\hat{\\lambda}_i$ and $I_i(\\lambda)=\\infty$ if $\\lambda\\ge\\hat{\\lambda}_i$. Note that, due to the symmetry of $J_i$, the function $I_i$ is even, and it is also strictly convex in $(-\\hat{\\lambda}_i,\\hat{\\lambda}_i)$. Also, we introduce the quantity\n\\beaa\ns^*:=\\inf_{\\lambda\\in(0,\\hat{\\lambda}_2)}\\frac{d[I_2(\\lambda)-1]+b}{\\lambda}.\n\\eeaa\nNote that $s^*$ is well-defined, the infimum is reached, and $s^*>0$.\n\nWe now state our main theorem as follows.\n\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\n\nThe main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \\eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \\eqref{pp} is called a traveling wave solution of \\eqref{pp}, if there exist a constant $s\\in\\R$ ({\\it the wave speed}) and a function $(\\phi,\\psi)$ ({\\it the wave profile}) of class $C^1(\\R)$ such that\n$$(U,V)(x,t)=(\\phi,\\psi)( z),\\ \\ z:=x-st.$$\nWe are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\\{s,\\phi,\\psi\\}$ satisfies the following system of equations:\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n& \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z) =0, &\\; z\\in \\mathbb{R},\\\\\n& d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\ntogether with the following asymptotic boundary conditions\n\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}\n\nWe now state our main theorem as follows.\n\nHereafter, a function $(\\phi,\\psi)$ is positive if $\\phi,\\psi>0$ in $\\bR$. A corresponding theorem to Theorem~\\ref{th:main}, under the condition $a\\ge 4$, on the traveling waves in the classical diffusion case was derived in \\cite{CGS21}. Theorem~\\ref{th:main} characterizes the minimal speed $s^*$ in the nonlocal dispersal case, but it needs the extra condition $d<b$ in the construction of lower and upper solutions to~\\eqref{TWS}.\n\nThe existence of wave profiles $(\\phi,\\psi)$ to \\eqref{TWS}-\\eqref{BC} for all speeds $s>s^*$ is based on the construction of lower and upper solutions, which is new for this singular nonlocal system. The case $s=s^*$ is more involved, it requires a special care and uses the boundedness of the support of $J_2$. The characterization of the limiting state behind the front, that is, $(\\phi,\\psi)(-\\infty)=(a^*,a^*)$ is carried out with a squeezing method and an argument by contradiction. The proof of the nonexistence of wave profiles for all speeds $s<s^*$ is also done by contradiction. Section~\\ref{sec2} is devoted to the existence part in Theorem~\\ref{th:main}, while the nonexistence is shown in Section~\\ref{sec3}.\n\n\\begin{proposition}\\label{le6}\nLet $s>0$ be given. Let $(\\overline{\\phi },\\overline{\\psi })$ and $(\\underline{\\phi},\\underline{\\psi})$ be a pair of bounded upper and lower solutions of \\eqref {TWS}. Then system \\eqref {TWS} admits a positive solution $(\\phi,\\psi)$ of class $C^1(\\R)$ such that\n\\begin{equation*}\n\\underline{\\phi }( z )\\leq \\phi(z)\\leq \\overline{\\phi}( z ),\\; \\upsi(z)\\leq \\psi(z)\\leq\\overline{\\psi}( z ),\\; z \\in \\mathbb{R}.\n\\end{equation*}\n\\end{proposition}\n\nFor a given $s>s^*$, we consider the quantity\n\\beaa\nA(\\lambda)=A(\\lambda;s):=d[I_2(\\lambda)-1]-s\\lambda+b,\\; \\lambda>0.\n\\eeaa\nIt follows from the definition of $s^*$ and the strict convexity of $A(\\lambda)$ that there are two positive constants $\\lambda_1<\\lambda_2<\\hat{\\ld}_2$ such that\n$$A(\\lambda_1)=A(\\ld_2)=0$$\nand $A(\\lambda)<0$ for all $\\lambda\\in(\\lambda_1,\\lambda_2)$. Also, set \n$$B(\\ld)=B(\\ld;s):=[I_1(\\ld)-1]-s\\ld.$$ \nSince $B(0)=0$ and $B'(0)=-s<0$, we can choose a constant $\\ld_0\\in(0,\\min\\{\\ld_1,\\hat{\\ld}_1\\})$ small enough such that\n$$B(\\ld_0)<0.$$\nNow, for a fixed constant $\\mu\\in(1,\\min{\\{ \\lambda_2/ \\lambda_1,2\\}})$, we choose \n\\be\\label{q0}\nq>\\max\\Big\\{1,\\frac{2b}{-A(\\mu\\ld_1)}\\Big\\}.\n\\ee\nThen, the function $f(z) := e^{ -\\lambda_1z}-qe^{-\\mu \\lambda_1z}$ has exactly one zero $z_0>0$ and exactly one maximum point $z_M\\in(z_0,\\infty)$, and there holds $f(z_M)>0$. Thus, using $a\\ge4>1$ and $d<b$, we can choose $\\delta$ such that\n$$0<\\delta<\\min\\Big\\{f(z_M),\\underbrace{1-\\frac{1}{a}}_{=a^*},\\frac12\\Big(1-\\frac{d}{b}\\Big)\\Big\\},$$\nand a point $z_1\\in(z_0,z_M)$ such that $f(z_1)=\\delta$. Note that\n\\be\\label{delta}\nd<b(1-2\\delta).\n\\ee\nFurthermore, the inequalities $z_1>z_0>0$, $\\ld_1>0$ and $\\mu>1$ imply\n\\be\\label{q}\n{e^{(\\mu-1)\\ld_1z_1}>e^{(\\mu-1)\\ld_1z_0}=q.}\n\\ee\nWith these choices of $\\mu,q,\\delta$, we finally choose $\\e$ such that\n\\be\\label{ep1} \n0<\\e<\\min{\\left\\{\\frac{\\delta}{1+s\\ld_1+a},\\frac{e^{(\\mu-1)\\lambda_1z_1}-q}{(1+s\\lambda_1+a)e^{(\\mu-1)\\lambda_1z_1}} \\right\\}}.\n\\ee\nNote that the constant $\\e$ is admissible, due to \\eqref{q}, and that $0<\\e<\\delta<1/2$.\n\n\\begin{lemma}\\label{la:upper-lower2}\nAssume that $a\\ge 4$, $d<b$, and that $J_2$ has a compact support such that its support is contained in $[-S,S]$ for some $S\\in(0,\\infty)$. For $s=s^{*}$, the functions $(\\overline{\\phi},\\overline{\\psi})$ and~$(\\underline{\\phi},\\underline{\\psi})$ defined in \\eqref{uplosol2} are a pair of upper and lower solutions of \\eqref{TWS}.\n\\end{lemma}\n\nFirst, for any $s\\ge s^*$, the existence of a $C^1(\\R)$ solution $(\\phi,\\psi)$ to~\\eqref{TWS} such that $1/2\\le\\underline{\\phi}\\le\\phi\\le\\overline{\\phi}<1$ and $0<\\underline{\\psi}\\le\\psi\\le\\overline{\\psi}\\le1$ in $\\R$ follows from Lemmas~\\ref{la:upper-lower1}-\\ref{la:upper-lower2} and Proposition~\\ref{le6}. Moreover, by the construction of upper and lower solutions, it is clear that $(\\phi,\\psi)(\\infty)=(1,0)$.\n\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\n\n\\begin{equation}\\label{pp}\n\\begin{cases}\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-V](x,t), \\; x\\in\\bR,\\, t>0,\\\\\nV_t(x,t)=d \\bN_2[V(\\cdot,t)](x)+[bV(1-V/U)](x,t), \\; x\\in\\bR,\\, t>0,\n\\end{cases}\n\\end{equation}\n\n\\begin{theorem}\\label{th:main}\nLet $a$, $b$ and $d$ be given positive constants such that $a\\ge 4$ and $d<b$. Then there exists a positive solution $(\\phi,\\psi)$ of \\eqref{TWS}-\\eqref{BC} for each $s > s^{*}$. This existence also holds for $s=s^*$, if we further assume that $J_2$ has a compact support. Moreover, there exist no positive solutions of \\eqref{TWS}-\\eqref{BC} for $s < s^{*}$.\n\\end{theorem}", "post_theorem_intro_text_len": 1964, "post_theorem_intro_text": "Hereafter, a function $(\\phi,\\psi)$ is positive if $\\phi,\\psi>0$ in $\\bR$. A corresponding theorem to Theorem~\\ref{th:main}, under the condition $a\\ge 4$, on the traveling waves in the classical diffusion case was derived in \\cite{CGS21}. Theorem~\\ref{th:main} characterizes the minimal speed $s^*$ in the nonlocal dispersal case, but it needs the extra condition $d<b$ in the construction of lower and upper solutions to~\\eqref{TWS}.\n\nIt is worth to note that there are some difficulties in the study of system \\eqref{pp}. It is a non-monotone system which is lack of comparison principle and a singularity may occur in the $V$-equation when $U$ reaches zero in a finite time which has been confirmed numerically in \\cite{CLS}. Thus system \\eqref{pp} is named as a {\\it singular} prey-predator model due to the latter fact. This is in contrast to the system with $U$-equation in \\eqref{pp} replaced by\n\\beaa\nU_t(x,t)= \\bN_1[U(\\cdot,t)](x)+[aU(1-U)-kUV](x,t), \\; x\\in\\bR,\\, t>0,\n\\eeaa\nin which the functional response of predation is a linear function of prey. In this case, the strong maximum principle for the scalar equation gives that $U>0$ for all $t>0$. Hence no singularity can occur for the nonlinear term $V/U$ in the $V$-equation.\n\nThe existence of wave profiles $(\\phi,\\psi)$ to \\eqref{TWS}-\\eqref{BC} for all speeds $s>s^*$ is based on the construction of lower and upper solutions, which is new for this singular nonlocal system. The case $s=s^*$ is more involved, it requires a special care and uses the boundedness of the support of $J_2$. The characterization of the limiting state behind the front, that is, $(\\phi,\\psi)(-\\infty)=(a^*,a^*)$ is carried out with a squeezing method and an argument by contradiction. The proof of the nonexistence of wave profiles for all speeds $s<s^*$ is also done by contradiction. Section~\\ref{sec2} is devoted to the existence part in Theorem~\\ref{th:main}, while the nonexistence is shown in Section~\\ref{sec3}.", "sketch": "The post-theorem introduction outlines the proof of Theorem~\\ref{th:main} as follows:\n\n- **Existence for $s>s^*$:** “based on the construction of lower and upper solutions,” which the authors note “is new for this singular nonlocal system,” and the extra condition $d<b$ is needed “in the construction of lower and upper solutions to~\\eqref{TWS}.”\n\n- **Existence at $s=s^*$:** “more involved,” requiring “special care” and using “the boundedness of the support of $J_2$” (i.e., compact support).\n\n- **Limiting state behind the front:** the claim “$(\\phi,\\psi)(-\\infty)=(a^*,a^*)$” is proved “with a squeezing method and an argument by contradiction.”\n\n- **Nonexistence for $s<s^*$:** proved “by contradiction.”\n\n- **Organization:** “Section~\\ref{sec2} is devoted to the existence part in Theorem~\\ref{th:main}, while the nonexistence is shown in Section~\\ref{sec3}.”", "expanded_sketch": "The post-theorem introduction outlines the proof strategy used to establish the main theorem as follows:\n\n- **Existence for $s>s^*$:** “based on the construction of lower and upper solutions,” which the authors note “is new for this singular nonlocal system,” and the extra condition $d<b$ is needed “in the construction of lower and upper solutions to\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}.”\n\n- **Existence at $s=s^*$:** “more involved,” requiring “special care” and using “the boundedness of the support of $J_2$” (i.e., compact support).\n\n- **Limiting state behind the front:** the claim “$(\\phi,\\psi)(-\\infty)=(a^*,a^*)$” is proved “with a squeezing method and an argument by contradiction.”\n\n- **Nonexistence for $s<s^*$:** proved “by contradiction.”\n\n- **Organization:** Next they prove the existence part in the main theorem, while the nonexistence is shown later.", "expanded_theorem": "\\label{th:main}\nLet $a$, $b$ and $d$ be given positive constants such that $a\\ge 4$ and $d<b$. Then there exists a positive solution $(\\phi,\\psi)$ of\n\\begin{equation}\\label{TWS}\n\\left\\{\n\\begin{aligned}\n&   \\bN_1[\\phi](z)+ s\\phi'(z) +a\\phi(z)\\left[1-\\phi(z)\\right]-\\psi(z)    =0, &\\; z\\in \\mathbb{R},\\\\\n&  d \\bN_2[\\psi](z)+ s\\psi'(z) +b\\psi(z)\\left[1-\\frac{\\psi(z)}{\\phi(z)}\\right] =0, & \\; z\\in \\mathbb{R},\n\\end{aligned}\n\\right.\n\\end{equation}\nwith boundary conditions\n\\begin{equation}\\label{BC}\n(\\phi,\\psi)(\\infty) = (1,0),\\quad (\\phi,\\psi)(-\\infty) = (a^{*},a^{*}).\n\\end{equation}\nfor each $s > s^{*}$. This existence also holds for $s=s^*$, if we further assume that $J_2$ has a compact support. Moreover, there exist no positive solutions of the system \\eqref{TWS} with boundary conditions \\eqref{BC} for $s < s^{*}$.", "theorem_type": ["Existence", "Nonexistence"], "mcq": {"question": "Let $a,b,d>0$ with $a\\ge 4$ and $d<b$. For $i=1,2$, define the nonlocal dispersal operators\n\\[\n\\mathcal N_i[u](z)=\\int_{\\mathbb R}J_i(y)u(z-y)\\,dy-u(z),\n\\]\nwhere $J_i$ are the dispersal kernels from the model, and set\n\\[\na^*=1-\\frac1a,\n\\qquad\nI_2(\\lambda)=\\int_{\\mathbb R}J_2(y)e^{\\lambda y}\\,dy,\n\\qquad\ns^*=\\inf_{\\lambda\\in(0,\\hat\\lambda_2)}\\frac{d\\,[I_2(\\lambda)-1]+b}{\\lambda}.\n\\]\nConsider positive profile pairs $(\\phi,\\psi)$, meaning $\\phi(z)>0$ and $\\psi(z)>0$ for all $z\\in\\mathbb R$, satisfying\n\\[\n\\begin{cases}\n\\mathcal N_1[\\phi](z)+s\\phi'(z)+a\\phi(z)(1-\\phi(z))-\\psi(z)=0, & z\\in\\mathbb R,\\\\\n d\\,\\mathcal N_2[\\psi](z)+s\\psi'(z)+b\\psi(z)\\left(1-\\dfrac{\\psi(z)}{\\phi(z)}\\right)=0, & z\\in\\mathbb R,\n\\end{cases}\n\\]\nwith asymptotic boundary conditions\n\\[\n(\\phi,\\psi)(\\infty)=(1,0),\\qquad (\\phi,\\psi)(-\\infty)=(a^*,a^*).\n\\]\nWhich statement holds for the existence and nonexistence of such positive solutions as the wave speed $s$ varies?", "correct_choice": {"label": "A", "text": "For every $s>s^*$, there exists a positive solution $(\\phi,\\psi)$ of the above system with the stated boundary conditions. The same existence conclusion also holds for $s=s^*$ if, in addition, $J_2$ has compact support. For every $s<s^*$, there is no positive solution of the system with those boundary conditions."}, "choices": [{"label": "B", "text": "For every $s\\ge s^*$, there exists a positive solution $(\\phi,\\psi)$ of the above system with the stated boundary conditions, without any additional assumption on $J_2$. For every $s<s^*$, there is no positive solution of the system with those boundary conditions."}, {"label": "C", "text": "For every $s>s^*$, there exists a positive solution $(\\phi,\\psi)$ of the above system with the stated boundary conditions. For every $s<s^*$, there is no positive solution of the system with those boundary conditions."}, {"label": "D", "text": "There exists a positive solution $(\\phi,\\psi)$ of the above system with the stated boundary conditions for every $s>s^*$, and there is no positive solution for $s\\le s^*$."}, {"label": "E", "text": "If $J_2$ has compact support, then there exists a positive solution $(\\phi,\\psi)$ of the above system with the stated boundary conditions for every $s\\ge s^*$. Moreover, for every $s<s^*$ there is no positive solution only for some choice of the parameters $a,b,d$ with $a\\ge 4$ and $d<b$."}], "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": "compact-support requirement at the critical speed", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "geometric_construction", "tampered_component": "critical-speed existence at $s=s^*$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "exceptional critical case $s=s^*$ with compact support", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "universal quantifier in the nonexistence statement", "template_used": "wildcard"}]}}
{"id": "2512.07773v1", "paper_link": "http://arxiv.org/abs/2512.07773v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{THM:main} \nLet $u(t,x)$ be solutions to \\eqref{weakNLS1} with initial data $u_0^\\omega$ given in \\eqref{random_data}.\nFor any $t = \\mathcal O ( \\varepsilon^{-1} |\\log \\varepsilon|)$ and $z_0 > 0$, we have that\n\\begin{align}\n\\label{THM_LDP4}\n\\lim_{\\varepsilon \\to 0^+} \\varepsilon \\log \\mathbb{P} \\Big( \\|u(t,x) \\|_{\\mathcal{F} L^1 (\\mathbb{T})} > z_0\\varepsilon^{-1/2} \\Big) = - \\frac{z_0^2}{\\sum_k c_k^2},\n\\end{align}\n\n\\noindent \nholds if and only if $(c_k) \\in \\ell^1$.", "start_pos": 16675, "end_pos": 17112, "label": "THM:main"}, "ref_dict": {"minp": "\\begin{align}\\label{minp}\n\\theta^* (z) = {\\rm argmin}\\,\\, \\mathcal I (\\theta)\n\\end{align}", "decay": "\\begin{align}\\label{decay}\nc_k = a e^{-b|k|} \n\\quad \\text{or} \\quad \nc_k = a e^{-b|k|^2},\n\\end{align}", "weakNLS1": "\\begin{align}\n\\label{weakNLS1}\ni\\pa_t u + \\Delta u = \\varepsilon^{2} |u|^2 u.\n\\end{align}", "THM_LDP3": "\\begin{align}\n\\label{THM_LDP3}\n\\lim_{\\varepsilon \\to 0^+} \n\\varepsilon \\log \\PP \\Big( \\sup_{x} |u(t,x)| > z_0 \\varepsilon^{-1/2} \\Big)\n= - \\frac{z_0^2}{\\sum_k c_k^2},\n\\end{align}", "COR:main": "\\begin{corollary}\\label{COR:main} \nLet $u(t,x)$ be solutions to \\eqref{weakNLS1} with initial data $u_0^\\o$ given in \\eqref{random_data}.\nFor any $t = \\mathcal O ( \\eps^{-1} |\\log \\eps|)$ and $z_0 > 0$, we have that \\eqref{THM_LDP3} holds provided $(c_k) \\in \\l^1$.\n\\end{corollary}", "random_data": "\\begin{equation}\\label{random_data}\nu_0 (x)= \\sum_{k\\in\\Z} c_k g_k e^{ikx},\n\\end{equation}", "PROP:linear2": "\\begin{proposition}\n\\label{PROP:linear2}\nConsider the linear Schr\\\"odinger equation on the torus $\\T = [0,2\\pi]$ as in \\eqref{linear}, with random initial data $u_0^\\o$ given by \\eqref{random_data}.\nThen,\n\\begin{align}\n\\label{LDP_linear2}\n\\lim_{\\eps\\to 0^+} \\eps \\log \\PP \\big( \\|u(t)\\|_{L^\\infty (\\T)} \\ge z_0 \\eps^{-1/2} \\big) = - \\frac{z_0^2}{\\sum_{k \\in \\Z} c_k^2}, \n\\end{align}\n\n\\noi \nprovided $(c_k) \\in \\l^1$.\n\\end{proposition}", "obser": "\\begin{align}\\label{obser}\n\\lim_{\\eps \\to 0^+} \\eps \\log \\PP \\Big( \\sup_{x} |u(t,x) | > z_0\\eps^{-1/2} \\Big) = \\lim_{\\eps \\to 0^+} \\eps \\log \\PP \\Big( \\|u(t,x) \\|_{\\FL^1} > z_0\\eps^{-1/2} \\Big),\n\\end{align}", "THM_LDP4": "\\begin{align}\n\\label{THM_LDP4}\n\\lim_{\\eps \\to 0^+} \\eps \\log \\PP \\Big( \\|u(t,x) \\|_{\\FL^1 (\\T)} > z_0\\eps^{-1/2} \\Big) = - \\frac{z_0^2}{\\sum_k c_k^2},\n\\end{align}", "THM:main": "\\begin{theorem}\n\\label{THM:main} \nLet $u(t,x)$ be solutions to \\eqref{weakNLS1} with initial data $u_0^\\o$ given in \\eqref{random_data}.\nFor any $t = \\mathcal O ( \\eps^{-1} |\\log \\eps|)$ and $z_0 > 0$, we have that\n\\begin{align}\n\\label{THM_LDP4}\n\\lim_{\\eps \\to 0^+} \\eps \\log \\PP \\Big( \\|u(t,x) \\|_{\\FL^1 (\\T)} > z_0\\eps^{-1/2} \\Big) = - \\frac{z_0^2}{\\sum_k c_k^2},\n\\end{align}\n\n\\noi \nholds if and only if $(c_k) \\in \\l^1$.\n\\end{theorem}", "NLS": "\\begin{equation}\n\\label{NLS}\n\\begin{cases}\ni\\pa_t u + \\partial_x^2 u = \\pm \\,|u|^2\\, u,\\\\\nu(t,x)|_{t=0}=u_0.\n\\end{cases}\n\\end{equation}", "LDP2": "\\begin{align}\n\\label{LDP2}\n\\log \\PP \\Big( \\sup_{x} |u(t,x)| > z \\Big) = - I(z) + o(1), \\quad z \\to \\infty,\n\\end{align}", "LDP1": "\\begin{align}\n\\label{LDP1}\n\\log \\PP (\\mathcal D(t,z)) = - \\mathcal I (\\theta^* (z)) + o(1),\n\\end{align}", "random1": "\\begin{align}\n\\label{random1}\nu_0^N (x) = \\sum_{|k| \\le N} \\theta_k e^{ikx},\n\\end{align}"}, "pre_theorem_intro_text_len": 7821, "pre_theorem_intro_text": "In this paper, we consider the following Cauchy problem of the one-dimensional cubic nonlinear Schr\\\"odinger equation (NLS) on the torus $\\mathbb{T}= \\mathbb{R} / (2 \\pi \\mathbb{Z})$: \n\\begin{equation}\n\\label{NLS}\n\\begin{cases}\ni\\pa_t u + \\partial_x^2 u = \\pm \\,|u|^2\\, u,\\\\\nu(t,x)|_{t=0}=u_0.\n\\end{cases}\n\\end{equation}\n\n\\noindent \nThe equation \\eqref{NLS} is fundamental in theoretical physics and applied mathematics, for describing the evolution of wave packets in various physical systems, such as nonlinear optics, fluids, and plasmas; see \\cite{SS99} for a review.\nBourgain \\cite{BO93} proved \\eqref{NLS} is deterministically global well-posed in $L^2 (\\mathbb{T})$.\nThe probabilistic study of \\eqref{NLS} has emerging importance since the seminal work of Bourgain on invariant measures \\cite{BO94,BO96}.\n\nMore recently, probabilistic analyses of \\eqref{NLS} have also found applications in oceanography, particularly in modelling rare extreme events such as the formation of {\\it rogue waves} in deep-sea dynamics~\\cite{DGV18,DGV19,GGKS21}. In particular, Garrido et al.~\\cite{GGKS21} established a large deviations principle (LDP) describing the probability of observing a wave of unusually large height in the weakly nonlinear regime. Their result relied on a strong exponential decay assumption on the Fourier coefficients of the random initial data, which plays a crucial role in their analysis.\nIn this paper, we improve the exponential decay condition on coefficients of initial data in \\cite{GGKS21} to a $\\ell^1$ decay condition. \n\nWe consider \nrandom initial data of the form:\n\\begin{equation}\\label{random_data}\nu_0 (x)= \\sum_{k\\in\\mathbb{Z}} c_k g_k e^{ikx},\n\\end{equation}\nwhere $c_k \\in \\mathcal{C}$ and $\\{g_k\\}_{k\\in\\mathbb{Z}}$ are independent, identically distributed, complex Gaussian random variables with $\\mathbb{E} g_k=0$, $\\mathbb{E} g_k g_j=0$ and $\\mathbb{E} g_k \\overline{g}_j =\\delta_{kj}$ for $k,j\\in\\mathbb{Z}$.  \nWithout loss of any generality, we only consider nonnegative real coefficients $c_k \\ge 0$, due to the rotation invariance of complex Gaussian random variables. \n\nThe study of the large deviation principle for solutions to \\eqref{NLS} was initiated by \\cite{DGV18,DGV19}, where the authors conjectured that the existence of an LDP can be used to predict the formation of rogue waves.\nIn particular, they considered \\eqref{NLS} with random initial data $u_0^N$ of the form\n\\begin{align}\n\\label{random1}\nu_0^N (x) = \\sum_{|k| \\le N} \\theta_k e^{ikx},\n\\end{align}\n\n\\noindent\nwhere the initial data is parametrized by a random vector $\\theta = (\\theta_k)_{|k| \\le N} \\in \\mathbb C^{2N+1}$.\nHere $(\\theta_k)$ are independent complex Gaussian random variables with $\\mathbb{E} \\theta_k = \\mathbb{E} \\theta_k^2 = 0$ and $\\mathbb{E} |\\theta_k|^2 = c_k^2$, for some fast-decaying $c_k > 0$.\nThen, the set of initial data that generates a rouge wave of height at least $z>0$ at time $t>0$ is given by\n\\begin{align}\n\\label{Dtz}\n\\mathcal D (t,z) : = \\bigg\\{ (\\theta_k)_{|k| \\le N} \\in \\mathcal{C}^{2N+1} \\Big| \\sup_x |u (t,x| \\theta)| > z\\bigg\\},\n\\end{align}\n\n\\noindent \nwhere $u$ is the solution to NLS \\eqref{NLS} with the truncated initial data \\eqref{random1}.\nDematteis et al, then proposed a theoretical framework of a large deviations principle (LDP) to quantify the likelihood of $\\mathcal D (t,z)$.\nIn particular, consider the minimization problem\n\\begin{align}\\label{minp}\n\\theta^* (z) = {\\rm argmin}\\,\\, \\mathcal I (\\theta)\n\\end{align}\n\n\\noindent \nwhere\n\\[\n\\mathcal I (\\theta) = \\max_{y \\in \\mathcal{C}^{2N+1}} [\\langle y,\\theta \\rangle - S(y)] \\quad \\textup{ and } \\quad S(y) = \\log \\mathbb{E} e^{\\langle y, \\theta \\rangle}.\n\\]\n\n\\noindent\nThen Dematteis et al. claim that for $t >0$ \n\\begin{align}\n\\label{LDP1}\n\\log \\mathbb{P} (\\mathcal D(t,z)) = - \\mathcal I (\\theta^* (z)) + o(1),\n\\end{align}\n\n\\noindent \nas $z \\to \\infty$, provided \nthe minimization problem \\eqref{minp} has a unique solution.\n\nGarrido et al.~\\cite{GGKS21} rigorously proved the conjectured LDP~\\eqref{LDP1} locally in time for the nonlinear Schr\\\"odinger equation with weak nonlinearity. \nA major difficulty in analysing the LDP problem~\\eqref{LDP1} lies in estimating the gradient \n$\\nabla_{\\theta} \\sup_{x} |u(t,x|\\theta)|$ \nand in verifying the convexity of the set $\\mathcal{D}(t,z)$, \nboth of which are extremely challenging to establish in practice. \nTo overcome these obstacles, Garrido et al.~\\cite{GGKS21} introduced a new formulation and considered the following LDP problem:\n\\begin{align}\n\\label{LDP2}\n\\log \\mathbb{P} \\Big( \\sup_{x} |u(t,x)| > z \\Big) = - I(z) + o(1), \\quad z \\to \\infty,\n\\end{align}\nwhere $I$ denotes the corresponding rate function. \nThe advantage of this reformulation is that it circumvents the need to solve the minimization problem in~\\eqref{minp}. \nWithin this framework, they established~\\eqref{LDP2} for initial data of the form~\\eqref{random1} with infinitely many Fourier modes ($N = \\infty$) satisfying a \\emph{strong decay condition} on the coefficients.\n\nIn particular, consider the weakly nonlinear Schr\\\"odinger equation on the circle $\\mathbb{T} = [0,2\\pi]$:\n\\begin{align}\n\\label{weakNLS1}\ni\\pa_t u + \\Delta u = \\varepsilon^{2} |u|^2 u.\n\\end{align}\nIt was proved in~\\cite{GGKS21} that\n\\begin{align}\n\\label{THM_LDP3}\n\\lim_{\\varepsilon \\to 0^+} \n\\varepsilon \\log \\mathbb{P} \\Big( \\sup_{x} |u(t,x)| > z_0 \\varepsilon^{-1/2} \\Big)\n= - \\frac{z_0^2}{\\sum_k c_k^2},\n\\end{align}\nfor $0 < t \\lesssim \\varepsilon^{-1}$ and $z_0 > 0$,\nwhere $u(t,x)$ denotes the solution to~\\eqref{weakNLS1} with random initial data $u_0^\\omega$ given by~\\eqref{random_data} and coefficients satisfying\n\\begin{align}\\label{decay}\nc_k = a e^{-b|k|} \n\\quad \\text{or} \\quad \nc_k = a e^{-b|k|^2},\n\\end{align}\nfor some fixed $a,b > 0$. \n\nThe proof of~\\eqref{THM_LDP3} is based on the G\\\"artner--Ellis theorem combined with resonant approximation techniques. \nIn their analysis, the rapid decay condition~\\eqref{decay} plays a crucial role in both the linear and nonlinear estimates. \nThey further observed that alternative families of coefficients $(c_k)$ might also yield~\\eqref{LDP2}, and posed an \\textit{open question} concerning the identification of the optimal decay condition under which~\\eqref{LDP2} holds; see~\\cite[Remark~1.3]{GGKS21}. \nOne of the principal aims of the present work is to address this question.\n\nTo this purpose, we consider a LDP under Fourier-Lebesgue norm, which is stronger than the $L^\\infty (\\mathbb{T})$ norm used in \\eqref{THM_LDP3}.\nOur conventions for the Fourier transform are as follows:\n\\[\n\\widehat f (n) = \\frac1{{2\\pi}} \\int_0^{2\\pi} f(x) e^{ix n}dx \\quad \\textup{ and } \\quad f(x) = \\sum_{n\\in \\mathbb{Z}} \\widehat f(n) e^{ixn}\n\\]\n\n\\noindent \nfor functions on the circle $\\mathbb{T} = \\mathbb{R}/(2\\pi \\mathbb{Z})$.\nWith the above notations, we define the Fourier-Lebesgue norm by\n\\[\n\\begin{split} \n\\|f\\|_{\\mathcal{F} L^1 (\\mathbb{T})} = \\sum_{n \\in \\mathbb{Z}} |\\widehat f(n)|.\n\\end{split}\n\\]\n\n\\noindent \nIt is easy to see that\n\\[\n\\sup_{x \\in \\mathbb{T}} |f(x)| \\le \\|f\\|_{\\mathcal{F} L^1 (\\mathbb{T})}.\n\\]\n\n\\noindent \nWe note that $L^\\infty$ and $\\mathcal{F} L^1$ scales the same, and furthermore, the LDPs under $L^\\infty$ and $\\mathcal{F} L^1$ share the same rate function.\nIn particular, for $t \\lesssim \\varepsilon^{-1}$, we have\n\\begin{align}\\label{obser}\n\\lim_{\\varepsilon \\to 0^+} \\varepsilon \\log \\mathbb{P} \\Big( \\sup_{x} |u(t,x) | > z_0\\varepsilon^{-1/2} \\Big) = \\lim_{\\varepsilon \\to 0^+} \\varepsilon \\log \\mathbb{P} \\Big( \\|u(t,x) \\|_{\\mathcal{F} L^1} > z_0\\varepsilon^{-1/2} \\Big),\n\\end{align}\n\n\\noindent \nwhere $u(t,x)$ is the solution to \\eqref{weakNLS1} with initial data \\eqref{random1} in $\\mathcal{F}L^1$.\nThis motivates us to consider the LDP under the $\\mathcal{F} L^1$ norm.\nWe are ready to state our first main result.", "context": "We consider \nrandom initial data of the form:\n\\begin{equation}\\label{random_data}\nu_0 (x)= \\sum_{k\\in\\mathbb{Z}} c_k g_k e^{ikx},\n\\end{equation}\nwhere $c_k \\in \\mathcal{C}$ and $\\{g_k\\}_{k\\in\\mathbb{Z}}$ are independent, identically distributed, complex Gaussian random variables with $\\mathbb{E} g_k=0$, $\\mathbb{E} g_k g_j=0$ and $\\mathbb{E} g_k \\overline{g}_j =\\delta_{kj}$ for $k,j\\in\\mathbb{Z}$. \nWithout loss of any generality, we only consider nonnegative real coefficients $c_k \\ge 0$, due to the rotation invariance of complex Gaussian random variables.\n\n\\noindent\nwhere the initial data is parametrized by a random vector $\\theta = (\\theta_k)_{|k| \\le N} \\in \\mathbb C^{2N+1}$.\nHere $(\\theta_k)$ are independent complex Gaussian random variables with $\\mathbb{E} \\theta_k = \\mathbb{E} \\theta_k^2 = 0$ and $\\mathbb{E} |\\theta_k|^2 = c_k^2$, for some fast-decaying $c_k > 0$.\nThen, the set of initial data that generates a rouge wave of height at least $z>0$ at time $t>0$ is given by\n\\begin{align}\n\\label{Dtz}\n\\mathcal D (t,z) : = \\bigg\\{ (\\theta_k)_{|k| \\le N} \\in \\mathcal{C}^{2N+1} \\Big| \\sup_x |u (t,x| \\theta)| > z\\bigg\\},\n\\end{align}\n\nGarrido et al.~\\cite{GGKS21} rigorously proved the conjectured LDP~\\eqref{LDP1} locally in time for the nonlinear Schr\\\"odinger equation with weak nonlinearity. \nA major difficulty in analysing the LDP problem~\\eqref{LDP1} lies in estimating the gradient \n$\\nabla_{\\theta} \\sup_{x} |u(t,x|\\theta)|$ \nand in verifying the convexity of the set $\\mathcal{D}(t,z)$, \nboth of which are extremely challenging to establish in practice. \nTo overcome these obstacles, Garrido et al.~\\cite{GGKS21} introduced a new formulation and considered the following LDP problem:\n\\begin{align}\n\\label{LDP2}\n\\log \\mathbb{P} \\Big( \\sup_{x} |u(t,x)| > z \\Big) = - I(z) + o(1), \\quad z \\to \\infty,\n\\end{align}\nwhere $I$ denotes the corresponding rate function. \nThe advantage of this reformulation is that it circumvents the need to solve the minimization problem in~\\eqref{minp}. \nWithin this framework, they established~\\eqref{LDP2} for initial data of the form~\\eqref{random1} with infinitely many Fourier modes ($N = \\infty$) satisfying a \\emph{strong decay condition} on the coefficients.\n\nIn particular, consider the weakly nonlinear Schr\\\"odinger equation on the circle $\\mathbb{T} = [0,2\\pi]$:\n\\begin{align}\n\\label{weakNLS1}\ni\\pa_t u + \\Delta u = \\varepsilon^{2} |u|^2 u.\n\\end{align}\nIt was proved in~\\cite{GGKS21} that\n\\begin{align}\n\\label{THM_LDP3}\n\\lim_{\\varepsilon \\to 0^+} \n\\varepsilon \\log \\mathbb{P} \\Big( \\sup_{x} |u(t,x)| > z_0 \\varepsilon^{-1/2} \\Big)\n= - \\frac{z_0^2}{\\sum_k c_k^2},\n\\end{align}\nfor $0 < t \\lesssim \\varepsilon^{-1}$ and $z_0 > 0$,\nwhere $u(t,x)$ denotes the solution to~\\eqref{weakNLS1} with random initial data $u_0^\\omega$ given by~\\eqref{random_data} and coefficients satisfying\n\\begin{align}\\label{decay}\nc_k = a e^{-b|k|} \n\\quad \\text{or} \\quad \nc_k = a e^{-b|k|^2},\n\\end{align}\nfor some fixed $a,b > 0$.\n\n\\noindent \nWe note that $L^\\infty$ and $\\mathcal{F} L^1$ scales the same, and furthermore, the LDPs under $L^\\infty$ and $\\mathcal{F} L^1$ share the same rate function.\nIn particular, for $t \\lesssim \\varepsilon^{-1}$, we have\n\\begin{align}\\label{obser}\n\\lim_{\\varepsilon \\to 0^+} \\varepsilon \\log \\mathbb{P} \\Big( \\sup_{x} |u(t,x) | > z_0\\varepsilon^{-1/2} \\Big) = \\lim_{\\varepsilon \\to 0^+} \\varepsilon \\log \\mathbb{P} \\Big( \\|u(t,x) \\|_{\\mathcal{F} L^1} > z_0\\varepsilon^{-1/2} \\Big),\n\\end{align}\n\n\\noindent \nwhere $u(t,x)$ is the solution to \\eqref{weakNLS1} with initial data \\eqref{random1} in $\\mathcal{F}L^1$.\nThis motivates us to consider the LDP under the $\\mathcal{F} L^1$ norm.\nWe are ready to state our first main result.\n\n\\begin{align}\n\\label{LDP1}\n\\log \\PP (\\mathcal D(t,z)) = - \\mathcal I (\\theta^* (z)) + o(1),\n\\end{align}\n\n\\begin{align}\n\\label{LDP2}\n\\log \\PP \\Big( \\sup_{x} |u(t,x)| > z \\Big) = - I(z) + o(1), \\quad z \\to \\infty,\n\\end{align}\n\n\\begin{align}\\label{minp}\n\\theta^* (z) = {\\rm argmin}\\,\\, \\mathcal I (\\theta)\n\\end{align}\n\n\\begin{align}\n\\label{random1}\nu_0^N (x) = \\sum_{|k| \\le N} \\theta_k e^{ikx},\n\\end{align}\n\n\\begin{equation}\\label{random_data}\nu_0 (x)= \\sum_{k\\in\\Z} c_k g_k e^{ikx},\n\\end{equation}", "full_context": "We consider \nrandom initial data of the form:\n\\begin{equation}\\label{random_data}\nu_0 (x)= \\sum_{k\\in\\mathbb{Z}} c_k g_k e^{ikx},\n\\end{equation}\nwhere $c_k \\in \\mathcal{C}$ and $\\{g_k\\}_{k\\in\\mathbb{Z}}$ are independent, identically distributed, complex Gaussian random variables with $\\mathbb{E} g_k=0$, $\\mathbb{E} g_k g_j=0$ and $\\mathbb{E} g_k \\overline{g}_j =\\delta_{kj}$ for $k,j\\in\\mathbb{Z}$. \nWithout loss of any generality, we only consider nonnegative real coefficients $c_k \\ge 0$, due to the rotation invariance of complex Gaussian random variables.\n\n\\noindent\nwhere the initial data is parametrized by a random vector $\\theta = (\\theta_k)_{|k| \\le N} \\in \\mathbb C^{2N+1}$.\nHere $(\\theta_k)$ are independent complex Gaussian random variables with $\\mathbb{E} \\theta_k = \\mathbb{E} \\theta_k^2 = 0$ and $\\mathbb{E} |\\theta_k|^2 = c_k^2$, for some fast-decaying $c_k > 0$.\nThen, the set of initial data that generates a rouge wave of height at least $z>0$ at time $t>0$ is given by\n\\begin{align}\n\\label{Dtz}\n\\mathcal D (t,z) : = \\bigg\\{ (\\theta_k)_{|k| \\le N} \\in \\mathcal{C}^{2N+1} \\Big| \\sup_x |u (t,x| \\theta)| > z\\bigg\\},\n\\end{align}\n\nGarrido et al.~\\cite{GGKS21} rigorously proved the conjectured LDP~\\eqref{LDP1} locally in time for the nonlinear Schr\\\"odinger equation with weak nonlinearity. \nA major difficulty in analysing the LDP problem~\\eqref{LDP1} lies in estimating the gradient \n$\\nabla_{\\theta} \\sup_{x} |u(t,x|\\theta)|$ \nand in verifying the convexity of the set $\\mathcal{D}(t,z)$, \nboth of which are extremely challenging to establish in practice. \nTo overcome these obstacles, Garrido et al.~\\cite{GGKS21} introduced a new formulation and considered the following LDP problem:\n\\begin{align}\n\\label{LDP2}\n\\log \\mathbb{P} \\Big( \\sup_{x} |u(t,x)| > z \\Big) = - I(z) + o(1), \\quad z \\to \\infty,\n\\end{align}\nwhere $I$ denotes the corresponding rate function. \nThe advantage of this reformulation is that it circumvents the need to solve the minimization problem in~\\eqref{minp}. \nWithin this framework, they established~\\eqref{LDP2} for initial data of the form~\\eqref{random1} with infinitely many Fourier modes ($N = \\infty$) satisfying a \\emph{strong decay condition} on the coefficients.\n\nIn particular, consider the weakly nonlinear Schr\\\"odinger equation on the circle $\\mathbb{T} = [0,2\\pi]$:\n\\begin{align}\n\\label{weakNLS1}\ni\\pa_t u + \\Delta u = \\varepsilon^{2} |u|^2 u.\n\\end{align}\nIt was proved in~\\cite{GGKS21} that\n\\begin{align}\n\\label{THM_LDP3}\n\\lim_{\\varepsilon \\to 0^+} \n\\varepsilon \\log \\mathbb{P} \\Big( \\sup_{x} |u(t,x)| > z_0 \\varepsilon^{-1/2} \\Big)\n= - \\frac{z_0^2}{\\sum_k c_k^2},\n\\end{align}\nfor $0 < t \\lesssim \\varepsilon^{-1}$ and $z_0 > 0$,\nwhere $u(t,x)$ denotes the solution to~\\eqref{weakNLS1} with random initial data $u_0^\\omega$ given by~\\eqref{random_data} and coefficients satisfying\n\\begin{align}\\label{decay}\nc_k = a e^{-b|k|} \n\\quad \\text{or} \\quad \nc_k = a e^{-b|k|^2},\n\\end{align}\nfor some fixed $a,b > 0$.\n\n\\noindent \nWe note that $L^\\infty$ and $\\mathcal{F} L^1$ scales the same, and furthermore, the LDPs under $L^\\infty$ and $\\mathcal{F} L^1$ share the same rate function.\nIn particular, for $t \\lesssim \\varepsilon^{-1}$, we have\n\\begin{align}\\label{obser}\n\\lim_{\\varepsilon \\to 0^+} \\varepsilon \\log \\mathbb{P} \\Big( \\sup_{x} |u(t,x) | > z_0\\varepsilon^{-1/2} \\Big) = \\lim_{\\varepsilon \\to 0^+} \\varepsilon \\log \\mathbb{P} \\Big( \\|u(t,x) \\|_{\\mathcal{F} L^1} > z_0\\varepsilon^{-1/2} \\Big),\n\\end{align}\n\n\\noindent \nwhere $u(t,x)$ is the solution to \\eqref{weakNLS1} with initial data \\eqref{random1} in $\\mathcal{F}L^1$.\nThis motivates us to consider the LDP under the $\\mathcal{F} L^1$ norm.\nWe are ready to state our first main result.\n\n\\begin{align}\n\\label{LDP1}\n\\log \\PP (\\mathcal D(t,z)) = - \\mathcal I (\\theta^* (z)) + o(1),\n\\end{align}\n\n\\begin{align}\n\\label{LDP2}\n\\log \\PP \\Big( \\sup_{x} |u(t,x)| > z \\Big) = - I(z) + o(1), \\quad z \\to \\infty,\n\\end{align}\n\n\\begin{align}\\label{minp}\n\\theta^* (z) = {\\rm argmin}\\,\\, \\mathcal I (\\theta)\n\\end{align}\n\n\\begin{align}\n\\label{random1}\nu_0^N (x) = \\sum_{|k| \\le N} \\theta_k e^{ikx},\n\\end{align}\n\n\\begin{equation}\\label{random_data}\nu_0 (x)= \\sum_{k\\in\\Z} c_k g_k e^{ikx},\n\\end{equation}\n\nIn particular, consider the weakly nonlinear Schr\\\"odinger equation on the circle $\\T = [0,2\\pi]$:\n\\begin{align}\n\\label{weakNLS1}\ni\\pa_t u + \\Delta u = \\varepsilon^{2} |u|^2 u.\n\\end{align}\nIt was proved in~\\cite{GGKS21} that\n\\begin{align}\n\\label{THM_LDP3}\n\\lim_{\\varepsilon \\to 0^+} \n\\varepsilon \\log \\PP \\Big( \\sup_{x} |u(t,x)| > z_0 \\varepsilon^{-1/2} \\Big)\n= - \\frac{z_0^2}{\\sum_k c_k^2},\n\\end{align}\nfor $0 < t \\lesssim \\varepsilon^{-1}$ and $z_0 > 0$,\nwhere $u(t,x)$ denotes the solution to~\\eqref{weakNLS1} with random initial data $u_0^\\omega$ given by~\\eqref{random_data} and coefficients satisfying\n\\begin{align}\\label{decay}\nc_k = a e^{-b|k|} \n\\quad \\text{or} \\quad \nc_k = a e^{-b|k|^2},\n\\end{align}\nfor some fixed $a,b > 0$.\n\n\\noi \nWe note that $L^\\infty$ and $\\FL^1$ scales the same, and furthermore, the LDPs under $L^\\infty$ and $\\FL^1$ share the same rate function.\nIn particular, for $t \\les \\eps^{-1}$, we have\n\\begin{align}\\label{obser}\n\\lim_{\\eps \\to 0^+} \\eps \\log \\PP \\Big( \\sup_{x} |u(t,x) | > z_0\\eps^{-1/2} \\Big) = \\lim_{\\eps \\to 0^+} \\eps \\log \\PP \\Big( \\|u(t,x) \\|_{\\FL^1} > z_0\\eps^{-1/2} \\Big),\n\\end{align}\n\n\\begin{remark}\n\\rm \nWe remark that the time scale $t = \\mathcal O ( \\eps^{-1} |\\log \\eps|)$ is critical;\nsee \\cite[Remark 1.7]{GGKS21} for more discussion on critical time.\nIn the limit $\\eps \\to 0^+$, the large deviation principle \\eqref{THM_LDP4} may hold for any fixed time $t > 0$.\n\\end{remark}\n\n\\begin{corollary}\\label{COR:main} \nLet $u(t,x)$ be solutions to \\eqref{weakNLS1} with initial data $u_0^\\o$ given in \\eqref{random_data}.\nFor any $t = \\mathcal O ( \\eps^{-1} |\\log \\eps|)$ and $z_0 > 0$, we have that \\eqref{THM_LDP3} holds provided $(c_k) \\in \\l^1$.\n\\end{corollary}\n\n\\begin{proposition}\n\\label{PROP:linear1}\nConsider the linear Schr\\\"odinger equation on the torus $\\T = [0,2\\pi]$ as in \\eqref{linear}, with random initial data $u_0^\\o$ given by \\eqref{random_data}.\nThen,\n\\begin{align}\n\\label{LDP_linear1}\n\\lim_{\\eps\\to 0^+} \\eps \\log \\P \\big( \\|u(t)\\|_{\\FL^1(\\T)} \\ge z_0 \\eps^{-1/2} \\big) = - \\frac{z_0^2}{\\sum_{k \\in \\Z} c_k^2},\n\\end{align}\n\nWe turn to the summation in \\eqref{main} over $k \\in B$. \nWe note that\n\\begin{align}\n\\label{main2}\n\\begin{split}\n\\eps & \\sum_{k \\in B} \\log \\left( 1+ \\sqrt{\\pi}\\varepsilon^{-1/2}\\, c_k\\, \\exp\\left(\\frac{\\varepsilon^{-1} \\, c_k^2}{4}\\right) \\right) \\\\\n& \\le \\eps \\sum_{k \\in B} \\log \\left( \\big( 1+ \\sqrt{\\pi}\\varepsilon^{-1/2}\\, c_k\\, \\big) \\exp\\left(\\frac{\\varepsilon^{-1} \\, c_k^2}{4}\\right) \\right) \\\\\n& \\le \\eps \\sum_{k \\in B} \\log \\big( 1+ \\sqrt{\\pi}\\varepsilon^{-1/2}\\, c_k\\, \\big) + \\eps \\sum_{k \\in B} \\left(\\frac{\\varepsilon^{-1} \\, c_k^2}{4}\\right) \\\\\n& \\le \\sqrt{\\pi}\\varepsilon^{1/2} \\sum_{k \\in B} \\, c_k\\, + \\frac14 \\sum_{k \\in B} c_k^2,\n\\end{split}\n\\end{align}\nwhich is again uniformly bounded as $(c_k) \\in \\l^1 \\subset \\l^2$, where we used $\\eps^{1/2} \\le c_k$ for $k \\in B$.\n\\end{proof}\n\n\\subsection{Linear LDP II - \\texorpdfstring{$L^\\infty$}{Lg} norm}\nIn this subsection,\nwe prove an LDP for the linear solution to \\eqref{linear} under $L^\\infty$ norm.\nIn particular, we shall prove\n\\begin{proposition}\n\\label{PROP:linear2}\nConsider the linear Schr\\\"odinger equation on the torus $\\T = [0,2\\pi]$ as in \\eqref{linear}, with random initial data $u_0^\\o$ given by \\eqref{random_data}.\nThen,\n\\begin{align}\n\\label{LDP_linear2}\n\\lim_{\\eps\\to 0^+} \\eps \\log \\PP \\big( \\|u(t)\\|_{L^\\infty (\\T)} \\ge z_0 \\eps^{-1/2} \\big) = - \\frac{z_0^2}{\\sum_{k \\in \\Z} c_k^2}, \n\\end{align}\n\n\\noi \nwhere $\\mathcal B_\\eps$ is defined in \\eqref{Beps}.\nWe first note, see also \\cite[Proposition 4.1]{GGKS21},\\footnote{Please note that \\cite[Proposition 4.1]{GGKS21} only requires that $\\|c_k\\|_{\\l^1_k} < \\infty$.} that \n\\begin{align}\n\\label{pp400}\n\\lim_{\\eps \\to 0^+} \\eps \\log \\PP \\big( \\| u_{\\rm app} (t) \\|_{L^\\infty (\\T)} > z_0 (\\eps^{-1/2} + \\eps^{-1/2 + \\dl}) \\big) = - \\frac{z_0^2}{\\sum_{k \\in \\Z} c_k^2}.\n\\end{align}\n\n\\begin{align}\n\\label{THM_LDP4}\n\\lim_{\\eps \\to 0^+} \\eps \\log \\PP \\Big( \\|u(t,x) \\|_{\\FL^1 (\\T)} > z_0\\eps^{-1/2} \\Big) = - \\frac{z_0^2}{\\sum_k c_k^2},\n\\end{align}\n\n\\begin{equation}\\label{random_data}\nu_0 (x)= \\sum_{k\\in\\Z} c_k g_k e^{ikx},\n\\end{equation}", "post_theorem_intro_text_len": 2107, "post_theorem_intro_text": "\\begin{remark}\n\\rm \nWe remark that the time scale $t = \\mathcal O ( \\varepsilon^{-1} |\\log \\varepsilon|)$ is critical;\nsee \\cite[Remark 1.7]{GGKS21} for more discussion on critical time.\nIn the limit $\\varepsilon \\to 0^+$, the large deviation principle \\eqref{THM_LDP4} may hold for any fixed time $t > 0$.\n\\end{remark}\n\n\\begin{remark}\n\\rm \nTo prove Theorem~\\ref{THM:main}, we first develop a novel approach to establish\nthe LDP for the linear Schr\\\"odinger equation under the sole assumption that\nthe initial data lies in $\\mathcal{F} L^1$. We then carry out the nonlinear analysis,\ntogether with a perturbative argument, entirely within the $\\mathcal{F} L^1$ framework.\n\\end{remark}\n\nAs a consequence of Theorem \\ref{THM:main} and the observation \\eqref{obser}, we can improve \\eqref{THM_LDP3} in terms of the decay of $c_k$.\n\n\\begin{corollary}\\label{COR:main} \nLet $u(t,x)$ be solutions to \\eqref{weakNLS1} with initial data $u_0^\\omega$ given in \\eqref{random_data}.\nFor any $t = \\mathcal O ( \\varepsilon^{-1} |\\log \\varepsilon|)$ and $z_0 > 0$, we have that \\eqref{THM_LDP3} holds provided $(c_k) \\in \\ell^1$.\n\\end{corollary} \n\n\\begin{remark}\n\\rm \nWe note that Corollary \\ref{COR:main} improves \\cite[Theorem 1.1]{GGKS21} in the following sense.\nThe coefficients $(c_k)$ in \\cite[Theorem 1.1]{GGKS21} is required to decay condition \\eqref{decay}.\nIn Proposition \\ref{PROP:linear2}, it requires only $(c_k) \\in \\ell^1$.\nIt is expected that \\eqref{THM_LDP3} holds for initial data $u_0 \\in L^2 \\cap L^\\infty$, where $\\mathcal{F} L^1 \\subset L^2 \\cap L^\\infty$.\nHowever, our current argument can only handle $\\mathcal{F} L^1$ data.\n\\end{remark}\n\n\\begin{remark}\n\\rm \nRecently, in \\cite{BGMS25}, the authors studied rogue waves and large deviations\nfor two-dimensional pure-gravity deep-water waves, assuming initial data with\nexponentially decaying Fourier coefficients; see \\cite[(1.14)]{BGMS25}. It is\nexpected that our argument also applies in their setting and allows one to\nrelax this assumption to merely $\\ell^1$ decay of the Fourier coefficients.\nWe plan to pursue this in future work.\n\\end{remark}", "sketch": "To prove Theorem~\\ref{THM:main}, the authors say they (i) “first develop a novel approach to establish the LDP for the linear Schr\\\"odinger equation under the sole assumption that the initial data lies in $\\mathcal{F} L^1$,” and then (ii) “carry out the nonlinear analysis, together with a perturbative argument, entirely within the $\\mathcal{F} L^1$ framework.”", "expanded_sketch": "To prove the main theorem, the authors say they (i) “first develop a novel approach to establish the LDP for the linear Schr\\\"odinger equation under the sole assumption that the initial data lies in $\\mathcal{F} L^1$,” and then (ii) “carry out the nonlinear analysis, together with a perturbative argument, entirely within the $\\mathcal{F} L^1$ framework.”,", "expanded_theorem": "\\label{THM:main} \nLet $u(t,x)$ be solutions to \\begin{align}\n\\label{weakNLS1}\ni\\pa_t u + \\Delta u = \\varepsilon^{2} |u|^2 u.\n\\end{align} with initial data $u_0^\\omega$ given in \\begin{equation}\\label{random_data}\nu_0 (x)= \\sum_{k\\in\\Z} c_k g_k e^{ikx},\n\\end{equation}.\nFor any $t = \\mathcal O ( \\varepsilon^{-1} |\\log \\varepsilon|)$ and $z_0 > 0$, we have that\n\\begin{align}\n\\label{THM_LDP4}\n\\lim_{\\varepsilon \\to 0^+} \\varepsilon \\log \\mathbb{P} \\Big( \\|u(t,x) \\|_{\\mathcal{F} L^1 (\\mathbb{T})} > z_0\\varepsilon^{-1/2} \\Big) = - \\frac{z_0^2}{\\sum_k c_k^2},\n\\end{align}\n\n\\noindent \nholds if and only if $(c_k) \\in \\ell^1.\\,", "theorem_type": ["Biconditional or Equivalence", "Asymptotic or Limit"], "mcq": {"question": "Let $u(t,x)$ solve the weakly nonlinear Schr\\\"odinger equation on the torus $\\mathbb T$,\n\\[\ni\\partial_t u+\\Delta u=\\varepsilon^2|u|^2u,\n\\]\nwith random initial data\n\\[\nu_0(x)=\\sum_{k\\in\\mathbb Z} c_k g_k e^{ikx},\n\\]\nwhere $\\{g_k\\}_{k\\in\\mathbb Z}$ are independent identically distributed complex Gaussian random variables and $(c_k)$ is a sequence of coefficients. For any $t=\\mathcal O(\\varepsilon^{-1}|\\log\\varepsilon|)$ and any $z_0>0$, suppose one considers the large-deviation asymptotic\n\\[\n\\lim_{\\varepsilon\\to0^+}\\varepsilon\\log\\mathbb P\\Big(\\|u(t,\\cdot)\\|_{\\mathcal F L^1(\\mathbb T)}>z_0\\varepsilon^{-1/2}\\Big)\n= -\\frac{z_0^2}{\\sum_k c_k^2}.\n\\]\nWhich of the following statements is equivalent to this asymptotic formula?", "correct_choice": {"label": "A", "text": "The coefficient sequence satisfies $(c_k)\\in \\ell^1$."}, "choices": [{"label": "B", "text": "The coefficient sequence satisfies $(c_k)\\in \\ell^2$."}, {"label": "C", "text": "The coefficient sequence satisfies $\\sum_k c_k^2<\\infty$."}, {"label": "D", "text": "For each fixed $z_0>0$, there exists a time range $t=\\mathcal O(\\varepsilon^{-1}|\\log\\varepsilon|)$ on which the asymptotic formula holds whenever $(c_k)\\in \\ell^1\\cap \\ell^2$."}, {"label": "E", "text": "The asymptotic formula holds if and only if the coefficient sequence satisfies $(c_k)\\in \\mathcal F L^1(\\mathbb T)$."}], "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": "sharp \\ell^1 threshold replaced by weaker square-summability", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped the full equivalence and kept only the weaker consequence implied by $(c_k)\\in\\ell^1$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "replaced the iff criterion by a merely sufficient mixed assumption with altered quantifier structure", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "confused coefficient summability on $(c_k)$ with function-space membership notation", "template_used": "wildcard"}]}}
{"id": "2512.07839v1", "paper_link": "http://arxiv.org/abs/2512.07839v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm: new theorem}\nFor any $m \\equiv 3 \\pmod{4}$ there exists $N$ such that for all $n \\geq N$ there exists an equilateral polygon of size n.", "start_pos": 4407, "end_pos": 4585, "label": "thm: new theorem"}, "ref_dict": {}, "pre_theorem_intro_text_len": 1699, "pre_theorem_intro_text": "The study of equilateral polygons within lattices combines elements of geometry, number theory, and discrete mathematics. In particular, the problem of characterizing which polygons can be embedded in given planar lattices has drawn sustained interest. A central focus has been on determining for which values of $n$ there exists a convex equilateral $n$-gon whose vertices lie in a specified lattice.\n\nRecent results by Maehara \\cite{maehara2019planar} established that every planar integral lattice contains convex equilateral $n$-gons for all even $n \\geq 4$, and for certain odd values of $n$, depending on a number-theoretic invariant of the lattice. In particular, Maehara showed that a planar integral lattice $L$ contains some equilateral polygon with an odd number of sides if and only if the square-free part of the square of the determinant of $L$, denoted $\\nu(L)$, satisfies $\\nu(L) \\equiv 3 \\pmod{4}$.\n\nBuilding on this, Iino and Sakiyama \\cite{iino2025planarlatticesequilateraloddgons} studied rectangular lattices of the form $\\Lambda(m) = L[(1,0),(0,\\sqrt{m})]$ for square-free integers $m \\equiv 3 \\pmod{4}$, and provided both necessary and sufficient conditions for the existence of equilateral polygons with a given odd number of sides. Among their key findings is that for such lattices, if a convex equilateral $n$-gon exists, then $n$ must be at least as large as every prime dividing $m$. They also proved that this is a sufficient condition when the largest prime factor of $m$ is less than 29 by a brute-force computer search.\n\nIn this paper, we extend the classification of equilateral polygons in rectangular lattices of the form $\\Lambda(m)$, with $m \\equiv 3 \\pmod{4}$.", "context": "The study of equilateral polygons within lattices combines elements of geometry, number theory, and discrete mathematics. In particular, the problem of characterizing which polygons can be embedded in given planar lattices has drawn sustained interest. A central focus has been on determining for which values of $n$ there exists a convex equilateral $n$-gon whose vertices lie in a specified lattice.\n\nRecent results by Maehara \\cite{maehara2019planar} established that every planar integral lattice contains convex equilateral $n$-gons for all even $n \\geq 4$, and for certain odd values of $n$, depending on a number-theoretic invariant of the lattice. In particular, Maehara showed that a planar integral lattice $L$ contains some equilateral polygon with an odd number of sides if and only if the square-free part of the square of the determinant of $L$, denoted $\\nu(L)$, satisfies $\\nu(L) \\equiv 3 \\pmod{4}$.\n\nBuilding on this, Iino and Sakiyama \\cite{iino2025planarlatticesequilateraloddgons} studied rectangular lattices of the form $\\Lambda(m) = L[(1,0),(0,\\sqrt{m})]$ for square-free integers $m \\equiv 3 \\pmod{4}$, and provided both necessary and sufficient conditions for the existence of equilateral polygons with a given odd number of sides. Among their key findings is that for such lattices, if a convex equilateral $n$-gon exists, then $n$ must be at least as large as every prime dividing $m$. They also proved that this is a sufficient condition when the largest prime factor of $m$ is less than 29 by a brute-force computer search.\n\nIn this paper, we extend the classification of equilateral polygons in rectangular lattices of the form $\\Lambda(m)$, with $m \\equiv 3 \\pmod{4}$.", "full_context": "The study of equilateral polygons within lattices combines elements of geometry, number theory, and discrete mathematics. In particular, the problem of characterizing which polygons can be embedded in given planar lattices has drawn sustained interest. A central focus has been on determining for which values of $n$ there exists a convex equilateral $n$-gon whose vertices lie in a specified lattice.\n\nRecent results by Maehara \\cite{maehara2019planar} established that every planar integral lattice contains convex equilateral $n$-gons for all even $n \\geq 4$, and for certain odd values of $n$, depending on a number-theoretic invariant of the lattice. In particular, Maehara showed that a planar integral lattice $L$ contains some equilateral polygon with an odd number of sides if and only if the square-free part of the square of the determinant of $L$, denoted $\\nu(L)$, satisfies $\\nu(L) \\equiv 3 \\pmod{4}$.\n\nBuilding on this, Iino and Sakiyama \\cite{iino2025planarlatticesequilateraloddgons} studied rectangular lattices of the form $\\Lambda(m) = L[(1,0),(0,\\sqrt{m})]$ for square-free integers $m \\equiv 3 \\pmod{4}$, and provided both necessary and sufficient conditions for the existence of equilateral polygons with a given odd number of sides. Among their key findings is that for such lattices, if a convex equilateral $n$-gon exists, then $n$ must be at least as large as every prime dividing $m$. They also proved that this is a sufficient condition when the largest prime factor of $m$ is less than 29 by a brute-force computer search.\n\nIn this paper, we extend the classification of equilateral polygons in rectangular lattices of the form $\\Lambda(m)$, with $m \\equiv 3 \\pmod{4}$.\n\n\\begin{abstract}\nWe study the existence of equilateral polygons in planar integer lattices. Maehara showed that it's sufficient to work with rectangular lattices $\\Lambda(m) = L[(1,0),(0,\\sqrt{m})]$ with $m \\equiv 3 \\pmod{4}$. Building on results of Maehara and of Iino and Sakiyama, we show that for every such $m$ there exists $N$ such that for all $n \\geq N$, the lattice $\\Lambda(m)$ contains an equilateral $n$-gon. This extends previous classifications of equilateral polygons in planar lattices.\n\\end{abstract}\n\nRecent results by Maehara \\cite{maehara2019planar} established that every planar integral lattice contains convex equilateral $n$-gons for all even $n \\geq 4$, and for certain odd values of $n$, depending on a number-theoretic invariant of the lattice. In particular, Maehara showed that a planar integral lattice $L$ contains some equilateral polygon with an odd number of sides if and only if the square-free part of the square of the determinant of $L$, denoted $\\nu(L)$, satisfies $\\nu(L) \\equiv 3 \\pmod{4}$.\n\nBuilding on this, Iino and Sakiyama \\cite{iino2025planarlatticesequilateraloddgons} studied rectangular lattices of the form $\\Lambda(m) = L[(1,0),(0,\\sqrt{m})]$ for square-free integers $m \\equiv 3 \\pmod{4}$, and provided both necessary and sufficient conditions for the existence of equilateral polygons with a given odd number of sides. Among their key findings is that for such lattices, if a convex equilateral $n$-gon exists, then $n$ must be at least as large as every prime dividing $m$. They also proved that this is a sufficient condition when the largest prime factor of $m$ is less than 29 by a brute-force computer search.\n\nIn this paper, we extend the classification of equilateral polygons in rectangular lattices of the form $\\Lambda(m)$, with $m \\equiv 3 \\pmod{4}$.\n\nWe will give an existence proof in \\Cref{Section: Existence} followed by a explicit construction in \\Cref{Section: Construction}.\n\nFor any $m \\equiv 3 \\pmod{4}$ does there exist $N$ such that for all $n \\geq N$ there exists a \\textbf{convex} equilateral polygon of size n?\n\n\\begin{open} \nFor any $m \\equiv 3 \\pmod{4}$ does there exist an equilateral polygon of size $n'$, where $n'$ is the largest prime factor of m?\n\\end{open}\n\nFor any $m \\equiv 3 \\pmod{4}$ does there exist a \\textbf{convex} equilateral polygon of size $n'$, where $n'$ is the largest prime factor of m?", "post_theorem_intro_text_len": 129, "post_theorem_intro_text": "We will give an existence proof in \\Cref{Section: Existence} followed by a explicit construction in \\Cref{Section: Construction}.", "sketch": "The post-theorem introduction indicates the proof strategy: it will present \"an existence proof\" (in \\Cref{Section: Existence}) and then an \"explicit construction\" (in \\Cref{Section: Construction}).", "expanded_sketch": "The post-theorem introduction indicates the proof strategy: it will present \"an existence proof\" (in \\Cref{Section: Existence}) and then an \"explicit construction\" (in \\Cref{Section: Construction}).", "expanded_theorem": "\\label{thm: new theorem}\nFor any $m \\equiv 3 \\pmod{4}$ there exists $N$ such that for all $n \\geq N$ there exists an equilateral polygon of size n.,", "theorem_type": "unknown", "mcq": {"question": "Let \\(\\Lambda(m)=L[(1,0),(0,\\sqrt m)]=\\{(a,b\\sqrt m):a,b\\in \\mathbb Z\\}\\) be the planar rectangular lattice associated to an integer \\(m\\equiv 3\\pmod 4\\). An equilateral polygon of size \\(n\\) means an \\(n\\)-gon whose vertices lie in \\(\\Lambda(m)\\) and whose side lengths are all equal. Which statement holds for such lattices?", "correct_choice": {"label": "A", "text": "For every integer \\(m\\equiv 3\\pmod 4\\), there exists an integer \\(N\\) such that for every integer \\(n\\ge N\\), the lattice \\(\\Lambda(m)\\) contains an equilateral polygon of size \\(n\\)."}, "choices": [{"label": "B", "text": "For every integer \\(m\\equiv 3\\pmod 4\\), there exists an integer \\(N\\) such that for every integer \\(n\\ge N\\), the lattice \\(\\Lambda(m)\\) contains a convex equilateral polygon of size \\(n\\)."}, {"label": "C", "text": "For every integer \\(m\\equiv 3\\pmod 4\\), there exists an integer \\(n\\) such that the lattice \\(\\Lambda(m)\\) contains an equilateral polygon of size \\(n\\)."}, {"label": "D", "text": "There exists an integer \\(N\\) such that for every integer \\(m\\equiv 3\\pmod 4\\) and every integer \\(n\\ge N\\), the lattice \\(\\Lambda(m)\\) contains an equilateral polygon of size \\(n\\)."}, {"label": "E", "text": "For every integer \\(m\\equiv 3\\pmod 4\\), there exists an integer \\(N\\) such that for every odd integer \\(n\\ge N\\), the lattice \\(\\Lambda(m)\\) contains an equilateral polygon of size \\(n\\), and one may take \\(N\\) to be the largest prime factor of \\(m\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "geometric_construction", "tampered_component": "equilateral_vs_convex_conclusion", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped_eventual_all_n_conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "dependence_of_N_on_m", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "computational_check", "tampered_component": "threshold_equals_largest_prime_factor", "template_used": "boundary_range"}]}}
{"id": "2512.07967v1", "paper_link": "http://arxiv.org/abs/2512.07967v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:mainthm}\nLet $S$ be a complex algebraic variety admitting a trim resolution. Then\n\\begin{itemize}\n\\item\nFor $s\\in S$, the local Euler obstruction $\\Eu_S(s)$ equals the Euler characteristic of\nthe fiber over $s$ in any trim resolution of $S$.\n\\item\nThe {\\em stringy\\/} Chern class of $S$ equals its Chern-Mather class:  \n$\\cstr(S)=\\cma(S)$.\n\\item\nThe characteristic cycle of the intersection cohomology sheaf of $S$ is irreducible.\n\\end{itemize}", "start_pos": 4984, "end_pos": 5468, "label": "thm:mainthm"}, "ref_dict": {"thm:mainthm": "\\begin{theorem}\\label{thm:mainthm}\nLet $S$ be a complex algebraic variety admitting a trim resolution. Then\n\\begin{itemize}\n\\item\nFor $s\\in S$, the local Euler obstruction $\\Eu_S(s)$ equals the Euler characteristic of\nthe fiber over $s$ in any trim resolution of $S$.\n\\item\nThe {\\em stringy\\/} Chern class of $S$ equals its Chern-Mather class:  \n$\\cstr(S)=\\cma(S)$.\n\\item\nThe characteristic cycle of the intersection cohomology sheaf of $S$ is irreducible.\n\\end{itemize}\n\\end{theorem}", "Lagrapf": "\\begin{equation}\\label{eq:strMa}\nf_*(c(TY)\\cap [Y]) = \\cma(S)\n\\end{equation}\nin the Chow group of $S$. In general, stringy Chern classes are defined for\nnormal varieties with a $\\Qbb$-Cartier canonical divisor and at worst log-terminal\nsingularities, and take value in the Chow group with rational coefficients. These\nadditional stipulations are not needed for the notion considered in this note; we can \nadopt the left-hand side of~\\eqref{eq:strMa} as the definition of stringy Chern class\nin the (integral) Chow group of $S$, compatibly with the more general definition.\nThe fact that the class is independent of the choice of trim resolution is\nalso a consequence of Theorem~\\ref{thm:mainthm}.\n\nWe note that Theorem~\\ref{thm:mainthm} implies that Batyrev's {\\em stringy Euler\nnumber\\/} (\\cite{MR2001j:14018}) of a variety admitting a trim resolution equals \nits Euler characteristic weighted by the local Euler obstruction.\n\nTheorem~\\ref{thm:mainthm} is a consequence of considerations concerning Sabbah's\nformalism of {\\em conical Lagrangian cycles,\\/} to which local Euler obstructions\nand Chern-Mather classes relate directly (see~\\S\\ref{Lagrapf}). Concerning intersection \ncohomology, recall that if $\\pi:Y \\to S$ is a small resolution, then the \nintersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant \nsheaf on~$Y$ (\\cite[\\S6.2]{MR696691}). We evaluate the corresponding push-forward \nat the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$. \nMore precisely, we prove (Proposition~\\ref{prop:Sabpf}) that, for trim resolutions, the \nLagrangian push-forward of the zero-section~$T^*_YY$ of the cotangent bundle \n$T^* Y$ equals the conormal cycle $T^*_S X$. The theorem follows from this \nmore basic observation, as we show in~\\S\\ref{thmpf}. \n\nThere is considerable interest in conditions implying that the characteristic cycle of\nthe intersection cohomology sheaf is irreducible. Lusztig (\\cite[13.7, p.~414]{MR1088333}) \nexpressed the `hope' that this may be the case for Schubert varieties in flag manifolds of \ntype A, D, E. A counterexample was constructed by Kashiwara and Saito for type A in \n$F\\ell(8)$ (\\cite{MR1458969}, \\cite{MR1896039}), while it holds in $F\\ell(n)$ for $n\\le 7$.\nIrreducibility is also known for all Schubert \nvarieties in the standard Grassmannian (\\cite{MR1084458}), and more generally \nfor all Schubert varieties in cominuscule Grassmannians of \ntypes A, D, and E, see~\\cite{MR1451256, MihalceaSingh}.\n\nWe hope that Theorem~\\ref{thm:mainthm} may help in streamlining such verifications. \nFor instance, since the Abel-Jacobi resolution of the theta divisor of a non-hyperelliptic \ncurve is trim, the irreducibility of the $\\IC$ characteristic cycle in this case (first established \nin \\cite{MR1642745}) follows directly from~Theorem~\\ref{thm:mainthm}.\n\nThe connection between the irreducibility of the $\\IC$ characteristic cycle and \nthe equality of Chern-Mather and stringy Chern classes was pointed out by B.~Jones\nin~\\cite[Remark 3.3.2]{MR2628830}, ultimately as an application of the microlocal \nindex formula of Dubson and Kashiwara. \nWe freely borrow ideas from~\\cite{MR2628830} \nin~\\S\\ref{thmpf}.\n\nA condition on fibers of small resolutions is considered in~\\cite{graham}, including a proof\nthat the condition implies the irreducibility of the IC characteristic cycles.\\medskip\n\n{\\em Acknowledgments.} This work was supported in part by an award from the Simons\nFoundation, SFI-MPS-TSM-00013681. The author also thanks David Massey and\nLeonardo Mihalcea for useful conversations, and Caltech for hospitality as most of this \nwork was carried out.\n\n\\section{Lagrangian push-forward}\\label{Lagrapf}\nLet $X$ be a nonsingular variety and denote by $T^* X$ the cotangent bundle of $X$.\nThe {\\em conormal variety\\/} $T^*_W X$ of a closed subvariety $W\\subseteq X$ is the \nclosure in $T^* X$ of the conormal variety to the nonsingular part $W^\\circ$ of $V$: \n$T^*_W X = \\overline{T^*_{W^\\circ} X}$. The conormal variety of $X$ itself is the\nzero-section $T^*_XX$ of the cotangent bundle. All conormal varieties have dimension\n$\\dim X$; they determine conormal {\\em cycles\\/} in $Z_{\\dim X} T^* X$.\nIt will also be convenient to take the projective completion of these constructions:\nwe will denote by $\\Tbb^* X$ the projective completion $\\Pbb(T^* X\\oplus \\one)$ \nof $T^* X$ (here, $\\Pbb$ denotes the projective bundle of lines) and by~$\\Tbb^*_WX$ \nthe closure of the conormal variety in $\\Tbb^* X$. \n\nFor a nonsingular variety $X$, we denote by $\\cL (X)$ the free abelian group of \n{\\em conical Lagrangian cycles\\/} in the cotangent bundle $T^* X$.\nConormal cycles are conical Lagrangian, and in fact (cf.~\\cite[Lemma~3]{MR1063344}) \n$\\cL (X)$ may be realized as the free abelian group on conormal cycles. \nFor a closed (and possibly singular) subvariety $V\\subseteq X$, we denote by\n$\\cL(V)$ the subgroup of $\\cL(X)$ generated by the conormal cycles $T^*_W X$\nwith $W\\subseteq V$. Thus, elements of~$\\cL (V)$ may be viewed as finite integer linear \ncombinations $\\sum_W m_W T^*_WX$ ranging over closed subvarieties $W$ of $V$.\nClearly $\\cL(V)$ is isomorphic to the group of algebraic cycles of~$V$, and in particular\nit is independent of the ambient nonsingular variety $X$.\n\nImportant invariants of $V$ may be expressed directly in terms of Lagrangian cycles \nby means of intersection-theoretic operations, after taking the projective completion. \nAs above, realize $V$ as a closed subvariety of a\nnonsingular variety $X$; the results will be independent of the choice of $X$. \nLet $\\pi: \\Tbb^*_V X\\to V$ be the natural projection; and let $\\cO(1)$ be the \ntautological line bundle on $\\Tbb^* X=\\Pbb(T^* X\\oplus \\one)$.\n\\begin{itemize}\n\\item[---]\nThe {\\em local Euler obstruction\\/} $\\Eu_V: V\\to \\Zbb$ is\n\\[\n\\Eu_V(p)= (-1)^{\\dim X-\\dim V} \\int c(\\pi^* TX|_V) c(\\cO(1))^{-1}\\cap s(\\pi^{-1}(p), \\Tbb^*_V X)\n\\]\nwhere $s(\\pi^{-1}(p), \\Tbb^*_V X)$ denotes the {\\em Segre class\\/} in the sense \nof~\\cite[Chapter~4]{85k:14004};\n\\item[---]\nThe {\\em Chern-Mather class\\/} $\\cma(V)\\in A_*V$ is\n\\[\n\\cma(V)=(-1)^{\\dim X-\\dim V} c(TX|_V)\\cap \\pi_* \\left(c(\\cO(1))^{-1}\\cap [\\Tbb^*_V X]\\right)\\saf.\n\\]\n\\end{itemize}\nEquivalent results were established by C.~Sabbah (\\cite[1.2.1, 1.2.2]{MR804052}); we also\naddress the reader to~\\cite{MR1063344} and~\\cite{MR2002g:14005} for clear treatments of these\nformulas. \n\nLet $\\cF(V)$ denote the abelian group of constructible functions $V\\to \\Zbb$.\nFollowing~\\cite[\\S1]{MR2002g:14005}, the relation between conormal cycles and local\nEuler obstructions is recorded by the homomorphism\n\\[\n\\CC: \\cF(V) \\to \\cL(V)\n\\]\ndefined by prescribing\n\\[\n\\Eu_W \\mapsto (-1)^{\\dim W} T^*_W X\n\\]\nfor all closed subvarieties $W$ of $V$. \nIn fact, $\\CC$ is an {\\em isomorphism:\\/} it simply matches a basis of $\\cF(V)$ with a \nbasis of $\\cL(V)$.\n\n\\begin{defin}\\label{def:CC}\nThe {\\em characteristic cycle\\/} $\\CC(\\alpha)$ of a constructible function $\\alpha$ on $V$ \nis the image of $\\alpha$ in $\\cL(V)$ under this isomorphism. \n\\qede\\end{defin}\n\nLikewise, the formula for Chern-Mather classes motivates the introduction of a homomorphism\n\\[\nc_*:\\cL(V) \\to A_*(V)\n\\]\ndefined on generators by\n\\[\nT^*_W X \\mapsto (-1)^{\\dim W} \\cma(W)\\saf.\n\\]\nThen the composition $c_*\\circ \\CC:\\cF(V) \\to A_*(V)$ agrees with the value at $V$ \nof MacPherson's natural transformation $\\cF \\leadsto A_*$, where $\\cF$ is taken as \na functor with push-forward defined by Euler characteristics of fibers; \nsee~\\cite{MR0361141} and~\\cite[Example 19.1.7]{85k:14004}. \nFor instance, the {\\em Chern-Schwartz-MacPherson class\\/} of a (possibly singular)\nalgebraic variety $V$ is the image $c_*(\\CC(\\one_V))\\in A_*V$ of the characteristic \ncycle in~$\\cL(V)$ of the constant function $\\one_V$.\nSabbah provides an alternative proof of the naturality of this assignment, which is\nthe main result of~\\cite{MR0361141}, by defining a covariant {\\em push-forward\\/}\n\\[\n\\varphi_*: \\cL(V') \\to \\cL(V'')\n\\]\nfor every proper map $\\varphi: V' \\to V''$, making $\\cL$ into a functor, in such a way \nthat the above homomorphisms define natural transformations\n\\[\n\\cF \\leadsto \\cL\\quad, \\quad \\cL \\leadsto A_*\n\\]\nwhose composition agrees with MacPherson's natural transformation.\n\nWe are interested in explicit formulas for this {\\em Lagrangian push-forward.\\/}\nAfter embedding~$V''$ in a nonsingular variety $X$ and replacing $V'$ by a resolution $Y$,\nwe can reduce to the case of a proper morphism $f:Y \\to X$ of nonsingular varieties. \nWe are specifically interested in the image $f_*(T^*_YY)$ of the zero-section in this\nsituation. \n\nTheorem~\\ref{thm:mainthm} will be a consequence of the following result. We recall the \ndefinition of `trim' given in the introduction.\n\n\\begin{defin}\\label{def:contained}\nLet $\\pi:Y \\to S$ be a proper birational morphism of varieties, with $Y$ nonsingular.\nFor $0\\le d\\le \\dim Y$, let $Y_d$ denote the locus where the rank of the differential\n$d\\pi$ equals $d$. Then $\\pi$ is a {\\em trim resolution\\/} if $\\dim Y_d<d$\nfor all $0\\le d<\\dim Y$.\n\\qede\\end{defin}\n\n\\begin{prop}\\label{prop:Sabpf}\nLet $f: Y\\to X$ be a proper morphism of nonsingular varieties, such that $Y \\to f(Y)$ is\na trim resolution. Then $f_* (T^*_Y Y)=T^*_{f(Y)} X$.\n\\end{prop}\n\n\\begin{remark}\\label{rem:YtoS}\nLet $\\pi: Y\\to S$ be a proper surjective morphism, with $Y$ nonsingular, and \n$\\iota:S\\hookrightarrow X$ a closed embedding, with $X$ nonsingular; and let \n$f=\\iota\\circ \\pi: Y \\to X$ be the composition. \nAs explained above, $T^*_{f(Y)} X=T^*_S X$ may be viewed as an element of $\\cL(S)$;  \nProposition~\\ref{prop:Sabpf} states that if $\\pi$ is trim, then this is the image $\\pi_*(T^*_YY)$ \nunder $\\pi_*: \\cL(Y)\\to \\cL(S)$.\n\\qede\\end{remark}\n\nThe rest of this section is devoted to the proof of Proposition~\\ref{prop:Sabpf}.\\smallskip\n\n{\\em A priori,\\/} $f_*(T^*_Y Y)$ is an integer linear combination of conormal cycles\n$\\sum_Z m_Z T^*_ZX$, with $Z\\subseteq f(Y)$. One of the summands is $T^*_{f(Y)}X$.\nThe task is to show that if $Y\\to f(Y)$ is trim and $Z\\subsetneq f(Y)$ is a {\\em proper\\/} \nsubvariety of $f(Y)$, then the coefficient of $T^*_Z X$ in $f_* (T^*_Y Y)$ is~$0$.\nFollowing Sabbah, we write $f$ as the composition\n\\[\n\\xymatrix{\nY \\ar[r]^-\\gamma & Y\\times X \\ar[r]^-\\rho & X\n}\n\\]\nwhere $\\gamma$ is the graph of $f$ and $\\rho$ is the (smooth) projection.\nLet $p: \\rho^*(T^* X) \\to T^* X$ be the natural mophism. Then  (cf.~\\cite[\\S2]{MR804052})\nthe components $T^*_ZX$ appearing in the decomposition of $f_*(T^*_YY)$ are the\nirreducible components of the image\n\\begin{equation}\\label{eq:pint}\np \\left(\\rho^*(T^* X) \\cap T^*_{\\gamma(Y)}(Y\\times X)\\right)\n\\end{equation}", "eq:strMa": "\\begin{equation}\\label{eq:strMa}\nf_*(c(TY)\\cap [Y]) = \\cma(S)\n\\end{equation}", "thmpf": "\\label{thmpf}\n\nLet $S$ be a variety admitting a trim resolution $\\pi: Y\\to S$; let $\\iota: S\\hookrightarrow X$\nbe an embedding in a nonsingular variety, and let $f=\\iota\\circ \\pi:Y \\to X$ be the compo", "Furrem": "\\label{Furrem}\n\n1. In this note we have considered the notion of `trim' only for proper birational morphisms  \nbecause that is the case relevant to our application in Theorem~\\ref{thm:mainthm}. \nIt co", "prop:Sabpf": "\\begin{prop}\\label{prop:Sabpf}\nLet $f: Y\\to X$ be a proper morphism of nonsingular varieties, such that $Y \\to f(Y)$ is\na trim resolution. Then $f_* (T^*_Y Y)=T^*_{f(Y)} X$.\n\\end{prop}", "prop:consma": "\\begin{prop}\\label{prop:consma}\nLet $\\pi: Y \\to S$ be a trim proper birational map. Then $\\pi$ is small.\n\\end{prop}", "def:contained": "\\begin{defin}\\label{def:contained}\nLet $\\pi:Y \\to S$ be a proper birational morphism of varieties, with $Y$ nonsingular.\nFor $0\\le d\\le \\dim Y$, let $Y_d$ denote the locus where the rank of the differential\n$d\\pi$ equals $d$. Then $\\pi$ is a {\\em trim resolution\\/} if $\\dim Y_d<d$\nfor all $0\\le d<\\dim Y$.\n\\qede\\end{defin}"}, "pre_theorem_intro_text_len": 1160, "pre_theorem_intro_text": "\\label{intro}\nRecall that a proper surjective stratified map $\\pi: Y\\to S$ of complex algebraic varieties\nof the same dimension is {\\em small\\/} if for all proper strata $Z$ of $S$, $2d(Z)< \\codim_Z S$, \nwhere $d(Z)$ denotes the (common) dimension of the fibers of $\\pi$ over points of $Z$. \nWe say that $\\pi$ is a {\\em small resolution\\/} if $Y$ is nonsingular and $\\pi$ is birational \nand small.\n\nWe introduce a strengthening of this condition, which we name `trim'\n(Definition~\\ref{def:contained}). \nAssume $\\pi:Y\\to S$ is a proper birational map, with $Y$ nonsingular. \nFor $d\\ge 0$, denote by $Y_d\\subseteq Y$ the locally closed subset along which \nthe rank of the differential of $\\pi$ is~$d$. We say that $\\pi$ is {\\em trim\\/} if \n$\\dim Y_d<d$ for all $d<\\dim Y$.\n\nTrim maps are small (Proposition~\\ref{prop:consma}).\nIf $\\pi:Y\\to S$ is small, and for all strata~$Z$ of~$S$ the restriction $\\pi^{-1}(Z)\\to Z$ is a smooth \nmorphism, then $\\pi$ is trim (\\S\\ref{Furrem}). \nFor instance, the standard Abel-Jacobi resolution of the theta divisor of a non-hyperelliptic \ncurve (which is small, cf.~\\cite{MR1642745}) is trim.\n\nOur main result is the following.", "context": "\\label{intro}\nRecall that a proper surjective stratified map $\\pi: Y\\to S$ of complex algebraic varieties\nof the same dimension is {\\em small\\/} if for all proper strata $Z$ of $S$, $2d(Z)< \\codim_Z S$, \nwhere $d(Z)$ denotes the (common) dimension of the fibers of $\\pi$ over points of $Z$. \nWe say that $\\pi$ is a {\\em small resolution\\/} if $Y$ is nonsingular and $\\pi$ is birational \nand small.\n\nWe introduce a strengthening of this condition, which we name `trim'\n(Definition~\\ref{def:contained}). \nAssume $\\pi:Y\\to S$ is a proper birational map, with $Y$ nonsingular. \nFor $d\\ge 0$, denote by $Y_d\\subseteq Y$ the locally closed subset along which \nthe rank of the differential of $\\pi$ is~$d$. We say that $\\pi$ is {\\em trim\\/} if \n$\\dim Y_d<d$ for all $d<\\dim Y$.\n\nTrim maps are small (Proposition~\\ref{prop:consma}).\nIf $\\pi:Y\\to S$ is small, and for all strata~$Z$ of~$S$ the restriction $\\pi^{-1}(Z)\\to Z$ is a smooth \nmorphism, then $\\pi$ is trim (\\S\\ref{Furrem}). \nFor instance, the standard Abel-Jacobi resolution of the theta divisor of a non-hyperelliptic \ncurve (which is small, cf.~\\cite{MR1642745}) is trim.\n\nOur main result is the following.\n\n\\label{Furrem}\n\n1. In this note we have considered the notion of `trim' only for proper birational morphisms  \nbecause that is the case relevant to our application in Theorem~\\ref{thm:mainthm}. \nIt co\n\n\\begin{defin}\\label{def:contained}\nLet $\\pi:Y \\to S$ be a proper birational morphism of varieties, with $Y$ nonsingular.\nFor $0\\le d\\le \\dim Y$, let $Y_d$ denote the locus where the rank of the differential\n$d\\pi$ equals $d$. Then $\\pi$ is a {\\em trim resolution\\/} if $\\dim Y_d<d$\nfor all $0\\le d<\\dim Y$.\n\\qede\\end{defin}\n\n\\begin{prop}\\label{prop:consma}\nLet $\\pi: Y \\to S$ be a trim proper birational map. Then $\\pi$ is small.\n\\end{prop}", "full_context": "\\label{intro}\nRecall that a proper surjective stratified map $\\pi: Y\\to S$ of complex algebraic varieties\nof the same dimension is {\\em small\\/} if for all proper strata $Z$ of $S$, $2d(Z)< \\codim_Z S$, \nwhere $d(Z)$ denotes the (common) dimension of the fibers of $\\pi$ over points of $Z$. \nWe say that $\\pi$ is a {\\em small resolution\\/} if $Y$ is nonsingular and $\\pi$ is birational \nand small.\n\nWe introduce a strengthening of this condition, which we name `trim'\n(Definition~\\ref{def:contained}). \nAssume $\\pi:Y\\to S$ is a proper birational map, with $Y$ nonsingular. \nFor $d\\ge 0$, denote by $Y_d\\subseteq Y$ the locally closed subset along which \nthe rank of the differential of $\\pi$ is~$d$. We say that $\\pi$ is {\\em trim\\/} if \n$\\dim Y_d<d$ for all $d<\\dim Y$.\n\nTrim maps are small (Proposition~\\ref{prop:consma}).\nIf $\\pi:Y\\to S$ is small, and for all strata~$Z$ of~$S$ the restriction $\\pi^{-1}(Z)\\to Z$ is a smooth \nmorphism, then $\\pi$ is trim (\\S\\ref{Furrem}). \nFor instance, the standard Abel-Jacobi resolution of the theta divisor of a non-hyperelliptic \ncurve (which is small, cf.~\\cite{MR1642745}) is trim.\n\nOur main result is the following.\n\n\\label{Furrem}\n\n1. In this note we have considered the notion of `trim' only for proper birational morphisms  \nbecause that is the case relevant to our application in Theorem~\\ref{thm:mainthm}. \nIt co\n\n\\begin{defin}\\label{def:contained}\nLet $\\pi:Y \\to S$ be a proper birational morphism of varieties, with $Y$ nonsingular.\nFor $0\\le d\\le \\dim Y$, let $Y_d$ denote the locus where the rank of the differential\n$d\\pi$ equals $d$. Then $\\pi$ is a {\\em trim resolution\\/} if $\\dim Y_d<d$\nfor all $0\\le d<\\dim Y$.\n\\qede\\end{defin}\n\n\\begin{prop}\\label{prop:consma}\nLet $\\pi: Y \\to S$ be a trim proper birational map. Then $\\pi$ is small.\n\\end{prop}\n\n\\begin{abstract}\nWe introduce {\\em trim\\/} resolutions of complex algebraic varieties, a strengthening \nof the notion of small resolution. We prove that the characteristic cycle of the intersection \ncohomology sheaf of a variety admitting a trim resolution is irreducible and that for \nsuch varieties the stringy and Chern-Mather classes coincide.\n\\end{abstract}\n\nOur main result is the following.\n\nThe definition of characteristic cycle will be recalled below (\\S\\ref{Lagrapf}).\n`Stringy' Chern classes were defined in~\\cite{MR2183846, MR2304329} \n(see~\\cite{MR2280127} for a lean account). If $\\pi: Y\\to S$ is a crepant resolution, \n$\\cstr(S)$ equals the push-forward of the total Chern class of the tangent bundle\nof~$Y$; the equality stated in the theorem is\n\\begin{equation}\\label{eq:strMa}\nf_*(c(TY)\\cap [Y]) = \\cma(S)\n\\end{equation}\nin the Chow group of $S$. In general, stringy Chern classes are defined for\nnormal varieties with a $\\Qbb$-Cartier canonical divisor and at worst log-terminal\nsingularities, and take value in the Chow group with rational coefficients. These\nadditional stipulations are not needed for the notion considered in this note; we can \nadopt the left-hand side of~\\eqref{eq:strMa} as the definition of stringy Chern class\nin the (integral) Chow group of $S$, compatibly with the more general definition.\nThe fact that the class is independent of the choice of trim resolution is\nalso a consequence of Theorem~\\ref{thm:mainthm}.\n\nWe note that Theorem~\\ref{thm:mainthm} implies that Batyrev's {\\em stringy Euler\nnumber\\/} (\\cite{MR2001j:14018}) of a variety admitting a trim resolution equals \nits Euler characteristic weighted by the local Euler obstruction.\n\nTheorem~\\ref{thm:mainthm} is a consequence of considerations concerning Sabbah's\nformalism of {\\em conical Lagrangian cycles,\\/} to which local Euler obstructions\nand Chern-Mather classes relate directly (see~\\S\\ref{Lagrapf}). Concerning intersection \ncohomology, recall that if $\\pi:Y \\to S$ is a small resolution, then the \nintersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant \nsheaf on~$Y$ (\\cite[\\S6.2]{MR696691}). We evaluate the corresponding push-forward \nat the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$. \nMore precisely, we prove (Proposition~\\ref{prop:Sabpf}) that, for trim resolutions, the \nLagrangian push-forward of the zero-section~$T^*_YY$ of the cotangent bundle \n$T^* Y$ equals the conormal cycle $T^*_S X$. The theorem follows from this \nmore basic observation, as we show in~\\S\\ref{thmpf}.\n\nImportant invariants of $V$ may be expressed directly in terms of Lagrangian cycles \nby means of intersection-theoretic operations, after taking the projective completion. \nAs above, realize $V$ as a closed subvariety of a\nnonsingular variety $X$; the results will be independent of the choice of $X$. \nLet $\\pi: \\Tbb^*_V X\\to V$ be the natural projection; and let $\\cO(1)$ be the \ntautological line bundle on $\\Tbb^* X=\\Pbb(T^* X\\oplus \\one)$.\n\\begin{itemize}\n\\item[---]\nThe {\\em local Euler obstruction\\/} $\\Eu_V: V\\to \\Zbb$ is\n\\[\n\\Eu_V(p)= (-1)^{\\dim X-\\dim V} \\int c(\\pi^* TX|_V) c(\\cO(1))^{-1}\\cap s(\\pi^{-1}(p), \\Tbb^*_V X)\n\\]\nwhere $s(\\pi^{-1}(p), \\Tbb^*_V X)$ denotes the {\\em Segre class\\/} in the sense \nof~\\cite[Chapter~4]{85k:14004};\n\\item[---]\nThe {\\em Chern-Mather class\\/} $\\cma(V)\\in A_*V$ is\n\\[\n\\cma(V)=(-1)^{\\dim X-\\dim V} c(TX|_V)\\cap \\pi_* \\left(c(\\cO(1))^{-1}\\cap [\\Tbb^*_V X]\\right)\\saf.\n\\]\n\\end{itemize}\nEquivalent results were established by C.~Sabbah (\\cite[1.2.1, 1.2.2]{MR804052}); we also\naddress the reader to~\\cite{MR1063344} and~\\cite{MR2002g:14005} for clear treatments of these\nformulas.\n\n{\\em A priori,\\/} $f_*(T^*_Y Y)$ is an integer linear combination of conormal cycles\n$\\sum_Z m_Z T^*_ZX$, with $Z\\subseteq f(Y)$. One of the summands is $T^*_{f(Y)}X$.\nThe task is to show that if $Y\\to f(Y)$ is trim and $Z\\subsetneq f(Y)$ is a {\\em proper\\/} \nsubvariety of $f(Y)$, then the coefficient of $T^*_Z X$ in $f_* (T^*_Y Y)$ is~$0$.\nFollowing Sabbah, we write $f$ as the composition\n\\[\n\\xymatrix{\nY \\ar[r]^-\\gamma & Y\\times X \\ar[r]^-\\rho & X\n}\n\\]\nwhere $\\gamma$ is the graph of $f$ and $\\rho$ is the (smooth) projection.\nLet $p: \\rho^*(T^* X) \\to T^* X$ be the natural mophism. Then (cf.~\\cite[\\S2]{MR804052})\nthe components $T^*_ZX$ appearing in the decomposition of $f_*(T^*_YY)$ are the\nirreducible components of the image\n\\begin{equation}\\label{eq:pint}\np \\left(\\rho^*(T^* X) \\cap T^*_{\\gamma(Y)}(Y\\times X)\\right)\n\\end{equation}\nwhere we view both $\\rho^*(T^* X)$ and $T^*_{\\gamma(Y)}(Y\\times X)$\nas subschemes of $T^* (Y\\times X)$. \nLet $T^*_ZX$ be one such component, and let $z$ be a general point of $Z$\nand $\\xi$ a general covector in the fiber $(T^*_ZX)_z$. For $(z,\\xi)$ to be in\nthe image~\\eqref{eq:pint}, there must be a point $(y,z)\\in \\gamma(Y)$ mapping to \n$z$, such that \n$\n(0,\\xi)\\in T^*_y Y\\oplus T^*_z X\\cong T^*_{(y,z)} (Y\\times X)\n$\nvanishes on $T_{(y,z)}\\gamma(Y)$.\n\nBy Proposition~\\ref{prop:consma}, $\\pi$ is small. It follows (cf.~\\cite[\\S6.2]{MR696691})\nthat the intersection cohomology sheaf $\\IC^\\bullet_S$ is the direct image of the intersection \ncohomology sheaf of $Y$. Since $Y$ is nonsingular, the latter is a shift of the constant\nsheaf. Therefore,\n\\[\n\\IC^\\bullet_S = R\\pi_* \\Qbb_Y[\\dim Y]\\saf.\n\\]\n(Concerning the shift, we follow the modern convention as in e.g., \n\\cite[Remark~4.2.4]{MR2525735}.)\nThe following formula for the stalk Euler characteristic of $\\IC^\\bullet_S$ at $z\\in S$\nis a consequence of standard properties of (derived) direct images:\n\\[\n\\chi_z(\\IC^\\bullet_S) = \n\\chi_z(R\\pi_* \\Qbb_Y[\\dim Y]) =\n(-1)^{\\dim S} \\sum_i (-1)^i \\dim H^i(\\pi^{-1}(z);\\Qbb) \n= (-1)^{\\dim S}\\chi(\\pi^{-1}(z))\\saf.\n\\]\nBy the first assertion in Theorem~\\ref{thm:mainthm}, this implies \n\\[\n\\chi_z(\\IC^\\bullet_S) = (-1)^{\\dim S}\\Eu_S(z)\\saf.\n\\]\nNow embed $S$ in a nonsingular variety $X$. \nThe characteristic cycle of $\\IC^\\bullet_S$ is\n\\[\nCC(\\chi_z(\\IC^\\bullet_S)) = CC((-1)^{\\dim S}\\Eu_S(z)) = T^*_SX\\saf,\n\\]\nand this verifies it is irreducible, completing the proof of \nTheorem~\\ref{thm:mainthm}.", "post_theorem_intro_text_len": 4013, "post_theorem_intro_text": "The definition of characteristic cycle will be recalled below (\\S\\ref{Lagrapf}).\n`Stringy' Chern classes were defined in~\\cite{MR2183846, MR2304329} \n(see~\\cite{MR2280127} for a lean account). If $\\pi: Y\\to S$ is a crepant resolution, \n$\\cstr(S)$ equals the push-forward of the total Chern class of the tangent bundle\nof~$Y$; the equality stated in the theorem is\n\\begin{equation}\\label{eq:strMa}\nf_*(c(TY)\\cap [Y]) = \\cma(S)\n\\end{equation}\nin the Chow group of $S$. In general, stringy Chern classes are defined for\nnormal varieties with a $\\Qbb$-Cartier canonical divisor and at worst log-terminal\nsingularities, and take value in the Chow group with rational coefficients. These\nadditional stipulations are not needed for the notion considered in this note; we can \nadopt the left-hand side of~\\eqref{eq:strMa} as the definition of stringy Chern class\nin the (integral) Chow group of $S$, compatibly with the more general definition.\nThe fact that the class is independent of the choice of trim resolution is\nalso a consequence of Theorem~\\ref{thm:mainthm}.\n\nWe note that Theorem~\\ref{thm:mainthm} implies that Batyrev's {\\em stringy Euler\nnumber\\/} (\\cite{MR2001j:14018}) of a variety admitting a trim resolution equals \nits Euler characteristic weighted by the local Euler obstruction.\n\nTheorem~\\ref{thm:mainthm} is a consequence of considerations concerning Sabbah's\nformalism of {\\em conical Lagrangian cycles,\\/} to which local Euler obstructions\nand Chern-Mather classes relate directly (see~\\S\\ref{Lagrapf}). Concerning intersection \ncohomology, recall that if $\\pi:Y \\to S$ is a small resolution, then the \nintersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant \nsheaf on~$Y$ (\\cite[\\S6.2]{MR696691}). We evaluate the corresponding push-forward \nat the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$. \nMore precisely, we prove (Proposition~\\ref{prop:Sabpf}) that, for trim resolutions, the \nLagrangian push-forward of the zero-section~$T^*_YY$ of the cotangent bundle \n$T^* Y$ equals the conormal cycle $T^*_S X$. The theorem follows from this \nmore basic observation, as we show in~\\S\\ref{thmpf}. \n\nThere is considerable interest in conditions implying that the characteristic cycle of\nthe intersection cohomology sheaf is irreducible. Lusztig (\\cite[13.7, p.~414]{MR1088333}) \nexpressed the `hope' that this may be the case for Schubert varieties in flag manifolds of \ntype A, D, E. A counterexample was constructed by Kashiwara and Saito for type A in \n$F\\ell(8)$ (\\cite{MR1458969}, \\cite{MR1896039}), while it holds in $F\\ell(n)$ for $n\\le 7$.\nIrreducibility is also known for all Schubert \nvarieties in the standard Grassmannian (\\cite{MR1084458}), and more generally \nfor all Schubert varieties in cominuscule Grassmannians of \ntypes A, D, and E, see~\\cite{MR1451256, MihalceaSingh}.\n\nWe hope that Theorem~\\ref{thm:mainthm} may help in streamlining such verifications. \nFor instance, since the Abel-Jacobi resolution of the theta divisor of a non-hyperelliptic \ncurve is trim, the irreducibility of the $\\IC$ characteristic cycle in this case (first established \nin \\cite{MR1642745}) follows directly from~Theorem~\\ref{thm:mainthm}.\n\nThe connection between the irreducibility of the $\\IC$ characteristic cycle and \nthe equality of Chern-Mather and stringy Chern classes was pointed out by B.~Jones\nin~\\cite[Remark 3.3.2]{MR2628830}, ultimately as an application of the microlocal \nindex formula of Dubson and Kashiwara. \nWe freely borrow ideas from~\\cite{MR2628830} \nin~\\S\\ref{thmpf}.\n\nA condition on fibers of small resolutions is considered in~\\cite{graham}, including a proof\nthat the condition implies the irreducibility of the IC characteristic cycles.\\medskip\n\n{\\em Acknowledgments.} This work was supported in part by an award from the Simons\nFoundation, SFI-MPS-TSM-00013681. The author also thanks David Massey and\nLeonardo Mihalcea for useful conversations, and Caltech for hospitality as most of this \nwork was carried out.", "sketch": "Theorem~\\ref{thm:mainthm} is said to be “a consequence of considerations concerning Sabbah's formalism of {\\em conical Lagrangian cycles,\\/}” (to which “local Euler obstructions and Chern-Mather classes relate directly”). For intersection cohomology, using that for a small resolution $\\pi:Y\\to S$ “the intersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant sheaf on~$Y$,” the paper “evaluate[s] the corresponding push-forward at the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$.” More precisely, it proves (Proposition~\\ref{prop:Sabpf}) that for trim resolutions “the Lagrangian push-forward of the zero-section~$T^*_Y Y$ of the cotangent bundle $T^*Y$ equals the conormal cycle $T^*_S X$.” “The theorem follows from this more basic observation,” as shown in~\\S\\ref{thmpf}.", "expanded_sketch": "To prove the main theorem, it is said to be “a consequence of considerations concerning Sabbah's formalism of {\\em conical Lagrangian cycles,\\/}” (to which “local Euler obstructions and Chern-Mather classes relate directly”). For intersection cohomology, using that for a small resolution $\\pi:Y\\to S$ “the intersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant sheaf on~$Y$,” the paper “evaluate[s] the corresponding push-forward at the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$.” More precisely, it proves the following proposition.\n\n\\begin{prop}\\label{prop:Sabpf}\nLet $f: Y\\to X$ be a proper morphism of nonsingular varieties, such that $Y \\to f(Y)$ is\na trim resolution. Then $f_* (T^*_Y Y)=T^*_{f(Y)} X$.\n\\end{prop}\n\nThat is, for trim resolutions “the Lagrangian push-forward of the zero-section~$T^*_Y Y$ of the cotangent bundle $T^*Y$ equals the conormal cycle $T^*_S X$.” “The theorem follows from this more basic observation,” as shown later.\n\\label{thmpf}\n\nLet $S$ be a variety admitting a trim resolution $\\pi: Y\\to S$; let $\\iota: S\\hookrightarrow X$\nbe an embedding in a nonsingular variety, and let $f=\\iota\\circ \\pi:Y \\to X$ be the compo", "expanded_theorem": "\\label{thm:mainthm}\nLet $S$ be a complex algebraic variety admitting a trim resolution. Then\n\\begin{itemize}\n\\item\nFor $s\\in S$, the local Euler obstruction $\\Eu_S(s)$ equals the Euler characteristic of\nthe fiber over $s$ in any trim resolution of $S$.\n\\item\nThe {\\em stringy\\/} Chern class of $S$ equals its Chern-Mather class:  \n$\\cstr(S)=\\cma(S)$.\n\\item\nThe characteristic cycle of the intersection cohomology sheaf of $S$ is irreducible.\n\\end{itemize}", "theorem_type": ["Universal", "Implication"], "mcq": {"question": "Let $S$ be a complex algebraic variety admitting a trim resolution, meaning that there exists a proper birational morphism $\\pi:Y\\to S$ with $Y$ nonsingular such that, for each $0\\le d\\le \\dim Y$, if $Y_d\\subseteq Y$ denotes the locus where the differential $d\\pi$ has rank $d$, then $\\dim Y_d<d$ for every $d<\\dim Y$. For such a variety, let $\\Eu_S:S\\to \\mathbb Z$ be the local Euler obstruction, and in this setting let the stringy Chern class be given by $\\cstr(S)=\\pi_*(c(TY)\\cap [Y])\\in A_*S$ for a trim resolution $\\pi$, while $\\cma(S)$ denotes the Chern-Mather class of $S$. Which of the following conclusions about $S$ holds under these hypotheses?", "correct_choice": {"label": "A", "text": "For every $s\\in S$, the local Euler obstruction satisfies $\\Eu_S(s)=\\chi(\\pi^{-1}(s))$ for the fiber over $s$ in any trim resolution $\\pi:Y\\to S$; moreover $\\cstr(S)=\\cma(S)$ in the Chow group of $S$, and the characteristic cycle of the intersection cohomology sheaf of $S$ is irreducible."}, "choices": [{"label": "B", "text": "For every $s\\in S$, the local Euler obstruction satisfies $\\Eu_S(s)=\\chi(\\pi^{-1}(s))$ for the fiber over $s$ in some trim resolution $\\pi:Y\\to S$; moreover $\\cstr(S)=\\cma(S)$ in the Chow group of $S$, and the characteristic cycle of the intersection cohomology sheaf of $S$ is supported on a single conormal component, though not necessarily irreducible."}, {"label": "C", "text": "For every $s\\in S$, the local Euler obstruction satisfies $\\Eu_S(s)=\\chi(\\pi^{-1}(s))$ for the fiber over $s$ in any trim resolution $\\pi:Y\\to S$, and moreover $\\cstr(S)=\\cma(S)$ in the Chow group of $S$."}, {"label": "D", "text": "For every $s\\in S$, the local Euler obstruction satisfies $\\Eu_S(s)=\\chi(\\pi^{-1}(s))$ for the fiber over $s$ in any small resolution $\\pi:Y\\to S$; moreover whenever $S$ admits a small resolution one has $\\cstr(S)=\\cma(S)$ in the Chow group of $S$, and the characteristic cycle of the intersection cohomology sheaf of $S$ is irreducible."}, {"label": "E", "text": "For every $s\\in S$, the local Euler obstruction satisfies $\\Eu_S(s)=\\chi(\\pi^{-1}(s))$ for the fiber over $s$ in any trim resolution $\\pi:Y\\to S$; moreover $\\cstr(S)=\\cma(S)$ in the Chow group of $S$, and the characteristic cycle of the constant sheaf on $S$ is irreducible."}], "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": "irreducibility_of_characteristic_cycle", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_irreducibility_of_IC_characteristic_cycle", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "trim_vs_small_hypothesis", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "IC_sheaf_replaced_by_constant_sheaf", "template_used": "property_confusion"}]}}
{"id": "2512.08275v1", "paper_link": "http://arxiv.org/abs/2512.08275v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{main theorem}\nLet $\\Omega\\subset\\mathbb{C}^{n}$, with $n>1$,  be a (possibly unbounded) pseudoconvex domain.  \nIf $\\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point, \\ then the Bergman metric of $\\Omega$ is well-defined in a nonempty open subset of $\\Omega$, denoted by $\\Omega^* $,  and the Bergman metric  cannot be  Einstein on any open subset of  $\\Omega^*$.", "start_pos": 7135, "end_pos": 7553, "label": "main theorem"}, "ref_dict": {"main theorem": "\\begin{theorem}\\label{main theorem}\nLet $\\Omega\\subset\\mathbb{C}^{n}$, with $n>1$,  be a (possibly unbounded) pseudoconvex domain.  \nIf $\\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point, \\ then the Bergman metric of $\\Omega$ is well-defined in a nonempty open subset of $\\Omega$, denoted by $\\Omega^* $,  and the Bergman metric  cannot be  Einstein on any open subset of  $\\Omega^*$.\n\\end{theorem}", "def1-9-24": "\\begin{definition}\\label{def1-9-24}\nLet $\\Omega$ be a possibly unbounded domain in $\\mathbb{C}^{n}$  with $n>1$ and let $p\\in\\partial\\Omega$.  \nWe say that $p$  is  a  strongly pseudoconvex polyhedral boundary  point if there exists a neighborhood $U$ of $p$ in $\\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\\rho_{1},\\dots,\\rho_{m}\\colon U\\to\\mathbb{R}$ with $m>1$  such that\n$\\Omega\\cap U=\\{z\\in U:\\rho_{1}(z)<0,\\dots,\\rho_{m}(z)<0\\}$  and \n$\\{\\partial\\rho_{1}|_p,\\dots,\\partial\\rho_{m}|_p\\}$  are linearly independent over $\\mathbb{C}$.\n\\end{definition}"}, "pre_theorem_intro_text_len": 2517, "pre_theorem_intro_text": "For any bounded domain $D\\subset \\mathbb{C}^n$, its Bergman metric is an  invariant Kähler metric.\nCheng and Yau \\cite{CY80} proved  that every bounded pseudoconvex domain in $\\mathbb{C}^n$ with a $C^2$-smooth boundary admits a unique complete K\\\"ahler–Einstein metric (up to a scaling factor)  which is also biholomorphically invariant.\nLater,  Mok and Yau \\cite{MY80}  removed the  boundary regularity assumption and proved the existence of such a metric for arbitrary bounded pseudoconvex domains.\n\nA natural problem arising from these works is to determine under what circumstances these two important  invariant metrics  coincide. A classical conjecture  of Yau \\cite{Yau} states that the Bergman metric of a bounded pseudoconvex domain is Einstein if and only if it is biholomorphic to a bounded homogeneous domain.  Earlier,\nCheng \\cite{C79}  had conjectured that the Bergman metric of a smoothly bounded strongly pseudoconvex domain is K\\\"ahler–Einstein if and only if the domain is biholomorphic to the unit ball.\nCheng's conjecture was confirmed in dimension two by Fu–Wong \\cite{FW97} and Nemirovski--Shafikov \\cite{NS06}, and was resolved in all dimensions by Huang–Xiao \\cite{HX16} based on earlier work of many authors.\nSubsequent generalizations were obtained for Stein manifolds and Stein spaces with compact strongly pseudoconvex boundaries; see Huang–Li \\cite{HL23}, Ebenfelt–Xiao–Xu \\cite{EXX22,EXX24}, and references therein.\nRelated variations of Cheng’s conjecture were  also discussed by S. Li in his papers \\cite{L1, L2, L3}.\n\nIn a more recent development, Savale and Xiao \\cite{SX23} investigated Bergman –Einstein metrics on smoothly bounded  pseudoconvex domains in $\\mathbb{C}^2$.\nThey proved that a smoothly bounded pseudoconvex domain of finite type in  $\\mathbb{C}^2$, whose Bergman metric is Einstein, must be biholomorphic to the unit ball in $\\mathbb{C}^2$.\nA prior result by Fu–Wong \\cite{FW97} established an analogous statement for smoothly bounded complete Reinhardt pseudoconvex domains of finite type in  $\\mathbb{C}^2$.\n\nDespite these advances, not much is known about the Einstein property of Bergman metrics on unbounded pseudoconvex domains or in bounded pseudoconvex domains with  rough boundary points. In this paper, we aim to conduct a study along these lines.\nWe will show that the Bergman metric of a pseudoconvex domain, possibly unbounded, which possesses a strongly pseudoconvex polyhedral boundary point (as defined in Definition \\ref{def1-9-24}) is not Einstein.", "context": "For any bounded domain $D\\subset \\mathbb{C}^n$, its Bergman metric is an invariant Kähler metric.\nCheng and Yau \\cite{CY80} proved that every bounded pseudoconvex domain in $\\mathbb{C}^n$ with a $C^2$-smooth boundary admits a unique complete K\\\"ahler–Einstein metric (up to a scaling factor) which is also biholomorphically invariant.\nLater, Mok and Yau \\cite{MY80} removed the boundary regularity assumption and proved the existence of such a metric for arbitrary bounded pseudoconvex domains.\n\nA natural problem arising from these works is to determine under what circumstances these two important invariant metrics coincide. A classical conjecture of Yau \\cite{Yau} states that the Bergman metric of a bounded pseudoconvex domain is Einstein if and only if it is biholomorphic to a bounded homogeneous domain. Earlier,\nCheng \\cite{C79} had conjectured that the Bergman metric of a smoothly bounded strongly pseudoconvex domain is K\\\"ahler–Einstein if and only if the domain is biholomorphic to the unit ball.\nCheng's conjecture was confirmed in dimension two by Fu–Wong \\cite{FW97} and Nemirovski--Shafikov \\cite{NS06}, and was resolved in all dimensions by Huang–Xiao \\cite{HX16} based on earlier work of many authors.\nSubsequent generalizations were obtained for Stein manifolds and Stein spaces with compact strongly pseudoconvex boundaries; see Huang–Li \\cite{HL23}, Ebenfelt–Xiao–Xu \\cite{EXX22,EXX24}, and references therein.\nRelated variations of Cheng’s conjecture were also discussed by S. Li in his papers \\cite{L1, L2, L3}.\n\nIn a more recent development, Savale and Xiao \\cite{SX23} investigated Bergman –Einstein metrics on smoothly bounded pseudoconvex domains in $\\mathbb{C}^2$.\nThey proved that a smoothly bounded pseudoconvex domain of finite type in $\\mathbb{C}^2$, whose Bergman metric is Einstein, must be biholomorphic to the unit ball in $\\mathbb{C}^2$.\nA prior result by Fu–Wong \\cite{FW97} established an analogous statement for smoothly bounded complete Reinhardt pseudoconvex domains of finite type in $\\mathbb{C}^2$.\n\nDespite these advances, not much is known about the Einstein property of Bergman metrics on unbounded pseudoconvex domains or in bounded pseudoconvex domains with rough boundary points. In this paper, we aim to conduct a study along these lines.\nWe will show that the Bergman metric of a pseudoconvex domain, possibly unbounded, which possesses a strongly pseudoconvex polyhedral boundary point (as defined in Definition \\ref{def1-9-24}) is not Einstein.\n\n\\begin{definition}\\label{def1-9-24}\nLet $\\Omega$ be a possibly unbounded domain in $\\mathbb{C}^{n}$  with $n>1$ and let $p\\in\\partial\\Omega$.  \nWe say that $p$  is  a  strongly pseudoconvex polyhedral boundary  point if there exists a neighborhood $U$ of $p$ in $\\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\\rho_{1},\\dots,\\rho_{m}\\colon U\\to\\mathbb{R}$ with $m>1$  such that\n$\\Omega\\cap U=\\{z\\in U:\\rho_{1}(z)<0,\\dots,\\rho_{m}(z)<0\\}$  and \n$\\{\\partial\\rho_{1}|_p,\\dots,\\partial\\rho_{m}|_p\\}$  are linearly independent over $\\mathbb{C}$.\n\\end{definition}", "full_context": "For any bounded domain $D\\subset \\mathbb{C}^n$, its Bergman metric is an invariant Kähler metric.\nCheng and Yau \\cite{CY80} proved that every bounded pseudoconvex domain in $\\mathbb{C}^n$ with a $C^2$-smooth boundary admits a unique complete K\\\"ahler–Einstein metric (up to a scaling factor) which is also biholomorphically invariant.\nLater, Mok and Yau \\cite{MY80} removed the boundary regularity assumption and proved the existence of such a metric for arbitrary bounded pseudoconvex domains.\n\nA natural problem arising from these works is to determine under what circumstances these two important invariant metrics coincide. A classical conjecture of Yau \\cite{Yau} states that the Bergman metric of a bounded pseudoconvex domain is Einstein if and only if it is biholomorphic to a bounded homogeneous domain. Earlier,\nCheng \\cite{C79} had conjectured that the Bergman metric of a smoothly bounded strongly pseudoconvex domain is K\\\"ahler–Einstein if and only if the domain is biholomorphic to the unit ball.\nCheng's conjecture was confirmed in dimension two by Fu–Wong \\cite{FW97} and Nemirovski--Shafikov \\cite{NS06}, and was resolved in all dimensions by Huang–Xiao \\cite{HX16} based on earlier work of many authors.\nSubsequent generalizations were obtained for Stein manifolds and Stein spaces with compact strongly pseudoconvex boundaries; see Huang–Li \\cite{HL23}, Ebenfelt–Xiao–Xu \\cite{EXX22,EXX24}, and references therein.\nRelated variations of Cheng’s conjecture were also discussed by S. Li in his papers \\cite{L1, L2, L3}.\n\nIn a more recent development, Savale and Xiao \\cite{SX23} investigated Bergman –Einstein metrics on smoothly bounded pseudoconvex domains in $\\mathbb{C}^2$.\nThey proved that a smoothly bounded pseudoconvex domain of finite type in $\\mathbb{C}^2$, whose Bergman metric is Einstein, must be biholomorphic to the unit ball in $\\mathbb{C}^2$.\nA prior result by Fu–Wong \\cite{FW97} established an analogous statement for smoothly bounded complete Reinhardt pseudoconvex domains of finite type in $\\mathbb{C}^2$.\n\nDespite these advances, not much is known about the Einstein property of Bergman metrics on unbounded pseudoconvex domains or in bounded pseudoconvex domains with rough boundary points. In this paper, we aim to conduct a study along these lines.\nWe will show that the Bergman metric of a pseudoconvex domain, possibly unbounded, which possesses a strongly pseudoconvex polyhedral boundary point (as defined in Definition \\ref{def1-9-24}) is not Einstein.\n\n\\begin{definition}\\label{def1-9-24}\nLet $\\Omega$ be a possibly unbounded domain in $\\mathbb{C}^{n}$  with $n>1$ and let $p\\in\\partial\\Omega$.  \nWe say that $p$  is  a  strongly pseudoconvex polyhedral boundary  point if there exists a neighborhood $U$ of $p$ in $\\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\\rho_{1},\\dots,\\rho_{m}\\colon U\\to\\mathbb{R}$ with $m>1$  such that\n$\\Omega\\cap U=\\{z\\in U:\\rho_{1}(z)<0,\\dots,\\rho_{m}(z)<0\\}$  and \n$\\{\\partial\\rho_{1}|_p,\\dots,\\partial\\rho_{m}|_p\\}$  are linearly independent over $\\mathbb{C}$.\n\\end{definition}\n\nOne of the main tools used in the proof of Theorem \\ref{main theorem} is the rescaling argument, which has been used to work on many related \nproblems. In particular, in connection with our present work, we \nmention the papers by Wong \\cite{W77},\nKim \\cite{Kim}, Kim-Yu \\cite{KY}, Krantz-Yu \\cite{KYu}\nand Boas-Straube-Yu \\cite{BSY}, where the rescaling method has been used to study the boundary limit of various quantities associated with the Bergman metric. Indeed, our current work has benefited from their studies. A recent application of the rescaling method can also be found in Huang-Zhu \\cite{HZ}, where it is employed in solving a CR transversality problem. Another recent application of the rescaling method was used in working on the pinched properties of a K\\\"ahler metric is a recent paper of Bracci-Gauthier-Zimmer \\cite{BGZ}.\n\nWe next recall the definition of strongly pseudoconvex polyhedral boundary points for a domain $\\Omega\\subset{\\mathbb C}^n$.\n\\begin{definition}\\label{def1-9-24}\nLet $\\Omega$ be a possibly unbounded domain in $\\mathbb{C}^{n}$ with $n>1$ and let $p\\in\\partial\\Omega$. \nWe say that $p$ is a strongly pseudoconvex polyhedral boundary point if there exists a neighborhood $U$ of $p$ in $\\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\\rho_{1},\\dots,\\rho_{m}\\colon U\\to\\mathbb{R}$ with $m>1$ such that\n$\\Omega\\cap U=\\{z\\in U:\\rho_{1}(z)<0,\\dots,\\rho_{m}(z)<0\\}$ and \n$\\{\\partial\\rho_{1}|_p,\\dots,\\partial\\rho_{m}|_p\\}$ are linearly independent over $\\mathbb{C}$.\n\\end{definition}\n\n\\begin{proposition}\\label{prop1-4-3}\nLet $\\Omega$ be a possibly unbounded pseudoconvex domain in $\\mathbb C^n$ which is stronlgy pseudoconvex polyhedral at some boundary point $p\\in\\partial\\Omega$. Let $U$ be a neighborhood of $p$ such that $U\\cap\\Omega$ is connected, and on which the Bergman metric $g_{\\Omega}$ is well defined. Then its Bergman metric $g_\\Omega$ is K\\\"ahler-Einstein on $U\\cap\\Omega$ if and only if its Bergman canonical invariant $J_{\\Omega}\\equiv (n+1)^n\\frac{\\pi^n}{n!}$ on $U\\cap\\Omega$. \n\\end{proposition}\n\\begin{proof}\nBy Corollary \\ref{coro1-3-30}, for any smooth boundary point $q\\in U\\cap\\partial\\Omega$ near which $\\partial\\Omega$ is strongly pseudoconvex, one has \n \\begin{equation}\\label{e2-3-31}\n \\lim_{z\\rightarrow q}J_{\\Omega}(z)=\\frac{(n+1)^n\\pi^n}{n!}.\n \\end{equation}\n The Bergman metric $g_{\\Omega}$ {is} K\\\"ahler-Einstein if its Ricci curvature $R_{\\Omega}=c g_{\\Omega}$ for some constant $c$. By Corollary \\ref{coro1-3-30}, one has $c=-1$. Consequently, the K\\\"ahler-Einstein assumption implies that $\\log J_{\\Omega}$ is a pluriharmonic function on $U\\cap\\Omega$. Now, for any attached holomorphic disk $\\phi:\\Delta\\rightarrow\\Omega$ where $\\phi$ is holomorphic in $\\Delta:=\\{t\\in {\\mathbb C}: |t|<1\\}$, continuous up to $\\overline {\\Delta}$, and $\\phi(\\partial\\Delta)$ is contained in the smooth part of $ U\\cap \\partial\\Omega$, we have that $\\log\n J_{\\Omega}(\\phi(t))$ is harmonic. Since it is constant on the strongly pseudoconvex part of the boundary by \\eqref{e2-3-31}, it assumes the value\n\\[\n\\log\\frac{(n+1)^n\\pi^n}{n!}\n\\]\neverywhere on $\\Delta$. Now, since $\\partial \\Omega$ is strongly pseudoconvex near $q$, the union of such disks fills up an open subset of $\\partial\\Omega$ near $q$. Since $\\log\n J_{\\Omega}$ is well defined in $U\\cap\\Omega$ on which it is real analytic, we conclude that $\\log J_{\\Omega}\\equiv\n \\log\\frac{(n+1)^n\\pi^n}{n!}$ over $U\\cap\\Omega$ as $U\\cap\\Omega$ is connected by definition. \nConversely, if $J_\\Omega(z)$ takes a constant value near $p$, then the Bergman metric is obviously K\\\"ahler-Einstien. Thus, we have the conclusion of the proposition.\n\\end{proof}\n\\begin{remark}\\label{remark1-10-21}\n\nNote that the zero set of the Bergman kernel function, denoted by $E$, is a complex analytic variety in $\\Omega$. \nThus, $J_\\Omega$ is a well-defined real-analytic function on $\\Omega\\setminus E$. Since $\\Omega\\setminus E$ is connected, $J_\\Omega$ is constant if and only if it is constant on some nonempty open subset of $\\Omega$. In particular, when $\\Omega$ contains a $C^2$-smooth strongly pseudoconvex boundary point, the Bergman metric of the domain $\\Omega$ is Kähler–Einstein wherever it is well-defined if and only if $J_\\Omega=c$ is a constant on a certain open subset of $\\Omega\\setminus E$ .\n In this case, $c=\\frac{(n+1)^n\\pi^n}{n!}$, and the Bergman space $A^2(\\Omega)$ separates holomorphic directions at any point in $\\Omega\\setminus E$ and thus the Bergman metric is well-defined in $\\Omega\\setminus E$.\n\\end{remark}\n\\section{Stability of the Bergman kernels}\\label{sec2}\n\n\\begin{proof}[Proof of Theorem \\ref{main theorem}]\nLet $\\Omega$ be a pseudoconvex domain which is strongly pseudoconvex polyhedral at a boundary point $p\\in\\partial\\Omega$ as defined in Definition~\\ref{def1-9-24}. After shrinking $U$ if necessary, we may assume that\n\\begin{enumerate}\n\\item[(i)] there are $C^{2}$-smooth strongly plurisubharmonic functions $\\{\\rho_{j}\\}_{j=1}^{m}$ with $m>1$ on $U$ such that\n\\begin{equation}\\label{9-24-a2}\n U\\cap\\Omega=\\{z\\in U:\\rho_{1}(z)<0,\\dots,\\rho_{m}(z)<0\\};\n\\end{equation}\n\\item[(ii)] the vectors $\\partial\\rho_{1}(q),\\dots,\\partial\\rho_{m}(q)$ are linearly independent over $\\mathbb C$ for $q\\in U\\cap \\overline\\Omega$.\n\\end{enumerate}\nBy Lemma \\ref{25-9-23a1}, after a suitable change of coordinates on a small neighborhood $V\\Subset U$ of $p$, we may assume that $p=0$ and \n\\begin{equation}\\label{9-24-a1}\nV\\cap\\Omega=\\{(z_1,\\cdots, z_n)\\in V: {\\rm Im~ }z_1>\\Phi_1(z, \\overline z), \\cdots, {\\rm Im}~ z_m>\\Phi_m(z, \\overline z)\\}\n\\end{equation}\nwhere $$\\Phi_1(z, \\overline z)=|z|^2+R_1(z), \\Phi_j(z, \\overline z)=\\sum a^j_{\\alpha\\overline\\beta} z_{\\alpha}\\overline z_{\\beta}+R_j(z), \\quad 2\\leq j\\leq m$$\nwith each remainder $R_j=\\mathcal O(|z|^3)$. Write $$U_0=\\{z\\in\\mathbb C^n: |z_j|<\\varepsilon_0, 1\\leq j\\leq n\\}$$ with $\\varepsilon_0\\ll 1$ such that $U_0\\Subset V$ and on $U_0$ one has\n$$|R_j(z)|\\leq \\frac{A_0}{2}|z|^2, \\forall z\\in U_0, 1\\leq j\\leq m,$$\nwhere $A_0$ is defined in (\\ref{10-27-a3}).\nWe first construct a bounded continuous plurisubharmonic function $\\psi$ in $\\Omega$ where $\\psi$ is strictly plurisubharmonic near $p=0$ as follows: \n\\begin{lemma}\\label{le3-28-1}\nAfter shrinking $U_0$ if necessary, there exists a plurisubharmonic function $\\psi:\\Omega\\rightarrow (-\\infty, 0)$ such that $$\\psi (z)>-c_0, ~\\left(\\frac{\\partial^2\\psi(z)}{\\partial z_j\\partial\\overline z_k}\\right)\\geq c ~I_{n}, ~~z\\in U_0\\cap\\Omega$$for some constants $c_0>0,~c>0$.\n\\end{lemma}\n\\begin{proof}\nLet $(V, z)$ be the coordinates given as in (\\ref{9-24-a1}) and set $\\varphi=\\frac{A_0}4|z|^2-y_1$. Then $\\varphi$ is strictly plurisubharmonic on $V$ and satisfies $\\varphi(0)=0, \\varphi(z)<0$ when $z\\in U_0\\cap\\overline\\Omega\\setminus \\{0\\}$. Take $r>0$ such that $\\overline {\\mathbb B^n(0, r)}\\Subset U_0$. Set $M=\\max\\{\\varphi(z): z\\in \\partial \\mathbb B^n(0, r)\\cap\\overline \\Omega\\}$. Then $M<0$. Now we define $\\psi$ as follows:\n\\begin{equation}\n \\psi=\\begin{cases}\n \\max\\{\\varphi(z), M\\}, &z\\in \\mathbb B^n(0, r)\\cap\\Omega\\\\\n M, &z\\in\\Omega\\setminus \\mathbb B^n(0, r).\n \\end{cases}\n\\end{equation}\nThen $\\psi$ is a bounded and continuous plurisubharmonic function on $\\Omega$ with $$\\psi(0)=0, ~~\\psi(z)<0, \\forall z\\in\\overline\\Omega\\setminus\\{0\\}.$$\nFurthermore, $\\psi$ is equal to $\\varphi$ near $0$ with $$\\left(\\frac{\\partial^2\\psi}{\\partial z_i\\partial\\overline z_j}\\right)= \\frac{A_0}4 I_n $$\nin some neighborhood $U$ of $0$ in $\\mathbb C^n$. Moreover, $\\psi>-c_0$ on $U$ for some positive constant $c_0$.\n\\end{proof}", "post_theorem_intro_text_len": 2280, "post_theorem_intro_text": "\\begin{corollary}\nLet $\\Omega\\subset\\mathbb{C}^{n}$, with $n>1$,  be a bounded pseudoconvex domain.  If $\\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point,then its  Bergman metric cannot be Einstein.\n\\end{corollary}\n\nOne of the main tools used in the proof of Theorem \\ref{main theorem} is the rescaling argument, which has been  used  to work on many related \nproblems. In particular, in connection with our present work, we \nmention   the papers by Wong \\cite{W77},\nKim \\cite{Kim},  Kim-Yu \\cite{KY}, Krantz-Yu \\cite{KYu}\nand Boas-Straube-Yu \\cite{BSY},  where the rescaling method has been used to study the boundary limit of  various quantities associated with  the Bergman metric. Indeed, our current work  has benefited   from  their studies.  A recent application of the rescaling method can also be found in Huang-Zhu \\cite{HZ}, where it is employed in solving a CR transversality problem. Another recent  application of the rescaling method was used in working on the pinched  properties of a K\\\"ahler metric is a recent  paper of Bracci-Gauthier-Zimmer \\cite{BGZ}.\n\nThe  ideas of our proof of the main theorem can be stated briefly as follows: First, we   show that if the Bergman metric of our domain is Einstein, then its Bergman invariant function is constant and equals that of the unit ball. Next we carefully construct a special sequence of points approaching a strongly pseudoconvex polyhedral boundary point  tangentially such that the limit domain is equivalent to the product of a ball and a bidisk of lower dimension.  \nTo obtain such a sequence, we assign weight $2$ to one of the complex normal directions, weight $1.5$  to the other normal directions, and weight $1$  to the remaining CR directions. The main part of the paper is then devoted to showing that the Bergman invariant function of this product domain coincides with that of  $\\Omega$.\nA direct computation shows that the Bergman invariant function of the unit ball differs from that of the aforementioned  product domain, leading to a contradiction.  In this respect, our proof departs from earlier approaches to the Cheng conjecture and its generalizations, where the contradiction is derived via spherical CR geometry and the Qi-Keng Lu uniformization theorem \\cite{HX20}.", "sketch": "The ideas of the proof of Theorem~\\ref{main theorem} are stated as follows. One main tool is a “rescaling argument.” First, “we show that if the Bergman metric of our domain is Einstein, then its Bergman invariant function is constant and equals that of the unit ball.” Next, “we carefully construct a special sequence of points approaching a strongly pseudoconvex polyhedral boundary point tangentially such that the limit domain is equivalent to the product of a ball and a bidisk of lower dimension.” To obtain such a sequence, “we assign weight $2$ to one of the complex normal directions, weight $1.5$ to the other normal directions, and weight $1$ to the remaining CR directions.” The main part then shows “that the Bergman invariant function of this product domain coincides with that of $\\Omega$.” Finally, “a direct computation shows that the Bergman invariant function of the unit ball differs from that of the aforementioned product domain,” yielding “a contradiction.”", "expanded_sketch": "The ideas of the proof, in establishing the main theorem, are stated as follows. One main tool is a “rescaling argument.” First, “we show that if the Bergman metric of our domain is Einstein, then its Bergman invariant function is constant and equals that of the unit ball.” Next, “we carefully construct a special sequence of points approaching a strongly pseudoconvex polyhedral boundary point tangentially such that the limit domain is equivalent to the product of a ball and a bidisk of lower dimension.” To obtain such a sequence, “we assign weight $2$ to one of the complex normal directions, weight $1.5$ to the other normal directions, and weight $1$ to the remaining CR directions.” The main part then shows “that the Bergman invariant function of this product domain coincides with that of $\\Omega$.” Finally, “a direct computation shows that the Bergman invariant function of the unit ball differs from that of the aforementioned product domain,” yielding “a contradiction.”", "expanded_theorem": "\\label{main theorem}\nLet $\\Omega\\subset\\mathbb{C}^{n}$, with $n>1$,  be a (possibly unbounded) pseudoconvex domain.  \nIf $\\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point, \\ then the Bergman metric of $\\Omega$ is well-defined in a nonempty open subset of $\\Omega$, denoted by $\\Omega^* $,  and the Bergman metric  cannot be  Einstein on any open subset of  $\\Omega^*$.", "theorem_type": ["Implication", "Nonexistence"], "mcq": {"question": "Let \\(\\Omega\\subset \\mathbb C^n\\) with \\(n>1\\) be a pseudoconvex domain, possibly unbounded. Assume that \\(\\Omega\\) has a strongly pseudoconvex polyhedral boundary point \\(p\\in \\partial\\Omega\\), meaning that there exist a neighborhood \\(U\\) of \\(p\\) in \\(\\mathbb C^n\\) and \\(C^2\\)-smooth strongly plurisubharmonic functions \\(\\rho_1,\\dots,\\rho_m:U\\to\\mathbb R\\) with \\(m>1\\) such that\n\\[\n\\Omega\\cap U=\\{z\\in U: \\rho_1(z)<0,\\dots,\\rho_m(z)<0\\},\n\\]\nand the complex differentials \\(\\{\\partial\\rho_1|_p,\\dots,\\partial\\rho_m|_p\\}\\) are linearly independent over \\(\\mathbb C\\). Under these assumptions, which statement about the Bergman metric of \\(\\Omega\\) holds?", "correct_choice": {"label": "A", "text": "There exists a nonempty open subset \\(\\Omega^*\\subset \\Omega\\) on which the Bergman metric of \\(\\Omega\\) is well-defined, and the Bergman metric cannot be Einstein on any open subset of \\(\\Omega^*\\)."}, "choices": [{"label": "B", "text": "There exists a nonempty open subset \\(\\Omega^*\\subset \\Omega\\) on which the Bergman metric of \\(\\Omega\\) is well-defined, and if the Bergman metric is Einstein on some open subset of \\(\\Omega^*\\), then \\(\\Omega\\) is biholomorphic to the unit ball."}, {"label": "C", "text": "There exists a nonempty open subset \\(\\Omega^*\\subset \\Omega\\) on which the Bergman metric of \\(\\Omega\\) is well-defined."}, {"label": "D", "text": "There exists a nonempty open subset \\(\\Omega^*\\subset \\Omega\\) on which the Bergman metric of \\(\\Omega\\) is well-defined, and the Bergman metric is not Einstein on some nonempty open subset of \\(\\Omega^*\\)."}, {"label": "E", "text": "There exists a nonempty open subset \\(\\Omega^*\\subset \\Omega\\) on which the Bergman metric of \\(\\Omega\\) is well-defined, and the Bergman invariant function on \\(\\Omega^*\\) is constant and equal to that of the unit ball."}], "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": "conclusion-from-contradiction replaced by global classification", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped the non-Einstein conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "quantifier on where Einstein fails", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "computational_check", "tampered_component": "constant-ball invariant asserted unconditionally instead of under Einstein hypothesis", "template_used": "wildcard"}]}}
{"id": "2512.08275v1", "paper_link": "http://arxiv.org/abs/2512.08275v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{main theorem}\nLet $\\Omega\\subset\\mathbb{C}^{n}$, with $n>1$,  be a (possibly unbounded) pseudoconvex domain.  \nIf $\\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point, \\ then the Bergman metric of $\\Omega$ is well-defined in a nonempty open subset of $\\Omega$, denoted by $\\Omega^* $,  and the Bergman metric  cannot be  Einstein on any open subset of  $\\Omega^*$.", "start_pos": 7135, "end_pos": 7553, "label": "main theorem"}, "ref_dict": {"main theorem": "\\begin{theorem}\\label{main theorem}\nLet $\\Omega\\subset\\mathbb{C}^{n}$, with $n>1$,  be a (possibly unbounded) pseudoconvex domain.  \nIf $\\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point, \\ then the Bergman metric of $\\Omega$ is well-defined in a nonempty open subset of $\\Omega$, denoted by $\\Omega^* $,  and the Bergman metric  cannot be  Einstein on any open subset of  $\\Omega^*$.\n\\end{theorem}", "def1-9-24": "\\begin{definition}\\label{def1-9-24}\nLet $\\Omega$ be a possibly unbounded domain in $\\mathbb{C}^{n}$  with $n>1$ and let $p\\in\\partial\\Omega$.  \nWe say that $p$  is  a  strongly pseudoconvex polyhedral boundary  point if there exists a neighborhood $U$ of $p$ in $\\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\\rho_{1},\\dots,\\rho_{m}\\colon U\\to\\mathbb{R}$ with $m>1$  such that\n$\\Omega\\cap U=\\{z\\in U:\\rho_{1}(z)<0,\\dots,\\rho_{m}(z)<0\\}$  and \n$\\{\\partial\\rho_{1}|_p,\\dots,\\partial\\rho_{m}|_p\\}$  are linearly independent over $\\mathbb{C}$.\n\\end{definition}"}, "pre_theorem_intro_text_len": 2517, "pre_theorem_intro_text": "For any bounded domain $D\\subset \\mathbb{C}^n$, its Bergman metric is an  invariant Kähler metric.\nCheng and Yau \\cite{CY80} proved  that every bounded pseudoconvex domain in $\\mathbb{C}^n$ with a $C^2$-smooth boundary admits a unique complete K\\\"ahler–Einstein metric (up to a scaling factor)  which is also biholomorphically invariant.\nLater,  Mok and Yau \\cite{MY80}  removed the  boundary regularity assumption and proved the existence of such a metric for arbitrary bounded pseudoconvex domains.\n\nA natural problem arising from these works is to determine under what circumstances these two important  invariant metrics  coincide. A classical conjecture  of Yau \\cite{Yau} states that the Bergman metric of a bounded pseudoconvex domain is Einstein if and only if it is biholomorphic to a bounded homogeneous domain.  Earlier,\nCheng \\cite{C79}  had conjectured that the Bergman metric of a smoothly bounded strongly pseudoconvex domain is K\\\"ahler–Einstein if and only if the domain is biholomorphic to the unit ball.\nCheng's conjecture was confirmed in dimension two by Fu–Wong \\cite{FW97} and Nemirovski--Shafikov \\cite{NS06}, and was resolved in all dimensions by Huang–Xiao \\cite{HX16} based on earlier work of many authors.\nSubsequent generalizations were obtained for Stein manifolds and Stein spaces with compact strongly pseudoconvex boundaries; see Huang–Li \\cite{HL23}, Ebenfelt–Xiao–Xu \\cite{EXX22,EXX24}, and references therein.\nRelated variations of Cheng’s conjecture were  also discussed by S. Li in his papers \\cite{L1, L2, L3}.\n\nIn a more recent development, Savale and Xiao \\cite{SX23} investigated Bergman –Einstein metrics on smoothly bounded  pseudoconvex domains in $\\mathbb{C}^2$.\nThey proved that a smoothly bounded pseudoconvex domain of finite type in  $\\mathbb{C}^2$, whose Bergman metric is Einstein, must be biholomorphic to the unit ball in $\\mathbb{C}^2$.\nA prior result by Fu–Wong \\cite{FW97} established an analogous statement for smoothly bounded complete Reinhardt pseudoconvex domains of finite type in  $\\mathbb{C}^2$.\n\nDespite these advances, not much is known about the Einstein property of Bergman metrics on unbounded pseudoconvex domains or in bounded pseudoconvex domains with  rough boundary points. In this paper, we aim to conduct a study along these lines.\nWe will show that the Bergman metric of a pseudoconvex domain, possibly unbounded, which possesses a strongly pseudoconvex polyhedral boundary point (as defined in Definition \\ref{def1-9-24}) is not Einstein.", "context": "For any bounded domain $D\\subset \\mathbb{C}^n$, its Bergman metric is an invariant Kähler metric.\nCheng and Yau \\cite{CY80} proved that every bounded pseudoconvex domain in $\\mathbb{C}^n$ with a $C^2$-smooth boundary admits a unique complete K\\\"ahler–Einstein metric (up to a scaling factor) which is also biholomorphically invariant.\nLater, Mok and Yau \\cite{MY80} removed the boundary regularity assumption and proved the existence of such a metric for arbitrary bounded pseudoconvex domains.\n\nA natural problem arising from these works is to determine under what circumstances these two important invariant metrics coincide. A classical conjecture of Yau \\cite{Yau} states that the Bergman metric of a bounded pseudoconvex domain is Einstein if and only if it is biholomorphic to a bounded homogeneous domain. Earlier,\nCheng \\cite{C79} had conjectured that the Bergman metric of a smoothly bounded strongly pseudoconvex domain is K\\\"ahler–Einstein if and only if the domain is biholomorphic to the unit ball.\nCheng's conjecture was confirmed in dimension two by Fu–Wong \\cite{FW97} and Nemirovski--Shafikov \\cite{NS06}, and was resolved in all dimensions by Huang–Xiao \\cite{HX16} based on earlier work of many authors.\nSubsequent generalizations were obtained for Stein manifolds and Stein spaces with compact strongly pseudoconvex boundaries; see Huang–Li \\cite{HL23}, Ebenfelt–Xiao–Xu \\cite{EXX22,EXX24}, and references therein.\nRelated variations of Cheng’s conjecture were also discussed by S. Li in his papers \\cite{L1, L2, L3}.\n\nIn a more recent development, Savale and Xiao \\cite{SX23} investigated Bergman –Einstein metrics on smoothly bounded pseudoconvex domains in $\\mathbb{C}^2$.\nThey proved that a smoothly bounded pseudoconvex domain of finite type in $\\mathbb{C}^2$, whose Bergman metric is Einstein, must be biholomorphic to the unit ball in $\\mathbb{C}^2$.\nA prior result by Fu–Wong \\cite{FW97} established an analogous statement for smoothly bounded complete Reinhardt pseudoconvex domains of finite type in $\\mathbb{C}^2$.\n\nDespite these advances, not much is known about the Einstein property of Bergman metrics on unbounded pseudoconvex domains or in bounded pseudoconvex domains with rough boundary points. In this paper, we aim to conduct a study along these lines.\nWe will show that the Bergman metric of a pseudoconvex domain, possibly unbounded, which possesses a strongly pseudoconvex polyhedral boundary point (as defined in Definition \\ref{def1-9-24}) is not Einstein.\n\n\\begin{definition}\\label{def1-9-24}\nLet $\\Omega$ be a possibly unbounded domain in $\\mathbb{C}^{n}$  with $n>1$ and let $p\\in\\partial\\Omega$.  \nWe say that $p$  is  a  strongly pseudoconvex polyhedral boundary  point if there exists a neighborhood $U$ of $p$ in $\\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\\rho_{1},\\dots,\\rho_{m}\\colon U\\to\\mathbb{R}$ with $m>1$  such that\n$\\Omega\\cap U=\\{z\\in U:\\rho_{1}(z)<0,\\dots,\\rho_{m}(z)<0\\}$  and \n$\\{\\partial\\rho_{1}|_p,\\dots,\\partial\\rho_{m}|_p\\}$  are linearly independent over $\\mathbb{C}$.\n\\end{definition}", "full_context": "For any bounded domain $D\\subset \\mathbb{C}^n$, its Bergman metric is an invariant Kähler metric.\nCheng and Yau \\cite{CY80} proved that every bounded pseudoconvex domain in $\\mathbb{C}^n$ with a $C^2$-smooth boundary admits a unique complete K\\\"ahler–Einstein metric (up to a scaling factor) which is also biholomorphically invariant.\nLater, Mok and Yau \\cite{MY80} removed the boundary regularity assumption and proved the existence of such a metric for arbitrary bounded pseudoconvex domains.\n\nA natural problem arising from these works is to determine under what circumstances these two important invariant metrics coincide. A classical conjecture of Yau \\cite{Yau} states that the Bergman metric of a bounded pseudoconvex domain is Einstein if and only if it is biholomorphic to a bounded homogeneous domain. Earlier,\nCheng \\cite{C79} had conjectured that the Bergman metric of a smoothly bounded strongly pseudoconvex domain is K\\\"ahler–Einstein if and only if the domain is biholomorphic to the unit ball.\nCheng's conjecture was confirmed in dimension two by Fu–Wong \\cite{FW97} and Nemirovski--Shafikov \\cite{NS06}, and was resolved in all dimensions by Huang–Xiao \\cite{HX16} based on earlier work of many authors.\nSubsequent generalizations were obtained for Stein manifolds and Stein spaces with compact strongly pseudoconvex boundaries; see Huang–Li \\cite{HL23}, Ebenfelt–Xiao–Xu \\cite{EXX22,EXX24}, and references therein.\nRelated variations of Cheng’s conjecture were also discussed by S. Li in his papers \\cite{L1, L2, L3}.\n\nIn a more recent development, Savale and Xiao \\cite{SX23} investigated Bergman –Einstein metrics on smoothly bounded pseudoconvex domains in $\\mathbb{C}^2$.\nThey proved that a smoothly bounded pseudoconvex domain of finite type in $\\mathbb{C}^2$, whose Bergman metric is Einstein, must be biholomorphic to the unit ball in $\\mathbb{C}^2$.\nA prior result by Fu–Wong \\cite{FW97} established an analogous statement for smoothly bounded complete Reinhardt pseudoconvex domains of finite type in $\\mathbb{C}^2$.\n\nDespite these advances, not much is known about the Einstein property of Bergman metrics on unbounded pseudoconvex domains or in bounded pseudoconvex domains with rough boundary points. In this paper, we aim to conduct a study along these lines.\nWe will show that the Bergman metric of a pseudoconvex domain, possibly unbounded, which possesses a strongly pseudoconvex polyhedral boundary point (as defined in Definition \\ref{def1-9-24}) is not Einstein.\n\n\\begin{definition}\\label{def1-9-24}\nLet $\\Omega$ be a possibly unbounded domain in $\\mathbb{C}^{n}$  with $n>1$ and let $p\\in\\partial\\Omega$.  \nWe say that $p$  is  a  strongly pseudoconvex polyhedral boundary  point if there exists a neighborhood $U$ of $p$ in $\\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\\rho_{1},\\dots,\\rho_{m}\\colon U\\to\\mathbb{R}$ with $m>1$  such that\n$\\Omega\\cap U=\\{z\\in U:\\rho_{1}(z)<0,\\dots,\\rho_{m}(z)<0\\}$  and \n$\\{\\partial\\rho_{1}|_p,\\dots,\\partial\\rho_{m}|_p\\}$  are linearly independent over $\\mathbb{C}$.\n\\end{definition}\n\nOne of the main tools used in the proof of Theorem \\ref{main theorem} is the rescaling argument, which has been used to work on many related \nproblems. In particular, in connection with our present work, we \nmention the papers by Wong \\cite{W77},\nKim \\cite{Kim}, Kim-Yu \\cite{KY}, Krantz-Yu \\cite{KYu}\nand Boas-Straube-Yu \\cite{BSY}, where the rescaling method has been used to study the boundary limit of various quantities associated with the Bergman metric. Indeed, our current work has benefited from their studies. A recent application of the rescaling method can also be found in Huang-Zhu \\cite{HZ}, where it is employed in solving a CR transversality problem. Another recent application of the rescaling method was used in working on the pinched properties of a K\\\"ahler metric is a recent paper of Bracci-Gauthier-Zimmer \\cite{BGZ}.\n\nWe next recall the definition of strongly pseudoconvex polyhedral boundary points for a domain $\\Omega\\subset{\\mathbb C}^n$.\n\\begin{definition}\\label{def1-9-24}\nLet $\\Omega$ be a possibly unbounded domain in $\\mathbb{C}^{n}$ with $n>1$ and let $p\\in\\partial\\Omega$. \nWe say that $p$ is a strongly pseudoconvex polyhedral boundary point if there exists a neighborhood $U$ of $p$ in $\\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\\rho_{1},\\dots,\\rho_{m}\\colon U\\to\\mathbb{R}$ with $m>1$ such that\n$\\Omega\\cap U=\\{z\\in U:\\rho_{1}(z)<0,\\dots,\\rho_{m}(z)<0\\}$ and \n$\\{\\partial\\rho_{1}|_p,\\dots,\\partial\\rho_{m}|_p\\}$ are linearly independent over $\\mathbb{C}$.\n\\end{definition}\n\n\\begin{proposition}\\label{prop1-4-3}\nLet $\\Omega$ be a possibly unbounded pseudoconvex domain in $\\mathbb C^n$ which is stronlgy pseudoconvex polyhedral at some boundary point $p\\in\\partial\\Omega$. Let $U$ be a neighborhood of $p$ such that $U\\cap\\Omega$ is connected, and on which the Bergman metric $g_{\\Omega}$ is well defined. Then its Bergman metric $g_\\Omega$ is K\\\"ahler-Einstein on $U\\cap\\Omega$ if and only if its Bergman canonical invariant $J_{\\Omega}\\equiv (n+1)^n\\frac{\\pi^n}{n!}$ on $U\\cap\\Omega$. \n\\end{proposition}\n\\begin{proof}\nBy Corollary \\ref{coro1-3-30}, for any smooth boundary point $q\\in U\\cap\\partial\\Omega$ near which $\\partial\\Omega$ is strongly pseudoconvex, one has \n \\begin{equation}\\label{e2-3-31}\n \\lim_{z\\rightarrow q}J_{\\Omega}(z)=\\frac{(n+1)^n\\pi^n}{n!}.\n \\end{equation}\n The Bergman metric $g_{\\Omega}$ {is} K\\\"ahler-Einstein if its Ricci curvature $R_{\\Omega}=c g_{\\Omega}$ for some constant $c$. By Corollary \\ref{coro1-3-30}, one has $c=-1$. Consequently, the K\\\"ahler-Einstein assumption implies that $\\log J_{\\Omega}$ is a pluriharmonic function on $U\\cap\\Omega$. Now, for any attached holomorphic disk $\\phi:\\Delta\\rightarrow\\Omega$ where $\\phi$ is holomorphic in $\\Delta:=\\{t\\in {\\mathbb C}: |t|<1\\}$, continuous up to $\\overline {\\Delta}$, and $\\phi(\\partial\\Delta)$ is contained in the smooth part of $ U\\cap \\partial\\Omega$, we have that $\\log\n J_{\\Omega}(\\phi(t))$ is harmonic. Since it is constant on the strongly pseudoconvex part of the boundary by \\eqref{e2-3-31}, it assumes the value\n\\[\n\\log\\frac{(n+1)^n\\pi^n}{n!}\n\\]\neverywhere on $\\Delta$. Now, since $\\partial \\Omega$ is strongly pseudoconvex near $q$, the union of such disks fills up an open subset of $\\partial\\Omega$ near $q$. Since $\\log\n J_{\\Omega}$ is well defined in $U\\cap\\Omega$ on which it is real analytic, we conclude that $\\log J_{\\Omega}\\equiv\n \\log\\frac{(n+1)^n\\pi^n}{n!}$ over $U\\cap\\Omega$ as $U\\cap\\Omega$ is connected by definition. \nConversely, if $J_\\Omega(z)$ takes a constant value near $p$, then the Bergman metric is obviously K\\\"ahler-Einstien. Thus, we have the conclusion of the proposition.\n\\end{proof}\n\\begin{remark}\\label{remark1-10-21}\n\nNote that the zero set of the Bergman kernel function, denoted by $E$, is a complex analytic variety in $\\Omega$. \nThus, $J_\\Omega$ is a well-defined real-analytic function on $\\Omega\\setminus E$. Since $\\Omega\\setminus E$ is connected, $J_\\Omega$ is constant if and only if it is constant on some nonempty open subset of $\\Omega$. In particular, when $\\Omega$ contains a $C^2$-smooth strongly pseudoconvex boundary point, the Bergman metric of the domain $\\Omega$ is Kähler–Einstein wherever it is well-defined if and only if $J_\\Omega=c$ is a constant on a certain open subset of $\\Omega\\setminus E$ .\n In this case, $c=\\frac{(n+1)^n\\pi^n}{n!}$, and the Bergman space $A^2(\\Omega)$ separates holomorphic directions at any point in $\\Omega\\setminus E$ and thus the Bergman metric is well-defined in $\\Omega\\setminus E$.\n\\end{remark}\n\\section{Stability of the Bergman kernels}\\label{sec2}\n\n\\begin{proof}[Proof of Theorem \\ref{main theorem}]\nLet $\\Omega$ be a pseudoconvex domain which is strongly pseudoconvex polyhedral at a boundary point $p\\in\\partial\\Omega$ as defined in Definition~\\ref{def1-9-24}. After shrinking $U$ if necessary, we may assume that\n\\begin{enumerate}\n\\item[(i)] there are $C^{2}$-smooth strongly plurisubharmonic functions $\\{\\rho_{j}\\}_{j=1}^{m}$ with $m>1$ on $U$ such that\n\\begin{equation}\\label{9-24-a2}\n U\\cap\\Omega=\\{z\\in U:\\rho_{1}(z)<0,\\dots,\\rho_{m}(z)<0\\};\n\\end{equation}\n\\item[(ii)] the vectors $\\partial\\rho_{1}(q),\\dots,\\partial\\rho_{m}(q)$ are linearly independent over $\\mathbb C$ for $q\\in U\\cap \\overline\\Omega$.\n\\end{enumerate}\nBy Lemma \\ref{25-9-23a1}, after a suitable change of coordinates on a small neighborhood $V\\Subset U$ of $p$, we may assume that $p=0$ and \n\\begin{equation}\\label{9-24-a1}\nV\\cap\\Omega=\\{(z_1,\\cdots, z_n)\\in V: {\\rm Im~ }z_1>\\Phi_1(z, \\overline z), \\cdots, {\\rm Im}~ z_m>\\Phi_m(z, \\overline z)\\}\n\\end{equation}\nwhere $$\\Phi_1(z, \\overline z)=|z|^2+R_1(z), \\Phi_j(z, \\overline z)=\\sum a^j_{\\alpha\\overline\\beta} z_{\\alpha}\\overline z_{\\beta}+R_j(z), \\quad 2\\leq j\\leq m$$\nwith each remainder $R_j=\\mathcal O(|z|^3)$. Write $$U_0=\\{z\\in\\mathbb C^n: |z_j|<\\varepsilon_0, 1\\leq j\\leq n\\}$$ with $\\varepsilon_0\\ll 1$ such that $U_0\\Subset V$ and on $U_0$ one has\n$$|R_j(z)|\\leq \\frac{A_0}{2}|z|^2, \\forall z\\in U_0, 1\\leq j\\leq m,$$\nwhere $A_0$ is defined in (\\ref{10-27-a3}).\nWe first construct a bounded continuous plurisubharmonic function $\\psi$ in $\\Omega$ where $\\psi$ is strictly plurisubharmonic near $p=0$ as follows: \n\\begin{lemma}\\label{le3-28-1}\nAfter shrinking $U_0$ if necessary, there exists a plurisubharmonic function $\\psi:\\Omega\\rightarrow (-\\infty, 0)$ such that $$\\psi (z)>-c_0, ~\\left(\\frac{\\partial^2\\psi(z)}{\\partial z_j\\partial\\overline z_k}\\right)\\geq c ~I_{n}, ~~z\\in U_0\\cap\\Omega$$for some constants $c_0>0,~c>0$.\n\\end{lemma}\n\\begin{proof}\nLet $(V, z)$ be the coordinates given as in (\\ref{9-24-a1}) and set $\\varphi=\\frac{A_0}4|z|^2-y_1$. Then $\\varphi$ is strictly plurisubharmonic on $V$ and satisfies $\\varphi(0)=0, \\varphi(z)<0$ when $z\\in U_0\\cap\\overline\\Omega\\setminus \\{0\\}$. Take $r>0$ such that $\\overline {\\mathbb B^n(0, r)}\\Subset U_0$. Set $M=\\max\\{\\varphi(z): z\\in \\partial \\mathbb B^n(0, r)\\cap\\overline \\Omega\\}$. Then $M<0$. Now we define $\\psi$ as follows:\n\\begin{equation}\n \\psi=\\begin{cases}\n \\max\\{\\varphi(z), M\\}, &z\\in \\mathbb B^n(0, r)\\cap\\Omega\\\\\n M, &z\\in\\Omega\\setminus \\mathbb B^n(0, r).\n \\end{cases}\n\\end{equation}\nThen $\\psi$ is a bounded and continuous plurisubharmonic function on $\\Omega$ with $$\\psi(0)=0, ~~\\psi(z)<0, \\forall z\\in\\overline\\Omega\\setminus\\{0\\}.$$\nFurthermore, $\\psi$ is equal to $\\varphi$ near $0$ with $$\\left(\\frac{\\partial^2\\psi}{\\partial z_i\\partial\\overline z_j}\\right)= \\frac{A_0}4 I_n $$\nin some neighborhood $U$ of $0$ in $\\mathbb C^n$. Moreover, $\\psi>-c_0$ on $U$ for some positive constant $c_0$.\n\\end{proof}", "post_theorem_intro_text_len": 2280, "post_theorem_intro_text": "\\begin{corollary}\nLet $\\Omega\\subset\\mathbb{C}^{n}$, with $n>1$,  be a bounded pseudoconvex domain.  If $\\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point,then its  Bergman metric cannot be Einstein.\n\\end{corollary}\n\nOne of the main tools used in the proof of Theorem \\ref{main theorem} is the rescaling argument, which has been  used  to work on many related \nproblems. In particular, in connection with our present work, we \nmention   the papers by Wong \\cite{W77},\nKim \\cite{Kim},  Kim-Yu \\cite{KY}, Krantz-Yu \\cite{KYu}\nand Boas-Straube-Yu \\cite{BSY},  where the rescaling method has been used to study the boundary limit of  various quantities associated with  the Bergman metric. Indeed, our current work  has benefited   from  their studies.  A recent application of the rescaling method can also be found in Huang-Zhu \\cite{HZ}, where it is employed in solving a CR transversality problem. Another recent  application of the rescaling method was used in working on the pinched  properties of a K\\\"ahler metric is a recent  paper of Bracci-Gauthier-Zimmer \\cite{BGZ}.\n\nThe  ideas of our proof of the main theorem can be stated briefly as follows: First, we   show that if the Bergman metric of our domain is Einstein, then its Bergman invariant function is constant and equals that of the unit ball. Next we carefully construct a special sequence of points approaching a strongly pseudoconvex polyhedral boundary point  tangentially such that the limit domain is equivalent to the product of a ball and a bidisk of lower dimension.  \nTo obtain such a sequence, we assign weight $2$ to one of the complex normal directions, weight $1.5$  to the other normal directions, and weight $1$  to the remaining CR directions. The main part of the paper is then devoted to showing that the Bergman invariant function of this product domain coincides with that of  $\\Omega$.\nA direct computation shows that the Bergman invariant function of the unit ball differs from that of the aforementioned  product domain, leading to a contradiction.  In this respect, our proof departs from earlier approaches to the Cheng conjecture and its generalizations, where the contradiction is derived via spherical CR geometry and the Qi-Keng Lu uniformization theorem \\cite{HX20}.", "sketch": "The ideas of the proof of Theorem~\\ref{main theorem} are stated as follows. One main tool is a “rescaling argument.” First, “we show that if the Bergman metric of our domain is Einstein, then its Bergman invariant function is constant and equals that of the unit ball.” Next, “we carefully construct a special sequence of points approaching a strongly pseudoconvex polyhedral boundary point tangentially such that the limit domain is equivalent to the product of a ball and a bidisk of lower dimension.” To obtain such a sequence, “we assign weight $2$ to one of the complex normal directions, weight $1.5$ to the other normal directions, and weight $1$ to the remaining CR directions.” The main part then shows “that the Bergman invariant function of this product domain coincides with that of $\\Omega$.” Finally, “a direct computation shows that the Bergman invariant function of the unit ball differs from that of the aforementioned product domain,” yielding “a contradiction.”", "expanded_sketch": "The ideas of the proof, in establishing the main theorem, are stated as follows. One main tool is a “rescaling argument.” First, “we show that if the Bergman metric of our domain is Einstein, then its Bergman invariant function is constant and equals that of the unit ball.” Next, “we carefully construct a special sequence of points approaching a strongly pseudoconvex polyhedral boundary point tangentially such that the limit domain is equivalent to the product of a ball and a bidisk of lower dimension.” To obtain such a sequence, “we assign weight $2$ to one of the complex normal directions, weight $1.5$ to the other normal directions, and weight $1$ to the remaining CR directions.” The main part then shows “that the Bergman invariant function of this product domain coincides with that of $\\Omega$.” Finally, “a direct computation shows that the Bergman invariant function of the unit ball differs from that of the aforementioned product domain,” yielding “a contradiction.”", "expanded_theorem": "\\label{main theorem}\nLet $\\Omega\\subset\\mathbb{C}^{n}$, with $n>1$,  be a (possibly unbounded) pseudoconvex domain.  \nIf $\\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point, \\ then the Bergman metric of $\\Omega$ is well-defined in a nonempty open subset of $\\Omega$, denoted by $\\Omega^* $,  and the Bergman metric  cannot be  Einstein on any open subset of  $\\Omega^*$.", "theorem_type": ["Implication", "Nonexistence"], "mcq": {"question": "Let \\(\\Omega\\subset \\mathbb{C}^n\\) with \\(n>1\\) be a pseudoconvex domain, possibly unbounded. Suppose \\(\\Omega\\) has a strongly pseudoconvex polyhedral boundary point \\(p\\in \\partial\\Omega\\), meaning that there exist a neighborhood \\(U\\) of \\(p\\) in \\(\\mathbb C^n\\) and \\(C^2\\)-smooth strongly plurisubharmonic functions \\(\\rho_1,\\dots,\\rho_m:U\\to\\mathbb R\\) with \\(m>1\\) such that\n\\[\n\\Omega\\cap U=\\{z\\in U: \\rho_1(z)<0,\\dots,\\rho_m(z)<0\\},\n\\]\nand the complex differentials \\(\\partial \\rho_1|_p,\\dots,\\partial \\rho_m|_p\\) are linearly independent over \\(\\mathbb C\\). The Bergman metric is the invariant K\\\"ahler metric associated to \\(\\Omega\\) when it is defined, and a K\\\"ahler metric is called Einstein if its Ricci form is a constant multiple of the metric. Which statement holds for \\(\\Omega\\)?", "correct_choice": {"label": "A", "text": "There exists a nonempty open subset \\(\\Omega^*\\subset \\Omega\\) on which the Bergman metric of \\(\\Omega\\) is well defined, and that Bergman metric is not Einstein on any open subset of \\(\\Omega^*\\)."}, "choices": [{"label": "B", "text": "There exists a nonempty open subset \\(\\Omega^*\\subset \\Omega\\) on which the Bergman metric of \\(\\Omega\\) is well defined, and if that Bergman metric is Einstein on some open subset of \\(\\Omega^*\\), then \\(\\Omega\\) is biholomorphic to the unit ball."}, {"label": "C", "text": "There exists a nonempty open subset \\(\\Omega^*\\subset \\Omega\\) on which the Bergman metric of \\(\\Omega\\) is well defined."}, {"label": "D", "text": "If the Bergman metric of \\(\\Omega\\) is well defined on a nonempty open subset \\(\\Omega^*\\subset \\Omega\\), then there exists an open subset of \\(\\Omega^*\\) on which the Bergman metric is Einstein."}, {"label": "E", "text": "There exists a nonempty open subset \\(\\Omega^*\\subset \\Omega\\) on which the Bergman metric of \\(\\Omega\\) is well defined, and its Bergman invariant function agrees on \\(\\Omega^*\\) with that of the unit ball."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "computational_check", "tampered_component": "contradiction_target_replaced_by_ball_rigidity", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped_non_Einstein_conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "trace_identity", "tampered_component": "quantifier_reversal_on_Einstein_locus", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "computational_check", "tampered_component": "ball_invariant_function_equality_without_Einstein_hypothesis", "template_used": "wildcard"}]}}
{"id": "2512.08562v1", "paper_link": "http://arxiv.org/abs/2512.08562v1", "theorems_cnt": 6, "theorem": {"env_name": "theorem", "content": "[\\textbf{Dynamical stability of $n$-solitons}]\\label{thm1.1}\nGiven $ n\\in \\mathbb{N} $, $ n\\geq 1 $, a collection of wave speeds $\\textbf{c} = (c_{1},\\dots,c_{n})$ with $ 0 < c_{1} < \\dots < c_{n} $ and a collection of space transitions $\\textbf{x} = (x_{1},\\dots,x_{n})\\in \\mathbb{R}^{n}$, let $ U^{(n)}(\\cdot,\\cdot;\\textbf{c},\\textbf{x}) $ be the corresponding $n$-soliton solutions of (\\ref{ILW}). Then for any $ \\epsilon > 0 $, there exists $ \\delta > 0 $ such that for any $ u_{0}\\in H^{\\frac{n}{2}}(\\mathbb{R}) $, the following stability property holds, if\n\n\\begin{align}\n\t&\\bigl\\lVert u_{0} - U^{(n)}(0,\\cdot;\\textbf{c},\\textbf{x}) \\bigr\\rVert_{H^{\\frac{n}{2}}} < \\delta,\n\\end{align}\nthen for any $t \\in \\mathbb{R}$, the corresponding solution of (\\ref{ILW}) verifies\n\\begin{align}\n\\inf_{\\tau \\in \\mathbb{R},\\  \\textbf{y} \\in \\mathbb{R}^{n}} \\bigl\\lVert u(t) - U^{(n)}(\\tau,\\cdot;\\textbf{c},\\textbf{y}) \\bigr\\rVert_{H^{\\frac{n}{2}}} < \\epsilon.\n\\end{align}", "start_pos": 22335, "end_pos": 23327, "label": "thm1.1"}, "ref_dict": {"1.9": "\\begin{align}\n\t\\langle f_{12},g_{12}\\rangle\n\t&:= \\int_{\\mathbb{R}^{2}}f_{12}g_{12}^{*} \\mathrm{d}x_{1}\\mathrm{d}x_{2},\\label{1.9}\n\\end{align}", "thm1.1": "\\begin{theorem}[\\textbf{Dynamical stability of $n$-solitons}]\\label{thm1.1}\nGiven $ n\\in \\mathbb{N} $, $ n\\geq 1 $, a collection of wave speeds $\\textbf{c} = (c_{1},\\dots,c_{n})$ with $ 0 < c_{1} < \\dots < c_{n} $ and a collection of space transitions $\\textbf{x} = (x_{1},\\dots,x_{n})\\in \\mathbb{R}^{n}$, let $ U^{(n)}(\\cdot,\\cdot;\\textbf{c},\\textbf{x}) $ be the corresponding $n$-soliton solutions of (\\ref{ILW}). Then for any $ \\epsilon > 0 $, there exists $ \\delta > 0 $ such that for any $ u_{0}\\in H^{\\frac{n}{2}}(\\mathbb{R}) $, the following stability property holds, if\n\n\\begin{align}\n\t&\\bigl\\lVert u_{0} - U^{(n)}(0,\\cdot;\\textbf{c},\\textbf{x}) \\bigr\\rVert_{H^{\\frac{n}{2}}} < \\delta,\n\\end{align}\nthen for any $t \\in \\mathbb{R}$, the corresponding solution of (\\ref{ILW}) verifies\n\\begin{align}\n\\inf_{\\tau \\in \\mathbb{R},\\  \\textbf{y} \\in \\mathbb{R}^{n}} \\bigl\\lVert u(t) - U^{(n)}(\\tau,\\cdot;\\textbf{c},\\textbf{y}) \\bigr\\rVert_{H^{\\frac{n}{2}}} < \\epsilon.\n\\end{align}\n\\end{theorem}", "th1.3": "\\begin{theorem}[\\textbf{The negative eigenvalues of $\\mathcal{L}_n$}]\\label{th1.3}\n\tThe linearized operator $\\mathcal{L}_{n}$ about the $n$-soliton solution possesses exactly $\\left\\lfloor\\frac{n+1}{2}\\right\\rfloor$ negative eigenvalues $\\lambda_{k}$, where $k=1,2,\\cdots,\\left\\lfloor\\frac{n+1}{2}\\right\\rfloor$ and $\\lfloor x\\rfloor$ denotes the floor function. Furthermore, the negative eigenvalues of $\\mathcal{L}_n(n\\geq 1$)  have the following expression\n\t\\begin{align}\\label{negetive}\n\t\t\\lambda_{k} = -M^\\delta_k (c_{2k-1}-\\frac{1}{\\delta})\\prod_{ j\\neq 2k-1}^{n}(c_{j}-c_{2k-1}),\\quad k=1,2,\\cdots,\\left\\lfloor\\frac{n+1}{2}\\right\\rfloor,\n\t\\end{align}\n\twhere $M^\\delta_k$ are positive constants independent of the wave speeds $c_1,\\cdots, c_n$.\n\\end{theorem}", "thm1.2": "\\begin{theorem}[\\textbf{Orbital stability of double solitons}]\\label{thm1.2}\n\tThe double solitons $U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)$ of ILW equation (\\ref{ILW}) with $0<c_1<c_2$ are orbitally stable in $H^1(\\mathbb{R})$ in the following sense.\nThere exist parameters $\\epsilon_0$ and $A_0$, depending on $c_1$ and $c_2$, if there exists $\\epsilon \\in (0,\\epsilon_0)$ such that for any $u_0\\in H^1(\\mathbb{R})$,\n\\begin{align}\\label{u02}\n\t\\bigl\\lVert u_0 - U^{(2)}_{c_1,c_2}(0;0,0) \\bigr\\rVert_{H^1(\\mathbb{R})} < \\epsilon,\n\\end{align}\nthen there exist $x_1(t), x_2(t) \\in \\mathbb{R}$, such that the corresponding solution $u(t,x)$ of the ILW equation (\\ref{ILW}) with the initial data $u(0)=u_0$ satisfies $u_t \\in C([0,\\infty),H^1(\\mathbb{R}))$ and\n\\begin{align}\\label{1.22}\n\t\\sup_{t\\in (0,\\infty)}|\\!|u(t)-U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)|\\!|_{H^1(\\mathbb{R})}<A_0\\epsilon,\n\\end{align}\nwith\n\\begin{align}\n\t\\sup_{t\\in (0,\\infty)}(|x_1'(t)|+|x_2'(t)|)\\leq CA_0\\epsilon.\n\\end{align}\n\\end{theorem}", "1.15": "\\begin{equation}\n\t\\mathcal{R}_{12} := (\\mathcal{J}_{12}^{(1)})^{-1}\\mathcal{J}_{12}^{(2)},\\quad\n\t\\mathcal{R}_{12}^{\\star} := \\mathcal{J}_{12}^{(2)}(\\mathcal{J}_{12}^{(1)})^{-1} = i\\theta_{12}^{-}T^\\delta_{12} - \\theta_{12}^{+},\\label{1.15}\n\\end{equation}", "ILW": "\\begin{align}\n\tu_t+\\frac{1}{\\delta}u_x+2uu_x+T^\\delta u_{xx}=0,\\quad  (t,x)\\in \\mathbb{R} \\times \\mathbb{R},\\label{ILW}\n\\end{align}", "T1": "\\begin{align}\n\tv = \\frac{3}{\\delta} u\\left( \\frac{3}{\\delta}t,x\\right), \\label{T1}\n\\end{align}", "lagrange": "\\begin{align}\n\t\\mathcal{S}_{n}(u) = H_{n+1}(u) + \\sum_{m=1}^{n} \\mu_{m} H_{m}(u),\\label{lagrange}\n\\end{align}", "1.16": "\\begin{align}\n\tu(x,t)=Q_c(x-ct-x_0), \\quad Q_c(s)=\\frac{a\\sin(a\\delta)}{\\cosh(as)+\\cos(a\\delta)},\\quad c>0,~ x_0\\in \\mathbb{R}, \\label{1.16}\n\\end{align}"}, "pre_theorem_intro_text_len": 16852, "pre_theorem_intro_text": "\\label{sec.1}\nIn this work, we consider the stability of $n$-soliton solutions for the intermediate long wave (ILW) equation\\cite{J,KAS,KKD}\n\\begin{align}\n\tu_t+\\frac{1}{\\delta}u_x+2uu_x+T^\\delta u_{xx}=0,\\quad  (t,x)\\in \\mathbb{R} \\times \\mathbb{R},\\label{ILW}\n\\end{align}\nwhere $u=u(t,x)\\in\\mathbb{R}$ is a real-valued function, $T^\\delta $ is the inverse of Tilbert  transform $T$ with a large scale parameter $\\delta>0$. These two operators are defined as\n\\begin{align}\n\t(T^\\delta f)(x)&=\\frac{1}{2\\delta}\\text{P.V.}\\int_{\\mathbb{R}}\\coth \\frac{\\pi (y-x)}{2\\delta}f(y)dy,\\\\\\nonumber\n\t\t(Tf)(x)&=\\frac{1}{2\\delta}\\text{P.V.}\\int_{\\mathbb{R}}\\mathrm{cosech} \\frac{\\pi (y-x)}{2\\delta}f(y)dy,\n\\end{align}\nand P.V. indicates that the integral is to be computed in the principle value sense. Moreover, $T^\\delta$ is a zero Fourier multiplier, in the sense that $\\partial_x T^\\delta$ is the multiplier with symbol\n\\begin{align}\n\t\\sigma(\\partial_x T^\\delta)=\\widehat{\\partial_x T^\\delta}=-2\\pi\\xi\\coth(2\\pi\\delta \\xi).\n\\end{align}\n\nThe ILW equation (\\ref{ILW}) describes weakly nonlinear internal wave propagation in stratified fluids of finite depth $\\delta$. First derived by Kubota, Ko and Dobbs \\cite{KKD}, this model captures essential nonlinear and dispersive characteristics of wave dynamics. Joseph \\cite{J} subsequently formalized the derivation of the ILW equation by integrating the linear dispersion relation from \\cite{P} within Whitham's nonlocal framework \\cite{W}, and explicitly constructed its solitary wave solution. This represents a key advancement in establishing the integrability of the equation. Furthermore, the ILW equation itself arises in the context of the two-layer internal wave system \\cite{BLS, CGK}. A rigorous derivation of its system form in one or two spatial dimensions, including the case with a free upper surface, can be found in \\cite{Xu}.\n\nIn addition, the ILW equation (\\ref{ILW}) establishes a fundamental connection between two canonical models in water wave theory: the Korteweg-de Vries (KdV) equation for shallow water and the Benjamin-Ono (BO) equation for deep water \\cite{ABFS, AFS,HKO,SAF}, with shared initial data in appropriate Sobolev spaces. Specifically, the asymptotic behavior of the ILW solution is governed by the depth parameter $\\delta$. Let $u(t,x)$ be the solution of the ILW equation (\\ref{ILW}). Then, under the scaling transformation\n\\begin{align}\n\tv = \\frac{3}{\\delta} u\\left( \\frac{3}{\\delta}t,x\\right), \\label{T1}\n\\end{align}\nthe rescaled function $v$ converges (in a suitable functional sense, typically distributional or weak convergence) to the solution of the KdV equation\n\\begin{align}\n\tu_t + 2uu_x + \\frac{\\delta}{3}u_{xxx} = 0, \\label{KdV}\n\\end{align}\nin the shallow-water limit ($\\delta \\to 0$) and in the deep-water limit ($\\delta \\to \\infty$), (\\ref{T1}) will converge to the the solution of the BO equation\n\\begin{align}\n\tu_t + 2uu_x + Hu_{xx} = 0, \\label{BO}\n\\end{align}\nwhere $H$ denotes the Hilbert transform, defined by the principal value integral\n\\begin{align*}\n\t(Hf)(x)=\\frac{1}{\\pi}\\text{P.V.}\\int_{\\mathbb{R}}\\frac{f(y)}{y-x}\\mathrm{d}y.\n\\end{align*}\n\nAs a fundamental integrable system \\cite{AC,FA,KS}, the ILW equation (\\ref{ILW}) possesses an infinite-dimensional completely integrable Hamiltonian structure \\cite{KAS, Sa}. This integrability is manifested in the existence of an infinite hierarchy of conservation laws and a Lax-pair formulation.\nA key distinction between the ILW equation and the KdV equation lies in their dispersion operators \\cite{MPS,Sa,Xu}. Specifically, the ILW equation incorporates a singular nonlocal integro-differential operator $T^\\delta$, which fundamentally alters the asymptotic properties of its solutions. In particular, the soliton solutions of the ILW equation exhibit algebraic decay, in contrast to the exponential decay characteristic of KdV solitons. This difference in decay rates represents a key qualitative difference between the two models. Formally, the following quantities are conserved along the flow of the ILW equation \\cite{LR}\n\\begin{align}\n\tH_0(u)&=:\\int_{\\mathbb{R}}u\\text{d}x, \\\\\n    H_1(u)&=:\\frac{1}{2}\\int_{\\mathbb{R}}u^2\\text{d}x, \\\\\n\tH_2(u)&=:-\\int_{\\mathbb{R}}\\frac{1}{3}u^3+\\frac{1}{2}uT^\\delta u_x +\\frac{u^2}{2\\delta}\\text{d}x,\\\\\n\tH_3(u)&=:\\int_{\\mathbb{R}}\\frac{1}{4}u^4+\\frac{3}{4}u^2T^\\delta u_x+\\frac{3}{8}(T^\\delta u_x)^2+\\frac{1}{3\\delta}u^3+\\frac{1}{2\\delta}uT^\\delta u_x+\\frac{u^2}{8\\delta^2}\\text{d}x.\n\\end{align}\n\nIt is noteworthy that the energy space for the ILW equation, defined as the domain of the Hamiltonian $H_2(u)$, is $H^{\\frac{1}{2}}(\\mathbb{R})$.\nThis functional framework provides the natural setting for investigating solution properties.\nIn the context of weak solutions, Ginibre and Velo \\cite{GV} established the existence of global solutions satisfying\n$u \\in L^\\infty(\\mathbb{R}, H^1(\\mathbb{R})) \\cap L^2_{\\mathrm{loc}}(\\mathbb{R}, H^{\\frac{3}{2}}_{\\mathrm{loc}}(\\mathbb{R}))$.\nFor strong solutions, the first global well-posedness result in $H^s(\\mathbb{R})$ was obtained by \\cite{ABFS} for $s > \\frac{3}{2}$.\nSubsequent developments progressively lowered the regularity threshold: Molinet and Vento \\cite{MV} extended well-posedness to $s > \\frac{1}{2}$\n(where unconditional uniqueness holds), while Molinet, Pilod, and Vento \\cite{MPV} advanced it to $s > \\frac{1}{4}$.\nA significant breakthrough in the sharp low-regularity theory was achieved by Ifrim and Saut \\cite{IS}, who proved global well-posedness\nin $L^2(\\mathbb{R})$ for small initial data. Most recently, Gassot and Laurens \\cite{GAL} established sharp global well-posedness\nfor the ILW equation in $H^s(\\mathbb{T})$ for all $s > -\\frac{1}{2}$, thereby\ncompleting the well-posedness theory on the torus. Additionally, they proved the\ncontinuous convergence of solutions to those of the Benjamin-Ono equation in the\ndeep-water limit $\\delta \\to \\infty$.\n\nBased upon the facts above, the (\\ref{ILW}) equation can naturally be viewed as a Hamiltonian system of the form \\cite{KSA, LR}\n\\begin{align}\n\tu_t=\\mathcal{J}\\frac{\\delta H_2(u)}{\\delta u},\n\\end{align}\nwhere $\\mathcal{J}=\\partial_x$, and $\\frac{\\delta H_2(u)}{\\delta u}$ (subsequently simplified to $H_2'(u)$) refers to the variational derivative of $H_2(u)$ can be written as\n\\begin{align*}\n\t(\\frac{\\partial}{\\partial \\epsilon}H_2(u+\\epsilon f))\\Big|_{\\epsilon=0}=\\int_{\\mathbb{R}}\\frac{\\delta H_{2}}{\\delta u}(x)f(x)\\text{d}x.\n\\end{align*}\n\nMoreover, the ILW equation possesses a bi-Hamiltonian structure \\cite{Sa}. However, unlike the KdV equation, the bi-Hamiltonian structure of the ILW equation is highly nontrivial due to its formulation involving both the spatial derivative operator $\\partial_x$ and the singular integral operator $T^\\delta$. This operator composition aligns the ILW equation with structural features characteristic of completely integrable systems in two spatial dimensions.\n\nTo formalize this connection, let subscripts $1, 2$ denote dependence on the independent variables $x:= x_1$ and $x_2$. For arbitrary functions $f_{12}$ and $g_{12}$, we define the associated bilinear form by\n\\begin{align}\n\t\\langle f_{12},g_{12}\\rangle\n\t&:= \\int_{\\mathbb{R}^{2}}f_{12}g_{12}^{*} \\mathrm{d}x_{1}\\mathrm{d}x_{2},\\label{1.9}\n\\end{align}\nwhere the asterisk superscript $^*$ denotes complex conjugation. Define operators in $L^{2}(\\mathbb{R}^{2},\\mathbb{C})$ (with domain $H^{1}(\\mathbb{R}^{2},\\mathbb{C})$)\n\\begin{align}\n\t\\theta_{12}^{\\pm}\n\t&:= u_{1}\\pm u_{2} + i(\\partial_{x_{1}} \\mp \\partial_{x_{2}}), \\quad u_{j}=u(x_{j},t), \\quad j=1,2.\\label{1.10}\n\\end{align}\nThen the two compatible Hamiltonian operators for the ILW equation are given by\n\\begin{align}\n\t\\mathcal{J}_{12}^{(1)}\n\t:= \\theta_{12}^{-}, \\quad\n\t\\mathcal{J}_{12}^{(2)}\n\t:= (i\\theta_{12}^{-}T^\\delta_{12} - \\theta_{12}^{+})\\theta_{12}^{-},\n\\end{align}\nwhere $T^\\delta_{12}$ is an extended operator defined as\n\\begin{align}\n\t(T^\\delta_{12}f_{12})(x_{1},x_{2}) =\\frac{1}{2\\delta}\\mathrm{P.V.}\\int_{\\mathbb{R}}\\coth \\left(\\frac{\\pi}{2\\delta}[\\xi-(x_1+x_2)]\\right)F(\\xi,x_1-x_2)\\mathrm{d}\\xi,\\label{1.13}\n\\end{align}\nand\n\\begin{align}\n\tf(x_1,x_2)=F(x_1+x_2,x_1-x_2).\n\\end{align}\nThen the ILW hierarchy can be expressed in the form\n\\begin{align}\\label{1.14}\n\tu_t&=\\frac{i}{2^n}\\int_{\\mathbb{R}}\\delta(x_1-x_2)(\\mathcal{R}_{12}^\\star)^n\\theta_{12}^-\\cdot1\\mathrm{d}x_2\\\\\\nonumber\n\t&=\\frac{i}{2^n}\\int_{\\mathbb{R}}\\delta(x_1-x_2)\\theta_{12}^-\\mathcal{R}_{12}^n\\cdot1\\mathrm{d}x_2=\\mathcal{J}\\frac{\\delta H_n(u)}{\\delta u},\\quad \\text{for all} ~n\\in \\mathbb{N},\n\\end{align}\nwhere $\\star$ denotes the adjoint with respect to the bilinear form (\\ref{1.9}). The recursion operator $\\mathcal{R}_{12}$ and its adjoint $\\mathcal{R}_{12}^{\\star}$ are defined by\n\\begin{equation}\n\t\\mathcal{R}_{12} := (\\mathcal{J}_{12}^{(1)})^{-1}\\mathcal{J}_{12}^{(2)},\\quad\n\t\\mathcal{R}_{12}^{\\star} := \\mathcal{J}_{12}^{(2)}(\\mathcal{J}_{12}^{(1)})^{-1} = i\\theta_{12}^{-}T^\\delta_{12} - \\theta_{12}^{+},\\label{1.15}\n\\end{equation}\nand in view of (\\ref{1.15}), they satisfy the compatibility condition\n\\begin{equation*}\n\t\\mathcal{R}_{12}^{\\star}\\mathcal{J}_{12}^{(1)} = \\mathcal{J}_{12}^{(1)}\\mathcal{R}_{12}.\n\\end{equation*}\nConsequently, the first few equations of the ILW hierarchy are\n\\begin{align*}\n\t&u_t-u_x=0, \\quad \\text{for}\\quad n=1, \\\\\n\t&u_t+\\frac{1}{\\delta}u_x+2uu_x+T^\\delta u_{xx}=0,\\quad \\text{for}\\quad n=2;\\\\\n\t&u_t-\\left(\\frac{1}{4}({T^{\\delta}}^2-1)u_{xx}+ u^3+\\frac{3}{2}(uT^\\delta u_x+T^\\delta uu_{xx})\\right)_x=0, \\quad \\text{for}\\quad n=3.\n\\end{align*}\n\nIn common with the classical KdV and BO equations, between which the ILW equation (\\ref{ILW}) serves as a model-theoretical bridge and admits a family of unique exact solitary-wave solutions \\cite{AT,SA,SAK} of the form\n\\begin{align}\n\tu(x,t)=Q_c(x-ct-x_0), \\quad Q_c(s)=\\frac{a\\sin(a\\delta)}{\\cosh(as)+\\cos(a\\delta)},\\quad c>0,~ x_0\\in \\mathbb{R}, \\label{1.16}\n\\end{align}\nwhere $c$ represents the wave speed and $a$ is the unique solution of the transcendental equation\n\\begin{align*}\n\ta\\delta \\cot(a\\delta)=1-c\\delta, \\quad a\\in \\left(0, \\frac{\\pi}{\\delta}\\right).\n\\end{align*}\nSubstituting (\\ref{1.16}) into equation (\\ref{ILW}) yields\n\\begin{align}\n\tT^\\delta \\partial_x Q_c +\\left(\\frac{1}{\\delta}-c\\right)Q_c+Q_c^2=0, \\quad c>0.\\label{1.17}\n\\end{align}\nBeyond these fundamental solitary waves, the ILW equation (\\ref{ILW}) also supports more complex solutions, including multi-solitons \\cite{JE} that admit parametric representations similar to the single-soliton case \\cite{AT,LP,M,PD}.\nThe ILW $n$-soliton solution $U^{(n)}_\\mathbf{c}$ is characterized by a collection of wave speeds $\\mathbf{c}=(c_1, c_2, \\cdots, c_n)$ and initial positions $\\mathbf{x}=(x_1, x_2, \\cdots, x_n)$, where the wave speeds satisfy $c_j>0$ with $c_j\\neq c_k$ for $j\\neq k$ ($j,k=1, 2, \\cdots, n$). Moreover, in the long-time limit, these $n$-soliton solutions decompose into a superposition of $n$ individual solitons as follows\n\\begin{align}\n\t\\lim_{|t| \\to +\\infty} \\left\\| U^{(n)}(t,\\cdot;\\mathbf{c},\\mathbf{x}) - \\sum_{j=1}^{n} Q_{c_{j}}(\\cdot - c_{j}t - x_{j}) \\right\\|_{H^{s}(\\mathbb{R})} = 0, \\quad s \\in \\mathbb{N}.\\label{resolution}\n\\end{align}\n\nIn recent years, the stability theory for solitary waves and multi-solitons has emerged as a prominent research direction. Within soliton theory, stability concepts are systematically categorized into four distinct types based on analytical methodology and robustness guarantees: (i) linear (spectral) stability, concerning the eigenvalue distribution of linearized operators; (ii) Lyapunov (dynamical) stability, established via positive definiteness of the second variation of Lyapunov functionals at soliton solutions; (iii) orbital (nonlinear) stability, which requires that solutions remain within a neighborhood of the soliton orbit under finite-amplitude disturbances; and (iv) asymptotic stability, demanding convergence to specific soliton profiles.\n\nAs fundamental models in integrable systems, the stability of solitary waves for both KdV and BO equations has been extensively investigated. For instance, Maddocks and Sachs \\cite{MS} established orbital stability of KdV multi-solitons in the energy space defined by conservation laws. Killip and Visan \\cite{KV} proved orbital stability of KdV multi-solitons in $H^{-1}(\\mathbb{R})$ using low-regularity conservation laws in \\cite{BP}. For the BO equation, Wang \\cite{LW} and Matsuno \\cite{Mat} studied dynamical stability of multi-soliton solutions. Additionally, Badreddine, Killip and Visan \\cite{BKP} proved that the multi-soliton solutions to the BO equation are uniformly orbitally stable in $H^s(\\mathbb{R})$ with $-\\frac{1}{2}<s<\\frac{1}{2}$.\n\nIn contrast to the extensive results for KdV and BO equations, stability analysis for solitary wave solutions of the ILW equation remains comparatively underdeveloped. Existing studies primarily focus on orbital stability. Albert and Bona \\cite{AB} employed a rigorous variational approach combined with spectral analysis to establish orbital stability of ILW solitary waves in its energy space $\\mathcal{X} = \\{g \\in L^2(\\mathbb{R})| \\int_\\mathbb{R}(1+\\alpha(k))|\\hat{g}|^2\\mathrm{d}k < \\infty\\}$. Subsequent work \\cite{A} extended these results to spaces $H^{\\frac{1}{2}}(\\mathbb{R})$, while \\cite{PCA} addressed linear and orbital stability of periodic wave solutions in periodic Sobolev spaces.\n\nConsidering the growing importance of understanding coherent structures in nonlocal dispersive equations, a systematic analysis of soliton and multi-soliton stability for the ILW equation represents a significant open research direction. Hence, in this work, we aim to establish dynamical stability results of $n$-soliton solutions to the ILW equation (\\ref{ILW}) in an appropriate Sobolev space.\n\nOur strategy is to adapt the methods of Maddocks and Sachs \\cite{MS}.\nIn accordance with the ideas in \\cite{MS}, we construct an appropriate  Lyapunov functional $\\mathcal{S}_{n}$ of the ILW $n$-solitons is given by\n\\begin{align}\n\t\\mathcal{S}_{n}(u) = H_{n+1}(u) + \\sum_{m=1}^{n} \\mu_{m} H_{m}(u),\\label{lagrange}\n\\end{align}\nand $\\mu_{m}$ are Lagrange multipliers which will be expressed in terms of the elementary symmetric functions of $c_{1},c_{2},\\dots,c_{n}$. Using (\\ref{lagrange}), this condition can be written as the following Euler-Lagrange equation\n\\begin{align}\n\t\\frac{\\delta H_{n+1}(u)}{\\delta u} + \\sum_{m=1}^{n} \\mu_{m} \\frac{\\delta H_{m}(u)}{\\delta u} = 0,\\quad \\text{at } u = U^{(n)}.\\label{1.20}\n\\end{align}\nThe dynamical stability of $U^{(n)}$ is implied by the fact that $U^{(n)}(x)$ is a minimizer of the functional $H_{n+1}$ under the following $n$ constraints\n\\begin{align}\n\tH_{m}(u) = H_{m}\\left(U^{(n)}\\right),\\quad m = 1,2,\\dots,n,\n\\end{align}\nwhich requires that the self-adjoint second variation of the operator $\\mathcal{S}_n$,\n\\begin{align}\n\t\\mathcal{L}_n:=\\mathcal{S}''_{n}(U^{(n)}), \\label{VL}\n\\end{align}\nis strictly positive if one modulates the directions given by the constraints.\n\nThe demonstration of dynamical stability for the ILW equation relies on the recursion operators and the construction of a Lyapunov functional. This approach utilizes the trace formula derived from the inverse scattering transform (IST) framework to forge a link with conservation laws, in a manner analogous to techniques applied in the study of the Camassa-Holm equation \\cite{AYP, CGI, DKS}. The stability proof centers on a spectral analysis of the second variation $\\mathcal{L}_{n} = \\mathcal{S}_{n}''(U^{(n)})$ evaluated at a smooth $n$-soliton $U^{(n)}$, coupled with an eigenvalue computation for the Hessian matrix $D = \\left\\{\\frac{\\partial^2 \\mathcal{S}_{n}}{\\partial \\mu_{i} \\partial \\mu_{j}} \\right\\}$.\n\nThe spectral characterization of the linearized operator $\\mathcal{L}_{n}$ around the $n$-soliton $U^{(n)}$ is central to this endeavor. In the case of a single soliton, the operator $L_n = H''_{n+1}(Q_c) + c H''_{n}(Q_c)$ is analyzed. The inertia index $\\mathrm{in}(L_n)$ is determined by constructing the derivative operators $\\mathcal{J}L_n$ and $L_n\\mathcal{J}$ and leveraging IST theory, specifically employing Jost solutions and square eigenfunctions. For the general $n$-soliton, the recursive operator $\\mathcal{R}$ originating from the bi-Hamiltonian structure is applied to compute $\\mathrm{in}(\\mathcal{L}_n)$.\n\nFinally, we conclude by proving the orbital stability of double solitons. This final result is obtained through a spectral analysis of the linearized operator $\\mathcal{L}_{2}$ around the double soliton $U^{(2)}$, supported by a contradictory argument. Our analysis shows that $\\mathcal{L}_{2}$ has one simple negative eigenvalue and one double eigenvalue at zero.\n\nIn what follows, we will present our main results. The first is the dynamical stability of the multi-solitons to the ILW equation (\\ref{ILW}).", "context": "To formalize this connection, let subscripts $1, 2$ denote dependence on the independent variables $x:= x_1$ and $x_2$. For arbitrary functions $f_{12}$ and $g_{12}$, we define the associated bilinear form by\n\\begin{align}\n \\langle f_{12},g_{12}\\rangle\n &:= \\int_{\\mathbb{R}^{2}}f_{12}g_{12}^{*} \\mathrm{d}x_{1}\\mathrm{d}x_{2},\\label{1.9}\n\\end{align}\nwhere the asterisk superscript $^*$ denotes complex conjugation. Define operators in $L^{2}(\\mathbb{R}^{2},\\mathbb{C})$ (with domain $H^{1}(\\mathbb{R}^{2},\\mathbb{C})$)\n\\begin{align}\n \\theta_{12}^{\\pm}\n &:= u_{1}\\pm u_{2} + i(\\partial_{x_{1}} \\mp \\partial_{x_{2}}), \\quad u_{j}=u(x_{j},t), \\quad j=1,2.\\label{1.10}\n\\end{align}\nThen the two compatible Hamiltonian operators for the ILW equation are given by\n\\begin{align}\n \\mathcal{J}_{12}^{(1)}\n := \\theta_{12}^{-}, \\quad\n \\mathcal{J}_{12}^{(2)}\n := (i\\theta_{12}^{-}T^\\delta_{12} - \\theta_{12}^{+})\\theta_{12}^{-},\n\\end{align}\nwhere $T^\\delta_{12}$ is an extended operator defined as\n\\begin{align}\n (T^\\delta_{12}f_{12})(x_{1},x_{2}) =\\frac{1}{2\\delta}\\mathrm{P.V.}\\int_{\\mathbb{R}}\\coth \\left(\\frac{\\pi}{2\\delta}[\\xi-(x_1+x_2)]\\right)F(\\xi,x_1-x_2)\\mathrm{d}\\xi,\\label{1.13}\n\\end{align}\nand\n\\begin{align}\n f(x_1,x_2)=F(x_1+x_2,x_1-x_2).\n\\end{align}\nThen the ILW hierarchy can be expressed in the form\n\\begin{align}\\label{1.14}\n u_t&=\\frac{i}{2^n}\\int_{\\mathbb{R}}\\delta(x_1-x_2)(\\mathcal{R}_{12}^\\star)^n\\theta_{12}^-\\cdot1\\mathrm{d}x_2\\\\\\nonumber\n &=\\frac{i}{2^n}\\int_{\\mathbb{R}}\\delta(x_1-x_2)\\theta_{12}^-\\mathcal{R}_{12}^n\\cdot1\\mathrm{d}x_2=\\mathcal{J}\\frac{\\delta H_n(u)}{\\delta u},\\quad \\text{for all} ~n\\in \\mathbb{N},\n\\end{align}\nwhere $\\star$ denotes the adjoint with respect to the bilinear form (\\ref{1.9}). The recursion operator $\\mathcal{R}_{12}$ and its adjoint $\\mathcal{R}_{12}^{\\star}$ are defined by\n\\begin{equation}\n \\mathcal{R}_{12} := (\\mathcal{J}_{12}^{(1)})^{-1}\\mathcal{J}_{12}^{(2)},\\quad\n \\mathcal{R}_{12}^{\\star} := \\mathcal{J}_{12}^{(2)}(\\mathcal{J}_{12}^{(1)})^{-1} = i\\theta_{12}^{-}T^\\delta_{12} - \\theta_{12}^{+},\\label{1.15}\n\\end{equation}\nand in view of (\\ref{1.15}), they satisfy the compatibility condition\n\\begin{equation*}\n \\mathcal{R}_{12}^{\\star}\\mathcal{J}_{12}^{(1)} = \\mathcal{J}_{12}^{(1)}\\mathcal{R}_{12}.\n\\end{equation*}\nConsequently, the first few equations of the ILW hierarchy are\n\\begin{align*}\n &u_t-u_x=0, \\quad \\text{for}\\quad n=1, \\\\\n &u_t+\\frac{1}{\\delta}u_x+2uu_x+T^\\delta u_{xx}=0,\\quad \\text{for}\\quad n=2;\\\\\n &u_t-\\left(\\frac{1}{4}({T^{\\delta}}^2-1)u_{xx}+ u^3+\\frac{3}{2}(uT^\\delta u_x+T^\\delta uu_{xx})\\right)_x=0, \\quad \\text{for}\\quad n=3.\n\\end{align*}\n\nIn common with the classical KdV and BO equations, between which the ILW equation (\\ref{ILW}) serves as a model-theoretical bridge and admits a family of unique exact solitary-wave solutions \\cite{AT,SA,SAK} of the form\n\\begin{align}\n u(x,t)=Q_c(x-ct-x_0), \\quad Q_c(s)=\\frac{a\\sin(a\\delta)}{\\cosh(as)+\\cos(a\\delta)},\\quad c>0,~ x_0\\in \\mathbb{R}, \\label{1.16}\n\\end{align}\nwhere $c$ represents the wave speed and $a$ is the unique solution of the transcendental equation\n\\begin{align*}\n a\\delta \\cot(a\\delta)=1-c\\delta, \\quad a\\in \\left(0, \\frac{\\pi}{\\delta}\\right).\n\\end{align*}\nSubstituting (\\ref{1.16}) into equation (\\ref{ILW}) yields\n\\begin{align}\n T^\\delta \\partial_x Q_c +\\left(\\frac{1}{\\delta}-c\\right)Q_c+Q_c^2=0, \\quad c>0.\\label{1.17}\n\\end{align}\nBeyond these fundamental solitary waves, the ILW equation (\\ref{ILW}) also supports more complex solutions, including multi-solitons \\cite{JE} that admit parametric representations similar to the single-soliton case \\cite{AT,LP,M,PD}.\nThe ILW $n$-soliton solution $U^{(n)}_\\mathbf{c}$ is characterized by a collection of wave speeds $\\mathbf{c}=(c_1, c_2, \\cdots, c_n)$ and initial positions $\\mathbf{x}=(x_1, x_2, \\cdots, x_n)$, where the wave speeds satisfy $c_j>0$ with $c_j\\neq c_k$ for $j\\neq k$ ($j,k=1, 2, \\cdots, n$). Moreover, in the long-time limit, these $n$-soliton solutions decompose into a superposition of $n$ individual solitons as follows\n\\begin{align}\n \\lim_{|t| \\to +\\infty} \\left\\| U^{(n)}(t,\\cdot;\\mathbf{c},\\mathbf{x}) - \\sum_{j=1}^{n} Q_{c_{j}}(\\cdot - c_{j}t - x_{j}) \\right\\|_{H^{s}(\\mathbb{R})} = 0, \\quad s \\in \\mathbb{N}.\\label{resolution}\n\\end{align}\n\nOur strategy is to adapt the methods of Maddocks and Sachs \\cite{MS}.\nIn accordance with the ideas in \\cite{MS}, we construct an appropriate Lyapunov functional $\\mathcal{S}_{n}$ of the ILW $n$-solitons is given by\n\\begin{align}\n \\mathcal{S}_{n}(u) = H_{n+1}(u) + \\sum_{m=1}^{n} \\mu_{m} H_{m}(u),\\label{lagrange}\n\\end{align}\nand $\\mu_{m}$ are Lagrange multipliers which will be expressed in terms of the elementary symmetric functions of $c_{1},c_{2},\\dots,c_{n}$. Using (\\ref{lagrange}), this condition can be written as the following Euler-Lagrange equation\n\\begin{align}\n \\frac{\\delta H_{n+1}(u)}{\\delta u} + \\sum_{m=1}^{n} \\mu_{m} \\frac{\\delta H_{m}(u)}{\\delta u} = 0,\\quad \\text{at } u = U^{(n)}.\\label{1.20}\n\\end{align}\nThe dynamical stability of $U^{(n)}$ is implied by the fact that $U^{(n)}(x)$ is a minimizer of the functional $H_{n+1}$ under the following $n$ constraints\n\\begin{align}\n H_{m}(u) = H_{m}\\left(U^{(n)}\\right),\\quad m = 1,2,\\dots,n,\n\\end{align}\nwhich requires that the self-adjoint second variation of the operator $\\mathcal{S}_n$,\n\\begin{align}\n \\mathcal{L}_n:=\\mathcal{S}''_{n}(U^{(n)}), \\label{VL}\n\\end{align}\nis strictly positive if one modulates the directions given by the constraints.\n\nFinally, we conclude by proving the orbital stability of double solitons. This final result is obtained through a spectral analysis of the linearized operator $\\mathcal{L}_{2}$ around the double soliton $U^{(2)}$, supported by a contradictory argument. Our analysis shows that $\\mathcal{L}_{2}$ has one simple negative eigenvalue and one double eigenvalue at zero.\n\nIn what follows, we will present our main results. The first is the dynamical stability of the multi-solitons to the ILW equation (\\ref{ILW}).", "full_context": "To formalize this connection, let subscripts $1, 2$ denote dependence on the independent variables $x:= x_1$ and $x_2$. For arbitrary functions $f_{12}$ and $g_{12}$, we define the associated bilinear form by\n\\begin{align}\n \\langle f_{12},g_{12}\\rangle\n &:= \\int_{\\mathbb{R}^{2}}f_{12}g_{12}^{*} \\mathrm{d}x_{1}\\mathrm{d}x_{2},\\label{1.9}\n\\end{align}\nwhere the asterisk superscript $^*$ denotes complex conjugation. Define operators in $L^{2}(\\mathbb{R}^{2},\\mathbb{C})$ (with domain $H^{1}(\\mathbb{R}^{2},\\mathbb{C})$)\n\\begin{align}\n \\theta_{12}^{\\pm}\n &:= u_{1}\\pm u_{2} + i(\\partial_{x_{1}} \\mp \\partial_{x_{2}}), \\quad u_{j}=u(x_{j},t), \\quad j=1,2.\\label{1.10}\n\\end{align}\nThen the two compatible Hamiltonian operators for the ILW equation are given by\n\\begin{align}\n \\mathcal{J}_{12}^{(1)}\n := \\theta_{12}^{-}, \\quad\n \\mathcal{J}_{12}^{(2)}\n := (i\\theta_{12}^{-}T^\\delta_{12} - \\theta_{12}^{+})\\theta_{12}^{-},\n\\end{align}\nwhere $T^\\delta_{12}$ is an extended operator defined as\n\\begin{align}\n (T^\\delta_{12}f_{12})(x_{1},x_{2}) =\\frac{1}{2\\delta}\\mathrm{P.V.}\\int_{\\mathbb{R}}\\coth \\left(\\frac{\\pi}{2\\delta}[\\xi-(x_1+x_2)]\\right)F(\\xi,x_1-x_2)\\mathrm{d}\\xi,\\label{1.13}\n\\end{align}\nand\n\\begin{align}\n f(x_1,x_2)=F(x_1+x_2,x_1-x_2).\n\\end{align}\nThen the ILW hierarchy can be expressed in the form\n\\begin{align}\\label{1.14}\n u_t&=\\frac{i}{2^n}\\int_{\\mathbb{R}}\\delta(x_1-x_2)(\\mathcal{R}_{12}^\\star)^n\\theta_{12}^-\\cdot1\\mathrm{d}x_2\\\\\\nonumber\n &=\\frac{i}{2^n}\\int_{\\mathbb{R}}\\delta(x_1-x_2)\\theta_{12}^-\\mathcal{R}_{12}^n\\cdot1\\mathrm{d}x_2=\\mathcal{J}\\frac{\\delta H_n(u)}{\\delta u},\\quad \\text{for all} ~n\\in \\mathbb{N},\n\\end{align}\nwhere $\\star$ denotes the adjoint with respect to the bilinear form (\\ref{1.9}). The recursion operator $\\mathcal{R}_{12}$ and its adjoint $\\mathcal{R}_{12}^{\\star}$ are defined by\n\\begin{equation}\n \\mathcal{R}_{12} := (\\mathcal{J}_{12}^{(1)})^{-1}\\mathcal{J}_{12}^{(2)},\\quad\n \\mathcal{R}_{12}^{\\star} := \\mathcal{J}_{12}^{(2)}(\\mathcal{J}_{12}^{(1)})^{-1} = i\\theta_{12}^{-}T^\\delta_{12} - \\theta_{12}^{+},\\label{1.15}\n\\end{equation}\nand in view of (\\ref{1.15}), they satisfy the compatibility condition\n\\begin{equation*}\n \\mathcal{R}_{12}^{\\star}\\mathcal{J}_{12}^{(1)} = \\mathcal{J}_{12}^{(1)}\\mathcal{R}_{12}.\n\\end{equation*}\nConsequently, the first few equations of the ILW hierarchy are\n\\begin{align*}\n &u_t-u_x=0, \\quad \\text{for}\\quad n=1, \\\\\n &u_t+\\frac{1}{\\delta}u_x+2uu_x+T^\\delta u_{xx}=0,\\quad \\text{for}\\quad n=2;\\\\\n &u_t-\\left(\\frac{1}{4}({T^{\\delta}}^2-1)u_{xx}+ u^3+\\frac{3}{2}(uT^\\delta u_x+T^\\delta uu_{xx})\\right)_x=0, \\quad \\text{for}\\quad n=3.\n\\end{align*}\n\nIn common with the classical KdV and BO equations, between which the ILW equation (\\ref{ILW}) serves as a model-theoretical bridge and admits a family of unique exact solitary-wave solutions \\cite{AT,SA,SAK} of the form\n\\begin{align}\n u(x,t)=Q_c(x-ct-x_0), \\quad Q_c(s)=\\frac{a\\sin(a\\delta)}{\\cosh(as)+\\cos(a\\delta)},\\quad c>0,~ x_0\\in \\mathbb{R}, \\label{1.16}\n\\end{align}\nwhere $c$ represents the wave speed and $a$ is the unique solution of the transcendental equation\n\\begin{align*}\n a\\delta \\cot(a\\delta)=1-c\\delta, \\quad a\\in \\left(0, \\frac{\\pi}{\\delta}\\right).\n\\end{align*}\nSubstituting (\\ref{1.16}) into equation (\\ref{ILW}) yields\n\\begin{align}\n T^\\delta \\partial_x Q_c +\\left(\\frac{1}{\\delta}-c\\right)Q_c+Q_c^2=0, \\quad c>0.\\label{1.17}\n\\end{align}\nBeyond these fundamental solitary waves, the ILW equation (\\ref{ILW}) also supports more complex solutions, including multi-solitons \\cite{JE} that admit parametric representations similar to the single-soliton case \\cite{AT,LP,M,PD}.\nThe ILW $n$-soliton solution $U^{(n)}_\\mathbf{c}$ is characterized by a collection of wave speeds $\\mathbf{c}=(c_1, c_2, \\cdots, c_n)$ and initial positions $\\mathbf{x}=(x_1, x_2, \\cdots, x_n)$, where the wave speeds satisfy $c_j>0$ with $c_j\\neq c_k$ for $j\\neq k$ ($j,k=1, 2, \\cdots, n$). Moreover, in the long-time limit, these $n$-soliton solutions decompose into a superposition of $n$ individual solitons as follows\n\\begin{align}\n \\lim_{|t| \\to +\\infty} \\left\\| U^{(n)}(t,\\cdot;\\mathbf{c},\\mathbf{x}) - \\sum_{j=1}^{n} Q_{c_{j}}(\\cdot - c_{j}t - x_{j}) \\right\\|_{H^{s}(\\mathbb{R})} = 0, \\quad s \\in \\mathbb{N}.\\label{resolution}\n\\end{align}\n\nOur strategy is to adapt the methods of Maddocks and Sachs \\cite{MS}.\nIn accordance with the ideas in \\cite{MS}, we construct an appropriate Lyapunov functional $\\mathcal{S}_{n}$ of the ILW $n$-solitons is given by\n\\begin{align}\n \\mathcal{S}_{n}(u) = H_{n+1}(u) + \\sum_{m=1}^{n} \\mu_{m} H_{m}(u),\\label{lagrange}\n\\end{align}\nand $\\mu_{m}$ are Lagrange multipliers which will be expressed in terms of the elementary symmetric functions of $c_{1},c_{2},\\dots,c_{n}$. Using (\\ref{lagrange}), this condition can be written as the following Euler-Lagrange equation\n\\begin{align}\n \\frac{\\delta H_{n+1}(u)}{\\delta u} + \\sum_{m=1}^{n} \\mu_{m} \\frac{\\delta H_{m}(u)}{\\delta u} = 0,\\quad \\text{at } u = U^{(n)}.\\label{1.20}\n\\end{align}\nThe dynamical stability of $U^{(n)}$ is implied by the fact that $U^{(n)}(x)$ is a minimizer of the functional $H_{n+1}$ under the following $n$ constraints\n\\begin{align}\n H_{m}(u) = H_{m}\\left(U^{(n)}\\right),\\quad m = 1,2,\\dots,n,\n\\end{align}\nwhich requires that the self-adjoint second variation of the operator $\\mathcal{S}_n$,\n\\begin{align}\n \\mathcal{L}_n:=\\mathcal{S}''_{n}(U^{(n)}), \\label{VL}\n\\end{align}\nis strictly positive if one modulates the directions given by the constraints.\n\nFinally, we conclude by proving the orbital stability of double solitons. This final result is obtained through a spectral analysis of the linearized operator $\\mathcal{L}_{2}$ around the double soliton $U^{(2)}$, supported by a contradictory argument. Our analysis shows that $\\mathcal{L}_{2}$ has one simple negative eigenvalue and one double eigenvalue at zero.\n\nIn what follows, we will present our main results. The first is the dynamical stability of the multi-solitons to the ILW equation (\\ref{ILW}).\n\nThen we will provide a more precise description of the orbital stability of the double solitons to the ILW equation.\n\n\\begin{theorem}[\\textbf{Orbital stability of double solitons}]\\label{thm1.2}\n The double solitons $U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)$ of ILW equation (\\ref{ILW}) with $0<c_1<c_2$ are orbitally stable in $H^1(\\mathbb{R})$ in the following sense.\nThere exist parameters $\\epsilon_0$ and $A_0$, depending on $c_1$ and $c_2$, if there exists $\\epsilon \\in (0,\\epsilon_0)$ such that for any $u_0\\in H^1(\\mathbb{R})$,\n\\begin{align}\\label{u02}\n \\bigl\\lVert u_0 - U^{(2)}_{c_1,c_2}(0;0,0) \\bigr\\rVert_{H^1(\\mathbb{R})} < \\epsilon,\n\\end{align}\nthen there exist $x_1(t), x_2(t) \\in \\mathbb{R}$, such that the corresponding solution $u(t,x)$ of the ILW equation (\\ref{ILW}) with the initial data $u(0)=u_0$ satisfies $u_t \\in C([0,\\infty),H^1(\\mathbb{R}))$ and\n\\begin{align}\\label{1.22}\n \\sup_{t\\in (0,\\infty)}|\\!|u(t)-U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)|\\!|_{H^1(\\mathbb{R})}<A_0\\epsilon,\n\\end{align}\nwith\n\\begin{align}\n \\sup_{t\\in (0,\\infty)}(|x_1'(t)|+|x_2'(t)|)\\leq CA_0\\epsilon.\n\\end{align}\n\\end{theorem}\n\n\\begin{align*}\n (\\mathcal{R}^\\star(u))^m u_x &= \\mathcal{J} (\\mathcal{R} (u))^m H_1'(u) = \\mathcal{J} (\\mathcal{R} (u))^m u, \\quad m \\in \\mathbb{N}.\n\\end{align*}\nFor subsequent applications, establishing the uniqueness and differentiability of $\\mathcal{R}(u)$ becomes essential. In contrast, the Korteweg-de Vries system ($\\delta = 3$) exhibits explicit structure. For Schwartz-class functions $u \\in \\mathcal{S}(\\mathbb{R})$, its recursion operator is $\\mathcal{R}_K(u) = -\\partial_x^2 - \\tfrac{2}{3}u - \\tfrac{2}{3}\\partial_x^{-1} u \\partial_x$, then $\\mathcal{R}_K'(u) = -\\tfrac{2}{3} - \\tfrac{2}{3}\\partial_x^{-1}(\\cdot\\partial_x)$.\n\\begin{prop}\\label{prop2.2}\n Given $u \\in H^{k+1}(\\mathbb{R})$ with $k \\geq 0$, there exists a unique linear operator\n \\[\n \\mathcal{R}(u): H^{k+1}(\\mathbb{R}) \\to H^{k}(\\mathbb{R}),\n \\]\n such that relations (\\ref{2.39}) and (\\ref{2.41}) hold true. Moreover, $\\mathcal{R}(u)$ is differentiable with respect to $u$.\n\\end{prop}\n\\begin{proof}\n The fundamental approach involves establishing a connection between the recursion operators $\\mathcal{R}(u)$ and $\\mathcal{R}_{12}$ (\\ref{1.15}). Assuming $u \\in \\mathcal{S}(\\mathbb{R})$, it follows from (\\ref{1.14}) and (\\ref{2.39}) that\n \\begin{align*}\n \\mathcal{S}(\\mathbb{R}) \\ni H_{m+1}^{\\prime}(u) & = \\frac{i}{2^{m+1}}\\mathcal{J}^{-1}\\int_{\\mathbb{R}} \\delta(x_{1} - x_{2}) \\theta_{12}^{-} \\mathcal{R}_{12}^{m+1} \\cdot 1 \\mathrm{d}x_{2} \\\\\n & = \\mathcal{R}(u)H_{m}^{\\prime}(u) \\\\\n & = \\mathcal{R}(u) \\frac{i}{2^m}\\mathcal{J}^{-1}\\int_{\\mathbb{R}} \\delta(x_{1} - x_{2}) \\theta_{12}^{-} \\mathcal{R}_{12}^{m} \\cdot 1 \\mathrm{d}x_{2}.\n \\end{align*}\n The uniqueness of $\\mathcal{R}(u)$ follows by an induction argument over $m$. Furthermore, the asymptotic behavior of the operator is characterized by\n \\[\n \\mathcal{R}(u) \\sim -T^\\delta\\partial_{x} + L(u),\n \\]\n where $L(u): \\mathcal{S}(\\mathbb{R}) \\mapsto \\mathcal{S}(\\mathbb{R})$ represents a differentiable higher-order remainder term. By standard density argument, $\\mathcal{R}(u)$ itself is differentiable and satisfies $\\mathcal{R}^{\\prime}(u) \\sim L^{\\prime}(u)$.\n\\end{proof}\n\\begin{re}\nThe propositions above will be shown in Section \\ref{sec.3} that understanding the spectral information of the (adjoint) recursion operators $\\mathcal{R} (u)$ and $\\mathcal{R}^\\star(u)$ is essential in proving the (spectral) stability of the ILW multi-solitons.\n\\end{re}\n\nNext, we will proceed to prove Theorem \\ref{th1.3}.\n\\subsection{The negative eigenvalues of $\\mathcal{L}_n$ :Proof of Theorem \\ref{th1.3}}\nIn this subsection, our goal is to prove the Theorem \\ref{th1.3}.\n\\begin{proof}[\\textbf{Proof of Theorem \\ref{th1.3}}]\n The linearized operators around the $n$-solitons $\\mathcal{L}_{n} = S_{n}^{\\prime\\prime}(U^{(n)})$ possess $\\left[\\frac{n+1}{2}\\right]$ negative eigenvalues, as previously verified in (\\ref{3.47}). We now prove the expression (\\ref{negetive}) for these eigenvalues. As $t\\to \\infty$, the spectrum $\\sigma(\\mathcal{L}_{n}(t))$ converges to the union of the spectra $\\sigma(\\mathcal{L}_{n,j})$ where $\\mathcal{L}_{n,j} = S_{n}^{\\prime\\prime}(Q_{c_{j}})$, that is\n \\begin{align}\n \\sigma(\\mathcal{L}_{n}(t)) \\rightarrow \\bigcup_{j=1}^{n} \\sigma(\\mathcal{L}_{n,j}), \\quad \\text{as} \\quad t \\rightarrow +\\infty.\n \\end{align}\n Since the operators $\\mathcal{L}_{n}(t)$ are isoinertial for each $n$, the spectrum $\\sigma(\\mathcal{L}_{n}(t))$ is independent of $t$. Thus, the negative eigenvalues of $\\mathcal{L}_{n}$ match those of $\\mathcal{L}_{n,j}$ for all $j = 1, 2, \\dots, n$. According to Lemma \\ref{lem3.5}, $\\mathcal{L}_{n,j}$ has negative eigenvalues when $j = 2k-1$ and $1 \\leq k \\leq \\left[\\frac{n+1}{2}\\right]$. Specially, when $n=1$, one has $\\mathcal{L}_1=L_1$. Therefore, (\\ref{n=1}) indicates that $\\mathcal{L}_1$ has the negative $\\lambda_1<0$. With the methods in \\cite{BBSSB}, we could compute $\\lambda_1$, which is\n \\begin{align}\\label{1.30}\n \\lambda_1=-\\frac{c-\\frac{1}{\\delta}}{2a^2\\sin^2(a\\delta)}<0,\n \\end{align}\n Then for $n=2$, we have\n \\begin{align}\\label{lamb}\n \\mathcal{L}_{2,1}=(\\mathcal{R}(Q_{c_1})+c_2)\\mathcal{S}_1''(Q_{c_1}).\n \\end{align}\nIn terms of Lemma \\ref{lem3.2}, the operator $(\\mathcal{R}(Q_{c_1}) + c_{2})$ has an eigenvalue $c_{2} - c_{1} > 0$, with continuous spectrum $(c_2, +\\infty)$ whose generalized eigenfunctions are not in $L^{2}(\\mathbb{R})$. Therefore, the unique negative eigenvalues of $\\mathcal{L}_{2,1}$ is $(c_2-c_1)\\lambda_1<0$.\\\\\nAssuming the result holds for $n = p$, namely, the $\\left[\\frac{p+1}{2}\\right]$-th negative eigenvalue of $\\mathcal{L}_{p}$ is\n \\begin{align}\\label{4.48}\n \\lambda_{k}^{(p)} = -M^\\delta (c_{2k-1}-\\frac{1}{\\delta}) \\prod_{j \\neq 2k-1}^{p} (c_{j} - c_{2k-1}), \\quad k = 1, 2, \\dots, \\left[\\frac{p+1}{2}\\right].\n \\end{align}\nThen for $n = p+1$ is even, we have $\\left[\\frac{p+1}{2}\\right] = \\left[\\frac{p+2}{2}\\right]$ and for $k = 1, 2, \\dots, \\left[\\frac{p+1}{2}\\right]$, we have\n \\begin{align}\n \\mathcal{L}_{p+1,2k-1} =(\\mathcal{R}(Q_{c_{2k-1}}) + c_{p+1}) \\mathcal{S}_{p}^{\\prime\\prime}(Q_{c_{2k-1}}).\n \\end{align}\n By Lemma \\ref{lem3.2}, the operator $(\\mathcal{R}(Q_{c_{2k-1}}) + c_{p+1})$ has an eigenvalue $c_{p+1} - c_{2k-1} > 0$, with continuous spectrum $(c_{p+1}, +\\infty)$ whose generalized eigenfunctions are not in $L^{2}(\\mathbb{R})$. Therefore, the $\\left[\\frac{p+2}{2}\\right]$-th negative eigenvalues of $\\mathcal{L}_{p+1,2k-1}$ is\n \\begin{align}\\label{4.50}\n \\lambda_{k}^{(p+1)}= -M^\\delta (c_{2k-1}-\\frac{1}{\\delta}) \\prod_{j \\neq 2k-1}^{p+1} (c_{j} - c_{2k-1}), \\quad k = 1, 2, \\dots, \\left[\\frac{p+1}{2}\\right].\n \\end{align}\nFor the case where $n = p+1$ is odd, we have $\\left[\\frac{p+1}{2}\\right] + 1 = \\left[\\frac{p+2}{2}\\right]$. The first $\\left[\\frac{p+1}{2}\\right]$ negative eigenvalues of $\\mathcal{L}_{p+1}$ are given by expression (\\ref{4.50}). We now compute the final negative eigenvalue. Consider\n\\begin{align}\n \\mathcal{L}_{p+1,p+1} =\\left( \\mathcal{R}(Q_{c_{p+1}}) + c_{j} \\right) \\tilde{S}_{p}^{\\prime\\prime}(Q_{c_{p+1}})\n\\end{align}\nwhere $\\tilde{S}_{p}$ denotes the action with wave speed $c_j$ in $S_p$ replaced by $c_{p+1}$ for some $1 \\leq j \\leq p$. By assumption (\\ref{4.48}), the discrete eigenvalue of $\\tilde{S}_{p}^{\\prime\\prime}(Q_{c_{p+1}})$ is\n\\begin{align}\n -M^\\delta (c_{p+1}-\\frac{1}{\\delta})\\prod_{l \\neq j}^{p} (c_l - c_{p+1}).\n\\end{align}\nThe operator $\\mathcal{R}(Q_{c_{p+1}}) + c_j$ has eigenvalue $c_j - c_{p+1} < 0$ by Lemma \\ref{lem3.2}, with continuous spectrum $(c_j, +\\infty)$ whose generalized eigenfunctions are not in $L^2(\\mathbb{R})$. Therefore, the last negative eigenvalue of $\\mathcal{L}_{p+1}$ is\n\\begin{align*}\n \\lambda_{\\left[\\frac{p+2}{2}\\right]}^{(p+1)} = -M^\\delta (c_{p+1}-\\frac{1}{\\delta}) \\prod_{l=1}^{p} (c_l - c_{p+1}).\n\\end{align*}\nThe proof of Theorem \\ref{th1.3} is completed by combining the eigenvalue expressions.\n\\end{proof}", "post_theorem_intro_text_len": 4644, "post_theorem_intro_text": "Since the stability result in \\cite{MS} coincides with orbital stability theorem in \\cite{GSS} for $n=1$, we obtain the following corollary regarding the orbital stability of ILW soliton solutions.\n\\begin{cor}\nThe single soliton solution of the ILW equation is orbital stable in $H^{\\frac{1}{2}}(\\mathbb{R})$.\n\\end{cor}\n\\begin{re}\nFor $n \\geq 2$, it should be noted that the stability result in \\cite{MS} aligns with the notion of orbital stability in \\cite{GSS} when the stability framework of \\cite{MS} is suitably extended. In Hamiltonian systems, invariant functionals (i.e., integrals of motion) generate Hamiltonian flows that commute with the original time evolution.\n\\end{re}\n\nThen we will provide a more precise description of the orbital stability of the double solitons to the ILW equation.\n\n\\begin{theorem}[\\textbf{Orbital stability of double solitons}]\\label{thm1.2}\n\tThe double solitons $U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)$ of ILW equation (\\ref{ILW}) with $0<c_1<c_2$ are orbitally stable in $H^1(\\mathbb{R})$ in the following sense.\nThere exist parameters $\\epsilon_0$ and $A_0$, depending on $c_1$ and $c_2$, if there exists $\\epsilon \\in (0,\\epsilon_0)$ such that for any $u_0\\in H^1(\\mathbb{R})$,\n\\begin{align}\\label{u02}\n\t\\bigl\\lVert u_0 - U^{(2)}_{c_1,c_2}(0;0,0) \\bigr\\rVert_{H^1(\\mathbb{R})} < \\epsilon,\n\\end{align}\nthen there exist $x_1(t), x_2(t) \\in \\mathbb{R}$, such that the corresponding solution $u(t,x)$ of the ILW equation (\\ref{ILW}) with the initial data $u(0)=u_0$ satisfies $u_t \\in C([0,\\infty),H^1(\\mathbb{R}))$ and\n\\begin{align}\\label{1.22}\n\t\\sup_{t\\in (0,\\infty)}|\\!|u(t)-U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)|\\!|_{H^1(\\mathbb{R})}<A_0\\epsilon,\n\\end{align}\nwith\n\\begin{align}\n\t\\sup_{t\\in (0,\\infty)}(|x_1'(t)|+|x_2'(t)|)\\leq CA_0\\epsilon.\n\\end{align}\n\\end{theorem}\n\nMoreover, something is interesting is that one can express the negative eigenvalues of the isoinertial operator $\\mathcal{L}_n$ in terms of the wave speeds $\\{c_j\\}^n_{j=1}$ as follows.\n\\begin{theorem}[\\textbf{The negative eigenvalues of $\\mathcal{L}_n$}]\\label{th1.3}\n\tThe linearized operator $\\mathcal{L}_{n}$ about the $n$-soliton solution possesses exactly $\\left\\lfloor\\frac{n+1}{2}\\right\\rfloor$ negative eigenvalues $\\lambda_{k}$, where $k=1,2,\\cdots,\\left\\lfloor\\frac{n+1}{2}\\right\\rfloor$ and $\\lfloor x\\rfloor$ denotes the floor function. Furthermore, the negative eigenvalues of $\\mathcal{L}_n(n\\geq 1$)  have the following expression\n\t\\begin{align}\\label{negetive}\n\t\t\\lambda_{k} = -M^\\delta_k (c_{2k-1}-\\frac{1}{\\delta})\\prod_{ j\\neq 2k-1}^{n}(c_{j}-c_{2k-1}),\\quad k=1,2,\\cdots,\\left\\lfloor\\frac{n+1}{2}\\right\\rfloor,\n\t\\end{align}\n\twhere $M^\\delta_k$ are positive constants independent of the wave speeds $c_1,\\cdots, c_n$.\n\\end{theorem}\n\\begin{re}\n\tSince the ILW equation serves as the bridge connecting KdV equation and BO equation. The stability results above are also valid for both equations. The specific results can be found in \\cite{WA} and \\cite{LW}, respectively.\n\\end{re}\n\nThe proof of our main stability result, Theorem \\ref{thm1.1}, proceeds by transforming the stability problem into a spectral analysis of an explicitly constructed linearized operator. Through the IST method, we establish a fundamental connection between the Hamiltonians and scattering data. Similarly, the proof of Theorem \\ref{thm1.2} on orbital stability is reduced to a detailed spectral analysis of the linearized operator $\\mathcal{L}_n$ arising from the second variation of the Lyapunov functional. Finally, the proof of Theorem \\ref{th1.3} relies on the application of spectral analysis results. The proof strategy unfolds as follows.\n\nIn Section \\ref{sec.2}, we begin by establishing the properties of the singular integral operators $T^\\delta$ and $T^\\delta_{12}$. We then introduce the IST framework for the ILW equation, from which the $n$-soliton solutions are formally derived.\n\nSection \\ref{sec.3} is devoted to a detailed spectral analysis of the linearized operator $\\mathcal{L}_n = \\mathcal{S}_n''(U^{(n)})$. We demonstrate that its inertia index is $\\left( \\left\\lfloor \\frac{n+1}{2} \\right\\rfloor, n \\right)$, thereby satisfying the crucial stability condition.\n\nIn Section \\ref{sec.4}, we apply the framework of constrained variational principles to establish the dynamical stability result in Theorem \\ref{thm1.1}. We then prove Theorem \\ref{thm1.2} on orbital stability of double solitons ($n=2$) through a refined spectral analysis of the linearized operator $\\mathcal{L}_2$, yielding a direct proof of orbital stability. Finally, we establish Theorem \\ref{th1.3} based on the spectral analysis results from Section \\ref{sec.3}.", "sketch": "The proof of Theorem~\\ref{thm1.1} “proceeds by transforming the stability problem into a spectral analysis of an explicitly constructed linearized operator.” “Through the IST method, we establish a fundamental connection between the Hamiltonians and scattering data.”\n\nMore concretely, the stated strategy is organized by sections as follows:\n\\begin{itemize}\n\\item \\emph{Section~\\ref{sec.2}:} establish “the properties of the singular integral operators $T^\\delta$ and $T^\\delta_{12}$,” and “introduce the IST framework for the ILW equation, from which the $n$-soliton solutions are formally derived.”\n\\item \\emph{Section~\\ref{sec.3}:} carry out “a detailed spectral analysis of the linearized operator $\\mathcal{L}_n = \\mathcal{S}_n''(U^{(n)})$,” showing “its inertia index is $\\left( \\left\\lfloor \\frac{n+1}{2} \\right\\rfloor, n \\right)$, thereby satisfying the crucial stability condition.”\n\\item \\emph{Section~\\ref{sec.4}:} “apply the framework of constrained variational principles to establish the dynamical stability result in Theorem~\\ref{thm1.1}.”\n\\end{itemize}\n\n(Additionally, the introduction notes that the orbital stability theorem (Theorem~\\ref{thm1.2}) is “reduced to a detailed spectral analysis of the linearized operator $\\mathcal{L}_n$ arising from the second variation of the Lyapunov functional,” and that Theorem~\\ref{th1.3} “relies on the application of spectral analysis results.”)", "expanded_sketch": "The proof of the main theorem proceeds by transforming the stability problem into a spectral analysis of an explicitly constructed linearized operator. Through the IST method, we establish a fundamental connection between the Hamiltonians and scattering data.\n\nMore concretely, the stated strategy is organized by sections as follows:\n\\begin{itemize}\n\\item \\emph{Section~\\ref{sec.2}:} establish “the properties of the singular integral operators $T^\\delta$ and $T^\\delta_{12}$,” and “introduce the IST framework for the ILW equation, from which the $n$-soliton solutions are formally derived.”\n\\item \\emph{Section~\\ref{sec.3}:} carry out “a detailed spectral analysis of the linearized operator $\\mathcal{L}_n = \\mathcal{S}_n''(U^{(n)})$,” showing “its inertia index is $\\left( \\left\\lfloor \\frac{n+1}{2} \\right\\rfloor, n \\right)$, thereby satisfying the crucial stability condition.”\n\\item \\emph{Section~\\ref{sec.4}:} “apply the framework of constrained variational principles to establish the dynamical stability result in the main theorem.”\n\\end{itemize}\n\nAdditionally, the introduction notes that the orbital stability theorem\n\\begin{theorem}[\\textbf{Orbital stability of double solitons}]\\label{thm1.2}\n\tThe double solitons $U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)$ of ILW equation (\\ref{ILW}) with $0<c_1<c_2$ are orbitally stable in $H^1(\\mathbb{R})$ in the following sense.\nThere exist parameters $\\epsilon_0$ and $A_0$, depending on $c_1$ and $c_2$, if there exists $\\epsilon \\in (0,\\epsilon_0)$ such that for any $u_0\\in H^1(\\mathbb{R})$,\n\\begin{align}\\label{u02}\n\t\\bigl\\lVert u_0 - U^{(2)}_{c_1,c_2}(0;0,0) \\bigr\\rVert_{H^1(\\mathbb{R})} < \\epsilon,\n\\end{align}\nthen there exist $x_1(t), x_2(t) \\in \\mathbb{R}$, such that the corresponding solution $u(t,x)$ of the ILW equation (\\ref{ILW}) with the initial data $u(0)=u_0$ satisfies $u_t \\in C([0,\\infty),H^1(\\mathbb{R}))$ and\n\\begin{align}\\label{1.22}\n\t\\sup_{t\\in (0,\\infty)}|\\!|u(t)-U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)|\\!|_{H^1(\\mathbb{R})}<A_0\\epsilon,\n\\end{align}\nwith\n\\begin{align}\n\t\\sup_{t\\in (0,\\infty)}(|x_1'(t)|+|x_2'(t)|)\\leq CA_0\\epsilon.\n\\end{align}\n\\end{theorem}\n\nis “reduced to a detailed spectral analysis of the linearized operator $\\mathcal{L}_n$ arising from the second variation of the Lyapunov functional,” and that the following theorem “relies on the application of spectral analysis results”:\n\\begin{theorem}[\\textbf{The negative eigenvalues of $\\mathcal{L}_n$}]\\label{th1.3}\n\tThe linearized operator $\\mathcal{L}_{n}$ about the $n$-soliton solution possesses exactly $\\left\\lfloor\\frac{n+1}{2}\\right\\rfloor$ negative eigenvalues $\\lambda_{k}$, where $k=1,2,\\cdots,\\left\\lfloor\\frac{n+1}{2}\\right\\rfloor$ and $\\lfloor x\\rfloor$ denotes the floor function. Furthermore, the negative eigenvalues of $\\mathcal{L}_n(n\\geq 1$)  have the following expression\n\t\\begin{align}\\label{negetive}\n\t\t\\lambda_{k} = -M^\\delta_k (c_{2k-1}-\\frac{1}{\\delta})\\prod_{ j\\neq 2k-1}^{n}(c_{j}-c_{2k-1}),\\quad k=1,2,\\cdots,\\left\\lfloor\\frac{n+1}{2}\\right\\rfloor,\n\t\\end{align}\n\twhere $M^\\delta_k$ are positive constants independent of the wave speeds $c_1,\\cdots, c_n$.\n\\end{theorem}", "expanded_theorem": "[\\textbf{Dynamical stability of $n$-solitons}]\\label{thm1.1}\nGiven $ n\\in \\mathbb{N} $, $ n\\geq 1 $, a collection of wave speeds $\\textbf{c} = (c_{1},\\dots,c_{n})$ with $ 0 < c_{1} < \\dots < c_{n} $ and a collection of space transitions $\\textbf{x} = (x_{1},\\dots,x_{n})\\in \\mathbb{R}^{n}$, let $ U^{(n)}(\\cdot,\\cdot;\\textbf{c},\\textbf{x}) $ be the corresponding $n$-soliton solutions of\n\\begin{align}\n\\tu_t+\\frac{1}{\\delta}u_x+2uu_x+T^\\delta u_{xx}=0,\\quad  (t,x)\\in \\mathbb{R} \\times \\mathbb{R},\\label{ILW}\n\\end{align}\nThen for any $ \\epsilon > 0 $, there exists $ \\delta > 0 $ such that for any $ u_{0}\\in H^{\\frac{n}{2}}(\\mathbb{R}) $, the following stability property holds, if\n\n\\begin{align}\n\t&\\bigl\\lVert u_{0} - U^{(n)}(0,\\cdot;\\textbf{c},\\textbf{x}) \\bigr\\rVert_{H^{\\frac{n}{2}}} < \\delta,\n\\end{align}\nthen for any $t \\in \\mathbb{R}$, the corresponding solution of the equation above verifies\n\\begin{align}\n\\inf_{\\tau \\in \\mathbb{R},\\  \\textbf{y} \\in \\mathbb{R}^{n}} \\bigl\\lVert u(t) - U^{(n)}(\\tau,\\cdot;\\textbf{c},\\textbf{y}) \\bigr\\rVert_{H^{\\frac{n}{2}}} < \\epsilon.\n\\end{align}", "theorem_type": ["Implication", "Existential–Universal"], "mcq": {"question": "Let \\(n\\in\\mathbb N\\) with \\(n\\ge 1\\), let \\(\\mathbf c=(c_1,\\dots,c_n)\\) satisfy \\(0<c_1<\\cdots<c_n\\), and let \\(\\mathbf x=(x_1,\\dots,x_n)\\in\\mathbb R^n\\). Let \\(U^{(n)}(t,x;\\mathbf c,\\mathbf x)\\) be the corresponding \\(n\\)-soliton solution of the ILW equation\n\\[\nu_t+\\frac{1}{\\delta}u_x+2uu_x+T^\\delta u_{xx}=0,\\qquad (t,x)\\in\\mathbb R\\times\\mathbb R,\n\\]\nwhere \\(T^\\delta\\) is the ILW nonlocal operator. Suppose \\(u(t)\\) is the solution with initial data \\(u(0)=u_0\\in H^{n/2}(\\mathbb R)\\). Which of the following conclusions holds about the orbit of the \\(n\\)-soliton family \\(\\{U^{(n)}(\\tau,\\cdot;\\mathbf c,\\mathbf y):\\tau\\in\\mathbb R,\\ \\mathbf y=(y_1,\\dots,y_n)\\in\\mathbb R^n\\}\\) in the \\(H^{n/2}(\\mathbb R)\\) norm?", "correct_choice": {"label": "A", "text": "For every \\(\\epsilon>0\\), there exists \\(\\eta>0\\) such that if\n\\[\n\\bigl\\lVert u_0-U^{(n)}(0,\\cdot;\\mathbf c,\\mathbf x)\\bigr\\rVert_{H^{n/2}}<\\eta,\n\\]\nthen the corresponding solution satisfies\n\\[\n\\inf_{\\tau\\in\\mathbb R,\\ \\mathbf y\\in\\mathbb R^n}\\bigl\\lVert u(t)-U^{(n)}(\\tau,\\cdot;\\mathbf c,\\mathbf y)\\bigr\\rVert_{H^{n/2}}<\\epsilon\n\\quad\\text{for every } t\\in\\mathbb R.\n\\]\nThus the \\(n\\)-soliton family is orbitally stable in \\(H^{n/2}(\\mathbb R)\\)."}, "choices": [{"label": "B", "text": "For every \\(\\epsilon>0\\), there exists \\(\\eta>0\\) such that if\n\\[\n\\bigl\\lVert u_0-U^{(n)}(0,\\cdot;\\mathbf c,\\mathbf x)\\bigr\\rVert_{H^{n/2}}<\\eta,\n\\]\nthen the corresponding solution satisfies\n\\[\n\\inf_{\\mathbf y\\in\\mathbb R^n}\\bigl\\lVert u(t)-U^{(n)}(t,\\cdot;\\mathbf c,\\mathbf y)\\bigr\\rVert_{H^{n/2}}<\\epsilon\n\\quad\\text{for every } t\\in\\mathbb R.\n\\]\nThus the \\(n\\)-soliton family is orbitally stable in \\(H^{n/2}(\\mathbb R)\\) with the time parameter fixed."}, {"label": "C", "text": "For every \\(\\epsilon>0\\), there exists \\(\\eta>0\\) such that if\n\\[\n\\bigl\\lVert u_0-U^{(n)}(0,\\cdot;\\mathbf c,\\mathbf x)\\bigr\\rVert_{H^{n/2}}<\\eta,\n\\]\nthen the corresponding solution satisfies\n\\[\n\\inf_{\\mathbf y\\in\\mathbb R^n}\\bigl\\lVert u(t)-U^{(n)}(t,\\cdot;\\mathbf c,\\mathbf y)\\bigr\\rVert_{H^{n/2}}<\\epsilon\n\\quad\\text{for every } t\\in\\mathbb R.\n\\]\nIn particular, the orbit remains close to the \\(n\\)-soliton family in \\(H^{n/2}(\\mathbb R)\\)."}, {"label": "D", "text": "There exists \\(\\eta>0\\) such that for every \\(\\epsilon>0\\), if\n\\[\n\\bigl\\lVert u_0-U^{(n)}(0,\\cdot;\\mathbf c,\\mathbf x)\\bigr\\rVert_{H^{n/2}}<\\eta,\n\\]\nthen the corresponding solution satisfies\n\\[\n\\inf_{\\tau\\in\\mathbb R,\\ \\mathbf y\\in\\mathbb R^n}\\bigl\\lVert u(t)-U^{(n)}(\\tau,\\cdot;\\mathbf c,\\mathbf y)\\bigr\\rVert_{H^{n/2}}<\\epsilon\n\\quad\\text{for every } t\\in\\mathbb R.\n\\]\nThus the \\(n\\)-soliton family is orbitally stable in \\(H^{n/2}(\\mathbb R)\\) with a uniform stability radius independent of \\(\\epsilon\\)."}, {"label": "E", "text": "For every \\(\\epsilon>0\\), there exists \\(\\eta>0\\) such that if\n\\[\n\\bigl\\lVert u_0-U^{(n)}(0,\\cdot;\\mathbf c,\\mathbf x)\\bigr\\rVert_{H^{n/2}}<\\eta,\n\\]\nthen the corresponding solution satisfies\n\\[\n\\bigl\\lVert u(t)-U^{(n)}(t,\\cdot;\\mathbf c,\\mathbf x)\\bigr\\rVert_{H^{n/2}}<\\epsilon\n\\quad\\text{for every } t\\in\\mathbb R.\n\\]\nThus the given \\(n\\)-soliton solution is asymptotically stable in \\(H^{n/2}(\\mathbb R)\\)."}], "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": "time-phase modulation parameter", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "free time-shift parameter \\(\\tau\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "quantifier order between \\(\\epsilon\\) and neighborhood size", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "orbital stability modulo symmetries replaced by fixed-profile/asymptotic stability", "template_used": "stronger_trap"}]}}
{"id": "2512.08599v1", "paper_link": "http://arxiv.org/abs/2512.08599v1", "theorems_cnt": 3, "theorem": {"env_name": "thm", "content": "\\label{thm:main} Let $F$ be a totally real field,\nand $F'$ be a real quadratic extension of $F$. Assume that $p$ is\nunramified in both $F$ and $F'$. Let $f$ be a Hilbert newform of\nparallel even weight $k$ over $F$ with trivial central character and\nlevel $\\mathfrak{n}_f$. Let $f'$ be the base change of $f$ to $F'$.\nLet $\\mathfrak{n}_{f'}$ be the level of $f'$.\n\nLet $p$ be a prime number satisfying $p\\geq\\mathrm{max}(k+2,7)$. We\nassume that $\\mathfrak{n}_f$ is prime to $p$ and $f$ is ordinary at\neach prime above $p$. When $[F:\\mathbb{Q}]$ is odd, we assume that\nthere exists at least one prime $\\mathfrak{q}$ such that\n$\\mathfrak{q}|| \\mathfrak{n}_f$, and $\\mathfrak{q}$ is split in\n$F'$.\n\nSuppose the following conditions hold.\n\\begin{enumerate}\n\\item The restrictions of $\\bar{\\rho}_f$ to $G_{F'(\\xi_p)}$ and $G_{F'(\n\\sqrt{ p^*})}$ are absolutely irreducible, where\n$p^*=(-1)^{\\frac{p-1}{2}}p$.\n\\item In the case of $k=2$, for each place $v$ of $F$ above $p$ we\nhave  $a_v^2(f)\\ {\\backslash\\hskip -10pt \\equiv } 1 \\ (\\mathrm{mod}\n\\ p)$; similarly, for each place $v'$ of $F'$ above $p$ we have\n$a_{v'}^2(f')\\ {\\backslash\\hskip -10pt \\equiv } 1 \\ (\\mathrm{mod} \\\np)$.\n\\item For each $\\mathfrak{l}|\\mathfrak{n}$, if\n$\\bar{\\rho}_f|_{I_{F_\\mathfrak{l}}}$ is absolutely irreducible, then\n$\\mathrm{N}(\\mathfrak{l})\\equiv  {\\hskip -10pt /}-1 \\ (\\mathrm{mod\n}\\ p)$. Similarly, for each $\\mathfrak{l}'|\\mathfrak{n}'$, if\n$\\bar{\\rho}_{f'}|_{I_{F'_{\\mathfrak{l}'}}}$ is absolutely\nirreducible, then $\\mathrm{N}(\\mathfrak{l}')\\equiv  {\\hskip -10pt\n/}-1 \\ (\\mathrm{mod }\\ p)$.\n\\item $\\rho_f$ is a minimal modular lifting of $\\bar{\\rho}_f$.\n\\end{enumerate} Then $$\n\\Omega_{f'}^\\mathrm{can}=(\\Omega_f^\\mathrm{can})^2 $$ up to a\n$p$-adic unit.", "start_pos": 9356, "end_pos": 11112, "label": "thm:main"}, "ref_dict": {"eq:con": "\\begin{equation}\\label{eq:con}(L_{p}(K'K_{\\infty},f'))\\supseteq\n\\mathrm{char}\\ \\mathrm{Sel}(K'K_{\\infty},A_{\\rho})^{\\vee}.\n\\end{equation}", "thm:Iw-main-int": "\\begin{thm}\\label{thm:Iw-main-int}\nAssume that $f$ satisfies conditions $(\\mathrm{CR}^+)$,\n$(\\mathfrak{n}^+\\text{-}\\mathrm{DT})$, $(\\mathrm{PO})$ and $(\\mathrm{Fuji}1$-$4)$.\nThen we have $$(L_p(K_\\infty, f))=\\mathrm{char}\\ \\mathrm{Sel}\n(K_\\infty, A_{\\rho})^\\vee .$$\n\\end{thm}", "eq:Lp-eq": "\\begin{equation}\\label{eq:Lp-eq} (L_{p}(K_{\\infty},f)\\cdot\nL_{p}(K_{\\infty},f\\otimes\\chi_{F'/F}))= \\mathrm{char}\\\n\\mathrm{Sel}(K'K_{\\infty},A_{\\rho})^{\\vee}. \\end{equation}", "eq:main": "\\begin{equation}\\label{eq:main}\n(L_{p}(K'K_{\\infty} ,f'))=(L_{p}(K_{\\infty} ,f))\\cdot\n(L_{p}(K_{\\infty} ,f\\otimes\\chi_{F'/F}))  \\end{equation}", "eq:contain": "\\begin{equation}\\label{eq:contain} (L_{p}(K'K_{\\infty},f'))\\supseteq (L_{p}(K_{\\infty},f)\\cdot\nL_{p}(K_{\\infty},f\\otimes\\chi_{F'/F})) . \\end{equation}", "thm:main": "\\begin{thm}\\label{thm:main} Let $F$ be a totally real field,\nand $F'$ be a real quadratic extension of $F$. Assume that $p$ is\nunramified in both $F$ and $F'$. Let $f$ be a Hilbert newform of\nparallel even weight $k$ over $F$ with trivial central character and\nlevel $\\mathfrak{n}_f$. Let $f'$ be the base change of $f$ to $F'$.\nLet $\\mathfrak{n}_{f'}$ be the level of $f'$.\n\nLet $p$ be a prime number satisfying $p\\geq\\mathrm{max}(k+2,7)$. We\nassume that $\\mathfrak{n}_f$ is prime to $p$ and $f$ is ordinary at\neach prime above $p$. When $[F:\\mathbb{Q}]$ is odd, we assume that\nthere exists at least one prime $\\mathfrak{q}$ such that\n$\\mathfrak{q}|| \\mathfrak{n}_f$, and $\\mathfrak{q}$ is split in\n$F'$.\n\nSuppose the following conditions hold.\n\\begin{enumerate}\n\\item The restrictions of $\\bar{\\rho}_f$ to $G_{F'(\\xi_p)}$ and $G_{F'(\n\\sqrt{ p^*})}$ are absolutely irreducible, where\n$p^*=(-1)^{\\frac{p-1}{2}}p$.\n\\item In the case of $k=2$, for each place $v$ of $F$ above $p$ we\nhave  $a_v^2(f)\\ {\\backslash\\hskip -10pt \\equiv } 1 \\ (\\mathrm{mod}\n\\ p)$; similarly, for each place $v'$ of $F'$ above $p$ we have\n$a_{v'}^2(f')\\ {\\backslash\\hskip -10pt \\equiv } 1 \\ (\\mathrm{mod} \\\np)$.\n\\item For each $\\mathfrak{l}|\\mathfrak{n}$, if\n$\\bar{\\rho}_f|_{I_{F_\\mathfrak{l}}}$ is absolutely irreducible, then\n$\\mathrm{N}(\\mathfrak{l})\\equiv  {\\hskip -10pt /}-1 \\ (\\mathrm{mod\n}\\ p)$. Similarly, for each $\\mathfrak{l}'|\\mathfrak{n}'$, if\n$\\bar{\\rho}_{f'}|_{I_{F'_{\\mathfrak{l}'}}}$ is absolutely\nirreducible, then $\\mathrm{N}(\\mathfrak{l}')\\equiv  {\\hskip -10pt\n/}-1 \\ (\\mathrm{mod }\\ p)$.\n\\item $\\rho_f$ is a minimal modular lifting of $\\bar{\\rho}_f$.\n\\end{enumerate} Then $$\n\\Omega_{f'}^\\mathrm{can}=(\\Omega_f^\\mathrm{can})^2 $$ up to a\n$p$-adic unit.\n\\end{thm}"}, "pre_theorem_intro_text_len": 4409, "pre_theorem_intro_text": "\\subsection{Special value formula, Gross period and Hida canonical period}\n\nIn \\cite{Gro87} Gross provided a formula expressing the special\nvalues of $L$-functions via Heegner points. To describe his result,\nlet $f$ be a newform on $\\Gamma_0(N)$  of even weight $k$, and let $K$ be an imaginary\nquadratic field of discriminant $D$ such that $D$ is prime to $N$.\n\nAssume $k=2$ and $N$ is a prime inert in $K$. Let $B$ be the definite quaternion algebra that is ramified exactly at $N$. Fix a prime number $p$. Let $H_n$ be the ring class field of $K$ of conductor $p^n$. When $\\chi$ is a character of the Galois group $\\mathrm{Gal}(H_n/K)$, one forms the Rankin-Selberg $L$-function $L(f,\\chi,s)$.\nGross \\cite{Gro87} showed that there exists a period $\\Omega_{f,K}$\ndepending on $f$ and $K$ but independent of $n$ and $\\chi$, called Gross\nperiod, such that $\\frac{L(f,\\chi,1)}{\\Omega_{f,K}}$ are algebraic\nand they satisfy\n$$\\frac{L(f,\\chi,1)}{\\Omega_{f,K}}\\cdot\\sqrt{D}p^{n}=|\\sum\\limits_{\\sigma\\in\\mathrm{Gal}(H_n/K)}\\chi(\\sigma)\\psi(P^{\\sigma})|^{2},$$\nwhere $P$ is the Heegner point on the Shimura set attached to $B$.\n\nGross formula is generalized by Shouwu Zhang \\cite[Theorem 7.1]{Zha04} for $k=2$ and general $N$, and by Haiping Yuan \\cite{Yua05} for $k>2$.\n\nIn \\cite{BD07}  Bertolini and Darmon used Gross' special value\nformula to construct anticyclotomic $p$-adic $L$-functions, i.e.\n$p$-adic $L$-functions for anticyclotomic $\\BZ_p$-extension instead\nof the cyclotomic $\\BZ_p$-extension. In the same paper they proved\none divisibility for Iwasawa main conjecture for elliptic curves in\nthe setting of anticyclotomic $\\BZ_p$-extension (\\cite[Theorem\n1]{BD07}), and also gave new evidence for Birch and Swinnerton-Dyer\nconjecture (\\cite[Corollary 4]{BD07}).\n\nGross period serves as a bridge between the complex $L$-function and\nthe $p$-adic $L$-function. But it depends on the choice of an imaginary\nquadratic extension $K$.\n\nHida \\cite{Hid81, Hid81.2} introduced a period that is independent\nof $K$ and called the {\\it canonical period}. Attached to $f$ there\nexists a homomorphism $\\lambda_f:T\\longrightarrow \\mathcal{O}_f$\nfrom the Hecke algebra $T$ of level $N$ to a discrete valuation ring\n$\\mathcal{O}_f$ that is finite over $\\BZ_p$. Let $\\eta $ be the\ncongruence number for $\\lambda_f$ defined by\n$$\\eta =\\lambda_f(\\mathrm{Ann}(\\mathrm{Ker}(\\lambda_f))).$$ Then the\ncanonical period is defined by\n$$\\Omega_{f}^{\\mathrm{can}}=\\frac{\\langle f, f\n\\rangle_{\\mathrm{Pet}}}{\\eta }.$$ The reader may consult\n\\cite[Section 2]{Vat03} about more knowledge on Gross period and the\ncanonical period.\n\nVatsal \\cite{Vat03} showed that\n$\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}$ lies in\n$\\mathcal{O}_f$. Pollack and Weston \\cite{PW11} pointed out that\n$\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}\\in\\mathcal{O}_f^{\\times}$\nis equivalent to the freeness of spaces of modular forms on $B$ over\nthe associated Hecke algebra and the vanishing of certain local\nTamagawa components. In \\cite{CH18} Chida and Hsieh actually proved\nthat $\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}$ is a $p$-adic\nunit, i.e.\n$\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}\\in\\mathcal{O}_f^{\\times}$\nunder a condition ($\\mathrm{CR^{+}}$). Wang \\cite{Wang} generalized Chida and Hsieh's result to the setting of Hilbert\nmodular forms under similar hypothesis. \nThere are two normalizations of anticyclotomic $p$-adic\n$L$-functions depending on which period one uses. However, in the\npresent paper we assume those hypothesis in \\cite{Wang} in order\nthat there is essentially no difference between these two\nnormalizations.\n\n\\subsection{Comparison of periods under base change}\n\nThe goal of the present paper is to compare the canonical period of\na Hilbert modular form and that of its base change to a (totally)\nreal quadratic extension.\n\nTo make it precise, let $F$ be a totally real number field, and  $f$\nbe a Hilbert newform of parallel even weight over $F$. Let $F'$ be a\ntotally real quadratic extension of $F$, and $f'$ be the base change\nof $f$ to $F'$. One may expect the following holds.\n\n\\begin{conj} We have\n$$ \\Omega_{f'}^\\mathrm{can}=(\\Omega_f^\\mathrm{can})^2 $$ up to a\n$p$-adic unit.\n\\end{conj}\n\nThe main result of this paper is the following theorem. Let $\\rho_f$ be $p$-adic Galois representation attached to $f$ (see \\cite{Wil88,Tay}), and let $\\bar{\\rho}_f$ denote the residue representation of $\\rho_f$.", "context": "Assume $k=2$ and $N$ is a prime inert in $K$. Let $B$ be the definite quaternion algebra that is ramified exactly at $N$. Fix a prime number $p$. Let $H_n$ be the ring class field of $K$ of conductor $p^n$. When $\\chi$ is a character of the Galois group $\\mathrm{Gal}(H_n/K)$, one forms the Rankin-Selberg $L$-function $L(f,\\chi,s)$.\nGross \\cite{Gro87} showed that there exists a period $\\Omega_{f,K}$\ndepending on $f$ and $K$ but independent of $n$ and $\\chi$, called Gross\nperiod, such that $\\frac{L(f,\\chi,1)}{\\Omega_{f,K}}$ are algebraic\nand they satisfy\n$$\\frac{L(f,\\chi,1)}{\\Omega_{f,K}}\\cdot\\sqrt{D}p^{n}=|\\sum\\limits_{\\sigma\\in\\mathrm{Gal}(H_n/K)}\\chi(\\sigma)\\psi(P^{\\sigma})|^{2},$$\nwhere $P$ is the Heegner point on the Shimura set attached to $B$.\n\nHida \\cite{Hid81, Hid81.2} introduced a period that is independent\nof $K$ and called the {\\it canonical period}. Attached to $f$ there\nexists a homomorphism $\\lambda_f:T\\longrightarrow \\mathcal{O}_f$\nfrom the Hecke algebra $T$ of level $N$ to a discrete valuation ring\n$\\mathcal{O}_f$ that is finite over $\\BZ_p$. Let $\\eta $ be the\ncongruence number for $\\lambda_f$ defined by\n$$\\eta =\\lambda_f(\\mathrm{Ann}(\\mathrm{Ker}(\\lambda_f))).$$ Then the\ncanonical period is defined by\n$$\\Omega_{f}^{\\mathrm{can}}=\\frac{\\langle f, f\n\\rangle_{\\mathrm{Pet}}}{\\eta }.$$ The reader may consult\n\\cite[Section 2]{Vat03} about more knowledge on Gross period and the\ncanonical period.\n\nVatsal \\cite{Vat03} showed that\n$\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}$ lies in\n$\\mathcal{O}_f$. Pollack and Weston \\cite{PW11} pointed out that\n$\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}\\in\\mathcal{O}_f^{\\times}$\nis equivalent to the freeness of spaces of modular forms on $B$ over\nthe associated Hecke algebra and the vanishing of certain local\nTamagawa components. In \\cite{CH18} Chida and Hsieh actually proved\nthat $\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}$ is a $p$-adic\nunit, i.e.\n$\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}\\in\\mathcal{O}_f^{\\times}$\nunder a condition ($\\mathrm{CR^{+}}$). Wang \\cite{Wang} generalized Chida and Hsieh's result to the setting of Hilbert\nmodular forms under similar hypothesis. \nThere are two normalizations of anticyclotomic $p$-adic\n$L$-functions depending on which period one uses. However, in the\npresent paper we assume those hypothesis in \\cite{Wang} in order\nthat there is essentially no difference between these two\nnormalizations.\n\nTo make it precise, let $F$ be a totally real number field, and $f$\nbe a Hilbert newform of parallel even weight over $F$. Let $F'$ be a\ntotally real quadratic extension of $F$, and $f'$ be the base change\nof $f$ to $F'$. One may expect the following holds.\n\n\\begin{conj} We have\n$$ \\Omega_{f'}^\\mathrm{can}=(\\Omega_f^\\mathrm{can})^2 $$ up to a\n$p$-adic unit.\n\\end{conj}\n\nThe main result of this paper is the following theorem. Let $\\rho_f$ be $p$-adic Galois representation attached to $f$ (see \\cite{Wil88,Tay}), and let $\\bar{\\rho}_f$ denote the residue representation of $\\rho_f$.", "full_context": "Assume $k=2$ and $N$ is a prime inert in $K$. Let $B$ be the definite quaternion algebra that is ramified exactly at $N$. Fix a prime number $p$. Let $H_n$ be the ring class field of $K$ of conductor $p^n$. When $\\chi$ is a character of the Galois group $\\mathrm{Gal}(H_n/K)$, one forms the Rankin-Selberg $L$-function $L(f,\\chi,s)$.\nGross \\cite{Gro87} showed that there exists a period $\\Omega_{f,K}$\ndepending on $f$ and $K$ but independent of $n$ and $\\chi$, called Gross\nperiod, such that $\\frac{L(f,\\chi,1)}{\\Omega_{f,K}}$ are algebraic\nand they satisfy\n$$\\frac{L(f,\\chi,1)}{\\Omega_{f,K}}\\cdot\\sqrt{D}p^{n}=|\\sum\\limits_{\\sigma\\in\\mathrm{Gal}(H_n/K)}\\chi(\\sigma)\\psi(P^{\\sigma})|^{2},$$\nwhere $P$ is the Heegner point on the Shimura set attached to $B$.\n\nHida \\cite{Hid81, Hid81.2} introduced a period that is independent\nof $K$ and called the {\\it canonical period}. Attached to $f$ there\nexists a homomorphism $\\lambda_f:T\\longrightarrow \\mathcal{O}_f$\nfrom the Hecke algebra $T$ of level $N$ to a discrete valuation ring\n$\\mathcal{O}_f$ that is finite over $\\BZ_p$. Let $\\eta $ be the\ncongruence number for $\\lambda_f$ defined by\n$$\\eta =\\lambda_f(\\mathrm{Ann}(\\mathrm{Ker}(\\lambda_f))).$$ Then the\ncanonical period is defined by\n$$\\Omega_{f}^{\\mathrm{can}}=\\frac{\\langle f, f\n\\rangle_{\\mathrm{Pet}}}{\\eta }.$$ The reader may consult\n\\cite[Section 2]{Vat03} about more knowledge on Gross period and the\ncanonical period.\n\nVatsal \\cite{Vat03} showed that\n$\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}$ lies in\n$\\mathcal{O}_f$. Pollack and Weston \\cite{PW11} pointed out that\n$\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}\\in\\mathcal{O}_f^{\\times}$\nis equivalent to the freeness of spaces of modular forms on $B$ over\nthe associated Hecke algebra and the vanishing of certain local\nTamagawa components. In \\cite{CH18} Chida and Hsieh actually proved\nthat $\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}$ is a $p$-adic\nunit, i.e.\n$\\frac{\\Omega_{f,K}}{\\Omega_{f}^{\\mathrm{can}}}\\in\\mathcal{O}_f^{\\times}$\nunder a condition ($\\mathrm{CR^{+}}$). Wang \\cite{Wang} generalized Chida and Hsieh's result to the setting of Hilbert\nmodular forms under similar hypothesis. \nThere are two normalizations of anticyclotomic $p$-adic\n$L$-functions depending on which period one uses. However, in the\npresent paper we assume those hypothesis in \\cite{Wang} in order\nthat there is essentially no difference between these two\nnormalizations.\n\nTo make it precise, let $F$ be a totally real number field, and $f$\nbe a Hilbert newform of parallel even weight over $F$. Let $F'$ be a\ntotally real quadratic extension of $F$, and $f'$ be the base change\nof $f$ to $F'$. One may expect the following holds.\n\n\\begin{conj} We have\n$$ \\Omega_{f'}^\\mathrm{can}=(\\Omega_f^\\mathrm{can})^2 $$ up to a\n$p$-adic unit.\n\\end{conj}\n\nThe main result of this paper is the following theorem. Let $\\rho_f$ be $p$-adic Galois representation attached to $f$ (see \\cite{Wil88,Tay}), and let $\\bar{\\rho}_f$ denote the residue representation of $\\rho_f$.\n\nThe main result of this paper is the following theorem. Let $\\rho_f$ be $p$-adic Galois representation attached to $f$ (see \\cite{Wil88,Tay}), and let $\\bar{\\rho}_f$ denote the residue representation of $\\rho_f$.\n\nWe choose an imaginary quadratic extension $K$ of $F$ that is\ninertia at $\\mathfrak{q}$ when $[F:\\mathbb{Q}]$ is odd. Denote $KF'$\nby $K'$. Fix a place $\\mathfrak{p}$ of $F$ above $p$. Let $K_\\infty$ be the $\\mathfrak{p}$-anticyclotomic\n$\\mathbb{Z}_p$-extension of $K$.\n\n\\begin{lem}\\label{lem:fitt} Let $S$ be the set of primes dividing $\\mathfrak{n}^-$. Then\n$$ \\mathrm{Fitt}(\\prod_{\\mathfrak{l}|\n\\mathfrak{n}^- }H^1((\\mathcal{K}_\\infty)_\\mathfrak{l},\nF^-_\\mathfrak{l}A_\\rho)^\\vee) =\\left(\\frac{L^S_p (\\mathcal{K}_\\infty,\nf)}{L_p (\\mathcal{K}_\\infty, f)}\\right).\n$$\n\\end{lem} This should be well known.\nHowever, the authors do not have a reference about it.\n\\begin{proof}\nBy definition (see Section \\ref{ss:selmer}) $H^{1}((\\mathcal{K}_{\\infty})_{\\mathfrak{l}},F_{\\mathfrak{l}}^{-}A_{\\rho})$ is the inductive limit $\\lim\\limits_{\\overrightarrow{\\;\\;L\\;\\;}} H^{1}(L_{\\mathfrak{l}},F_{\\mathfrak{l}}^{-}A_{\\rho}),$ where $L$ runs through all finite extensions of $K$ contained in $\\mathcal{K}_\\infty$. For each $L$, $H^{1}(L_{\\mathfrak{l}},F_{\\mathfrak{l}}^{-}A_{\\rho})$ is isomorphic to\n$\\bigoplus_{\\mathfrak{L}|\\mathfrak{l}} H^1(G_{L_\\mathfrak{L}}, F_{\\mathfrak{l}}^{-}A_{\\rho})$ where $\\mathfrak{L}$ runs over all places of $L$ above $\\mathfrak{l}$,\nand sits in the following exact sequence\n{\\small $$ \\bigoplus_{\\mathfrak{L}|\\mathfrak{l}} H^1(G_{L_\\mathfrak{L}}/I_{L_\\mathfrak{L}}, (F_{\\mathfrak{l}}^{-}A_{\\rho})^{I_{L_\\mathfrak{L}}})\\rightarrowtail \\bigoplus_{\\mathfrak{L}|\\mathfrak{l}} H^1(G_{L_\\mathfrak{L}}, F_{\\mathfrak{l}}^{-}A_{\\rho})\\twoheadrightarrow \\bigoplus_{\\mathfrak{L}|\\mathfrak{l}} H^1(I_{L_\\mathfrak{L}}, F_{\\mathfrak{l}}^{-}A_{\\rho})^{G_{L_\\mathfrak{L}}/I_{L_\\mathfrak{L}}} . $$}\nAs $G_{L_{\\mathfrak{L}}}$ acts trivially on $F_\\mathfrak{l}^-A_\\rho$, we have\n$$ H^1(G_{L_\\mathfrak{L}}/I_{L_\\mathfrak{L}}, (F_{\\mathfrak{l}}^{-}A_{\\rho})^{I_{L_\\mathfrak{L}}}) \\cong F_{\\mathfrak{l}}^{-}A_{\\rho}. $$ If $L'$ is a finite extension of $L$ contained in $\\mathcal{K}_\\infty$, and $\\mathfrak{L}'$ is a place of $L'$ above $\\mathfrak{L}$, we have the following commutative diagram\n\\[ \\xymatrix{ H^1(G_{L_\\mathfrak{L}}/I_{L_\\mathfrak{L}}, (F_{\\mathfrak{l}}^{-}A_{\\rho})^{I_{L_\\mathfrak{L}}})\\ar[rr]^{\\hskip45pt \\simeq}\\ar[d] && F_{\\mathfrak{l}}^{-}A_{\\rho} \\ar[d]^{\\times[G_{L_\\mathfrak{L}}/I_{L_\\mathfrak{L}}:G_{L'_{\\mathfrak{L}'}}/I_{L'_{\\mathfrak{L}'}}]} \\\\ H^1(G_{L'_{\\mathfrak{L}'}}/I_{L'_{\\mathfrak{L}'}}, (F_{\\mathfrak{l}}^{-}A_{\\rho})^{I_{L'_{\\mathfrak{L}'}}})\\ar[rr]^{\\hskip45pt\\simeq} && F_{\\mathfrak{l}}^{-}A_{\\rho}, } \\] where the left vertical arrow is the restriction map. Since both $L_{\\mathfrak{L}}$ and $L'_{\\mathfrak{L}'}$ are unramified over $K_\\mathfrak{l}$, $$[G_{L_\\mathfrak{L}}/I_{L_\\mathfrak{L}}:G_{L'_{\\mathfrak{L}'}}/I_{L'_{\\mathfrak{L}'}}]\n=|\\mathrm{Gal}(L'_{\\mathfrak{L}'}/L_{\\mathfrak{L}})|=[L'_{\\mathfrak{L}'}:L_{\\mathfrak{L}}]$$ is a power of $p$. Thus taking limit we have $$\\lim\\limits_{\\overrightarrow{\\;\\;L\\;\\;}}\\bigoplus_{\\mathfrak{L}: \\text{places of } L \\text{ above }\\mathfrak{l}} H^1(G_{L_\\mathfrak{L}}/I_{L_\\mathfrak{L}}, (F_{\\mathfrak{l}}^{-}A_{\\rho})^{I_{L_\\mathfrak{L}}})=0.$$\n\n\\begin{proof} The representation $\\bar{\\rho}_f$ factors through\n$\\mathrm{Gal}(F_1/F)$ for a finite Galois extension $F_1$ of $F$.\nTake a finite place $\\mathfrak{q}\\nmid p\\mathfrak{n}$. For each\n$\\mathfrak{l}|p\\mathfrak{n}$ take a unit $u_\\mathfrak{l}$ of\n$\\mathcal{O}_{F_{\\mathfrak{l}}}$ such that the image\n$u_\\mathfrak{l}$ in $k_{\\mathfrak{l}}$ by the natural homomorphism\n$$ \\mathcal{O}_{F_\\mathfrak{l}}\\rightarrow\n\\mathcal{O}_{F_\\mathfrak{l}}/\\mathfrak{l}\\cong k_\\mathfrak{l} $$\ngenerates the multiplicative group $k_\\mathfrak{l}^\\times$. Take a\nuniformizing element $\\omega_\\mathfrak{q}$ of\n$\\mathcal{O}_{F_{\\mathfrak{q}}}$. By the weak approximation theorem\nthere exists a totally positive element $a$ of $F$ such that\n$a_{\\mathfrak{l}}-u_{\\mathfrak{l}} \\in\n\\mathfrak{l}\\mathcal{O}_{F_\\mathfrak{l}}$ for each\n$\\mathfrak{l}|p\\mathfrak{n}$, and\n$a_\\mathfrak{q}-\\omega_\\mathfrak{q}\\in\n\\mathfrak{q}^2\\mathcal{O}_{F_\\mathfrak{q}}$. Put $F'=F(\\sqrt{a})$.\nThen each place dividing $p\\mathfrak{n}$ splits in $F'$. As $a$ is\ntotally positive, $F'$ is totally real. Let $f'$ be the base change\nof $f$ to $F'$.\n\n\\begin{thm}\\label{thm:Iw-main} Assume $f$ satisfies\n$(\\mathrm{CR}^+)$, $(\\mathfrak{n}^+\\text{-}\\mathrm{DT})$,\n$(\\mathrm{PO})$ and $(\\mathrm{Fuji}1$-$4)$.\n\\begin{enumerate}\n\\item\\label{it:Iw-main-a}\nWe have \\begin{equation}\\label{eq:char} \\mathrm{char}\\ \\mathrm{Sel}\n(K_\\infty, A_{\\rho})^\\vee = (L_p(K_\\infty, f)). \\end{equation}\n\\item\\label{it:Iw-main-b} Let $\\mathcal{L}$ be $\\mathcal{K}_\\infty,\n\\mathcal{K}^-_\\infty$ or $K^-_J$. If $[F:\\BQ]$ is even, then\n$$ \\mathrm{char}\\ \\mathrm{Sel}(\\mathcal{L}, A_\\rho)^\\vee = (L_p(\\mathcal{L}, f))\n$$ in $\\mathcal{O}_\\mathfrak{P}[[\\mathrm{Gal}(\\mathcal{L}/K)]]_E$.\n\\end{enumerate}\n\\end{thm}\n\\begin{proof} Let $F'$ be a totally real quadratic extension of $F$, as in Lemma \\ref{lem:quad-ext}. Then $[F':\\BQ]$ is even.\nLet $\\chi_{F'/F}$ be the quadratic character corresponding to the\nextension $F'/F$. Put $K'=F'K$. Let $\\mathcal{K}'^-_\\infty$ be the\nmaximal abelian anticyclotomic $\\BZ_p$-extension of $K'$ unramified\noutside $p$. Put $\\mathcal{K}'^+_\\infty=F_\\infty K'$,\n$\\mathcal{K}'_\\infty=\\mathcal{K}'^+_\\infty\\mathcal{K}'^-_\\infty$ and\n$\\Gamma_{K'}=\\mathrm{Gal}(\\mathcal{K}'_\\infty/K')$. Then\n$K'\\mathcal{K}_\\infty=F'\\mathcal{K}_\\infty$ is contained in\n$\\mathcal{K}'_\\infty$.\n\n\\begin{thm}\\label{thm:main-stronger} Let $F$ be a totally real field,\nand $F'$ be a real quadratic extension of $F$. Assume that $p$ is\nunramified in both $F$ and $F'$. Let $f$ be a Hilbert newform of\nparallel even weight $k$ over $F$ with trivial central character and\nlevel $\\mathfrak{n}_f$. Let $f'$ be the base change of $f$ to $F'$.\nLet $\\mathfrak{n}_{f'}$ be the level of $f'$.\n\nWe assume that $\\mathfrak{n}_f$ is prime to $p$ and $f$ is ordinary\nat each place above $p$. When $[F:\\mathbb{Q}]$ is odd, we assume\nthat there exists at least one prime $\\mathfrak{q}$ such that\n$\\mathfrak{q}|| \\mathfrak{n}_f$, $\\mathfrak{q}$ is split in $F'$,\nand $\\bar{\\rho}_{f}$ is ramified at $\\mathfrak{q}$. Suppose that, if\n$\\mathfrak{l}'||\\mathfrak{n}_{f'}$ and\n$\\mathrm{N}(\\mathfrak{l}')\\equiv 1 \\ (\\mathrm{mod} \\ p)$, then\n$\\bar{\\rho}_{f'}$ is ramified at $\\mathfrak{l}'$.\n\n\\begin{conj}\\label{conj-mu} Let $F$ be a totally real number field, $K$ a quadratic imaginary\nextension of $F$. Let $F'$ be a finite totally real extension of\n$F$, and $f_{F'}$ be a Hilbert modular form over $F'$ that is\nordinary at each place of $F'$ above $p$. Then\n$$\\mu(L_p(F'K_\\infty, f_{F'}))=0.$$\n\\end{conj}\nCorollary \\ref{thm:mu} says that Conjecture \\ref{conj-mu} holds when\n$F'$ is a quadratic real extension of $F$ and $f_{F'}$ is the base\nchange of a Hilbert modular form over $F$ (that satisfies some\nconditions). It seems impossible to use the method in \\cite{CH18,\nHung} to prove Conjecture \\ref{conj-mu}, though it works when\n$F'=F$.", "post_theorem_intro_text_len": 5953, "post_theorem_intro_text": "We choose an imaginary quadratic extension $K$ of $F$ that is\ninertia at $\\mathfrak{q}$ when $[F:\\mathbb{Q}]$ is odd. Denote $KF'$\nby $K'$. Fix a place $\\mathfrak{p}$ of $F$  above $p$. Let $K_\\infty$ be the $\\mathfrak{p}$-anticyclotomic\n$\\mathbb{Z}_p$-extension of $K$.\n\nUsing the canonical periods one attaches to $f$,\n$f\\otimes\\chi_{F'/F}$ and $f'$ the $p$-adic $L$-functions\n$$L_{p}(K_{\\infty} ,f), \\\nL_{p}(K_{\\infty} ,f\\otimes\\chi_{F'/F})  \\ \\text{ and }\nL_{p}(K'K_{\\infty} ,f').$$ All of them can be considered as elements\nof $\\mathcal{O}_{\\mathfrak{P}}[[\\mathrm{Gal}(K_{\\infty} /K)]]$. The\nring $\\mathcal{O}_{\\mathfrak{P}}\\supset \\mathcal{O}_f$ is some\ncoefficient ring that is clearly defined in our context. To prove\nTheorem \\ref{thm:main} we only need to show\n\\begin{equation}\\label{eq:main}\n(L_{p}(K'K_{\\infty} ,f'))=(L_{p}(K_{\\infty} ,f))\\cdot\n(L_{p}(K_{\\infty} ,f\\otimes\\chi_{F'/F}))  \\end{equation} in\n$\\mathcal{O}_{\\mathfrak{P}}[[\\mathrm{Gal}(K_{\\infty} /K)]].$\n\nOur strategy is to use Iwasawa main conjecture that provides an\nequality between a quantity measuring Selmer groups and $p$-adic\n$L$-functions. Actually what we need is Iwasawa main conjecture for\nHilbert modular forms in the anticyclotomic setting.\n\nIts proof is divided into two parts, one part proving one\ndivisibility by Ribet's method, and the other proving the converse\ndivisibility by Euler systems. The former divisibility was proved by\nSkinner and Urban \\cite{S-U} for elliptic modular forms, and was\nproved by Wan \\cite{Wan} for Hilbert modular forms. When\n$[F:\\mathbb{Q}]$ is odd, Wan needs the condition that Ihara Lemma\nfor Shimura curves holds. The latter divisibility was proved by\nBertolini and Darmon \\cite{BD07} for elliptic curves, and by Chida\nand Hsieh \\cite{CH15} for elliptic modular forms. It was proved by\nLongo \\cite{Longo} and Wang \\cite{Wang} for Hilbert modular forms\nassuming Ihara Lemma for Shimura curves. This condition was removed\nby the second author \\cite{Xie}.\n\nCombining results in \\cite{Wan} and \\cite{Xie} we obtain the\nfollowing theorem. Remark that we do not need Ihara Lemma for\nShimura curves even when $[F:\\mathbb{Q}]$ is odd, though such a\ncondition is needed in \\cite{Wan}.\n\n\\begin{thm}\\label{thm:Iw-main-int}\nAssume that $f$ satisfies conditions $(\\mathrm{CR}^+)$,\n$(\\mathfrak{n}^+\\text{-}\\mathrm{DT})$, $(\\mathrm{PO})$ and $(\\mathrm{Fuji}1$-$4)$.\nThen we have $$(L_p(K_\\infty, f))=\\mathrm{char}\\ \\mathrm{Sel}\n(K_\\infty, A_{\\rho})^\\vee .$$\n\\end{thm} See our context for Conditions $(\\mathrm{CR}^+)$,\n$(\\mathfrak{n}^+\\text{-}\\mathrm{DT})$, $(\\mathrm{PO})$ and\n$(\\mathrm{Fuji}1$-$4)$.\n\nBy Theorem \\ref{thm:Iw-main-int} we have $$\n(L_{p}(K_{\\infty},f))=\\mathrm{char}\\\n\\mathrm{Sel}(K_{\\infty},A_{\\rho})^{\\vee} $$ and\n    $$  (L_{p}(K_{\\infty},f\\otimes\\chi_{F'/F}))=\\mathrm{char}\\ \\mathrm{Sel}(K_{\\infty},A_{\\rho\\otimes\\chi_{F'/F}})^{\\vee}    .$$\nUsing the relation\n$$\\mathrm{Sel}(K'K_{\\infty},A_{\\rho})=\\mathrm{Sel}(K_{\\infty},A_{\\rho})\\bigoplus\\mathrm{Sel}(K_{\\infty},A_{\\rho\\otimes\\chi_{F'/F}})$$\nwe obtain \\begin{equation}\\label{eq:Lp-eq} (L_{p}(K_{\\infty},f)\\cdot\nL_{p}(K_{\\infty},f\\otimes\\chi_{F'/F}))= \\mathrm{char}\\\n\\mathrm{Sel}(K'K_{\\infty},A_{\\rho})^{\\vee}. \\end{equation}\n\nLet $\\mathfrak{p}'$ be a place of $F'$ above $\\mathfrak{p}$. Let\n$K'_\\infty$ be the $\\mathfrak{p}'$-anticyclotomic\n$\\mathbb{Z}_p$-extension of $K'$.    With $K'_\\infty$ instead of\n$K_\\infty$ we also have $$ (L_{p}(K_{\\infty}',f'))=\\mathrm{char}\\\n\\mathrm{Sel}(K_{\\infty}',A_{\\rho})^{\\vee},$$ from which we deduce\nthat \\begin{equation}\\label{eq:con}(L_{p}(K'K_{\\infty},f'))\\supseteq\n\\mathrm{char}\\ \\mathrm{Sel}(K'K_{\\infty},A_{\\rho})^{\\vee}.\n\\end{equation} Combining (\\ref{eq:Lp-eq}) and (\\ref{eq:con}) we\nobtain\n\\begin{equation}\\label{eq:contain} (L_{p}(K'K_{\\infty},f'))\\supseteq (L_{p}(K_{\\infty},f)\\cdot\nL_{p}(K_{\\infty},f\\otimes\\chi_{F'/F})) . \\end{equation} As\n(\\ref{eq:main}) holds in\n$\\mathcal{O}_{\\mathfrak{P}}[[\\mathrm{Gal}(K_{\\infty}\n/K)]][\\frac{1}{p}]$, and as the $\\mu$-invariants of\n$L_{p}(K_{\\infty} ,f)$ and $L_{p}(K_{\\infty} ,f\\otimes\\chi_{F'/F})$\nare zero, (\\ref{eq:contain}) implies (\\ref{eq:main}).\n\nThe paper is organized as follows. In Section \\ref{sec:a} we collect\nsome basic facts about automorphic forms on definite quaternion\nalgebras. In Section \\ref{sec:b} we recall the construction of\nanticyclotomic $p$-adic $L$-functions via Theta elements. In Section\n\\ref{sec:d}, we combine Wan's result in \\cite{Wan} with the result\nin \\cite{Xie} to prove Iwasawa main conjecture for Hilbert modular\nform in the anticyclotomic setting (under certain conditions).\nFinally we prove Theorem \\ref{thm:main} in Section \\ref{sec:e}.\n\nIn Sections \\ref{sec:a}-\\ref{sec:d}, the imaginary quadratic\nextension $K$ of $F$ is fixed. In Section \\ref{sec:e}, we will\nchoose a suitable $K$.\n\nThe authors thank B.H. Gross for his helpful advice.\n\n\\subsection*{Notations}\n\nWe fix a totally real number field \\(F\\) over \\(\\mathbb{Q}\\). Let\n\\(\\Sigma_{F}\\) be the set of all real embeddings of \\(F\\) and let\n\\(\\Sigma_{p}\\) be the set of all places of \\(F\\) above \\(p\\).\n\nFor a number field \\(L\\), and each place \\(v\\) of \\(L\\) above a\nprime number \\(l\\), let \\(|\\cdot|_{v}\\) or \\(|\\cdot|_{L_{v}}\\) be\nthe absolute value on \\(L_{v}\\) defined by\n\\(|x|_{v}=|N_{L_{v}/\\mathbb{Q}_{l}}(x)|_{l}\\), and  let\n\\(\\varpi_{v}\\) be a uniformizer of \\(\\mathcal{O}_{L_{v}}\\).\n\nLet $E$ be a finite extension of $\\mathbb{Q}$, $\\mathcal{O}$ be the ring of\nintegers in $E$ and $\\mathfrak{P}$ be a prime  of $\\mathcal{O}$\nabove $p$ such that $\\mathcal{O}_\\mathfrak{P}$ contains\n$\\mathcal{O}_f$. Let $\\omega$ be a uniformizer of\n$\\mathcal{O}_\\mathfrak{P}$. For each positive integer $n$ we put\n$\\mathcal{O}_{n}=\\mathcal{O}_\\mathfrak{P}/\\omega^n$. We consider\n$E_\\mathfrak{P}$, $\\mathcal{O}_\\mathfrak{P}$ and $\\mathcal{O}_n$ as\ncoefficient rings, and let $G_F=\\mathrm{Gal}(\\overline{F}/F)$ act trivially on them.\n\nLet $\\epsilon$ be the $p$-adic cyclotomic character of $G_F$.", "sketch": "Choose an imaginary quadratic extension $K/F$ (inert at $\\mathfrak{q}$ when $[F:\\mathbb{Q}]$ is odd) and put $K'=KF'$. Fix $\\mathfrak{p}\\mid p$ of $F$ and let $K_\\infty$ be the $\\mathfrak{p}$-anticyclotomic $\\mathbb{Z}_p$-extension of $K$. Using the canonical periods, attach $p$-adic $L$-functions $L_p(K_\\infty,f)$, $L_p(K_\\infty,f\\otimes\\chi_{F'/F})$ and $L_p(K'K_\\infty,f')$ in $\\mathcal{O}_{\\mathfrak{P}}[[\\mathrm{Gal}(K_\\infty/K)]]$. To prove Theorem~\\ref{thm:main} it suffices to show the ideal identity\n\\[\\tag{\\ref{eq:main}} (L_{p}(K'K_{\\infty} ,f'))=(L_{p}(K_{\\infty} ,f))\\cdot (L_{p}(K_{\\infty} ,f\\otimes\\chi_{F'/F}))\\]\ninside $\\mathcal{O}_{\\mathfrak{P}}[[\\mathrm{Gal}(K_\\infty/K)]]$.\n\nThe strategy is to use the anticyclotomic Iwasawa main conjecture, i.e. an equality between $p$-adic $L$-functions and characteristic ideals of Selmer groups. By Theorem~\\ref{thm:Iw-main-int},\n\\[(L_p(K_\\infty,f))=\\mathrm{char}\\,\\mathrm{Sel}(K_\\infty,A_{\\rho})^\\vee,\\qquad (L_p(K_\\infty,f\\otimes\\chi_{F'/F}))=\\mathrm{char}\\,\\mathrm{Sel}(K_\\infty,A_{\\rho\\otimes\\chi_{F'/F}})^\\vee.\n\\]\nUsing the Selmer decomposition\n\\[\\mathrm{Sel}(K'K_{\\infty},A_{\\rho})=\\mathrm{Sel}(K_{\\infty},A_{\\rho})\\bigoplus\\mathrm{Sel}(K_{\\infty},A_{\\rho\\otimes\\chi_{F'/F}}),\\]\none gets\n\\[\\tag{\\ref{eq:Lp-eq}} (L_p(K_\\infty,f)\\cdot L_p(K_\\infty,f\\otimes\\chi_{F'/F}))=\\mathrm{char}\\,\\mathrm{Sel}(K'K_\\infty,A_{\\rho})^\\vee.\n\\]\nThen, letting $\\mathfrak{p}'\\mid\\mathfrak{p}$ in $F'$ and $K'_\\infty$ be the $\\mathfrak{p}'$-anticyclotomic $\\mathbb{Z}_p$-extension of $K'$, one also has $(L_p(K'_\\infty,f'))=\\mathrm{char}\\,\\mathrm{Sel}(K'_\\infty,A_{\\rho})^\\vee$, which implies the containment\n\\[\\tag{\\ref{eq:con}} (L_p(K'K_\\infty,f'))\\supseteq \\mathrm{char}\\,\\mathrm{Sel}(K'K_\\infty,A_{\\rho})^\\vee.\n\\]\nCombining \\eqref{eq:Lp-eq} and \\eqref{eq:con} yields\n\\[\\tag{\\ref{eq:contain}} (L_{p}(K'K_{\\infty},f'))\\supseteq (L_{p}(K_{\\infty},f)\\cdot L_{p}(K_{\\infty},f\\otimes\\chi_{F'/F})).\n\\]\nFinally, since \\eqref{eq:main} holds after inverting $p$ and the $\\mu$-invariants of $L_p(K_\\infty,f)$ and $L_p(K_\\infty,f\\otimes\\chi_{F'/F})$ are zero, the containment \\eqref{eq:contain} implies the equality \\eqref{eq:main}, hence Theorem~\\ref{thm:main}.", "expanded_sketch": "Choose an imaginary quadratic extension $K/F$ (inert at $\\mathfrak{q}$ when $[F:\\mathbb{Q}]$ is odd) and put $K'=KF'$. Fix $\\mathfrak{p}\\mid p$ of $F$ and let $K_\\infty$ be the $\\mathfrak{p}$-anticyclotomic $\\mathbb{Z}_p$-extension of $K$. Using the canonical periods, attach $p$-adic $L$-functions $L_p(K_\\infty,f)$, $L_p(K_\\infty,f\\otimes\\chi_{F'/F})$ and $L_p(K'K_\\infty,f')$ in $\\mathcal{O}_{\\mathfrak{P}}[[\\mathrm{Gal}(K_\\infty/K)]]$. To prove the main theorem, it suffices to show the ideal identity\n\\begin{equation}\\label{eq:main}\n(L_{p}(K'K_{\\infty} ,f'))=(L_{p}(K_{\\infty} ,f))\\cdot\n(L_{p}(K_{\\infty} ,f\\otimes\\chi_{F'/F}))  \\end{equation}\ninside $\\mathcal{O}_{\\mathfrak{P}}[[\\mathrm{Gal}(K_\\infty/K)]]$.\n\nThe strategy is to use the anticyclotomic Iwasawa main conjecture, i.e. an equality between $p$-adic $L$-functions and characteristic ideals of Selmer groups. We first use the following theorem.\n\\begin{thm}\\label{thm:Iw-main-int}\nAssume that $f$ satisfies conditions $(\\mathrm{CR}^+)$,\n$(\\mathfrak{n}^+\\text{-}\\mathrm{DT})$, $(\\mathrm{PO})$ and $(\\mathrm{Fuji}1$-$4)$.\nThen we have $$(L_p(K_\\infty, f))=\\mathrm{char}\\ \\mathrm{Sel}\n(K_\\infty, A_{\\rho})^\\vee .$$\n\\end{thm}\nThus\n\\[(L_p(K_\\infty,f))=\\mathrm{char}\\,\\mathrm{Sel}(K_\\infty,A_{\\rho})^\\vee,\\qquad (L_p(K_\\infty,f\\otimes\\chi_{F'/F}))=\\mathrm{char}\\,\\mathrm{Sel}(K_\\infty,A_{\\rho\\otimes\\chi_{F'/F}})^\\vee.\n\\]\nUsing the Selmer decomposition\n\\[\\mathrm{Sel}(K'K_{\\infty},A_{\\rho})=\\mathrm{Sel}(K_{\\infty},A_{\\rho})\\bigoplus\\mathrm{Sel}(K_{\\infty},A_{\\rho\\otimes\\chi_{F'/F}}),\\]\none gets\n\\begin{equation}\\label{eq:Lp-eq} (L_{p}(K_{\\infty},f)\\cdot\nL_{p}(K_{\\infty},f\\otimes\\chi_{F'/F}))= \\mathrm{char}\\\n\\mathrm{Sel}(K'K_{\\infty},A_{\\rho})^{\\vee}. \\end{equation}\nThen, letting $\\mathfrak{p}'\\mid\\mathfrak{p}$ in $F'$ and $K'_\\infty$ be the $\\mathfrak{p}'$-anticyclotomic $\\mathbb{Z}_p$-extension of $K'$, one also has $(L_p(K'_\\infty,f'))=\\mathrm{char}\\,\\mathrm{Sel}(K'_\\infty,A_{\\rho})^\\vee$, which implies the containment\n\\begin{equation}\\label{eq:con}(L_{p}(K'K_{\\infty},f'))\\supseteq\n\\mathrm{char}\\ \\mathrm{Sel}(K'K_{\\infty},A_{\\rho})^{\\vee}.\n\\end{equation}\nCombining the equation above and the containment above yields\n\\begin{equation}\\label{eq:contain} (L_{p}(K'K_{\\infty},f'))\\supseteq (L_{p}(K_{\\infty},f)\\cdot\nL_{p}(K_{\\infty},f\\otimes\\chi_{F'/F})) . \\end{equation}\nFinally, since the displayed ideal identity above holds after inverting $p$ and the $\\mu$-invariants of $L_p(K_\\infty,f)$ and $L_p(K_\\infty,f\\otimes\\chi_{F'/F})$ are zero, the containment above implies the ideal identity\n\\begin{equation}\\label{eq:main}\n(L_{p}(K'K_{\\infty} ,f'))=(L_{p}(K_{\\infty} ,f))\\cdot\n(L_{p}(K_{\\infty} ,f\\otimes\\chi_{F'/F}))  \\end{equation}\nand this completes the proof of the main theorem.", "expanded_theorem": "\\label{thm:main} Let $F$ be a totally real field,\nand $F'$ be a real quadratic extension of $F$. Assume that $p$ is\nunramified in both $F$ and $F'$. Let $f$ be a Hilbert newform of\nparallel even weight $k$ over $F$ with trivial central character and\nlevel $\\mathfrak{n}_f$. Let $f'$ be the base change of $f$ to $F'$.\nLet $\\mathfrak{n}_{f'}$ be the level of $f'$.\n\nLet $p$ be a prime number satisfying $p\\geq\\mathrm{max}(k+2,7)$. We\nassume that $\\mathfrak{n}_f$ is prime to $p$ and $f$ is ordinary at\neach prime above $p$. When $[F:\\mathbb{Q}]$ is odd, we assume that\nthere exists at least one prime $\\mathfrak{q}$ such that\n$\\mathfrak{q}|| \\mathfrak{n}_f$, and $\\mathfrak{q}$ is split in\n$F'$.\n\nSuppose the following conditions hold.\n\\begin{enumerate}\n\\item The restrictions of $\\bar{\\rho}_f$ to $G_{F'(\\xi_p)}$ and $G_{F'(\n\\sqrt{ p^*})}$ are absolutely irreducible, where\n$p^*=(-1)^{\\frac{p-1}{2}}p$.\n\\item In the case of $k=2$, for each place $v$ of $F$ above $p$ we\nhave  $a_v^2(f)\\ {\\backslash\\hskip -10pt \\equiv } 1 \\ (\\mathrm{mod}\n\\ p)$; similarly, for each place $v'$ of $F'$ above $p$ we have\n$a_{v'}^2(f')\\ {\\backslash\\hskip -10pt \\equiv } 1 \\ (\\mathrm{mod} \\\n p)$.\n\\item For each $\\mathfrak{l}|\\mathfrak{n}$, if\n$\\bar{\\rho}_f|_{I_{F_\\mathfrak{l}}}$ is absolutely irreducible, then\n$\\mathrm{N}(\\mathfrak{l})\\equiv  {\\hskip -10pt /}-1 \\ (\\mathrm{mod\n}\\ p)$. Similarly, for each $\\mathfrak{l}'|\\mathfrak{n}'$, if\n$\\bar{\\rho}_{f'}|_{I_{F'_{\\mathfrak{l}'}}}$ is absolutely\nirreducible, then $\\mathrm{N}(\\mathfrak{l}')\\equiv  {\\hskip -10pt\n/}-1 \\ (\\mathrm{mod }\\ p)$.\n\\item $\\rho_f$ is a minimal modular lifting of $\\bar{\\rho}_f$.\n\\end{enumerate} Then $$\n\\Omega_{f'}^\\mathrm{can}=(\\Omega_f^\\mathrm{can})^2 $$ up to a\n$p$-adic unit.", "theorem_type": ["Implication", "Equality or Bound"], "mcq": {"question": "Let $F$ be a totally real field and let $F'/F$ be a real quadratic extension. Let $f$ be a Hilbert newform over $F$ of parallel even weight $k$, trivial central character, and level $\\mathfrak{n}_f$, and let $f'$ be the base change of $f$ to $F'$ with level $\\mathfrak{n}_{f'}$. Let $p$ be a prime such that $p\\ge \\max(k+2,7)$ and $p$ is unramified in both $F$ and $F'$. Assume that $\\mathfrak{n}_f$ is prime to $p$ and that $f$ is ordinary at every prime of $F$ above $p$. If $[F:\\mathbb{Q}]$ is odd, assume there exists a prime $\\mathfrak{q}$ such that $\\mathfrak{q}\\parallel \\mathfrak{n}_f$ (that is, $\\mathfrak{q}$ divides $\\mathfrak{n}_f$ exactly once) and $\\mathfrak{q}$ is split in $F'$. Let $\\rho_f$ be the $p$-adic Galois representation attached to $f$, and let $\\bar\\rho_f$ and $\\bar\\rho_{f'}$ denote the residual Galois representations attached to $f$ and $f'$, respectively. Suppose moreover that:\n\n1. the restrictions of $\\bar\\rho_f$ to $G_{F'(\\xi_p)}$ and to $G_{F'(\\sqrt{p^*})}$ are absolutely irreducible, where $p^*=(-1)^{\\frac{p-1}{2}}p$;\n2. if $k=2$, then for every place $v$ of $F$ above $p$, $a_v(f)^2\\not\\equiv 1\\pmod p$, and for every place $v'$ of $F'$ above $p$, $a_{v'}(f')^2\\not\\equiv 1\\pmod p$;\n3. for every prime $\\mathfrak{l}\\mid \\mathfrak{n}_f$, if $\\bar\\rho_f|_{I_{F_{\\mathfrak{l}}}}$ is absolutely irreducible, then $\\mathrm N(\\mathfrak{l})\\not\\equiv -1\\pmod p$, and for every prime $\\mathfrak{l}'\\mid \\mathfrak{n}_{f'}$, if $\\bar\\rho_{f'}|_{I_{F'_{\\mathfrak{l}'}}}$ is absolutely irreducible, then $\\mathrm N(\\mathfrak{l}')\\not\\equiv -1\\pmod p$;\n4. $\\rho_f$ is a minimal modular lifting of $\\bar\\rho_f$.\n\nFor a modular form $g$, write its canonical period as $\\Omega_g^{\\mathrm{can}}=\\langle g,g\\rangle_{\\mathrm{Pet}}/\\eta_g$, where $\\langle g,g\\rangle_{\\mathrm{Pet}}$ is the Petersson inner product and $\\eta_g$ is the associated congruence number. Under these hypotheses, which statement about the canonical periods of $f$ and $f'$ is valid?", "correct_choice": {"label": "A", "text": "The canonical periods satisfy\n$$\\Omega_{f'}^{\\mathrm{can}}=(\\Omega_f^{\\mathrm{can}})^2$$\nup to a $p$-adic unit; equivalently, the ratio $\\Omega_{f'}^{\\mathrm{can}}/(\\Omega_f^{\\mathrm{can}})^2$ is a $p$-adic unit."}, "choices": [{"label": "B", "text": "The canonical periods satisfy\n$$\\Omega_{f'}^{\\mathrm{can}}=(\\Omega_f^{\\mathrm{can}})^2$$\nexactly in the coefficient ring, i.e. the ratio $\\Omega_{f'}^{\\mathrm{can}}/(\\Omega_f^{\\mathrm{can}})^2$ is equal to $1$."}, {"label": "C", "text": "The canonical periods satisfy\n$$\\Omega_{f'}^{\\mathrm{can}}=u\\, (\\Omega_f^{\\mathrm{can}})^2$$\nfor some nonzero element $u$ in the coefficient field; in particular, $\\Omega_{f'}^{\\mathrm{can}}$ and $(\\Omega_f^{\\mathrm{can}})^2$ agree up to multiplication by a scalar."}, {"label": "D", "text": "The canonical periods satisfy\n$$\\Omega_{f'}^{\\mathrm{can}}=(\\Omega_f^{\\mathrm{can}})^2$$\nup to a $p$-adic unit whenever the restrictions of $\\bar\\rho_f$ to $G_{F(\\xi_p)}$ and to $G_{F(\\sqrt{p^*})}$ are absolutely irreducible; no hypothesis involving the corresponding restrictions over $F'$ is needed."}, {"label": "E", "text": "The canonical periods satisfy\n$$\\Omega_{f'}^{\\mathrm{can}}=(\\Omega_f^{\\mathrm{can}})^2$$\nup to multiplication by an arbitrary $p$-power; equivalently, the ratio $\\Omega_{f'}^{\\mathrm{can}}/(\\Omega_f^{\\mathrm{can}})^2$ is merely $p$-adically integral."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "uniformity", "tampered_component": "unit-ambiguity from ideal identity and period normalization", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the requirement that the scalar be a p-adic unit", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "characteristic", "tampered_component": "base-change irreducibility hypotheses over F'", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "mu=0 used to upgrade containment to equality up to unit", "template_used": "uniformity_effectivity"}]}}
{"id": "2512.08676v1", "paper_link": "http://arxiv.org/abs/2512.08676v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "[Kang--Koh--Tran {\\cite{KKT25}}]\\label{thm:KKT-circle}\nLet $r,t\\in\\mathbb{N}$ with $r,t\\ge2$. There exists a measurable partition\n\\[\nC = \\bigcup_{i=1}^r E_i\n\\]\nsuch that for every finite measurable cover\n\\[\nC \\subset F_1 \\cup \\cdots \\cup F_t,\n\\]\nthere exist an index $m \\in \\{1,\\dots,t\\}$ and a rotation $R_{\\theta}$ satisfying\n\\begin{align*}\n    \\mu_1\\big(R_{\\theta}(F_m) \\cap E_i\\big) > 0\n\\quad\\text{for all } 1\\le i\\le r.\n\\end{align*}", "start_pos": 6333, "end_pos": 6799, "label": "thm:KKT-circle"}, "ref_dict": {"thm:KKT-circle": "\\begin{theorem}[Kang--Koh--Tran {\\cite{KKT25}}]\\label{thm:KKT-circle}\nLet $r,t\\in\\mathbb{N}$ with $r,t\\ge2$. There exists a measurable partition\n\\[\nC = \\bigcup_{i=1}^r E_i\n\\]\nsuch that for every finite measurable cover\n\\[\nC \\subset F_1 \\cup \\cdots \\cup F_t,\n\\]\nthere exist an index $m \\in \\{1,\\dots,t\\}$ and a rotation $R_{\\theta}$ satisfying\n\\begin{align*}\n    \\mu_1\\big(R_{\\theta}(F_m) \\cap E_i\\big) > 0\n\\quad\\text{for all } 1\\le i\\le r.\n\\end{align*}\n\\end{theorem}", "definition-rotation": "\\begin{align}\\label{definition-rotation}\nR_\\theta(x_1,x_2,x'',x_n)\n :=\\bigl(\\,\n        x_1\\cos(2\\pi\\theta) - x_2\\sin(2\\pi\\theta),\\ \n        x_1\\sin(2\\pi\\theta) + x_2\\cos(2\\pi\\theta),\\ \n        x'',\\ x_n\n      \\,\\bigr),\n\\end{align}"}, "pre_theorem_intro_text_len": 2055, "pre_theorem_intro_text": "Classical Ramsey theory typically asks which structured subsets must appear in any finite colouring of the natural numbers. Raimi proposed a complementary point of view: he asked which partitions of $\\mathbb{N}$ cannot be avoided by any finite colouring of $\\mathbb{N}$, even after allowing a shift.\n\nMore precisely, given a partition $\\mathbb{N}=E_1\\cup E_2$, we say that $(E_1,E_2)$ is \\emph{unavoidable} if for every finite partition $\\mathbb{N}=\\bigcup\\limits_{j=1}^t F_j$ there exist $j\\in\\{1,\\dots,t\\}$ and $k\\in\\mathbb{N}$ such that both $(F_j+k)\\cap E_1$ and $(F_j+k)\\cap E_2$ are infinite. This formalises Raimi’s viewpoint on unavoidable partitions under shifts.\n\nA classical theorem of Raimi \\cite{Raimi} shows that such partitions do exist: there is a partition $\\mathbb{N}=E_1\\cup E_2$ with the property that for every finite partition $\\mathbb{N}=\\bigcup\\limits_{j=1}^t F_j$ with $t\\in\\mathbb{N}$, one can find $j\\in\\{1,\\dots,t\\}$ and $k\\in\\mathbb{N}$ such that both $(F_j+k)\\cap E_1$ and $(F_j+k)\\cap E_2$ are infinite. Raimi’s original proof used topological methods. Hindman later gave an elementary proof \\cite[p.~180, Theorem~11.15]{HM79} and showed that one may take $E_1$ to be the set of natural numbers whose last non-zero digit in the ternary expansion is $1$, and $E_2=\\mathbb{N}\\setminus E_1$.\n\nStrengthenings of this phenomenon, imposing density conditions on the partition sets or guaranteeing positive densities in the conclusion, were obtained by Hegyv\\'ari \\cite{NH} and by Bergelson and Weiss \\cite{Bergelson2}. More recently, Hegyv\\'ari, Pach, and Pham \\cite{HPP25} introduced a powerful and flexible framework, combining tools from harmonic analysis, additive combinatorics, and group theory, which yields polynomial and finite-group extensions of Raimi’s theorem and makes its connection to Ramsey theory explicit. Their beautiful construction in the finite-group setting has since been extended successfully to the continuous setting for circles by Kang, Koh, and the author in \\cite{KKT25}, which is stated as follows.", "context": "Classical Ramsey theory typically asks which structured subsets must appear in any finite colouring of the natural numbers. Raimi proposed a complementary point of view: he asked which partitions of $\\mathbb{N}$ cannot be avoided by any finite colouring of $\\mathbb{N}$, even after allowing a shift.\n\nMore precisely, given a partition $\\mathbb{N}=E_1\\cup E_2$, we say that $(E_1,E_2)$ is \\emph{unavoidable} if for every finite partition $\\mathbb{N}=\\bigcup\\limits_{j=1}^t F_j$ there exist $j\\in\\{1,\\dots,t\\}$ and $k\\in\\mathbb{N}$ such that both $(F_j+k)\\cap E_1$ and $(F_j+k)\\cap E_2$ are infinite. This formalises Raimi’s viewpoint on unavoidable partitions under shifts.\n\nA classical theorem of Raimi \\cite{Raimi} shows that such partitions do exist: there is a partition $\\mathbb{N}=E_1\\cup E_2$ with the property that for every finite partition $\\mathbb{N}=\\bigcup\\limits_{j=1}^t F_j$ with $t\\in\\mathbb{N}$, one can find $j\\in\\{1,\\dots,t\\}$ and $k\\in\\mathbb{N}$ such that both $(F_j+k)\\cap E_1$ and $(F_j+k)\\cap E_2$ are infinite. Raimi’s original proof used topological methods. Hindman later gave an elementary proof \\cite[p.~180, Theorem~11.15]{HM79} and showed that one may take $E_1$ to be the set of natural numbers whose last non-zero digit in the ternary expansion is $1$, and $E_2=\\mathbb{N}\\setminus E_1$.\n\nStrengthenings of this phenomenon, imposing density conditions on the partition sets or guaranteeing positive densities in the conclusion, were obtained by Hegyv\\'ari \\cite{NH} and by Bergelson and Weiss \\cite{Bergelson2}. More recently, Hegyv\\'ari, Pach, and Pham \\cite{HPP25} introduced a powerful and flexible framework, combining tools from harmonic analysis, additive combinatorics, and group theory, which yields polynomial and finite-group extensions of Raimi’s theorem and makes its connection to Ramsey theory explicit. Their beautiful construction in the finite-group setting has since been extended successfully to the continuous setting for circles by Kang, Koh, and the author in \\cite{KKT25}, which is stated as follows.", "full_context": "Classical Ramsey theory typically asks which structured subsets must appear in any finite colouring of the natural numbers. Raimi proposed a complementary point of view: he asked which partitions of $\\mathbb{N}$ cannot be avoided by any finite colouring of $\\mathbb{N}$, even after allowing a shift.\n\nMore precisely, given a partition $\\mathbb{N}=E_1\\cup E_2$, we say that $(E_1,E_2)$ is \\emph{unavoidable} if for every finite partition $\\mathbb{N}=\\bigcup\\limits_{j=1}^t F_j$ there exist $j\\in\\{1,\\dots,t\\}$ and $k\\in\\mathbb{N}$ such that both $(F_j+k)\\cap E_1$ and $(F_j+k)\\cap E_2$ are infinite. This formalises Raimi’s viewpoint on unavoidable partitions under shifts.\n\nA classical theorem of Raimi \\cite{Raimi} shows that such partitions do exist: there is a partition $\\mathbb{N}=E_1\\cup E_2$ with the property that for every finite partition $\\mathbb{N}=\\bigcup\\limits_{j=1}^t F_j$ with $t\\in\\mathbb{N}$, one can find $j\\in\\{1,\\dots,t\\}$ and $k\\in\\mathbb{N}$ such that both $(F_j+k)\\cap E_1$ and $(F_j+k)\\cap E_2$ are infinite. Raimi’s original proof used topological methods. Hindman later gave an elementary proof \\cite[p.~180, Theorem~11.15]{HM79} and showed that one may take $E_1$ to be the set of natural numbers whose last non-zero digit in the ternary expansion is $1$, and $E_2=\\mathbb{N}\\setminus E_1$.\n\nStrengthenings of this phenomenon, imposing density conditions on the partition sets or guaranteeing positive densities in the conclusion, were obtained by Hegyv\\'ari \\cite{NH} and by Bergelson and Weiss \\cite{Bergelson2}. More recently, Hegyv\\'ari, Pach, and Pham \\cite{HPP25} introduced a powerful and flexible framework, combining tools from harmonic analysis, additive combinatorics, and group theory, which yields polynomial and finite-group extensions of Raimi’s theorem and makes its connection to Ramsey theory explicit. Their beautiful construction in the finite-group setting has since been extended successfully to the continuous setting for circles by Kang, Koh, and the author in \\cite{KKT25}, which is stated as follows.\n\nA classical theorem of Raimi \\cite{Raimi} shows that such partitions do exist: there is a partition $\\mathbb{N}=E_1\\cup E_2$ with the property that for every finite partition $\\mathbb{N}=\\bigcup\\limits_{j=1}^t F_j$ with $t\\in\\mathbb{N}$, one can find $j\\in\\{1,\\dots,t\\}$ and $k\\in\\mathbb{N}$ such that both $(F_j+k)\\cap E_1$ and $(F_j+k)\\cap E_2$ are infinite. Raimi’s original proof used topological methods. Hindman later gave an elementary proof \\cite[p.~180, Theorem~11.15]{HM79} and showed that one may take $E_1$ to be the set of natural numbers whose last non-zero digit in the ternary expansion is $1$, and $E_2=\\mathbb{N}\\setminus E_1$.\n\nBefore stating the main results, we need to introduce some notation.\n\nFor every finite measurable cover\n\\[\n\\mathbb{S}^{n-1} \\subset F_1\\cup\\cdots\\cup F_t,\n\\]\nthere exist an index $m_0\\in\\{1,\\dots,t\\}$ and $\\theta_0\\in C$ such that\n\\[\n\\sigma_{n-1}\\big(R_{\\theta_0}(F_{m_0})\n \\cap E_i^{\\mathbb{S}^{n-1}}\\big)>0\n\\qquad\\text{for all }1\\le i\\le r.\n\\]\nwhere $R_\\theta$ denotes the rotation defined in \\eqref{definition-rotation} as restricted to $\\mathbb{S}^{n-1}$.\n\\end{theorem}\n\n\\begin{theorem}[Rotational power surfaces]\\label{thm:power-surface-Raimi}\nLet $n\\ge 3$ and $k>0$. Define\n\\[\n\\mathcal{S}_{k,R}\n := \\big\\{(x',x_n)\\in\\mathbb{R}^{n-1}\\times\\mathbb{R}\n : x_n = |x'|^{\\,k},\\ 0<|x'|\\le R\\big\\}.\n\\]\nLet\n$\\sigma_{k,R}$ be the normalized surface measure on $\\mathcal{S}_{k,R}$.\nThen there exists a measurable partition\n$\\{E_i^{\\mathcal{S}_{k,R}}\\}_{i=1}^r$ of $\\mathcal{S}_{k,R}$ with the following property:\nFor every finite measurable cover\n\\[\n\\mathcal{S}_{k,R} \\subset F_1\\cup\\cdots\\cup F_t\n\\]\nthere exist an index $m_0\\in\\{1,\\dots,t\\}$ and $\\theta_0\\in C$ such that\n\\[\n\\sigma_{k,R}\\big(R_{\\theta_0}(F_{m_0})\n \\cap E_i^{\\mathcal{S}_{k,R}}\\big)>0\n\\qquad\\text{for all }1\\le i\\le r.\n\\]\nIn particular, $k=1$ yields the cone, and $k=2$ yields the paraboloid.\n\\end{theorem}\n\nFor every finite measurable cover\n\\[\n\\mathcal{C}_{R,\\Omega} \\subset F_1\\cup\\cdots\\cup F_t\n\\]\nthere exist an index $m_0\\in\\{1,\\dots,t\\}$ and $\\theta_0\\in C$ such that\n\\[\n\\mu_{\\text{\\tiny $R,\\Omega$}}\\big(R_{\\theta_0}(F_{m_0})\\cap E_i^{\\mathcal{C}_{R,\\Omega}}\\big)>0\n\\qquad\\text{for all }1\\le i\\le r,\n\\]\nwhere $R_\\theta$ denotes the rotation defined in \\eqref{definition-rotation} as restricted to $\\mathcal{C}_{R,\\Omega}$.\n\\end{theorem}\n\nWe now present the following general theorem, of which the spherical, cylindrical, and rotational power surface cases are specific instances.\n\\begin{theorem}\\label{thm:general-circle-bundle}\nLet $(X,\\mu)$ be a probability space equipped with a measurable, measure-preserving action\n\\[\n\\{R_\\theta\\}_{\\theta\\in C},\\qquad C:=\\mathbb{R}/\\mathbb{Z},\n\\]\nof the circle group $C$. Assume there exist\n\\begin{itemize}\n \\item a probability space $(Y,\\nu)$,\n \\item a measurable set $N\\subset X$ with $\\mu(N)=0$,\n \\item a measurable bijection\n \\[\n \\Phi : C\\times Y \\longrightarrow X\\setminus N,\n \\]\n\\end{itemize}\nsatisfying:\n\\begin{enumerate}\n \\item[(i)] (\\emph{Equivariance}) For all $\\theta,\\alpha\\in C$ and $y\\in Y$,\n \\[\n \\Phi(\\theta+\\alpha,y) = R_\\alpha(\\Phi(\\theta,y)).\n \\]\n \\item[(ii)] (\\emph{Product disintegration}) For every bounded measurable function $f:X\\to\\mathbb{R}$,\n \\[\n \\int_X f(x)\\,d\\mu(x)\n = \\int_C \\int_Y f(\\Phi(\\theta,y))\\,d\\nu(y)\\,d\\mu_1(\\theta),\n \\]\n where $\\mu_1$ is the normalized Lebesgue measure on $C$.\n\\end{enumerate}\nLet $\\{E_i^C\\}_{i=1}^r$ be a measurable partition of $C$ with the Raimi property from Theorem~\\ref{thm:KKT-circle}. Define \n\\begin{equation*}\n \\begin{cases}\n E_1^X := \\Phi(E_1^C\\times Y) \\cup N,\\\\\n E_i^X:=\\Phi(E_i^C\\times Y), ~~~~~~~~2 \\leq i\\leq r.\n \\end{cases}\n\\end{equation*}\nThen $\\{E_i^X\\}_{i=1}^r$ is a measurable partition of $X$ with the following property:\n\nNow define a measurable partition of $C$ by\n\\[\n C_m := \\{\\theta\\in C : m(\\theta)=m\\},\\qquad 1\\le m\\le t.\n\\]\nBy construction, for every $\\theta\\in C_m$ we have\n\\begin{equation}\\label{eq:Am-lower-bound}\n \\nu(A_m(\\theta))\\ge \\frac1t.\n\\end{equation}\nBy applying Theorem~ \\ref{thm:KKT-circle} to the partition $\\{C_m\\}_{m=1}^t$ of $C$, there exist an index $m_0\\in\\{1,\\dots,t\\}$ and a rotation\n\\[\n R_{\\theta_0}: C\\to C,\\qquad R_{\\theta_0}(\\theta)=\\theta+\\theta_0,\n\\]\nsuch that\n\\begin{equation}\\label{eq:KKT-circle-hit}\n \\mu_1\\big(R_{\\theta_0}(C_{m_0})\\cap E_i^C\\big)>0,\n \\qquad\\, \\forall\\, 1\\le i\\le r.\n\\end{equation}\n\nTo complete the proof of the theorem, it remains to use \\eqref{eq:KKT-circle-hit} to show that\n\\begin{align}\\label{ineq:intersection-rotation-E_i^X}\n \\mu\\bigl(R_{\\theta_0}(F_{m_0}) \\cap E_i^X\\bigr) > 0\n \\qquad \\text{for all } 1 \\le i \\le r.\n\\end{align}\nLet \n\\[\n R_{\\theta_0}(\\theta,y):=(\\theta+\\theta_0,y)\n\\]\nbe the induced rotation on $C\\times Y$. By equivariance, we have \n\\[\n R_{\\theta_0}(F_{m_0})\\cap (X\\setminus N)\n = R_{\\theta_0}\\big(\\Phi(F_{m_0}^{\\mathrm e})\\big)\n = \\Phi\\big(R_{\\theta_0}(F_{m_0}^{\\mathrm e})\\big),\n\\]\nwhich, together with the definition of $E_i^X$, gives\n\\[\n R_{\\theta_0}(F_{m_0})\\cap E_i^X\n = \\Phi\\Big(R_{\\theta_0}(F_{m_0}^{\\mathrm e})\\cap (E_i^C\\times Y)\\Big)\n \\cup \\Big(R_{\\theta_0}(F_{m_0})\\cap N\\cap E_i^X\\Big).\n\\]\nBy the assumption on $N$, the definition of definition of $F_{m_0}^{\\mathrm e}$, and the rotation on $C\\times Y$, we have\n\\begin{align*}\n \\mu\\big(R_{\\theta_0}(F_{m_0})\\cap E_i^X\\big)\n =& (\\mu_1\\times\\nu)\\big(R_{\\theta_0}(F_{m_0}^{\\mathrm e})\\cap (E_i^C\\times Y)\\big) \\\\\n = &\\int_{C}\\int_Y \n \\mathbf{1}_{R_{\\theta_0}(F_{m_0}^{\\mathrm e})}(\\theta,y)\n \\mathbf{1}_{E_i^C}(\\theta)\\,d\\nu(y)\\,d\\mu_1(\\theta) \\\\\n =& \\int_{C}\\int_Y \n \\mathbf{1}_{A_{m_0}(\\theta-\\theta_0)}(y)\\mathbf{1}_{E_i^C}(\\theta)\n \\,d\\nu(y)\\,d\\mu_1(\\theta) \\\\\n =& \\int_{C} \\nu\\big(A_{m_0}(\\theta-\\theta_0)\\big)\\,\\mathbf{1}_{E_i^C}(\\theta)\\,d\\mu_1(\\theta).\n\\end{align*}\nBy restricting the outer integral to those $\\theta$ satisfying \n$\\theta-\\theta_0\\in C_{m_0}$, i.e.\\ $\\theta\\in R_{\\theta_0}(C_{m_0})$, and using \\eqref{eq:Am-lower-bound} together with \\eqref{eq:KKT-circle-hit}, we obtain\n\\begin{align*}\n \\mu\\big(R_{\\theta_0}(F_{m_0})\\cap E_i^X\\big)\n \\geq & \\int_{R_{\\theta_0}(C_{m_0})\\cap E_i^C} \\nu\\big(A_{m_0}(\\theta-\\theta_0)\\big)\\,d\\mu_1(\\theta) \\\\\n &\\ge \\frac1t\\,\\mu_1\\big(R_{\\theta_0}(C_{m_0})\\cap E_i^C\\big)>0\n\\end{align*}\nThis is precisely \\eqref{ineq:intersection-rotation-E_i^X}. This completes the proof.\n\\end{proof}\n\n\\begin{align}\\label{definition-rotation}\nR_\\theta(x_1,x_2,x'',x_n)\n :=\\bigl(\\,\n        x_1\\cos(2\\pi\\theta) - x_2\\sin(2\\pi\\theta),\\ \n        x_1\\sin(2\\pi\\theta) + x_2\\cos(2\\pi\\theta),\\ \n        x'',\\ x_n\n      \\,\\bigr),\n\\end{align}\n\n\\begin{theorem}[Kang--Koh--Tran {\\cite{KKT25}}]\\label{thm:KKT-circle}\nLet $r,t\\in\\mathbb{N}$ with $r,t\\ge2$. There exists a measurable partition\n\\[\nC = \\bigcup_{i=1}^r E_i\n\\]\nsuch that for every finite measurable cover\n\\[\nC \\subset F_1 \\cup \\cdots \\cup F_t,\n\\]\nthere exist an index $m \\in \\{1,\\dots,t\\}$ and a rotation $R_{\\theta}$ satisfying\n\\begin{align*}\n    \\mu_1\\big(R_{\\theta}(F_m) \\cap E_i\\big) > 0\n\\quad\\text{for all } 1\\le i\\le r.\n\\end{align*}\n\\end{theorem}", "post_theorem_intro_text_len": 5949, "post_theorem_intro_text": "The theorem provides a partition of the circle with the property that every finite measurable cover admits a translate meeting each partition element in positive measure.  This paper shows that this phenomenon extends beyond compact abelian groups to a wide class of non-group geometric surfaces that still exhibit \\textit{a hidden one-dimensional symmetry}. This answers a question raised in \\cite{KKT25} concerning the extension\nfrom the circle $C$ to the unit sphere $\\mathbb{S}^{n-1}\\subset \\mathbb{R}^n$.\n\nBefore stating the main results, we need to introduce some notation.\n\nFor notational convenience, given a point \n$x=(x_1,\\dots,x_n)\\in\\mathbb{R}^n$, we write\n\\[\nx' := (x_1,\\dots,x_{n-1}) \\in \\mathbb{R}^{\\,n-1},\n\\qquad\nx'' := (x_3,\\dots,x_{n-1}) \\in \\mathbb{R}^{\\,n-3}.\n\\]\nThroughout the paper, we denote by $C=\\mathbb{R}/\\mathbb{Z}$ the circle group. Let $\\{E_i^C\\}_{i=1}^r$ be the measurable partition of $C$ provided by Theorem~\\ref{thm:KKT-circle}.  \n\nWe also introduce the rotation\n\\begin{align}\\label{definition-rotation}\nR_\\theta(x_1,x_2,x'',x_n)\n :=\\bigl(\\,\n        x_1\\cos(2\\pi\\theta) - x_2\\sin(2\\pi\\theta),\\ \n        x_1\\sin(2\\pi\\theta) + x_2\\cos(2\\pi\\theta),\\ \n        x'',\\ x_n\n      \\,\\bigr),\n\\end{align}\nthat is, rotation by angle $2\\pi\\theta$ in the $(x_1,x_2)$–plane.\n\n\\begin{theorem}[Spheres]\\label{thm:sphere-Raimi}\nLet $n\\ge 3$,\n\\[\n\\mathbb{S}^{n-1} := \\{x\\in\\mathbb{R}^n : |x|=1\\}\n\\]\nbe the unit sphere equipped with the normalized surface measure $\\sigma_{n-1}$.\nThen there exists a measurable partition $\\{E_i^{\\mathbb{S}^{n-1}}\\}_{i=1}^r$ of $\\mathbb{S}^{n-1}$ such that:\n\nFor every finite measurable cover\n\\[\n\\mathbb{S}^{n-1} \\subset F_1\\cup\\cdots\\cup F_t,\n\\]\nthere exist an index $m_0\\in\\{1,\\dots,t\\}$ and $\\theta_0\\in C$ such that\n\\[\n\\sigma_{n-1}\\big(R_{\\theta_0}(F_{m_0})\n    \\cap E_i^{\\mathbb{S}^{n-1}}\\big)>0\n\\qquad\\text{for all }1\\le i\\le r.\n\\]\nwhere $R_\\theta$ denotes the rotation defined in \\eqref{definition-rotation} as restricted to $\\mathbb{S}^{n-1}$.\n\\end{theorem}\n\n\\begin{theorem}[Rotational power surfaces]\\label{thm:power-surface-Raimi}\nLet $n\\ge 3$ and $k>0$. Define\n\\[\n\\mathcal{S}_{k,R}\n := \\big\\{(x',x_n)\\in\\mathbb{R}^{n-1}\\times\\mathbb{R}\n        : x_n = |x'|^{\\,k},\\ 0<|x'|\\le R\\big\\}.\n\\]\nLet\n$\\sigma_{k,R}$ be the normalized surface measure on $\\mathcal{S}_{k,R}$.\nThen there exists a measurable partition\n$\\{E_i^{\\mathcal{S}_{k,R}}\\}_{i=1}^r$ of $\\mathcal{S}_{k,R}$ with the following property:\nFor every finite measurable cover\n\\[\n\\mathcal{S}_{k,R} \\subset F_1\\cup\\cdots\\cup F_t\n\\]\nthere exist an index $m_0\\in\\{1,\\dots,t\\}$ and $\\theta_0\\in C$ such that\n\\[\n\\sigma_{k,R}\\big(R_{\\theta_0}(F_{m_0})\n       \\cap E_i^{\\mathcal{S}_{k,R}}\\big)>0\n\\qquad\\text{for all }1\\le i\\le r.\n\\]\nIn particular, $k=1$ yields the cone, and $k=2$ yields the paraboloid.\n\\end{theorem}\n\n\\begin{theorem}[Cylindrical surface]\\label{thm:cylinder-Raimi}\nLet $n\\ge 3$, $R>0$, and $\\Omega\\subset\\mathbb{R}^{n-2}$ be a\nbounded Borel set. Define the cylindrical surface\n\\[\n\\mathcal{C}_{R,\\Omega}\n:= \\big\\{(x_1,x_2, x'', x_n)\\in\\mathbb{R}^2\\times\\mathbb{R}^{n-2}\n : x_1^2+x_2^2=R^2,\\ (x'', x_n)\\in\\Omega\\big\\}.\n\\]\nLet $\\mu_{\\text{\\tiny $R,\\Omega$}}$ be the normalized surface measure on $\\mathcal{C}_{R,\\Omega}$.\nThen there exists a measurable partition $\\{E_i^{\\mathcal{C}_{R,\\Omega}}\\}_{i=1}^r$ of $\\mathcal{C}_{R,\\Omega}$ with the following property:\n\nFor every finite measurable cover\n\\[\n\\mathcal{C}_{R,\\Omega} \\subset F_1\\cup\\cdots\\cup F_t\n\\]\nthere exist an index $m_0\\in\\{1,\\dots,t\\}$ and $\\theta_0\\in C$ such that\n\\[\n\\mu_{\\text{\\tiny $R,\\Omega$}}\\big(R_{\\theta_0}(F_{m_0})\\cap E_i^{\\mathcal{C}_{R,\\Omega}}\\big)>0\n\\qquad\\text{for all }1\\le i\\le r,\n\\]\nwhere $R_\\theta$ denotes the rotation defined in \\eqref{definition-rotation} as restricted to $\\mathcal{C}_{R,\\Omega}$.\n\\end{theorem}\n\n\\medskip\n\\paragraph{Sketch of proof.}\nThe essential observation is that spheres, rotational power surfaces, and cylinders all carry a natural measure-preserving action of the circle $C$ given by rotation in the $(x_1,x_2)$-plane.  Moreover, each of these surfaces admits a measurable, measure-preserving trivialization\n\\[\n\\Phi : C\\times Y \\longrightarrow X\\setminus N,\n\\]\nwhere $Y$ is a suitable parameter space and $N$ is a lower-dimensional set of measure zero.  This allows the surface measure on $X$ to disintegrate as\n\\[\nd\\mu_X(x)\n = d\\mu_1(\\theta)\\, d\\nu_Y(y),\n\\]\nmirroring the product structure of $C\\times Y$.\nThe equivariance identity\n\\[\n\\Phi(\\theta+\\alpha,y)=R_\\alpha(\\Phi(\\theta,y))\n\\]\nthen enables the measurable Raimi partition on the base circle to be lifted directly to a partition of $X$.\n\nThe proofs of our three geometric theorems reduce to verifying the structural hypotheses of the general circle-bundle theorem introduced in Section~\\ref{section-main-result}.  Once this framework is in place, the partition constructed in the previous work of Kang, Koh, and the author on the circle $C$ automatically induces the desired Raimi-type partitions on $\\mathbb{S}^{n-1}$, on rotational power surfaces, and on circular cylindrical surfaces.\n\nOur approach naturally leads to the following open question: \nLet $M$ be a compact hyperbolic surface with its normalized area measure. \nDoes $M$ admit a measurable Raimi--type partition (in the sense of this paper), even though it has no circle action and therefore lies outside our circle--bundle framework?\n\n{\\bf The paper is organized as follows.}\nIn Section~\\ref{section-main-result}, we establish the general circle-bundle theorem, which serves as the main structural tool of the paper.  Sections~\\ref{section-sphere}, \\ref{section-power-surface}, and \\ref{section-cylinder} are devoted to the proofs of the three principal applications: the sphere, the rotational power surfaces, and the circular cylindrical surfaces, respectively.  Each result follows by verifying the hypotheses of the general theorem and applying the measurable Raimi partition on the base circle.", "sketch": "The post-theorem text does not sketch a proof of Theorem~\\ref{thm:KKT-circle}; it only states that Theorem~\\ref{thm:KKT-circle} provides a partition of the circle with a translate/rotation intersection property and then uses it as an input for later results. The only explicit \\paragraph{Sketch of proof.} concerns the later geometric theorems (sphere/power surfaces/cylinders): it observes these surfaces have a measure-preserving circle action by rotation in the $(x_1,x_2)$-plane, admits a measurable measure-preserving trivialization \\(\\Phi: C\\times Y\\to X\\setminus N\\) leading to disintegration \\(d\\mu_X=d\\mu_1(\\theta)\\,d\\nu_Y(y)\\), uses equivariance \\(\\Phi(\\theta+\\alpha,y)=R_\\alpha(\\Phi(\\theta,y))\\) to lift the partition from the base circle, and reduces the proofs to verifying hypotheses of a general circle-bundle theorem and then applying the partition from Kang--Koh--Tran on \\(C\\).", "expanded_sketch": "The post-theorem text does not sketch a proof of the main theorem; it only states that the main theorem provides a partition of the circle with a translate/rotation intersection property and then uses it as an input for later results. The only explicit \\paragraph{Sketch of proof.} concerns the later geometric theorems (sphere/power surfaces/cylinders): it observes these surfaces have a measure-preserving circle action by rotation in the $(x_1,x_2)$-plane, admits a measurable measure-preserving trivialization \\(\\Phi: C\\times Y\\to X\\setminus N\\) leading to disintegration \\(d\\mu_X=d\\mu_1(\\theta)\\,d\\nu_Y(y)\\), uses equivariance \\(\\Phi(\\theta+\\alpha,y)=R_\\alpha(\\Phi(\\theta,y))\\) to lift the partition from the base circle, and reduces the proofs to verifying hypotheses of a general circle-bundle theorem and then applying the partition from the main theorem on \\(C\\).", "expanded_theorem": "[Kang--Koh--Tran {\\cite{KKT25}}]\\label{thm:KKT-circle}\nLet $r,t\\in\\mathbb{N}$ with $r,t\\ge2$. There exists a measurable partition\n\\[\nC = \\bigcup_{i=1}^r E_i\n\\]\nsuch that for every finite measurable cover\n\\[\nC \\subset F_1 \\cup \\cdots \\cup F_t,\n\\]\nthere exist an index $m \\in \\{1,\\dots,t\\}$ and a rotation $R_{\\theta}$ satisfying\n\\begin{align*}\n    \\mu_1\\big(R_{\\theta}(F_m) \\cap E_i\\big) > 0\n\\quad\\text{for all } 1\\le i\\le r.\n\\end{align*}", "theorem_type": ["Existential–Universal", "Universal–Existential"], "mcq": {"question": "Let $C:=\\mathbb{R}/\\mathbb{Z}$ be the circle group, let $\\mu_1$ denote its normalized Haar (Lebesgue) probability measure, and for $\\theta\\in C$ let the rotation $R_\\theta:C\\to C$ be given by $R_\\theta(x)=x+\\theta\\pmod 1$. Fix integers $r,t\\in\\mathbb{N}$ with $r,t\\ge 2$. Which statement holds?", "correct_choice": {"label": "A", "text": "There exists a measurable partition $C=\\bigcup_{i=1}^r E_i$ such that for every measurable cover $C\\subset F_1\\cup\\cdots\\cup F_t$, there exist an index $m\\in\\{1,\\dots,t\\}$ and some rotation $R_\\theta$ with $\\mu_1\\big(R_\\theta(F_m)\\cap E_i\\big)>0$ for every $1\\le i\\le r$."}, "choices": [{"label": "B", "text": "There exists a measurable partition $C=\\bigcup_{i=1}^r E_i$ such that for every measurable cover $C\\subset F_1\\cup\\cdots\\cup F_t$, there exists a single rotation $R_\\theta$ for which, for every $m\\in\\{1,\\dots,t\\}$, one has $\\mu_1\\big(R_\\theta(F_m)\\cap E_i\\big)>0$ for every $1\\le i\\le r$."}, {"label": "C", "text": "There exists a measurable partition $C=\\bigcup_{i=1}^r E_i$ such that for every measurable cover $C\\subset F_1\\cup\\cdots\\cup F_t$, there exist an index $m\\in\\{1,\\dots,t\\}$ and some rotation $R_\\theta$ with $\\mu_1\\big(R_\\theta(F_m)\\cap E_i\\big)>0$ for at least one $i\\in\\{1,\\dots,r\\}$."}, {"label": "D", "text": "For every measurable partition $C=\\bigcup_{i=1}^r E_i$, and for every measurable cover $C\\subset F_1\\cup\\cdots\\cup F_t$, there exist an index $m\\in\\{1,\\dots,t\\}$ and some rotation $R_\\theta$ with $\\mu_1\\big(R_\\theta(F_m)\\cap E_i\\big)>0$ for every $1\\le i\\le r$."}, {"label": "E", "text": "There exists a measurable partition $C=\\bigcup_{i=1}^r E_i$ such that for every measurable cover $C\\subset F_1\\cup\\cdots\\cup F_t$, there exist an index $m\\in\\{1,\\dots,t\\}$ and some rotation $R_\\theta$ with $\\mu_1\\big(R_\\theta(F_m)\\cap E_i\\big)=\\mu_1(E_i)$ for every $1\\le i\\le r$."}], "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": "existential choice of a single favorable cover element", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the requirement that all partition cells be hit positively", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "special existence of the partition", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "positive-measure intersection replaced by full-measure capture of each partition cell", "template_used": "wildcard"}]}}
{"id": "2512.09203v1", "paper_link": "http://arxiv.org/abs/2512.09203v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{themain}\nLet $f$ be a cuspidal Hecke eigenform of level $1$ with Hecke eigenvalues $\\lambda(n)\\ll \\tau(n)n^{\\theta_f}$. If $f$ is Maa\\ss~form we further assume  that its root number satisfies $\\ve(f)=1$. For $q\\nequiv 2 \\pmod4$ and $1\\le a,b\\le q$ satisfying $(a,b)=(ab,q)=1$, we have\n\\begin{align*}\nM_{f,E}(q;a,b)=&c_f\\prod_{p\\mid qab}\\left(1-\\frac{\\lambda(p)}{p}+\\frac1{p^2}\\r)\\left(1-\\frac1{p^2}\\r)^{-2}\\\\\n&\\times\\frac{c_{a,b}+c_{b,a}}{(ab)^{\\frac12}}\\frac{L(1,f)^2}{\\zeta(2)} +O\\left(q^{-\\frac{1}{20}+\\ve}a^{\\frac{3}{10}}+q^{-\\frac{1-2\\theta_f}{22+16\\theta_f}+\\ve}b^{\\frac{3+2\\theta_f }{11+8\\theta_f}}\\r),\n\\end{align*}\nwhere the coefficients $c_f$, $c_{a,b}$ are defined by\n\\begin{align}\nc_f=\n\\begin{cases}\n1/2,  &\\text{if} \\ f\\ \\text{is holomorphic},\\\\\n1, &\\text{if}\\ f\\ \\text{is a Maa\\ss\\ form},\n\\end{cases}\n\\quad c_{a,b}=\\sum_{a_1\\mid a^\\infty}\\sum_{b_1\\mid b^\\infty} \\frac{\\lambda(aa_1b_1)\\tau(ba_1b_1)}{a_1b_1}. \\label{eqc1c2}\n\\end{align}", "start_pos": 10873, "end_pos": 11856, "label": "themain"}, "ref_dict": {"eqgmoment": "\\begin{align}\\label{eqgmoment}\nM_{f,g}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\ppmod q}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,g\\otimes\\chi\\r)},\n\\end{align}", "eq+1": "\\begin{align}\\label{eq+1}\n\\sum_{\\substack{bm\\equiv \\pm an\\ppmod d\\\\(mn,q)=1}}\\lambda(m) \\tau(n)W\\left(\\frac mM\\right)W\\left(\\frac {n}{N}\\right),\n\\end{align}", "eqtbB": "\\begin{align}\n&E_{M,N}\\ll q^{-1+\\ve}(MN)^{\\frac12}+q^\\ve(M/N)^{\\frac12}, \\label{eqtbA}\\\\\n&E_{M,N}\\ll M^{\\theta_f} \\left(q^{-1+\\ve}(MN)^{\\frac12}+q^\\ve(N/M)^{\\frac12}\\r). \\label{eqtbB}\n\\end{align}", "rem1": "\\begin{remark}\\label{rem1}\nObserve that $\\ve(f,\\chi)$ is independent of $\\chi$ when $f$ is a Maa\\ss~form.\nWhen $\\ve(f,\\chi)=\\ve(f)=-1$, summing both sides of equation \\eqref{eqfe} over all primitive characters leads to the vanishing of $M_{f,E}(q;a,b)$ by symmetry. For holomorphic $f$, the root number $\\ve(f,\\chi)$ depends on $\\chi$ at most though its parity $\\chi(-1)$. The symmetry also implies that the total contribution of such characters $\\chi$ satisfying $\\ve(f,\\chi)=-1$ to $M_{f,E}(q;a,b)$ is zero. Therefore, we restrict our consideration to the following cases:\n\\begin{itemize}\n  \\item for a Maa\\ss~form $f$, we take $\\ve(f)=1$,\n  \\item for a holomorphic $f$, we consider only characters with $\\chi(-1)=\\ve(f)$.\n\\end{itemize}\nIn both cases, it holds that\n\\[\n\\ve(f,\\chi)=1,\n\\]\nwhich we assume henceforth.\n\\end{remark}", "themain": "\\begin{theorem}\\label{themain}\nLet $f$ be a cuspidal Hecke eigenform of level $1$ with Hecke eigenvalues $\\lambda(n)\\ll \\tau(n)n^{\\theta_f}$. If $f$ is Maa\\ss~form we further assume  that its root number satisfies $\\ve(f)=1$. For $q\\nequiv 2 \\pmod4$ and $1\\le a,b\\le q$ satisfying $(a,b)=(ab,q)=1$, we have\n\\begin{align*}\nM_{f,E}(q;a,b)=&c_f\\prod_{p\\mid qab}\\left(1-\\frac{\\lambda(p)}{p}+\\frac1{p^2}\\r)\\left(1-\\frac1{p^2}\\r)^{-2}\\\\\n&\\times\\frac{c_{a,b}+c_{b,a}}{(ab)^{\\frac12}}\\frac{L(1,f)^2}{\\zeta(2)} +O\\left(q^{-\\frac{1}{20}+\\ve}a^{\\frac{3}{10}}+q^{-\\frac{1-2\\theta_f}{22+16\\theta_f}+\\ve}b^{\\frac{3+2\\theta_f }{11+8\\theta_f}}\\r),\n\\end{align*}\nwhere the coefficients $c_f$, $c_{a,b}$ are defined by\n\\begin{align}\nc_f=\n\\begin{cases}\n1/2,  &\\text{if} \\ f\\ \\text{is holomorphic},\\\\\n1, &\\text{if}\\ f\\ \\text{is a Maa\\ss\\ form},\n\\end{cases}\n\\quad c_{a,b}=\\sum_{a_1\\mid a^\\infty}\\sum_{b_1\\mid b^\\infty} \\frac{\\lambda(aa_1b_1)\\tau(ba_1b_1)}{a_1b_1}. \\label{eqc1c2}\n\\end{align}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 4881, "pre_theorem_intro_text": "\\subsection{Moments of twisted $L$-functions}\nThe study of moments in families of $L$-functions constitutes a central problem in modern number theory. As emphasized in the elegant work of Young \\cite{You11}, these moments are not only pivotal for their wide-ranging applications but also serve as fundamental objects that unveil deep structural properties and inherent symmetries within the family.\n\nAsymptotic formulae for moments of $L$-function with a power-saving error term are particularly crucial, as they play vital roles in amplification techniques, mollification methods, and resonance techniques. The complexity of such moment calculations can be quantified by the ratio $r=\\log \\mathcal{C}/\\log |\\mathcal{F}|$, where $|\\mathcal{F}|$ denotes the size of the family and $\\mathcal{C}$ its analytic conductor. Notably, computational difficulty increases with growing complexity. The threshold $r=4$ is the precise boundary, where most current analytic techniques fall just short of producing an asymptotic formula. In the few successful cases, some deep input is typically indispensable; see \\cite{CLMR24} \\cite{CIS12} \\cite{IS00} \\cite{Kha12} \\cite{KMV00} \\cite{Li24} for example.\n\nLet $f$ and $g$ denote two fixed (holomorphic or non-holomorphic) Hecke eigenforms of level $1$, not necessarily cuspidal. The following second moment is defined in the context $r=4$: for $q\\nequiv 2\\pmod4$,\n\\begin{align}\\label{eqgmoment}\nM_{f,g}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,g\\otimes\\chi\\r)},\n\\end{align}\nwhere the sum runs over all primitive characters modulo $q$, and $\\vp^*(q)$ denotes the number of such characters. Asymptotics for the moment \\eqref{eqgmoment} with a power-saving error term are crucial to amplification and related analytic techniques in problems such as subconvexity, nonvanishing, and extreme values of $L$-functions; see \\cite{BFKMMS23} for a rich sample of applications.\n\nWhen both $f$ and $g$ correspond to non-cuspidal Eisenstein series, the moment \\eqref{eqgmoment} reduces to\n\\[\nM_{E,E}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q}\\left|L\\left(\\tfrac12,\\chi\\r)\\right|^4,\n\\]\nthe fourth moment of Dirichlet $L$-functions, which has a very long history and has been extensively studied; see \\cite{BFK+17a} \\cite{HB81} \\cite{Sou07} \\cite{Wu23} \\cite{You11} for example. The first asymptotic formula with a power-savings error term was established by Young \\cite{You11} for prime moduli. Wu \\cite{Wu23} proved the asymptotic formula for all admissible moduli, and the best-known error term to date is $O\\left(q^{-\\frac1{20}+\\ve}\\r)$; see \\cite{BFK+17b} \\cite{GWZ25}.\n\nWhen both $f$ and $g$ are cuspidal (holomorphic or Maa\\ss) in \\eqref{eqgmoment}, some special cases (ie. $f=g$) were studied earlier by Stefanicki \\cite{Ste96} and Gao, Khan, and  Ricotta \\cite{GKR09}. Blomer and Mili\\'cevi\\'c \\cite{BM15} established a power-saving asymptotic for most moduli, specifically when $q$ is not close to a prime or to a product of two primes of comparable size. The main obstacle in these cases has been the lack of power-saving estimates for bilinear forms involving Kloosterman sums in the P\\'olya-Vinogradov range.\nThe case of prime moduli was later addressed by Kowalski, Michel, and Sawin \\cite{KMS17}, who proved the asymptotic with a power-saving error term $O\\left(p^{-\\frac1{144}+\\ve}\\r)$. The remaining case was recently resolved by  Mili\\'cevi\\'c, Qin, and Wu \\cite{MQW25} and independently by  Pascadi~\\cite{Pascadi2025}. Both works \\cite{MQW25} and \\cite{Pascadi2025} established a power-saving asymptotic for all admissible moduli $q$, with the former also removing the dependence on the Ramanujan--Petersson conjecture and achieving a sharper error term $O\\left(q^{-\\frac1{216}+\\ve}\\r)$.\n\nThese are termed mixed moments when one eigenform is cuspidal while the other corresponds to a non-cuspidal Eisenstein series. The first power-saving asymptotic formula for this case was established by Blomer, Fouvry, Kowalski, Michel, and Mili\\'cevi\\'c \\cite{BFK+17a}, who proved that for prime moduli $p$\n\\[\nM_{f,E}(p)=\\frac1{\\vp^*(p)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod p}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,\\chi\\r)}^2=\\frac{L(1,f)^2}{\\zeta(2)}+O\\left(p^{-\\frac1{68}+\\ve}\\r).\n\\]\nThe error term was subsequently improved to $O\\left(p^{-\\frac1{64}+\\ve}\\r)$ by Shparlinski \\cite{Shp19}, and more recently to $O\\left(p^{-\\frac1{22}+\\ve}\\r)$ for holomorphic $f$ and to $O\\left(p^{-\\frac5{152}+\\ve}\\r)$ for Maa\\ss\\ forms $f$ by Khan and Zhang \\cite{KZ23}.\n\nIn this paper, we investigate the mixed moment for arbitrary moduli. More precisely, we study the following twisted mixed moment\n\\[\nM_{f,E}(q;a,b)=\\frac1{\\varphi^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q} L\\left(\\tfrac12,f\\otimes\\chi\\r)L\\left(\\tfrac12,\\ol\\chi\\r)^2\\chi(\\ol{a}b),\n\\]\nfor any integers $a,b\\le q$ satisfying $(a,b)=(ab,q)=1$.", "context": "Asymptotic formulae for moments of $L$-function with a power-saving error term are particularly crucial, as they play vital roles in amplification techniques, mollification methods, and resonance techniques. The complexity of such moment calculations can be quantified by the ratio $r=\\log \\mathcal{C}/\\log |\\mathcal{F}|$, where $|\\mathcal{F}|$ denotes the size of the family and $\\mathcal{C}$ its analytic conductor. Notably, computational difficulty increases with growing complexity. The threshold $r=4$ is the precise boundary, where most current analytic techniques fall just short of producing an asymptotic formula. In the few successful cases, some deep input is typically indispensable; see \\cite{CLMR24} \\cite{CIS12} \\cite{IS00} \\cite{Kha12} \\cite{KMV00} \\cite{Li24} for example.\n\nLet $f$ and $g$ denote two fixed (holomorphic or non-holomorphic) Hecke eigenforms of level $1$, not necessarily cuspidal. The following second moment is defined in the context $r=4$: for $q\\nequiv 2\\pmod4$,\n\\begin{align}\\label{eqgmoment}\nM_{f,g}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,g\\otimes\\chi\\r)},\n\\end{align}\nwhere the sum runs over all primitive characters modulo $q$, and $\\vp^*(q)$ denotes the number of such characters. Asymptotics for the moment \\eqref{eqgmoment} with a power-saving error term are crucial to amplification and related analytic techniques in problems such as subconvexity, nonvanishing, and extreme values of $L$-functions; see \\cite{BFKMMS23} for a rich sample of applications.\n\nWhen both $f$ and $g$ correspond to non-cuspidal Eisenstein series, the moment \\eqref{eqgmoment} reduces to\n\\[\nM_{E,E}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q}\\left|L\\left(\\tfrac12,\\chi\\r)\\right|^4,\n\\]\nthe fourth moment of Dirichlet $L$-functions, which has a very long history and has been extensively studied; see \\cite{BFK+17a} \\cite{HB81} \\cite{Sou07} \\cite{Wu23} \\cite{You11} for example. The first asymptotic formula with a power-savings error term was established by Young \\cite{You11} for prime moduli. Wu \\cite{Wu23} proved the asymptotic formula for all admissible moduli, and the best-known error term to date is $O\\left(q^{-\\frac1{20}+\\ve}\\r)$; see \\cite{BFK+17b} \\cite{GWZ25}.\n\nWhen both $f$ and $g$ are cuspidal (holomorphic or Maa\\ss) in \\eqref{eqgmoment}, some special cases (ie. $f=g$) were studied earlier by Stefanicki \\cite{Ste96} and Gao, Khan, and Ricotta \\cite{GKR09}. Blomer and Mili\\'cevi\\'c \\cite{BM15} established a power-saving asymptotic for most moduli, specifically when $q$ is not close to a prime or to a product of two primes of comparable size. The main obstacle in these cases has been the lack of power-saving estimates for bilinear forms involving Kloosterman sums in the P\\'olya-Vinogradov range.\nThe case of prime moduli was later addressed by Kowalski, Michel, and Sawin \\cite{KMS17}, who proved the asymptotic with a power-saving error term $O\\left(p^{-\\frac1{144}+\\ve}\\r)$. The remaining case was recently resolved by Mili\\'cevi\\'c, Qin, and Wu \\cite{MQW25} and independently by Pascadi~\\cite{Pascadi2025}. Both works \\cite{MQW25} and \\cite{Pascadi2025} established a power-saving asymptotic for all admissible moduli $q$, with the former also removing the dependence on the Ramanujan--Petersson conjecture and achieving a sharper error term $O\\left(q^{-\\frac1{216}+\\ve}\\r)$.\n\nThese are termed mixed moments when one eigenform is cuspidal while the other corresponds to a non-cuspidal Eisenstein series. The first power-saving asymptotic formula for this case was established by Blomer, Fouvry, Kowalski, Michel, and Mili\\'cevi\\'c \\cite{BFK+17a}, who proved that for prime moduli $p$\n\\[\nM_{f,E}(p)=\\frac1{\\vp^*(p)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod p}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,\\chi\\r)}^2=\\frac{L(1,f)^2}{\\zeta(2)}+O\\left(p^{-\\frac1{68}+\\ve}\\r).\n\\]\nThe error term was subsequently improved to $O\\left(p^{-\\frac1{64}+\\ve}\\r)$ by Shparlinski \\cite{Shp19}, and more recently to $O\\left(p^{-\\frac1{22}+\\ve}\\r)$ for holomorphic $f$ and to $O\\left(p^{-\\frac5{152}+\\ve}\\r)$ for Maa\\ss\\ forms $f$ by Khan and Zhang \\cite{KZ23}.\n\nIn this paper, we investigate the mixed moment for arbitrary moduli. More precisely, we study the following twisted mixed moment\n\\[\nM_{f,E}(q;a,b)=\\frac1{\\varphi^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q} L\\left(\\tfrac12,f\\otimes\\chi\\r)L\\left(\\tfrac12,\\ol\\chi\\r)^2\\chi(\\ol{a}b),\n\\]\nfor any integers $a,b\\le q$ satisfying $(a,b)=(ab,q)=1$.\n\n\\begin{align}\\label{eqgmoment}\nM_{f,g}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\ppmod q}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,g\\otimes\\chi\\r)},\n\\end{align}", "full_context": "Asymptotic formulae for moments of $L$-function with a power-saving error term are particularly crucial, as they play vital roles in amplification techniques, mollification methods, and resonance techniques. The complexity of such moment calculations can be quantified by the ratio $r=\\log \\mathcal{C}/\\log |\\mathcal{F}|$, where $|\\mathcal{F}|$ denotes the size of the family and $\\mathcal{C}$ its analytic conductor. Notably, computational difficulty increases with growing complexity. The threshold $r=4$ is the precise boundary, where most current analytic techniques fall just short of producing an asymptotic formula. In the few successful cases, some deep input is typically indispensable; see \\cite{CLMR24} \\cite{CIS12} \\cite{IS00} \\cite{Kha12} \\cite{KMV00} \\cite{Li24} for example.\n\nLet $f$ and $g$ denote two fixed (holomorphic or non-holomorphic) Hecke eigenforms of level $1$, not necessarily cuspidal. The following second moment is defined in the context $r=4$: for $q\\nequiv 2\\pmod4$,\n\\begin{align}\\label{eqgmoment}\nM_{f,g}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,g\\otimes\\chi\\r)},\n\\end{align}\nwhere the sum runs over all primitive characters modulo $q$, and $\\vp^*(q)$ denotes the number of such characters. Asymptotics for the moment \\eqref{eqgmoment} with a power-saving error term are crucial to amplification and related analytic techniques in problems such as subconvexity, nonvanishing, and extreme values of $L$-functions; see \\cite{BFKMMS23} for a rich sample of applications.\n\nWhen both $f$ and $g$ correspond to non-cuspidal Eisenstein series, the moment \\eqref{eqgmoment} reduces to\n\\[\nM_{E,E}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q}\\left|L\\left(\\tfrac12,\\chi\\r)\\right|^4,\n\\]\nthe fourth moment of Dirichlet $L$-functions, which has a very long history and has been extensively studied; see \\cite{BFK+17a} \\cite{HB81} \\cite{Sou07} \\cite{Wu23} \\cite{You11} for example. The first asymptotic formula with a power-savings error term was established by Young \\cite{You11} for prime moduli. Wu \\cite{Wu23} proved the asymptotic formula for all admissible moduli, and the best-known error term to date is $O\\left(q^{-\\frac1{20}+\\ve}\\r)$; see \\cite{BFK+17b} \\cite{GWZ25}.\n\nWhen both $f$ and $g$ are cuspidal (holomorphic or Maa\\ss) in \\eqref{eqgmoment}, some special cases (ie. $f=g$) were studied earlier by Stefanicki \\cite{Ste96} and Gao, Khan, and Ricotta \\cite{GKR09}. Blomer and Mili\\'cevi\\'c \\cite{BM15} established a power-saving asymptotic for most moduli, specifically when $q$ is not close to a prime or to a product of two primes of comparable size. The main obstacle in these cases has been the lack of power-saving estimates for bilinear forms involving Kloosterman sums in the P\\'olya-Vinogradov range.\nThe case of prime moduli was later addressed by Kowalski, Michel, and Sawin \\cite{KMS17}, who proved the asymptotic with a power-saving error term $O\\left(p^{-\\frac1{144}+\\ve}\\r)$. The remaining case was recently resolved by Mili\\'cevi\\'c, Qin, and Wu \\cite{MQW25} and independently by Pascadi~\\cite{Pascadi2025}. Both works \\cite{MQW25} and \\cite{Pascadi2025} established a power-saving asymptotic for all admissible moduli $q$, with the former also removing the dependence on the Ramanujan--Petersson conjecture and achieving a sharper error term $O\\left(q^{-\\frac1{216}+\\ve}\\r)$.\n\nThese are termed mixed moments when one eigenform is cuspidal while the other corresponds to a non-cuspidal Eisenstein series. The first power-saving asymptotic formula for this case was established by Blomer, Fouvry, Kowalski, Michel, and Mili\\'cevi\\'c \\cite{BFK+17a}, who proved that for prime moduli $p$\n\\[\nM_{f,E}(p)=\\frac1{\\vp^*(p)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod p}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,\\chi\\r)}^2=\\frac{L(1,f)^2}{\\zeta(2)}+O\\left(p^{-\\frac1{68}+\\ve}\\r).\n\\]\nThe error term was subsequently improved to $O\\left(p^{-\\frac1{64}+\\ve}\\r)$ by Shparlinski \\cite{Shp19}, and more recently to $O\\left(p^{-\\frac1{22}+\\ve}\\r)$ for holomorphic $f$ and to $O\\left(p^{-\\frac5{152}+\\ve}\\r)$ for Maa\\ss\\ forms $f$ by Khan and Zhang \\cite{KZ23}.\n\nIn this paper, we investigate the mixed moment for arbitrary moduli. More precisely, we study the following twisted mixed moment\n\\[\nM_{f,E}(q;a,b)=\\frac1{\\varphi^*(q)}\\ssum_{\\chi\\!\\!\\!\\!\\!\\pmod q} L\\left(\\tfrac12,f\\otimes\\chi\\r)L\\left(\\tfrac12,\\ol\\chi\\r)^2\\chi(\\ol{a}b),\n\\]\nfor any integers $a,b\\le q$ satisfying $(a,b)=(ab,q)=1$.\n\n\\begin{align}\\label{eqgmoment}\nM_{f,g}(q)=\\frac1{\\vp^*(q)}\\ssum_{\\chi\\ppmod q}L\\left(\\tfrac12,f\\otimes\\chi\\r)\\ol{L\\left(\\tfrac12,g\\otimes\\chi\\r)},\n\\end{align}\n\n\\subsection{Twisted $L$-functions}\nWe review some basic facts on twisted $L$-functions in this section; for further details, see \\cite[Section 2.3]{BFK+17a}.\nLet $\\chi$ be a primitive Dirichlet character modulo $q$ and let $f$ be a cuspidal Hecke eigenform with Hecke eigenvalues $\\lambda(n)$. The twisted form $f\\otimes\\chi$ remains cuspidal for the congruence subgroup $\\Gamma_0(q^2)$ with nebentypus $\\chi^2$ (see e.g \\cite[Propositions 14.19 \\& 14.20]{IK04}). Its $L$-function is given by\n\\[\nL(s,f\\otimes\\chi)=\\sum_{n\\ge1}\\frac{\\lambda(n)\\chi(n)}{n^{s}}=\\prod_p\\left(1-\\frac{\\lambda(p)\\chi(p)}{p^s}+\\frac{\\chi^2(p)}{p^{2s}}\\r)^{-1},\\ \\ \\text{for}\\ \\ \\re(s)>1,\n\\]\nwhere the product is over primes $p$.\nThe completed $L$-function is defined by\n\\begin{align*}\n\\Lambda(s,f\\otimes\\chi)=q^s L_\\infty(s,f\\otimes\\chi)L(s,f\\otimes\\chi),\n\\end{align*}\nwith\n\\begin{align*}\nL_\\infty(s,f\\otimes\\chi)=\\begin{cases}\n(2\\pi)^{-\\frac{k-1}2-s}\\Gamma\\left(\\frac{k-1}2+s\\r), &\\text{if} \\ f\\ \\text{is holomorphic of weight}\\ k,\\\\\n\\pi^{-s-\\ma}\\Gamma\\left(\\frac{s+i\\kappa+\\ma}2\\r)\\Gamma\\left(\\frac{s-i\\kappa+\\ma}2\\r), &\\text{if}\\ f\\ \\text{is a Maa\\ss\\ form with eigenvalue}\\ \\tfrac14+\\kappa^2.\n\\end{cases}\n\\end{align*}\nThis completed $L$-function admits analytic continuation to $\\mathbb{C}$ and satisfies the functional equation (see e.g \\cite[Theorem 14.17, Proposition 14.20]{IK04}):\n\\[\n\\Lambda(s,f\\otimes\\chi)=\\ve(f\\otimes\\chi)\\Lambda(1-s,f\\otimes\\ol{\\chi}),\n\\]\nwhere the root number is given by\n\\begin{align*}\n\\ve(f\\otimes\\chi)=\\begin{cases}\n\\ve(f)\\ve_\\chi^2, &\\text{if} \\ f\\ \\text{is holomorphic},\\\\\n\\chi(-1)\\ve(f)\\ve_\\chi^2, &\\text{if}\\ f\\ \\text{is a Maa\\ss\\ form},\n\\end{cases}\n\\end{align*}\nand $\\ve(f)=\\pm 1$ denotes the root number of $L(s,f)$.\n\nUsing the identity\n\\[\nS(hq\\ol{b}, m;a_1k)=e\\left(-m\\frac{\\ol{a}_1}{b}\\right)S_{\\infty,1/a_1}(hq,m;\\gammaup),\n\\]\nwhere $\\gammaup=a_1\\ssqrt{b}k$ and $Q=a_1b$, we derive from \\eqref{eqdefmS} that\n\\begin{align*}\n\\mS(a_1,r)=&\\frac{\\sqrt{b}}{aN_1}\\sum_{h\\sim H}\\sum_{m\\sim M^*} \\lambda(m)e\\left(-m\\frac{\\ol{a}_1}{b}\\right)\\\\\n&\\times\\sum_{\\gammaup}\\frac 1{\\gammaup}S_{\\infty,1/a_1}(hq, m;\\gammaup)W\\left(\\frac {r\\gammaup}{a_1\\ssqrt{b}N_1}\\r)\\mathring{V}_{+}\\left(\\frac{bm}{\\gammaup^2},h\\right).\n\\end{align*}\nPerforming the variable substitution $(bx-hq)/N\\rightarrow x$ in \\eqref{eqV0} yields\n\\[\n\\mathring{V}_{+}\\left(\\frac{bm}{\\gammaup^2},h\\right)=\\frac{N}{b}\\Omega\\Bigg(\\frac{4\\pi\\ssqrt{hqm}}{\\gammaup},\\frac{4\\pi\\ssqrt{mN}}{\\gammaup},h,m\\Bigg),\n\\]\nwhere\n\\[\n\\Omega(y,z,h,m)=\\int_{0}^\\infty W(x)W\\left(\\frac {hq+xN}M\\r) \\mJ_{+}\\left(\\ssqrt{y^2+xz^2}\\right)\\d x\n\\]\nwith\n\\begin{equation}\\label{eqyz}\ny\\asymp Y:=\\frac{r\\ssqrt{MM^*}}{a_1\\ssqrt{b}N_1}\\gg q^\\ve,\\quad z\\asymp \\frac{r\\ssqrt{M^*N}}{a_1\\ssqrt{b}N_1},\\quad h\\asymp H,\\quad m\\asymp M^*.\n\\end{equation}\nFor $M\\gg q^\\ve N$, it is easy to see from \\eqref{eqlargeM} and \\eqref{eqyz} that\n\\[\ny\\gg z\\quad \\text{and}\\quad y\\gg z^2q^{-\\ve}.\n\\]\nThen Lemma \\ref{lemdecomJ} yields the decomposition\n\\[\n\\Omega(y,z,h,m)=\\Omega_+(y,z,h,m)e^{iy}+\\Omega_-(y,z,h,m)e^{-iy}+O_A\\left(q^{-A}\\r),\n\\]\nwhere the functions $\\Omega_{\\pm}$ satisfy\n\\begin{align}\\label{eqWpmd}\ny^{j_1}z^{j_2}h^{j_3}m^{j_4}\\frac{\\partial^{j_1+j_2+j_3+j_4}}{\\partial y^{j_1}\\partial z^{j_2}\\partial h^{j_3}\\partial m^{j_4}}\\Omega_{\\pm}(y,z,h,m)\\ll_{j_1,j_2,j_3,j_4}q^{(j_1+j_2+j_3+j_4)\\ve} Y^{-\\frac12}.\n\\end{align}\nAfter separating variables in $\\Omega_\\pm$ via the Mellin transform, we bound $\\mS(a_1,r)$ by sums of the form\n\\begin{equation*}\n\\mS(a_1,r)\\ll q^\\ve\\frac{N Y^{-\\frac12}}{a\\ssqrt{b}N_1}\\B|\\sum_{H<h\\le 2H}\\alpha_h\\sum_{M^*<m\\le 2M^*} \\beta_m\\lambda(m)\\sum_{\\gammaup}\\frac1{\\gammaup}S_{\\infty,1/a_1}(hq,m;\\gammaup)\\Theta \\B(\\frac{4\\pi\\ssqrt{hqm}}{\\gammaup}\\B)\\B|\n\\end{equation*}\nfor some coefficients satisfying $\\alpha_h\\ll h^\\ve$, $\\beta_m\\ll m^\\ve$ and a test function\n\\[\n\\Theta (y)=W\\B(\\frac yY\\B)\\exp(\\pm iy)y^{-u}\n\\]\nwith $\\re(u)=\\ve$.\n\nAn application of the Kuznetsov formula (Lemma \\ref{lemKf}) followed by spectral truncation at parameter $Y^{\\frac12+\\ve}$ ( Lemma \\ref{lemBe}) produces\n\\begin{equation*}\n\\mS(a_1,r)\\ll q^\\ve\\frac{N Y^{-\\frac12}}{a\\ssqrt{b}N_1}\\B|\\sum_{H<h\\le 2H}\\alpha_h\\sum_{M^*<m\\le 2M^*} \\beta_m\\lambda(m)\\left(\\mH+\\mM+\\mE\\r)\\B|,\n\\end{equation*}\nwhere the spectral contributions decompose as\n\\[\n\\mH=\\sum_{\\substack{2\\le k\\le Y^{\\frac12+\\ve}\\\\ k ~\\text{even}}}\\sum_{f\\in \\mathcal{B}_k(Q)}\\tilde{\\Theta} (k)\\ssqrt{hqm}~ \\bar\\rho_f(hq)~\\rho_f(a_1^{-1},m),\n\\]\n\\[\n\\mM=\\sum_{\\substack{f\\in \\mathcal{B}(Q)\\\\ \\kappa_f\\ll Y^{\\frac12+\\ve}}}\\hat{\\Theta} (\\kappa_f)\\frac{\\ssqrt{hqm}~ \\bar\\rho_f(hq)~\\rho_f(a_1^{-1},m)}{\\cosh(\\pi \\kappa_f)},\n\\]\n\\[\n\\mE=\\frac1{4\\pi}\\sum_{\\mc}\\int_{|\\kappa|\\ll Y^{\\frac12+\\ve}}\\hat{\\Theta} (\\kappa)\\ssqrt{hqm} ~\\frac{\\bar\\rho_{\\mc}(hq,\\kappa)~\\rho_{1/{a_1},\\mc}(m,\\kappa)}{\\cosh(\\pi\\kappa)}\\d \\kappa,\n\\]\nrepresenting the holomorphic cusp forms, Maa\\ss\\ cusp forms, and the Eisenstein series respectively.\nWe foucus our detailed analysis on the Maa\\ss\\ case (the holomorphic and Eisenstein cases are identical). Since $\\hat{\\Theta} (\\kappa_f)\\ll Y^{-\\frac12+\\ve}$ (see \\eqref{eqphi1}), the contribution of the Maa\\ss\\ case is bounded by\n\\[\n\\ll q^\\ve\\frac{NY^{-1}}{a\\ssqrt{b}N_1}\\sum_{\\substack{f\\in \\mathcal{B}(Q)\\\\ \\kappa_f\\ll Y^{\\frac12+\\ve}}}\\B|\\sum_{H<h\\le 2H}\\alpha_h\\sum_{M^*<m\\le 2M^*} \\beta_m\\lambda(m)\\frac{\\ssqrt{hqm} ~\\bar\\rho_f(hq)~\\rho_f(a_1^{-1},m)}{\\cosh(\\pi \\kappa_f)}\\B|,\n\\]\nwhich, after applying the Cauchy-Schwarz inequality, can be bounded by\n\\begin{align*}\n\\ll q^\\ve\\frac{NY^{-1}}{a\\ssqrt{b}N_1}&\\B(\\sum_{\\substack{f\\in \\mathcal{B}(Q)\\\\ \\kappa_f\\ll Y^{\\frac12+\\ve}}}\\frac{1}{\\cosh(\\pi \\kappa_f)}\\B|\\sum_{H<h\\le 2H}\\alpha_h \\ssqrt{hq} ~\\rho_f(hq)\\B|^2\\B)^{\\frac12}\\\\\n&\\times \\B(\\sum_{\\substack{f\\in \\mathcal{B}(Q)\\\\ \\kappa_f\\ll Y^{\\frac12+\\ve}}}\\frac{1}{\\cosh(\\pi \\kappa_f)}\\B|\\sum_{M^*<m\\le 2M^*}\\beta_m\\lambda(m)\\ssqrt{m} ~\\rho_f(a_1^{-1},m)\\B|^2\\B)^{\\frac12}.\n\\end{align*}\nWe may handle the second spectral sum via Lemma \\ref{lemsls}, but for the first spectral sum, an application of Lemma \\ref{lemsls} leads to an extra factor of $q$ to the final estimate. To address this issue, we apply the following estimate, established in \\cite[Theorems 11.3-11.6]{GWZ25}.\n\\begin{lemma}\\label{thmsls}\nLet $Q,s\\in\\mathbb{N}$, $K, N\\ge 1$, and let $(a_n)_{n\\in[N,2N]}$ be complex coefficients. Then we have\n\\[\n\\sum_{\\substack{2\\le k\\le K\\\\ k~ \\text{even}}}\\Gamma(k)\\sum_{f\\in\\mathcal{B}_k(Q)}\\B|\\sum_{n}a_n\\ssqrt{sn}~\\rho_f(sn)\\B|^2, \\ \\ \\ \\sum_{\\mc}\\int_{-K}^K\\frac1{\\cosh(\\pi\\kappa)}\\B|\\sum_n a_n\\ssqrt{sn}~\\rho_{\\mc}(\\pm sn,\\kappa)\\B|^2\\d\\kappa,\n\\]\n\\[\n\\sum_{\\substack{f\\in\\mathcal{B}(Q)\\\\ |\\kappa_f|\\le K}}\\frac1{\\cosh(\\pi\\kappa_f)}\\B|\\sum_{n}a_n\\ssqrt{sn}~\\rho_f(sn)\\B|^2\n\\]\nare bounded by\n\\[\n\\ll (sQKN)^\\ve\\B(K+\\frac{s^{\\frac12}}{Q^{\\frac12}}\\B)\\B(K+\\frac{(s,Q)^{\\frac12}N}{Q^{\\frac12}}\\B)N\\|a_n\\|_\\infty.\n\\]\n\\end{lemma}\nApplying Lemma \\ref{thmsls} to the first spectral sum and Lemma \\ref{lemsls} to the second one reduces the Maa\\ss\\ contribution to\n\\begin{equation*}\n\\ll q^\\ve\\frac{NY^{-1}}{a\\ssqrt{b}N_1}\\B(H^{\\frac12}{M^*}^{\\frac12}\\left(Y+\\frac{q}{a_1b}\\r)^{\\frac14} \\left(Y+\\frac{(a_1b,q)H^2}{a_1b}\\r)^{\\frac14}\\left(Y+\\frac{M^*}{a_1b}\\r)^{\\frac12}\\B).\n\\end{equation*}\nTaking\n\\[\n H=\\frac{M}q,\\ \\ \\ M^*= q^\\ve\\frac{a_1^2N_1^2}{r^2}\\frac{bM}{N^2}, \\ \\ \\ Y=\\frac{r\\ssqrt{MM^*}}{a_1\\ssqrt{b}N_1}=\\frac{M}{N},\n\\]\nand observing that\n$\\frac{M^*}{a_1b}\\ll Y$ ( since $N_1^2\\ll N/a$), we obtain\nfor the Maa\\ss\\ contribution\n\\begin{align*}\n&\\ll q^\\ve\\frac{N}{r\\ssqrt{q}}\\left(Y+\\frac{q}{ab}\\r)^{\\frac14} \\left(Y+\\frac{(ab,q)H^2}{ab}\\r)^{\\frac14}Y^{\\frac12}\\\\\n&\\ll q^\\ve r^{-1}\\B(\\frac{M}{q^{\\frac12}}+\\frac{(ab,q)^{\\frac14}M^{\\frac 54}N^{\\frac14}}{(ab)^{\\frac14}q} +\\frac{M^{\\frac 34}N^{\\frac14}}{(ab)^{\\frac14}q^{\\frac14}}+\\frac{(ab,q)^{\\frac14}MN^{\\frac12}}{(ab)^{\\frac12}q^{\\frac 34}}\\B).\n\\end{align*}", "post_theorem_intro_text_len": 5321, "post_theorem_intro_text": "\\begin{remark}\nThe dependence on  $\\theta_f$ in the error term of Theorem \\ref{themain} arises from the trivial estimate \\eqref{eqtbB}, which constitutes the sole component requiring an upper bound for the Hecke eigenvalues of $f$. The assumption $\\ve(f)=1$ for a Maa\\ss~form is natural, as otherwise $M_{f,E}(q;a,b)=0$ for parity reasons (see Remark \\ref{rem1}). In the Maa\\ss~case, the coefficient $c_f$ addresses a literature discrepancy originating from a typographical oversight in \\cite[Theorem 1.2]{BFK+17a} and was subsequently replicated in \\cite[Thoerem 1.1]{KZ23}.\n\\end{remark}\n\nSetting $a=b=1$ in Theorem \\ref{themain} results in a power-saving asymptotic for $M_{f,E}(q)$ across all admissible moduli $q$.\n\n\\begin{corollary}\\label{cor}\nLet $f$ be a cuspidal Hecke eigenform of level $1$ with Hecke eigenvalues $\\lambda(n)\\ll \\tau(n)n^{\\theta_f}$. If $f$ is Maa\\ss~form we further assume that its root number satisfies $\\ve(f)=1$. For $q\\nequiv 2 \\pmod4$, we have\n\\begin{align*}\nM_{f,E}(q)=c_f\\prod_{p\\mid q}\\left(1-\\frac{\\lambda(p)}{p}+\\frac1{p^2}\\r)\\left(1-\\frac1{p^2}\\r)^{-2}\\frac{L(1,f)^2}{\\zeta(2)}+O\\left(q^{-\\frac{1-2\\theta_f }{22+16\\theta_f }+\\ve}\\r),\n\\end{align*}\nwhere $c_f$ is the same constant as previously defined.\n\\end{corollary}\nFor the holomorphic case, Deligne's well-known work \\cite{Del74} gives $\\theta_f=0$, whereas for the Maa\\ss~form we may take $\\theta_f= 7/{64}$ due to Kim--Sarnak \\cite{Kim03}. This yields an error term as sharp as the best bound \\cite{KZ23} known for prime moduli.\n\nThrough the functional equations of $L$-functions and the orthogonality of characters, the evaluation of the moment $M_{f,E}(q,a,b)$ reduces to analyzing the shifted convolution sum\n\\begin{align}\\label{eq+1}\n\\sum_{\\substack{bm\\equiv \\pm an\\!\\!\\!\\!\\!\\pmod d\\\\(mn,q)=1}}\\lambda(m) \\tau(n)W\\left(\\frac mM\\right)W\\left(\\frac {n}{N}\\right),\n\\end{align}\nwhere $a, b,M,N\\ge1$ and $W$ is smooth test functions. The classical approach to such shifted convolution sum involves either the $\\delta$-method or Jutila's circle method to eliminate the congruence condition (see \\cite{BM15}). For \\eqref{eq+1}, a more efficient approach is to expand the divisor function using the multiplicative decomposition $\\tau(n)=\\sum_{n_1n_2=n}1$, which decomposes the $n$-sum into two sums with smooth coefficients, making them amenable to Fourier-analytic techniques.\n\nThe diagonal terms (where $bm=an$) in \\eqref{eq+1} contribute to the main term, which can be evaluated explicitly using standard techniques. The primary challenge arise from the off-diagonal terms, which we classify into two distinct cases based on the relative magnitudes of $bM$ and $aN$: the balanced case and the unbalanced case. These two cases demand fundamentally different approaches.\n\nIn the balanced case where $bM$ and $aN$ are of comparable size, we first remove the coprimality condition and reduce the problem to estimating the sum\n\\begin{align}\\label{eqAdef}\nA_q(a,b,M,N)=\\sum_{\\substack{bm\\equiv \\pm an \\!\\!\\!\\!\\!\\pmod q\\\\ bm\\neq an}}\\lambda(m) \\tau(n) W\\left(\\frac {bm}M\\r) W\\left(\\frac {an}N\\r),\n\\end{align}\nwhere $W$ are some smooth functions.\nFor $M\\ge N$, we establish the bound\n\\[\nA_q(a,b,M,N)\\ll  q^\\ve\\Bigg(\\frac{M}{q^{\\frac12}}+\\frac{(ab,q)^{\\frac14}M^{\\frac54}N^{\\frac14}}{(ab)^{\\frac14}q} +\\frac{M^{\\frac34}N^{\\frac14}}{(ab)^{\\frac14}q^{\\frac14}} +\\frac{(ab,q)^{\\frac14}MN^{\\frac12}}{(ab)^{\\frac12}q^{\\frac34}}\\Bigg)\n\\]\nwith an analogous result holding for $N\\ge M$. Compared to the bound in \\cite[(8.13)]{BM15}, our result achieves two key improvements: reducing the influence of the factor $(ab,q)^{\\frac12}$ and recovering a saving $(ab)^{-\\frac14}$ in the third term. This improvements stem from two innovation: the elimination of dependence on the Ramanujan--Petersson conjecture and the minimization of the impact of $a, b$ through new upper bounds for sums involving Kloosterman sums, as established in the recent work of Gao, Wu, and Zhao \\cite{GWZ25}.\n\nFor the unbalanced case where $BM$ and $AN$ differ significantly in size, the theory of algebraic trace functions becomes inapplicable due to its restriction to prime moduli. To address this, we employ the multiplicative decomposition of the divisor function $\\tau(n)$, which allows us to reformulate the problem as analyzing a bilinear sum involving incomplete Kloosterman sum. The estimate of this bilinear form relies primarily on a recent result by Kerr, Shparlinski, Wu, and Xi \\cite{KSWX23} concerning arbitrary moduli.\nThe approach also requires the Weil bound for Kloosterman sums and several Fourier-analytic tools, including the Poisson summation formula and the Voronoi summation formula.\n\n\\subsection{Notation}\nWe adopt the standard convention that $\\ve$ denotes an arbitrarily small positive constant whose values may vary from one occurrence to another. All implied constants in our estimates may depend on $\\ve$ as well as on the fixed cusp form $f$.\nThroughout this paper, $W(x)$ denotes a smooth complex-valued function with compact support in $[1/2,3]$, satisfying\n\\begin{align}\\label{eqW}\nW^{(j)}(x)\\ll_{j}1,\n\\end{align}\nfor all integers $j\\ge0$. The function $W(x)$ may also vary at each occurrence. We call a modulus $q$ admissible if $q\\nequiv 2\\pmod4$, which is a natural assumption since no primitive characters exist modulo $q$ otherwise.", "sketch": "Through the functional equations of $L$-functions and the orthogonality of characters, the evaluation of the moment $M_{f,E}(q,a,b)$ (hence Theorem~\\ref{themain}) “reduces to analyzing the shifted convolution sum”\n\\[\n\\sum_{\\substack{bm\\equiv \\pm an\\!\\!\\!\\!\\!\\pmod d\\\\(mn,q)=1}}\\lambda(m) \\, \\tau(n)\\,W\\!\\left(\\frac mM\\right)W\\!\\left(\\frac {n}{N}\\right).\n\\]\nInstead of using “the $\\delta$-method or Jutila's circle method,” the argument uses “a more efficient approach” by expanding the divisor function via “the multiplicative decomposition $\\tau(n)=\\sum_{n_1n_2=n}1$,” which splits the $n$-sum into two smooth sums “amenable to Fourier-analytic techniques.”\n\nThe “diagonal terms (where $bm=an$)” give the main term and “can be evaluated explicitly using standard techniques.” The remaining “off-diagonal terms” are treated by splitting into two regimes “based on the relative magnitudes of $bM$ and $aN$”:\n\n1) **Balanced case** ($bM$ comparable to $aN$): “we first remove the coprimality condition and reduce the problem” to bounding\n\\[\nA_q(a,b,M,N)=\\sum_{\\substack{bm\\equiv \\pm an \\!\\!\\!\\!\\!\\pmod q\\\\ bm\\neq an}}\\lambda(m)\\tau(n) W\\!\\left(\\frac {bm}M\\right) W\\!\\left(\\frac {an}N\\right),\n\\]\nand establish the stated upper bound for $A_q(a,b,M,N)$ (and its analogue when $N\\ge M$). The text attributes the improvement to “two innovation: the elimination of dependence on the Ramanujan--Petersson conjecture and the minimization of the impact of $a, b$ through new upper bounds for sums involving Kloosterman sums” from \\cite{GWZ25}.\n\n2) **Unbalanced case** ($BM$ and $AN$ very different): since “the theory of algebraic trace functions becomes inapplicable due to its restriction to prime moduli,” one again uses the decomposition of $\\tau(n)$ to “reformulate the problem as analyzing a bilinear sum involving incomplete Kloosterman sum.” This bilinear form is estimated using \\cite{KSWX23} (for “arbitrary moduli”), together with “the Weil bound for Kloosterman sums” and Fourier-analytic tools including “the Poisson summation formula and the Voronoi summation formula.”\n\nFinally, the remark notes that the $\\theta_f$-dependence in the error term “arises from the trivial estimate \\eqref{eqtbB},” which is “the sole component requiring an upper bound for the Hecke eigenvalues of $f$.”", "expanded_sketch": "Through the functional equations of $L$-functions and the orthogonality of characters, the evaluation of the moment $M_{f,E}(q,a,b)$ (hence, in establishing the main theorem) “reduces to analyzing the shifted convolution sum”\n\\[\n\\sum_{\\substack{bm\\equiv \\pm an\\!\\!\\!\\!\\pmod d\\\\(mn,q)=1}}\\lambda(m) \\, \\tau(n)\\,W\\!\\left(\\frac mM\\right)W\\!\\left(\\frac {n}{N}\\right).\n\\]\nInstead of using “the $\\delta$-method or Jutila's circle method,” the argument uses “a more efficient approach” by expanding the divisor function via “the multiplicative decomposition $\\tau(n)=\\sum_{n_1n_2=n}1$,” which splits the $n$-sum into two smooth sums “amenable to Fourier-analytic techniques.”\n\nThe “diagonal terms (where $bm=an$)” give the main term and “can be evaluated explicitly using standard techniques.” The remaining “off-diagonal terms” are treated by splitting into two regimes “based on the relative magnitudes of $bM$ and $aN$”:\n\n1) **Balanced case** ($bM$ comparable to $aN$): “we first remove the coprimality condition and reduce the problem” to bounding\n\\[\nA_q(a,b,M,N)=\\sum_{\\substack{bm\\equiv \\pm an \\!\\!\\!\\!\\pmod q\\\\ bm\\neq an}}\\lambda(m)\\tau(n) W\\!\\left(\\frac {bm}M\\right) W\\!\\left(\\frac {an}N\\right),\n\\]\nand establish the stated upper bound for $A_q(a,b,M,N)$ (and its analogue when $N\\ge M$). The text attributes the improvement to “two innovation: the elimination of dependence on the Ramanujan--Petersson conjecture and the minimization of the impact of $a, b$ through new upper bounds for sums involving Kloosterman sums” from \\cite{GWZ25}.\n\n2) **Unbalanced case** ($BM$ and $AN$ very different): since “the theory of algebraic trace functions becomes inapplicable due to its restriction to prime moduli,” one again uses the decomposition of $\\tau(n)$ to “reformulate the problem as analyzing a bilinear sum involving incomplete Kloosterman sum.” This bilinear form is estimated using \\cite{KSWX23} (for “arbitrary moduli”), together with “the Weil bound for Kloosterman sums” and Fourier-analytic tools including “the Poisson summation formula and the Voronoi summation formula.”\n\nFinally, the remark notes that the $\\theta_f$-dependence in the error term “arises from the trivial estimate \n\\begin{align}\n&E_{M,N}\\\\ll q^{-1+\\ve}(MN)^{\\frac12}+q^\\ve(M/N)^{\\frac12}, \\label{eqtbA}\\\\\n&E_{M,N}\\\\ll M^{\\theta_f} \\left(q^{-1+\\ve}(MN)^{\\frac12}+q^\\ve(N/M)^{\\frac12}\\r). \\label{eqtbB}\n\\end{align}\n,” which is “the sole component requiring an upper bound for the Hecke eigenvalues of $f$.”", "expanded_theorem": "\\label{themain}\nLet $f$ be a cuspidal Hecke eigenform of level $1$ with Hecke eigenvalues $\\lambda(n)\\ll \\tau(n)n^{\\theta_f}$. If $f$ is Maa\\ss~form we further assume  that its root number satisfies $\\ve(f)=1$. For $q\\nequiv 2 \\pmod4$ and $1\\le a,b\\le q$ satisfying $(a,b)=(ab,q)=1$, we have\n\\begin{align*}\nM_{f,E}(q;a,b)=&c_f\\prod_{p\\mid qab}\\left(1-\\frac{\\lambda(p)}{p}+\\frac1{p^2}\\r)\\left(1-\\frac1{p^2}\\r)^{-2}\\\\\n&\\times\\frac{c_{a,b}+c_{b,a}}{(ab)^{\\frac12}}\\frac{L(1,f)^2}{\\zeta(2)} +O\\left(q^{-\\frac{1}{20}+\\ve}a^{\\frac{3}{10}}+q^{-\\frac{1-2\\theta_f}{22+16\\theta_f}+\\ve}b^{\\frac{3+2\\theta_f }{11+8\\theta_f}}\\r),\n\\end{align*}\nwhere the coefficients $c_f$, $c_{a,b}$ are defined by\n\\begin{align}\nc_f=\n\\begin{cases}\n1/2,  &\\text{if} \\ f\\ \\text{is holomorphic},\\\\\n1, &\\text{if}\\ f\\ \\text{is a Maa\\ss\\ form},\n\\end{cases}\n\\quad c_{a,b}=\\sum_{a_1\\mid a^\\infty}\\sum_{b_1\\mid b^\\infty} \\frac{\\lambda(aa_1b_1)\\tau(ba_1b_1)}{a_1b_1}. \\label{eqc1c2}\n\\end{align}", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Let \\(f\\) be a level-\\(1\\) cuspidal Hecke eigenform with Hecke eigenvalues \\(\\lambda(n)\\) satisfying \\(\\lambda(n)\\ll \\tau(n)n^{\\theta_f}\\), where \\(\\tau(n)\\) is the divisor function. If \\(f\\) is a Maa\\ss{} form, also assume its root number satisfies \\(\\varepsilon(f)=1\\). For an integer \\(q\\not\\equiv 2\\pmod 4\\), and integers \\(1\\le a,b\\le q\\) with \\((a,b)=(ab,q)=1\\), define the twisted mixed moment\n\\[\nM_{f,E}(q;a,b)=\\frac1{\\varphi^*(q)}\\sum_{\\chi\\, (\\mathrm{mod}\\, q)}^{\\mathrm{primitive}} L\\!\\left(\\tfrac12,f\\otimes\\chi\\right)L\\!\\left(\\tfrac12,\\overline\\chi\\right)^2\\chi(\\overline a b),\n\\]\nwhere \\(\\varphi^*(q)\\) is the number of primitive Dirichlet characters modulo \\(q\\), and \\(\\overline a\\) denotes the inverse of \\(a\\) modulo \\(q\\). Also define\n\\[\nc_f=\n\\begin{cases}\n1/2,& \\text{if }f\\text{ is holomorphic},\\\\\n1,& \\text{if }f\\text{ is a Maa\\ss{} form},\n\\end{cases}\n\\qquad\nc_{a,b}=\\sum_{a_1\\mid a^\\infty}\\sum_{b_1\\mid b^\\infty} \\frac{\\lambda(aa_1b_1)\\tau(ba_1b_1)}{a_1b_1},\n\\]\nwhere \\(a_1\\mid a^\\infty\\) and \\(b_1\\mid b^\\infty\\) mean that every prime divisor of \\(a_1\\) divides \\(a\\), and every prime divisor of \\(b_1\\) divides \\(b\\). Under these assumptions, which explicit asymptotic formula holds for \\(M_{f,E}(q;a,b)\\)?", "correct_choice": {"label": "A", "text": "For every \\(\\varepsilon>0\\),\n\\[\n\\begin{aligned}\nM_{f,E}(q;a,b)=&\\;c_f\\prod_{p\\mid qab}\\left(1-\\frac{\\lambda(p)}{p}+\\frac1{p^2}\\right)\\left(1-\\frac1{p^2}\\right)^{-2}\n\\times \\frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}}\\frac{L(1,f)^2}{\\zeta(2)} \\\\\n&\\;+O\\!\\left(q^{-\\frac1{20}+\\varepsilon}a^{\\frac3{10}}+q^{-\\frac{1-2\\theta_f}{22+16\\theta_f}+\\varepsilon}b^{\\frac{3+2\\theta_f}{11+8\\theta_f}}\\right),\n\\end{aligned}\n\\]\nwhere the product is over primes \\(p\\) dividing \\(qab\\)."}, "choices": [{"label": "B", "text": "For every \\(\\varepsilon>0\\),\n\\[\n\\begin{aligned}\nM_{f,E}(q;a,b)=&\\;c_f\\prod_{p\\mid qab}\\left(1-\\frac{\\lambda(p)}{p}+\\frac1{p^2}\\right)\\left(1-\\frac1{p^2}\\right)^{-2}\n\\times \\frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}}\\frac{L(1,f)^2}{\\zeta(2)} \\\\\n&\\;+O\\!\\left(q^{-\\frac1{20}+\\varepsilon}a^{\\frac3{10}}+q^{-\\frac1{22}+\\varepsilon}b^{\\frac3{11}}\\right),\n\\end{aligned}\n\\]\nwhere the product is over primes \\(p\\) dividing \\(qab\\)."}, {"label": "C", "text": "For every \\(\\varepsilon>0\\),\n\\[\nM_{f,E}(q;a,b)=c_f\\prod_{p\\mid qab}\\left(1-\\frac{\\lambda(p)}{p}+\\frac1{p^2}\\right)\\left(1-\\frac1{p^2}\\right)^{-2}\n\\times \\frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}}\\frac{L(1,f)^2}{\\zeta(2)}+O_{f,\\varepsilon,a,b}\\!\n\\left(q^{-\\delta}\\right)\n\\]\nfor some \\(\\delta=\\delta(f)>0\\)."}, {"label": "D", "text": "For every \\(\\varepsilon>0\\),\n\\[\n\\begin{aligned}\nM_{f,E}(q;a,b)=&\\;c_f\\prod_{p\\mid qab}\\left(1-\\frac{\\lambda(p)}{p}+\\frac1{p^2}\\right)\\left(1-\\frac1{p^2}\\right)^{-2}\n\\times \\frac{c_{a,b}}{(ab)^{1/2}}\\frac{L(1,f)^2}{\\zeta(2)} \\\\\n&\\;+O\\!\\left(q^{-\\frac1{20}+\\varepsilon}a^{\\frac3{10}}+q^{-\\frac{1-2\\theta_f}{22+16\\theta_f}+\\varepsilon}b^{\\frac{3+2\\theta_f}{11+8\\theta_f}}\\right),\n\\end{aligned}\n\\]\nwhere the product is over primes \\(p\\) dividing \\(qab\\)."}, {"label": "E", "text": "For every \\(\\varepsilon>0\\),\n\\[\n\\begin{aligned}\nM_{f,E}(q;a,b)=&\\;c_f\\prod_{p\\mid qab}\\left(1-\\frac{\\lambda(p)}{p}+\\frac1{p^2}\\right)\\left(1-\\frac1{p^2}\\right)^{-2}\n\\times \\frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}}\\frac{L(1,f)^2}{\\zeta(2)} \\\\\n&\\;+O\\!\\left(q^{-\\frac1{20}+\\varepsilon}(ab)^{\\frac3{10}}+q^{-\\frac{1-2\\theta_f}{22+16\\theta_f}+\\varepsilon}(ab)^{\\frac{3+2\\theta_f}{11+8\\theta_f}}\\right),\n\\end{aligned}\n\\]\nwhere the product is over primes \\(p\\) dividing \\(qab\\)."}], "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": "theta_f-dependent exponent in the second error term", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "explicit error exponents and explicit dependence on a,b", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "symmetric main-term contribution c_{a,b}+c_{b,a}", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "separate a- and b-dependence coming from different off-diagonal regimes", "template_used": "quantifier_dependence"}]}}
{"id": "2512.09598v1", "paper_link": "http://arxiv.org/abs/2512.09598v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{t1}\n    For any $k\\in \\mathbb{N}$ and $2$-coloring of $\\mathbb{N},$ there exist $x,y\\in\\mathbb{N}$ such that $\\{x,y,xy, x+iy: i\\leq k\\}$ is monochromatic.", "start_pos": 105649, "end_pos": 105840, "label": "t1"}, "ref_dict": {"t1": "\\begin{theorem}\\label{t1}\n    For any $k\\in \\mathbb{N}$ and $2$-coloring of $\\mathbb{N},$ there exist $x,y\\in\\mathbb{N}$ such that $\\{x,y,xy, x+iy: i\\leq k\\}$ is monochromatic. \n\\end{theorem}"}, "pre_theorem_intro_text_len": 1869, "pre_theorem_intro_text": "In this paper we are interested in the following well-known conjecture of Hindman.\n\n\\begin{conj}[\\cite{hindman.conjecture}]\\label{hindman conjecture}\n    Any finite coloring of $\\mathbb{N}$ contains monochromatic sets of the form $\\{x,y,xy,x+y\\}.$\n\\end{conj}\n\nDespite its simplicity, Hindman's conjecture has remained recalcitrant for decades, and even special cases and relaxations of it have been the subject of much recent interest \\cite{shkredov,cilleruelo2012combinatorial,green2016monochromatic,moreira2017monochromatic,bowen2025monochromatic,bowen.sabok,alweiss2022monochromatic,alweiss2023monochromatic,kousek2024revisiting,richter2025sums,green2025bounds}.  \n\nMost relevant to the present paper, Moreira \\cite{moreira2017monochromatic} has shown that any finite coloring of $\\mathbb{N}$ contains monochromatic sets $\\{x,xy,x+y\\}$, i.e., Hindman's conjecture is true if we do not require that the step size $y$ is also the desired color, and the first named author has recently given an alternative proof \\cite{alweiss2022monochromatic}.  The second author \\cite{bowen2025monochromatic} has shown that Hindman's conjecture is true for colorings of $\\mathbb{N}$ into two colors, the second and the third authors  have shown \\cite{bowen.sabok} that Hindman's conjecture holds for arbitrary finite colorings of $\\mathbb{Q}$ and the first author has shown \\cite{alweiss2023monochromatic} that the general version of the Hindman conjecture with more than two variables also holds for arbitrary finite colorings of $\\mathbb{Q}$.\n\nIn the present paper we return to the two-color case analyzed in \\cite{bowen2025monochromatic}. In particular, we give a simpler proof of the main result from \\cite{bowen2025monochromatic} that generalizes to deal with more complicated configurations.  Our main result is the following, which was previously only known in the case $k=1.$", "context": "\\begin{conj}[\\cite{hindman.conjecture}]\\label{hindman conjecture}\n Any finite coloring of $\\mathbb{N}$ contains monochromatic sets of the form $\\{x,y,xy,x+y\\}.$\n\\end{conj}\n\nDespite its simplicity, Hindman's conjecture has remained recalcitrant for decades, and even special cases and relaxations of it have been the subject of much recent interest \\cite{shkredov,cilleruelo2012combinatorial,green2016monochromatic,moreira2017monochromatic,bowen2025monochromatic,bowen.sabok,alweiss2022monochromatic,alweiss2023monochromatic,kousek2024revisiting,richter2025sums,green2025bounds}.\n\nMost relevant to the present paper, Moreira \\cite{moreira2017monochromatic} has shown that any finite coloring of $\\mathbb{N}$ contains monochromatic sets $\\{x,xy,x+y\\}$, i.e., Hindman's conjecture is true if we do not require that the step size $y$ is also the desired color, and the first named author has recently given an alternative proof \\cite{alweiss2022monochromatic}. The second author \\cite{bowen2025monochromatic} has shown that Hindman's conjecture is true for colorings of $\\mathbb{N}$ into two colors, the second and the third authors have shown \\cite{bowen.sabok} that Hindman's conjecture holds for arbitrary finite colorings of $\\mathbb{Q}$ and the first author has shown \\cite{alweiss2023monochromatic} that the general version of the Hindman conjecture with more than two variables also holds for arbitrary finite colorings of $\\mathbb{Q}$.\n\nIn the present paper we return to the two-color case analyzed in \\cite{bowen2025monochromatic}. In particular, we give a simpler proof of the main result from \\cite{bowen2025monochromatic} that generalizes to deal with more complicated configurations. Our main result is the following, which was previously only known in the case $k=1.$", "full_context": "\\begin{conj}[\\cite{hindman.conjecture}]\\label{hindman conjecture}\n Any finite coloring of $\\mathbb{N}$ contains monochromatic sets of the form $\\{x,y,xy,x+y\\}.$\n\\end{conj}\n\nDespite its simplicity, Hindman's conjecture has remained recalcitrant for decades, and even special cases and relaxations of it have been the subject of much recent interest \\cite{shkredov,cilleruelo2012combinatorial,green2016monochromatic,moreira2017monochromatic,bowen2025monochromatic,bowen.sabok,alweiss2022monochromatic,alweiss2023monochromatic,kousek2024revisiting,richter2025sums,green2025bounds}.\n\nMost relevant to the present paper, Moreira \\cite{moreira2017monochromatic} has shown that any finite coloring of $\\mathbb{N}$ contains monochromatic sets $\\{x,xy,x+y\\}$, i.e., Hindman's conjecture is true if we do not require that the step size $y$ is also the desired color, and the first named author has recently given an alternative proof \\cite{alweiss2022monochromatic}. The second author \\cite{bowen2025monochromatic} has shown that Hindman's conjecture is true for colorings of $\\mathbb{N}$ into two colors, the second and the third authors have shown \\cite{bowen.sabok} that Hindman's conjecture holds for arbitrary finite colorings of $\\mathbb{Q}$ and the first author has shown \\cite{alweiss2023monochromatic} that the general version of the Hindman conjecture with more than two variables also holds for arbitrary finite colorings of $\\mathbb{Q}$.\n\nIn the present paper we return to the two-color case analyzed in \\cite{bowen2025monochromatic}. In particular, we give a simpler proof of the main result from \\cite{bowen2025monochromatic} that generalizes to deal with more complicated configurations. Our main result is the following, which was previously only known in the case $k=1.$\n\nIn the present paper we return to the two-color case analyzed in \\cite{bowen2025monochromatic}. In particular, we give a simpler proof of the main result from \\cite{bowen2025monochromatic} that generalizes to deal with more complicated configurations. Our main result is the following, which was previously only known in the case $k=1.$\n\nIn addition to proving Theorem \\ref{t1}, a primary goal of this paper is to highlight that its basic proof strategy is fairly robust and allows for many alterations and adaptations. Indeed, the second author has recently used a similar strategy to give a new proof and generalizations of the non-commuting Schur theorem for finite colorings of amenable groups \\cite{bowen.non.commuting}. The basic strategy used in the proof of Theorem \\ref{t1} and the related results \\cite{bowen2025monochromatic,bowen.sabok,bowen.non.commuting} is essentially as follows:\n\n\\begin{theorem}\\label{t2}\n For any $k\\in \\mathbb{N}$ and any $2$-coloring of $\\mathbb{N},$ there exist $x,y\\in\\mathbb{N}$ with $\\{x,y,x^y, xy^i: i\\leq k\\}$ monochromatic. \n\\end{theorem}\n\n\\begin{theorem}\\label{rational.moreira}\n Let $P\\subseteq\\pol$ be finite. For any finite coloring of $\\mathbb{N}$ there exist $x,y\\in\\mathbb{N}$ and a color class $C$ such that $\\{x,xy,x+p(y):p\\in P\\}\\subseteq C$.\n\\end{theorem}\n\n\\begin{prop}\\label{central*}\n For any finite set of polynomials $P\\subseteq\\pol$ and any finite coloring of $\\mathbb{N}$ the set $$\\{y: \\exists x \\textnormal{ with } \\{x,xy,x+p(y):p\\in P\\} \\textnormal{ monochromatic} \\} $$\n\n\\begin{proof}[Second proof of Corollary \\ref{all_syndetic}]\n Fix a finite set $P\\subseteq \\pol.$ By definition, for each $i$ there is a finite set $F_i\\subset \\mathbb{N}$ such that $S_i/F_i\\supseteq \\mathbb{N}.$ Let $k=\\max_{i\\leq r}|F_i|$ and consider the coloring $c:\\mathbb{N}\\rightarrow [k]^r$ where an $n\\in\\mathbb{N}$ is colored based on the $r$-tuple listing the minimal $f_i\\in F_i$ such that $f_in\\in S_i.$ Let $P'=\\{p/f: p\\in P, f\\in \\bigcup_{i\\leq r}F_i\\}$. By Theorem \\ref{rational.moreira}, there are integers $x',y$ such that $S=\\{x', x'y, x'+p'(y): p'\\in P'\\}$ is monochromatic according to the coloring $c.$ Now if $y\\in S_i$, by the definition of $c$ there is an $f_i$ such that $f_iS\\subseteq S_i.$ Consequently, setting $x=x'f_i$ is as desired. \n\\end{proof}\n\n\\begin{lemma}\\label{all thick}\n Suppose $k\\in\\mathbb{N}$ and $\\mathbb{N}=R\\cup B$ with both $R$ and $B$ multiplicatively thick. Then one of the colors contains a set $\\{x,y,xy,x+iy:i\\leq k\\}$.\n\\end{lemma}\n\\begin{proof}\nSuppose for the sake of contradiction that the lemma is false.\n Without loss of generality assume that $R$ is additively piecewise syndetic. Choose $N\\in\\mathbb{N}$ large enough depending on $k$.\n\n\\begin{lemma}\\label{log reduction}\n If any finite coloring of $\\mathbb{N}$ contains a monochromatic set $\\{x,y, x2^y, x+iy: i\\leq k\\}$ then any finite coloring contains a monochromatic set of the form $\\{x,y,x^y,xy^i: i\\leq k\\}.$\n\\end{lemma}\n\n\\begin{theorem}\\label{t1}\n    For any $k\\in \\mathbb{N}$ and $2$-coloring of $\\mathbb{N},$ there exist $x,y\\in\\mathbb{N}$ such that $\\{x,y,xy, x+iy: i\\leq k\\}$ is monochromatic. \n\\end{theorem}", "post_theorem_intro_text_len": 2146, "post_theorem_intro_text": "In addition to proving Theorem \\ref{t1}, a primary goal of this paper is to highlight that its basic proof strategy is fairly robust and allows for many alterations and adaptations.  Indeed, the second author has recently used a similar strategy to give a new proof and generalizations of the non-commuting Schur theorem for finite colorings of amenable groups \\cite{bowen.non.commuting}.  The basic strategy used in the proof of Theorem \\ref{t1} and the related results \\cite{bowen2025monochromatic,bowen.sabok,bowen.non.commuting} is essentially as follows:\n\n\\begin{enumerate}\n    \\item First, show that the result holds when the color of the step size $y$ is ignored.  In this case, we are interested in finding monochromatic sets $\\{x,xy,x+iy: i\\leq k\\},$ which can be done either through Moreira's work \\cite{moreira2017monochromatic} or the first author's \\cite{alweiss2022monochromatic}.\n\n    \\item Define an appropriate structure vs randomness dichotomy, and argue that a $2$-coloring must either be structured or random.\n\n    \\item Argue that in either case the extra information we obtain can be used to upgrade the proof from step (1) to control the color of the $y$ term as well. \\end{enumerate}\n\nIn order to illustrate this technique we give two proofs of Theorem \\ref{t1} based on the two ways of handling step (1) and using two different structure vs randomness dichotomies.  These are presented in the two subsections of Section \\ref{section +}.\n\nWe also show that the same basic approach can be adapted to prove the following exponential version of the result.\n\n\\begin{theorem}\\label{t2}\n    For any $k\\in \\mathbb{N}$ and any $2$-coloring of $\\mathbb{N},$ there exist $x,y\\in\\mathbb{N}$ with $\\{x,y,x^y, xy^i: i\\leq k\\}$ monochromatic. \n\\end{theorem}\n\nThe $k=1$ case of this result was proven by Sahasrabudhe \\cite{sahasrabudhe2018monochromatic}, even for arbitrary finite colorings.  However, the result seems to be new for $k>1.$  Along the way, we also prove that any finite coloring of $\\mathbb{N}$ contains monochromatic sets of the form $\\{x,x^y,xy^i: i\\leq k\\},$ which gives a new exponential version of Moreira's theorem.", "sketch": "The authors state that “the basic strategy used in the proof of Theorem \\ref{t1} … is essentially as follows:”\n\\begin{enumerate}\n\\item “First, show that the result holds when the color of the step size $y$ is ignored.” Then one finds monochromatic sets $\\{x,xy,x+iy: i\\leq k\\}$, “which can be done either through Moreira's work … or the first author's …”.\n\\item “Define an appropriate structure vs randomness dichotomy, and argue that a $2$-coloring must either be structured or random.”\n\\item “Argue that in either case the extra information we obtain can be used to upgrade the proof from step (1) to control the color of the $y$ term as well.”\n\\end{enumerate}\nThey further note that they “give two proofs of Theorem \\ref{t1} based on the two ways of handling step (1) and using two different structure vs randomness dichotomies.”", "expanded_sketch": "The authors state that “the basic strategy used in the proof of the main theorem … is essentially as follows:”\n\\begin{enumerate}\n\\item “First, show that the result holds when the color of the step size $y$ is ignored.” Then one finds monochromatic sets $\\{x,xy,x+iy: i\\leq k\\}$, “which can be done either through Moreira's work … or the first author's …”.\n\\item “Define an appropriate structure vs randomness dichotomy, and argue that a $2$-coloring must either be structured or random.”\n\\item “Argue that in either case the extra information we obtain can be used to upgrade the proof from step (1) to control the color of the $y$ term as well.”\n\\end{enumerate}\nThey further note that they “give two proofs of the main theorem based on the two ways of handling step (1) and using two different structure vs randomness dichotomies.”", "expanded_theorem": "\\label{t1}\n    For any $k\\in \\mathbb{N}$ and $2$-coloring of $\\mathbb{N},$ there exist $x,y\\in\\mathbb{N}$ such that $\\{x,y,xy, x+iy: i\\leq k\\}$ is monochromatic.,", "theorem_type": ["Universal–Existential"], "mcq": {"question": "Let $k\\in\\mathbb{N}$, and let $\\mathbb{N}$ be colored with two colors (equivalently, let $c:\\mathbb{N}\\to\\{1,2\\}$ be any 2-coloring). Which statement holds?", "correct_choice": {"label": "A", "text": "There exist $x,y\\in\\mathbb{N}$ such that all elements of the set $\\{x,\\, y,\\, xy,\\, x+iy: i\\le k\\}$ have the same color; equivalently, $\\{x,y,xy,x+y,x+2y,\\dots,x+ky\\}$ is monochromatic."}, "choices": [{"label": "B", "text": "There exist $x,y\\in\\mathbb{N}$ such that all elements of the set $\\{x,\\, xy,\\, x+iy: i\\le k\\}$ have the same color; equivalently, $\\{x,xy,x+y,x+2y,\\dots,x+ky\\}$ is monochromatic."}, {"label": "C", "text": "There exist $x,y\\in\\mathbb{N}$ such that all elements of the set $\\{x,\\, y,\\, x+y,\\, x+2y,\\dots,x+ky\\}$ have the same color."}, {"label": "D", "text": "For every color class $C$ of the 2-coloring, there exist $x,y\\in\\mathbb{N}$ such that $\\{x,\\, y,\\, xy,\\, x+iy: i\\le k\\}\\subseteq C$."}, {"label": "E", "text": "There exist $x,y\\in\\mathbb{N}$ such that all elements of the set $\\{x,\\, y,\\, x^y,\\, xy^i: i\\le k\\}$ have the same color; equivalently, $\\{x,y,x^y,xy,xy^2,\\dots,xy^k\\}$ is monochromatic."}], "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": "control_of_y_term", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_multiplicative_term_xy", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "existential_color_class_replaced_by_every_color_class", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "log_reduction_conclusion_substituted_for_main_theorem", "template_used": "wildcard"}]}}
{"id": "2512.09683v1", "paper_link": "http://arxiv.org/abs/2512.09683v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm:backbone}\nThere exist constants $C_1,C_2>0$ such that for any $\\varepsilon\\in(0,\\frac{1}{10})$,\n\\begin{equation}\\label{eq:bb-proba}\n\\frac{C_1}{\\log|\\log\\varepsilon|}\\le\\mathbbm{P}[\\Bac_\\varepsilon]\\le\\frac{C_2}{\\log|\\log\\varepsilon|}.\n\\end{equation}", "start_pos": 32731, "end_pos": 33011, "label": "thm:backbone"}, "ref_dict": {"eq:p_e": "\\begin{equation}\\label{eq:p_e}\n\\P[E]=2^{\\frac{c+1}{2}}\\sqrt{\\frac{6}{1-c}}\\left(\\int_0^\\infty\\tau^{-1-\\frac{c}{2}}e^{\\frac{1-c}{6}\\pi\\tau}\\eta(2i\\tau)^{1-c}d\\tau\\right)^{-1}\n\\end{equation}", "thm:backbone": "\\begin{theorem}\\label{thm:backbone}\nThere exist constants $C_1,C_2>0$ such that for any $\\varepsilon\\in(0,\\frac{1}{10})$,\n\\begin{equation}\\label{eq:bb-proba}\n\\frac{C_1}{\\log|\\log\\varepsilon|}\\le\\P[\\Bac_\\varepsilon]\\le\\frac{C_2}{\\log|\\log\\varepsilon|}.\n\\end{equation}\n\\end{theorem}", "eq:def-backbone": "\\begin{equation}\\label{eq:def-backbone}\n\\Bac_\\varepsilon=\\{\\exists\\ \\text{two disjoint subpaths on the trajectory } B[0,\\tau_{\\D}] \\text{ joining } \\varepsilon\\S^1 \\text{ and } \\frac{1}{2}\\S^1\\}.\n\\end{equation}", "eq:bb-proba": "\\begin{equation}\\label{eq:bb-proba}\n\\frac{C_1}{\\log|\\log\\varepsilon|}\\le\\P[\\Bac_\\varepsilon]\\le\\frac{C_2}{\\log|\\log\\varepsilon|}.\n\\end{equation}", "appendix": "\\begin{aligned}\nF(\\varepsilon)&\\le \\frac{C_0C'}{1-q_0}\\frac{1}{\\log|\\log(\\varepsilon)|-n\\log2}+q_0^n\\\\\n&\\le  \\frac{C_0C'}{1-q_0}\\frac{1}{\\log|\\log(\\varepsilon)|-\\frac{\\log2}{|\\log q_0|}(\\log\\log|\\log\\varepsilon|-\\log C')}+\\frac{C'}{\\log|\\log\\varepsilon|}\n\\le\\frac{C_2}{\\log|\\log\\varepsilon|}\n\\end{aligned}\n$$\nfor some positive constant $C_2$, as desired.\n\\end{proof}\n\n\\appendix\n\\section{Proof of Theorem~\\ref{thm-cr-mod-relation}}\n\\label{appendix}\n\nIn this appendix, we provide the proof of Theorem~\\ref{thm-cr-mod-relation}, which is implicit in~\\cite{ARS-Annulus}. For convenience, we consider the upper half plane $\\hH:=\\{z\\in\\C:\\text{Im}(z)>0\\}$, and let $\\sm$ be the restriction of the $\\SLE_{8/3}$ loop measure on $\\C$ to the loops that are contained in $\\hH$ and surround $i$. According to the conformal restriction~\\cite{werner-loops}, if $\\Phi:\\D\\to\\hH$ is a conformal map with $\\Phi(0)=i$, then $\\sm=\\Phi_*\\SLE_{8/3,\\D}^{\\lp}$. Therefore, to prove Theorem~\\ref{thm-cr-mod-relation}, it suffices to show the following\n\\begin{theorem}\n\\label{thm-appendix}\nThere is a constant $C>0$ such that for any measurable function $f:\\R_+\\to\\R_+$ and $\\lambda\\ge0$, we have\n$$\n\\int f\\left(\\Mod(\\hH\\backslash\\overline{D(\\eta)})\\right)\\psi_\\eta'(i)^\\lambda\\sm(d\\eta)=\nC\\frac{\\sqrt{12\\lambda-1}}{\\sinh\\left(\\frac{\\pi}{3}\\sqrt{12\\lambda-1}\\right)}\\int_0^\\infty e^{-(2\\lambda-\\frac{1}{6})\\pi\\tau}\\eta(2i\\tau)f(\\tau)d \\tau.\n$$\nHere $\\psi_\\eta$ is the conformal map from $\\hH$ to $D(\\eta)$ such that $\\psi_\\eta(i)=i$ and $\\psi_\\eta'(i)>0$.\n\\end{theorem}\n\nThe proof of Theorem~\\ref{thm-appendix} is based on the SLE-coupled Liouville quantum gravity with parameter $\\sqrt{\\frac{8}{3}}$. In the following we fix $\\gamma=\\sqrt{\\frac{8}{3}}$ and let $Q=\\frac{\\gamma}{2}+\\frac{2}{\\gamma}$. Let $\\P_{\\hH}$ be the law of the free boundary Gaussian free field (GFF) on $\\hH$ with mean zero on the upper semi-circle $\\S^1\\cap\\hH$. \n\n\\begin{definition}[Liouville field on $\\hH$]\n\\label{def-LF-H}\nLet $(h,c)$ be sampled from $\\P_{\\hH}\\times[e^{-Qc}dc]$, and let $\\phi(z)=h(z)-2Q\\log|z|_++c$ (here $|z|_+:=\\max\\{|z|,1\\}$). Denote the law of $\\phi$ by $\\LF_{\\hH}$.\n\nFor $\\alpha\\in\\R$, let $\\LF_{\\hH}^{(\\alpha,i)}(d\\phi):=\\lim_{\\varepsilon\\to0}\\varepsilon^{\\frac{\\alpha^2}{2}}e^{\\alpha\\phi_\\varepsilon(i)}\\LF_{\\hH}(d\\phi)$ (the limit exists in the vague topology, see~\\cite[Lemma 2.2]{ARS-FZZ}). Here $\\phi_\\varepsilon(i)$ is the average of $\\phi$ over the circle $\\partial B_\\varepsilon(i)$.\n\\end{definition}\n\nFor a sample $\\phi$ from $\\LF_{\\hH}^{(\\alpha,i)}$, we can define its boundary Gaussian multiplicative chaos (GMC) measure $\\nu_{\\phi}:=\\lim\\limits_{\\varepsilon\\to0}\\varepsilon^{\\frac{\\gamma^2}{4}}e^{\\frac{\\gamma}{2}\\phi_\\varepsilon(x)}dx$, where $\\phi_\\varepsilon(x)$ is the average of $\\phi$ over the semi-circle $\\partial B_\\varepsilon(x)\\cap\\hH$.\nThen denote $\\{\\LF_{\\hH}^{(\\alpha,i)}(\\ell)\\}_{\\ell>0}$ to be the disintegration of $\\LF_{\\hH}^{(\\alpha,i)}$ over $\\nu_\\phi(\\R)$; i.e.~$\\LF_{\\hH}^{(\\alpha,i)}=\\int_0^\\infty\\LF_{\\hH}^{(\\alpha,i)}(\\ell)d\\ell$. We refer readers to see e.g.~\\cite[Section 2.2]{ARS-FZZ} for further details. In particular, by~\\cite[Lemma 2.7]{ARS-FZZ}, for $\\alpha>\\frac{\\gamma}{2}$,\nthere are constants $C_{\\alpha,\\gamma}>0$ such that $|\\LF_{\\hH}^{(\\alpha,i)}(\\ell)|=C_{\\alpha,\\gamma}\\ell^{\\frac{2}{\\gamma}(\\alpha-Q)-1}$ for any $\\ell>0$.\n\nWe also need the Liouville field on the annulus. For $\\tau>0$, let $\\mathcal{C}_\\tau=[0,\\tau]\\times[0,1]/\\mathord\\sim$ be the horizontal cylinder with modulus $\\tau$, where under $\\sim$ we identify $(x,0)$ and $(x,1)$ for each $x\\in[0,\\tau]$. Let $\\P_{\\tau}$ be the free boundary GFF on $\\mathcal{C}_\\tau$\nwith zero mean on the circle $\\{\\frac{\\tau}{2}\\}\\times[0,1]/\\mathord\\sim$.\n\\begin{definition}[{\\cite[Definition 2.2]{ARS-Annulus}}]\n\\label{def-LF-annulus}\nLet $(h,c)$ be sampled from $\\P_\\tau\\times dc$. Then denote the law of $\\phi=h+c$ by $\\LF_\\tau$ (which is an infinite measure on $H^{-1}(\\mathcal{C}_\\tau)$).\n\\end{definition}\n\nFor a sample $\\phi$ from $\\LF_\\tau$,\nwe can similarly define its boundary GMC measures $\\nu^1_\\phi, \\nu_\\phi^2$ on the two boundaries $\\partial_1\\mathcal{C}_\\tau=\\{0\\}\\times[0,1]\\mathord\\sim$ and $\\partial_2\\mathcal{C}_\\tau=\\{1\\}\\times[0,1]\\mathord\\sim$, respectively. We also let $\\{\\LF_\\tau(\\ell_1,\\ell_2)\\}_{\\ell_1,\\ell_2>0}$ be the disintegration of $\\LF_\\tau$ over $\\nu_\\phi^1(\\partial\\cC_1)$ and $\\nu_\\phi^2(\\partial\\cC_2)$, i.e.~$\\LF_\\tau=\\iint_{\\R_+^2}\\LF_\\tau(\\ell_1,\\ell_2)d\\ell_1d\\ell_2$. The following result from~\\cite{ARS-Annulus} gives the exact solvability of $|\\LF_\\tau(\\ell_1,\\ell_2)|$, which is based on~\\cite{wu-annulus}.\n\n\\begin{proposition}[{\\cite[Equation (3.6)]{ARS-Annulus}}]\n \\label{prop-LF-integrability}\nFor $\\tau>0$ and $y\\in(-1,\\frac{4}{\\gamma^2})$, we have\n$$\n\\iint_{\\R_+^2}\\ell_1e^{-\\ell_1}\\ell_2^{y}|\\LF_\\tau(\\ell_1,\\ell_2)|d\\ell_1d\\ell_2=\\frac{\\pi\\gamma y\\Gamma(1+y)}{2\\sin(\\frac{\\gamma^2}{4}\\pi y)}e^{\\frac{\\pi}{4}\\gamma^2\\tau y^2}.\n$$\n\\end{proposition}\n\nFor two domains $D,\\wt{D}\\subset\\C$ and a conformal map $g:D\\to\\wt{D}$, when $h$ is a distribution on $D$, we define $g\\bullet h:=h\\circ g^{-1}+Q\\log|(g^{-1})'|$ (which is a distribution on $\\wt D$). Let $\\Omega$ be the space of simple loops in $\\hH$ surrounding $i$, and $\\Conf(\\hH,i)$ be the group of conformal automorphisms of $\\hH$ that fix $i$. For $(\\phi,\\eta,g,\\theta)\\in H^{-1}(\\hH)\\times\\Omega\\times \\Conf(\\hH,i)\\times[0,1]$, define a measurable map $F$ by\n$$\nF(\\phi,\\eta,g,\\theta):=\\left((g\\circ \\psi_\\eta^{-1})\\bullet \\phi|_{D(\\eta)}, f_{\\eta}^\\theta\\bullet \\phi|_{\\hH\\backslash\\overline{D(\\eta)}},\\Mod(\\hH\\backslash\\overline{D(\\eta)})\\right).\n$$\nHere $\\psi_\\eta:\\hH\\to D(\\eta)$ is the conformal map such that $\\psi_\\eta(i)=i$ and $\\psi_\\eta'(i)>0$, and\n$f_{\\eta}^\\theta:\\hH\\backslash\\overline{D(\\eta)}\\to\\mathcal{C}_\\tau$ is the conformal map such that $f_{\\eta}^\\theta(0)=\\theta i$ with $\\tau=\\Mod(\\hH\\backslash\\overline{D(\\eta)})$.\n\nLet $\\Haar_{(\\hH,i)}$ be the Haar measure on $\\Conf(\\hH,i)$ such that $|\\Haar_{(\\hH,i)}|=1$, and let $\\Unif_{[0,1]}$ be the uniform measure on $[0,1]$.\nThe following proposition, which is essentially from~\\cite{ARS-Annulus}, describes the law of $\\LF_\\hH^{(\\gamma,i)}$ when cut by a simple loop sampled from $\\sm$.\n\n\\begin{proposition}\\label{prop-sle-weld}\nThere is a constant $C>0$ such that for any $\\ell_1>0$,\n$$\nF_*\\left(\\LF_{\\hH}^{(\\gamma,i)}(\\ell_1)\\times\\sm\\times\\Haar_{(\\hH,i)}\\times\\Unif_{[0,1]}\\right)\n=C\\int_0^\\infty \\left(\\int_0^\\infty \\ell_2\\LF_{\\hH}^{(\\gamma,i)}(\\ell_2)(d\\phi_1)\\times\\LF_\\tau(\\ell_2,\\ell_1)(d\\phi_2)d\\ell_2\\right)\n\\eta(2i\\tau)d\\tau.\n$$\nHere $F_*$ stands for the pushforward of measures, and we view the right side as a measure on $(\\phi_1,\\phi_2,\\tau)$.\n\\end{proposition}\n\\begin{proof}\nBy~\\cite{ARS-Annulus}, the $\\SLE_{8/3}$ loop cut the Brownian disk into an independent pair of a (smaller) Brownian disk and a Brownian annulus; see~\\cite[Proposition 4.5]{cfsx} for the precise statement and proof. The result then follows from that the uniform embedding of the Brownian disk on $\\hH$ gives the Liouville field on $\\hH$~\\cite[Theorem 3.4]{ARS-FZZ}.\n\\end{proof}\n\nFor $\\alpha\\in\\R$, let $\\Delta_\\alpha=\\frac{\\alpha}{2}(Q-\\frac{\\alpha}{2})$. Define the measure $\\sm^\\alpha$ via $\\frac{d\\sm^\\alpha}{d\\sm}(\\eta)=\\psi_\\eta'(i)^{2\\Delta_\\alpha-2}$, where $\\psi_\\eta:\\hH\\to D(\\eta)$ is the conformal map as above. The following proposition is obtained from Proposition~\\ref{prop-sle-weld} by a standard reweighting argument. Such an argument was first developed in~\\cite{AHS-SLE-integrability}, and appeared in many recent papers on the integrability of SLE/CLE, see e.g.~\\cite{ARS-FZZ,ACSW24b,ARS-Annulus,nolin2024backboneexponenttwodimensionalpercolation}. \n\n\\begin{proposition}\n \\label{prop-sle-reweight}\nThere is a constant $C>0$ such that for any $\\ell_1>0$ and $\\alpha\\in\\R$,\n$$\nF_*\\left(\\LF_{\\hH}^{(\\alpha,i)}(\\ell_1)\\times\\sm^\\alpha\\times\\Haar_{(\\hH,i)}\\times\\Unif_{[0,1]}\\right)\n=C\\int_0^\\infty \\left(\\int_0^\\infty \\ell_2\\LF_{\\hH}^{(\\alpha,i)}(\\ell_2)(d\\phi_1)\\times\\LF_\\tau(\\ell_2,\\ell_1)(d\\phi_2)d\\ell_2\\right)\n\\eta(2i\\tau)d\\tau.\n$$\nAs in Proposition~\\ref{prop-sle-weld}, we view the right side above as a measure on $(\\phi_1,\\phi_2,\\tau)$.\n\\end{proposition}\n\n\\begin{proof}\nFor $\\phi\\in H^{-1}(\\hH)$, $\\eta\\in\\Omega$ and $g\\in\\Conf(\\hH,i)$, let $\\phi_1=(g\\circ \\psi_\\eta^{-1})\\bullet \\phi|_{D(\\eta)}$. Recall that for $\\varepsilon>0$, $(\\phi_1)_\\varepsilon(i)$ is the average of $\\phi_1$ over $\\partial B_\\varepsilon(i)$.\nBy~\\cite[Lemma 4.8]{ARS-FZZ},\nas $\\varepsilon\\to0$, we have\n$$\n\\begin{aligned}\n&F_*\\left(\\varepsilon^{\\frac{1}{2}(\\alpha^2-\\gamma^2)}e^{(\\alpha-\\gamma)(\\phi_1)_\\varepsilon(i)}\\LF_{\\hH}^{(\\gamma,i)}(\\ell_2)(d\\phi)\\times\\sm(d\\eta)\\times\\Haar_{(\\hH,i)}\\times\\Unif_{[0,1]}\\right)\\\\\n=~&F_*\\left(\\varepsilon^{\\frac{1}{2}(\\alpha^2-\\gamma^2)}e^{(\\alpha-\\gamma)(\\phi\\circ\\psi_\\eta\\circ g^{-1})_\\varepsilon(i)}\\LF_{\\hH}^{(\\gamma,i)}(\\ell_2)(d\\phi)\\times|\\psi_\\eta'(i)|^{(\\alpha-\\gamma)Q}\\sm(d\\eta)\\times\\Haar_{(\\hH,i)}\\times\\Unif_{[0,1]}\\right)\\\\\n\\to~& F_*\\left(\\LF_{\\hH}^{(\\alpha,i)}(\\ell_2)\\times\\sm^\\alpha\\times\\Haar_{(\\hH,i)}\\times\\Unif_{[0,1]}\\right).\n\\end{aligned}"}, "pre_theorem_intro_text_len": 2839, "pre_theorem_intro_text": "\\label{section-intro}\n\\subsection{Overview and the main result}\n\nThere is a strong relation between the planar Brownian motion and critical percolation. In particular, for $n\\ge 1$, the Brownian intersection exponent $\\zeta_n=\\frac{4n^2-1}{12}$~\\cite{lsw-bm-exponents1,lsw-bm-exponents2,lsw-bm-exponents3} is the same as the alternating $2n$-arm exponent of critical percolation~\\cite{smirnov-werner-percolation}. This can be well understood in the view of conformal restriction~\\cite{lsw-restriction}: both of the \\emph{hulls} of Brownian motion and percolation cluster satisfy the restriction property, and therefore, their outer boundaries can both be described by Schramm-Loewner evolution (SLE) with parameter $\\kappa=\\frac{8}{3}$.\n\nNote that in the critical percolation, one can also define the \\emph{monochromatic} arm exponents corresponding to that there exist $n$ disjoint arms of the same color joining two boundaries of an annulus. Based on the relation between Brownian motion and critical percolation, it is natural to explore the analog of such monochromatic arm exponents in the context of Brownian motion. Recently,~\\cite{nolin2024backboneexponenttwodimensionalpercolation} derives the exact value of percolation monochromatic 2-arm exponent, namely the \\emph{backbone exponent}. In this paper, we investigate its Brownian counterpart, and find that such Brownian backbone probability indeed has an iterated logarithmic decay (so the ``Brownian backbone exponent\" is equal to 0). This also shows a big difference between Brownian motion and critical percolation, although their outer boundaries (or hulls) are the same.\n\nTo be precise, let $(B_t)_{t\\ge0}$ be a planar Brownian motion starting from $0$. Denote $\\mathbbm{S}^1:=\\{z\\in\\mathbbm{C}:|z|=1\\}$ and $\\mathbbm{D}:=\\{z\\in\\mathbbm{C}:|z|<1\\}$ to be the unit circle and the unit disk, respectively. Let $\\tau_\\mathbbm{D}$ be the first hitting time of $\\mathbbm{S}^1$ for $(B_t)_{t\\ge0}$.\nFor each $\\varepsilon\\in(0,\\frac{1}{2})$, consider the event\n\\begin{equation}\\label{eq:def-backbone}\n\\Bac_\\varepsilon=\\{\\exists\\ \\text{two disjoint subpaths on the trajectory } B[0,\\tau_{\\mathbbm{D}}] \\text{ joining } \\varepsilon\\mathbbm{S}^1 \\text{ and } \\frac{1}{2}\\mathbbm{S}^1\\}.\n\\end{equation}\nNamely, $\\Bac_\\varepsilon$ happens if there are two continuous curves $\\gamma^1,\\gamma^2:[0,1]\\to\\mathbbm{C}$ such that $\\gamma^1[0,1]\\cap\\gamma^2[0,1]=\\emptyset$, $\\gamma^i[0,1]\\subset B[0,\\tau_{\\mathbbm{D}}]$, $\\gamma^i(0)\\in\\varepsilon\\mathbbm{S}^1$, and $\\gamma^i(1)\\in\\frac{1}{2}\\mathbbm{S}^1$ for $i=1,2$. Using terminology from percolation, we call $\\Bac_\\varepsilon$ the \\emph{backbone event} for the Brownian trajectory $B[0,\\tau_\\mathbbm{D}]$.\n\nThe main result in this paper is the following iterated logarithmic decay for the probability $\\mathbbm{P}[\\Bac_\\varepsilon]$.", "context": "\\label{section-intro}\n\\subsection{Overview and the main result}\n\nThere is a strong relation between the planar Brownian motion and critical percolation. In particular, for $n\\ge 1$, the Brownian intersection exponent $\\zeta_n=\\frac{4n^2-1}{12}$~\\cite{lsw-bm-exponents1,lsw-bm-exponents2,lsw-bm-exponents3} is the same as the alternating $2n$-arm exponent of critical percolation~\\cite{smirnov-werner-percolation}. This can be well understood in the view of conformal restriction~\\cite{lsw-restriction}: both of the \\emph{hulls} of Brownian motion and percolation cluster satisfy the restriction property, and therefore, their outer boundaries can both be described by Schramm-Loewner evolution (SLE) with parameter $\\kappa=\\frac{8}{3}$.\n\nNote that in the critical percolation, one can also define the \\emph{monochromatic} arm exponents corresponding to that there exist $n$ disjoint arms of the same color joining two boundaries of an annulus. Based on the relation between Brownian motion and critical percolation, it is natural to explore the analog of such monochromatic arm exponents in the context of Brownian motion. Recently,~\\cite{nolin2024backboneexponenttwodimensionalpercolation} derives the exact value of percolation monochromatic 2-arm exponent, namely the \\emph{backbone exponent}. In this paper, we investigate its Brownian counterpart, and find that such Brownian backbone probability indeed has an iterated logarithmic decay (so the ``Brownian backbone exponent\" is equal to 0). This also shows a big difference between Brownian motion and critical percolation, although their outer boundaries (or hulls) are the same.\n\nTo be precise, let $(B_t)_{t\\ge0}$ be a planar Brownian motion starting from $0$. Denote $\\mathbbm{S}^1:=\\{z\\in\\mathbbm{C}:|z|=1\\}$ and $\\mathbbm{D}:=\\{z\\in\\mathbbm{C}:|z|<1\\}$ to be the unit circle and the unit disk, respectively. Let $\\tau_\\mathbbm{D}$ be the first hitting time of $\\mathbbm{S}^1$ for $(B_t)_{t\\ge0}$.\nFor each $\\varepsilon\\in(0,\\frac{1}{2})$, consider the event\n\\begin{equation}\\label{eq:def-backbone}\n\\Bac_\\varepsilon=\\{\\exists\\ \\text{two disjoint subpaths on the trajectory } B[0,\\tau_{\\mathbbm{D}}] \\text{ joining } \\varepsilon\\mathbbm{S}^1 \\text{ and } \\frac{1}{2}\\mathbbm{S}^1\\}.\n\\end{equation}\nNamely, $\\Bac_\\varepsilon$ happens if there are two continuous curves $\\gamma^1,\\gamma^2:[0,1]\\to\\mathbbm{C}$ such that $\\gamma^1[0,1]\\cap\\gamma^2[0,1]=\\emptyset$, $\\gamma^i[0,1]\\subset B[0,\\tau_{\\mathbbm{D}}]$, $\\gamma^i(0)\\in\\varepsilon\\mathbbm{S}^1$, and $\\gamma^i(1)\\in\\frac{1}{2}\\mathbbm{S}^1$ for $i=1,2$. Using terminology from percolation, we call $\\Bac_\\varepsilon$ the \\emph{backbone event} for the Brownian trajectory $B[0,\\tau_\\mathbbm{D}]$.\n\nThe main result in this paper is the following iterated logarithmic decay for the probability $\\mathbbm{P}[\\Bac_\\varepsilon]$.", "full_context": "\\label{section-intro}\n\\subsection{Overview and the main result}\n\nThere is a strong relation between the planar Brownian motion and critical percolation. In particular, for $n\\ge 1$, the Brownian intersection exponent $\\zeta_n=\\frac{4n^2-1}{12}$~\\cite{lsw-bm-exponents1,lsw-bm-exponents2,lsw-bm-exponents3} is the same as the alternating $2n$-arm exponent of critical percolation~\\cite{smirnov-werner-percolation}. This can be well understood in the view of conformal restriction~\\cite{lsw-restriction}: both of the \\emph{hulls} of Brownian motion and percolation cluster satisfy the restriction property, and therefore, their outer boundaries can both be described by Schramm-Loewner evolution (SLE) with parameter $\\kappa=\\frac{8}{3}$.\n\nNote that in the critical percolation, one can also define the \\emph{monochromatic} arm exponents corresponding to that there exist $n$ disjoint arms of the same color joining two boundaries of an annulus. Based on the relation between Brownian motion and critical percolation, it is natural to explore the analog of such monochromatic arm exponents in the context of Brownian motion. Recently,~\\cite{nolin2024backboneexponenttwodimensionalpercolation} derives the exact value of percolation monochromatic 2-arm exponent, namely the \\emph{backbone exponent}. In this paper, we investigate its Brownian counterpart, and find that such Brownian backbone probability indeed has an iterated logarithmic decay (so the ``Brownian backbone exponent\" is equal to 0). This also shows a big difference between Brownian motion and critical percolation, although their outer boundaries (or hulls) are the same.\n\nTo be precise, let $(B_t)_{t\\ge0}$ be a planar Brownian motion starting from $0$. Denote $\\mathbbm{S}^1:=\\{z\\in\\mathbbm{C}:|z|=1\\}$ and $\\mathbbm{D}:=\\{z\\in\\mathbbm{C}:|z|<1\\}$ to be the unit circle and the unit disk, respectively. Let $\\tau_\\mathbbm{D}$ be the first hitting time of $\\mathbbm{S}^1$ for $(B_t)_{t\\ge0}$.\nFor each $\\varepsilon\\in(0,\\frac{1}{2})$, consider the event\n\\begin{equation}\\label{eq:def-backbone}\n\\Bac_\\varepsilon=\\{\\exists\\ \\text{two disjoint subpaths on the trajectory } B[0,\\tau_{\\mathbbm{D}}] \\text{ joining } \\varepsilon\\mathbbm{S}^1 \\text{ and } \\frac{1}{2}\\mathbbm{S}^1\\}.\n\\end{equation}\nNamely, $\\Bac_\\varepsilon$ happens if there are two continuous curves $\\gamma^1,\\gamma^2:[0,1]\\to\\mathbbm{C}$ such that $\\gamma^1[0,1]\\cap\\gamma^2[0,1]=\\emptyset$, $\\gamma^i[0,1]\\subset B[0,\\tau_{\\mathbbm{D}}]$, $\\gamma^i(0)\\in\\varepsilon\\mathbbm{S}^1$, and $\\gamma^i(1)\\in\\frac{1}{2}\\mathbbm{S}^1$ for $i=1,2$. Using terminology from percolation, we call $\\Bac_\\varepsilon$ the \\emph{backbone event} for the Brownian trajectory $B[0,\\tau_\\mathbbm{D}]$.\n\nThe main result in this paper is the following iterated logarithmic decay for the probability $\\mathbbm{P}[\\Bac_\\varepsilon]$.\n\nThe main result in this paper is the following iterated logarithmic decay for the probability $\\P[\\Bac_\\varepsilon]$.\n\nLet $f$ be the conformal map from $D(\\ell_1)$ to $\\D$ with $f(0)=0$, $f'(0)>0$ and let $\\rho=f(\\wt\\ell_1)$. Note that $\\CR(\\rho,0)=\\frac{\\CR(\\wt\\ell_1,0)}{\\CR(\\ell_1,0)}$. \nThe main result of this section is the following exact law of $\\CR(\\rho,0)$, which is crucial to the final proof of Theorem~\\ref{thm:backbone} in Section~\\ref{section-backbone-2}.\n\\begin{proposition}\n\\label{prop-cr-rho}\nFor $\\lambda\\ge0$, we have\n\\begin{equation}\\label{eq:cr-rho}\n\\E[\\CR(\\rho,0)^\\lambda]=\\frac{\\sinh(\\frac{\\pi}{2}\\sqrt{12\\lambda-1})}{\\sqrt{12\\lambda-1}}-\\frac{2\\sqrt{3}\\sinh(\\frac{\\pi}{3}\\sqrt{12\\lambda-1})}{\\sqrt{12\\lambda-1}\\log\\left(\\frac{(2+\\sqrt{3})^{2}+\\tanh^{2}\\left(\\frac{\\pi}{12}\\sqrt{12\\lambda-1}\\right)}{1+(2+\\sqrt{3})^{2}\\tanh^{2}\\left(\\frac{\\pi}{12}\\sqrt{12\\lambda-1}\\right)}\\right)}.\n\\end{equation}\nWhen $\\lambda\\in[0,\\frac{1}{12})$, the right side of~\\eqref{eq:cr-rho} is defined by analytic continuation; see the end of Section~\\ref{section-intro}.\n\\end{proposition}\n\nLet $\\nu_\\D$ be the law of the loop chosen from the counting measure over $\\{\\ell_i\\}_{i\\ge1}$; namely, $\\nu_\\D$ is such that $\\int F(\\eta)\\nu_\\D(d\\eta)=\\P\\left[\\sum_{i\\ge1}F(\\ell_i)\\right]$ for any bounded measurable function $F$. The following result, based on~\\cite{cfsx} and Theorem~\\ref{thm-cr-mod-relation}, gives the explicit law of the conformal radius under $\\nu_\\D$.\n\\begin{lemma}\\label{lem-CR-BM-counting-ell}\nThere is a constant $C>0$ such that for $\\lambda\\ge0$, we have\n\\begin{equation}\n \\label{eq-CR-BM-counting-ell}\n \\int\\CR(\\eta,0)^\\lambda\\nu_{\\D}(d\\eta)=C\\frac{\\sqrt{12\\lambda-1}}{\\sinh(\\frac{\\pi}{3}\\pi\\sqrt{12\\lambda-1})}\\log\\left(\\frac{(2+\\sqrt{3})^{2}+\\tanh^{2}\\left(\\frac{\\pi}{12}\\sqrt{12\\lambda-1}\\right)}{1+(2+\\sqrt{3})^{2}\\tanh^{2}\\left(\\frac{\\pi}{12}\\sqrt{12\\lambda-1}\\right)}\\right).\n \\end{equation}\nWhen $\\lambda\\in(0,\\frac{1}{12})$, the right side of~\\eqref{eq-CR-BM-counting-ell} is defined via analytic continuation as before.\n\\end{lemma}\n\\begin{proof}\nBy combining~\\cite[Lemma 5.3]{cfsx} with~\\cite[Theorem 1.3]{cfsx}, we have $\\frac{d\\nu_{\\D}}{d\\SLE^{\\lp}_{8/3,\\D}}(\\eta)=\\frac{C_1}{\\Mod(\\eta,\\S^1)}$ for some constant $C_1>0$.\nThen by Theorem~\\ref{thm-cr-mod-relation}, we find\n\\begin{equation}\\label{eq:cr-integral}\n\\int\\CR(\\eta,0)^\\lambda\\nu_{\\D}(d\\eta)=C\n \\frac{\\sqrt{12\\lambda-1}}{\\sinh(\\frac{\\pi}{3}\\sqrt{12\\lambda-1})}\\int_0^\\infty e^{-(2\\lambda-\\frac{1}{6})\\pi\\tau}\\frac{\\eta(2i\\tau)}{\\tau}d\\tau.\n\\end{equation}\nfor some constant $C>0$. \nNote that for $a>-\\frac{\\pi}{6}$, we have $\\int_0^\\infty e^{-a\\tau}\\eta(2i\\tau)d\\tau=\\sqrt{\\frac{\\pi}{2a}}\\frac{\\sinh\\left(\\sqrt{\\frac{2}{3}\\pi a}\\right)}{\\cosh\\left(\\sqrt{\\frac{3}{2}\\pi a}\\right)}$ (see e.g.~\\cite[Equation (A.4)]{ARS-Annulus}) and\n $$\n \\frac{d}{da}\\left[-\\log\\left(\\frac{(2+\\sqrt{3})^{2}+\\tanh^{2}\\left(\\frac{1}{2}\\sqrt{\\frac{\\pi}{6}a}\\right)}{(2-\\sqrt{3})^2+\\tanh^{2}\\left(\\frac{1}{2}\\sqrt{\\frac{\\pi}{6}a}\\right)}\\right)\\right]=\\sqrt{\\frac{\\pi}{2a}}\\frac{\\sinh\\left(\\sqrt{\\frac{2}{3}\\pi a}\\right)}{\\cosh\\left(\\sqrt{\\frac{3}{2}\\pi a}\\right)}.\n $$\nTherefore, we obtain\n $$\n \\int_0^\\infty e^{-a\\tau}\\frac{\\eta(2i\\tau)}{\\tau}d\\tau=\\log\\left(\\frac{(2+\\sqrt{3})^{2}+\\tanh^{2}\\left(\\frac{1}{2}\\sqrt{\\frac{\\pi}{6}a}\\right)}{1+(2+\\sqrt{3})^{2}\\tanh^{2}\\left(\\frac{1}{2}\\sqrt{\\frac{\\pi}{6}a}\\right)}\\right).\n $$\nfor $a>-\\frac{\\pi}{6}$. Combined with the right side of~\\eqref{eq:cr-integral}, we conclude.\n\\end{proof}\n\nThe upper bound of $\\P[\\mathsf{Bac}_\\varepsilon]$ then relies on the following iterative inequality.\n\\begin{proposition}\n \\label{prop-upper-bound-backbone}\n Let $q_0=\\P[\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset]\\in(0,1)$. Then\n there is a constant $C_0>0$ such that for any $\\varepsilon\\in(0,10^{-4})$,\n $$\n \\P[\\Bac_\\varepsilon]\\le \\frac{C_0}{\\log|\\log\\varepsilon|}+q_0\\P[\\Bac_{5\\sqrt{\\varepsilon}}].\n $$\n\\end{proposition}\n\\begin{proof}\nNote that\n \\begin{align}\n \\P[\\Bac_\\varepsilon]&\\le\\P[\\ell_1\\cap5\\sqrt{\\varepsilon}\\D\\neq\\emptyset]+\\P[\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset,\\Bac_\\varepsilon]\\nonumber\\\\\n &\\le \\P[\\ell_1\\cap5\\sqrt{\\varepsilon}\\D\\neq\\emptyset]+\\P[\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D\\neq\\emptyset]+\\P[\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset,\\Bac_\\varepsilon]\\nonumber\\\\\n &= \\P[\\ell_1\\cap5\\sqrt{\\varepsilon}\\D\\neq\\emptyset]+\\P[\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D\\neq\\emptyset]+\\P[\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset,\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset, \\Bac_\\varepsilon]\\label{eq:upper-1}\n \\end{align}\nHere the last line follows from that $(\\wt\\ell_1\\cap(5\\sqrt{\\varepsilon}\\D\\cup\\A_{1/2})=\\emptyset)\\cap\\Bac_\\varepsilon=\\emptyset$.\nBy Lemma~\\ref{lem-CR-BM-ell1}, Corollary~\\ref{cor-estimate-cr-rho} and using Koebe's 1/4 theorem, there exist some positive constants $C,C'$ and $C_0$ such that\n\\begin{align}\n\\P[\\ell_1\\cap5\\sqrt{\\varepsilon}\\D\\neq\\emptyset]+\\P[\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D\\neq\\emptyset]&\\le\\P[\\CR(\\ell_1,0)<20\\sqrt{\\varepsilon}]+\\P[\\CR(\\wt\\ell_1,0)<20\\sqrt{\\varepsilon}]\\nonumber\\\\\n & \\le C\\varepsilon^{\\frac{1}{8}}+C'\\frac{1}{\\log|\\log\\varepsilon|}\\le \\frac{C_0}{\\log|\\log\\varepsilon|}.\\label{eq:upper-2}\n\\end{align}\n\nLet $q_0:=\\P[\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset]\\in(0,1)$. Then\n\\begin{equation}\\label{eq:upper-3}\n\\P[\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset,\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset,\\Bac_\\varepsilon]\\le q_0\\P[\\Bac_\\varepsilon|\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset,\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset].\n\\end{equation}\nLet $g:D(\\wt\\ell_1)\\to\\D$ be the conformal map such that $g(0)=0,g(B_{s_1})=1$.\nBy Lemma~\\ref{lem-independent-decomposition-BM}, we see that $(g(B_t))_{t\\in[0,s_1]}$ is independent of $B[s_1,\\tau_\\D]$, and has the same law (up to a time change) as $B[0,\\tau_{\\D}]$ conditioned on $B_{\\tau_\\D}=1$. Hence, if we denote $\\wt\\P$ to be the law of $(g(B_t))_{t\\in[0,s_1]}$ conditioned on the event $\\{\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset\\}\\cap\\{\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset\\}$, then $\\wt\\P$ is still the same as (up to a time change) the law of $B[0,\\tau_{\\D}]$ conditioned on $B_{\\tau_\\D}=1$.\nBy Lemma~\\ref{lem-variant-Koebe-1/4}, \non the event $\\{\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset\\}\\cap\\{\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset\\}$,\nwe have $\\overline{g(\\varepsilon\\D)}\\subset5\\sqrt{\\varepsilon}\\D$ and \n$g(\\frac{1}{2}\\D\\cap D(\\wt\\ell_1))\\supset \\frac{1}{2}\\D$. Then we find\n\\begin{equation}\\label{eq:upper-4}\n\\P[\\Bac_\\varepsilon|\\wt\\ell_1\\cap5\\sqrt{\\varepsilon}\\D=\\emptyset,\\wt\\ell_1\\cap\\A_{1/2}\\neq\\emptyset]\\le \\wt\\P[\\Bac_{5\\sqrt{\\varepsilon}}]=\\P[\\Bac_{5\\sqrt{\\varepsilon}}],\n\\end{equation}\nwhere the last equality follows from the rotational invariance of $B[0,\\tau_\\D]$. The result then follows from combining~\\eqref{eq:upper-1},~\\eqref{eq:upper-2},~\\eqref{eq:upper-3} and~\\eqref{eq:upper-4}.\n\\end{proof}\n\nNow we prove the upper bound for $\\P[\\Bac_\\varepsilon]$, finishing the proof of Theorem~\\ref{thm:backbone}.\n\\begin{proof}[Proof of Theorem~\\ref{thm:backbone}, the upper bound]\nLet $\\varepsilon\\in(0,10^{-4})$ and $F(\\varepsilon):=\\P[\\Bac_\\varepsilon]$. Note that the function $x\\mapsto\\frac{1}{\\log|\\log x|}$ is increasing on $(0,10^{-4})$ and there is a constant $C>0$ such that $\\frac{1}{\\log|\\log(25x)|}\\le\\frac{C}{\\log|\\log x|}$ for any $x\\in(0,10^{-4})$. Then for each $n\\in\\mathbbm{Z}_+$ with $25\\varepsilon^{\\frac{1}{2^n}}<10^{-4}$, by applying Proposition~\\ref{prop-upper-bound-backbone} for $n$ times, we find\n\\begin{align}\n F(\\varepsilon)\n &\\le C_0\\sum\\limits_{k=0}^{n-1}\\frac{q_0^k}{\\log|\\log(5^{2-\\frac{1}{2^{k-1}}}\\varepsilon^{\\frac{1}{2^k}})|}+q_0^nF(5^{2-\\frac{1}{2^{n-1}}}\\varepsilon^{\\frac{1}{2^n}})\\nonumber\\\\\n &\\le \\frac{C_0}{1-q_0}\\frac{1}{\\log|\\log(25\\varepsilon^{\\frac{1}{2^n}})|}+q_0^n\\le\\frac{C_0C}{1-q_0}\\frac{1}{\\log|\\log(\\varepsilon^{\\frac{1}{2^n}})|}+q_0^n\\nonumber\\\\\n &=\\frac{C_0C}{1-q_0}\\frac{1}{\\log|\\log(\\varepsilon)|-n\\log2}+q_0^n.\\label{eq-proof-upper-bound-backbone}\n\\end{align}", "post_theorem_intro_text_len": 7739, "post_theorem_intro_text": "\\begin{remark}\nThe same result also holds when $\\frac{1}{2}\\mathbbm{S}^1$ in~\\eqref{eq:def-backbone} is changed to any fixed $r\\mathbbm{S}^1$ with $r\\in(0,1)$ (and for sufficiently small $\\varepsilon$), except that the corresponding constants in~\\eqref{eq:bb-proba} will depend on $r$.\n\\end{remark}\n\nNote that the backbone event $\\Bac_\\varepsilon$ is closely related to the spatial distribution of the \\emph{cut points} of $B[0,\\tau_\\mathbbm{D}]$. Indeed, our proof of Theorem~\\ref{thm:backbone} is based on our recent paper with Fu and Sun~\\cite{cfsx}, from which we can define a layer structure for the cut points of $B[0,\\tau_\\mathbbm{D}]$ and then solve it explicitly; see Section~\\ref{section-backbone-1}. We mention that~\\cite{cfsx} heavily relies on the connection between planar Brownian motion and $\\SLE_{8/3}$, and especially their coupling with Liouville quantum gravity (LQG). It would be quite interesting to see if there is a derivation of Theorem~\\ref{thm:backbone} without relying on LQG.\n\nTheorem~\\ref{thm:backbone} is related to a certain kind of special points of Brownian motion. Namely, let\n\\[\n\\mathcal{B}:=\\{B_s:s\\in[0,\\tau_\\mathbbm{D}) \\text{ and there exists } \\varepsilon>0 \\text{ such that } (B_{s+u})_{0\\le u\\le\\varepsilon} \\text{ does not have a cut point}\\}.\n\\]\nThen Theorem~\\ref{thm:backbone} indicates that $\\mathcal{B}$ is non-empty and has Hausdorff dimension 2. Furthermore, it also suggests that the Hausdorff measure of $\\mathcal{B}$ with the gauge $r\\mapsto r^2\\log\\frac{1}{r}\\log\\log\\frac{1}{r}\\log\\log\\log\\frac{1}{r}$~would exist and be non-trivial (since the Hausdorff gauge of $B[0,\\tau_\\mathbbm{D}]$ itself is $r\\mapsto r^2\\log\\frac{1}{r}\\log\\log\\log\\frac{1}{r}$, see e.g.~\\cite{LG85}). In particular, due to that planar Brownian motion a.s.~has no double cut points~\\cite{BL}, we know that $\\mathcal{B}$ contains the set of double points of $(B_t)_{0\\le t\\le\\tau_\\mathbbm{D}}$, which has the Hausdorff gauge $r\\mapsto r^2(\\log\\frac{1}{r}\\log\\log\\log\\frac{1}{r})^2$~\\cite{LG87}.\n\n\\subsection{Outlook and discussions}\n\\label{section-discussion}\n\nHere we give several remarks and related questions before going into the proof.\n\n\\begin{itemize}\n\\setlength{\\itemsep}{0pt}\n\\setlength{\\parskip}{0pt}\n\\setlength{\\parsep}{0pt}\n\n\\item One can similarly consider the probability that there exist $2n$ disjoint subpaths on $B[0,\\tau_\\mathbbm{D}]$ joining $\\varepsilon\\mathbbm{S}^1$ and $\\frac{1}{2}\\mathbbm{S}^1$ with $n\\ge2$. Note that for $n\\ge2$, such $2n$-arm probability is not straightforwardly related to the cut points of $B[0,\\tau_\\mathbbm{D}]$. Instead, one need to consider the $(2n-1)$-tuples of the local cut points such that removing these $(2n-1)$ cut points from the trajectory $B[0,\\tau_\\mathbbm{D}]$, $0$ and $B_{\\tau_\\mathbbm{D}}$ are not in the same connected component of the remaining set. However, analyzing such tuples of local cut points (e.g.~the counterpart of the layer structure in Section~\\ref{section-backbone-1}) becomes much more complicated, and it seems difficult to solve them explicitly. We mention that similar difficulty also appears in deriving the monochromatic $k$-arm exponents of critical planar percolation for $k\\ge3$, see~\\cite[Remark 2.3]{nolin2024backboneexponenttwodimensionalpercolation}.\n\n\\item \nThere is also a natural half-plane variant of our setup. Namely, let $(e_t)_{t\\ge0}$ be the Brownian excursion on the upper half plane $\\mathbbm{H}$ from $0$ to $\\infty$, and define $R\\mathbbm{S}^+:=R\\mathbbm{S}^1\\cap\\mathbbm{H}$ for $R>0$. Then for $n\\ge1$, consider the asymptotic probability that \nthere exists $(2n+1)$ disjoint subpaths on the trajectory of $(e_t)$ joining $\\mathbbm{S}^+$ and $R\\mathbbm{S}^+$ as $R\\to\\infty$.\nNote that the case $n=1$ can be similarly related to the cut points of $(e_t)$, which has been proven in~\\cite[Theorem 4]{brownian-beads}.\n\n\\item In our forthcoming work~\\cite{cx}, we will extend the result of this paper to the Brownian loop soup cases. Let $(B_t)_{0\\le t\\le\\tau_\\mathbbm{D}}$ be as before, and let $\\mathcal{L}$ be an independent Brownian loop soup on $\\mathbbm{D}$ with intensity $\\frac{c}{2}$ for $c\\in(0,1]$. \nFor a subset $A\\subset\\ol\\mathbbm{D}$, denote $\\mathcal{C}(A)$ to be the union of $A$ and all loop-soup clusters in $\\mathcal{L}$ intersecting with $A$, \nand let $H:=\\mathcal{C}(B[0,\\tau_\\mathbbm{D}])$. \nHowever, unlike the Brownian motion case, \nthere is a positive probability that $0$ itself can have two disjoint subpaths (except on $0$) on $H$\nconnected to $\\frac{1}{2}\\mathbbm{S}^1$. Moreover, in~\\cite{cx} we will show the following\n\\begin{theorem}\nWe say $t\\in(0,\\tau_\\mathbbm{D}]$ is a cut time of $H$ if $\\mathcal{C}(B[0,t])\\cap \\mathcal{C}(B[t,\\tau_\\mathbbm{D}])=B_t$. Let $t_1$ be the largest cut time of $H$ such that the boundary of $\\mathcal{C}(B[0,t_1])$ is a simple loop, and denote $E$ to be the event that $t_1$ is the smallest cut time of $H$. \nThen for $c\\in(0,1)$,\n\\begin{equation}\\label{eq:p_e}\n\\mathbbm{P}[E]=2^{\\frac{c+1}{2}}\\sqrt{\\frac{6}{1-c}}\\left(\\int_0^\\infty\\tau^{-1-\\frac{c}{2}}e^{\\frac{1-c}{6}\\pi\\tau}\\eta(2i\\tau)^{1-c}d\\tau\\right)^{-1}\n\\end{equation}\nwhere $\\eta(2i\\tau):=e^{-\\frac{\\pi}{6}\\tau}\\prod_{n=1}^{\\infty}\\left(1-e^{-4\\pi n\\tau}\\right)$ is the Dedekind eta function.\n\\end{theorem}\nNote that the right side of~\\eqref{eq:p_e} tends to $1$ as $c\\uparrow1$, and tends to $0$ as $c\\downarrow0$.\n\n\\end{itemize}\n\n\\medskip\n\\noindent\\textbf{Organization of the paper.} In Section~\\ref{section-SLE loop}, we review the explicit relation between the laws of conformal radii and moduli for $\\SLE_{8/3}$-type loops. Such relation was implicitly established in~\\cite{ARS-Annulus}, and we will provide its proof in Appendix~\\ref{appendix} for completeness. Then in Section~\\ref{section-backbone-1}, we define a layer structure for the cut points of $B[0,\\tau_\\mathbbm{D}]$ and solve it explicitly, based on the results of~\\cite{cfsx} and Section~\\ref{section-SLE loop}. Finally, we finish the proof of Theorem~\\ref{thm:backbone} in Section~\\ref{section-backbone-2}.\n\n\\medskip\n\\noindent\\textbf{Basic notations.}\nFor two compact sets $A,B\\subset\\mathbbm{C}$, let $\\mathrm{dist}(A,B):=\\inf\\{|a-b|:a\\in A,b\\in B\\}$.\nFor a simple loop $\\ell\\subset\\mathbbm{C}$, let $D(\\ell)$ be the bounded connected component of $\\mathbbm{C}\\backslash\\ell$. For a simple loop $\\ell\\subset\\mathbbm{C}$ with $0\\in D(\\ell)$, we denote the conformal radius of $D(\\ell)$ seen from 0 by $\\mathrm{CR}(\\ell,0)$, i.e. if $f:\\mathbbm{D}\\to D(\\ell)$ is a conformal map that fixes the origin, then $\\mathrm{CR}(\\ell,0)=|f'(0)|$.\n\nFor $0<r<1$, let $\\A_r$ be the standard annulus $\\{z\\in\\mathbbm{C}: r<|z|<1\\}$. For each annular domain $A\\subset\\mathbbm{C}$,\nthere exists a unique $\\tau>0$ such that $A$ and $\\A_{e^{-2\\pi\\tau}}$ are conformally equivalent. We call $\\tau$ the \\textit{modulus} of $A$ and write $\\mathrm{Mod}(A):=\\tau$. For two simple loops $\\eta_1,\\eta_2$ such that $\\overline{D(\\eta_1)}\\subset D(\\eta_2)$, we also write the modulus of the annular domain $D(\\eta_2)\\setminus\\overline{D(\\eta_1)}$ as $\\mathrm{Mod}(\\eta_1,\\eta_2)$ for simplicity.\n\nWe will frequently deal with elementary functions such as $\\sqrt{x},\\sinh(\\sqrt{x}),\\cosh(\\sqrt{x})$, and $\\tanh(\\sqrt{x})$. In the following, we view them as functions defined on $\\mathbbm{R}$ by taking $i\\sqrt{|x|},i\\sin(\\sqrt{|x|}),\\cos(\\sqrt{|x|})$, and $i\\tan(\\sqrt{|x|})$ for $x<0$, respectively.\n\n\\medskip\n\\noindent\\textbf{Acknowledgment.}\nWe are grateful to Xin Sun for many helpful discussions and suggestions on the early draft of this paper.\nG.C. and Z.X.\\ were partially supported by National Key R\\&D Program of China (No.\\ 2023YFA1010700). G.C. was partially supported by National Key R\\&D Program of China (No. 2021YFA1002700).", "sketch": "Our proof of Theorem~\\ref{thm:backbone} is described as follows: it is “based on our recent paper with Fu and Sun~\\cite{cfsx}, from which we can define a layer structure for the cut points of $B[0,\\tau_\\mathbbm{D}]$ and then solve it explicitly; see Section~\\ref{section-backbone-1}.” The approach “heavily relies on the connection between planar Brownian motion and $\\SLE_{8/3}$, and especially their coupling with Liouville quantum gravity (LQG).” In terms of organization: Section~\\ref{section-SLE loop} reviews an “explicit relation between the laws of conformal radii and moduli for $\\SLE_{8/3}$-type loops”; Section~\\ref{section-backbone-1} then defines the layer structure and “solve[s] it explicitly, based on the results of~\\cite{cfsx} and Section~\\ref{section-SLE loop}”; finally “we finish the proof of Theorem~\\ref{thm:backbone} in Section~\\ref{section-backbone-2}.”", "expanded_sketch": "Our proof of the main theorem is described as follows: it is “based on our recent paper with Fu and Sun~\\cite{cfsx}, from which we can define a layer structure for the cut points of $B[0,\\tau_\\mathbbm{D}]$ and then solve it explicitly; see Section~\\ref{section-backbone-1}.” The approach “heavily relies on the connection between planar Brownian motion and $\\SLE_{8/3}$, and especially their coupling with Liouville quantum gravity (LQG).” In terms of organization: Next we review an “explicit relation between the laws of conformal radii and moduli for $\\SLE_{8/3}$-type loops”; we then define the layer structure and “solve[s] it explicitly, based on the results of~\\cite{cfsx} and the preceding review”; finally “we finish the proof of the main theorem later.”", "expanded_theorem": "\\label{thm:backbone}\nThere exist constants $C_1,C_2>0$ such that for any $\\varepsilon\\in(0,\\frac{1}{10})$,\n\\begin{equation}\\label{eq:bb-proba}\n\\frac{C_1}{\\log|\\log\\varepsilon|}\\le\\mathbbm{P}[\\Bac_\\varepsilon]\\le\\frac{C_2}{\\log|\\log\\varepsilon|}.\n\\end{equation}", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Let $(B_t)_{t\\ge 0}$ be planar Brownian motion started at $0$, let $\\mathbb D=\\{z\\in\\mathbb C:|z|<1\\}$, let $\\mathbb S^1=\\{z\\in\\mathbb C:|z|=1\\}$, and let $\\tau_{\\mathbb D}$ be the first time that $B_t$ hits $\\mathbb S^1$. For $\\varepsilon\\in(0,\\tfrac12)$, define the backbone event $\\mathrm{Bac}_\\varepsilon$ to be the event that there exist two disjoint subpaths of the Brownian trajectory $B[0,\\tau_{\\mathbb D}]$ joining $\\varepsilon\\mathbb S^1$ to $\\tfrac12\\mathbb S^1$; equivalently, there exist continuous curves $\\gamma^1,\\gamma^2:[0,1]\\to\\mathbb C$ with disjoint images, each image contained in $B[0,\\tau_{\\mathbb D}]$, such that $\\gamma^i(0)\\in \\varepsilon\\mathbb S^1$ and $\\gamma^i(1)\\in \\tfrac12\\mathbb S^1$ for $i=1,2$. Which explicit statement holds for the probability of $\\mathrm{Bac}_\\varepsilon$?", "correct_choice": {"label": "A", "text": "There exist constants $C_1,C_2>0$ such that for every $\\varepsilon\\in(0,\\tfrac{1}{10})$,\\[\\frac{C_1}{\\log|\\log\\varepsilon|}\\le \\mathbb P[\\mathrm{Bac}_\\varepsilon]\\le \\frac{C_2}{\\log|\\log\\varepsilon|}.\\]"}, "choices": [{"label": "B", "text": "There exist constants $C_1,C_2>0$ such that for every $\\varepsilon\\in(0,\\tfrac{1}{10})$,\\[\\frac{C_1}{|\\log\\varepsilon|}\\le \\mathbb P[\\mathrm{Bac}_\\varepsilon]\\le \\frac{C_2}{|\\log\\varepsilon|}.\\]"}, {"label": "C", "text": "There exists a constant $C_2>0$ such that for every $\\varepsilon\\in(0,\\tfrac{1}{10})$,\\[\\mathbb P[\\mathrm{Bac}_\\varepsilon]\\le \\frac{C_2}{\\log|\\log\\varepsilon|}.\\]"}, {"label": "D", "text": "There exist constants $C_1,C_2>0$ such that for every $\\varepsilon\\in(0,\\tfrac{1}{2})$,\\[\\frac{C_1}{\\log|\\log\\varepsilon|}\\le \\mathbb P[\\mathrm{Bac}_\\varepsilon]\\le \\frac{C_2}{\\log|\\log\\varepsilon|}.\\]"}, {"label": "E", "text": "There exist constants $C_1,C_2>0$ such that for every $\\varepsilon\\in(0,\\tfrac{1}{10})$,\\[\\frac{C_1}{\\log|\\log\\varepsilon|}\\le \\mathbb P[\\mathrm{Bac}_\\varepsilon]\\le \\frac{C_2}{\\log|\\log\\varepsilon|^2}.\\]"}], "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": "iterated-log decay replaced by single-log decay", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the lower bound while keeping the proved upper bound scale", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "validity range in $\\varepsilon$ extended from $(0,\\tfrac{1}{10})$ to $(0,\\tfrac{1}{2})$", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "asymmetric strengthening of the upper bound to a smaller order than the theorem gives", "template_used": "wildcard"}]}}
{"id": "2512.09839v1", "paper_link": "http://arxiv.org/abs/2512.09839v1", "theorems_cnt": 1, "theorem": {"env_name": "thm", "content": "[Main Theorem]\n\\label{mainThm}\nFor some $a_0 \\ge 1$ and $N \\in (0, 2)$, let $V : \\R^2 \\to \\R$ satisfy \n\\begin{equation}\n\\label{Vbound}\n\\left\\vertV(w)\\right\\vert \\le a_0^2 \\left< w \\right>^{-N}.\n\\end{equation}\nAssume that $u : \\R^2 \\to \\R$ is a solution to \n\\begin{equation}\n\\label{ellipEq}\n- \\Delta u + V u = 0 \\, \\text{ in } \\R^2\n\\end{equation}\nwith the properties that \n\\begin{equation}\n\\label{solNorm}\n\\left\\vertu(0)\\right\\vert = 1\n\\end{equation}\nand for each $w \\in \\R^2$,\n\\begin{equation}\n\\label{uBound}\n\\left\\vertu(w)\\right\\vert \\le \\exp\\pr{c_0 \\left\\vertw\\right\\vert^{1 - \\frac N 2}}.\n\\end{equation}\nFor every $\\varepsilon \\in \\left( 0, \\frac N 2 \\right) $, there exists ${R}_0(N, a_0, c_0, \\varepsilon) > 0$ so that whenever $R \\ge {R}_0$, it holds that\n\\begin{equation}\n\\label{uLower}\n\\inf_{\\left\\vertw_0\\right\\vert = R} \\left\\| u\\right\\|_{L^\\infty\\left( B_1(w_0) \\right) } \\ge \\exp\\pr{-R^{1 - \\frac N 2 + \\varepsilon}}.\n\\end{equation}", "start_pos": 21228, "end_pos": 22053, "label": "mainThm"}, "ref_dict": {"mainThm": "\\begin{thm}[Main Theorem]\n\\label{mainThm}\nFor some $a_0 \\ge 1$ and $N \\in (0, 2)$, let $V : \\R^2 \\to \\R$ satisfy \n\\begin{equation}\n\\label{Vbound}\n\\abs{V(w)} \\le a_0^2 \\innp{w}^{-N}.\n\\end{equation}\nAssume that $u : \\R^2 \\to \\R$ is a solution to \n\\begin{equation}\n\\label{ellipEq}\n- \\LP u + V u = 0 \\, \\text{ in } \\R^2\n\\end{equation}\nwith the properties that \n\\begin{equation}\n\\label{solNorm}\n\\abs{u(0)} = 1\n\\end{equation}\nand for each $w \\in \\R^2$,\n\\begin{equation}\n\\label{uBound}\n\\abs{u(w)} \\le \\exp\\pr{c_0 \\abs{w}^{1 - \\frac N 2}}.\n\\end{equation}\nFor every $\\eps \\in \\pr{0, \\frac N 2}$, there exists ${R}_0(N, a_0, c_0, \\eps) > 0$ so that whenever $R \\ge {R}_0$, it holds that\n\\begin{equation}\n\\label{uLower}\n\\inf_{\\abs{w_0} = R} \\norm{u}_{L^\\iny\\pr{B_1(w_0)}} \\ge \\exp\\pr{-R^{1 - \\frac N 2 + \\eps}}.\n\\end{equation}\n\\end{thm}", "ellipEq": "\\begin{equation}\n\\label{ellipEq}\n- \\LP u + V u = 0 \\, \\text{ in } \\R^2\n\\end{equation}", "TransMaps": "\\begin{align}\n\\label{propApp}\n\\sup_{B_{\\tilde r} \\setminus \\bigcup 3 \\WT D_j} \\abs{\\tilde h}\n\\ge \\pr{\\frac{16 {\\tilde r}} {\\WT R}}^{\\tilde \\kappa} \\sup_{B_{\\WT T} \\setminus \\bigcup 3 \\WT D_j} \\abs{\\tilde h},\n\\end{align}\nwhere\n\\begin{align*}\n\\tilde \\kappa\n:= \\kappa\\pr{\\WT R, \\WT S, M}\n= \\max \\set{6C_H C_1 R \\sqrt{\\log R}, 2^{13} \\pr{c_1 R + L + \\frac {c_d}{\\log R}} RS^{-1}}.\n\\end{align*}\nSince $R \\ge 2^{10}$ implies $\\frac R 8 > 1$, while $R \\ge R_2$ implies $\\sqrt{\\frac{98 \\cdot 16 R}{C_1 \\sqrt{\\log R}}} > 1$, then $r_0 := \\min\\set{\\frac R 8, \\sqrt{\\frac{98 \\cdot 16 R}{C_1 \\sqrt{\\log R}}}} > 1$.\nAssume that $R \\ge \\hat R_0$ and $r < r_0$.\nCombining \\eqref{ufComparison}, \\eqref{smallBallBound}, \\eqref{propApp}, \\eqref{scaleBounds}, and \\eqref{midBallBound} shows that\n\\begin{align*}\n\\frac 1{1 - c_b \\eps^2} \\sup_{B_r} \\abs{v}\n&\\ge \\sup_{B_r} \\abs{f}\n\\ge \\sup_{B_{\\tilde r} \\setminus \\bigcup 3 \\WT D_j} \\abs{\\tilde h}\n\\ge \\pr{\\frac{16 {\\tilde r}} {\\WT R}}^{\\tilde \\kappa} \\sup_{B_{\\WT T} \\setminus \\bigcup 3 \\WT D_j} \\abs{\\tilde h}\n\\ge \\pr{\\frac{r} {R}}^{2\\tilde \\kappa} \\frac {e^{- L}} {1 + c_b \\eps^2} .\n\\end{align*}\nSince $C_1 =  \\frac {a} {32} \\sqrt{\\frac{2 C_K }{\\ln 2}}$, $\\disp \\frac{1 - c_b \\eps^2}{1 + c_b \\eps^2} \\ge e^{- \\frac{c_d}{\\log R}}$, and $e < 4 \\le \\pr{2^{10}}^{\\frac 1 5} \\le \\pr{\\frac{R}{r}}^{\\frac 1 5}$, then \\eqref{propConc} follows, as required. \n\\end{proof}\n\n\\section{Transformation maps}\n\\label{TransMaps}\n\nIn this section, we introduce the transformation maps that serve as real-valued versions of the conformal maps $z \\mapsto z^\\al$.\nOnce the maps are defined, we show how they transform solutions to elliptic PDEs and how they transform balls.\nThese results will be used at the beginning and at the end of the proof of Theorem \\ref{mainThm}.\n\nGiven $z = (x,y) \\in \\R^2$ in Cartesian coordinates, the polar coordinates for $z$ are $(r, \\te) \\in \\R_+ \\times (-\\pi, \\pi]$, where $r = \\sqrt{x^2 + y^2}$ and $\\te = \\sgn(y) \\arccos\\pr{\\frac x r}$.\nLet $\\R^2_+ = \\set{z = (r,\\te) : r > 0, \\te \\in (- \\frac \\pi 2, \\frac \\pi 2)}$ denote the right half-plane.\nFor $\\al \\in (1, \\iny)$, we define $T_\\al : \\R^2 \\to \\R^2$ to be the map associated to the conformal transformation on $\\C$ given by $z \\mapsto z^\\al$. \nIn polar coordinates, $T_\\al$ is described as\n\\begin{equation}\n\\label{TalDefn}\nT_\\al(r, \\te) = (r^\\al, \\al \\te).\n\\end{equation}\nTo avoid continuity issues at $\\te = \\pi$, we restrict the domain and only consider $T_\\al : \\R^2_+ \\to \\R^2$.\n\n\\begin{lem}[Transformation of PDEs]\n\\label{transformEqLemma}\nIf $u : \\R^2 \\to \\R$ is a solution to \\eqref{ellipEq}, then with $v : \\R^2_+ \\to \\R$ and $W : \\R^2_+ \\to \\R$ defined by $v(z) = u(T_\\al(z))$ and $W(z) = \\al^2 \\abs{z}^{2\\al - 2} V(T_\\al(z))$, it holds that\n$$-\\LP v + W v = 0 \\; \\text{ in } \\R^2_+.$$\n\\end{lem}\n\n\\begin{proof}\nLet $w := T_\\al(z)$ be given in polar coordinates by $\\pr{\\rho, \\vp}$.\nSince $u$ is defined on $\\R^2$, then in polar coordinates, $u(\\rho, \\vp)$ can be defined for all $\\rho \\in \\R^+$ and all $\\vp \\in \\R$ using periodicity.\nWe then see that $v(z) = v(r, \\te) = u(r^\\al, \\al \\te) = u(T_\\al(z))$ is well-defined on $\\R^2_+$. \nSince\n\\begin{align*}\n\\del_r v(r, \\te) &= \\al r^{\\al - 1} \\del_\\rho u(\\rho, \\vp) \\\\\n\\del_{r}^2 v(r, \\te) &= \\al^2 r^{2\\al - 2} \\del_{\\rho}^2 u(\\rho, \\vp) + \\al \\pr{\\al - 1} r^{\\al - 2} \\del_\\rho u(\\rho, \\vp) \\\\\n\\del_{\\te}^2 v(r, \\te) &= \\al^2 \\del_{\\vp}^2 u(\\rho, \\vp),\n\\end{align*}\nthen\n\\begin{align*}\n\\LP v(z)\n&= \\del_{r}^2 v(r, \\te) + \\frac 1 r \\del_{r} v(r, \\te) + \\frac 1 {r^2} \\del_{\\te}^2 v(r, \\te) \\\\\n&= \\al^2 r^{2\\al - 2} \\del_{\\rho}^2 u(\\rho, \\vp) + \\al \\pr{\\al - 1} r^{\\al - 2} \\del_\\rho u(\\rho, \\vp)\n+ \\al r^{\\al - 2} \\del_\\rho u(\\rho, \\vp)\n+ r^{-2} \\al^2 \\del_{\\vp}^2 u(\\rho, \\vp) \\\\\n&= \\al^2 r^{2\\al - 2} \\brac{\\del_\\rho^2 u(\\rho, \\vp) + \\rho^{-1} \\del_\\rho u(\\rho, \\vp) + \\rho^{-2}\\del_{\\vp}^2 u(\\rho, \\vp)} \\\\\n&= \\al^2 r^{2\\al - 2} \\LP u(w) \n=  \\al^2 r^{2\\al - 2} V(w) u(w),\n\\end{align*}\nwhere we have used \\eqref{ellipEq}.\nWith $W(z) = \\al^2 \\abs{z}^{2\\al - 2} V(T_\\al(z)) = \\al^2 r^{2\\al - 2} V(w)$ as given, the conclusion follows.\n\\end{proof}\n\n\\begin{lem}[Ball containment]\n\\label{ballContainLemma}\nLet $z_0 = (r_0, 0)$ and set $\\disp \\tilde r = \\frac{r_0^{1 - \\al}}{2 \\sqrt 3 \\al}$.\nThere exists $r_\\al > 0$ so that whenever  with $r_0 \\ge r_\\al$, it holds that $T_\\al(B_{\\tilde r}(z_0)) \\su B_1(T_\\al(z_0))$ and $B_1(T_\\al(z_0)) \\su T_\\al\\pr{B_1(z_0)}$.\n\\end{lem}\n\n\\begin{proof}\nWith $\\tilde r$ as given, it can be shown that\n\\begin{equation}\n\\label{ballIn} \n\\begin{aligned}\nB_{\\tilde r}(z_0) \n&\\su \\set{r \\in [r_0 - \\tilde r, r_0 + \\tilde r], \\abs{\\te} \\le \\frac {2\\tilde r} {\\sqrt 3 r_0}} \\\\\n&= \\set{r \\in \\brac{r_0\\pr{1 -  \\frac{r_0^{- \\al}}{2 \\sqrt 3 \\al}}, r_0\\pr{1 +  \\frac{r_0^{- \\al}}{2 \\sqrt 3 \\al}}}, \\abs{\\te} \\le \\frac {1} {3 \\al r_0^\\al}}.\n\\end{aligned}\n\\end{equation}\nWith $w_0 = T_\\al(z_0)= \\pr{r_0^\\al, 0} = \\pr{\\rho_0, 0}$, it follows from \\eqref{ballIn} that\n\\begin{align*}\nT_\\al(B_{\\tilde r}(z_0)) \n&\\su \\set{r^\\al \\in \\brac{r_0^\\al\\pr{1 -  \\frac{r_0^{- \\al}}{2 \\sqrt 3 \\al}}^\\al, r_0^\\al \\pr{1 +  \\frac{r_0^{- \\al}}{2 \\sqrt 3 \\al}}^\\al}, \\abs{\\al \\te} \\le \\frac {\\al} {3 \\al r_0^\\al}} \\\\\n&= \\set{\\rho \\in \\brac{\\rho_0 \\pr{1 - \\frac{\\rho_0^{- 1}}{2 \\sqrt 3 \\al}}^\\al, \\rho_0 \\pr{1 + \\frac{\\rho_0^{- 1}}{2 \\sqrt 3 \\al} }^\\al}, \\abs{\\vp} \\le \\frac {1} {3 \\rho_0} }.\n\\end{align*}\nSince\n\\begin{align*}\n\\rho_0 \\pr{1 \\pm \\frac{\\rho_0^{- 1}}{2 \\sqrt 3 \\al} }^\\al\n&= \\rho_0 \\pm \\frac{1}{2 \\sqrt 3} + \\frac{(1 - \\frac 1 \\al)}{24} r_0^{- \\al} + \\ldots \n\\end{align*}\nthen there exists $r_1(\\al) \\gg 1$ so that whenever $r_0 \\ge r_1$, we have $\\rho_0 - \\frac 1 2 \\le \\rho_0 \\pr{1 - \\frac{\\rho_0^{- 1}}{2 \\sqrt 3 \\al} }^\\al$ and $\\rho_0 \\pr{1 + \\frac{\\rho_0^{- 1}}{2 \\sqrt 3 \\al} }^\\al \\le \\rho_0 +\\frac 1 2$.\nBecause,\n\\begin{align}\n&B_1(w_0)  \\supset \\set{\\rho \\in \\brac{\\rho_0 - \\frac 1 2, \\rho_0 + \\frac 1 2}, \\abs{\\vp} \\le \\frac 1 {3\\rho_0} },\n\\label{ballOf}\n\\end{align}", "harmonic": "\\begin{equation}\n\\label{uLower}\n\\inf_{\\abs{w_0} = R} \\norm{u}_{L^\\iny\\pr{B_1(w_0)}} \\ge \\exp\\pr{-R^{1 - \\frac N 2 + \\eps}}.\n\\end{equation}\n\\end{thm}\n\nThe results of \\cite{Dav14} prove an estimate of the form \\eqref{est} with $\\be = \\frac{4 - 2N}{3} = \\frac 4 3 \\pr{1 - \\frac N 2} > 1 - \\frac N 2 + \\eps$.\nIn that article, the assumptions are the same as those in Theorem \\ref{mainThm}, except that $u$ may be complex-valued.\nThus, as in the case of bounded $V$, Theorem \\ref{mainThm} illustrates that better estimates hold in the real-valued planar setting.\n\nAs illustrated by the following example, Theorem \\ref{mainThm} is sharp (up to $\\eps$) for all $N \\in (0, 2)$.\nFix $N \\in (0, 2)$, then set $u(z) = \\exp\\pr{- \\abs{z}^{1 - \\frac N 2}}$.\nA computation shows that $u$ satisfies \\eqref{ellipEq} where\n$$V(z) := \\pr{1 - \\frac N 2}^2 \\pr{\\abs{z}^{1 - \\frac N 2} - 1}\\abs{z}^{-1 - \\frac N 2}$$ \nsatisfies $\\abs{V(z)} \\lesssim \\abs{z}^{-N}$.\nOn the other hand, for any $\\be > 0$, with $u(z) = \\abs{z}^{-\\be}$ on $\\abs{z} > 1$, we see that $\\disp \\LP u = \\be^2 \\abs{z}^{-\\be-2}$ and therefore $u$ satisfies \\eqref{ellipEq} on an exterior domain with $V(z) := \\be^2\\abs{z}^{-2}$.\nIn particular, we may not have exponential behavior when $V$ decays fast enough, which explains why we restrict ourselves to $N < 2$.\n\nTo prove Theorem \\ref{mainThm}, we use an iterative argument that is reminiscent of the one in \\cite{Dav14}, see also \\cite{LW14, DKW19, Dav20a, Dav25}.\nTo initialize the iteration, we apply a quantitative estimate of the form \\eqref{est} with $\\be = 1$.\nThis result, which verifies Landis' conjecture in the real-valued planar setting, was originally proved by Logunov, Malinnikova, Nadirashvili, and Nazarov in \\cite{LMNN20}, and we formulate it in Theorem \\ref{LandisGrowth} below.\nThe iteration argument then relies on repeated applications of Proposition \\ref{InductiveProp} which is proved using the ideas from \\cite{LMNN20}.\nRoughly speaking, Proposition \\ref{InductiveProp} shows that if an estimate like \\eqref{est} holds with $\\be = \\be_0$, then for some $x_1$ with $\\abs{x_1} \\gg \\abs{x_0}$, another estimate like \\eqref{est} holds with $x_0$ replaced by $x_1$ and $\\be = \\be_1 \\in \\brac{1, \\be_0}$.\nWhen $\\be_0 = 1$, Proposition \\ref{InductiveProp} isn't useful, but when $\\be_0 > 1$, we can decrease the exponent, i.e., make $\\be_1 < \\be_0$.\nTherefore, to benefit from the iteration argument, we need to transform to a situation where $\\be_0 > 1$.\nWe observe that if $u$ is composed with the real-variable version of the conformal transformation $z \\mapsto z^\\al$, then the new function also satisfies a Schr\\\"odinger equation.\nBy choosing $\\al > 1$ appropriately, we can ensure that the new potential function is bounded and that the new solution function satisfies a version of \\eqref{est} with $\\be > 1$.\nBy repeatedly applying Proposition \\ref{InductiveProp} to the transformed equation, we can make $\\be$ arbitrarily close to $1$.\nFinally, to reach the conclusion, we undo the change of variables.\n\nWe use the notation $B_r(z)$ to denote a ball of radius $r > 0$ centered at the point $z$, abbreviated by $B_r$ when the center is clear.\nGeneric constants are denoted by $c, C$ and may change from line to line without comment.\nSpecific constants will be indicated by subscripts.\n\nThe article is organized as follows.\nIn Section \\ref{harmonic}, we present a unique continuation theorem for harmonic functions in punctured domains.\nThe content of this section is very similar to \\cite[Section 5]{LMNN20} and \\cite[Section 2]{Dav24}.\nThe iterative result described by Proposition \\ref{InductiveProp} is the content of Section \\ref{localProof}.\nProposition \\ref{InductiveProp} is a three-ball inequality for solutions to Schr\\\"odinger equations and its proof relies on the results from Section \\ref{harmonic}. In Section \\ref{TransMaps}, we introduce the real-valued versions of $z \\mapsto z^\\al$ and record some of their properties.\nIn particular, we show how solutions behave when they are composed with these transformations.\nFinally, the proof of Theorem \\ref{mainThm} is presented in Section \\ref{MainProof}.\n\n\\section{Decay properties of harmonic functions in punctured domains}\n\\label{harmonic}\n\nIn this section, we present and prove quantitative unique continuation results (in the form of three-ball inequalities) for harmonic functions in punctured domains.\nWe recall the following application of the Harnack inequality which appears in \\cite[Claim 5.2]{LMNN20} and is repeated in \\cite[Lemma 2.1]{Dav24}.\n\n\\begin{lem}[Harnack application]\n\\label{discBounds}\nLet $\\set{D_j}$ be a finite collection of $100$-separated unit disks in the plane.\nAssume that $h$ is real-valued and harmonic in $\\R^2 \\setminus \\bigcup D_j$ and that for each index $j$, $h$ doesn't change sign in $5D_j \\setminus D_j$.\nThere exists an absolute constant $C_H \\ge 10$ for which\n\\begin{enumerate}\n\\item $\\disp \\max_{\\del \\pr{3 D_j}} \\abs{h} \\le C_H \\min_{\\del \\pr{3 D_j}} \\abs{h}$\n\\item $\\disp \\max_{\\del \\pr{3 D_j}} \\abs{\\gr h} \\le C_H \\min_{\\del \\pr{3 D_j}} \\abs{h}$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nAn application of the Harnack inequality shows that there exists $C_H > 0$ so that for every $j$\n\\begin{align*}\n\\max_{\\del \\pr{3 D_j}} \\abs{h}\n&\\le \\sup_{4 D_j \\setminus 2 D_j} \\abs{h} \n\\le C_H \\inf_{4 D_j \\setminus 2 D_j} \\abs{h}\n\\le C_H \\min_{\\del \\pr{3 D_j}} \\abs{h}.\n\\end{align*}\nFor each $z \\in \\del \\pr{3 D_j}$, since $h$ doesn't change signs in $B_2(z)$, an application of Cauchy's inequality as in \\cite[Lemma 1.11]{HL11} shows that $\\abs{\\gr h(z)} \\le \\abs{h(z)}$ and the conclusion follows.\n\\end{proof}\n\nNow we state and prove the main result of this section.\nThe following is a slight modification of \\cite[Lemma 2.1]{Dav24}, which resembles the result \\cite[Theorem 5.3]{LMNN20}.\n\n\\begin{prop}[Three-ball inequality for harmonic function in punctured domain]\n\\label{harmonicProp}\nLet $\\set{D_j}$ be a finite collection of $100$-separated unit disks in the plane for which $0 \\notin \\bigcup 3 D_j$.\nFor some $R \\ge 2^{10}$, let $h$ be a harmonic function in $B_R \\setminus \\bigcup D_j$ with the property that for each index $j$, $h$ doesn't change sign in $\\pr{5D_j \\setminus D_j} \\cap B_R$.\nAssume that for some $S \\in \\pr{2^8, \\frac R 2}$ and some $M > \\log\\pr{16R}$, with $T := R - \\frac S {32}$, \nit holds that\n\\begin{equation}\n\\label{normalization}\n\\sup_{B_T \\setminus \\bigcup 3 D_j} \\abs{h}  \\ge e^{-M} \\sup_{B_R \\setminus \\bigcup 3 D_j} \\abs{h}.\n\\end{equation}", "localProof": "\\begin{align}\n\\label{j2j0Comp}\nC_Hm_{j_2} \\abs{z_{j_2}}^{-k} \n&\\ge \\sup_{\\del \\pr{3 D_{j_2}}} \\abs{h} \\abs{z_{j_2}}^{-k}\n\\ge \\abs{h(s z_{j_0})} \\abs{s z_{j_0}}^{-k}\n\\ge \\frac{m_{j_0}}{2} s^{-k} \\abs{z_{j_0}}^{-k}.\n\\end{align}\nSince $z_{j_0} \\in \\del \\pr{3 D_{j_0}}$ and $s z_{j_0} \\in \\del \\pr{3 D_{j_2}}$ where $j_0 \\ne j_2$, and the balls $\\set{D_j}$ are of unit radius and $100$-separated, then $\\abs{z_{j_0} - s z_{j_0}} \\ge 96$.\nAfter rearrangement, we see that $s^{-k} \\ge \\pr{1 - \\frac{96}{R}}^{-k}$.\nSince $10 \\le C_H$, $2C_HR \\le k$, and $\\frac{96}{R} < - \\log\\pr{1 - \\frac{96}{R}}$, then \n\\begin{align*}\n\\log\\pr{2C_H}\n&< 96 \\cdot 2 C_H\n\\le \\frac{96}{R} k\n< - k \\log\\pr{1 - \\frac{96}{R}}\n\\le - k \\log s,\n\\end{align*}\nfrom which it follows that $s^{-k} > 2C_H$.\nWe then conclude from \\eqref{j2j0Comp} that $m_{j_2} \\abs{z_{j_2}}^{-k} > m_{j_0} \\abs{z_{j_0}}^{-k}$ which contradicts \\eqref{j0Defn} and gives the desired contradiction.\nIn other words, \\eqref{contraBound} fails to hold and we see that\n\\begin{align*}\n\\sup_{B_r \\setminus \\bigcup 3 D_j} \\abs{h} \n&> \\pr{\\frac{16r} R}^{3k}\n= \\pr{\\frac {16r} R}^{3k} \\sup_{B_T \\setminus \\bigcup 3 D_j} \\abs{h},\n\\end{align*}\nwhich implies \\eqref{lowerBound} by our choice of $k$.\n\\end{proof}\n\n\\section{The iterative proposition}\n\\label{localProof}\n\nHere we present the proposition that is used repeatedly in the iteration argument.\nThis unique continuation result takes the form of a three-ball inequality for solutions to Schr\\\"odinger equations.\nThe techniques used to prove this theorem are very similar to those that appear in the proofs of \\cite[Theorem 2.2]{LMNN20} and \\cite[Theorem 1.2]{Dav24}.\n\n\\begin{prop}[Iterative Proposition]\n\\label{InductiveProp}\nGiven $a > 0$ and $R \\ge \\hat R_0(a)$, let $W : B_R \\to \\R$ satisfy $\\norm{W}_{L^\\iny(B_R)} \\le a^2$ and let $v : B_R \\to \\R$ be a solution to \n$$-\\LP v + W v = 0 \\; \\text{ in } B_R$$\nwith\n\\begin{equation}\n\\label{propUB}\n\\norm{v}_{L^\\iny(B_R)} \\le e^{c_1 R}\n\\end{equation}\nfor some $c_1 \\ge 1$.\nAssume that for some $S \\in \\pr{2^8, \\frac R 2}$, there exists $z_0 \\in \\overline{B}_{R-S}$ and $L \\ge 0$ such that\n\\begin{equation}\n\\label{propLB}\n\\abs{v(z_0)} \\ge e^{- L}.\n\\end{equation}\nThen there exists $r_0(a, R) > 1$ so that whenever $r \\in (0, r_0)$, it holds that\n\\begin{align}\n\\label{propConc}\n\\norm{v}_{L^\\iny(B_r)}\n&\\ge \\pr{\\frac{r} {R}}^{\\tau},\n\\end{align}", "InductiveProp": "\\begin{prop}[Iterative Proposition]\n\\label{InductiveProp}\nGiven $a > 0$ and $R \\ge \\hat R_0(a)$, let $W : B_R \\to \\R$ satisfy $\\norm{W}_{L^\\iny(B_R)} \\le a^2$ and let $v : B_R \\to \\R$ be a solution to \n$$-\\LP v + W v = 0 \\; \\text{ in } B_R$$\nwith\n\\begin{equation}\n\\label{propUB}\n\\norm{v}_{L^\\iny(B_R)} \\le e^{c_1 R}\n\\end{equation}\nfor some $c_1 \\ge 1$.\nAssume that for some $S \\in \\pr{2^8, \\frac R 2}$, there exists $z_0 \\in \\overline{B}_{R-S}$ and $L \\ge 0$ such that\n\\begin{equation}\n\\label{propLB}\n\\abs{v(z_0)} \\ge e^{- L}.\n\\end{equation}\nThen there exists $r_0(a, R) > 1$ so that whenever $r \\in (0, r_0)$, it holds that\n\\begin{align}\n\\label{propConc}\n\\norm{v}_{L^\\iny(B_r)}\n&\\ge \\pr{\\frac{r} {R}}^{\\tau},\n\\end{align}\nwith\n$$\\tau(R, S, L, a, c_1) = \\max \\set{3 a C_H\\sqrt{\\frac{C_K }{32\\ln 2}} R \\sqrt{\\log R}, 2^{14} \\pr{c_1 R + L + \\frac {c_d}{\\log R}} RS^{-1}} +\\frac 1 5 \\pr{L + \\frac{c_d}{\\log R}},$$\nwhere $C_K$ and $c_d$ are universal constants, and $C_H$ is the Harnack constant from Lemma \\ref{discBounds}.\n\\end{prop}", "est": "\\begin{equation}\n \\inf_{|x_0| = R}\\norm{u}_{L^\\iny\\pr{B_1(x_0)}} \\ge \\exp{(-CR^{\\be}\\log R)},\n\\label{est}\n\\end{equation}", "ePDE": "\\begin{equation}\n\\label{ePDE}\n-\\LP u + V u = 0 \\; \\text{ in } \\, \\R^n,\n\\end{equation}", "MainProof": "\\begin{align}\n&B_1(w_0)  \\supset \\set{\\rho \\in \\brac{\\rho_0 - \\frac 1 2, \\rho_0 + \\frac 1 2}, \\abs{\\vp} \\le \\frac 1 {3\\rho_0} },\n\\label{ballOf}\n\\end{align}\nthen it follows that $T_\\al(B_{\\tilde r}(z_0))  \\su B_1(w_0)$.\n\nOn the other hand, with $T_\\al^{-1}(B_1(w_0)) = \\set{z : T_\\al(z) \\in B_1(w_0)}$, \\eqref{ballIn} (with $\\tilde r$ replaced by $1$) implies that \n\\begin{align*}\nT_\\al^{-1}(B_1(w_0))\n&\\su \\set{r^\\al \\in [\\rho_0 - 1, \\rho_0 + 1], \\abs{\\al \\te} \\le \\frac {2} {\\sqrt 3 \\rho_0}} \\\\\n&= \\set{r \\in \\brac{\\pr{r_0^\\al - 1}^{\\frac 1 \\al}, \\pr{r_0^\\al + 1}^{\\frac 1 \\al}}, \\abs{\\te} \\le \\frac {2} {\\sqrt 3 \\al r_0^\\al}}.\n\\end{align*}\nSince\n\\begin{align*}\n\\pr{r_0^\\al \\pm 1}^{\\frac 1 \\al}\n&= r_0 \\pr{1 \\pm \\frac 1 {r_0^\\al}}^{\\frac 1 \\al}\n= r_0 \\pm \\frac 1 {\\al r_0^{\\al-1}} - \\frac{1 - \\frac 1 \\al}{2 \\al r_0^{2\\al-1}}  + \\ldots \n\\end{align*}\nthen there exist $r_2(\\al) \\gg 1$ so that whenever $r_0 \\ge r_2$, we have $\\pr{r_0^\\al + 1}^{\\frac 1 \\al} \\le r_0 + \\frac 1 2$ and $\\pr{r_0^\\al - 1}^{\\frac 1 \\al} \\ge r_0 - \\frac 1 2$.\nIf $r_0 \\ge r_3 := \\pr{\\frac {2 \\sqrt 3} { \\al }}^{\\frac 1 {\\al - 1}}$, then\n\\begin{align*}\nT_\\al^{-1}(B_1(w_0))\n&\\su \\set{r \\in \\brac{r_0 - \\frac 1 2, r_0 + \\frac 1 2}, \\abs{\\te} \\le \\frac {1} {3 r_0}}\n\\su B_1(z_0),\n\\end{align*}\nwhere we have used \\eqref{ballOf}.\nIt follows that $B_1(w_0) \\su T^\\al(B_1(z_0))$.\nTo complete the proof, choose $r_\\al = \\max\\set{r_1, r_2, r_3}$.\n\\end{proof}\n\n\\section{Proof of Theorem \\ref{mainThm}}\n\\label{MainProof}\n\nIn this section, we prove the main result, Theorem \\ref{mainThm}.\nBefore proceeding to the rigorous details, we describe the main steps:\n\\begin{itemize}\n\\item Initialize the iterative argument by applying Theorem \\ref{LandisGrowth} to the solution function $u$.\n\\item Compose the solution with the transformation map given in \\eqref{TalDefn} to get a solution $v$ to a different Schr\\\"odinger equation, see Lemma \\ref{transformEqLemma}.\n\\item Apply the iterative result, Proposition \\ref{InductiveProp}, to $v$ many times to reduce the exponent.\n\\item Undo the change of variables to get the desired estimate for $u$.\n\\end{itemize}\nWe recall the following result originally proved in \\cite{LMNN20}, see also \\cite[Theorem 1.1]{Dav24} in the case where $N = 0$ for this formulation.\nThis estimate serves as the initialization step in our iteration argument.\n\n\\begin{thm}[Initialization step]\n\\label{LandisGrowth}\nFor some $a_0 > 0$, let $V : \\R^2 \\to \\R$ satisfy $\\norm{V}_{L^\\iny} \\le a_0^2$.\nAssume that $u : \\R^2 \\to \\R$ is a solution to \\eqref{ellipEq} that satisfies \\eqref{solNorm} and for each $w \\in \\R^2$,\n$$\\abs{u(w)} \\le \\exp\\pr{c_0 \\abs{w}}.$$\nThen there exists constants $\\overline{C}_0 = \\overline{C}_0\\pr{a_0, c_0} > 0$ and $\\overline{R}_0 > 0$ so that whenever $R \\ge \\overline{R}_0$, it holds that\n\\begin{equation}\n\\label{uLower0}\n\\inf_{\\abs{w_0} = R}\\norm{u}_{L^\\iny\\pr{B_1(w_0)}} \\ge \\exp\\pr{- \\overline{C}_0 R \\log^{\\frac 3 2} R}.\n\\end{equation}\n\\end{thm}\n\nObserve that if $u$ satisfies the hypothesis of Theorem \\ref{mainThm}, then it also satisfies the hypotheses of Theorem \\ref{LandisGrowth}.\nWe now have all of the tools we need to prove Theorem \\ref{mainThm}.\n\n\\begin{proof}[Proof of Theorem \\ref{mainThm}]\n\nWith $N \\in (0, 2)$ as given, let $\\al =\\pr{1 - \\frac N 2}^{-1} = \\frac 2 {2-N} \\in (1, \\iny)$.\nSet $\\de = \\min\\set{\\frac{\\eps}{2 - N}, 1}$.\nWith $r_\\al$ from Lemma \\ref{ballContainLemma}, $\\overline{C}_0(a_0, c_0)$ and $\\overline{R}_0$ from Theorem \\ref{LandisGrowth}, and $\\hat R_0(a)$ from Proposition \\ref{InductiveProp}, define $\\bar{r}_1 \\ge \\max\\set{r_\\al, \\overline{R}_0^{\\frac 1 \\al}, \\hat R_0\\pr{a_0 \\al}}$ as small as possible so that each of the following conditions hold:\n\\begin{align}\n& r^{\\de} \\ge \\overline{C}_0 \\al^{\\frac 3 2} \\log^{\\frac 3 2} r\n\\label{rBig1} \\\\\n&r \\log r \\ge c_d \n\\label{rBig2} \\\\\n& r^{\\frac \\de {1 + \\de}} \\ge \\frac{3 a_0 \\al C_H}{2^{14} \\pr{5c_0 + 4} } \\sqrt{\\frac{C_K }{32\\ln 2}} \\sqrt{\\log r}\n\\label{rBig3} \\\\\n& r^{\\frac {\\de^2} 2 } \\ge \\brac{2^{14} \\pr{5c_0 + 4} + \\frac 2 5 }\\log r\n\\label{rBig4} \\\\\n& r^{\\frac {5\\de^2} 6 } \\ge \\brac{2^{14} \\pr{5c_0 + 4} + \\frac 2 5}\\log \\brac{\\frac{4\\al}{ \\sqrt 3}  \\pr{\\frac 3 2 r}^{\\al} }\n\\label{rBig5}.\n\\end{align}", "LandisGrowth": "\\begin{thm}[Initialization step]\n\\label{LandisGrowth}\nFor some $a_0 > 0$, let $V : \\R^2 \\to \\R$ satisfy $\\norm{V}_{L^\\iny} \\le a_0^2$.\nAssume that $u : \\R^2 \\to \\R$ is a solution to \\eqref{ellipEq} that satisfies \\eqref{solNorm} and for each $w \\in \\R^2$,\n$$\\abs{u(w)} \\le \\exp\\pr{c_0 \\abs{w}}.$$\nThen there exists constants $\\overline{C}_0 = \\overline{C}_0\\pr{a_0, c_0} > 0$ and $\\overline{R}_0 > 0$ so that whenever $R \\ge \\overline{R}_0$, it holds that\n\\begin{equation}\n\\label{uLower0}\n\\inf_{\\abs{w_0} = R}\\norm{u}_{L^\\iny\\pr{B_1(w_0)}} \\ge \\exp\\pr{- \\overline{C}_0 R \\log^{\\frac 3 2} R}.\n\\end{equation}\n\\end{thm}"}, "pre_theorem_intro_text_len": 5108, "pre_theorem_intro_text": "This article is concerned with the quantitative unique continuation of solutions to elliptic Schr\\\"odinger equations in the plane.\nWe consider equations with potentials that exhibits pointwise decay at infinity.\nThe main result is a nearly sharp estimate for the optimal rate of decay at infinity for real-valued solutions.\nThis theorem may be interpreted as a quantitative Landis-type theorem.\n\nIn the late 1960s, E.~M.~Landis \\cite{KL88} conjectured that if $u$ and $V$ are bounded functions that satisfy\n\\begin{equation}\n\\label{ePDE}\n-\\Delta u + V u = 0 \\; \\text{ in } \\, \\R^n,\n\\end{equation}\nand $u$ decays faster than exponentially, i.e., $\\left\\vertu(x)\\right\\vert \\lesssim \\exp\\pr{- c \\left\\vertx\\right\\vert^{1+}}$, then it follows that $u \\equiv 0$.\nThis conjecture was later disproved by Meshkov \\cite{M92} who constructed non-trivial complex-valued functions $u$ and $V$ that solve \\eqref{ePDE} in $\\R^2$, where $V$ is bounded and $\\left\\vertu(x)\\right\\vert \\lesssim \\exp\\pr{- c \\left\\vertx\\right\\vert^{4/3}}$. \nUsing Carleman estimate techniques, Meshkov also proved a \\textit{qualitative unique continuation} result: \nIf \\eqref{ePDE} holds, where $V$ is bounded and $u$ satisfies a decay estimate of the form $\\abs{u\\left( x \\right) } \\lesssim \\exp\\pr{- c \\left\\vertx\\right\\vert^{4/3+}}$, then necessarily $u \\equiv 0$.\n\nIn their work on Anderson localization \\cite{BK05}, Bourgain and Kenig established a quantitative version of Meshkov's result. \nAs a first step in their proof, they used three-ball inequalities derived from Carleman estimates to establish \\textit{order of vanishing} estimates for local solutions to Schr\\\"odinger equations of the form \\eqref{ePDE}.\nThen, through a scaling argument, they proved a \\textit{quantitative unique continuation} result.\nMore specifically, they showed that if $u$ and $V$ are bounded and satisfy \\eqref{ePDE}, and $u$ is normalized so that $\\left\\vertu(0)\\right\\vert \\ge 1$, then for sufficiently large values of $R$,\n\\begin{equation}\n \\inf_{|x_0| = R}\\left\\| u\\right\\|_{L^\\infty\\left( B_1(x_0) \\right) } \\ge \\exp{(-CR^{\\beta}\\log R)},\n\\label{est}\n\\end{equation} \nwhere $\\beta = \\frac 4 3$.\nSince $ \\frac 4 3 > 1$, the constructions of Meshkov, in combination with the qualitative and quantitative unique continuation theorems just described, indicate that Landis' conjecture cannot be true for complex-valued solutions in $\\R^2$.\nHowever, at the time, Landis' conjecture still remained open in the real-valued and higher-dimensional settings.\n\nThe first significant step towards resolving Landis' conjecture in the real-valued planar setting was made by Kenig, Silvestre and Wang in \\cite{KSW15} where they proved a quantitative form of Landis' conjecture under the assumption that the zeroth-order term satisfies $V \\ge 0$ a.e.\nThe techniques in \\cite{KSW15} exploit the relationship between real-valued solutions to second-order elliptic PDEs in the plane and solutions to complex-valued Beltrami equations.\nUsing similar ideas, analogous results were established in the settings with drift terms \\cite{KW15}, variable coefficients \\cite{DKW17}, singular lower-order terms \\cite{DW20}, and when $V_-$ exhibits decay at infinity \\cite{DKW19, Dav20a}.\nThen, in \\cite{LMNN20}, Logunov, Malinnikova, Nadirashvili, and Nazarov proved Landis' conjecture in the real-valued planar setting.\nTheir proof uses the nodal structure of the domain along with a domain reduction technique to eliminate any sign condition on the zeroth-order term.\nMany of the ideas from \\cite{LMNN20} are used in this article.\n\nIn \\cite{Dav14}, we studied the quantitative unique continuation properties of solutions to elliptic equations of the form \n$$\\Delta u + W \\cdot \\nabla u + V u = \\lambda u \\; \\text{ in } \\; \\R^n,$$\nwhere $V$ and $W$ exhibit pointwise decay at infinity, and $\\lambda \\in \\C$.\nWith $\\left< x \\right> = \\sqrt{1 + \\left\\vertx\\right\\vert^2}$, it was shown that if $\\abs{V\\left( x \\right) } \\lesssim \\left< x \\right>^{-N}$ and $\\abs{W\\left( x \\right) } \\lesssim \\left< x \\right>^{-P}$ for $N, P \\ge 0$, then the quantitative estimate \\eqref{est} holds with $\\beta = \\max \\set{1, \\frac{4-2N}{3}, 2 - 2P}$ and $\\log R$ replaced by a different slowly-decaying function.\nThese quantitative estimates were generalized in \\cite{LW14}, where Lin and Wang proved analogous estimates for solutions to the corresponding equations with variable-coefficient leading terms.\nQualitative estimates for similar equations are given in \\cite{CS97}.\nThe constructions presented in \\cite{Dav14, Dav15} show that the estimates described in this paragraph are sharp.\n\nIn \\cite{Dav24}, we studied quantitative unique continuation at infinity for real-valued solutions to \\eqref{ePDE} when $n = 2$ and the potential exhibits growth at infinity, i.e., $\\left\\vertV(z)\\right\\vert \\lesssim \\left\\vertz\\right\\vert^N$ for $N > 0$.\nThe techniques in that article rely heavily on the ideas in \\cite{LMNN20} and careful scaling arguments.\nHere, we address the more difficult setting where the potential exhibits decay at infinity.\nThe precise statement of the main theorem is as follows.", "context": "This article is concerned with the quantitative unique continuation of solutions to elliptic Schr\\\"odinger equations in the plane.\nWe consider equations with potentials that exhibits pointwise decay at infinity.\nThe main result is a nearly sharp estimate for the optimal rate of decay at infinity for real-valued solutions.\nThis theorem may be interpreted as a quantitative Landis-type theorem.\n\nIn the late 1960s, E.~M.~Landis \\cite{KL88} conjectured that if $u$ and $V$ are bounded functions that satisfy\n\\begin{equation}\n\\label{ePDE}\n-\\Delta u + V u = 0 \\; \\text{ in } \\, \\R^n,\n\\end{equation}\nand $u$ decays faster than exponentially, i.e., $\\left\\vertu(x)\\right\\vert \\lesssim \\exp\\pr{- c \\left\\vertx\\right\\vert^{1+}}$, then it follows that $u \\equiv 0$.\nThis conjecture was later disproved by Meshkov \\cite{M92} who constructed non-trivial complex-valued functions $u$ and $V$ that solve \\eqref{ePDE} in $\\R^2$, where $V$ is bounded and $\\left\\vertu(x)\\right\\vert \\lesssim \\exp\\pr{- c \\left\\vertx\\right\\vert^{4/3}}$. \nUsing Carleman estimate techniques, Meshkov also proved a \\textit{qualitative unique continuation} result: \nIf \\eqref{ePDE} holds, where $V$ is bounded and $u$ satisfies a decay estimate of the form $\\abs{u\\left( x \\right) } \\lesssim \\exp\\pr{- c \\left\\vertx\\right\\vert^{4/3+}}$, then necessarily $u \\equiv 0$.\n\nIn their work on Anderson localization \\cite{BK05}, Bourgain and Kenig established a quantitative version of Meshkov's result. \nAs a first step in their proof, they used three-ball inequalities derived from Carleman estimates to establish \\textit{order of vanishing} estimates for local solutions to Schr\\\"odinger equations of the form \\eqref{ePDE}.\nThen, through a scaling argument, they proved a \\textit{quantitative unique continuation} result.\nMore specifically, they showed that if $u$ and $V$ are bounded and satisfy \\eqref{ePDE}, and $u$ is normalized so that $\\left\\vertu(0)\\right\\vert \\ge 1$, then for sufficiently large values of $R$,\n\\begin{equation}\n \\inf_{|x_0| = R}\\left\\| u\\right\\|_{L^\\infty\\left( B_1(x_0) \\right) } \\ge \\exp{(-CR^{\\beta}\\log R)},\n\\label{est}\n\\end{equation} \nwhere $\\beta = \\frac 4 3$.\nSince $ \\frac 4 3 > 1$, the constructions of Meshkov, in combination with the qualitative and quantitative unique continuation theorems just described, indicate that Landis' conjecture cannot be true for complex-valued solutions in $\\R^2$.\nHowever, at the time, Landis' conjecture still remained open in the real-valued and higher-dimensional settings.\n\nThe first significant step towards resolving Landis' conjecture in the real-valued planar setting was made by Kenig, Silvestre and Wang in \\cite{KSW15} where they proved a quantitative form of Landis' conjecture under the assumption that the zeroth-order term satisfies $V \\ge 0$ a.e.\nThe techniques in \\cite{KSW15} exploit the relationship between real-valued solutions to second-order elliptic PDEs in the plane and solutions to complex-valued Beltrami equations.\nUsing similar ideas, analogous results were established in the settings with drift terms \\cite{KW15}, variable coefficients \\cite{DKW17}, singular lower-order terms \\cite{DW20}, and when $V_-$ exhibits decay at infinity \\cite{DKW19, Dav20a}.\nThen, in \\cite{LMNN20}, Logunov, Malinnikova, Nadirashvili, and Nazarov proved Landis' conjecture in the real-valued planar setting.\nTheir proof uses the nodal structure of the domain along with a domain reduction technique to eliminate any sign condition on the zeroth-order term.\nMany of the ideas from \\cite{LMNN20} are used in this article.\n\nIn \\cite{Dav14}, we studied the quantitative unique continuation properties of solutions to elliptic equations of the form \n$$\\Delta u + W \\cdot \\nabla u + V u = \\lambda u \\; \\text{ in } \\; \\R^n,$$\nwhere $V$ and $W$ exhibit pointwise decay at infinity, and $\\lambda \\in \\C$.\nWith $\\left< x \\right> = \\sqrt{1 + \\left\\vertx\\right\\vert^2}$, it was shown that if $\\abs{V\\left( x \\right) } \\lesssim \\left< x \\right>^{-N}$ and $\\abs{W\\left( x \\right) } \\lesssim \\left< x \\right>^{-P}$ for $N, P \\ge 0$, then the quantitative estimate \\eqref{est} holds with $\\beta = \\max \\set{1, \\frac{4-2N}{3}, 2 - 2P}$ and $\\log R$ replaced by a different slowly-decaying function.\nThese quantitative estimates were generalized in \\cite{LW14}, where Lin and Wang proved analogous estimates for solutions to the corresponding equations with variable-coefficient leading terms.\nQualitative estimates for similar equations are given in \\cite{CS97}.\nThe constructions presented in \\cite{Dav14, Dav15} show that the estimates described in this paragraph are sharp.\n\nIn \\cite{Dav24}, we studied quantitative unique continuation at infinity for real-valued solutions to \\eqref{ePDE} when $n = 2$ and the potential exhibits growth at infinity, i.e., $\\left\\vertV(z)\\right\\vert \\lesssim \\left\\vertz\\right\\vert^N$ for $N > 0$.\nThe techniques in that article rely heavily on the ideas in \\cite{LMNN20} and careful scaling arguments.\nHere, we address the more difficult setting where the potential exhibits decay at infinity.\nThe precise statement of the main theorem is as follows.\n\n\\begin{equation}\n\\label{ePDE}\n-\\LP u + V u = 0 \\; \\text{ in } \\, \\R^n,\n\\end{equation}\n\n\\begin{equation}\n \\inf_{|x_0| = R}\\norm{u}_{L^\\iny\\pr{B_1(x_0)}} \\ge \\exp{(-CR^{\\be}\\log R)},\n\\label{est}\n\\end{equation}", "full_context": "This article is concerned with the quantitative unique continuation of solutions to elliptic Schr\\\"odinger equations in the plane.\nWe consider equations with potentials that exhibits pointwise decay at infinity.\nThe main result is a nearly sharp estimate for the optimal rate of decay at infinity for real-valued solutions.\nThis theorem may be interpreted as a quantitative Landis-type theorem.\n\nIn the late 1960s, E.~M.~Landis \\cite{KL88} conjectured that if $u$ and $V$ are bounded functions that satisfy\n\\begin{equation}\n\\label{ePDE}\n-\\Delta u + V u = 0 \\; \\text{ in } \\, \\R^n,\n\\end{equation}\nand $u$ decays faster than exponentially, i.e., $\\left\\vertu(x)\\right\\vert \\lesssim \\exp\\pr{- c \\left\\vertx\\right\\vert^{1+}}$, then it follows that $u \\equiv 0$.\nThis conjecture was later disproved by Meshkov \\cite{M92} who constructed non-trivial complex-valued functions $u$ and $V$ that solve \\eqref{ePDE} in $\\R^2$, where $V$ is bounded and $\\left\\vertu(x)\\right\\vert \\lesssim \\exp\\pr{- c \\left\\vertx\\right\\vert^{4/3}}$. \nUsing Carleman estimate techniques, Meshkov also proved a \\textit{qualitative unique continuation} result: \nIf \\eqref{ePDE} holds, where $V$ is bounded and $u$ satisfies a decay estimate of the form $\\abs{u\\left( x \\right) } \\lesssim \\exp\\pr{- c \\left\\vertx\\right\\vert^{4/3+}}$, then necessarily $u \\equiv 0$.\n\nIn their work on Anderson localization \\cite{BK05}, Bourgain and Kenig established a quantitative version of Meshkov's result. \nAs a first step in their proof, they used three-ball inequalities derived from Carleman estimates to establish \\textit{order of vanishing} estimates for local solutions to Schr\\\"odinger equations of the form \\eqref{ePDE}.\nThen, through a scaling argument, they proved a \\textit{quantitative unique continuation} result.\nMore specifically, they showed that if $u$ and $V$ are bounded and satisfy \\eqref{ePDE}, and $u$ is normalized so that $\\left\\vertu(0)\\right\\vert \\ge 1$, then for sufficiently large values of $R$,\n\\begin{equation}\n \\inf_{|x_0| = R}\\left\\| u\\right\\|_{L^\\infty\\left( B_1(x_0) \\right) } \\ge \\exp{(-CR^{\\beta}\\log R)},\n\\label{est}\n\\end{equation} \nwhere $\\beta = \\frac 4 3$.\nSince $ \\frac 4 3 > 1$, the constructions of Meshkov, in combination with the qualitative and quantitative unique continuation theorems just described, indicate that Landis' conjecture cannot be true for complex-valued solutions in $\\R^2$.\nHowever, at the time, Landis' conjecture still remained open in the real-valued and higher-dimensional settings.\n\nThe first significant step towards resolving Landis' conjecture in the real-valued planar setting was made by Kenig, Silvestre and Wang in \\cite{KSW15} where they proved a quantitative form of Landis' conjecture under the assumption that the zeroth-order term satisfies $V \\ge 0$ a.e.\nThe techniques in \\cite{KSW15} exploit the relationship between real-valued solutions to second-order elliptic PDEs in the plane and solutions to complex-valued Beltrami equations.\nUsing similar ideas, analogous results were established in the settings with drift terms \\cite{KW15}, variable coefficients \\cite{DKW17}, singular lower-order terms \\cite{DW20}, and when $V_-$ exhibits decay at infinity \\cite{DKW19, Dav20a}.\nThen, in \\cite{LMNN20}, Logunov, Malinnikova, Nadirashvili, and Nazarov proved Landis' conjecture in the real-valued planar setting.\nTheir proof uses the nodal structure of the domain along with a domain reduction technique to eliminate any sign condition on the zeroth-order term.\nMany of the ideas from \\cite{LMNN20} are used in this article.\n\nIn \\cite{Dav14}, we studied the quantitative unique continuation properties of solutions to elliptic equations of the form \n$$\\Delta u + W \\cdot \\nabla u + V u = \\lambda u \\; \\text{ in } \\; \\R^n,$$\nwhere $V$ and $W$ exhibit pointwise decay at infinity, and $\\lambda \\in \\C$.\nWith $\\left< x \\right> = \\sqrt{1 + \\left\\vertx\\right\\vert^2}$, it was shown that if $\\abs{V\\left( x \\right) } \\lesssim \\left< x \\right>^{-N}$ and $\\abs{W\\left( x \\right) } \\lesssim \\left< x \\right>^{-P}$ for $N, P \\ge 0$, then the quantitative estimate \\eqref{est} holds with $\\beta = \\max \\set{1, \\frac{4-2N}{3}, 2 - 2P}$ and $\\log R$ replaced by a different slowly-decaying function.\nThese quantitative estimates were generalized in \\cite{LW14}, where Lin and Wang proved analogous estimates for solutions to the corresponding equations with variable-coefficient leading terms.\nQualitative estimates for similar equations are given in \\cite{CS97}.\nThe constructions presented in \\cite{Dav14, Dav15} show that the estimates described in this paragraph are sharp.\n\nIn \\cite{Dav24}, we studied quantitative unique continuation at infinity for real-valued solutions to \\eqref{ePDE} when $n = 2$ and the potential exhibits growth at infinity, i.e., $\\left\\vertV(z)\\right\\vert \\lesssim \\left\\vertz\\right\\vert^N$ for $N > 0$.\nThe techniques in that article rely heavily on the ideas in \\cite{LMNN20} and careful scaling arguments.\nHere, we address the more difficult setting where the potential exhibits decay at infinity.\nThe precise statement of the main theorem is as follows.\n\n\\begin{equation}\n\\label{ePDE}\n-\\LP u + V u = 0 \\; \\text{ in } \\, \\R^n,\n\\end{equation}\n\n\\begin{equation}\n \\inf_{|x_0| = R}\\norm{u}_{L^\\iny\\pr{B_1(x_0)}} \\ge \\exp{(-CR^{\\be}\\log R)},\n\\label{est}\n\\end{equation}\n\nIn \\cite{Dav24}, we studied quantitative unique continuation at infinity for real-valued solutions to \\eqref{ePDE} when $n = 2$ and the potential exhibits growth at infinity, i.e., $\\abs{V(z)} \\lesssim \\abs{z}^N$ for $N > 0$.\nThe techniques in that article rely heavily on the ideas in \\cite{LMNN20} and careful scaling arguments.\nHere, we address the more difficult setting where the potential exhibits decay at infinity.\nThe precise statement of the main theorem is as follows.\n\nThe results of \\cite{Dav14} prove an estimate of the form \\eqref{est} with $\\be = \\frac{4 - 2N}{3} = \\frac 4 3 \\pr{1 - \\frac N 2} > 1 - \\frac N 2 + \\eps$.\nIn that article, the assumptions are the same as those in Theorem \\ref{mainThm}, except that $u$ may be complex-valued.\nThus, as in the case of bounded $V$, Theorem \\ref{mainThm} illustrates that better estimates hold in the real-valued planar setting.\n\n\\begin{prop}[Iterative Proposition]\n\\label{InductiveProp}\nGiven $a > 0$ and $R \\ge \\hat R_0(a)$, let $W : B_R \\to \\R$ satisfy $\\norm{W}_{L^\\iny(B_R)} \\le a^2$ and let $v : B_R \\to \\R$ be a solution to \n$$-\\LP v + W v = 0 \\; \\text{ in } B_R$$\nwith\n\\begin{equation}\n\\label{propUB}\n\\norm{v}_{L^\\iny(B_R)} \\le e^{c_1 R}\n\\end{equation}\nfor some $c_1 \\ge 1$.\nAssume that for some $S \\in \\pr{2^8, \\frac R 2}$, there exists $z_0 \\in \\overline{B}_{R-S}$ and $L \\ge 0$ such that\n\\begin{equation}\n\\label{propLB}\n\\abs{v(z_0)} \\ge e^{- L}.\n\\end{equation}\nThen there exists $r_0(a, R) > 1$ so that whenever $r \\in (0, r_0)$, it holds that\n\\begin{align}\n\\label{propConc}\n\\norm{v}_{L^\\iny(B_r)}\n&\\ge \\pr{\\frac{r} {R}}^{\\tau},\n\\end{align}\nwith\n$$\\tau(R, S, L, a, c_1) = \\max \\set{3 a C_H\\sqrt{\\frac{C_K }{32\\ln 2}} R \\sqrt{\\log R}, 2^{14} \\pr{c_1 R + L + \\frac {c_d}{\\log R}} RS^{-1}} +\\frac 1 5 \\pr{L + \\frac{c_d}{\\log R}},$$\nwhere $C_K$ and $c_d$ are universal constants, and $C_H$ is the Harnack constant from Lemma \\ref{discBounds}.\n\\end{prop}\n\nDefine $\\Om = B_R \\setminus \\pr{F_0 \\bigcup F_1}$ and $\\Om_1 = B_R \\setminus F_1$.\nAs shown in \\cite[\\S 3.1]{LMNN20}, there exists a universal constant $c_P$ (that depends on $c_s$) so that $\\Om$ has Poincar\\'e constant bounded above by $c_P \\rho^2$.\nIn particular, since $c_P \\rho^2 \\norm{W}_{L^\\iny(B_R)} \\le c_P \\rho^2 a^2$, then by choosing $\\rho \\le \\rho_0$, a universal constant, we can apply \\cite[Lemma 3.2]{LMNN20}.\nFor $\\eps \\ll 1$ to be defined later on, let \n\\begin{equation}\n\\label{rhoDef}\n\\rho = \\eps a^{-1}.\n\\end{equation}\nAn application of the arguments in \\cite[\\S 3.2]{LMNN20} then shows that there exists $\\phi : \\Om \\to \\R$ with the properties that\n\\begin{align}\n&\\LP \\phi - W \\phi = 0 \\text{ in } \\Om\n\\nonumber \\\\\n&\\phi -1 \\in W^{1,2}_0(\\Om)\n\\nonumber \\\\\n&\\norm{\\phi -1}_\\iny \\le c_b \\pr{\\rho a}^2 = c_b \\eps^2,\n\\label{vpuBound}\n\\end{align}\nwhere $c_b$ is a universal constant that depends on $c_P$, and we have used \\eqref{rhoDef}.\nBy extending $\\phi$ to equal $1$ across $F_0 \\bigcup F_1$, it is then shown in \\cite[Lemma 4.1]{LMNN20} that $\\disp f := \\frac v \\phi \\in W^{1,2}_{\\loc}(B_R)$ is a weak solution to the divergence-form equation\n$$\\di\\pr{\\phi^2 \\gr f} = 0 \\; \\text{ in } \\Om_1.$$\nMoreover, the bound in \\eqref{vpuBound} implies that for any $z \\in B_R$,\n\\begin{equation}\n\\label{ufComparison}\n\\pr{1 - c_b \\eps^2} \\abs{f(z)} \\le \\abs{v(z)} \\le \\pr{1 + c_b \\eps^2} \\abs{f(z)}.\n\\end{equation}\n\nWe introduce the Beltrami coefficient $\\mu$, defined as follows:\n\\begin{equation*}\n\\mu = \\left\\{ \\begin{array}{ll}\n\\frac{1 - \\phi^2}{1 + \\phi^2} \\frac{f_x + i f_y}{f_x - i f_y} & \\text{ in } \\Om_1 \\text{ when } \\, \\gr f \\ne 0 \\\\\n0 & \\text{ otherwise}\n\\end{array} \\right..\n\\end{equation*}\nSince $\\abs{\\mu} \\lesssim \\eps^2$, then as shown in \\cite{AIM09}, there exists a $K$-quasiconformal homeomorphism of the complex plane where $K \\le 1 + C_K \\eps^2$ and $C_K$ depends on $c_b$.\nThat is, there exists some $w \\in W^{1,2}_{\\loc}$ that satisfies the Beltrami equation $\\disp \\frac{\\del w}{\\del \\overline{z}} = \\mu \\frac{\\del w}{\\del z}$. \nIn fact, an application of the Riemann uniformization theorem shows that there exists a $K$-quasiconformal homeomorphism $g: B_R \\to B_R$ that is onto with $g(0) = 0$.\nMoreover, the function $h : = f \\circ g^{-1}$ is harmonic in $g(\\Om_1)$.\n\n\\begin{thm}[Initialization step]\n\\label{LandisGrowth}\nFor some $a_0 > 0$, let $V : \\R^2 \\to \\R$ satisfy $\\norm{V}_{L^\\iny} \\le a_0^2$.\nAssume that $u : \\R^2 \\to \\R$ is a solution to \\eqref{ellipEq} that satisfies \\eqref{solNorm} and for each $w \\in \\R^2$,\n$$\\abs{u(w)} \\le \\exp\\pr{c_0 \\abs{w}}.$$\nThen there exists constants $\\overline{C}_0 = \\overline{C}_0\\pr{a_0, c_0} > 0$ and $\\overline{R}_0 > 0$ so that whenever $R \\ge \\overline{R}_0$, it holds that\n\\begin{equation}\n\\label{uLower0}\n\\inf_{\\abs{w_0} = R}\\norm{u}_{L^\\iny\\pr{B_1(w_0)}} \\ge \\exp\\pr{- \\overline{C}_0 R \\log^{\\frac 3 2} R}.\n\\end{equation}\n\\end{thm}\n\nNow assume that \\eqref{claim} holds for $k-1 \\in \\N$.\nDefine $S_k = \\frac 1 2 \\abs{z_{k-1}}$ and $R_k = \\abs{z_{k-1}}^{\\al_{k-1} +\\de}$. \nComparing with \\eqref{zkNorm}, we see that $\\abs{z_k} = R_k + S_k - 1$.\nSince $B_1(z_{k-1}) \\su B_{R_k - S_k}(z_k)$, then the inductive hypothesis shows that there exists $z_0 \\in \\overline{B}_{R_k - S_k}(z_k)$ so that\n\\begin{align*}\n\\abs{v(z_0)}\n\\ge \\exp\\pr{- \\abs{z_{k-1}}^{\\al_{k-1} + \\de}}\n= \\exp\\pr{- R_k}.\n\\end{align*}\nThe bound in \\eqref{uBound} shows that $\\disp \\abs{v(z)} = \\abs{u(T_\\al(z))} \\le \\exp\\pr{c_0 \\abs{z}^{\\al\\pr{1 - \\frac N 2}}} = \\exp\\pr{c_0 \\abs{z}}$ from which it follows that\n\\begin{align*}\n\\norm{v}_{L^\\iny\\pr{B_{R_k}(z_k)}}\n\\le \\exp\\pr{c_0 \\pr{\\abs{z_k} + R_k}}\n\\le \\exp\\pr{\\frac {5c_0} 2 R_k}.\n\\end{align*}\nLet $W : \\R^2_+ \\to \\R$ be given by $W(z) = \\al^2 \\abs{z}^{2\\al - 2} V(T_\\al(z))= \\al^2 \\abs{w}^{2 - \\frac 2 \\al} V(w)$ so that by \\eqref{Vbound},\n\\begin{align*}\n\\abs{W(z)} = \\al^2 \\abs{w}^{2 - \\frac 2 \\al} \\abs{V(w)} \\le a_0^2 \\al^2 \\abs{w}^{2 - N - \\frac 2 \\al} = \\pr{a_0 \\al}^2.\n\\end{align*}\nAn application of Lemma \\ref{transformEqLemma} with assumption \\eqref{ellipEq} shows that\n\\begin{equation*}\n- \\LP v + W v = 0 \\; \\text{ in } \\R^2_+.\n\\end{equation*}\nAs $R_k \\ge \\bar{r}_1$ implies that $R_k \\ge \\hat R_0(a_0 \\al)$ and $S_k \\in \\pr{2^8, \\frac {R_k} 2}$, Proposition \\ref{InductiveProp} is applicable with $a = a_0 \\al$, $R= R_k$, $S= S_k$, $L = R_k$, and $c_1 = \\frac {5c_0} 2$.\nWith $r = 1$, Proposition \\ref{InductiveProp} shows that\n\\begin{align}\n\\label{vkBd}\n\\norm{v}_{L^\\iny(B_1(z_k))}\n&\\ge \\exp\\pr{-\\tau_k(R_k) \\log R_k},\n\\end{align}\nwhere\n\\begin{equation}\n\\label{taukDefn}\n\\begin{aligned}\n\\tau_k(R_k) &= \\max \\set{3 a_0 \\al C_H\\sqrt{\\frac{C_K }{32\\ln 2}} R_k \\sqrt{\\log R_k}, 2^{15} \\pr{\\frac {5c_0} 2 + 1 + \\frac {c_d}{R_k \\log R_k}} R_k^{2 - \\frac 1 {\\al_{k-1} + \\de}}} \\\\\n&+\\frac 1 5 \\pr{R_k + \\frac{c_d}{\\log R_k}}.\n\\end{aligned}\n\\end{equation}\nBecause $R_k \\ge \\bar{r}_1$ and $\\al_{k - 1} > 1$, conditions \\eqref{rBig2} and \\eqref{rBig3} imply that\n\\begin{align}\n\\label{tauKBound}\n\\tau_k(R_k) &\\le \\brac{2^{14} \\pr{5c_0 + 4} + \\frac 2 5} R_k^{2 - \\frac 1 {\\al_{k-1} + \\de}} \n\\end{align}\nwhile condition \\eqref{rBig4} implies that\n$$ \\brac{2^{14} \\pr{5c_0 + 4} + \\frac 2 5} \\log R_k \\le R_k^{\\frac {\\de^2} 2}.$$\nSince $\\al > 1$ and $\\de \\le 1$ implies that $2 - \\frac{1}{\\al + \\de} + \\frac{\\de^2}2 < 2 - \\frac 1 \\al + \\de$, then combining these bounds shows that\n$$\\tau_k(R_k) \\log R_k \\le R_k^{2 - \\frac 1 {\\al_{k-1}} + \\de} = R_k^{\\al_{k} + \\de}.$$\nReturning to \\eqref{vkBd}, we conclude that\n\\begin{align*}\n\\norm{v}_{L^\\iny(B_1(z_k))}\n&\\ge \\exp\\pr{-R_k^{\\al_{k} + \\de}}\n\\ge \\exp\\pr{-\\abs{z_k}^{\\al_{k} + \\de}},\n\\end{align*}\nestablishing the claim given by \\eqref{claim}.", "post_theorem_intro_text_len": 4239, "post_theorem_intro_text": "The results of \\cite{Dav14} prove an estimate of the form \\eqref{est} with $\\beta = \\frac{4 - 2N}{3} = \\frac 4 3 \\left( 1 - \\frac N 2 \\right)  > 1 - \\frac N 2 + \\varepsilon$.\nIn that article, the assumptions are the same as those in Theorem \\ref{mainThm}, except that $u$ may be complex-valued.\nThus, as in the case of bounded $V$, Theorem \\ref{mainThm} illustrates that better estimates hold in the real-valued planar setting.\n\nAs illustrated by the following example, Theorem \\ref{mainThm} is sharp (up to $\\varepsilon$) for all $N \\in (0, 2)$.\nFix $N \\in (0, 2)$, then set $u(z) = \\exp\\pr{- \\left\\vertz\\right\\vert^{1 - \\frac N 2}}$.\nA computation shows that $u$ satisfies \\eqref{ellipEq} where\n$$V(z) := \\left( 1 - \\frac N 2 \\right) ^2 \\pr{\\left\\vertz\\right\\vert^{1 - \\frac N 2} - 1}\\left\\vertz\\right\\vert^{-1 - \\frac N 2}$$ \nsatisfies $\\left\\vertV(z)\\right\\vert \\lesssim \\left\\vertz\\right\\vert^{-N}$.\nOn the other hand, for any $\\beta > 0$, with $u(z) = \\left\\vertz\\right\\vert^{-\\beta}$ on $\\left\\vertz\\right\\vert > 1$, we see that $\\displaystyle \\Delta u = \\beta^2 \\left\\vertz\\right\\vert^{-\\beta-2}$ and therefore $u$ satisfies \\eqref{ellipEq} on an exterior domain with $V(z) := \\beta^2\\left\\vertz\\right\\vert^{-2}$.\nIn particular, we may not have exponential behavior when $V$ decays fast enough, which explains why we restrict ourselves to $N < 2$.\n\nTo prove Theorem \\ref{mainThm}, we use an iterative argument that is reminiscent of the one in \\cite{Dav14}, see also \\cite{LW14, DKW19, Dav20a, Dav25}.\nTo initialize the iteration, we apply a quantitative estimate of the form \\eqref{est} with $\\beta = 1$.\nThis result, which verifies Landis' conjecture in the real-valued planar setting, was originally proved by Logunov, Malinnikova, Nadirashvili, and Nazarov in \\cite{LMNN20}, and we formulate it in Theorem \\ref{LandisGrowth} below.\nThe iteration argument then relies on repeated applications of Proposition \\ref{InductiveProp} which is proved using the ideas from \\cite{LMNN20}.\nRoughly speaking, Proposition \\ref{InductiveProp} shows that if an estimate like \\eqref{est} holds with $\\beta = \\be_0$, then for some $x_1$ with $\\left\\vertx_1\\right\\vert \\gg \\left\\vertx_0\\right\\vert$, another estimate like \\eqref{est} holds with $x_0$ replaced by $x_1$ and $\\beta = \\be_1 \\in \\left[1, \\be_0\\right]$.\nWhen $\\be_0 = 1$, Proposition \\ref{InductiveProp} isn't useful, but when $\\be_0 > 1$, we can decrease the exponent, i.e., make $\\be_1 < \\be_0$.\nTherefore, to benefit from the iteration argument, we need to transform to a situation where $\\be_0 > 1$.\nWe observe that if $u$ is composed with the real-variable version of the conformal transformation $z \\mapsto z^\\alpha$, then the new function also satisfies a Schr\\\"odinger equation.\nBy choosing $\\alpha > 1$ appropriately, we can ensure that the new potential function is bounded and that the new solution function satisfies a version of \\eqref{est} with $\\beta > 1$.\nBy repeatedly applying Proposition \\ref{InductiveProp} to the transformed equation, we can make $\\beta$ arbitrarily close to $1$.\nFinally, to reach the conclusion, we undo the change of variables.\n\nWe use the notation $B_r(z)$ to denote a ball of radius $r > 0$ centered at the point $z$, abbreviated by $B_r$ when the center is clear.\nGeneric constants are denoted by $c, C$ and may change from line to line without comment.\nSpecific constants will be indicated by subscripts.\n\nThe article is organized as follows.\nIn Section \\ref{harmonic}, we present a unique continuation theorem for harmonic functions in punctured domains.\nThe content of this section is very similar to \\cite[Section 5]{LMNN20} and \\cite[Section 2]{Dav24}.\nThe iterative result described by Proposition \\ref{InductiveProp} is the content of Section \\ref{localProof}.\nProposition \\ref{InductiveProp} is a three-ball inequality for solutions to Schr\\\"odinger equations and its proof relies on the results from Section \\ref{harmonic}. In Section \\ref{TransMaps}, we introduce the real-valued versions of $z \\mapsto z^\\alpha$ and record some of their properties.\nIn particular, we show how solutions behave when they are composed with these transformations.\nFinally, the proof of Theorem \\ref{mainThm} is presented in Section \\ref{MainProof}.", "sketch": "To prove Theorem~\\ref{mainThm}, the paper uses “an iterative argument that is reminiscent of the one in \\cite{Dav14}.”\n\nKey steps (as described):\n\\begin{itemize}\n\\item \\textbf{Initialize the iteration.} “To initialize the iteration, we apply a quantitative estimate of the form \\eqref{est} with $\\beta = 1$,” namely the real-valued planar Landis growth result proved in \\cite{LMNN20} and stated as Theorem~\\ref{LandisGrowth}.\n\\item \\textbf{Inductive improvement via a three-ball type step.} The iteration “relies on repeated applications of Proposition~\\ref{InductiveProp}.” Roughly, this proposition shows that if an estimate like \\eqref{est} holds with exponent $\\beta=\\be_0$, then for some $x_1$ with “$|x_1|\\gg |x_0|$,” another estimate like \\eqref{est} holds with $x_0$ replaced by $x_1$ and “$\\beta=\\be_1\\in[1,\\be_0]$.” Moreover, “when $\\be_0>1$, we can decrease the exponent, i.e., make $\\be_1<\\be_0$.”\n\\item \\textbf{Create a regime with $\\beta>1$ via a power map.} Since “when $\\be_0=1$, Proposition~\\ref{InductiveProp} isn’t useful,” they “transform to a situation where $\\be_0>1$” by composing $u$ with “the real-variable version of the conformal transformation $z\\mapsto z^\\alpha$,” noting the composed function still “satisfies a Schr\\\"odinger equation.” Choosing “$\\alpha>1$ appropriately,” they ensure “the new potential function is bounded” and the new solution satisfies a version of \\eqref{est} “with $\\beta>1$.”\n\\item \\textbf{Iterate to push $\\beta$ down to $1$.} “By repeatedly applying Proposition~\\ref{InductiveProp} to the transformed equation, we can make $\\beta$ arbitrarily close to $1$.”\n\\item \\textbf{Conclude by reversing the change of variables.} “Finally, to reach the conclusion, we undo the change of variables.”\n\\end{itemize}\n", "expanded_sketch": "To prove the main theorem, the paper uses “an iterative argument that is reminiscent of the one in \\cite{Dav14}.”\n\nKey steps (as described):\n\\begin{itemize}\n\\item \\textbf{Initialize the iteration.} “To initialize the iteration, we apply a quantitative estimate of the form\n\\begin{equation}\n \\inf_{|x_0| = R}\\norm{u}_{L^\\iny\\pr{B_1(x_0)}} \\ge \\exp{(-CR^{\\be}\\log R)},\n\\label{est}\n\\end{equation}\nwith $\\beta = 1$,” namely the real-valued planar Landis growth result proved in \\cite{LMNN20} and stated as\n\\begin{thm}[Initialization step]\n\\label{LandisGrowth}\nFor some $a_0 > 0$, let $V : \\R^2 \\to \\R$ satisfy $\\norm{V}_{L^\\iny} \\le a_0^2$.\nAssume that $u : \\R^2 \\to \\R$ is a solution to \\eqref{ellipEq} that satisfies \\eqref{solNorm} and for each $w \\in \\R^2$,\n$$\\abs{u(w)} \\le \\exp\\pr{c_0 \\abs{w}}.$$\nThen there exists constants $\\overline{C}_0 = \\overline{C}_0\\pr{a_0, c_0} > 0$ and $\\overline{R}_0 > 0$ so that whenever $R \\ge \\overline{R}_0$, it holds that\n\\begin{equation}\n\\label{uLower0}\n\\inf_{\\abs{w_0} = R}\\norm{u}_{L^\\iny\\pr{B_1(w_0)}} \\ge \\exp\\pr{- \\overline{C}_0 R \\log^{\\frac 3 2} R}.\n\\end{equation}\n\\end{thm}\n\n\\item \\textbf{Inductive improvement via a three-ball type step.} The iteration “relies on repeated applications of\n\\begin{prop}[Iterative Proposition]\n\\label{InductiveProp}\nGiven $a > 0$ and $R \\ge \\hat R_0(a)$, let $W : B_R \\to \\R$ satisfy $\\norm{W}_{L^\\iny(B_R)} \\le a^2$ and let $v : B_R \\to \\R$ be a solution to \n$$-\\LP v + W v = 0 \\; \\text{ in } B_R$$\nwith\n\\begin{equation}\n\\label{propUB}\n\\norm{v}_{L^\\iny(B_R)} \\le e^{c_1 R}\n\\end{equation}\nfor some $c_1 \\ge 1$.\nAssume that for some $S \\in \\pr{2^8, \\frac R 2}$, there exists $z_0 \\in \\overline{B}_{R-S}$ and $L \\ge 0$ such that\n\\begin{equation}\n\\label{propLB}\n\\abs{v(z_0)} \\ge e^{- L}.\n\\end{equation}\nThen there exists $r_0(a, R) > 1$ so that whenever $r \\in (0, r_0)$, it holds that\n\\begin{align}\n\\label{propConc}\n\\norm{v}_{L^\\iny(B_r)}\n&\\ge \\pr{\\frac{r} {R}}^{\\tau},\n\\end{align}\nwith\n$$\\tau(R, S, L, a, c_1) = \\max \\set{3 a C_H\\sqrt{\\frac{C_K }{32\\ln 2}} R \\sqrt{\\log R}, 2^{14} \\pr{c_1 R + L + \\frac {c_d}{\\log R}} RS^{-1}} +\\frac 1 5 \\pr{L + \\frac{c_d}{\\log R}},$$\nwhere $C_K$ and $c_d$ are universal constants, and $C_H$ is the Harnack constant from Lemma \\ref{discBounds}.\n\\end{prop}\nRoughly, this proposition shows that if an estimate like the equation above holds with exponent $\\beta=\\be_0$, then for some $x_1$ with “$|x_1|\\gg |x_0|$,” another estimate like the equation above holds with $x_0$ replaced by $x_1$ and “$\\beta=\\be_1\\in[1,\\be_0]$.” Moreover, “when $\\be_0>1$, we can decrease the exponent, i.e., make $\\be_1<\\be_0$.”\n\n\\item \\textbf{Create a regime with $\\beta>1$ via a power map.} Since “when $\\be_0=1$, the iterative proposition above isn’t useful,” they “transform to a situation where $\\be_0>1$” by composing $u$ with “the real-variable version of the conformal transformation $z\\mapsto z^\\alpha$,” noting the composed function still “satisfies a Schr\\\"odinger equation.” Choosing “$\\alpha>1$ appropriately,” they ensure “the new potential function is bounded” and the new solution satisfies a version of the equation above “with $\\beta>1$.”\n\n\\item \\textbf{Iterate to push $\\beta$ down to $1$.} “By repeatedly applying the iterative proposition above to the transformed equation, we can make $\\beta$ arbitrarily close to $1$.”\n\n\\item \\textbf{Conclude by reversing the change of variables.} “Finally, to reach the conclusion, we undo the change of variables.” This completes the proof of the main theorem.\n\\end{itemize}\n", "expanded_theorem": "[Main Theorem]\n\\label{mainThm}\nFor some $a_0 \\ge 1$ and $N \\in (0, 2)$, let $V : \\R^2 \\to \\R$ satisfy \n\\begin{equation}\n\\label{Vbound}\n\\left\\vertV(w)\\right\\vert \\le a_0^2 \\left< w \\right>^{-N}.\n\\end{equation}\nAssume that $u : \\R^2 \\to \\R$ is a solution to \n\\begin{equation}\n\\label{ellipEq}\n- \\Delta u + V u = 0 \\, \\text{ in } \\R^2\n\\end{equation}\nwith the properties that \n\\begin{equation}\n\\label{solNorm}\n\\left\\vertu(0)\\right\\vert = 1\n\\end{equation}\nand for each $w \\in \\R^2$,\n\\begin{equation}\n\\label{uBound}\n\\left\\vertu(w)\\right\\vert \\le \\exp\\pr{c_0 \\left\\vertw\\right\\vert^{1 - \\frac N 2}}.\n\\end{equation}\nFor every $\\varepsilon \\in \\left( 0, \\frac N 2 \\right) $, there exists ${R}_0(N, a_0, c_0, \\varepsilon) > 0$ so that whenever $R \\ge {R}_0$, it holds that\n\\begin{equation}\n\\label{uLower}\n\\inf_{\\left\\vertw_0\\right\\vert = R} \\left\\| u\\right\\|_{L^\\infty\\left( B_1(w_0) \\right) } \\ge \\exp\\pr{-R^{1 - \\frac N 2 + \\varepsilon}}.\n\\end{equation}", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Let $a_0\\ge 1$ and $N\\in(0,2)$. Suppose $V:\\mathbb{R}^2\\to\\mathbb{R}$ satisfies\n\\[\n|V(w)|\\le a_0^2\\langle w\\rangle^{-N}\\quad\\text{for all }w\\in\\mathbb{R}^2,\n\\]\nwhere $\\langle w\\rangle=\\sqrt{1+|w|^2}$. Let $u:\\mathbb{R}^2\\to\\mathbb{R}$ solve\n\\[\n-\\Delta u+Vu=0\\quad\\text{in }\\mathbb{R}^2,\n\\]\nwith\n\\[\n|u(0)|=1\n\\]\nand the growth bound\n\\[\n|u(w)|\\le \\exp\\big(c_0|w|^{1-N/2}\\big)\\quad\\text{for every }w\\in\\mathbb{R}^2.\n\\]\nHere $B_1(w_0)$ denotes the unit ball centered at $w_0$. Under these assumptions, which quantitative estimate holds?", "correct_choice": {"label": "A", "text": "For every $\\varepsilon\\in(0,N/2)$, there exists $R_0=R_0(N,a_0,c_0,\\varepsilon)>0$ such that for every $R\\ge R_0$,\n\\[\n\\inf_{|w_0|=R}\\|u\\|_{L^\\infty(B_1(w_0))}\\ge \\exp\\big(-R^{1-N/2+\\varepsilon}\\big).\n\\]"}, "choices": [{"label": "B", "text": "For every $\\varepsilon\\in(0,N/2)$, there exists $R_0=R_0(N,a_0,c_0)>0$ such that for every $R\\ge R_0$,\n\\[\n\\inf_{|w_0|=R}\\|u\\|_{L^\\infty(B_1(w_0))}\\ge \\exp\\big(-R^{1-N/2}\\big).\n\\]"}, {"label": "C", "text": "There exists $R_0=R_0(N,a_0,c_0)>0$ such that for every $R\\ge R_0$,\n\\[\n\\inf_{|w_0|=R}\\|u\\|_{L^\\infty(B_1(w_0))}\\ge \\exp\\big(-R\\big).\n\\]"}, {"label": "D", "text": "For every $\\varepsilon\\in(0,N/2)$ and every $R_0>0$, there exists $R\\ge R_0$ such that\n\\[\n\\inf_{|w_0|=R}\\|u\\|_{L^\\infty(B_1(w_0))}\\ge \\exp\\big(-R^{1-N/2+\\varepsilon}\\big).\n\\]"}, {"label": "E", "text": "For every $\\varepsilon\\in(0,N/2)$, there exists $R_0=R_0(N,a_0,c_0,\\varepsilon)>0$ such that for every $R\\ge R_0$,\n\\[\n\\inf_{|w_0|\\le R}\\|u\\|_{L^\\infty(B_1(w_0))}\\ge \\exp\\big(-R^{1-N/2+\\varepsilon}\\big).\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": "epsilon-loss in final exponent", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "replace sharp exponent $1-N/2+\\varepsilon$ by weaker linear exponent", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "eventual-for-all-$R$ conclusion weakened to infinitely-many-$R$ statement", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "sphere condition $|w_0|=R$ replaced by ball condition $|w_0|\\le R$", "template_used": "wildcard"}]}}
{"id": "2512.09873v1", "paper_link": "http://arxiv.org/abs/2512.09873v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm: main-ucp}\nLet $T>0$ and $G\\subset [0,T]\\times {\\mathbb T}$ be a spacetime measurable set. The following statements are equivalent.\n\\begin{itemize}\n\\item [(1)] UCP holds on $G$.\n\n\\item [(2)] $G$ satisfies \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} and does not obey \\blackhyperref{def:OSC}{$({\\rm OSC})$} for any non-trivial pair $(A, B)$\\footnote{Reminder: for simplicity, when there is no risk of confusion,  sometimes we simply write (OSC) indicating that the set $G$ does not obey the observable symmetry condition for any non-trivial pair $(A, B)$.}.\n\\end{itemize}", "start_pos": 32586, "end_pos": 33189, "label": "thm: main-ucp"}, "ref_dict": {"eq: wave-eq-0": "\\begin{equation}\\label{eq: wave-eq-0}\n(\\partial_t^2-\\partial_x^2)u=0,\\quad (u,\\partial_tu)\\big|_{t=0}=(u_0,u_1)\\in \\dot H^1(\\T)\\times L^2(\\T),\n\\end{equation}", "wave:nece:2": "\\begin{equation}\\label{wave:nece:2}\n \\mbox{meas}_{\\R^2} \\Big( [G\\cap L_{\\xi\\in A}] \\, \\Delta \\, [G\\cap L_{\\eta\\in B}]\\Big)=0. \n\\end{equation}", "def:classSc": "\\begin{definition}[Symmetric function pair]\\label{def:classSc}\nWe say \n$(f,g)$ is a symmetric  function pair if  there exists a decomposition pair $\\{(A_k, B_k),\\ k = 1,\\dots, K\\}$, a sequence of different numbers $\\{s_k\\}_{k= 1}^K$ such that \n\\begin{equation*}\n f=\\sum_{1\\leq k\\leq K}s_k\\chi_{A_k}, \\quad g=\\sum_{1\\leq k\\leq K}s_k\\chi_{B_k},\n\\end{equation*}\nand \n\\begin{equation*}\n    \\int_{\\T} (f+ g)(x) \\d x= 0.\n\\end{equation*}\nThe set of all symmetric function pairs is denoted by $\\mathcal{S}^2_c$.\n\\end{definition}", "fig:GCCfail1": "\\begin{tikzpicture}[scale=1.0]\n\n  \\draw[->] (-0.2,0) -- (6.2,0) node[right] {$x$};\n  \\draw[->] (0,-0.2) -- (0,6.2) node[above] {$t$};\n\n  \\draw (3,0) node[below] {$\\pi$} -- (3,0.1);\n  \\draw (6,0) node[below] {$2\\pi$} -- (6,0.1);\n  \\draw (0,3) node[left] {$\\pi$} -- (0.1,3);\n  \\draw (0,6) node[left] {$2\\pi$} -- (0.1,6);\n\n  \\draw (0,0) -- (6,0) -- (6,6) -- (0,6) -- cycle;\n\nll[blue!30] (0,0) coordinate (A1) -- (3,0) coordinate (B1) -- (1.5,1.5) coordinate (C1)-- cycle;\n\n\\node at (barycentric cs:A1=1,B1=1,C1=1) {\\textcolor{white}{-1}};\n\nll[red!30] (3,0) coordinate (A2) -- (6,0) coordinate (B2) -- (4.5,1.5) coordinate (C2)-- cycle;\n\n\\node at (barycentric cs:A2=1,B2=1,C2=1) {\\textcolor{white}{1}};\n\nll[red!30] (0,3) coordinate (A3) -- (1.5,1.5) coordinate (B3) -- (3,3) coordinate (C3)--(1.5,4.5) coordinate (D3)-- cycle;\n\n\\node at (barycentric cs:A3=1,B3=1,C3=1,D3=1) {\\textcolor{white}{1}};\n\nll[blue!30] (3,3) coordinate (A4) -- (4.5,1.5) coordinate (B4) -- (6,3) coordinate (C4)--(4.5,4.5) coordinate (D4)-- cycle;\n\n\\node at (barycentric cs:A4=1,B4=1,C4=1,D4=1) {\\textcolor{white}{-1}};\n\nll[blue!30] (0,6) coordinate (A5) -- (1.5,4.5) coordinate (B5) -- (3,6) coordinate (C5)-- cycle;\n\n\\node at (barycentric cs:A5=1,B5=1,C5=1) {\\textcolor{white}{-1}};\n\nll[red!30] (3,6) coordinate (A6) -- (4.5,4.5) coordinate (B6) -- (6,6) coordinate (C6)-- cycle;\n\n\\node at (barycentric cs:A6=1,B6=1,C6=1) {\\textcolor{white}{1}};\n\n  \\draw[dashed] (-0.5,3.5) -- (3.5,-0.5);   \\draw[dashed] (0,6) -- (6,0);   \\draw[dashed] (2.5,6.5) -- (6.5,2.5); \n  \\draw[dashed] (-0.5,2.5) -- (3.5,6.5);   \\draw[dashed] (0,0) -- (6,6);   \\draw[dashed] (2.5,-0.5) -- (6.5,3.5);   \n\n\\node at (10,4) {$G=G_{0}\\cup G_1$};  \n\n\\node at (10, 3) {$G_{0}=\\{(x,t): u_\\xi=u_\\eta=-1\\}$: blue part};\n\n\\node at (10, 2) {$G_1=\\{(x,t): u_\\xi=u_\\eta=1\\}$: red part};\n\n\\end{tikzpicture}\n    \\caption{Observability fails on $G$, though $G$ satisfies GCC. }\n    \\label{fig:GCCfail1}\n\\end{figure}\n\n\\vspace{2mm}\n\\noindent {\\bf The counterexample.} \n Let $T= 2\\pi$. Let $G\\subset [0,2\\pi]\\times \\T$ be given in Fig. \\ref{fig:GCCfail1}. Clearly $G$ satisfies \\blackhyperref{def:GCC}{({\\rm GCC})}.  However, the observable symmetry condition holds for $A=B=(0,\\pi)$ or $A=B=(\\pi,2\\pi)$. Then one can show that the observability inequality \\eqref{eq: wave-ob}  fails on $G$, see Section \\ref{sec-osc-nece}. \n\n\\subsection{Necessary and sufficient unique continuation conditon}\nThe second result is about the unique continuation property (UCP).  Let $T>0$ and let $G\\in [0, T]\\times \\T$ be a measurable set.  UCP is the qualitative version of the observability and asks:\n$$\n\\mbox{(UCP) \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; Let $u$ be a solution of \\eqref{eq: wave-eq-0} and }   u_t =0 \\mbox{ a.e. in } G \\Longrightarrow u \\equiv 0.  \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n$$\n\nMotivated by the geometric features of the system, we formulate the following minimal geometric assumption on the observation region, which is required for unique continuation.\n\n\\vspace{2mm}\n\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak  GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$,  \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s>0.   \\notag\n\\end{equation}\n\\end{enumerate}\n\n\\vspace{2mm}\n\nThis condition is equivalent to \n\\begin{equation}\n      \\textrm{ for } a.e. \\; x\\in \\T, \\mbox{meas}_\\R(G\\cap L_{\\eta= x})>0, \\;\\;  \n  \\mbox{meas}_\\R(G\\cap L_{\\xi=x})>0. \\notag\n\\end{equation} \nCompared to the standard \\blackhyperref{def:GCC}{({\\rm GCC})}, it is not required a uniform positive lower bound.\n\n\\begin{theorem}\\label{thm: main-ucp}\nLet $T>0$ and $G\\subset [0,T]\\times \\T$ be a spacetime measurable set. The following statements are equivalent.\n\\begin{itemize}\n\\item [(1)] UCP holds on $G$.\n\n\\item [(2)] $G$ satisfies \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} and does not obey \\blackhyperref{def:OSC}{$({\\rm OSC})$} for any non-trivial pair $(A, B)$\\footnote{Reminder: for simplicity, when there is no risk of confusion,  sometimes we simply write (OSC) indicating that the set $G$ does not obey the observable symmetry condition for any non-trivial pair $(A, B)$.}.\n\\end{itemize}\n\\end{theorem}\n\\begin{remark}\n    The essence of this result lies in the role of the observable symmetry condition. Under the minimal \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} assumption, unique continuation can be established only up to a symmetric function pair belonging to the class $\\mathcal{S}^2$, which is generated precisely by the observable symmetry condition (see Definitions~\\ref{def:classS}). In other words, the \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} eliminates all obstructions except those arising from the intrinsic symmetry group, and full uniqueness is recovered once solutions are taken modulo this symmetry.\n\\end{remark}\n\n\\begin{figure}[htp]\n    \\centering\n\n\\tikzset{every picture/.style={line width=0.75pt}} \n\\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]\n\n\\draw   (141,130) -- (292,130) -- (292,170) -- (141,170) -- cycle ;\n\\draw   (425,130) -- (539,130) -- (539,170) -- (425,170) -- cycle ;\n\\draw    (303,146.5) -- (415,146.5) ;\n\\draw [shift={(417,146.5)}, rotate = 180] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\\draw    (416,154.5) -- (305,153.52) ;\n\\draw [shift={(303,153.5)}, rotate = 0.51] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\n\\draw (149,143) node [anchor=north west][inner sep=0.75pt]   [align=left] {\\blackhyperref{def:OSC}{${\\rm OSC}$} \\ + \\ \\blackhyperref{def:weakGCC}{{\\rm weak GCC}}};\n\\draw (462,142) node [anchor=north west][inner sep=0.75pt]   [align=left] {UCP};\n\\draw (337,130) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize Sec \\ref{sec: OSC+WGCC>UCP}}};\n\\draw (320,156) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize Sec \\ref{sec:neosctoucp}, Sec \\ref{sec: ucp-wgcc}}};\n\n\\end{tikzpicture}", "def:classS": "\\begin{definition}[Weak symmetric function pair]\\label{def:classS}\nWe say \n$(f,g)$ is a weak symmetric  function pair if  there exists a weak decomposition pair $\\{(A_k, B_k), k\\in I\\}$, a sequence of different  numbers $\\{s_k\\}_{k\\in I}$ such that \n\\begin{equation*}\n f=\\sum_{k\\in I}s_k\\chi_{A_k}, \\quad g=\\sum_{k\\in I}s_k\\chi_{B_k} \\quad \\mbox{ in } L^2(\\T)\n\\end{equation*}\nand \n\\begin{equation*}\n    \\int_{\\T} (f+ g)(x) \\d x= 0.\n\\end{equation*}\nThe set of all weak symmetric function pairs is denoted by $\\mathcal{S}^2$.\n\\end{definition}", "eq:GCC-wave:0": "\\begin{equation}\\label{eq:GCC-wave:0}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s\\geq c_0.\\footnote{\\text{This integral is defined in the sense of \\eqref{eq:GCC:measure}.}}\n\\end{equation}", "eq:GCC:measure": "\\begin{equation}\\label{eq:GCC:measure}\n      \\textrm{ for } a.e. \\; x\\in \\T, \\mbox{meas}_\\R(G\\cap L_{\\eta= x})\\geq c_0, \\;\\;  \n  \\mbox{meas}_\\R(G\\cap L_{\\xi=x})\\geq c_0. \n\\end{equation}", "eq: wave-ob": "\\begin{equation}\\label{eq: wave-ob}\n\\|u_0\\|^2_{\\dot H^1(\\T)}+\\|u_1\\|^2_{L^2(\\T)}\\leq C\\int_G|\\partial_tu(t,x)|^2\\d t\\d x.\n\\end{equation}", "eq: wave-eq": "\\begin{equation}\\label{eq: wave-eq}\n(\\partial_t^2-\\partial_x^2)u =f\\mathbf{1}_G,\\quad (u, \\partial_t u)\\big|_{t=0}=(u_0,u_1)\\in \\dot H^1(\\T)\\times L^2(\\T).\n\\end{equation}", "def:GCC": "\\begin{enumerate}\n\\item[({\\bf GCC})]\\label{def:GCC} \nLet $T>0$. A measurable set $G \\subset [0, T] \\times \\mathbb{T}$ is said to satisfy \nthe GCC if there exists a constant \n$c_0 > 0$ such that for almost every $x \\in \\mathbb{T}$,\n\\begin{equation}\\label{eq:GCC-wave:0}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s\\geq c_0.\\footnote{\\text{This integral is defined in the sense of \\eqref{eq:GCC:measure}.}}\n\\end{equation} \n\\end{enumerate}", "def:weakGCC": "\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak  GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$,  \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s>0.   \\notag\n\\end{equation}\n\\end{enumerate}", "fig:OSC-role": "\\label{fig:OSC-role}\n\\end{figure}\n\n\\vspace{3mm}\nThe first result is a new conservation law of the forced equation \\eqref{eq: wave-eq}. \n\\begin{proposition}[Conservation law]\n\\label{prop-sym-conser}\nLe", "def:OSC": "\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset \\T$ be two measurable subsets. $G\\in [0, T]\\times \\T$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\nExpression \\eqref{wave:nece:2} means that $G\\cap L_{\\xi\\in A} = G\\cap L_{\\eta\\in B}$ modulo zero measure set in $\\R^2$.}  \n\\begin{equation}\\label{wave:nece:2}\n \\mbox{meas}_{\\R^2} \\Big( [G\\cap L_{\\xi\\in A}] \\, \\Delta \\, [G\\cap L_{\\eta\\in B}]\\Big)=0. \n\\end{equation} \n\\end{enumerate}"}, "pre_theorem_intro_text_len": 17122, "pre_theorem_intro_text": "Let $T > 0$. Let $G \\subset [0,T] \\times \\mathbb{T}$ be a spacetime measurable set with positive measure.\nConsider the {\\it observability problem} of the wave equation: whether there exists some $C= C(G)>0$ such that every solution $u$ to \n\\begin{equation}\\label{eq: wave-eq-0}\n(\\partial_t^2-\\partial_x^2)u=0,\\quad (u,\\partial_tu)\\big|_{t=0}=(u_0,u_1)\\in \\dot H^1({\\mathbb T})\\times L^2({\\mathbb T}),\n\\end{equation}\nsatisfies\n\\begin{equation}\\label{eq: wave-ob}\n\\|u_0\\|^2_{\\dot H^1({\\mathbb T})}+\\|u_1\\|^2_{L^2({\\mathbb T})}\\leq C\\int_G|\\partial_tu(t,x)|^2{\\,\\rm d} t{\\,\\rm d} x.\n\\end{equation}\n\nThanks to the classical Hilbert uniqueness method and an argument to raise regularity (see Appendix \\ref{sec: HUM-app} for details on this reduction), the observability of \\eqref{eq: wave-ob} is equivalent to the {\\it exact controllability} of\nthe controlled wave equation on $[0,T]\\times{\\mathbb T}$ with control force $f\\in L^2(G)$:\n\\begin{equation}\\label{eq: wave-eq}\n(\\partial_t^2-\\partial_x^2)u =f\\mathbf{1}_G,\\quad (u, \\partial_t u)\\big|_{t=0}=(u_0,u_1)\\in \\dot H^1({\\mathbb T})\\times L^2({\\mathbb T}).\n\\end{equation}\n\nOur primary objective is to seek the sufficient and necessary geometric conditions on $G$ such that the unique continuation and observability hold.\n\n\\subsection{History and setting}\nFrom the early work of Russell \\cite{Russell-1}, the studies of the controllability and observability for wave equations have been central topics in control theory. \n\n\\subsubsection{Geometric control condition}\nIn the pioneering work of Rauch and Taylor \\cite{RT}, they first related the observability to a geometric condition on the damped region $\\omega$ and the rays of geometric optics in the boundaryless case. \nLater, in another pioneering work of Bardos-Lebeau-Rauch \\cite{BLR-gcc}, the well-known \\textit{Geometric control condition} is introduced: for an open observed region $\\omega$, every generalized ray should meet $\\omega$ in a finite time. \n\nSince then, it has become one of the most natural assumptions for the controlled waves. In the existing literature, the observation is most often made on cylindrical domains $G=(0,T)\\times \\omega$, with $\\omega$ being an open subset. Under suitable smooth conditions, it is well-known that GCC is sufficient, and depending on the domain, necessary for the observability in the cylindrical domains $(0,T)\\times \\omega$, see \\cite{BLR-gcc,BG-97}. When considering stabilization problems, GCC is also a useful condition for the exponential decay of energy. We refer to \\cite{RT,Haraux}. Otherwise, one may have logarithmic type of energy decay results \\cite{LR-97, Burq-98}. GCC also plays a role in practical issues, such as sensor designs, tomography techniques used for imaging bodies (see \\cite{RouLebeauAnalPDE2017} for example), etc. For a comprehensive reference of the numerical study, we refer to \\cite{zuazua-review} and its references therein. \n\nFinally, we give a very brief overview of the boundary control case. There are also fruitful results in this direction. For related GCC, we refer to \\cite{Lebeau-boundary}. In particular, for 1D wave equations, one can find many nonlinear results \\cite{Li-1,LY-2} and the references therein.\n\n\\subsubsection{Unique continuation}\nA qualitative version of observability is the unique continuation property.\nThe easiest way to ensure this property is to apply the analyticity based on Holmgren's theorem. Besides, H\\\"ormander's pseudo-convexity condition and Carleman estimates are powerful tools dealing with the unique continuation problem. In this direction, there is a large literature such as \\cite{RZ-98,Tataru,Hormander-92,Hormander-96} and more recent work includes \\cite{Laurent-Leautaud-2019, MS-21,FCL,Shao}. Here we point out that to construct the appropriate weight to apply Carleman estimates, it is crucial to understand the behavior across the suitable spacetime surfaces, which is more delicate than considering a cylindrical region.\n\n\\subsubsection{Our setting: spacetime measurable observable region}\nRecently, researchers started to focus on \n the spacetime setting \\cite{Castro-Cindea-Munch-2014, RouLebeauAnalPDE2017,Shao,PK}. Meanwhile, the study of the case where $G$ is a spacetime region is far from complete, even for open regions.  \n\n In this paper, we focus on the setting that $G\\subset[0,T]\\times{\\mathbb T}$ is measurable of positive measure. The following condition may be viewed as the natural analogue of the standard geometric control condition in this spacetime measurable setting. \n\n\\vspace{2mm}\n\\begin{enumerate}\n\\item[({\\bf GCC})]\\label{def:GCC} \nLet $T>0$. A measurable set $G \\subset [0, T] \\times \\mathbb{T}$ is said to satisfy \nthe GCC if there exists a constant \n$c_0 > 0$ such that for almost every $x \\in \\mathbb{T}$,\n\\begin{equation}\\label{eq:GCC-wave:0}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s){\\,\\rm d} s\\geq c_0.\\footnote{\\text{This integral is defined in the sense of \\eqref{eq:GCC:measure}.}}\n\\end{equation} \n\\end{enumerate}\n\\vspace{2mm}\n\nFrom now on, in this paper, \\blackhyperref{def:GCC}{({\\rm GCC})}  refers to the preceding definition with respect to \\eqref{eq:GCC-wave:0}. \nIt is well-known that under the standard setting, namely the observable region is a cylinder $[0, T]\\times \\omega$, GCC yields several important properties: weak observability and the necessity of observability. These results can be generalized to the spacetime \\blackhyperref{def:GCC}{({\\rm GCC})} for a spacetime measurable observable region. We put the former result in Section \\ref{sec: weak-ob-sec} and the later in Appendix \\ref{sec:app:3}.  \n\n\\subsection{A new symmetry condition}\n\nExtend the system $2\\pi-$periodically to $x\\in \\mathbb{R}$,  and introduce the null coordinate:\n\\begin{equation}\n    \\xi=x+t \\; \\textrm{ and } \\; \\eta=x-t. \\notag\n\\end{equation}\nUnder this new coordinate,  \n\\begin{equation*}\n  2  u_{\\xi}= u_{x}+ u_t \\; \\textrm{ and } \\;  2  u_{\\eta}= u_{x}- u_t\n\\end{equation*}\nand the wave equation \\eqref{eq: wave-eq} becomes\\footnote{We considered the equation on $U(\\xi, \\eta):= u(t, x)$ and $F(\\xi, \\eta)= f(t, x)$. For ease of notation, we still denote $U$ by $u$ and $F$ by $f$.  }\n\\begin{equation}\n    4 \\partial_{\\xi} \\partial_{\\eta} u=  f \\mathbf{1}_G. \\notag\n\\end{equation}\n\nFor any $\\xi_0\\in\\mathbb{R}$, we denote the  line $\\{(t,x):x+t=\\xi_0\\}$ by $L_{\\xi=\\xi_0}$,  and call it the $\\xi$-characteristic. Similarly, we denote the  line $\\{(t,x):x- t=\\eta_0\\}$ by $L_{\\eta=\\eta_0}$ and  call it $L_{\\eta=\\eta_0}$ the $\\eta$-characteristic. Thus the \\blackhyperref{def:GCC}{({\\rm GCC})} is equivalent to\n\\begin{equation}\\label{eq:GCC:measure}\n      \\textrm{ for } a.e. \\; x\\in {\\mathbb T}, \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\eta= x})\\geq c_0, \\;\\;  \n  \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\xi=x})\\geq c_0. \n\\end{equation} \nWe further define measurable cylinders for any measurable sets $A, B\\subset [0, 2\\pi]$ as\n\\begin{equation}\\label{eq: cylinder}\nL_{\\xi\\in A}:=\\bigcup_{\\xi_0\\in A}L_{\\xi=\\xi_0} \\; \\textrm{ and } \\; L_{\\eta\\in B}:=\\bigcup_{\\eta_0\\in B}L_{\\eta=\\eta_0}.\n\\end{equation}\n\nIntroduce the following symmetry condition:\n\n\\vspace{2mm}\n\n\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset {\\mathbb T}$ be two measurable subsets. $G\\in [0, T]\\times {\\mathbb T}$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\nExpression \\eqref{wave:nece:2} means that $G\\cap L_{\\xi\\in A} = G\\cap L_{\\eta\\in B}$ modulo zero measure set in $\\mathbb{R}^2$.}  \n\\begin{equation}\\label{wave:nece:2}\n \\mbox{meas}_{\\mathbb{R}^2} \\Big( [G\\cap L_{\\xi\\in A}] \\, \\Delta \\, [G\\cap L_{\\eta\\in B}]\\Big)=0. \n\\end{equation} \n\\end{enumerate}\n\n\\vspace{2mm}\n\nWe call a pair $(A, B)$ trivial, if $|A|= |B|= 0$ or $|A|= |B|= 2\\pi$.  In principle, we are interested in {\\it non-trivial} pairs $(A, B)$. Since the observable symmetry condition is automatically satisfied for trivial pairs ($A, B$). \n\nThe observable symmetry condition is very important in our work, see Fig. \\ref{fig:OSC-role}.\n\n\\begin{figure}[htp]\n    \\centering\n\n\\tikzset{every picture/.style={line width=0.75pt}} \n\\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]\n\n\\draw   (192,38.4) .. controls (192,32.38) and (196.88,27.5) .. (202.9,27.5) -- (452.1,27.5) .. controls (458.12,27.5) and (463,32.38) .. (463,38.4) -- (463,71.1) .. controls (463,77.12) and (458.12,82) .. (452.1,82) -- (202.9,82) .. controls (196.88,82) and (192,77.12) .. (192,71.1) -- cycle ;\n\\draw   (72,158.7) .. controls (72,152.51) and (77.01,147.5) .. (83.2,147.5) -- (199.8,147.5) .. controls (205.99,147.5) and (211,152.51) .. (211,158.7) -- (211,192.3) .. controls (211,198.49) and (205.99,203.5) .. (199.8,203.5) -- (83.2,203.5) .. controls (77.01,203.5) and (72,198.49) .. (72,192.3) -- cycle ;\n\\draw   (254,151.3) .. controls (254,143.13) and (260.63,136.5) .. (268.8,136.5) -- (393.2,136.5) .. controls (401.37,136.5) and (408,143.13) .. (408,151.3) -- (408,195.7) .. controls (408,203.87) and (401.37,210.5) .. (393.2,210.5) -- (268.8,210.5) .. controls (260.63,210.5) and (254,203.87) .. (254,195.7) -- cycle ;\n\\draw   (436,158.7) .. controls (436,152.51) and (441.01,147.5) .. (447.2,147.5) -- (563.8,147.5) .. controls (569.99,147.5) and (575,152.51) .. (575,158.7) -- (575,192.3) .. controls (575,198.49) and (569.99,203.5) .. (563.8,203.5) -- (447.2,203.5) .. controls (441.01,203.5) and (436,198.49) .. (436,192.3) -- cycle ;\n\\draw    (226,82.5) -- (144.57,146.27) ;\n\\draw [shift={(143,147.5)}, rotate = 321.93] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\\draw    (329,82.5) -- (329,134.5) ;\n\\draw [shift={(329,136.5)}, rotate = 270] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\\draw    (430,82.5) -- (509.44,146.25) ;\n\\draw [shift={(511,147.5)}, rotate = 218.75] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\n\\draw (219,47) node [anchor=north west][inner sep=0.75pt]   [align=left] {Observable symmetry condition};\n\\draw (81,167.5) node [anchor=north west][inner sep=0.75pt]   [align=left] {Conservation law};\n\\draw (282,146.5) node [anchor=north west][inner sep=0.75pt]   [align=left] {\\begin{minipage}[lt]{66.78pt}\\setlength\\topsep{0pt}\n\\begin{center}\nControllability \\\\Observability \\\\UCP\n\\end{center}\n\n\\end{minipage}};\n\\draw (455,167.5) node [anchor=north west][inner sep=0.75pt]   [align=left] {Symetric \\ \\ \\  class };\n\\draw (515,167.9) node [anchor=north west][inner sep=0.75pt]  [font=\\small]  {$\\mathcal{S}^{2}$};\n\\draw (128,106) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize Noether}};\n\\draw (333,106) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize necessary}};\n\\draw (486,106) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize generate}};\n\n\\end{tikzpicture}\n\n    \\caption{The role of (OSC) in this paper}\n    \\label{fig:OSC-role}\n\\end{figure}\n\n\\vspace{3mm}\nThe first result is a new conservation law of the forced equation \\eqref{eq: wave-eq}. \n\\begin{proposition}[Conservation law]\n\\label{prop-sym-conser}\nLet $A,B\\subset {\\mathbb T}$. Assume that $G$ satisfies the ($A, B$)-{\\it observable symmetry condition}.\nLet $(u,u_t)\\in C([0,T];\\dot{H}^1({\\mathbb T})\\times L^2({\\mathbb T}))$ be a solution to the forced equation \\eqref{eq: wave-eq} and define the energy \\footnote{Indeed, under the $(\\xi, \\eta)-$coordinate this conservation is write as\n\\begin{equation*}\n \\frac{1}{2} I(t)=   \\int_{A-t}u_{\\xi}(t,x){\\,\\rm d} x+\\int_{B+t}u_{\\eta}(t,x){\\,\\rm d} x. \n\\end{equation*}\n}\n$$\nI(t)= \\int_{A-t}(u_x+u_t)(t,x){\\,\\rm d} x+\\int_{B+t}(u_x-u_t)(t,x){\\,\\rm d} x.\n$$\nThen we have \n$$\nI(t)=I(0), \\;  \\forall t\\in [0, T].\n$$\nIn the case $|A|= |B|= 2\\pi$, this conservation law is exactly the condition $\\int_{{\\mathbb T}} u_x (t, x) {\\,\\rm d} x= 0$.\n\\end{proposition}\n\n\\begin{remark}\n  This proposition can be understood as a variant of Noether’s theorem\\footnote{Noether’s theorem: ``Every continuous symmetry of the action (or the equations of motion, in a suitable sense) implies a conserved quantity.''} adapted to a controlled wave equation: a geometric symmetry of the forcing region yields a conserved functional of the solution. Indeed,   \nthe key hypothesis is that the region $G$ satisfies the  {\\it observable symmetry condition} for ($A, B$). The conclusion shows that the energy $I(t)$ is invariant in time, regardless of the force term. \n\\end{remark}\n\nThis new type of symmetry provides a {\\it necessary condition} for the controllability and observability of \\eqref{eq: wave-eq}, and even a {\\it necessary condition} for the unique continuation of  \\eqref{eq: wave-ob}. See Section \\ref{sec:symmetry} for the detailed construction of counterexamples.  Moreover, this construction gives rise to two classes of function pairs satisfying the symmetry condition, $\\mathcal{S}^2_c$ and $\\mathcal{S}^2$, in Definitions \\ref{def:classSc} and \\ref{def:classS}.\n\n\\begin{figure}[htp]\n    \\centering\n\\begin{tikzpicture}[scale=1.0]\n\n  \\draw[->] (-0.2,0) -- (6.2,0) node[right] {$x$};\n  \\draw[->] (0,-0.2) -- (0,6.2) node[above] {$t$};\n\n  \\draw (3,0) node[below] {$\\pi$} -- (3,0.1);\n  \\draw (6,0) node[below] {$2\\pi$} -- (6,0.1);\n  \\draw (0,3) node[left] {$\\pi$} -- (0.1,3);\n  \\draw (0,6) node[left] {$2\\pi$} -- (0.1,6);\n\n  \\draw (0,0) -- (6,0) -- (6,6) -- (0,6) -- cycle;\n\nll[blue!30] (0,0) coordinate (A1) -- (3,0) coordinate (B1) -- (1.5,1.5) coordinate (C1)-- cycle;\n\n\\node at (barycentric cs:A1=1,B1=1,C1=1) {\\textcolor{white}{-1}};\n\nll[red!30] (3,0) coordinate (A2) -- (6,0) coordinate (B2) -- (4.5,1.5) coordinate (C2)-- cycle;\n\n\\node at (barycentric cs:A2=1,B2=1,C2=1) {\\textcolor{white}{1}};\n\nll[red!30] (0,3) coordinate (A3) -- (1.5,1.5) coordinate (B3) -- (3,3) coordinate (C3)--(1.5,4.5) coordinate (D3)-- cycle;\n\n\\node at (barycentric cs:A3=1,B3=1,C3=1,D3=1) {\\textcolor{white}{1}};\n\nll[blue!30] (3,3) coordinate (A4) -- (4.5,1.5) coordinate (B4) -- (6,3) coordinate (C4)--(4.5,4.5) coordinate (D4)-- cycle;\n\n\\node at (barycentric cs:A4=1,B4=1,C4=1,D4=1) {\\textcolor{white}{-1}};\n\nll[blue!30] (0,6) coordinate (A5) -- (1.5,4.5) coordinate (B5) -- (3,6) coordinate (C5)-- cycle;\n\n\\node at (barycentric cs:A5=1,B5=1,C5=1) {\\textcolor{white}{-1}};\n\nll[red!30] (3,6) coordinate (A6) -- (4.5,4.5) coordinate (B6) -- (6,6) coordinate (C6)-- cycle;\n\n\\node at (barycentric cs:A6=1,B6=1,C6=1) {\\textcolor{white}{1}};\n\n  \\draw[dashed] (-0.5,3.5) -- (3.5,-0.5);   \\draw[dashed] (0,6) -- (6,0);   \\draw[dashed] (2.5,6.5) -- (6.5,2.5); \n  \\draw[dashed] (-0.5,2.5) -- (3.5,6.5);   \\draw[dashed] (0,0) -- (6,6);   \\draw[dashed] (2.5,-0.5) -- (6.5,3.5);   \n\n\\node at (10,4) {$G=G_{0}\\cup G_1$};  \n\n\\node at (10, 3) {$G_{0}=\\{(x,t): u_\\xi=u_\\eta=-1\\}$: blue part};\n\n\\node at (10, 2) {$G_1=\\{(x,t): u_\\xi=u_\\eta=1\\}$: red part};\n\n\\end{tikzpicture}\n    \\caption{Observability fails on $G$, though $G$ satisfies GCC. }\n    \\label{fig:GCCfail1}\n\\end{figure}\n\n\\vspace{2mm}\n\\noindent {\\bf The counterexample.} \n Let $T= 2\\pi$. Let $G\\subset [0,2\\pi]\\times {\\mathbb T}$ be given in Fig. \\ref{fig:GCCfail1}. Clearly $G$ satisfies \\blackhyperref{def:GCC}{({\\rm GCC})}.  However, the observable symmetry condition holds for $A=B=(0,\\pi)$ or $A=B=(\\pi,2\\pi)$. Then one can show that the observability inequality \\eqref{eq: wave-ob}  fails on $G$, see Section \\ref{sec-osc-nece}. \n\n\\subsection{Necessary and sufficient unique continuation conditon}\nThe second result is about the unique continuation property (UCP).  Let $T>0$ and let $G\\in [0, T]\\times {\\mathbb T}$ be a measurable set.  UCP is the qualitative version of the observability and asks:\n$$\n\\mbox{(UCP) \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; Let $u$ be a solution of \\eqref{eq: wave-eq-0} and }   u_t =0 \\mbox{ a.e. in } G \\Longrightarrow u \\equiv 0.  \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n$$\n\nMotivated by the geometric features of the system, we formulate the following minimal geometric assumption on the observation region, which is required for unique continuation.\n\n\\vspace{2mm}\n\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak  GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$,  \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s){\\,\\rm d} s>0.   \\notag\n\\end{equation}\n\\end{enumerate}\n\n\\vspace{2mm}\n\nThis condition is equivalent to \n\\begin{equation}\n      \\textrm{ for } a.e. \\; x\\in {\\mathbb T}, \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\eta= x})>0, \\;\\;  \n  \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\xi=x})>0. \\notag\n\\end{equation} \nCompared to the standard \\blackhyperref{def:GCC}{({\\rm GCC})}, it is not required a uniform positive lower bound.", "context": "\\vspace{2mm}\n\\begin{enumerate}\n\\item[({\\bf GCC})]\\label{def:GCC} \nLet $T>0$. A measurable set $G \\subset [0, T] \\times \\mathbb{T}$ is said to satisfy \nthe GCC if there exists a constant \n$c_0 > 0$ such that for almost every $x \\in \\mathbb{T}$,\n\\begin{equation}\\label{eq:GCC-wave:0}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s){\\,\\rm d} s\\geq c_0.\\footnote{\\text{This integral is defined in the sense of \\eqref{eq:GCC:measure}.}}\n\\end{equation} \n\\end{enumerate}\n\\vspace{2mm}\n\n\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset {\\mathbb T}$ be two measurable subsets. $G\\in [0, T]\\times {\\mathbb T}$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\n\\vspace{3mm}\nThe first result is a new conservation law of the forced equation \\eqref{eq: wave-eq}. \n\\begin{proposition}[Conservation law]\n\\label{prop-sym-conser}\nLet $A,B\\subset {\\mathbb T}$. Assume that $G$ satisfies the ($A, B$)-{\\it observable symmetry condition}.\nLet $(u,u_t)\\in C([0,T];\\dot{H}^1({\\mathbb T})\\times L^2({\\mathbb T}))$ be a solution to the forced equation \\eqref{eq: wave-eq} and define the energy \\footnote{Indeed, under the $(\\xi, \\eta)-$coordinate this conservation is write as\n\\begin{equation*}\n \\frac{1}{2} I(t)= \\int_{A-t}u_{\\xi}(t,x){\\,\\rm d} x+\\int_{B+t}u_{\\eta}(t,x){\\,\\rm d} x. \n\\end{equation*}\n}\n$$\nI(t)= \\int_{A-t}(u_x+u_t)(t,x){\\,\\rm d} x+\\int_{B+t}(u_x-u_t)(t,x){\\,\\rm d} x.\n$$\nThen we have \n$$\nI(t)=I(0), \\; \\forall t\\in [0, T].\n$$\nIn the case $|A|= |B|= 2\\pi$, this conservation law is exactly the condition $\\int_{{\\mathbb T}} u_x (t, x) {\\,\\rm d} x= 0$.\n\\end{proposition}\n\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$, \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s){\\,\\rm d} s>0. \\notag\n\\end{equation}\n\\end{enumerate}\n\n\\vspace{2mm}\n\nThis condition is equivalent to \n\\begin{equation}\n \\textrm{ for } a.e. \\; x\\in {\\mathbb T}, \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\eta= x})>0, \\;\\; \n \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\xi=x})>0. \\notag\n\\end{equation} \nCompared to the standard \\blackhyperref{def:GCC}{({\\rm GCC})}, it is not required a uniform positive lower bound.\n\n\\begin{enumerate}\n\\item[({\\bf GCC})]\\label{def:GCC} \nLet $T>0$. A measurable set $G \\subset [0, T] \\times \\mathbb{T}$ is said to satisfy \nthe GCC if there exists a constant \n$c_0 > 0$ such that for almost every $x \\in \\mathbb{T}$,\n\\begin{equation}\\label{eq:GCC-wave:0}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s\\geq c_0.\\footnote{\\text{This integral is defined in the sense of \\eqref{eq:GCC:measure}.}}\n\\end{equation} \n\\end{enumerate}\n\n\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset \\T$ be two measurable subsets. $G\\in [0, T]\\times \\T$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\nExpression \\eqref{wave:nece:2} means that $G\\cap L_{\\xi\\in A} = G\\cap L_{\\eta\\in B}$ modulo zero measure set in $\\R^2$.}  \n\\begin{equation}\\label{wave:nece:2}\n \\mbox{meas}_{\\R^2} \\Big( [G\\cap L_{\\xi\\in A}] \\, \\Delta \\, [G\\cap L_{\\eta\\in B}]\\Big)=0. \n\\end{equation} \n\\end{enumerate}\n\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak  GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$,  \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s>0.   \\notag\n\\end{equation}\n\\end{enumerate}\n\n\\begin{equation}\\label{eq: wave-eq}\n(\\partial_t^2-\\partial_x^2)u =f\\mathbf{1}_G,\\quad (u, \\partial_t u)\\big|_{t=0}=(u_0,u_1)\\in \\dot H^1(\\T)\\times L^2(\\T).\n\\end{equation}\n\n\\begin{equation}\\label{eq:GCC:measure}\n      \\textrm{ for } a.e. \\; x\\in \\T, \\mbox{meas}_\\R(G\\cap L_{\\eta= x})\\geq c_0, \\;\\;  \n  \\mbox{meas}_\\R(G\\cap L_{\\xi=x})\\geq c_0. \n\\end{equation}", "full_context": "\\vspace{2mm}\n\\begin{enumerate}\n\\item[({\\bf GCC})]\\label{def:GCC} \nLet $T>0$. A measurable set $G \\subset [0, T] \\times \\mathbb{T}$ is said to satisfy \nthe GCC if there exists a constant \n$c_0 > 0$ such that for almost every $x \\in \\mathbb{T}$,\n\\begin{equation}\\label{eq:GCC-wave:0}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s){\\,\\rm d} s\\geq c_0.\\footnote{\\text{This integral is defined in the sense of \\eqref{eq:GCC:measure}.}}\n\\end{equation} \n\\end{enumerate}\n\\vspace{2mm}\n\n\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset {\\mathbb T}$ be two measurable subsets. $G\\in [0, T]\\times {\\mathbb T}$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\n\\vspace{3mm}\nThe first result is a new conservation law of the forced equation \\eqref{eq: wave-eq}. \n\\begin{proposition}[Conservation law]\n\\label{prop-sym-conser}\nLet $A,B\\subset {\\mathbb T}$. Assume that $G$ satisfies the ($A, B$)-{\\it observable symmetry condition}.\nLet $(u,u_t)\\in C([0,T];\\dot{H}^1({\\mathbb T})\\times L^2({\\mathbb T}))$ be a solution to the forced equation \\eqref{eq: wave-eq} and define the energy \\footnote{Indeed, under the $(\\xi, \\eta)-$coordinate this conservation is write as\n\\begin{equation*}\n \\frac{1}{2} I(t)= \\int_{A-t}u_{\\xi}(t,x){\\,\\rm d} x+\\int_{B+t}u_{\\eta}(t,x){\\,\\rm d} x. \n\\end{equation*}\n}\n$$\nI(t)= \\int_{A-t}(u_x+u_t)(t,x){\\,\\rm d} x+\\int_{B+t}(u_x-u_t)(t,x){\\,\\rm d} x.\n$$\nThen we have \n$$\nI(t)=I(0), \\; \\forall t\\in [0, T].\n$$\nIn the case $|A|= |B|= 2\\pi$, this conservation law is exactly the condition $\\int_{{\\mathbb T}} u_x (t, x) {\\,\\rm d} x= 0$.\n\\end{proposition}\n\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$, \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s){\\,\\rm d} s>0. \\notag\n\\end{equation}\n\\end{enumerate}\n\n\\vspace{2mm}\n\nThis condition is equivalent to \n\\begin{equation}\n \\textrm{ for } a.e. \\; x\\in {\\mathbb T}, \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\eta= x})>0, \\;\\; \n \\mbox{meas}_\\mathbb{R}(G\\cap L_{\\xi=x})>0. \\notag\n\\end{equation} \nCompared to the standard \\blackhyperref{def:GCC}{({\\rm GCC})}, it is not required a uniform positive lower bound.\n\n\\begin{enumerate}\n\\item[({\\bf GCC})]\\label{def:GCC} \nLet $T>0$. A measurable set $G \\subset [0, T] \\times \\mathbb{T}$ is said to satisfy \nthe GCC if there exists a constant \n$c_0 > 0$ such that for almost every $x \\in \\mathbb{T}$,\n\\begin{equation}\\label{eq:GCC-wave:0}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s\\geq c_0.\\footnote{\\text{This integral is defined in the sense of \\eqref{eq:GCC:measure}.}}\n\\end{equation} \n\\end{enumerate}\n\n\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset \\T$ be two measurable subsets. $G\\in [0, T]\\times \\T$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\nExpression \\eqref{wave:nece:2} means that $G\\cap L_{\\xi\\in A} = G\\cap L_{\\eta\\in B}$ modulo zero measure set in $\\R^2$.}  \n\\begin{equation}\\label{wave:nece:2}\n \\mbox{meas}_{\\R^2} \\Big( [G\\cap L_{\\xi\\in A}] \\, \\Delta \\, [G\\cap L_{\\eta\\in B}]\\Big)=0. \n\\end{equation} \n\\end{enumerate}\n\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak  GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$,  \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s>0.   \\notag\n\\end{equation}\n\\end{enumerate}\n\n\\begin{equation}\\label{eq: wave-eq}\n(\\partial_t^2-\\partial_x^2)u =f\\mathbf{1}_G,\\quad (u, \\partial_t u)\\big|_{t=0}=(u_0,u_1)\\in \\dot H^1(\\T)\\times L^2(\\T).\n\\end{equation}\n\n\\begin{equation}\\label{eq:GCC:measure}\n      \\textrm{ for } a.e. \\; x\\in \\T, \\mbox{meas}_\\R(G\\cap L_{\\eta= x})\\geq c_0, \\;\\;  \n  \\mbox{meas}_\\R(G\\cap L_{\\xi=x})\\geq c_0. \n\\end{equation}\n\n\\begin{figure}[htp]\n \\centering\n\n\\subsection{Necessary and sufficient observability/controllability condition}\nFinally, we obtain the following sharp and complete geometric characterization of observable regions.\n\\begin{theorem}\\label{thm: main-ob}\nLet $T>0$ and let $G\\subset [0, T]\\times \\T$ be a measurable set. Then the following two statements are equivalent.\n\\begin{itemize}\n \\item[(1)] The observability inequality \\eqref{eq: wave-ob} holds on $G$;\n \\item [(2)] $G$ satisfies \\blackhyperref{def:GCC}{{\\rm (GCC)}} and does not obey \\blackhyperref{def:OSC}{$({\\rm OSC})$} for any non-tirvial pair ($A, B$). \n\\end{itemize}\n\\end{theorem}\n\n\\begin{proposition}[Unique continuation implies weak GCC]\nLet $T>0$ and $G\\in [0, T]\\times \\T$. Assume that the following unique continuation property holds\n$$\n\\mbox{ Let u be a solution of \\eqref{eq: wave-eq-0} and } u_t =0 \\mbox{ a.e. in } G \\Longrightarrow u \\equiv 0. \n$$\nThen $G$ satisfies the \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}}. \n\\end{proposition}\n\\begin{proof}\nWe argue by contradiction. Suppose that $G$ does not satisfy the \\blackhyperref{def:weakGCC}{({\\rm weak GCC})}, there exists a subset $A\\subset \\T$ such that $\\mbox{meas}_\\R(A)>0$ and\n\\begin{align*}\n\\mbox{meas}_\\R(G\\cap L_{\\eta\\in A})=0 \\quad \\mbox{ or } \\quad \n \\mbox{meas}_\\R(G\\cap L_{\\xi\\in A})=0.\n\\end{align*}\nWithout loss of generality, we only consider the case\n\\begin{align}\\label{equ-ucp-wgcc-1}\n\\mbox{meas}_\\R(G\\cap L_{\\eta\\in A})=0. \n\\end{align}\nSince $A$ has positive measure, one can find a non-zero function $\\phi$ such that\n\\begin{align}\\label{equ-ucp-wgcc-2}\n \\int_A \\phi(x)\\d x=0, \\quad \\int_A|\\phi| \\d x=\\mbox{meas}_{\\R}(A)>0. \n\\end{align}\nSet \n$$\nu_{0x}=-u_1=\\chi_A \\phi.\n$$\nThis is possible since $\\int_\\T u_{0,x}\\d x =\\int_A\\phi \\d x =0$. Then we have\n$$\n\\partial_\\xi u|_{t=0}=u_{0x}+u_1=0, \\quad \\partial_\\eta u|_{t=0}=u_{0x}-u_1=2\\chi_A\\phi.\n$$\nIt follows that\n$$\n\\partial_\\xi u=0 \\quad \\mbox{ on } [0,T]\\times \\T\n$$\nand\n$$\n\\partial_\\eta u= 0 \\quad \\mbox{ on } L_{\\eta\\in \\T\\backslash A}.\n$$\nThus we have\n$$\nu_t=\\frac{1}{2}(\\partial_\\xi u-\\partial_\\eta u)=0 \\quad \\mbox{ on } L_{\\eta\\in \\T\\backslash A}.\n$$\nThanks to \\eqref{equ-ucp-wgcc-1}, we see $G\\subset L_{\\eta\\in \\T\\backslash A}$, and thus $u_t=0$ on $G$. By the UCP, we must have $\\partial_\\eta u|_{t=0}=2\\chi_A\\phi\\equiv 0$, which leads a contradiction with \\eqref{equ-ucp-wgcc-2}.\n\\end{proof}\n\n\\subsection{OSC and weak GCC imply unique continuation }\\label{sec: OSC+WGCC>UCP}\nIn this sequel, we generalize the result in Proposition \\ref{prop-ucp-S2}, which forms the following proposition.\n\\begin{proposition}[Weak GCC implies UCP up to $\\mathcal{S}^2$]\\label{prop-weakGCC-ucp-S2}\n Let $T>0$. Let $G\\in [0, T]\\times \\T$ satisfy the \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}}.\n If $u$ solves the wave equation \\eqref{eq: wave-eq-0} and $u_t = 0$ on $G$, then its initial state $(\\partial_\\xi u, \\partial_\\eta u)|_{t=0}$ belongs to the symmetry function pair class $\\mathcal{S}^2$.\n\\end{proposition}\n\\begin{proof}\nWe use the same notation in the proof of Proposition \\ref{prop-ucp-S2}. Recall that we have the relation \\eqref{equ-xi-eta-exchange}, namely \n\\begin{equation*} \\partial_{\\eta}u|_{\\{t= 0\\}\\cap L_{\\eta= \\eta_0}}=\\partial_{\\eta}u(\\xi_0,\\eta_0)= \\partial_{\\xi}u|_{\\{t= 0\\}\\cap L_{\\xi\\in \\mathcal{U}_{\\eta= \\eta_0}}}\n\\end{equation*}\nup to sets with zero measure.\nNamely, the value of $\\partial_{\\eta}u(\\xi_0,\\eta_0)$ determines almost every value of $\\partial_{\\xi}u|_{\\{t= 0\\}\\cap L_{\\xi\\in \\mathcal{U}_{\\eta= \\eta_0}}}$. Since $G$ satisfies the \\blackhyperref{def:weakGCC}{({\\rm weak GCC})}, we know \n$$\n\\mbox{meas}_\\R(\\{t= 0\\}\\cap L_{\\xi\\in \\mathcal{U}_{\\eta= \\eta_0}})>0.\n$$\nIn other words, $\\partial_{\\xi}u|_{\\{t= 0\\}}$ equals to $\\partial_{\\eta}u(\\xi_0,\\eta_0)$ at least on a positive measure subset set of $\\T$. Thus the set\n$$\n\\mathcal{U}_{(\\xi_0,\\eta_0)}:=\\{x_0\\in \\T: \\partial_{\\xi}u|_{\\{t= 0,x=x_0\\}}=\\partial_{\\eta}u(\\xi_0,\\eta_0)\\}\n$$\nhas positive measure. Repeating this process for other point $(\\xi_0',\\eta_0')\\in [0,T]\\times \\T$, we shall find that, there exists a family sets $\\mathcal{U}_\\ell (\\ell\\in I)$, $I$ is a index set, such that\n\\begin{align}\\label{equ-Ul-129}\n \\mathcal{U}_\\ell\\cap \\mathcal{U}_{\\ell'}=\\emptyset \\mbox{ for } \\ell\\neq \\ell', \\quad \\mbox{meas}_\\R(\\mathcal{U}_\\ell)>0 \\mbox{ for } \\ell\\in I , \\quad \\T=\\cup_{\\ell\\in I}\\mathcal{U}_\\ell \n\\end{align} \nand $\\partial_{\\xi}u|_{\\{t= 0\\}}$ is a constant on each $\\mathcal{U}_\\ell$.\nOne can show that, under the restrictions \\eqref{equ-Ul-129}, the index set $I$ is at most countable, thus, after relabeling if necessary, $\\{\\mathcal{U}_\\ell, \\ell\\ge 1\\}$ is a weak decomposition of $\\T$. Moreover, there exists a sequence of different numbers $\\{s_\\ell\\}$ such that\n$$\n\\partial_{\\xi}u|_{\\{t= 0\\}}=s_\\ell \\mbox{ on } \\mathcal{U}_\\ell.\n$$\nNote that $\\partial_{\\xi}u|_{\\{t= 0\\}}\\in L^2(\\T)$, and $\\mathcal{U}_\\ell$ are disjoint for different $\\ell$, we find\n$$\n\\|\\partial_{\\xi}u|_{\\{t= 0\\}}\\|_{L^2(\\T)}^2=\\sum_\\ell |s_\\ell|^2\\mbox{meas}_\\R(\\mathcal{U}_\\ell)<\\infty.\n$$\nThus we obtain\n$$\n\\partial_{\\xi}u|_{\\{t= 0\\}} = \\sum_{\\ell\\geq 1} s_\\ell \\chi_{\\mathcal{U}_\\ell} \\quad \\mbox{ in } L^2(\\T).\n$$\nSimilarly, we can find a weak decomposition $\\mathcal{V}_{\\ell}(\\ell\\geq 1)$ such that\n$$\n\\partial_{\\eta}u|_{\\{t= 0\\}} = \\sum_{\\ell\\geq 1} s_\\ell \\chi_{\\mathcal{V}_\\ell} \\quad \\mbox{ in } L^2(\\T).\n$$\nIn summary, we find that $(\\partial_{\\xi}u|_{\\{t= 0\\}}, \\partial_{\\eta}u|_{\\{t= 0\\}})\\in \\mathcal{S}^2$.\n\\end{proof}\n\nThe following theorem gives a sufficient and necessary condition of the observable set for the transport equation. We only state the result for \\eqref{eq: transport-eq}, the reader easily figures out the the necessary modifications for \\eqref{eq: transport-eq-2}.\n\\begin{theorem}\\label{thm-tran-suff-nece}\nLet $T>0$ and $G$ be a measurable subset of $[0,T]\\times \\T$ with positive measure. Then the observability inequality\n\\begin{align}\\label{equ-tran-suff-necc-1}\n \\|u_0\\|^2_{L^2(\\T)}\\leq C\\int_G|u(t,x)|^2\\d x \\d t \n\\end{align}\nholds with a positive constant $C>0$ for all solutions to the transport equation \\eqref{eq: transport-eq} if and only if there exists a constant $c_0>0$ such that \n\\begin{equation}\\label{equ-tran-suff-necc-2}\n\\int_0^T\\mathbf{1}_G(s,x- s)\\d s\\geq c_0 \\quad \\mbox{ for a.e. } x\\in\\T \n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe direction $\\eqref{equ-tran-suff-necc-2}\\Longrightarrow \\eqref{equ-tran-suff-necc-1}$ follows clearly from Proposition \\ref{prop: tran-eq-ob}. To show the inverse direction, namely $\\eqref{equ-tran-suff-necc-1}\\Longrightarrow \\eqref{equ-tran-suff-necc-2}$, we use the contradiction argument. Suppose that \\eqref{equ-tran-suff-necc-2} is not true, then for any $\\varepsilon>0$, there exists a set $E_\\varepsilon\\subset \\T$ with positive measure $|E_\\varepsilon|>0$ such that\n\\begin{align}\\label{equ-tran-suff-necc-3} \n\\int_0^T\\mathbf{1}_G(s,x- s)\\d s<\\varepsilon \\quad \\mbox{ for a.e. } x\\in E_\\varepsilon. \n\\end{align}\nLet $u_{0\\varepsilon}(x)=1_{E_\\varepsilon}(x)$. Then $u_{0\\varepsilon}\\in L^2(\\T)$ and its support is contained in $E_\\varepsilon$. Then, similar to the proof in Proposition \\ref{prop: tran-eq-ob}, we have\n\\begin{align*}\n\\int_G|u(t,x)|^2\\d x \\d t =\\int_{\\T}|u_{0\\varepsilon}(x)|^2\\int_{\\T}\\mathbf{1}_G(t,x-t)\\d t\\d x\\leq \\varepsilon \\|u_{0\\varepsilon}\\|^2_{L^2(\\T)}. \n\\end{align*}\nTaking $\\varepsilon>0$ small enough, say $\\varepsilon=C/2$, we obtain a contradiction with \\eqref{equ-tran-suff-necc-1}. \n\\end{proof}\n\\subsection{GCC is necessary for the wave observability}\\label{sec:app:3}\n\\begin{corollary}\\label{wave-GCC-nece}\nIf the observability \\eqref{eq: wave-ob} holds on some observation set $G\\subset [0,T]\\times \\T$, then $G$ needs to satisfies \\blackhyperref{def:GCC}{{\\rm (GCC)}}.\n\\end{corollary}\n\\begin{proof}", "post_theorem_intro_text_len": 6409, "post_theorem_intro_text": "\\begin{remark}\n    The essence of this result lies in the role of the observable symmetry condition. Under the minimal \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} assumption, unique continuation can be established only up to a symmetric function pair belonging to the class $\\mathcal{S}^2$, which is generated precisely by the observable symmetry condition (see Definitions~\\ref{def:classS}). In other words, the \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} eliminates all obstructions except those arising from the intrinsic symmetry group, and full uniqueness is recovered once solutions are taken modulo this symmetry.\n\\end{remark}\n\n\\begin{figure}[htp]\n    \\centering\n\n\\tikzset{every picture/.style={line width=0.75pt}} \n\\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]\n\n\\draw   (141,130) -- (292,130) -- (292,170) -- (141,170) -- cycle ;\n\\draw   (425,130) -- (539,130) -- (539,170) -- (425,170) -- cycle ;\n\\draw    (303,146.5) -- (415,146.5) ;\n\\draw [shift={(417,146.5)}, rotate = 180] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\\draw    (416,154.5) -- (305,153.52) ;\n\\draw [shift={(303,153.5)}, rotate = 0.51] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\n\\draw (149,143) node [anchor=north west][inner sep=0.75pt]   [align=left] {\\blackhyperref{def:OSC}{${\\rm OSC}$} \\ + \\ \\blackhyperref{def:weakGCC}{{\\rm weak GCC}}};\n\\draw (462,142) node [anchor=north west][inner sep=0.75pt]   [align=left] {UCP};\n\\draw (337,130) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize Sec \\ref{sec: OSC+WGCC>UCP}}};\n\\draw (320,156) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize Sec \\ref{sec:neosctoucp}, Sec \\ref{sec: ucp-wgcc}}};\n\n\\end{tikzpicture}\n\n    \\caption{Outline of the proof to the equivalence.}\n    \\label{fig:OSC+WGCC=UCP}\n\\end{figure}\n\n\\subsection{Necessary and sufficient observability/controllability condition}\nFinally, we obtain the following sharp and complete geometric characterization of observable regions.\n\\begin{theorem}\\label{thm: main-ob}\nLet $T>0$ and let $G\\subset [0, T]\\times {\\mathbb T}$ be a measurable set. Then the following two statements are equivalent.\n\\begin{itemize}\n    \\item[(1)] The observability inequality \\eqref{eq: wave-ob}  holds on $G$;\n    \\item [(2)] $G$ satisfies \\blackhyperref{def:GCC}{{\\rm (GCC)}} and does not obey \\blackhyperref{def:OSC}{$({\\rm OSC})$} for any non-tirvial pair ($A, B$). \n\\end{itemize}\n\\end{theorem}\n\nTo the best of our knowledge, this work provides the first necessary and sufficient geometric condition for observability of the wave equation on spacetime region. The observable symmetry condition plays a central role: it completes the classical GCC by capturing the additional geometric structure. This viewpoint also leads to necessary and sufficient conditions in a variety of other important geometric configurations, such as $[0, T]\\times \\omega$,  measurable Cartesian products  $E_t\\times F_{x}$, and general spacetime open sets.  A detailed comparison is presented in Section~\\ref{sec:example}.\n\n\\begin{figure}[htp]\n    \\centering\n\n\\tikzset{every picture/.style={line width=0.75pt}} \n\\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]\n\n\\draw   (178,130) -- (292,130) -- (292,170) -- (178,170) -- cycle ;\n\\draw   (425,130) -- (539,130) -- (539,170) -- (425,170) -- cycle ;\n\\draw    (303,146.5) -- (415,146.5) ;\n\\draw [shift={(417,146.5)}, rotate = 180] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\\draw    (416,154.5) -- (305,153.52) ;\n\\draw [shift={(303,153.5)}, rotate = 0.51] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;\n\n\\draw (188,143) node [anchor=north west][inner sep=0.75pt]   [align=left] {\\blackhyperref{def:OSC}{${\\rm OSC}$} \\ + \\ \\blackhyperref{def:GCC}{{\\rm GCC}}};\n\\draw (439,142) node [anchor=north west][inner sep=0.75pt]   [align=left] {Observability};\n\\draw (343,130) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize Sec \\ref{sec:sharp:obser}}};\n\\draw (320,156) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\\footnotesize Sec \\ref{sec-osc-nece}, Sec \\ref{sec:app:3}}};\n\n\\end{tikzpicture}\n    \\caption{Outline of the proof to the equivalence.}\n    \\label{fig:OSC+GCC=OB}\n\\end{figure}\n\n\\begin{remark}\nThe main theorem reveals two structural features of observability that distinguish the wave equation from other important models, for example, the heat equation,  \\cite{AEWZ,WWZZ}. First, the family of observable regions $\\mathcal{O}(T)$ depends intrinsically on $T$; non-observable sets arise from geometric obstructions that vary with $T$, leading to a genuinely time-dependent structure for $\\{\\mathcal{O}(T)\\}_{T>0}$.\nSecond, our characterization also suggests that no waiting time is needed for observability, indicating that, for wave equations, the spacetime geometry of the observation region may play a more decisive role than the duration of control. See Section~\\ref{sec:furthercomment}.\n\\end{remark}\n\nMoreover, we believe that the symmetry mechanism uncovered in this work may extend to other instances of wave equations and to a broader range of models, including coupled systems, semilinear equations, and geometric wave equations. These ideas may also provide insight for higher-dimensional problems, where geometric propagation and microlocal structures are considerably more intricate. In addition, the geometric perspectives here may prove useful in questions of sensor placement, optimization of observation regions, and numerical implementations of control and observability.\n\n\\subsection*{Acknowledgements}\nThe authors would like to thank Nicolas Burq for valuable and useful discussions during the preparation of this manuscript.  Shengquan Xiang is partially supported by the NSFC under grants 12571474 and 12301562.   Ming Wang was partially supported by the NSFC under grants 12571260, 12171442 and 12171178.", "sketch": "A proof sketch for Theorem~\\ref{thm: main-ucp} is indicated as follows.\n\n- The key mechanism is the \\blackhyperref{def:OSC}{(OSC)} (\\emph{observable symmetry condition}): “Under the minimal \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} assumption, unique continuation can be established only up to a symmetric function pair belonging to the class $\\mathcal{S}^2$, which is generated precisely by the observable symmetry condition (see Definitions~\\ref{def:classS}).”\n\n- Conceptual structure: “the \\blackhyperref{def:weakGCC}{{\\rm (weak GCC)}} eliminates all obstructions except those arising from the intrinsic symmetry group, and full uniqueness is recovered once solutions are taken modulo this symmetry.”\n\n- The introduction also provides an explicit “Outline of the proof to the equivalence” (Figure~\\ref{fig:OSC+WGCC=UCP}), depicting the equivalence between “\\blackhyperref{def:OSC}{{\\rm OSC}} + \\blackhyperref{def:weakGCC}{{\\rm weak GCC}}” and “UCP,” with the relevant parts located in “Sec \\ref{sec: OSC+WGCC>UCP}” and “Sec \\ref{sec:neosctoucp}, Sec \\ref{sec: ucp-wgcc}.”", "expanded_sketch": "A proof sketch for the main theorem is indicated as follows.\n\n- The key mechanism is the following observable symmetry condition:\n\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset \\T$ be two measurable subsets. $G\\in [0, T]\\times \\T$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\nExpression \\eqref{wave:nece:2} means that $G\\cap L_{\\xi\\in A} = G\\cap L_{\\eta\\in B}$ modulo zero measure set in $\\R^2$.}  \n\\begin{equation}\\label{wave:nece:2}\n \\mbox{meas}_{\\R^2} \\Big( [G\\cap L_{\\xi\\in A}] \\, \\Delta \\, [G\\cap L_{\\eta\\in B}]\\Big)=0. \n\\end{equation} \n\\end{enumerate}\nUnder the minimal assumption\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak  GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$,  \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s>0.   \\notag\n\\end{equation}\n\\end{enumerate}\nunique continuation can be established only up to a symmetric function pair belonging to the class $\\mathcal{S}^2$, where\n\\begin{definition}[Weak symmetric function pair]\\label{def:classS}\nWe say \n$(f,g)$ is a weak symmetric  function pair if  there exists a weak decomposition pair $\\{(A_k, B_k), k\\in I\\}$, a sequence of different  numbers $\\{s_k\\}_{k\\in I}$ such that \n\\begin{equation*}\n f=\\sum_{k\\in I}s_k\\chi_{A_k}, \\quad g=\\sum_{k\\in I}s_k\\chi_{B_k} \\quad \\mbox{ in } L^2(\\T)\n\\end{equation*}\nand \n\\begin{equation*}\n    \\int_{\\T} (f+ g)(x) \\d x= 0.\n\\end{equation*}\nThe set of all weak symmetric function pairs is denoted by $\\mathcal{S}^2$.\n\\end{definition}\nIn particular, this class is generated precisely by the observable symmetry condition above.\n\n- Conceptual structure: the weak GCC condition stated above eliminates all obstructions except those arising from the intrinsic symmetry group, and full uniqueness is recovered once solutions are taken modulo this symmetry.\n\n- The introduction also provides an explicit “Outline of the proof to the equivalence” (Figure~\\ref{fig:OSC+WGCC=UCP}), depicting the equivalence between “the observable symmetry condition above + the weak GCC condition above” and “UCP,” with the relevant parts located later and in \\ref{sec: OSC+WGCC>UCP} and \\ref{sec:neosctoucp}, \\ref{sec: ucp-wgcc}.", "expanded_theorem": "\\label{thm: main-ucp}\nLet $T>0$ and $G\\subset [0,T]\\times {\\mathbb T}$ be a spacetime measurable set. The following statements are equivalent.\n\\begin{itemize}\n\\item [(1)] UCP holds on $G$.\n\n\\item [(2)] $G$ satisfies\n\\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]\n\\item[({\\bf Weak  GCC})]\\label{def:weakGCC} \nLet $T>0$. A measurable set $G\\subset [0, T]\\times \\mathbb{T}$ is said to satisfy the \n weak GCC if for almost every \n$x\\in\\mathbb{T}$,  \n\\begin{equation}\n\\int_0^T\\mathbf{1}_G(s,x\\pm s)\\d s>0.   \\notag\n\\end{equation}\n\\end{enumerate}\nand does not obey\n\\begin{enumerate}\n\\item[({\\bf OSC})]\\label{def:OSC} \nLet $A, B\\subset \\T$ be two measurable subsets. $G\\in [0, T]\\times \\T$ is said to obey the {\\it observable symmetry condition} for ($A, B$) if\\footnote{As usual, $A \\Delta B = (A\\backslash B)\\cup (B\\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.\n\nExpression \\eqref{wave:nece:2} means that $G\\cap L_{\\xi\\in A} = G\\cap L_{\\eta\\in B}$ modulo zero measure set in $\\R^2$.}  \n\\begin{equation}\\label{wave:nece:2}\n \\mbox{meas}_{\\R^2} \\Big( [G\\cap L_{\\xi\\in A}] \\, \\Delta \\, [G\\cap L_{\\eta\\in B}]\\Big)=0. \n\\end{equation} \n\\end{enumerate}\nfor any non-trivial pair $(A, B)$\\footnote{Reminder: for simplicity, when there is no risk of confusion,  sometimes we simply write (OSC) indicating that the set $G$ does not obey the observable symmetry condition for any non-trivial pair $(A, B)$.}.\n\\end{itemize}", "theorem_type": ["Biconditional or Equivalence", "Classification or Bijection"], "mcq": {"question": "Let $T>0$ and let $G\\subset [0,T]\\times \\mathbb T$ be a measurable spacetime set, where $\\mathbb T$ is the one-dimensional torus. Which explicit geometric condition on $G$ is equivalent to the unique continuation property (UCP) on $G$ for the wave setting?", "correct_choice": {"label": "A", "text": "$G$ satisfies the weak geometric control condition, meaning that for almost every $x\\in\\mathbb T$ one has both $\\int_0^T \\mathbf 1_G(s,x+s)\\,ds>0$ and $\\int_0^T \\mathbf 1_G(s,x-s)\\,ds>0$, and moreover $G$ does not obey the observable symmetry condition for any non-trivial pair $(A,B)$ of measurable subsets of $\\mathbb T$; equivalently, there is no non-trivial pair $(A,B)$ such that $$\\operatorname{meas}_{\\mathbb R^2}\\Big(\\,[G\\cap\\{(t,x):x+t\\in A\\}]\\,\\Delta\\,[G\\cap\\{(t,x):x-t\\in B\\}]\\,\\Big)=0,$$ where $\\Delta$ denotes symmetric difference."}, "choices": [{"label": "B", "text": "$G$ satisfies the geometric control condition in the strong sense that there exists a constant $c_0>0$ such that for almost every $x\\in\\mathbb T$ one has both $\\int_0^T \\mathbf 1_G(s,x+s)\\,ds\\ge c_0$ and $\\int_0^T \\mathbf 1_G(s,x-s)\\,ds\\ge c_0$, and moreover $G$ does not obey the observable symmetry condition for any non-trivial pair $(A,B)$ of measurable subsets of $\\mathbb T$; equivalently, there is no non-trivial pair $(A,B)$ such that $$\\operatorname{meas}_{\\mathbb R^2}\\Big(\\,[G\\cap\\{(t,x):x+t\\in A\\}]\\,\\Delta\\,[G\\cap\\{(t,x):x-t\\in B\\}]\\,\\Big)=0.$$"}, {"label": "C", "text": "$G$ satisfies the weak geometric control condition, meaning that for almost every $x\\in\\mathbb T$ one has both $\\int_0^T \\mathbf 1_G(s,x+s)\\,ds>0$ and $\\int_0^T \\mathbf 1_G(s,x-s)\\,ds>0$."}, {"label": "D", "text": "$G$ satisfies the weak geometric control condition, meaning that for almost every $x\\in\\mathbb T$ one has both $\\int_0^T \\mathbf 1_G(s,x+s)\\,ds>0$ and $\\int_0^T \\mathbf 1_G(s,x-s)\\,ds>0$, and moreover $G$ does not obey the observable symmetry condition for any pair $(A,B)$ of measurable subsets of $\\mathbb T$, including the trivial pair; equivalently, for every measurable pair $(A,B)$ one has $$\\operatorname{meas}_{\\mathbb R^2}\\Big(\\,[G\\cap\\{(t,x):x+t\\in A\\}]\\,\\Delta\\,[G\\cap\\{(t,x):x-t\\in B\\}]\\,\\Big)>0.$$"}, {"label": "E", "text": "$G$ satisfies the weak geometric control condition, meaning that for almost every $x\\in\\mathbb T$ at least one of the two integrals $\\int_0^T \\mathbf 1_G(s,x+s)\\,ds$ or $\\int_0^T \\mathbf 1_G(s,x-s)\\,ds$ is strictly positive, and moreover $G$ does not obey the observable symmetry condition for any non-trivial pair $(A,B)$ of measurable subsets of $\\mathbb T$; equivalently, there is no non-trivial pair $(A,B)$ such that $$\\operatorname{meas}_{\\mathbb R^2}\\Big(\\,[G\\cap\\{(t,x):x+t\\in A\\}]\\,\\Delta\\,[G\\cap\\{(t,x):x-t\\in B\\}]\\,\\Big)=0.$$"}], "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": "weak-vs-strong GCC lower bound", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the no-OSC condition", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "restriction to non-trivial symmetry pairs", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "both characteristic directions required in weak GCC", "template_used": "wildcard"}]}}
{"id": "2512.10455v1", "paper_link": "http://arxiv.org/abs/2512.10455v1", "theorems_cnt": 4, "theorem": {"env_name": "bigthm", "content": "\\label{thm:charac-Markov}\n  Let $X_0$ be a smooth affine surface over an algebraically closed field $K$ with a non-elementary automorphism group.\n  If $X_0$ is completable by a cycle of rational curves, then we have two mutually exclusive possibilities. \n  \\begin{enumerate}\n    \\item $X_0 = \\G_m^2$. \n    \\item $X_0$ is a cubic affine surface of Markov type.\n  \\end{enumerate}\n  The distinction between the two cases comes from whether $X_0$ admits non-constant invertible regular functions.", "start_pos": 22363, "end_pos": 22882, "label": "thm:charac-Markov"}, "ref_dict": {}, "pre_theorem_intro_text_len": 4061, "pre_theorem_intro_text": "\\label{sec:intro}\n\nLet $K$ be an algebraically closed field and let $X_0$ be an irreducible smooth affine surface. A \\emph{completion} of $X_0$ is a\nsmooth projective surface $X$ that contains $X_0$ a Zariski dense open subset. If $f$ is an automorphism of $X_0$, the\n\\emph{dynamical degree} $\\lambda (f)$ of $f$ is defined as follows: take a completion $X$ of $X_0$ and $H$ an ample divisor over $X$,\nthen\n\\begin{equation}\n  \\lambda (f) = \\lim_n \\left( (f^n)^* H \\cdot H \\right)^{1/n}.\n  \\label{}\n\\end{equation}\nThe limit exists, it does not depend on $X$ or on the choice of the ample divisor $H$. We always have that $\\lambda\n(f) \\geq 1$.\nwe say that an automorphism of $X_0$ is \\emph{loxodromic} if its dynamical degree is $>1$. It follows from the author's\nwork in \\cite{abboudDynamicsEndomorphismsAffine2023} that if $K=\\mathbf{C}$ the topological entropy of $f$ is equal to $\\log\n\\lambda(f)$ so loxodromic automorphisms are exactly the ones with positive entropy. We say that $\\Aut (X_0)$ is\n\\emph{non-elementary} if it contains two loxodromic automorphisms with no common iterates.\n\nIn \\cite{abboudDynamicsEndomorphismsAffine2023}, the author studied the dynamics of loxodromic automorphisms of normal\naffine surfaces. Using Gizatullin's work on affine surfaces, we showed that there is a dichotomy. If $X_0$ is a normal\naffine surface with a loxodromic automorphism, then either $X_0$ is\ncompletable by a zigzag of rational curves or by a cycle of rational curves. We have the following equivalent\nconditions. We denote by $\\overline \\kappa (X_0)$ the log Kodaira dimension of $X_0$.\n\n\\begin{prop}\\label{prop:charac-cycle}\n  Let $X_0$ be a normal affine surface with a loxodromic automorphism, then we have the following dichotomy: either\n  $X_0$ is completable by a tree of rational curves or by a cycle of rational curves and\n  the following are equivalent. \n  \\begin{enumerate}\n    \\item $X_0$ is completable by a cycle of rational curves. \n    \\item $\\overline \\kappa (X_0) = 0$. \n    \\item For every loxodromic automorphism $f$ of $X_0$, $\\lambda (f)$ is a quadratic integer.\n  \\end{enumerate}\n Or\n \\begin{enumerate}\n   \\item $X_0$ is completable by a tree of rational curves. \n   \\item $\\overline \\kappa (X_0) = - \\infty$.\n   \\item For every loxodromic automorphism $f$ of $X_0$, $\\lambda (f)$ is an integer.\n \\end{enumerate}\n\\end{prop}\nThe case of a tree of rational curve is quite rich, the affine plane $\\mathbf{A}^2$ is the main example of such an affine surface but there are\nmany other examples of non isomorphic affine surfaces. In \\cite{blancAffineSurfacesHuge2013}, Blanc and Dubouloz showed\nthat there are affine surfaces completable by a zigzag with a huge automorphism group. In\n\\cite{botSmoothComplexRational2023}, Bot showed that there are moduli spaces of such surfaces which gives a smooth rational\ncomplex affine surface with uncountably many non-isomorphic real forms. \n\nFor the cycle case, the main example is the algebraic torus $\\G_m^2$ and cubic affine surfaces of Markov type, i.e the\ncomplement of a triangle of lines $\\Delta$ in a smooth cubic projective surface $S$ in $\\mathbf{P}^3$, (see\n\\cite{el-hutiCubicSurfacesMarkov1974}). If $q$ is one of the three intersection points of the triangle of lines in\n$S$, it defines an involution $\\sigma_p$ as follows: if $p \\in S \\setminus \\Delta$, the line between $q$ and $p$\nintersects $S \\setminus \\Delta$ in one other point which is $\\sigma_q (p)$. Up to finite index the automorphism group of\nsuch an affine surface is generated\nby the three involutions $\\sigma_{q_1}, \\sigma_{q_2}, \\sigma_{q_3}$ where $q_1, q_2, q_3$ are the three intersection\npoints in $\\Delta$. These families of cubic affine surfaces have been studied extensively as they appear in different\nareas of mathematics. They are related to the Painlevé Equation, to the character varieties of the 4-punctured sphere or\nthe once punctured torus, see \\cite{cantatHolomorphicDynamicsPainleve2007}.\nThe purpose of this paper is to prove the following theorem which states that these are the only smooth examples.", "context": "\\label{sec:intro}\n\nLet $K$ be an algebraically closed field and let $X_0$ be an irreducible smooth affine surface. A \\emph{completion} of $X_0$ is a\nsmooth projective surface $X$ that contains $X_0$ a Zariski dense open subset. If $f$ is an automorphism of $X_0$, the\n\\emph{dynamical degree} $\\lambda (f)$ of $f$ is defined as follows: take a completion $X$ of $X_0$ and $H$ an ample divisor over $X$,\nthen\n\\begin{equation}\n \\lambda (f) = \\lim_n \\left( (f^n)^* H \\cdot H \\right)^{1/n}.\n \\label{}\n\\end{equation}\nThe limit exists, it does not depend on $X$ or on the choice of the ample divisor $H$. We always have that $\\lambda\n(f) \\geq 1$.\nwe say that an automorphism of $X_0$ is \\emph{loxodromic} if its dynamical degree is $>1$. It follows from the author's\nwork in \\cite{abboudDynamicsEndomorphismsAffine2023} that if $K=\\mathbf{C}$ the topological entropy of $f$ is equal to $\\log\n\\lambda(f)$ so loxodromic automorphisms are exactly the ones with positive entropy. We say that $\\Aut (X_0)$ is\n\\emph{non-elementary} if it contains two loxodromic automorphisms with no common iterates.\n\nIn \\cite{abboudDynamicsEndomorphismsAffine2023}, the author studied the dynamics of loxodromic automorphisms of normal\naffine surfaces. Using Gizatullin's work on affine surfaces, we showed that there is a dichotomy. If $X_0$ is a normal\naffine surface with a loxodromic automorphism, then either $X_0$ is\ncompletable by a zigzag of rational curves or by a cycle of rational curves. We have the following equivalent\nconditions. We denote by $\\overline \\kappa (X_0)$ the log Kodaira dimension of $X_0$.\n\n\\begin{prop}\\label{prop:charac-cycle}\n Let $X_0$ be a normal affine surface with a loxodromic automorphism, then we have the following dichotomy: either\n $X_0$ is completable by a tree of rational curves or by a cycle of rational curves and\n the following are equivalent. \n \\begin{enumerate}\n \\item $X_0$ is completable by a cycle of rational curves. \n \\item $\\overline \\kappa (X_0) = 0$. \n \\item For every loxodromic automorphism $f$ of $X_0$, $\\lambda (f)$ is a quadratic integer.\n \\end{enumerate}\n Or\n \\begin{enumerate}\n \\item $X_0$ is completable by a tree of rational curves. \n \\item $\\overline \\kappa (X_0) = - \\infty$.\n \\item For every loxodromic automorphism $f$ of $X_0$, $\\lambda (f)$ is an integer.\n \\end{enumerate}\n\\end{prop}\nThe case of a tree of rational curve is quite rich, the affine plane $\\mathbf{A}^2$ is the main example of such an affine surface but there are\nmany other examples of non isomorphic affine surfaces. In \\cite{blancAffineSurfacesHuge2013}, Blanc and Dubouloz showed\nthat there are affine surfaces completable by a zigzag with a huge automorphism group. In\n\\cite{botSmoothComplexRational2023}, Bot showed that there are moduli spaces of such surfaces which gives a smooth rational\ncomplex affine surface with uncountably many non-isomorphic real forms.\n\nFor the cycle case, the main example is the algebraic torus $\\G_m^2$ and cubic affine surfaces of Markov type, i.e the\ncomplement of a triangle of lines $\\Delta$ in a smooth cubic projective surface $S$ in $\\mathbf{P}^3$, (see\n\\cite{el-hutiCubicSurfacesMarkov1974}). If $q$ is one of the three intersection points of the triangle of lines in\n$S$, it defines an involution $\\sigma_p$ as follows: if $p \\in S \\setminus \\Delta$, the line between $q$ and $p$\nintersects $S \\setminus \\Delta$ in one other point which is $\\sigma_q (p)$. Up to finite index the automorphism group of\nsuch an affine surface is generated\nby the three involutions $\\sigma_{q_1}, \\sigma_{q_2}, \\sigma_{q_3}$ where $q_1, q_2, q_3$ are the three intersection\npoints in $\\Delta$. These families of cubic affine surfaces have been studied extensively as they appear in different\nareas of mathematics. They are related to the Painlevé Equation, to the character varieties of the 4-punctured sphere or\nthe once punctured torus, see \\cite{cantatHolomorphicDynamicsPainleve2007}.\nThe purpose of this paper is to prove the following theorem which states that these are the only smooth examples.", "full_context": "\\label{sec:intro}\n\nLet $K$ be an algebraically closed field and let $X_0$ be an irreducible smooth affine surface. A \\emph{completion} of $X_0$ is a\nsmooth projective surface $X$ that contains $X_0$ a Zariski dense open subset. If $f$ is an automorphism of $X_0$, the\n\\emph{dynamical degree} $\\lambda (f)$ of $f$ is defined as follows: take a completion $X$ of $X_0$ and $H$ an ample divisor over $X$,\nthen\n\\begin{equation}\n \\lambda (f) = \\lim_n \\left( (f^n)^* H \\cdot H \\right)^{1/n}.\n \\label{}\n\\end{equation}\nThe limit exists, it does not depend on $X$ or on the choice of the ample divisor $H$. We always have that $\\lambda\n(f) \\geq 1$.\nwe say that an automorphism of $X_0$ is \\emph{loxodromic} if its dynamical degree is $>1$. It follows from the author's\nwork in \\cite{abboudDynamicsEndomorphismsAffine2023} that if $K=\\mathbf{C}$ the topological entropy of $f$ is equal to $\\log\n\\lambda(f)$ so loxodromic automorphisms are exactly the ones with positive entropy. We say that $\\Aut (X_0)$ is\n\\emph{non-elementary} if it contains two loxodromic automorphisms with no common iterates.\n\nIn \\cite{abboudDynamicsEndomorphismsAffine2023}, the author studied the dynamics of loxodromic automorphisms of normal\naffine surfaces. Using Gizatullin's work on affine surfaces, we showed that there is a dichotomy. If $X_0$ is a normal\naffine surface with a loxodromic automorphism, then either $X_0$ is\ncompletable by a zigzag of rational curves or by a cycle of rational curves. We have the following equivalent\nconditions. We denote by $\\overline \\kappa (X_0)$ the log Kodaira dimension of $X_0$.\n\n\\begin{prop}\\label{prop:charac-cycle}\n Let $X_0$ be a normal affine surface with a loxodromic automorphism, then we have the following dichotomy: either\n $X_0$ is completable by a tree of rational curves or by a cycle of rational curves and\n the following are equivalent. \n \\begin{enumerate}\n \\item $X_0$ is completable by a cycle of rational curves. \n \\item $\\overline \\kappa (X_0) = 0$. \n \\item For every loxodromic automorphism $f$ of $X_0$, $\\lambda (f)$ is a quadratic integer.\n \\end{enumerate}\n Or\n \\begin{enumerate}\n \\item $X_0$ is completable by a tree of rational curves. \n \\item $\\overline \\kappa (X_0) = - \\infty$.\n \\item For every loxodromic automorphism $f$ of $X_0$, $\\lambda (f)$ is an integer.\n \\end{enumerate}\n\\end{prop}\nThe case of a tree of rational curve is quite rich, the affine plane $\\mathbf{A}^2$ is the main example of such an affine surface but there are\nmany other examples of non isomorphic affine surfaces. In \\cite{blancAffineSurfacesHuge2013}, Blanc and Dubouloz showed\nthat there are affine surfaces completable by a zigzag with a huge automorphism group. In\n\\cite{botSmoothComplexRational2023}, Bot showed that there are moduli spaces of such surfaces which gives a smooth rational\ncomplex affine surface with uncountably many non-isomorphic real forms.\n\nFor the cycle case, the main example is the algebraic torus $\\G_m^2$ and cubic affine surfaces of Markov type, i.e the\ncomplement of a triangle of lines $\\Delta$ in a smooth cubic projective surface $S$ in $\\mathbf{P}^3$, (see\n\\cite{el-hutiCubicSurfacesMarkov1974}). If $q$ is one of the three intersection points of the triangle of lines in\n$S$, it defines an involution $\\sigma_p$ as follows: if $p \\in S \\setminus \\Delta$, the line between $q$ and $p$\nintersects $S \\setminus \\Delta$ in one other point which is $\\sigma_q (p)$. Up to finite index the automorphism group of\nsuch an affine surface is generated\nby the three involutions $\\sigma_{q_1}, \\sigma_{q_2}, \\sigma_{q_3}$ where $q_1, q_2, q_3$ are the three intersection\npoints in $\\Delta$. These families of cubic affine surfaces have been studied extensively as they appear in different\nareas of mathematics. They are related to the Painlevé Equation, to the character varieties of the 4-punctured sphere or\nthe once punctured torus, see \\cite{cantatHolomorphicDynamicsPainleve2007}.\nThe purpose of this paper is to prove the following theorem which states that these are the only smooth examples.\n\nLet $K$ be an algebraically closed field and let $X_0$ be an irreducible smooth affine surface. A \\emph{completion} of $X_0$ is a\nsmooth projective surface $X$ that contains $X_0$ a Zariski dense open subset. If $f$ is an automorphism of $X_0$, the\n\\emph{dynamical degree} $\\lambda (f)$ of $f$ is defined as follows: take a completion $X$ of $X_0$ and $H$ an ample divisor over $X$,\nthen\n\\begin{equation}\n \\lambda (f) = \\lim_n \\left( (f^n)^* H \\cdot H \\right)^{1/n}.\n \\label{}\n\\end{equation}\nThe limit exists, it does not depend on $X$ or on the choice of the ample divisor $H$. We always have that $\\lambda\n(f) \\geq 1$.\nwe say that an automorphism of $X_0$ is \\emph{loxodromic} if its dynamical degree is $>1$. It follows from the author's\nwork in \\cite{abboudDynamicsEndomorphismsAffine2023} that if $K=\\C$ the topological entropy of $f$ is equal to $\\log\n\\lambda(f)$ so loxodromic automorphisms are exactly the ones with positive entropy. We say that $\\Aut (X_0)$ is\n\\emph{non-elementary} if it contains two loxodromic automorphisms with no common iterates.\n\n\\begin{rmq}\\label{rmq:}\n If we do not suppose that the automorphism group is non-elementary then other examples can arise. For example, if $C$\n is a curve of degree 3 in $\\P^2$ with a nodal singularity, then $X_0 := \\P^2 \\setminus C$ is a smooth affine surface\n completable by a cycle of rational curves. Indeed, if we blow up the nodal singularity we get a completion $X$ of\n $X_0$ such that $\\Delta_X = C_1 \\cup C_2$ which are two smooth rational curves that meet at two points. Here\n $\\Aut (X_0)$ is virtually cyclic with the generator being a loxodromic automorphism so the automorphism group is\n elementary.\n\\end{rmq}\n\nLet $X_0$ be a normal affine surface, we write $K[X_0]$ for its ring of regular functions. A $K$-valuation over\n$K[X_0]$ is a map $v : K[X_0] \\rightarrow \\R \\cup \\left\\{ \\infty \\right\\}$ such that \n\\begin{enumerate}\n \\item $v_{|K^\\times} = 0$. \n \\item $v(0) = \\infty$.\n \\item $\\forall P,Q \\in K[X_0], \\quad v(PQ) = v(P) + v (Q)$. \n \\item $\\forall P,Q \\in K[X_0], \\quad v(P+Q) \\geq \\min \\left( v(P), v(Q) \\right)$.\n\\end{enumerate}\nWe give two examples that will be useful for this note. First we have \\emph{divisorial} valuations. Let $X$ be a\ncompletion of $X_0$ and $E$ a prime divisor at infinity, then $\\ord_E$ the order of vanishing along $E$ is a valuations\nover $K[X_0]$. The second example is as follows. Let $E,F$ be two divisors at infinity in $X$ intersecting at a point\n$p$, then we can find local coordinates $(x,y)$ at $p$ such that $x = 0 $ is a local equation of $E$ and $y = 0$ is a\nlocal equation of $F$. For $\\alpha, \\beta > 0$ we define $v_{\\alpha,\\beta}$ as follows. The completion of the local ring\nat $p$ with respect to its maximal ideal is $K \\left[ [ x,y ] \\right]$ the ring of formal power series in $x,y$. Define \n\\begin{equation}\n v_{\\alpha,\\beta} \\left( \\sum_{i,j} a_{ij} x^i y^j \\right) = \\min \\left( \\alpha i + \\beta j : a_{ij} \\neq 0 \\right).\n \\label{<+label+>}\n\\end{equation}\nThis defines a valuation over $K[X_0]$ because every $P \\in K[X_0]$ is in the fraction field of $\\OO_{X,p}$ the local\nring at $p$ and this fraction field embeds into $K [ [ x ,y ] ]$. We have two possibilities, if $\\alpha / \\beta \\in \\Q$,\nthen $v_{\\alpha,\\beta} = \\ord_G$ is also divisorial but we need to blow up $p$ and infinitely near points to make $G$\nappear. Otherwise $v_{\\alpha,\\beta}$ is \\emph{irrational}.\n\n\\section{Proof of Theorem \\ref{thm:charac-Markov}}\\label{sec:}\nLet $X_0$ be a smooth affine surface completable by a cycle of rational curves and assume that $\\Aut (X_0)$ is not elementary.\n\\begin{prop}\\label{prop:vanishing-self-intersection}\n For any $[v] \\in \\cC_\\infty$, we have $Z_v^2 = 0$.\n\\end{prop}\n\\begin{proof}\n Let $f,g $ be two loxodromic automorphisms generating a non-elementary subgroup. Write $v_f^\\pm, v_g^\\pm$ for the\n eigenvaluations of $f$ and $g$ respectively. Recall that they are all different by Proposition\n \\ref{prop:different-eigenvaluations}. From Proposition \\ref{prop:eigenvaluations-divisor}, we have that \n \\begin{equation}\n Z_{v^\\pm_f}^2 = Z_{v^\\pm_g}^2 = 0.\n \\label{<+label+>}\n \\end{equation}\n Let $Y$ be a cyclic completion of $X_0$ such that the centers of these four eigenvaluations are distinct. Recall that\n they are irrational valuations so their centers is always a satellite point. Let\n $p = E \\cap F = c_Y (v_f^+)$. We look at the monomial valuations centered at $p$. The function \n \\begin{equation}\n \\phi: s \\in [0, +\\infty) \\mapsto Z_{v_{1,s}}^2 \n \\label{<+label+>}\n \\end{equation}\n is a polynomial map of $s$ of degree $\\leq 2$ by Lemma \\ref{lemme:self-intersection-irrational}. We have that $v_f^+ = v_{1, s_0}$ for some $s_0 > 0$. Now, we know\n that $f^n_* [v_g^+] \\rightarrow [v_f^+]$ by Theorem \\ref{thm:dynamics-loxodromic-automorphisms}. This implies that after a finite number of steps we have up to\n renormalisation\n \\begin{equation}\n f^n_* v_g^+ = v_{1, s_n}\n \\label{<+label+>}\n \\end{equation}\nfor some $s_n > 0$ with $s_n \\neq s_0$ and $s_n \\rightarrow s_0$. But since $Z_{v_g^+}^2 = 0$ and this is invariant by the action of $f$\nwe have that $Z_{v_{1,s_n}}^2 = 0$ so the function $\\phi$ is zero. Since for any valuation $[v] \\in \\cC_\\infty \\setminus\n\\left\\{ [v_f^-] \\right\\}, f^n_* [v] \\rightarrow [v_f^+]$ we have that $Z_v^2 = 0$.\n\\end{proof}\nThis implies by Proposition \\ref{prop:nef-valuation} that for every $v \\in \\cC_\\infty, Z_v$ is nef.\n\\begin{cor}\\label{cor:intersection-positive}\n If $E,F$ are different prime divisors at infinity of a cyclic completion $X$ of $X_0$, then \n \\begin{equation}\n \\hat E \\cdot \\hat F > 0.\n \\label{<+label+>}\n \\end{equation}\n\\end{cor}\n\\begin{proof}\n We have by Proposition \\ref{prop:vanishing-self-intersection} that $(\\hat E)^2 = (\\hat F)^2 = 0$ so that they are nef\n and effective divisors by Proposition \\ref{prop:nef-valuation}. This implies that $\\hat E \\cdot \\hat F \\geq 0$.\n If the intersection number is zero, then by the Hodge index theorem we must have that $\\hat E = \\hat F$ but this is a\n contradiction.\n\\end{proof}\n\n\\begin{prop}\\label{prop:trivial-k-plus-delta}\n Let $X_0$ be a smooth affine surface completable by a cycle of rational curves with a non-elementary automorphism\n group, then the class $K+\\Delta \\in \\cNS_{cyc} (X_0)$ is equal to $0$.\n\\end{prop}\n\\begin{proof}\nRecall by Proposition \\ref{prop:Cartier-canonical-class} that $(K+ \\Delta) \\in \\Cartier_{cyc}(X_0)_\\R$ and it is defined\n by $K_X + \\Delta_X$ for any cyclic completion $X$ of $X_0$. Furthermore, it is fixed by $\\Aut (X_0)$. We show that\n \\begin{equation}\n \\forall v \\in \\cC_\\infty, \\quad Z_v \\cdot (K+\\Delta) = 0.\n \\end{equation}\n Let $f,g \\in \\Aut (X_0)$ be two loxodromic automorphisms not sharing a common iterate with their eigenvaluations\n $v_h^\\pm$ for $h = f,g$. We have that \n \\begin{equation}\n Z_{v_h^\\pm} \\cdot (K+\\Delta) = (h^{\\pm 1})^* \\left( Z_{v_h^\\pm} \\cdot (K+\\Delta) \\right) = \\lambda \\left(h^{\\pm 1}\\right) Z_{v_h^\\pm}\n \\cdot (K+\\Delta).\n \\label{<+label+>}\n \\end{equation}\n This implies that $Z_{v_h^\\pm} \\cdot (K+\\Delta) = 0$. Now, let $X$ be a cyclic completion of $X_0$ and let $p = E \\cap F = c_X (v_f^+)$. The function \n \\begin{equation}\n \\phi: s > 0 \\mapsto Z_{v_{1,s}} \\cdot (K+\\Delta)\n \\label{<+label+>}\n \\end{equation}\n is a polynomial of degree at most 1 because we have \n \\begin{equation}\n Z_{v_{1,s}} \\cdot (K+\\Delta) = (Z_{v_{1,s}})_X \\cdot (K+\\Delta)_X = (\\hat E + s \\hat F) \\cdot (K_X + \\Delta_X).\n \\label{<+label+>}\n \\end{equation}\n Now we have that $v_f^+ = v_{1,s_0}$ for some $s_0 > 0$. As in the proof of Proposition\n \\ref{prop:vanishing-self-intersection}, for $n$ large enough, $f^n_* v_g^+ = v_{1,s_n}$ and we have $\\phi(s_n) = 0$.\n So that the function $\\phi$ is 0. Now, for any $[v] \\in \\cC_\\infty \\setminus \\left\\{ [v_{-,f}] \\right\\}$ we have that\n $f^n_* [v] \\rightarrow [v_{+, f}]$ so it will belong to $[v_E, v_F[$ after finitely many iterations. So that we get\n $Z_v \\cdot (K+\\Delta) = (f^n)_* (Z_v \\cdot (K+\\Delta)) = Z_{f^n_* v} \\cdot (K+\\Delta) = 0 $.", "post_theorem_intro_text_len": 3141, "post_theorem_intro_text": "\\begin{rmq}\\label{rmq:}\n  If we do not suppose that the automorphism group is non-elementary then other examples can arise. For example, if $C$\n  is a curve of degree 3 in $\\mathbf{P}^2$ with a nodal singularity, then $X_0 := \\mathbf{P}^2 \\setminus C$ is a smooth affine surface\n  completable by a cycle of rational curves. Indeed, if we blow up the nodal singularity we get a completion $X$ of\n  $X_0$ such that $\\Delta_X = C_1 \\cup C_2$ which are two smooth rational curves that meet at two points. Here\n  $\\Aut (X_0)$ is virtually cyclic with the generator being a loxodromic automorphism so the automorphism group is\n  elementary.\n\\end{rmq}\n\n\\subsection{Idea of the proof}\\label{subsec:}\nThe proof goes as follows. We use the valuative techniques from \\cite{abboudDynamicsEndomorphismsAffine2023}. In the\ncase of an affine surface completable by a cycle of rational curves, the space of valuations centered at infinity of\n$X_0$ contains an $\\Aut(X_0)$-equivariant set which is homeomorphic to a circle. This circle is homeomorphic to the\ncompletion of the inverse limits of the dual graphs of cyclic completions. If $\\Aut(X_0)$ is non-elementary, then its\naction on this circle has very large orbits. This imposes strong constraints on the intersection form on the space of\ndivisors supported at infinity. We then conclude that $X_0$ has to be the complement of a triangle of lines in a smooth\ncubic surface unless $X_0 = \\G_m^2$.\n\n\\subsection{Study of endomorphisms of affine surfaces of Markov type}\\label{subsec:endomorphisms-intro}\nWe fully classify the dynamics of cubic affine surfaces of Markov type in characteristic zero by showing that they do not have\nnon-invertible dominant endomorphisms. \n\n\\begin{bigthm}\\label{thm:endomorphisms-affine-surface}\n  Let $K$ be a field of characteristic zero. If $X_0$ is a smooth cubic affine surface of Markov type over $K$ and $f$ is a\n  dominant endomorphism of $X_0$, then $f$ is an automorphism.\n\\end{bigthm}\nTo show this result we can assume that $K = \\mathbf{C}$, then using valuative techniques and the geometry of the space of\nvaluations centered at infinity we show that $f$ must be proper and unramified. Thus, it is a covering of the complex\nmanifold $X_0 (\\mathbf{C})$ which is simply connected (see eg \\cite[Lemma 3.10]{cantatHolomorphicDynamicsPainleve2007}). Hence\n$f$ must be a homeomorphism and therefore an automorphism.\n\nNotice that this is not true for singular cubic affine surfaces. Indeed, let $\\sC \\subset \\mathbf{A}^3$ be the Cayley cubic defined by \n\\begin{equation}\n  x^2 + y^2 + z^2 = xyz + 4.\n  \\label{<+label+>}\n\\end{equation}\nIt is the quotient of $\\G_m^2$ by the involution $(u,v) \\mapsto (u^{-1}, v^{-1})$. The Cayley cubic has four singular\npoints which are orbifold singularities of order 2. It has many endomorphisms, namely every monomial endomorphism of\n$\\G_m^2$ descends to an endomorphism of $\\sC$. The proof fails for this surface because it is not simply connected as an\norbifold. Indeed, its orbifold fundamental group is not trivial.\n\n\\subsection*{Acknowledgements} I thank Serge Cantat and Matteo Ruggiero for related discussions on this problem.", "sketch": "The proof of Theorem~\\ref{thm:charac-Markov} is described in the subsection \\emph{Idea of the proof}. One \"use[s] the valuative techniques from \\cite{abboudDynamicsEndomorphismsAffine2023}\" and, for an affine surface completable by a cycle of rational curves, considers the space of valuations centered at infinity of $X_0$, which \"contains an $\\Aut(X_0)$-equivariant set which is homeomorphic to a circle.\" This circle is identified as \"homeomorphic to the completion of the inverse limits of the dual graphs of cyclic completions.\" If $\\Aut(X_0)$ is non-elementary, then \"its action on this circle has very large orbits,\" which \"imposes strong constraints on the intersection form on the space of divisors supported at infinity.\" From these constraints one concludes that \"$X_0$ has to be the complement of a triangle of lines in a smooth cubic surface unless $X_0 = \\G_m^2$.\"", "expanded_sketch": "The proof of the main theorem is described in the subsection \\emph{Idea of the proof}. One \"use[s] the valuative techniques from \\cite{abboudDynamicsEndomorphismsAffine2023}\" and, for an affine surface completable by a cycle of rational curves, considers the space of valuations centered at infinity of $X_0$, which \"contains an $\\Aut(X_0)$-equivariant set which is homeomorphic to a circle.\" This circle is identified as \"homeomorphic to the completion of the inverse limits of the dual graphs of cyclic completions.\" If $\\Aut(X_0)$ is non-elementary, then \"its action on this circle has very large orbits,\" which \"imposes strong constraints on the intersection form on the space of divisors supported at infinity.\" From these constraints one concludes that \"$X_0$ has to be the complement of a triangle of lines in a smooth cubic surface unless $X_0 = \\G_m^2$.\"", "expanded_theorem": "\\label{thm:charac-Markov}\n  Let $X_0$ be a smooth affine surface over an algebraically closed field $K$ with a non-elementary automorphism group.\n  If $X_0$ is completable by a cycle of rational curves, then we have two mutually exclusive possibilities. \n  \\begin{enumerate}\n    \\item $X_0 = \\G_m^2$. \n    \\item $X_0$ is a cubic affine surface of Markov type.\n  \\end{enumerate}\n  The distinction between the two cases comes from whether $X_0$ admits non-constant invertible regular functions.", "theorem_type": ["Classification or Bijection", "Implication"], "mcq": {"question": "Let $X_0$ be a smooth affine surface over an algebraically closed field $K$. Assume that $\\operatorname{Aut}(X_0)$ is non-elementary, meaning it contains two loxodromic automorphisms with no common iterate, where loxodromic means dynamical degree $>1$. Also assume that $X_0$ admits a smooth projective completion whose boundary divisor is a cycle of rational curves. Which description gives the complete classification of such surfaces?", "correct_choice": {"label": "A", "text": "Exactly two mutually exclusive possibilities occur: either $X_0\\cong \\mathbb{G}_m^2$, or $X_0$ is a cubic affine surface of Markov type; the distinction between these two cases is given by whether $X_0$ admits non-constant invertible regular functions."}, "choices": [{"label": "B", "text": "Exactly two mutually exclusive possibilities occur: either $X_0\\cong \\mathbb{G}_m^2$, or $X_0$ is the complement of a cycle of rational curves in an arbitrary smooth del Pezzo surface; the distinction between these two cases is given by whether $X_0$ admits non-constant invertible regular functions."}, {"label": "C", "text": "One always has the following dichotomy: either $X_0\\cong \\mathbb{G}_m^2$, or $X_0$ is a cubic affine surface of Markov type."}, {"label": "D", "text": "If $X_0$ is completable by a cycle of rational curves and $\\operatorname{Aut}(X_0)$ contains a loxodromic automorphism, then exactly two mutually exclusive possibilities occur: either $X_0\\cong \\mathbb{G}_m^2$, or $X_0$ is a cubic affine surface of Markov type; the distinction between these two cases is given by whether $X_0$ admits non-constant invertible regular functions."}, {"label": "E", "text": "Exactly two mutually exclusive possibilities occur: either $X_0\\cong \\mathbb{G}_m^2$, or $X_0$ is the complement of a triangle of lines in a cubic projective surface of Markov type; the distinction between these two cases is given by whether $X_0$ admits non-constant invertible regular functions."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "geometric_construction", "tampered_component": "specific ambient cubic-surface conclusion", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "criterion via non-constant invertible regular functions", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "non-elementary hypothesis weakened to existence of one loxodromic automorphism", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "smoothness of the cubic surface", "template_used": "property_confusion"}]}}
{"id": "2512.10820v1", "paper_link": "http://arxiv.org/abs/2512.10820v1", "theorems_cnt": 1, "theorem": {"env_name": "thm", "content": "\\label{vbint-}\nLet $0<r\\leq\\infty$. \nLet $M$ be a domain in ${\\bf C}^n$ with $\\partial M\\in C^2$. Assume that   the Levi form of $\\partial M$ on $T^{(1,0)}\\partial M$ has either $n-1$ positive or at least three negative eigenvalues at  $\\zeta_0\\in\\partial M$.  \nLet $E$ be a complex vector bundle of class $Lip$ $($resp. $\\Lambda^{r+1})$ over $\\overline M$. Assume that a connection  $D\\colon Lip(M,E)\\to L_{(1)}^\\infty(M,E)$  $($resp. $\\Lambda^{r+1}(M,E)\\to \\Lambda^{r}_{(1)}(M,E))$ satisfies $\\Omega^{(0,2)}=0$ on $ M$ in the sense of distributions. Then $E|_U$ has a    holomorphic vector bundle structure, where $U$ is some open subset of $\\overline M$ and $\\zeta_0\\in U$. Furthermore, the holomorphic structure is of class $C^{1/2}$ $($resp. $\\Lambda^{r+1/2})$ with respect to the original $Lip$ $($resp. $\\Lambda^{r+1})$ structure of $E|_U$.", "start_pos": 10767, "end_pos": 11596, "label": "vbint-"}, "ref_dict": {"d2=0": "\\begin{prop}\\label{d2=0}\nLet $D$ be a connection for a complex vector bundle $E$. Let $e=(e_1,\\dots, e_{k_0})^t$ be a local frame field of $E$ over $U$. Assume that $De=\\om \\otimes e$ and $\\om\\in L^2_{loc}(U)$ and $d\\om\\in L^1_{loc}(U)$. If $A$ is admissible, then $\\hat\\om=(dA)A^{-1}+A\\om A^{-1}$ is still in $L^2_{loc}(U)$ and  the curvature  forms    satisfy $A\\Om A^{-1}=\\hat\\Om$. \n\\end{prop}", "vbint-": "\\begin{thm}\\label{vbint-}\nLet $0<r\\leq\\infty$. \nLet $M$ be a domain in $\\cc^n$ with $\\pd M\\in C^2$. Assume that   the Levi form of $\\pd M$ on $T^{(1,0)}\\pd M$ has either $n-1$ positive or at least three negative eigenvalues at  $\\zeta_0\\in\\pd M$.  \nLet $E$ be a complex vector bundle of class $Lip$ $($resp. $\\Lambda^{r+1})$ over $\\ov M$. Assume that a connection  $D\\colon Lip(M,E)\\to L_{(1)}^\\infty(M,E)$  $($resp. $\\Lambda^{r+1}(M,E)\\to \\Lambda^{r}_{(1)}(M,E))$ satisfies $\\Om^{(0,2)}=0$ on $ M$ in the sense of distributions. Then $E|_U$ has a    holomorphic vector bundle structure, where $U$ is some open subset of $\\ov M$ and $\\zeta_0\\in U$. Furthermore, the holomorphic structure is of class $C^{1/2}$ $($resp. $\\Lambda^{r+1/2})$ with respect to the original $Lip$ $($resp. $\\Lambda^{r+1})$ structure of $E|_U$.\n\\end{thm}", "vbint": "\\begin{thm}\\label{vbint} \nLet $M$ be a domain in $\\mathbb C^n$ with $\\pd M\\in C^2$. Assume that    the Levi-form of $M$ has\neither $(n-1)$ positive or at least $3$ negative Levi eigenvalues at $\\zeta_0\\in\\pd M$. Let $U$ be an open set in $\\ov M$ and $\\zeta_0\\in U$.    Assume that $\\om$ is a $k_0\\times k_0$ matrix of $(0,1)$ forms on $U$ such that $\\db\\om=\\om\\wedge\\om$ holds on $U\\setminus\\pd M$ in the sense of distributions. Assume that $\\om$ is in $L^\\infty(U)$ $($resp. $\\Lambda^r(U)$ with $r>0)$. Then there is an invertible matrix $A$   of class $\\Lambda^{1/2}$ $($resp. $\\Lambda^{r+1/2})$ on some neighborhood $U'$ of $\\zeta_0$ in $\\ov M$ such that \n$\n\\db A+A\\om=0\n$\nin the sense of distributions on $U'\\setminus\\pd M$.\n\\end{thm}"}, "pre_theorem_intro_text_len": 2656, "pre_theorem_intro_text": "\\label{sec1} \n Let $M\\subset{\\bf C}^n$ be a relatively compact domain  with boundary $\\partial M\\in C^2$. Let $E$ be a smooth (i.e. $C^\\infty$) complex vector bundle   over the closure $\\overline M$. Denote by $C^\\infty(\\overline M, E), C_{(k)}^\\infty(E,\\overline M)$ the  spaces of smooth sections of $E$ and $E$-valued smooth $k$-forms on $\\overline M$, respectively.\nA Koszul {\\it connection} $D$ in $E$ of is a ${\\bf C}$-linear map \nfrom $\n C_{(k)}^\\infty(\\overline M,E)$ to $ C^\\infty_{(k+1)}(\\overline M,E)\n$\nsatisfying,  for any   $f\\in C_{(k)}^\\infty(\\overline M)$ and     $s\\in C^\\infty(\\overline M,E)$,  \n$$\nD(fs)=df\\otimes s +(-1)^kf\\wedge Ds.\n$$     The curvature $R$ of $D$ is defined as\n$\nR=D\\circ D\\colon C^\\infty_{(k)}(\\overline M,E)\\to C^\\infty_{(k+2)}(\\overline M,E).\n$\n  Let $e=(e_1,\\dots, e_{k_0})^t$ be a smooth local frame of $E$. Then\n$$\nDe_i=\\sum \\omega^j_i\\otimes e_j, \\ De=\\omega e;\\quad  D^2e_i=\\sum \\Omega^j_i\\otimes e_j, \\ D^2e=\\Omega e,\n$$\nwhere $\\omega$ and $\\Omega$ are  the connection and curvature forms.  \nIf $\\hat e=Ae$ is another frame, then\n\\begin{equation}\\label{hatomom}\n\\hat \\omega =(dA)A^{-1} +A\\omega A^{-1}, \\quad  \\widehat\\Omega=A\\Omega A^{-1}.\n\\end{equation}\nDecomposing into types,  $\\omega=\\omega^{(1,0)}+\\omega^{(0,1)}$ and\n$\\Omega=\\Omega^{(2,0)}+\\Omega^{(1,1)}+\\Omega^{(0,2)}$, yields  \n\\begin{equation}\\label{om02}\n \\Omega^{(0,2)}=\\dbar\\omega^{(0,1)}-\\omega^{(0,1)}\\wedge\\omega^{(0,1)}.\n\\end{equation}\n\nThe {\\it  integrability problem} studied in this paper is to find a frame   $\\hat e=Ae$ for which $\\hat\\omega^{(0,1)}=0$, which is equivalent to solving\n\\begin{equation}\\label{dbAA}\nA^{-1}\\dbar A+\\omega^{(0,1)}=0.\n\\end{equation}\nIf such frames $\\hat e_k$ exist on an open covering $\\{U_k\\}$ of $\\overline M$, then on overlap $U_j\\cap U_k$ their transition matrices satisfy    $\\dbar  A_{kj}=0$. In this case we say that $(E,\\overline M,D)$ carries a   holomorphic vector bundle structure.    By (\\ref{hatomom})-(\\ref{om02}),  the solvability of (\\ref{dbAA}) requires the {\\it formal} integrability condition\n\\begin{equation}\\label{Om02=0}\n\\dbar\\omega^{(0,1)}=\\omega^{(0,1)}\\wedge\\omega^{(0,1)},\\quad \\text{i.e.}\\quad \\Omega^{(0,2)}=0.\n\\end{equation}\nKoszul and Malgrange~~\\cite{MR131882} proved that when the boundary is not considered,   the formal integrability condition is sufficient for the integrality; see also Kobayashi~~\\cite{MR0909698}*{Prop.~3.7, p.~17} for a proof using the Newlander-Nirenberg theorem.\n\nOur main result establishes integrability and optimal boundary regularity  for the H\\\"older-Zygmund class (see Section 3 for definition) under the formal integrability condition on $M$.", "context": "\\label{sec1} \n Let $M\\subset{\\bf C}^n$ be a relatively compact domain with boundary $\\partial M\\in C^2$. Let $E$ be a smooth (i.e. $C^\\infty$) complex vector bundle over the closure $\\overline M$. Denote by $C^\\infty(\\overline M, E), C_{(k)}^\\infty(E,\\overline M)$ the spaces of smooth sections of $E$ and $E$-valued smooth $k$-forms on $\\overline M$, respectively.\nA Koszul {\\it connection} $D$ in $E$ of is a ${\\bf C}$-linear map \nfrom $\n C_{(k)}^\\infty(\\overline M,E)$ to $ C^\\infty_{(k+1)}(\\overline M,E)\n$\nsatisfying, for any $f\\in C_{(k)}^\\infty(\\overline M)$ and $s\\in C^\\infty(\\overline M,E)$, \n$$\nD(fs)=df\\otimes s +(-1)^kf\\wedge Ds.\n$$ The curvature $R$ of $D$ is defined as\n$\nR=D\\circ D\\colon C^\\infty_{(k)}(\\overline M,E)\\to C^\\infty_{(k+2)}(\\overline M,E).\n$\n Let $e=(e_1,\\dots, e_{k_0})^t$ be a smooth local frame of $E$. Then\n$$\nDe_i=\\sum \\omega^j_i\\otimes e_j, \\ De=\\omega e;\\quad D^2e_i=\\sum \\Omega^j_i\\otimes e_j, \\ D^2e=\\Omega e,\n$$\nwhere $\\omega$ and $\\Omega$ are the connection and curvature forms. \nIf $\\hat e=Ae$ is another frame, then\n\\begin{equation}\\label{hatomom}\n\\hat \\omega =(dA)A^{-1} +A\\omega A^{-1}, \\quad \\widehat\\Omega=A\\Omega A^{-1}.\n\\end{equation}\nDecomposing into types, $\\omega=\\omega^{(1,0)}+\\omega^{(0,1)}$ and\n$\\Omega=\\Omega^{(2,0)}+\\Omega^{(1,1)}+\\Omega^{(0,2)}$, yields \n\\begin{equation}\\label{om02}\n \\Omega^{(0,2)}=\\dbar\\omega^{(0,1)}-\\omega^{(0,1)}\\wedge\\omega^{(0,1)}.\n\\end{equation}\n\nThe {\\it integrability problem} studied in this paper is to find a frame $\\hat e=Ae$ for which $\\hat\\omega^{(0,1)}=0$, which is equivalent to solving\n\\begin{equation}\\label{dbAA}\nA^{-1}\\dbar A+\\omega^{(0,1)}=0.\n\\end{equation}\nIf such frames $\\hat e_k$ exist on an open covering $\\{U_k\\}$ of $\\overline M$, then on overlap $U_j\\cap U_k$ their transition matrices satisfy $\\dbar A_{kj}=0$. In this case we say that $(E,\\overline M,D)$ carries a holomorphic vector bundle structure. By (\\ref{hatomom})-(\\ref{om02}), the solvability of (\\ref{dbAA}) requires the {\\it formal} integrability condition\n\\begin{equation}\\label{Om02=0}\n\\dbar\\omega^{(0,1)}=\\omega^{(0,1)}\\wedge\\omega^{(0,1)},\\quad \\text{i.e.}\\quad \\Omega^{(0,2)}=0.\n\\end{equation}\nKoszul and Malgrange~~\\cite{MR131882} proved that when the boundary is not considered, the formal integrability condition is sufficient for the integrality; see also Kobayashi~~\\cite{MR0909698}*{Prop.~3.7, p.~17} for a proof using the Newlander-Nirenberg theorem.\n\nOur main result establishes integrability and optimal boundary regularity for the H\\\"older-Zygmund class (see Section 3 for definition) under the formal integrability condition on $M$.", "full_context": "\\label{sec1} \n Let $M\\subset{\\bf C}^n$ be a relatively compact domain with boundary $\\partial M\\in C^2$. Let $E$ be a smooth (i.e. $C^\\infty$) complex vector bundle over the closure $\\overline M$. Denote by $C^\\infty(\\overline M, E), C_{(k)}^\\infty(E,\\overline M)$ the spaces of smooth sections of $E$ and $E$-valued smooth $k$-forms on $\\overline M$, respectively.\nA Koszul {\\it connection} $D$ in $E$ of is a ${\\bf C}$-linear map \nfrom $\n C_{(k)}^\\infty(\\overline M,E)$ to $ C^\\infty_{(k+1)}(\\overline M,E)\n$\nsatisfying, for any $f\\in C_{(k)}^\\infty(\\overline M)$ and $s\\in C^\\infty(\\overline M,E)$, \n$$\nD(fs)=df\\otimes s +(-1)^kf\\wedge Ds.\n$$ The curvature $R$ of $D$ is defined as\n$\nR=D\\circ D\\colon C^\\infty_{(k)}(\\overline M,E)\\to C^\\infty_{(k+2)}(\\overline M,E).\n$\n Let $e=(e_1,\\dots, e_{k_0})^t$ be a smooth local frame of $E$. Then\n$$\nDe_i=\\sum \\omega^j_i\\otimes e_j, \\ De=\\omega e;\\quad D^2e_i=\\sum \\Omega^j_i\\otimes e_j, \\ D^2e=\\Omega e,\n$$\nwhere $\\omega$ and $\\Omega$ are the connection and curvature forms. \nIf $\\hat e=Ae$ is another frame, then\n\\begin{equation}\\label{hatomom}\n\\hat \\omega =(dA)A^{-1} +A\\omega A^{-1}, \\quad \\widehat\\Omega=A\\Omega A^{-1}.\n\\end{equation}\nDecomposing into types, $\\omega=\\omega^{(1,0)}+\\omega^{(0,1)}$ and\n$\\Omega=\\Omega^{(2,0)}+\\Omega^{(1,1)}+\\Omega^{(0,2)}$, yields \n\\begin{equation}\\label{om02}\n \\Omega^{(0,2)}=\\dbar\\omega^{(0,1)}-\\omega^{(0,1)}\\wedge\\omega^{(0,1)}.\n\\end{equation}\n\nThe {\\it integrability problem} studied in this paper is to find a frame $\\hat e=Ae$ for which $\\hat\\omega^{(0,1)}=0$, which is equivalent to solving\n\\begin{equation}\\label{dbAA}\nA^{-1}\\dbar A+\\omega^{(0,1)}=0.\n\\end{equation}\nIf such frames $\\hat e_k$ exist on an open covering $\\{U_k\\}$ of $\\overline M$, then on overlap $U_j\\cap U_k$ their transition matrices satisfy $\\dbar A_{kj}=0$. In this case we say that $(E,\\overline M,D)$ carries a holomorphic vector bundle structure. By (\\ref{hatomom})-(\\ref{om02}), the solvability of (\\ref{dbAA}) requires the {\\it formal} integrability condition\n\\begin{equation}\\label{Om02=0}\n\\dbar\\omega^{(0,1)}=\\omega^{(0,1)}\\wedge\\omega^{(0,1)},\\quad \\text{i.e.}\\quad \\Omega^{(0,2)}=0.\n\\end{equation}\nKoszul and Malgrange~~\\cite{MR131882} proved that when the boundary is not considered, the formal integrability condition is sufficient for the integrality; see also Kobayashi~~\\cite{MR0909698}*{Prop.~3.7, p.~17} for a proof using the Newlander-Nirenberg theorem.\n\nOur main result establishes integrability and optimal boundary regularity for the H\\\"older-Zygmund class (see Section 3 for definition) under the formal integrability condition on $M$.\n\n\\begin{thm}[convex configuration estimate]\\label{conv-est} Let $r>0$ and $1\\leq q\\leq n-1$.\nLet $(D^1,D_{r_2}^2)$ be an $(n-q)$-convex configuration. Let $D^{12}_r=D^1\\cap D^2_r$.\nThe homotopy operators $H_q, H_{q+1}$ in \\rta{hf-c} satisfies\n\\aln{}\\|H_s\\var\\|_{\\Lambda^{r+1/2}(D^{1 2}_{(1-\\theta)r_2})}&\\leq \\f{C_r( \\nabla\\rho^1,\\nabla^2\\rho^1)}{\\theta^{3r+2r_0}}\\|\\var\\|_{\\Lambda^r(D^{1 2}_{r_2})},\\quad \\forall\\theta\\in(0,1). \n\\end{align*}\nMoreover, $C_r(\\nabla\\rho^1,\\nabla^2\\rho^1)$ is stable under small $C^2$ perturbations of $\\rho^1$.\n\\end{thm}\n\\begin{proof}Recall from \\re{hq1}-\\re{nhq2-} that $H_s =H^{(1)}_s + H_s^{(2)}$ with \n\\aln{} \n H^{(1)}_s f&:=R_{ D^{2}; s-1}^0 \\cL E_{D^{12}} f+R_{U^1;s-1 }^{01}[\\db,\\cL E_{D^{12}}] f,\\\\\nH^{(2)}_sf&=L_{1^+;{s-1}\n}^{01} \\cL E_{D^{12}} f +L_{2;s-1}^{02} f+L_{12;s-1}^{012}f.\n\\end{align*}\nTake a cut-off function $\\chi\\in C^\\infty_0(D^{2}_{(1-\\theta/3)r_2})$ with $\\chi=1$ on $D^{2}_{(1-\\theta/2)r_2}$, and decompose\n$\n H^{(1)}_s f= H^{(1)}_s(\\chi f)+ H^{(1)}_s( (1-\\chi)f).\n$\nThen the estimate in~\\cite{gong-shi-nn} says that \n$$\n\\|H_s^{(1)}(\\chi \\var)\\|_{\\Lambda^{r+1/2}(D^{1 2}_{r_2})}\\leq C_r \\|\\chi\\var\\|_{\\Lambda^r(D^{1 2}_{r_2})}.\n$$\nWe can choose $\\chi$ such that $\\|\\chi\\|_{C^r}\\leq {C_r}{\\theta^{-r-1}}$. Thus, \n$$\n\\|\\chi\\var\\|_{\\Lambda^r(D^{12}_{r_2})}\\leq {C_r }{\\theta^{-r-r_0}}\\|\\var\\|_{\\Lambda^r(D^{12}_{r_2})}.\n$$\nSince $(1-\\chi)\\var$ vanishes on $D_{(1-\\theta/2)r}^{12}$ and\n\\ga{}\\label{giest}\n|g^i(z,\\zeta)\\cdot(\\zeta-z)|\\geq |\\zeta-z|^2/C,\\quad z\\in D^{1}\\cap D^2,\\zeta\\in D^2\\setminus D^1, i=0,1,2,\n\\end{gather}\nthen we can obtain $$\n\\|H_s^{(1)}((1-\\chi) \\var)\\|_{\\Lambda^{r+1/2}(D^{12}_{(1-\\theta)r_2})}\\leq C_r\\theta^{-2r-r_0} \\|(1-\\chi)\\var\\|_{\\Lambda^r(D^{12}_{r_2})}.\n$$\nThe $H_s^{(2)}$ is given by boundary integrals on $S^1_+, S^2$ and $ S^{12}$ respectively. The estimate \\re{giest} also yields \n$\\|H_s^{(2)}\\var\\|_{\\Lambda^{r+1/2}(D^{12}_{(1-\\theta)r_2})}\\leq C_r \\theta^{-2r-r_0} \\|\\var\\|_{\\Lambda^r(D^{12}_{r_2})}.\n$\nWe have obtained the desired estimate of $H_s$. \n\\end{proof}\nWe now derive estimates for the concave case. We still have $H_s=H_s^{(1)}+H_s^{(2)}$, where $H_s^{(1)}$ has the same form as in the convex case. However, \n$$\nH_{s}^{(2)}f=\\sum_{i=1}^2L_{i3,s}^{0i3}f- L^{01}_{1^+,s-1}\\cL Ef +L_{12;s-1}^{123}f-\\hat T_{ D^{23}_{(1-\\theta)r}, s}L^{23}_{12;s} f.\n$$\nBy \\re{g3theta}, we have $\n\\RE\\{g^3(z,\\zeta)\\cdot(\\zeta-z)\\} \\geq\\theta r_3^2$ for $z\\in D^{23}_{(1-\\theta)r}$ and $\\zeta\\in S^{13}\\cup S^{23}\\cup S^{12}\\subset\\pd D^{123}_r$. Some kernels in $H^{(2)}_s$ involve first-order $z$-derivatives of $\\nabla\\rho^1(z)$.\nAs in the convex case, we can obtain \n$$\\|H^{(2)}_sf\\|_{\\Lambda^{r+1/2}(D^{123}_{(1-\\theta)r})}\\leq {C_r(\\nabla\\rho^1 )}{\\theta^{-r-r_0}}\\|\\rho^1\\|_{\\Lambda^{3/2}}\\|f\\|_{\\Lambda^\\e}.\n$$The kernels of $H^{(1)}f$ involve the second-order $z$-derivatives $ \\nabla\\rho^2(z)$.\nThen by ~\\cite{gong-shi-nn}*{Thm. 5.12}, we have \n$$\\|H^{(1)}_sf\n\\|_{\\Lambda^{r+\\del}(D^{123}_{(1-\\theta)r})}\\leq {C_r }{\\theta^{-r-r_0}}\\left(\\|\\rho^1\\|_{\\Lambda^{r+2+\\del}}\\|f\\|_{\\Lambda^\\e}+ \\|f\\|_{\\Lambda^r}\\right).\n$$Therefore, we have obtained part $(a)$ below.\n\\begin{thm}[concave configuration estimate]\\label{concave-est} Let $\\e>0,r>0$ and $1\\leq q\\leq n-3$. \nAssume that $\\rho^1\\in C^{2+\\e}$. The following hold.\n\\bpp\n\\item The operators $H_q, H_{q+1}$ in Theorem~$\\ref{cchf}$ satisfy\n\\al{}\n\\label{hqfr12}\n\\|H_sf\\|_{\\Lambda^{r+\\del}(D^{123}_{(1-\\theta)r})}&\\leq \\f{C_{r,\\e}}{\\theta^{3r+r_0}} (\\|\\rho^1\\|_{\\Lambda^{r+2+\\del}}\\|f\\|_{\\Lambda^\\epsilon({D_{r}^{123}})}+ \\|f\\|_{\\Lambda^r(D^{123}_{r})})\n\\end{align}\nfor $0\\leq\\delta\\leq1/2$ and $0<\\theta<1$. Furthermore, $C_{r,\\e}=C_{r,\\e}(\\nabla\\rho^1,\\nabla^2\\rho^1)$ is stable under small $C^2$ perturbations of $\\rho^1$;\n the same estimate holds for the $\\db$ solution operator $H_q$ in \\rta{cchf-closed}.\n \\item \nThe homotopy formula \\rea{tsqf+-cv} in Theorem~$\\ref{cchf}$ holds for $f\\in \\Lambda^{\\e}_{(0,q)}(D^{123})$ when $\\db f\\in \\Lambda^{\\e}_{(0,q)}(D^{123})$, and the $\\db$-solution operator $H_q$ in \\rta{cchf-closed} is valid for all $\\db$-closed $f\\in\\Lambda^{\\e}_{(0,q)}(D^{123})$ and satisfies \\rea{hqfr12}.\n\\epp\n\\end{thm}\n\\begin{proof}We verify $(b)$ using $(a)$. We know that $f=\\db H_qf+H_{q+1}\\db f$ when $f\\in C^{1+\\e}$. Let us write $H_q$ as $H_{D^{123}_r}$ to indicate the dependent on the domain. Now assume $f\\in \\Lambda^r_{(0,q)}(D^{123})$ with $\\db f\\in\\Lambda^r$. We find a sequence $f_j\\in C^{\\infty}_{(0,q)}(D^{123})$ such that $f_j $ and $\\db f_j $ converge to $f,\\db f$ in $\\Lambda^{r'}(D^{123})$ as $j\\to\\infty$ for any $r'<r$. Furthermore, $$\\|f_j\\|_{\\Lambda^{r}(D^{123})}\\leq C_r\\|f\\|_{\\Lambda^{r}(D^{123})}, \\quad \\|\\db f_j\\|_{\\Lambda^{r}(D^{123})}\\leq C_r\\|\\db f\\|_{\\Lambda^{r}(D^{123})}.$$\nThen $H_qf=\\lim H_qf_j$ and $H_{q+1}\\db f=\\lim H_{q+1}\\db f_j$ gives us \\rea{tsqf+-cv}.\nAssume now $f$ is $\\db$-closed. Set $L_t(z)=z+tie_{q+2}$ with $t>0$, where $e_{q+2}$ is the $(q+2)$-th unit vector of $\\cc^n$. Fix $\\theta\\in(0,1/2)$. When $0<t<t_\\theta$, we have $L_t(D^{123}_{(1-\\theta)r})\\subset D^{123}_r$. Using a mollifier, we find $\\db$-closed $f_j\\in C^\\infty(D^{123}_{(1-\\theta)r})$ such that $f_j$ converges to $f$ in $\\Lambda^{r'}(D^{123}_{(1-\\theta)r})$-norm for any $r'<r$, whereas $\\|f_j\\|_{\\Lambda^{r}(D^{123}_{(1-\\theta)r})}\\leq C_r\\|f\\|_{\\Lambda^{r}(D^{123}_{r})}$. We have $f_j=\\db H_{D^{123}_{(1-\\theta')r},q}f_j$ on $D^{123}_{(1-\\theta)r}$ for $0<\\theta'<\\theta/2$.\nLetting $j\\to\\infty$, we obtain $f=\\db H_{D^{123}_{(1-\\theta')r},q}f$ on $D^{123}_{(1-\\theta)r}$ and\n\\eq{Hth''}\n\\|H_{D^{123}_{(1-\\theta')r},q}f\\|_{\\Lambda^{r+\\del}(D^{123}_{(1-\\theta)r})}\\leq \\f{C_{r,\\e}}{\\theta^{3r+r_0}} (\\|\\rho^1\\|_{\\Lambda^{r+2+\\del}}\\|f\\|_{\\Lambda^\\epsilon }+ \\|f\\|_{\\Lambda^r }).\n\\eeq\nBy the Arzel\\`{a}-Ascoli theorem, $H_{D^{123}_{(1-\\theta'_j)r},q}f$ converges to $H_{D^{123}_{r},q}f$ in $\\Lambda^{r'+\\delta}(D^{123}_{(1-\\theta)r})$ norm for a sequence $\\theta'_j\\to0$ for any $r'<r$. Then \\re{Hth''} yield\n\\re{hqfr12} for $H_q$.\n\\end{proof}\n\n\\setcounter{thm}{0}\\setcounter{equation}{0}\n\\section{Integrability of complex vector bundles}\\label{sec:NM}\nIn Section 2, we have seen that \\rt{vbint-} follows from the following result.\n\\begin{thm}\\label{vbint} \nLet $M$ be a domain in $\\mathbb C^n$ with $\\pd M\\in C^2$. Assume that the Levi-form of $M$ has\neither $(n-1)$ positive or at least $3$ negative Levi eigenvalues at $\\zeta_0\\in\\pd M$. Let $U$ be an open set in $\\ov M$ and $\\zeta_0\\in U$. Assume that $\\om$ is a $k_0\\times k_0$ matrix of $(0,1)$ forms on $U$ such that $\\db\\om=\\om\\wedge\\om$ holds on $U\\setminus\\pd M$ in the sense of distributions. Assume that $\\om$ is in $L^\\infty(U)$ $($resp. $\\Lambda^r(U)$ with $r>0)$. Then there is an invertible matrix $A$ of class $\\Lambda^{1/2}$ $($resp. $\\Lambda^{r+1/2})$ on some neighborhood $U'$ of $\\zeta_0$ in $\\ov M$ such that \n$\n\\db A+A\\om=0\n$\nin the sense of distributions on $U'\\setminus\\pd M$.\n\\end{thm} \n\\begin{proof}The proof is based on a KAM-type iteration method. Such an approach was developed by Webster~\\ci{MR999729} for an interior version of Newlander-Nirenberg theorem. It was also used in~\\cites{MR2742034,MR2829316} to establish the integrability of CR vector bundles on strictly pseudoconvex hypersurfaces in $\\cc^n$ for $n\\geq4$.", "post_theorem_intro_text_len": 5320, "post_theorem_intro_text": "Our theorem concerns the integrability of $(E,D)$ in the H\\\"older-Zygmund spaces $\\Lambda^r$, which is the standard H\\\"older spaces $C^r$ when $r$ is not an integer; see Section 5 for definition. The theorem also treats  a Koszul connection $D$ that maps the space $ Lip(M,E)$ of Lipschitz sections  \n into  $ L^\\infty_{(1)}( M,E)$, the space  of $E$-valued $1$-forms of class $L^\\infty$ on $M$. Such a connection extends trivially to a Koszul connection on $E$ over $\\overline M$.   The theory of $L^\\infty$ connections, which is of independent interest, is required  even when the original connections are $C^\\infty$ for the (strictly) $3$-concave case. Theorem~\\ref{vbint-}  is reduced to Theorem~\\ref{vbint}, and the reduction requires a careful discussion in Section~\\ref{sec:int-formal} of   $L^\\infty$ connections on Lipschitz bundles  and of  {\\it admissible} frame changes for which the transformation law (\\ref{hatomom}) remains valid; see Proposition~\\ref{d2=0}.\n\n\\medskip\n\nWe now outline our approach. Equation (\\ref{dbAA}) may be viewed as a nonlinear $\\dbar$-equation, and the integrability condition (\\ref{Om02=0}) is also nonlinear. \n We   apply homotopy formulas to the study of (\\ref{dbAA}). \nThe use of integral representations to establish regularity of $\\overline\\partial$-solutions on strongly pseudoconvex domains in ${\\bf C}^n$ has a long history. Sup-norm estimate for $\\dbar$-solutions to $(0,1)$-forms was proved by  Grauert and Lieb~\\cite{MR273057} and  Henkin~\\cite{MR0249660}.   Kerzman~\\cite{MR0281944} obtained  $L^p$ and $C^{\\beta}$ estimates of $\\dbar$-solutions for $(0,1)$-forms and all $\\beta<1/2$.\n  Lieb~~\\cite{MR283235} obtained the $L^\\infty$ and the $C^\\beta$ estimates of $\\dbar$-solutions for $(0,q)$-forms.  \nHenkin and Romanov~\\cite{MR0293121}  proved the sharp $C^{1/2}$ estimate of \n$\\dbar$-solutions for continuous $(0,1)$-forms. \n\nWe recall several important developments on integral representation formulas for  $\\overline\\partial$   that lead to derivative estimates on strongly pseudoconvex domains $D\\subset{\\bf C}^n$. When $\\partial D$ is of class $ C^{k+2}$,   Siu~\\cite{MR330515} showed that the Henkin solution operator satisfies a $C^{k+1/2}$ estimate  for  $\\dbar$-closed $(0,1)$ forms of class $C^k$, and for  $\\dbar$-closed $(0,q)$-forms   with $q\\geq1$,  Lieb and Range~\\cite{MR597825} constructed a new  $\\dbar$ solution operator   and established  the same estimate. When $\\partial D$ is only $ C^2$,  a homotopy formula was constructed  in~\\cite{MR3961327} using the Stein extension, producing   homotopy operators with $\\Lambda^{r+1/2}$ estimate for $r>1$.    Shi and Yao~\\cites{MR4688544,MR4861589} later developed   a homotopy formula   employing the Rychkov extension and obtained $\\Lambda^{r+1/2}$ estimates for all $r>0$, as well as $H^{s+1/2,p}$ estimates  for $1<p<\\infty$ and $s>1/p$ when $\\partial D\\in C^2$; they also showed that the  estimates hold  for all $s\\in{\\bf R}$ when $\\partial D$ is sufficiently smooth.  Very recently, Yao~~\\cite{yaoc2} further reduced the boundary smoothness assumption to $ C^2$ for all $s\\in{\\bf R}$. We also refer the reader to references cited in~\\cites{MR3961327,MR4688544,MR4861589,yaoc2}.\n\nWe also note that   Range and Siu~~\\cite{MR338450} proved the $C^\\beta$ estimate for all $\\beta<1/2$ for $\\dbar$-solutions for continuous $(0,q)$-forms on the (real) transversal intersection of\n  strictly pseudoconvex domains. Whether the optimal gain of $1/2$  derivative   estimates  holds for $\\dbar$-solutions on such transversal intersections remains an open problem.\n  Higher-order derivative estimates in this setting were obtained by Brinkmann~~\\cite{Br84}, Michel~~\\cite{MR928297}, and Michel-Perotti~~\\cite{MR1038709}.\n Peters~~\\cite{MR1135535}  constructed a homotopy formula for the weakly transversal intersection of strictly pseudoconvex domains and established higher-order estimates, though with a loss of derivatives.\n It would be interesting to determine whether the results mentioned above can be applied to the integrability problem for intersections of strictly pseudoconvex domains. We also remark that the integrability of almost CR vector bundles on strongly pseudoconvex hypersurfaces in $\\mathbb C^n$ with $n\\geq4$ was proved by Webster~~\\cite{MR1128608}, and sharper regularity results in the finitely smooth case were obtained by Gong-Webster~\\cites{MR2742034,MR2829316}.\nIt remains an open question whether Theorem~\\ref{vbint-} holds if $\\partial M$ has two negative Levi eigenvalues. This appears to be related to the unsettled embedding problem for strongly pseudoconvex local CR structures in ${\\bf R}^5$, investigated by Webster~\\cite{MR0995504} and Gong-Webster~~\\cite{MR2868966}.\n\nWe organize the paper as follows. In Section 2, we examine the formal integrability condition in detail for connections on Lipschitz complex vector bundles and introduce the notion of admissible frame changes. Sections 3 and 4 reformulate the local homotopy formulas in~\\cites{MR986248,MR4866351} \n for shrinking domains. In Section 5, we recall  estimates established in \\cites{MR986248,gong-shi-nn} and derive  new estimates on shrinking domains. Finally, in Section 6, we complete the proof of Theorem~\\ref{vbint-}  by establishing Theorem~\\ref{vbint}.\n\n\\setcounter{thm}{0}\\setcounter{equation}{0}", "sketch": "Theorem~\\ref{vbint-} is reduced to Theorem~\\ref{vbint}, and the reduction requires a careful discussion in Section~\\ref{sec:int-formal} of $L^\\infty$ connections on Lipschitz bundles and of \\textit{admissible} frame changes for which the transformation law (\\ref{hatomom}) remains valid; see Proposition~\\ref{d2=0}.\n\n“We now outline our approach. Equation (\\ref{dbAA}) may be viewed as a nonlinear $\\dbar$-equation, and the integrability condition (\\ref{Om02=0}) is also nonlinear. We apply homotopy formulas to the study of (\\ref{dbAA}).”\n\nThe proof is then completed as follows: Sections 3 and 4 “reformulate the local homotopy formulas … for shrinking domains,” Section 5 recalls and derives “estimates … on shrinking domains,” and “Finally, in Section 6, we complete the proof of Theorem~\\ref{vbint-} by establishing Theorem~\\ref{vbint}.”", "expanded_sketch": "To prove the main theorem, we reduce it to the following theorem.\n\n\\begin{thm}\\label{vbint} \nLet $M$ be a domain in $\\mathbb C^n$ with $\\pd M\\in C^2$. Assume that    the Levi-form of $M$ has\neither $(n-1)$ positive or at least $3$ negative Levi eigenvalues at $\\zeta_0\\in\\pd M$. Let $U$ be an open set in $\\ov M$ and $\\zeta_0\\in U$.    Assume that $\\om$ is a $k_0\\times k_0$ matrix of $(0,1)$ forms on $U$ such that $\\db\\om=\\om\\wedge\\om$ holds on $U\\setminus\\pd M$ in the sense of distributions. Assume that $\\om$ is in $L^\\infty(U)$ $($resp. $\\Lambda^r(U)$ with $r>0)$. Then there is an invertible matrix $A$   of class $\\Lambda^{1/2}$ $($resp. $\\Lambda^{r+1/2})$ on some neighborhood $U'$ of $\\zeta_0$ in $\\ov M$ such that \n$\n\\db A+A\\om=0\n$\nin the sense of distributions on $U'\\setminus\\pd M$.\n\\end{thm}\n\nThe reduction requires a careful discussion of $L^\\infty$ connections on Lipschitz bundles and of \\textit{admissible} frame changes for which the transformation law (\\ref{hatomom}) remains valid; for this we use the following proposition.\n\n\\begin{prop}\\label{d2=0}\nLet $D$ be a connection for a complex vector bundle $E$. Let $e=(e_1,\\dots, e_{k_0})^t$ be a local frame field of $E$ over $U$. Assume that $De=\\om \\otimes e$ and $\\om\\in L^2_{loc}(U)$ and $d\\om\\in L^1_{loc}(U)$. If $A$ is admissible, then $\\hat\\om=(dA)A^{-1}+A\\om A^{-1}$ is still in $L^2_{loc}(U)$ and  the curvature  forms    satisfy $A\\Om A^{-1}=\\hat\\Om$. \n\\end{prop}\n\n“We now outline our approach. Equation (\\ref{dbAA}) may be viewed as a nonlinear $\\dbar$-equation, and the integrability condition (\\ref{Om02=0}) is also nonlinear. We apply homotopy formulas to the study of (\\ref{dbAA}).”\n\nThe proof is then completed as follows: Sections 3 and 4 “reformulate the local homotopy formulas … for shrinking domains,” Section 5 recalls and derives “estimates … on shrinking domains,” and finally we complete the proof of the main theorem by establishing the theorem above.", "expanded_theorem": "\\label{vbint-}\nLet $0<r\\leq\\infty$. \nLet $M$ be a domain in ${\\bf C}^n$ with $\\partial M\\in C^2$. Assume that   the Levi form of $\\partial M$ on $T^{(1,0)}\\partial M$ has either $n-1$ positive or at least three negative eigenvalues at  $\\zeta_0\\in\\partial M$.  \nLet $E$ be a complex vector bundle of class $Lip$ $($resp. $\\Lambda^{r+1})$ over $\\overline M$. Assume that a connection  $D\\colon Lip(M,E)\\to L_{(1)}^\\infty(M,E)$  $($resp. $\\Lambda^{r+1}(M,E)\\to \\Lambda^{r}_{(1)}(M,E))$ satisfies $\\Omega^{(0,2)}=0$ on $ M$ in the sense of distributions. Then $E|_U$ has a    holomorphic vector bundle structure, where $U$ is some open subset of $\\overline M$ and $\\zeta_0\\in U$. Furthermore, the holomorphic structure is of class $C^{1/2}$ $($resp. $\\Lambda^{r+1/2})$ with respect to the original $Lip$ $($resp. $\\Lambda^{r+1})$ structure of $E|_U$.", "theorem_type": ["Existence", "Implication"], "mcq": {"question": "Let $0<r\\le \\infty$. Let $M\\subset \\mathbb C^n$ be a domain with $\\partial M\\in C^2$, and let $\\zeta_0\\in \\partial M$. Assume that the Levi form of $\\partial M$ on $T^{(1,0)}\\partial M$ has either $n-1$ positive eigenvalues or at least three negative eigenvalues at $\\zeta_0$. Let $E$ be a complex vector bundle over $\\overline M$ of class $Lip$ (respectively of Hölder-Zygmund class $\\Lambda^{r+1}$). Suppose $D$ is a connection on $E$ of the corresponding regularity, namely $D\\colon Lip(M,E)\\to L^{\\infty}_{(1)}(M,E)$ (respectively $D\\colon \\Lambda^{r+1}(M,E)\\to \\Lambda^r_{(1)}(M,E)$), and that its curvature has vanishing $(0,2)$-part on $M$ in the sense of distributions. In a local frame $e$, if $De=\\omega e$ and $D^2e=\\Omega e$, this means $\\Omega^{(0,2)}=0$, equivalently $\\bar\\partial\\omega^{(0,1)}=\\omega^{(0,1)}\\wedge \\omega^{(0,1)}$. A holomorphic vector bundle structure on $E|_U$ means local frames on an open cover of $U$ whose transition matrices are holomorphic (equivalently, local frame changes can be made so that the $(0,1)$-part of the connection vanishes). Under these assumptions, which existence statement holds?", "correct_choice": {"label": "A", "text": "There exists an open subset $U\\subset \\overline M$ with $\\zeta_0\\in U$ such that $E|_U$ admits a holomorphic vector bundle structure. Moreover, relative to the original $Lip$ structure on $E|_U$ this holomorphic structure is of class $C^{1/2}$, and relative to the original $\\Lambda^{r+1}$ structure on $E|_U$ it is of class $\\Lambda^{r+1/2}$."}, "choices": [{"label": "B", "text": "There exists an open subset $U\\subset \\overline M$ with $\\zeta_0\\in U$ such that $E|_U$ admits a holomorphic vector bundle structure. Moreover, relative to the original $Lip$ structure on $E|_U$ this holomorphic structure is of class $C^{1}$, and relative to the original $\\Lambda^{r+1}$ structure on $E|_U$ it is of class $\\Lambda^{r+1}$."}, {"label": "C", "text": "There exists an open subset $U\\subset \\overline M$ with $\\zeta_0\\in U$ such that $E|_U$ admits a holomorphic vector bundle structure."}, {"label": "D", "text": "There exists an open subset $U\\subset \\overline M$ with $\\zeta_0\\in U$ such that $E|_U$ admits a holomorphic vector bundle structure, provided the identity $\\bar\\partial\\omega^{(0,1)}=\\omega^{(0,1)}\\wedge\\omega^{(0,1)}$ holds on all of $U$, including across $\\partial M$, in the sense of distributions."}, {"label": "E", "text": "For every neighborhood $U$ of $\\zeta_0$ in $\\overline M$, the restriction $E|_U$ admits a holomorphic vector bundle structure. Moreover, one can choose a local frame change $A$ of the same regularity as the original bundle structure, namely $A\\in Lip(U)$ in the $Lip$ case and $A\\in \\Lambda^{r+1}(U)$ in the $\\Lambda^{r+1}$ case, such that $\\bar\\partial A+A\\omega^{(0,1)}=0$ on $U\\setminus\\partial M$."}], "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": "half-derivative gain only", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "boundary regularity conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "equation required only on interior/distribution on $U\\setminus\\partial M$", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "local existence on some neighborhood and reduced regularity of frame change", "template_used": "quantifier_dependence"}]}}
{"id": "2512.10845v1", "paper_link": "http://arxiv.org/abs/2512.10845v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{Thm for E}\n    If $E$ is uniformly weakly RC-positive over a compact K\\\"ahler manifold $X$, then $S^kE\\otimes \\det E$ is uniformly RC-positive for any $k\\geq 0$, and $S^kE$ is uniformly RC-positive for $k$ large.", "start_pos": 62389, "end_pos": 62637, "label": "Thm for E"}, "ref_dict": {"Thm for E": "\\begin{theorem}\\label{Thm for E}\n    If $E$ is uniformly weakly RC-positive over a compact K\\\"ahler manifold $X$, then $S^kE\\otimes \\det E$ is uniformly RC-positive for any $k\\geq 0$, and $S^kE$ is uniformly RC-positive for $k$ large.\n\\end{theorem}", "lemma abc": "\\begin{lemma}\\label{lemma abc}\n  Assume the curvature $\\Theta$ of $h$ is positive on every fiber $\\mathcal{X}_t$. For $1\\leq k\\leq m$, the following are equivalent. \n\\begin{enumerate}[label=\\Alph*.]\n    \\item\\label{A} The curvature $\\Theta$ has at least $n+k$ positive eigenvalues at every point in $\\mathcal{X}$.\n    \\item\\label{B} The horizontal component $\\Theta_\\mathcal{H}$ has at least $k$ positive eigenvalues at every point in $\\mathcal{X}$.\n    \\item\\label{C}  There exists a positive $\\beta\\in C^{\\infty}(\\mathcal{X}, p^*(\\wedge^{1,1}T^*Y))$ such that the sum of any $m-k+1$ eigenvalues of $\\Theta_\\mathcal{H}$ with respect to $\\beta$ is positive.\n\\end{enumerate}\nMoreover, when $k=1$, statement C can be rephrased as $\\Theta_\\mathcal{H}\\wedge \\beta^{m-1}>0$ or equivalently $\\Theta^{n+1}\\wedge \\beta^{m-1}>0$.\n\\end{lemma}", "Thm 1": "\\begin{theorem}\\label{Thm 1}\n    If the curvature $\\Theta$ of $h$ is positive on every fiber, and for any point $t\\in Y$, there exists a nonzero tangent vector $v\\in T_tY$, such that $\\Theta(\\Tilde{v},\\bar{\\Tilde{v}})|_{(t,z)}>0$ for any lift $\\Tilde{v}$ of $v$ to $T_{(t,z)}\\mathcal{X}$, then the Hermitian bundle $(V,H)$ is uniformly RC-positive.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 4709, "pre_theorem_intro_text": "In \\cite{YangCamb}, Yang introduces the notion of RC-positivity as a differential geometric counterpart of rational connectedness.  RC-positivity plays a crucial role in Yang's proof of a conjecture of Yau: If a compact K\\\"ahler manifold has positive holomorphic sectional curvature, then the manifold is projective and rationally connected. A stronger notion called uniform RC-positivity is introduced by Yang in \\cite{YangForum} which also can be used to prove the same conjecture of  Yau. For the semipositive case of Yau's conjecture, see \\cite{HeierWong,MatsumuraPAMQ,MatsumuraAJM}.\n\nLet us recall the definition of RC-positivity and uniform RC-positivity. Let $E$ be a holomorphic vector bundle of rank $r$ over a compact complex manifold $X$ of dimension $n$. Given a Hermitian metric $H$ on $E$, we denote the Chern curvature of $H$ by $\\Theta^H$, which is an $\\text{End}E$-valued $(1,1)$-form. We denote by $TX$ the holomorphic tangent bundle of $X$. For a vector $u\\in E_t$ and a tangent vector $v\\in T_tX$ with $t\\in X$, we define the expression $$H(\\Theta^H u,u)(v,\\bar{v})$$ to be  $\\sum_{j,k} H(\\Theta^H_{j\\bar{k}}u,u)v_j \\bar{v}_k$ locally where we write the curvature $\\Theta^H=\\sum_{j,k}\\Theta^H_{j\\bar{k}}dt_j\\wedge d\\bar{t}_k$ and  $v=\\sum_j v_j\\partial/\\partial t_j$. \n\n\\begin{definition} \n\nA Hermitian metric $H$ on a holomorphic vector bundle $E\\to X$ is called RC-positive if for any $t\\in X$ and any nonzero $u\\in E_t$, there is a nonzero tangent vector $v\\in T_tX$ such that $H(\\Theta^Hu,u)(v,\\bar v)>0$. On the other hand, a Hermitian metric $H$ is called uniformly RC-positive if for any $t\\in X$, there is a nonzero tangent vector $v\\in T_tX$ such that for any nonzero $u\\in E_t$, we have $H(\\Theta^H u,u)(v,\\bar v)>0$.\n\nA holomorphic vector bundle $E\\to X$ is called (uniformly) RC-positive if it admits a (uniformly) RC-positive Hermitian metric.\n\\end{definition}\nIt is clear that uniform RC-positivity implies RC-positivity. To motivate the definition of (uniform) weak RC-positivity, let us consider a Hermitian metric $H$ on $E$ and the induced metric $h$ on the line bundle $O_{P(E^*)}(1)$ over the projectivized bundle $P(E^*)$. By a standard computation (for example, see \\cite[Formula (4.5)]{YangCamb}), we know that if $(E,H)$ is RC-positive, then the curvature $\\Theta$ of $h$ is positive on every fiber and has at least $r$ positive eigenvalues at every point in $P(E^*)$. The existence of such a metric $h$ on $O_{P(E^*)}(1)$ is called weak RC-positivity of $E$  (\\cite[Definition 3.3]{YangCamb}).\n\nSimilarly, using the same computation, we see that if $(E,H)$ is uniformly RC-positive, then the curvature $\\Theta$ of the induced metric $h$ satisfies\n\\begin{enumerate}[label=\\alph*.]\n    \\item\\label{a} $\\Theta$ is positive on every fiber. \n    \\item $\\Theta$ has at least $r$ positive eigenvalues at every point in $P(E^*)$.\n    \\item For any point $t\\in X$, there exists a nonzero tangent vector $v\\in T_tX$, such that $\\Theta(\\Tilde{v},\\bar{\\Tilde{v}})|_{(t,[\\zeta])}>0$ for any lift $\\Tilde{v}$ of $v$ to $T_{(t,[\\zeta])}P(E^*)$.\n\\end{enumerate}\nFollowing Yang, we call the existence of such a metric $h$ on $O_{P(E^*)}(1)$ uniform weak RC-positivity of $E$. Note that in the third condition, we consider the lifts to the tangent space $T_{(t,[\\zeta])}P(E^*)$ for any point $[\\zeta]$ in the fiber $P(E_t^*)$ not just one point $[\\zeta]$. The second condition is implied by the first and the third, so we will omit it later on. Let us summarize the definition.\n\\begin{definition} The bundle $E$ is called weakly RC-positive if there exists a metric $h$ on $O_{P(E^*)}(1)$ with properties a and b. The bundle $E$ is called uniformly weakly RC-positive if there exists a metric $h$ on $O_{P(E^*)}(1)$ with properties a and c.\n\\end{definition}\nIn Yang's solution to Yau's conjecture, two main theorems \\cite[Theroem 1.3 and Theorem 1.4]{YangCamb}, although formulated in terms of RC-positivity,  hold under weak RC-positivity. So, it is natural to ask if weak RC-positivity of $E$ implies RC-positivity of $E$ (\\cite[Question 7.11]{YangCamb} and \\cite[Problem 13]{Inayama}). This question has the same flavor as a conjecture of Griffiths \\cite{Griff69}: If $E$ is ample, then $E$ is Griffiths positive. For the developments on the Griffiths conjecture, see \\cite{Umemura,CampanaFlenner,Berndtsson09,MourouganeTaka,positivityandvanishingthmliu,liu2014curvatures,FengLiuWan,demailly2020hermitianyangmills,pingali2021note,finski2020monge,Finskichara,wu_2022,wupositivelyII,wuIII,Mazhang,lempert2024two,murakami2025analytic,wu2025mean}.\n\nIn this paper, we make some progress in this direction. In particular, we prove the following theorem regarding uniform RC-positivity.", "context": "In \\cite{YangCamb}, Yang introduces the notion of RC-positivity as a differential geometric counterpart of rational connectedness. RC-positivity plays a crucial role in Yang's proof of a conjecture of Yau: If a compact K\\\"ahler manifold has positive holomorphic sectional curvature, then the manifold is projective and rationally connected. A stronger notion called uniform RC-positivity is introduced by Yang in \\cite{YangForum} which also can be used to prove the same conjecture of Yau. For the semipositive case of Yau's conjecture, see \\cite{HeierWong,MatsumuraPAMQ,MatsumuraAJM}.\n\nLet us recall the definition of RC-positivity and uniform RC-positivity. Let $E$ be a holomorphic vector bundle of rank $r$ over a compact complex manifold $X$ of dimension $n$. Given a Hermitian metric $H$ on $E$, we denote the Chern curvature of $H$ by $\\Theta^H$, which is an $\\text{End}E$-valued $(1,1)$-form. We denote by $TX$ the holomorphic tangent bundle of $X$. For a vector $u\\in E_t$ and a tangent vector $v\\in T_tX$ with $t\\in X$, we define the expression $$H(\\Theta^H u,u)(v,\\bar{v})$$ to be $\\sum_{j,k} H(\\Theta^H_{j\\bar{k}}u,u)v_j \\bar{v}_k$ locally where we write the curvature $\\Theta^H=\\sum_{j,k}\\Theta^H_{j\\bar{k}}dt_j\\wedge d\\bar{t}_k$ and $v=\\sum_j v_j\\partial/\\partial t_j$.\n\nA Hermitian metric $H$ on a holomorphic vector bundle $E\\to X$ is called RC-positive if for any $t\\in X$ and any nonzero $u\\in E_t$, there is a nonzero tangent vector $v\\in T_tX$ such that $H(\\Theta^Hu,u)(v,\\bar v)>0$. On the other hand, a Hermitian metric $H$ is called uniformly RC-positive if for any $t\\in X$, there is a nonzero tangent vector $v\\in T_tX$ such that for any nonzero $u\\in E_t$, we have $H(\\Theta^H u,u)(v,\\bar v)>0$.\n\nA holomorphic vector bundle $E\\to X$ is called (uniformly) RC-positive if it admits a (uniformly) RC-positive Hermitian metric.\n\\end{definition}\nIt is clear that uniform RC-positivity implies RC-positivity. To motivate the definition of (uniform) weak RC-positivity, let us consider a Hermitian metric $H$ on $E$ and the induced metric $h$ on the line bundle $O_{P(E^*)}(1)$ over the projectivized bundle $P(E^*)$. By a standard computation (for example, see \\cite[Formula (4.5)]{YangCamb}), we know that if $(E,H)$ is RC-positive, then the curvature $\\Theta$ of $h$ is positive on every fiber and has at least $r$ positive eigenvalues at every point in $P(E^*)$. The existence of such a metric $h$ on $O_{P(E^*)}(1)$ is called weak RC-positivity of $E$ (\\cite[Definition 3.3]{YangCamb}).\n\nSimilarly, using the same computation, we see that if $(E,H)$ is uniformly RC-positive, then the curvature $\\Theta$ of the induced metric $h$ satisfies\n\\begin{enumerate}[label=\\alph*.]\n \\item\\label{a} $\\Theta$ is positive on every fiber. \n \\item $\\Theta$ has at least $r$ positive eigenvalues at every point in $P(E^*)$.\n \\item For any point $t\\in X$, there exists a nonzero tangent vector $v\\in T_tX$, such that $\\Theta(\\Tilde{v},\\bar{\\Tilde{v}})|_{(t,[\\zeta])}>0$ for any lift $\\Tilde{v}$ of $v$ to $T_{(t,[\\zeta])}P(E^*)$.\n\\end{enumerate}\nFollowing Yang, we call the existence of such a metric $h$ on $O_{P(E^*)}(1)$ uniform weak RC-positivity of $E$. Note that in the third condition, we consider the lifts to the tangent space $T_{(t,[\\zeta])}P(E^*)$ for any point $[\\zeta]$ in the fiber $P(E_t^*)$ not just one point $[\\zeta]$. The second condition is implied by the first and the third, so we will omit it later on. Let us summarize the definition.\n\\begin{definition} The bundle $E$ is called weakly RC-positive if there exists a metric $h$ on $O_{P(E^*)}(1)$ with properties a and b. The bundle $E$ is called uniformly weakly RC-positive if there exists a metric $h$ on $O_{P(E^*)}(1)$ with properties a and c.\n\\end{definition}\nIn Yang's solution to Yau's conjecture, two main theorems \\cite[Theroem 1.3 and Theorem 1.4]{YangCamb}, although formulated in terms of RC-positivity, hold under weak RC-positivity. So, it is natural to ask if weak RC-positivity of $E$ implies RC-positivity of $E$ (\\cite[Question 7.11]{YangCamb} and \\cite[Problem 13]{Inayama}). This question has the same flavor as a conjecture of Griffiths \\cite{Griff69}: If $E$ is ample, then $E$ is Griffiths positive. For the developments on the Griffiths conjecture, see \\cite{Umemura,CampanaFlenner,Berndtsson09,MourouganeTaka,positivityandvanishingthmliu,liu2014curvatures,FengLiuWan,demailly2020hermitianyangmills,pingali2021note,finski2020monge,Finskichara,wu_2022,wupositivelyII,wuIII,Mazhang,lempert2024two,murakami2025analytic,wu2025mean}.\n\nIn this paper, we make some progress in this direction. In particular, we prove the following theorem regarding uniform RC-positivity.", "full_context": "In \\cite{YangCamb}, Yang introduces the notion of RC-positivity as a differential geometric counterpart of rational connectedness. RC-positivity plays a crucial role in Yang's proof of a conjecture of Yau: If a compact K\\\"ahler manifold has positive holomorphic sectional curvature, then the manifold is projective and rationally connected. A stronger notion called uniform RC-positivity is introduced by Yang in \\cite{YangForum} which also can be used to prove the same conjecture of Yau. For the semipositive case of Yau's conjecture, see \\cite{HeierWong,MatsumuraPAMQ,MatsumuraAJM}.\n\nLet us recall the definition of RC-positivity and uniform RC-positivity. Let $E$ be a holomorphic vector bundle of rank $r$ over a compact complex manifold $X$ of dimension $n$. Given a Hermitian metric $H$ on $E$, we denote the Chern curvature of $H$ by $\\Theta^H$, which is an $\\text{End}E$-valued $(1,1)$-form. We denote by $TX$ the holomorphic tangent bundle of $X$. For a vector $u\\in E_t$ and a tangent vector $v\\in T_tX$ with $t\\in X$, we define the expression $$H(\\Theta^H u,u)(v,\\bar{v})$$ to be $\\sum_{j,k} H(\\Theta^H_{j\\bar{k}}u,u)v_j \\bar{v}_k$ locally where we write the curvature $\\Theta^H=\\sum_{j,k}\\Theta^H_{j\\bar{k}}dt_j\\wedge d\\bar{t}_k$ and $v=\\sum_j v_j\\partial/\\partial t_j$.\n\nA Hermitian metric $H$ on a holomorphic vector bundle $E\\to X$ is called RC-positive if for any $t\\in X$ and any nonzero $u\\in E_t$, there is a nonzero tangent vector $v\\in T_tX$ such that $H(\\Theta^Hu,u)(v,\\bar v)>0$. On the other hand, a Hermitian metric $H$ is called uniformly RC-positive if for any $t\\in X$, there is a nonzero tangent vector $v\\in T_tX$ such that for any nonzero $u\\in E_t$, we have $H(\\Theta^H u,u)(v,\\bar v)>0$.\n\nA holomorphic vector bundle $E\\to X$ is called (uniformly) RC-positive if it admits a (uniformly) RC-positive Hermitian metric.\n\\end{definition}\nIt is clear that uniform RC-positivity implies RC-positivity. To motivate the definition of (uniform) weak RC-positivity, let us consider a Hermitian metric $H$ on $E$ and the induced metric $h$ on the line bundle $O_{P(E^*)}(1)$ over the projectivized bundle $P(E^*)$. By a standard computation (for example, see \\cite[Formula (4.5)]{YangCamb}), we know that if $(E,H)$ is RC-positive, then the curvature $\\Theta$ of $h$ is positive on every fiber and has at least $r$ positive eigenvalues at every point in $P(E^*)$. The existence of such a metric $h$ on $O_{P(E^*)}(1)$ is called weak RC-positivity of $E$ (\\cite[Definition 3.3]{YangCamb}).\n\nSimilarly, using the same computation, we see that if $(E,H)$ is uniformly RC-positive, then the curvature $\\Theta$ of the induced metric $h$ satisfies\n\\begin{enumerate}[label=\\alph*.]\n \\item\\label{a} $\\Theta$ is positive on every fiber. \n \\item $\\Theta$ has at least $r$ positive eigenvalues at every point in $P(E^*)$.\n \\item For any point $t\\in X$, there exists a nonzero tangent vector $v\\in T_tX$, such that $\\Theta(\\Tilde{v},\\bar{\\Tilde{v}})|_{(t,[\\zeta])}>0$ for any lift $\\Tilde{v}$ of $v$ to $T_{(t,[\\zeta])}P(E^*)$.\n\\end{enumerate}\nFollowing Yang, we call the existence of such a metric $h$ on $O_{P(E^*)}(1)$ uniform weak RC-positivity of $E$. Note that in the third condition, we consider the lifts to the tangent space $T_{(t,[\\zeta])}P(E^*)$ for any point $[\\zeta]$ in the fiber $P(E_t^*)$ not just one point $[\\zeta]$. The second condition is implied by the first and the third, so we will omit it later on. Let us summarize the definition.\n\\begin{definition} The bundle $E$ is called weakly RC-positive if there exists a metric $h$ on $O_{P(E^*)}(1)$ with properties a and b. The bundle $E$ is called uniformly weakly RC-positive if there exists a metric $h$ on $O_{P(E^*)}(1)$ with properties a and c.\n\\end{definition}\nIn Yang's solution to Yau's conjecture, two main theorems \\cite[Theroem 1.3 and Theorem 1.4]{YangCamb}, although formulated in terms of RC-positivity, hold under weak RC-positivity. So, it is natural to ask if weak RC-positivity of $E$ implies RC-positivity of $E$ (\\cite[Question 7.11]{YangCamb} and \\cite[Problem 13]{Inayama}). This question has the same flavor as a conjecture of Griffiths \\cite{Griff69}: If $E$ is ample, then $E$ is Griffiths positive. For the developments on the Griffiths conjecture, see \\cite{Umemura,CampanaFlenner,Berndtsson09,MourouganeTaka,positivityandvanishingthmliu,liu2014curvatures,FengLiuWan,demailly2020hermitianyangmills,pingali2021note,finski2020monge,Finskichara,wu_2022,wupositivelyII,wuIII,Mazhang,lempert2024two,murakami2025analytic,wu2025mean}.\n\nIn this paper, we make some progress in this direction. In particular, we prove the following theorem regarding uniform RC-positivity.\n\nIn this paper, we obtain results in this direction. In particular, we show that if a vector bundle $E$ is uniformly weakly RC-positive, then $S^kE\\otimes \\det E$ is uniformly RC-positive for any $k\\geq 0$, and $S^kE$ is uniformly RC-positive for $k$ large. We also discuss an approach that might lead to a solution to the question of whether weak RC-positivity of $E$ implies RC-positivity of $E$.\n\nIn this paper, we make some progress in this direction. In particular, we prove the following theorem regarding uniform RC-positivity.\n\nAnother motivation for establishing Theorem \\ref{Thm for E} is to understand the relation between rational connectedness of $X$ and (uniform) RC-positivity of the holomorphic tangent bundle $TX$. According to Yang \\cite[Theorem 1.4]{YangCamb} and \\cite[Theorem 1.3]{YangForum}, for a compact K\\\"ahler manifold $X^n$, if one of the following is true, then $X$ is projective and rationally connected.\n\\begin{enumerate}\n \\item The holomorphic tangent bundle $TX$ is uniformly RC-positive.\n \\item The exterior power $\\wedge^p TX$ is RC-positive for $1\\leq p\\leq n$.\n \\end{enumerate}\n One can ask if the converse is true (\\cite[Problem 4.15]{YangForum}). A partial converse is proved in \\cite[Theorem 1.4]{YangForum}: if $X$ is projective and rationally connected, then the line bundle $O_{\\wedge^p TX}(-1)$ is RC-positive for $1\\leq p \\leq n$. So, Theorem \\ref{Thm for E} can be viewed as a step towards this converse problem: constructing uniformly RC-positive Hermitian metrics out of metrics on the line bundle $O_{P(E^*)}(1)$.\n\nFor the proof of Theorem \\ref{Thm for E}, instead of the fibration $p:P(E^*)\\to X$, we will work on a more general fibration and prove a general theorem which contains Theorem \\ref{Thm for E}\nas a special case. We consider a proper holomorphic surjection $p:\\mathcal{X}^{n+m}\\to Y^m$ between two complex manifolds with $\\mathcal{X}$ K\\\"ahler, $Y$ compact, and the differential $dp$ surjective at every point. We denote the fibers $p^{-1}(t)$ by $\\mathcal{X}_t$ for $t\\in Y$. Let $(L,h)$ be a Hermitian line bundle over $\\mathcal{X}$. Let $$V_t=H^0(\\mathcal{X}_t, L|_{\\mathcal{X}_t}\\otimes K_{\\mathcal{X}_t}).$$ We assume that $\\dim V_t$ is independent of $t\\in Y$. So, the direct image of the sheaf of sections of $L\\otimes K_{\\mathcal{X}/Y}$ is locally free by Grauert's direct image theorem, where $K_{\\mathcal{X}/Y}$ is the relative canonical bundle. We denote by $V$ the associated vector bundle over $Y$. There is a naturally defined Hermitian metric $H$ on $V$. For $u$ in $V_t$ with $t\\in Y$, \n\\begin{equation}\\label{metric}\n H(u,u):=\\int_{\\mathcal{X}_t}h(u,u). \n\\end{equation}\nWe extend the metric $h$ to act on sections $u$ of $L|_{\\mathcal{X}_t}\\otimes K_{\\mathcal{X}_t}$ so that $h(u,u)$ is an $(n,n)$-form on $\\mathcal{X}_t$. In terms of local coordinates, if $u=u'\\otimes e$ with $u'$ an $(n,0)$-form and $e$ a frame of $L|_{\\mathcal{X}_t}$, then $h(u,u)=c_n u' \\wedge \\overline{u'} h(e,e)$ where $c_n=i^{n^2}$. Under this more general fibration, we can show \n\\begin{theorem}\\label{Thm 1}\n If the curvature $\\Theta$ of $h$ is positive on every fiber, and for any point $t\\in Y$, there exists a nonzero tangent vector $v\\in T_tY$, such that $\\Theta(\\Tilde{v},\\bar{\\Tilde{v}})|_{(t,z)}>0$ for any lift $\\Tilde{v}$ of $v$ to $T_{(t,z)}\\mathcal{X}$, then the Hermitian bundle $(V,H)$ is uniformly RC-positive.\n\\end{theorem}\n\nNow, we consider the fibration $p:P(E^*)\\to X$, and we assume $X$ is K\\\"ahler to make sure $P(E^*)$ is K\\\"ahler (see \\cite[Subsection 5.2]{wu2025mean}). Therefore, by Theorem \\ref{Thm 1}, we have Theorem \\ref{Thm for E}. Indeed, the vector bundle $V$ in Theorem \\ref{Thm 1} is associated with the direct image of $L\\otimes K_{\\mathcal{X}/Y}$. In the present situation, the relative canonical bundle\n$K_{P(E^*)/X}$ is isomorphic to $O_{P(E^*)}(-r)\\otimes p^*\\det E$. If we choose $O_{P(E^*)}(r+k)$ for the line bundle $L$, then $V$ is $S^k\\otimes \\det E$. On the other hand, if we choose $O_{P(E^*)}(k)\\otimes K^{-1}_{P(E^*)/X}$ for $L$, then $V$ is $S^kE$ (we use an arbitrary metric $g$ on $K^{-1}_{P(E^*)/X}$, and the effect of $g$ can be absorbed by taking $k$ large).\n\nWe choose a coordinate system $(t_1,\\ldots,t_m)$ around the point $t_0$ in $Y$ such that $v_0=\\partial/\\partial t_1$ at $t_0$. Consider a fixed $u_0\\neq 0$ in $V_{t_0}$. A standard argument allows us to extend $u_0$ to a local holomorphic section $u$ of $V$ such that $D'u=0$ at $t_0$ and $u(t_0)=u_0\\neq0$. A straightforward computation gives\n\\begin{equation}\\label{standard} \\partial \\bar{\\partial} H(u,u)=- H(\\Theta^V u,u) \\text{ at } t_0. \n\\end{equation}\nOn the other hand, if we let $\\mathbf{u}$ be a representative of $u$ and write $\\mathbf{u}=u'\\otimes e$ with $u'$ an $(n,0)$-form and $e$ some local frame of $L$, then \n\\begin{equation}\n H(u,u)=p_*(c_n u'\\wedge \\overline{u'} e^{-\\phi}) \n\\end{equation}\nwhere $e^{-\\phi}=h(e,e)$ and $c_n=i^{n^2}$. According to \\cite[Proposition 4.2]{Berndtsson09}, we can choose a representative $\\mathbf{u}$ such that in $\\bar{\\partial}\\mathbf{u}= \\sum \\eta_j\\wedge dt_j$, the $\\eta_j$ is primitive on $\\mathcal{X}_{t_0}$. Moreover, \n$\\partial^\\phi u'=0$ at $t_0$. After using such a representative, we obtain\n\\begin{equation}\\label{4.4}\n\\partial\\bar{\\partial}H(u,u)\n = \n -c_n p_* ( u'\\wedge\\overline{u'}\\wedge \\partial\\bar{\\partial} \\phi e^{-\\phi})\n +\n (-1)^n c_np_* (\\bar{\\partial}u'\\wedge \\overline{\\bar{\\partial}u'}e^{-\\phi}) \\text{ at } t_0. \n\\end{equation}\nWe apply the above $(1,1)$-form to the tangent vector $v_0=\\partial/\\partial t_1$ and get \n\\begin{equation}\\label{4.6}\n\\partial\\bar{\\partial}H(u,u)(v_0,\\bar{v}_0)\n = \n -c_n p_* ( u'\\wedge\\overline{u'}\\wedge \\partial\\bar{\\partial} \\phi e^{-\\phi})(v_0,\\bar{v}_0)\n +\n (-1)^n c_np_* (\\bar{\\partial}u'\\wedge \\overline{\\bar{\\partial}u'}e^{-\\phi}) (v_0,\\bar{v}_0)\\text{ at } t_0.\n\\end{equation}\nBecause $\\bar{\\partial}\\mathbf{u}= \\sum \\eta_j\\wedge dt_j$ and $\\mathbf{u}=u'\\otimes e$, we see $\\sum \\eta_j\\wedge dt_j=\\bar{\\partial}\\mathbf{u}=\\bar{\\partial}u'\\otimes e $. If we write $\\eta_j=\\eta_j'\\otimes e$, then $\\bar{\\partial}u'=\\sum \\eta_j'\\wedge dt_j$. So the last term in (\\ref{4.6}) is equal to \\begin{equation}\\label{4.7}\n (-1)^n c_n\\int_{\\mathcal{X}_{t_0}}(-1)^{n}\\sum \\eta'_j\\wedge \n\\overline{\\eta'}_k \\wedge dt_j\\wedge d\\bar{t}_k e^{-\\phi}(v_0,\\bar{v}_0)= c_n \\int_{\\mathcal{X}_{t_0}} \\eta'_1\\wedge \n\\overline{\\eta'}_1 e^{-\\phi}\\leq 0;\n\\end{equation}\nthe last inequality is by the fact that the $\\eta_1$ is primitive on $\\mathcal{X}_{t_0}$.\n\nThe case we care about most in this paper is when $k=1$ in Lemma \\ref{lemma abc} because it corresponds to weak RC-positivity. The difficulty in proving a theorem like Theorem \\ref{Thm for E} or Theorem \\ref{Thm 1} for weak RC-positivity is that the $\\beta$ in Lemma \\ref{lemma abc} is on $\\mathcal{X}$, so it does not quite fit into Berndtsson's computation, especially formula (\\ref{4.11}).\nSo, we raise the question:\n\\begin{question}\n Is it possible to choose $\\beta$ in Lemma \\ref{lemma abc} so that $\\beta=p^*\\alpha$ for some Hermitian metric $\\alpha$ on $Y$? \n\\end{question}\nThis question is somewhat bold because if it is possible to choose $\\beta=p^*\\alpha$, then we can use \\cite[Theorem 4]{wu2025mean} to deduce that if $E$ is weakly RC-positive, then $S^k\\otimes \\det E$ has positive mean curvature for $k\\geq 0$. Moreover, it is even possible to use \\cite[Theorem 5]{wu2025mean} to deduce that if $E$ is weakly RC-positive, then $E$ has positive mean curvature. Since positive mean curvature implies RC-positivity (\\cite[Theorem 3.6]{YangCamb}), this would mean that RC-positivity, weak RC-positivity, and mean curvature positivity are all equivalent. Such an equivalence is conjectured for tangent bundle $TX$ in\n\\cite[Problems 4.15 and 4.17]{YangForum}.", "post_theorem_intro_text_len": 5285, "post_theorem_intro_text": "Another motivation for establishing Theorem \\ref{Thm for E} is to understand the relation between rational connectedness of $X$ and (uniform) RC-positivity of the holomorphic tangent bundle $TX$. According to Yang \\cite[Theorem 1.4]{YangCamb} and \\cite[Theorem 1.3]{YangForum}, for a compact K\\\"ahler manifold $X^n$, if one of the following is true, then $X$ is projective and rationally connected.\n\\begin{enumerate}\n     \\item The holomorphic tangent bundle $TX$ is uniformly RC-positive.\n     \\item The exterior power $\\wedge^p TX$ is RC-positive for $1\\leq p\\leq n$.\n \\end{enumerate}\n One can ask if the converse is true (\\cite[Problem 4.15]{YangForum}). A partial converse is proved in \\cite[Theorem 1.4]{YangForum}: if $X$ is projective and rationally connected, then the line bundle $O_{\\wedge^p TX}(-1)$ is RC-positive for $1\\leq p \\leq n$. So, Theorem \\ref{Thm for E} can be viewed as a step towards this converse problem: constructing uniformly RC-positive Hermitian metrics out of metrics on the line bundle $O_{P(E^*)}(1)$.\n\nWe also prove a lemma (Lemma \\ref{lemma abc} in Section \\ref{section ?}) and discuss how a variant of this lemma might lead to a solution to the original question of Yang, namely, weak RC-positivity of $E$ implying RC positivity of $E$.\n\nFor the proof of Theorem \\ref{Thm for E}, instead of the fibration $p:P(E^*)\\to X$, we will work on a more general fibration and prove a general theorem which contains Theorem \\ref{Thm for E}\nas a special case. We consider a proper holomorphic surjection $p:\\mathcal{X}^{n+m}\\to Y^m$ between two complex manifolds with $\\mathcal{X}$ K\\\"ahler, $Y$ compact, and the differential $dp$ surjective at every point. We denote the fibers $p^{-1}(t)$ by $\\mathcal{X}_t$ for $t\\in Y$. Let $(L,h)$ be a Hermitian line bundle over $\\mathcal{X}$. Let $$V_t=H^0(\\mathcal{X}_t, L|_{\\mathcal{X}_t}\\otimes K_{\\mathcal{X}_t}).$$ We assume that $\\dim V_t$ is independent of $t\\in Y$. So, the direct image of the sheaf of sections of $L\\otimes K_{\\mathcal{X}/Y}$ is locally free by Grauert's direct image theorem, where $K_{\\mathcal{X}/Y}$ is the relative canonical bundle. We denote by $V$ the associated vector bundle over $Y$. There is a naturally defined Hermitian metric $H$ on $V$. For $u$ in $V_t$ with $t\\in Y$, \n\\begin{equation}\\label{metric}\n  H(u,u):=\\int_{\\mathcal{X}_t}h(u,u).  \n\\end{equation}\nWe extend the metric $h$ to act on sections $u$ of $L|_{\\mathcal{X}_t}\\otimes K_{\\mathcal{X}_t}$ so that $h(u,u)$ is an $(n,n)$-form on $\\mathcal{X}_t$. In terms of local coordinates, if $u=u'\\otimes e$ with $u'$ an $(n,0)$-form and $e$ a frame of $L|_{\\mathcal{X}_t}$, then $h(u,u)=c_n u' \\wedge \\overline{u'} h(e,e)$ where $c_n=i^{n^2}$. Under this more general fibration, we can show \n\\begin{theorem}\\label{Thm 1}\n    If the curvature $\\Theta$ of $h$ is positive on every fiber, and for any point $t\\in Y$, there exists a nonzero tangent vector $v\\in T_tY$, such that $\\Theta(\\Tilde{v},\\bar{\\Tilde{v}})|_{(t,z)}>0$ for any lift $\\Tilde{v}$ of $v$ to $T_{(t,z)}\\mathcal{X}$, then the Hermitian bundle $(V,H)$ is uniformly RC-positive.\n\\end{theorem}\n\nActually, the precise statement we prove in Theorem \\ref{Thm 1} is: For a fixed point $t_0 \\in Y$, if the curvature $\\Theta$ of $h$ is positive on the fiber $\\mathcal{X}_{t_0}$ , and  there exists a nonzero tangent vector $v\\in T_{t_0}Y$, such that $\\Theta(\\Tilde{v},\\bar{\\Tilde{v}})|_{(t_0,z)}>0$ for any lift $\\Tilde{v}$ of $v$ to $T_{(t_0,z)}\\mathcal{X}$, then the Hermitian bundle $(V,H)$ is uniformly RC-positive at $t_0$.\n\nNow, we consider the fibration $p:P(E^*)\\to X$, and we assume $X$ is K\\\"ahler to make sure $P(E^*)$ is K\\\"ahler (see \\cite[Subsection 5.2]{wu2025mean}). Therefore, by Theorem \\ref{Thm 1}, we have Theorem \\ref{Thm for E}. Indeed, the vector bundle $V$ in Theorem \\ref{Thm 1} is associated with the direct image of $L\\otimes K_{\\mathcal{X}/Y}$. In the present situation, the relative canonical bundle\n$K_{P(E^*)/X}$ is isomorphic to $O_{P(E^*)}(-r)\\otimes p^*\\det E$. If we choose $O_{P(E^*)}(r+k)$ for the line bundle $L$, then $V$ is $S^k\\otimes \\det E$. On the other hand, if we choose $O_{P(E^*)}(k)\\otimes K^{-1}_{P(E^*)/X}$ for $L$, then $V$ is $S^kE$ (we use an arbitrary metric $g$ on $K^{-1}_{P(E^*)/X}$, and the effect of $g$ can be absorbed by taking $k$ large). \n\nThe proof of Theorem \\ref{Thm 1} is an adaptation of \\cite[Section 3]{wu2025mean}, but we still include the details for completeness  (the original argument is due to Berndtsson in \\cite[Section 4]{Berndtsson09} and \\cite[Section 2]{BoMathz}. See also \\cite{CampanaCaoMihai}).\n\nThis paper is organized as follows. In Section \\ref{section prelim}, we give a local expression for the assumption (uniform weak RC-positivity) in Theorem \\ref{Thm 1} which will be used in the proof of the main theorem. In Section \\ref{section proof}, we prove Theorem \\ref{Thm 1}. In Section \\ref{section ?}, we discuss\na characterization of weak RC-positivity and its possible application.\n\nI would like to thank Shin-ichi Matsumura for bringing to my attention the question of RC-positivity and weak RC-positivity. I am grateful to L\\'aszl\\'o Lempert, Siarhei Finski, and Xiaokui Yang for their interest in the paper. Thanks are also due to the Erd\\H{o}s Center, Budapest for the support.", "sketch": "To prove Theorem~\\ref{Thm for E}, the text says that “instead of the fibration $p:P(E^*)\\to X$, we will work on a more general fibration and prove a general theorem which contains Theorem~\\ref{Thm for E}$ as a special case.” Concretely, one considers a proper holomorphic submersion $p:\\mathcal{X}^{n+m}\\to Y^m$ with $\\mathcal{X}$ K\\\"ahler and $Y$ compact, a Hermitian line bundle $(L,h)$ on $\\mathcal{X}$, and the direct-image bundle $V$ with fibers\n\\[\nV_t=H^0(\\mathcal{X}_t,\\, L|_{\\mathcal{X}_t}\\otimes K_{\\mathcal{X}_t}),\n\\]\nendowed with the “naturally defined Hermitian metric”\n\\[\nH(u,u):=\\int_{\\mathcal{X}_t} h(u,u).\\tag{\\ref{metric}}\n\\]\nUnder the hypothesis (in Theorem~\\ref{Thm 1}) that “the curvature $\\Theta$ of $h$ is positive on every fiber” and that for each $t\\in Y$ there is a nonzero $v\\in T_tY$ with “$\\Theta(\\widetilde v,\\overline{\\widetilde v})|_{(t,z)}>0$ for any lift $\\widetilde v$ of $v$,” one concludes that “the Hermitian bundle $(V,H)$ is uniformly RC-positive.”\n\nThen Theorem~\\ref{Thm for E} is obtained by applying Theorem~\\ref{Thm 1} to $p:P(E^*)\\to X$ (with $X$ K\\\"ahler so $P(E^*)$ is K\\\"ahler). In this specialization, one uses that\n\\[\nK_{P(E^*)/X}\\simeq O_{P(E^*)}(-r)\\otimes p^*\\det E.\n\\]\nChoosing $L=O_{P(E^*)}(r+k)$ gives that the resulting direct image bundle $V$ is “$S^kE\\otimes\\det E$.” Alternatively, choosing $L=O_{P(E^*)}(k)\\otimes K^{-1}_{P(E^*)/X}$ gives $V=S^kE$, and “the effect of [a metric on $K^{-1}_{P(E^*)/X}$] can be absorbed by taking $k$ large.”\n\nFinally, the proof of Theorem~\\ref{Thm 1} is described as “an adaptation of \\cite[Section 3]{wu2025mean}” and attributed to Berndtsson (with references).", "expanded_sketch": "To prove the main theorem, the text says that “instead of the fibration $p:P(E^*)\\to X$, we will work on a more general fibration and prove a general theorem which contains the main theorem as a special case.” Concretely, one considers a proper holomorphic submersion $p:\\mathcal{X}^{n+m}\\to Y^m$ with $\\mathcal{X}$ K\\\"ahler and $Y$ compact, a Hermitian line bundle $(L,h)$ on $\\mathcal{X}$, and the direct-image bundle $V$ with fibers\n\\[\nV_t=H^0(\\mathcal{X}_t,\\, L|_{\\mathcal{X}_t}\\otimes K_{\\mathcal{X}_t}),\n\\]\nendowed with the “naturally defined Hermitian metric”\n\\[\nH(u,u):=\\int_{\\mathcal{X}_t} h(u,u).\\tag{\\ref{metric}}\n\\]\nWe first prove the following theorem.\n\\begin{theorem}\\label{Thm 1}\n    If the curvature $\\Theta$ of $h$ is positive on every fiber, and for any point $t\\in Y$, there exists a nonzero tangent vector $v\\in T_tY$, such that $\\Theta(\\Tilde{v},\\bar{\\Tilde{v}})|_{(t,z)}>0$ for any lift $\\Tilde{v}$ of $v$ to $T_{(t,z)}\\mathcal{X}$, then the Hermitian bundle $(V,H)$ is uniformly RC-positive.\n\\end{theorem}\nUnder these hypotheses, one concludes that “the Hermitian bundle $(V,H)$ is uniformly RC-positive.”\n\nThen the main theorem is obtained by applying the preceding theorem to $p:P(E^*)\\to X$ (with $X$ K\\\"ahler so $P(E^*)$ is K\\\"ahler). In this specialization, one uses that\n\\[\nK_{P(E^*)/X}\\simeq O_{P(E^*)}(-r)\\otimes p^*\\det E.\n\\]\nChoosing $L=O_{P(E^*)}(r+k)$ gives that the resulting direct image bundle $V$ is “$S^kE\\otimes\\det E$.” Alternatively, choosing $L=O_{P(E^*)}(k)\\otimes K^{-1}_{P(E^*)/X}$ gives $V=S^kE$, and “the effect of [a metric on $K^{-1}_{P(E^*)/X}$] can be absorbed by taking $k$ large.”\n\nFinally, the proof of the preceding theorem is described as “an adaptation of \\cite[Section 3]{wu2025mean}” and attributed to Berndtsson (with references).", "expanded_theorem": "\\label{Thm for E}\n    If $E$ is uniformly weakly RC-positive over a compact K\\\"ahler manifold $X$, then $S^kE\\otimes \\det E$ is uniformly RC-positive for any $k\\geq 0$, and $S^kE$ is uniformly RC-positive for $k$ large.", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let $E\\to X$ be a holomorphic vector bundle over a compact K\\\"ahler manifold $X$. Assume that $E$ is uniformly weakly RC-positive in the following sense: there exists a Hermitian metric $h$ on the line bundle $O_{P(E^*)}(1)$ over the projectivized bundle $P(E^*)$ such that, if $\\Theta$ denotes the curvature of $h$, then (i) $\\Theta$ is positive on every fiber of $P(E^*)\\to X$, and (ii) for every point $t\\in X$ there exists a nonzero tangent vector $v\\in T_tX$ such that for every point $[\\zeta]\\in P(E_t^*)$ and every lift $\\widetilde v\\in T_{(t,[\\zeta])}P(E^*)$ of $v$, one has $\\Theta(\\widetilde v,\\overline{\\widetilde v})>0$ at $(t,[\\zeta])$. Here $S^kE$ denotes the $k$th symmetric power of $E$, and $\\det E$ its determinant line bundle. A holomorphic vector bundle $F\\to X$ is called uniformly RC-positive if it admits a Hermitian metric $H$ such that for every $t\\in X$ there exists a nonzero vector $v\\in T_tX$ with $H(\\Theta^H u,u)(v,\\bar v)>0$ for every nonzero $u\\in F_t$, where $\\Theta^H$ is the Chern curvature of $H$. Under these assumptions, which statement holds?", "correct_choice": {"label": "A", "text": "For every integer $k\\ge 0$, the bundle $S^kE\\otimes \\det E$ is uniformly RC-positive, and the bundle $S^kE$ is uniformly RC-positive for all sufficiently large $k$."}, "choices": [{"label": "B", "text": "For every integer $k\\ge 0$, the bundle $S^kE\\otimes \\det E$ is uniformly RC-positive, and the bundle $S^kE$ is uniformly RC-positive for every integer $k\\ge 0$."}, {"label": "C", "text": "For every integer $k\\ge 0$, the bundle $S^kE\\otimes \\det E$ is uniformly RC-positive."}, {"label": "D", "text": "For every integer $k\\ge 0$, the bundle $S^kE\\otimes \\det E$ is RC-positive, and the bundle $S^kE$ is RC-positive for all sufficiently large $k$."}, {"label": "E", "text": "There exists an integer $k_0\\ge 0$, depending only on $\\operatorname{rk}(E)$, such that for every integer $k\\ge k_0$, both $S^kE\\otimes \\det E$ and $S^kE$ are uniformly RC-positive."}], "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": "large-k requirement for absorbing the metric on $K^{-1}_{P(E^*)/X}$", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "the additional conclusion about $S^kE$ for sufficiently large $k$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "uniformity conclusion of the direct-image theorem", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "non-uniform dependence of the threshold for large $k$", "template_used": "quantifier_dependence"}]}}
{"id": "2512.11246v1", "paper_link": "http://arxiv.org/abs/2512.11246v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{t:mainthm1} Fix $(M^{2n}, J)$ an OT manifold and $g_0$ a pluriclosed metric on $M$.  The solution to pluriclosed flow with initial data $g_0$ exists on $[0, \\infty)$.", "start_pos": 8094, "end_pos": 8289, "label": "t:mainthm1"}, "ref_dict": {"c:OT_PCF": "\\begin{conj} \\label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\\omega_0$, the normalized pluriclosed flow \\ref{e:PCF_normalized} with initial metric $\\omega_0$ exists on $[0, \\infty)$, and converges to $(\\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.\n\\end{conj}"}, "pre_theorem_intro_text_len": 1275, "pre_theorem_intro_text": "In recent years the pluriclosed flow \\cite{PCF, PCFReg} and generalized K\\\"ahler-Ricci flow \\cite{apostolov2022generalized,StreetsSTB,GKRF} have been developed as a tool for understanding the geometry of complex, especially non-K\\\"ahler, manifolds \\cite{barbaro2023bismut,barbaro2025global,barbaro2025pluriclosed,fino2024pluriclosed,fusi2024pluriclosed,garcia2023non, ye2024pluriclosed}.  A natural class of non-K\\\"ahler manifolds are the Oeljeklaus-Toma (OT) manifolds \\cite{oeljeklaus2005non}, whose geometry is linked to the structure of number fields, and which are natural higher dimensional generalizations of Inoue surfaces \\cite{inoue1974surfaces}.  In \\cite{fusi2024pluriclosed} a complete description of the pluriclosed flow with left-invariant initial data on OT manifolds was obtained, in particular showing that the solution exists for all time and collapses after blowdown to a torus in the Gromov-Hausdorff sense.  Moreover the blowdown on the universal cover converges in the Cheeger-Gromov sense to a soliton.  It is natural to conjecture that these statements hold for arbitrary initial data (cf. Conjecture \\ref{c:OT_PCF}).  In this work we confirm some aspects of this conjecture.  \n\nThe first main result is to establish the global existence of the flow:", "context": "In recent years the pluriclosed flow \\cite{PCF, PCFReg} and generalized K\\\"ahler-Ricci flow \\cite{apostolov2022generalized,StreetsSTB,GKRF} have been developed as a tool for understanding the geometry of complex, especially non-K\\\"ahler, manifolds \\cite{barbaro2023bismut,barbaro2025global,barbaro2025pluriclosed,fino2024pluriclosed,fusi2024pluriclosed,garcia2023non, ye2024pluriclosed}. A natural class of non-K\\\"ahler manifolds are the Oeljeklaus-Toma (OT) manifolds \\cite{oeljeklaus2005non}, whose geometry is linked to the structure of number fields, and which are natural higher dimensional generalizations of Inoue surfaces \\cite{inoue1974surfaces}. In \\cite{fusi2024pluriclosed} a complete description of the pluriclosed flow with left-invariant initial data on OT manifolds was obtained, in particular showing that the solution exists for all time and collapses after blowdown to a torus in the Gromov-Hausdorff sense. Moreover the blowdown on the universal cover converges in the Cheeger-Gromov sense to a soliton. It is natural to conjecture that these statements hold for arbitrary initial data (cf. Conjecture \\ref{c:OT_PCF}). In this work we confirm some aspects of this conjecture.\n\nThe first main result is to establish the global existence of the flow:\n\n\\begin{conj} \\label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\\omega_0$, the normalized pluriclosed flow \\ref{e:PCF_normalized} with initial metric $\\omega_0$ exists on $[0, \\infty)$, and converges to $(\\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.\n\\end{conj}", "full_context": "In recent years the pluriclosed flow \\cite{PCF, PCFReg} and generalized K\\\"ahler-Ricci flow \\cite{apostolov2022generalized,StreetsSTB,GKRF} have been developed as a tool for understanding the geometry of complex, especially non-K\\\"ahler, manifolds \\cite{barbaro2023bismut,barbaro2025global,barbaro2025pluriclosed,fino2024pluriclosed,fusi2024pluriclosed,garcia2023non, ye2024pluriclosed}. A natural class of non-K\\\"ahler manifolds are the Oeljeklaus-Toma (OT) manifolds \\cite{oeljeklaus2005non}, whose geometry is linked to the structure of number fields, and which are natural higher dimensional generalizations of Inoue surfaces \\cite{inoue1974surfaces}. In \\cite{fusi2024pluriclosed} a complete description of the pluriclosed flow with left-invariant initial data on OT manifolds was obtained, in particular showing that the solution exists for all time and collapses after blowdown to a torus in the Gromov-Hausdorff sense. Moreover the blowdown on the universal cover converges in the Cheeger-Gromov sense to a soliton. It is natural to conjecture that these statements hold for arbitrary initial data (cf. Conjecture \\ref{c:OT_PCF}). In this work we confirm some aspects of this conjecture.\n\nThe first main result is to establish the global existence of the flow:\n\n\\begin{conj} \\label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\\omega_0$, the normalized pluriclosed flow \\ref{e:PCF_normalized} with initial metric $\\omega_0$ exists on $[0, \\infty)$, and converges to $(\\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.\n\\end{conj}\n\nThe first main result is to establish the global existence of the flow:\n\nThe expected qualitative behavior of the normalized pluriclosed flow on OT manifolds is captured by solutions with initial data the model metrics $\\gw_{h}^{a,b}$. Straightforward computations show that the normalized pluriclosed flow with this initial data is\n\\begin{align}\\label{e:normalized equation}\n \\omega^{a,b}_h(t) = \\sum_{i=1}^s \\sqrt{-1}((1-e^{-t})\\frac{3}{4}+e^{-t}a_i)\\frac{1}{(\\operatorname{Im}w_i)^2}dw_i\\wedge d\\bar{w}_i + \\sqrt{-1}e^{-t}b_i\\operatorname{Im}w_i dz_i\\wedge d\\bar{z}_i.\n\\end{align}\nLater, we shall denote the time-dependent normalized model metric starting with $\\omega^{a,b}_h$ as $\\omega^{a,b}_h(t)$. We shall write $\\omega_h(t)$ for short if $a_i$, $b_i=1$, for all $1\\leq i\\leq s$. \nObserve that for these model solutions the Chern torsion $T$ is uniformly bounded in time, i.e., $|T(t)|\\leq C$. Moreover, it follows that\n\\begin{align*}\n \\frac{\\omega_h^{a,b}(t)}{t+1}\\to \\sum_{i=1}^s \\frac{3}{4(\\operatorname{Im}w_i)^2}dw_i\\wedge d\\bar{w}_i,\n\\end{align*}\nwhich can be considered as a degenerate metric on $X(K, U)$. As explained in \\cite{fusi2024pluriclosed}, the blowdown manifolds converge in Gromov-Hausdorff sense to a torus $\\mathbb T^s$ with a canonical flat metric $d(K, U)$ depending only on the algebraic field $K$ and the rank $s$ subgroup $U$. We recall the result here, noting that OT manifolds are compact solvmanifolds and the model metrics are left-invariant:\n\\begin{thm} \\label{t:homogOT} (\\cite{fusi2024pluriclosed}) Let $\\omega_0$ be a left-invariant pluriclosed metric on an OT manifold $M$, then the normalized pluriclosed flow starting with $\\omega_0$ converges to $(\\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.\n\\end{thm}\n\n\\noindent It is reasonable to conjecture that this behavior holds in the general case:\n\\begin{conj} \\label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\\omega_0$, the normalized pluriclosed flow \\ref{e:PCF_normalized} with initial metric $\\omega_0$ exists on $[0, \\infty)$, and converges to $(\\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.\n\\end{conj}\n\nWe shall prove the first part of the conjecture in Section \\ref{s:long time section}. For the second half of the conjecture, we have the following sufficiency condition.\n\\begin{prop}\\label{p:GH convergence argument}\n Let $\\omega(t)$ be the normalized pluriclosed flow solution on the OT manifold $X(K,U)$. Suppose that there exists a constant $C>0$ such that \n \\begin{itemize}\n \\item $C^{-1}\\leq \\operatorname{tr}_{\\omega(t)}\\omega_h(t)\\leq C$,\n \\item $\\lim\\limits_{t\\to\\infty}\\operatorname{tr}_{\\omega(t)}\\omega_h(t) = 1$ for all $x\\in X(K,U)$.\n \\end{itemize}\n Then, we have\n \\begin{align*}\n (X(K,U),\\omega(t))\\to (\\mathbb{T}^s,d(K, U)),\n\\end{align*}\nto the flat metric $d(K,U)$ in the Gromov-Hausdorff sense.\n\\end{prop}\n\\begin{proof} We give a brief sketch as the result is already essentially contained in e.g. \\cite{fusi2024pluriclosed}, \\cite{ZhengOT}. Note that for OT manifolds, there is a canonical fibration map:\n \\begin{align*}\n F:X(K,U)\\to \\mathbb{T}^s\n \\end{align*}\n where the fiber is diffeomorphic to $\\mathbb{T}^{3s}$. In particular, $E_\\mathbb{C}$ will be in the kernel of $dF$. Since the quotient of $\\{z\\}\\times \\mathbb{C}^s$ is dense in the $\\mathbb{T}^{3s}$ fiber (See Section 2 of \\cite{VerbiOT}), the degenerate metric $\\sum_{i=1}^s \\frac{3}{4(\\operatorname{Im}w_i)^2}dw_i\\wedge d\\bar{w}_i$ will induce a metric on the base $\\mathbb{T}^s$. Let $d(K,U)$ be the metric on $\\mathbb{T}^s$ induced from $\\sum_{i=1}^s \\frac{3}{4(\\operatorname{Im}w_i)^2}dw_i\\wedge d\\bar{w}_i$, which is flat.\n\nSecondly, when $s = 1$, then $E_\\mathbb{C}$ and $E_\\mathbb{H}$ are holomorphic line bundles. Thus, the AM-GM inequality becomes equality in this case, and we can have the following lower bound of $\\dot{\\phi}$.\n\\begin{lemma}\\label{l:potential derivative lower bound}\n On Inoue surface $S_M$, for the potential $\\phi$, there exists a constant $C>0$, such that:\n \\begin{align*}\n \\dot{\\phi}\\geq -C.\n \\end{align*}\n\\end{lemma}\n\\begin{proof}\n Note $s=1$. Choose a large constant $\\Lambda>0$, consider the quantity $\\dot{\\phi} + \\phi+\\frac{1}{\\Lambda}\\phi$. Then\n \\begin{align*}\n (\\frac{\\partial}{\\partial t} - \\Delta)(\\dot{\\phi}+\\phi+\\frac{1}{\\Lambda}\\phi) = 1 +\\frac{1}{\\Lambda}\\dot{\\phi}+ \\frac{1}{\\Lambda}e^{-t}\\operatorname{tr}_{g_\\mathbb{C}}g^0_\\mathbb{C} - \\frac{1}{\\Lambda}e^{-t}\\operatorname{tr}_{g_\\mathbb{H}}g^0_\\mathbb{H} - (\\frac{1}{\\Lambda}(1-e^{-t})+1)\\frac{3}{4}\\operatorname{tr}_{g_\\mathbb{H}}h_\\mathbb{H}.\n \\end{align*}\n Now, for $A>0$, a very large constant, such that $\\dot{\\phi}+\\phi +\\frac{1}{\\Lambda}\\phi < -A$. By \\eqref{e:potential eq}, notice that in this case $s=1$, we know:\n \\begin{align*}\n \\dot{\\phi} +\\phi = \\log\\frac{\\det g_\\mathbb{C}/c_1}{\\det e^{-t}h_\\mathbb{C}} - \\log\\frac{\\det g_\\mathbb{H}}{\\det \\frac{3}{4}h_\\mathbb{H}} = -\\log(c_1e^{-t}\\operatorname{tr}_{g_\\mathbb{C}}h_\\mathbb{C}) + \\log (\\frac{3}{4}\\operatorname{tr}_{g_\\mathbb{H}}h_\\mathbb{H}).\n \\end{align*}\n By Corollary \\ref{c:Inoue C0}, in particular $|\\phi|\\leq C_0$. Then, there is a large constant $A' = A- C_0>0$ such that $\\dot{\\phi}\\leq -A'$, and \n \\begin{align*}\n \\operatorname{tr}_{g_\\mathbb{H}}h_\\mathbb{H}\\leq \\frac{4}{3}e^{\\frac{C_0}{\\Lambda}-A}e^{-t}c_1 \\operatorname{tr}_{g_\\mathbb{C}}h_\\mathbb{C}\\leq \\frac{4}{3}e^{\\frac{C_0}{\\Lambda}-A}c_0e^{-t}\\operatorname{tr}_{g_\\mathbb{C}}g^0_\\mathbb{C},\n \\end{align*}\n for some constant $c_0$, only depends on the initial metric.\n By choosing $\\Lambda$ large, such that $1-\\frac{C_0}{\\Lambda}>\\frac{2}{3}$, fixed, when $A$ is large enough, $c_0e^{\\frac{C_0}{\\Lambda}-A} \\leq \\frac{1}{\\Lambda}$. Thus, choose an $A = \\frac{\\Lambda}{2}$ large enough, such that $\\dot{\\phi}(t)\\geq -\\frac{1}{2}A$, for $t\\in[0,T]$, where $T$ is large enough. Then for the first time $\\dot{\\phi}+\\phi +\\frac{1}{\\Lambda}\\phi = -A$, we have:\n \\begin{align*}\n (\\frac{\\partial}{\\partial t}-\\Delta)(\\dot{\\phi} + \\phi +\\frac{1}{\\Lambda}\\phi)&\\geq (\\frac{1}{\\Lambda } - \\frac{C_1e^{-t}}{\\Lambda}-c_0e^{\\frac{C_0}{\\Lambda}-A})e^{-t}\\operatorname{tr}_{g_\\mathbb{C}}h_\\mathbb{C} + 1+ \\frac{1}{\\Lambda}\\dot{\\phi}\\\\\n &>0.\n \\end{align*}\n By the maximum principle, the result follows.\n\\end{proof}\nCombine with the potential derivative upper bound (Lemma \\ref{l:potential derivative upper bound}), we have the following metric lower bound.\n\\begin{thm}\\label{t:Inoue metric lower bound}\n On an Inoue surface $S_M$, for generalized K\\\"ahler metric $\\omega_0$, there is a constant $C$, such that the normalized pluriclosed flow solution $\\omega(t)$with initial data $\\gw_0$ will satisfy\n \\begin{align*}\n \\omega(t)\\geq C\\omega_h(t),\n \\end{align*}\n where $\\omega_h(t)$ is the model flow.\n\\end{thm}\n\\begin{proof}\n By Lemma \\ref{l:potential derivative upper bound} and Lemma \\ref{l:potential derivative lower bound}, we have that:\n \\begin{align*}\n -C\\leq\\dot{\\phi} +\\phi \\leq C\n \\end{align*}\n for some constant $C$. Now, from the equation \\eqref{e:potential eq} and $s=1$, we have:\n \\begin{align*}\n -C\\leq \\operatorname{tr}_{g_\\mathbb{H}}h_\\mathbb{H}/\\operatorname{tr}_{g_\\mathbb{C}}e^{-t}h_\\mathbb{C}\\leq C.\n \\end{align*}\n Note that in the normalized pluriclosed flow model case, \\eqref{e:normalized equation}, the $E_\\mathbb{C}$ part will shrink at the rate of $e^{-t}$, while the $E_\\mathbb{H}$ part will be equivalent to $h_\\mathbb{H}$. Thus, by Corollary \\ref{c:lower z}, we have the desired lower bound for $E_\\mathbb{C}$.\n\\end{proof}\n\\begin{proof} [Proof of Theorem \\ref{t:mainthm2}]\nCombine Lemma \\ref{l:C0 bound}, Corollary \\ref{c:imporved C0}, Lemma \\ref{l:potential derivative upper bound}, Corollary \\ref{c:Inoue C0}, Lemma \\ref{l:potential derivative lower bound}, and Theorem \\ref{t:Inoue metric lower bound}, the results follow.\n\\end{proof}", "post_theorem_intro_text_len": 1301, "post_theorem_intro_text": "\\noindent The proof exploits natural class of background metrics arising from the homogeneous structure on OT-manifolds.  In the case of Inoue surfaces these are known as Tricerri metrics \\cite{tricerri1982some}.  In the next theorem we give some refined estimates in the special case of generalized K\\\"ahler-Ricci flow (GKRF) \\cite{GKRF}.  Oeljeklaus-Toma manifolds admit natural classes of generalized K\\\"ahler structures with split tangent bundle, and for such metrics the GKRF reduces to a scalar parabolic flow of Monge-Amp\\`ere type \\cite{StreetsSTB}.\n\n\\begin{thm} \\label{t:mainthm2} For generalized K\\\"ahler-Ricci flow on an Oeljeklaus-Toma manifold\n\\begin{enumerate}\n\\item The scalar potential $\\phi$ satisfies\n\\begin{align*}\n    - C \\leq \\phi \\leq C e^{-t}(1 + t).\n\\end{align*}\n\\item Assuming there exists a Tricerri-type metric in $[\\gw_0]$, we have\n\\begin{align*}\n    - C e^{-t}(1 + t) \\leq \\phi \\leq C e^{-t}(1 + t).\n\\end{align*}\n\\item On Inoue surfaces of type $S_M$, the estimate of item (2) holds. In addition:\n\\begin{align*}\n    -C\\leq \\dot{\\phi}\\leq C.\n\\end{align*}\n\\item On Inoue surfaces of type $S_M$, we have:\n\\begin{align*}\n        \\omega(t)\\geq C\\omega_h(t),\n\\end{align*}\nwhere $\\omega_h(t)$ is the model flow with initial data the Tricerri metric $h$.\n\\end{enumerate}\n\\end{thm}", "sketch": "The proof of Theorem~\\ref{t:mainthm1} is described only at a high level: it “exploits [a] natural class of background metrics arising from the homogeneous structure on OT-manifolds.” In the case of Inoue surfaces, “these are known as Tricerri metrics.”", "expanded_sketch": "The proof of To prove the main theorem, is described only at a high level: it “exploits [a] natural class of background metrics arising from the homogeneous structure on OT-manifolds.” In the case of Inoue surfaces, “these are known as Tricerri metrics.”", "expanded_theorem": "\\label{t:mainthm1} Fix $(M^{2n}, J)$ an OT manifold and $g_0$ a pluriclosed metric on $M$. The solution to pluriclosed flow with initial data $g_0$ exists on $[0, \\infty)$.,", "theorem_type": ["Universal", "Existence"], "mcq": {"question": "Which statement holds for every Oeljeklaus--Toma (OT) complex manifold $(M^{2n},J)$ and every pluriclosed Hermitian metric $g_0$ on $M$ (that is, the associated fundamental $(1,1)$-form $\\omega_0$ satisfies $\\partial\\bar\\partial\\omega_0=0$)?", "correct_choice": {"label": "A", "text": "The pluriclosed flow with initial metric $g_0$ exists for all forward time; equivalently, its solution is defined on the entire interval $[0,\\infty)$."}, "choices": [{"label": "B", "text": "The pluriclosed flow with initial metric $g_0$ exists for all forward time and, for every such initial metric, converges as $t\\to\\infty$ in the Gromov--Hausdorff sense to a flat torus canonically associated to $M$."}, {"label": "C", "text": "The pluriclosed flow with initial metric $g_0$ exists on some maximal time interval $[0,T)$ with $T>0$."}, {"label": "D", "text": "There exists a time $T=T(M,J)>0$, depending only on the OT complex manifold, such that for every pluriclosed Hermitian metric $g_0$ on $M$, the pluriclosed flow with initial metric $g_0$ exists on $[0,T]$."}, {"label": "E", "text": "For every OT complex manifold $(M^{2n},J)$ and every pluriclosed Hermitian metric $g_0$ on $M$, the normalized pluriclosed flow with initial metric $g_0$ exists on $[0,\\infty)$ and is generated by one of the homogeneous background metrics arising from the OT-manifold structure."}], "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": "adds asymptotic Gromov--Hausdorff convergence not asserted in the theorem", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped the global-in-time conclusion $[0,\\infty)$ to mere short-time existence on a maximal interval", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "replaces infinite-time existence by a uniform manifold-dependent finite existence time", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "confuses use of homogeneous background metrics in the proof with the actual initial-data/general-flow statement", "template_used": "wildcard"}]}}
{"id": "2512.11246v1", "paper_link": "http://arxiv.org/abs/2512.11246v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{t:mainthm1} Fix $(M^{2n}, J)$ an OT manifold and $g_0$ a pluriclosed metric on $M$.  The solution to pluriclosed flow with initial data $g_0$ exists on $[0, \\infty)$.", "start_pos": 8094, "end_pos": 8289, "label": "t:mainthm1"}, "ref_dict": {"c:OT_PCF": "\\begin{conj} \\label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\\omega_0$, the normalized pluriclosed flow \\ref{e:PCF_normalized} with initial metric $\\omega_0$ exists on $[0, \\infty)$, and converges to $(\\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.\n\\end{conj}"}, "pre_theorem_intro_text_len": 1275, "pre_theorem_intro_text": "In recent years the pluriclosed flow \\cite{PCF, PCFReg} and generalized K\\\"ahler-Ricci flow \\cite{apostolov2022generalized,StreetsSTB,GKRF} have been developed as a tool for understanding the geometry of complex, especially non-K\\\"ahler, manifolds \\cite{barbaro2023bismut,barbaro2025global,barbaro2025pluriclosed,fino2024pluriclosed,fusi2024pluriclosed,garcia2023non, ye2024pluriclosed}.  A natural class of non-K\\\"ahler manifolds are the Oeljeklaus-Toma (OT) manifolds \\cite{oeljeklaus2005non}, whose geometry is linked to the structure of number fields, and which are natural higher dimensional generalizations of Inoue surfaces \\cite{inoue1974surfaces}.  In \\cite{fusi2024pluriclosed} a complete description of the pluriclosed flow with left-invariant initial data on OT manifolds was obtained, in particular showing that the solution exists for all time and collapses after blowdown to a torus in the Gromov-Hausdorff sense.  Moreover the blowdown on the universal cover converges in the Cheeger-Gromov sense to a soliton.  It is natural to conjecture that these statements hold for arbitrary initial data (cf. Conjecture \\ref{c:OT_PCF}).  In this work we confirm some aspects of this conjecture.  \n\nThe first main result is to establish the global existence of the flow:", "context": "In recent years the pluriclosed flow \\cite{PCF, PCFReg} and generalized K\\\"ahler-Ricci flow \\cite{apostolov2022generalized,StreetsSTB,GKRF} have been developed as a tool for understanding the geometry of complex, especially non-K\\\"ahler, manifolds \\cite{barbaro2023bismut,barbaro2025global,barbaro2025pluriclosed,fino2024pluriclosed,fusi2024pluriclosed,garcia2023non, ye2024pluriclosed}. A natural class of non-K\\\"ahler manifolds are the Oeljeklaus-Toma (OT) manifolds \\cite{oeljeklaus2005non}, whose geometry is linked to the structure of number fields, and which are natural higher dimensional generalizations of Inoue surfaces \\cite{inoue1974surfaces}. In \\cite{fusi2024pluriclosed} a complete description of the pluriclosed flow with left-invariant initial data on OT manifolds was obtained, in particular showing that the solution exists for all time and collapses after blowdown to a torus in the Gromov-Hausdorff sense. Moreover the blowdown on the universal cover converges in the Cheeger-Gromov sense to a soliton. It is natural to conjecture that these statements hold for arbitrary initial data (cf. Conjecture \\ref{c:OT_PCF}). In this work we confirm some aspects of this conjecture.\n\nThe first main result is to establish the global existence of the flow:\n\n\\begin{conj} \\label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\\omega_0$, the normalized pluriclosed flow \\ref{e:PCF_normalized} with initial metric $\\omega_0$ exists on $[0, \\infty)$, and converges to $(\\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.\n\\end{conj}", "full_context": "In recent years the pluriclosed flow \\cite{PCF, PCFReg} and generalized K\\\"ahler-Ricci flow \\cite{apostolov2022generalized,StreetsSTB,GKRF} have been developed as a tool for understanding the geometry of complex, especially non-K\\\"ahler, manifolds \\cite{barbaro2023bismut,barbaro2025global,barbaro2025pluriclosed,fino2024pluriclosed,fusi2024pluriclosed,garcia2023non, ye2024pluriclosed}. A natural class of non-K\\\"ahler manifolds are the Oeljeklaus-Toma (OT) manifolds \\cite{oeljeklaus2005non}, whose geometry is linked to the structure of number fields, and which are natural higher dimensional generalizations of Inoue surfaces \\cite{inoue1974surfaces}. In \\cite{fusi2024pluriclosed} a complete description of the pluriclosed flow with left-invariant initial data on OT manifolds was obtained, in particular showing that the solution exists for all time and collapses after blowdown to a torus in the Gromov-Hausdorff sense. Moreover the blowdown on the universal cover converges in the Cheeger-Gromov sense to a soliton. It is natural to conjecture that these statements hold for arbitrary initial data (cf. Conjecture \\ref{c:OT_PCF}). In this work we confirm some aspects of this conjecture.\n\nThe first main result is to establish the global existence of the flow:\n\n\\begin{conj} \\label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\\omega_0$, the normalized pluriclosed flow \\ref{e:PCF_normalized} with initial metric $\\omega_0$ exists on $[0, \\infty)$, and converges to $(\\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.\n\\end{conj}\n\nThe first main result is to establish the global existence of the flow:\n\nThe expected qualitative behavior of the normalized pluriclosed flow on OT manifolds is captured by solutions with initial data the model metrics $\\gw_{h}^{a,b}$. Straightforward computations show that the normalized pluriclosed flow with this initial data is\n\\begin{align}\\label{e:normalized equation}\n \\omega^{a,b}_h(t) = \\sum_{i=1}^s \\sqrt{-1}((1-e^{-t})\\frac{3}{4}+e^{-t}a_i)\\frac{1}{(\\operatorname{Im}w_i)^2}dw_i\\wedge d\\bar{w}_i + \\sqrt{-1}e^{-t}b_i\\operatorname{Im}w_i dz_i\\wedge d\\bar{z}_i.\n\\end{align}\nLater, we shall denote the time-dependent normalized model metric starting with $\\omega^{a,b}_h$ as $\\omega^{a,b}_h(t)$. We shall write $\\omega_h(t)$ for short if $a_i$, $b_i=1$, for all $1\\leq i\\leq s$. \nObserve that for these model solutions the Chern torsion $T$ is uniformly bounded in time, i.e., $|T(t)|\\leq C$. Moreover, it follows that\n\\begin{align*}\n \\frac{\\omega_h^{a,b}(t)}{t+1}\\to \\sum_{i=1}^s \\frac{3}{4(\\operatorname{Im}w_i)^2}dw_i\\wedge d\\bar{w}_i,\n\\end{align*}\nwhich can be considered as a degenerate metric on $X(K, U)$. As explained in \\cite{fusi2024pluriclosed}, the blowdown manifolds converge in Gromov-Hausdorff sense to a torus $\\mathbb T^s$ with a canonical flat metric $d(K, U)$ depending only on the algebraic field $K$ and the rank $s$ subgroup $U$. We recall the result here, noting that OT manifolds are compact solvmanifolds and the model metrics are left-invariant:\n\\begin{thm} \\label{t:homogOT} (\\cite{fusi2024pluriclosed}) Let $\\omega_0$ be a left-invariant pluriclosed metric on an OT manifold $M$, then the normalized pluriclosed flow starting with $\\omega_0$ converges to $(\\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.\n\\end{thm}\n\n\\noindent It is reasonable to conjecture that this behavior holds in the general case:\n\\begin{conj} \\label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\\omega_0$, the normalized pluriclosed flow \\ref{e:PCF_normalized} with initial metric $\\omega_0$ exists on $[0, \\infty)$, and converges to $(\\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.\n\\end{conj}\n\nWe shall prove the first part of the conjecture in Section \\ref{s:long time section}. For the second half of the conjecture, we have the following sufficiency condition.\n\\begin{prop}\\label{p:GH convergence argument}\n Let $\\omega(t)$ be the normalized pluriclosed flow solution on the OT manifold $X(K,U)$. Suppose that there exists a constant $C>0$ such that \n \\begin{itemize}\n \\item $C^{-1}\\leq \\operatorname{tr}_{\\omega(t)}\\omega_h(t)\\leq C$,\n \\item $\\lim\\limits_{t\\to\\infty}\\operatorname{tr}_{\\omega(t)}\\omega_h(t) = 1$ for all $x\\in X(K,U)$.\n \\end{itemize}\n Then, we have\n \\begin{align*}\n (X(K,U),\\omega(t))\\to (\\mathbb{T}^s,d(K, U)),\n\\end{align*}\nto the flat metric $d(K,U)$ in the Gromov-Hausdorff sense.\n\\end{prop}\n\\begin{proof} We give a brief sketch as the result is already essentially contained in e.g. \\cite{fusi2024pluriclosed}, \\cite{ZhengOT}. Note that for OT manifolds, there is a canonical fibration map:\n \\begin{align*}\n F:X(K,U)\\to \\mathbb{T}^s\n \\end{align*}\n where the fiber is diffeomorphic to $\\mathbb{T}^{3s}$. In particular, $E_\\mathbb{C}$ will be in the kernel of $dF$. Since the quotient of $\\{z\\}\\times \\mathbb{C}^s$ is dense in the $\\mathbb{T}^{3s}$ fiber (See Section 2 of \\cite{VerbiOT}), the degenerate metric $\\sum_{i=1}^s \\frac{3}{4(\\operatorname{Im}w_i)^2}dw_i\\wedge d\\bar{w}_i$ will induce a metric on the base $\\mathbb{T}^s$. Let $d(K,U)$ be the metric on $\\mathbb{T}^s$ induced from $\\sum_{i=1}^s \\frac{3}{4(\\operatorname{Im}w_i)^2}dw_i\\wedge d\\bar{w}_i$, which is flat.\n\nSecondly, when $s = 1$, then $E_\\mathbb{C}$ and $E_\\mathbb{H}$ are holomorphic line bundles. Thus, the AM-GM inequality becomes equality in this case, and we can have the following lower bound of $\\dot{\\phi}$.\n\\begin{lemma}\\label{l:potential derivative lower bound}\n On Inoue surface $S_M$, for the potential $\\phi$, there exists a constant $C>0$, such that:\n \\begin{align*}\n \\dot{\\phi}\\geq -C.\n \\end{align*}\n\\end{lemma}\n\\begin{proof}\n Note $s=1$. Choose a large constant $\\Lambda>0$, consider the quantity $\\dot{\\phi} + \\phi+\\frac{1}{\\Lambda}\\phi$. Then\n \\begin{align*}\n (\\frac{\\partial}{\\partial t} - \\Delta)(\\dot{\\phi}+\\phi+\\frac{1}{\\Lambda}\\phi) = 1 +\\frac{1}{\\Lambda}\\dot{\\phi}+ \\frac{1}{\\Lambda}e^{-t}\\operatorname{tr}_{g_\\mathbb{C}}g^0_\\mathbb{C} - \\frac{1}{\\Lambda}e^{-t}\\operatorname{tr}_{g_\\mathbb{H}}g^0_\\mathbb{H} - (\\frac{1}{\\Lambda}(1-e^{-t})+1)\\frac{3}{4}\\operatorname{tr}_{g_\\mathbb{H}}h_\\mathbb{H}.\n \\end{align*}\n Now, for $A>0$, a very large constant, such that $\\dot{\\phi}+\\phi +\\frac{1}{\\Lambda}\\phi < -A$. By \\eqref{e:potential eq}, notice that in this case $s=1$, we know:\n \\begin{align*}\n \\dot{\\phi} +\\phi = \\log\\frac{\\det g_\\mathbb{C}/c_1}{\\det e^{-t}h_\\mathbb{C}} - \\log\\frac{\\det g_\\mathbb{H}}{\\det \\frac{3}{4}h_\\mathbb{H}} = -\\log(c_1e^{-t}\\operatorname{tr}_{g_\\mathbb{C}}h_\\mathbb{C}) + \\log (\\frac{3}{4}\\operatorname{tr}_{g_\\mathbb{H}}h_\\mathbb{H}).\n \\end{align*}\n By Corollary \\ref{c:Inoue C0}, in particular $|\\phi|\\leq C_0$. Then, there is a large constant $A' = A- C_0>0$ such that $\\dot{\\phi}\\leq -A'$, and \n \\begin{align*}\n \\operatorname{tr}_{g_\\mathbb{H}}h_\\mathbb{H}\\leq \\frac{4}{3}e^{\\frac{C_0}{\\Lambda}-A}e^{-t}c_1 \\operatorname{tr}_{g_\\mathbb{C}}h_\\mathbb{C}\\leq \\frac{4}{3}e^{\\frac{C_0}{\\Lambda}-A}c_0e^{-t}\\operatorname{tr}_{g_\\mathbb{C}}g^0_\\mathbb{C},\n \\end{align*}\n for some constant $c_0$, only depends on the initial metric.\n By choosing $\\Lambda$ large, such that $1-\\frac{C_0}{\\Lambda}>\\frac{2}{3}$, fixed, when $A$ is large enough, $c_0e^{\\frac{C_0}{\\Lambda}-A} \\leq \\frac{1}{\\Lambda}$. Thus, choose an $A = \\frac{\\Lambda}{2}$ large enough, such that $\\dot{\\phi}(t)\\geq -\\frac{1}{2}A$, for $t\\in[0,T]$, where $T$ is large enough. Then for the first time $\\dot{\\phi}+\\phi +\\frac{1}{\\Lambda}\\phi = -A$, we have:\n \\begin{align*}\n (\\frac{\\partial}{\\partial t}-\\Delta)(\\dot{\\phi} + \\phi +\\frac{1}{\\Lambda}\\phi)&\\geq (\\frac{1}{\\Lambda } - \\frac{C_1e^{-t}}{\\Lambda}-c_0e^{\\frac{C_0}{\\Lambda}-A})e^{-t}\\operatorname{tr}_{g_\\mathbb{C}}h_\\mathbb{C} + 1+ \\frac{1}{\\Lambda}\\dot{\\phi}\\\\\n &>0.\n \\end{align*}\n By the maximum principle, the result follows.\n\\end{proof}\nCombine with the potential derivative upper bound (Lemma \\ref{l:potential derivative upper bound}), we have the following metric lower bound.\n\\begin{thm}\\label{t:Inoue metric lower bound}\n On an Inoue surface $S_M$, for generalized K\\\"ahler metric $\\omega_0$, there is a constant $C$, such that the normalized pluriclosed flow solution $\\omega(t)$with initial data $\\gw_0$ will satisfy\n \\begin{align*}\n \\omega(t)\\geq C\\omega_h(t),\n \\end{align*}\n where $\\omega_h(t)$ is the model flow.\n\\end{thm}\n\\begin{proof}\n By Lemma \\ref{l:potential derivative upper bound} and Lemma \\ref{l:potential derivative lower bound}, we have that:\n \\begin{align*}\n -C\\leq\\dot{\\phi} +\\phi \\leq C\n \\end{align*}\n for some constant $C$. Now, from the equation \\eqref{e:potential eq} and $s=1$, we have:\n \\begin{align*}\n -C\\leq \\operatorname{tr}_{g_\\mathbb{H}}h_\\mathbb{H}/\\operatorname{tr}_{g_\\mathbb{C}}e^{-t}h_\\mathbb{C}\\leq C.\n \\end{align*}\n Note that in the normalized pluriclosed flow model case, \\eqref{e:normalized equation}, the $E_\\mathbb{C}$ part will shrink at the rate of $e^{-t}$, while the $E_\\mathbb{H}$ part will be equivalent to $h_\\mathbb{H}$. Thus, by Corollary \\ref{c:lower z}, we have the desired lower bound for $E_\\mathbb{C}$.\n\\end{proof}\n\\begin{proof} [Proof of Theorem \\ref{t:mainthm2}]\nCombine Lemma \\ref{l:C0 bound}, Corollary \\ref{c:imporved C0}, Lemma \\ref{l:potential derivative upper bound}, Corollary \\ref{c:Inoue C0}, Lemma \\ref{l:potential derivative lower bound}, and Theorem \\ref{t:Inoue metric lower bound}, the results follow.\n\\end{proof}", "post_theorem_intro_text_len": 1301, "post_theorem_intro_text": "\\noindent The proof exploits natural class of background metrics arising from the homogeneous structure on OT-manifolds.  In the case of Inoue surfaces these are known as Tricerri metrics \\cite{tricerri1982some}.  In the next theorem we give some refined estimates in the special case of generalized K\\\"ahler-Ricci flow (GKRF) \\cite{GKRF}.  Oeljeklaus-Toma manifolds admit natural classes of generalized K\\\"ahler structures with split tangent bundle, and for such metrics the GKRF reduces to a scalar parabolic flow of Monge-Amp\\`ere type \\cite{StreetsSTB}.\n\n\\begin{thm} \\label{t:mainthm2} For generalized K\\\"ahler-Ricci flow on an Oeljeklaus-Toma manifold\n\\begin{enumerate}\n\\item The scalar potential $\\phi$ satisfies\n\\begin{align*}\n    - C \\leq \\phi \\leq C e^{-t}(1 + t).\n\\end{align*}\n\\item Assuming there exists a Tricerri-type metric in $[\\gw_0]$, we have\n\\begin{align*}\n    - C e^{-t}(1 + t) \\leq \\phi \\leq C e^{-t}(1 + t).\n\\end{align*}\n\\item On Inoue surfaces of type $S_M$, the estimate of item (2) holds. In addition:\n\\begin{align*}\n    -C\\leq \\dot{\\phi}\\leq C.\n\\end{align*}\n\\item On Inoue surfaces of type $S_M$, we have:\n\\begin{align*}\n        \\omega(t)\\geq C\\omega_h(t),\n\\end{align*}\nwhere $\\omega_h(t)$ is the model flow with initial data the Tricerri metric $h$.\n\\end{enumerate}\n\\end{thm}", "sketch": "The proof of Theorem~\\ref{t:mainthm1} is described only at a high level: it “exploits [a] natural class of background metrics arising from the homogeneous structure on OT-manifolds.” In the case of Inoue surfaces, “these are known as Tricerri metrics.”", "expanded_sketch": "The proof of To prove the main theorem, is described only at a high level: it “exploits [a] natural class of background metrics arising from the homogeneous structure on OT-manifolds.” In the case of Inoue surfaces, “these are known as Tricerri metrics.”", "expanded_theorem": "\\label{t:mainthm1} Fix $(M^{2n}, J)$ an OT manifold and $g_0$ a pluriclosed metric on $M$. The solution to pluriclosed flow with initial data $g_0$ exists on $[0, \\infty)$.,", "theorem_type": ["Universal", "Existence"], "mcq": {"question": "Let $(M^{2n},J)$ be an OeljeklausToma (OT) manifold, and let $g_0$ be a pluriclosed Hermitian metric on $M$ (equivalently, if $\\omega_0$ is its associated $(1,1)$-form, then $\\partial\\bar\\partial\\omega_0=0$). Which existence statement holds for the pluriclosed flow on $M$ starting from $g_0$?", "correct_choice": {"label": "A", "text": "There exists a solution $g(t)$ to the pluriclosed flow on $M$ with initial condition $g(0)=g_0$, and this solution is defined for all times $t\\in[0,\\infty)$."}, "choices": [{"label": "B", "text": "There exists a solution $g(t)$ to the pluriclosed flow on $M$ with initial condition $g(0)=g_0$, and there is a maximal time $T<\\infty$ such that the solution is defined precisely for $t\\in[0,T)$."}, {"label": "C", "text": "There exists a solution $g(t)$ to the pluriclosed flow on $M$ with initial condition $g(0)=g_0$, and this solution is defined on some interval $t\\in[0,T)$ for some $T>0$."}, {"label": "D", "text": "For every pluriclosed Hermitian metric $g_0$ on $M$, there exists a constant $T=T(M)>0$ such that the pluriclosed flow with initial condition $g(0)=g_0$ is defined for all times $t\\in[0,T]$."}, {"label": "E", "text": "There exists a solution $g(t)$ to the pluriclosed flow on $M$ with initial condition $g(0)=g_0$, and this solution is unique and converges as $t\\to\\infty$ to a canonical homogeneous background metric on $M$."}], "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": "infinite-time interval replaced by finite maximal existence time", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped global-in-time conclusion, retaining only existence on some positive time interval", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "uniform dependence of existence time only on the manifold rather than global existence for each initial metric", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "imports convergence to a canonical homogeneous background metric from the proof strategy instead of the theorem statement", "template_used": "wildcard"}]}}
{"id": "2512.11432v1", "paper_link": "http://arxiv.org/abs/2512.11432v1", "theorems_cnt": 2, "theorem": {"env_name": "theo", "content": "Let $f \\in \\mathcal{E}^\\infty(\\mathbb{R}^2)$ be such that $Z(f)$ is the Hawaiian earring. Then $f$ is flat at the origin:\n\\[\n\\frac{\\partial^{\\alpha +\\beta}f}{\\partial x_1^\\alpha \\partial x_2^\\beta}(0,0)\n = 0 \n\\qquad \\text{for all } \\alpha,\\beta \\in\\mathbb{N}.\n\\]", "start_pos": 6484, "end_pos": 6754, "label": null}, "ref_dict": {}, "pre_theorem_intro_text_len": 2627, "pre_theorem_intro_text": "The study of zero sets of smooth functions and ideals in the ring \n$\\mathcal{E}(\\Omega)$ of infinitely differentiable  functions on an open set \n$\\Omega \\subset \\mathbb{R}^n$ lies at the intersection of real analytic geometry, \nsingularity theory, and differential topology. \nA central tool in this context is the {\\L}ojasiewicz inequality, which provides a \nquantitative relation between a function and its gradient, and plays a fundamental \nrole in the analysis of subanalytic sets, stratifications, and resolution of singularities.\n\nA finitely generated ideal $I = (f_1, \\ldots, f_k)\\mathcal{E}(\\Omega)$ is said\nto be a {\\L}ojasiewicz ideal if there exists a function $g \\in I$ that satisfies\na {\\L}ojasiewicz inequality with respect to\n$\nZ(I) := \\{ x \\in \\Omega \\mid f_1(x) = f_2(x) = \\cdots = f_k(x) = 0 \\}.\n$\nSee the definition below.\\\\\nA classical result of Ren\\'e Thom \\cite{thom1967some} asserts that if $I$ is a finitely generated  Lojasiewicz ideal, \nthen its zero set $Z(I)$ contains an open dense subset of smooth points.\\\\  In analytic geometry this is expected: analytic ideals naturally enforce \ngeometric regularity on their zero sets.  \nHowever, in the smooth category such regularity properties are more subtle, since \narbitrary smooth functions may exhibit behavior impossible in the analytic setting.\n\nThis raises a natural question: \n\\emph{does the converse hold?}  \nDoes the presence of an open dense set of smooth points in $Z(I)$ imply that \nthe ideal $I$ is \\L{}ojasiewicz?  \nThe purpose of the present work is to examine this question and to identify \nmechanisms by which such a converse can fail.\\\\\nTo illustrate this phenomenon, we focus on a classical topological example:  \nthe \\emph{Hawaiian earring}, defined as the union of the circles\n\\[\nC_n = \\left\\{ (x_1,x_2) \\in \\mathbb{R}^2 : \n\\left(x_1 - \\frac{1}{n}\\right)^2 + x_2^2 = \\frac{1}{n^2} \\right\\}, \\qquad n \\ge 1.\n\\]\nThis compact set consists of infinitely many circles tangent at the origin and \naccumulating there.  \nIt is well known that the Hawaiian earring has highly pathological local \ntopology at the origin: it is not locally contractible, not semianalytic, and \ncannot arise as the zero set of any nontrivial real analytic function.\n\nWe consider smooth functions with real values $f \\in \\mathcal{E}^\\infty(\\mathbb{R}^2)$ whose zero \nset $Z(f)$ is \\emph{exactly} the Hawaiian earring.  \nSuch functions exist by Whitney's extension theorem, see theorem 3.1, Ch IV, \\cite{Tougeron1972}, yet they exhibit extremely \ndegenerate behavior at the origin.  \nOur first main result states that any such function must vanish to infinite order.", "context": "The study of zero sets of smooth functions and ideals in the ring \n$\\mathcal{E}(\\Omega)$ of infinitely differentiable functions on an open set \n$\\Omega \\subset \\mathbb{R}^n$ lies at the intersection of real analytic geometry, \nsingularity theory, and differential topology. \nA central tool in this context is the {\\L}ojasiewicz inequality, which provides a \nquantitative relation between a function and its gradient, and plays a fundamental \nrole in the analysis of subanalytic sets, stratifications, and resolution of singularities.\n\nA finitely generated ideal $I = (f_1, \\ldots, f_k)\\mathcal{E}(\\Omega)$ is said\nto be a {\\L}ojasiewicz ideal if there exists a function $g \\in I$ that satisfies\na {\\L}ojasiewicz inequality with respect to\n$\nZ(I) := \\{ x \\in \\Omega \\mid f_1(x) = f_2(x) = \\cdots = f_k(x) = 0 \\}.\n$\nSee the definition below.\\\\\nA classical result of Ren\\'e Thom \\cite{thom1967some} asserts that if $I$ is a finitely generated Lojasiewicz ideal, \nthen its zero set $Z(I)$ contains an open dense subset of smooth points.\\\\ In analytic geometry this is expected: analytic ideals naturally enforce \ngeometric regularity on their zero sets. \nHowever, in the smooth category such regularity properties are more subtle, since \narbitrary smooth functions may exhibit behavior impossible in the analytic setting.\n\nThis raises a natural question: \n\\emph{does the converse hold?} \nDoes the presence of an open dense set of smooth points in $Z(I)$ imply that \nthe ideal $I$ is \\L{}ojasiewicz? \nThe purpose of the present work is to examine this question and to identify \nmechanisms by which such a converse can fail.\\\\\nTo illustrate this phenomenon, we focus on a classical topological example: \nthe \\emph{Hawaiian earring}, defined as the union of the circles\n\\[\nC_n = \\left\\{ (x_1,x_2) \\in \\mathbb{R}^2 : \n\\left(x_1 - \\frac{1}{n}\\right)^2 + x_2^2 = \\frac{1}{n^2} \\right\\}, \\qquad n \\ge 1.\n\\]\nThis compact set consists of infinitely many circles tangent at the origin and \naccumulating there. \nIt is well known that the Hawaiian earring has highly pathological local \ntopology at the origin: it is not locally contractible, not semianalytic, and \ncannot arise as the zero set of any nontrivial real analytic function.\n\nWe consider smooth functions with real values $f \\in \\mathcal{E}^\\infty(\\mathbb{R}^2)$ whose zero \nset $Z(f)$ is \\emph{exactly} the Hawaiian earring. \nSuch functions exist by Whitney's extension theorem, see theorem 3.1, Ch IV, \\cite{Tougeron1972}, yet they exhibit extremely \ndegenerate behavior at the origin. \nOur first main result states that any such function must vanish to infinite order.", "full_context": "The study of zero sets of smooth functions and ideals in the ring \n$\\mathcal{E}(\\Omega)$ of infinitely differentiable functions on an open set \n$\\Omega \\subset \\mathbb{R}^n$ lies at the intersection of real analytic geometry, \nsingularity theory, and differential topology. \nA central tool in this context is the {\\L}ojasiewicz inequality, which provides a \nquantitative relation between a function and its gradient, and plays a fundamental \nrole in the analysis of subanalytic sets, stratifications, and resolution of singularities.\n\nA finitely generated ideal $I = (f_1, \\ldots, f_k)\\mathcal{E}(\\Omega)$ is said\nto be a {\\L}ojasiewicz ideal if there exists a function $g \\in I$ that satisfies\na {\\L}ojasiewicz inequality with respect to\n$\nZ(I) := \\{ x \\in \\Omega \\mid f_1(x) = f_2(x) = \\cdots = f_k(x) = 0 \\}.\n$\nSee the definition below.\\\\\nA classical result of Ren\\'e Thom \\cite{thom1967some} asserts that if $I$ is a finitely generated Lojasiewicz ideal, \nthen its zero set $Z(I)$ contains an open dense subset of smooth points.\\\\ In analytic geometry this is expected: analytic ideals naturally enforce \ngeometric regularity on their zero sets. \nHowever, in the smooth category such regularity properties are more subtle, since \narbitrary smooth functions may exhibit behavior impossible in the analytic setting.\n\nThis raises a natural question: \n\\emph{does the converse hold?} \nDoes the presence of an open dense set of smooth points in $Z(I)$ imply that \nthe ideal $I$ is \\L{}ojasiewicz? \nThe purpose of the present work is to examine this question and to identify \nmechanisms by which such a converse can fail.\\\\\nTo illustrate this phenomenon, we focus on a classical topological example: \nthe \\emph{Hawaiian earring}, defined as the union of the circles\n\\[\nC_n = \\left\\{ (x_1,x_2) \\in \\mathbb{R}^2 : \n\\left(x_1 - \\frac{1}{n}\\right)^2 + x_2^2 = \\frac{1}{n^2} \\right\\}, \\qquad n \\ge 1.\n\\]\nThis compact set consists of infinitely many circles tangent at the origin and \naccumulating there. \nIt is well known that the Hawaiian earring has highly pathological local \ntopology at the origin: it is not locally contractible, not semianalytic, and \ncannot arise as the zero set of any nontrivial real analytic function.\n\nWe consider smooth functions with real values $f \\in \\mathcal{E}^\\infty(\\mathbb{R}^2)$ whose zero \nset $Z(f)$ is \\emph{exactly} the Hawaiian earring. \nSuch functions exist by Whitney's extension theorem, see theorem 3.1, Ch IV, \\cite{Tougeron1972}, yet they exhibit extremely \ndegenerate behavior at the origin. \nOur first main result states that any such function must vanish to infinite order.\n\nThis raises a natural question: \n\\emph{does the converse hold?} \nDoes the presence of an open dense set of smooth points in $Z(I)$ imply that \nthe ideal $I$ is \\L{}ojasiewicz? \nThe purpose of the present work is to examine this question and to identify \nmechanisms by which such a converse can fail.\\\\\nTo illustrate this phenomenon, we focus on a classical topological example: \nthe \\emph{Hawaiian earring}, defined as the union of the circles\n\\[\nC_n = \\left\\{ (x_1,x_2) \\in \\mathbb{R}^2 : \n\\left(x_1 - \\frac{1}{n}\\right)^2 + x_2^2 = \\frac{1}{n^2} \\right\\}, \\qquad n \\ge 1.\n\\]\nThis compact set consists of infinitely many circles tangent at the origin and \naccumulating there. \nIt is well known that the Hawaiian earring has highly pathological local \ntopology at the origin: it is not locally contractible, not semianalytic, and \ncannot arise as the zero set of any nontrivial real analytic function.\n\nWe then formulate a general geometric criterion for flatness, valid for arbitrary \ncollections of smooth arcs tangent at a point with unbounded curvature or \ninfinitely varying radius of osculation. \nThis leads to a \\emph{degenerate \\L{}ojasiewicz inequality} adapted to situations \nwhere the classical inequality necessarily fails.\n\n\\begin{theo}\nLet $\\Gamma = \\bigcup\\limits_{n \\ge 1} \\gamma_n$ be a union of smooth embedded arcs \nmeeting at a common point $p$ with curvature tending to $\\infty$ or oscillating \nwithout bound. \nIf $f \\in \\mathcal{E}^\\infty(\\Omega)$ vanishes on $\\Gamma$, then $f$ satisfies a flatness \nestimate of the form\n\\[\n|f(x)| \\le C_N\\, d(x,\\Gamma)^N,\n\\qquad \\text{for all } N \\ge 1 \\text{ and all } x \\text{ near } p,\n\\]\nfor suitable constants $C_N>0$. \nIn particular, $f$ is flat at $p$.\n\\end{theo}\n\n\\begin{rem}\nIn this case, for any system of generators $g_1,\\ldots,g_p$ of $I$, \nthe functions $\\sum\\limits_{j=1}^p g_j^2$ and $\\sum\\limits_{j=1}^p |g_j|$ both satisfy \nthe {\\L}ojasiewicz inequality with respect to $Z(I)$.\n\\end{rem}\nIt should be noted that the property of an ideal to be {\\L}ojasiewicz is a \nlocal one: if an ideal is {\\L}ojasiewicz on an open set $U$, then the induced \nideal on any smaller open subset is also {\\L}ojasiewicz.\\\\\nAny analytic ideal is {\\L}ojasiewicz, as this follows from the fundamental \n{\\L}ojasiewicz inequality for analytic functions \\cite{Lojasiewicz1965}. Moreover, any finitely \ngenerated ideal that is closed in $\\mathcal{E}^\\infty(\\Omega)$ is also a \n{\\L}ojasiewicz ideal, see Corollary 4.4, Ch V, \\cite{Tougeron1972}. Indeed, by Whitney's spectral theorem, a smooth function \nbelongs to a closed finitely generated ideal $I=(f_1,\\ldots,f_k)$ if and only if, \nfor every point $x\\in\\Omega$, its Taylor expansion at $x$ lies in the ideal \ngenerated by the Taylor expansions $T_x f_1,\\ldots,T_x f_k$ in the formal power \nseries algebra $\\mathbb{R}[[X-x]]$. \nHowever, it should be emphasized that there exist {\\L}ojasiewicz ideals \nwhich are not closed, in dimension $n >1$, see Examples 4.8, Ch V, \\cite{Tougeron1972}.\n\\subsection{Smooth points of the locus of zeros}\n\\begin{defn}[Smooth point]\nLet $E \\subset \\mathbb{R}^n$ be closed set. \nA point $x \\in E$ is said to be smooth if, in a neighborhood of $x$, \nthe set $E$ coincides with a $k$-dimensional embedded submanifold of $\\mathbb{R}^n$, \nwhere $k$ is its local dimension at $x$ and may vary with $x$.\n\\end{defn}\n\\begin{Ex}\n The \\emph{Hawaiian earring}, \n defined as the union of circles \n\\[\n\\bigcup_{n=1}^{\\infty} \\left\\{(x,y)\\in \\mathbb{R}^2 : \\left(x-\\frac{1}{n}\\right)^2 + y^2 = \\frac{1}{n^2} \\right\\},\n\\] \nis smooth at every point except the origin, where infinitely many circles accumulate. \nHence, the set of smooth points of the Hawaiian earring is dense in $E= \\bigcup_{n=1}^{\\infty} \\left\\{(x,y)\\in \\mathbb{R}^2 : \\left(x-\\frac{1}{n}\\right)^2 + y^2 = \\frac{1}{n^2} \\right\\}$.\n\n\\begin{theo}\nLet $f:\\mathbb{R}^2\\to\\mathbb{R}$ be a $C^\\infty$ function. \nAssume that $f$ vanishes on the Hawaiian earring\n\\[\n\\mathcal H=\\bigcup_{n\\ge1} C_n, \\qquad \nC_n=\\left\\{(x,y):\\left(x-\\tfrac1n\\right)^2+y^2=\\tfrac1{n^2}\\right\\}.\n\\]\nThen $f$ is flat at the origin: \n\\[\n\\frac{\\partial^{\\alpha_1 +\\alpha_2 f}}{\\partial x_1^{\\alpha_1} \\partial x^{\\alpha_2}_2}(0,0)\n = 0 \n\\qquad \\text{for all } \\alpha =(\\alpha_1,\\alpha_2) \\in\\N^2.\n\\]\n\\end{theo}\n\nThus there is no nonzero homogeneous term in the Taylor expansion of $f$ at $(0,0)$, \nso all derivatives of $f$ at the origin vanish. \nHence $f$ is flat at $(0,0)$.\n\\section{Łojasiewicz Inequality and the Hawaiian Earring}\n\\begin{theo}\nLet $f \\in C^\\infty(\\mathbb{R}^2)$ and assume its zero set is the Hawaiian earring\n\\[\n\\mathcal H:= \\bigcup_{k=1}^{\\infty} \\left\\{ (x,y) \\in \\mathbb{R}^2 : \\left(x - \\frac{1}{k}\\right)^2 + y^2 = \\frac{1}{k^2} \\right\\}.\n\\]\nThen $f$ is flat at the origin and does \\emph{not} satisfy a Łojasiewicz inequality with respect to $H$ at $0$; that is, there exist no constants $C>0$ and $\\theta>0$ such that\n\\[\n|f(x)| \\ge C \\, \\mathrm{dist}(x,H)^\\theta\n\\]\nfor all $x$ in a neighborhood of $0$.\n\\end{theo}\n{\\bf{Proof.}}\nSince $f \\in C^\\infty$ vanishes on the Hawaiian earring, which has tangent directions at $0$ dense in the unit circle $S^1$, Theorem 4 implies that $f$ is flat at the origin:\n\\[\nD^\\alpha f(0) = 0 \\quad \\text{for all multi-indices } \\alpha.\n\\]\nLet $x \\in \\mathbb{R}^2$ be close to $0$. Consider the sequence of circles in $H$ with radii $r_k = 1/k$ and centers $c_k = (1/k,0)$. \nChoosing $k \\sim 1/\\|x\\|$, the circle $C_k$ closest to $x$ has radius $r_k \\sim \\|x\\|$. \nHence, for points near $0$,\n\\[\n\\mathrm{dist}(x,H) \\sim \\|x\\|.\n\\]\nAssume, for contradiction, that there exist constants $C>0$ and $\\theta>0$ such that\n\\[\n|f(x)| \\ge C \\, \\mathrm{dist}(x,H)^\\theta\n\\]\nfor all $x$ near $0$.\n\nChoosing $N > \\theta$ gives\n\\[\n|f(x)| \\ll \\mathrm{dist}(x,H)^\\theta\n\\]\nfor $x$ sufficiently close to $0$, which contradicts the assumed Łojasiewicz inequality. \nTherefore, the ideal generated by\n the function $f$ in the the ring $\\mathcal{E}^\\infty(\\R^2)$ is not a \n Lojasiewicz ideal.\n\\section{{Jet-Determination of Smooth Functions via Families of Arcs}}\n\\begin{theo}\nLet $\\mathcal A = \\{\\gamma_i\\}_{i \\in I}$ be a family of smooth arcs\n\\[\n\\gamma_i : (-\\varepsilon,\\varepsilon) \\to \\mathbb{R}^n, \\qquad \\gamma_i(0)=0, \\quad \\gamma_i \\not\\equiv \\text{constant}.\n\\]\nFor each arc, denote its $m$-jet at $0$ by\n\\[\nj^m \\gamma_i(0) = (\\gamma_i'(0), \\gamma_i''(0), \\dots, \\gamma_i^{(m)}(0)).\n\\]\nThe following assertions are equivalent:\n\\begin{enumerate}\n\\item[(A)] \\textbf{Jet–Determination Property:} For every function $f \\in C^\\infty(\\mathbb{R}^n)$,\n\\[\nf(\\gamma_i(s)) \\equiv 0 \\ \\text{for all } i \\quad \\Longrightarrow \\quad D^\\alpha f(0)=0 \\ \\text{for all multi-indices } \\alpha.\n\\]\n\\item[(B)] \\textbf{Jet–Nondegeneracy Condition:} For every integer $m \\ge 1$, there exists no nonzero homogeneous polynomial $P_m$ of degree $m$ such that\n\\[\nP_m(j^m \\gamma_i(0))=0 \\qquad \\text{for all } i \\in I.\n\\]\n\\end{enumerate}\n\\end{theo}\n{\\bf{Proof.}}\n\\textbf{(A) $\\implies$ (B).} \\\\Suppose (A) holds and, for contradiction, that (B) fails for some $m$. \nThen there exists a nonzero homogeneous polynomial $P_m$ of degree $m$ such that\n\\[\nP_m(j^m \\gamma_i(0)) = 0 \\quad \\text{for all } i.\n\\]\n\nThen $E_n$ is jet-determining at $0$: if $f \\in C^\\infty(\\mathbb{R}^n)$ vanishes on $E_n$, then $f$ is flat at the origin:\n\\[\nD^\\alpha f(0) = 0 \\quad \\text{for all multi-indices } \\alpha.\n\\]\n\\end{cor}", "post_theorem_intro_text_len": 1547, "post_theorem_intro_text": "The proof relies on purely geometric considerations: any nonzero term of the \nTaylor expansion of $f$ at $(0,0)$ would define an algebraic curve of finite \norder, and no such curve can contain infinitely many smooth arcs with tangency \nand curvature behavior comparable to the circles $C_n$.  \nThus the geometry of the Hawaiian earring forces infinite-order vanishing.\n\nWe then formulate a general geometric criterion for flatness, valid for arbitrary \ncollections of smooth arcs tangent at a point with unbounded curvature or \ninfinitely varying radius of osculation.  \nThis leads to a \\emph{degenerate \\L{}ojasiewicz inequality} adapted to situations \nwhere the classical inequality necessarily fails.\n\n\\begin{theo}\nLet $\\Gamma = \\bigcup\\limits_{n \\ge 1} \\gamma_n$ be a union of smooth embedded arcs \nmeeting at a common point $p$ with curvature tending to $\\infty$ or oscillating \nwithout bound.  \nIf $f \\in \\mathcal{E}^\\infty(\\Omega)$ vanishes on $\\Gamma$, then $f$ satisfies a flatness \nestimate of the form\n\\[\n|f(x)| \\le C_N\\, d(x,\\Gamma)^N,\n\\qquad \\text{for all } N \\ge 1 \\text{ and all } x \\text{ near } p,\n\\]\nfor suitable constants $C_N>0$.  \nIn particular, $f$ is flat at $p$.\n\\end{theo}\n\nThis result demonstrates that certain pathological smooth zero sets necessarily \nforce infinite-order degeneracy, and therefore cannot arise from \\L{}ojasiewicz \nideals.  \nIn particular, the ideal generated by a function defining the Hawaiian earring is \nnever \\L{}ojasiewicz, despite the fact that its zero set is smooth away from a \nsingle point.", "sketch": "The proof is described as relying on “purely geometric considerations”: if the Taylor expansion of $f$ at $(0,0)$ had “any nonzero term,” it would “define an algebraic curve of finite order,” but “no such curve can contain infinitely many smooth arcs with tangency and curvature behavior comparable to the circles $C_n$.” Hence, “the geometry of the Hawaiian earring forces infinite-order vanishing,” i.e., flatness at the origin.\n\nThe introduction then says it “formulate[s] a general geometric criterion for flatness” for “collections of smooth arcs tangent at a point with unbounded curvature or infinitely varying radius of osculation,” yielding a “degenerate \\L{}ojasiewicz inequality” (a flatness estimate $|f(x)|\\le C_N d(x,\\Gamma)^N$) in settings “where the classical inequality necessarily fails.”", "expanded_sketch": "The proof is described as relying on “purely geometric considerations”: if the Taylor expansion of $f$ at $(0,0)$ had “any nonzero term,” it would “define an algebraic curve of finite order,” but “no such curve can contain infinitely many smooth arcs with tangency and curvature behavior comparable to the circles $C_n$.” Hence, “the geometry of the Hawaiian earring forces infinite-order vanishing,” i.e., flatness at the origin.\n\nThe introduction then says it “formulate[s] a general geometric criterion for flatness” for “collections of smooth arcs tangent at a point with unbounded curvature or infinitely varying radius of osculation,” yielding a “degenerate \\L{}ojasiewicz inequality” (a flatness estimate $|f(x)|\\le C_N d(x,\\Gamma)^N$) in settings “where the classical inequality necessarily fails.”", "expanded_theorem": "Let $f \\in \\mathcal{E}^\\infty(\\mathbb{R}^2)$ be such that $Z(f)$ is the Hawaiian earring. Then $f$ is flat at the origin:\n\\[\n\\frac{\\partial^{\\alpha +\\beta}f}{\\partial x_1^\\alpha \\partial x_2^\\beta}(0,0)\n = 0 \n\\qquad \\text{for all } \\alpha,\\beta \\in\\mathbb{N}.\n\\],", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let \\(f \\in \\mathcal{E}^\\infty(\\mathbb{R}^2)\\), where \\(\\mathcal{E}^\\infty(\\mathbb{R}^2)\\) denotes the ring of smooth real-valued functions on \\(\\mathbb{R}^2\\). Assume that the zero set\n\\[\nZ(f):=\\{(x,y)\\in \\mathbb{R}^2: f(x,y)=0\\}\n\\]\nis exactly the Hawaiian earring\n\\[\n\\mathcal H= \\bigcup_{k=1}^{\\infty} \\left\\{ (x,y) \\in \\mathbb{R}^2 : \\left(x - \\frac{1}{k}\\right)^2 + y^2 = \\frac{1}{k^2} \\right\\}.\n\\]\nUnder these assumptions, which statement about the behavior of \\(f\\) at the origin holds?", "correct_choice": {"label": "A", "text": "\\(f\\) is flat at the origin; equivalently,\n\\[\n\\frac{\\partial^{\\alpha+\\beta}f}{\\partial x_1^\\alpha\\partial x_2^\\beta}(0,0)=0\n\\qquad \\text{for all } \\alpha,\\beta\\in\\mathbb{N}.\n\\]"}, "choices": [{"label": "B", "text": "\\(f\\) vanishes to finite order at the origin; equivalently, there exists an integer \\(m\\ge 1\\) such that\n\\[\n\\frac{\\partial^{\\alpha+\\beta}f}{\\partial x_1^\\alpha\\partial x_2^\\beta}(0,0)=0\n\\quad \\text{for all } \\alpha+\\beta<m,\n\\qquad \\text{and} \\qquad\n\\frac{\\partial^{m}f}{\\partial x_1^\\alpha\\partial x_2^\\beta}(0,0)\\neq 0\n\\]\nfor some \\(\\alpha,\\beta\\in\\mathbb N\\) with \\(\\alpha+\\beta=m\\)."}, {"label": "C", "text": "\\(f\\) vanishes at the origin together with all first-order partial derivatives; that is,\n\\[\nf(0,0)=0,\n\\qquad\n\\frac{\\partial f}{\\partial x_1}(0,0)=\\frac{\\partial f}{\\partial x_2}(0,0)=0.\n\\]"}, {"label": "D", "text": "\\(f\\) is flat only along the tangent directions determined by the circles of \\(\\mathcal H\\) at the origin; equivalently, every directional derivative of every order of \\(f\\) along such tangent directions vanishes at \\((0,0)\\), but there may exist mixed partial derivatives\n\\[\n\\frac{\\partial^{\\alpha+\\beta}f}{\\partial x_1^\\alpha\\partial x_2^\\beta}(0,0)\\neq 0\n\\]\nfor some \\(\\alpha,\\beta\\in\\mathbb N\\)."}, {"label": "E", "text": "There exists an integer \\(N\\ge 1\\) such that \\(f\\) satisfies a finite-order flatness estimate near the origin,\n\\[\n|f(x,y)|\\le C\\,\\operatorname{dist}\\bigl((x,y),\\mathcal H\\bigr)^N,\n\\]\nfor some constant \\(C>0\\); equivalently, \\(f\\) need not be flat at the origin, but only vanish to sufficiently high finite order there."}], "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": "exclusion_of_any_nonzero_lowest_Taylor_term", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "geometric_construction", "tampered_component": "all_higher_order_vanishing dropped to first-order vanishing only", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "from full jet flatness to directional-flatness-only", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "infinite-order flatness estimate replaced by one fixed finite exponent", "template_used": "uniformity_effectivity"}]}}
{"id": "2512.11523v1", "paper_link": "http://arxiv.org/abs/2512.11523v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{quantization of MA energy} If $L$ is a big and semipositive line bundle on $X$, then for any  $u\\in \\PSH(X,\\theta)\\cap L^\\infty$, we have\n\\[\n\\lim_{k\\to\\infty}E_k(u)=E_\\theta(u).\n\\]", "start_pos": 11715, "end_pos": 11923, "label": "quantization of MA energy"}, "ref_dict": {"epsilon twisted local holomorphic morse": "\\begin{equation}\\label{epsilon twisted local holomorphic morse}\n    \\frac{n!}{k^n}B_{k,\\epsilon}\\leq \\frac{e^{\\rho_k+\\epsilon\\td{\\rho}_k}}{(1-e^{-\\epsilon(\\log k)^2})^n}(\\theta+\\epsilon\\omega)^n,\n\\end{equation}", "Pointwise Semi-Classical OT": "\\begin{thm}\\label{Pointwise Semi-Classical OT} There exists $C>0$ and $p_1\\in \\N$ such that for any $p\\geq p_1$, any $x\\in X$, and any $v\\in (A^p)_x$,  one can find a global holomorphic section $a=a_{p,x,v}\\in H^0(X,A^p)$ with\n\\begin{equation}\\label{OT bound}\n    a(x)=v,\\quad |v|_{(h_0^A)^p}\\leq \\sup_X|a|_{(h_0^A)^p}\\leq \\left(1+\\frac{C}{p}\\right)|v|_{(h_0^A)^p},\n\\end{equation}\nwhere $C=C(n,X,\\omega)$ depends only on dimension $n$, manifold $X$, and K\\\"ahler form $\\omega=c_1(A,h_0^A)$.\n\\end{thm}", "convergence of Bergman measures": "\\begin{cor}\\label{convergence of Bergman measures}  Let $u\\in \\PSH(X,\\theta)\\cap L^\\infty(X)$. Then $\\beta_{L^k,u}$ converges weakly to non-pluripolar measure $\\frac{1}{\\vol L}\\theta_{u}^n$ as $k\\to\\infty$, where $\\vol L$ is the volume of the line bundle $L$, see \\eqref{volume of a line bundle}.\n\\end{cor}", "Bergman kernel comparison": "\\begin{thm}\\label{Bergman kernel comparison} For $\\epsilon\\in \\Q^+$ and admissible $u$, and for sufficiently divisible $k\\gg 1$, we have\n\\[\nB_{L^k,u}\\leq \\left(1+\\frac{C}{\\epsilon k}\\right)^2B_{(L\\otimes A^{\\epsilon})^k,u},\n\\]\nwhere $C=C(n,X,\\omega)$ depends only on dimension $n$, manifold $X$, and K\\\"ahler form $\\omega=c_1(A,h_0^A)$.\n\\end{thm}", "Bergman Measures, Positivity of Direct Images, and Equilibrium": "\\begin{equation}\n    \\frac{d}{dt}\\Big|_{t=0}E_\\theta(P_\\theta(u+tf))=\\frac{1}{\\vol(\\{\\theta\\})}\\int_Xf\\theta_u^n.\n\\end{equation}\n\\end{lem}\n\\section{Bergman Measures, Positivity of Direct Images, and Equilibrium}\\label{Bergman Measures, Positivity of Direct Images, and Equilibrium}\n\\subsection*{Bergman Measure and Quantized Monge--Amp\\`ere Energy}\nLet $V$ be a finite-dimensional complex vector space of dimension $N$. We denote $\\H_V$ by the space of positive-definite hermitian inner products on $V$. For $H,H'\\in \\H_V$, there exists an $H$-hermitian endomorphism $A\\in \\End(V)$ such that\n\\begin{equation}\\label{transfer map}\n(s_1,s_2)_{H'}=(e^As_1,s_2)_H,\\quad s_1,s_2\\in V.\n\\end{equation}", "quantization of MA energy": "\\begin{thm}\\label{quantization of MA energy} If $L$ is a big and semipositive line bundle on $X$, then for any  $u\\in \\PSH(X,\\theta)\\cap L^\\infty$, we have\n\\[\n\\lim_{k\\to\\infty}E_k(u)=E_\\theta(u).\n\\]\n\\end{thm}", "MA envelope": "\\begin{equation}\\label{MA envelope}\n\t\tP_\\theta(f):=usc(\\sup\\{u\\in \\PSH(X,\\theta):u\\leq f\\})\n\t\\end{equation}", "pitfall for finite energy potentials": "\\begin{rmk}\\label{pitfall for finite energy potentials}\nThe argument proving \\eqref{lower bound on energy} fails for general \n\\(u\\in \\mathcal{E}^1(X,\\theta)\\), since one cannot in general guarantee that the \ninitial tangent \\(\\dot u_0\\) of the weak geodesic \\(t\\mapsto u_t\\) in \n\\(\\PSH(X,\\theta)\\) is integrable with respect to \\(\\omega_\\epsilon^n\\) in \n\\eqref{key estimate}, even for arbitrarily small \\(\\epsilon>0\\). One obstruction comes from a counterexample of Di Nezza \n\\cite[Example 4.5]{DN15}, which shows that for any \\(\\epsilon>0\\),\n\\[\n\\mathcal{E}^1(X,\\theta)\\not\\subset \\mathcal{E}^1(X,\\omega_\\epsilon),\n\\]\nIndeed, let \\(u\\in \\mathcal{E}^1(X,\\theta)\\setminus \n\\mathcal{E}^1(X,\\omega_\\epsilon)\\) with \\(u\\le0\\). From \n\\(\\PSH(X,\\theta)\\subset \\PSH(X,\\omega_\\epsilon)\\) and envelope definition of the weak geodesic, the weak geodesic \n\\(v_t\\) joining \\(0\\) and \\(u\\) in \\(\\PSH(X,\\omega_\\epsilon)\\) satisfies\n\\[\nv_t \\le u_t,\\qquad t\\in[0,1],\n\\]\nby the envelope construction. Since \\(u\\notin \\mathcal{E}^1(X,\\omega_\\epsilon)\\), one can  shows that\n\\[\n\\int_X \\dot v_0\\,\\omega_\\epsilon^n = -\\infty.\n\\]\nAs \\(u_t\\) and \\(v_t\\) agree at the endpoints, \nwe get $\n\\dot u_0 \\le \\dot v_0 \\le 0$. This implies that $\\dot{u}_0\\notin L^1(X,\\omega_\\epsilon^n)$ since \n\\[\n\\int_X \\dot u_0\\,\\omega_\\epsilon^n\n   \\;\\le\\; \\int_X \\dot v_0\\,\\omega_\\epsilon^n\n   \\;=\\; -\\infty,\n\\]\nThus, a necessary condition for the integrability of \\(\\dot u_0\\) is that  \n\\(u\\in \\mathcal{E}^1(X,\\omega_\\epsilon)\\). To author's current knowledge, we do not aware of any  general sufficient condition.\n\\end{rmk}", "Hilbert norm": "\\begin{equation}\\label{Hilbert norm}\n\t\t(s_1,s_2)_{Hilb_k(u)}:=\\int_X \\la s_1,s_2\\ra_{(h_0^L)^k}e^{-ku},\\quad k\\in \\N,\n\\end{equation}", "volume of a line bundle": "\\begin{equation}\\label{volume of a line bundle}\n\t\t\\vol(\\{L\\})=\\limsup_{k\\to\\infty}\\frac{\\dim_\\C H^0(X,L^k)}{k^n/n!},\n\t\\end{equation}", "finite energy space and weak geodesic": "\\begin{equation}\\label{volume of a line bundle}\n\t\t\\vol(\\{L\\})=\\limsup_{k\\to\\infty}\\frac{\\dim_\\C H^0(X,L^k)}{k^n/n!},\n\t\\end{equation}\n\twhich we  denote by $\\vol(L)$. Hence, $L$ is big iff the right-hand side of \\eqref{volume of a line bundle} is positive and furthermore $\\limsup$ in \\eqref{volume of a line bundle} is actually a limit by a Theorem of Fujita \\cite{Fujita} (see also  \\cite[\\S 2.2C]{LazarsfeldI}).\n\n\\subsection*{Finite Energy Space and Weak Geodesics}\\label{finite energy space and weak geodesic}\n\tLet $\\theta$ be a closed, smooth $(1,1)$-form so that $\\{\\theta\\}$ is big. If $u\\in\\PSH(X,\\theta)$ has minimal singularity type, then we define \\textit{Monge--Amp\\`ere energy} by\n\t\\begin{equation}\\label{MA energy}\n\tE_\\theta(u):=\\frac{1}{(n+1)\\vol(\\{\\theta\\})}\\sum_{j=0}^n\\int_X (u-V_\\theta) \\theta_u^j\\wedge \\theta^{n-j}_{V_\\theta}\n\t\\end{equation}", "on diagonal Bergman kernel": "\\begin{equation}\\label{on diagonal Bergman kernel}\nK_p(x):=K_p(x,x)=\\sum_{j=1}^{d_p}|S_j(x)|^2_{(h_0^A)^p}.    \n\\end{equation}"}, "pre_theorem_intro_text_len": 5031, "pre_theorem_intro_text": "Given an ample line bundle \\(L\\) over a complex projective manifold \\(X\\), endowed with the Kähler form \\(\\omega\\) polarized by \\(L\\), a central theme in Kähler geometry—first articulated by Yau \\cite[p.~139]{Yau}—is the approximation of transcendental geometric objects on \\((X,\\omega)\\) by asymptotic algebraic data coming from the powers \\(L^k:=L^{\\otimes k}\\). This is the philosophy of \\emph{Kähler quantization}.\n\nThe initial proposal is to approximate the infinite-dimensional space\n\\[\n   \\mathcal{H}_\\omega := \\{\\,\\phi\\in C^\\infty(X) : \\omega + dd^c \\phi > 0\\,\\}\n\\]\nof Kähler potentials by the finite-dimensional spaces \\(\\mathcal{H}_k\\) of positive Hermitian forms on \\(H^0(X,L^k)\\). Via the Kodaira embedding, elements of \\(\\mathcal{H}_k\\) correspond to Fubini–Study metrics and thus form a distinguished  subspace of \\(\\mathcal{H}_\\omega\\). This was established in the seminal works of Tian \\cite{Tian}, Bouche \\cite{Bouche}, Ruan \\cite{Ruan}, Catlin \\cite{Catlin}, Zelditch \\cite{Zelditch}, Lu \\cite{Lu}, among many others, using asymptotics of the Bergman kernel (cf. Theorem~\\ref{on diagonal Bergman kernel}).\n\nDonaldson later proposed that not only the metrics but also the \\emph{geometry} of \\(\\mathcal{H}_\\omega\\) should be captured by the finite-dimensional geometries of \\(\\mathcal{H}_k\\) \\cite[p.~483]{DonaldsonI}. This principle has since been realized in various settings and has had deep impacts on problems concerning canonical metrics and stability; see, for example, \\cite{PS,SongZelditchqu,CS,BerndtssonProb,DLR,DonaldsonI,DonaldsonScalII,RTZ,Zhang} among a rapidly expanding body of literature. We refer to \\cite{Mama} for a detailed exposition of some classical developments in this direction.\n\nA natural question is whether this quantization picture extends beyond the ample case. This has become increasingly relevant in view of recent progress on canonical metrics in degenerate or singular settings \\cite{BBJ,LTW,Li,DZ24,Dervan,Xu,PT25-1,PTT23,PT25-2}, where the positivity of \\(L\\)  only holds in a weak sense.  The present note fits into this broader direction, establishing a quantization result for Monge--Amp\\`ere energy in big and semipositive setting.\n\n\\subsection*{Statement of Main Result}\nWe now describe our results, referring to  Section~\\ref{finite energy space and weak geodesic} and Section~\\ref{Bergman Measures, Positivity of Direct Images, and Equilibrium} for details. Let $L$ be holomorphic line bundle over a complex projective manifold $X$ of dimension $n$. We assume that $L$ is big and semipositive.  Let $h_0^L$ be a smooth metric on $L$ and let $\\theta:=c_1(L,h_0^L)$ denote the Chern curvature of $h_0^L$ so that $\\theta$ is a smooth semipositive $(1,1)$-form.  \n\nOne of the difficulties in the degenerate setting is the lack of smooth potentials. Instead, we work with  the set of $\\theta$-plurisubharmonic functions on $X$, denoted by $\\PSH(X,\\theta)$. This parametrizes singular hermitian metrics on $L$ whose curvature $\\theta_u:=\\theta+dd^cu$ is positive in the current sense. Following the pioneering work of Boucksom et al. \\cite{BEGZ}, for $u\\in \\PSH(X,\\theta)$, one can define the \\textit{non-pluripolar Monge--Amp\\`ere measure} $\\theta_u^n$, which is a Borel measure on $X$ that puts no mass on pluripolar sets. Moreover, we denote by\n\\[\nE_\\theta:\\PSH(X,\\theta)\\to [-\\infty,\\infty)\n\\]\nthe \\textit{Monge--Amp\\`ere energy} or \\textit{Aubin--Yau energy}. It is the primitive of the Monge--Amp\\`ere operator and plays an important role in the variational approach to canonical K\\\"ahler metrics.\n\nOn the other hand, let $K_X:=\\Omega^{n,0}_X$ be the canonical bundle on $X$. For $k\\in\\mathbb N$, we consider the adjoint line bundle $L^k\\otimes K_X$. For global sections $s_1,s_2\\in H^0(X,L^k\\otimes K_X)$, locally written  as\n\\[\ns_j=\\sigma_j\\, dz_1\\wedge \\cdots \\wedge dz_n,\\quad j=1,2,\n\\]\nwhere $\\sigma_j$ are local holomorphic sections of $L^k$, we set\n\\[\n\\langle s_1,s_2\\ra_{(h_0^L)^k}:=\\langle \\sigma_1,\\sigma_2\\ra_{(h_0^L)^k}\\left(\\frac{i}{2}\\right)^n\ndz_1\\wedge d\\bar{z}_1\\wedge \\cdots \\wedge dz_n\\wedge d\\bar{z}_n,\n\\]\nyielding  a smooth volume form on $X$. For a function $u:X\\to [-\\infty,\\infty)$, we define the \\textit{Hilbert norm}\n\\begin{equation}\\label{Hilbert norm}\n\t\t(s_1,s_2)_{Hilb_k(u)}:=\\int_X \\langle s_1,s_2\\ra_{(h_0^L)^k}e^{-ku},\\quad k\\in \\mathbb N,\n\\end{equation}\nfor sections $s_1,s_2\\in H^0(X,L^k\\otimes K_X)$ such that \\eqref{Hilbert norm} is finite. We say $u$ is \\textit{admissible} if $Hilb_k(u)$ is finite on the whole space $H^0(X,L^k\\otimes K_X)$ for any $k\\in \\mathbb N$. Clearly, any bounded function is admissible. For an admissible function $u$, we define the \\textit{quantized Monge--Amp\\`ere energy} by\n\\begin{equation}\\label{quantized MA-intro}\n    E_k(u):=-\\frac{1}{kN_k}\\log\\left(\\frac{\\det Hilb_k(u)}{\\det Hilb_k(0)}\\right),\n\\end{equation}\nwhich is first introduced by Donaldson \\cite{DonaldsonScalII} (see also Berman–Boucksom \\cite{BB10}).\n\nOur main result is the  quantization of Monge--Amp\\`ere energy in adjoint twisting.", "context": "The initial proposal is to approximate the infinite-dimensional space\n\\[\n \\mathcal{H}_\\omega := \\{\\,\\phi\\in C^\\infty(X) : \\omega + dd^c \\phi > 0\\,\\}\n\\]\nof Kähler potentials by the finite-dimensional spaces \\(\\mathcal{H}_k\\) of positive Hermitian forms on \\(H^0(X,L^k)\\). Via the Kodaira embedding, elements of \\(\\mathcal{H}_k\\) correspond to Fubini–Study metrics and thus form a distinguished subspace of \\(\\mathcal{H}_\\omega\\). This was established in the seminal works of Tian \\cite{Tian}, Bouche \\cite{Bouche}, Ruan \\cite{Ruan}, Catlin \\cite{Catlin}, Zelditch \\cite{Zelditch}, Lu \\cite{Lu}, among many others, using asymptotics of the Bergman kernel (cf. Theorem~\\ref{on diagonal Bergman kernel}).\n\nA natural question is whether this quantization picture extends beyond the ample case. This has become increasingly relevant in view of recent progress on canonical metrics in degenerate or singular settings \\cite{BBJ,LTW,Li,DZ24,Dervan,Xu,PT25-1,PTT23,PT25-2}, where the positivity of \\(L\\) only holds in a weak sense. The present note fits into this broader direction, establishing a quantization result for Monge--Amp\\`ere energy in big and semipositive setting.\n\n\\subsection*{Statement of Main Result}\nWe now describe our results, referring to Section~\\ref{finite energy space and weak geodesic} and Section~\\ref{Bergman Measures, Positivity of Direct Images, and Equilibrium} for details. Let $L$ be holomorphic line bundle over a complex projective manifold $X$ of dimension $n$. We assume that $L$ is big and semipositive. Let $h_0^L$ be a smooth metric on $L$ and let $\\theta:=c_1(L,h_0^L)$ denote the Chern curvature of $h_0^L$ so that $\\theta$ is a smooth semipositive $(1,1)$-form.\n\nOne of the difficulties in the degenerate setting is the lack of smooth potentials. Instead, we work with the set of $\\theta$-plurisubharmonic functions on $X$, denoted by $\\PSH(X,\\theta)$. This parametrizes singular hermitian metrics on $L$ whose curvature $\\theta_u:=\\theta+dd^cu$ is positive in the current sense. Following the pioneering work of Boucksom et al. \\cite{BEGZ}, for $u\\in \\PSH(X,\\theta)$, one can define the \\textit{non-pluripolar Monge--Amp\\`ere measure} $\\theta_u^n$, which is a Borel measure on $X$ that puts no mass on pluripolar sets. Moreover, we denote by\n\\[\nE_\\theta:\\PSH(X,\\theta)\\to [-\\infty,\\infty)\n\\]\nthe \\textit{Monge--Amp\\`ere energy} or \\textit{Aubin--Yau energy}. It is the primitive of the Monge--Amp\\`ere operator and plays an important role in the variational approach to canonical K\\\"ahler metrics.\n\nOn the other hand, let $K_X:=\\Omega^{n,0}_X$ be the canonical bundle on $X$. For $k\\in\\mathbb N$, we consider the adjoint line bundle $L^k\\otimes K_X$. For global sections $s_1,s_2\\in H^0(X,L^k\\otimes K_X)$, locally written as\n\\[\ns_j=\\sigma_j\\, dz_1\\wedge \\cdots \\wedge dz_n,\\quad j=1,2,\n\\]\nwhere $\\sigma_j$ are local holomorphic sections of $L^k$, we set\n\\[\n\\langle s_1,s_2\\ra_{(h_0^L)^k}:=\\langle \\sigma_1,\\sigma_2\\ra_{(h_0^L)^k}\\left(\\frac{i}{2}\\right)^n\ndz_1\\wedge d\\bar{z}_1\\wedge \\cdots \\wedge dz_n\\wedge d\\bar{z}_n,\n\\]\nyielding a smooth volume form on $X$. For a function $u:X\\to [-\\infty,\\infty)$, we define the \\textit{Hilbert norm}\n\\begin{equation}\\label{Hilbert norm}\n (s_1,s_2)_{Hilb_k(u)}:=\\int_X \\langle s_1,s_2\\ra_{(h_0^L)^k}e^{-ku},\\quad k\\in \\mathbb N,\n\\end{equation}\nfor sections $s_1,s_2\\in H^0(X,L^k\\otimes K_X)$ such that \\eqref{Hilbert norm} is finite. We say $u$ is \\textit{admissible} if $Hilb_k(u)$ is finite on the whole space $H^0(X,L^k\\otimes K_X)$ for any $k\\in \\mathbb N$. Clearly, any bounded function is admissible. For an admissible function $u$, we define the \\textit{quantized Monge--Amp\\`ere energy} by\n\\begin{equation}\\label{quantized MA-intro}\n E_k(u):=-\\frac{1}{kN_k}\\log\\left(\\frac{\\det Hilb_k(u)}{\\det Hilb_k(0)}\\right),\n\\end{equation}\nwhich is first introduced by Donaldson \\cite{DonaldsonScalII} (see also Berman–Boucksom \\cite{BB10}).\n\nOur main result is the quantization of Monge--Amp\\`ere energy in adjoint twisting.\n\n\\begin{equation}\n    \\frac{d}{dt}\\Big|_{t=0}E_\\theta(P_\\theta(u+tf))=\\frac{1}{\\vol(\\{\\theta\\})}\\int_Xf\\theta_u^n.\n\\end{equation}\n\\end{lem}\n\\section{Bergman Measures, Positivity of Direct Images, and Equilibrium}\\label{Bergman Measures, Positivity of Direct Images, and Equilibrium}\n\\subsection*{Bergman Measure and Quantized Monge--Amp\\`ere Energy}\nLet $V$ be a finite-dimensional complex vector space of dimension $N$. We denote $\\H_V$ by the space of positive-definite hermitian inner products on $V$. For $H,H'\\in \\H_V$, there exists an $H$-hermitian endomorphism $A\\in \\End(V)$ such that\n\\begin{equation}\\label{transfer map}\n(s_1,s_2)_{H'}=(e^As_1,s_2)_H,\\quad s_1,s_2\\in V.\n\\end{equation}\n\n\\begin{equation}\\label{Hilbert norm}\n\t\t(s_1,s_2)_{Hilb_k(u)}:=\\int_X \\la s_1,s_2\\ra_{(h_0^L)^k}e^{-ku},\\quad k\\in \\N,\n\\end{equation}\n\n\\begin{equation}\\label{volume of a line bundle}\n\t\t\\vol(\\{L\\})=\\limsup_{k\\to\\infty}\\frac{\\dim_\\C H^0(X,L^k)}{k^n/n!},\n\t\\end{equation}\n\twhich we  denote by $\\vol(L)$. Hence, $L$ is big iff the right-hand side of \\eqref{volume of a line bundle} is positive and furthermore $\\limsup$ in \\eqref{volume of a line bundle} is actually a limit by a Theorem of Fujita \\cite{Fujita} (see also  \\cite[\\S 2.2C]{LazarsfeldI}).\n\n\\subsection*{Finite Energy Space and Weak Geodesics}\\label{finite energy space and weak geodesic}\n\tLet $\\theta$ be a closed, smooth $(1,1)$-form so that $\\{\\theta\\}$ is big. If $u\\in\\PSH(X,\\theta)$ has minimal singularity type, then we define \\textit{Monge--Amp\\`ere energy} by\n\t\\begin{equation}\\label{MA energy}\n\tE_\\theta(u):=\\frac{1}{(n+1)\\vol(\\{\\theta\\})}\\sum_{j=0}^n\\int_X (u-V_\\theta) \\theta_u^j\\wedge \\theta^{n-j}_{V_\\theta}\n\t\\end{equation}\n\n\\begin{equation}\\label{on diagonal Bergman kernel}\nK_p(x):=K_p(x,x)=\\sum_{j=1}^{d_p}|S_j(x)|^2_{(h_0^A)^p}.    \n\\end{equation}", "full_context": "The initial proposal is to approximate the infinite-dimensional space\n\\[\n \\mathcal{H}_\\omega := \\{\\,\\phi\\in C^\\infty(X) : \\omega + dd^c \\phi > 0\\,\\}\n\\]\nof Kähler potentials by the finite-dimensional spaces \\(\\mathcal{H}_k\\) of positive Hermitian forms on \\(H^0(X,L^k)\\). Via the Kodaira embedding, elements of \\(\\mathcal{H}_k\\) correspond to Fubini–Study metrics and thus form a distinguished subspace of \\(\\mathcal{H}_\\omega\\). This was established in the seminal works of Tian \\cite{Tian}, Bouche \\cite{Bouche}, Ruan \\cite{Ruan}, Catlin \\cite{Catlin}, Zelditch \\cite{Zelditch}, Lu \\cite{Lu}, among many others, using asymptotics of the Bergman kernel (cf. Theorem~\\ref{on diagonal Bergman kernel}).\n\nA natural question is whether this quantization picture extends beyond the ample case. This has become increasingly relevant in view of recent progress on canonical metrics in degenerate or singular settings \\cite{BBJ,LTW,Li,DZ24,Dervan,Xu,PT25-1,PTT23,PT25-2}, where the positivity of \\(L\\) only holds in a weak sense. The present note fits into this broader direction, establishing a quantization result for Monge--Amp\\`ere energy in big and semipositive setting.\n\n\\subsection*{Statement of Main Result}\nWe now describe our results, referring to Section~\\ref{finite energy space and weak geodesic} and Section~\\ref{Bergman Measures, Positivity of Direct Images, and Equilibrium} for details. Let $L$ be holomorphic line bundle over a complex projective manifold $X$ of dimension $n$. We assume that $L$ is big and semipositive. Let $h_0^L$ be a smooth metric on $L$ and let $\\theta:=c_1(L,h_0^L)$ denote the Chern curvature of $h_0^L$ so that $\\theta$ is a smooth semipositive $(1,1)$-form.\n\nOne of the difficulties in the degenerate setting is the lack of smooth potentials. Instead, we work with the set of $\\theta$-plurisubharmonic functions on $X$, denoted by $\\PSH(X,\\theta)$. This parametrizes singular hermitian metrics on $L$ whose curvature $\\theta_u:=\\theta+dd^cu$ is positive in the current sense. Following the pioneering work of Boucksom et al. \\cite{BEGZ}, for $u\\in \\PSH(X,\\theta)$, one can define the \\textit{non-pluripolar Monge--Amp\\`ere measure} $\\theta_u^n$, which is a Borel measure on $X$ that puts no mass on pluripolar sets. Moreover, we denote by\n\\[\nE_\\theta:\\PSH(X,\\theta)\\to [-\\infty,\\infty)\n\\]\nthe \\textit{Monge--Amp\\`ere energy} or \\textit{Aubin--Yau energy}. It is the primitive of the Monge--Amp\\`ere operator and plays an important role in the variational approach to canonical K\\\"ahler metrics.\n\nOn the other hand, let $K_X:=\\Omega^{n,0}_X$ be the canonical bundle on $X$. For $k\\in\\mathbb N$, we consider the adjoint line bundle $L^k\\otimes K_X$. For global sections $s_1,s_2\\in H^0(X,L^k\\otimes K_X)$, locally written as\n\\[\ns_j=\\sigma_j\\, dz_1\\wedge \\cdots \\wedge dz_n,\\quad j=1,2,\n\\]\nwhere $\\sigma_j$ are local holomorphic sections of $L^k$, we set\n\\[\n\\langle s_1,s_2\\ra_{(h_0^L)^k}:=\\langle \\sigma_1,\\sigma_2\\ra_{(h_0^L)^k}\\left(\\frac{i}{2}\\right)^n\ndz_1\\wedge d\\bar{z}_1\\wedge \\cdots \\wedge dz_n\\wedge d\\bar{z}_n,\n\\]\nyielding a smooth volume form on $X$. For a function $u:X\\to [-\\infty,\\infty)$, we define the \\textit{Hilbert norm}\n\\begin{equation}\\label{Hilbert norm}\n (s_1,s_2)_{Hilb_k(u)}:=\\int_X \\langle s_1,s_2\\ra_{(h_0^L)^k}e^{-ku},\\quad k\\in \\mathbb N,\n\\end{equation}\nfor sections $s_1,s_2\\in H^0(X,L^k\\otimes K_X)$ such that \\eqref{Hilbert norm} is finite. We say $u$ is \\textit{admissible} if $Hilb_k(u)$ is finite on the whole space $H^0(X,L^k\\otimes K_X)$ for any $k\\in \\mathbb N$. Clearly, any bounded function is admissible. For an admissible function $u$, we define the \\textit{quantized Monge--Amp\\`ere energy} by\n\\begin{equation}\\label{quantized MA-intro}\n E_k(u):=-\\frac{1}{kN_k}\\log\\left(\\frac{\\det Hilb_k(u)}{\\det Hilb_k(0)}\\right),\n\\end{equation}\nwhich is first introduced by Donaldson \\cite{DonaldsonScalII} (see also Berman–Boucksom \\cite{BB10}).\n\nOur main result is the quantization of Monge--Amp\\`ere energy in adjoint twisting.\n\n\\begin{equation}\n    \\frac{d}{dt}\\Big|_{t=0}E_\\theta(P_\\theta(u+tf))=\\frac{1}{\\vol(\\{\\theta\\})}\\int_Xf\\theta_u^n.\n\\end{equation}\n\\end{lem}\n\\section{Bergman Measures, Positivity of Direct Images, and Equilibrium}\\label{Bergman Measures, Positivity of Direct Images, and Equilibrium}\n\\subsection*{Bergman Measure and Quantized Monge--Amp\\`ere Energy}\nLet $V$ be a finite-dimensional complex vector space of dimension $N$. We denote $\\H_V$ by the space of positive-definite hermitian inner products on $V$. For $H,H'\\in \\H_V$, there exists an $H$-hermitian endomorphism $A\\in \\End(V)$ such that\n\\begin{equation}\\label{transfer map}\n(s_1,s_2)_{H'}=(e^As_1,s_2)_H,\\quad s_1,s_2\\in V.\n\\end{equation}\n\n\\begin{equation}\\label{Hilbert norm}\n\t\t(s_1,s_2)_{Hilb_k(u)}:=\\int_X \\la s_1,s_2\\ra_{(h_0^L)^k}e^{-ku},\\quad k\\in \\N,\n\\end{equation}\n\n\\begin{equation}\\label{volume of a line bundle}\n\t\t\\vol(\\{L\\})=\\limsup_{k\\to\\infty}\\frac{\\dim_\\C H^0(X,L^k)}{k^n/n!},\n\t\\end{equation}\n\twhich we  denote by $\\vol(L)$. Hence, $L$ is big iff the right-hand side of \\eqref{volume of a line bundle} is positive and furthermore $\\limsup$ in \\eqref{volume of a line bundle} is actually a limit by a Theorem of Fujita \\cite{Fujita} (see also  \\cite[\\S 2.2C]{LazarsfeldI}).\n\n\\subsection*{Finite Energy Space and Weak Geodesics}\\label{finite energy space and weak geodesic}\n\tLet $\\theta$ be a closed, smooth $(1,1)$-form so that $\\{\\theta\\}$ is big. If $u\\in\\PSH(X,\\theta)$ has minimal singularity type, then we define \\textit{Monge--Amp\\`ere energy} by\n\t\\begin{equation}\\label{MA energy}\n\tE_\\theta(u):=\\frac{1}{(n+1)\\vol(\\{\\theta\\})}\\sum_{j=0}^n\\int_X (u-V_\\theta) \\theta_u^j\\wedge \\theta^{n-j}_{V_\\theta}\n\t\\end{equation}\n\n\\begin{equation}\\label{on diagonal Bergman kernel}\nK_p(x):=K_p(x,x)=\\sum_{j=1}^{d_p}|S_j(x)|^2_{(h_0^A)^p}.    \n\\end{equation}\n\nRescaling $w=\\sqrt{k}z$ on the standard shrinking polydisc gives the uniform approximation\n\\[\nk\\Phi_\\epsilon(w/\\sqrt{k})\n=\\Phi_\\epsilon^0(w)+\\rho_k+\\epsilon\\widetilde\\rho_k,\n\\qquad \n\\rho_k,\\widetilde\\rho_k\\to0.\n\\]\nRepeating the submean-value argument used for Theorem~\\ref{Berman's local Morse inequality} then yields\n\\[\n\\frac{n!|s|^2(x)}{k^n\\|s\\|^2}\n\\le\ne^{\\rho_k+\\epsilon\\widetilde\\rho_k}\n\\frac{n!dV_n}{(2\\pi)^n\\prod_{j=1}^n\\int_0^{\\log k} e^{-(\\lambda_j+\\epsilon)r^2}r\\,dr}.\n\\]\nSince the quadratic model satisfies $\\lambda_j+\\epsilon\\ge\\epsilon>0$, the Gaussian integrals in the denominator behave exactly as in the proof of Theorem~\\ref{Berman's local Morse inequality}, with $\\lambda_j$ replaced by $\\lambda_j+\\epsilon$. \nUsing the identity\n\\[\n(\\theta+\\epsilon\\omega)^n\n= \\frac{n!}{\\pi^n}\\!\\left(\\prod_{j=1}^n (\\lambda_j+\\epsilon)\\right)dV_n,\n\\]\nthe same computation as in the untwisted case yields\n\\[\n\\frac{n!|s|^2(x)}{k^n\\|s\\|^2}\n\\;\\le\\;\ne^{\\rho_k+\\epsilon\\widetilde\\rho_k}\n\\Biggl(\\prod_{j=1}^n\n \\frac{1}{1-e^{-(\\lambda_j+\\epsilon)(\\log k)^2}}\\Biggr)\n(\\theta+\\epsilon\\omega)^n(x)\\leq {(1-e^{-\\epsilon(\\log k)^2})^n}(\\theta+\\epsilon\\omega)^n(x),\n\\]\nsince $\\lambda_j(x)+\\epsilon\\ge\\epsilon$ for all $j$ and $x\\in X$. \\eqref{epsilon twisted local holomorphic morse} then follows from extremal characterization \\eqref{extremal for twisted Bergman measure}.\n\\end{proof}\n\\section{Proof of Theorem \\ref{quantization of MA energy} and Corollary \\ref{convergence of Bergman measures}}\nRecall that $X$ is a compact projective manifold of dimension $n$, $L$ be a big and semipositive line bundle over $X$ with fixed smooth metric $h_0^L$ on $L$ with Chern curvature form $\\theta:=c_1(L,h^L_0)\\geq 0$. Let us first deduce Corollary \\ref{convergence of Bergman measures} from Theorem \\ref{quantization of MA energy}.\n\\begin{proof}[Proof of Corollary \\ref{convergence of Bergman measures}] For $f\\in C^0(X)$ and $t\\in \\R$, $u+tf$ is bounded and thus is admissible. We set \n\\[\nf_k(t):=E_k(u+tf),\\quad g(t):=E_\\theta(P_\\theta(u+tf)).\n\\]\nNotice that $u+tf\\geq P_\\theta(u+tf)\\geq m:=\\inf_X(u+tf)>-\\infty$ by definition of envelope and thus, $P_\\theta(u+tf)\\in \\PSH(X,\\theta)\\cap L^\\infty$. By Theorem \\ref{quantization of MA energy}, we obtain that for any $t\\in \\R$,\n\\[\n\\liminf_{k\\to\\infty}f_k(t)\\geq \\lim E_k(P_\\theta(u+tf))=g(t),\\quad \\lim_{k\\to\\infty}f_k(0)=\\lim_{k\\to\\infty}E_k(u)=E_\\theta(u)=g(0).\n\\]\nBy Lemma \\ref{variational formula for QMA along affine path}, $f_k(t)$ is concave and is differentiable at $t=0$:\n\\[\nf_k'(0)=\\int_X f\\beta_{L^k,u}.\n\\]\nOn the other hand, by Lemma \\ref{variation formula of MA energy}, $g(t)$ is differentiable at $t=0$ and\n\\[\ng'(0)=\\int_Xf\\frac{\\theta_u^n}{\\vol(\\{\\theta\\})}.\n\\]\nThe result follows from the elementary lemma proved in \\cite[Lemma 3.1]{BBWN}.\n\\end{proof}\n\\begin{rmk}\n In fact, the arguments in Theorem \\ref{upper bound estimate} below shows that $\\lim_{k\\to\\infty}f_k(t)=g(t)$.\n\\end{rmk}\n\nWe now begin with the upper bound in Theorem \\ref{quantization of MA energy} which holds for any finite-energy potentials.\n\\begin{thm}\\label{upper bound estimate} For $u\\in \\E^1(X,\\theta)$, $\\limsup_{k\\to\\infty}E_k(u)\\leq E_\\theta(u)$.\n\\end{thm}\n\\begin{proof} The proof goes exactly as in \\cite[Theorem 3.5]{BF14} and \\cite[Theorem A]{BB10}. Since $u$ is upper semi-continuous, we can find a sequence $\\phi_j\\in C^\\infty(X)$ such that $\\phi_j\\downarrow u$. Clearly, $\\phi_j$ is admissible and $E_k(u)\\leq E_k(\\phi_j)$ by Lemma \\ref{properties of Hilb and QMA} (ii). By Corollary \\ref{Quantization of MOnge-Ampere energy for smooth functions}, \n\\[\n\\limsup_{k\\to\\infty}E_k(u)\\leq \\lim_{k\\to\\infty}E_k(\\phi_j)=E_\\theta(P_\\theta(\\phi_j)),\\quad \\forall j\\in \\N.\n\\]\nOn the other hand, by definition of envelope, $u\\leq P_\\theta(\\phi_j)\\leq \\phi_j$ and thus $P_\\theta(\\phi_j)\\downarrow u$. Then\n\\[\n\\lim_{k\\to\\infty}E_k(u)\\leq\\lim_{j\\to\\infty}E_\\theta(P_\\theta(\\phi_j))=E_\\theta(u),\n\\]\nby continuity of Monge--Amp\\`ere energy along decreasing sequences. \n\\end{proof}\n\\begin{rmk} The proof actually shows that Theorem \\ref{upper bound estimate} holds for any big and nef line bundle $L$.\n\\end{rmk}\n\\begin{proof}[Proof of Theorem \\ref{quantization of MA energy}] It remains to prove that for any $u\\in \\PSH(X,\\theta)\\cap L^\\infty$, we have\n\\begin{equation}\\label{lower bound on energy}\n \\liminf_{{k\\to\\infty}}E_k(u)\\geq E_\\theta(u).\n\\end{equation} \nFollowing \\cite{BF14}, we consider the following functional on $\\E^1(X,\\theta)$:\n\\[\n\\mathcal{F}_k(u):=E_k(u)-E_\\theta(u),\\quad u\\in \\E^1(X,\\theta).\n\\]\nBy Lemma \\ref{Properties on Energy and E1} (iii) and Lemma \\ref{properties of Hilb and QMA} (ii), $\\F_k$ is continuous along decreasing sequences in $\\E^1(X,\\theta)$. Also, by definition \\eqref{MA energy} and Lemma \\ref{properties of Hilb and QMA} (iii), $\\F_k(u+C)=\\F_k(u)$ for $C\\in \\R$. For fixed $u\\in \\PSH(X,\\theta)\\cap L^\\infty$, we may assume that $u\\leq 0$ by addition of a constant. By Lemma \\ref{minimal singular/finite energy geodesic} (iii), $\\dot{u}_0\\in L^\\infty(X)$.\n\nNow, for fixed $\\epsilon\\in \\Q^+$ and an positive line bundle $(A,h_0^A)$ with curvature $\\omega$, we set $\\omega_\\epsilon:=\\theta+\\epsilon\\omega$. By Theorem \\ref{Bergman kernel comparison} and \\eqref{epsilon twisted local holomorphic morse}, \n\\[\n\\frac{1}{N_k}B_{L^k,0}\\leq N_k^{-1}\\left(1+\\frac\n{C}{\\epsilon k}\\right)^2B_{k,\\epsilon}\\leq \\frac{k^n/n!}{N_k}\\frac{e^{\\rho_k+\\epsilon\\td{\\rho}_k}}{(1-e^{-\\epsilon(\\log k)^2})^n}\\omega_\\epsilon^n\n\\]\nAs $\\dot{u}_0\\leq 0$, we obtain\n\\begin{equation}\\label{key estimate}\n\\F_k(u)\\geq \\int_X\\dot{u}_0\\left(\\left(1+\\frac{C}{\\epsilon k}\\right)^2\\frac{k^n/n!}{N_k}\\frac{e^{\\rho_k+\\epsilon\\td{\\rho}_k}}{(1-e^{-\\epsilon(\\log k)^2})^n}\\omega_\\epsilon^n-\\frac{1}{\\vol L}\\theta^n\\right). \n\\end{equation}\nTaking $k\\to \\infty$, we see that for fixed $\\epsilon\\in \\Q^+$, since $\\rho_k,\\td{\\rho}_k\\to 0$ and from \\eqref{volume of a line bundle} (since $L$ is big):\n\\[\n1+\\frac{C}{\\epsilon k}\\to 1,\\quad \\frac{k^n/n!}{N_k}\\to \\frac{1}{\\vol L},\\quad\\frac{e^{\\rho_k+\\epsilon\\td{\\rho}_k}}{(1-e^{-\\epsilon(\\log k)^2})^n}\\to 1.\n\\]\nAs a result, we obtain for fixed $\\epsilon\\in \\Q^+$,\n\\[\n\\liminf_{k\\to\\infty}\\F_k(u)\\geq \\frac{1}{\\vol L}\\int_X \\dot{u}_0\\left(\\omega_\\epsilon^n-\\theta^n\\right)=\\frac{1}{\\vol L}\\sum_{j=1}^n\\int_X\\dot{u}_0\\theta^{n-j}\\wedge (\\epsilon \\omega)^j.\n\\]\nTaking $\\epsilon\\to 0+$, we obtain the result.\n\\end{proof}\\begin{rmk}\\label{pitfall for finite energy potentials}\nThe argument proving \\eqref{lower bound on energy} fails for general \n\\(u\\in \\mathcal{E}^1(X,\\theta)\\), since one cannot in general guarantee that the \ninitial tangent \\(\\dot u_0\\) of the weak geodesic \\(t\\mapsto u_t\\) in \n\\(\\PSH(X,\\theta)\\) is integrable with respect to \\(\\omega_\\epsilon^n\\) in \n\\eqref{key estimate}, even for arbitrarily small \\(\\epsilon>0\\). One obstruction comes from a counterexample of Di Nezza \n\\cite[Example 4.5]{DN15}, which shows that for any \\(\\epsilon>0\\),\n\\[\n\\mathcal{E}^1(X,\\theta)\\not\\subset \\mathcal{E}^1(X,\\omega_\\epsilon),\n\\]\nIndeed, let \\(u\\in \\mathcal{E}^1(X,\\theta)\\setminus \n\\mathcal{E}^1(X,\\omega_\\epsilon)\\) with \\(u\\le0\\). From \n\\(\\PSH(X,\\theta)\\subset \\PSH(X,\\omega_\\epsilon)\\) and envelope definition of the weak geodesic, the weak geodesic \n\\(v_t\\) joining \\(0\\) and \\(u\\) in \\(\\PSH(X,\\omega_\\epsilon)\\) satisfies\n\\[\nv_t \\le u_t,\\qquad t\\in[0,1],\n\\]\nby the envelope construction. Since \\(u\\notin \\mathcal{E}^1(X,\\omega_\\epsilon)\\), one can shows that\n\\[\n\\int_X \\dot v_0\\,\\omega_\\epsilon^n = -\\infty.\n\\]\nAs \\(u_t\\) and \\(v_t\\) agree at the endpoints, \nwe get $\n\\dot u_0 \\le \\dot v_0 \\le 0$. This implies that $\\dot{u}_0\\notin L^1(X,\\omega_\\epsilon^n)$ since \n\\[\n\\int_X \\dot u_0\\,\\omega_\\epsilon^n\n \\;\\le\\; \\int_X \\dot v_0\\,\\omega_\\epsilon^n\n \\;=\\; -\\infty,\n\\]\nThus, a necessary condition for the integrability of \\(\\dot u_0\\) is that \n\\(u\\in \\mathcal{E}^1(X,\\omega_\\epsilon)\\). To author's current knowledge, we do not aware of any general sufficient condition.\n\\end{rmk}\n\\printbibliography", "post_theorem_intro_text_len": 6334, "post_theorem_intro_text": "A result such as Theorem~\\ref{quantization of MA energy} was raised as an open problem in Berman–Freixas i Montplet \\cite[Remark~3.6]{BF14}, and our theorem provides a partial answer in the big and semipositive setting.\nWhen $L$ is ample and no adjoint twisting is present, Donaldson \\cite{DonaldsonScalII} first established the convergence for smooth Kähler potentials; this was later extended to continuous potentials by Berman–Boucksom \\cite[Theorem~A]{BB10} and to all finite-energy potentials by Darvas–Lu–Rubinstein \\cite{DLR}.\nIn the adjoint setting, Berman–Freixas i Montplet \\cite[Theorem~3.5]{BF14} proved the quantization of energy for finite-energy potentials, and subsequently Darvas–Xia \\cite{DX22} obtained the result for arbitrary twisting.\nIn the big, untwisted case, Berman–Boucksom \\cite[Theorem~A]{BB10} showed convergence for all continuous metrics.\nMore recently, Darvas–Xia \\cite{DX24} proved a “partial quantization’’ theorem for continuous metrics under arbitrary twisting on a pseudoeffective line bundle.\n\nTheorem \\ref{quantization of MA energy} has an immediate application to quantization of non-pluripolar measure. For an admissible $u$, we fix a basis $\\{S_1,\\dots,S_{N_k}\\}$ of $H^0(X,L^k\\otimes K_X)$ with $N_k=\\dim_\\mathbb C H^0(X,L^k\\otimes K_X)$. Then we define \\textit{Bergman measure} and its normalization by\n\t\\begin{equation}\\label{Bergman measure}\n\tB_{L^k,u}(x)=\\sum_{j=1}^{N_k}|S_j(x)|^2_{(h_u^L)^k}=\\sum_{j=1}^{N_k}|S_j(x)|^2_{(h_0^L)^k}e^{-ku(x)},\\quad \\beta_{L^k,u}=\\frac{1}{N_k}B_{L^k,u}.\n\t\\end{equation}\nCoupling with  a standard trick in Berman--Boucksom--Witt Nystr\\\"om \\cite{BBWN}, Theorem \\ref{quantization of MA energy} yields the following weak convergence of normalized Bergman measures to non-pluripolar measures\n\\begin{cor}\\label{convergence of Bergman measures}  Let $u\\in \\PSH(X,\\theta)\\cap L^\\infty(X)$. Then $\\beta_{L^k,u}$ converges weakly to non-pluripolar measure $\\frac{1}{\\vol L}\\theta_{u}^n$ as $k\\to\\infty$, where $\\vol L$ is the volume of the line bundle $L$, see \\eqref{volume of a line bundle}.\n\\end{cor}\nOur approach refines the arguments in \\cite{BF14} as follows. Fix an ample line bundle $A$ on $X$ and $\\varepsilon\\in \\mathbb Q^+$. We prove a comparison Theorem for Bergman measure to $L$ and to small ample twist $L\\otimes A^\\varepsilon$ with almost optimal asymptotic bound (Theorem \\ref{Bergman kernel comparison}). This allows us to apply Berman's local Morse inequality \\cite[Proposition 2.5]{BF14} to the twisted bundle with explicit dependence on $\\varepsilon$, see  \\eqref{epsilon twisted local holomorphic morse}. Combining this with the standard variational relations between the Monge–Ampère energy and non-pluripolar measures, the quantized energy and Bergman measures, and the convexity of the quantized Monge–Ampère energy along weak geodesics,  we can carry out the proof of Theorem \\ref{quantization of MA energy} for bounded potentials following standard argument as in \\cite[Theorem A]{BB10} and \\cite[Theorem 3.5]{BF14}. However, as we will explain in Remark \\ref{pitfall for finite energy potentials}, our method does not presently extend to arbitrary finite-energy potentials, due to integrability issues for initial tangent vectors of weak geodesics. \n\nAs this work was being finalized, Siarhei Finski kindly informed us of his related work \\cite{FinskiBig}, in which he quantizes the Darvas–Mabuchi $d_p$ distances on big line bundles and for potentials of the type $P_\\theta(f)$ (defined below \\ref{MA envelope}), where $f$ is continuous. In particular, for $p=1$ and $\\theta$ semipositive, his result implies the version of Theorem~\\ref{quantization of MA energy} for such potentials,  a subset of the class $\\PSH(X,\\theta)\\cap L^\\infty$. Our setting and approach differ from his in two main aspects. His quantization is formulated using the untwisted section spaces $H^0(X,L^k)$, whereas we work in the adjoint setting. This allows us to apply the Berndtsson–P\\u{a}un positivity theorem directly. Additionally, he reduces the problem to the ample case via passing to certain birational models over $X$, while our reduction is by twisting $L^k\\otimes K_X$ by an ample line bundle on $X$ as explained above.\n\n\\subsection*{Organization}\nWe briefly describe the structure of the paper.\n\nIn Section~2 we collect the necessary background from pluripotential theory,  following \\cite{BEGZ,BBGZ,DDL18,DDL18L1}. We also recall the basic variational formulas for \\(E_\\theta\\).\n\nSection~3 is devoted to Bergman measures and quantized Monge--Amp\\`ere energies in the adjoint setting. We prove basic properties of $B_{L^k,u}$ and $E_k(u)$ and discuss their variational formula. Next, we recall the positivity of direct images à la Berndtsson–P\\u{a}un \\cite{BerndtssonPositivity1,BerndtssonPositivity2,BP08,PT18} and the relation with convexity of $E_k$ along weak geodesics. Finally, we recall Berman’s local holomorphic Morse inequalities and the convergence of Bergman measures at equilibrium, adapting the arguments of \\cite{Ber09,BB10} to our adjoint framework.\n\nSection 4 is the new input in our argument.  We prove the the comparison Theorem  (Theorem \\ref{Bergman kernel comparison}).    This follows from choosing peak sections with asymptotically optimal $L^\\infty$-control from a pointwise semi-classical Ohsawa--Takegoshi type extension theorem for powers of an ample line bundle (Theorem \\ref{Pointwise Semi-Classical OT}). We also discuss the relation of Theorem \\ref{Pointwise Semi-Classical OT} with recent work of Finski \\cite[Theorem 1.10]{Finski1}. We also present a  local Morse inequality in ample twisting setting.\n\nFinally, in Section~5 we combine these ingredients to prove Theorems~\\ref{quantization of MA energy} and Corollary  \\ref{convergence of Bergman measures} and conclude with obstacles to aribtrary finite-energy potentials in Remark \\ref{pitfall for finite energy potentials}.\n\\subsection*{Acknowledgment}  This work was supported by NSF Grant DMS-2405274 and the Hauptman Fellowship. The author is deeply grateful to his advisor, Tamás Darvas, for his constant support and guidance. The author also wishes to thank Siarhei Finski for his enlightening lecture and discussions at the summer school at the Rényi Institute, Budapest, and for kindly sharing his preprint \\cite{FinskiBig} along with related insights.", "sketch": "Fix an ample line bundle $A$ on $X$ and $\\varepsilon\\in\\mathbb Q^+$. The approach is to (i) prove a comparison theorem for Bergman measures for $L$ versus the small ample twist $L\\otimes A^\\varepsilon$ with an almost optimal asymptotic bound (Theorem~\\ref{Bergman kernel comparison}); (ii) use this to apply Berman's local Morse inequality to the twisted bundle with explicit dependence on $\\varepsilon$ (see \\eqref{epsilon twisted local holomorphic morse}); (iii) combine this with “the standard variational relations between the Monge–Ampère energy and non-pluripolar measures, the quantized energy and Bergman measures,” together with “the convexity of the quantized Monge–Ampère energy along weak geodesics,” to “carry out the proof of Theorem~\\ref{quantization of MA energy} for bounded potentials following standard argument as in \\cite[Theorem A]{BB10} and \\cite[Theorem 3.5]{BF14}.” The method is noted not to extend to arbitrary finite-energy potentials because of “integrability issues for initial tangent vectors of weak geodesics” (Remark~\\ref{pitfall for finite energy potentials}).", "expanded_sketch": "Fix an ample line bundle $A$ on $X$ and $\\varepsilon\\in\\mathbb Q^+$. The approach is to (i) prove a comparison theorem for Bergman measures for $L$ versus the small ample twist $L\\otimes A^\\varepsilon$ with an almost optimal asymptotic bound. We first prove the following theorem. \n\n\\begin{thm}\\label{Bergman kernel comparison} For $\\epsilon\\in \\Q^+$ and admissible $u$, and for sufficiently divisible $k\\gg 1$, we have\n\\[\nB_{L^k,u}\\leq \\left(1+\\frac{C}{\\epsilon k}\\right)^2B_{(L\\otimes A^{\\epsilon})^k,u},\n\\]\nwhere $C=C(n,X,\\omega)$ depends only on dimension $n$, manifold $X$, and K\\\"ahler form $\\omega=c_1(A,h_0^A)$.\n\\end{thm}\n\n(ii) use this to apply Berman's local Morse inequality to the twisted bundle with explicit dependence on $\\varepsilon$, namely\n\\begin{equation}\\label{epsilon twisted local holomorphic morse}\n    \\frac{n!}{k^n}B_{k,\\epsilon}\\leq \\frac{e^{\\rho_k+\\epsilon\\td{\\rho}_k}}{(1-e^{-\\epsilon(\\log k)^2})^n}(\\theta+\\epsilon\\omega)^n,\n\\end{equation}\n\n(iii) combine this with “the standard variational relations between the Monge–Ampère energy and non-pluripolar measures, the quantized energy and Bergman measures,” together with “the convexity of the quantized Monge–Ampère energy along weak geodesics,” to carry out the proof of the main theorem for bounded potentials following the standard argument as in Berman–Boucksom, Theorem A in \\cite[Theorem A]{BB10} and Berman–Fujita, Theorem 3.5 in \\cite[Theorem 3.5]{BF14}.\n\nThe method is noted not to extend to arbitrary finite-energy potentials because of “integrability issues for initial tangent vectors of weak geodesics.” More precisely:\n\n\\begin{rmk}\\label{pitfall for finite energy potentials}\nThe argument proving \\eqref{lower bound on energy} fails for general \n\\(u\\in \\mathcal{E}^1(X,\\theta)\\), since one cannot in general guarantee that the \ninitial tangent \\(\\dot u_0\\) of the weak geodesic \\(t\\mapsto u_t\\) in \n\\(\\PSH(X,\\theta)\\) is integrable with respect to \\(\\omega_\\epsilon^n\\) in \n\\eqref{key estimate}, even for arbitrarily small \\(\\epsilon>0\\). One obstruction comes from a counterexample of Di Nezza \n\\cite[Example 4.5]{DN15}, which shows that for any \\(\\epsilon>0\\),\n\\[\n\\mathcal{E}^1(X,\\theta)\\not\\subset \\mathcal{E}^1(X,\\omega_\\epsilon),\n\\]\nIndeed, let \\(u\\in \\mathcal{E}^1(X,\\theta)\\setminus \n\\mathcal{E}^1(X,\\omega_\\epsilon)\\) with \\(u\\le0\\). From \n\\(\\PSH(X,\\theta)\\subset \\PSH(X,\\omega_\\epsilon)\\) and envelope definition of the weak geodesic, the weak geodesic \n\\(v_t\\) joining \\(0\\) and \\(u\\) in \\(\\PSH(X,\\omega_\\epsilon)\\) satisfies\n\\[\nv_t \\le u_t,\\qquad t\\in[0,1],\n\\]\nby the envelope construction. Since \\(u\\notin \\mathcal{E}^1(X,\\omega_\\epsilon)\\), one can  shows that\n\\[\n\\int_X \\dot v_0\\,\\omega_\\epsilon^n = -\\infty.\n\\]\nAs \\(u_t\\) and \\(v_t\\) agree at the endpoints, \nwe get $\n\\dot u_0 \\le \\dot v_0 \\le 0$. This implies that $\\dot{u}_0\\notin L^1(X,\\omega_\\epsilon^n)$ since \n\\[\n\\int_X \\dot u_0\\,\\omega_\\epsilon^n\n   \\;\\le\\; \\int_X \\dot v_0\\,\\omega_\\epsilon^n\n   \\;=\\; -\\infty,\n\\]\nThus, a necessary condition for the integrability of \\(\\dot u_0\\) is that  \n\\(u\\in \\mathcal{E}^1(X,\\omega_\\epsilon)\\). To author's current knowledge, we do not aware of any  general sufficient condition.\n\\end{rmk}", "expanded_theorem": "\\label{quantization of MA energy} If $L$ is a big and semipositive line bundle on $X$, then for any  $u\\in \\PSH(X,\\theta)\\cap L^\\infty$, we have\n\\[\n\\lim_{k\\to\\infty}E_k(u)=E_\\theta(u).\n\\],", "theorem_type": ["Asymptotic or Limit", "Implication"], "mcq": {"question": "Let $X$ be a complex projective manifold, let $L$ be a big and semipositive holomorphic line bundle on $X$, and choose a smooth Hermitian metric $h_0^L$ on $L$ with semipositive Chern curvature $\\theta:=c_1(L,h_0^L)$. For $u\\in \\PSH(X,\\theta)\\cap L^\\infty(X)$, where $\\PSH(X,\\theta)$ denotes the $\\theta$-plurisubharmonic functions (so $\\theta+dd^c u\\ge 0$ in the sense of currents), define for each $k\\in\\mathbb N$ the Hilbert norm on $H^0(X,L^k\\otimes K_X)$ by\n\\[\n(s_1,s_2)_{\\mathrm{Hilb}_k(u)}:=\\int_X \\langle s_1,s_2\\rangle_{(h_0^L)^k}e^{-ku},\n\\]\nand the quantized Monge--Amp\\`ere energy by\n\\[\nE_k(u):=-\\frac{1}{kN_k}\\log\\left(\\frac{\\det \\mathrm{Hilb}_k(u)}{\\det \\mathrm{Hilb}_k(0)}\\right),\n\\]\nwhere $N_k=\\dim_\\mathbb C H^0(X,L^k\\otimes K_X)$. Let $E_\\theta(u)$ denote the Monge--Amp\\`ere (Aubin--Yau) energy on $\\PSH(X,\\theta)$, i.e. the primitive of the non-pluripolar Monge--Amp\\`ere operator. Under these assumptions, which statement about $E_k(u)$ holds as $k\\to\\infty$?", "correct_choice": {"label": "A", "text": "One has\n\\[\n\\lim_{k\\to\\infty} E_k(u)=E_\\theta(u).\n\\]"}, "choices": [{"label": "B", "text": "One has\n\\[\n\\limsup_{k\\to\\infty} E_k(u)\\le E_\\theta(u),\n\\]\nand in fact equality holds only when the semipositive form \\(\\theta\\) is K\\\"ahler."}, {"label": "C", "text": "One has\n\\[\n\\limsup_{k\\to\\infty} E_k(u)\\le E_\\theta(u).\n\\]"}, {"label": "D", "text": "For every \\(u\\in \\PSH(X,\\theta)\\cap L^\\infty(X)\\), there exists a subsequence \\(k_j\\to\\infty\\) such that\n\\[\n\\lim_{j\\to\\infty} E_{k_j}(u)=E_\\theta(u),\n\\]\nand the convergence is uniform in \\(u\\) over bounded subsets of \\(\\PSH(X,\\theta)\\cap L^\\infty(X)\\).\n\\]"}, {"label": "E", "text": "If \\(L\\) is big and semipositive, then for any \\(u\\in \\PSH(X,\\theta)\\cap \\mathcal E^1(X,\\theta)\\), one has\n\\[\n\\lim_{k\\to\\infty} E_k(u)=E_\\theta(u).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "semipositive-nonample equality case", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "full limit equality weakened to upper-bound asymptotics", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "pointwise convergence upgraded to uniform-in-u subsequential convergence", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "boundedness hypothesis replaced by finite-energy class", "template_used": "boundary_range"}]}}
{"id": "2512.11595v1", "paper_link": "http://arxiv.org/abs/2512.11595v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{th:cfs}\nFor a fixed value of $x$, we consider the function which maps the tree to the set of values given by evaluating every vertex at this value of $x$.\n    \\begin{enumerate}\n        \\item Setting a value $x>0$ injectively maps the tree to a dense subset of $(0,1)$.\n        \\item Setting a value $x>0$ bijectively maps the tree to $\\mathbb{Q} \\cap (0,1)$ if and only if $x \\in \\mathbb{Z}^+$.\n    \\end{enumerate}", "start_pos": 21673, "end_pos": 22123, "label": "th:cfs"}, "ref_dict": {"fig:FareyPoly": "\\begin{tikzpicture}[\n    level 1/.style = {sibling distance=6cm},\n    level 2/.style = {sibling distance=3cm},\n    level 3/.style = {sibling distance=1.5cm},\n    level distance          = 1cm,\n    edge from parent/.style = {draw},\n    scale=1.2\n    ]\n\n    \\node {$\\frac{x}{x+1}$}\n    child{\n      node {$\\frac{x}{x+2}$}\n      child{\n        node {$\\frac{x}{x+3}$}\n        child{\n          node {$\\frac{x}{x+4}$}\n        }\n        child{\n          node {$\\frac{x+3}{x+4}$}\n        }\n      }\n      child{\n        node {$\\frac{x+2}{x+3}$}\n        child{\n          node {$\\frac{x^2+2x}{x^2+4x+2}$}\n        }\n        child{\n          node {$\\frac{x^2+3x}{x^2+4x+2}$}\n        }\n      }\n    }\n    child{\n      node {$\\frac{x+1}{x+2}$}\n      child{\n        node {$\\frac{x^2+x}{x^2+3x+1}$}\n        child{\n          node {$\\frac{x^2+x}{x^2+4x+2}$}\n        }\n        child{\n          node {$\\frac{x^2+3x+1}{x^2+4x+2}$}\n        }\n      }\n      child{\n        node {$\\frac{x^2+2x}{x^2+3x+1}$}\n        child{\n          node {$\\frac{x^2+2x}{x^2+4x+3}$}\n        }\n        child{\n          node {$\\frac{x^2+3x+1}{x^2+4x+3}$}\n        }\n      }\n    };\n\\end{tikzpicture}\n\n        \\caption{The first four levels of the Farey polynomial tree.}\\label{fig:FareyPoly}\n    \\end{center}\n\\end{figure}\n\nWe label each vertex in the tree with notation inspired by continued fractions. For $a_i \\in \\mathbb{Z}^+$:\n\\begin{equation}\\label{eqn: vertex labels}\n[a_1,a_2,\\ldots,a_k]_x = \\Phi_1^{a_1-1} \\circ \\Phi_0 \\circ \\Phi_1^{a_2-1} \\circ \\Phi_0 \\circ \\cdots \\circ \\Phi_1^{a_{k-1}-1}\\circ \\Phi_0 \\circ \\Phi_1^{a_k-1} \\left( \\frac{1}{1}\\right)\n\\end{equation}\n\nIn this way the root of our tree is the vertex $[2]_x=[1,1]_x$, and in general $[a_1,\\ldots,a_{k-1},a_k+1]_x=[a_1,\\ldots,a_{k-1},a_k,1]_x$. Observe also that\n\\begin{align*}\n    \\Phi_0\\left( [a_1,\\ldots,a_k]_x\\right) &= [1,a_1,\\ldots,a_k]_x\\\\\n    \\Phi_1\\left( [a_1,\\ldots,a_k]_x\\right) &= [a_1+1,a_2,\\ldots,a_k]_x\\\\\n\\end{align*}The tree we present here has some very nice properties. We highlight some of them with the following two theorems; by ``subtree at vertex $[a_1,\\ldots,a_k]_x$\" we mean the tree generated by $\\Phi_1$ and $\\Phi_0$ rooted at vertex $[a_1,\\ldots,a_k]_x$.\n\nWe have the following two main theorems.\n\n\\begin{theorem}\\label{th:cfs}\nFor a fixed value of $x$, we consider the function which maps the tree to the set of values given by evaluating every vertex at this value of $x$.\n    \\begin{enumerate}\n        \\item Setting a value $x>0$ injectively maps the tree to a dense subset of $(0,1)$.\n        \\item Setting a value $x>0$ bijectively maps the tree to $\\mathbb{Q} \\cap (0,1)$ if and only if $x \\in \\mathbb{Z}^+$.\n    \\end{enumerate}\n\\end{theorem}\nNote that from (1) in Theorem \\ref{th:cfs} it follows that the rational functions in the tree don't intersect on $(0,\\infty)$.\n\n\\begin{theorem}\\label{th:dense}\n    For the subtree $\\Omega$ rooted at any vertex $[a_1,\\ldots,a_k]_x$, if we set\n    \\[R=\\left\\{ x \\, | \\, \\textrm{there is a } p(x)/q(x) \\in \\Omega \\textrm{ for which }q(x)=0\\right\\}\\]\n    then $R$ is dense in $(-\\infty,-1]$.\n\\end{theorem}\n\nWe begin \\Cref{sec:cfs} with a derivation of this tree. For Theorem \\ref{th:cfs} we use the close relationship between the tree and a family of continued fractions explained in Section \\ref{sec:cfresults}. Theorem \\ref{th:dense} is proven in \\Cref{sec:dense} by inductively selecting a path in the tree whose vertices have poles converging to an arbitrary $\\alpha<-1$; comments are included regarding the possibility of expanding this result to $\\alpha \\in (-1,0)$. A similar tree derived from the theory of backwards continued fractions is presented in \\Cref{sec:other trees}.\nWe will see a similar behaviour in view of Theorem \\ref{th:cfs} and a very different one in view of Theorem \\ref{th:dense} for this tree.\n\n\\section{Forward Farey Tree}\\label{sec:cfs}\n\\subsection{Background and Definition}\nThe \\textit{Gauss map} $T:[0,1) \\mapsto [0,1)$, given by\n\\[T(t) = \\begin{cases}\n    \\frac{1}{t} - \\floor{\\frac{1}{t}} & t\\neq 0\\\\\n    0 & t=0,\n\\end{cases}\\]\nmay be used to generate the \\textit{regular continued fraction} expansion of $t$:\n\\[ t = \\cfrac{1}{a_1+\\cfrac{1}{a_2+\\cfrac{1}{a_3+\\ddots}}},\\]\nwhere $a_i =a(T^{i-1}(t))= \\floor{1/T^{i-1}(t)}$. This map is the `sped-up' version of the \\textit{Farey map} $F$, given by\n\\[ F(t) = \\begin{cases}\n    F_0(t) = \\frac{1-t}{t} & x \\geq 1/2\\\\\n    F_1(t) = \\frac{t}{1-t} & x \\leq 1/2\n\\end{cases}\\]\nwith the observation that \n\\begin{equation}\\label{eqn: RCF functional}T(t) = F_0 \\circ F_1 ^{a(t)-1} (t).\\end{equation}\nNote that $a(t)-1=\\min(n\\in\\mathbb{N}_0 : T^n(t)\\in [\\frac{1}{2},1])$ i.e. it is first hitting time of $x$ to the interval $[\\frac{1}{2},1]$ (possibly zero) and $T$ can be understood as the composition of $F_0$ and the induced transformation on this interval (where a return time of zero is allowed).\n\nA generalization of regular continued fractions, $N$-continued fractions, allow for all numerators to be some fixed $N \\in \\mathbb{Z}\\setminus\\{0\\}$. These were first studied in  \\cite{AW11,BGRKWY208} and later in many other papers such as \\cite{CK,DO18,DKW13,JKN,KL} . Most papers study positive values of $N$ and in the remainder of this section we will also take $N>0$. For negative $N$, there is a relation with the backward continued fractions in Section \\ref{sec:other trees}. One of the key differences with regular continued fractions is that there is not a unique continued fraction for every $t\\in(0,1)$. For some studied algorithms generating $N$-continued fractions it is proven that there are quadratic irrationals without periodic expansion (and even rationals with aperiodic expansions), see \\cite{KL}. For the greedy $N$-continued fractions, which we have here, there is strong numeric evidence that there are quadratic irrationals with an aperiodic expansion (see \\cite{DKW13}), though this problem is still open. The greedy $N$-continued fractions can be generated with a similar map as the Gauss map namely by \\[ T_N(t) = \\begin{cases}\n   \\left(\\frac{N}{t}-N\\right) - \\floor{\\frac{N}{t}-N} & t \\neq 0\\\\\n     0 & t=0,\n\\end{cases}\\]\nwhich generates\n\\[ t= \\cfrac{N}{N+(a_1-1)+\\cfrac{N}{N+(a_2-1)+\\cfrac{N}{N+(a_3-1)+\\ddots}}},\\]\nwhere now $a_i=a(N,t)= \\floor{N/T_N^{i-1}(t)-N}+1$. Then $T_N(x)$ is a sped-up version of the associated $N$-Farey map\n\\[F_N(t) = \\begin{cases}\n    F_{N,0}(t) = \\frac{N(1-t)}{t} &  t \\geq \\frac{N}{N+1} \\\\\n    F_{N,1}(t) = \\frac{Nt}{N-t} & t \\leq \\frac{N}{N+1}\n\\end{cases}\\]\nwith the observation that\n\\begin{equation}\\label{eqn: N-CF functional}T_N(t) = F_{N,0} \\circ F_{N,1}^{a(N,t)-1}(t).\\end{equation}\nSee Figure \\ref{fig:Fareymap} (right) for an example of $N=2$. It is important to point out that this definition of $a_i=a(N,t)$ is not typical: more typical, e.g. in the above references for N-continued fractions, is that $a(N,t)=\\floor{N/t}$ and $T_N(t)$ is equivalent but without the `$-N$' both inside and outside the integer part function. Our definition here differs only by an integer constant for $a$, is identical for $T_N$, gives a similar presentation between \\Cref{eqn: RCF functional} and (\\ref{eqn: N-CF functional}), and is what we will now generalize. \n\nNote that from a dynamical point of view, there is no necessity of taking $N\\in\\mathbb{Z}^+$ in the definition of the map $F_N(t)$. To this end, let us define a map $F_x:[0,1]\\rightarrow[0,1]$ with $x\\in(0,\\infty)$\\footnote{note that for other values of $x$, we have that $\\frac{x}{x+1}$ is not in $[0,1]$  and the map cannot be used to generate continued fractions.} as\n\\begin{equation}\\label{eq:F_x}\n  F_x(t)=\n\\begin{cases}\nF_{x,0}(t)=\\frac{x(1-t)}{t}   & t \\geq \\frac{x}{x+1}\\\\[1ex]\nF_{x,1}(t)=\\frac{xt}{x-t}, & t \\leq \\frac{x}{x+1},\n\\end{cases}  \n\\end{equation}\nsee Figure \\ref{fig:Fareymap} (left) for $x=1/3$.\n\\begin{figure}[ht]\n\t\t\\centering\n\n\t\t\\subfigure{\\begin{tikzpicture}[scale=5]\n\t\t\t\t\\draw[white] (-0.25,0)--(1,0);\n\t\t\t\t\\draw(0,0)node[below]{\\small $0$}--(1,0)node[below]{\\small $1$}--(1,1)--(0,1)node[left]{\\small $1$}--(0,0);\n\n\t\t\t\t\\draw[thick, blue, smooth, samples =20, domain=1/4:1] plot(\\x,{1/(3*\\x)-1/3});\n\t\t\t\t\\draw[thick,blue, smooth, samples =20, domain=0:1/4] plot(\\x,{\\x/(1-3*\\x)});\n\n\t\t\t\t\\draw[dotted](1/4,0)node[below]{\\small $\\frac{1}{4}$}--(1/4,1);\n\n\t\t\\end{tikzpicture}", "th:cfs": "\\begin{theorem}\\label{th:cfs}\nFor a fixed value of $x$, we consider the function which maps the tree to the set of values given by evaluating every vertex at this value of $x$.\n    \\begin{enumerate}\n        \\item Setting a value $x>0$ injectively maps the tree to a dense subset of $(0,1)$.\n        \\item Setting a value $x>0$ bijectively maps the tree to $\\mathbb{Q} \\cap (0,1)$ if and only if $x \\in \\mathbb{Z}^+$.\n    \\end{enumerate}\n\\end{theorem}", "th:dense": "\\begin{theorem}\\label{th:dense}\n    For the subtree $\\Omega$ rooted at any vertex $[a_1,\\ldots,a_k]_x$, if we set\n    \\[R=\\left\\{ x \\, | \\, \\textrm{there is a } p(x)/q(x) \\in \\Omega \\textrm{ for which }q(x)=0\\right\\}\\]\n    then $R$ is dense in $(-\\infty,-1]$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 3793, "pre_theorem_intro_text": "Let us first define the tree of interest which we will call the \\textit{Farey polynomial tree} (an explanation of the derivation of this tree will be given in the beginning of \\Cref{sec:cfs}). On the nodes we have rational functions $v(x)=p(x)/q(x)$. The root of the tree is $\\frac{x}{x+1}$ and the two offspring of each node are found using the following two functions:\n\\begin{equation}\\label{eqn: Phi}\\begin{split}\n\\Phi_0\\left( \\frac{p(x)}{q(x)}\\right) &= \\begin{cases}\n\\frac{xq(x)}{xq(x)+p(x)} & p(0)\\neq 0\\\\\n\\frac{q(x)}{q(x)+p(x)/x}& p(0)=0\\end{cases}\\\\\n\\\\\n\\Phi_1\\left( \\frac{p(x)}{q(x)}\\right) &= \\begin{cases}\n\\frac{xp(x)}{xq(x)+p(x)} & p(0)\\neq 0\\\\\n\\frac{p(x)}{q(x)+p(x)/x}& p(0)=0.\\end{cases}\\end{split}\\end{equation}\n\nSee Figure \\ref{fig:FareyPoly} for the first few levels. These two functions are the inverse branches of a generalized Farey map and the tree generalizes the classical Farey tree which is found for $x=1$. Note that we define it by using inverse images of the Farey map and not by using mediants of neighboring fractions, but the trees are intimately related (see \\cite{BI09}). The Farey tree is well studied and appears in many different branches of mathematics (see for example~\\cite{AK22,BBDG24,BDM21,DKS25,DS07,KO86,LRS17} and the references therein). Curiously, Farey himself (a geologist) did not publish anything significant on the matter. It was Cauchy who proved one of the basic ideas of the Farey sequence and attributed it to Farey (see \\cite[Notes on Chapter~3]{HW79}).\n\n\\begin{figure}[ht]\n    \\begin{center}\n\n\\begin{tikzpicture}[\n    level 1/.style = {sibling distance=6cm},\n    level 2/.style = {sibling distance=3cm},\n    level 3/.style = {sibling distance=1.5cm},\n    level distance          = 1cm,\n    edge from parent/.style = {draw},\n    scale=1.2\n    ]\n\n    \\node {$\\frac{x}{x+1}$}\n    child{\n      node {$\\frac{x}{x+2}$}\n      child{\n        node {$\\frac{x}{x+3}$}\n        child{\n          node {$\\frac{x}{x+4}$}\n        }\n        child{\n          node {$\\frac{x+3}{x+4}$}\n        }\n      }\n      child{\n        node {$\\frac{x+2}{x+3}$}\n        child{\n          node {$\\frac{x^2+2x}{x^2+4x+2}$}\n        }\n        child{\n          node {$\\frac{x^2+3x}{x^2+4x+2}$}\n        }\n      }\n    }\n    child{\n      node {$\\frac{x+1}{x+2}$}\n      child{\n        node {$\\frac{x^2+x}{x^2+3x+1}$}\n        child{\n          node {$\\frac{x^2+x}{x^2+4x+2}$}\n        }\n        child{\n          node {$\\frac{x^2+3x+1}{x^2+4x+2}$}\n        }\n      }\n      child{\n        node {$\\frac{x^2+2x}{x^2+3x+1}$}\n        child{\n          node {$\\frac{x^2+2x}{x^2+4x+3}$}\n        }\n        child{\n          node {$\\frac{x^2+3x+1}{x^2+4x+3}$}\n        }\n      }\n    };\n\\end{tikzpicture}\n\n        \\caption{The first four levels of the Farey polynomial tree.}\\label{fig:FareyPoly}\n    \\end{center}\n\\end{figure}\n\nWe label each vertex in the tree with notation inspired by continued fractions. For $a_i \\in \\mathbb{Z}^+$:\n\\begin{equation}\\label{eqn: vertex labels}\n[a_1,a_2,\\ldots,a_k]_x = \\Phi_1^{a_1-1} \\circ \\Phi_0 \\circ \\Phi_1^{a_2-1} \\circ \\Phi_0 \\circ \\cdots \\circ \\Phi_1^{a_{k-1}-1}\\circ \\Phi_0 \\circ \\Phi_1^{a_k-1} \\left( \\frac{1}{1}\\right)\n\\end{equation}\n\nIn this way the root of our tree is the vertex $[2]_x=[1,1]_x$, and in general $[a_1,\\ldots,a_{k-1},a_k+1]_x=[a_1,\\ldots,a_{k-1},a_k,1]_x$. Observe also that\n\\begin{align*}\n    \\Phi_0\\left( [a_1,\\ldots,a_k]_x\\right) &= [1,a_1,\\ldots,a_k]_x\\\\\n    \\Phi_1\\left( [a_1,\\ldots,a_k]_x\\right) &= [a_1+1,a_2,\\ldots,a_k]_x\\\\\n\\end{align*}The tree we present here has some very nice properties. We highlight some of them with the following two theorems; by ``subtree at vertex $[a_1,\\ldots,a_k]_x$\" we mean the tree generated by $\\Phi_1$ and $\\Phi_0$ rooted at vertex $[a_1,\\ldots,a_k]_x$.\n\nWe have the following two main theorems.", "context": "Let us first define the tree of interest which we will call the \\textit{Farey polynomial tree} (an explanation of the derivation of this tree will be given in the beginning of \\Cref{sec:cfs}). On the nodes we have rational functions $v(x)=p(x)/q(x)$. The root of the tree is $\\frac{x}{x+1}$ and the two offspring of each node are found using the following two functions:\n\\begin{equation}\\label{eqn: Phi}\\begin{split}\n\\Phi_0\\left( \\frac{p(x)}{q(x)}\\right) &= \\begin{cases}\n\\frac{xq(x)}{xq(x)+p(x)} & p(0)\\neq 0\\\\\n\\frac{q(x)}{q(x)+p(x)/x}& p(0)=0\\end{cases}\\\\\n\\\\\n\\Phi_1\\left( \\frac{p(x)}{q(x)}\\right) &= \\begin{cases}\n\\frac{xp(x)}{xq(x)+p(x)} & p(0)\\neq 0\\\\\n\\frac{p(x)}{q(x)+p(x)/x}& p(0)=0.\\end{cases}\\end{split}\\end{equation}\n\nSee Figure \\ref{fig:FareyPoly} for the first few levels. These two functions are the inverse branches of a generalized Farey map and the tree generalizes the classical Farey tree which is found for $x=1$. Note that we define it by using inverse images of the Farey map and not by using mediants of neighboring fractions, but the trees are intimately related (see \\cite{BI09}). The Farey tree is well studied and appears in many different branches of mathematics (see for example~\\cite{AK22,BBDG24,BDM21,DKS25,DS07,KO86,LRS17} and the references therein). Curiously, Farey himself (a geologist) did not publish anything significant on the matter. It was Cauchy who proved one of the basic ideas of the Farey sequence and attributed it to Farey (see \\cite[Notes on Chapter~3]{HW79}).\n\n\\caption{The first four levels of the Farey polynomial tree.}\\label{fig:FareyPoly}\n \\end{center}\n\\end{figure}\n\nWe label each vertex in the tree with notation inspired by continued fractions. For $a_i \\in \\mathbb{Z}^+$:\n\\begin{equation}\\label{eqn: vertex labels}\n[a_1,a_2,\\ldots,a_k]_x = \\Phi_1^{a_1-1} \\circ \\Phi_0 \\circ \\Phi_1^{a_2-1} \\circ \\Phi_0 \\circ \\cdots \\circ \\Phi_1^{a_{k-1}-1}\\circ \\Phi_0 \\circ \\Phi_1^{a_k-1} \\left( \\frac{1}{1}\\right)\n\\end{equation}\n\nIn this way the root of our tree is the vertex $[2]_x=[1,1]_x$, and in general $[a_1,\\ldots,a_{k-1},a_k+1]_x=[a_1,\\ldots,a_{k-1},a_k,1]_x$. Observe also that\n\\begin{align*}\n \\Phi_0\\left( [a_1,\\ldots,a_k]_x\\right) &= [1,a_1,\\ldots,a_k]_x\\\\\n \\Phi_1\\left( [a_1,\\ldots,a_k]_x\\right) &= [a_1+1,a_2,\\ldots,a_k]_x\\\\\n\\end{align*}The tree we present here has some very nice properties. We highlight some of them with the following two theorems; by ``subtree at vertex $[a_1,\\ldots,a_k]_x$\" we mean the tree generated by $\\Phi_1$ and $\\Phi_0$ rooted at vertex $[a_1,\\ldots,a_k]_x$.\n\nWe have the following two main theorems.", "full_context": "Let us first define the tree of interest which we will call the \\textit{Farey polynomial tree} (an explanation of the derivation of this tree will be given in the beginning of \\Cref{sec:cfs}). On the nodes we have rational functions $v(x)=p(x)/q(x)$. The root of the tree is $\\frac{x}{x+1}$ and the two offspring of each node are found using the following two functions:\n\\begin{equation}\\label{eqn: Phi}\\begin{split}\n\\Phi_0\\left( \\frac{p(x)}{q(x)}\\right) &= \\begin{cases}\n\\frac{xq(x)}{xq(x)+p(x)} & p(0)\\neq 0\\\\\n\\frac{q(x)}{q(x)+p(x)/x}& p(0)=0\\end{cases}\\\\\n\\\\\n\\Phi_1\\left( \\frac{p(x)}{q(x)}\\right) &= \\begin{cases}\n\\frac{xp(x)}{xq(x)+p(x)} & p(0)\\neq 0\\\\\n\\frac{p(x)}{q(x)+p(x)/x}& p(0)=0.\\end{cases}\\end{split}\\end{equation}\n\nSee Figure \\ref{fig:FareyPoly} for the first few levels. These two functions are the inverse branches of a generalized Farey map and the tree generalizes the classical Farey tree which is found for $x=1$. Note that we define it by using inverse images of the Farey map and not by using mediants of neighboring fractions, but the trees are intimately related (see \\cite{BI09}). The Farey tree is well studied and appears in many different branches of mathematics (see for example~\\cite{AK22,BBDG24,BDM21,DKS25,DS07,KO86,LRS17} and the references therein). Curiously, Farey himself (a geologist) did not publish anything significant on the matter. It was Cauchy who proved one of the basic ideas of the Farey sequence and attributed it to Farey (see \\cite[Notes on Chapter~3]{HW79}).\n\n\\caption{The first four levels of the Farey polynomial tree.}\\label{fig:FareyPoly}\n \\end{center}\n\\end{figure}\n\nWe label each vertex in the tree with notation inspired by continued fractions. For $a_i \\in \\mathbb{Z}^+$:\n\\begin{equation}\\label{eqn: vertex labels}\n[a_1,a_2,\\ldots,a_k]_x = \\Phi_1^{a_1-1} \\circ \\Phi_0 \\circ \\Phi_1^{a_2-1} \\circ \\Phi_0 \\circ \\cdots \\circ \\Phi_1^{a_{k-1}-1}\\circ \\Phi_0 \\circ \\Phi_1^{a_k-1} \\left( \\frac{1}{1}\\right)\n\\end{equation}\n\nIn this way the root of our tree is the vertex $[2]_x=[1,1]_x$, and in general $[a_1,\\ldots,a_{k-1},a_k+1]_x=[a_1,\\ldots,a_{k-1},a_k,1]_x$. Observe also that\n\\begin{align*}\n \\Phi_0\\left( [a_1,\\ldots,a_k]_x\\right) &= [1,a_1,\\ldots,a_k]_x\\\\\n \\Phi_1\\left( [a_1,\\ldots,a_k]_x\\right) &= [a_1+1,a_2,\\ldots,a_k]_x\\\\\n\\end{align*}The tree we present here has some very nice properties. We highlight some of them with the following two theorems; by ``subtree at vertex $[a_1,\\ldots,a_k]_x$\" we mean the tree generated by $\\Phi_1$ and $\\Phi_0$ rooted at vertex $[a_1,\\ldots,a_k]_x$.\n\nWe have the following two main theorems.\n\n\\begin{theorem}\\label{th:dense}\n For the subtree $\\Omega$ rooted at any vertex $[a_1,\\ldots,a_k]_x$, if we set\n \\[R=\\left\\{ x \\, | \\, \\textrm{there is a } p(x)/q(x) \\in \\Omega \\textrm{ for which }q(x)=0\\right\\}\\]\n then $R$ is dense in $(-\\infty,-1]$.\n\\end{theorem}\n\n\\section{Forward Farey Tree}\\label{sec:cfs}\n\\subsection{Background and Definition}\nThe \\textit{Gauss map} $T:[0,1) \\mapsto [0,1)$, given by\n\\[T(t) = \\begin{cases}\n \\frac{1}{t} - \\floor{\\frac{1}{t}} & t\\neq 0\\\\\n 0 & t=0,\n\\end{cases}\\]\nmay be used to generate the \\textit{regular continued fraction} expansion of $t$:\n\\[ t = \\cfrac{1}{a_1+\\cfrac{1}{a_2+\\cfrac{1}{a_3+\\ddots}}},\\]\nwhere $a_i =a(T^{i-1}(t))= \\floor{1/T^{i-1}(t)}$. This map is the `sped-up' version of the \\textit{Farey map} $F$, given by\n\\[ F(t) = \\begin{cases}\n F_0(t) = \\frac{1-t}{t} & x \\geq 1/2\\\\\n F_1(t) = \\frac{t}{1-t} & x \\leq 1/2\n\\end{cases}\\]\nwith the observation that \n\\begin{equation}\\label{eqn: RCF functional}T(t) = F_0 \\circ F_1 ^{a(t)-1} (t).\\end{equation}\nNote that $a(t)-1=\\min(n\\in\\mathbb{N}_0 : T^n(t)\\in [\\frac{1}{2},1])$ i.e. it is first hitting time of $x$ to the interval $[\\frac{1}{2},1]$ (possibly zero) and $T$ can be understood as the composition of $F_0$ and the induced transformation on this interval (where a return time of zero is allowed).\n\nA generalization of regular continued fractions, $N$-continued fractions, allow for all numerators to be some fixed $N \\in \\mathbb{Z}\\setminus\\{0\\}$. These were first studied in \\cite{AW11,BGRKWY208} and later in many other papers such as \\cite{CK,DO18,DKW13,JKN,KL} . Most papers study positive values of $N$ and in the remainder of this section we will also take $N>0$. For negative $N$, there is a relation with the backward continued fractions in Section \\ref{sec:other trees}. One of the key differences with regular continued fractions is that there is not a unique continued fraction for every $t\\in(0,1)$. For some studied algorithms generating $N$-continued fractions it is proven that there are quadratic irrationals without periodic expansion (and even rationals with aperiodic expansions), see \\cite{KL}. For the greedy $N$-continued fractions, which we have here, there is strong numeric evidence that there are quadratic irrationals with an aperiodic expansion (see \\cite{DKW13}), though this problem is still open. The greedy $N$-continued fractions can be generated with a similar map as the Gauss map namely by \\[ T_N(t) = \\begin{cases}\n \\left(\\frac{N}{t}-N\\right) - \\floor{\\frac{N}{t}-N} & t \\neq 0\\\\\n 0 & t=0,\n\\end{cases}\\]\nwhich generates\n\\[ t= \\cfrac{N}{N+(a_1-1)+\\cfrac{N}{N+(a_2-1)+\\cfrac{N}{N+(a_3-1)+\\ddots}}},\\]\nwhere now $a_i=a(N,t)= \\floor{N/T_N^{i-1}(t)-N}+1$. Then $T_N(x)$ is a sped-up version of the associated $N$-Farey map\n\\[F_N(t) = \\begin{cases}\n F_{N,0}(t) = \\frac{N(1-t)}{t} & t \\geq \\frac{N}{N+1} \\\\\n F_{N,1}(t) = \\frac{Nt}{N-t} & t \\leq \\frac{N}{N+1}\n\\end{cases}\\]\nwith the observation that\n\\begin{equation}\\label{eqn: N-CF functional}T_N(t) = F_{N,0} \\circ F_{N,1}^{a(N,t)-1}(t).\\end{equation}\nSee Figure \\ref{fig:Fareymap} (right) for an example of $N=2$. It is important to point out that this definition of $a_i=a(N,t)$ is not typical: more typical, e.g. in the above references for N-continued fractions, is that $a(N,t)=\\floor{N/t}$ and $T_N(t)$ is equivalent but without the `$-N$' both inside and outside the integer part function. Our definition here differs only by an integer constant for $a$, is identical for $T_N$, gives a similar presentation between \\Cref{eqn: RCF functional} and (\\ref{eqn: N-CF functional}), and is what we will now generalize.\n\n\\begin{proposition}\\label{prop: elementary vertex properties without proof}\n At every vertex $[a_1,\\ldots,a_k]_x=p/q$, we have:\n \\begin{itemize}\n \\item $p,q$ are polynomials of the same degree.\n \\item All coefficients of each are positive integers, except the constant term of $p(x)$ which may be zero.\n \\item Term-by-term, the coefficients of $q$ are strictly larger than those of $p$, except for the leading coefficients which are both one.\n \\end{itemize}\n \\begin{proof}\n All properties are preserved by both $\\Phi_0$ and $\\Phi_1$, and are true for the root vertex $p(x)=x$, $q(x)=x+1$.\n \\end{proof}\n\\end{proposition}\n\\begin{proposition}\\label{prop: increasing functions}\n For every vertex $v=[a_1,\\ldots,a_k]_x$, we have both\n \\begin{align}\n v'(x)&> 0 &\\textrm{(wherever defined)}\\\\\n xv'(x) &\\leq v(x) & \\textrm{(with equality if and only if $x=0$ and $v(0)=0$)}\n \\end{align}\n \\begin{proof}\n We induct on $k$, the length of the expansion of $v$. For $k=1$ we have \n \\[v=\\frac{x}{x+a_1-1}\\]\n (where $a_1 \\geq 2$) for which both claims may be trivially verified. Now assume both properties hold for $v$, and for some $a \\in \\mathbb{Z}^+$ set \n \\[V = [a,v]_x = \\begin{cases}\\frac{x}{x+a-1+v} & a \\geq 2 \\textrm{ or } v(0) \\neq 0\\\\ \\frac{1}{1+v/x} & a=1 \\textrm{ and } v(0)=0. \n \\end{cases}\\]\n Note that by \\Cref{prop: elementary vertex properties without proof} $x=0$ is at most a simple root of $v$, so the second case may be properly defined even for $x=0$, and if we establish $v'>0$, then $xv'$ also has at most a simple root at $x=0$.\n In the first case, we compute\n \\begin{align*}\n V'&=\\frac{a-1+v-xv'}{(x+a-1+v)^2}.\\\\\n \\intertext{Since we have assumed $xv'<v$, the first claim now follows.}\n xV'-V&= \\frac{x((a-1)+v-xv')-x(x+a-1+v)}{(x+a-1+v)^2}\\\\\n &=-x^2\\frac{v'+1}{(x+a-1+v)^2}.\\\\\n \\end{align*}\n Which is negative unless $x=0$, in which case it follows that $V(0)=0$ as well.\n\n\\section{Backward Farey Tree}\\label{sec:other trees}\nThe results we have on the Farey polynomial tree raise the question of how general these results are: there are many other different families of continued fraction algorithms. In this section we take the backward continued fractions as a starting point and compare it with Theorem \\ref{th:cfs} and Theorem \\ref{th:dense}. \nJust as for regular continued fractions, the backward continued fractions appears in different fields of mathematics. There are many nice papers studying backwards continued fractions, but we want to highlight \\cite{MO19,MO20} where backwards (and regular) continued fractions are used to draw connections between graph theory, combinatorics, number theory, and algebra. \nFor backward continued fractions, a parametrised family is already studied in the form of $ 1/(u(1-x)) \\mod 1$ where $u\\in(0,4)$, introduced and studied in \\cite{GH96a,GH96b} and more recently \\cite{LS20, LS22}. In contrast to those papers, in this article we will make the branches full again by shifting the digit set in order to get a better comparison with the Farey polynomial tree. Of course, instead of having $ 1/(u(1-x)) \\mod 1$, we can also put the parameter in the numerator. This leads to the fast map \n\\begin{equation}\n T_x(t)=\\frac{x}{1-t}- x -\\left\\lfloor \\frac{x}{1-t}-x \\right\\rfloor.\n\\end{equation}\nThe continued fractions are of the form \n\\[\nt=1-\\cfrac{x}{x + a_1-\\cfrac{x}{x+ a_2 -\\ddots}}\n\\]\nwhere $a_i\\in\\mathbb{Z^+}$, given by\n\\[a(t)=1+\\floor{\\frac{x}{1-t}-x}.\\] Note that, when taking $x\\in \\mathbb{Z^+}$, you essentially get $N$-continued fractions with a negative value for $N$. \nThe slow map is given by\n\\begin{equation}\\label{eq:F_x backwards}\n F_x(t)=\n\\begin{cases}\n\\dfrac{xt}{1-t} & t\\in [0,\\frac{1}{x+1})\\\\[1ex]\n1-\\displaystyle\\frac{x(t-1)}{1-t-x}, & t\\in [\\frac{1}{x+1},1],\n\\end{cases} \n\\end{equation}\n see Figure \\ref{fig:Fareymapbackward}.\n\n\\begin{theorem}\\label{th:backwardcf}\n For a fixed value of $x$, we consider the function which maps the backward tree to the set of values given by evaluating every vertex at this value of $x$.\n \\begin{enumerate}\n \\item Setting a value $x\\geq 1$ injectively maps the backward tree to a dense subset of $(0,1)$. \\\\For $x\\in(0,1)$, the mapping is injective and maps to a subset of $(0,1)$ but the image is not dense.\n \\item Setting a value $x \\geq 1$ bijectively maps the tree to $\\mathbb{Q} \\cap (0,1)$ if and only if $x \\in \\mathbb{Z}^+$.\n \\end{enumerate}\n\\end{theorem}", "post_theorem_intro_text_len": 1118, "post_theorem_intro_text": "Note that from (1) in Theorem \\ref{th:cfs} it follows that the rational functions in the tree don't intersect on $(0,\\infty)$.\n\n\\begin{theorem}\\label{th:dense}\n    For the subtree $\\Omega$ rooted at any vertex $[a_1,\\ldots,a_k]_x$, if we set\n    \\[R=\\left\\{ x \\, | \\, \\textrm{there is a } p(x)/q(x) \\in \\Omega \\textrm{ for which }q(x)=0\\right\\}\\]\n    then $R$ is dense in $(-\\infty,-1]$.\n\\end{theorem}\n\nWe begin \\Cref{sec:cfs} with a derivation of this tree. For Theorem \\ref{th:cfs} we use the close relationship between the tree and a family of continued fractions explained in Section \\ref{sec:cfresults}. Theorem \\ref{th:dense} is proven in \\Cref{sec:dense} by inductively selecting a path in the tree whose vertices have poles converging to an arbitrary $\\alpha<-1$; comments are included regarding the possibility of expanding this result to $\\alpha \\in (-1,0)$. A similar tree derived from the theory of backwards continued fractions is presented in \\Cref{sec:other trees}.\nWe will see a similar behaviour in view of Theorem \\ref{th:cfs} and a very different one in view of Theorem \\ref{th:dense} for this tree.", "sketch": "For Theorem~\\ref{th:cfs} the introduction says it is proved by “us[ing] the close relationship between the tree and a family of continued fractions explained in Section~\\ref{sec:cfresults}.”\n\n(No further proof structure for Theorem~\\ref{th:cfs} is described here; the only other proof outline given is for Theorem~\\ref{th:dense}, which is “proven … by inductively selecting a path in the tree whose vertices have poles converging to an arbitrary $\\alpha<-1$.”)", "expanded_sketch": "To prove the main theorem, the introduction says it is proved by “us[ing] the close relationship between the tree and a family of continued fractions explained later.”\n\n(No further proof structure for the main theorem is described here; the only other proof outline given is for \\begin{theorem}\\label{th:dense}\n    For the subtree $\\Omega$ rooted at any vertex $[a_1,\\ldots,a_k]_x$, if we set\n    \\[R=\\left\\{ x \\, | \\, \\textrm{there is a } p(x)/q(x) \\in \\Omega \\textrm{ for which }q(x)=0\\right\\}\\]\n    then $R$ is dense in $(-\\infty,-1]$.\n\\end{theorem}, which is “proven … by inductively selecting a path in the tree whose vertices have poles converging to an arbitrary $\\alpha<-1$.”)", "expanded_theorem": "\\label{th:cfs}\nFor a fixed value of $x$, we consider the function which maps the tree to the set of values given by evaluating every vertex at this value of $x$.\n    \\begin{enumerate}\n        \\item Setting a value $x>0$ injectively maps the tree to a dense subset of $(0,1)$.\n        \\item Setting a value $x>0$ bijectively maps the tree to $\\mathbb{Q} \\cap (0,1)$ if and only if $x \\in \\mathbb{Z}^+$.\n    \\end{enumerate}", "theorem_type": "unknown", "mcq": {"question": "Consider the Farey polynomial tree whose vertices are rational functions in a variable $X$, with root $\\dfrac{X}{X+1}$, and where each vertex $\\dfrac{p(X)}{q(X)}$ has two children given by\n\\[\n\\Phi_0\\!\\left(\\frac{p(X)}{q(X)}\\right)=\n\\begin{cases}\n\\dfrac{Xq(X)}{Xq(X)+p(X)} & \\text{if } p(0)\\neq 0,\\\\[1ex]\n\\dfrac{q(X)}{q(X)+p(X)/X} & \\text{if } p(0)=0,\n\\end{cases}\n\\qquad\n\\Phi_1\\!\\left(\\frac{p(X)}{q(X)}\\right)=\n\\begin{cases}\n\\dfrac{Xp(X)}{Xq(X)+p(X)} & \\text{if } p(0)\\neq 0,\\\\[1ex]\n\\dfrac{p(X)}{q(X)+p(X)/X} & \\text{if } p(0)=0.\n\\end{cases}\n\\]\nFor a fixed real number $x$, evaluate every vertex at $X=x$, thereby obtaining a map from the vertices of the tree to real numbers. Which statement holds about this evaluation map?", "correct_choice": {"label": "A", "text": "For every fixed $x>0$, the evaluation map is injective and its image is a dense subset of $(0,1)$. Moreover, this map is a bijection onto $\\mathbb{Q}\\cap(0,1)$ if and only if $x\\in\\mathbb{Z}^+$."}, "choices": [{"label": "B", "text": "For every fixed $x\\ge 1$, the evaluation map is injective and its image is a dense subset of $(0,1)$. Moreover, this map is a bijection onto $\\mathbb{Q}\\cap(0,1)$ if and only if $x\\in\\mathbb{Z}^+$."}, {"label": "C", "text": "For every fixed $x\\in\\mathbb{Z}^+$, the evaluation map is injective and its image is a dense subset of $(0,1)$, and in fact it is a bijection onto $\\mathbb{Q}\\cap(0,1)$."}, {"label": "D", "text": "For every fixed $x>0$, the evaluation map is injective and its image is all of $(0,1)$. Moreover, this map is a bijection onto $\\mathbb{Q}\\cap(0,1)$ if and only if $x\\in\\mathbb{Z}^+$."}, {"label": "E", "text": "For every fixed $x>0$, the evaluation map has dense image in $(0,1)$, and it is injective if and only if $x\\in\\mathbb{Z}^+$. Moreover, in that case the map is a bijection onto $\\mathbb{Q}\\cap(0,1)$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "parameter range $x>0$ replaced by $x\\ge 1$", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "restricts the universal quantifier from all $x>0$ to integer parameters only", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dense image strengthened to surjective image $(0,1)$", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "injectivity made dependent on integrality rather than holding for all $x>0$", "template_used": "wildcard"}]}}
{"id": "2512.11601v1", "paper_link": "http://arxiv.org/abs/2512.11601v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm:K2}\n  The set of $\\mathcal{P}$-positions of the variant of Wythoff game where the set of terminal positions is $\\{(x,y)\\mid x+y\\le 2\\}$,  is exactly \\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}\nwhere $g:\\mathbb{N}\\to\\mathbb{N}$ is the function given in Definition~\\ref{def:fg}.", "start_pos": 103245, "end_pos": 103752, "label": "thm:K2"}, "ref_dict": {"cor:behave": "\\begin{corollary}\\label{cor:behave}\n  For the sequences defined by \\eqref{eq:mexdef}, \n  there exist a bounded function $\\lambda_\\ell:\\mathbb{N}\\to\\mathbb{Z}$ such that, for all $n$,   \n  \\[\n   a_n=\\lfloor (n+\\ell)\\phi \\rfloor +\\lambda_\\ell(n).\n \\]\n So, in particular, \\(b_n=\\lfloor (n+\\ell)\\phi^2 \\rfloor + \\lambda_\\ell(n) +1\\).\n  \\end{corollary}", "thm:K2": "\\begin{theorem}\\label{thm:K2}\n  The set of $\\mathcal{P}$-positions of the variant of Wythoff game where the set of terminal positions is $\\{(x,y)\\mid x+y\\le 2\\}$,  is exactly \\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}\nwhere $g:\\mathbb{N}\\to\\mathbb{N}$ is the function given in Definition~\\ref{def:fg}.\n\\end{theorem}", "thm:rec_car": "\\begin{theorem}\\label{thm:rec_car}\n  Let $\\ell\\ge 1$ and define recursively $(a_n,b_n)_{n\\ge 0}$ by\n\\begin{equation}\\label{eq:mexdef}\n  (a_0,b_0)=(\\ell+1,2\\ell+2) \\quad\\text{ and }\\quad \\forall n\\ge 1, \\left\\{\n    \\begin{array}{l}\na_n =\\mex (\\{ a_i,b_i \\mid i<n \\}\\cup\\{0,1,\\ldots,\\ell\\}),\\\\\nb_n =a_n+n+\\ell+1.\\\\\n    \\end{array}\n  \\right. \n\\end{equation} \nThe set of non-terminal $\\mathcal{P}$-positions of $K^\\ell$ is $\\{(a_n,b_n)\\mid n\\ge 0\\} \\cup \\{(b_n,a_n)\\mid n\\ge 0\\} $.\n\\end{theorem}", "eq:Ppos": "\\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}", "def:fg": "\\begin{definition}\\label{def:fg}\n  We let $g(0)=1$, $g(1)=0$ and\n  \\[\n    g(n)=\\left\\{\n      \\begin{array}{ll}\n        1-g(m),& \\text{ if }\\lfloor n\\phi\\rfloor= \\lfloor m(\\phi+1)\\rfloor+1 \\text{ for some }m\\ge 0;\\\\\n        1,& \\text{ otherwise}.\n      \\end{array}\\right.\n  \\]\n\n\\end{definition}", "ssec:autom_proof": "\\begin{array}{c|llll}\n    &g_n & \\mu(g_n) & \\mu^2(g_n) & \\mu^3(g_n) \\\\\n    \\hline\n 0&1& 0& 1& 10\\\\\n 1&0& 1& 10& 011\\\\\n 2&1& 10& 011& 11010\\\\\n 3&1& 01& 110& 01011\\\\\n 4&0& 11& 010& 11011\\\\\n 5&1& 0& 11& 010\\\\\n  \\end{array}\n\\]\n  \\caption{The different $3$-types when applying a $\\varphi$-morphism to $g$.}\n  \\label{tab:types}\n\\end{table}\n\nNow we associate each symbol $g_n$ with its $3$-type and get the infinite word\n\\[  \n  01|2|31|45|2|35|4|31|45|4|35|45|\\cdots\n\\]\nwhere the vertical bars indicate the factorization of the Fibonacci word $ab|a|ab|ab|a|\\cdots$ with factors $\\sigma(a)=ab$ and $\\sigma(b)=a$, the images of the letters by the Fibonacci morphism. From this, we deduce the morphism $\\mu$ given in Example~\\ref{exa:mu}: the image of the $n$th letter has to be the $n$th block in the factorization. So $0\\mapsto 01$, then $1\\mapsto 2$ and so on. From the morphism, it is a routine to obtain the DFA in Figure~\\ref{fig:automG}, see Definition~\\ref{def:dfao}.\n\nFinally, one may define a coding~$\\rho$ to recover the original sequence where each type is mapped to the first element of the tuple. So, $0,2,3,5\\mapsto 1$ and $1,4\\mapsto 0$. This can equivalently be performed by comparing the morphic word and the original sequence\n\\begin{align*}\n  &01231452354314543545\\cdots\\\\\n  &10110011110100101101\\cdots\n\\end{align*}\nand matching corresponding elements.\n\\subsection{An automatic proof for $K^2$}\\label{ssec:autom_proof}\n\nNow that $g$ has been defined by a DFA fed with Fibonacci representations (and stored as {\\tt G.txt} in {\\tt Word Automata Lib}), the set of $\\mathcal{P}$-positions \\eqref{eq:Ppos} obtained by Komak et al. can be described by the following commands:\n\\begin{verbatim}\ndef pposK2_asym \"?msd_fib (a+b<=1) | En,x $phin(n,x) & a+1=x+G[n]\n                                                     & b=x+n+G[n]\":\ndef pposK2 \"?msd_fib $pposK2_asym(a,b) | $pposK2_asym(b,a) \":\n\\end{verbatim}\nThe resulting automaton has $27$ states.\n\nFor the sake of presentation, let us recall some basics on combinatorial games.\n\\begin{definition}\\label{def:stab}\nA set $S$ of positions is {\\em stable} if, for all $s,t\\in S$, $s\\neq t$, there is no move between $s$ and $t$. A set $S$ of positions is {\\em absorbing}, if for all $t\\not\\in S$, there exists $s\\in S$ such that there is a move from $t$ to $s$ (or, $t$ has an option in $S$).  \n\\end{definition}\nIt is a well-known property in graph theory that every acyclic graph has a unique kernel, i.e., a stable and absorbing set of vertices. In a take-away game where tokens are removed from piles, the corresponding game graph is acyclic: it is not possible to visit twice the same position during a game.\n\\begin{proposition}[Folklore]\\label{pro:kernel}\n  The sets of $\\mathcal{P}$- and $\\mathcal{N}$-positions of an\n  impartial acyclic game are uniquely determined by the following two\n  properties:\n\\begin{enumerate}\n\\item Every move from a $\\mathcal{P}$-position leads to an\n  $\\mathcal{N}$-position; equivalently there is no move between two $\\mathcal{P}$-positions (stability property of $\\mathcal{P}(G)$).\n  \\item From every $\\mathcal{N}$-position, there exists a move leading to a $\\mathcal{P}$-position (absorbing property of $\\mathcal{P}(G)$).\n\\end{enumerate}\n\\end{proposition}\n\nAs observed in \\cite{MRRW}, it is then a routine test to check stability and absorbing property:\n\\begin{verbatim}\neval stableK2 \"?msd_fib Ap,q,r,s (($pposK2(p,q) & $pposK2(r,s)\n     & p>=r & q>=s & p+q>2) => ((p=r & q=s) | (p>r & q>s & p+s!=q+r)))\":\n\neval absorbingK2 \"?msd_fib Ap,q (~$pposK2(p,q) => Ex,y\n     (x<=p & y<=q & $pposK2(x,y) & (p+y=q+x | p=x | q=y))) \":\n\\end{verbatim}\nThese two commands return {\\tt TRUE} resulting in the following result, \\cite[Thm.~3]{Komak}. From Proposition~\\ref{pro:kernel}, we immediately get an alternative proof of Theorem~\\ref{thm:K2}.\n\n\\section{Towards an algebraic characterization for $K^\\ell$}\\label{sec:newGame}\n\nIn \\cite{Komak}, Komak et al. chose $\\{(x,y)\\mid x+y\\le 2\\}$ as set of terminal positions. It is natural to consider a parameterized version where the set of terminal states is\n\\[\n  \\{(x,y)\\in\\mathbb{N}^2\\mid x+y\\le \\ell\\}\n\\]\nfor some $\\ell\\in\\mathbb{N}$. The corresponding game is denoted by $K^\\ell$. Note that the game $K^0$ is the classic Wythoff game. In this section, we obtain new results concerning the games $K^1$ and~$K^3$.\n\n\\subsection{The $K^1$-game}\nUp to our knowledge, the game $K^1$ does not seem to have been studied in the literature. Computing the first $\\mathcal{P}$-positions of $K^1$ and writing their Fibonacci expansions permitted us to state a conjecture that can be tested with {\\tt Walnut}. This result is quite similar to the famous characterization of the $\\mathcal{P}$-positions of Wythoff game obtained by Fraenkel \\cite{Fraenkel3}.\n\\begin{table}[h!t]\n  \\[\n    \\begin{array}{c|r|c|r}\n      a_n & \\rep_F(a_n) & b_n & \\rep_F(b_n) \\\\\n      \\hline\n 0&\\varepsilon  & 1 &1 \\\\\n 2&10 & 4 &101 \\\\\n 3&100 & 6 & 1001 \\\\\n 5&1000 & 9 & 10001 \\\\\n 7&1010 & 12 & 10101 \\\\\n 8&10000 & 14 & 100001 \\\\\n 10&10010 & 17 &100101 \\\\\n 11&10100 & 19 & 101001 \\\\\n 13&100000 & 22 & 1000001 \\\\\n 15&100010 & 25 & 1000101 \\\\\n  \\end{array}", "def:stab": "\\begin{definition}\\label{def:stab}\nA set $S$ of positions is {\\em stable} if, for all $s,t\\in S$, $s\\neq t$, there is no move between $s$ and $t$. A set $S$ of positions is {\\em absorbing}, if for all $t\\not\\in S$, there exists $s\\in S$ such that there is a move from $t$ to $s$ (or, $t$ has an option in $S$).  \n\\end{definition}", "thm:carK1": "\\begin{theorem}\\label{thm:carK1}\n  A position $(a,b)$, with $a\\le b$, is a $\\mathcal{P}$-position of the game $K^1$, with a set of terminal positions being $\\{(x,y)\\mid x+y\\le 1\\}$ and Wythoff's moves, if and only if $\\rep_F(a)$ ends with $0$ and $\\rep_F(b)=\\rep_F(a)1$. \n\\end{theorem}", "tab:gh": "\\begin{array}{c|ccccccccccccccccccccc}\n      & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10  & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 &  19 & 20 \\\\\n      \\hline\n      h& 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 & 6 & 6 &  7 & 8 & 8 & 9 & 9 & 10 & 11 & 11 & 12 &  12 \\\\\n      g&1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 &  1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\\\\n    \\end{array}\n  \\]\n  \\caption{First values of the sequence $g$ and Hofstadter sequence $h$.}\n  \\label{tab:gh}\n\\end{table}\n\nThe main result of Komak {\\em et al.} provides an algebraic description of set of $\\mathcal{P}$-positions of $K^2$:\n\\begin{theorem}\\label{thm:K2}\n  The set of $\\mathcal{P}$-positions of the variant of Wythoff game where the set of terminal positions is $\\{(x,y)\\mid x+y\\le 2\\}$,  is exactly \\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}\nwhere $g:\\mathbb{N}\\to\\mathbb{N}$ is the function given in Definition~\\ref{def:fg}.\n\\end{theorem}\n\nThe authors provide a classical proof of their result by showing that the set of $\\mathcal{P}$-positions described by \\eqref{eq:Ppos} is both stable and absorbing (see Definition~\\ref{def:stab}). The proof is quite long and requires a detailed case analysis. Without diminishing their achievement, once the set \\eqref{eq:Ppos} has been conjectured, if the set can be handled by {\\tt Walnut}, then it becomes straightforward to produce an automated proof in just a few lines. Our approach therefore complements the work carried out in \\cite{Komak}.\n\nIn \\cite{block}, other variants of Wythoff game are studied, in which some blocking maneuvers are added. Let $k\\ge 1$. In the game denoted by $W^k$, for each move, before the next player (i.e., the player who is about to play) moves, the previous player (i.e., the one who has just played) may declare at most $k-1$ of the options as forbidden. When the next player has moved, any blocking maneuver is forgotten and has no further incidence on the game. This terminology of previous and next player explains why we speak of $\\mathcal{N}$- and $\\mathcal{P}$-positions, respectively. For $k=1$, this is the classical game of Wythoff. When $k=2,3$, we reconsider Larsson's result and obtain an automated proof, replacing a 2-page-long analysis by some easy-to-describe first-order formulas. \n\nAs Shallit has already noted on several occasions, the two approaches are complementary. A classical, purely combinatorial proof usually provides structural insights, whereas the use of automated provers helps avoid lengthy case analyses and enables the exploration of directions that are difficult to access through traditional techniques. In particular, in this article, we easily obtain a variety of results that would otherwise require significantly more effort using ``classical'' methods. Moreover, these results lead us to formulate general theorems.\n\\smallskip\n\nThis article is organized as follows. Section~\\ref{sec:2} --- which was the starting point of this article --- aims to give an automatic proof of the algebraic characterization \\eqref{eq:Ppos} of the $\\mathcal{P}$-positions of $K^2$. We begin with preliminary results about Fibonacci-automatic sequences. We define a notion of $\\varphi$-morphism and describe a heuristic that we extensively use throughout the paper to construct such $\\varphi$-morphisms. Given a sufficiently long prefix of an infinite word, we obtain morphisms that can be used in our automatic proofs (and we can therefore prove the correctness of the procedure). Basic results from combinatorial game theory are given in Section~\\ref{ssec:autom_proof}.\n\nIn Section~\\ref{sec:newGame}, we obtain results about new games: a parameterized version of a variant of Wythoff game denoted by $K^\\ell$, where the set of terminal states is\n\\[\n  \\{(x,y)\\in\\mathbb{N}^2\\mid x+y\\le \\ell\\}\n\\]\nfor some $\\ell\\in\\mathbb{N}$. We study the games $K^1$, $K^3$ and $K^4$ and obtain algebraic characterizations of the $\\mathcal{P}$-positions similar to \\cite{Komak}. In particular, as for the classical Wythoff game $K^0$, the game $K^1$ has a nice set of $\\mathcal{P}$-positions: a pair $(a,b)$ of integers such that $a\\le b$ is a $\\mathcal{P}$-position of $K^1$ if and only if its Fibonacci representation $\\rep_F(a)$ ends with zero and $\\rep_F(b)$ is of the form $\\rep_F(a)1$, see Theorem~\\ref{thm:carK1}. \n\nIn Section~\\ref{sec:6}, we go further in the analysis of the games $K^\\ell$ for an arbitrary $\\ell\\ge 1$. We provide a recursive characterization of the set of $\\mathcal{P}$-positions. The non-terminal $\\mathcal{P}$-positions $(a_n,b_n)_{n\\ge 0}$ of $K^\\ell$ are given, in Theorem~\\ref{thm:rec_car}, by\n\\[\n  (a_0,b_0)=(\\ell+1,2\\ell+2) \\quad\\text{ and }\\quad \\forall n\\ge 1, \\left\\{\n    \\begin{array}{l}\na_n =\\mex (\\{ a_i,b_i \\mid i<n \\}\\cup\\{0,1,\\ldots,\\ell\\}) \\\\\nb_n =a_n+n+\\ell+1.\\\\\n    \\end{array}"}, "pre_theorem_intro_text_len": 3799, "pre_theorem_intro_text": "In combinatorial game theory, proofs of many results are obtained by using a suitable numeration system \\cite{Duchene,Fraenkel1,Fraenkel2,RigoLNM}. As a typical example, the Zeckendorf numeration based on the Fibonacci sequence is used to characterize the $\\mathcal{P}$-positions (i.e., losing positions) of the Wythoff game \\cite{Fraenkel3}. Fraenkel showed that a pair $(a,b)$ of integers such that $a\\le b$ is a $\\mathcal{P}$-position of Wythoff game if and only if $\\rep_F(a)$ ends with an even number of zeroes and $\\rep_F(b)$ is the left-shift of $\\rep_F(a)$, i.e., $\\rep_F(b)=\\rep_F(a)0$.\n\nWhen the numeration system has good properties (i.e., it is an {\\em addable} system as discussed in \\cite{ShallitBook}), the formalism of first-order logic can be used to express certain properties of combinatorial games. We have recently exploited this approach and used {\\tt Walnut} to reprove classical results in combinatorial games theory, as well as to obtain new ones \\cite{MRRW}. {\\tt Walnut} is a free, open-source prover for first-order statements dealing with automatic sequences \\cite{Mousavi,ShallitBook}. \n\nWe assume that the reader is sufficiently familiar with the use of {\\tt Walnut}. For references on combinatorial game theory, we refer to \\cite{RigoLNM} and \\cite{MRRW}.\n\nThe Wythoff game is a {\\em take-away} game: two players alternately remove tokens from two piles. One can remove a positive number of token from one pile or, remove the same positive number of token from both piles. The Wythoff game is {\\em impartial}: the set of options (i.e., positions reachable in one legal move) of a given position does not depend on which player is in turn to move. Wythoff game can be seen as a queen on an infinite chessboard $\\mathbb{N}^2$: in a position $(x,y)\\neq(0,0)$ the queen can move horizontally to $(x-n,y)$ with $0<n\\le x$, vertically to $(x,y-n)$ with $0<n\\le y$ or diagonally to $(x-n,y-n)$ with $0<n\\le \\min\\{x,y\\}$. According to the {\\em normal convention}, a player who is unable to move loses the game; passing is not allowed. Otherwise stated, the player who takes the last token wins or, in an equivalent way, when the queen reaches $(0,0)$.\n\nIn this article, we are mainly interested in two kinds of variations of the Wythoff game. \n\nIn \\cite{Komak}, a variant of Wythoff game has been introduced: the set $\\{(x,y)\\mid x+y\\le 2\\}$ is declared to be the set of {\\em terminal} positions. If a player moves the queen into this terminal set with the usual Wythoff moves, that player wins the game. Let us call this game $K^2$. We chose this notation because we will later introduce a generalization of the game obtained by varying the set of terminal positions. Let $\\phi$ be the golden ratio~$(1+\\sqrt{5})/2$. We first define a sequence~$(g(n))_{n\\ge 0}$ as follows.\n\\begin{definition}\\label{def:fg}\n  We let $g(0)=1$, $g(1)=0$ and\n  \\[\n    g(n)=\\left\\{\n      \\begin{array}{ll}\n        1-g(m),& \\text{ if }\\lfloor n\\phi\\rfloor= \\lfloor m(\\phi+1)\\rfloor+1 \\text{ for some }m\\ge 0;\\\\\n        1,& \\text{ otherwise}.\n      \\end{array}\\right.\n  \\]\n\n\\end{definition}\nThe first values of the sequence~$g$ are given in Table~\\ref{tab:gh}. \n\\begin{table}[h!tb]\n  \\centering\n  \\[\n    \\begin{array}{c|ccccccccccccccccccccc}\n      & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10  & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 &  19 & 20 \\\\\n      \\hline\n      h& 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 & 6 & 6 &  7 & 8 & 8 & 9 & 9 & 10 & 11 & 11 & 12 &  12 \\\\\n      g&1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 &  1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\\\\n    \\end{array}\n  \\]\n  \\caption{First values of the sequence $g$ and Hofstadter sequence $h$.}\n  \\label{tab:gh}\n\\end{table}\n\nThe main result of Komak {\\em et al.} provides an algebraic description of set of $\\mathcal{P}$-positions of $K^2$:", "context": "In combinatorial game theory, proofs of many results are obtained by using a suitable numeration system \\cite{Duchene,Fraenkel1,Fraenkel2,RigoLNM}. As a typical example, the Zeckendorf numeration based on the Fibonacci sequence is used to characterize the $\\mathcal{P}$-positions (i.e., losing positions) of the Wythoff game \\cite{Fraenkel3}. Fraenkel showed that a pair $(a,b)$ of integers such that $a\\le b$ is a $\\mathcal{P}$-position of Wythoff game if and only if $\\rep_F(a)$ ends with an even number of zeroes and $\\rep_F(b)$ is the left-shift of $\\rep_F(a)$, i.e., $\\rep_F(b)=\\rep_F(a)0$.\n\nThe Wythoff game is a {\\em take-away} game: two players alternately remove tokens from two piles. One can remove a positive number of token from one pile or, remove the same positive number of token from both piles. The Wythoff game is {\\em impartial}: the set of options (i.e., positions reachable in one legal move) of a given position does not depend on which player is in turn to move. Wythoff game can be seen as a queen on an infinite chessboard $\\mathbb{N}^2$: in a position $(x,y)\\neq(0,0)$ the queen can move horizontally to $(x-n,y)$ with $0<n\\le x$, vertically to $(x,y-n)$ with $0<n\\le y$ or diagonally to $(x-n,y-n)$ with $0<n\\le \\min\\{x,y\\}$. According to the {\\em normal convention}, a player who is unable to move loses the game; passing is not allowed. Otherwise stated, the player who takes the last token wins or, in an equivalent way, when the queen reaches $(0,0)$.\n\nIn this article, we are mainly interested in two kinds of variations of the Wythoff game.\n\nIn \\cite{Komak}, a variant of Wythoff game has been introduced: the set $\\{(x,y)\\mid x+y\\le 2\\}$ is declared to be the set of {\\em terminal} positions. If a player moves the queen into this terminal set with the usual Wythoff moves, that player wins the game. Let us call this game $K^2$. We chose this notation because we will later introduce a generalization of the game obtained by varying the set of terminal positions. Let $\\phi$ be the golden ratio~$(1+\\sqrt{5})/2$. We first define a sequence~$(g(n))_{n\\ge 0}$ as follows.\n\\begin{definition}\\label{def:fg}\n We let $g(0)=1$, $g(1)=0$ and\n \\[\n g(n)=\\left\\{\n \\begin{array}{ll}\n 1-g(m),& \\text{ if }\\lfloor n\\phi\\rfloor= \\lfloor m(\\phi+1)\\rfloor+1 \\text{ for some }m\\ge 0;\\\\\n 1,& \\text{ otherwise}.\n \\end{array}\\right.\n \\]\n\n\\end{definition}\nThe first values of the sequence~$g$ are given in Table~\\ref{tab:gh}. \n\\begin{table}[h!tb]\n \\centering\n \\[\n \\begin{array}{c|ccccccccccccccccccccc}\n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\\\\n \\hline\n h& 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 & 6 & 6 & 7 & 8 & 8 & 9 & 9 & 10 & 11 & 11 & 12 & 12 \\\\\n g&1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\\\\n \\end{array}\n \\]\n \\caption{First values of the sequence $g$ and Hofstadter sequence $h$.}\n \\label{tab:gh}\n\\end{table}\n\nThe main result of Komak {\\em et al.} provides an algebraic description of set of $\\mathcal{P}$-positions of $K^2$:", "full_context": "In combinatorial game theory, proofs of many results are obtained by using a suitable numeration system \\cite{Duchene,Fraenkel1,Fraenkel2,RigoLNM}. As a typical example, the Zeckendorf numeration based on the Fibonacci sequence is used to characterize the $\\mathcal{P}$-positions (i.e., losing positions) of the Wythoff game \\cite{Fraenkel3}. Fraenkel showed that a pair $(a,b)$ of integers such that $a\\le b$ is a $\\mathcal{P}$-position of Wythoff game if and only if $\\rep_F(a)$ ends with an even number of zeroes and $\\rep_F(b)$ is the left-shift of $\\rep_F(a)$, i.e., $\\rep_F(b)=\\rep_F(a)0$.\n\nThe Wythoff game is a {\\em take-away} game: two players alternately remove tokens from two piles. One can remove a positive number of token from one pile or, remove the same positive number of token from both piles. The Wythoff game is {\\em impartial}: the set of options (i.e., positions reachable in one legal move) of a given position does not depend on which player is in turn to move. Wythoff game can be seen as a queen on an infinite chessboard $\\mathbb{N}^2$: in a position $(x,y)\\neq(0,0)$ the queen can move horizontally to $(x-n,y)$ with $0<n\\le x$, vertically to $(x,y-n)$ with $0<n\\le y$ or diagonally to $(x-n,y-n)$ with $0<n\\le \\min\\{x,y\\}$. According to the {\\em normal convention}, a player who is unable to move loses the game; passing is not allowed. Otherwise stated, the player who takes the last token wins or, in an equivalent way, when the queen reaches $(0,0)$.\n\nIn this article, we are mainly interested in two kinds of variations of the Wythoff game.\n\nIn \\cite{Komak}, a variant of Wythoff game has been introduced: the set $\\{(x,y)\\mid x+y\\le 2\\}$ is declared to be the set of {\\em terminal} positions. If a player moves the queen into this terminal set with the usual Wythoff moves, that player wins the game. Let us call this game $K^2$. We chose this notation because we will later introduce a generalization of the game obtained by varying the set of terminal positions. Let $\\phi$ be the golden ratio~$(1+\\sqrt{5})/2$. We first define a sequence~$(g(n))_{n\\ge 0}$ as follows.\n\\begin{definition}\\label{def:fg}\n We let $g(0)=1$, $g(1)=0$ and\n \\[\n g(n)=\\left\\{\n \\begin{array}{ll}\n 1-g(m),& \\text{ if }\\lfloor n\\phi\\rfloor= \\lfloor m(\\phi+1)\\rfloor+1 \\text{ for some }m\\ge 0;\\\\\n 1,& \\text{ otherwise}.\n \\end{array}\\right.\n \\]\n\n\\end{definition}\nThe first values of the sequence~$g$ are given in Table~\\ref{tab:gh}. \n\\begin{table}[h!tb]\n \\centering\n \\[\n \\begin{array}{c|ccccccccccccccccccccc}\n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\\\\n \\hline\n h& 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 & 6 & 6 & 7 & 8 & 8 & 9 & 9 & 10 & 11 & 11 & 12 & 12 \\\\\n g&1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\\\\n \\end{array}\n \\]\n \\caption{First values of the sequence $g$ and Hofstadter sequence $h$.}\n \\label{tab:gh}\n\\end{table}\n\nThe main result of Komak {\\em et al.} provides an algebraic description of set of $\\mathcal{P}$-positions of $K^2$:\n\n\\end{definition}\nThe first values of the sequence~$g$ are given in Table~\\ref{tab:gh}. \n\\begin{table}[h!tb]\n \\centering\n \\[\n \\begin{array}{c|ccccccccccccccccccccc}\n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\\\\n \\hline\n h& 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 & 6 & 6 & 7 & 8 & 8 & 9 & 9 & 10 & 11 & 11 & 12 & 12 \\\\\n g&1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\\\\n \\end{array}\n \\]\n \\caption{First values of the sequence $g$ and Hofstadter sequence $h$.}\n \\label{tab:gh}\n\\end{table}\n\nThe authors provide a classical proof of their result by showing that the set of $\\mathcal{P}$-positions described by \\eqref{eq:Ppos} is both stable and absorbing (see Definition~\\ref{def:stab}). The proof is quite long and requires a detailed case analysis. Without diminishing their achievement, once the set \\eqref{eq:Ppos} has been conjectured, if the set can be handled by {\\tt Walnut}, then it becomes straightforward to produce an automated proof in just a few lines. Our approach therefore complements the work carried out in \\cite{Komak}.\n\nIn Section~\\ref{sec:6}, we go further in the analysis of the games $K^\\ell$ for an arbitrary $\\ell\\ge 1$. We provide a recursive characterization of the set of $\\mathcal{P}$-positions. The non-terminal $\\mathcal{P}$-positions $(a_n,b_n)_{n\\ge 0}$ of $K^\\ell$ are given, in Theorem~\\ref{thm:rec_car}, by\n\\[\n (a_0,b_0)=(\\ell+1,2\\ell+2) \\quad\\text{ and }\\quad \\forall n\\ge 1, \\left\\{\n \\begin{array}{l}\na_n =\\mex (\\{ a_i,b_i \\mid i<n \\}\\cup\\{0,1,\\ldots,\\ell\\}) \\\\\nb_n =a_n+n+\\ell+1.\\\\\n \\end{array}\n \\right.\n\\]\nWe prove this result in an automatic way for $\\ell=1,2$, then extend it for all values of the parameter using classical arguments. With Corollary~\\ref{cor:behave}, we generalize Theorem~\\ref{thm:K2} and obtain that $(a_n,b_n)_{n\\ge 0}$ are of the form\n\\[ a_n=\\lfloor (n+\\ell)\\phi\\rfloor + \\lambda_\\ell(n) \\quad\\text{ and }\\quad\n b_n=\\lfloor (n+\\ell)\\phi^2\\rfloor + \\lambda_\\ell(n)+1\\]\nfor some bounded integer-valued function $\\lambda_\\ell$. Our result can be related to the recent paper \\cite{Letouzey} where discrepancy of generalized Hofstadter functions is studied (and formalized in {\\tt Coq}). Our methods are inspired by \\cite{Fraenkel5,Kimberling2,Larsson}.\n\nThe first non-terminal $\\mathcal{P}$-positions $(a_n,b_n)_{n\\ge 0}$ are given in Table~\\ref{tab:k3}. From now on, we take this convention: in the indexing of the $P$-positions, we do not take terminal positions into account. Thus, $(a_0,b_0)$ represents the first non-terminal $\\mathcal{P}$-position. Due to the symmetry of the game, we only consider sequences where $a_n<b_n$. \n\\begin{table}[h!t]\n\\[\n \\begin{array}{c|cccccccccccccccccccccccccccccc}\n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 \\\\\n \\hline\n a_n& 4 & 5 & 6 & 7 & 9 & 11 & 13 & 15 & 16 & 18 & 19 & 21 & 22 & 24 & 25 & 27 &\n 29 & 30 \\\\\n b_n & 8 & 10 & 12 & 14 & 17 & 20 & 23 & 26 & 28 & 31 & 33 & 36 & 38 & 41 & 43 & 46\n & 49 & 51 & \\\\\n \\end{array}\n \\]\n \\caption{First (non-terminal) $\\mathcal{P}$-positions of $K^3$, $(n\\ge 0)$.}\n \\label{tab:k3}\n\\end{table}\n\\begin{theorem}\\label{thm:fibaut2}\nThere exists a Fibonacci-automatic function $g_3:\\mathbb{N}\\to\\{0,1,2\\}$ such that, for all $n\\ge 0$, the $n$th non-terminal $\\mathcal{P}$-position $(a_n,b_n)$ of $K^3$ is given by \n\\begin{equation}\\label{eq:K3}\n a_n=\\lfloor (n+2)\\phi \\rfloor+g_3(n+1)-1 \\quad \\text{ and }\\quad \n b_n=\\lfloor (n+2)\\phi^2 \\rfloor+g_3(n+1)+1.\n\\end{equation}\n\\end{theorem}\nThe first few values of $g_3(n)$ are\n\\[\n 1221011221211111111121121121211211112111121111111211112\\cdots\n\\]\nUsing the heuristic described in Section~\\ref{ssec:heuristic}, assuming that the above infinite word is generated by a $\\varphi$-morphism, we obtain the morphism \n\\begin{align*}\n \\mu_3:\\;&\n0 \\mapsto 01,\n1 \\mapsto 2,\n2 \\mapsto 34,\n3 \\mapsto 56,\n4 \\mapsto 7,\n5 \\mapsto 78,\\\\ \n& 6 \\mapsto 9,\n7 \\mapsto ab,\n8 \\mapsto a,\n9 \\mapsto 56,\na \\mapsto ab, \nb \\mapsto 7\n\\end{align*}\nand the coding\n\\[\\rho_3:0,3,5,6,8,a,b\\mapsto 1,\\quad 1,2,7,9\\mapsto 2,\\quad 4\\mapsto 0.\\]\nWe thus define $g_3(n)$ as the $n$th symbol in $\\rho_3(\\mu_3^\\omega(1))$. Note that to produce enough value of $g_3(n)$, we have to compute sufficiently many $\\mathcal{P}$-positions of $W^3$.\n\n\\subsection{Even further}\nFor the game~$K^4$, we have applied the same procedure. The first non-terminal $\\mathcal{P}$-positions $(a_n,b_n)_{n\\ge 0}$ are given in Table~\\ref{tab:k4}.\n\\begin{table}[h!t]\n\\[\n\\begin{array}{c|cccccccccccccccccc}\n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 \\\\\n \\hline\n a_n &5 & 6 & 7 & 8 & 9 & 11 & 13 & 15 & 17 & 19 & 20 & 22 & 23 & 25 & 26 & 28 & 29 & 31 \\\\\n b_n & 10 & 12 & 14 & 16 & 18 & 21 & 24 & 27 & 30 & 33 & 35 & 38 & 40 & 43 & 45 & 48 & 50 & 53 \\\\\n\\end{array}\n \\]\n \\caption{First (non-terminal) $\\mathcal{P}$-positions of $K^4$, $(n\\ge 0)$.}\n \\label{tab:k4}\n\\end{table}\nTo get a long enough prefix, we have computed the first $600$ $\\mathcal{P}$-positions $(a_n,b_n)$ with $a_n\\le b_n$ (up to $a_n$ close to $1000$). From this, we conjectured that\n\\begin{equation}\\label{eq:K4}\n a_n=\\lfloor (n+2)\\phi \\rfloor+g_4(n+1) \\quad \\text{ and }\\quad \n b_n=\\lfloor (n+2)\\phi^2 \\rfloor+g_4(n+1)+3,\n\\end{equation}\nwhere the first few values of $g_4(n)$ are\n\\[\n 12210001112111111010110110111111111111111111\\cdots.\n\\]\nHaving the first $600$ values of $g_4(n)$, we apply the heuristic of Section~\\ref{ssec:heuristic}. Note that here, to be able to discern types correctly, we have to consider $4$-types (there are distinct $4$-types whose restrictions to $3$-types are equal). Since we have to take fourth iterations, this explains why we need a longer prefix. We obtain $18$ different $4$-types and the heuristic gives the following morphisms:\n\\begin{align*}\n \\mu_4:\\;&0 \\mapsto 01,\\ 1 \\mapsto 2,\\ 2 \\mapsto 34,\\ 3 \\mapsto 56,\\ 4 \\mapsto 7,\\ 5 \\mapsto 89,\\ 6 \\mapsto a,\\ 7 \\mapsto bc,\\ 8 \\mapsto dc,\\\\\n &9 \\mapsto d,\\ a \\mapsto ef,\\ b \\mapsto ef,\\ c \\mapsto e,\\ d \\mapsto 7g,\\ e \\mapsto eh,\\ f \\mapsto b,\\ g \\mapsto d,\\ h \\mapsto b \n\\end{align*}\n\\[\n \\rho_4: 0,3,7,8,9,b,c,d,e,h \\mapsto 1,\\ 1,2,a \\mapsto 2,\\ 4,5,6,f,g \\mapsto 0\n\\]\nWith $g_4(n)$ being the $n$th symbol of $\\rho_4(\\mu_4^\\omega(1))$, we have a $18$-state DFAO. Formulas \\eqref{eq:K4} translates in {\\tt Walnut} into\n\\begin{verbatim}\ndef pposK4_asym \"?msd_fib (a+b<=4) | (En,x $phin(n+2,x) & (a=x+K4[n+1])\n & (b=x+K4[n+1]+n+5))\":\n\\end{verbatim}\nand we then follow exactly the same proof to get the result.\n\nLet us begin with some classical and well-known results on Wythoff game before moving on to the games discussed in this article. We recall the {\\em recursive definition} of the $\\mathcal{P}$-positions of Wythoff game~$K^0$, with our convention:\n\\[\n (a_0,b_0)=(1,2) \\quad\\text{ and }\\quad \\forall n\\ge 1, \\left\\{\n \\begin{array}{l}\na_n =\\mex (\\{ a_i,b_i \\mid i<n \\}\\cup\\{0\\}),\\\\\nb_n =a_n+n+1,\\\\\n \\end{array}\n \\right.\n\\]\nwhere $\\mex$ stands for {\\em minimum excluded value}, i.e. the least non-negative integer not in the set. In other words, $\\mex S=\\min(\\mathbb{N}\\setminus S)$. So $a_1=\\mex\\{0,1,2\\}=3$ and $b_1=a_1+2=5$, then $a_2=\\mex\\{0,1,2,3,5\\}=4$ and $b_2=a_2+3=7$ and so on.\n\nLarsson characterized \\cite[Thm.~1.1]{block} the set of $\\mathcal{P}$-positions of $W^2$, a variant of Wythoff game where one option may be blocked, see Figure~\\ref{fig:w2w3}. We provide an automatic proof of this result. \n\\begin{theorem}\nThe set of $\\mathcal{P}$-positions of $W^2$ is given by\n\\[\n \\mathcal{R}_2=\\{(0,0)\\}\\cup \\{ \\{n,2n+1\\}\\mid n\\ge 0\\}\\cup \\{\\{2\\lfloor n\\phi\\rfloor+2,2\\lfloor n\\phi^2\\rfloor+2\\}\\mid n\\ge 0\\}.\n\\] \n\\end{theorem}", "post_theorem_intro_text_len": 6231, "post_theorem_intro_text": "The authors provide a classical proof of their result by showing that the set of $\\mathcal{P}$-positions described by \\eqref{eq:Ppos} is both stable and absorbing (see Definition~\\ref{def:stab}). The proof is quite long and requires a detailed case analysis. Without diminishing their achievement, once the set \\eqref{eq:Ppos} has been conjectured, if the set can be handled by {\\tt Walnut}, then it becomes straightforward to produce an automated proof in just a few lines. Our approach therefore complements the work carried out in \\cite{Komak}.\n\nIn \\cite{block}, other variants of Wythoff game are studied, in which some blocking maneuvers are added. Let $k\\ge 1$. In the game denoted by $W^k$, for each move, before the next player (i.e., the player who is about to play) moves, the previous player (i.e., the one who has just played) may declare at most $k-1$ of the options as forbidden. When the next player has moved, any blocking maneuver is forgotten and has no further incidence on the game. This terminology of previous and next player explains why we speak of $\\mathcal{N}$- and $\\mathcal{P}$-positions, respectively. For $k=1$, this is the classical game of Wythoff. When $k=2,3$, we reconsider Larsson's result and obtain an automated proof, replacing a 2-page-long analysis by some easy-to-describe first-order formulas. \n\nAs Shallit has already noted on several occasions, the two approaches are complementary. A classical, purely combinatorial proof usually provides structural insights, whereas the use of automated provers helps avoid lengthy case analyses and enables the exploration of directions that are difficult to access through traditional techniques. In particular, in this article, we easily obtain a variety of results that would otherwise require significantly more effort using ``classical'' methods. Moreover, these results lead us to formulate general theorems.\n\\smallskip\n\nThis article is organized as follows. Section~\\ref{sec:2} --- which was the starting point of this article --- aims to give an automatic proof of the algebraic characterization \\eqref{eq:Ppos} of the $\\mathcal{P}$-positions of $K^2$. We begin with preliminary results about Fibonacci-automatic sequences. We define a notion of $\\varphi$-morphism and describe a heuristic that we extensively use throughout the paper to construct such $\\varphi$-morphisms. Given a sufficiently long prefix of an infinite word, we obtain morphisms that can be used in our automatic proofs (and we can therefore prove the correctness of the procedure). Basic results from combinatorial game theory are given in Section~\\ref{ssec:autom_proof}.\n\nIn Section~\\ref{sec:newGame}, we obtain results about new games: a parameterized version of a variant of Wythoff game denoted by $K^\\ell$, where the set of terminal states is\n\\[\n  \\{(x,y)\\in\\mathbb{N}^2\\mid x+y\\le \\ell\\}\n\\]\nfor some $\\ell\\in\\mathbb{N}$. We study the games $K^1$, $K^3$ and $K^4$ and obtain algebraic characterizations of the $\\mathcal{P}$-positions similar to \\cite{Komak}. In particular, as for the classical Wythoff game $K^0$, the game $K^1$ has a nice set of $\\mathcal{P}$-positions: a pair $(a,b)$ of integers such that $a\\le b$ is a $\\mathcal{P}$-position of $K^1$ if and only if its Fibonacci representation $\\rep_F(a)$ ends with zero and $\\rep_F(b)$ is of the form $\\rep_F(a)1$, see Theorem~\\ref{thm:carK1}. \n\nIn Section~\\ref{sec:6}, we go further in the analysis of the games $K^\\ell$ for an arbitrary $\\ell\\ge 1$. We provide a recursive characterization of the set of $\\mathcal{P}$-positions. The non-terminal $\\mathcal{P}$-positions $(a_n,b_n)_{n\\ge 0}$ of $K^\\ell$ are given, in Theorem~\\ref{thm:rec_car}, by\n\\[\n  (a_0,b_0)=(\\ell+1,2\\ell+2) \\quad\\text{ and }\\quad \\forall n\\ge 1, \\left\\{\n    \\begin{array}{l}\na_n =\\mex (\\{ a_i,b_i \\mid i<n \\}\\cup\\{0,1,\\ldots,\\ell\\}) \\\\\nb_n =a_n+n+\\ell+1.\\\\\n    \\end{array}\n  \\right.\n\\]\nWe prove this result in an automatic way for $\\ell=1,2$, then extend it for all values of the parameter using classical arguments. With Corollary~\\ref{cor:behave}, we generalize Theorem~\\ref{thm:K2} and obtain that $(a_n,b_n)_{n\\ge 0}$ are of the form\n\\[ a_n=\\lfloor (n+\\ell)\\phi\\rfloor + \\lambda_\\ell(n) \\quad\\text{ and }\\quad\n  b_n=\\lfloor (n+\\ell)\\phi^2\\rfloor + \\lambda_\\ell(n)+1\\]\nfor some bounded integer-valued function $\\lambda_\\ell$. Our result can be related to the recent paper \\cite{Letouzey} where discrepancy of generalized Hofstadter functions is studied (and formalized in {\\tt Coq}). Our methods are inspired by   \\cite{Fraenkel5,Kimberling2,Larsson}.\n\nIt is well-known that the set of $\\mathcal{P}$-positions of the Wythoff game are coded by the Fibonacci word~$\\mathbf{f}=abaababa\\cdots$, see \\cite{Vision,Duchene}. If we start indexing letters of~$\\mathbf{f}$ with $1$, the indices of the letters $a$ (resp. $b$) in~$\\mathbf{f}$ correspond to the sequence $(a_n)_{n\\ge 0}$ (resp. $(b_n)_{n\\ge 0}$). Using the recursive definition from the previous section, we obtain  similar results with $K^1,K^2,K^3$: their $\\mathcal{P}$-positions are encoded by morphic words obtained by $\\varphi$-morphisms and codings. \n\nIn Section~\\ref{sec:3}, we obtain automatic proofs of Larsson's result about $W^2$ and $W^3$. Interestingly, the automata arising in the computations are much larger than those for $K^2$.\n\nIn Section~\\ref{sec:redundant}, we are concerned with redundant moves of the games discussed so far. A move is considered {\\em redundant} if, upon its removal from the rule-set, then the set of $\\mathcal{P}$-positions remains unchanged. It is a classical question in combinatorial game theory: do distinct rule-sets yield the same set of $\\mathcal{P}$-positions? In particular, can certain moves be added or removed without altering the $\\mathcal{P}$-positions?\n\n\\begin{remark}\n  A {\\tt Jupyter} notebook recording all the {\\tt Walnut} computations is available (it has been produced using Ollinger's {\\tt Walnut-Kernel}. \n  The {\\tt Mathematica} code and the various {\\tt Walnut} files are also available online\\footnote{at {\\tt https://hdl.handle.net/2268/338482}, you may also download a standalone Wolfram Player.}. In particular, the package permits us to compute $\\mathcal{P}$-positions and $\\varphi$-morphisms.\n\\end{remark}", "sketch": "To prove Theorem~\\ref{thm:K2}, the authors “provide a classical proof … by showing that the set of $\\mathcal{P}$-positions described by \\eqref{eq:Ppos} is both stable and absorbing (see Definition~\\ref{def:stab}).” They note that “the proof is quite long and requires a detailed case analysis.” They also indicate that, “once the set \\eqref{eq:Ppos} has been conjectured,” an alternative route is to “produce an automated proof” (e.g., “if the set can be handled by {\\tt Walnut}”).", "expanded_sketch": "To prove the main theorem, the authors “provide a classical proof … by showing that the set of $\\mathcal{P}$-positions described by \\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}\n is both stable and absorbing (see \\begin{definition}\\label{def:stab}\nA set $S$ of positions is {\\em stable} if, for all $s,t\\in S$, $s\\neq t$, there is no move between $s$ and $t$. A set $S$ of positions is {\\em absorbing}, if for all $t\\not\\in S$, there exists $s\\in S$ such that there is a move from $t$ to $s$ (or, $t$ has an option in $S$).  \n\\end{definition}).” They note that “the proof is quite long and requires a detailed case analysis.” They also indicate that, “once the set in the equation above has been conjectured,” an alternative route is to “produce an automated proof” (e.g., “if the set can be handled by {\\tt Walnut}”).", "expanded_theorem": "\\label{thm:K2}\n  The set of $\\mathcal{P}$-positions of the variant of Wythoff game where the set of terminal positions is $\\{(x,y)\\mid x+y\\le 2\\}$,  is exactly \\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}\nwhere $g:\\mathbb{N}\\to\\mathbb{N}$ is defined as follows: \\begin{definition}\\label{def:fg}\n  We let $g(0)=1$, $g(1)=0$ and\n  \\[\n    g(n)=\\left\\{\n      \\begin{array}{ll}\n        1-g(m),& \\text{ if }\\lfloor n\\phi\\rfloor= \\lfloor m(\\phi+1)\\rfloor+1 \\text{ for some }m\\ge 0;\\\\\n        1,& \\text{ otherwise}.\n      \\end{array}\\right.\n  \\]\n\n\\end{definition}", "theorem_type": ["Classification or Bijection", "Equivalence"], "mcq": {"question": "Consider the Wythoff-type game on positions (x,y) ∈ ℕ² in which a move from (x,y) consists of replacing it by (x-n,y), (x,y-n), or (x-n,y-n), where n is any positive integer allowed by the coordinates. The terminal positions are exactly those with x+y ≤ 2, and under normal play a P-position means a losing position. Let φ=(1+√5)/2, and define g:ℕ→ℕ by g(0)=1, g(1)=0, and for n≥0,\n\n g(n)=1-g(m) if floor(nφ)=floor(m(φ+1))+1 for some m≥0, and g(n)=1 otherwise.\n\nWhich statement gives exactly the set of P-positions of this game?", "correct_choice": {"label": "A", "text": "The set of P-positions is exactly\n\n{ (⌊nφ⌋+g(n)-1, ⌊nφ²⌋+g(n)) : n≥0 } ∪ { (⌊nφ²⌋+g(n), ⌊nφ⌋+g(n)-1) : n≥0 }."}, "choices": [{"label": "B", "text": "The set of P-positions is exactly\n\n{ (⌊nφ⌋+g(n), ⌊nφ²⌋+g(n)) : n≥0 } ∪ { (⌊nφ²⌋+g(n), ⌊nφ⌋+g(n)) : n≥0 }."}, {"label": "C", "text": "Every position of the form\n\n(⌊nφ⌋+g(n)-1, ⌊nφ²⌋+g(n))\n\nwith n≥0 is a P-position."}, {"label": "D", "text": "The set of P-positions is exactly\n\n{ (⌊nφ⌋+g(n)-1, ⌊nφ²⌋+g(n)) : n≥1 } ∪ { (⌊nφ²⌋+g(n), ⌊nφ⌋+g(n)-1) : n≥1 }."}, {"label": "E", "text": "The set of P-positions is exactly\n\n{ (⌊nφ⌋+g(n)-1, ⌊nφ²⌋+g(n)) : n≥0 }."}], "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": "offset-by-1 needed for stability against terminal region and moves within the candidate set", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "case_split", "tampered_component": "dropped the converse exactness/union description, keeping only one included family of P-positions", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "boundary index n=0", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "symmetry of coordinates required for absorbing all non-candidate positions", "template_used": "wildcard"}]}}
{"id": "2512.11601v1", "paper_link": "http://arxiv.org/abs/2512.11601v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm:K2}\n  The set of $\\mathcal{P}$-positions of the variant of Wythoff game where the set of terminal positions is $\\{(x,y)\\mid x+y\\le 2\\}$,  is exactly \\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}\nwhere $g:\\mathbb{N}\\to\\mathbb{N}$ is the function given in Definition~\\ref{def:fg}.", "start_pos": 103245, "end_pos": 103752, "label": "thm:K2"}, "ref_dict": {"cor:behave": "\\begin{corollary}\\label{cor:behave}\n  For the sequences defined by \\eqref{eq:mexdef}, \n  there exist a bounded function $\\lambda_\\ell:\\mathbb{N}\\to\\mathbb{Z}$ such that, for all $n$,   \n  \\[\n   a_n=\\lfloor (n+\\ell)\\phi \\rfloor +\\lambda_\\ell(n).\n \\]\n So, in particular, \\(b_n=\\lfloor (n+\\ell)\\phi^2 \\rfloor + \\lambda_\\ell(n) +1\\).\n  \\end{corollary}", "thm:K2": "\\begin{theorem}\\label{thm:K2}\n  The set of $\\mathcal{P}$-positions of the variant of Wythoff game where the set of terminal positions is $\\{(x,y)\\mid x+y\\le 2\\}$,  is exactly \\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}\nwhere $g:\\mathbb{N}\\to\\mathbb{N}$ is the function given in Definition~\\ref{def:fg}.\n\\end{theorem}", "thm:rec_car": "\\begin{theorem}\\label{thm:rec_car}\n  Let $\\ell\\ge 1$ and define recursively $(a_n,b_n)_{n\\ge 0}$ by\n\\begin{equation}\\label{eq:mexdef}\n  (a_0,b_0)=(\\ell+1,2\\ell+2) \\quad\\text{ and }\\quad \\forall n\\ge 1, \\left\\{\n    \\begin{array}{l}\na_n =\\mex (\\{ a_i,b_i \\mid i<n \\}\\cup\\{0,1,\\ldots,\\ell\\}),\\\\\nb_n =a_n+n+\\ell+1.\\\\\n    \\end{array}\n  \\right. \n\\end{equation} \nThe set of non-terminal $\\mathcal{P}$-positions of $K^\\ell$ is $\\{(a_n,b_n)\\mid n\\ge 0\\} \\cup \\{(b_n,a_n)\\mid n\\ge 0\\} $.\n\\end{theorem}", "eq:Ppos": "\\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}", "def:fg": "\\begin{definition}\\label{def:fg}\n  We let $g(0)=1$, $g(1)=0$ and\n  \\[\n    g(n)=\\left\\{\n      \\begin{array}{ll}\n        1-g(m),& \\text{ if }\\lfloor n\\phi\\rfloor= \\lfloor m(\\phi+1)\\rfloor+1 \\text{ for some }m\\ge 0;\\\\\n        1,& \\text{ otherwise}.\n      \\end{array}\\right.\n  \\]\n\n\\end{definition}", "ssec:autom_proof": "\\begin{array}{c|llll}\n    &g_n & \\mu(g_n) & \\mu^2(g_n) & \\mu^3(g_n) \\\\\n    \\hline\n 0&1& 0& 1& 10\\\\\n 1&0& 1& 10& 011\\\\\n 2&1& 10& 011& 11010\\\\\n 3&1& 01& 110& 01011\\\\\n 4&0& 11& 010& 11011\\\\\n 5&1& 0& 11& 010\\\\\n  \\end{array}\n\\]\n  \\caption{The different $3$-types when applying a $\\varphi$-morphism to $g$.}\n  \\label{tab:types}\n\\end{table}\n\nNow we associate each symbol $g_n$ with its $3$-type and get the infinite word\n\\[  \n  01|2|31|45|2|35|4|31|45|4|35|45|\\cdots\n\\]\nwhere the vertical bars indicate the factorization of the Fibonacci word $ab|a|ab|ab|a|\\cdots$ with factors $\\sigma(a)=ab$ and $\\sigma(b)=a$, the images of the letters by the Fibonacci morphism. From this, we deduce the morphism $\\mu$ given in Example~\\ref{exa:mu}: the image of the $n$th letter has to be the $n$th block in the factorization. So $0\\mapsto 01$, then $1\\mapsto 2$ and so on. From the morphism, it is a routine to obtain the DFA in Figure~\\ref{fig:automG}, see Definition~\\ref{def:dfao}.\n\nFinally, one may define a coding~$\\rho$ to recover the original sequence where each type is mapped to the first element of the tuple. So, $0,2,3,5\\mapsto 1$ and $1,4\\mapsto 0$. This can equivalently be performed by comparing the morphic word and the original sequence\n\\begin{align*}\n  &01231452354314543545\\cdots\\\\\n  &10110011110100101101\\cdots\n\\end{align*}\nand matching corresponding elements.\n\\subsection{An automatic proof for $K^2$}\\label{ssec:autom_proof}\n\nNow that $g$ has been defined by a DFA fed with Fibonacci representations (and stored as {\\tt G.txt} in {\\tt Word Automata Lib}), the set of $\\mathcal{P}$-positions \\eqref{eq:Ppos} obtained by Komak et al. can be described by the following commands:\n\\begin{verbatim}\ndef pposK2_asym \"?msd_fib (a+b<=1) | En,x $phin(n,x) & a+1=x+G[n]\n                                                     & b=x+n+G[n]\":\ndef pposK2 \"?msd_fib $pposK2_asym(a,b) | $pposK2_asym(b,a) \":\n\\end{verbatim}\nThe resulting automaton has $27$ states.\n\nFor the sake of presentation, let us recall some basics on combinatorial games.\n\\begin{definition}\\label{def:stab}\nA set $S$ of positions is {\\em stable} if, for all $s,t\\in S$, $s\\neq t$, there is no move between $s$ and $t$. A set $S$ of positions is {\\em absorbing}, if for all $t\\not\\in S$, there exists $s\\in S$ such that there is a move from $t$ to $s$ (or, $t$ has an option in $S$).  \n\\end{definition}\nIt is a well-known property in graph theory that every acyclic graph has a unique kernel, i.e., a stable and absorbing set of vertices. In a take-away game where tokens are removed from piles, the corresponding game graph is acyclic: it is not possible to visit twice the same position during a game.\n\\begin{proposition}[Folklore]\\label{pro:kernel}\n  The sets of $\\mathcal{P}$- and $\\mathcal{N}$-positions of an\n  impartial acyclic game are uniquely determined by the following two\n  properties:\n\\begin{enumerate}\n\\item Every move from a $\\mathcal{P}$-position leads to an\n  $\\mathcal{N}$-position; equivalently there is no move between two $\\mathcal{P}$-positions (stability property of $\\mathcal{P}(G)$).\n  \\item From every $\\mathcal{N}$-position, there exists a move leading to a $\\mathcal{P}$-position (absorbing property of $\\mathcal{P}(G)$).\n\\end{enumerate}\n\\end{proposition}\n\nAs observed in \\cite{MRRW}, it is then a routine test to check stability and absorbing property:\n\\begin{verbatim}\neval stableK2 \"?msd_fib Ap,q,r,s (($pposK2(p,q) & $pposK2(r,s)\n     & p>=r & q>=s & p+q>2) => ((p=r & q=s) | (p>r & q>s & p+s!=q+r)))\":\n\neval absorbingK2 \"?msd_fib Ap,q (~$pposK2(p,q) => Ex,y\n     (x<=p & y<=q & $pposK2(x,y) & (p+y=q+x | p=x | q=y))) \":\n\\end{verbatim}\nThese two commands return {\\tt TRUE} resulting in the following result, \\cite[Thm.~3]{Komak}. From Proposition~\\ref{pro:kernel}, we immediately get an alternative proof of Theorem~\\ref{thm:K2}.\n\n\\section{Towards an algebraic characterization for $K^\\ell$}\\label{sec:newGame}\n\nIn \\cite{Komak}, Komak et al. chose $\\{(x,y)\\mid x+y\\le 2\\}$ as set of terminal positions. It is natural to consider a parameterized version where the set of terminal states is\n\\[\n  \\{(x,y)\\in\\mathbb{N}^2\\mid x+y\\le \\ell\\}\n\\]\nfor some $\\ell\\in\\mathbb{N}$. The corresponding game is denoted by $K^\\ell$. Note that the game $K^0$ is the classic Wythoff game. In this section, we obtain new results concerning the games $K^1$ and~$K^3$.\n\n\\subsection{The $K^1$-game}\nUp to our knowledge, the game $K^1$ does not seem to have been studied in the literature. Computing the first $\\mathcal{P}$-positions of $K^1$ and writing their Fibonacci expansions permitted us to state a conjecture that can be tested with {\\tt Walnut}. This result is quite similar to the famous characterization of the $\\mathcal{P}$-positions of Wythoff game obtained by Fraenkel \\cite{Fraenkel3}.\n\\begin{table}[h!t]\n  \\[\n    \\begin{array}{c|r|c|r}\n      a_n & \\rep_F(a_n) & b_n & \\rep_F(b_n) \\\\\n      \\hline\n 0&\\varepsilon  & 1 &1 \\\\\n 2&10 & 4 &101 \\\\\n 3&100 & 6 & 1001 \\\\\n 5&1000 & 9 & 10001 \\\\\n 7&1010 & 12 & 10101 \\\\\n 8&10000 & 14 & 100001 \\\\\n 10&10010 & 17 &100101 \\\\\n 11&10100 & 19 & 101001 \\\\\n 13&100000 & 22 & 1000001 \\\\\n 15&100010 & 25 & 1000101 \\\\\n  \\end{array}", "def:stab": "\\begin{definition}\\label{def:stab}\nA set $S$ of positions is {\\em stable} if, for all $s,t\\in S$, $s\\neq t$, there is no move between $s$ and $t$. A set $S$ of positions is {\\em absorbing}, if for all $t\\not\\in S$, there exists $s\\in S$ such that there is a move from $t$ to $s$ (or, $t$ has an option in $S$).  \n\\end{definition}", "thm:carK1": "\\begin{theorem}\\label{thm:carK1}\n  A position $(a,b)$, with $a\\le b$, is a $\\mathcal{P}$-position of the game $K^1$, with a set of terminal positions being $\\{(x,y)\\mid x+y\\le 1\\}$ and Wythoff's moves, if and only if $\\rep_F(a)$ ends with $0$ and $\\rep_F(b)=\\rep_F(a)1$. \n\\end{theorem}", "tab:gh": "\\begin{array}{c|ccccccccccccccccccccc}\n      & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10  & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 &  19 & 20 \\\\\n      \\hline\n      h& 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 & 6 & 6 &  7 & 8 & 8 & 9 & 9 & 10 & 11 & 11 & 12 &  12 \\\\\n      g&1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 &  1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\\\\n    \\end{array}\n  \\]\n  \\caption{First values of the sequence $g$ and Hofstadter sequence $h$.}\n  \\label{tab:gh}\n\\end{table}\n\nThe main result of Komak {\\em et al.} provides an algebraic description of set of $\\mathcal{P}$-positions of $K^2$:\n\\begin{theorem}\\label{thm:K2}\n  The set of $\\mathcal{P}$-positions of the variant of Wythoff game where the set of terminal positions is $\\{(x,y)\\mid x+y\\le 2\\}$,  is exactly \\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}\nwhere $g:\\mathbb{N}\\to\\mathbb{N}$ is the function given in Definition~\\ref{def:fg}.\n\\end{theorem}\n\nThe authors provide a classical proof of their result by showing that the set of $\\mathcal{P}$-positions described by \\eqref{eq:Ppos} is both stable and absorbing (see Definition~\\ref{def:stab}). The proof is quite long and requires a detailed case analysis. Without diminishing their achievement, once the set \\eqref{eq:Ppos} has been conjectured, if the set can be handled by {\\tt Walnut}, then it becomes straightforward to produce an automated proof in just a few lines. Our approach therefore complements the work carried out in \\cite{Komak}.\n\nIn \\cite{block}, other variants of Wythoff game are studied, in which some blocking maneuvers are added. Let $k\\ge 1$. In the game denoted by $W^k$, for each move, before the next player (i.e., the player who is about to play) moves, the previous player (i.e., the one who has just played) may declare at most $k-1$ of the options as forbidden. When the next player has moved, any blocking maneuver is forgotten and has no further incidence on the game. This terminology of previous and next player explains why we speak of $\\mathcal{N}$- and $\\mathcal{P}$-positions, respectively. For $k=1$, this is the classical game of Wythoff. When $k=2,3$, we reconsider Larsson's result and obtain an automated proof, replacing a 2-page-long analysis by some easy-to-describe first-order formulas. \n\nAs Shallit has already noted on several occasions, the two approaches are complementary. A classical, purely combinatorial proof usually provides structural insights, whereas the use of automated provers helps avoid lengthy case analyses and enables the exploration of directions that are difficult to access through traditional techniques. In particular, in this article, we easily obtain a variety of results that would otherwise require significantly more effort using ``classical'' methods. Moreover, these results lead us to formulate general theorems.\n\\smallskip\n\nThis article is organized as follows. Section~\\ref{sec:2} --- which was the starting point of this article --- aims to give an automatic proof of the algebraic characterization \\eqref{eq:Ppos} of the $\\mathcal{P}$-positions of $K^2$. We begin with preliminary results about Fibonacci-automatic sequences. We define a notion of $\\varphi$-morphism and describe a heuristic that we extensively use throughout the paper to construct such $\\varphi$-morphisms. Given a sufficiently long prefix of an infinite word, we obtain morphisms that can be used in our automatic proofs (and we can therefore prove the correctness of the procedure). Basic results from combinatorial game theory are given in Section~\\ref{ssec:autom_proof}.\n\nIn Section~\\ref{sec:newGame}, we obtain results about new games: a parameterized version of a variant of Wythoff game denoted by $K^\\ell$, where the set of terminal states is\n\\[\n  \\{(x,y)\\in\\mathbb{N}^2\\mid x+y\\le \\ell\\}\n\\]\nfor some $\\ell\\in\\mathbb{N}$. We study the games $K^1$, $K^3$ and $K^4$ and obtain algebraic characterizations of the $\\mathcal{P}$-positions similar to \\cite{Komak}. In particular, as for the classical Wythoff game $K^0$, the game $K^1$ has a nice set of $\\mathcal{P}$-positions: a pair $(a,b)$ of integers such that $a\\le b$ is a $\\mathcal{P}$-position of $K^1$ if and only if its Fibonacci representation $\\rep_F(a)$ ends with zero and $\\rep_F(b)$ is of the form $\\rep_F(a)1$, see Theorem~\\ref{thm:carK1}. \n\nIn Section~\\ref{sec:6}, we go further in the analysis of the games $K^\\ell$ for an arbitrary $\\ell\\ge 1$. We provide a recursive characterization of the set of $\\mathcal{P}$-positions. The non-terminal $\\mathcal{P}$-positions $(a_n,b_n)_{n\\ge 0}$ of $K^\\ell$ are given, in Theorem~\\ref{thm:rec_car}, by\n\\[\n  (a_0,b_0)=(\\ell+1,2\\ell+2) \\quad\\text{ and }\\quad \\forall n\\ge 1, \\left\\{\n    \\begin{array}{l}\na_n =\\mex (\\{ a_i,b_i \\mid i<n \\}\\cup\\{0,1,\\ldots,\\ell\\}) \\\\\nb_n =a_n+n+\\ell+1.\\\\\n    \\end{array}"}, "pre_theorem_intro_text_len": 3799, "pre_theorem_intro_text": "In combinatorial game theory, proofs of many results are obtained by using a suitable numeration system \\cite{Duchene,Fraenkel1,Fraenkel2,RigoLNM}. As a typical example, the Zeckendorf numeration based on the Fibonacci sequence is used to characterize the $\\mathcal{P}$-positions (i.e., losing positions) of the Wythoff game \\cite{Fraenkel3}. Fraenkel showed that a pair $(a,b)$ of integers such that $a\\le b$ is a $\\mathcal{P}$-position of Wythoff game if and only if $\\rep_F(a)$ ends with an even number of zeroes and $\\rep_F(b)$ is the left-shift of $\\rep_F(a)$, i.e., $\\rep_F(b)=\\rep_F(a)0$.\n\nWhen the numeration system has good properties (i.e., it is an {\\em addable} system as discussed in \\cite{ShallitBook}), the formalism of first-order logic can be used to express certain properties of combinatorial games. We have recently exploited this approach and used {\\tt Walnut} to reprove classical results in combinatorial games theory, as well as to obtain new ones \\cite{MRRW}. {\\tt Walnut} is a free, open-source prover for first-order statements dealing with automatic sequences \\cite{Mousavi,ShallitBook}. \n\nWe assume that the reader is sufficiently familiar with the use of {\\tt Walnut}. For references on combinatorial game theory, we refer to \\cite{RigoLNM} and \\cite{MRRW}.\n\nThe Wythoff game is a {\\em take-away} game: two players alternately remove tokens from two piles. One can remove a positive number of token from one pile or, remove the same positive number of token from both piles. The Wythoff game is {\\em impartial}: the set of options (i.e., positions reachable in one legal move) of a given position does not depend on which player is in turn to move. Wythoff game can be seen as a queen on an infinite chessboard $\\mathbb{N}^2$: in a position $(x,y)\\neq(0,0)$ the queen can move horizontally to $(x-n,y)$ with $0<n\\le x$, vertically to $(x,y-n)$ with $0<n\\le y$ or diagonally to $(x-n,y-n)$ with $0<n\\le \\min\\{x,y\\}$. According to the {\\em normal convention}, a player who is unable to move loses the game; passing is not allowed. Otherwise stated, the player who takes the last token wins or, in an equivalent way, when the queen reaches $(0,0)$.\n\nIn this article, we are mainly interested in two kinds of variations of the Wythoff game. \n\nIn \\cite{Komak}, a variant of Wythoff game has been introduced: the set $\\{(x,y)\\mid x+y\\le 2\\}$ is declared to be the set of {\\em terminal} positions. If a player moves the queen into this terminal set with the usual Wythoff moves, that player wins the game. Let us call this game $K^2$. We chose this notation because we will later introduce a generalization of the game obtained by varying the set of terminal positions. Let $\\phi$ be the golden ratio~$(1+\\sqrt{5})/2$. We first define a sequence~$(g(n))_{n\\ge 0}$ as follows.\n\\begin{definition}\\label{def:fg}\n  We let $g(0)=1$, $g(1)=0$ and\n  \\[\n    g(n)=\\left\\{\n      \\begin{array}{ll}\n        1-g(m),& \\text{ if }\\lfloor n\\phi\\rfloor= \\lfloor m(\\phi+1)\\rfloor+1 \\text{ for some }m\\ge 0;\\\\\n        1,& \\text{ otherwise}.\n      \\end{array}\\right.\n  \\]\n\n\\end{definition}\nThe first values of the sequence~$g$ are given in Table~\\ref{tab:gh}. \n\\begin{table}[h!tb]\n  \\centering\n  \\[\n    \\begin{array}{c|ccccccccccccccccccccc}\n      & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10  & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 &  19 & 20 \\\\\n      \\hline\n      h& 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 & 6 & 6 &  7 & 8 & 8 & 9 & 9 & 10 & 11 & 11 & 12 &  12 \\\\\n      g&1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 &  1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\\\\n    \\end{array}\n  \\]\n  \\caption{First values of the sequence $g$ and Hofstadter sequence $h$.}\n  \\label{tab:gh}\n\\end{table}\n\nThe main result of Komak {\\em et al.} provides an algebraic description of set of $\\mathcal{P}$-positions of $K^2$:", "context": "In combinatorial game theory, proofs of many results are obtained by using a suitable numeration system \\cite{Duchene,Fraenkel1,Fraenkel2,RigoLNM}. As a typical example, the Zeckendorf numeration based on the Fibonacci sequence is used to characterize the $\\mathcal{P}$-positions (i.e., losing positions) of the Wythoff game \\cite{Fraenkel3}. Fraenkel showed that a pair $(a,b)$ of integers such that $a\\le b$ is a $\\mathcal{P}$-position of Wythoff game if and only if $\\rep_F(a)$ ends with an even number of zeroes and $\\rep_F(b)$ is the left-shift of $\\rep_F(a)$, i.e., $\\rep_F(b)=\\rep_F(a)0$.\n\nThe Wythoff game is a {\\em take-away} game: two players alternately remove tokens from two piles. One can remove a positive number of token from one pile or, remove the same positive number of token from both piles. The Wythoff game is {\\em impartial}: the set of options (i.e., positions reachable in one legal move) of a given position does not depend on which player is in turn to move. Wythoff game can be seen as a queen on an infinite chessboard $\\mathbb{N}^2$: in a position $(x,y)\\neq(0,0)$ the queen can move horizontally to $(x-n,y)$ with $0<n\\le x$, vertically to $(x,y-n)$ with $0<n\\le y$ or diagonally to $(x-n,y-n)$ with $0<n\\le \\min\\{x,y\\}$. According to the {\\em normal convention}, a player who is unable to move loses the game; passing is not allowed. Otherwise stated, the player who takes the last token wins or, in an equivalent way, when the queen reaches $(0,0)$.\n\nIn this article, we are mainly interested in two kinds of variations of the Wythoff game.\n\nIn \\cite{Komak}, a variant of Wythoff game has been introduced: the set $\\{(x,y)\\mid x+y\\le 2\\}$ is declared to be the set of {\\em terminal} positions. If a player moves the queen into this terminal set with the usual Wythoff moves, that player wins the game. Let us call this game $K^2$. We chose this notation because we will later introduce a generalization of the game obtained by varying the set of terminal positions. Let $\\phi$ be the golden ratio~$(1+\\sqrt{5})/2$. We first define a sequence~$(g(n))_{n\\ge 0}$ as follows.\n\\begin{definition}\\label{def:fg}\n We let $g(0)=1$, $g(1)=0$ and\n \\[\n g(n)=\\left\\{\n \\begin{array}{ll}\n 1-g(m),& \\text{ if }\\lfloor n\\phi\\rfloor= \\lfloor m(\\phi+1)\\rfloor+1 \\text{ for some }m\\ge 0;\\\\\n 1,& \\text{ otherwise}.\n \\end{array}\\right.\n \\]\n\n\\end{definition}\nThe first values of the sequence~$g$ are given in Table~\\ref{tab:gh}. \n\\begin{table}[h!tb]\n \\centering\n \\[\n \\begin{array}{c|ccccccccccccccccccccc}\n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\\\\n \\hline\n h& 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 & 6 & 6 & 7 & 8 & 8 & 9 & 9 & 10 & 11 & 11 & 12 & 12 \\\\\n g&1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\\\\n \\end{array}\n \\]\n \\caption{First values of the sequence $g$ and Hofstadter sequence $h$.}\n \\label{tab:gh}\n\\end{table}\n\nThe main result of Komak {\\em et al.} provides an algebraic description of set of $\\mathcal{P}$-positions of $K^2$:", "full_context": "In combinatorial game theory, proofs of many results are obtained by using a suitable numeration system \\cite{Duchene,Fraenkel1,Fraenkel2,RigoLNM}. As a typical example, the Zeckendorf numeration based on the Fibonacci sequence is used to characterize the $\\mathcal{P}$-positions (i.e., losing positions) of the Wythoff game \\cite{Fraenkel3}. Fraenkel showed that a pair $(a,b)$ of integers such that $a\\le b$ is a $\\mathcal{P}$-position of Wythoff game if and only if $\\rep_F(a)$ ends with an even number of zeroes and $\\rep_F(b)$ is the left-shift of $\\rep_F(a)$, i.e., $\\rep_F(b)=\\rep_F(a)0$.\n\nThe Wythoff game is a {\\em take-away} game: two players alternately remove tokens from two piles. One can remove a positive number of token from one pile or, remove the same positive number of token from both piles. The Wythoff game is {\\em impartial}: the set of options (i.e., positions reachable in one legal move) of a given position does not depend on which player is in turn to move. Wythoff game can be seen as a queen on an infinite chessboard $\\mathbb{N}^2$: in a position $(x,y)\\neq(0,0)$ the queen can move horizontally to $(x-n,y)$ with $0<n\\le x$, vertically to $(x,y-n)$ with $0<n\\le y$ or diagonally to $(x-n,y-n)$ with $0<n\\le \\min\\{x,y\\}$. According to the {\\em normal convention}, a player who is unable to move loses the game; passing is not allowed. Otherwise stated, the player who takes the last token wins or, in an equivalent way, when the queen reaches $(0,0)$.\n\nIn this article, we are mainly interested in two kinds of variations of the Wythoff game.\n\nIn \\cite{Komak}, a variant of Wythoff game has been introduced: the set $\\{(x,y)\\mid x+y\\le 2\\}$ is declared to be the set of {\\em terminal} positions. If a player moves the queen into this terminal set with the usual Wythoff moves, that player wins the game. Let us call this game $K^2$. We chose this notation because we will later introduce a generalization of the game obtained by varying the set of terminal positions. Let $\\phi$ be the golden ratio~$(1+\\sqrt{5})/2$. We first define a sequence~$(g(n))_{n\\ge 0}$ as follows.\n\\begin{definition}\\label{def:fg}\n We let $g(0)=1$, $g(1)=0$ and\n \\[\n g(n)=\\left\\{\n \\begin{array}{ll}\n 1-g(m),& \\text{ if }\\lfloor n\\phi\\rfloor= \\lfloor m(\\phi+1)\\rfloor+1 \\text{ for some }m\\ge 0;\\\\\n 1,& \\text{ otherwise}.\n \\end{array}\\right.\n \\]\n\n\\end{definition}\nThe first values of the sequence~$g$ are given in Table~\\ref{tab:gh}. \n\\begin{table}[h!tb]\n \\centering\n \\[\n \\begin{array}{c|ccccccccccccccccccccc}\n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\\\\n \\hline\n h& 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 & 6 & 6 & 7 & 8 & 8 & 9 & 9 & 10 & 11 & 11 & 12 & 12 \\\\\n g&1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\\\\n \\end{array}\n \\]\n \\caption{First values of the sequence $g$ and Hofstadter sequence $h$.}\n \\label{tab:gh}\n\\end{table}\n\nThe main result of Komak {\\em et al.} provides an algebraic description of set of $\\mathcal{P}$-positions of $K^2$:\n\n\\end{definition}\nThe first values of the sequence~$g$ are given in Table~\\ref{tab:gh}. \n\\begin{table}[h!tb]\n \\centering\n \\[\n \\begin{array}{c|ccccccccccccccccccccc}\n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\\\\n \\hline\n h& 0 & 1 & 1 & 2 & 3 & 3 & 4 & 4 & 5 & 6 & 6 & 7 & 8 & 8 & 9 & 9 & 10 & 11 & 11 & 12 & 12 \\\\\n g&1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\\\\n \\end{array}\n \\]\n \\caption{First values of the sequence $g$ and Hofstadter sequence $h$.}\n \\label{tab:gh}\n\\end{table}\n\nThe authors provide a classical proof of their result by showing that the set of $\\mathcal{P}$-positions described by \\eqref{eq:Ppos} is both stable and absorbing (see Definition~\\ref{def:stab}). The proof is quite long and requires a detailed case analysis. Without diminishing their achievement, once the set \\eqref{eq:Ppos} has been conjectured, if the set can be handled by {\\tt Walnut}, then it becomes straightforward to produce an automated proof in just a few lines. Our approach therefore complements the work carried out in \\cite{Komak}.\n\nIn Section~\\ref{sec:6}, we go further in the analysis of the games $K^\\ell$ for an arbitrary $\\ell\\ge 1$. We provide a recursive characterization of the set of $\\mathcal{P}$-positions. The non-terminal $\\mathcal{P}$-positions $(a_n,b_n)_{n\\ge 0}$ of $K^\\ell$ are given, in Theorem~\\ref{thm:rec_car}, by\n\\[\n (a_0,b_0)=(\\ell+1,2\\ell+2) \\quad\\text{ and }\\quad \\forall n\\ge 1, \\left\\{\n \\begin{array}{l}\na_n =\\mex (\\{ a_i,b_i \\mid i<n \\}\\cup\\{0,1,\\ldots,\\ell\\}) \\\\\nb_n =a_n+n+\\ell+1.\\\\\n \\end{array}\n \\right.\n\\]\nWe prove this result in an automatic way for $\\ell=1,2$, then extend it for all values of the parameter using classical arguments. With Corollary~\\ref{cor:behave}, we generalize Theorem~\\ref{thm:K2} and obtain that $(a_n,b_n)_{n\\ge 0}$ are of the form\n\\[ a_n=\\lfloor (n+\\ell)\\phi\\rfloor + \\lambda_\\ell(n) \\quad\\text{ and }\\quad\n b_n=\\lfloor (n+\\ell)\\phi^2\\rfloor + \\lambda_\\ell(n)+1\\]\nfor some bounded integer-valued function $\\lambda_\\ell$. Our result can be related to the recent paper \\cite{Letouzey} where discrepancy of generalized Hofstadter functions is studied (and formalized in {\\tt Coq}). Our methods are inspired by \\cite{Fraenkel5,Kimberling2,Larsson}.\n\nThe first non-terminal $\\mathcal{P}$-positions $(a_n,b_n)_{n\\ge 0}$ are given in Table~\\ref{tab:k3}. From now on, we take this convention: in the indexing of the $P$-positions, we do not take terminal positions into account. Thus, $(a_0,b_0)$ represents the first non-terminal $\\mathcal{P}$-position. Due to the symmetry of the game, we only consider sequences where $a_n<b_n$. \n\\begin{table}[h!t]\n\\[\n \\begin{array}{c|cccccccccccccccccccccccccccccc}\n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 \\\\\n \\hline\n a_n& 4 & 5 & 6 & 7 & 9 & 11 & 13 & 15 & 16 & 18 & 19 & 21 & 22 & 24 & 25 & 27 &\n 29 & 30 \\\\\n b_n & 8 & 10 & 12 & 14 & 17 & 20 & 23 & 26 & 28 & 31 & 33 & 36 & 38 & 41 & 43 & 46\n & 49 & 51 & \\\\\n \\end{array}\n \\]\n \\caption{First (non-terminal) $\\mathcal{P}$-positions of $K^3$, $(n\\ge 0)$.}\n \\label{tab:k3}\n\\end{table}\n\\begin{theorem}\\label{thm:fibaut2}\nThere exists a Fibonacci-automatic function $g_3:\\mathbb{N}\\to\\{0,1,2\\}$ such that, for all $n\\ge 0$, the $n$th non-terminal $\\mathcal{P}$-position $(a_n,b_n)$ of $K^3$ is given by \n\\begin{equation}\\label{eq:K3}\n a_n=\\lfloor (n+2)\\phi \\rfloor+g_3(n+1)-1 \\quad \\text{ and }\\quad \n b_n=\\lfloor (n+2)\\phi^2 \\rfloor+g_3(n+1)+1.\n\\end{equation}\n\\end{theorem}\nThe first few values of $g_3(n)$ are\n\\[\n 1221011221211111111121121121211211112111121111111211112\\cdots\n\\]\nUsing the heuristic described in Section~\\ref{ssec:heuristic}, assuming that the above infinite word is generated by a $\\varphi$-morphism, we obtain the morphism \n\\begin{align*}\n \\mu_3:\\;&\n0 \\mapsto 01,\n1 \\mapsto 2,\n2 \\mapsto 34,\n3 \\mapsto 56,\n4 \\mapsto 7,\n5 \\mapsto 78,\\\\ \n& 6 \\mapsto 9,\n7 \\mapsto ab,\n8 \\mapsto a,\n9 \\mapsto 56,\na \\mapsto ab, \nb \\mapsto 7\n\\end{align*}\nand the coding\n\\[\\rho_3:0,3,5,6,8,a,b\\mapsto 1,\\quad 1,2,7,9\\mapsto 2,\\quad 4\\mapsto 0.\\]\nWe thus define $g_3(n)$ as the $n$th symbol in $\\rho_3(\\mu_3^\\omega(1))$. Note that to produce enough value of $g_3(n)$, we have to compute sufficiently many $\\mathcal{P}$-positions of $W^3$.\n\n\\subsection{Even further}\nFor the game~$K^4$, we have applied the same procedure. The first non-terminal $\\mathcal{P}$-positions $(a_n,b_n)_{n\\ge 0}$ are given in Table~\\ref{tab:k4}.\n\\begin{table}[h!t]\n\\[\n\\begin{array}{c|cccccccccccccccccc}\n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 \\\\\n \\hline\n a_n &5 & 6 & 7 & 8 & 9 & 11 & 13 & 15 & 17 & 19 & 20 & 22 & 23 & 25 & 26 & 28 & 29 & 31 \\\\\n b_n & 10 & 12 & 14 & 16 & 18 & 21 & 24 & 27 & 30 & 33 & 35 & 38 & 40 & 43 & 45 & 48 & 50 & 53 \\\\\n\\end{array}\n \\]\n \\caption{First (non-terminal) $\\mathcal{P}$-positions of $K^4$, $(n\\ge 0)$.}\n \\label{tab:k4}\n\\end{table}\nTo get a long enough prefix, we have computed the first $600$ $\\mathcal{P}$-positions $(a_n,b_n)$ with $a_n\\le b_n$ (up to $a_n$ close to $1000$). From this, we conjectured that\n\\begin{equation}\\label{eq:K4}\n a_n=\\lfloor (n+2)\\phi \\rfloor+g_4(n+1) \\quad \\text{ and }\\quad \n b_n=\\lfloor (n+2)\\phi^2 \\rfloor+g_4(n+1)+3,\n\\end{equation}\nwhere the first few values of $g_4(n)$ are\n\\[\n 12210001112111111010110110111111111111111111\\cdots.\n\\]\nHaving the first $600$ values of $g_4(n)$, we apply the heuristic of Section~\\ref{ssec:heuristic}. Note that here, to be able to discern types correctly, we have to consider $4$-types (there are distinct $4$-types whose restrictions to $3$-types are equal). Since we have to take fourth iterations, this explains why we need a longer prefix. We obtain $18$ different $4$-types and the heuristic gives the following morphisms:\n\\begin{align*}\n \\mu_4:\\;&0 \\mapsto 01,\\ 1 \\mapsto 2,\\ 2 \\mapsto 34,\\ 3 \\mapsto 56,\\ 4 \\mapsto 7,\\ 5 \\mapsto 89,\\ 6 \\mapsto a,\\ 7 \\mapsto bc,\\ 8 \\mapsto dc,\\\\\n &9 \\mapsto d,\\ a \\mapsto ef,\\ b \\mapsto ef,\\ c \\mapsto e,\\ d \\mapsto 7g,\\ e \\mapsto eh,\\ f \\mapsto b,\\ g \\mapsto d,\\ h \\mapsto b \n\\end{align*}\n\\[\n \\rho_4: 0,3,7,8,9,b,c,d,e,h \\mapsto 1,\\ 1,2,a \\mapsto 2,\\ 4,5,6,f,g \\mapsto 0\n\\]\nWith $g_4(n)$ being the $n$th symbol of $\\rho_4(\\mu_4^\\omega(1))$, we have a $18$-state DFAO. Formulas \\eqref{eq:K4} translates in {\\tt Walnut} into\n\\begin{verbatim}\ndef pposK4_asym \"?msd_fib (a+b<=4) | (En,x $phin(n+2,x) & (a=x+K4[n+1])\n & (b=x+K4[n+1]+n+5))\":\n\\end{verbatim}\nand we then follow exactly the same proof to get the result.\n\nLet us begin with some classical and well-known results on Wythoff game before moving on to the games discussed in this article. We recall the {\\em recursive definition} of the $\\mathcal{P}$-positions of Wythoff game~$K^0$, with our convention:\n\\[\n (a_0,b_0)=(1,2) \\quad\\text{ and }\\quad \\forall n\\ge 1, \\left\\{\n \\begin{array}{l}\na_n =\\mex (\\{ a_i,b_i \\mid i<n \\}\\cup\\{0\\}),\\\\\nb_n =a_n+n+1,\\\\\n \\end{array}\n \\right.\n\\]\nwhere $\\mex$ stands for {\\em minimum excluded value}, i.e. the least non-negative integer not in the set. In other words, $\\mex S=\\min(\\mathbb{N}\\setminus S)$. So $a_1=\\mex\\{0,1,2\\}=3$ and $b_1=a_1+2=5$, then $a_2=\\mex\\{0,1,2,3,5\\}=4$ and $b_2=a_2+3=7$ and so on.\n\nLarsson characterized \\cite[Thm.~1.1]{block} the set of $\\mathcal{P}$-positions of $W^2$, a variant of Wythoff game where one option may be blocked, see Figure~\\ref{fig:w2w3}. We provide an automatic proof of this result. \n\\begin{theorem}\nThe set of $\\mathcal{P}$-positions of $W^2$ is given by\n\\[\n \\mathcal{R}_2=\\{(0,0)\\}\\cup \\{ \\{n,2n+1\\}\\mid n\\ge 0\\}\\cup \\{\\{2\\lfloor n\\phi\\rfloor+2,2\\lfloor n\\phi^2\\rfloor+2\\}\\mid n\\ge 0\\}.\n\\] \n\\end{theorem}", "post_theorem_intro_text_len": 6231, "post_theorem_intro_text": "The authors provide a classical proof of their result by showing that the set of $\\mathcal{P}$-positions described by \\eqref{eq:Ppos} is both stable and absorbing (see Definition~\\ref{def:stab}). The proof is quite long and requires a detailed case analysis. Without diminishing their achievement, once the set \\eqref{eq:Ppos} has been conjectured, if the set can be handled by {\\tt Walnut}, then it becomes straightforward to produce an automated proof in just a few lines. Our approach therefore complements the work carried out in \\cite{Komak}.\n\nIn \\cite{block}, other variants of Wythoff game are studied, in which some blocking maneuvers are added. Let $k\\ge 1$. In the game denoted by $W^k$, for each move, before the next player (i.e., the player who is about to play) moves, the previous player (i.e., the one who has just played) may declare at most $k-1$ of the options as forbidden. When the next player has moved, any blocking maneuver is forgotten and has no further incidence on the game. This terminology of previous and next player explains why we speak of $\\mathcal{N}$- and $\\mathcal{P}$-positions, respectively. For $k=1$, this is the classical game of Wythoff. When $k=2,3$, we reconsider Larsson's result and obtain an automated proof, replacing a 2-page-long analysis by some easy-to-describe first-order formulas. \n\nAs Shallit has already noted on several occasions, the two approaches are complementary. A classical, purely combinatorial proof usually provides structural insights, whereas the use of automated provers helps avoid lengthy case analyses and enables the exploration of directions that are difficult to access through traditional techniques. In particular, in this article, we easily obtain a variety of results that would otherwise require significantly more effort using ``classical'' methods. Moreover, these results lead us to formulate general theorems.\n\\smallskip\n\nThis article is organized as follows. Section~\\ref{sec:2} --- which was the starting point of this article --- aims to give an automatic proof of the algebraic characterization \\eqref{eq:Ppos} of the $\\mathcal{P}$-positions of $K^2$. We begin with preliminary results about Fibonacci-automatic sequences. We define a notion of $\\varphi$-morphism and describe a heuristic that we extensively use throughout the paper to construct such $\\varphi$-morphisms. Given a sufficiently long prefix of an infinite word, we obtain morphisms that can be used in our automatic proofs (and we can therefore prove the correctness of the procedure). Basic results from combinatorial game theory are given in Section~\\ref{ssec:autom_proof}.\n\nIn Section~\\ref{sec:newGame}, we obtain results about new games: a parameterized version of a variant of Wythoff game denoted by $K^\\ell$, where the set of terminal states is\n\\[\n  \\{(x,y)\\in\\mathbb{N}^2\\mid x+y\\le \\ell\\}\n\\]\nfor some $\\ell\\in\\mathbb{N}$. We study the games $K^1$, $K^3$ and $K^4$ and obtain algebraic characterizations of the $\\mathcal{P}$-positions similar to \\cite{Komak}. In particular, as for the classical Wythoff game $K^0$, the game $K^1$ has a nice set of $\\mathcal{P}$-positions: a pair $(a,b)$ of integers such that $a\\le b$ is a $\\mathcal{P}$-position of $K^1$ if and only if its Fibonacci representation $\\rep_F(a)$ ends with zero and $\\rep_F(b)$ is of the form $\\rep_F(a)1$, see Theorem~\\ref{thm:carK1}. \n\nIn Section~\\ref{sec:6}, we go further in the analysis of the games $K^\\ell$ for an arbitrary $\\ell\\ge 1$. We provide a recursive characterization of the set of $\\mathcal{P}$-positions. The non-terminal $\\mathcal{P}$-positions $(a_n,b_n)_{n\\ge 0}$ of $K^\\ell$ are given, in Theorem~\\ref{thm:rec_car}, by\n\\[\n  (a_0,b_0)=(\\ell+1,2\\ell+2) \\quad\\text{ and }\\quad \\forall n\\ge 1, \\left\\{\n    \\begin{array}{l}\na_n =\\mex (\\{ a_i,b_i \\mid i<n \\}\\cup\\{0,1,\\ldots,\\ell\\}) \\\\\nb_n =a_n+n+\\ell+1.\\\\\n    \\end{array}\n  \\right.\n\\]\nWe prove this result in an automatic way for $\\ell=1,2$, then extend it for all values of the parameter using classical arguments. With Corollary~\\ref{cor:behave}, we generalize Theorem~\\ref{thm:K2} and obtain that $(a_n,b_n)_{n\\ge 0}$ are of the form\n\\[ a_n=\\lfloor (n+\\ell)\\phi\\rfloor + \\lambda_\\ell(n) \\quad\\text{ and }\\quad\n  b_n=\\lfloor (n+\\ell)\\phi^2\\rfloor + \\lambda_\\ell(n)+1\\]\nfor some bounded integer-valued function $\\lambda_\\ell$. Our result can be related to the recent paper \\cite{Letouzey} where discrepancy of generalized Hofstadter functions is studied (and formalized in {\\tt Coq}). Our methods are inspired by   \\cite{Fraenkel5,Kimberling2,Larsson}.\n\nIt is well-known that the set of $\\mathcal{P}$-positions of the Wythoff game are coded by the Fibonacci word~$\\mathbf{f}=abaababa\\cdots$, see \\cite{Vision,Duchene}. If we start indexing letters of~$\\mathbf{f}$ with $1$, the indices of the letters $a$ (resp. $b$) in~$\\mathbf{f}$ correspond to the sequence $(a_n)_{n\\ge 0}$ (resp. $(b_n)_{n\\ge 0}$). Using the recursive definition from the previous section, we obtain  similar results with $K^1,K^2,K^3$: their $\\mathcal{P}$-positions are encoded by morphic words obtained by $\\varphi$-morphisms and codings. \n\nIn Section~\\ref{sec:3}, we obtain automatic proofs of Larsson's result about $W^2$ and $W^3$. Interestingly, the automata arising in the computations are much larger than those for $K^2$.\n\nIn Section~\\ref{sec:redundant}, we are concerned with redundant moves of the games discussed so far. A move is considered {\\em redundant} if, upon its removal from the rule-set, then the set of $\\mathcal{P}$-positions remains unchanged. It is a classical question in combinatorial game theory: do distinct rule-sets yield the same set of $\\mathcal{P}$-positions? In particular, can certain moves be added or removed without altering the $\\mathcal{P}$-positions?\n\n\\begin{remark}\n  A {\\tt Jupyter} notebook recording all the {\\tt Walnut} computations is available (it has been produced using Ollinger's {\\tt Walnut-Kernel}. \n  The {\\tt Mathematica} code and the various {\\tt Walnut} files are also available online\\footnote{at {\\tt https://hdl.handle.net/2268/338482}, you may also download a standalone Wolfram Player.}. In particular, the package permits us to compute $\\mathcal{P}$-positions and $\\varphi$-morphisms.\n\\end{remark}", "sketch": "To prove Theorem~\\ref{thm:K2}, the authors “provide a classical proof … by showing that the set of $\\mathcal{P}$-positions described by \\eqref{eq:Ppos} is both stable and absorbing (see Definition~\\ref{def:stab}).” They note that “the proof is quite long and requires a detailed case analysis.” They also indicate that, “once the set \\eqref{eq:Ppos} has been conjectured,” an alternative route is to “produce an automated proof” (e.g., “if the set can be handled by {\\tt Walnut}”).", "expanded_sketch": "To prove the main theorem, the authors “provide a classical proof … by showing that the set of $\\mathcal{P}$-positions described by \\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}\n is both stable and absorbing (see \\begin{definition}\\label{def:stab}\nA set $S$ of positions is {\\em stable} if, for all $s,t\\in S$, $s\\neq t$, there is no move between $s$ and $t$. A set $S$ of positions is {\\em absorbing}, if for all $t\\not\\in S$, there exists $s\\in S$ such that there is a move from $t$ to $s$ (or, $t$ has an option in $S$).  \n\\end{definition}).” They note that “the proof is quite long and requires a detailed case analysis.” They also indicate that, “once the set in the equation above has been conjectured,” an alternative route is to “produce an automated proof” (e.g., “if the set can be handled by {\\tt Walnut}”).", "expanded_theorem": "\\label{thm:K2}\n  The set of $\\mathcal{P}$-positions of the variant of Wythoff game where the set of terminal positions is $\\{(x,y)\\mid x+y\\le 2\\}$,  is exactly \\begin{equation}\\label{eq:Ppos}\n  \\left\\{ (\\lfloor n\\phi\\rfloor +g(n)-1, \\lfloor n \\phi^2\\rfloor +g(n))\\mid n\\ge 0\\right\\} \\cup\n  \\left\\{ (\\lfloor n \\phi^2\\rfloor +g(n), (\\lfloor n\\phi\\rfloor +g(n)-1)\\mid n\\ge 0\\right\\}\n\\end{equation}\nwhere $g:\\mathbb{N}\\to\\mathbb{N}$ is defined as follows: \\begin{definition}\\label{def:fg}\n  We let $g(0)=1$, $g(1)=0$ and\n  \\[\n    g(n)=\\left\\{\n      \\begin{array}{ll}\n        1-g(m),& \\text{ if }\\lfloor n\\phi\\rfloor= \\lfloor m(\\phi+1)\\rfloor+1 \\text{ for some }m\\ge 0;\\\\\n        1,& \\text{ otherwise}.\n      \\end{array}\\right.\n  \\]\n\n\\end{definition}", "theorem_type": ["Classification or Bijection", "Equivalence"], "mcq": {"question": "Consider the variant of Wythoff game played on positions \\((x,y)\\in\\mathbb N^2\\) (with \\(\\mathbb N=\\{0,1,2,\\dots\\}\\)), where a move from \\((x,y)\\neq(0,0)\\) consists of replacing it by \\((x-n,y)\\), \\((x,y-n)\\), or \\((x-n,y-n)\\) for some positive integer \\(n\\) with the coordinates remaining nonnegative. In this variant, every position with \\(x+y\\le 2\\) is terminal, and moving into that terminal set wins. Which set is exactly the set of \\(\\mathcal P\\)-positions (losing positions for the player to move)? Here \\(\\phi=(1+\\sqrt5)/2\\), and \\(g:\\mathbb N\\to\\mathbb N\\) is defined by \\(g(0)=1\\), \\(g(1)=0\\), and for \\(n\\ge 2\\),\n\\[\n g(n)=\\begin{cases}\n 1-g(m), & \\text{if } \\lfloor n\\phi\\rfloor=\\lfloor m(\\phi+1)\\rfloor+1 \\text{ for some } m\\ge 0,\\\\\n 1, & \\text{otherwise.}\n \\end{cases}\n\\]\nWhich explicit description gives all such \\(\\mathcal P\\)-positions?", "correct_choice": {"label": "A", "text": "\\[\n\\left\\{\\bigl(\\lfloor n\\phi\\rfloor+g(n)-1,\\ \\lfloor n\\phi^2\\rfloor+g(n)\\bigr)\\mid n\\ge 0\\right\\}\n\\ \\cup\\ \n\\left\\{\\bigl(\\lfloor n\\phi^2\\rfloor+g(n),\\ \\lfloor n\\phi\\rfloor+g(n)-1\\bigr)\\mid n\\ge 0\\right\\}.\n\\]"}, "choices": [{"label": "B", "text": "\\[\n\\left\\{\\bigl(\\lfloor n\\phi\\rfloor+g(n),\\ \\lfloor n\\phi^2\\rfloor+g(n)\\bigr)\\mid n\\ge 0\\right\\}\n\\ \\cup\\ \n\\left\\{\\bigl(\\lfloor n\\phi^2\\rfloor+g(n),\\ \\lfloor n\\phi\\rfloor+g(n)\\bigr)\\mid n\\ge 0\\right\\}.\n\\]"}, {"label": "C", "text": "\\[\n\\left\\{\\bigl(\\lfloor n\\phi\\rfloor+g(n)-1,\\ \\lfloor n\\phi^2\\rfloor+g(n)\\bigr)\\mid n\\ge 0\\right\\}.\n\\]"}, {"label": "D", "text": "\\[\n\\left\\{\\bigl(\\lfloor n\\phi\\rfloor+g(n)-1,\\ \\lfloor n\\phi^2\\rfloor+g(n)\\bigr)\\mid n\\ge 1\\right\\}\n\\ \\cup\\ \n\\left\\{\\bigl(\\lfloor n\\phi^2\\rfloor+g(n),\\ \\lfloor n\\phi\\rfloor+g(n)-1\\bigr)\\mid n\\ge 1\\right\\}.\n\\]"}, {"label": "E", "text": "\\[\n\\left\\{\\bigl(\\lfloor n\\phi\\rfloor+g(n)-1,\\ \\lfloor n\\phi^2\\rfloor+g(n)\\bigr)\\mid n\\ge 0\\right\\}\n\\ \\cap\\ \n\\left\\{\\bigl(\\lfloor n\\phi^2\\rfloor+g(n),\\ \\lfloor n\\phi\\rfloor+g(n)-1\\bigr)\\mid n\\ge 0\\right\\}.\n\\]"}], "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": "offset_-1_in_first_coordinate", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "case_split", "tampered_component": "dropped_symmetric_second_family", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "inclusion_of_base_case_n0", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "union_of_two_oriented_families", "template_used": "wildcard"}]}}
{"id": "2512.12077v1", "paper_link": "http://arxiv.org/abs/2512.12077v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:square}\nWe have $|\\cS_2(n)|=(\\frac{1}{2}-o(n^{-1/15})) 2^{2n}$.", "start_pos": 9002, "end_pos": 9105, "label": "thm:square"}, "ref_dict": {"corollary:lt": "\\begin{corollary}\\label{corollary:lt} For almost all $s\\in\\{0,1\\}^{n}$, $\\rm{LT}(s)\\geq \\sceil{\\frac{n}{2}}-1$.\n\\end{corollary}", "thm:square": "\\begin{theorem}\\label{thm:square}\nWe have $|\\cS_2(n)|=(\\frac{1}{2}-o(n^{-1/15})) 2^{2n}$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 1922, "pre_theorem_intro_text": "Let $[k]^{*}$ be the collection of words of finite length over the finite alphabet $[k] = \\{1,\\ldots, k\\}$, and say $s'\\in [k]^*$ is a \\textit{subword} of $s \\in [k]^*$ if $s'$ is a (not necessarily contiguous) subsequence of $s$. A \\textit{shuffle square} is a word $s\\in [k]^*$ that can be partitioned into two disjoint, identical subwords. Shuffle squares were first studied by Henshall, Rampersad, and Shallit~\\cite{henshall}, in the theory of formal languages. In this area, Boulteau and Vialette~\\cite{binarynphard} proved that recognizing binary shuffle squares is NP-complete, building on work of Rizzi and Vialette~\\cite{unshufflelargek} and Buss and Soltys~\\cite{unshuffle7buss} on larger alphabets.\n\nAs shuffle squares must have even length, it is convenient to say that a word of length $2n$ has \\textit{semi-length} $n$. Let $\\cS_k(n)$ be the set of shuffle squares of semi-length $n$ over the alphabet $[k]$. Henshall, Rampersad, and Shallit asked for the asymptotics of $|\\cS_k(n)|$, and gave conjectural asymptotic formulas for $n$ fixed and $k\\rightarrow\\infty$. These formulas were verified by the first author, Huang, Nam, and Thaper~\\cite{HHNT}, who also posed the following surprising conjecture in the opposite regime, where $k=2$ is fixed and $n\\rightarrow \\infty$, supported by numerical evidence from the OEIS~\\cite{oeis}.\n\n\\begin{conjecture}[\\cite{HHNT}]\nAsymptotically half of all even-length binary words are shuffle squares, that is, $|\\cS_2(n)| = (\\frac{1}{2}-o(1)) 2^{2n}$.\n\\end{conjecture}\n\nNote that shuffle squares must have an even count of each letter, so this conjecture states that almost all words satisfying this obvious parity constraint are actually shuffle squares. Towards this, they proved using a greedy algorithm that $|\\cS_2(n)| \\ge \\binom{2n}{n} = \\Theta(n^{-1/2}2^{2n})$. Our main theorem resolves this conjecture in the affirmative with an explicit polynomial error term.", "context": "Let $[k]^{*}$ be the collection of words of finite length over the finite alphabet $[k] = \\{1,\\ldots, k\\}$, and say $s'\\in [k]^*$ is a \\textit{subword} of $s \\in [k]^*$ if $s'$ is a (not necessarily contiguous) subsequence of $s$. A \\textit{shuffle square} is a word $s\\in [k]^*$ that can be partitioned into two disjoint, identical subwords. Shuffle squares were first studied by Henshall, Rampersad, and Shallit~\\cite{henshall}, in the theory of formal languages. In this area, Boulteau and Vialette~\\cite{binarynphard} proved that recognizing binary shuffle squares is NP-complete, building on work of Rizzi and Vialette~\\cite{unshufflelargek} and Buss and Soltys~\\cite{unshuffle7buss} on larger alphabets.\n\nAs shuffle squares must have even length, it is convenient to say that a word of length $2n$ has \\textit{semi-length} $n$. Let $\\cS_k(n)$ be the set of shuffle squares of semi-length $n$ over the alphabet $[k]$. Henshall, Rampersad, and Shallit asked for the asymptotics of $|\\cS_k(n)|$, and gave conjectural asymptotic formulas for $n$ fixed and $k\\rightarrow\\infty$. These formulas were verified by the first author, Huang, Nam, and Thaper~\\cite{HHNT}, who also posed the following surprising conjecture in the opposite regime, where $k=2$ is fixed and $n\\rightarrow \\infty$, supported by numerical evidence from the OEIS~\\cite{oeis}.\n\n\\begin{conjecture}[\\cite{HHNT}]\nAsymptotically half of all even-length binary words are shuffle squares, that is, $|\\cS_2(n)| = (\\frac{1}{2}-o(1)) 2^{2n}$.\n\\end{conjecture}\n\nNote that shuffle squares must have an even count of each letter, so this conjecture states that almost all words satisfying this obvious parity constraint are actually shuffle squares. Towards this, they proved using a greedy algorithm that $|\\cS_2(n)| \\ge \\binom{2n}{n} = \\Theta(n^{-1/2}2^{2n})$. Our main theorem resolves this conjecture in the affirmative with an explicit polynomial error term.", "full_context": "Let $[k]^{*}$ be the collection of words of finite length over the finite alphabet $[k] = \\{1,\\ldots, k\\}$, and say $s'\\in [k]^*$ is a \\textit{subword} of $s \\in [k]^*$ if $s'$ is a (not necessarily contiguous) subsequence of $s$. A \\textit{shuffle square} is a word $s\\in [k]^*$ that can be partitioned into two disjoint, identical subwords. Shuffle squares were first studied by Henshall, Rampersad, and Shallit~\\cite{henshall}, in the theory of formal languages. In this area, Boulteau and Vialette~\\cite{binarynphard} proved that recognizing binary shuffle squares is NP-complete, building on work of Rizzi and Vialette~\\cite{unshufflelargek} and Buss and Soltys~\\cite{unshuffle7buss} on larger alphabets.\n\nAs shuffle squares must have even length, it is convenient to say that a word of length $2n$ has \\textit{semi-length} $n$. Let $\\cS_k(n)$ be the set of shuffle squares of semi-length $n$ over the alphabet $[k]$. Henshall, Rampersad, and Shallit asked for the asymptotics of $|\\cS_k(n)|$, and gave conjectural asymptotic formulas for $n$ fixed and $k\\rightarrow\\infty$. These formulas were verified by the first author, Huang, Nam, and Thaper~\\cite{HHNT}, who also posed the following surprising conjecture in the opposite regime, where $k=2$ is fixed and $n\\rightarrow \\infty$, supported by numerical evidence from the OEIS~\\cite{oeis}.\n\n\\begin{conjecture}[\\cite{HHNT}]\nAsymptotically half of all even-length binary words are shuffle squares, that is, $|\\cS_2(n)| = (\\frac{1}{2}-o(1)) 2^{2n}$.\n\\end{conjecture}\n\nNote that shuffle squares must have an even count of each letter, so this conjecture states that almost all words satisfying this obvious parity constraint are actually shuffle squares. Towards this, they proved using a greedy algorithm that $|\\cS_2(n)| \\ge \\binom{2n}{n} = \\Theta(n^{-1/2}2^{2n})$. Our main theorem resolves this conjecture in the affirmative with an explicit polynomial error term.\n\n\\begin{conjecture}[\\cite{HHNT}]\nAsymptotically half of all even-length binary words are shuffle squares, that is, $|\\cS_2(n)| = (\\frac{1}{2}-o(1)) 2^{2n}$.\n\\end{conjecture}\n\nNote that shuffle squares must have an even count of each letter, so this conjecture states that almost all words satisfying this obvious parity constraint are actually shuffle squares. Towards this, they proved using a greedy algorithm that $|\\cS_2(n)| \\ge \\binom{2n}{n} = \\Theta(n^{-1/2}2^{2n})$. Our main theorem resolves this conjecture in the affirmative with an explicit polynomial error term.\n\nThe exponent $-1/15$ is not optimized, but within a modest factor of the best possible using our arguments. It would be interesting to decide if the error term is actually polynomial or exponential.\n\nWhile previous work dealt with the worst-case behavior of $\\rm{LT}(s)$, \\cref{thm:square} implies the best possible bound for its average-case behavior. \n\\begin{corollary}\\label{corollary:lt} For almost all $s\\in\\{0,1\\}^{n}$, $\\rm{LT}(s)\\geq \\sceil{\\frac{n}{2}}-1$.\n\\end{corollary}\n\n\\begin{lemma}\\label{lem:pt_decomposition} Let $s\\in\\{0,1\\}^\\omega$ be uniformly random. Let $\\cev q_t(w;v)=q_t(\\rev(w);\\rev(v))$.\nThen for all $v\\in\\{0,1\\}^*$,\n\\begin{equation}\\label{eq:pt_as_qt}\n\\PP[v\\in B_n(s)]=\\cev q_n(\\emptyword;v)+\\frac 12\\sum_{t=0}^{n-1}\\sum_{w\\in \\{0,1\\}^*}\\Big(\\delta_t(1w1,w)\\cev q_{n-t-1}(w1;v)+\\delta_t(0w0,w)\\cev q_{n-t-1}(w0;v)\\Big).\n\\end{equation}\nIn particular, we have\n\\begin{equation}\\label{eq:pt_as_qt_beta}\n\\PP[\\emptyword\\in B_n(s)]=q_n(\\emptyword)+\\sum_{t=0}^{n-1}\\sum_{w1\\in \\Sigma_2^{(01)}}\\delta_t(1w1,w) q_{n-t-1}(\\rev(w1)).\n\\end{equation}\n\\end{lemma}\nRoughly speaking, \\eqref{eq:pt_as_qt_beta} captures the fact that any successful buffer thread $(w_j)_j$ has a critical first time $n-t-1$ at word $w_{n-t-1}=1w$ where it diverges from the greedy algorithm.\n\\begin{proof}\nWe prove \\eqref{eq:pt_as_qt} by induction on $n$. By construction, $\\PP[v\\in B_0]=\\bbone_{v=\\emptyword}=q_0(\\emptyword;\\rev(v))$. Observe that for any word $v1$ ending in $1$, we have\n\\begin{align*}\n\\PP[v1\\in B_{n+1}]&=\\PP[s(n+1)=1]\\PP[(v\\in B_n)\\cup (1v1\\in B_n)]+\\PP[s(n+1)=0]\\PP[0v1\\in B_n]\\\\\n&=\\frac 12\\Big(\\PP[v\\in B_n]+\\delta_n(1v1,v)+\\PP[0v1\\in B_n]\\Big).\n\\end{align*}\nObserve that $q_n$ satisfies a similar recurrence: for each initializiation of the form $1v'$, we can condition on the first bit of $s$ to get\n\\begin{align*}\nq_n(w;1v')&=\\PP[s(1)=1]\\PP[\\greedy(s[2,n],v')=w]+\\PP[s(1)=0]\\PP[\\greedy(s[2,n],1v'0)=w]\\\\\n&=\\frac 12(q_{n-1}(w;v')+q_{n-1}(w;1v'0)).\n\\end{align*}\nReversing the words, we obtain\n\\[\n\\cev q_n(w;v1)=\\frac 12(\\cev q_{n-1}(w;v)+\\cev q_{n-1}(w;0v1)).\n\\]\nDenote the RHS of~\\eqref{eq:pt_as_qt} by $Q_n(v)$. Since the above is a linear recurrence relation, $Q_n$ obeys the same equation, plus the added terms at time $n+1$:\n\\[\nQ_{n+1}(v1)=\\frac 12(Q_n(v)+Q_n(0v1))+\\frac 12\\sum_{w\\in \\{0,1\\}^*}\\Big(\\delta_n(1w1,w)\\cev q_{0}(w1;v1)+\\delta_n(0w0,w)\\cev q_{0}(w0;v1)\\Big)\n\\]\nRecall that $\\cev q_0(w;v)=\\bbone_{w=v}$, so the summation evaluates to $\\frac 12\\delta_n(1v1,v)$. Thus by induction, $\\PP[v1\\in B_{n+1}]=Q_{n+1}(v1)$. By symmetry, $\\PP[v0\\in B_{n+1}]=Q_{n+1}(v0)$, showing \\eqref{eq:pt_as_qt} holds for all nonempty words $v$ at time $n+1$.\n\nWe prove that $X^*_1$ has the desired statistics and $X^*_2$ is a negligible perturbation. First, consider $X^*_1$. Since $X,Z$ are independent, we immediately have $\\Ex[(X-1)\\bbone_{Z=0}]=\\frac12 (k-1)$, and $\\Ex[(X-1-k)^2\\bbone_{Z=0}]=\\frac 12(2k+1)$. Next we control the second term in $X_1^*$; both technical claims below are proven in Section~\\ref{section:numericalclaims}.\n\\begin{claim}\\label{claim:Ybounds} (Proved in~\\cref{section:numericalclaims}.) With $X,Y,Z$ as defined above, $\\Ex[(Y-Z)\\bbone_{Z>0}]=\\frac 12(k-3+2^{-k})$ and $\\Ex[(Y-Z-k)^2\\bbone_{Z>0}]<4k+9$.\n\\end{claim}\nTogether, we have\n\\begin{align*}\n\\Ex[X^*_1-k]&=\\frac 12(k-1)+\\frac 12(k-3+2^{-k})-k=-2+2^{-k-1},\\quad\\text{and}\\\\\n\\Ex[(X^*_1-k)^2]&=\\Ex[(X^*_1-k)^2\\bbone_{Z=0}]+\\Ex[(X^*_1-k)^2\\bbone_{Z>0}]<\\frac 12(2k+1)+\\frac 12(4k+9)=3k+5.\n\\end{align*}\nNext, we control $X^*_2$. Because $\\PP[X=0]\n=2^{-k}$, it is simple to check that $\\Ex[\\max(Z,1)\\bbone_{X=0}]=\\frac 32\\cdot 2^{-k}$ and $\\Ex[\\max(Z,1)^2\\bbone_{X=0}]=\\frac 72\\cdot 2^{-k}$, which are very small. The next claim bounds the $2C'$ term.\n\\begin{claim}\\label{claim:C'bounds}(Proved in~\\cref{section:numericalclaims}.) With $M,Z,C'$ as defined above, $\\Ex[C']\\leq 2(7/8)^{k'}$ and $\\Ex[(C')^2]\\leq 6(7/8)^{k'}$. \n\\end{claim}\nWe conclude that $\\Ex[X^*_2],\\Ex[(X^*_2)^2]=O((7/8)^k)$. Therefore,\n\\[\n\\Ex[X^*-k]=\\Ex[X^*_1-k]+\\Ex[X^*_2]=-2+O((7/8)^k),\n\\]\nand similarly by the Minkowski inequality,\n\\[\n\\Ex[(X^*-k)^2]\\leq \\Big(\\sqrt{\\Ex[(X^*_1-k)^2]}+\\sqrt{\\Ex[(X^*_2)^2]}\\Big)^2=3k+5+o(1)\n\\]\nThis proves~\\eqref{eq:algorithmstep2}. To obtain tail bounds~\\eqref{eq:algorithmstep3} and~\\eqref{eq:algorithmstep4}, first note that \n\\begin{equation}\\label{eq:T*expection}\nT^*\\leq T_3= (k+X)+(Z+1)+(X-1+Y),\n\\end{equation}\nand thus $\\Ex[T^*]\\leq 4k$. By considering cases, we have\n\\[\\PP[T^* \\geq (4+5\\eps)k]\\leq \\PP[X\\geq (1+\\eps)k]+\\PP[Y\\geq (1+2\\eps)k\\mid X<(1+\\eps)k]+\\PP[Z\\geq \\eps k]\\]\nBy the definition of $Z$, we have $\\PP[Z\\geq \\eps k]=2^{-\\eps k}$. For the other two terms, recall that $X\\sim \\rm{NB}(k)$, and the conditional distribution $\\big(Y\\mid X< (1+\\eps)k\\big)$ is stochastically dominated by $\\rm{NB}((1+\\eps)k)$. Both terms can be bounded by Proposition \\ref{prop:nbchernoff}, so \n\\[\n\\PP[T^*\\geq (4+5\\eps)k]\\leq 3e^{-\\Omega(\\eps^2k/(1+\\eps))},\n\\]\nwhich gives~\\eqref{eq:algorithmstep3}. We obtain tail bounds on $X^*$ similarly. Using~\\eqref{eq:X*expression}, we concentrate $X^*$ by bounding $X,Y$, and $Z$, noting that $|2C'-Z|\\leq |Z|$:\n\\begin{multline*}\n\\PP[|X^*-k|\\geq 3\\eps k]\\leq \\PP[Z\\geq \\eps k]+\\PP[|X-k-1|\\geq \\eps k]+\n\\PP\\big[|Y-k|\\geq 2\\eps k\\ \\big|\\ |X-k-1|<\\eps k\\big].\n\\end{multline*}\nVerifying that each of these is bounded by $e^{-\\Omega(\\eps^2k/(1+\\eps))}$, we conclude~\\eqref{eq:algorithmstep4}.\n\\end{proof}\n\n\\begin{problem}\\label{prob:ternary} Is it true that $|\\cS_3(n)|=(9-o(1))^n$? \n\\end{problem}\n\\noindent This problem is closely related to the extremal problem of determining whether $\\rm{LT}(3,n)=(\\frac 12-o(1))n$. The following one directional implication follows from work of Bukh and Zhou.\n\\begin{theorem}[{\\cite[Theorem 4]{bukhzhou}}]\\label{prop:alphatwins} If $\\cS_k(n)\\leq (b_k)^{2n+o(n)}$ and there exists $\\alpha_k\\in (2/k,1)$ such that\n\\[\n\\alpha_k\\log\\Big(\\frac{b_k}{\\alpha_kk}\\Big)+(1-\\alpha_k)\\log \\Big(\\frac{k-1}{k(1-\\alpha_k)}\\Big)<0,\n\\]\nthen $\\rm{LT}(k,n)< \\frac 12\\alpha_kn$ for sufficiently large $n$. In particular, if $b_k<k$, then we can take $\\alpha_k<1$.\n\\end{theorem}\n\\noindent This can be obtained by a union bound over all ``leftmost\" embeddings of shuffle squares in a random word $s$. Though we do not offer a conjecture on Problem~\\ref{prob:ternary}, numerical evidence suggests that $|\\cS_3(n)|=o(9^n)$.", "post_theorem_intro_text_len": 3114, "post_theorem_intro_text": "The exponent $-1/15$ is not optimized, but within a modest factor of the best possible using our arguments. It would be interesting to decide if the error term is actually polynomial or exponential. \n\nClosely related to the shuffle square problem is the ``twins'' problem introduced by Axenovich, Person, and Puzynina~\\cite{Axenovich}. For a word $s\\in [k]^*$, let $\\rm{LT}(s)$ be the maximum semi-length of a subword of $s$ that is a shuffle square, so $s\\in \\cS_k(n)$ if and only if $\\rm{LT}(s)=n$. Equivalently, $\\rm{LT}(s)$ is the largest $m$ such that $s$ contains ``twins'' of length $m$, i.e. disjoint identical subwords of length $m$. Axenovich, Person, and Puzynina proved that for any $s\\in \\{0,1\\}^{2n}$, we have $\\rm{LT}(s)=(1-o(1))n$. More progress has been made on twins for various alphabets~\\cite{basu,bukhzhou}, and the twins problem is related to regularity lemmas for words, the longest common subsequence problem \\cite{bukhhogenson,bukhma,ChvatalSankoff,He-Li}, and codes for correcting deletion errors \\cite{Guruswami-He-Li}.\n\nWhile previous work dealt with the worst-case behavior of $\\rm{LT}(s)$, \\cref{thm:square} implies the best possible bound for its average-case behavior. \n\\begin{corollary}\\label{corollary:lt} For almost all $s\\in\\{0,1\\}^{n}$, $\\rm{LT}(s)\\geq \\sceil{\\frac{n}{2}}-1$.\n\\end{corollary}\n\n\\begin{proof} Let $s\\in \\{0,1\\}^{2n}$ be uniformly random among words with an even number of $0$'s and $1$'s. If $\\rm{LT}(s)=n$, then by deleting one or two bits, we immediately have $\\rm{LT}(s[1,2n-1])=n-1$ and $\\rm{LT}(s[1,2n-2])\\geq n-2$. Thus by \\cref{thm:square},\n\\[\n\\PP[\\rm{LT}(s[1,2n-2])\\geq n-2]\\geq \\PP[\\rm{LT}(s[1,2n-1])=n-1]\\geq \\PP[s\\in \\cS_2(n)]=1-o(1).\n\\]\nMoreover, $s[1,2n-1]$ is uniformly random in $\\{0,1\\}^{2n-1}$ and $s[1,2n-2]$ is uniformly random in $\\{0,1\\}^{2n-2}$, implying the corollary.\n\\end{proof}\n\nIn \\cite{HHNT}, it was conjectured that almost every binary word can be made into a shuffle square by deleting at most three bits - \\cref{corollary:lt} shows that it actually suffices to delete two (which is best possible).\n\n\\par{\\bf Organization.} In Section~\\ref{section:preliminaries}, we introduce the notion of a \\textit{buffer} for a binary word, thereby reformulating the problem as a discrete stochastic process on subsets of $\\{0,1\\}^*$. We define and analyze the greedy algorithm using buffers and obtain asymptotic results on its behavior. In Section~\\ref{section:buffersetanalysis}, we analyze the evolution of the set of all buffers to show that a random word $s$ can ``absorb\" small defects in an almost-perfect partition. In Section~\\ref{section:boostedgreedy}, we give an improved algorithm which produces a shuffle square with constant probability. This boosted greedy algorithm stores local information about the word and occasionally backtracks for an improvement. Together with the buffer set analysis, this proves~\\cref{thm:square}. In Section~\\ref{section:conclusion}, we discuss the shuffle square problem on larger alphabets, and we provide a boosted greedy algorithm for $k$-ary words which gives a new lower bound for $|\\cS_k(n)|$.", "sketch": "In the \".\\par{\\bf Organization.}\" paragraph, the authors outline how they prove Theorem~\\ref{thm:square}: (1) they \"introduce the notion of a \\textit{buffer} for a binary word, thereby reformulating the problem as a discrete stochastic process on subsets of $\\{0,1\\}^*$\"; (2) they \"define and analyze the greedy algorithm using buffers and obtain asymptotic results on its behavior\"; (3) they \"analyze the evolution of the set of all buffers to show that a random word $s$ can \\`\\`absorb\\\" small defects in an almost-perfect partition\"; (4) they \"give an improved algorithm which produces a shuffle square with constant probability\"—a \"boosted greedy algorithm\" that \"stores local information about the word and occasionally backtracks for an improvement\"; (5) \"[t]ogether with the buffer set analysis, this proves~\\cref{thm:square}.\"", "expanded_sketch": "In the \".\\par{\\bf Organization.}\" paragraph, the authors outline how they prove the main theorem: (1) they \"introduce the notion of a \\textit{buffer} for a binary word, thereby reformulating the problem as a discrete stochastic process on subsets of $\\{0,1\\}^*$\"; (2) they \"define and analyze the greedy algorithm using buffers and obtain asymptotic results on its behavior\"; (3) they \"analyze the evolution of the set of all buffers to show that a random word $s$ can \\`\\`absorb\\\" small defects in an almost-perfect partition\"; (4) they \"give an improved algorithm which produces a shuffle square with constant probability\"—a \"boosted greedy algorithm\" that \"stores local information about the word and occasionally backtracks for an improvement\"; (5) \"[t]ogether with the buffer set analysis, this proves the main theorem.\"", "expanded_theorem": "\\label{thm:square}\nWe have $|\\cS_2(n)|=(\\frac{1}{2}-o(n^{-1/15})) 2^{2n}$.", "theorem_type": ["Asymptotic or Limit", "Inequality or Bound"], "mcq": {"question": "Let $[2]^*$ be the set of all finite binary words over $\\{0,1\\}$. A word $s'\\in[2]^*$ is a subword of $s\\in[2]^*$ if $s'$ is a subsequence of $s$ (not necessarily contiguous). A binary word $s$ of length $2n$ is called a shuffle square if its positions can be partitioned into two disjoint subwords that are identical as words. Let $\\mathcal S_2(n)$ denote the set of shuffle squares of semi-length $n$, i.e. binary shuffle squares of length $2n$. Which asymptotic estimate holds for $|\\mathcal S_2(n)|$ as $n\\to\\infty$?", "correct_choice": {"label": "A", "text": "$|\\mathcal S_2(n)|=\\left(\\frac{1}{2}-o\\!ig(n^{-1/15}\\big)\\right)2^{2n}$."}, "choices": [{"label": "B", "text": "$|\\mathcal S_2(n)|=\\left(\\frac{1}{2}+o\\!\\big(n^{-1/15}\\big)\\right)2^{2n}$."}, {"label": "C", "text": "$|\\mathcal S_2(n)|=\\left(\\frac{1}{2}-o(1)\\right)2^{2n}$."}, {"label": "D", "text": "$|\\mathcal S_2(n)|=\\left(\\frac{1}{2}-O\\!\\big(n^{-1/15}\\big)\\right)2^{2n}$."}, {"label": "E", "text": "$|\\mathcal S_2(n)|\\geq \\left(\\frac{1}{2}-o\\!\\big(n^{-1/15}\\big)\\right)2^{2n}$."}], "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": "defect-absorption yields a deficit from $1/2$, not an excess", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "explicit polynomial error term $o(n^{-1/15})$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "non-effective little-o error replaced by stronger big-O control", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "exact asymptotic equality replaced by one-sided lower bound at the same scale", "template_used": "wildcard"}]}}
{"id": "2512.12321v1", "paper_link": "http://arxiv.org/abs/2512.12321v1", "theorems_cnt": 1, "theorem": {"env_name": "thm", "content": "\\label{thm:Kitaev.vanishing}\nIf $U,V\\in\\mathcal{L}^\\times$ satisfy\n\\begin{equation}\\label{eqn:Kitaev.class}\n(U-\\mathbf{1})(V-\\mathbf{1})\\in\\mathcal{L}^1\\quad\\mathrm{and}\\quad (V-\\mathbf{1})(U-\\mathbf{1})\\quad \\in\\mathcal{L}^1,\n\\end{equation}\nthen $\\det(UVU^{-1}V^{-1})=1$.", "start_pos": 2998, "end_pos": 3291, "label": "thm:Kitaev.vanishing"}, "ref_dict": {"thm:Kitaev.vanishing": "\\begin{thm}\\label{thm:Kitaev.vanishing}\nIf $U,V\\in\\mathcal{L}^\\times$ satisfy\n\\begin{equation}\\label{eqn:Kitaev.class}\n(U-\\mathbf{1})(V-\\mathbf{1})\\in\\mathcal{L}^1\\quad\\mathrm{and}\\quad (V-\\mathbf{1})(U-\\mathbf{1})\\quad \\in\\mathcal{L}^1,\n\\end{equation}\nthen $\\det(UVU^{-1}V^{-1})=1$.\n\\end{thm}", "prop:algebraic.step": "\\begin{prop}\\label{prop:algebraic.step}\nLet $U,V\\in\\mathcal{L}^\\times$ satisfy the Kitaev condition . Then\n\\begin{equation}\\label{eqn:algebraic.simplification}\n\\det(UVU^{-1}V^{-1})={\\det}^{(2)}\\big(e_{12}(U-\\mathbf{1})e_{21}(V-\\mathbf{1})e_{12}(\\mathbf{1}-U)e_{21}(\\mathbf{1}-V)\\big).\n\\end{equation}\n\\end{prop}", "eqn:Kitaev.class": "\\begin{equation}\\label{eqn:Kitaev.class}\n(U-\\mathbf{1})(V-\\mathbf{1})\\in\\mathcal{L}^1\\quad\\mathrm{and}\\quad (V-\\mathbf{1})(U-\\mathbf{1})\\quad \\in\\mathcal{L}^1,\n\\end{equation}", "eqn:HH.formula": "\\begin{equation}\\label{eqn:HH.formula}\n\\det(e^Ae^Be^{-A}e^{-B})=\\exp(\\mathrm{Tr}[A,B]).\n\\end{equation}"}, "pre_theorem_intro_text_len": 436, "pre_theorem_intro_text": "\\label{sec:introduction}\nWe write $\\mathcal{L}\\equiv \\mathcal{L}(\\mathcal{H})$ for the ring of bounded linear operators on a Hilbert space $\\mathcal{H}$, and $\\mathcal{L}^1\\equiv \\mathcal{L}^1(\\mathcal{H})$ for the ideal of \\emph{trace class} operators. The group of invertibles in $\\mathcal{L}$ is denoted $\\mathcal{L}^\\times$, and $\\mathbf{1}$ denotes the identity operator on $\\mathcal{H}$.\n\nThe following Theorem is our main result.", "context": "\\label{sec:introduction}\nWe write $\\mathcal{L}\\equiv \\mathcal{L}(\\mathcal{H})$ for the ring of bounded linear operators on a Hilbert space $\\mathcal{H}$, and $\\mathcal{L}^1\\equiv \\mathcal{L}^1(\\mathcal{H})$ for the ideal of \\emph{trace class} operators. The group of invertibles in $\\mathcal{L}$ is denoted $\\mathcal{L}^\\times$, and $\\mathbf{1}$ denotes the identity operator on $\\mathcal{H}$.\n\nThe following Theorem is our main result.", "full_context": "\\label{sec:introduction}\nWe write $\\mathcal{L}\\equiv \\mathcal{L}(\\mathcal{H})$ for the ring of bounded linear operators on a Hilbert space $\\mathcal{H}$, and $\\mathcal{L}^1\\equiv \\mathcal{L}^1(\\mathcal{H})$ for the ideal of \\emph{trace class} operators. The group of invertibles in $\\mathcal{L}$ is denoted $\\mathcal{L}^\\times$, and $\\mathbf{1}$ denotes the identity operator on $\\mathcal{H}$.\n\nThe following Theorem is our main result.\n\n\\section{Introduction}\\label{sec:introduction}\nWe write $\\mathcal{L}\\equiv \\mathcal{L}(\\mathcal{H})$ for the ring of bounded linear operators on a Hilbert space $\\mathcal{H}$, and $\\mathcal{L}^1\\equiv \\mathcal{L}^1(\\mathcal{H})$ for the ideal of \\emph{trace class} operators. The group of invertibles in $\\mathcal{L}$ is denoted $\\mathcal{L}^\\times$, and $\\mathbf{1}$ denotes the identity operator on $\\mathcal{H}$.\n\nThe Fredholm determinant is conjugation invariant, and is a group homomorphism from the invertible determinant class operators to the non-zero complex numbers. From these algebraic properties alone, it is straightforward to show that $\\det(UVU^{-1}V^{-1})=1$ whenever any of $U,V$ or $UV$ is determinant class. Nevertheless, in general, $\\det(UVU^{-1}V^{-1})$ can take \\emph{any} non-zero value \\cite{AV,EF}, and it is closely related to the rich subject of traces of commutators \\cite{HH}, as well as $K$-theory \\cite{Brown}.\nNow, Kitaev's condition \\eqref{eqn:Kitaev.class} implies\n\\begin{equation}\\label{eqn:Kitaev.implies.det.class}\n\\begin{aligned}\nUVU^{-1}V^{-1}-\\mathbf{1}&=(UV-VU)U^{-1}V^{-1}\\\\\n&=\\big((U-\\mathbf{1})(V-\\mathbf{1})-(V-\\mathbf{1})(U-\\mathbf{1})\\big)U^{-1}V^{-1}\\in\\mathcal{L}^1,\n\\end{aligned}\n\\end{equation}\nso that $\\det(UVU^{-1}V^{-1})$ is automatically well-defined, but it is not at all obvious that it must be trivial. Furthermore, because Kitaev's condition occurs naturally in geometric and physics problems (see Section \\ref{sec:quantized.traces}), the veracity of Theorem \\ref{thm:Kitaev.vanishing} is of considerable general interest for applications.\n\n\\section{Proof of determinant trivialization theorem}\\label{sec:proof}\nFor each $n\\geq 1$, we write $M_n(-)$ for the $n\\times n$ matrix ring with entries in $(-)$, and $\\GL_n(\\mathcal{L})$ for the group of invertibles in $M_n(\\mathcal{L})$. We may identify $M_n(\\mathcal{L})$ with $\\mathcal{L}(\\mathcal{H}^{\\oplus n})$, and $M_n(\\mathcal{L}^1)$ with the ideal $\\mathcal{L}^1(\\mathcal{H}^{\\oplus n})$. Thus, there are Fredholm determinant homomorphisms\n\\[\n{\\det}^{(n)}:\\{U\\in \\GL_n(\\mathcal{L})\\,:\\,U-\\mathbf{1}^{\\oplus n}\\in M_n(\\mathcal{L}^1)\\}\\to\\mathbb{C}^\\times,\\qquad n\\geq 1,\n\\]\nand they are compatible with $\\det=\\det^{(1)}$ in the sense that\n\\begin{equation}\\label{eqn.det.stable}\n{\\det}^{(n)}\\begin{pmatrix} U & 0 \\\\ 0 & \\mathbf{1}^{\\oplus (n-1)}\\end{pmatrix} = {\\det}^{(1)}\\, U,\\qquad n\\geq 1.\n\\end{equation}\n\nBy combining the Steinberg relations with the Kitaev condition \\eqref{eqn:Kitaev.class}, and exploiting the redundancy Lemma \\ref{lem:redundant.elementary}, we can rewrite $\\det(UVU^{-1}V^{-1})$ as the determinant of a \\emph{single} commutator of \\emph{elementary matrices}:\n\\begin{prop}\\label{prop:algebraic.step}\nLet $U,V\\in\\mathcal{L}^\\times$ satisfy the Kitaev condition . Then\n\\begin{equation}\\label{eqn:algebraic.simplification}\n\\det(UVU^{-1}V^{-1})={\\det}^{(2)}\\big(e_{12}(U-\\mathbf{1})e_{21}(V-\\mathbf{1})e_{12}(\\mathbf{1}-U)e_{21}(\\mathbf{1}-V)\\big).\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nAccording to the identities \\eqref{eqn:mult.commutator.stabalized}--\\eqref{eqn:Whitehead.big.UV},\n\\begin{equation}\\label{eqn:det.alg.calc.first.factorization}\n\\det(UVU^{-1}V^{-1})={\\det}^{(3)} \\big(d_{12}(U)d_{13}(V)d_{12}(U)^{-1}d_{13}(V)^{-1}\\big),\n\\end{equation}\nwhere\n\\begin{equation}\\label{eqn:for.shuffling}\n\\begin{aligned}\nd_{12}(U) & = e_{21}(U^{-1})e_{12}(\\mathbf{1}-U)e_{21}(-\\mathbf{1})e_{12}(\\mathbf{1}-U^{-1})\\\\\nd_{13}(V) & = e_{31}(V^{-1})e_{13}(\\mathbf{1}-V)e_{31}(-\\mathbf{1})e_{13}(\\mathbf{1}-V^{-1}).\n\\end{aligned}\n\\end{equation}\nWe would like to swap the positions of $d_{12}(U)$ and $d_{13}(V)$ in \\eqref{eqn:det.alg.calc.first.factorization}, in order to cancel $d_{12}(U)^{-1}d_{13}(V)^{-1}$. This entails moving the elementary matrix factors of $d_{12}(U)$ past those of $d_{13}(V)$. Because of \\eqref{eqn:Steinberg.commutator.123}, we will pick up some commutator terms of the form $e_{ik}(ST)$ or $e_{ki}(-TS)$ along the way.\nIn general, a complicated collection of commutator terms will be acquired.\n\nFortunately, for the purposes of computing the determinant \\eqref{eqn:det.alg.calc.first.factorization}, Lemma \\ref{lem:redundant.elementary} tells us that we can discard any $e_{ik}(ST)$ or $e_{ki}(-TS)$ term with $ST\\in\\mathcal{L}^1$ or $TS\\in \\mathcal{L}^1$, since this kind of elementary matrix has trivial determinant. The Kitaev condition \\eqref{eqn:Kitaev.class} enters here: Because $(U-\\mathbf{1})(V-\\mathbf{1})$ is trace class, for $i\\neq k$,\n\\begin{align*}\n&{\\det}^{(3)}\\big(W e_{ij}(U-\\mathbf{1})e_{jk}(V-\\mathbf{1}) W^\\prime\\big)\\\\\n&={\\det}^{(3)}\\big(W e_{ik}((U-\\mathbf{1})(V-\\mathbf{1}))e_{jk}(V-\\mathbf{1})e_{ij}(U-\\mathbf{1}) W^\\prime\\big) & (\\mathrm{Eq.}\\,\\eqref{eqn:Steinberg.commutator.123})\\\\\n&={\\det}^{(3)}\\big(W e_{jk}(V-\\mathbf{1})e_{ij}(U-\\mathbf{1}) W^\\prime\\big) & (\\mathrm{Lemma}\\;\\ref{lem:redundant.elementary})\n\\end{align*}\nholds whenever $W,W^\\prime$ are invertible operators such that the above determinants are defined. Similarly with $U$ and $V$ swapped. The above property is conveniently expressed as an extra ``commutativity'' relation,\n\\[\ne_{ij}(U-\\mathbf{1})e_{jk}(V-\\mathbf{1})\\simeq e_{jk}(V-\\mathbf{1})e_{ij}(U-\\mathbf{1}),\\qquad i\\neq k,\n\\]\n\\emph{where ``$\\simeq$'' indicates equality after taking a determinant}. \nKitaev's condition \\eqref{eqn:Kitaev.class} also implies\n\\[\n(U-\\mathbf{1})V=U-\\mathbf{1}=V(U-\\mathbf{1}),\\qquad\n(V-\\mathbf{1})U=V-\\mathbf{1}=U(V-\\mathbf{1})\\qquad \\mathrm{mod}\\;\\mathcal{L}^1,\n\\]\nThe extra relations made available by Kitaev's condition \\eqref{eqn:Kitaev.class} are summarized below:\n\\begin{equation}\\label{eqn:Steinberg.Kitaev.extra.identity.2}\n\\begin{aligned}\ne_{ij}(U-\\mathbf{1})e_{jk}(V-\\mathbf{1}) &\\simeq e_{jk}(V-\\mathbf{1})e_{ij}(U-\\mathbf{1}), & i\\neq k,\\\\\ne_{ij}(V-\\mathbf{1})e_{jk}(U-\\mathbf{1}) &\\simeq e_{jk}(U-\\mathbf{1})e_{ij}(V-\\mathbf{1}), & i\\neq k;\\\\\ne_{ij}((U-\\mathbf{1})V)&\\simeq e_{ij}(U-\\mathbf{1})\\simeq e_{ij}(V(U-\\mathbf{1})),\\\\\ne_{ij}((V-\\mathbf{1})U)&\\simeq e_{ij}(V-\\mathbf{1})\\simeq e_{ij}(U(V-\\mathbf{1})).\n\\end{aligned}\n\\end{equation}\nAs Kitaev's condition \\eqref{eqn:Kitaev.class} holds with $U$ replaced by $U^{-1}$ and/or $V$ replaced by $V^{-1}$, the same replacements can be made in \\eqref{eqn:Steinberg.Kitaev.extra.identity.2}.\n\nNow specialize to the case $R=\\mathcal{L}$ and $I=\\mathcal{L}^1$. For $U,V\\in \\mathcal{L}^\\times$ with determinant class $UVU^{-1}V^{-1}$, the above general definitions give\n\\begin{equation}\\label{eqn:Brown.observation}\n\\begin{aligned}\n\\check{\\det}\\circ\\partial\\{u,v\\}\n&=\\check{\\det}\\big[d_{12}(U)d_{13}(V)d_{12}(U)^{-1}d_{13}(V)^{-1}\\big]_{K_1(\\mathcal{L},\\mathcal{L}^1)}\\\\\n&=\\check{\\det}\\left[\\begin{pmatrix}UVU^{-1}V^{-1} & 0 & 0 \\\\ 0 & \\mathbf{1} & 0 \\\\ 0 & 0 & \\mathbf{1}\\end{pmatrix}\\right]_{K_1(\\mathcal{L},\\mathcal{L}^1)}\\\\\n&=\\det(UVU^{-1}V^{-1}).\n\\end{aligned}\n\\end{equation}\nThis $K$-theoretic nature of $\\det(UVU^{-1}V^{-1})$ was first observed by Brown \\cite{Brown}.\n\n\\subsection{Application to quantized traces}\\label{sec:quantized.traces}\\label{sec:application.quantized.traces}\nLet $A,B\\in\\mathcal{L}$, and suppose \n\\begin{equation}\\label{eqn:quantized.trace.hypothesis}\n[A,B]\\in \\mathcal{L}^1\\;\\;\\mathrm{and}\\;\\;(e^A-\\mathbf{1})(e^B-\\mathbf{1})\\in\\mathcal{L}^1.\n\\end{equation}\nThen, as in \\eqref{eqn:Kitaev.implies.det.class}, $(e^B-\\mathbf{1})(e^A-\\mathbf{1})$ is trace class as well. Thus, $e^A,e^B$ satisfy Kitaev's condition \\eqref{eqn:Kitaev.class}, so Theorem \\ref{thm:Kitaev.vanishing} guarantees that $\\det(e^Ae^Be^{-A}e^{-B})=1$. By the Pincus--Helton--Howe formula \\eqref{eqn:HH.formula}, we learn that the condition \\eqref{eqn:quantized.trace.hypothesis} implies the trace quantization,\n\\[\n\\frac{1}{2\\pi i}\\mathrm{Tr}[A,B]\\in\\mathbb{Z}.\n\\]", "post_theorem_intro_text_len": 4114, "post_theorem_intro_text": "This result was first suggested by A.~Kitaev in his influential paper \\cite[pp.~56]{Kitaev}, for the purpose of demonstrating the integrality of certain trace formulae appearing in modern quantum physics (see Section \\ref{sec:quantized.traces}). We shall therefore refer to \\eqref{eqn:Kitaev.class} as the \\emph{Kitaev condition} on an invertible pair $U,V\\in \\mathcal{L}^\\times$. \n\n\\medskip\nRecall that an operator $T\\in\\mathcal{L}$ is \\emph{trace class} if for an(y) orthonormal basis $\\mathcal{B}$, the sum $\\sum_{e\\in \\mathcal{B}}\\big\\langle e\\big||T|e\\big\\rangle$ is finite. In this case, its \\emph{trace}\n\\[\n\\mathrm{Tr}(T)=\\sum_{e\\in \\mathcal{B}}\\big\\langle e\\big|Te\\big\\rangle \\in\\mathbb{C}\n\\]\nis well-defined, independently of the choice of $\\mathcal{B}$. An operator $U\\in\\mathcal{L}$ is \\emph{determinant class} if $U-\\mathbf{1}\\in \\mathcal{L}^1$, in which case, its \\emph{(Fredholm) determinant} $\\det(U)\\in\\mathbb{C}$ is well-defined. See \\cite[Chapter 3]{Simon} for a detailed treatment. The identity\n\\[\n\\det(\\exp(T))=\\exp(\\mathrm{Tr}(T)),\\qquad T\\in\\mathcal{L}^1,\n\\]\nshows that a trivial determinant is intimately related to integrality of a corresponding trace (up to a $2\\pi i$ factor).\n\nThe Fredholm determinant is conjugation invariant, and is a group homomorphism from the invertible determinant class operators to the non-zero complex numbers. From these algebraic properties alone, it is straightforward to show that $\\det(UVU^{-1}V^{-1})=1$ whenever any of $U,V$ or $UV$ is determinant class. Nevertheless, in general, $\\det(UVU^{-1}V^{-1})$ can take \\emph{any} non-zero value \\cite{AV,EF}, and it is closely related to the rich subject of traces of commutators \\cite{HH}, as well as $K$-theory \\cite{Brown}.\nNow, Kitaev's condition \\eqref{eqn:Kitaev.class} implies\n\\begin{equation}\\label{eqn:Kitaev.implies.det.class}\n\\begin{aligned}\nUVU^{-1}V^{-1}-\\mathbf{1}&=(UV-VU)U^{-1}V^{-1}\\\\\n&=\\big((U-\\mathbf{1})(V-\\mathbf{1})-(V-\\mathbf{1})(U-\\mathbf{1})\\big)U^{-1}V^{-1}\\in\\mathcal{L}^1,\n\\end{aligned}\n\\end{equation}\nso that $\\det(UVU^{-1}V^{-1})$ is automatically well-defined, but it is not at all obvious that it must be trivial. Furthermore, because Kitaev's condition occurs naturally in geometric and physics problems (see Section \\ref{sec:quantized.traces}), the veracity of Theorem \\ref{thm:Kitaev.vanishing} is of considerable general interest for applications. \n\nMore than fifteen years after Kitaev's paper \\cite{Kitaev} appeared, some progress was made in \\cite{EF}, where Theorem \\ref{thm:Kitaev.vanishing} was proved under the extra assumption that $(U^*-\\mathbf{1})(V-\\mathbf{1})$ and $(V-\\mathbf{1})(U^*-\\mathbf{1})$ are also trace class. Borel functional calculus played a key role in \\cite{EF}, and it is unclear whether the techniques there can be improved to address the full \\emph{adjoint-free} Theorem \\ref{thm:Kitaev.vanishing}. Nevertheless, the result of \\cite{EF} shows that there are no easy counterexamples to Theorem \\ref{thm:Kitaev.vanishing}.\n\nOur approach to Theorem \\ref{thm:Kitaev.vanishing} is completely different, and we believe that it gets to the heart of the problem. Our proof is presented in Section \\ref{sec:proof}, and comprises two main steps, both of which use the Kitaev condition \\eqref{eqn:Kitaev.class} in crucial ways.\n\\begin{enumerate}\n\\item An \\emph{algebraic} step, Prop.~\\ref{prop:algebraic.step}, simplifies $\\det(UVU^{-1}V^{-1})$ to $\\det(E_1E_2E_1^{-1}E_2^{-1})$, where $E_1,E_2$ are so-called elementary matrices. This is the most novel part of the proof. Experts in algebraic $K$-theory may notice that the abstract concept of Steinberg symbols in $K_2$-theory is guiding our manipulations. We choose to keep the presentation concrete, and defer the $K$-theoretic interpretation to Section \\ref{sec:K.perspective}.\n\\item An \\emph{analytic} step computes $\\det(E_1E_2E_1^{-1}E_2^{-1})=1$ using a formula of Pincus--Helton--Howe (Eq.~\\eqref{eqn:HH.formula}), and Lidskii's trace theorem \\cite{Lidskii}.\n\\end{enumerate}\nFinally, the physical context for Theorem \\ref{thm:Kitaev.vanishing} is discussed in Section \\ref{sec:quantized.traces}.", "sketch": "The proof of Theorem~\\ref{thm:Kitaev.vanishing} (given in Section~\\ref{sec:proof}) has two main steps, both using the Kitaev condition \\eqref{eqn:Kitaev.class} in crucial ways:\n\\begin{enumerate}\n\\item An \\emph{algebraic} step (Prop.~\\ref{prop:algebraic.step}) that “simplifies $\\det(UVU^{-1}V^{-1})$ to $\\det(E_1E_2E_1^{-1}E_2^{-1})$, where $E_1,E_2$ are so-called elementary matrices.” The text notes that “Steinberg symbols in $K_2$-theory [are] guiding our manipulations,” though the presentation is kept concrete.\n\\item An \\emph{analytic} step that “computes $\\det(E_1E_2E_1^{-1}E_2^{-1})=1$ using a formula of Pincus--Helton--Howe (Eq.~\\eqref{eqn:HH.formula}), and Lidskii's trace theorem.”\n\\end{enumerate}\nAdditionally, the introduction observes that Kitaev's condition implies the commutator is determinant-class via \\eqref{eqn:Kitaev.implies.det.class}, so $\\det(UVU^{-1}V^{-1})$ is well-defined, though “it is not at all obvious that it must be trivial.”", "expanded_sketch": "The proof of \\ref{thm:Kitaev.vanishing} (given later) has two main steps, both using the Kitaev condition\n\\begin{equation}\\label{eqn:Kitaev.class}\n(U-\\mathbf{1})(V-\\mathbf{1})\\in\\mathcal{L}^1\\quad\\mathrm{and}\\quad (V-\\mathbf{1})(U-\\mathbf{1})\\quad \\in\\mathcal{L}^1,\n\\end{equation}\nin crucial ways:\n\\begin{enumerate}\n\\item An \\emph{algebraic} step. We first prove the following proposition.\n\\begin{prop}\\label{prop:algebraic.step}\nLet $U,V\\in\\mathcal{L}^\\times$ satisfy the Kitaev condition . Then\n\\begin{equation}\\label{eqn:algebraic.simplification}\n\\det(UVU^{-1}V^{-1})={\\det}^{(2)}\\big(e_{12}(U-\\mathbf{1})e_{21}(V-\\mathbf{1})e_{12}(\\mathbf{1}-U)e_{21}(\\mathbf{1}-V)\\big).\n\\end{equation}\n\\end{prop}\nThis “simplifies $\\det(UVU^{-1}V^{-1})$ to $\\det(E_1E_2E_1^{-1}E_2^{-1})$, where $E_1,E_2$ are so-called elementary matrices.” The text notes that “Steinberg symbols in $K_2$-theory [are] guiding our manipulations,” though the presentation is kept concrete.\n\\item An \\emph{analytic} step that “computes $\\det(E_1E_2E_1^{-1}E_2^{-1})=1$ using a formula of Pincus--Helton--Howe,\n\\begin{equation}\\label{eqn:HH.formula}\n\\det(e^Ae^Be^{-A}e^{-B})=\\exp(\\mathrm{Tr}[A,B]).\n\\end{equation}\nand Lidskii's trace theorem.”\n\\end{enumerate}\nAdditionally, the introduction observes that Kitaev's condition implies the commutator is determinant-class via \\eqref{eqn:Kitaev.implies.det.class}, so $\\det(UVU^{-1}V^{-1})$ is well-defined, though “it is not at all obvious that it must be trivial.”", "expanded_theorem": "\\label{thm:Kitaev.vanishing}\nIf $U,V\\in\\mathcal{L}^\\times$ satisfy\n\\begin{equation}\\label{eqn:Kitaev.class}\n(U-\\mathbf{1})(V-\\mathbf{1})\\in\\mathcal{L}^1\\quad\\mathrm{and}\\quad (V-\\mathbf{1})(U-\\mathbf{1})\\quad \\in\\mathcal{L}^1,\n\\end{equation}\nthen $\\det(UVU^{-1}V^{-1})=1$.", "theorem_type": ["Implication"], "mcq": {"question": "Let mathcal{H} be a Hilbert space, let \\(\\mathcal{L}(\\mathcal{H})\\) be the algebra of bounded linear operators on \\(\\mathcal{H}\\), let \\(\\mathcal{L}^1(\\mathcal{H})\\) be the ideal of trace-class operators, let \\(\\mathcal{L}(\\mathcal{H})^\\times\\) denote the invertible bounded operators, and let \\(\\mathbf{1}\\) be the identity operator. Suppose \\(U,V\\in \\mathcal{L}(\\mathcal{H})^\\times\\) satisfy\n\\[\n(U-\\mathbf{1})(V-\\mathbf{1})\\in \\mathcal{L}^1(\\mathcal{H})\n\\quad\\text{and}\\quad\n(V-\\mathbf{1})(U-\\mathbf{1})\\in \\mathcal{L}^1(\\mathcal{H}).\n\\]\nFor an invertible operator of the form \\(\\mathbf{1}+K\\) with \\(K\\in \\mathcal{L}^1(\\mathcal{H})\\), write \\(\\det\\) for its Fredholm determinant. Under these hypotheses, which statement about the multiplicative commutator \\(UVU^{-1}V^{-1}\\) holds?", "correct_choice": {"label": "A", "text": "The Fredholm determinant of the commutator is trivial: \\(\\det(UVU^{-1}V^{-1})=1\\)."}, "choices": [{"label": "B", "text": "The multiplicative commutator is always determinant class under these hypotheses, and its Fredholm determinant is given by the trace formula\n\\[\n\\det(UVU^{-1}V^{-1})=\\exp\\!\\big(\\operatorname{Tr}((U-\\mathbf{1})(V-\\mathbf{1})-(V-\\mathbf{1})(U-\\mathbf{1}))\\big).\n\\]"}, {"label": "C", "text": "The multiplicative commutator is determinant class, i.e.\n\\[\nUVU^{-1}V^{-1}-\\mathbf{1}\\in \\mathcal{L}^1(\\mathcal{H}).\n\\]"}, {"label": "D", "text": "If \\((U-\\mathbf{1})(V-\\mathbf{1})\\in \\mathcal{L}^1(\\mathcal{H})\\), then automatically\n\\[\n\\det(UVU^{-1}V^{-1})=1,\n\\]\nwithout any need to assume \\((V-\\mathbf{1})(U-\\mathbf{1})\\in \\mathcal{L}^1(\\mathcal{H})\\)."}, {"label": "E", "text": "Under these hypotheses the commutator itself is trace class relative to the identity in the stronger sense that\n\\[\nUVU^{-1}V^{-1}=\\mathbf{1},\n\\]\nso in particular its Fredholm determinant equals \\(1\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "trace_identity", "tampered_component": "analytic-step trace formula applied directly to U,V rather than logarithmic elementary-matrix commutator", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "drop the final trivialization conclusion and keep only determinant-class/well-definedness", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "two-sided Kitaev condition reduced to one-sided trace-class product hypothesis", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "algebraic.step", "tampered_component": "replace determinant triviality of the commutator by actual operator triviality", "template_used": "stronger_trap"}]}}
{"id": "2512.13450v1", "paper_link": "http://arxiv.org/abs/2512.13450v1", "theorems_cnt": 2, "theorem": {"env_name": "Theoremstar", "content": "For almost all irrational $\\theta\\in [0,1]$, if we denote by $q_k/p_k$ the sequence of convergents of the continued fraction expansion of $\\theta$, then one has\n$$\\lim_{k\\to \\infty}\\frac{\\sigma_2(q_k/p_k)}{p_k^2}=\\frac{16}{\\pi^3}\\sum_{n\\ge 1 \\textrm{ odd}}\\frac{1}{n^3\\sin(n\\pi\\theta)}.$$", "start_pos": 45297, "end_pos": 45623, "label": null}, "ref_dict": {"fig:sigma2": "\\begin{center}\n\\includegraphics[width=10cm]{genre2.png}\n\\caption{Graph of the signature function $\\sigma_2$}\\label{fig:sigma2}\n\\end{center}", "sigma2": "\\begin{Theorem}\\label{sigma2}\n  For $0<q<p$ coprime odd integers, we have\n\\begin{equation}\\nonumber \\sigma_2({{\\textstyle\\frac q p}})= \\frac {1-p^2}{6p^2} +\n  \\frac 1 {4p^2}\\sum_{n=1,\\, \\mathrm{ odd}}^{p-2}\\frac{f(n,p,q)}{\\sin^3(\\frac{n\\pi}{2p})\\sin^2(\\frac{qn\\pi}{p})}\n\\end{equation}\nwhere \n\\begin{eqnarray*} f(n,p,q)&=(3p-3)\\sin(\\frac{(2q-1)n\\pi}{2p})+(p+1)\\sin(\\frac{(2q-3)n\\pi}{2p})\\\\\n&+(p-1)\\sin(\\frac{(2q+3)n\\pi}{2p})+(3p+3)\\sin(\\frac{(2q+1)n\\pi}{2p}).\n\\end{eqnarray*}\n\\end{Theorem}", "defsigma2": "\\begin{equation}\\label{defsigma2}\n  \\sigma_2({\\textstyle{\\frac{q}{p}}})=\\sum_{(j,k,\\ell)\\in \\Delta_p}\n (-1)^{\\lfloor {jq}/{p}\\rfloor +\\lfloor {kq}/{p}\\rfloor+\\lfloor {\\ell q}/{p}\\rfloor}\n\\end{equation}", "genre2": "\\begin{Remark}\n  {\\em\nThe $\\mathrm{SU}_2$-TQFT $\\boV_p$ corresponds in Conformal Field Theory to the Wess-Zumino-Witten model at odd level $\\ell=p-2$. Similar $\\mathrm{SU}_2$-TQFTs exist at  even levels, that is, when $p$ is even. Their signature functions can again be computed using the Frobenius algebra formalism of Theorem~\\ref{DM1}.  But we won't consider even level $\\mathrm{SU}_2$-TQFTs in this paper.\n}\\end{Remark}\n\n\\section{A trigonometric formula for the signature in genus 2}\\label{genre2}\nLet us now start our investigation of the asymptotic behavior of the signature function. The first result of this paper is the following trigonometric formula for the signature in genus $2$.\n\n\\begin{Theorem}\\label{sigma2}\n  For $0<q<p$ coprime odd integers, we have\n\\begin{equation}\\nonumber \\sigma_2({{\\textstyle\\frac q p}})= \\frac {1-p^2}{6p^2} +\n  \\frac 1 {4p^2}\\sum_{n=1,\\, \\mathrm{ odd}}^{p-2}\\frac{f(n,p,q)}{\\sin^3(\\frac{n\\pi}{2p})\\sin^2(\\frac{qn\\pi}{p})}\n\\end{equation}\nwhere \n\\begin{eqnarray*} f(n,p,q)&=(3p-3)\\sin(\\frac{(2q-1)n\\pi}{2p})+(p+1)\\sin(\\frac{(2q-3)n\\pi}{2p})\\\\\n&+(p-1)\\sin(\\frac{(2q+3)n\\pi}{2p})+(3p+3)\\sin(\\frac{(2q+1)n\\pi}{2p}).\n\\end{eqnarray*}\n\\end{Theorem}\n\n\\begin{proof}\nBy a finite Fourier transform over $\\Z/2p\\Z$, we can write for $m\\in\\Z$ :\n\\begin{equation}\\label{FFT}\n  (-1)^{\\lfloor m/p\\rfloor}=\n  \\frac 1 p\n  \\sum_{n=1,\\text{ odd}}^{2p-1}\\frac{\\sin(\\pi n(2m+1)/2p)}{\\sin(\\pi n/2p)}\n  =\\frac 1 p  \\sum_{n=1,\\text{ odd}}^{2p-1} \\frac {t^{2m+1}-t^{-2m-1}}{t-t^{-1}}\n    \\Big|_{t=e^{\\frac {i\\pi n}{2p}}~.\n      }\n\\end{equation}\nIn the sequel, we set $\\O_p=\\{1,3,\\ldots,2p-1\\}$ and $\\O_p^+=\\{1,3,\\ldots,p\\}$. Using Formula \\eqref{FFT} for each sign $\\epsilon_{j+1}=(-1)^{\\lfloor q(j+1)/p\\rfloor}$ and plugging it into the formula of Lemma \\ref{triplepsilon}, we get \n\\begin{equation}\\nonumber\\sigma_2({\\textstyle{\\frac q p}})\\,\\,\n  =\\sum_{(j,k,\\ell)\\in T_p} \\epsilon_{j+1}\\epsilon_{k+1}\\epsilon_{\\ell+1}\n  =\\frac1 {p^3} \\sum_{(a,b,c)\\in \\O_p^3}\\psi(a,b,c)\\end{equation} \nwhere\n\\begin{equation}\n\\psi(a,b,c)=\\!\\!\\!\\!\\sum\\limits_{(j,k,l)\\in T_p}\\!\\!\\!\\!\n\\frac{(x^{2q(j+1)+1}-x^{-2q(j+1)-1})(y^{2q(k+1)+1}-y^{-2q(k+1)-1})(z^{2q(\\ell+1)+1}-z^{-2q(\\ell+1)-1})}{(x-x^{-1})(y-y^{-1})(z-z^{-1})}\\nonumber\n\\end{equation}\nwith $x=e^{\\frac{i\\pi a}{2p}}$,  $y=e^{\\frac{i\\pi b}{2p}}$ and $z=e^{\\frac{i\\pi c}{2p}}$.\n\nThe function $\\psi$ is clearly symmetric in its three variables and satisfies also $\\psi(2p-a,b,c)=\\psi(a,b,c)$. We can hence rewrite $\\sigma_2(\\frac q p)$  as \n\\begin{equation}\\label{eq4n}\\sigma_2({\\textstyle{\\frac q p}})=\n  \\frac1 {p^3}\n  \\sum_{a,b,c\\in \\O_p^+} 2^{m_a+m_b+m_c}\\psi(a,b,c)\\end{equation}\nwhere $m_a=1$ if $a\\ne p$ and $m_p=0$.\n\nIn order to compute $\\psi(a,b,c)$, we rewrite it as follows:\n\\begin{equation} \\psi(a,b,c)= \\sum\\limits_{\\epsilon,\\mu,\\nu=\\pm 1}\\epsilon\\mu\\nu \\frac{ x^{\\epsilon}y^{\\mu}z^{\\nu}F(x^{2q\\epsilon},y^{2q\\mu},z^{2q\\nu})}{(x-x^{-1})(y-y^{-1})(z-z^{-1})}\\label{eq5n}\n\\end{equation} where $F$ is the polynomial defined by $F(X,Y,Z)=\\sum_{(j,k,l)\\in T_p}X^{j+1}Y^{k+1}Z^{\\ell +1}$.\n\nThe computation of $F(x^{2q\\epsilon},y^{2q\\mu},z^{2q\\nu})$ is simplified by the following observation:  we have $$F(X,Y,Z)=X^pY^p F(X^{-1}, Y^{-1},Z)~,$$ since the involution $(j,k,\\ell ) \\mapsto (p-2-j,p-2-k,\\ell)$ preserves the set $T_p$. Therefore, since $F$ is symmetric in its three variables,  whenever $F(X,Y,Z)$ is evaluated at numbers $X=x^{2q}$, $Y=y^{2q}$, and $Z=z^{2q}$\nsatisfying $X^p=Y^p=Z^p=-1$, then for  any $\\epsilon,\\mu,\\nu=\\pm 1$, we have that $F(X^\\epsilon, Y^\\mu,Z^\\nu)$ is equal to either $F(X,Y,Z)$ (when $\\epsilon\\mu\\nu =1$) or to  $F(X^{-1}, Y^{-1},Z^{-1})= \\overline {F(X,Y,Z)}$ (when $\\epsilon\\mu\\nu =-1$).\n\\begin{Remark}\\label{rk33}{\\em For later use, we note the following special case: when $F(x^{2q},y^{2q},z^{2q})$ happens to be real, then $F(x^{2q\\epsilon},y^{2q\\mu},z^{2q\\nu})$ \n    does not depend at all on $\\epsilon,\\mu$, or $\\nu$, and formula \\eqref{eq5n} for $\\psi(a,b,c)$ simply becomes $\\psi(a,b,c)=F(x^{2q},y^{2q},z^{2q})$.}\n    \\end{Remark}"}, "pre_theorem_intro_text_len": 2862, "pre_theorem_intro_text": "Let $p$ be an odd positive integer and $\\mathbb{Q}(\\zeta)$ be the cyclotomic field generated by a primitive root of unity $\\zeta$ of order $2p$. The SU$_2$-TQFT constructs for any surface $S_g$ of genus $g$ a finite dimensional $\\mathbb{Q}(\\zeta)$-vector space $\\boV_p(S_g)$ endowed with an Hermitian form $h_g$ defined over $\\mathbb{Q}(\\zeta)$. These vector spaces are part of a modular functor, meaning that they support a projective unitary action of the corresponding mapping class groups  and satisfy a list of compatibilities when cutting the surfaces along simple closed curves.\n\nThe dimension of $\\boV_p(S_g)$, given by the celebrated Verlinde formula, has received considerable attention since the nineties. On the other hand, taking $0<q<p$ two coprime odd integers, one can embed $\\mathbb{Q}(\\zeta)$ into $\\mathbb{C}$ by sending $\\zeta$ to $e^{i\\pi q/p}$. The Hermitian form $h_g$ has a signature that we denote by  \n$$\\sigma_g(\\textstyle{\\frac{q}{p}})=\\Sign\\Big(\\boV_p(S_g)\\!\\!\\underset{\\zeta=e^{i\\pi q/p}}{\\otimes}\\!\\!\\mathbb{C},\\,h_g\\Big)$$\nand which has been very little studied. It was shown in \\cite{DM} that for a fixed $\\zeta=e^{i\\pi q/p}$, these signatures have cut and paste properties very similar to those of the dimensions, which allows to encode them in a single Frobenius algebra $V_{\\frac q p}$. Later it was proved in \\cite{S2B} that $V_{\\frac q p}$ is closely related to the hyperbolic geometry of the two-bridge knot $K(p,q)$. The purpose of this paper is to study signatures in a different direction, by analyzing the asymptotic behavior of $\\sigma_{g}(\\frac{q}{p})$ when $\\frac{q}{p}\\to \\theta$ for a fixed $\\theta\\in [0,1]$. \n\nIf we take $q=1$, the Hermitian form $h_g$ becomes positive definite,\nso that $\\sigma_g(\\frac 1 p)$ coincides with the dimension of $\\boV_p(S_g)$.  The problem of computing the asymptotic behavior of the dimensions when $p\\to \\infty$ is well-known, and related to considerations of semi-classical analysis on character varieties $\\mathcal{M}_g=\\Hom(\\pi_1(S_g),\\SU_2)/\\SU_2$. This led Witten  \\cite{Witten} to show that for $g>1$,  \n$$\\lim_{p\\to \\infty}\\frac{\\dim \\boV_p(S_g)}{p^{3g-3}}=\\operatorname{Vol}(\\mathcal{M}_g)=2(2\\pi^2)^{1-g}\\zeta(2g-2).$$\n\nIn genus $0$ and $1$, the hermitian form is positive definite for all values of $q$, so let us consider the first non trivial case $g=2$. In this case, one can write a simple formula for  $\\sigma_2(\\frac q p)$:\n\\begin{equation}\\label{defsigma2}\n  \\sigma_2({\\textstyle{\\frac{q}{p}}})=\\sum_{(j,k,\\ell)\\in \\Delta_p}\n (-1)^{\\lfloor {jq}/{p}\\rfloor +\\lfloor {kq}/{p}\\rfloor+\\lfloor {\\ell q}/{p}\\rfloor}\n\\end{equation}\n where the sum runs over the set $\\Delta_p$ of triples of integers satisfying \n$$\\begin{cases}0<j,k,\\ell<p, \\\\\nj<k+\\ell,k<j+\\ell,\\ell<j+k,\\\\\nj+k+\\ell<2p,\\textrm{ and } j+k+\\ell\\textrm{ odd}.\n\\end{cases}$$\n\nOur main result is the following", "context": "Let $p$ be an odd positive integer and $\\mathbb{Q}(\\zeta)$ be the cyclotomic field generated by a primitive root of unity $\\zeta$ of order $2p$. The SU$_2$-TQFT constructs for any surface $S_g$ of genus $g$ a finite dimensional $\\mathbb{Q}(\\zeta)$-vector space $\\boV_p(S_g)$ endowed with an Hermitian form $h_g$ defined over $\\mathbb{Q}(\\zeta)$. These vector spaces are part of a modular functor, meaning that they support a projective unitary action of the corresponding mapping class groups and satisfy a list of compatibilities when cutting the surfaces along simple closed curves.\n\nThe dimension of $\\boV_p(S_g)$, given by the celebrated Verlinde formula, has received considerable attention since the nineties. On the other hand, taking $0<q<p$ two coprime odd integers, one can embed $\\mathbb{Q}(\\zeta)$ into $\\mathbb{C}$ by sending $\\zeta$ to $e^{i\\pi q/p}$. The Hermitian form $h_g$ has a signature that we denote by \n$$\\sigma_g(\\textstyle{\\frac{q}{p}})=\\Sign\\Big(\\boV_p(S_g)\\!\\!\\underset{\\zeta=e^{i\\pi q/p}}{\\otimes}\\!\\!\\mathbb{C},\\,h_g\\Big)$$\nand which has been very little studied. It was shown in \\cite{DM} that for a fixed $\\zeta=e^{i\\pi q/p}$, these signatures have cut and paste properties very similar to those of the dimensions, which allows to encode them in a single Frobenius algebra $V_{\\frac q p}$. Later it was proved in \\cite{S2B} that $V_{\\frac q p}$ is closely related to the hyperbolic geometry of the two-bridge knot $K(p,q)$. The purpose of this paper is to study signatures in a different direction, by analyzing the asymptotic behavior of $\\sigma_{g}(\\frac{q}{p})$ when $\\frac{q}{p}\\to \\theta$ for a fixed $\\theta\\in [0,1]$.\n\nIf we take $q=1$, the Hermitian form $h_g$ becomes positive definite,\nso that $\\sigma_g(\\frac 1 p)$ coincides with the dimension of $\\boV_p(S_g)$. The problem of computing the asymptotic behavior of the dimensions when $p\\to \\infty$ is well-known, and related to considerations of semi-classical analysis on character varieties $\\mathcal{M}_g=\\Hom(\\pi_1(S_g),\\SU_2)/\\SU_2$. This led Witten \\cite{Witten} to show that for $g>1$, \n$$\\lim_{p\\to \\infty}\\frac{\\dim \\boV_p(S_g)}{p^{3g-3}}=\\operatorname{Vol}(\\mathcal{M}_g)=2(2\\pi^2)^{1-g}\\zeta(2g-2).$$\n\nIn genus $0$ and $1$, the hermitian form is positive definite for all values of $q$, so let us consider the first non trivial case $g=2$. In this case, one can write a simple formula for $\\sigma_2(\\frac q p)$:\n\\begin{equation}\\label{defsigma2}\n \\sigma_2({\\textstyle{\\frac{q}{p}}})=\\sum_{(j,k,\\ell)\\in \\Delta_p}\n (-1)^{\\lfloor {jq}/{p}\\rfloor +\\lfloor {kq}/{p}\\rfloor+\\lfloor {\\ell q}/{p}\\rfloor}\n\\end{equation}\n where the sum runs over the set $\\Delta_p$ of triples of integers satisfying \n$$\\begin{cases}0<j,k,\\ell<p, \\\\\nj<k+\\ell,k<j+\\ell,\\ell<j+k,\\\\\nj+k+\\ell<2p,\\textrm{ and } j+k+\\ell\\textrm{ odd}.\n\\end{cases}$$\n\nOur main result is the following", "full_context": "Let $p$ be an odd positive integer and $\\mathbb{Q}(\\zeta)$ be the cyclotomic field generated by a primitive root of unity $\\zeta$ of order $2p$. The SU$_2$-TQFT constructs for any surface $S_g$ of genus $g$ a finite dimensional $\\mathbb{Q}(\\zeta)$-vector space $\\boV_p(S_g)$ endowed with an Hermitian form $h_g$ defined over $\\mathbb{Q}(\\zeta)$. These vector spaces are part of a modular functor, meaning that they support a projective unitary action of the corresponding mapping class groups and satisfy a list of compatibilities when cutting the surfaces along simple closed curves.\n\nThe dimension of $\\boV_p(S_g)$, given by the celebrated Verlinde formula, has received considerable attention since the nineties. On the other hand, taking $0<q<p$ two coprime odd integers, one can embed $\\mathbb{Q}(\\zeta)$ into $\\mathbb{C}$ by sending $\\zeta$ to $e^{i\\pi q/p}$. The Hermitian form $h_g$ has a signature that we denote by \n$$\\sigma_g(\\textstyle{\\frac{q}{p}})=\\Sign\\Big(\\boV_p(S_g)\\!\\!\\underset{\\zeta=e^{i\\pi q/p}}{\\otimes}\\!\\!\\mathbb{C},\\,h_g\\Big)$$\nand which has been very little studied. It was shown in \\cite{DM} that for a fixed $\\zeta=e^{i\\pi q/p}$, these signatures have cut and paste properties very similar to those of the dimensions, which allows to encode them in a single Frobenius algebra $V_{\\frac q p}$. Later it was proved in \\cite{S2B} that $V_{\\frac q p}$ is closely related to the hyperbolic geometry of the two-bridge knot $K(p,q)$. The purpose of this paper is to study signatures in a different direction, by analyzing the asymptotic behavior of $\\sigma_{g}(\\frac{q}{p})$ when $\\frac{q}{p}\\to \\theta$ for a fixed $\\theta\\in [0,1]$.\n\nIf we take $q=1$, the Hermitian form $h_g$ becomes positive definite,\nso that $\\sigma_g(\\frac 1 p)$ coincides with the dimension of $\\boV_p(S_g)$. The problem of computing the asymptotic behavior of the dimensions when $p\\to \\infty$ is well-known, and related to considerations of semi-classical analysis on character varieties $\\mathcal{M}_g=\\Hom(\\pi_1(S_g),\\SU_2)/\\SU_2$. This led Witten \\cite{Witten} to show that for $g>1$, \n$$\\lim_{p\\to \\infty}\\frac{\\dim \\boV_p(S_g)}{p^{3g-3}}=\\operatorname{Vol}(\\mathcal{M}_g)=2(2\\pi^2)^{1-g}\\zeta(2g-2).$$\n\nIn genus $0$ and $1$, the hermitian form is positive definite for all values of $q$, so let us consider the first non trivial case $g=2$. In this case, one can write a simple formula for $\\sigma_2(\\frac q p)$:\n\\begin{equation}\\label{defsigma2}\n \\sigma_2({\\textstyle{\\frac{q}{p}}})=\\sum_{(j,k,\\ell)\\in \\Delta_p}\n (-1)^{\\lfloor {jq}/{p}\\rfloor +\\lfloor {kq}/{p}\\rfloor+\\lfloor {\\ell q}/{p}\\rfloor}\n\\end{equation}\n where the sum runs over the set $\\Delta_p$ of triples of integers satisfying \n$$\\begin{cases}0<j,k,\\ell<p, \\\\\nj<k+\\ell,k<j+\\ell,\\ell<j+k,\\\\\nj+k+\\ell<2p,\\textrm{ and } j+k+\\ell\\textrm{ odd}.\n\\end{cases}$$\n\nOur main result is the following\n\nOur main result is the following\n\nFigure \\ref{fig:sigma2} suggests that the convergence is rapid: the graph of the map on the right hand side is shown in green whereas red dots represent the points $(q/p,\\sigma_2(q/p)/p^2)$ for $0<q<p<31$.\n\nWe briefly review the properties of the convergents $q_k/p_k$ of $\\theta$ that we will need. See {\\em e.g.} Perron \\cite{P} or Khinchin \\cite{Kh1} for more details\\footnote{Unfortunately, \\cite{Kh1} writes $p_k$ for our $q_k$ and {\\em vice versa.}}. Each irrational $\\theta\\in {\\mathbb R}$ has a unique regular continued fraction expansion $$\\theta = [a_0; a_1, a_2, \\ldots ] = a_0+\\cfrac{1}{a_1+\\cfrac{1}{a_2+\\dots}} $$ where regular means that $a_0\\in \\Z$ and $a_i\\in \\Z_{\\geq 1}$ for $i\\geq 1$. Note that in our case $a_0=0$ since\n$\\theta \\in [0,1]\\setminus \\Q$.\nThe convergents\n$$\\frac{q_k}{p_k} = [a_0; a_1, a_2, \\ldots, a_k] = a_0+\\cfrac{1}{a_1+\\cfrac{1}{\\dots+ \\frac{1}{a_k}}} $$ can also be defined recursively by\n\\begin{eqnarray} q_k&=a_k q_{k-1} + q_{k-2}, \\ \\ \\ \\ q_0=a_0, \\ \\ \\ \\ q_{-1}=1 \\label{rec5}\\\\\n p_k&=a_k p_{k-1} + p_{k-2}, \\ \\ \\ \\ p_0=1, \\ \\ \\ \\ p_{-1}=0 \\label{rec6}\n\\end{eqnarray}\nThe following classical facts are well-known. \n\\begin{Proposition}\\label{qpfacts}\n For all irrational $\\theta$, the convergents $q_k/p_k$ of $\\theta$ satisfy the following properties:\n \\begin{enumerate} \n \\item[(i)] One has $1=p_0\\leq p_1<p_2<\\ldots$, {\\em i.e.} the denominators $p_k$ are positive integers and moreover strictly increasing from $p_1$ onwards.\n \\item[(ii)] For all $k\\geq 1$, $q_k$ and $p_k$ are coprime.\n \\item[(iii)] $\\lim_{k\\rightarrow\\infty} q_k/p_k = \\theta$.\n \\item[(iv)]\n For all $i\\geq 1$, we have\n \\begin{equation}\\nonumber \n \\frac 1 {p_i(p_i+p_{i+1})} <\n \\Big|\\theta - \\frac {q_i}{p_i} \\Big| < \\frac 1 {p_ip_{i+1}} < \\frac 1 {p_i^2}~.\n \\end{equation}\n \\item[(v)] {\\em [Lagrange]}\\footnote{For a proof, see Perron \\cite[\\S 15, Satz 2.17]{P}.} For all $1\\leq i<k$, and any $n,\\ell \\in\\Z$ with $1\\leq n<p_{i+1}$, we have\n \\begin{equation}\\nonumber \n \\Big| n\\frac {q_k}{p_k} -\\ell \\Big|\n \\geq \\Big|p_i\\frac {q_k}{p_k} - q_i\\Big| \\geq \\frac{1}{p_i + p_{i+1}} ~.\n \\end{equation}\n \\end{enumerate}\n \\end{Proposition}\n\n\\begin{proof}[\\bf{Proof of Theorem~\\ref{asympto}}] We start from the trigonometric formula for $\\sigma_2(\\frac{q}{p})$ given in Theorem~\\ref{sigma2}. The first observation is that the term $(1-p^2)/(6p^2)$ in this formula stays bounded as $p\\rightarrow \\infty$ and so does not contribute to the limit of $\\sigma_2(\\frac{q_k}{p_k})/ p_k^2$. Thus it suffices to show\n \\begin{equation}\\label{eq4}\n \\lim_{k\\rightarrow \\infty} \\frac{1}{4p_k^4} \\sum_{n=1,\\textrm{ odd}}^{p_k-2}\\frac{f(n,p_k,q_k)}{\\sin^3({n\\pi}/{2p_k})\\sin^2({q_kn\\pi}/{p_k})} =\\frac{16}{\\pi^3}\\sum_{n\\ge 1, \\textrm{odd}}\\frac{1}{n^3\\sin(n\\pi\\theta)}\n \\end{equation} with $f(n,p,q)$ given as in Theorem~\\ref{sigma2}.\n\nLet $S^{(1)}(q_k,p_k,\\theta)$ denote the sum $S^{(1)}(q_k,p_k)$ but with $\\sin(n\\pi\\theta)$ in the denominator in place of $\\sin({q_kn\\pi}/{p_k})$. Using (\\ref{sin-ineq}), we see that the $n$-th summand of $S^{(1)}(q_k,p_k,\\theta)$ is bounded from above by $1/(n^3\\sin(n\\pi\\theta))$. So we can apply the Lebesgue dominated convergence theorem to this sum. We get\n $$\\lim_{k\\rightarrow \\infty}S^{(1)}(q_k,p_k,\\theta) =\\frac{8}{\\pi^3}\\sum_{n\\ge 1, \\textrm{odd}}\\frac{1}{n^3\\sin(n\\pi\\theta)}~.$$\n Thus to prove Lemma~\\ref{3.3} it suffices to show that the difference $d_k=S^{(1)}(q_k,p_k) -S^{(1)}(q_k,p_k,\\theta)$ goes to zero as $k\\rightarrow \\infty$. We have \n $$d_k= \\frac{1}{p_k^3} \\sum_{n=1,\\textrm{ odd}}^{p_k-2}\\frac{\\cos (n\\pi/2p_k)}{\\sin^3({n\\pi}/{2p_k}) }\\Biggl (\\frac {1} {\\sin({q_kn\\pi}/{p_k})} - \\frac {1} {\\sin(n\\pi\\theta)}\\Biggr)~.$$ We can bound the denominators from below using (\\ref{sin-ineq}), (\\ref{sinn-inequ}), and the trivial lower bound $|\\sin({q_kn\\pi}/{p_k})| \\geq \\sin (\\pi/p_k) \\geq 2/p_k$. Thus for all $\\epsilon >0$ we have\n\n\\begin{proof}[\\bf{Proof of Lemma~\\ref{3.4}}]\n We assume that $\\mu(\\theta)=2$ and that $\\theta$ satisfies the conditions\n in Theorem~\\ref{ThKL}.\n Recall $S^{(2)}(q_k,p_k)$, defined in (\\ref{sum2}), is a sum from $n=1$ to $n=p_k-2$. We cut\nit into two parts, as follows. Put\n\\begin{equation}\\label{floor} N(k)= \\lfloor 3k/4 \\rfloor\n\\end{equation} and write $$S^{(2)}(q_k,p_k) = S^{(2.1)}(q_k,p_k)\n+S^{(2.2)}(q_k,p_k)~,$$ where $S^{(2.1)}(q_k,p_k)$ is the sum from\n$n=1$ to $n=p_{N(k)+1}-1$ and $S^{(2.2)}(q_k,p_k)$ is the sum from\n$n=p_{N(k)+1}$ to $p_k-2$. The following two lemmas provide upper bounds for these two sums.\n\\begin{Lemma}\\label{3.9} We have $$|S^{(2.1)}(q_k,p_k) |\\leq \\frac\n{p_{N(k)}} {p_k} F_7(k)$$ where $F_7(k) ={\\mathcal O}(k^7)$.\n\\end{Lemma}\n\\begin{Lemma}\\label{3.10} We have $$|S^{(2.2)}(q_k,p_k)| \\leq \\frac{1}{2} \\zeta(2)\n\\frac {p_{k}} {p_{N(k)+1}^2}~.$$\n\\end{Lemma}\nAssuming these two lemmas for the moment, since $\\lim_{m\\rightarrow\\infty} p_m^{1/m}= L>1$ by the Khinchin-L{\\'e}vy Theorem~\\ref{ThKL}(ii), $p_k$ grows like $L^k$ as $k\\rightarrow \\infty$. It is not hard to deduce from this that both $ S^{(2.1)}(q_k,p_k)$ and $ S^{(2.2)}(q_k,p_k)$ converge to zero.~\\footnote{Here is how to see this for $ S^{(2.1)}(q_k,p_k)$. Theorem~\\ref{ThKL}(ii) gives that for all $\\epsilon>0$ there exist $m_0(\\epsilon)$ so that $L^{1-\\epsilon} \\leq p_m^{1/m} \\leq L^{1+\\epsilon}$ holds for for all $m\\geq m_0(\\epsilon)$. Thus also $L^{m-m\\epsilon} \\leq p_m\\leq L^{m+m\\epsilon}$ for all $m\\geq m_0(\\epsilon)$. Applying now the first inequality with $m=k$ and the second one with $m=N(k)=\\lfloor 3k/4\\rfloor$ and taking $\\epsilon$ small enough, it follows that $p_{N(k)}/p_k \\leq L^{-\\delta k}$ for some $\\delta>0$ as soon as $k\\geq 2 m_0(\\epsilon)$. Thus $p_{N(k)} F_7(k)/ {p_k} \\leq k^7 L^{-\\delta k}$ for $k$ big enough, proving that the limit as $k\\rightarrow\\infty$ is zero, as asserted. }\nThis proves Lemma~\\ref{3.4}. \n\\end{proof}\nIt remains to prove Lemmas~\\ref{3.9} and~\\ref{3.10}. Let us first deal\nwith Lemma~\\ref{3.10} whose proof is easy.\n\\begin{proof}[\\bf{Proof of Lemma~\\ref{3.10}}] Since $n\\geq p_{N(k)+1}$\n in this sum,\n we can bound the\n$\\sin^2(n\\pi/2p_k)$ term in the denominator from below by\n$p_{N(k)+1}^2/p_k^2$. Thus\n\\begin{equation}\\label{sum3}|S^{(2.2)}(q_k,p_k)| \\leq \\frac 1 {p_k\np_{N(k)+1}^2}\\sum_{n=1,\\textrm{ odd}}^{p_k-2} \\frac 1\n{\\sin^2(q_kn\\pi/p_k)}~.\\end{equation}\n\n\\section{The higher genus case}\\label{sec5}\nIn this section, we briefly discuss how the asymptotic behaviour of the genus $2$ signature might generalize to higher genus. Numerical experiments (see below) indicate that\n $\\sigma_{g,n}(\\frac{q}{p};\\lambda)$ might grow like $p^{\\max\\{g,2g-2\\}}$ when $\\frac q p$ goes to an irrational $\\theta\\in[0,1]$. So we ask the following question.\n\\begin{Question}\\label{51}\n Given $g,n\\in \\Z_{\\geq 0}$ and $n$-tuples of integers $\\lambda=(\\lambda_1,\\ldots,\\lambda_n)\\in \\Z^n_{\\geq 0}$,\n is there a function $F_{g;\\lambda}:\n [0,1]\n \\to \\R$ defined almost everywhere satisfying the following: \n For almost all irrational $\\theta\\in[0,1]$,\n if we write $\\frac{q_k}{p_k}$ for the convergents of $\\theta$, one has \n$$ \\lim_{k\\to\\infty}\\frac{\\sigma_{g,n}(\\frac{q_k}{p_k};\\lambda)}{p_k^{\\max\\{g,2g-2\\}}}=F_{g;\\lambda}(\\theta)$$\n\\end{Question}\nTheorem \\ref{asympto} shows that the answer is yes for $g=2$ and $n=0$. Let us\ncompare this to the asymptotics of the dimension\nof the TQFT vector spaces\nwhich \nis well-known: it has a semi-classical interpretation for which we refer to \\cite[Section 3]{Witten}. Precisely:\n$$ \\lim_{p\\to\\infty}\\frac{\\dim \\boV_p(S_g)}{p^{\\max\\{g,3g-3\\}}}=\\operatorname{Vol}(\\mathcal{M}_g)=2(2\\pi^2)^{1-g}\\zeta(2g-2)$$\nwhere the last equality holds only for $g>0$ and $\\mathcal{M}_g$ is the character variety of representations of $\\pi_1(S_g)$ into SU$_2$, endowed with its Liouville measure. We have no conceptual explanation why the order of growth $3g-3$\nfor the dimension\nshould be replaced by $2g-2$\nfor the signature.", "post_theorem_intro_text_len": 4698, "post_theorem_intro_text": "Figure \\ref{fig:sigma2}  suggests that the convergence is rapid: the graph of the map on the right hand side is shown in green whereas red dots represent the points $(q/p,\\sigma_2(q/p)/p^2)$ for $0<q<p<31$. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=10cm]{genre2.png}\n\\caption{Graph of the signature function $\\sigma_2$}\\label{fig:sigma2}\n\\end{center}\n\\end{figure}\n\nThe proof of this theorem involves several steps and occupies Sections~\\ref{genre2} and~\\ref{sec2}. We first apply in Section~\\ref{genre2} a finite Fourier transform in Equation \\eqref{defsigma2} to transform this expression into a sum of evaluations of a rational function at roots of unity indexed by $\\Delta_p$. We then apply Brion's theorem to get an explicit trigonometric sum computing $\\sigma_2(q/p)$ (see Theorem \\ref{sigma2} for the precise statement): this step already presents miraculous cancellations for which we do not have any conceptual explanation. \n\nThen in Section~\\ref{sec2}, we analyse the asymptotic behavior of the trigonometric sum in the most technical part of the paper. In order to control its denominators, we need to impose that the coefficients of the continued fraction expansion of $\\theta$ do not grow too quickly. We use for that  some classical results of Khinchin on metric Diophantine approximation and a variation on a technique due to Zaremba.\n\nIn higher genus one can also write down a formula for\n  $\\sigma_g(q/p)$ as the sum of signs indexed by lattice points inside some polytope (see \\cite[Remark 4.12]{BHMV}~\\footnote{Our $\\boV_p$ corresponds to the $V_{2p}$ of \\cite{BHMV}.}.) But the signs are much more complicated than in Formula \\eqref{defsigma2} which prevents us from generalizing our theorem. We briefly discuss in Section~\\ref{sec5} how the asymptotics might look like for general surfaces including marked points, and illustrate this by some numerical experiments.\n\nIn the last section of the paper, we discuss some modular properties of the signature. We observe that the function \n$$\\Lambda(\\theta)=\\frac{16}{\\pi^3}\\sum_{n\\ge 1\\textrm{ odd}}\\frac{1}{n^3\\sin(n\\pi\\theta)}$$\nis the boundary value of an Eichler integral of a modular form of weight $4$ for the level $2$ group $\\Gamma(2)$. In addition to the obvious property $\\Lambda(\\theta+1)=-\\Lambda(\\theta)$, it satisfies the following transformation law:\n\\begin{equation}\\label{Lambdamod}\n\\Lambda\\Big(\\frac{\\theta}{2\\theta+1}\\Big)(2\\theta+1)^2-\\Lambda(\\theta)=2\\theta^2+2\\theta+1.\n\\end{equation}\nThis functional equation suggests that a similar integral version holds for signatures as stated in the following conjecture, checked by computer for all $0<q<p<100$. \n\\begin{Conjecturestar}\nFor any odd coprime integers $0<q<p$ one has\n$$\\sigma_2\\Big(\\frac{q}{2q+p}\\Big)-\\sigma_2\\Big(\\frac{q}{p}\\Big)=2q^2+2qp+p^2-1.$$\n\\end{Conjecturestar}\n\nThis formula is reminiscent of the reciprocity formula of Dedekind sums: in fact we will make this analogy more precise in Section~\\ref{SectionS}. We prove that the signature at $\\zeta=e^{i\\pi q/p}$ of  the TQFT-vector space associated to the 4 times punctured sphere (with  colors $(p-1)/2$  at the marked points)  is actually equal to a 2-smoothed version, $S(q/p)$,  of the Dedekind sum $s(q,p)$. It is then classical that  this function $S(q/p)$ is the boundary value of a modular form (this time of weight $1/2$) and that it satisfies the following transformation law\n\n$$S\\Big(\\frac{q}{2q+p}\\Big)-S\\Big(\\frac{q}{p}\\Big)=1.$$\nWe provide an elementary proof of these facts for the benefit of the non-specialist reader. \n\nThis very satisfactory state of things would hopefully generalize to higher genus as well. However, even in genus 2, a precise (non asymptotic) relation between the signature $\\sigma_g(q/p)$\nand the Eichler integral is still lacking. Such a relation would presumably lead to a proof  of the conjectured transformation law for $\\sigma_2$. \n\nTo end this introduction, we would like to mention the parallel paper \\cite{MY} in which the first author and S. Yoon compute the asymptotics of the function\n$\\sigma_g(q/p)$, but for sequences of the form $\\frac{q_n}{p_n}=\\frac{a+bn}{c+dn}$ where $\\left(\\begin{smallmatrix} a & b \\\\ c & d \\end{smallmatrix}\\right)\\in \\Gamma(2)$. This regime is somewhat orthogonal to the one described in the present paper. The techniques used and the results are also very different. A synthesis of the two papers has yet to be made.\n\\vskip 8pt\n\n{\\bf Acknowledgements:} It is our pleasure to thank Pierre Charollois for early discussions on many aspects of this paper, Gaëtan Chenevier for his expertise on modular forms, \nand Don Zagier for very helpful explanations about Eichler integrals and quantum modular forms.", "sketch": "The proof “involves several steps and occupies Sections~\\ref{genre2} and~\\ref{sec2}.” First, “we apply in Section~\\ref{genre2} a finite Fourier transform in Equation \\eqref{defsigma2} to transform this expression into a sum of evaluations of a rational function at roots of unity indexed by $\\Delta_p$.” Next, “we then apply Brion's theorem to get an explicit trigonometric sum computing $\\sigma_2(q/p)$ (see Theorem \\ref{sigma2} for the precise statement),” a step that “already presents miraculous cancellations.” Then, “in Section~\\ref{sec2}, we analyse the asymptotic behavior of the trigonometric sum in the most technical part of the paper.” To “control its denominators,” one “need[s] to impose that the coefficients of the continued fraction expansion of $\\theta$ do not grow too quickly,” using “classical results of Khinchin on metric Diophantine approximation and a variation on a technique due to Zaremba.”", "expanded_sketch": "The proof “involves several steps and occupies” the part starting at\n\n\\begin{Remark}\n  {\\em\nThe $\\mathrm{SU}_2$-TQFT $\\boV_p$ corresponds in Conformal Field Theory to the Wess-Zumino-Witten model at odd level $\\ell=p-2$. Similar $\\mathrm{SU}_2$-TQFTs exist at  even levels, that is, when $p$ is even. Their signature functions can again be computed using the Frobenius algebra formalism of Theorem~\\ref{DM1}.  But we won't consider even level $\\mathrm{SU}_2$-TQFTs in this paper.\n}\\end{Remark}\n\n\\section{A trigonometric formula for the signature in genus 2}\\label{genre2}\nLet us now start our investigation of the asymptotic behavior of the signature function. The first result of this paper is the following trigonometric formula for the signature in genus $2$.\n\n\\begin{Theorem}\\label{sigma2}\n  For $0<q<p$ coprime odd integers, we have\n\\begin{equation}\\nonumber \\sigma_2({{\\textstyle\\frac q p}})= \\frac {1-p^2}{6p^2} +\n  \\frac 1 {4p^2}\\sum_{n=1,\\, \\mathrm{ odd}}^{p-2}\\frac{f(n,p,q)}{\\sin^3(\\frac{n\\pi}{2p})\\sin^2(\\frac{qn\\pi}{p})}\n\\end{equation}\nwhere \n\\begin{eqnarray*} f(n,p,q)&=(3p-3)\\sin(\\frac{(2q-1)n\\pi}{2p})+(p+1)\\sin(\\frac{(2q-3)n\\pi}{2p})\\\\\n&+(p-1)\\sin(\\frac{(2q+3)n\\pi}{2p})+(3p+3)\\sin(\\frac{(2q+1)n\\pi}{2p}).\n\\end{eqnarray*}\n\\end{Theorem}\n\nand the later section \\ref{sec2}.\n\nFirst, “we apply” a finite Fourier transform in the defining expression\n\\begin{equation}\\label{defsigma2}\n  \\sigma_2({\\textstyle{\\frac{q}{p}}})=\\sum_{(j,k,\\ell)\\in \\Delta_p}\n (-1)^{\\lfloor {jq}/{p}\\rfloor +\\lfloor {kq}/{p}\\rfloor+\\lfloor {\\ell q}/{p}\\rfloor}\n\\end{equation}\nto transform this expression into a sum of evaluations of a rational function at roots of unity indexed by $\\Delta_p$.\n\nNext, “we then apply Brion's theorem to get an explicit trigonometric sum computing $\\sigma_2(q/p)$,” namely the formula stated in\n\\begin{Theorem}\\label{sigma2}\n  For $0<q<p$ coprime odd integers, we have\n\\begin{equation}\\nonumber \\sigma_2({{\\textstyle\\frac q p}})= \\frac {1-p^2}{6p^2} +\n  \\frac 1 {4p^2}\\sum_{n=1,\\, \\mathrm{ odd}}^{p-2}\\frac{f(n,p,q)}{\\sin^3(\\frac{n\\pi}{2p})\\sin^2(\\frac{qn\\pi}{p})}\n\\end{equation}\nwhere \n\\begin{eqnarray*} f(n,p,q)&=(3p-3)\\sin(\\frac{(2q-1)n\\pi}{2p})+(p+1)\\sin(\\frac{(2q-3)n\\pi}{2p})\\\\\n&+(p-1)\\sin(\\frac{(2q+3)n\\pi}{2p})+(3p+3)\\sin(\\frac{(2q+1)n\\pi}{2p}).\n\\end{eqnarray*}\n\\end{Theorem}\n\na step that “already presents miraculous cancellations.” Then, “we analyse the asymptotic behavior of the trigonometric sum in the most technical part of the paper.” To “control its denominators,” one “need[s] to impose that the coefficients of the continued fraction expansion of $\\theta$ do not grow too quickly,” using “classical results of Khinchin on metric Diophantine approximation and a variation on a technique due to Zaremba.”", "expanded_theorem": "For almost all irrational $\\theta\\in [0,1]$, if we denote by $q_k/p_k$ the sequence of convergents of the continued fraction expansion of $\\theta$, then one has\n$$\\lim_{k\\to \\infty}\\frac{\\sigma_2(q_k/p_k)}{p_k^2}=\\frac{16}{\\pi^3}\\sum_{n\\ge 1 \\textrm{ odd}}\\frac{1}{n^3\\sin(n\\pi\\theta)}.$$", "theorem_type": ["Asymptotic or Limit", "Universal"], "mcq": {"question": "For a reduced rational number \\(q/p\\in(0,1)\\), define\n\\[\n\\sigma_2\\!\\left(\\frac qp\\right)=\\sum_{(a,b,c)\\in \\Delta_p}(-1)^{\\lfloor aq/p\\rfloor+\\lfloor bq/p\\rfloor+\\lfloor cq/p\\rfloor},\n\\]\nwhere \\(\\Delta_p\\) is the set of triples \\((a,b,c)\\) of integers such that\n\\[\n0<a,b,c<p,\\qquad a<b+c,\\quad b<a+c,\\quad c<a+b,\n\\]\n\\[\na+b+c<2p,\\qquad a+b+c\\text{ is odd}.\n\\]\nNow let \\(\\theta\\in[0,1]\\) be irrational, and let \\(q_k/p_k\\) be the convergents of the regular continued-fraction expansion of \\(\\theta\\). Which statement holds for Lebesgue-almost every irrational \\(\\theta\\in[0,1]\\)?", "correct_choice": {"label": "A", "text": "The limit exists and is given by\n\\[\n\\lim_{k\\to\\infty}\\frac{\\sigma_2(q_k/p_k)}{p_k^2}\n=\\frac{16}{\\pi^3}\\sum_{n\\ge 1\\,\\mathrm{odd}}\\frac{1}{n^3\\sin(n\\pi\\theta)}.\n\\]"}, "choices": [{"label": "B", "text": "The limit exists and is given by\n\\[\n\\lim_{k\\to\\infty}\\frac{\\sigma_2(q_k/p_k)}{p_k^2}\n=\\frac{16}{\\pi^3}\\sum_{n\\ge 1}\\frac{1}{n^3\\sin(n\\pi\\theta)}.\n\\]"}, {"label": "C", "text": "For Lebesgue-almost every irrational \\(\\theta\\in[0,1]\\), the limit\n\\[\n\\lim_{k\\to\\infty}\\frac{\\sigma_2(q_k/p_k)}{p_k^2}\n\\]\nexists."}, {"label": "D", "text": "The limit exists and is given by\n\\[\n\\lim_{k\\to\\infty}\\frac{\\sigma_2(q_k/p_k)}{p_k^2}\n=\\frac{16}{\\pi^3}\\sum_{n\\ge 1\\,\\mathrm{odd}}\\frac{1}{n^3\\,|\\sin(n\\pi\\theta)|}.\n\\]"}, {"label": "E", "text": "For every irrational \\(\\theta\\in[0,1]\\), if \\(q_k/p_k\\) are the convergents of its regular continued-fraction expansion, then\n\\[\n\\lim_{k\\to\\infty}\\frac{\\sigma_2(q_k/p_k)}{p_k^2}\n=\\frac{16}{\\pi^3}\\sum_{n\\ge 1\\,\\mathrm{odd}}\\frac{1}{n^3\\sin(n\\pi\\theta)}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "odd-index restriction in the trigonometric expansion", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "explicit identification of the limiting value by the odd-series formula", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "signed denominator \\(\\sin(n\\pi\\theta)\\) replaced by absolute value, destroying the exact Fourier-cancellation structure", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "almost-everywhere metric Diophantine hypothesis strengthened to all irrationals", "template_used": "stronger_trap"}]}}
{"id": "2512.13952v1", "paper_link": "http://arxiv.org/abs/2512.13952v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{dimension bound theorem}\n    Let \\(M\\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \\(k\\), \\(\\ell\\) be nonnegative integers. Then\n    \\begin{equation}\\label{equation in the first statement of the main theorem}\n        \\dim \\mathcal{P}_{4k, 4\\ell}(M) \\le \\begin{cases}\n             \\displaystyle \\sum_{i = 0}^{k} \\dim \\mathcal{H}_{4(k - i), 4(\\ell - i)}(M) & \\quad k \\le \\ell + 1, \\\\ \\displaystyle\n            1 + \\sum_{i = 0}^{\\ell} \\dim \\mathcal{H}_{4(k - i), 4(\\ell - i)}(M) & \\quad k > \\ell + 1\n        \\end{cases}\n    \\end{equation}\n    Moreover, these inequalities are sharp in \\(\\mathbb{R}^n\\).", "start_pos": 7841, "end_pos": 8549, "label": "dimension bound theorem"}, "ref_dict": {"definition of polynomially bounded growth in introduction": "\\begin{equation}\\label{definition of polynomially bounded growth in introduction}\n    \\sup_{B_R(p) \\times [-R^4, 0]}|u(x, t)| \\le C(1 + R)^d, \\quad \\sup_{B_R(p) \\times [-R^4, 0]}|\\nabla u(x, t)| \\le C'(1 + R)^{d'}\n\\end{equation}", "dimension bound theorem": "\\begin{theorem}\\label{dimension bound theorem}\n    Let \\(M\\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \\(k\\), \\(\\ell\\) be nonnegative integers. Then\n    \\begin{equation}\\label{equation in the first statement of the main theorem}\n        \\dim \\mathcal{P}_{4k, 4\\ell}(M) \\le \\begin{cases}\n             \\displaystyle \\sum_{i = 0}^{k} \\dim \\mathcal{H}_{4(k - i), 4(\\ell - i)}(M) & \\quad k \\le \\ell + 1, \\\\ \\displaystyle\n            1 + \\sum_{i = 0}^{\\ell} \\dim \\mathcal{H}_{4(k - i), 4(\\ell - i)}(M) & \\quad k > \\ell + 1\n        \\end{cases}\n    \\end{equation}\n    Moreover, these inequalities are sharp in \\(\\mathbb{R}^n\\).\n\\end{theorem}"}, "pre_theorem_intro_text_len": 4953, "pre_theorem_intro_text": "The relationship between the geometry of manifolds and the analytic properties of functions on manifolds is a defining theme of geometric analysis. Our direction starts with the Liouville theorems for harmonic functions on \\(\\mathbb{R}^n\\) and Yau's generalization.\n\nYau proved in \\cite{yau1975harmonic} that a bounded harmonic function on a complete manifold with nonnegative Ricci curvature is a constant. In 1974, he conjectured that a more general result should hold: on a complete manifold \\(M\\) with nonnegative Ricci curvature, the space \\(\\mathcal{H}_d(M)\\) of harmonic functions with polynomially bounded growth should have finite dimension. Colding and Minicozzi proved his conjecture in \\cite{colding1997harmonic}. \n\nA natural generalization is to try to show this result for solutions of the heat equation. However, the heat equation is very flexible compared to the Laplace equation, and since there are bounded solutions to the heat equation, no Liouville theorem is possible in general.\n\nDespite this, if we restrict attention to specifically \\textit{ancient} solutions of the heat equation, that is, solutions which are defined for all time going back to \\(-\\infty\\), then Liouville theorems actually do become possible. Indeed, in \\cite{colding2021optimal}, Colding and Minicozzi generalize \\cite{colding1997harmonic} to show that the space \\(\\mathcal{P}_d(M)\\) of ancient solutions of the heat equation with polynomially bounded growth also has finite dimension. In (\\cite{colding2020complexity}, \\cite{colding2019search}, \\cite{colding2019liouville}), Colding and Minicozzi show how the spaces \\(\\mathcal{P}_d(M)\\) are relevant to geometric flows.\n\nContinuing to more types of equations, Wang and Zhu recently generalized the result of \\cite{colding1997harmonic} to biharmonic functions \\cite{wang2025qualitativebehaviorbiharmonicfunctions}, i.e., functions \\(u: M \\rightarrow \\mathbb{R}\\) solving\n\\begin{equation*}\n    \\Delta\\Delta u = 0.\n\\end{equation*}\n\nThis equation is also more flexible than the Laplace equation (indeed, any harmonic function is biharmonic), and we cannot prove as general a Liouville theorem as for harmonic functions. To find a Liouville theorem, rather than restricting attention to a subclass of biharmonic functions as in \\cite{colding2021optimal}, Wang and Zhu instead restrict attention to a subclass of manifolds with polynomial volume growth and Ricci curvature bounded below at infinity.\n\nOur goal in this paper is to generalize Wang and Zhu's result to ancient solutions of the biharmonic heat equation, following the strategy of Colding and Minicozzi in \\cite{colding2021optimal}. Our main result is\n\\begin{theorem*}\n    Let \\(M\\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \\(u: M \\times (-\\infty, 0] \\rightarrow \\mathbb{R}\\) be an ancient solution of\n    \\begin{equation*}\n        \\partial_t u(x, t) + \\Delta\\Delta u(x, t) = 0\n    \\end{equation*}\n    such that \\(|u(x, t)|\\) and \\(|\\nabla u(x, t)|\\) have polynomially bounded growth in the heat balls \\(B_R(x) \\times [-R^4, 0]\\). The space of all such solutions \\(u(x, t)\\) is finite dimensional.\n\\end{theorem*}\n\n\\subsection{Definitions and Notation}\nWe now give more precise definitions and statements. Given a manifold \\(M\\) and an interval \\(I \\subset \\mathbb{R}\\), a function \\(u: M \\times I \\rightarrow \\mathbb{R}\\) satisfies the biharmonic heat equation if\n\\begin{equation}\\label{first instance of biharmonic heat equation}\n    \\partial_t u(x, t) + \\Delta\\Delta u(x, t) = 0.\n\\end{equation}\nWe will call such a function ``bicaloric'' for brevity. A bicaloric function \\(u\\) is ancient if it can be defined on an interval extending infinitely backwards in time, i.e. for \\(t \\in (-\\infty, 0]\\). We say that \\(u \\in \\mathcal{P}_{d, d'}(M)\\) for \\(d, d' > 0\\) if \\(\\partial_t u + \\Delta\\Delta u = 0\\), \\(u\\) is ancient, and for some constants \\(C, C' > 0\\),\n\\begin{equation}\\label{definition of polynomially bounded growth in introduction}\n    \\sup_{B_R(p) \\times [-R^4, 0]}|u(x, t)| \\le C(1 + R)^d, \\quad \\sup_{B_R(p) \\times [-R^4, 0]}|\\nabla u(x, t)| \\le C'(1 + R)^{d'}\n\\end{equation}\nfor any \\(p \\in M\\) and \\(R > 0\\). We similarly say that \\(u \\in \\mathcal{H}_{d, d'}(M)\\) if \\(\\Delta\\Delta u = 0\\) and the same bounds in (\\ref{definition of polynomially bounded growth in introduction}) hold, where we take the supremum over only the ball \\(B_R(p)\\).\n\nA manifold \\(M\\) is said to have polynomial volume growth if there are constants \\(C, d_V > 0\\) and some \\(p \\in M\\) such that \\(\\operatorname{Vol}(B_R(p)) \\le C(1 + R)^{d_V}\\) for all \\(R> 0\\). Furthermore, we say that the Ricci curvature tensor is bounded below quadratically with constant \\(K\\) if for some \\(p \\in M\\) and all \\(R > 0\\),\n\\begin{equation}\n    \\sup_{v \\in TB_{R}(p)}\\frac{\\Ric(v, v)}{|v|^2} \\ge -\\frac{K}{R^2}.\n\\end{equation}\n\nWith these definitions, our main results are more precisely stated as", "context": "Yau proved in \\cite{yau1975harmonic} that a bounded harmonic function on a complete manifold with nonnegative Ricci curvature is a constant. In 1974, he conjectured that a more general result should hold: on a complete manifold \\(M\\) with nonnegative Ricci curvature, the space \\(\\mathcal{H}_d(M)\\) of harmonic functions with polynomially bounded growth should have finite dimension. Colding and Minicozzi proved his conjecture in \\cite{colding1997harmonic}.\n\nThis equation is also more flexible than the Laplace equation (indeed, any harmonic function is biharmonic), and we cannot prove as general a Liouville theorem as for harmonic functions. To find a Liouville theorem, rather than restricting attention to a subclass of biharmonic functions as in \\cite{colding2021optimal}, Wang and Zhu instead restrict attention to a subclass of manifolds with polynomial volume growth and Ricci curvature bounded below at infinity.\n\nOur goal in this paper is to generalize Wang and Zhu's result to ancient solutions of the biharmonic heat equation, following the strategy of Colding and Minicozzi in \\cite{colding2021optimal}. Our main result is\n\\begin{theorem*}\n Let \\(M\\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \\(u: M \\times (-\\infty, 0] \\rightarrow \\mathbb{R}\\) be an ancient solution of\n \\begin{equation*}\n \\partial_t u(x, t) + \\Delta\\Delta u(x, t) = 0\n \\end{equation*}\n such that \\(|u(x, t)|\\) and \\(|\\nabla u(x, t)|\\) have polynomially bounded growth in the heat balls \\(B_R(x) \\times [-R^4, 0]\\). The space of all such solutions \\(u(x, t)\\) is finite dimensional.\n\\end{theorem*}\n\n\\subsection{Definitions and Notation}\nWe now give more precise definitions and statements. Given a manifold \\(M\\) and an interval \\(I \\subset \\mathbb{R}\\), a function \\(u: M \\times I \\rightarrow \\mathbb{R}\\) satisfies the biharmonic heat equation if\n\\begin{equation}\\label{first instance of biharmonic heat equation}\n \\partial_t u(x, t) + \\Delta\\Delta u(x, t) = 0.\n\\end{equation}\nWe will call such a function ``bicaloric'' for brevity. A bicaloric function \\(u\\) is ancient if it can be defined on an interval extending infinitely backwards in time, i.e. for \\(t \\in (-\\infty, 0]\\). We say that \\(u \\in \\mathcal{P}_{d, d'}(M)\\) for \\(d, d' > 0\\) if \\(\\partial_t u + \\Delta\\Delta u = 0\\), \\(u\\) is ancient, and for some constants \\(C, C' > 0\\),\n\\begin{equation}\\label{definition of polynomially bounded growth in introduction}\n \\sup_{B_R(p) \\times [-R^4, 0]}|u(x, t)| \\le C(1 + R)^d, \\quad \\sup_{B_R(p) \\times [-R^4, 0]}|\\nabla u(x, t)| \\le C'(1 + R)^{d'}\n\\end{equation}\nfor any \\(p \\in M\\) and \\(R > 0\\). We similarly say that \\(u \\in \\mathcal{H}_{d, d'}(M)\\) if \\(\\Delta\\Delta u = 0\\) and the same bounds in (\\ref{definition of polynomially bounded growth in introduction}) hold, where we take the supremum over only the ball \\(B_R(p)\\).\n\nA manifold \\(M\\) is said to have polynomial volume growth if there are constants \\(C, d_V > 0\\) and some \\(p \\in M\\) such that \\(\\operatorname{Vol}(B_R(p)) \\le C(1 + R)^{d_V}\\) for all \\(R> 0\\). Furthermore, we say that the Ricci curvature tensor is bounded below quadratically with constant \\(K\\) if for some \\(p \\in M\\) and all \\(R > 0\\),\n\\begin{equation}\n \\sup_{v \\in TB_{R}(p)}\\frac{\\Ric(v, v)}{|v|^2} \\ge -\\frac{K}{R^2}.\n\\end{equation}\n\nWith these definitions, our main results are more precisely stated as\n\n\\begin{equation}\\label{definition of polynomially bounded growth in introduction}\n    \\sup_{B_R(p) \\times [-R^4, 0]}|u(x, t)| \\le C(1 + R)^d, \\quad \\sup_{B_R(p) \\times [-R^4, 0]}|\\nabla u(x, t)| \\le C'(1 + R)^{d'}\n\\end{equation}", "full_context": "Yau proved in \\cite{yau1975harmonic} that a bounded harmonic function on a complete manifold with nonnegative Ricci curvature is a constant. In 1974, he conjectured that a more general result should hold: on a complete manifold \\(M\\) with nonnegative Ricci curvature, the space \\(\\mathcal{H}_d(M)\\) of harmonic functions with polynomially bounded growth should have finite dimension. Colding and Minicozzi proved his conjecture in \\cite{colding1997harmonic}.\n\nThis equation is also more flexible than the Laplace equation (indeed, any harmonic function is biharmonic), and we cannot prove as general a Liouville theorem as for harmonic functions. To find a Liouville theorem, rather than restricting attention to a subclass of biharmonic functions as in \\cite{colding2021optimal}, Wang and Zhu instead restrict attention to a subclass of manifolds with polynomial volume growth and Ricci curvature bounded below at infinity.\n\nOur goal in this paper is to generalize Wang and Zhu's result to ancient solutions of the biharmonic heat equation, following the strategy of Colding and Minicozzi in \\cite{colding2021optimal}. Our main result is\n\\begin{theorem*}\n Let \\(M\\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \\(u: M \\times (-\\infty, 0] \\rightarrow \\mathbb{R}\\) be an ancient solution of\n \\begin{equation*}\n \\partial_t u(x, t) + \\Delta\\Delta u(x, t) = 0\n \\end{equation*}\n such that \\(|u(x, t)|\\) and \\(|\\nabla u(x, t)|\\) have polynomially bounded growth in the heat balls \\(B_R(x) \\times [-R^4, 0]\\). The space of all such solutions \\(u(x, t)\\) is finite dimensional.\n\\end{theorem*}\n\n\\subsection{Definitions and Notation}\nWe now give more precise definitions and statements. Given a manifold \\(M\\) and an interval \\(I \\subset \\mathbb{R}\\), a function \\(u: M \\times I \\rightarrow \\mathbb{R}\\) satisfies the biharmonic heat equation if\n\\begin{equation}\\label{first instance of biharmonic heat equation}\n \\partial_t u(x, t) + \\Delta\\Delta u(x, t) = 0.\n\\end{equation}\nWe will call such a function ``bicaloric'' for brevity. A bicaloric function \\(u\\) is ancient if it can be defined on an interval extending infinitely backwards in time, i.e. for \\(t \\in (-\\infty, 0]\\). We say that \\(u \\in \\mathcal{P}_{d, d'}(M)\\) for \\(d, d' > 0\\) if \\(\\partial_t u + \\Delta\\Delta u = 0\\), \\(u\\) is ancient, and for some constants \\(C, C' > 0\\),\n\\begin{equation}\\label{definition of polynomially bounded growth in introduction}\n \\sup_{B_R(p) \\times [-R^4, 0]}|u(x, t)| \\le C(1 + R)^d, \\quad \\sup_{B_R(p) \\times [-R^4, 0]}|\\nabla u(x, t)| \\le C'(1 + R)^{d'}\n\\end{equation}\nfor any \\(p \\in M\\) and \\(R > 0\\). We similarly say that \\(u \\in \\mathcal{H}_{d, d'}(M)\\) if \\(\\Delta\\Delta u = 0\\) and the same bounds in (\\ref{definition of polynomially bounded growth in introduction}) hold, where we take the supremum over only the ball \\(B_R(p)\\).\n\nA manifold \\(M\\) is said to have polynomial volume growth if there are constants \\(C, d_V > 0\\) and some \\(p \\in M\\) such that \\(\\operatorname{Vol}(B_R(p)) \\le C(1 + R)^{d_V}\\) for all \\(R> 0\\). Furthermore, we say that the Ricci curvature tensor is bounded below quadratically with constant \\(K\\) if for some \\(p \\in M\\) and all \\(R > 0\\),\n\\begin{equation}\n \\sup_{v \\in TB_{R}(p)}\\frac{\\Ric(v, v)}{|v|^2} \\ge -\\frac{K}{R^2}.\n\\end{equation}\n\nWith these definitions, our main results are more precisely stated as\n\n\\begin{equation}\\label{definition of polynomially bounded growth in introduction}\n    \\sup_{B_R(p) \\times [-R^4, 0]}|u(x, t)| \\le C(1 + R)^d, \\quad \\sup_{B_R(p) \\times [-R^4, 0]}|\\nabla u(x, t)| \\le C'(1 + R)^{d'}\n\\end{equation}\n\nA manifold \\(M\\) is said to have polynomial volume growth if there are constants \\(C, d_V > 0\\) and some \\(p \\in M\\) such that \\(\\operatorname{Vol}(B_R(p)) \\le C(1 + R)^{d_V}\\) for all \\(R> 0\\). Furthermore, we say that the Ricci curvature tensor is bounded below quadratically with constant \\(K\\) if for some \\(p \\in M\\) and all \\(R > 0\\),\n\\begin{equation}\n \\sup_{v \\in TB_{R}(p)}\\frac{\\Ric(v, v)}{|v|^2} \\ge -\\frac{K}{R^2}.\n\\end{equation}\n\nCombining this with Wang and Zhu's result \\cite{wang2025qualitativebehaviorbiharmonicfunctions}, we have the following corollary:\n\n\\begin{corollary}\n Let \\(M\\) be a Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Then for \\(k, \\ell \\ge 0\\) the spaces \\(\\mathcal{P}_{4k, 4\\ell}(M)\\) are finite dimensional.\n\\end{corollary}\n\nSince there are \\(d + 1\\) vectors \\((1, t_i, \\ldots, t_i^{d + 1})\\), they span \\(\\mathbb{R}^{d + 1}\\), and so there are constants \\(b_i^j\\) such that\n \\begin{equation}\\label{bdefinitions}\n e_j = b_i^j(1, t_i, \\ldots, t_i^{d}).\n \\end{equation}\n It now follows that\n \\begin{equation}\\label{expression of p in b}\n p_j(x) = b_i^ju(x, t_i), \\quad \\nabla p_j(x) = b_i^j\\nabla u(x, t_i),\n \\end{equation}\n and we conclude that \\(p_j\\) can grow at most polynomially of degree \\(4k\\) and \\(\\nabla p_j\\) can grow at most polynomially of degree \\(4\\ell\\). Because \\(p_j\\) vanishes when \\(j > k\\) and \\(\\nabla p_j\\) vanishes when \\(j > \\ell + 1\\), we have\n \\begin{equation}\n u = p_0 + t p_1 + \\ldots + t^{k}p_{k} \\quad \\text{and} \\quad \\nabla u = \\nabla p_0 + t\\nabla p_1 + \\ldots + t^{\\ell}\\nabla p_{\\ell},\n \\end{equation}\n it follows that\n \\begin{equation}\n \\begin{split}\n |u(x, t)| \\le C(1 + |t|^k + |x|^{4k}) \\quad \\text{and} \\quad |\\nabla u(x, t)| \\le C(1 + |t|^{\\ell} + |x|^{4\\ell})\n \\end{split}\n \\end{equation}\n From equation (\\ref{bdefinitions}) we have\n \\begin{equation}\n \\begin{split}\n \\sum_i b_i^ju(x, R^4t_i) & = \\sum_i\\sum_{m}b_i^jp_{m}(x)R^{4j}t_i^{m} = R^{4j}\\sum_mp_m(x)\\left(\\sum_ib_{i}^jt_i^m\\right) \\\\ & = \\sum R^{4j}\\sum_m p_m(x)\\delta_{mj} \\\\ & = R^{4j}p_j(x).\n \\end{split}\n \\end{equation}\n Similarly,\n \\begin{equation}\n \\sum_i b_i^j\\nabla u(x, R^4t_i) = R^{4j}\\nabla p_j(x)\n \\end{equation}\n Thus\n \\begin{equation}\n \\begin{split}\n |R^{4j}p_j(x)| & = \\left|\\sum_i b_i^ju(x, R^4t_i)\\right| \\le A\\sum_i\\left|u(x, R^4t_i)\\right| \\\\ & \\le A(1 + |x|^{4k} + \\sum_i|Rt_i|^{4d}) \\le AR^{4k}\n \\end{split}\n \\end{equation}\n and similarly\n \\begin{equation}\n \\begin{split}\n |R^{4j}\\nabla p_j(x)| & \\le A'(1 + |x|^{4\\ell} + \\sum_i|Rt_i|^{4\\ell}) \\le A'R^{4\\ell},\n \\end{split}\n \\end{equation}\n so that \\(|p_j(x)| \\le A_jR^{4(k - j)}\\) and \\(|p_j(x)| \\le A_j'R^{4(\\ell - j)}\\).\n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{dimension bound theorem}]\n Choose some \\(u \\in {P}_{4k, 4\\ell}(M)\\) and suppose \\(u = p_0(x) + tp_1(x) + \\cdots + t^dp_d(x)\\), as in lemma \\ref{coefficientbound}. Then \\(\\Delta\\Delta p_{d} = 0\\) and for \\(j < d\\), \\(\\Delta\\Delta p_j = -(j+1)p_{j + 1}\\). Thus, there a linear map \\(\\Psi_0: \\mathcal{P}_{4k, 4\\ell} \\rightarrow \\mathcal{H}_{4k, 4\\ell}\\) defined by \\(\\Psi_0u = p_{d}\\) (here we use the coefficient estimate in (\\ref{coefficients in polynomial expression bounds})). If we let \\(\\mathcal{K}_0 = \\ker \\Psi_0\\) we find\n \\begin{equation}\n \\dim \\mathcal{P}_{4k, 4\\ell} \\le \\dim \\mathcal{K}_0 + \\dim \\mathcal{H}_{4k, 4\\ell}\n \\end{equation}\n If \\(u \\in \\mathcal{K}_0\\), then \\(p_{d} = 0\\) and \\(\\Delta\\Delta p_{d - 1} = -dp_{d} = 0\\), and so we have a map \\(\\Psi_1: \\mathcal{K}_0 \\rightarrow \\mathcal{H}_{4(k - 1), 4(\\ell - 1)}\\) defined by setting \\(\\Psi_1u = p_{d - 1}\\). Letting \\(\\ker \\Psi_1 = \\mathcal{K}_1\\) then\n \\begin{equation}\n \\dim \\mathcal{K}_0 \\le \\dim \\mathcal{K}_1 + \\dim \\mathcal{H}_{4k, 4\\ell}\n \\end{equation}\n When \\(k \\le \\ell + 1\\), we can repeat this \\(k\\) times to get\n \\begin{equation}\n \\dim \\mathcal{P}_{4k, 4\\ell}(M) \\le \\sum_{i = 0}^{k} \\dim \\mathcal{H}_{4(k - i), 4(\\ell - i)}(M)\n \\end{equation}\n When \\(k > \\ell + 1\\), we have from lemma \\ref{coefficientbound} that \\(\\nabla p_{\\ell + 1} = 0\\), and so \\(p_{\\ell + 1} = p_d\\) is a constant. Thus in this case \\(p_d\\) lies in a one dimensional subspace of \\(H_{4k, 4\\ell}(M)\\), so that\n \\begin{equation}\n \\dim \\mathcal{P}_{4k, 4\\ell} = \\dim \\mathcal{K}_0 + 1.\n \\end{equation}\n We then iterate the same argument as before, \\(\\ell\\) times, to get the second inequality in (\\ref{equation in the first statement of the main theorem}).\n\n\\end{proof}\n\\subsection{Bicaloric Polynomials in \\(\\mathbb{R}^n\\)}\nNow we consider polynomially bounded solutions to the biharmonic heat equation in \\(\\mathbb{R}^n\\).\n\\begin{proposition}\\label{bicaloric functions in Rn are polynomials}\n Let \\(u \\in \\mathcal{P}_{k, \\ell}(\\mathbb{R}^n)\\). Then \\(u\\) is a polynomial in \\(x_i\\) and \\(t\\).\n\\end{proposition}\n\\begin{proof}\n As before, this follows from the reverse Poincaré estimate and the fact that the operators \\(\\partial_{x_i}\\), \\(\\partial_t\\), and \\(\\Delta\\) commute in \\(\\mathbb{R}^n\\).\n\\end{proof}\n\\begin{corollary}\n If \\(u \\in \\mathcal{P}_{k, \\ell}(\\mathbb{R}^n)\\), then there is some \\(d\\) such that \\(u \\in \\mathcal{P}_{d, d-1}(\\mathbb{R}^n)\\).\n\\end{corollary}\\label{fewer caloric spaces in R^n}\n\\begin{proof}\n This follows the same way as before.\n\\end{proof}\nGiven a monomial in \\(x_i\\) and \\(t\\), we define its biparabolic degree as follows: for \\(t^{n_0}\\prod x_i^{n_i}\\), the biparabolic degree is \\(4n_0 + \\sum n_i\\). The degree of a polynomial is then the maximal degree of the monomials summing to it. Let \\(\\mathcal{A}_j^n\\) be the set of homogeneous polynomials in \\(\\mathbb{R}^n\\) of biparabolic degree \\(j\\). We have\n\\begin{equation}\n \\mathcal{A}_d^n = A_d^n \\oplus t A_{d - 4}^n \\oplus t^2A_{d - 8}^n \\oplus \\cdots\n\\end{equation}\n\\begin{lemma}\\label{dim for bicaloric polynomials}\n For \\(d > 0\\) we have \\(\\dim(\\mathcal{P}_{d, d-1}(\\mathbb{R}^n) \\cap \\mathcal{A}_d^n) = \\dim A_d^n\\) and\n \\begin{equation}\n \\dim \\mathcal{P}_{d, d-1}(\\mathbb{R}^n) = \\sum_{j = 0}^d\\dim A_j^n.\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\n Both \\(\\partial_t\\) and \\(\\Delta\\Delta\\) map \\(\\mathcal{A}_d^n\\) to \\(\\mathcal{A}_{d-4}^n\\). We note that for \\(u \\in \\mathcal{A}_{d-4}^n\\) we have\n \\begin{equation}\n (\\partial_t + \\Delta\\Delta)\\left(tu - \\frac{1}{2}t^2(\\partial_t + \\Delta\\Delta)u + \\frac{1}{6}t^3(\\partial_t + \\Delta\\Delta)^2 u - \\cdots\\right) = u,\n \\end{equation}\n so that \\(\\partial_t + \\Delta\\Delta\\) is surjective. Thus,\n \\begin{equation}\n \\dim (\\mathcal{P}_{d, d-1}(\\mathbb{R}^n) \\cap \\mathcal{A}_d^n) = \\dim \\mathcal{A}_d^n - \\dim\\mathcal{A}_{d-4}^n = \\dim A_d^n\n \\end{equation}\n since \\(\\mathcal{P}_{d, d-1}(\\mathbb{R}^n) \\cap \\mathcal{A}_d^n\\) is the kernel of \\(\\partial_t + \\Delta\\Delta\\) restricted to \\(\\mathcal{A}_d^n\\). Summing gives the second claim.\n\n\\end{proof}\nNow finally we show that the estimate in Theorem \\ref{dimension bound theorem} is sharp in \\(\\mathbb{R}^n\\).\n\\begin{corollary}\\label{sharp in Rn}\n For positive integers \\(d > 0\\)\n \\begin{equation}\n \\dim \\mathcal{P}_{4d, 4d - 1}(\\mathbb{R}^n) = \\sum_{i = 0}^{d} \\dim \\mathcal{H}_{4(d - i), 4(d - i) - 1}(\\mathbb{R}^n).\n \\end{equation}\n\\end{corollary}\n\\begin{proof}\n From lemmas \\ref{dim for bicaloric polynomials} and \\ref{dim for biharmonic polynomials} we have\n \\begin{equation}\n \\begin{split}\n \\dim \\mathcal{P}_{4d, 4d-1}(\\mathbb{R}^n) & = \\sum_{j = 0}^{4d}\\dim A_j^n = \\sum_{j = 0}^d(\\dim A_{4j}^n + \\dim A_{4j - 1}^n + \\dim A_{4j - 2}^n + \\dim A_{4j - 3}^n) \\\\ & = \\sum_{j = 0}^d\\dim \\mathcal{H}_{4d, 4d - 1}(\\mathbb{R}^n).\n \\end{split}\n \\end{equation}\n Setting \\(k = \\ell = d\\) in Theorem \\ref{dimension bound theorem} and noting \\(\\mathcal{P}_{4d, 4d}(\\mathbb{R}^n) = \\mathcal{P}_{4d, 4d-1}(\\mathbb{R}^n)\\) and \\(\\mathcal{H}_{4d, 4d}(\\mathbb{R}^n) = \\mathcal{H}_{4d, 4d-1}(\\mathbb{R}^n)\\) shows that the estimate in Theorem \\ref{dimension bound theorem} is sharp.", "post_theorem_intro_text_len": 4214, "post_theorem_intro_text": "Combining this with Wang and Zhu's result \\cite{wang2025qualitativebehaviorbiharmonicfunctions}, we have the following corollary:\n\n\\begin{corollary}\n    Let \\(M\\) be a Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Then for \\(k, \\ell \\ge 0\\) the spaces \\(\\mathcal{P}_{4k, 4\\ell}(M)\\) are finite dimensional.\n\\end{corollary}\n\n\\subsection{Harmonic and biharmonic functions}\n\nBiharmonic functions arise in several variational problems. Just as minimizing \\(\\int|\\nabla u|^2\\) leads one to the Laplace and heat equations, minimizing \\(\\int |\\Delta u|^2\\) leads to the biharmonic and biharmonic heat equations.\n\nIn general, fourth order elliptic operators arise naturally when taking variations involving second order objects, one major example being variations of metrics in conformal geometry (see \\cite{chang1995extremal}, \\cite{Lin1998classification}). They also arise in the study of the Willmore energy. For an immersed surface \\(\\phi: M^2 \\rightarrow \\mathbb{R}^3\\), the Willmore energy is defined as\n\\begin{equation}\\label{Willmore functional}\n    \\mathcal{W}(\\phi) = \\int_{M}H^2\\,dA\n\\end{equation}\nwhere \\(dA\\) is the induced volume element and \\(H\\) is the mean curvature \\cite{willmore2000surfaces}. In studying critical points of this functional one arrives at the Euler-Lagrange equation\n\\begin{equation}\\label{euler-lagrange equation}\n    \\Delta H + 2H(H^2 - K) = 0,\n\\end{equation}\na fourth order elliptic operator. The biharmonic heat equation similarly arises when studying the gradient flow of the Willmore energy (\\cite{kuwert2002gradient}, \\cite{lamm2005biharmonic}). Ancient solutions to heat equations often appear when doing blowup analysis of general solutions to a variational problem. See \\cite{kuwert2004removable} for blowup analysis of singularities of Willmore flows. We also again reference (\\cite{colding2020complexity}, \\cite{colding2019search}, \\cite{colding2019liouville}) for more on how ancient solutions to heat equations with polynomially bounded growth are relevant to geometric flows.\n\nAlthough both arise from variational problems, biharmonic functions in general differ significantly from harmonic functions, because no maximum principle holds for biharmonic functions. This limits the kinds of estimates we can find for biharmonic functions. In particular, the usual pointwise derivative estimates one can find for harmonic functions on a ball cannot be found for a biharmonic function.\n\nOn the bright side, energy methods for harmonic and caloric functions seem to have analogs for biharmonic and bicaloric functions, which we will see as we prove Theorem \\ref{dimension bound theorem}. We are still limited to some extent, however, because when performing integrations by parts we are forced to use the Bochner formula\n\\begin{equation*}\n    \\frac{1}{2}\\Delta|\\nabla u|^2 = |\\nabla^2u| + \\langle \\nabla \\Delta u, \\nabla u\\rangle + \\Ric(\\nabla u, \\nabla u)\n\\end{equation*}\nto control the factor \\(\\langle \\nabla \\Delta u, \\nabla u\\rangle\\). It is the appearance of the Ricci term here that makes the decay on Ricci curvature crucial for our result.\n\nOur methodology is inspired by Colding and Minicozzi's in \\cite{colding2021optimal}. We will show a reverse Poincaré inequality for bicaloric functions on ``heat balls'' \\(B_R(p) \\times [-R^4, 0]\\). Because we are considering ancient bicaloric functions, we will be able to apply the inequality as \\(R \\rightarrow \\infty\\) to get strong, global control of their behavior. In particular we will see that high order time derivatives \\(\\partial_t^ku\\) must vanish identically, allowing us to write for some finite \\(d\\):\n\\[u(x, t) = p_d(x)t^d + \\cdots + p_1(x)t + p_0(x)\\]\nwith \\(\\Delta \\Delta p_d = 0\\) and \\(\\Delta \\Delta p_j = -(j + 1)p_{j + 1}\\). This will allow us to directly compare the spaces \\(\\mathcal{H}_{4k, 4\\ell}(M)\\) with \\(\\mathcal{P}_{4k, 4\\ell}(M)\\).\n\nTo show the dimension estimates are sharp in \\(\\mathbb{R}^n\\), we will consider biharmonic and bicaloric polynomials (analogs of the harmonic polynomials), enabling us to explicitly compute the dimensions of the spaces \\(\\mathcal{H}_{4k, 4\\ell}(\\mathbb{R}^n)\\) and \\(\\mathcal{P}_{4k, 4\\ell}(\\mathbb{R}^n)\\).", "sketch": "The introduction indicates the proof strategy for Theorem~\\ref{dimension bound theorem} as follows.\n\n1. **Use energy methods and Bochner to handle fourth order terms.** It says that while working \"as we prove Theorem \\ref{dimension bound theorem}\" one uses analogs of energy methods for harmonic/caloric functions, but when integrating by parts one is \"forced to use the Bochner formula\" \n\\[\n\\frac{1}{2}\\Delta|\\nabla u|^2 = |\\nabla^2u| + \\langle \\nabla \\Delta u, \\nabla u\\rangle + \\Ric(\\nabla u, \\nabla u)\n\\]\n\"to control the factor \\(\\langle \\nabla \\Delta u, \\nabla u\\rangle\\),\" and \"the appearance of the Ricci term here\" is why the quadratic lower bound/decay assumption on Ricci is crucial.\n\n2. **Establish a reverse Poincar\\'e inequality on heat balls for bicaloric functions.** Inspired by Colding--Minicozzi, the paper will \"show a reverse Poincar\\'e inequality for bicaloric functions on `heat balls' \\(B_R(p)\\times[-R^4,0]\\).\"\n\n3. **Send \\(R\\to\\infty\\) using the ancient assumption to get global control.** Because the functions are ancient bicaloric, \"we will be able to apply the inequality as \\(R\\to\\infty\\) to get strong, global control of their behavior.\"\n\n4. **Deduce vanishing of high time derivatives and a finite time-polynomial expansion.** From this global control, \"high order time derivatives \\(\\partial_t^k u\\) must vanish identically,\" so for some finite \\(d\\)\n\\[\nu(x,t)=p_d(x)t^d+\\cdots+p_1(x)t+p_0(x),\n\\]\nwith relations \"\\(\\Delta\\Delta p_d=0\\)\" and \"\\(\\Delta\\Delta p_j=-(j+1)p_{j+1}\\).\"\n\n5. **Compare \\(\\mathcal{H}_{4k,4\\ell}(M)\\) and \\(\\mathcal{P}_{4k,4\\ell}(M)\\) via the coefficient relations.** The stated expansion and recursions \"will allow us to directly compare the spaces \\(\\mathcal{H}_{4k,4\\ell}(M)\\) with \\(\\mathcal{P}_{4k,4\\ell}(M)\\),\" yielding the dimension bound.\n\n6. **Sharpness in \\(\\mathbb{R}^n\\) by explicit polynomial models.** To show sharpness, it says they will consider \"biharmonic and bicaloric polynomials (analogs of the harmonic polynomials), enabling us to explicitly compute the dimensions\" of \\(\\mathcal{H}_{4k,4\\ell}(\\mathbb{R}^n)\\) and \\(\\mathcal{P}_{4k,4\\ell}(\\mathbb{R}^n)\\).", "expanded_sketch": "The introduction indicates the proof strategy for the following theorem.\n\n\\begin{theorem}\\label{dimension bound theorem}\n    Let \\(M\\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \\(k\\), \\(\\ell\\) be nonnegative integers. Then\n    \\begin{equation}\\label{equation in the first statement of the main theorem}\n        \\dim \\mathcal{P}_{4k, 4\\ell}(M) \\le \\begin{cases}\n             \\displaystyle \\sum_{i = 0}^{k} \\dim \\mathcal{H}_{4(k - i), 4(\\ell - i)}(M) & \\quad k \\le \\ell + 1, \\\\ \\displaystyle\n            1 + \\sum_{i = 0}^{\\ell} \\dim \\mathcal{H}_{4(k - i), 4(\\ell - i)}(M) & \\quad k > \\ell + 1\n        \\end{cases}\n    \\end{equation}\n    Moreover, these inequalities are sharp in \\(\\mathbb{R}^n\\).\n\\end{theorem}\n\n1. **Use energy methods and Bochner to handle fourth order terms.** In establishing the main theorem, one uses analogs of energy methods for harmonic/caloric functions, but when integrating by parts one is forced to use the Bochner formula\n\\[\n\\frac{1}{2}\\Delta|\\nabla u|^2 = |\\nabla^2u| + \\langle \\nabla \\Delta u, \\nabla u\\rangle + \\Ric(\\nabla u, \\nabla u)\n\\]\nso as to control the factor \\(\\langle \\nabla \\Delta u, \\nabla u\\rangle\\); the appearance of the Ricci term here is why the quadratic lower bound/decay assumption on Ricci is crucial.\n\n2. **Establish a reverse Poincar\\'e inequality on heat balls for bicaloric functions.** Inspired by Colding--Minicozzi, the paper will show a reverse Poincar\\'e inequality for bicaloric functions on “heat balls” \\(B_R(p)\\times[-R^4,0]\\).\n\n3. **Send \\(R\\to\\infty\\) using the ancient assumption to get global control.** Because the functions are ancient bicaloric, one will be able to apply the inequality as \\(R\\to\\infty\\) to get strong, global control of their behavior.\n\n4. **Deduce vanishing of high time derivatives and a finite time-polynomial expansion.** From this global control, high order time derivatives \\(\\partial_t^k u\\) must vanish identically, so for some finite \\(d\\)\n\\[\nu(x,t)=p_d(x)t^d+\\cdots+p_1(x)t+p_0(x),\n\\]\nwith relations “\\(\\Delta\\Delta p_d=0\\)” and “\\(\\Delta\\Delta p_j=-(j+1)p_{j+1}\\).”\n\n5. **Compare \\(\\mathcal{H}_{4k,4\\ell}(M)\\) and \\(\\mathcal{P}_{4k,4\\ell}(M)\\) via the coefficient relations.** The stated expansion and recursions allow one to directly compare the spaces \\(\\mathcal{H}_{4k,4\\ell}(M)\\) with \\(\\mathcal{P}_{4k,4\\ell}(M)\\), yielding the dimension bound \\eqref{equation in the first statement of the main theorem}.\n\n6. **Sharpness in \\(\\mathbb{R}^n\\) by explicit polynomial models.** To show sharpness, one considers biharmonic and bicaloric polynomials (analogs of the harmonic polynomials), enabling one to explicitly compute the dimensions of \\(\\mathcal{H}_{4k,4\\ell}(\\mathbb{R}^n)\\) and \\(\\mathcal{P}_{4k,4\\ell}(\\mathbb{R}^n)\\).", "expanded_theorem": "\\label{dimension bound theorem}\n    Let \\(M\\) be a complete Riemannian manifold with polynomial volume growth and Ricci curvature bounded below quadratically. Let \\(k\\), \\(\\ell\\) be nonnegative integers. Then\n    \\begin{equation}\\label{equation in the first statement of the main theorem}\n        \\dim \\mathcal{P}_{4k, 4\\ell}(M) \\le \\begin{cases}\n             \\displaystyle \\sum_{i = 0}^{k} \\dim \\mathcal{H}_{4(k - i), 4(\\ell - i)}(M) & \\quad k \\le \\ell + 1, \\\\ \\displaystyle\n            1 + \\sum_{i = 0}^{\\ell} \\dim \\mathcal{H}_{4(k - i), 4(\\ell - i)}(M) & \\quad k > \\ell + 1\n        \\end{cases}\n    \\end{equation}\n    Moreover, these inequalities are sharp in \\(\\mathbb{R}^n\\).", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Let \\(M\\) be a complete Riemannian manifold with polynomial volume growth, meaning that there exist constants \\(C,d_V>0\\) and a point \\(p\\in M\\) such that \\(\\operatorname{Vol}(B_R(p))\\le C(1+R)^{d_V}\\) for all \\(R>0\\), and assume its Ricci curvature is bounded below quadratically, i.e. there exist \\(K>0\\) and \\(p\\in M\\) such that for all \\(R>0\\),\n\\[\n\\sup_{v\\in T B_R(p)} \\frac{\\operatorname{Ric}(v,v)}{|v|^2} \\ge -\\frac{K}{R^2}.\n\\]\nFor exponents \\(d,d'\\), let \\(\\mathcal P_{d,d'}(M)\\) denote the space of ancient solutions \\(u:M\\times(-\\infty,0]\\to\\mathbb R\\) of the biharmonic heat equation\n\\[\n\\partial_t u+\\Delta\\Delta u=0\n\\]\nsuch that for some constants \\(C,C'>0\\), for every point \\(q\\in M\\) and every \\(R>0\\),\n\\[\n\\sup_{B_R(q)\\times[-R^4,0]} |u(x,t)|\\le C(1+R)^d,\n\\qquad\n\\sup_{B_R(q)\\times[-R^4,0]} |\\nabla u(x,t)|\\le C'(1+R)^{d'}.\n\\]\nLet \\(\\mathcal H_{d,d'}(M)\\) denote the space of biharmonic functions \\(v:M\\to\\mathbb R\\) with \\(\\Delta\\Delta v=0\\) satisfying the analogous bounds over \\(B_R(q)\\). For nonnegative integers \\(k\\) and \\(\\ell\\), which dimension estimate (including the Euclidean sharpness statement) holds?", "correct_choice": {"label": "A", "text": "For every nonnegative integers \\(k,\\ell\\),\n\\[\n\\dim \\mathcal P_{4k,4\\ell}(M)\\le\n\\begin{cases}\n\\displaystyle \\sum_{i=0}^{k} \\dim \\mathcal H_{4(k-i),\\,4(\\ell-i)}(M), & \\text{if } k\\le \\ell+1,\\\\[1ex]\n\\displaystyle 1+\\sum_{i=0}^{\\ell} \\dim \\mathcal H_{4(k-i),\\,4(\\ell-i)}(M), & \\text{if } k>\\ell+1.\n\\end{cases}\n\\]\nMoreover, these inequalities are sharp when \\(M=\\mathbb R^n\\)."}, "choices": [{"label": "B", "text": "For every nonnegative integers \\(k,\\ell\\),\n\\[\n\\dim \\mathcal P_{4k,4\\ell}(M)\\le\n\\sum_{i=0}^{\\min\\{k,\\ell\\}} \\dim \\mathcal H_{4(k-i),\\,4(\\ell-i)}(M).\n\\]\nMoreover, this inequality is sharp when \\(M=\\mathbb R^n\\)."}, {"label": "C", "text": "For every nonnegative integers \\(k,\\ell\\),\n\\[\n\\dim \\mathcal P_{4k,4\\ell}(M)<\\infty.\n\\]\nMoreover, this finiteness statement holds when \\(M=\\mathbb R^n\\)."}, {"label": "D", "text": "For every nonnegative integers \\(k,\\ell\\),\n\\[\n\\dim \\mathcal P_{4k,4\\ell}(M)\\le\n\\begin{cases}\n\\displaystyle \\sum_{i=0}^{k} \\dim \\mathcal H_{4(k-i),\\,4(\\ell-i)}(M), & \\text{if } k\\le \\ell,\\\\[1ex]\n\\displaystyle 1+\\sum_{i=0}^{\\ell} \\dim \\mathcal H_{4(k-i),\\,4(\\ell-i)}(M), & \\text{if } k>\\ell.\n\\end{cases}\n\\]\nMoreover, these inequalities are sharp when \\(M=\\mathbb R^n\\)."}, {"label": "E", "text": "For every nonnegative integers \\(k,\\ell\\),\n\\[\n\\dim \\mathcal P_{4k,4\\ell}(M)=\n\\begin{cases}\n\\displaystyle \\sum_{i=0}^{k} \\dim \\mathcal H_{4(k-i),\\,4(\\ell-i)}(M), & \\text{if } k\\le \\ell+1,\\\\[1ex]\n\\displaystyle 1+\\sum_{i=0}^{\\ell} \\dim \\mathcal H_{4(k-i),\\,4(\\ell-i)}(M), & \\text{if } k>\\ell+1.\n\\end{cases}\n\\]\nIn particular, the same equalities hold for every such manifold \\(M\\), not only for \\(M=\\mathbb R^n\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "extra_constant_and_upper_limit_case_split", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "explicit_casewise_dimension_bound_and_sharpness_formula", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "threshold_k_leq_ell_plus_1", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "inequality_vs_equality_and_Euclidean_only_sharpness", "template_used": "stronger_trap"}]}}
{"id": "2512.14262v2", "paper_link": "http://arxiv.org/abs/2512.14262v2", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\nIf \\(n \\neq 6\\), then\n    $$ \\gcd(2^3, n) \\cdot H^3(\\Kum_{n-1}(A), \\mathbb Z)_{\\tors} = 0.$$\nIf instead \\(n=6\\), then\n$$6 \\cdot H^3(\\Kum_{5}(A), \\mathbb Z)_{\\tors} = 0.$$", "start_pos": 26104, "end_pos": 26321, "label": "thm:main"}, "ref_dict": {"que:hk_torsion": "\\begin{question}\\label{que:hk_torsion}\nLet $X$ be a \\hk \\  manifold. Is $H^*(X, \\mathbb Z)$ torsion-free?\n\\end{question}", "cor:nice_statement_even": "\\begin{corollary}\\label{cor:nice_statement_even}\n    The third integral cohomology group of a generalized Kummer variety of dimension divisible by four is torsion-free.\n\\end{corollary}"}, "pre_theorem_intro_text_len": 2102, "pre_theorem_intro_text": "The study of \\hk \\ manifolds is a central topic in the theory of compact Kähler manifolds: in virtue of the Beauville--Bogomolov decomposition theorem \\cite[Thm.\\ 1]{beauville}, they represent one of three ``fundamental\" types of compact Kähler manifolds with \\(c_1(K_X)=0\\), together with abelian varieties and strict Calabi-Yau manifolds.\nTheir topology is extremely interesting: on the one hand it is severely restricted compared to that of a general compact \\Kah \\ manifold, for example by the Fujiki relations, \\cite{Fujiki}. Moreover, endowed with the usual Hodge structure, the cohomology of \\hkm s largely determines their geometry by the Torelli Theorem, see for example \\cite{Markman_torelli}. On the other hand, the topology of \\hkm s remains very mysterious, with a lot of basic open questions. One of these questions is the following:\n\n\\begin{question}\\label{que:hk_torsion}\nLet $X$ be a \\hk \\  manifold. Is $H^*(X, \\mathbb Z)$ torsion-free?\n\\end{question}\nFor \\hk \\ manifolds of $K3^{[n]}$-type, the answer is positive by \\cite{Markman2007_integral}, which has then been further generalized by \\cite{Totaro_2020_integral} to the Hilbert scheme of points on any smooth projective surface with torsion-free cohomology. Already for the next most understood \\hk \\ deformation type, that of \\textit{generalized Kummer varieties} (or \\textit{Kummer manifolds}), denoted as \\(\\Kum_n(A)\\), this question is open in its full generality. In \\cite{kapfermenet}, however,  Kapfer and Menet give a positive answer in the case of Kummer fourfolds. \n\nThe abelian group $H^3(X, \\mathbb Z)_{\\tors}$ is of particular interest as it sits in the short exact sequence\n$$0 \\to H^2(X, \\mathbb Z) / \\NS(X, \\mathbb Z) \\otimes_\\Z \\Q/\\Z \\to \\Br(X) \\to H^3(X, \\mathbb Z)_{\\tors} \\to 0.$$\nFor this reason it is often called the \\textit{topological Brauer group}, see also \\cite[Sec.\\ 1]{grothendieck}, and it is the lowest degree cohomology group whose torsion part is not known for any \\hk \\ manifold other than those of K3\\(^{[n]}\\)-type and Kummer fourfolds.\n\nThe aim of this note is to prove the following result:", "context": "The study of \\hk \\ manifolds is a central topic in the theory of compact Kähler manifolds: in virtue of the Beauville--Bogomolov decomposition theorem \\cite[Thm.\\ 1]{beauville}, they represent one of three ``fundamental\" types of compact Kähler manifolds with \\(c_1(K_X)=0\\), together with abelian varieties and strict Calabi-Yau manifolds.\nTheir topology is extremely interesting: on the one hand it is severely restricted compared to that of a general compact \\Kah \\ manifold, for example by the Fujiki relations, \\cite{Fujiki}. Moreover, endowed with the usual Hodge structure, the cohomology of \\hkm s largely determines their geometry by the Torelli Theorem, see for example \\cite{Markman_torelli}. On the other hand, the topology of \\hkm s remains very mysterious, with a lot of basic open questions. One of these questions is the following:\n\n\\begin{question}\\label{que:hk_torsion}\nLet $X$ be a \\hk \\ manifold. Is $H^*(X, \\mathbb Z)$ torsion-free?\n\\end{question}\nFor \\hk \\ manifolds of $K3^{[n]}$-type, the answer is positive by \\cite{Markman2007_integral}, which has then been further generalized by \\cite{Totaro_2020_integral} to the Hilbert scheme of points on any smooth projective surface with torsion-free cohomology. Already for the next most understood \\hk \\ deformation type, that of \\textit{generalized Kummer varieties} (or \\textit{Kummer manifolds}), denoted as \\(\\Kum_n(A)\\), this question is open in its full generality. In \\cite{kapfermenet}, however, Kapfer and Menet give a positive answer in the case of Kummer fourfolds.\n\nThe abelian group $H^3(X, \\mathbb Z)_{\\tors}$ is of particular interest as it sits in the short exact sequence\n$$0 \\to H^2(X, \\mathbb Z) / \\NS(X, \\mathbb Z) \\otimes_\\Z \\Q/\\Z \\to \\Br(X) \\to H^3(X, \\mathbb Z)_{\\tors} \\to 0.$$\nFor this reason it is often called the \\textit{topological Brauer group}, see also \\cite[Sec.\\ 1]{grothendieck}, and it is the lowest degree cohomology group whose torsion part is not known for any \\hk \\ manifold other than those of K3\\(^{[n]}\\)-type and Kummer fourfolds.\n\nThe aim of this note is to prove the following result:", "full_context": "The study of \\hk \\ manifolds is a central topic in the theory of compact Kähler manifolds: in virtue of the Beauville--Bogomolov decomposition theorem \\cite[Thm.\\ 1]{beauville}, they represent one of three ``fundamental\" types of compact Kähler manifolds with \\(c_1(K_X)=0\\), together with abelian varieties and strict Calabi-Yau manifolds.\nTheir topology is extremely interesting: on the one hand it is severely restricted compared to that of a general compact \\Kah \\ manifold, for example by the Fujiki relations, \\cite{Fujiki}. Moreover, endowed with the usual Hodge structure, the cohomology of \\hkm s largely determines their geometry by the Torelli Theorem, see for example \\cite{Markman_torelli}. On the other hand, the topology of \\hkm s remains very mysterious, with a lot of basic open questions. One of these questions is the following:\n\n\\begin{question}\\label{que:hk_torsion}\nLet $X$ be a \\hk \\ manifold. Is $H^*(X, \\mathbb Z)$ torsion-free?\n\\end{question}\nFor \\hk \\ manifolds of $K3^{[n]}$-type, the answer is positive by \\cite{Markman2007_integral}, which has then been further generalized by \\cite{Totaro_2020_integral} to the Hilbert scheme of points on any smooth projective surface with torsion-free cohomology. Already for the next most understood \\hk \\ deformation type, that of \\textit{generalized Kummer varieties} (or \\textit{Kummer manifolds}), denoted as \\(\\Kum_n(A)\\), this question is open in its full generality. In \\cite{kapfermenet}, however, Kapfer and Menet give a positive answer in the case of Kummer fourfolds.\n\nThe abelian group $H^3(X, \\mathbb Z)_{\\tors}$ is of particular interest as it sits in the short exact sequence\n$$0 \\to H^2(X, \\mathbb Z) / \\NS(X, \\mathbb Z) \\otimes_\\Z \\Q/\\Z \\to \\Br(X) \\to H^3(X, \\mathbb Z)_{\\tors} \\to 0.$$\nFor this reason it is often called the \\textit{topological Brauer group}, see also \\cite[Sec.\\ 1]{grothendieck}, and it is the lowest degree cohomology group whose torsion part is not known for any \\hk \\ manifold other than those of K3\\(^{[n]}\\)-type and Kummer fourfolds.\n\nThe aim of this note is to prove the following result:\n\n\\begin{question}\\label{que:hk_torsion}\nLet $X$ be a \\hk \\ manifold. Is $H^*(X, \\mathbb Z)$ torsion-free?\n\\end{question}\nFor \\hk \\ manifolds of $K3^{[n]}$-type, the answer is positive by \\cite{Markman2007_integral}, which has then been further generalized by \\cite{Totaro_2020_integral} to the Hilbert scheme of points on any smooth projective surface with torsion-free cohomology. Already for the next most understood \\hk \\ deformation type, that of \\textit{generalized Kummer varieties} (or \\textit{Kummer manifolds}), denoted as \\(\\Kum_n(A)\\), this question is open in its full generality. In \\cite{kapfermenet}, however, Kapfer and Menet give a positive answer in the case of Kummer fourfolds.\n\nThe abelian group $H^3(X, \\mathbb Z)_{\\tors}$ is of particular interest as it sits in the short exact sequence\n$$0 \\to H^2(X, \\mathbb Z) / \\NS(X, \\mathbb Z) \\otimes_\\Z \\Q/\\Z \\to \\Br(X) \\to H^3(X, \\mathbb Z)_{\\tors} \\to 0.$$\nFor this reason it is often called the \\textit{topological Brauer group}, see also \\cite[Sec.\\ 1]{grothendieck}, and it is the lowest degree cohomology group whose torsion part is not known for any \\hk \\ manifold other than those of K3\\(^{[n]}\\)-type and Kummer fourfolds.\n\nThe strategy of the proof is inspired by and generalizes the approach of \\cite{kapfermenet}:\nwe first consider a rational cover of degree \\(n\\) by a smooth projective variety with no torsion in degree three cohomology, which forces \\(H^3(\\Kum_{n-1}(A),\\Z)_{\\tors}\\) to be entirely of \\(n\\)-torsion. By considering a double cover of a suitable open of \\(\\Kum_{n-1}(A)\\), we then show that \\(H^3(\\Kum_{n-1}(A),\\Z)_{\\tors}\\) is a $2$-group, via an analysis of the cohomology of the alternating group and the Cartan--Leray spectral sequence. When \\(n\\) is odd, we obtain Corollary \\ref{cor:nice_statement_even}.\nWe are then left with asking ourselves the following\n\n\\end{lemma}\n\\begin{proof}\nSince \\(H^3(-,\\Z)_{\\tors}\\) is a birational invariant, by passing to a resolution of indeterminacies we may assume that \\(f\\) is everywhere defined. The result then follows from the projection formula.\n\\end{proof}\n\\begin{corollary}\\label{cor:n-tors_of_Kum}\n The abelian group $H^3(\\Kum_{n-1}(A), \\mathbb Z)_{\\tors}$ is $n$-torsion.\n\\end{corollary}\n\\begin{proof}\n The map \\(K_{n-1}(A) \\dashrightarrow \\Kum_{n-1}(A)\\) factors through the quotient by \\(\\mathfrak{S}_{n-1}\\) acting on the first \\(n-1\\) entries of \\(A^n\\), yielding a map of degree \\(n\\). \n The quotient \\(K_{n-1}(A)/\\mathfrak{S}_{n-1}\\) is birational to \\(A^{[n-1]}\\), hence we have a generically finite rational map of degree \\(n\\)\n \\[A^{[n-1]} \\dashrightarrow \\Kum_{n-1}(A).\\] \nBy \\cite{Markman2007_integral} the group $H^3(A^{[n-1]},\\Z)$ is torsion-free. The result then follows from Lemma \\ref{lem:dn_tors} with \\(N=1\\).\n\\end{proof}\n\\section{Quotient by the alternating group}\nIdeally, one would like to apply the argument of the previous section to the rational double cover $$A^{n-1} / \\mathfrak A_n \\dashrightarrow \\Kum_{n-1}(A).$$ However, there is no smooth projective model \\(Z\\) of $A^{n-1} / \\alt_n$ for which \\(H^3(Z, \\Z)_{\\tors}\\) is known.\n\nLet $U \\coloneqq K_{n-1}(A) \\cap (A^n\\setminus \\Delta_2) \\subset A^n$ and $V \\coloneqq \\Kum_{n-1}(A) \\cap (A^{[n]} \\setminus D_2) \\subset A^{[n]}$.\nFor codimension reasons, avoiding the smaller diagonals does not change $H^3(-, \\mathbb Z)_{\\tors}$:\n\\begin{lemma}\\label{lem:torsvkum}\n We have\n $H^3(V, \\mathbb Z)_{\\tors} \\simeq H^3(\\Kum_{n-1}(A), \\mathbb Z)_{\\tors}.$\n\\end{lemma}\n\\begin{proof}\nWe consider the stratification \n\\[D_{2}= Z_0 \\supset Z_1\\supset \\cdots \\]\nwhere \\(Z_{i+1}\\) is the singular locus of \\(Z_i\\). By the long exact sequence of relative cohomology and the Thom isomorphism, we have\n\\[H^3(\\Kum_{n-1}\\setminus Z_{1},\\Z)\\simeq H^3(\\Kum_{n-1}\\setminus Z_{2},\\Z) \\simeq \\dots \\simeq H^3(\\Kum_{n-1}(A), \\mathbb Z),\\]\nsee \\cite[Lemma 11.13]{Voisin_book_I}. By the same argument we also have the exact sequence \n$$ 0\\to H^3(\\Kum_{n-1} \\setminus Z_1,\\mathbb Z) \\to H^3( \\Kum_{n-1}(A) \\setminus Z_0, \\mathbb Z) \\to \\mathbb Z^{\\pi_0(Z_0 \\setminus Z_1)}.$$\nThus \n\\[H^3(\\Kum_{n-1}(A),\\Z)_{\\tors}\\simeq H^3(\\Kum_{n-1}(A)\\setminus Z_0,\\Z)_{\\tors}.\\]\nSince $V = \\Kum_{n-1}(A) \\setminus Z_0$, we conclude.\n\\end{proof}\nSince the alternating group acts freely on $U$, the relation between the quotient $U / \\alt_n$ and the open subset $V \\subset \\Kum_{n-1}(A)$ can be made precise as follows:\n\\begin{lemma}\\label{lem:doublecover_V}\nThere exists a ramified double cover \n $$\\pi \\colon \\Bl_{\\overline{\\Delta}} (U/\\alt_n) \\to V,$$\n where \\(\\overline{\\Delta}\\) is the image in \\(U/\\alt_n\\) of \\(\\Delta\\subset U\\).\n\\end{lemma}\n\\begin{proof}\nBy \\cite[Sec.\\ 8, p.\\ 770]{beauville}, we have\n\\[(\\Bl_{\\Delta} U )/ \\mathfrak S_n \\simeq V\\]\nMoreover, since $\\mathfrak A_n$ acts freely on $U$ and stabilizes $\\Delta$, we have\n$$\\Bl_{\\Delta}(U) / \\mathfrak A_n \\simeq \\Bl_{\\overline{\\Delta}}(U / \\mathfrak A_n).$$\nThe action of \\(\\Z/2\\Z\\simeq\\mathfrak{S}_n/\\alt_n\\) induced onto the quotient \\(\\Bl_{\\Delta}(U) / \\mathfrak A_n \\) produces a double cover\n $$\\Bl_{\\overline\\Delta}(U / \\mathfrak A_n) \\simeq (\\Bl_{\\Delta} U ) / \\mathfrak A_n \\to (\\Bl_{\\Delta} U) / \\mathfrak S_n \\simeq V,$$\n which is what we wanted.\n \\end{proof}\n\n\\begin{proposition}\\label{prop:spectral_sequence_argument}\n If there are integers $N_p$ for $1 \\leq p \\leq 3$ such that $$N_p \\cdot H^p(\\mathfrak A_n, H^{3-p}(K_{n-1}(A), \\mathbb Z)) = 0,$$ then we have\n $$2N_1N_2N_3 \\cdot H^3(\\Kum_{n-1}(A), \\mathbb Z)_{\\tors} = 0.$$\n\\end{proposition}\n\\begin{proof}\nSince the action of $\\mathfrak A_n$ on $U$ is free, there is the Cartan--Leray spectral sequence\n$$H^p(\\mathfrak A_n, H^q(U, \\mathbb Z)) \\Rightarrow H^{p+q}(U / \\mathfrak A_n, \\mathbb Z).$$\nNote that for $p \\leq 6$, by the same argument as in Lemma \\ref{lem:torsvkum} we have $H^p(U, \\mathbb Z) \\simeq H^p(K_{n-1}(A), \\mathbb Z)$, as the complement of $U \\subset K_{n-1}(A)$ is of codimension four. In particular, $H^0(\\mathfrak A_n, H^3(U, \\mathbb Z)) \\simeq H^3(K_{n-1}(A))^{\\mathfrak A_n}$ is torsion-free. \nIt then follows that $H^3(U / \\mathfrak A_n, \\mathbb Z)_{\\tors}$ is an extension of subquotients of $H^{p}(\\mathfrak A_n, H^{3-p}(K_{n-1}(A, \\mathbb Z)))$ for $1 \\leq p \\leq 3$, which implies that \\(N_1 N_2 N_3 \\cdot H^3(U / \\mathfrak A_n, \\mathbb Z)_{\\tors} = 0\\) .\nAs blowing-up smooth subvarieties leaves $H^3(-, \\mathbb Z)_{\\tors}$ invariant, we have\n$$H^3(\\Bl_{\\overline{\\Delta}}(U / \\mathfrak A_n), \\mathbb Z)_{\\tors} \\simeq H^3(U / \\mathfrak A_n, \\mathbb Z)_{\\tors}.$$\nBy applying \\cite[Thm.\\ 5.4]{aguilarprieto} to the double cover of Lemma \\ref{lem:doublecover_V} $$\\pi \\colon \\Bl_{\\overline\\Delta}(U / \\mathfrak A_n) \\to V,$$\nwe have that \\(\\pi_* \\pi^* \\alpha =2 \\alpha\\) for any \\(\\alpha \\in H^*(V,\\Z)\\), which allows us to conclude by Lemma \\ref{lem:torsvkum}.\n\\end{proof}\n\n\\begin{theorem*}[Theorem \\ref{thm:main}]\nIf \\(n \\neq 6\\), then\n $$ \\gcd(2^3, n) \\cdot H^3(\\Kum_{n-1}(A), \\mathbb Z)_{\\tors} = 0.$$\nIf instead \\(n=6\\), then\n$$6 \\cdot H^3(\\Kum_{5}(A), \\mathbb Z)_{\\tors} = 0.$$\n\\end{theorem*}\n\n\\begin{proof}\nOn the one hand, we have $n \\cdot H^3(\\Kum_{n-1}(A), \\mathbb Z)_{\\tors} = 0$ by Corollary \\ref{cor:n-tors_of_Kum}. Note that this already implies the result for $n \\in \\{4, 6, 8\\}$. On the other hand, by combining Proposition \\ref{prop:spectral_sequence_argument} and Proposition \\ref{prop:spectralval}, we have $2^3 \\cdot H^3(\\Kum_{n-1}(A), \\mathbb Z)_{\\tors} = 0$ for $n = 3$ or $n \\geq 8$ and $2 \\cdot 12^3 \\cdot H^3(\\Kum_{n-1}(A), \\mathbb Z)_{\\tors} = 0$ for $n=5,7$. The claim then follows by taking the greatest common divisor of $n$ and $2^3$ (respectively \\ $2 \\cdot 12^3$).\n\\end{proof}\n\n\\begin{corollary}\\label{cor:nice_statement_even}\n    The third integral cohomology group of a generalized Kummer variety of dimension divisible by four is torsion-free.\n\\end{corollary}", "post_theorem_intro_text_len": 2336, "post_theorem_intro_text": "For odd values of \\(n\\), this amounts to the following:\n\\begin{corollary}\\label{cor:nice_statement_even}\n    The third integral cohomology group of a generalized Kummer variety of dimension divisible by four is torsion-free.\n\\end{corollary}\nAs already mentioned, the case \\(n=2\\) follows from \\cite{kapfermenet}. This result confirms folklore expectations, as stated for example in \\cite[Rem.\\ 2.1.(ii)]{huyb2024}\n\nThe strategy of the proof is inspired by and generalizes the approach of \\cite{kapfermenet}:\nwe first consider a rational cover of degree \\(n\\) by a smooth projective variety with no torsion in degree three cohomology, which forces \\(H^3(\\Kum_{n-1}(A),\\Z)_{\\tors}\\) to be entirely of \\(n\\)-torsion. By considering a double cover of a suitable open of \\(\\Kum_{n-1}(A)\\), we then show that \\(H^3(\\Kum_{n-1}(A),\\Z)_{\\tors}\\) is a $2$-group, via an analysis of the cohomology of the alternating group and the Cartan--Leray spectral sequence. When \\(n\\) is odd, we obtain Corollary \\ref{cor:nice_statement_even}.\nWe are then left with asking ourselves the following\n\n\\begin{question}\n    Does \\(H^3(\\Kum_{2n+1}(A),\\Z)_{\\tors}\\) vanish for \\(n \\geq 1\\) as well?\n\\end{question}\n\n\\subsection{Acknowledgements}\nThe authors would like to thank Daniel Huybrechts and Emanuele Macrì for their interest in the project, as well as Nick Addington for sharing his computations and suggesting that earlier bounds could be improved. The first author would like to thank Claire Voisin for the invitation to Paris, where this project was initiated. \nMany thanks to Daniel Huybrechts and Paolo Stellari, respectively, for bringing up Question \\ref{que:hk_torsion} at the ``K3 surfaces and friends\"\\footnote{Lorentz Center, Leiden, June 2025  (\\href{https://www.lorentzcenter.nl/k3-surfaces-en-friends-brauer-groups-and-moduli.html}{\\underline{link}}) }\nand PRAGMATIC\\footnote{University of Catania, Italy, September 2025 (\\href{https://www.dmi.unict.it/pragmatic/docs/Pragmatic2025.html}{\\underline{link}})} summer schools. We thank the organizers of said events for the wonderful research environment.\nBoth authors were supported by the ERC Synergy Grant 854361 HyperK.\nThe first author is grateful for the support provided by the International Max Planck Research School on Moduli Spaces at the Max Planck Institute for Mathematics in Bonn.", "sketch": "The strategy of the proof (inspired by and generalizing \\,\\cite{kapfermenet}) is as follows. To prove Theorem~\\ref{thm:main}, one first considers “a rational cover of degree \\(n\\) by a smooth projective variety with no torsion in degree three cohomology,” which “forces \\(H^3(\\Kum_{n-1}(A),\\Z)_{\\tors}\\) to be entirely of \\(n\\)-torsion.” Then, “by considering a double cover of a suitable open of \\(\\Kum_{n-1}(A)\\),” one shows that \\(H^3(\\Kum_{n-1}(A),\\Z)_{\\tors}\\) “is a \\(2\\)-group,” using “an analysis of the cohomology of the alternating group and the Cartan--Leray spectral sequence.” When \\(n\\) is odd, combining these conclusions yields Corollary~\\ref{cor:nice_statement_even}.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:main}\nIf \\(n \\neq 6\\), then\n    $$ \\gcd(2^3, n) \\cdot H^3(\\Kum_{n-1}(A), \\mathbb Z)_{\\tors} = 0.$$\nIf instead \\(n=6\\), then\n$$6 \\cdot H^3(\\Kum_{5}(A), \\mathbb Z)_{\\tors} = 0.$$,", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let \\(A\\) be an abelian surface, and for an integer \\(n\\ge 2\\) let \\(\\Kum_{n-1}(A)\\) denote the corresponding generalized Kummer variety. Write \\(H^3(\\Kum_{n-1}(A),\\mathbb Z)_{\\tors}\\) for the torsion subgroup of its third integral cohomology. Which annihilation statement holds for this torsion subgroup?", "correct_choice": {"label": "A", "text": "If \\(n\\neq 6\\), then \\[\\gcd(2^3,n)\\cdot H^3(\\Kum_{n-1}(A),\\mathbb Z)_{\\tors}=0.\\] If \\(n=6\\), then \\[6\\cdot H^3(\\Kum_{5}(A),\\mathbb Z)_{\\tors}=0.\\]"}, "choices": [{"label": "B", "text": "For every integer \\(n\\ge 2\\), one has\n\\[2^3\\cdot H^3(\\Kum_{n-1}(A),\\mathbb Z)_{\\tors}=0.\\]"}, {"label": "C", "text": "If \\(n\\) is odd, then\n\\[H^3(\\Kum_{n-1}(A),\\mathbb Z)_{\\tors}=0.\\]"}, {"label": "D", "text": "If \\(n\\neq 6\\), then\n\\[\\operatorname{lcm}(2^3,n)\\cdot H^3(\\Kum_{n-1}(A),\\mathbb Z)_{\\tors}=0.\\]\nIf \\(n=6\\), then\n\\[6\\cdot H^3(\\Kum_{5}(A),\\mathbb Z)_{\\tors}=0.\\]"}, {"label": "E", "text": "If \\(n\\neq 6\\), then\n\\[\\gcd(2^2,n)\\cdot H^3(\\Kum_{n-1}(A),\\mathbb Z)_{\\tors}=0.\\]\nIf \\(n=6\\), then\n\\[2\\cdot H^3(\\Kum_{5}(A),\\mathbb Z)_{\\tors}=0.\\]"}], "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": "exceptional_case_n_eq_6_and_gcd_dependence", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "full_annihilator_replaced_by_odd_n_consequence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "gcd_replaced_by_lcm", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "sharp_2_primary_exponent_and_special_n_6_constant", "template_used": "wildcard"}]}}
{"id": "2512.14581v1", "paper_link": "http://arxiv.org/abs/2512.14581v1", "theorems_cnt": 1, "theorem": {"env_name": "ex", "content": "Two amplitudes in the spectral action matrix model are\n\\begin{align*}\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (0.45,-0.7) to (1,0);\n\\draw (1.55,-0.7) to (1,0);\n\\draw (1,0) arc (-90:270:0.5cm);\n\\node at (1,-0.55) {$i$};\n\\node at (0.2,0.8) {$j$};\nlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n\\end{tikzpicture}\n}\n\\quad=\\quad\\sum_{k=1}^N\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (0.45,-.7) to (1,0);\n\t\\draw (1.55,-.7) to (1,0);\n\t\\draw (1,0) arc (-90:270:0.5cm);\n\t\\node at (1,-.55) {$i$};\n\t\\node at (0.2,0.8) {$j$};\n\tlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n        \\node at (1,.575) {$k$};\n\\end{tikzpicture}\n}\\quad=\\quad\\sum_{k=1}^N\\frac{f'[\\lambda_j,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]},\n\\end{align*}\nand\n\\begin{align*}\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (0.45,-.7) to (1,0);\n\t\\draw (1.55,-.7) to (1,0);\n\t\\draw (1,0.054) arc (-90:270:0.3cm);\n\t\\draw (1,-0.035) arc (-90:270:.7cm);\n\t\\node at (1,-.55) {$i$};\n\t\\node at (0,1) {$j$};\n\tlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n\\end{tikzpicture}\n}\n\\quad=\\quad\\sum_{k,l=1}^N\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (0.45,-.7) to (1,0);\n\t\\draw (1.55,-.7) to (1,0);\n\t\\draw (1,0.054) arc (-90:270:0.3cm);\n\t\\draw (1,-0.035) arc (-90:270:.7cm);\n\t\\node at (1,-.55) {$i$};\n\t\\node at (0,1) {$j$};\n\t\\node at (1,1) {$k$};\n\t\\node at (1,0.4) {$l$};\n\tlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n\\end{tikzpicture}\n}\\quad=\\quad\\sum_{k,l=1}^N\\frac{f'[\\lambda_j,\\lambda_k,\\lambda_l,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]f'[\\lambda_k,\\lambda_l]}.\n\\end{align*}", "start_pos": 75967, "end_pos": 77444, "label": null}, "ref_dict": {"sct:main2": "\\begin{tikzpicture}[thick]\n\\draw (-1.5,0) to (-1,0);\n\\draw (-1,0) to[out=90,in=90] (0,0);\n\\draw (-1,0) to[out=-90,in=-90] (0,0);\n\\draw (1,0) to[out=90,in=90] (0,0);\n\\draw (1,0) to[out=-90,in=-90] (0,0);\n\\draw (1,0) to (1.5,0);\n\\vertex{-1,0};\n\\vertex{0,0};\n\\vertex{1,0};\n\\node at (-1.65,0) {\\footnotesize $1$};\n\\node at (1.65,0) {\\footnotesize$2$};\n\\end{tikzpicture}$$\ndoes not appear, because its two unbroken faces do not share an edge.\n\\end{rem}\n\n\\section{Power counting when allowing eigenvalues with vanishing derivative}\\label{sct:main2}\nIn our first main theorem, Theorem \\ref{thm:main}, we assumed that $f'[\\lambda_{k_1},\\ldots,\\lambda_{k_n}]\\neq0$. With $\\{\\lambda_k\\}_{k\\geq1}$ in general position, this assumption is typically satisfied. However, there will be at least one~${x\\in\\R}$ such that $f'(x)=0$, which means that the assumption of nonvanishing divided differences is sensitive to infinitesimal changes in $\\{\\lambda_k\\}_{k\\geq1}$. In fact, we shall show that the actual order of divergence of many graphs is sensitive to the question of whether or not $f'$ vanishes on $\\{\\lambda_k\\}_{k\\geq1}$.\n\nWe will refer to faces with an index $i_0$ such that $f'(\\lambda_{i_0})=0$ as \\emph{0-faces} or \\emph{singular faces}, and similarly for indices and eigenvalues.\nAs in Section~\\ref{sect: lower}, given a subset $\\bfr\\subseteq\\mathcal U$ of the set of unbroken faces of a Feynman ribbon graph~$G$, we let~$G_{\\bfr}$ be the graph obtained from $G$ by artificially declaring the faces in $\\mathfrak{b}$ to be broken. Elements of $\\mathfrak{b}$ are called 0-faces.\\begin{thm}\\label{thm:main2}\nLet $f\\in C^{\\infty}(\\mathbb{R})_\\R$ be an even function with $f'$ of precise order $-p-1$ for some $p \\in\\mathbb{R}_{\\ge0}$.\nLet~$\\{\\lambda_k\\}_{k=1}^\\infty$ be a sequence of real numbers with constants $K,c_1,c_2>0$ such that $c_1k^{1/d}\\leq |\\lambda_k|\\leq c_2k^{1/d}$ for all $k\\geq K$. Assume that for all $k,l\\in\\N_{\\geq1}$ we have $f'[\\lambda_{k},\\lambda_{l}]\\neq0$. Then, for any Feynman ribbon graph $G=(G^0,n,G^1)$ whose vertices have valence $\\geq3$, for all external indices $i_1,\\ldots,i_n\\in\\N_{\\geq1}$, there exist $M,c_4>0$ such that for all~$N\\geq M$,\n$$|\\Ampl_{N,i_1,\\ldots,i_n}(G)|\\leq c_4 N^{\\tilde\\omega(G)},$$\n$$\\tilde \\omega(G):=\\max_{\\bfr\\subseteq\\mathcal U}\\omega_{\\bfr}(G_{\\bfr}):=\\max_{\\bfr\\subseteq\\mathcal U}(U^{\\bfr}+\\frac{p}{d}(E^{\\bfr}_{\\textnormal{fi}}-V^{\\bfr}_{\\textnormal{fi}})+\\frac{p+1}{d}(E^{\\bfr}_{10}-V^{\\bfr}_{10})),$$\nwhere $U^{\\bfr}$ is the number of unbroken faces of $G_{\\bfr}$, $E^{\\bfr}_{\\textnormal{fi}}$ is the number of fully internal edges of $G_{\\bfr}$ (propagators bordered on both sides by unbroken faces) and $V^{\\bfr}_{\\textnormal{fi}}$ is the number of fully internal vertices of $G_{\\bfr}$ (vertices bordered on all sides by unbroken faces). Respectively, $E^{\\bfr}_{10}$ and $V^{\\bfr}_{10}$ are the number of edges and vertices of~$G_{\\bfr}$ that border exactly one 0-face and for the rest unbroken faces.\n\\end{thm}\n\\begin{proof}\nUsing the precise order of $f'$, there exists $R>0$ such that, for all $x\\in\\mathbb{R}\\setminus[-R,R]$ and all~$0\\le k\\le \\max_{v\\in G^{0}}\\;\\deg(v)$, equation \\eqref{eqn: precise_order} is satisfied and the conclusion of Lemma \\ref{lem:zero-index bounds} holds. Let~$i_R$ be the lowest index such that $|\\lambda_i| > R$ for $i\\ge i_R$. For brevity we assume -- without loss of generality -- that $f'(\\lambda_0)=0$ and $f'(\\lambda_k)\\neq0$ for~${k\\geq 1}$. As in the proof of Lemma~\\ref{lem: restricted_lower_sufficient}, a sum splitting argument yields\n\\begin{equation*}\n    \\Ampl_{N,i_1,\\ldots,i_n}(G) = \\sum_{\\bfr\\subseteq \\mathcal{U}}\\sum_{\\gamma:\\bfr\\to\\{0,\\dotsc,i_R-1\\}}\\Ampl^{\\ge i_R}_{N,{i_1},\\dotsc,i_n,\\gamma}(G_{\\bfr}).\n\\end{equation*}\nWe claim that it suffices to show that \\begin{align}\\label{eqn: ampl_Gbgamma}\n|\\Ampl^{\\ge i_R}_{N,{i_1},\\dotsc,i_n,\\gamma}(G_{\\bfr})|&\\lesssim N^{\\omega_{\\bfr,\\gamma}(G_{\\bfr})},\\\\\n\\omega_{\\bfr,\\gamma}(G_{\\bfr})&:=(U^{\\bfr}+\\frac{p}{d}(E_{\\textnormal{fi}}^b-V_{\\textnormal{fi}}^b)+\\frac{p+1}{d}(E_{10}^{\\bfr,\\gamma}-V_{10}^{\\bfr,\\gamma})),\\nonumber\n\\end{align}\nas the following graph-theoretical argument shows that $\\omega_{\\bfr,\\gamma}(G_{\\bfr})\\le \\omega_{\\bfr}(G_{\\bfr})$ for all such $\\gamma$. In particular, if we let $\\gamma_0$ denote the map that sends all artificially broken faces to the zero index, then $\\omega_{\\bfr,\\gamma_0}(G_{\\bfr})= \\omega_{\\bfr}(G_{\\bfr})$, hence this bound is sharp.\n\nMore generally, we have $\\omega_{\\bfr}(G_{\\bfr,\\gamma})\\leq\\omega_{\\bfr}(G_{\\bfr,\\gamma'})$ whenever $\\gamma'$ is obtained from $\\gamma$ by setting \\begin{equation*}\n    \\gamma'(b)=\\begin{cases}\n        0,&b=b', \\\\\n        \\gamma(b),&\\text{else},\n    \\end{cases}\n\\end{equation*}\nfor some $b'$ with $\\gamma(b')\\neq0$. Indeed, as the values $U^{\\bfr}$, $\\Efi^b$, and $\\Vfi^b$ do not depend on $\\gamma$, it suffices to show \\begin{equation*}\nE_{10}^{\\bfr,\\gamma}-V_{10}^{\\bfr,\\gamma}\\leq E_{10}^{\\bfr,\\gamma'}-V_{10}^{\\bfr,\\gamma'}.\n\\end{equation*}\nTowards this, first note that $E_{10}^{\\bfr,\\gamma'}-E_{10}^{\\bfr,\\gamma}$ equals the number of edges that border the face $b'$ on the one side and border an unbroken face (i.e.\\ an element of $\\mathcal{U}\\setminus\\bfr$) on the other. Moreover, $V_{10}^{\\bfr,\\gamma'}-V_{10}^{\\bfr,\\gamma}$ equals the number of vertices that border $b'$ and for the rest only unbroken faces. For each distinct vertex of the latter kind there is at least one distinct edge of the former kind, implying that $$ V_{10}^{\\bfr,\\gamma'}-V_{10}^{\\bfr,\\gamma}\\leq E_{10}^{\\bfr,\\gamma'}-E_{10}^{\\bfr,\\gamma},$$which concludes the argument. \n\nIt remains to prove~\\eqref{eqn: ampl_Gbgamma}. The strategy is similar to the proof of Theorem \\ref{thm:upper}, but with different bounds for the edges and vertices, in which we distinguish three different types of indices and their corresponding eigenvalues.\nWe distinguish between (i) eigenvalues $\\lambda_0$ such that $f'(\\lambda_0)=0$, (ii) eigenvalues $\\lambda_i$ for which $f'(\\lambda_i)\\neq0$ and the index $i<R$ is fixed, and (iii) eigenvalues $\\lambda_k$ for which $f'(\\lambda_k)\\neq0$ and the index $k\\geq R$ is running. Analogously to the proof of Theorem \\ref{thm:upper}, noting Remark \\ref{rem:apparent shortcut} and its resolution, the above estimate follows from the bounds on edges and vertices that are summarized and proved in Lemma \\ref{lem:final bounds} below, where we stress that all constants that arise are independent of the running indices.\n\\end{proof}\n\n\\begin{lem}\\label{lem:final bounds}\nLet $n\\geq3$. Let $f,\\{\\lambda_k\\}_{k=1}^\\infty$ be as in Theorem \\ref{thm:main2}. Let $\\lambda_0\\in\\R$ be such that $f'(\\lambda_0)=0$. There exist $c_1,c_2>0$ such that we have\n\\begin{multicols}{2}\n\\begin{enumerate}\n\\item $|f'\\{\\lambda_{k_1},\\lambda_{k_2}\\}|>c_1|\\lambda_{k_1}|^{-p}$,\n\\item $|f'\\{\\lambda_{i},\\lambda_{k}\\}|>c_1$,\n\\item $|f'\\{\\lambda_{i_1},\\lambda_{i_2}\\}|>c_1$,\n\\item $|f'\\{\\lambda_0,\\lambda_k\\}|> c_1|\\lambda_k|^{-p-1}$,\n\\item $|f'\\{\\lambda_0,\\lambda_i\\}|>c_1$,\n\\item $|f'\\{\\lambda_0,\\lambda_0\\}|>c_1$,\n\\item $|f'\\{\\lambda_{k_1},\\ldots,\\lambda_{k_n}\\}|\\leq c_2|\\lambda_{k_1}|^{-p}$,\n\\item $|f'\\{\\lambda_0,\\lambda_{k_2},\\ldots,\\lambda_{k_n}\\}|\\leq c_2|\\lambda_{k_2}|^{-p-1}$,\n\\item $|f'\\{\\lambda_{j_1},\\ldots,\\lambda_{j_n}\\}|\\leq c_2$,\nll}] \nll}] \nll}]\n\\end{enumerate}\n\\end{multicols}\n\\noindent\nfor all $i,i_1,i_2\\leq i_R$, $k,k_1,\\ldots,k_n\\geq i_R$ such that $f'(\\lambda_j)\\neq0$ for $j\\in\\{i,i_1,i_2,k,k_1,\\ldots,k_n\\}$ and such that $|\\lambda_{k_1}|\\leq\\cdots\\leq|\\lambda_{k_n}|$, and for all $j_1,\\ldots,j_n\\in\\N$.\n\\end{lem}\n\n\\begin{proof}\nThe estimates 1.\\ and 7.\\ follow from Theorem~\\ref{thm:weighted_divdiff_bound}, 4.\\ and 8.\\ follow from Lemma~\\ref{lem:zero-index bounds}, and 9.\\ was shown in Corollary~\\ref{cor:wdivdifs are bounded}. Estimates 3., 5., and 6.\\ follow from the fact that there is a finite amount of indices $i<i_R$, and for each $i_1,i_2<i_R$ the (weighted) divided difference is nonzero. {Estimate 2.\\ follows from Theorem~\\ref{thm:no_vanishing_divdifs}.}\n\\end{proof}\nWe conclude this section with some remarks and examples illustrating the influence of 0-faces on the behaviour of amplitudes. \n\n\\begin{rem}\nThe following explains the absence of a lower bound in Theorem \\ref{thm:main2}. Suppose $i_1,i_2,i_3$ are such that $f'[i_1,i_2,i_3]=0$, $f'(\\lambda_{i_1})\\neq0$, and $f'(\\lambda_{i_3})\\neq0$. We then compute the amplitude\n\\begin{align*}\n\\raisebox{-19.5pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0.5) to (0,0);\n\\draw (-0.5,-.5) to (0,0);\n\\draw (0,0) to[out=60,in=120] (1,0);\n\\draw (0,0) to[out=-60,in=-120] (1,0);\n\\draw (1,0) to (1.5,0);\n\\node at (0,0.5) {$i_1$};\n\\node at (-.5,0) {$i_2$};\n\\node at (0,-.5) {$i_3$};\n\\vertex{0,0};\n\\vertex{1,0};\n\\end{tikzpicture}", "sct:Definitions": "\\begin{align}\\label{eq:power counting formula 2}\n\\tilde \\omega(G)=\\max(U^{\\bfr}+\\frac{p}{d}(E^{\\bfr}_{\\textnormal{fi}}-V^{\\bfr}_{\\textnormal{fi}})+\\frac{p+1}{d}(E^{\\bfr}_{10}-V^{\\bfr}_{10})),\n\\end{align}\nin which the maximum is taken over all subsets $\\bfr$ of unbroken faces, $U^{\\bfr},\\Efi^b,\\Vfi^b$ are as before when designating the elements of $\\bfr$ as broken, and $V^{\\bfr}_{10}/E^{\\bfr}_{10}$ denotes the number of vertices/edges bordering only unbroken faces except for either exactly one element of $\\bfr$ or exactly one external index $i_0$ satisfying~${f'(\\lambda_{i_0})=0}$.\n\n\\paragraph{Consequences of \\eqref{eq:power counting formula}}\nNote that $\\Efi-\\Vfi\\geq 0$ for any graph. A remarkable result is that a function~$f$ with \\textit{faster} decay results in a \\textit{higher} degree of divergence of the graph. This is not obvious from the definition of the Feynman rules, even for simple graphs like in Example 1, and even for concrete functions like $f(x)=x^{-p}$.\nBecause divided differences of such functions may well be negative, one might \\textit{a priori} expect cancellations of terms damping the degree of divergence, but this does not happen.\n\nAnother corollary of our power counting formula is the following observation. Among the connected graphs with~$n\\geq1$ external edges and $L$ loops, the maximal value of $U$ is $L-1$, and the maximal value of $\\Efi-\\Vfi$ is $L-1$. The maximal value $\\omega(G)=L-1+\\frac{p}{d}(L-1)$ is attained by a nonempty set of diagrams, by virtue of our lower bound. These diagrams with maximal divergence are precisely the planar diagrams that cannot be split into two connected components by removing one vertex and all external edges, and moreover have only one unbroken face -- cf.\\ Remark \\ref{rem:2-point 2-loop}.\n\\paragraph{Consequences of \\eqref{eq:power counting formula 2}}\nThe situation where divided differences of $f'$ may vanish is more complicated (cf. Section \\ref{sct:main2}), but \\eqref{eq:power counting formula 2} leads us to conjecture that the diagrams with maximal divergence are of the same planar form as before -- cf.\\ Remark \\ref{rem:conjectures}. As we discuss in Remark \\ref{rem:UV/IR}, the influence of modes $\\lambda_i$ with $f'(\\lambda_i)=0$ is reminiscent of the UV/IR behavior of scalar field theories on noncommutative spacetime, which underlines the need for a rigorous renormalization analysis of the spectral action beyond the weak field approximation.\n\n\\paragraph{Techniques} To prove the above power counting formulas, we introduce several techniques in the asymptotic analysis of divided differences, and prove upper and lower bounds for them, which appear to be novel. A pivotal concept throughout the proof is the weighted divided difference\n$$f'\\{x_1,\\ldots,x_n\\}:=(-1)^n x_1\\cdots x_n f'[x_1,\\ldots,x_n].$$\nIts asymptotic behavior is more easily understood, because sending any of its variables to infinity yields another weighted divided difference.\nFurthermore, a key result is that weighted divided differences are positive on large enough subsets of $\\R^n$ for the functions $f$ we are considering. We thus generalize and give a new proof for Hunter's positivity theorem \\cite{Hunter1977}, which is recovered by taking $f(x)=x^{-p}$ ($p\\in 2\\N$). Moreover, we show that for functions bounded above or below by $x^{-p}$, with similar bounds on their derivatives, the asymptotic behavior of the weighed divided differences away from the origin is determined by the variable with the smallest modulus, which is crucial in the proof of our main theorem.\n\nOur main theorem is proved by combining this positivity and the mentioned upper and lower bounds with an interesting graph-theoretical argument.\n\n\\paragraph{Acknowledgements}\nWe are grateful to Martijn Caspers, Séverin Charbonnier, Harald Grosse, and Walter van Suijlekom for useful discussions. TvN thanks the Erwin Schr\\\"odinger Institute and the organizers and participants of the April 2023 conference 'Non-commutative Geometry meets Topological Recursion', where the motivation for this paper originated.\nTvN was supported by NWO project ‘Noncommutative multi-linear harmonic analysis and higher order spectral shift’,\nOCENW.M.22.070. EMH thanks the Max Planck Institute for Mathematics in Bonn for its financial support.\n\n\\section{Definitions}\n\\label{sct:Definitions}\n\nThroughout we fix a positive real number $d\\in\\R_{>0}$, and a sequence $\\{\\lambda_k\\}_{k=1}^\\infty\\subseteq\\R$ with the property that there exist numbers $K\\in\\N$, $c_1,c_2\\in\\R_{>0}$ such that\n\\begin{align}\\label{eq:eigenvalues asymptotics assumption}\n    c_1 k^{1/d}\\leq|\\lambda_k|\\leq c_2 k^{1/d}\n\\end{align}", "eq:power counting formula": "\\begin{align}\\label{eq:power counting formula}\n\\omega(G)= U+\\frac{p}{d}(E_{\\f}-V_{\\f}).\n\\end{align}", "rem:2-point 2-loop": "\\begin{rem}\\label{rem:2-point 2-loop}\nFrom Theorem \\ref{thm:main} we may derive, at each loop order, the set of relevant diagrams. For example, the set of 2-point 2-loop diagrams (with vertices of valence $\\geq3$) with maximal order of divergence is\n\\begin{align*}\n~&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.3,.6) arc (180:0:.3cm);\n\\draw (0,0) to[out=110,in=-90] (-.3,.6);\n\\draw (0,0) to[out=70,in=-90] (.3,.6);\n\\vertex{0,0};\n\\node at (-1.15,0) {\\footnotesize $1$};\n\\node at (1.15,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (0,0) to (0,.5);\n\\draw (0,.5) arc (-90:270:.3cm);\n\\vertex{0,0};\n\\vertex{0,.5};\n\\node at (-1.15,0) {\\footnotesize $1$};\n\\node at (1.15,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.3,.8) arc (180:360:.3cm);\n\\draw (0,1.4) to[out=-160,in=90] (-.3,.8);\n\\draw (0,1.4) to[out=-20,in=90] (.3,.8);\n\\vertex{0,0};\n\\vertex{0,1.4};\n\\node at (-1.15,0) {\\footnotesize $1$};\n\\node at (1.15,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (0,1.4) to (0,.9);\n\\draw (0,.9) arc (90:450:.3cm);\n\\vertex{0,0};\n\\vertex{0,1.4};\n\\vertex{0,.9};\n\\node at (-1.15,0) {\\footnotesize $1$};\n\\node at (1.15,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n\\\\\n+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,-.5) to (1,-.5);\n\\draw (0,-.5) to (0,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.3,.6) arc (180:0:.3cm);\n\\draw (0,0) to[out=110,in=-90] (-.3,.6);\n\\draw (0,0) to[out=70,in=-90] (.3,.6);\n\\vertex{0,0};\n\\vertex{0,-.5};\n\\node at (-1.15,-.5) {\\footnotesize $1$};\n\\node at (1.15,-.5) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,-.5) to (1,-.5);\n\\draw (0,-.5) to (0,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (0,0) to (0,.5);\n\\draw (0,.5) arc (-90:270:.3cm);\n\\vertex{0,0};\n\\vertex{0,.5};\n\\vertex{0,-.5};\n\\node at (-1.15,-.5) {\\footnotesize $1$};\n\\node at (1.15,-.5) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,-.5) to (1,-.5);\n\\draw (0,-.5) to (0,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.3,.8) arc (180:360:.3cm);\n\\draw (0,1.4) to[out=-160,in=90] (-.3,.8);\n\\draw (0,1.4) to[out=-20,in=90] (.3,.8);\n\\vertex{0,0};\n\\vertex{0,1.4};\n\\vertex{0,-.5};\n\\node at (-1.15,-.5) {\\footnotesize $1$};\n\\node at (1.15,-.5) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,-.5) to (1,-.5);\n\\draw (0,-.5) to (0,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (0,1.4) to (0,.9);\n\\draw (0,.9) arc (90:450:.3cm);\n\\vertex{0,0};\n\\vertex{0,1.4};\n\\vertex{0,.9};\n\\vertex{0,-.5};\n\\node at (-1.15,-.5) {\\footnotesize $1$};\n\\node at (1.15,-.5) {\\footnotesize$2$};\n\\end{tikzpicture}}\n\\\\\n+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (0,1.4) to (0,0);\n\\vertex{0,0};\n\\vertex{0,1.4};\n\\node at (-1.15,0) {\\footnotesize $1$};\n\\node at (1.15,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.7,.7) to (.7,.7);\n\\vertex{0,0};\n\\vertex{-.7,.7};\n\\vertex{.7,.7};\n\\node at (-1.15,0) {\\footnotesize $1$};\n\\node at (1.15,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,-.5) to (1,-.5);\n\\draw (0,-.5) to (0,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (0,1.4) to (0,0);\n\\vertex{0,0};\n\\vertex{0,1.4};\n\\vertex{0,-.5};\n\\node at (-1.15,-.5) {\\footnotesize $1$};\n\\node at (1.15,-.5) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,-.5) to (1,-.5);\n\\draw (0,-.5) to (0,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.7,.7) to (.7,.7);\n\\vertex{0,0};\n\\vertex{0,-.5};\n\\vertex{-.7,.7};\n\\vertex{.7,.7};\n\\node at (-1.15,-.5) {\\footnotesize $1$};\n\\node at (1.15,-.5) {\\footnotesize$2$};\n\\end{tikzpicture}}\n\\\\\n+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (-.8,0) to[out=45,in=90] (0,0);\n\\draw (-.8,0) to[out=-45,in=-90] (0,0);\n\\draw (.8,0) to (1.2,0);\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\node at (-1.35,0) {\\footnotesize $1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (-.8,0) to (-0.3,0);\n\\draw (-.3,0) to[out=45,in=90] (.5,0);\n\\draw (-.3,0) to[out=-45,in=-90] (.5,0);\n\\draw (.8,0) to (1.2,0);\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\vertex{-.3,0};\n\\node at (-1.35,0) {\\footnotesize$1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (0,.5) to[out=-160,in=180] (0,-.1);\n\\draw (0,.5) to[out=-20,in=0] (0,-.1);\n\\draw (.8,0) to (1.2,0);\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\vertex{0,.5};\n\\node at (-1.35,0) {\\footnotesize$1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (0,.5) to (0,.05);\n\\draw (0,.05) arc (90:450:.2cm);\n\\draw (.8,0) to (1.2,0);\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\vertex{0,.5};\n\\vertex{0,.05};\n\\node at (-1.35,0) {\\footnotesize$1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n\\\\\n+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (-.8,0) to[out=0,in=-90] (0,.5);\n\\draw (.8,0) to (1.2,0);\n\\vertex{0,.5};\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\node at (-1.35,0) {\\footnotesize $1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (.8,0) to[out=180,in=-90] (0,.5);\n\\draw (.8,0) to (1.2,0);\n\\vertex{0,.5};\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\node at (-1.35,0) {\\footnotesize $1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (.8,0) to (1.2,0);\n\\draw (-.4,.4) arc (180:360:.4cm);\n\\vertex{-.4,.4};\n\\vertex{.4,.4};\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\node at (-1.35,0) {\\footnotesize $1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\quad\"1\\leftrightarrow2\"\n\\\\\n+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (-.8,0) to (.8,0);\n\\draw (.8,0) to (1.2,0);\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\node at (-1.35,0) {\\footnotesize $1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}\n&+&\n\\raisebox{-15pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.2,0) to (-.8,0);\n\\draw (-.8,0) to[out=90,in=90] (.8,0);\n\\draw (-.8,0) to[out=-90,in=-90] (.8,0);\n\\draw (.8,0) to (1.2,0);\n\\draw (0,.5) to (0,-.5);\n\\vertex{0,.5};\n\\vertex{0,-.5};\n\\vertex{-.8,0};\n\\vertex{.8,0};\n\\node at (-1.35,0) {\\footnotesize $1$};\n\\node at (1.35,0) {\\footnotesize$2$};\n\\end{tikzpicture}}.\n\\end{align*}\nEach has an order of divergence of $2+\\frac{p}{d}$ (and no less).\nOne notices these diagrams are automatically planar. They are moreover connected and, though not necessarily 1PI, satisfy a similar connectivity property. Namely, their dual graphs stay connected after removing the vertices that correspond to the broken faces of the original graph. For instance, the diagram\n$$\\begin{tikzpicture}[thick]\n\\draw (-1.5,0) to (-1,0);\n\\draw (-1,0) to[out=90,in=90] (0,0);\n\\draw (-1,0) to[out=-90,in=-90] (0,0);\n\\draw (1,0) to[out=90,in=90] (0,0);\n\\draw (1,0) to[out=-90,in=-90] (0,0);\n\\draw (1,0) to (1.5,0);\n\\vertex{-1,0};\n\\vertex{0,0};\n\\vertex{1,0};\n\\node at (-1.65,0) {\\footnotesize $1$};\n\\node at (1.65,0) {\\footnotesize$2$};\n\\end{tikzpicture}$$\ndoes not appear, because its two unbroken faces do not share an edge.\n\\end{rem}", "rem:UV/IR": "\\begin{rem}\\label{rem:UV/IR}\nSmooth even functions $f$ satisfy $f'(0)=0$, and if $\\{\\lambda_k\\}$ is the spectrum of a typical Dirac operator, these eigenvalues correspond to modes of zero momentum. The fact that UV-divergences can become higher as external momenta vanish is reminiscent of noncommutative quantum field theory, cf.\\ \\cite{MRS2000}. For instance, in the naive version of noncommutative $\\phi^4_4$, such behavior was shown to lead to UV/IR mixing and, consequently, nonrenormalizability \\cite{CR2000,CR2001,GW2005a}. This prompted the Grosse--Wulkenhaar model which solved the UV/IR problem \\cite{GW2005b} and proved incredibly successful \\cite{DGMR2007,GW2014}.\n\nIn light of this, it seems instructive to determine whether UV/IR-mixing is present in the spectral action matrix model and its relatives. This is a difficult question to answer,\nrequiring a careful renormalization analysis and the passage to continuous spectrum.\n\\end{rem}", "rem:conjectures": "\\begin{rem}\\label{rem:conjectures}\nIn the setting of Theorem \\ref{thm:main2} and for $d<3$, the second graph of \\eqref{eq:3-point 4-loop} has maximal order among the 3-point 4-loop diagrams, but not every 3-point 4-loop diagram that has maximal order in the setting of Theorem \\ref{thm:main} has maximal order in the setting of Theorem \\ref{thm:main2}. That being said, we conjecture that, among the $n$-points $L$-loop graphs, every graph with maximal order in the setting of Theorem \\ref{thm:main2} has maximal order in the setting of Theorem \\ref{thm:main}. We moreover conjecture that the Ward identity, in the sense of \\cite{vNvS22b}, holds in both cases when restricting to the graphs of maximal order of divergence.\nThis might help generalize the results of \\cite{vNvS22b} to higher loop, which is a pressing open problem.\n\\end{rem}", "eq:power counting formula 2": "\\begin{align}\\label{eq:power counting formula 2}\n\\tilde \\omega(G)=\\max(U^{\\bfr}+\\frac{p}{d}(E^{\\bfr}_{\\textnormal{fi}}-V^{\\bfr}_{\\textnormal{fi}})+\\frac{p+1}{d}(E^{\\bfr}_{10}-V^{\\bfr}_{10})),\n\\end{align}", "eq:path integral": "\\begin{align}\\label{eq:path integral}\n\\frac{\\int_{H_N}V_{i_1j_1}\\cdots V_{i_mj_m}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V}{\\int_{H_N}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V},\n\\end{align}"}, "pre_theorem_intro_text_len": 5228, "pre_theorem_intro_text": "\\label{sect: intro}\nMotivated by the quest for a QFT description of the spectral action~\\cite{ChamseddineConnes1997} we would like to analyze the large $N$ behavior of the correlation functions\n\\begin{align}\\label{eq:path integral}\n\\frac{\\int_{H_N}V_{i_1j_1}\\cdots V_{i_mj_m}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V}{\\int_{H_N}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V},\n\\end{align}\nwhere $\\hbar$ is a formal parameter, $f:\\mathbb{R}\\to\\mathbb{R}$ is a sufficiently regular function, \n$D=\\operatorname{diag}(\\lambda_k)_{k=1}^\\infty$ is a self-adjoint operator diagonalized with respect to some countable orthonormal basis, and $H_N$ is the space of hermitian matrices acting on the first $N$ basis elements\\footnote{$H_N$ is equipped with the canonical measure coming from the real and imaginary parts of the components of $V$. At this point the integral is only formal.}.\n\nFollowing \\cite{vNvS21,vNvS22b,vNvS23}, we write \n$$\\Tr(f(D+V)-f(D))=\\sum_{n=1}^\\infty\\sum_{i_1,\\ldots,i_n=1}^N \\frac{1}{n}f'[\\lambda_{i_1},\\ldots,\\lambda_{i_n}]V_{i_1i_2}\\cdots V_{i_n i_1},$$ in terms of the divided differences of the derivative of $f$, defined inductively\nby $f'[x]:=f'(x)$, $f'[x,y]:=\\frac{f'(x)-f'(y)}{x-y}$, $f'[x,y,z]:=\\frac{f'[x,z]-f'[y,z]}{x-y}=\\frac{f'(x)-f'(z)}{(x-y)(x-z)}-\\frac{f'(y)-f'(z)}{(x-y)(y-z)}$, \\textit{et cetera}. By designating the second order term $\\frac12\\sum_{k,l=1}^Nf'[\\lambda_k,\\lambda_l]|V_{kl}|^2$ as the free theory, we can apply standard methods of Gaussian integration in order to express \\eqref{eq:path integral} as a combinatorial series whose terms are ribbon graph amplitudes \\cite{Eynard2016,vNvS23}. The respective ribbon graphs are generalized Kontsevich graphs in the terminology of~\\cite{BCEG}. Although \\cite{BCEG} considers polynomials $f$ (the order of which limits the valency of the vertices), the corresponding amplitudes turn out to be formally the same as in \\cite{vNvS22b}: these amplitudes are certain fractions of divided differences of $f'$.\n\nA stunning feature of these graph amplitudes, noted independently by \\cite{BCEG,vNvS22b}, is the Ward--Takahashi identity, which \\cite{vNvS22b} showed to imply one-loop renormalizability of the spectral action matrix model in the Gomis--Weinberg sense, raising the question what algebraic relations govern the graphs at arbitrary loop order, and if their renormalization flow can be understood. These questions, besides being inherently interesting \\cite{Azarfar2024,BG2016,HKPV2022,Perez2022,Perez2025,Steinacker2010,tHooft1982}, prepare for replacing the integration space~$H_N$ in \\eqref{eq:path integral} by a space of fields incorporating physical content such as the spectral Standard Model \\cite{ChamseddineConnes1997,CC2012,CIS2020}, and beyond \\cite{AMST2016,CCM2015,CCS2013,DM2017,Suijlekom2015}. Moreover, their positive answer would bring the spectral action more in line with noncommutative quantum field theory \\cite{GW2005a,GW2005b,GW2014,Riv2007}.\n\n\\paragraph{Feynman rules}\nWe summarize the Feynman rules referred to above --\nprecise definitions are in Section \\ref{sct:Definitions}. The Feynman diagrams are ribbon graphs for which not the edges but the faces are labeled by indices.\nFor each vertex bordered by faces with indices $i_1,\\ldots,i_n$, we multiply by a factor $f'[\\lambda_{i_1},\\ldots,\\lambda_{i_n}]$, e.g.,\n\\begin{align*}\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (-0.5,-0.5) to (0,0);\n\t\\draw (-0.5,0.5) to (0,0);\n\t\\draw (0.5,0.5) to (0,0);\n\t\\draw (0.5,-0.5) to (0,0);\n\t\\node at (-0.5,0) {$i$};\n\t\\node at (0,0.5) {$j$};\n\t\\node at (0.5,0) {$k$};\n\t\\node at (0,-0.5) {$l$};\n\tlldraw (0,0) circle (4pt);\nlldraw[white] (0,0) circle (2pt)\n\t;\n\\end{tikzpicture}}\n\\,\\quad&=\\quad f'[\\lambda_i,\\lambda_j,\\lambda_k,\\lambda_l],\\\\\n\\intertext{for each internal edge bordered by $i$ and $j$, we divide by a factor $f'[\\lambda_i,\\lambda_j]$, i.e.,}\n\\raisebox{-14pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (-0.5,-0.5) to (0.5,0.5);\n\t\\node at (-0.25,0.25) {$i$};\n\t\\node at (0.25,-0.25) {$j$};\n\\end{tikzpicture}}\n\\,\\quad&=\\quad\\frac{1}{f'[\\lambda_i,\\lambda_j]},\\\\\n\\intertext{and, finally, we sum over each unbroken face (face without external edges), e.g.,}\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);\n\\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);\n\\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);\n\\draw (-0.5,0) to (-0.8,0);\n\\draw (0.5,0.5) to (0.7,0.7);\n\\draw (0.5,-0.5) to (0.7,-0.7);\nlldraw (-0.5,0) circle (4pt);\nlldraw[white] (-0.5,0) circle (2pt)\n\t;\nlldraw (0.5,0.5) circle (4pt);\nlldraw[white] (0.5,0.5) circle (2pt)\n\t;\nlldraw (0.5,-0.5) circle (4pt);\nlldraw[white] (0.5,-0.5) circle (2pt)\n\t;\n\\end{tikzpicture}}\n\\,\\quad&=\\quad\\sum_{k=1}^N\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);\n\\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);\n\\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);\n\\draw (-0.5,0) to (-0.8,0);\n\\draw (0.5,0.5) to (0.7,0.7);\n\\draw (0.5,-0.5) to (0.7,-0.7);\n\\node at (0.15,0) {$k$};\nlldraw (-0.5,0) circle (4pt);\nlldraw[white] (-0.5,0) circle (2pt)\n\t;\nlldraw (0.5,0.5) circle (4pt);\nlldraw[white] (0.5,0.5) circle (2pt)\n\t;\nlldraw (0.5,-0.5) circle (4pt);\nlldraw[white] (0.5,-0.5) circle (2pt)\n\t;\n\\end{tikzpicture}}\\,\\,.\n\\end{align*}", "context": "\\label{sect: intro}\nMotivated by the quest for a QFT description of the spectral action~\\cite{ChamseddineConnes1997} we would like to analyze the large $N$ behavior of the correlation functions\n\\begin{align}\\label{eq:path integral}\n\\frac{\\int_{H_N}V_{i_1j_1}\\cdots V_{i_mj_m}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V}{\\int_{H_N}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V},\n\\end{align}\nwhere $\\hbar$ is a formal parameter, $f:\\mathbb{R}\\to\\mathbb{R}$ is a sufficiently regular function, \n$D=\\operatorname{diag}(\\lambda_k)_{k=1}^\\infty$ is a self-adjoint operator diagonalized with respect to some countable orthonormal basis, and $H_N$ is the space of hermitian matrices acting on the first $N$ basis elements\\footnote{$H_N$ is equipped with the canonical measure coming from the real and imaginary parts of the components of $V$. At this point the integral is only formal.}.\n\nFollowing \\cite{vNvS21,vNvS22b,vNvS23}, we write \n$$\\Tr(f(D+V)-f(D))=\\sum_{n=1}^\\infty\\sum_{i_1,\\ldots,i_n=1}^N \\frac{1}{n}f'[\\lambda_{i_1},\\ldots,\\lambda_{i_n}]V_{i_1i_2}\\cdots V_{i_n i_1},$$ in terms of the divided differences of the derivative of $f$, defined inductively\nby $f'[x]:=f'(x)$, $f'[x,y]:=\\frac{f'(x)-f'(y)}{x-y}$, $f'[x,y,z]:=\\frac{f'[x,z]-f'[y,z]}{x-y}=\\frac{f'(x)-f'(z)}{(x-y)(x-z)}-\\frac{f'(y)-f'(z)}{(x-y)(y-z)}$, \\textit{et cetera}. By designating the second order term $\\frac12\\sum_{k,l=1}^Nf'[\\lambda_k,\\lambda_l]|V_{kl}|^2$ as the free theory, we can apply standard methods of Gaussian integration in order to express \\eqref{eq:path integral} as a combinatorial series whose terms are ribbon graph amplitudes \\cite{Eynard2016,vNvS23}. The respective ribbon graphs are generalized Kontsevich graphs in the terminology of~\\cite{BCEG}. Although \\cite{BCEG} considers polynomials $f$ (the order of which limits the valency of the vertices), the corresponding amplitudes turn out to be formally the same as in \\cite{vNvS22b}: these amplitudes are certain fractions of divided differences of $f'$.\n\nA stunning feature of these graph amplitudes, noted independently by \\cite{BCEG,vNvS22b}, is the Ward--Takahashi identity, which \\cite{vNvS22b} showed to imply one-loop renormalizability of the spectral action matrix model in the Gomis--Weinberg sense, raising the question what algebraic relations govern the graphs at arbitrary loop order, and if their renormalization flow can be understood. These questions, besides being inherently interesting \\cite{Azarfar2024,BG2016,HKPV2022,Perez2022,Perez2025,Steinacker2010,tHooft1982}, prepare for replacing the integration space~$H_N$ in \\eqref{eq:path integral} by a space of fields incorporating physical content such as the spectral Standard Model \\cite{ChamseddineConnes1997,CC2012,CIS2020}, and beyond \\cite{AMST2016,CCM2015,CCS2013,DM2017,Suijlekom2015}. Moreover, their positive answer would bring the spectral action more in line with noncommutative quantum field theory \\cite{GW2005a,GW2005b,GW2014,Riv2007}.\n\n\\paragraph{Feynman rules}\nWe summarize the Feynman rules referred to above --\nprecise definitions are in Section \\ref{sct:Definitions}. The Feynman diagrams are ribbon graphs for which not the edges but the faces are labeled by indices.\nFor each vertex bordered by faces with indices $i_1,\\ldots,i_n$, we multiply by a factor $f'[\\lambda_{i_1},\\ldots,\\lambda_{i_n}]$, e.g.,\n\\begin{align*}\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n \\draw (-0.5,-0.5) to (0,0);\n \\draw (-0.5,0.5) to (0,0);\n \\draw (0.5,0.5) to (0,0);\n \\draw (0.5,-0.5) to (0,0);\n \\node at (-0.5,0) {$i$};\n \\node at (0,0.5) {$j$};\n \\node at (0.5,0) {$k$};\n \\node at (0,-0.5) {$l$};\n lldraw (0,0) circle (4pt);\nlldraw[white] (0,0) circle (2pt)\n ;\n\\end{tikzpicture}}\n\\,\\quad&=\\quad f'[\\lambda_i,\\lambda_j,\\lambda_k,\\lambda_l],\\\\\n\\intertext{for each internal edge bordered by $i$ and $j$, we divide by a factor $f'[\\lambda_i,\\lambda_j]$, i.e.,}\n\\raisebox{-14pt}{\n\\begin{tikzpicture}[thick]\n \\draw (-0.5,-0.5) to (0.5,0.5);\n \\node at (-0.25,0.25) {$i$};\n \\node at (0.25,-0.25) {$j$};\n\\end{tikzpicture}}\n\\,\\quad&=\\quad\\frac{1}{f'[\\lambda_i,\\lambda_j]},\\\\\n\\intertext{and, finally, we sum over each unbroken face (face without external edges), e.g.,}\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);\n\\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);\n\\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);\n\\draw (-0.5,0) to (-0.8,0);\n\\draw (0.5,0.5) to (0.7,0.7);\n\\draw (0.5,-0.5) to (0.7,-0.7);\nlldraw (-0.5,0) circle (4pt);\nlldraw[white] (-0.5,0) circle (2pt)\n ;\nlldraw (0.5,0.5) circle (4pt);\nlldraw[white] (0.5,0.5) circle (2pt)\n ;\nlldraw (0.5,-0.5) circle (4pt);\nlldraw[white] (0.5,-0.5) circle (2pt)\n ;\n\\end{tikzpicture}}\n\\,\\quad&=\\quad\\sum_{k=1}^N\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);\n\\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);\n\\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);\n\\draw (-0.5,0) to (-0.8,0);\n\\draw (0.5,0.5) to (0.7,0.7);\n\\draw (0.5,-0.5) to (0.7,-0.7);\n\\node at (0.15,0) {$k$};\nlldraw (-0.5,0) circle (4pt);\nlldraw[white] (-0.5,0) circle (2pt)\n ;\nlldraw (0.5,0.5) circle (4pt);\nlldraw[white] (0.5,0.5) circle (2pt)\n ;\nlldraw (0.5,-0.5) circle (4pt);\nlldraw[white] (0.5,-0.5) circle (2pt)\n ;\n\\end{tikzpicture}}\\,\\,.\n\\end{align*}", "full_context": "\\label{sect: intro}\nMotivated by the quest for a QFT description of the spectral action~\\cite{ChamseddineConnes1997} we would like to analyze the large $N$ behavior of the correlation functions\n\\begin{align}\\label{eq:path integral}\n\\frac{\\int_{H_N}V_{i_1j_1}\\cdots V_{i_mj_m}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V}{\\int_{H_N}e^{-\\hbar^{-1}\\Tr(f(D+V)-f(D))}d V},\n\\end{align}\nwhere $\\hbar$ is a formal parameter, $f:\\mathbb{R}\\to\\mathbb{R}$ is a sufficiently regular function, \n$D=\\operatorname{diag}(\\lambda_k)_{k=1}^\\infty$ is a self-adjoint operator diagonalized with respect to some countable orthonormal basis, and $H_N$ is the space of hermitian matrices acting on the first $N$ basis elements\\footnote{$H_N$ is equipped with the canonical measure coming from the real and imaginary parts of the components of $V$. At this point the integral is only formal.}.\n\nFollowing \\cite{vNvS21,vNvS22b,vNvS23}, we write \n$$\\Tr(f(D+V)-f(D))=\\sum_{n=1}^\\infty\\sum_{i_1,\\ldots,i_n=1}^N \\frac{1}{n}f'[\\lambda_{i_1},\\ldots,\\lambda_{i_n}]V_{i_1i_2}\\cdots V_{i_n i_1},$$ in terms of the divided differences of the derivative of $f$, defined inductively\nby $f'[x]:=f'(x)$, $f'[x,y]:=\\frac{f'(x)-f'(y)}{x-y}$, $f'[x,y,z]:=\\frac{f'[x,z]-f'[y,z]}{x-y}=\\frac{f'(x)-f'(z)}{(x-y)(x-z)}-\\frac{f'(y)-f'(z)}{(x-y)(y-z)}$, \\textit{et cetera}. By designating the second order term $\\frac12\\sum_{k,l=1}^Nf'[\\lambda_k,\\lambda_l]|V_{kl}|^2$ as the free theory, we can apply standard methods of Gaussian integration in order to express \\eqref{eq:path integral} as a combinatorial series whose terms are ribbon graph amplitudes \\cite{Eynard2016,vNvS23}. The respective ribbon graphs are generalized Kontsevich graphs in the terminology of~\\cite{BCEG}. Although \\cite{BCEG} considers polynomials $f$ (the order of which limits the valency of the vertices), the corresponding amplitudes turn out to be formally the same as in \\cite{vNvS22b}: these amplitudes are certain fractions of divided differences of $f'$.\n\nA stunning feature of these graph amplitudes, noted independently by \\cite{BCEG,vNvS22b}, is the Ward--Takahashi identity, which \\cite{vNvS22b} showed to imply one-loop renormalizability of the spectral action matrix model in the Gomis--Weinberg sense, raising the question what algebraic relations govern the graphs at arbitrary loop order, and if their renormalization flow can be understood. These questions, besides being inherently interesting \\cite{Azarfar2024,BG2016,HKPV2022,Perez2022,Perez2025,Steinacker2010,tHooft1982}, prepare for replacing the integration space~$H_N$ in \\eqref{eq:path integral} by a space of fields incorporating physical content such as the spectral Standard Model \\cite{ChamseddineConnes1997,CC2012,CIS2020}, and beyond \\cite{AMST2016,CCM2015,CCS2013,DM2017,Suijlekom2015}. Moreover, their positive answer would bring the spectral action more in line with noncommutative quantum field theory \\cite{GW2005a,GW2005b,GW2014,Riv2007}.\n\n\\paragraph{Feynman rules}\nWe summarize the Feynman rules referred to above --\nprecise definitions are in Section \\ref{sct:Definitions}. The Feynman diagrams are ribbon graphs for which not the edges but the faces are labeled by indices.\nFor each vertex bordered by faces with indices $i_1,\\ldots,i_n$, we multiply by a factor $f'[\\lambda_{i_1},\\ldots,\\lambda_{i_n}]$, e.g.,\n\\begin{align*}\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n \\draw (-0.5,-0.5) to (0,0);\n \\draw (-0.5,0.5) to (0,0);\n \\draw (0.5,0.5) to (0,0);\n \\draw (0.5,-0.5) to (0,0);\n \\node at (-0.5,0) {$i$};\n \\node at (0,0.5) {$j$};\n \\node at (0.5,0) {$k$};\n \\node at (0,-0.5) {$l$};\n lldraw (0,0) circle (4pt);\nlldraw[white] (0,0) circle (2pt)\n ;\n\\end{tikzpicture}}\n\\,\\quad&=\\quad f'[\\lambda_i,\\lambda_j,\\lambda_k,\\lambda_l],\\\\\n\\intertext{for each internal edge bordered by $i$ and $j$, we divide by a factor $f'[\\lambda_i,\\lambda_j]$, i.e.,}\n\\raisebox{-14pt}{\n\\begin{tikzpicture}[thick]\n \\draw (-0.5,-0.5) to (0.5,0.5);\n \\node at (-0.25,0.25) {$i$};\n \\node at (0.25,-0.25) {$j$};\n\\end{tikzpicture}}\n\\,\\quad&=\\quad\\frac{1}{f'[\\lambda_i,\\lambda_j]},\\\\\n\\intertext{and, finally, we sum over each unbroken face (face without external edges), e.g.,}\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);\n\\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);\n\\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);\n\\draw (-0.5,0) to (-0.8,0);\n\\draw (0.5,0.5) to (0.7,0.7);\n\\draw (0.5,-0.5) to (0.7,-0.7);\nlldraw (-0.5,0) circle (4pt);\nlldraw[white] (-0.5,0) circle (2pt)\n ;\nlldraw (0.5,0.5) circle (4pt);\nlldraw[white] (0.5,0.5) circle (2pt)\n ;\nlldraw (0.5,-0.5) circle (4pt);\nlldraw[white] (0.5,-0.5) circle (2pt)\n ;\n\\end{tikzpicture}}\n\\,\\quad&=\\quad\\sum_{k=1}^N\n\\raisebox{-18pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);\n\\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);\n\\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);\n\\draw (-0.5,0) to (-0.8,0);\n\\draw (0.5,0.5) to (0.7,0.7);\n\\draw (0.5,-0.5) to (0.7,-0.7);\n\\node at (0.15,0) {$k$};\nlldraw (-0.5,0) circle (4pt);\nlldraw[white] (-0.5,0) circle (2pt)\n ;\nlldraw (0.5,0.5) circle (4pt);\nlldraw[white] (0.5,0.5) circle (2pt)\n ;\nlldraw (0.5,-0.5) circle (4pt);\nlldraw[white] (0.5,-0.5) circle (2pt)\n ;\n\\end{tikzpicture}}\\,\\,.\n\\end{align*}\n\nFor renormalization purposes, it is relevant \\cite{CR2000,CR2001,GW2005a,GW2005b,ILV2012,KLV2014,LOR2015,Riv2007,RVW} to know the asymptotic behavior of these amplitudes as $N\\to\\infty$. The main objective of this paper is to prove the following formulas describing this asymptotic behavior.\n\n\\begin{rem}\\label{rem:apparent shortcut}\nWe explain here a problem with an apparent shortcut to the above proof. Indeed, given positive~${n_1,\\ldots,n_U}$ it is not hard to show (using $\\lambda_j\\sim j^{1/d}$) that\n\\begin{align}\\label{eq:sums like integrals}\n \\sum_{j_1=1}^N\\cdots\\sum_{j_U=1}^N \\lambda_{j_1}^{n_1}\\cdots\\lambda_{j_U}^{n_U}=\\O(N^{U+(n_1+\\ldots+n_U)/d}),\n\\end{align} \nexactly as in the case with integrals instead of sums. If one skips the graph-theoretical Lemma \\ref{lem:injection fi vertices to fi edges} and applies the estimates of Theorem \\ref{thm:weighted_divdiff_bound} for arbitrary bordering indices of the fully internal vertices and edges, one can estimate the amplitude by the sum on the left-hand side of \\eqref{eq:sums like integrals} where indeed the powers automatically add up to $n_1+\\ldots+n_U=p(\\Efi-\\Vfi)$ as required! However, the $n_1,\\ldots,n_U$ may not be positive, which invalidates \\eqref{eq:sums like integrals}. The following example shows that such an estimate of the amplitude by \\eqref{eq:sums like integrals} really is too coarse in general. We surely have\n\\begin{align}\\label{eq:example diagram}\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n \\draw (0.45,-.7) to (1,0);\n \\draw (1.55,-.7) to (1,0);\n \\draw (1,0.054) arc (-90:270:0.3cm);\n \\draw (1,-0.035) arc (-90:270:.7cm);\n \\node at (1,-.55) {$i_1$};\n \\node at (0,1) {$i_2$};\n \\vertex{1,0};\n\\end{tikzpicture}\n}\n=\\lambda_{i_1}^{-1}\\lambda_{i_2}^{-1}\\sum_{k=1}^N\\sum_{l=1}^N\\frac{f'\\{i_1,k,l,k,i_1,i_2\\}}{f'\\{i_1,k\\}f'\\{k,l\\}}\\lesssim\n\\sum_{k=1}^N\\sum_{l=1}^N\\frac{l^{-p/d}}{k^{-p/d}k^{-p/d}}=\\sum_{k=1}^N k^{2p/d}\\sum_{l=1}^N l^{-p/d}.\n\\end{align}\nNaively adding up the orders gives the correct result, $\\O(N^{2+\\frac{p}{d}})$. But because $-p/d$ is negative, we cannot add up the powers: Assuming $p/d>1$, the sequence $(\\sum_{l=1}^N l^{-p/d})_{N\\in\\N}$ is convergent with nonzero limit. Hence, $\\sum_{l=1}^N l^{-p/d}$ is of order $N^0$, not of order $N^{1-p/d}$. The right-hand side of \\eqref{eq:example diagram} is therefore $\\O(N^{1+2p/d})$, and as we now know we have a better bound for the left-hand side. The trick is simply to choose the indices so that each vertex contribution is canceled by an edge contribution.\n\\end{rem}\n\n\\begin{rem}\nThe following explains the absence of a lower bound in Theorem \\ref{thm:main2}. Suppose $i_1,i_2,i_3$ are such that $f'[i_1,i_2,i_3]=0$, $f'(\\lambda_{i_1})\\neq0$, and $f'(\\lambda_{i_3})\\neq0$. We then compute the amplitude\n\\begin{align*}\n\\raisebox{-19.5pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0.5) to (0,0);\n\\draw (-0.5,-.5) to (0,0);\n\\draw (0,0) to[out=60,in=120] (1,0);\n\\draw (0,0) to[out=-60,in=-120] (1,0);\n\\draw (1,0) to (1.5,0);\n\\node at (0,0.5) {$i_1$};\n\\node at (-.5,0) {$i_2$};\n\\node at (0,-.5) {$i_3$};\n\\vertex{0,0};\n\\vertex{1,0};\n\\end{tikzpicture}\n}\n=-\\lambda_{i_1}^{-1}\\lambda_{i_2}^{-1}\\lambda_{i_3}^{-1}\\sum_{k=1}^N\\frac{f'\\{i_1,i_2,i_3,k\\}f'\\{i_1,i_3,k\\}}{f'\\{i_1,k\\}f'\\{i_3,k\\}}.\n\\end{align*}\nContrary to the situation where $f'[i_1,i_2,i_3]\\neq0$, the factor $|f'\\{i_1,i_2,i_3,k\\}|$ obtained from the 4-vertex is proportional to $|\\lambda_k|^{-1}$ as $k\\to\\infty$.\nThe other factors are proportional to $1$ as usual. We obtain\n\\begin{align*}\n\\raisebox{-19.5pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-0.5,0.5) to (0,0);\n\\draw (-0.5,-.5) to (0,0);\n\\draw (0,0) to[out=60,in=120] (1,0);\n\\draw (0,0) to[out=-60,in=-120] (1,0);\n\\draw (1,0) to (1.5,0);\n\\node at (0,0.5) {$i_1$};\n\\node at (-.5,0) {$i_2$};\n\\node at (0,-.5) {$i_3$};\n\\vertex{0,0};\n\\vertex{1,0};\n\\end{tikzpicture}}\n\\sim\\sum_{k=1}^N|\\lambda_k|^{-1}\\sim\\sum_{k=1}^Nk^{-\\frac{1}{d}}<\\O(N).\n\\end{align*}\nThe order depends on $d$ but (since $d\\geq0$) at least it is smaller than $\\O(N)$ in the sense that there exists no $c$ such that the lower bound is $\\geq cN$. For $d<1$ the graph is in fact finite.\n\\end{rem}\n\n\\begin{rem}\nEven though there exist only finitely many eigenmodes $\\lambda_k$ with $f'(\\lambda_k)=0$, these singular modes can boost the order of divergence not only when occurring as external indices. For instance, assuming non-singular external indices, we have the divergences\n\\begin{align*}\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.3,.6) arc (180:0:.3cm);\n\\draw (0,0) to[out=110,in=-90] (-.3,.6);\n\\draw (0,0) to[out=70,in=-90] (.3,.6);\n\\vertex{0,0};\n\\end{tikzpicture}}\n=\\O(N^{2+\\frac{p}{d}})\n\\quad\\text{and}\\quad\n\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1,0) to (1,0);\n\\draw (-.7,.7) arc (180:0:.7cm);\n\\draw (0,0) to[out=160,in=-90] (-.7,.7);\n\\draw (0,0) to[out=20,in=-90] (.7,.7);\n\\draw (-.3,.6) arc (180:0:.3cm);\n\\draw (0,0) to[out=110,in=-90] (-.3,.6);\n\\draw (0,0) to[out=70,in=-90] (.3,.6);\n\\vertex{0,0};\n\\node at (0,.5) {$0$};\n\\end{tikzpicture}}\n=\\O(N^{1+\\frac{p+1}{d}}).\n\\end{align*}\nThe latter is larger than the former precisely if $d<1$. More generally, at second loop order, the boost of UV-divergence by internal singular indices is not apparent for $d\\geq1$. Indeed, if $d\\geq1$, then the diagrams of Remark \\ref{rem:2-point 2-loop} remain precisely those of maximal order, also in the more general setting of Theorem \\ref{thm:main2}. \nHowever, for any $d\\in\\N$, at loop order $L=d+2$ the maximal diagrams become those where one of the faces is artificially broken by a singular index, such as\n\\begin{align}\n\\raisebox{-35pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.5,0) to (-1,0);\n\\draw (-1,0) arc (-180:180:1cm);\n\\draw (-1,0) to (0.5,.866);\n\\draw (0.5,.866) to (0.5,-.866);\n\\draw (-1,0) to (0.5,-.866);\n\\draw (0.5,.866) to (.75,1.3);\n\\draw (0.5,-.866) to (.75,-1.3);\n\\vertex{-1,0};\n\\vertex{0.5,.866};\n\\vertex{0.5,-.866};\n\\end{tikzpicture}}\n~~\n=\\O(N^{4+3p/d})\\quad\\text{and}\\quad\n\\raisebox{-35pt}{\n\\begin{tikzpicture}[thick]\n\\draw (-1.5,0) to (-1,0);\n\\draw (-1,0) arc (-180:180:1cm);\n\\draw (-1,0) to (0.5,.866);\n\\draw (0.5,.866) to (0.5,-.866);\n\\draw (-1,0) to (0.5,-.866);\n\\draw (0.5,.866) to (.75,1.3);\n\\draw (0.5,-.866) to (.75,-1.3);\n\\node at (0,0) {$0$};\n\\vertex{-1,0};\n\\vertex{0.5,.866};\n\\vertex{0.5,-.866};\n\\end{tikzpicture}}\n=\\O(N^{3+3(p+1)/d}).\\label{eq:3-point 4-loop}\n\\end{align}\nThe latter is larger than the former precisely if $d<3$. Graphs similar to the examples above yield divergences increased by breaking faces to singular modes for any $d<L-1$.\n\\end{rem}\n\\begin{rem}\\label{rem:conjectures}\nIn the setting of Theorem \\ref{thm:main2} and for $d<3$, the second graph of \\eqref{eq:3-point 4-loop} has maximal order among the 3-point 4-loop diagrams, but not every 3-point 4-loop diagram that has maximal order in the setting of Theorem \\ref{thm:main} has maximal order in the setting of Theorem \\ref{thm:main2}. That being said, we conjecture that, among the $n$-points $L$-loop graphs, every graph with maximal order in the setting of Theorem \\ref{thm:main2} has maximal order in the setting of Theorem \\ref{thm:main}. We moreover conjecture that the Ward identity, in the sense of \\cite{vNvS22b}, holds in both cases when restricting to the graphs of maximal order of divergence.\nThis might help generalize the results of \\cite{vNvS22b} to higher loop, which is a pressing open problem.\n\\end{rem}", "post_theorem_intro_text_len": 5794, "post_theorem_intro_text": "For renormalization purposes, it is relevant \\cite{CR2000,CR2001,GW2005a,GW2005b,ILV2012,KLV2014,LOR2015,Riv2007,RVW} to know the asymptotic behavior of these amplitudes as $N\\to\\infty$. The main objective of this paper is to prove the following formulas describing this asymptotic behavior. \n\n\\paragraph{Main results}\nSuppose that $\\{\\lambda_k\\}_{k=1}^\\infty\\subseteq\\mathbb{R}$ is a sequence of asymptotic behavior $\\lambda_k\\asymp k^{1/d}$ for $d\\in\\R_{>0}$. Suppose that $f$ is smooth, even, and satisfies, for $x\\to\\infty$, $f^{(n)}(x)\\asymp (-1)^nx^{-p-n}$ ($n\\in\\mathbb{N}$) for some~$p>0$. Suppose moreover that the divided differences of $f$ do not vanish on $\\{\\lambda_k\\}_{k=1}^\\infty$. For a graph $G$ with $U$ unbroken faces, $E_{\\textnormal{fi}}$ edges which do not border a broken face, and $V_{\\textnormal{fi}}$ vertices which do not border a broken face, the amplitude of $G$ is bounded from above and below by a constant times $N^{\\omega(G)}$, where\n\\begin{align}\\label{eq:power counting formula}\n\\omega(G)= U+\\frac{p}{d}(E_{\\textnormal{fi}}-V_{\\textnormal{fi}}).\n\\end{align}\n\nHowever, when the assumption of vanishing divided differences is not satisfied, the divergences become (for certain graphs \\textit{strictly}) larger. In this case, the amplitude is bounded from above by $N^{\\tilde\\omega(G)}$, where\n\\begin{align}\\label{eq:power counting formula 2}\n\\tilde \\omega(G)=\\max(U^{\\mathfrak{b}}+\\frac{p}{d}(E^{\\mathfrak{b}}_{\\textnormal{fi}}-V^{\\mathfrak{b}}_{\\textnormal{fi}})+\\frac{p+1}{d}(E^{\\mathfrak{b}}_{10}-V^{\\mathfrak{b}}_{10})),\n\\end{align}\nin which the maximum is taken over all subsets $\\mathfrak{b}$ of unbroken faces, $U^{\\mathfrak{b}},E_\\textnormal{fi}^b,V_\\textnormal{fi}^b$ are as before when designating the elements of $\\mathfrak{b}$ as broken, and $V^{\\mathfrak{b}}_{10}/E^{\\mathfrak{b}}_{10}$ denotes the number of vertices/edges bordering only unbroken faces except for either exactly one element of $\\mathfrak{b}$ or exactly one external index $i_0$ satisfying~${f'(\\lambda_{i_0})=0}$.\n\n\\paragraph{Consequences of \\eqref{eq:power counting formula}}\nNote that $E_\\textnormal{fi}-V_\\textnormal{fi}\\geq 0$ for any graph. A remarkable result is that a function~$f$ with \\textit{faster} decay results in a \\textit{higher} degree of divergence of the graph. This is not obvious from the definition of the Feynman rules, even for simple graphs like in Example 1, and even for concrete functions like $f(x)=x^{-p}$.\nBecause divided differences of such functions may well be negative, one might \\textit{a priori} expect cancellations of terms damping the degree of divergence, but this does not happen.\n\nAnother corollary of our power counting formula is the following observation. Among the connected graphs with~$n\\geq1$ external edges and $L$ loops, the maximal value of $U$ is $L-1$, and the maximal value of $E_\\textnormal{fi}-V_\\textnormal{fi}$ is $L-1$. The maximal value $\\omega(G)=L-1+\\frac{p}{d}(L-1)$ is attained by a nonempty set of diagrams, by virtue of our lower bound. These diagrams with maximal divergence are precisely the planar diagrams that cannot be split into two connected components by removing one vertex and all external edges, and moreover have only one unbroken face -- cf.\\ Remark \\ref{rem:2-point 2-loop}.\n\\paragraph{Consequences of \\eqref{eq:power counting formula 2}}\nThe situation where divided differences of $f'$ may vanish is more complicated (cf. Section \\ref{sct:main2}), but \\eqref{eq:power counting formula 2} leads us to conjecture that the diagrams with maximal divergence are of the same planar form as before -- cf.\\ Remark \\ref{rem:conjectures}. As we discuss in Remark \\ref{rem:UV/IR}, the influence of modes $\\lambda_i$ with $f'(\\lambda_i)=0$ is reminiscent of the UV/IR behavior of scalar field theories on noncommutative spacetime, which underlines the need for a rigorous renormalization analysis of the spectral action beyond the weak field approximation.\n\n\\paragraph{Techniques} To prove the above power counting formulas, we introduce several techniques in the asymptotic analysis of divided differences, and prove upper and lower bounds for them, which appear to be novel. A pivotal concept throughout the proof is the weighted divided difference\n$$f'\\{x_1,\\ldots,x_n\\}:=(-1)^n x_1\\cdots x_n f'[x_1,\\ldots,x_n].$$\nIts asymptotic behavior is more easily understood, because sending any of its variables to infinity yields another weighted divided difference.\nFurthermore, a key result is that weighted divided differences are positive on large enough subsets of $\\mathbb{R}^n$ for the functions $f$ we are considering. We thus generalize and give a new proof for Hunter's positivity theorem \\cite{Hunter1977}, which is recovered by taking $f(x)=x^{-p}$ ($p\\in 2\\mathbb{N}$). Moreover, we show that for functions bounded above or below by $x^{-p}$, with similar bounds on their derivatives, the asymptotic behavior of the weighed divided differences away from the origin is determined by the variable with the smallest modulus, which is crucial in the proof of our main theorem.\n\nOur main theorem is proved by combining this positivity and the mentioned upper and lower bounds with an interesting graph-theoretical argument.\n\n\\paragraph{Acknowledgements}\nWe are grateful to Martijn Caspers, Séverin Charbonnier, Harald Grosse, and Walter van Suijlekom for useful discussions. TvN thanks the Erwin Schr\\\"odinger Institute and the organizers and participants of the April 2023 conference 'Non-commutative Geometry meets Topological Recursion', where the motivation for this paper originated.\nTvN was supported by NWO project ‘Noncommutative multi-linear harmonic analysis and higher order spectral shift’,\nOCENW.M.22.070. EMH thanks the Max Planck Institute for Mathematics in Bonn for its financial support.", "sketch": "To prove the above power counting formulas, the authors “introduce several techniques in the asymptotic analysis of divided differences, and prove upper and lower bounds for them.” A “pivotal concept throughout the proof is the weighted divided difference\n$$f'\\{x_1,\\ldots,x_n\\}:=(-1)^n x_1\\cdots x_n f'[x_1,\\ldots,x_n],$$\nwhose asymptotic behavior is “more easily understood, because sending any of its variables to infinity yields another weighted divided difference.”\n\nA “key result” used in the proof is that “weighted divided differences are positive on large enough subsets of $\\mathbb{R}^n$ for the functions $f$ we are considering,” which generalizes Hunter’s positivity theorem. Moreover, for functions “bounded above or below by $x^{-p}$, with similar bounds on their derivatives,” they show that “the asymptotic behavior of the weighed divided differences away from the origin is determined by the variable with the smallest modulus,” and this is stated to be “crucial in the proof of our main theorem.”\n\nFinally, “our main theorem is proved by combining this positivity and the mentioned upper and lower bounds with an interesting graph-theoretical argument.”", "expanded_sketch": "To prove the above power counting formulas, the authors “introduce several techniques in the asymptotic analysis of divided differences, and prove upper and lower bounds for them.” A “pivotal concept throughout the proof is the weighted divided difference\n$$f'\\{x_1,\\ldots,x_n\\}:=(-1)^n x_1\\cdots x_n f'[x_1,\\ldots,x_n],$$\nwhose asymptotic behavior is “more easily understood, because sending any of its variables to infinity yields another weighted divided difference.”\n\nA “key result” used in the proof is that “weighted divided differences are positive on large enough subsets of $\\mathbb{R}^n$ for the functions $f$ we are considering,” which generalizes Hunter’s positivity theorem. Moreover, for functions “bounded above or below by $x^{-p}$, with similar bounds on their derivatives,” they show that “the asymptotic behavior of the weighed divided differences away from the origin is determined by the variable with the smallest modulus,” and this is stated to be “crucial in the proof of our main theorem.”\n\nFinally, “our main theorem is proved by combining this positivity and the mentioned upper and lower bounds with an interesting graph-theoretical argument.”,", "expanded_theorem": "Two amplitudes in the spectral action matrix model are\n\\begin{align*}\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\\draw (0.45,-0.7) to (1,0);\n\\draw (1.55,-0.7) to (1,0);\n\\draw (1,0) arc (-90:270:0.5cm);\n\\node at (1,-0.55) {$i$};\n\\node at (0.2,0.8) {$j$};\nlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n\\end{tikzpicture}\n}\n\\quad=\\quad\\sum_{k=1}^N\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (0.45,-.7) to (1,0);\n\t\\draw (1.55,-.7) to (1,0);\n\t\\draw (1,0) arc (-90:270:0.5cm);\n\t\\node at (1,-.55) {$i$};\n\t\\node at (0.2,0.8) {$j$};\n\tlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n        \\node at (1,.575) {$k$};\n\\end{tikzpicture}\n}\\quad=\\quad\\sum_{k=1}^N\\frac{f'[\\lambda_j,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]},\n\\end{align*}\nand\n\\begin{align*}\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (0.45,-.7) to (1,0);\n\t\\draw (1.55,-.7) to (1,0);\n\t\\draw (1,0.054) arc (-90:270:0.3cm);\n\t\\draw (1,-0.035) arc (-90:270:.7cm);\n\t\\node at (1,-.55) {$i$};\n\t\\node at (0,1) {$j$};\n\tlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n\\end{tikzpicture}\n}\n\\quad=\\quad\\sum_{k,l=1}^N\n\t\\raisebox{-20pt}{\n\\begin{tikzpicture}[thick]\n\t\\draw (0.45,-.7) to (1,0);\n\t\\draw (1.55,-.7) to (1,0);\n\t\\draw (1,0.054) arc (-90:270:0.3cm);\n\t\\draw (1,-0.035) arc (-90:270:.7cm);\n\t\\node at (1,-.55) {$i$};\n\t\\node at (0,1) {$j$};\n\t\\node at (1,1) {$k$};\n\t\\node at (1,0.4) {$l$};\n\tlldraw (1,0) circle (4pt);\nlldraw[white] (1,0) circle (2pt)\n\t;\n\\end{tikzpicture}\n}\\quad=\\quad\\sum_{k,l=1}^N\\frac{f'[\\lambda_j,\\lambda_k,\\lambda_l,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]f'[\\lambda_k,\\lambda_l]}.\n\\end{align*}", "theorem_type": ["Universal", "Classification or Bijection"], "mcq": {"question": "In the spectral action matrix model, let the ribbon-graph amplitude be computed by the rules: a vertex bordered by faces with labels \\(i_1,\\dots,i_n\\) contributes \\(f'[\\lambda_{i_1},\\dots,\\lambda_{i_n}]\\), an internal edge bordered by faces \\(a,b\\) contributes \\(1/f'[\\lambda_a,\\lambda_b]\\), and each unbroken internal face label is summed from \\(1\\) to \\(N\\). Here \\(f'[x_1,\\dots,x_n]\\) denotes the divided difference of \\(f'\\). What are the explicit amplitudes of the one-vertex 2-point ribbon graphs with external face label \\(i\\): (1) the graph with one internal loop and outer face label \\(j\\), and (2) the graph with two nested internal loops and outer face label \\(j\\)?", "correct_choice": {"label": "A", "text": "They are\n\\[\n\\sum_{k=1}^N \\frac{f'[\\lambda_j,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]}\n\\]\nfor the one-loop graph, and\n\\[\n\\sum_{k,l=1}^N \\frac{f'[\\lambda_j,\\lambda_k,\\lambda_l,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]f'[\\lambda_k,\\lambda_l]}\n\\]\nfor the graph with two nested loops."}, "choices": [{"label": "B", "text": "They are\n\\[\n\\sum_{k=1}^N \\frac{f'[\\lambda_j,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_i]}\n\\]\nfor the one-loop graph, and\n\\[\n\\sum_{k,l=1}^N \\frac{f'[\\lambda_j,\\lambda_k,\\lambda_l,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_i]f'[\\lambda_k,\\lambda_l]}\n\\]\nfor the graph with two nested loops."}, {"label": "C", "text": "They are obtained by summing a divided-difference numerator over the internal face labels, namely\n\\[\n\\sum_{k=1}^N \\frac{f'[\\lambda_j,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]}\n\\]\nfor the one-loop graph, and for the graph with two nested loops a double sum over \\(k,l\\) of a term with numerator\n\\[\nf'[\\lambda_j,\\lambda_k,\\lambda_l,\\lambda_k,\\lambda_j,\\lambda_i],\n\\]\ndivided by the product of the edge factors prescribed by the internal edges."}, {"label": "D", "text": "They are\n\\[\n\\sum_{k=1}^N \\frac{f'[\\lambda_j,\\lambda_k,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]}\n\\]\nfor the one-loop graph, and\n\\[\n\\sum_{k,l=1}^N \\frac{f'[\\lambda_j,\\lambda_k,\\lambda_l,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]f'[\\lambda_k,\\lambda_l]}\n\\]\nfor the graph with two nested loops."}, {"label": "E", "text": "They are\n\\[\n\\sum_{k=1}^N \\frac{f'[\\lambda_j,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]^2}\n\\]\nfor the one-loop graph, and\n\\[\n\\sum_{k,l=1}^N \\frac{f'[\\lambda_j,\\lambda_k,\\lambda_l,\\lambda_k,\\lambda_j,\\lambda_i]}{f'[\\lambda_j,\\lambda_k]f'[\\lambda_j,\\lambda_l]}\n\\]\nfor the graph with two nested loops."}], "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": "edge-factor depends on adjacent face labels, not external pair", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "geometric_construction", "tampered_component": "dropped the fully explicit denominator/product specification while retaining the correct summation and numerator pattern", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "cyclic repetition of face labels around the vertex", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "one denominator factor per internal edge with the correct adjacent labels", "template_used": "stronger_trap"}]}}
{"id": "2512.14627v1", "paper_link": "http://arxiv.org/abs/2512.14627v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\mathbb{R}^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\mathop{\\rm div}\\nolimits b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\mathbb{G}\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\eqref{drifteq} such that\n\\EQ{\\label{main1-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|\\mathbb{G}\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming \\eqref{main1-eq}.", "start_pos": 12114, "end_pos": 12850, "label": "main1"}, "ref_dict": {"main1-eq": "\\begin{theorem}\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\GG\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\eqref{drifteq} such that\n\\EQ{\\label{main1-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|\\GG\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming \\eqref{main1-eq}.\n\\end{theorem}", "change-var": "\\begin{proof}\nSimilar to \\cite{CT1} we will use the Gehring's lemma approach \\cite{Gehring, Gia}. Denote $\\Om_{x,R}=\\Om \\cap B(x,R)$ for $x\\in \\overline \\Om$ and $R>0$. Since $\\pd \\Om$ is Lipschitz and compact, there\nis a radius $R_0\\in (0,1]$ such that for every $x_0 \\in\\pd \\Om$,\n\\EQ{ \\label{R0.def}\n\\Om_{x_0,2R_0}= \\{ (x',x_n) \\in B_{2R_0}(0) \\subset \\R^2,\\ x_n>\\ga(x')\\},\n}\nafter suitable coordinate rotation and translation, where $\\ga(x')$ is a Lipschitz function defined for $x'\\in B_{2R_0}'(0) \\subset \\R$ with Lipschitz constant $L$ and $\\ga(0)=0$.\n\nTo prove the a priori estimate \\eqref{eq5.1}, we first consider the interior case with $x_0\\in\\Omega$ and $B_{\\rho}=B_{\\rho}(x_0)\\subset\\Omega$. We rescale it to the unit ball $B$ and denote $u_{\\rho}(x)=\\rho u(\\rho x+x_0),b_{\\rho}(x)=\\rho b(\\rho x+x_0),\\pi_{\\rho}(x)=\\rho^2\\pi(\\rho x+x_0)$ and $\\GG_{\\rho}(x)=\\rho^2 \\GG(\\rho x+x_0)$. Let $\\bar{u}_{\\rho}=u_{\\rho}-k$ where $k$ is a real constant to be choose later.  By Remark \\ref{Wolfpdecom}, we can decompose $\\pi_{\\rho} - (\\pi_{\\rho})_{B}=\\pi_1+\\pi_2+\\pi_3$ in $B$, where $\\pi_1,\\pi_2,\\pi_3$ are given by\n\\EQ{\n\\nabla\\pi_1=\\mathcal{W}_{4/3,B}(\\Delta \\bar{u}_{\\rho}),\\quad \\nabla\\pi_2=\\mathcal{W}_{4/3,B}(-\\div(b_{\\rho}\\otimes \\bar{u}_{\\rho})),\\quad \\nabla\\pi_3=\\mathcal{W}_{4/3,B}(\\div \\GG_{\\rho}),\n}\nwith $\\int_B \\pi_i=0$.\nTherefore, by Theorem \\ref{theoremL2wolf} we have the following pressure bound\n\\EQ{\\label{wolfbound}\n\\|\\pi_1\\|_{4/3,B}\\le c_1\\|\\nabla \\bar{u}_{\\rho}\\|_{4/3,B},\\quad\\|\\pi_2\\|_{4/3,B}\\le c_1 \\|b_{\\rho}\\bar{u}_{\\rho}\\|_{4/3,B},\\quad \\|\\pi_3\\|_{4/3,B}\\le c_1\\|\\GG_{\\rho}\\|_{4/3,B},\n}\nwhere $c_1$ is a global constant.\nFor simplicity of notations let us drop the subscript $\\rho$ for the further calculations, until we need to scale back to ball $B_{\\rho}(x_0)$.\n\nTake a smooth cutoff function $\\eta$ on $B$ with $\\eta=1$ in $B_{\\frac{1}{2}}$. Use the test function $\\zeta=\\eta^4 \\bar{u}$ in the weak form \\eqref{soln-pair} with $(u,\\pi)$ replaced by $(\\bar u, \\pi - (\\pi)_B)$ to get the energy estimate (using \\eqref{5.1} due to $b \\in L^{2+\\de}$)\n\\EQS{\\label{3.1}\n\\int_{B} |\\nabla (\\eta^2\\bar{u})| ^2\n\\le &\\int_{B} |\\bar{u}|^2(|\\nabla \\eta^2|^2) +\\int_{B} |\\bar{u}|^2|b||\\nabla\\eta|\\eta^3\n\\\\\n&+\\int_{B} (|\\pi_1|+|\\pi_2|+|\\pi_3|)|\\bar{u}||\\nabla\\eta|\\eta^3+\\bka{\\div \\GG_{\\rho}, \\bar{u}\\eta^4}_B.\n}\n\nWe will estimate each of the terms on the right hand side separately, using Lemmas \\ref{Holder} and \\ref{Sobolev} in ball $B$,\n\\EQS{\n\\int_{B} |\\bar{u}|^2|b||\\nabla \\eta|\\eta^3\n&\\le \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}^2\\eta^3\\|_{L^{2,1}(B)}\\le \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}\\eta^{3/2}\\|_{L^{4,2}(B)}^2\n\\\\\n&\\le C \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}\\eta^{3/2}\\|^2_{W^{1,4/3}(B)}.\n}\nFor pressure terms we use \\eqref{wolfbound} and get the following\n\\EQN{\n\\int_{B}|\\pi_1||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_1\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{L^4(B)}\n\\\\\n&\\le\nc\\|\\nabla \\bar{u}\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}\\le c\\|\\nabla \\bar{u}\\|_{4/3,B}^2+c\\|\\eta \\bar{u}\\|^2_{4/3,B},\n}\n\\EQN{\n\\int_{B}|\\pi_2||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_2\\|_{L^{4/3}(B)}\\|\\eta^2\\bar{u}\\|_{L^4(B)}\\le c\\|\\bar{u}\\|_{L^{4,4/3}(B)}\\|b\\|_{L^{2,\\infty}(B)}\\|\\eta^2\\bar{u}\\|_{L^{4}(B)}\n\\\\\n&\\le c(b)\\|\\nabla \\bar{u}\\|_{4/3,B}^2+c(b)\\| \\bar{u}\\|^2_{4/3,B}\n}\n\\EQN{\n\\int_{B}|\\pi_3||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_3\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{L^4(B)}\\le  c\\|\\GG\\|_{2,B}\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}\n\\\\\n&\\le \\frac 14 \\|\\GG\\|_{2,B}^2+c\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}^2.\n}\nLastly we estimate the term containing $\\GG$,\n\\EQN{\n\\bka{\\div \\GG_{\\rho}, \\bar{u}\\eta^4}_B\n&=-\\int_{B} \\GG :\\nabla(\\bar{u}\\eta^4)=-\\int_{B} \\GG: \\nabla(\\eta^2\\bar{u})\\eta^2-2\\int_{B} \\GG :\\bar{u}\\nabla\\eta\\eta^3\n\\\\\n&\\le \\frac{1}{16}\\|\\nabla(\\eta^2\\bar{u})\\|^2_{2,B}+c\\|\\eta^2 \\bar{u}\\|_{2,B}^2+ \\frac 14\\|\\GG\\|_{2,B}^2.\n}\nCombining these estimates with \\eqref{3.1} we get the following\n\\EQ{\n\\int |\\nabla (\\eta^2\\bar{u})|^2\\le c(b)\\|\\bar{u}\\|^2_{W^{1,4/3}(B)}+c(b)(\\|\\bar{u}\\|_{2,B}^2+\\|\\bar{u}\\|_{4/3,B}^2)+\\|\\GG\\|_{2,B}^2,\n}\nHere $c(b)$ depends on $b$ only through $\\norm{b}_{L^{2,\\infty}(\\Om)}$. Finally, since $\\eta=1$ on $B_{1/2}$ we get that\n\\EQ{\n\\int_{B_{1/2}} |\\nabla \\bar{u}|^2\\le c(b)\\|\\nabla \\bar{u}\\|^2_{4/3,B}+c(b)\\|\\bar{u}\\|_{2,B}^2+\\|\\GG\\|_{2,B}^2.\n}\nWe now choose $k=(u_{\\rho})_{B}$ as the constant in $\\bar{u}=u_{\\rho}-k$ and apply Poincar\\'e inequality to get\n\\EQ{\n\\int_{B_{1/2}} |\\nabla u_{\\rho}|^2\\le c(b)\\|\\nabla u_{\\rho}\\|^2_{4/3,B}+\\|\\GG\\|_{2,B}^2.\n}\nLastly we scale back to $B_{\\rho}(x_0)$ and get that\n\\EQ{\\label{5.12}\n\\frac{1}{|B_{\\rho/2}|}\\int_{B_{\\rho/2}} |\\nabla u|^2\n\\le c(b) \\Big(\\frac{1}{|B_{\\rho}|}\\int_{B_{\\rho}}|\\nabla u|^{4/3}\\Big)^{3/2}+\\frac{c}{|B_{\\rho}|}\\|\\GG\\|_{2,B_{\\rho}}^2.\n}\n\n\\medskip\n\nNext we consider the boundary case, $\\Omega_{\\rho}=\\Omega\\cap B_{\\rho}(x_0)$ with $x_0\\in\\pd\\Om$, and $0<\\rho<R_0$. The significant difference from the internal case is the pressure estimate: Unlike the internal case we can not use Wolf decomposition estimate in a uniform ball, therefore we need to track the dependence of constant $c_1$ when we apply Theorem \\ref{theoremL2wolf} to get estimates similar to \\eqref{pressure_bound_est}\n and make sure it is uniform across all  $\\Omega_{\\rho}(x_0)$.\n Rescale the domain as the interior case so that $x_0=0$ and $\\rho=1$.\n We will use change of coordinates\n \\EQ{\\label{change-var}\n y_1=x_1, \\quad y_2=x_2-\\gamma(x_1),\n }\n to straighten the boundary and map $\\Om_{5/8}\\subset\\Om_1$ into domains $\\Om_{5/8}'\\subset\\Om_1'$ with flat boundary on $y_2=0$. Since $\\gamma$ was a Lipschitz curve with sufficiently small Lipschitz constant we can assume that $\\Om_{5/8}'\\subset B_{6/8}^+\\subset B_{7/8}^+\\subset\\Om_1'$.\n  (This is the first place that specifies the smallness of the Lipschitz constant.) By Lemma \\ref{geometry} we can fix a smooth $V$ such that $B_{6/8}^+\\subset V\\subset B_{7/8}^+$. After rewriting the equation \\eqref{drifteq} in new variables we get the following system (with all derivatives in $y$) in $\\Om_1'$:\n\\EQS{\\label{changed_var}\n-\\Delta_y u+\\div (\\gamma' e_1\\otimes\\partial_{y_2}u)+\n\\gamma' \\pd_1\\pd_2 u - (\\gamma')^2\\partial_{y_2}^2u+\n\\\\\n+\\div_y(b\\otimes u)- \\gamma'\\partial_{y_2}(b\\otimes u)_1+\\nabla_y \\pi-\\gamma'\\partial_{y_2}\\pi e_1&=\\div\\GG-\\gamma'\\partial_2 \\GG_1 ,\n\\\\\n\\div_y u-\\gamma'\\partial_2 u_1&=0.\n}\nAbove, $(b\\otimes u)_1=(b_1u_1,b_1u_2)$ and $\\GG_1=(G_{11},G_{12})$ are their first rows. In the second term, $\\ga'$ is hidden behind $\\div$ to avoid $\\ga''$ which we do not assume any bound.\n\nWe apply Wolf's local pressure projection $\\mathcal{W}_{4/3,V}$ on \\eqref{changed_var} in the smooth domain $V$ and get that  $\\pi-(\\pi)_{V}=\\pi_1+\\pi_2+\\pi_3+\\pi_4$ where\n\\EQS{\\label{pressure_decomp_b}\n\\nabla\\pi_1&=\\mathcal{W}_{4/3,V}\\bkt{\\Delta_y u-\\div (\\gamma' e_1\\otimes\\partial_{y_2}u)-\\gamma' \\pd_1\\pd_2 u +(\\gamma')^2\\partial_{y_2}^2u}\n\\\\\n\\nabla\\pi_2&=\\mathcal{W}_{4/3,V}\\bkt{-\\div_y(b\\otimes u)+\\gamma'\\partial_{y_2}(b\\otimes u)_1}\n\\\\\n \\nabla\\pi_3&=\\mathcal{W}_{4/3,V}(\\gamma' \\partial_{y_2}\\pi e_1)\n \\\\\n \\nabla\\pi_4&= \\mathcal{W}_{4/3,V}(\\div \\GG-\\gamma'\\partial_2 \\GG_1).\n}\nFrom Theorem \\ref{theoremL2wolf} the local pressure projection  $\\mathcal{W}_{4/3,V}$ is well defined, and we have the bound \\eqref{pWolfbound} with global constant $c$. Also using $\\ga'\\pd_2 = \\pd_2 \\ga'$, and Lemmas \\ref{Holder} and \\ref{Sobolev} in $V$,\n\\EQS{\\label{pressure_bound_est}\n\\|\\pi_1\\|_{4/3,V}&\\le c(1+\\|\\gamma'\\|_{\\infty}+\\|\\gamma'\\|_{\\infty}^2)\\|\\nabla u\\|_{4/3,V}\n\\\\\n\\|\\pi_2\\|_{4/3,V}&\\le  c(1+\\|\\gamma'\\|_{\\infty})\\|u\\otimes b\\|_{4/3,V}\n\\\\\n\\|\\pi_3\\|_{4/3,V}&\\le  c\\|\\gamma'\\|_{\\infty}\\|\\pi\\|_{4/3,V}\\le c\\|\\gamma'\\|_{\\infty}\\|\\pi_3\\|_{4/3,V}+c\\|\\gamma'\\|_{\\infty}\\|\\pi_1+\\pi_2+\\pi_4\\|_{4/3,V}\n\\\\\n\\|\\pi_4\\|_{4/3,V}&\\le  c(1+\\|\\gamma'\\|_{\\infty})\\|\\GG\\|_{4/3,V}\n}\nHere we apply our choice of $\\gamma$ so that $\\|\\gamma'\\|_{\\infty}<1$ and\n$ c\\|\\gamma'\\|_{\\infty}<1/2$. (This is the second place that specifies the smallness of the Lipschitz constant.) After applying it to \\eqref{pressure_bound_est} we get the following\n\\EQS{\\label{pressure_3}\n\\|\\pi_3\\|_{4/3,V}\\le 2c\\|\\gamma'\\|_{\\infty}\\|\\pi_1+\\pi_2+\\pi_4\\|_{4/3,V}.\n}\n\nLet $\\Om^\\sharp$ be the pre-image of $V$ under the map \\eqref{change-var} in $x$-variables.\nThe pressure component $\\pi_i$ can be considered functions on $\\Om^\\sharp$ and since \\eqref{change-var} has determinant 1 and $\\|\\nabla_y u\\|_{4/3,V}\\le 2\\|\\nabla_x u\\|_{4/3,\\Omega^\\sharp}$, by assumption $\\|\\gamma'\\|_{\\infty}\\le 1$ we have\n\\EQS{\\label{pressure-est}\n\\norm{\\pi-(\\pi)_{V}}_{4/3, \\Om^\\sharp} &\\le {\\textstyle\\sum}_{i=1}^4 \\norm{\\pi_i}_{4/3, \\Om^\\sharp}\n={\\textstyle\\sum}_{i=1}^4 \\norm{\\pi_i}_{4/3, V}\n\\\\\n&\\le c\\|\\nabla _y u\\|_{4/3,V} +c \\|u\\otimes b\\|_{4/3,V} +c \\|\\GG\\|_{4/3,V}\n\\\\\n&\\le c(1+ \\| b\\|_{L^{2,\\infty}(V)}) (\\|\\nabla_y u\\|_{4/3,V} +\\| u\\|_{4/3,V} )+ c \\|\\GG\\|_{4/3,V}\n\\\\\n&\\le c(1+ \\| b\\|_{L^{2,\\infty}(\\Omega)}) (\\|\\nabla_x u\\|_{4/3,\\Omega^\\sharp} +\\| u\\|_{4/3,\\Omega^\\sharp} )+ c \\|\\GG\\|_{4/3,\\Omega^\\sharp}.\n}\nThis pressure estimate is the only purpose of our change of variables \\eqref{change-var}. Noting that $\\Om_{5/8}\\subset \\Om^\\sharp\\subset \\Om_1$, we then proceed the following.\n\n Let $\\eta$ be a cutoff function supported in $B_{5/8}(x_0)$ such that $\\eta=1$ on $B_{1/2}(x_0)$. Due to our boundary condition $u|_{\\partial\\Omega}=0$ we can use function $\\eta^4u$ as an admissible test function in \\eqref{drifteq}. Therefore we get the following local energy inequality\n\\EQS{\\label{eq5.19}\n\\int_{\\Om_1} |\\nabla (\\eta^2u)|^2\n&\\le \\int_{\\Om_1} |u|^2|\\nabla \\eta^2|^2+\\int_{\\Om_1} |u|^2|b||\\nabla \\eta|\\eta^3\\\\\n&\\quad +\\int_{\\Om_1} |\\pi-(\\pi)_{V}||u||\\nabla\\eta|\\eta^3-\\int_{\\Om_1} \\GG :\\nabla(u\\eta^4).\n}\nUsing \\eqref{pressure-est} to bound the pressure term and similarly to internal case\nusing Lemmas \\ref{Holder} and \\ref{Sobolev} in $\\Om_1$, with uniform constant $C_2$ in Lemma \\ref{Sobolev} using $\\eta^4u\\in W^{1,2}_0(\\Om_1)$, we get\n\\EQS{\\label{5.16}\n\\int_{\\Om_{1/2}} |\\nabla u|^2 &\\le C(b)(\\|\\nabla u\\|^2_{4/3,\\Om_1}+\\|u\\|_{2,\\Om_1}^2)+c\\|\\GG\\|_{2,\\Om_1}^2.\n}\nNotice that here the constant $C(b)$ does not depend on $\\Om_1$.\nFinally we extend function $u,\\GG$ by zero to  $B_1(x_0)\\setminus \\Omega$. Since $u=0$ on the set $E:=\\Om^c\\cap B_1(x_0)$ of non zero measure with the ratio $\\frac{|E|}{|B_{1}|}$ bounded from below, we can use Poincar\\'e inequality to bound $\\|u\\|_{2,\\Om_1}^2 \\le c \\|\\nabla u\\|^2_{4/3,\\Om_1}$, with a constant independent of $\\Om_1$ with the given lower bound of the ratio.\n(This version of Poincar\\'e inequality follows from Proposition 6.14 of \\cite[page 116]{Lieberman},  with $f=\\al 1_E$, nt_{B_1} f=1$. It gives $\\norm{u}_{4/3,B_1} \\le C\\al  \\norm{\\nabla u}_{4/3,B_1} $.\nWe also have\n$\\norm{u}_{4,B_1} \\le C \\norm{u}_{W^{1,4/3}(B_1)}$.)\n\nWe also re-scale the inequality for arbitrary $B_r(x_0)$ and get \\eqref{5.12}.\n\n\\medskip\nLastly we need to consider the case of arbitrary $x_0,r$ such that $\\Om^c\\cap B_{2r}(x_0)\\neq\\emptyset$. For any such $x_0$ we can find $y_0\\in\\partial\\Om\\cap B_{2r}(x_0)$ such that\n\\EQ{\nB_r(x_0)\\subset B_{3r}(y_0)\\subset B_{6r}(y_0)\\subset B_{8r}(x_0).\n}\nUsing \\eqref{5.12} for $B_{3r}(y_0)$ (assuming $6r<R_0$) we get\n\\EQS{\\label{5.17}\n\\frac{1}{|B_r|}\\int_{B_r(x_0)} |\\nabla u|^2\n&\\le \\frac{C}{|B_{3r}|}\\int_{B_{3r}(y_0)} |\\nabla u|^2\n\\\\\n&\\le c(b) \\Big(\\frac{1}{|B_{6r}|}\\int_{B_{6r}(y_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{6r}|}\\|\\GG\\|_{L^2(B_{6r}(y_0))}^2\n\\\\\n& \\le c(b) \\Big(\\frac{1}{|B_{8r}|}\\int_{B_{8r}(x_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{8r}|}\\|\\GG\\|_{L^2(B_{8r}(x_0))}^2.\n}\n\nFor Gehring's lemma we combine the internal estimate \\eqref{5.12} and the boundary estimate \\eqref{5.17} and conclude that\n\\EQ{\\label{5.19}\n\\frac{1}{|B_r|}\\int_{B_r(x_0)} |\\nabla u|^2 \\le\nc(b) \\Big(\\frac{1}{|B_{8r}|}\\int_{B_{8r}(x_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{8r}|}\\|\\GG\\|_{L^2(B_{8r}(x_0))}^2,\n}\nfor any $x_0$ in a big cube containing $\\overline \\Om$, and any $r\\le R_0/6$.\nWe can now apply Proposition 1.1 of Giaquinta \\cite[page 122]{Gia} (also see \\cite[Proposition 3.7]{DongKim}) to get that $\\nb u \\in L^p_\\loc$ for $p \\in (2,p_0)$ for some $p_0 >2$, and\n\\EQ{\\label{0619}\nnt_{B_r(x_0)} |\\nabla u|^p}^{\\frac 1p} \\le\nnt_{B_{8r}(x_0)} |\\GG|^p}^{\\frac 1p},\n}\nfor all $x_0 \\in \\overline \\Om$ and $r\\le R_0/6$, with\n$p_0$ and $c_1$ depending only on $c(b)$. Summing \\eqref{0619} over a finite cover of $\\overline \\Om$ of balls of radius $R_0/6$,\nwe get \\eqref{eq5.1}.\n\\end{proof}", "main2-eq": "\\begin{theorem}\\label{th1.2}\n  Let $\\Om$ be a bounded Lipschtiz domain in $\\RR^2$.  Assume $b\\in L^{2,\\infty}(\\Omega)$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\\in L^q(\\Om)$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ of \\eqref{scalar.eq} such that\n\\EQ{\\label{main2-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|G\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\\Om)$ without assuming \\eqref{main2-eq}.\n\\end{theorem}", "main1": "\\begin{theorem}\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\GG\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\eqref{drifteq} such that\n\\EQ{\\label{main1-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|\\GG\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming \\eqref{main1-eq}.\n\\end{theorem}", "scalar.eq": "\\begin{theorem}\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\GG\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\eqref{drifteq} such that\n\\EQ{\\label{main1-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|\\GG\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming \\eqref{main1-eq}.\n\\end{theorem}\n\n\\emph{Comments on Theorem \\ref{main1}:}\n\\EN{\n\\item This theorem extends the higher dimensional result \\cite[Theorem 1.5]{CT1} to the dimension 2 case.\nIn \\cite[Theorem 1.5]{CT1} we take $L_0=1/2$. In this paper $L_0$ is specified after \\eqref{change-var} and after \\eqref{pressure_bound_est}, and is computable.\nWe skip in this paper the analogous results for small $b$ case (but full range $1<q<\\infty$) as in \\cite[Proposition 1.3, Theorem 1.4]{CT1}.\n\n\\item  Since $\\Om$ is Lipschitz we can assume that there exists $R_0>0$ such that for any $x_0\\in \\partial\\Om$ we have\n\\EQ{\n\\Om\\cap B_{2R_0}(x_0)=\\{(x',x_n)\\in B_{2R_0}(0),x_n>\\gamma(x')\\}.\n}\nThe constant $C$ in \\eqref{main1-eq} will actually depend only on $R_0$, $\\text{diam}~\\Om$, and $\\|b\\|_{L^{2,\\infty}}$.\n\n\\item If we further assume $b \\in L^2(\\Om)$\nand $b$ has a divergence-free extension to $L^2(\\R^2)$,\nthen there is another way to prove uniqueness of the weak solution in $W^{1,2}_{0,\\si}(\\Om)$ by using Hardy-BMO duality, even when the Lipschitz constant of $\\Om$ is large.\nSee Theorem \\ref{energy_2} in Section \\ref{sec3}.\n\n\\item We may simply assume $\\Om$ is a bounded $C^1$ domain, as for\na bounded $C^1$ domain and any constant $\\lambda>0$, each point $x$ on $\\pd\\Om$ has a neighborhood $U_x$ such that $\\pd \\Om \\cap U_x$ is the graph of a Lipschitz function with a Lipschitz constant less than $\\lambda$.\n\n}\n\nFor corresponding results on scalar equations\n\\EQ{\\label{scalar.eq}\n-\\Delta u +b\\cdot\\nabla u=\\div  G, \\quad u|_{\\pd \\Om}=0,\n}\nwith critical drift coefficient $b \\in L^n$, see\nKim \\& Kim \\cite{KK}. When $b \\in L^{n,\\infty}(\\Om)$ for $n \\ge 3$, see \\cite{KiTs,KPT} which also cover more general assumptions on drift $b$ as well as \\cite{CS20,CS24} and Kwon \\cite{Kwon} for $n=2$ case or drift in Morrey spaces which include $L^{n,\\infty}$.\nThere are also results on H\\\"older continuity of the solutions, for example with $b\\in BMO^{-1}(\\R^3)$, $\\div b=0$, see Seregin-Sverak-Silvestre-Zlatos \\cite{SSSZ}.\n\nUsing the same proof scheme we can also prove the following theorem for the  scalar equation \\eqref{scalar.eq}.\n\\begin{theorem}\\label{th1.2}\n  Let $\\Om$ be a bounded Lipschtiz domain in $\\RR^2$.  Assume $b\\in L^{2,\\infty}(\\Omega)$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\\in L^q(\\Om)$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ of \\eqref{scalar.eq} such that\n\\EQ{\\label{main2-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|G\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\\Om)$ without assuming \\eqref{main2-eq}.\n\\end{theorem}", "drifteq": "\\label{drifteq}\n-\\Delta u +b\\cdot\\nabla u+\\nabla \\pi=\\div \\GG,\\quad \\div u=0, \\quad u|_{\\pd \\Om}=0,\n}\nwith given drift $b$ in the critical space, $b\\in L^{2,\\oo}(\\Omega)^2$, $\\div b=0$, and  force $\\G", "1.2": "\\label{1.2}\n\\De u = \\div (\\nb u),\\quad\nb\\cdot\\nabla u = \\div (b\\otimes u), \\quad \\int_\\Om  \\div \\GG \\cdot \\zeta = -\n\\int_\\Om  \\GG:\\nb \\zeta.\n}\nFor zero divergence spaces or spaces with zero average we", "energy_2": "\\begin{theorem}[Unique weak solution for $L^2$ drift]\\label{energy_2}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$.\nAssume $b\\in L^{2}(\\Om)^2$ has an extension\n$\\tilde{b}\\in L^{2}(\\R^2)^2$ with $\\div \\tilde{b}=0$ in $\\R^2$.\nThen for any $f \\in W^{-1,2}(\\Omega)^2$,\nthere exists a unique weak solution $u\\in W^{1,2}_{0,\\sigma}(\\Omega)$\nof \\eqref{drifteq}. Furthermore, there is $\\pi \\in L^2(\\Om)$ so that $(u,\\pi)$ solves \\eqref{drifteq} in distributional sense, $\\int_\\Om \\pi=0$, and\n\\EQ{\\label{eq6.1b}\n\\|u\\|_{ W^{1,2}(\\Omega)}+\\|\\pi\\|_{L^2(\\Omega)}\\le C\\|f\\|_{W^{-1,2}(\\Om)},\n}\nfor some constant $C=C(\\Om)$ independent of $b$.\n\\end{theorem}", "pressure_bound_est": "\\begin{proof}\nSimilar to \\cite{CT1} we will use the Gehring's lemma approach \\cite{Gehring, Gia}. Denote $\\Om_{x,R}=\\Om \\cap B(x,R)$ for $x\\in \\overline \\Om$ and $R>0$. Since $\\pd \\Om$ is Lipschitz and compact, there\nis a radius $R_0\\in (0,1]$ such that for every $x_0 \\in\\pd \\Om$,\n\\EQ{ \\label{R0.def}\n\\Om_{x_0,2R_0}= \\{ (x',x_n) \\in B_{2R_0}(0) \\subset \\R^2,\\ x_n>\\ga(x')\\},\n}\nafter suitable coordinate rotation and translation, where $\\ga(x')$ is a Lipschitz function defined for $x'\\in B_{2R_0}'(0) \\subset \\R$ with Lipschitz constant $L$ and $\\ga(0)=0$.\n\nTo prove the a priori estimate \\eqref{eq5.1}, we first consider the interior case with $x_0\\in\\Omega$ and $B_{\\rho}=B_{\\rho}(x_0)\\subset\\Omega$. We rescale it to the unit ball $B$ and denote $u_{\\rho}(x)=\\rho u(\\rho x+x_0),b_{\\rho}(x)=\\rho b(\\rho x+x_0),\\pi_{\\rho}(x)=\\rho^2\\pi(\\rho x+x_0)$ and $\\GG_{\\rho}(x)=\\rho^2 \\GG(\\rho x+x_0)$. Let $\\bar{u}_{\\rho}=u_{\\rho}-k$ where $k$ is a real constant to be choose later.  By Remark \\ref{Wolfpdecom}, we can decompose $\\pi_{\\rho} - (\\pi_{\\rho})_{B}=\\pi_1+\\pi_2+\\pi_3$ in $B$, where $\\pi_1,\\pi_2,\\pi_3$ are given by\n\\EQ{\n\\nabla\\pi_1=\\mathcal{W}_{4/3,B}(\\Delta \\bar{u}_{\\rho}),\\quad \\nabla\\pi_2=\\mathcal{W}_{4/3,B}(-\\div(b_{\\rho}\\otimes \\bar{u}_{\\rho})),\\quad \\nabla\\pi_3=\\mathcal{W}_{4/3,B}(\\div \\GG_{\\rho}),\n}\nwith $\\int_B \\pi_i=0$.\nTherefore, by Theorem \\ref{theoremL2wolf} we have the following pressure bound\n\\EQ{\\label{wolfbound}\n\\|\\pi_1\\|_{4/3,B}\\le c_1\\|\\nabla \\bar{u}_{\\rho}\\|_{4/3,B},\\quad\\|\\pi_2\\|_{4/3,B}\\le c_1 \\|b_{\\rho}\\bar{u}_{\\rho}\\|_{4/3,B},\\quad \\|\\pi_3\\|_{4/3,B}\\le c_1\\|\\GG_{\\rho}\\|_{4/3,B},\n}\nwhere $c_1$ is a global constant.\nFor simplicity of notations let us drop the subscript $\\rho$ for the further calculations, until we need to scale back to ball $B_{\\rho}(x_0)$.\n\nTake a smooth cutoff function $\\eta$ on $B$ with $\\eta=1$ in $B_{\\frac{1}{2}}$. Use the test function $\\zeta=\\eta^4 \\bar{u}$ in the weak form \\eqref{soln-pair} with $(u,\\pi)$ replaced by $(\\bar u, \\pi - (\\pi)_B)$ to get the energy estimate (using \\eqref{5.1} due to $b \\in L^{2+\\de}$)\n\\EQS{\\label{3.1}\n\\int_{B} |\\nabla (\\eta^2\\bar{u})| ^2\n\\le &\\int_{B} |\\bar{u}|^2(|\\nabla \\eta^2|^2) +\\int_{B} |\\bar{u}|^2|b||\\nabla\\eta|\\eta^3\n\\\\\n&+\\int_{B} (|\\pi_1|+|\\pi_2|+|\\pi_3|)|\\bar{u}||\\nabla\\eta|\\eta^3+\\bka{\\div \\GG_{\\rho}, \\bar{u}\\eta^4}_B.\n}\n\nWe will estimate each of the terms on the right hand side separately, using Lemmas \\ref{Holder} and \\ref{Sobolev} in ball $B$,\n\\EQS{\n\\int_{B} |\\bar{u}|^2|b||\\nabla \\eta|\\eta^3\n&\\le \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}^2\\eta^3\\|_{L^{2,1}(B)}\\le \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}\\eta^{3/2}\\|_{L^{4,2}(B)}^2\n\\\\\n&\\le C \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}\\eta^{3/2}\\|^2_{W^{1,4/3}(B)}.\n}\nFor pressure terms we use \\eqref{wolfbound} and get the following\n\\EQN{\n\\int_{B}|\\pi_1||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_1\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{L^4(B)}\n\\\\\n&\\le\nc\\|\\nabla \\bar{u}\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}\\le c\\|\\nabla \\bar{u}\\|_{4/3,B}^2+c\\|\\eta \\bar{u}\\|^2_{4/3,B},\n}\n\\EQN{\n\\int_{B}|\\pi_2||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_2\\|_{L^{4/3}(B)}\\|\\eta^2\\bar{u}\\|_{L^4(B)}\\le c\\|\\bar{u}\\|_{L^{4,4/3}(B)}\\|b\\|_{L^{2,\\infty}(B)}\\|\\eta^2\\bar{u}\\|_{L^{4}(B)}\n\\\\\n&\\le c(b)\\|\\nabla \\bar{u}\\|_{4/3,B}^2+c(b)\\| \\bar{u}\\|^2_{4/3,B}\n}\n\\EQN{\n\\int_{B}|\\pi_3||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_3\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{L^4(B)}\\le  c\\|\\GG\\|_{2,B}\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}\n\\\\\n&\\le \\frac 14 \\|\\GG\\|_{2,B}^2+c\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}^2.\n}\nLastly we estimate the term containing $\\GG$,\n\\EQN{\n\\bka{\\div \\GG_{\\rho}, \\bar{u}\\eta^4}_B\n&=-\\int_{B} \\GG :\\nabla(\\bar{u}\\eta^4)=-\\int_{B} \\GG: \\nabla(\\eta^2\\bar{u})\\eta^2-2\\int_{B} \\GG :\\bar{u}\\nabla\\eta\\eta^3\n\\\\\n&\\le \\frac{1}{16}\\|\\nabla(\\eta^2\\bar{u})\\|^2_{2,B}+c\\|\\eta^2 \\bar{u}\\|_{2,B}^2+ \\frac 14\\|\\GG\\|_{2,B}^2.\n}\nCombining these estimates with \\eqref{3.1} we get the following\n\\EQ{\n\\int |\\nabla (\\eta^2\\bar{u})|^2\\le c(b)\\|\\bar{u}\\|^2_{W^{1,4/3}(B)}+c(b)(\\|\\bar{u}\\|_{2,B}^2+\\|\\bar{u}\\|_{4/3,B}^2)+\\|\\GG\\|_{2,B}^2,\n}\nHere $c(b)$ depends on $b$ only through $\\norm{b}_{L^{2,\\infty}(\\Om)}$. Finally, since $\\eta=1$ on $B_{1/2}$ we get that\n\\EQ{\n\\int_{B_{1/2}} |\\nabla \\bar{u}|^2\\le c(b)\\|\\nabla \\bar{u}\\|^2_{4/3,B}+c(b)\\|\\bar{u}\\|_{2,B}^2+\\|\\GG\\|_{2,B}^2.\n}\nWe now choose $k=(u_{\\rho})_{B}$ as the constant in $\\bar{u}=u_{\\rho}-k$ and apply Poincar\\'e inequality to get\n\\EQ{\n\\int_{B_{1/2}} |\\nabla u_{\\rho}|^2\\le c(b)\\|\\nabla u_{\\rho}\\|^2_{4/3,B}+\\|\\GG\\|_{2,B}^2.\n}\nLastly we scale back to $B_{\\rho}(x_0)$ and get that\n\\EQ{\\label{5.12}\n\\frac{1}{|B_{\\rho/2}|}\\int_{B_{\\rho/2}} |\\nabla u|^2\n\\le c(b) \\Big(\\frac{1}{|B_{\\rho}|}\\int_{B_{\\rho}}|\\nabla u|^{4/3}\\Big)^{3/2}+\\frac{c}{|B_{\\rho}|}\\|\\GG\\|_{2,B_{\\rho}}^2.\n}\n\n\\medskip\n\nNext we consider the boundary case, $\\Omega_{\\rho}=\\Omega\\cap B_{\\rho}(x_0)$ with $x_0\\in\\pd\\Om$, and $0<\\rho<R_0$. The significant difference from the internal case is the pressure estimate: Unlike the internal case we can not use Wolf decomposition estimate in a uniform ball, therefore we need to track the dependence of constant $c_1$ when we apply Theorem \\ref{theoremL2wolf} to get estimates similar to \\eqref{pressure_bound_est}\n and make sure it is uniform across all  $\\Omega_{\\rho}(x_0)$.\n Rescale the domain as the interior case so that $x_0=0$ and $\\rho=1$.\n We will use change of coordinates\n \\EQ{\\label{change-var}\n y_1=x_1, \\quad y_2=x_2-\\gamma(x_1),\n }\n to straighten the boundary and map $\\Om_{5/8}\\subset\\Om_1$ into domains $\\Om_{5/8}'\\subset\\Om_1'$ with flat boundary on $y_2=0$. Since $\\gamma$ was a Lipschitz curve with sufficiently small Lipschitz constant we can assume that $\\Om_{5/8}'\\subset B_{6/8}^+\\subset B_{7/8}^+\\subset\\Om_1'$.\n  (This is the first place that specifies the smallness of the Lipschitz constant.) By Lemma \\ref{geometry} we can fix a smooth $V$ such that $B_{6/8}^+\\subset V\\subset B_{7/8}^+$. After rewriting the equation \\eqref{drifteq} in new variables we get the following system (with all derivatives in $y$) in $\\Om_1'$:\n\\EQS{\\label{changed_var}\n-\\Delta_y u+\\div (\\gamma' e_1\\otimes\\partial_{y_2}u)+\n\\gamma' \\pd_1\\pd_2 u - (\\gamma')^2\\partial_{y_2}^2u+\n\\\\\n+\\div_y(b\\otimes u)- \\gamma'\\partial_{y_2}(b\\otimes u)_1+\\nabla_y \\pi-\\gamma'\\partial_{y_2}\\pi e_1&=\\div\\GG-\\gamma'\\partial_2 \\GG_1 ,\n\\\\\n\\div_y u-\\gamma'\\partial_2 u_1&=0.\n}\nAbove, $(b\\otimes u)_1=(b_1u_1,b_1u_2)$ and $\\GG_1=(G_{11},G_{12})$ are their first rows. In the second term, $\\ga'$ is hidden behind $\\div$ to avoid $\\ga''$ which we do not assume any bound.\n\nWe apply Wolf's local pressure projection $\\mathcal{W}_{4/3,V}$ on \\eqref{changed_var} in the smooth domain $V$ and get that  $\\pi-(\\pi)_{V}=\\pi_1+\\pi_2+\\pi_3+\\pi_4$ where\n\\EQS{\\label{pressure_decomp_b}\n\\nabla\\pi_1&=\\mathcal{W}_{4/3,V}\\bkt{\\Delta_y u-\\div (\\gamma' e_1\\otimes\\partial_{y_2}u)-\\gamma' \\pd_1\\pd_2 u +(\\gamma')^2\\partial_{y_2}^2u}\n\\\\\n\\nabla\\pi_2&=\\mathcal{W}_{4/3,V}\\bkt{-\\div_y(b\\otimes u)+\\gamma'\\partial_{y_2}(b\\otimes u)_1}\n\\\\\n \\nabla\\pi_3&=\\mathcal{W}_{4/3,V}(\\gamma' \\partial_{y_2}\\pi e_1)\n \\\\\n \\nabla\\pi_4&= \\mathcal{W}_{4/3,V}(\\div \\GG-\\gamma'\\partial_2 \\GG_1).\n}\nFrom Theorem \\ref{theoremL2wolf} the local pressure projection  $\\mathcal{W}_{4/3,V}$ is well defined, and we have the bound \\eqref{pWolfbound} with global constant $c$. Also using $\\ga'\\pd_2 = \\pd_2 \\ga'$, and Lemmas \\ref{Holder} and \\ref{Sobolev} in $V$,\n\\EQS{\\label{pressure_bound_est}\n\\|\\pi_1\\|_{4/3,V}&\\le c(1+\\|\\gamma'\\|_{\\infty}+\\|\\gamma'\\|_{\\infty}^2)\\|\\nabla u\\|_{4/3,V}\n\\\\\n\\|\\pi_2\\|_{4/3,V}&\\le  c(1+\\|\\gamma'\\|_{\\infty})\\|u\\otimes b\\|_{4/3,V}\n\\\\\n\\|\\pi_3\\|_{4/3,V}&\\le  c\\|\\gamma'\\|_{\\infty}\\|\\pi\\|_{4/3,V}\\le c\\|\\gamma'\\|_{\\infty}\\|\\pi_3\\|_{4/3,V}+c\\|\\gamma'\\|_{\\infty}\\|\\pi_1+\\pi_2+\\pi_4\\|_{4/3,V}\n\\\\\n\\|\\pi_4\\|_{4/3,V}&\\le  c(1+\\|\\gamma'\\|_{\\infty})\\|\\GG\\|_{4/3,V}\n}\nHere we apply our choice of $\\gamma$ so that $\\|\\gamma'\\|_{\\infty}<1$ and\n$ c\\|\\gamma'\\|_{\\infty}<1/2$. (This is the second place that specifies the smallness of the Lipschitz constant.) After applying it to \\eqref{pressure_bound_est} we get the following\n\\EQS{\\label{pressure_3}\n\\|\\pi_3\\|_{4/3,V}\\le 2c\\|\\gamma'\\|_{\\infty}\\|\\pi_1+\\pi_2+\\pi_4\\|_{4/3,V}.\n}\n\nLet $\\Om^\\sharp$ be the pre-image of $V$ under the map \\eqref{change-var} in $x$-variables.\nThe pressure component $\\pi_i$ can be considered functions on $\\Om^\\sharp$ and since \\eqref{change-var} has determinant 1 and $\\|\\nabla_y u\\|_{4/3,V}\\le 2\\|\\nabla_x u\\|_{4/3,\\Omega^\\sharp}$, by assumption $\\|\\gamma'\\|_{\\infty}\\le 1$ we have\n\\EQS{\\label{pressure-est}\n\\norm{\\pi-(\\pi)_{V}}_{4/3, \\Om^\\sharp} &\\le {\\textstyle\\sum}_{i=1}^4 \\norm{\\pi_i}_{4/3, \\Om^\\sharp}\n={\\textstyle\\sum}_{i=1}^4 \\norm{\\pi_i}_{4/3, V}\n\\\\\n&\\le c\\|\\nabla _y u\\|_{4/3,V} +c \\|u\\otimes b\\|_{4/3,V} +c \\|\\GG\\|_{4/3,V}\n\\\\\n&\\le c(1+ \\| b\\|_{L^{2,\\infty}(V)}) (\\|\\nabla_y u\\|_{4/3,V} +\\| u\\|_{4/3,V} )+ c \\|\\GG\\|_{4/3,V}\n\\\\\n&\\le c(1+ \\| b\\|_{L^{2,\\infty}(\\Omega)}) (\\|\\nabla_x u\\|_{4/3,\\Omega^\\sharp} +\\| u\\|_{4/3,\\Omega^\\sharp} )+ c \\|\\GG\\|_{4/3,\\Omega^\\sharp}.\n}\nThis pressure estimate is the only purpose of our change of variables \\eqref{change-var}. Noting that $\\Om_{5/8}\\subset \\Om^\\sharp\\subset \\Om_1$, we then proceed the following.\n\n Let $\\eta$ be a cutoff function supported in $B_{5/8}(x_0)$ such that $\\eta=1$ on $B_{1/2}(x_0)$. Due to our boundary condition $u|_{\\partial\\Omega}=0$ we can use function $\\eta^4u$ as an admissible test function in \\eqref{drifteq}. Therefore we get the following local energy inequality\n\\EQS{\\label{eq5.19}\n\\int_{\\Om_1} |\\nabla (\\eta^2u)|^2\n&\\le \\int_{\\Om_1} |u|^2|\\nabla \\eta^2|^2+\\int_{\\Om_1} |u|^2|b||\\nabla \\eta|\\eta^3\\\\\n&\\quad +\\int_{\\Om_1} |\\pi-(\\pi)_{V}||u||\\nabla\\eta|\\eta^3-\\int_{\\Om_1} \\GG :\\nabla(u\\eta^4).\n}\nUsing \\eqref{pressure-est} to bound the pressure term and similarly to internal case\nusing Lemmas \\ref{Holder} and \\ref{Sobolev} in $\\Om_1$, with uniform constant $C_2$ in Lemma \\ref{Sobolev} using $\\eta^4u\\in W^{1,2}_0(\\Om_1)$, we get\n\\EQS{\\label{5.16}\n\\int_{\\Om_{1/2}} |\\nabla u|^2 &\\le C(b)(\\|\\nabla u\\|^2_{4/3,\\Om_1}+\\|u\\|_{2,\\Om_1}^2)+c\\|\\GG\\|_{2,\\Om_1}^2.\n}\nNotice that here the constant $C(b)$ does not depend on $\\Om_1$.\nFinally we extend function $u,\\GG$ by zero to  $B_1(x_0)\\setminus \\Omega$. Since $u=0$ on the set $E:=\\Om^c\\cap B_1(x_0)$ of non zero measure with the ratio $\\frac{|E|}{|B_{1}|}$ bounded from below, we can use Poincar\\'e inequality to bound $\\|u\\|_{2,\\Om_1}^2 \\le c \\|\\nabla u\\|^2_{4/3,\\Om_1}$, with a constant independent of $\\Om_1$ with the given lower bound of the ratio.\n(This version of Poincar\\'e inequality follows from Proposition 6.14 of \\cite[page 116]{Lieberman},  with $f=\\al 1_E$, nt_{B_1} f=1$. It gives $\\norm{u}_{4/3,B_1} \\le C\\al  \\norm{\\nabla u}_{4/3,B_1} $.\nWe also have\n$\\norm{u}_{4,B_1} \\le C \\norm{u}_{W^{1,4/3}(B_1)}$.)\n\nWe also re-scale the inequality for arbitrary $B_r(x_0)$ and get \\eqref{5.12}.\n\n\\medskip\nLastly we need to consider the case of arbitrary $x_0,r$ such that $\\Om^c\\cap B_{2r}(x_0)\\neq\\emptyset$. For any such $x_0$ we can find $y_0\\in\\partial\\Om\\cap B_{2r}(x_0)$ such that\n\\EQ{\nB_r(x_0)\\subset B_{3r}(y_0)\\subset B_{6r}(y_0)\\subset B_{8r}(x_0).\n}\nUsing \\eqref{5.12} for $B_{3r}(y_0)$ (assuming $6r<R_0$) we get\n\\EQS{\\label{5.17}\n\\frac{1}{|B_r|}\\int_{B_r(x_0)} |\\nabla u|^2\n&\\le \\frac{C}{|B_{3r}|}\\int_{B_{3r}(y_0)} |\\nabla u|^2\n\\\\\n&\\le c(b) \\Big(\\frac{1}{|B_{6r}|}\\int_{B_{6r}(y_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{6r}|}\\|\\GG\\|_{L^2(B_{6r}(y_0))}^2\n\\\\\n& \\le c(b) \\Big(\\frac{1}{|B_{8r}|}\\int_{B_{8r}(x_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{8r}|}\\|\\GG\\|_{L^2(B_{8r}(x_0))}^2.\n}\n\nFor Gehring's lemma we combine the internal estimate \\eqref{5.12} and the boundary estimate \\eqref{5.17} and conclude that\n\\EQ{\\label{5.19}\n\\frac{1}{|B_r|}\\int_{B_r(x_0)} |\\nabla u|^2 \\le\nc(b) \\Big(\\frac{1}{|B_{8r}|}\\int_{B_{8r}(x_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{8r}|}\\|\\GG\\|_{L^2(B_{8r}(x_0))}^2,\n}\nfor any $x_0$ in a big cube containing $\\overline \\Om$, and any $r\\le R_0/6$.\nWe can now apply Proposition 1.1 of Giaquinta \\cite[page 122]{Gia} (also see \\cite[Proposition 3.7]{DongKim}) to get that $\\nb u \\in L^p_\\loc$ for $p \\in (2,p_0)$ for some $p_0 >2$, and\n\\EQ{\\label{0619}\nnt_{B_r(x_0)} |\\nabla u|^p}^{\\frac 1p} \\le\nnt_{B_{8r}(x_0)} |\\GG|^p}^{\\frac 1p},\n}\nfor all $x_0 \\in \\overline \\Om$ and $r\\le R_0/6$, with\n$p_0$ and $c_1$ depending only on $c(b)$. Summing \\eqref{0619} over a finite cover of $\\overline \\Om$ of balls of radius $R_0/6$,\nwe get \\eqref{eq5.1}.\n\\end{proof}", "soln-pair": "\\label{soln-pair}\n\\int_{\\Om} (\\nb u - b \\otimes u): \\nb \\zeta - \\pi \\div \\zeta  = -\\int_\\Om \\GG : \\nb \\zeta,\\quad\n\\forall \\zeta\\in C_c^{\\infty}(\\Om)^2.\n}\nHere we used \\eqref{1.2} to integrate by parts", "th1.2": "\\begin{theorem}\\label{th1.2}\n  Let $\\Om$ be a bounded Lipschtiz domain in $\\RR^2$.  Assume $b\\in L^{2,\\infty}(\\Omega)$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\\in L^q(\\Om)$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ of \\eqref{scalar.eq} such that\n\\EQ{\\label{main2-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|G\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\\Om)$ without assuming \\eqref{main2-eq}.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 3896, "pre_theorem_intro_text": "\\label{sec1}\n\nFor a bounded Lipschitz domain $\\Om$  in $\\mathbb{R}^2$ we consider the solutions of the perturbed Stokes system $(u,\\pi):\\Om \\to \\mathbb{R}^2 \\times \\mathbb{R}$ that satisfy\n\\EQ{\\label{drifteq}\n-\\Delta u +b\\cdot\\nabla u+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{G},\\quad \\mathop{\\rm div}\\nolimits u=0, \\quad u|_{\\partial \\Om}=0,\n}\nwith given drift $b$ in the critical space, $b\\in L^{2,\\infty}(\\Omega)^2$, $\\mathop{\\rm div}\\nolimits b=0$, and  force $\\mathbb{G}\\in L^q(\\Om)^{2\\times 2}$, $1 <q<\\infty$. We denote by\n$L^{q,r}(\\Om)$ the Lorentz spaces. For derivatives of vectors and matrixes we use the following notations\n$(\\mathop{\\rm div}\\nolimits \\mathbb{G})_i=\\pd_j \\GG_{ji}$ and $(\\nabla \\zeta)_{ji} = \\pd_j \\zeta_i$. Since $\\mathop{\\rm div}\\nolimits b=0$  we can integrate by parts in the following terms\n\\EQ{\\label{1.2}\n\\De u = \\mathop{\\rm div}\\nolimits (\\nabla u),\\quad\nb\\cdot\\nabla u = \\mathop{\\rm div}\\nolimits (b\\otimes u), \\quad \\int_\\Om  \\mathop{\\rm div}\\nolimits \\mathbb{G} \\cdot \\zeta = -\n\\int_\\Om  \\mathbb{G}:\\nabla \\zeta.\n}\nFor zero divergence spaces or spaces with zero average we use the following notations\n(for $\\Om\\subset\\mathbb{R}^n$, $n\\ge2$)\n\\EQS{\nC^{\\infty}_{c}(\\Omega) &= \\big\\{ u \\in C^{\\infty}(\\Omega):~ \\supp\\{u\\}\\Subset \\Omega\\big\\},\\\\\nC^{\\infty}_{c,\\si}(\\Omega) &= \\big\\{ u \\in C^{\\infty}_c(\\Omega)^n:\\  \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\\nW^{1,q}_{0,\\si}(\\Omega) &= \\big\\{ u \\in W^{1,q}_0(\\Omega)^n:\\  \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\ \\quad L^q_0(\\Om)&=\\left \\{ \\pi \\in L^q(\\Om):\\  \\textstyle \\int_\\Om \\pi \\,dx=0 \\right \\}.\n}\n\nNext we consider $(u,\\pi)$ as a \\emph{weak solution pair} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$, $\\pi\\in L^q_0(\\Omega)$, and $(u,\\pi)$ satisfies  \\eqref{drifteq} in distributional sense,\\EQ{\\label{soln-pair}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta - \\pi \\mathop{\\rm div}\\nolimits \\zeta  = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_c^{\\infty}(\\Om)^2.\n}\nHere we used \\eqref{1.2} to integrate by parts.\nWe say $u$ is a \\emph{$q$-weak solution} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $u$ satisfies \\eqref{drifteq} in weak sense, i.e.,\n\\EQ{\\label{weak-soln}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta   = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_{c,\\sigma}^{\\infty}(\\Omega).\n}\nIn case $q=2$ we call $u$ a \\emph{weak solution}. In this paper we also consider a dual system of \\eqref{drifteq}\n\\EQ{\\label{drifteq_dual}\n-\\Delta v -\\nabla\\cdot(b\\otimes v)+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{F},\\quad \\mathop{\\rm div}\\nolimits v=0, \\quad v|_{\\partial \\Om}=0.\n}\nThis system is equivalent to \\eqref{drifteq} when the drift is divergence free, $\\mathop{\\rm div}\\nolimits b=0$, but in general case it is different.\n\nThe boundary value problem of the perturbed Stokes system \\eqref{drifteq} and the corresponding scalar equation \\eqref{scalar.eq}\nhas a huge amount of literature when we consider subcritical drift $b\\in L^p(\\Om)$, $\\Om \\subset \\mathbb{R}^n$, with $p> n \\ge2$. In the subcritical case we may treat the drift term $b\\cdot \\nabla u$ as a perturbation to $\\De u$.  The critical case is when $b\\in L^n(\\Om)$ or $b\\in L^{n,\\infty}(\\Om)$. System \\eqref{drifteq} in $\\Om \\subset \\mathbb{R}^3$ with critical drift $b\\in L^3(\\Om)$ has previously been studied;\nSee Section 4 of Kim \\cite{Kim09}, Theorem 18 in Choe \\& Kim \\cite{ChoeKim}, and Section 5 of\nAmrouche \\& Rodríguez-Bellido \\cite{AmRB}. They contain estimates similar to \\eqref{main1-eq}. A similar result for small $b\\in L^{n,\\infty}(\\Om)$, $n \\ge3$, is proved by the authors \\cite[Theorem 1.4]{CT1}.\n\nIn this paper we extend our previous work for   higher dimensions $n \\ge 3$ \\cite{CT1} to $n=2$. Our main theorem below extends \\cite[Theorem 1.5]{CT1} to the dimension 2 case and proves existence of the solutions with higher integrability of the gradient.", "context": "\\label{sec1}\n\nFor a bounded Lipschitz domain $\\Om$ in $\\mathbb{R}^2$ we consider the solutions of the perturbed Stokes system $(u,\\pi):\\Om \\to \\mathbb{R}^2 \\times \\mathbb{R}$ that satisfy\n\\EQ{\\label{drifteq}\n-\\Delta u +b\\cdot\\nabla u+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{G},\\quad \\mathop{\\rm div}\\nolimits u=0, \\quad u|_{\\partial \\Om}=0,\n}\nwith given drift $b$ in the critical space, $b\\in L^{2,\\infty}(\\Omega)^2$, $\\mathop{\\rm div}\\nolimits b=0$, and force $\\mathbb{G}\\in L^q(\\Om)^{2\\times 2}$, $1 <q<\\infty$. We denote by\n$L^{q,r}(\\Om)$ the Lorentz spaces. For derivatives of vectors and matrixes we use the following notations\n$(\\mathop{\\rm div}\\nolimits \\mathbb{G})_i=\\pd_j \\GG_{ji}$ and $(\\nabla \\zeta)_{ji} = \\pd_j \\zeta_i$. Since $\\mathop{\\rm div}\\nolimits b=0$ we can integrate by parts in the following terms\n\\EQ{\\label{1.2}\n\\De u = \\mathop{\\rm div}\\nolimits (\\nabla u),\\quad\nb\\cdot\\nabla u = \\mathop{\\rm div}\\nolimits (b\\otimes u), \\quad \\int_\\Om \\mathop{\\rm div}\\nolimits \\mathbb{G} \\cdot \\zeta = -\n\\int_\\Om \\mathbb{G}:\\nabla \\zeta.\n}\nFor zero divergence spaces or spaces with zero average we use the following notations\n(for $\\Om\\subset\\mathbb{R}^n$, $n\\ge2$)\n\\EQS{\nC^{\\infty}_{c}(\\Omega) &= \\big\\{ u \\in C^{\\infty}(\\Omega):~ \\supp\\{u\\}\\Subset \\Omega\\big\\},\\\\\nC^{\\infty}_{c,\\si}(\\Omega) &= \\big\\{ u \\in C^{\\infty}_c(\\Omega)^n:\\ \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\\nW^{1,q}_{0,\\si}(\\Omega) &= \\big\\{ u \\in W^{1,q}_0(\\Omega)^n:\\ \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\ \\quad L^q_0(\\Om)&=\\left \\{ \\pi \\in L^q(\\Om):\\ \\textstyle \\int_\\Om \\pi \\,dx=0 \\right \\}.\n}\n\nNext we consider $(u,\\pi)$ as a \\emph{weak solution pair} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$, $\\pi\\in L^q_0(\\Omega)$, and $(u,\\pi)$ satisfies \\eqref{drifteq} in distributional sense,\\EQ{\\label{soln-pair}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta - \\pi \\mathop{\\rm div}\\nolimits \\zeta = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_c^{\\infty}(\\Om)^2.\n}\nHere we used \\eqref{1.2} to integrate by parts.\nWe say $u$ is a \\emph{$q$-weak solution} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $u$ satisfies \\eqref{drifteq} in weak sense, i.e.,\n\\EQ{\\label{weak-soln}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_{c,\\sigma}^{\\infty}(\\Omega).\n}\nIn case $q=2$ we call $u$ a \\emph{weak solution}. In this paper we also consider a dual system of \\eqref{drifteq}\n\\EQ{\\label{drifteq_dual}\n-\\Delta v -\\nabla\\cdot(b\\otimes v)+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{F},\\quad \\mathop{\\rm div}\\nolimits v=0, \\quad v|_{\\partial \\Om}=0.\n}\nThis system is equivalent to \\eqref{drifteq} when the drift is divergence free, $\\mathop{\\rm div}\\nolimits b=0$, but in general case it is different.\n\nThe boundary value problem of the perturbed Stokes system \\eqref{drifteq} and the corresponding scalar equation \\eqref{scalar.eq}\nhas a huge amount of literature when we consider subcritical drift $b\\in L^p(\\Om)$, $\\Om \\subset \\mathbb{R}^n$, with $p> n \\ge2$. In the subcritical case we may treat the drift term $b\\cdot \\nabla u$ as a perturbation to $\\De u$. The critical case is when $b\\in L^n(\\Om)$ or $b\\in L^{n,\\infty}(\\Om)$. System \\eqref{drifteq} in $\\Om \\subset \\mathbb{R}^3$ with critical drift $b\\in L^3(\\Om)$ has previously been studied;\nSee Section 4 of Kim \\cite{Kim09}, Theorem 18 in Choe \\& Kim \\cite{ChoeKim}, and Section 5 of\nAmrouche \\& Rodríguez-Bellido \\cite{AmRB}. They contain estimates similar to \\eqref{main1-eq}. A similar result for small $b\\in L^{n,\\infty}(\\Om)$, $n \\ge3$, is proved by the authors \\cite[Theorem 1.4]{CT1}.\n\nIn this paper we extend our previous work for higher dimensions $n \\ge 3$ \\cite{CT1} to $n=2$. Our main theorem below extends \\cite[Theorem 1.5]{CT1} to the dimension 2 case and proves existence of the solutions with higher integrability of the gradient.", "full_context": "\\label{sec1}\n\nFor a bounded Lipschitz domain $\\Om$ in $\\mathbb{R}^2$ we consider the solutions of the perturbed Stokes system $(u,\\pi):\\Om \\to \\mathbb{R}^2 \\times \\mathbb{R}$ that satisfy\n\\EQ{\\label{drifteq}\n-\\Delta u +b\\cdot\\nabla u+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{G},\\quad \\mathop{\\rm div}\\nolimits u=0, \\quad u|_{\\partial \\Om}=0,\n}\nwith given drift $b$ in the critical space, $b\\in L^{2,\\infty}(\\Omega)^2$, $\\mathop{\\rm div}\\nolimits b=0$, and force $\\mathbb{G}\\in L^q(\\Om)^{2\\times 2}$, $1 <q<\\infty$. We denote by\n$L^{q,r}(\\Om)$ the Lorentz spaces. For derivatives of vectors and matrixes we use the following notations\n$(\\mathop{\\rm div}\\nolimits \\mathbb{G})_i=\\pd_j \\GG_{ji}$ and $(\\nabla \\zeta)_{ji} = \\pd_j \\zeta_i$. Since $\\mathop{\\rm div}\\nolimits b=0$ we can integrate by parts in the following terms\n\\EQ{\\label{1.2}\n\\De u = \\mathop{\\rm div}\\nolimits (\\nabla u),\\quad\nb\\cdot\\nabla u = \\mathop{\\rm div}\\nolimits (b\\otimes u), \\quad \\int_\\Om \\mathop{\\rm div}\\nolimits \\mathbb{G} \\cdot \\zeta = -\n\\int_\\Om \\mathbb{G}:\\nabla \\zeta.\n}\nFor zero divergence spaces or spaces with zero average we use the following notations\n(for $\\Om\\subset\\mathbb{R}^n$, $n\\ge2$)\n\\EQS{\nC^{\\infty}_{c}(\\Omega) &= \\big\\{ u \\in C^{\\infty}(\\Omega):~ \\supp\\{u\\}\\Subset \\Omega\\big\\},\\\\\nC^{\\infty}_{c,\\si}(\\Omega) &= \\big\\{ u \\in C^{\\infty}_c(\\Omega)^n:\\ \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\\nW^{1,q}_{0,\\si}(\\Omega) &= \\big\\{ u \\in W^{1,q}_0(\\Omega)^n:\\ \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\ \\quad L^q_0(\\Om)&=\\left \\{ \\pi \\in L^q(\\Om):\\ \\textstyle \\int_\\Om \\pi \\,dx=0 \\right \\}.\n}\n\nNext we consider $(u,\\pi)$ as a \\emph{weak solution pair} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$, $\\pi\\in L^q_0(\\Omega)$, and $(u,\\pi)$ satisfies \\eqref{drifteq} in distributional sense,\\EQ{\\label{soln-pair}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta - \\pi \\mathop{\\rm div}\\nolimits \\zeta = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_c^{\\infty}(\\Om)^2.\n}\nHere we used \\eqref{1.2} to integrate by parts.\nWe say $u$ is a \\emph{$q$-weak solution} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $u$ satisfies \\eqref{drifteq} in weak sense, i.e.,\n\\EQ{\\label{weak-soln}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_{c,\\sigma}^{\\infty}(\\Omega).\n}\nIn case $q=2$ we call $u$ a \\emph{weak solution}. In this paper we also consider a dual system of \\eqref{drifteq}\n\\EQ{\\label{drifteq_dual}\n-\\Delta v -\\nabla\\cdot(b\\otimes v)+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{F},\\quad \\mathop{\\rm div}\\nolimits v=0, \\quad v|_{\\partial \\Om}=0.\n}\nThis system is equivalent to \\eqref{drifteq} when the drift is divergence free, $\\mathop{\\rm div}\\nolimits b=0$, but in general case it is different.\n\nThe boundary value problem of the perturbed Stokes system \\eqref{drifteq} and the corresponding scalar equation \\eqref{scalar.eq}\nhas a huge amount of literature when we consider subcritical drift $b\\in L^p(\\Om)$, $\\Om \\subset \\mathbb{R}^n$, with $p> n \\ge2$. In the subcritical case we may treat the drift term $b\\cdot \\nabla u$ as a perturbation to $\\De u$. The critical case is when $b\\in L^n(\\Om)$ or $b\\in L^{n,\\infty}(\\Om)$. System \\eqref{drifteq} in $\\Om \\subset \\mathbb{R}^3$ with critical drift $b\\in L^3(\\Om)$ has previously been studied;\nSee Section 4 of Kim \\cite{Kim09}, Theorem 18 in Choe \\& Kim \\cite{ChoeKim}, and Section 5 of\nAmrouche \\& Rodríguez-Bellido \\cite{AmRB}. They contain estimates similar to \\eqref{main1-eq}. A similar result for small $b\\in L^{n,\\infty}(\\Om)$, $n \\ge3$, is proved by the authors \\cite[Theorem 1.4]{CT1}.\n\nIn this paper we extend our previous work for higher dimensions $n \\ge 3$ \\cite{CT1} to $n=2$. Our main theorem below extends \\cite[Theorem 1.5]{CT1} to the dimension 2 case and proves existence of the solutions with higher integrability of the gradient.\n\nThe boundary value problem of the perturbed Stokes system \\eqref{drifteq} and the corresponding scalar equation \\eqref{scalar.eq}\nhas a huge amount of literature when we consider subcritical drift $b\\in L^p(\\Om)$, $\\Om \\subset \\R^n$, with $p> n \\ge2$. In the subcritical case we may treat the drift term $b\\cdot \\nb u$ as a perturbation to $\\De u$. The critical case is when $b\\in L^n(\\Om)$ or $b\\in L^{n,\\infty}(\\Om)$. System \\eqref{drifteq} in $\\Om \\subset \\R^3$ with critical drift $b\\in L^3(\\Om)$ has previously been studied;\nSee Section 4 of Kim \\cite{Kim09}, Theorem 18 in Choe \\& Kim \\cite{ChoeKim}, and Section 5 of\nAmrouche \\& Rodríguez-Bellido \\cite{AmRB}. They contain estimates similar to \\eqref{main1-eq}. A similar result for small $b\\in L^{n,\\infty}(\\Om)$, $n \\ge3$, is proved by the authors \\cite[Theorem 1.4]{CT1}.\n\n\\emph{Comments on Theorem \\ref{main1}:}\n\\EN{\n\\item This theorem extends the higher dimensional result \\cite[Theorem 1.5]{CT1} to the dimension 2 case.\nIn \\cite[Theorem 1.5]{CT1} we take $L_0=1/2$. In this paper $L_0$ is specified after \\eqref{change-var} and after \\eqref{pressure_bound_est}, and is computable.\nWe skip in this paper the analogous results for small $b$ case (but full range $1<q<\\infty$) as in \\cite[Proposition 1.3, Theorem 1.4]{CT1}.\n\nUsing the same proof scheme we can also prove the following theorem for the scalar equation \\eqref{scalar.eq}.\n\\begin{theorem}\\label{th1.2}\n Let $\\Om$ be a bounded Lipschtiz domain in $\\RR^2$. Assume $b\\in L^{2,\\infty}(\\Omega)$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\\in L^q(\\Om)$, there exists a weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ of \\eqref{scalar.eq} such that\n\\EQ{\\label{main2-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|G\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\\Om)$ without assuming \\eqref{main2-eq}.\n\\end{theorem}\n\n\\section{Existence of weak solutions}\\label{sec3}\nIn this section we will prove existence of weak solutions to the perturbed Stokes system \\eqref{drifteq} for $b\\in L^2$ or $L^{2,\\infty}$, $\\div b=0$,\n\\EQ{ \\label{eq3.1}\n-\\Delta u+ b\\cdot\\nabla u + \\nb \\pi =f,\\quad \\div u=0, \\quad u|_{\\pd \\Om}=0,\n}\nin $\\Om \\subset \\R^2$. When $b\\in L^2$, we will further show uniqueness.\nRecall that a weak solution $u$ belongs to $ W^{1,2}_{0,\\sigma}(\\Omega)$ and satisfies a weak form of \\eqref{drifteq} similar to \\eqref{weak-soln}.\n\\begin{theorem}[Weak solutions for weak $L^2$ drift]\\label{energy}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$.\nAssume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nThen for any $f \\in W^{-1,2}(\\Omega)^2$,\nthere exists a weak solution $u\\in W^{1,2}_{0,\\sigma}(\\Omega)$\nof \\eqref{drifteq}. Moreover, there is $\\pi \\in L^2(\\Om)$ so that $(u,\\pi)$ solves \\eqref{drifteq} in distributional sense, $\\int_\\Om \\pi=0$, and\n\\EQ{\\label{eq6.1}\n\\|u\\|_{ W^{1,2}(\\Omega)}+\\|\\pi\\|_{L^2(\\Omega)}\\le C\\|f\\|_{W^{-1,2}(\\Om)},\n}\nfor some constant $C=C(\\Om)$ independent of $b$.\n\\end{theorem}\n\n\\begin{theorem}[Unique weak solution for $L^2$ drift]\\label{energy_2}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$.\nAssume $b\\in L^{2}(\\Om)^2$ has an extension\n$\\tilde{b}\\in L^{2}(\\R^2)^2$ with $\\div \\tilde{b}=0$ in $\\R^2$.\nThen for any $f \\in W^{-1,2}(\\Omega)^2$,\nthere exists a unique weak solution $u\\in W^{1,2}_{0,\\sigma}(\\Omega)$\nof \\eqref{drifteq}. Furthermore, there is $\\pi \\in L^2(\\Om)$ so that $(u,\\pi)$ solves \\eqref{drifteq} in distributional sense, $\\int_\\Om \\pi=0$, and\n\\EQ{\\label{eq6.1b}\n\\|u\\|_{ W^{1,2}(\\Omega)}+\\|\\pi\\|_{L^2(\\Omega)}\\le C\\|f\\|_{W^{-1,2}(\\Om)},\n}\nfor some constant $C=C(\\Om)$ independent of $b$.\n\\end{theorem}\n\nApproximating $u$ in $W^{1,2}_{0,\\si}(\\Om)$ by smooth $u_j$\nand using $\\div b=0$ conclude that\n\\EQ{\\label{eq3.4}\n\\int_\\Om (b\\cdot \\nb) u\\cdot u = \\lim_{j \\to \\infty} \\int_\\Om (b\\cdot \\nb) u_j\\cdot u_j = \\int_\\Om b\\cdot \\nb(\\frac12|u_j|^2) = 0.\n}\n Thus we get the coercivity of this bilinear form, indeed,\n\\EQ{\n\\mathcal{B}(u,u)=\\|\\nabla u\\|_{L^2(\\Om)}^2, \\quad \\forall u\\in W_{0,\\sigma}^{1,2}(\\Om).\n}\nTherefore, by Lax-Milgram theorem, for any $f \\in W^{-1,2}(\\Omega)^n\\subset (W_{0,\\sigma}^{1,2}(\\Om)^n)^*$, there exists a unique $u\\in W_{0,\\sigma}^{1,2}(\\Om)$ such that\n\\EQ{\n\\mathcal{B}(u,v)=\\langle f,v\\rangle,\\quad \\forall v\\in W_{0,\\sigma}^{1,2}(\\Om).\n}\nThis is the weak form of \\eqref{drifteq} and hence\n$u$ is the unique weak solution. By taking $v=u$, we get $\\norm{u}_{W^{1,2}_0(\\Om)} \\le C\\norm{f}_{W^{-1,2}(\\Om)}$.\nLastly, we apply Lemma \\ref{pressure} with $q=2$ (and using that $\\Om$ is bounded and Lipschitz) to get an associated pressure $\\pi\\in L^2_0(\\Om)$ that satisfies the bound.\n\\end{proof}\n\n\\begin{theorem}\\label{theoremL2wolf}\nAssume $p>1$ and $V\\subset \\R^n$, $n\\ge2$, is a bounded Lipschitz domain with Lipschitz constant less than a sufficiently small $L_0(p,n)>0$.\nThen the problem \\eqref{Stokes} has a unique weak solution pair $(v,\\pi)$ with $v \\in W^{1,p}_{0,\\si}(V)$ and $\\pi \\in L^p_0(V)$ for given $f=\\div \\FF$, $ \\FF \\in L^p(\\Om)^{n\\times n}$, and the local pressure projection in $V$,\n\\[\n\\mathcal{W}_{p,V}(\\div \\FF)=\\nabla\\pi,\n\\]\nsatisfies the following inequality\n\\EQ{\\label{pWolfbound}\n\\|\\pi\\|_{L^p(V)}\\le c_1\\|\\FF\\|_{L^p(V)},\n}\nwhere $c_1=c_1(n,V,p)$.\n\\end{theorem}\n\n\\begin{theorem}\\label{Capriori}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$ with Lipschitz constant $L \\le L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nAssume further $b \\in L^{2+\\delta}$ for some $\\delta>0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $2<p<p_0$,\nany weak solution $u\\in W^{1,2}_{0,\\si}(\\Omega)$ of \\eqref{drifteq} with force $\\GG\\in L^p(\\Om)^{2\\times 2}$\nis in $ W^{1,p}(\\Omega)$ and\n\\EQ{\\label{eq5.1}\n\\|u\\|_{W^{1,p}(\\Omega)}\\le c\\|u\\|_{W^{1,2}(\\Omega)}+ c\\|\\GG\\|_{L^p(\\Omega)}\n}\nfor some constant $c(\\Omega,\\|b\\|_{L^{2,\\infty}})>0$.\n\\end{theorem}", "post_theorem_intro_text_len": 7623, "post_theorem_intro_text": "\\emph{Comments on Theorem \\ref{main1}:}\n\\EN{\n\\item This theorem extends the higher dimensional result \\cite[Theorem 1.5]{CT1} to the dimension 2 case.\nIn \\cite[Theorem 1.5]{CT1} we take $L_0=1/2$. In this paper $L_0$ is specified after \\eqref{change-var} and after \\eqref{pressure_bound_est}, and is computable.\nWe skip in this paper the analogous results for small $b$ case (but full range $1<q<\\infty$) as in \\cite[Proposition 1.3, Theorem 1.4]{CT1}.\n\n\\item  Since $\\Om$ is Lipschitz we can assume that there exists $R_0>0$ such that for any $x_0\\in \\partial\\Om$ we have\n\\EQ{\n\\Om\\cap B_{2R_0}(x_0)=\\{(x',x_n)\\in B_{2R_0}(0),x_n>\\gamma(x')\\}.\n}\nThe constant $C$ in \\eqref{main1-eq} will actually depend only on $R_0$, $\\text{diam}~\\Om$, and $\\|b\\|_{L^{2,\\infty}}$.\n\n\\item If we further assume $b \\in L^2(\\Om)$\nand $b$ has a divergence-free extension to $L^2(\\mathbb{R}^2)$,\nthen there is another way to prove uniqueness of the weak solution in $W^{1,2}_{0,\\si}(\\Om)$ by using Hardy-BMO duality, even when the Lipschitz constant of $\\Om$ is large.\nSee Theorem \\ref{energy_2} in Section \\ref{sec3}.\n\n\\item We may simply assume $\\Om$ is a bounded $C^1$ domain, as for\na bounded $C^1$ domain and any constant $\\lambda>0$, each point $x$ on $\\partial\\Om$ has a neighborhood $U_x$ such that $\\partial \\Om \\cap U_x$ is the graph of a Lipschitz function with a Lipschitz constant less than $\\lambda$.\n\n}\n\nFor corresponding results on scalar equations\n\\EQ{\\label{scalar.eq}\n-\\Delta u +b\\cdot\\nabla u=\\mathop{\\rm div}\\nolimits  G, \\quad u|_{\\partial \\Om}=0,\n}\nwith critical drift coefficient $b \\in L^n$, see\nKim \\& Kim \\cite{KK}. When $b \\in L^{n,\\infty}(\\Om)$ for $n \\ge 3$, see \\cite{KiTs,KPT} which also cover more general assumptions on drift $b$ as well as \\cite{CS20,CS24} and Kwon \\cite{Kwon} for $n=2$ case or drift in Morrey spaces which include $L^{n,\\infty}$.\nThere are also results on H\\\"older continuity of the solutions, for example with $b\\in BMO^{-1}(\\mathbb{R}^3)$, $\\mathop{\\rm div}\\nolimits b=0$, see Seregin-Sverak-Silvestre-Zlatos \\cite{SSSZ}.\n\nUsing the same proof scheme we can also prove the following theorem for the  scalar equation \\eqref{scalar.eq}.\n\\begin{theorem}\\label{th1.2}\n  Let $\\Om$ be a bounded Lipschtiz domain in $\\mathbb{R}^2$.  Assume $b\\in L^{2,\\infty}(\\Omega)$ and $\\mathop{\\rm div}\\nolimits b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\\in L^q(\\Om)$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ of \\eqref{scalar.eq} such that\n\\EQ{\\label{main2-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|G\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\\Om)$ without assuming \\eqref{main2-eq}.\n\\end{theorem}\n\nThis theorem improves $\\alpha=0$ case of \\cite[Theorem 1.1]{CS20} and $n=2$ case of Kwon \\cite[Theorem 1.1]{Kwon} which show that for any $q>2$, there is $p \\in (2,q)$ depending on $q$, $\\left  \\| b  \\right \\|_{L^{2,\\infty}}$ and $\\Om$ such that for any $G \\in L^q(\\Om)$, there is a unique $p$-weak solution $u$ of \\eqref{scalar.eq} with force $G$. Moreover,\n\\[\n\\left  \\| \\nabla u  \\right \\|_{L^p(\\Om)} \\le C \\left  \\| G  \\right \\|_{L^q(\\Om)}.\n\\]\nTheorem \\ref{th1.2} improves this result by asserting that $p=q$ when $2<q<p_0$ is sufficiently close to $2$. This improvement further allows us to prove a priori bounds for $q$-weak solutions for $q \\in (p_0',2]$ by duality, which yields existence and uniqueness\n for $q \\in (p_0',2]$.\n\nWe skip the proof of Theoem \\ref{th1.2} since it is similar to Theoem \\ref{main1} and is easier, without the need of pressure estimate. Because it has no pressure estimate, the Lipschitz constant of the boundary $\\partial \\Om$ is allowed to be arbitrarily large.\n\n\\medskip\n\nWe now explain the new difficulties and ideas.\nTheorem \\ref{main1}\nextends the higher dimensional result \\cite[Theorem 1.5]{CT1} to the 2D case, and\nwe will still use Gehring's approach. In this approach, we need to do local energy estimate in every ball and prove a reversed H\\\"older inequality in a general ball with a uniform constant.\n\nThe first new difficulty is that we cannot use $\\zeta = u \\eta^4$ for some cut-off function $\\eta$ as the test function in the weak form \\eqref{soln-pair}, because $b \\otimes u$ is no longer in $L^2$ as in the 3D case, and\n the term\n\\EQ{\\label{drift-integral}\n\\int_\\Om b \\otimes u : \\nabla( u \\eta^4)\n}\nis not integrable. Hence we will mollify the drift $b$, so that we can derive an a priori bound. Since there is no theorem that extends a div-free vector field in $L^{2,\\infty}(\\Omega)$ to $L^{2,\\infty}(\\mathbb{R}^2)$ for a general bounded domain, we will consider the approximation problems in subdomains $\\Om_n$ that converge to $\\Om$.\n\nThe second new difficulty comes from the lower integrability of the pressure.\nWe will\nbound the pressure using Wolf's local pressure projection with a uniform constant as in \\cite{CT1}.\nDenote $\\Omega_R=\\Omega\\cap B_R(x_0)$ for some $x_0\\in \\overline \\Om$. In \\cite{CT1}, it suffices to bound $\\pi -(\\pi)_{\\Omega_R}$ in $L^2(\\Omega_R)$. However,  for dimension 2  we need to bound $\\pi -(\\pi)_{\\Omega\\cap B_R}$ in $L^q(\\Omega_R)$, $q<2$. (We choose $q=4/3$ for simplicity.)\nWe have to choose $q<2$ because\n\\[\n\\pi {\\ \\lesssim \\ } \\nabla u + b \\otimes u -\\mathbb{G}\n\\]\nand $b \\otimes u$ is at best in $L^{2-}$ for $u \\in W^{1,2}_\\mathrm{loc}$ even if we assume $b \\in L^2$. This would not be an issue if $b \\in L^{r}$, $r>2$, and we can only assume $u \\in W^{1,2}_\\mathrm{loc}$ which is the assumption of the reversed H\\\"older inequality.\n\nTo obtain a uniform constant of Wolf's local pressure projection in $L^q(\\Omega_R)$ for $q<2$ gives a significant higher restriction on the regularity of the domain. Instead of a Lipschitz domain with Lipschitz constant $L$ less than 1/2, we have to assume $L$ is sufficiently small and do boundary stretching in the proof for the 2D case.\n\nThe third difficulty is the approximation of a Lipschitz domain. In order to construct the solution we are using an approximation method, therefore the first logical idea would be to construct a sequence of $b_n\\in L^{2,\\infty}(\\Om)$, $\\mathop{\\rm div}\\nolimits b=0$, such that $b_n$ converge in a sense that allows us to prove that the limit will satisfy the equation, for example convergence almost everywhere is sufficient and $\\|b_n\\|_{L^{2,\\infty}(\\Om)}$ are bounded uniformly. For example in star-shaped domains it is trivial, but for arbitrary Lipschitz domains this question is not so clear. To deal with this issue, following Verchota \\cite{Verchota_1,Verchota_2} and later Kwon \\cite{Kwon}, Chernobai-Shilkin \\cite{CS24}, we instead approximate the domain by Lipschtiz sub-domains with controlled Lipschitz constant and on sub-domains we can use standard mollification of $b$ and construct an approximation sequence as zero extension of solutions in these sub-domains. See Section \\ref{sec6} for details.\n\nThe rest of the paper is organized as the following. In section 2 we have preliminary results concerning Lorentz spaces and the pressure, which will be used in the later sections. In section 3 we derive the existence of weak solutions in $W^{1,2}(\\Omega)$. Section 4 contains Wolf's local pressure projection method that will be used to estimate the pressure. In section 5, assuming higher integrability of the drift, we prove uniform local estimate for the gradient which will imply higher integrability due to Gehring's lemma. Section 6 contains proof of the main Theorem \\ref{main1}. In the Appendix we put a proof of geometrical lemma that was used in section 4.", "sketch": "We will still use Gehring's approach: “do local energy estimate in every ball and prove a reversed H\\\"older inequality in a general ball with a uniform constant.”\n\nNew issues/steps specific to proving Theorem~\\ref{main1} in 2D:\n\\begin{itemize}\n\\item \\textbf{Non-integrable drift term; mollify and approximate domains.} One “cannot use $\\zeta = u\\eta^4$ … as the test function … because $b\\otimes u$ is no longer in $L^2$ … and the term $\\int_\\Om b \\otimes u : \\nabla( u \\eta^4)$ is not integrable.” Thus “we will mollify the drift $b$, so that we can derive an a priori bound.” Since there is no general extension theorem to $L^{2,\\infty}(\\mathbb{R}^2)$, they “consider the approximation problems in subdomains $\\Om_n$ that converge to $\\Om$.”\n\\item \\textbf{Pressure control via Wolf projection in $L^q$, $q<2$.} “The second new difficulty comes from the lower integrability of the pressure.” They “bound the pressure using Wolf's local pressure projection with a uniform constant.” In 2D they need “to bound $\\pi-(\\pi)_{\\Omega\\cap B_R}$ in $L^q(\\Omega_R)$, $q<2$ (we choose $q=4/3$),” because “$\\pi\\lesssim \\nabla u + b\\otimes u -\\mathbb{G}$” and “$b\\otimes u$ is at best in $L^{2-}$ for $u\\in W^{1,2}_{\\mathrm{loc}}$.” Achieving a uniform constant for the projection in $L^q$, $q<2$ forces stronger boundary regularity: “we have to assume $L$ is sufficiently small and do boundary stretching in the proof for the 2D case.”\n\\item \\textbf{Approximation of Lipschitz domains for constructing solutions.} To construct solutions they use an approximation method, but approximating $b$ directly in arbitrary Lipschitz domains is “not so clear.” Instead, “we approximate the domain by Lipschtiz sub-domains with controlled Lipschitz constant,” on which “we can use standard mollification of $b$,” and then “construct an approximation sequence as zero extension of solutions in these sub-domains.”\n\\end{itemize}\n\nThey indicate the proof is completed by these ingredients and is carried out later: “See Section~\\ref{sec6} for details,” and “Section 6 contains proof of the main Theorem~\\ref{main1}.”", "expanded_sketch": "We will still use Gehring's approach: “do local energy estimate in every ball and prove a reversed H\\\"older inequality in a general ball with a uniform constant.”\n\nNew issues/steps specific to establishing the main theorem in 2D:\n\\begin{itemize}\n\\item \\textbf{Non-integrable drift term; mollify and approximate domains.} One “cannot use $\\zeta = u\\eta^4$ … as the test function … because $b\\otimes u$ is no longer in $L^2$ … and the term $\\int_\\Om b \\otimes u : \\nabla( u \\eta^4)$ is not integrable.” Thus “we will mollify the drift $b$, so that we can derive an a priori bound.” Since there is no general extension theorem to $L^{2,\\infty}(\\mathbb{R}^2)$, they “consider the approximation problems in subdomains $\\Om_n$ that converge to $\\Om$.”\n\\item \\textbf{Pressure control via Wolf projection in $L^q$, $q<2$.} “The second new difficulty comes from the lower integrability of the pressure.” They “bound the pressure using Wolf's local pressure projection with a uniform constant.” In 2D they need “to bound $\\pi-(\\pi)_{\\Omega\\cap B_R}$ in $L^q(\\Omega_R)$, $q<2$ (we choose $q=4/3$),” because “$\\pi\\lesssim \\nabla u + b\\otimes u -\\mathbb{G}$” and “$b\\otimes u$ is at best in $L^{2-}$ for $u\\in W^{1,2}_{\\mathrm{loc}}$.” Achieving a uniform constant for the projection in $L^q$, $q<2$ forces stronger boundary regularity: “we have to assume $L$ is sufficiently small and do boundary stretching in the proof for the 2D case.”\n\\item \\textbf{Approximation of Lipschitz domains for constructing solutions.} To construct solutions they use an approximation method, but approximating $b$ directly in arbitrary Lipschitz domains is “not so clear.” Instead, “we approximate the domain by Lipschtiz sub-domains with controlled Lipschitz constant,” on which “we can use standard mollification of $b$,” and then “construct an approximation sequence as zero extension of solutions in these sub-domains.”\n\\end{itemize}\n\nThey indicate the proof is completed by these ingredients and is carried out later: “See Section~\\ref{sec6} for details,” and “Section 6 contains proof of the main theorem.”", "expanded_theorem": "\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\mathbb{R}^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\mathop{\\rm div}\\nolimits b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\mathbb{G}\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\label{drifteq}\n-\\Delta u +b\\cdot\\nabla u+\\nabla \\pi=\\div \\GG,\\quad \\div u=0, \\quad u|_{\\pd \\Om}=0,\n}\nwith given drift $b$ in the critical space, $b\\in L^{2,\\oo}(\\Omega)^2$, $\\div b=0$, and  force $\\G such that\n\\begin{theorem}\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\GG\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\eqref{drifteq} such that\n\\EQ{\\label{main1-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|\\GG\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming \\eqref{main1-eq}.\n\\end{theorem}\n\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming the preceding estimate.", "theorem_type": ["Existential–Universal", "Uniqueness"], "mcq": {"question": "Let Ω ⊂ ℝ² be a bounded Lipschitz domain whose Lipschitz constant L satisfies L < L0 for some sufficiently small 0 < L0 < 1. Assume b ∈ L^{2,\\infty}(Ω)^2 and div b = 0. For q > 1, set W^{1,q}_{0,σ}(Ω) = {u ∈ W^{1,q}_0(Ω)^2 : div u = 0} and L^q_0(Ω) = {\\pi ∈ L^q(Ω) : \\int_Ω \\pi\\,dx = 0}. A weak solution pair (u,\\pi) of\n-Δu + b\\cdot∇u + ∇\\pi = div \\mathbb{G}, \\quad div\\,u = 0 \\text{ in } Ω, \\quad u|_{\\partial Ω}=0,\nis one with u ∈ W^{1,q}_{0,σ}(Ω), \\pi ∈ L^q_0(Ω), and\n\\int_Ω (∇u - b\\otimes u):∇\\zeta - \\pi\\,div\\,\\zeta = -\\int_Ω \\mathbb{G}:∇\\zeta \\quad \\text{for all } \\zeta \\in C_c^\\infty(Ω)^2.\nIf p0' denotes the Hölder conjugate p0/(p0-1), which statement holds for this problem?", "correct_choice": {"label": "A", "text": "There exists p0 = p0(\\|b\\|_{L^{2,\\infty}}) > 2 such that for every q with p0' < q < p0 and every \\mathbb{G} ∈ L^q(Ω)^{2\\times 2}, there exists a unique weak solution pair (u,\\pi) ∈ W^{1,q}_{0,σ}(Ω) \\times L^q_0(Ω) of\n-Δu + b\\cdot∇u + ∇\\pi = div \\mathbb{G}, \\quad div\\,u = 0 \\text{ in } Ω, \\quad u|_{\\partial Ω}=0,\nand this pair satisfies\n\\|u\\|_{W^{1,q}(Ω)} + \\|\\pi\\|_{L^q(Ω)} \\le C(Ω,\\|b\\|_{L^{2,\\infty}})\\,\\|\\mathbb{G}\\|_{L^q(Ω)}.\nEquivalently, for each such q, the solution pair is unique in W^{1,q}_{0,σ}(Ω) \\times L^q_0(Ω) even without assuming the estimate a priori."}, "choices": [{"label": "B", "text": "There exists p0 = p0(\\|b\\|_{L^{2,\\infty}}) > 2 such that for every q with 1 < q < p0 and every \\mathbb{G} \\in L^q(\\Omega)^{2\\times 2}, there exists a unique weak solution pair (u,\\pi) \\in W^{1,q}_{0,\\sigma}(\\Omega) \\times L^q_0(\\Omega) of\n-\\Delta u + b\\cdot\\nabla u + \\nabla\\pi = \\operatorname{div} \\mathbb{G}, \\quad \\operatorname{div} u = 0 \\text{ in } \\Omega, \\quad u|_{\\partial\\Omega}=0,\nand this pair satisfies\n\\|u\\|_{W^{1,q}(\\Omega)} + \\|\\pi\\|_{L^q(\\Omega)} \\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\,\\|\\mathbb{G}\\|_{L^q(\\Omega)}."}, {"label": "C", "text": "There exists p0 = p0(\\|b\\|_{L^{2,\\infty}}) > 2 such that for every q with p0' < q < p0 and every \\mathbb{G} \\in L^q(\\Omega)^{2\\times 2}, there exists a weak solution pair (u,\\pi) \\in W^{1,q}_{0,\\sigma}(\\Omega) \\times L^q_0(\\Omega) of\n-\\Delta u + b\\cdot\\nabla u + \\nabla\\pi = \\operatorname{div} \\mathbb{G}, \\quad \\operatorname{div} u = 0 \\text{ in } \\Omega, \\quad u|_{\\partial\\Omega}=0,\nsuch that\n\\|u\\|_{W^{1,q}(\\Omega)} + \\|\\pi\\|_{L^q(\\Omega)} \\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\,\\|\\mathbb{G}\\|_{L^q(\\Omega)}."}, {"label": "D", "text": "There exists p0 = p0(\\|b\\|_{L^{2,\\infty}}) > 2 such that for every q with p0' < q < p0 and every \\mathbb{G} \\in L^q(\\Omega)^{2\\times 2}, there exists a unique weak solution pair (u,\\pi) \\in W^{1,q}_{0,\\sigma}(\\Omega) \\times L^q_0(\\Omega) of\n-\\Delta u + b\\cdot\\nabla u + \\nabla\\pi = \\operatorname{div} \\mathbb{G}, \\quad \\operatorname{div} u = 0 \\text{ in } \\Omega, \\quad u|_{\\partial\\Omega}=0,\nand this pair satisfies\n\\|u\\|_{W^{1,q}(\\Omega)} + \\|\\pi\\|_{L^q(\\Omega)} \\le C(\\Omega)\\,\\|\\mathbb{G}\\|_{L^q(\\Omega)},\nwith a constant independent of b."}, {"label": "E", "text": "If \\Omega \\subset \\mathbb{R}^2 is a bounded Lipschitz domain and b \\in L^{2,\\infty}(\\Omega)^2 is divergence free, then for every q with p0' < q < p0 and every \\mathbb{G} \\in L^q(\\Omega)^{2\\times 2}, there exists a unique weak solution pair (u,\\pi) \\in W^{1,q}_{0,\\sigma}(\\Omega) \\times L^q_0(\\Omega) of\n-\\Delta u + b\\cdot\\nabla u + \\nabla\\pi = \\operatorname{div} \\mathbb{G}, \\quad \\operatorname{div} u = 0 \\text{ in } \\Omega, \\quad u|_{\\partial\\Omega}=0,\nand this pair satisfies\n\\|u\\|_{W^{1,q}(\\Omega)} + \\|\\pi\\|_{L^q(\\Omega)} \\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\,\\|\\mathbb{G}\\|_{L^q(\\Omega)}."}], "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": "lower q-threshold p0'<q", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "uniqueness clause", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dependence of constants on \\|b\\|_{L^{2,\\infty}}", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "small Lipschitz constant requirement for Wolf projection/boundary stretching", "template_used": "wildcard"}]}}
{"id": "2512.14627v1", "paper_link": "http://arxiv.org/abs/2512.14627v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\mathbb{R}^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\mathop{\\rm div}\\nolimits b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\mathbb{G}\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\eqref{drifteq} such that\n\\EQ{\\label{main1-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|\\mathbb{G}\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming \\eqref{main1-eq}.", "start_pos": 12114, "end_pos": 12850, "label": "main1"}, "ref_dict": {"main1-eq": "\\begin{theorem}\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\GG\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\eqref{drifteq} such that\n\\EQ{\\label{main1-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|\\GG\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming \\eqref{main1-eq}.\n\\end{theorem}", "change-var": "\\begin{proof}\nSimilar to \\cite{CT1} we will use the Gehring's lemma approach \\cite{Gehring, Gia}. Denote $\\Om_{x,R}=\\Om \\cap B(x,R)$ for $x\\in \\overline \\Om$ and $R>0$. Since $\\pd \\Om$ is Lipschitz and compact, there\nis a radius $R_0\\in (0,1]$ such that for every $x_0 \\in\\pd \\Om$,\n\\EQ{ \\label{R0.def}\n\\Om_{x_0,2R_0}= \\{ (x',x_n) \\in B_{2R_0}(0) \\subset \\R^2,\\ x_n>\\ga(x')\\},\n}\nafter suitable coordinate rotation and translation, where $\\ga(x')$ is a Lipschitz function defined for $x'\\in B_{2R_0}'(0) \\subset \\R$ with Lipschitz constant $L$ and $\\ga(0)=0$.\n\nTo prove the a priori estimate \\eqref{eq5.1}, we first consider the interior case with $x_0\\in\\Omega$ and $B_{\\rho}=B_{\\rho}(x_0)\\subset\\Omega$. We rescale it to the unit ball $B$ and denote $u_{\\rho}(x)=\\rho u(\\rho x+x_0),b_{\\rho}(x)=\\rho b(\\rho x+x_0),\\pi_{\\rho}(x)=\\rho^2\\pi(\\rho x+x_0)$ and $\\GG_{\\rho}(x)=\\rho^2 \\GG(\\rho x+x_0)$. Let $\\bar{u}_{\\rho}=u_{\\rho}-k$ where $k$ is a real constant to be choose later.  By Remark \\ref{Wolfpdecom}, we can decompose $\\pi_{\\rho} - (\\pi_{\\rho})_{B}=\\pi_1+\\pi_2+\\pi_3$ in $B$, where $\\pi_1,\\pi_2,\\pi_3$ are given by\n\\EQ{\n\\nabla\\pi_1=\\mathcal{W}_{4/3,B}(\\Delta \\bar{u}_{\\rho}),\\quad \\nabla\\pi_2=\\mathcal{W}_{4/3,B}(-\\div(b_{\\rho}\\otimes \\bar{u}_{\\rho})),\\quad \\nabla\\pi_3=\\mathcal{W}_{4/3,B}(\\div \\GG_{\\rho}),\n}\nwith $\\int_B \\pi_i=0$.\nTherefore, by Theorem \\ref{theoremL2wolf} we have the following pressure bound\n\\EQ{\\label{wolfbound}\n\\|\\pi_1\\|_{4/3,B}\\le c_1\\|\\nabla \\bar{u}_{\\rho}\\|_{4/3,B},\\quad\\|\\pi_2\\|_{4/3,B}\\le c_1 \\|b_{\\rho}\\bar{u}_{\\rho}\\|_{4/3,B},\\quad \\|\\pi_3\\|_{4/3,B}\\le c_1\\|\\GG_{\\rho}\\|_{4/3,B},\n}\nwhere $c_1$ is a global constant.\nFor simplicity of notations let us drop the subscript $\\rho$ for the further calculations, until we need to scale back to ball $B_{\\rho}(x_0)$.\n\nTake a smooth cutoff function $\\eta$ on $B$ with $\\eta=1$ in $B_{\\frac{1}{2}}$. Use the test function $\\zeta=\\eta^4 \\bar{u}$ in the weak form \\eqref{soln-pair} with $(u,\\pi)$ replaced by $(\\bar u, \\pi - (\\pi)_B)$ to get the energy estimate (using \\eqref{5.1} due to $b \\in L^{2+\\de}$)\n\\EQS{\\label{3.1}\n\\int_{B} |\\nabla (\\eta^2\\bar{u})| ^2\n\\le &\\int_{B} |\\bar{u}|^2(|\\nabla \\eta^2|^2) +\\int_{B} |\\bar{u}|^2|b||\\nabla\\eta|\\eta^3\n\\\\\n&+\\int_{B} (|\\pi_1|+|\\pi_2|+|\\pi_3|)|\\bar{u}||\\nabla\\eta|\\eta^3+\\bka{\\div \\GG_{\\rho}, \\bar{u}\\eta^4}_B.\n}\n\nWe will estimate each of the terms on the right hand side separately, using Lemmas \\ref{Holder} and \\ref{Sobolev} in ball $B$,\n\\EQS{\n\\int_{B} |\\bar{u}|^2|b||\\nabla \\eta|\\eta^3\n&\\le \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}^2\\eta^3\\|_{L^{2,1}(B)}\\le \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}\\eta^{3/2}\\|_{L^{4,2}(B)}^2\n\\\\\n&\\le C \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}\\eta^{3/2}\\|^2_{W^{1,4/3}(B)}.\n}\nFor pressure terms we use \\eqref{wolfbound} and get the following\n\\EQN{\n\\int_{B}|\\pi_1||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_1\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{L^4(B)}\n\\\\\n&\\le\nc\\|\\nabla \\bar{u}\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}\\le c\\|\\nabla \\bar{u}\\|_{4/3,B}^2+c\\|\\eta \\bar{u}\\|^2_{4/3,B},\n}\n\\EQN{\n\\int_{B}|\\pi_2||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_2\\|_{L^{4/3}(B)}\\|\\eta^2\\bar{u}\\|_{L^4(B)}\\le c\\|\\bar{u}\\|_{L^{4,4/3}(B)}\\|b\\|_{L^{2,\\infty}(B)}\\|\\eta^2\\bar{u}\\|_{L^{4}(B)}\n\\\\\n&\\le c(b)\\|\\nabla \\bar{u}\\|_{4/3,B}^2+c(b)\\| \\bar{u}\\|^2_{4/3,B}\n}\n\\EQN{\n\\int_{B}|\\pi_3||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_3\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{L^4(B)}\\le  c\\|\\GG\\|_{2,B}\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}\n\\\\\n&\\le \\frac 14 \\|\\GG\\|_{2,B}^2+c\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}^2.\n}\nLastly we estimate the term containing $\\GG$,\n\\EQN{\n\\bka{\\div \\GG_{\\rho}, \\bar{u}\\eta^4}_B\n&=-\\int_{B} \\GG :\\nabla(\\bar{u}\\eta^4)=-\\int_{B} \\GG: \\nabla(\\eta^2\\bar{u})\\eta^2-2\\int_{B} \\GG :\\bar{u}\\nabla\\eta\\eta^3\n\\\\\n&\\le \\frac{1}{16}\\|\\nabla(\\eta^2\\bar{u})\\|^2_{2,B}+c\\|\\eta^2 \\bar{u}\\|_{2,B}^2+ \\frac 14\\|\\GG\\|_{2,B}^2.\n}\nCombining these estimates with \\eqref{3.1} we get the following\n\\EQ{\n\\int |\\nabla (\\eta^2\\bar{u})|^2\\le c(b)\\|\\bar{u}\\|^2_{W^{1,4/3}(B)}+c(b)(\\|\\bar{u}\\|_{2,B}^2+\\|\\bar{u}\\|_{4/3,B}^2)+\\|\\GG\\|_{2,B}^2,\n}\nHere $c(b)$ depends on $b$ only through $\\norm{b}_{L^{2,\\infty}(\\Om)}$. Finally, since $\\eta=1$ on $B_{1/2}$ we get that\n\\EQ{\n\\int_{B_{1/2}} |\\nabla \\bar{u}|^2\\le c(b)\\|\\nabla \\bar{u}\\|^2_{4/3,B}+c(b)\\|\\bar{u}\\|_{2,B}^2+\\|\\GG\\|_{2,B}^2.\n}\nWe now choose $k=(u_{\\rho})_{B}$ as the constant in $\\bar{u}=u_{\\rho}-k$ and apply Poincar\\'e inequality to get\n\\EQ{\n\\int_{B_{1/2}} |\\nabla u_{\\rho}|^2\\le c(b)\\|\\nabla u_{\\rho}\\|^2_{4/3,B}+\\|\\GG\\|_{2,B}^2.\n}\nLastly we scale back to $B_{\\rho}(x_0)$ and get that\n\\EQ{\\label{5.12}\n\\frac{1}{|B_{\\rho/2}|}\\int_{B_{\\rho/2}} |\\nabla u|^2\n\\le c(b) \\Big(\\frac{1}{|B_{\\rho}|}\\int_{B_{\\rho}}|\\nabla u|^{4/3}\\Big)^{3/2}+\\frac{c}{|B_{\\rho}|}\\|\\GG\\|_{2,B_{\\rho}}^2.\n}\n\n\\medskip\n\nNext we consider the boundary case, $\\Omega_{\\rho}=\\Omega\\cap B_{\\rho}(x_0)$ with $x_0\\in\\pd\\Om$, and $0<\\rho<R_0$. The significant difference from the internal case is the pressure estimate: Unlike the internal case we can not use Wolf decomposition estimate in a uniform ball, therefore we need to track the dependence of constant $c_1$ when we apply Theorem \\ref{theoremL2wolf} to get estimates similar to \\eqref{pressure_bound_est}\n and make sure it is uniform across all  $\\Omega_{\\rho}(x_0)$.\n Rescale the domain as the interior case so that $x_0=0$ and $\\rho=1$.\n We will use change of coordinates\n \\EQ{\\label{change-var}\n y_1=x_1, \\quad y_2=x_2-\\gamma(x_1),\n }\n to straighten the boundary and map $\\Om_{5/8}\\subset\\Om_1$ into domains $\\Om_{5/8}'\\subset\\Om_1'$ with flat boundary on $y_2=0$. Since $\\gamma$ was a Lipschitz curve with sufficiently small Lipschitz constant we can assume that $\\Om_{5/8}'\\subset B_{6/8}^+\\subset B_{7/8}^+\\subset\\Om_1'$.\n  (This is the first place that specifies the smallness of the Lipschitz constant.) By Lemma \\ref{geometry} we can fix a smooth $V$ such that $B_{6/8}^+\\subset V\\subset B_{7/8}^+$. After rewriting the equation \\eqref{drifteq} in new variables we get the following system (with all derivatives in $y$) in $\\Om_1'$:\n\\EQS{\\label{changed_var}\n-\\Delta_y u+\\div (\\gamma' e_1\\otimes\\partial_{y_2}u)+\n\\gamma' \\pd_1\\pd_2 u - (\\gamma')^2\\partial_{y_2}^2u+\n\\\\\n+\\div_y(b\\otimes u)- \\gamma'\\partial_{y_2}(b\\otimes u)_1+\\nabla_y \\pi-\\gamma'\\partial_{y_2}\\pi e_1&=\\div\\GG-\\gamma'\\partial_2 \\GG_1 ,\n\\\\\n\\div_y u-\\gamma'\\partial_2 u_1&=0.\n}\nAbove, $(b\\otimes u)_1=(b_1u_1,b_1u_2)$ and $\\GG_1=(G_{11},G_{12})$ are their first rows. In the second term, $\\ga'$ is hidden behind $\\div$ to avoid $\\ga''$ which we do not assume any bound.\n\nWe apply Wolf's local pressure projection $\\mathcal{W}_{4/3,V}$ on \\eqref{changed_var} in the smooth domain $V$ and get that  $\\pi-(\\pi)_{V}=\\pi_1+\\pi_2+\\pi_3+\\pi_4$ where\n\\EQS{\\label{pressure_decomp_b}\n\\nabla\\pi_1&=\\mathcal{W}_{4/3,V}\\bkt{\\Delta_y u-\\div (\\gamma' e_1\\otimes\\partial_{y_2}u)-\\gamma' \\pd_1\\pd_2 u +(\\gamma')^2\\partial_{y_2}^2u}\n\\\\\n\\nabla\\pi_2&=\\mathcal{W}_{4/3,V}\\bkt{-\\div_y(b\\otimes u)+\\gamma'\\partial_{y_2}(b\\otimes u)_1}\n\\\\\n \\nabla\\pi_3&=\\mathcal{W}_{4/3,V}(\\gamma' \\partial_{y_2}\\pi e_1)\n \\\\\n \\nabla\\pi_4&= \\mathcal{W}_{4/3,V}(\\div \\GG-\\gamma'\\partial_2 \\GG_1).\n}\nFrom Theorem \\ref{theoremL2wolf} the local pressure projection  $\\mathcal{W}_{4/3,V}$ is well defined, and we have the bound \\eqref{pWolfbound} with global constant $c$. Also using $\\ga'\\pd_2 = \\pd_2 \\ga'$, and Lemmas \\ref{Holder} and \\ref{Sobolev} in $V$,\n\\EQS{\\label{pressure_bound_est}\n\\|\\pi_1\\|_{4/3,V}&\\le c(1+\\|\\gamma'\\|_{\\infty}+\\|\\gamma'\\|_{\\infty}^2)\\|\\nabla u\\|_{4/3,V}\n\\\\\n\\|\\pi_2\\|_{4/3,V}&\\le  c(1+\\|\\gamma'\\|_{\\infty})\\|u\\otimes b\\|_{4/3,V}\n\\\\\n\\|\\pi_3\\|_{4/3,V}&\\le  c\\|\\gamma'\\|_{\\infty}\\|\\pi\\|_{4/3,V}\\le c\\|\\gamma'\\|_{\\infty}\\|\\pi_3\\|_{4/3,V}+c\\|\\gamma'\\|_{\\infty}\\|\\pi_1+\\pi_2+\\pi_4\\|_{4/3,V}\n\\\\\n\\|\\pi_4\\|_{4/3,V}&\\le  c(1+\\|\\gamma'\\|_{\\infty})\\|\\GG\\|_{4/3,V}\n}\nHere we apply our choice of $\\gamma$ so that $\\|\\gamma'\\|_{\\infty}<1$ and\n$ c\\|\\gamma'\\|_{\\infty}<1/2$. (This is the second place that specifies the smallness of the Lipschitz constant.) After applying it to \\eqref{pressure_bound_est} we get the following\n\\EQS{\\label{pressure_3}\n\\|\\pi_3\\|_{4/3,V}\\le 2c\\|\\gamma'\\|_{\\infty}\\|\\pi_1+\\pi_2+\\pi_4\\|_{4/3,V}.\n}\n\nLet $\\Om^\\sharp$ be the pre-image of $V$ under the map \\eqref{change-var} in $x$-variables.\nThe pressure component $\\pi_i$ can be considered functions on $\\Om^\\sharp$ and since \\eqref{change-var} has determinant 1 and $\\|\\nabla_y u\\|_{4/3,V}\\le 2\\|\\nabla_x u\\|_{4/3,\\Omega^\\sharp}$, by assumption $\\|\\gamma'\\|_{\\infty}\\le 1$ we have\n\\EQS{\\label{pressure-est}\n\\norm{\\pi-(\\pi)_{V}}_{4/3, \\Om^\\sharp} &\\le {\\textstyle\\sum}_{i=1}^4 \\norm{\\pi_i}_{4/3, \\Om^\\sharp}\n={\\textstyle\\sum}_{i=1}^4 \\norm{\\pi_i}_{4/3, V}\n\\\\\n&\\le c\\|\\nabla _y u\\|_{4/3,V} +c \\|u\\otimes b\\|_{4/3,V} +c \\|\\GG\\|_{4/3,V}\n\\\\\n&\\le c(1+ \\| b\\|_{L^{2,\\infty}(V)}) (\\|\\nabla_y u\\|_{4/3,V} +\\| u\\|_{4/3,V} )+ c \\|\\GG\\|_{4/3,V}\n\\\\\n&\\le c(1+ \\| b\\|_{L^{2,\\infty}(\\Omega)}) (\\|\\nabla_x u\\|_{4/3,\\Omega^\\sharp} +\\| u\\|_{4/3,\\Omega^\\sharp} )+ c \\|\\GG\\|_{4/3,\\Omega^\\sharp}.\n}\nThis pressure estimate is the only purpose of our change of variables \\eqref{change-var}. Noting that $\\Om_{5/8}\\subset \\Om^\\sharp\\subset \\Om_1$, we then proceed the following.\n\n Let $\\eta$ be a cutoff function supported in $B_{5/8}(x_0)$ such that $\\eta=1$ on $B_{1/2}(x_0)$. Due to our boundary condition $u|_{\\partial\\Omega}=0$ we can use function $\\eta^4u$ as an admissible test function in \\eqref{drifteq}. Therefore we get the following local energy inequality\n\\EQS{\\label{eq5.19}\n\\int_{\\Om_1} |\\nabla (\\eta^2u)|^2\n&\\le \\int_{\\Om_1} |u|^2|\\nabla \\eta^2|^2+\\int_{\\Om_1} |u|^2|b||\\nabla \\eta|\\eta^3\\\\\n&\\quad +\\int_{\\Om_1} |\\pi-(\\pi)_{V}||u||\\nabla\\eta|\\eta^3-\\int_{\\Om_1} \\GG :\\nabla(u\\eta^4).\n}\nUsing \\eqref{pressure-est} to bound the pressure term and similarly to internal case\nusing Lemmas \\ref{Holder} and \\ref{Sobolev} in $\\Om_1$, with uniform constant $C_2$ in Lemma \\ref{Sobolev} using $\\eta^4u\\in W^{1,2}_0(\\Om_1)$, we get\n\\EQS{\\label{5.16}\n\\int_{\\Om_{1/2}} |\\nabla u|^2 &\\le C(b)(\\|\\nabla u\\|^2_{4/3,\\Om_1}+\\|u\\|_{2,\\Om_1}^2)+c\\|\\GG\\|_{2,\\Om_1}^2.\n}\nNotice that here the constant $C(b)$ does not depend on $\\Om_1$.\nFinally we extend function $u,\\GG$ by zero to  $B_1(x_0)\\setminus \\Omega$. Since $u=0$ on the set $E:=\\Om^c\\cap B_1(x_0)$ of non zero measure with the ratio $\\frac{|E|}{|B_{1}|}$ bounded from below, we can use Poincar\\'e inequality to bound $\\|u\\|_{2,\\Om_1}^2 \\le c \\|\\nabla u\\|^2_{4/3,\\Om_1}$, with a constant independent of $\\Om_1$ with the given lower bound of the ratio.\n(This version of Poincar\\'e inequality follows from Proposition 6.14 of \\cite[page 116]{Lieberman},  with $f=\\al 1_E$, nt_{B_1} f=1$. It gives $\\norm{u}_{4/3,B_1} \\le C\\al  \\norm{\\nabla u}_{4/3,B_1} $.\nWe also have\n$\\norm{u}_{4,B_1} \\le C \\norm{u}_{W^{1,4/3}(B_1)}$.)\n\nWe also re-scale the inequality for arbitrary $B_r(x_0)$ and get \\eqref{5.12}.\n\n\\medskip\nLastly we need to consider the case of arbitrary $x_0,r$ such that $\\Om^c\\cap B_{2r}(x_0)\\neq\\emptyset$. For any such $x_0$ we can find $y_0\\in\\partial\\Om\\cap B_{2r}(x_0)$ such that\n\\EQ{\nB_r(x_0)\\subset B_{3r}(y_0)\\subset B_{6r}(y_0)\\subset B_{8r}(x_0).\n}\nUsing \\eqref{5.12} for $B_{3r}(y_0)$ (assuming $6r<R_0$) we get\n\\EQS{\\label{5.17}\n\\frac{1}{|B_r|}\\int_{B_r(x_0)} |\\nabla u|^2\n&\\le \\frac{C}{|B_{3r}|}\\int_{B_{3r}(y_0)} |\\nabla u|^2\n\\\\\n&\\le c(b) \\Big(\\frac{1}{|B_{6r}|}\\int_{B_{6r}(y_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{6r}|}\\|\\GG\\|_{L^2(B_{6r}(y_0))}^2\n\\\\\n& \\le c(b) \\Big(\\frac{1}{|B_{8r}|}\\int_{B_{8r}(x_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{8r}|}\\|\\GG\\|_{L^2(B_{8r}(x_0))}^2.\n}\n\nFor Gehring's lemma we combine the internal estimate \\eqref{5.12} and the boundary estimate \\eqref{5.17} and conclude that\n\\EQ{\\label{5.19}\n\\frac{1}{|B_r|}\\int_{B_r(x_0)} |\\nabla u|^2 \\le\nc(b) \\Big(\\frac{1}{|B_{8r}|}\\int_{B_{8r}(x_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{8r}|}\\|\\GG\\|_{L^2(B_{8r}(x_0))}^2,\n}\nfor any $x_0$ in a big cube containing $\\overline \\Om$, and any $r\\le R_0/6$.\nWe can now apply Proposition 1.1 of Giaquinta \\cite[page 122]{Gia} (also see \\cite[Proposition 3.7]{DongKim}) to get that $\\nb u \\in L^p_\\loc$ for $p \\in (2,p_0)$ for some $p_0 >2$, and\n\\EQ{\\label{0619}\nnt_{B_r(x_0)} |\\nabla u|^p}^{\\frac 1p} \\le\nnt_{B_{8r}(x_0)} |\\GG|^p}^{\\frac 1p},\n}\nfor all $x_0 \\in \\overline \\Om$ and $r\\le R_0/6$, with\n$p_0$ and $c_1$ depending only on $c(b)$. Summing \\eqref{0619} over a finite cover of $\\overline \\Om$ of balls of radius $R_0/6$,\nwe get \\eqref{eq5.1}.\n\\end{proof}", "main2-eq": "\\begin{theorem}\\label{th1.2}\n  Let $\\Om$ be a bounded Lipschtiz domain in $\\RR^2$.  Assume $b\\in L^{2,\\infty}(\\Omega)$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\\in L^q(\\Om)$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ of \\eqref{scalar.eq} such that\n\\EQ{\\label{main2-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|G\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\\Om)$ without assuming \\eqref{main2-eq}.\n\\end{theorem}", "main1": "\\begin{theorem}\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\GG\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\eqref{drifteq} such that\n\\EQ{\\label{main1-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|\\GG\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming \\eqref{main1-eq}.\n\\end{theorem}", "scalar.eq": "\\begin{theorem}\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\GG\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\eqref{drifteq} such that\n\\EQ{\\label{main1-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|\\GG\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming \\eqref{main1-eq}.\n\\end{theorem}\n\n\\emph{Comments on Theorem \\ref{main1}:}\n\\EN{\n\\item This theorem extends the higher dimensional result \\cite[Theorem 1.5]{CT1} to the dimension 2 case.\nIn \\cite[Theorem 1.5]{CT1} we take $L_0=1/2$. In this paper $L_0$ is specified after \\eqref{change-var} and after \\eqref{pressure_bound_est}, and is computable.\nWe skip in this paper the analogous results for small $b$ case (but full range $1<q<\\infty$) as in \\cite[Proposition 1.3, Theorem 1.4]{CT1}.\n\n\\item  Since $\\Om$ is Lipschitz we can assume that there exists $R_0>0$ such that for any $x_0\\in \\partial\\Om$ we have\n\\EQ{\n\\Om\\cap B_{2R_0}(x_0)=\\{(x',x_n)\\in B_{2R_0}(0),x_n>\\gamma(x')\\}.\n}\nThe constant $C$ in \\eqref{main1-eq} will actually depend only on $R_0$, $\\text{diam}~\\Om$, and $\\|b\\|_{L^{2,\\infty}}$.\n\n\\item If we further assume $b \\in L^2(\\Om)$\nand $b$ has a divergence-free extension to $L^2(\\R^2)$,\nthen there is another way to prove uniqueness of the weak solution in $W^{1,2}_{0,\\si}(\\Om)$ by using Hardy-BMO duality, even when the Lipschitz constant of $\\Om$ is large.\nSee Theorem \\ref{energy_2} in Section \\ref{sec3}.\n\n\\item We may simply assume $\\Om$ is a bounded $C^1$ domain, as for\na bounded $C^1$ domain and any constant $\\lambda>0$, each point $x$ on $\\pd\\Om$ has a neighborhood $U_x$ such that $\\pd \\Om \\cap U_x$ is the graph of a Lipschitz function with a Lipschitz constant less than $\\lambda$.\n\n}\n\nFor corresponding results on scalar equations\n\\EQ{\\label{scalar.eq}\n-\\Delta u +b\\cdot\\nabla u=\\div  G, \\quad u|_{\\pd \\Om}=0,\n}\nwith critical drift coefficient $b \\in L^n$, see\nKim \\& Kim \\cite{KK}. When $b \\in L^{n,\\infty}(\\Om)$ for $n \\ge 3$, see \\cite{KiTs,KPT} which also cover more general assumptions on drift $b$ as well as \\cite{CS20,CS24} and Kwon \\cite{Kwon} for $n=2$ case or drift in Morrey spaces which include $L^{n,\\infty}$.\nThere are also results on H\\\"older continuity of the solutions, for example with $b\\in BMO^{-1}(\\R^3)$, $\\div b=0$, see Seregin-Sverak-Silvestre-Zlatos \\cite{SSSZ}.\n\nUsing the same proof scheme we can also prove the following theorem for the  scalar equation \\eqref{scalar.eq}.\n\\begin{theorem}\\label{th1.2}\n  Let $\\Om$ be a bounded Lipschtiz domain in $\\RR^2$.  Assume $b\\in L^{2,\\infty}(\\Omega)$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\\in L^q(\\Om)$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ of \\eqref{scalar.eq} such that\n\\EQ{\\label{main2-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|G\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\\Om)$ without assuming \\eqref{main2-eq}.\n\\end{theorem}", "drifteq": "\\label{drifteq}\n-\\Delta u +b\\cdot\\nabla u+\\nabla \\pi=\\div \\GG,\\quad \\div u=0, \\quad u|_{\\pd \\Om}=0,\n}\nwith given drift $b$ in the critical space, $b\\in L^{2,\\oo}(\\Omega)^2$, $\\div b=0$, and  force $\\G", "1.2": "\\label{1.2}\n\\De u = \\div (\\nb u),\\quad\nb\\cdot\\nabla u = \\div (b\\otimes u), \\quad \\int_\\Om  \\div \\GG \\cdot \\zeta = -\n\\int_\\Om  \\GG:\\nb \\zeta.\n}\nFor zero divergence spaces or spaces with zero average we", "energy_2": "\\begin{theorem}[Unique weak solution for $L^2$ drift]\\label{energy_2}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$.\nAssume $b\\in L^{2}(\\Om)^2$ has an extension\n$\\tilde{b}\\in L^{2}(\\R^2)^2$ with $\\div \\tilde{b}=0$ in $\\R^2$.\nThen for any $f \\in W^{-1,2}(\\Omega)^2$,\nthere exists a unique weak solution $u\\in W^{1,2}_{0,\\sigma}(\\Omega)$\nof \\eqref{drifteq}. Furthermore, there is $\\pi \\in L^2(\\Om)$ so that $(u,\\pi)$ solves \\eqref{drifteq} in distributional sense, $\\int_\\Om \\pi=0$, and\n\\EQ{\\label{eq6.1b}\n\\|u\\|_{ W^{1,2}(\\Omega)}+\\|\\pi\\|_{L^2(\\Omega)}\\le C\\|f\\|_{W^{-1,2}(\\Om)},\n}\nfor some constant $C=C(\\Om)$ independent of $b$.\n\\end{theorem}", "pressure_bound_est": "\\begin{proof}\nSimilar to \\cite{CT1} we will use the Gehring's lemma approach \\cite{Gehring, Gia}. Denote $\\Om_{x,R}=\\Om \\cap B(x,R)$ for $x\\in \\overline \\Om$ and $R>0$. Since $\\pd \\Om$ is Lipschitz and compact, there\nis a radius $R_0\\in (0,1]$ such that for every $x_0 \\in\\pd \\Om$,\n\\EQ{ \\label{R0.def}\n\\Om_{x_0,2R_0}= \\{ (x',x_n) \\in B_{2R_0}(0) \\subset \\R^2,\\ x_n>\\ga(x')\\},\n}\nafter suitable coordinate rotation and translation, where $\\ga(x')$ is a Lipschitz function defined for $x'\\in B_{2R_0}'(0) \\subset \\R$ with Lipschitz constant $L$ and $\\ga(0)=0$.\n\nTo prove the a priori estimate \\eqref{eq5.1}, we first consider the interior case with $x_0\\in\\Omega$ and $B_{\\rho}=B_{\\rho}(x_0)\\subset\\Omega$. We rescale it to the unit ball $B$ and denote $u_{\\rho}(x)=\\rho u(\\rho x+x_0),b_{\\rho}(x)=\\rho b(\\rho x+x_0),\\pi_{\\rho}(x)=\\rho^2\\pi(\\rho x+x_0)$ and $\\GG_{\\rho}(x)=\\rho^2 \\GG(\\rho x+x_0)$. Let $\\bar{u}_{\\rho}=u_{\\rho}-k$ where $k$ is a real constant to be choose later.  By Remark \\ref{Wolfpdecom}, we can decompose $\\pi_{\\rho} - (\\pi_{\\rho})_{B}=\\pi_1+\\pi_2+\\pi_3$ in $B$, where $\\pi_1,\\pi_2,\\pi_3$ are given by\n\\EQ{\n\\nabla\\pi_1=\\mathcal{W}_{4/3,B}(\\Delta \\bar{u}_{\\rho}),\\quad \\nabla\\pi_2=\\mathcal{W}_{4/3,B}(-\\div(b_{\\rho}\\otimes \\bar{u}_{\\rho})),\\quad \\nabla\\pi_3=\\mathcal{W}_{4/3,B}(\\div \\GG_{\\rho}),\n}\nwith $\\int_B \\pi_i=0$.\nTherefore, by Theorem \\ref{theoremL2wolf} we have the following pressure bound\n\\EQ{\\label{wolfbound}\n\\|\\pi_1\\|_{4/3,B}\\le c_1\\|\\nabla \\bar{u}_{\\rho}\\|_{4/3,B},\\quad\\|\\pi_2\\|_{4/3,B}\\le c_1 \\|b_{\\rho}\\bar{u}_{\\rho}\\|_{4/3,B},\\quad \\|\\pi_3\\|_{4/3,B}\\le c_1\\|\\GG_{\\rho}\\|_{4/3,B},\n}\nwhere $c_1$ is a global constant.\nFor simplicity of notations let us drop the subscript $\\rho$ for the further calculations, until we need to scale back to ball $B_{\\rho}(x_0)$.\n\nTake a smooth cutoff function $\\eta$ on $B$ with $\\eta=1$ in $B_{\\frac{1}{2}}$. Use the test function $\\zeta=\\eta^4 \\bar{u}$ in the weak form \\eqref{soln-pair} with $(u,\\pi)$ replaced by $(\\bar u, \\pi - (\\pi)_B)$ to get the energy estimate (using \\eqref{5.1} due to $b \\in L^{2+\\de}$)\n\\EQS{\\label{3.1}\n\\int_{B} |\\nabla (\\eta^2\\bar{u})| ^2\n\\le &\\int_{B} |\\bar{u}|^2(|\\nabla \\eta^2|^2) +\\int_{B} |\\bar{u}|^2|b||\\nabla\\eta|\\eta^3\n\\\\\n&+\\int_{B} (|\\pi_1|+|\\pi_2|+|\\pi_3|)|\\bar{u}||\\nabla\\eta|\\eta^3+\\bka{\\div \\GG_{\\rho}, \\bar{u}\\eta^4}_B.\n}\n\nWe will estimate each of the terms on the right hand side separately, using Lemmas \\ref{Holder} and \\ref{Sobolev} in ball $B$,\n\\EQS{\n\\int_{B} |\\bar{u}|^2|b||\\nabla \\eta|\\eta^3\n&\\le \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}^2\\eta^3\\|_{L^{2,1}(B)}\\le \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}\\eta^{3/2}\\|_{L^{4,2}(B)}^2\n\\\\\n&\\le C \\|b\\|_{L^{2,\\infty}(B)}\\|\\bar{u}\\eta^{3/2}\\|^2_{W^{1,4/3}(B)}.\n}\nFor pressure terms we use \\eqref{wolfbound} and get the following\n\\EQN{\n\\int_{B}|\\pi_1||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_1\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{L^4(B)}\n\\\\\n&\\le\nc\\|\\nabla \\bar{u}\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}\\le c\\|\\nabla \\bar{u}\\|_{4/3,B}^2+c\\|\\eta \\bar{u}\\|^2_{4/3,B},\n}\n\\EQN{\n\\int_{B}|\\pi_2||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_2\\|_{L^{4/3}(B)}\\|\\eta^2\\bar{u}\\|_{L^4(B)}\\le c\\|\\bar{u}\\|_{L^{4,4/3}(B)}\\|b\\|_{L^{2,\\infty}(B)}\\|\\eta^2\\bar{u}\\|_{L^{4}(B)}\n\\\\\n&\\le c(b)\\|\\nabla \\bar{u}\\|_{4/3,B}^2+c(b)\\| \\bar{u}\\|^2_{4/3,B}\n}\n\\EQN{\n\\int_{B}|\\pi_3||\\bar{u}||\\nabla\\eta|\\eta^3 &\\le \\|\\pi_3\\|_{4/3,B}\\|\\eta^2 \\bar{u}\\|_{L^4(B)}\\le  c\\|\\GG\\|_{2,B}\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}\n\\\\\n&\\le \\frac 14 \\|\\GG\\|_{2,B}^2+c\\|\\eta^2 \\bar{u}\\|_{W^{1,4/3}(B)}^2.\n}\nLastly we estimate the term containing $\\GG$,\n\\EQN{\n\\bka{\\div \\GG_{\\rho}, \\bar{u}\\eta^4}_B\n&=-\\int_{B} \\GG :\\nabla(\\bar{u}\\eta^4)=-\\int_{B} \\GG: \\nabla(\\eta^2\\bar{u})\\eta^2-2\\int_{B} \\GG :\\bar{u}\\nabla\\eta\\eta^3\n\\\\\n&\\le \\frac{1}{16}\\|\\nabla(\\eta^2\\bar{u})\\|^2_{2,B}+c\\|\\eta^2 \\bar{u}\\|_{2,B}^2+ \\frac 14\\|\\GG\\|_{2,B}^2.\n}\nCombining these estimates with \\eqref{3.1} we get the following\n\\EQ{\n\\int |\\nabla (\\eta^2\\bar{u})|^2\\le c(b)\\|\\bar{u}\\|^2_{W^{1,4/3}(B)}+c(b)(\\|\\bar{u}\\|_{2,B}^2+\\|\\bar{u}\\|_{4/3,B}^2)+\\|\\GG\\|_{2,B}^2,\n}\nHere $c(b)$ depends on $b$ only through $\\norm{b}_{L^{2,\\infty}(\\Om)}$. Finally, since $\\eta=1$ on $B_{1/2}$ we get that\n\\EQ{\n\\int_{B_{1/2}} |\\nabla \\bar{u}|^2\\le c(b)\\|\\nabla \\bar{u}\\|^2_{4/3,B}+c(b)\\|\\bar{u}\\|_{2,B}^2+\\|\\GG\\|_{2,B}^2.\n}\nWe now choose $k=(u_{\\rho})_{B}$ as the constant in $\\bar{u}=u_{\\rho}-k$ and apply Poincar\\'e inequality to get\n\\EQ{\n\\int_{B_{1/2}} |\\nabla u_{\\rho}|^2\\le c(b)\\|\\nabla u_{\\rho}\\|^2_{4/3,B}+\\|\\GG\\|_{2,B}^2.\n}\nLastly we scale back to $B_{\\rho}(x_0)$ and get that\n\\EQ{\\label{5.12}\n\\frac{1}{|B_{\\rho/2}|}\\int_{B_{\\rho/2}} |\\nabla u|^2\n\\le c(b) \\Big(\\frac{1}{|B_{\\rho}|}\\int_{B_{\\rho}}|\\nabla u|^{4/3}\\Big)^{3/2}+\\frac{c}{|B_{\\rho}|}\\|\\GG\\|_{2,B_{\\rho}}^2.\n}\n\n\\medskip\n\nNext we consider the boundary case, $\\Omega_{\\rho}=\\Omega\\cap B_{\\rho}(x_0)$ with $x_0\\in\\pd\\Om$, and $0<\\rho<R_0$. The significant difference from the internal case is the pressure estimate: Unlike the internal case we can not use Wolf decomposition estimate in a uniform ball, therefore we need to track the dependence of constant $c_1$ when we apply Theorem \\ref{theoremL2wolf} to get estimates similar to \\eqref{pressure_bound_est}\n and make sure it is uniform across all  $\\Omega_{\\rho}(x_0)$.\n Rescale the domain as the interior case so that $x_0=0$ and $\\rho=1$.\n We will use change of coordinates\n \\EQ{\\label{change-var}\n y_1=x_1, \\quad y_2=x_2-\\gamma(x_1),\n }\n to straighten the boundary and map $\\Om_{5/8}\\subset\\Om_1$ into domains $\\Om_{5/8}'\\subset\\Om_1'$ with flat boundary on $y_2=0$. Since $\\gamma$ was a Lipschitz curve with sufficiently small Lipschitz constant we can assume that $\\Om_{5/8}'\\subset B_{6/8}^+\\subset B_{7/8}^+\\subset\\Om_1'$.\n  (This is the first place that specifies the smallness of the Lipschitz constant.) By Lemma \\ref{geometry} we can fix a smooth $V$ such that $B_{6/8}^+\\subset V\\subset B_{7/8}^+$. After rewriting the equation \\eqref{drifteq} in new variables we get the following system (with all derivatives in $y$) in $\\Om_1'$:\n\\EQS{\\label{changed_var}\n-\\Delta_y u+\\div (\\gamma' e_1\\otimes\\partial_{y_2}u)+\n\\gamma' \\pd_1\\pd_2 u - (\\gamma')^2\\partial_{y_2}^2u+\n\\\\\n+\\div_y(b\\otimes u)- \\gamma'\\partial_{y_2}(b\\otimes u)_1+\\nabla_y \\pi-\\gamma'\\partial_{y_2}\\pi e_1&=\\div\\GG-\\gamma'\\partial_2 \\GG_1 ,\n\\\\\n\\div_y u-\\gamma'\\partial_2 u_1&=0.\n}\nAbove, $(b\\otimes u)_1=(b_1u_1,b_1u_2)$ and $\\GG_1=(G_{11},G_{12})$ are their first rows. In the second term, $\\ga'$ is hidden behind $\\div$ to avoid $\\ga''$ which we do not assume any bound.\n\nWe apply Wolf's local pressure projection $\\mathcal{W}_{4/3,V}$ on \\eqref{changed_var} in the smooth domain $V$ and get that  $\\pi-(\\pi)_{V}=\\pi_1+\\pi_2+\\pi_3+\\pi_4$ where\n\\EQS{\\label{pressure_decomp_b}\n\\nabla\\pi_1&=\\mathcal{W}_{4/3,V}\\bkt{\\Delta_y u-\\div (\\gamma' e_1\\otimes\\partial_{y_2}u)-\\gamma' \\pd_1\\pd_2 u +(\\gamma')^2\\partial_{y_2}^2u}\n\\\\\n\\nabla\\pi_2&=\\mathcal{W}_{4/3,V}\\bkt{-\\div_y(b\\otimes u)+\\gamma'\\partial_{y_2}(b\\otimes u)_1}\n\\\\\n \\nabla\\pi_3&=\\mathcal{W}_{4/3,V}(\\gamma' \\partial_{y_2}\\pi e_1)\n \\\\\n \\nabla\\pi_4&= \\mathcal{W}_{4/3,V}(\\div \\GG-\\gamma'\\partial_2 \\GG_1).\n}\nFrom Theorem \\ref{theoremL2wolf} the local pressure projection  $\\mathcal{W}_{4/3,V}$ is well defined, and we have the bound \\eqref{pWolfbound} with global constant $c$. Also using $\\ga'\\pd_2 = \\pd_2 \\ga'$, and Lemmas \\ref{Holder} and \\ref{Sobolev} in $V$,\n\\EQS{\\label{pressure_bound_est}\n\\|\\pi_1\\|_{4/3,V}&\\le c(1+\\|\\gamma'\\|_{\\infty}+\\|\\gamma'\\|_{\\infty}^2)\\|\\nabla u\\|_{4/3,V}\n\\\\\n\\|\\pi_2\\|_{4/3,V}&\\le  c(1+\\|\\gamma'\\|_{\\infty})\\|u\\otimes b\\|_{4/3,V}\n\\\\\n\\|\\pi_3\\|_{4/3,V}&\\le  c\\|\\gamma'\\|_{\\infty}\\|\\pi\\|_{4/3,V}\\le c\\|\\gamma'\\|_{\\infty}\\|\\pi_3\\|_{4/3,V}+c\\|\\gamma'\\|_{\\infty}\\|\\pi_1+\\pi_2+\\pi_4\\|_{4/3,V}\n\\\\\n\\|\\pi_4\\|_{4/3,V}&\\le  c(1+\\|\\gamma'\\|_{\\infty})\\|\\GG\\|_{4/3,V}\n}\nHere we apply our choice of $\\gamma$ so that $\\|\\gamma'\\|_{\\infty}<1$ and\n$ c\\|\\gamma'\\|_{\\infty}<1/2$. (This is the second place that specifies the smallness of the Lipschitz constant.) After applying it to \\eqref{pressure_bound_est} we get the following\n\\EQS{\\label{pressure_3}\n\\|\\pi_3\\|_{4/3,V}\\le 2c\\|\\gamma'\\|_{\\infty}\\|\\pi_1+\\pi_2+\\pi_4\\|_{4/3,V}.\n}\n\nLet $\\Om^\\sharp$ be the pre-image of $V$ under the map \\eqref{change-var} in $x$-variables.\nThe pressure component $\\pi_i$ can be considered functions on $\\Om^\\sharp$ and since \\eqref{change-var} has determinant 1 and $\\|\\nabla_y u\\|_{4/3,V}\\le 2\\|\\nabla_x u\\|_{4/3,\\Omega^\\sharp}$, by assumption $\\|\\gamma'\\|_{\\infty}\\le 1$ we have\n\\EQS{\\label{pressure-est}\n\\norm{\\pi-(\\pi)_{V}}_{4/3, \\Om^\\sharp} &\\le {\\textstyle\\sum}_{i=1}^4 \\norm{\\pi_i}_{4/3, \\Om^\\sharp}\n={\\textstyle\\sum}_{i=1}^4 \\norm{\\pi_i}_{4/3, V}\n\\\\\n&\\le c\\|\\nabla _y u\\|_{4/3,V} +c \\|u\\otimes b\\|_{4/3,V} +c \\|\\GG\\|_{4/3,V}\n\\\\\n&\\le c(1+ \\| b\\|_{L^{2,\\infty}(V)}) (\\|\\nabla_y u\\|_{4/3,V} +\\| u\\|_{4/3,V} )+ c \\|\\GG\\|_{4/3,V}\n\\\\\n&\\le c(1+ \\| b\\|_{L^{2,\\infty}(\\Omega)}) (\\|\\nabla_x u\\|_{4/3,\\Omega^\\sharp} +\\| u\\|_{4/3,\\Omega^\\sharp} )+ c \\|\\GG\\|_{4/3,\\Omega^\\sharp}.\n}\nThis pressure estimate is the only purpose of our change of variables \\eqref{change-var}. Noting that $\\Om_{5/8}\\subset \\Om^\\sharp\\subset \\Om_1$, we then proceed the following.\n\n Let $\\eta$ be a cutoff function supported in $B_{5/8}(x_0)$ such that $\\eta=1$ on $B_{1/2}(x_0)$. Due to our boundary condition $u|_{\\partial\\Omega}=0$ we can use function $\\eta^4u$ as an admissible test function in \\eqref{drifteq}. Therefore we get the following local energy inequality\n\\EQS{\\label{eq5.19}\n\\int_{\\Om_1} |\\nabla (\\eta^2u)|^2\n&\\le \\int_{\\Om_1} |u|^2|\\nabla \\eta^2|^2+\\int_{\\Om_1} |u|^2|b||\\nabla \\eta|\\eta^3\\\\\n&\\quad +\\int_{\\Om_1} |\\pi-(\\pi)_{V}||u||\\nabla\\eta|\\eta^3-\\int_{\\Om_1} \\GG :\\nabla(u\\eta^4).\n}\nUsing \\eqref{pressure-est} to bound the pressure term and similarly to internal case\nusing Lemmas \\ref{Holder} and \\ref{Sobolev} in $\\Om_1$, with uniform constant $C_2$ in Lemma \\ref{Sobolev} using $\\eta^4u\\in W^{1,2}_0(\\Om_1)$, we get\n\\EQS{\\label{5.16}\n\\int_{\\Om_{1/2}} |\\nabla u|^2 &\\le C(b)(\\|\\nabla u\\|^2_{4/3,\\Om_1}+\\|u\\|_{2,\\Om_1}^2)+c\\|\\GG\\|_{2,\\Om_1}^2.\n}\nNotice that here the constant $C(b)$ does not depend on $\\Om_1$.\nFinally we extend function $u,\\GG$ by zero to  $B_1(x_0)\\setminus \\Omega$. Since $u=0$ on the set $E:=\\Om^c\\cap B_1(x_0)$ of non zero measure with the ratio $\\frac{|E|}{|B_{1}|}$ bounded from below, we can use Poincar\\'e inequality to bound $\\|u\\|_{2,\\Om_1}^2 \\le c \\|\\nabla u\\|^2_{4/3,\\Om_1}$, with a constant independent of $\\Om_1$ with the given lower bound of the ratio.\n(This version of Poincar\\'e inequality follows from Proposition 6.14 of \\cite[page 116]{Lieberman},  with $f=\\al 1_E$, nt_{B_1} f=1$. It gives $\\norm{u}_{4/3,B_1} \\le C\\al  \\norm{\\nabla u}_{4/3,B_1} $.\nWe also have\n$\\norm{u}_{4,B_1} \\le C \\norm{u}_{W^{1,4/3}(B_1)}$.)\n\nWe also re-scale the inequality for arbitrary $B_r(x_0)$ and get \\eqref{5.12}.\n\n\\medskip\nLastly we need to consider the case of arbitrary $x_0,r$ such that $\\Om^c\\cap B_{2r}(x_0)\\neq\\emptyset$. For any such $x_0$ we can find $y_0\\in\\partial\\Om\\cap B_{2r}(x_0)$ such that\n\\EQ{\nB_r(x_0)\\subset B_{3r}(y_0)\\subset B_{6r}(y_0)\\subset B_{8r}(x_0).\n}\nUsing \\eqref{5.12} for $B_{3r}(y_0)$ (assuming $6r<R_0$) we get\n\\EQS{\\label{5.17}\n\\frac{1}{|B_r|}\\int_{B_r(x_0)} |\\nabla u|^2\n&\\le \\frac{C}{|B_{3r}|}\\int_{B_{3r}(y_0)} |\\nabla u|^2\n\\\\\n&\\le c(b) \\Big(\\frac{1}{|B_{6r}|}\\int_{B_{6r}(y_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{6r}|}\\|\\GG\\|_{L^2(B_{6r}(y_0))}^2\n\\\\\n& \\le c(b) \\Big(\\frac{1}{|B_{8r}|}\\int_{B_{8r}(x_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{8r}|}\\|\\GG\\|_{L^2(B_{8r}(x_0))}^2.\n}\n\nFor Gehring's lemma we combine the internal estimate \\eqref{5.12} and the boundary estimate \\eqref{5.17} and conclude that\n\\EQ{\\label{5.19}\n\\frac{1}{|B_r|}\\int_{B_r(x_0)} |\\nabla u|^2 \\le\nc(b) \\Big(\\frac{1}{|B_{8r}|}\\int_{B_{8r}(x_0)}|\\nabla u|^{\\frac{4}{3}}\\Big)^{\\frac{3}{2}}+\\frac{C}{|B_{8r}|}\\|\\GG\\|_{L^2(B_{8r}(x_0))}^2,\n}\nfor any $x_0$ in a big cube containing $\\overline \\Om$, and any $r\\le R_0/6$.\nWe can now apply Proposition 1.1 of Giaquinta \\cite[page 122]{Gia} (also see \\cite[Proposition 3.7]{DongKim}) to get that $\\nb u \\in L^p_\\loc$ for $p \\in (2,p_0)$ for some $p_0 >2$, and\n\\EQ{\\label{0619}\nnt_{B_r(x_0)} |\\nabla u|^p}^{\\frac 1p} \\le\nnt_{B_{8r}(x_0)} |\\GG|^p}^{\\frac 1p},\n}\nfor all $x_0 \\in \\overline \\Om$ and $r\\le R_0/6$, with\n$p_0$ and $c_1$ depending only on $c(b)$. Summing \\eqref{0619} over a finite cover of $\\overline \\Om$ of balls of radius $R_0/6$,\nwe get \\eqref{eq5.1}.\n\\end{proof}", "soln-pair": "\\label{soln-pair}\n\\int_{\\Om} (\\nb u - b \\otimes u): \\nb \\zeta - \\pi \\div \\zeta  = -\\int_\\Om \\GG : \\nb \\zeta,\\quad\n\\forall \\zeta\\in C_c^{\\infty}(\\Om)^2.\n}\nHere we used \\eqref{1.2} to integrate by parts", "th1.2": "\\begin{theorem}\\label{th1.2}\n  Let $\\Om$ be a bounded Lipschtiz domain in $\\RR^2$.  Assume $b\\in L^{2,\\infty}(\\Omega)$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\\in L^q(\\Om)$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ of \\eqref{scalar.eq} such that\n\\EQ{\\label{main2-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|G\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\\Om)$ without assuming \\eqref{main2-eq}.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 3896, "pre_theorem_intro_text": "\\label{sec1}\n\nFor a bounded Lipschitz domain $\\Om$  in $\\mathbb{R}^2$ we consider the solutions of the perturbed Stokes system $(u,\\pi):\\Om \\to \\mathbb{R}^2 \\times \\mathbb{R}$ that satisfy\n\\EQ{\\label{drifteq}\n-\\Delta u +b\\cdot\\nabla u+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{G},\\quad \\mathop{\\rm div}\\nolimits u=0, \\quad u|_{\\partial \\Om}=0,\n}\nwith given drift $b$ in the critical space, $b\\in L^{2,\\infty}(\\Omega)^2$, $\\mathop{\\rm div}\\nolimits b=0$, and  force $\\mathbb{G}\\in L^q(\\Om)^{2\\times 2}$, $1 <q<\\infty$. We denote by\n$L^{q,r}(\\Om)$ the Lorentz spaces. For derivatives of vectors and matrixes we use the following notations\n$(\\mathop{\\rm div}\\nolimits \\mathbb{G})_i=\\pd_j \\GG_{ji}$ and $(\\nabla \\zeta)_{ji} = \\pd_j \\zeta_i$. Since $\\mathop{\\rm div}\\nolimits b=0$  we can integrate by parts in the following terms\n\\EQ{\\label{1.2}\n\\De u = \\mathop{\\rm div}\\nolimits (\\nabla u),\\quad\nb\\cdot\\nabla u = \\mathop{\\rm div}\\nolimits (b\\otimes u), \\quad \\int_\\Om  \\mathop{\\rm div}\\nolimits \\mathbb{G} \\cdot \\zeta = -\n\\int_\\Om  \\mathbb{G}:\\nabla \\zeta.\n}\nFor zero divergence spaces or spaces with zero average we use the following notations\n(for $\\Om\\subset\\mathbb{R}^n$, $n\\ge2$)\n\\EQS{\nC^{\\infty}_{c}(\\Omega) &= \\big\\{ u \\in C^{\\infty}(\\Omega):~ \\supp\\{u\\}\\Subset \\Omega\\big\\},\\\\\nC^{\\infty}_{c,\\si}(\\Omega) &= \\big\\{ u \\in C^{\\infty}_c(\\Omega)^n:\\  \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\\nW^{1,q}_{0,\\si}(\\Omega) &= \\big\\{ u \\in W^{1,q}_0(\\Omega)^n:\\  \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\ \\quad L^q_0(\\Om)&=\\left \\{ \\pi \\in L^q(\\Om):\\  \\textstyle \\int_\\Om \\pi \\,dx=0 \\right \\}.\n}\n\nNext we consider $(u,\\pi)$ as a \\emph{weak solution pair} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$, $\\pi\\in L^q_0(\\Omega)$, and $(u,\\pi)$ satisfies  \\eqref{drifteq} in distributional sense,\\EQ{\\label{soln-pair}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta - \\pi \\mathop{\\rm div}\\nolimits \\zeta  = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_c^{\\infty}(\\Om)^2.\n}\nHere we used \\eqref{1.2} to integrate by parts.\nWe say $u$ is a \\emph{$q$-weak solution} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $u$ satisfies \\eqref{drifteq} in weak sense, i.e.,\n\\EQ{\\label{weak-soln}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta   = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_{c,\\sigma}^{\\infty}(\\Omega).\n}\nIn case $q=2$ we call $u$ a \\emph{weak solution}. In this paper we also consider a dual system of \\eqref{drifteq}\n\\EQ{\\label{drifteq_dual}\n-\\Delta v -\\nabla\\cdot(b\\otimes v)+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{F},\\quad \\mathop{\\rm div}\\nolimits v=0, \\quad v|_{\\partial \\Om}=0.\n}\nThis system is equivalent to \\eqref{drifteq} when the drift is divergence free, $\\mathop{\\rm div}\\nolimits b=0$, but in general case it is different.\n\nThe boundary value problem of the perturbed Stokes system \\eqref{drifteq} and the corresponding scalar equation \\eqref{scalar.eq}\nhas a huge amount of literature when we consider subcritical drift $b\\in L^p(\\Om)$, $\\Om \\subset \\mathbb{R}^n$, with $p> n \\ge2$. In the subcritical case we may treat the drift term $b\\cdot \\nabla u$ as a perturbation to $\\De u$.  The critical case is when $b\\in L^n(\\Om)$ or $b\\in L^{n,\\infty}(\\Om)$. System \\eqref{drifteq} in $\\Om \\subset \\mathbb{R}^3$ with critical drift $b\\in L^3(\\Om)$ has previously been studied;\nSee Section 4 of Kim \\cite{Kim09}, Theorem 18 in Choe \\& Kim \\cite{ChoeKim}, and Section 5 of\nAmrouche \\& Rodríguez-Bellido \\cite{AmRB}. They contain estimates similar to \\eqref{main1-eq}. A similar result for small $b\\in L^{n,\\infty}(\\Om)$, $n \\ge3$, is proved by the authors \\cite[Theorem 1.4]{CT1}.\n\nIn this paper we extend our previous work for   higher dimensions $n \\ge 3$ \\cite{CT1} to $n=2$. Our main theorem below extends \\cite[Theorem 1.5]{CT1} to the dimension 2 case and proves existence of the solutions with higher integrability of the gradient.", "context": "\\label{sec1}\n\nFor a bounded Lipschitz domain $\\Om$ in $\\mathbb{R}^2$ we consider the solutions of the perturbed Stokes system $(u,\\pi):\\Om \\to \\mathbb{R}^2 \\times \\mathbb{R}$ that satisfy\n\\EQ{\\label{drifteq}\n-\\Delta u +b\\cdot\\nabla u+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{G},\\quad \\mathop{\\rm div}\\nolimits u=0, \\quad u|_{\\partial \\Om}=0,\n}\nwith given drift $b$ in the critical space, $b\\in L^{2,\\infty}(\\Omega)^2$, $\\mathop{\\rm div}\\nolimits b=0$, and force $\\mathbb{G}\\in L^q(\\Om)^{2\\times 2}$, $1 <q<\\infty$. We denote by\n$L^{q,r}(\\Om)$ the Lorentz spaces. For derivatives of vectors and matrixes we use the following notations\n$(\\mathop{\\rm div}\\nolimits \\mathbb{G})_i=\\pd_j \\GG_{ji}$ and $(\\nabla \\zeta)_{ji} = \\pd_j \\zeta_i$. Since $\\mathop{\\rm div}\\nolimits b=0$ we can integrate by parts in the following terms\n\\EQ{\\label{1.2}\n\\De u = \\mathop{\\rm div}\\nolimits (\\nabla u),\\quad\nb\\cdot\\nabla u = \\mathop{\\rm div}\\nolimits (b\\otimes u), \\quad \\int_\\Om \\mathop{\\rm div}\\nolimits \\mathbb{G} \\cdot \\zeta = -\n\\int_\\Om \\mathbb{G}:\\nabla \\zeta.\n}\nFor zero divergence spaces or spaces with zero average we use the following notations\n(for $\\Om\\subset\\mathbb{R}^n$, $n\\ge2$)\n\\EQS{\nC^{\\infty}_{c}(\\Omega) &= \\big\\{ u \\in C^{\\infty}(\\Omega):~ \\supp\\{u\\}\\Subset \\Omega\\big\\},\\\\\nC^{\\infty}_{c,\\si}(\\Omega) &= \\big\\{ u \\in C^{\\infty}_c(\\Omega)^n:\\ \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\\nW^{1,q}_{0,\\si}(\\Omega) &= \\big\\{ u \\in W^{1,q}_0(\\Omega)^n:\\ \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\ \\quad L^q_0(\\Om)&=\\left \\{ \\pi \\in L^q(\\Om):\\ \\textstyle \\int_\\Om \\pi \\,dx=0 \\right \\}.\n}\n\nNext we consider $(u,\\pi)$ as a \\emph{weak solution pair} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$, $\\pi\\in L^q_0(\\Omega)$, and $(u,\\pi)$ satisfies \\eqref{drifteq} in distributional sense,\\EQ{\\label{soln-pair}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta - \\pi \\mathop{\\rm div}\\nolimits \\zeta = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_c^{\\infty}(\\Om)^2.\n}\nHere we used \\eqref{1.2} to integrate by parts.\nWe say $u$ is a \\emph{$q$-weak solution} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $u$ satisfies \\eqref{drifteq} in weak sense, i.e.,\n\\EQ{\\label{weak-soln}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_{c,\\sigma}^{\\infty}(\\Omega).\n}\nIn case $q=2$ we call $u$ a \\emph{weak solution}. In this paper we also consider a dual system of \\eqref{drifteq}\n\\EQ{\\label{drifteq_dual}\n-\\Delta v -\\nabla\\cdot(b\\otimes v)+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{F},\\quad \\mathop{\\rm div}\\nolimits v=0, \\quad v|_{\\partial \\Om}=0.\n}\nThis system is equivalent to \\eqref{drifteq} when the drift is divergence free, $\\mathop{\\rm div}\\nolimits b=0$, but in general case it is different.\n\nThe boundary value problem of the perturbed Stokes system \\eqref{drifteq} and the corresponding scalar equation \\eqref{scalar.eq}\nhas a huge amount of literature when we consider subcritical drift $b\\in L^p(\\Om)$, $\\Om \\subset \\mathbb{R}^n$, with $p> n \\ge2$. In the subcritical case we may treat the drift term $b\\cdot \\nabla u$ as a perturbation to $\\De u$. The critical case is when $b\\in L^n(\\Om)$ or $b\\in L^{n,\\infty}(\\Om)$. System \\eqref{drifteq} in $\\Om \\subset \\mathbb{R}^3$ with critical drift $b\\in L^3(\\Om)$ has previously been studied;\nSee Section 4 of Kim \\cite{Kim09}, Theorem 18 in Choe \\& Kim \\cite{ChoeKim}, and Section 5 of\nAmrouche \\& Rodríguez-Bellido \\cite{AmRB}. They contain estimates similar to \\eqref{main1-eq}. A similar result for small $b\\in L^{n,\\infty}(\\Om)$, $n \\ge3$, is proved by the authors \\cite[Theorem 1.4]{CT1}.\n\nIn this paper we extend our previous work for higher dimensions $n \\ge 3$ \\cite{CT1} to $n=2$. Our main theorem below extends \\cite[Theorem 1.5]{CT1} to the dimension 2 case and proves existence of the solutions with higher integrability of the gradient.", "full_context": "\\label{sec1}\n\nFor a bounded Lipschitz domain $\\Om$ in $\\mathbb{R}^2$ we consider the solutions of the perturbed Stokes system $(u,\\pi):\\Om \\to \\mathbb{R}^2 \\times \\mathbb{R}$ that satisfy\n\\EQ{\\label{drifteq}\n-\\Delta u +b\\cdot\\nabla u+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{G},\\quad \\mathop{\\rm div}\\nolimits u=0, \\quad u|_{\\partial \\Om}=0,\n}\nwith given drift $b$ in the critical space, $b\\in L^{2,\\infty}(\\Omega)^2$, $\\mathop{\\rm div}\\nolimits b=0$, and force $\\mathbb{G}\\in L^q(\\Om)^{2\\times 2}$, $1 <q<\\infty$. We denote by\n$L^{q,r}(\\Om)$ the Lorentz spaces. For derivatives of vectors and matrixes we use the following notations\n$(\\mathop{\\rm div}\\nolimits \\mathbb{G})_i=\\pd_j \\GG_{ji}$ and $(\\nabla \\zeta)_{ji} = \\pd_j \\zeta_i$. Since $\\mathop{\\rm div}\\nolimits b=0$ we can integrate by parts in the following terms\n\\EQ{\\label{1.2}\n\\De u = \\mathop{\\rm div}\\nolimits (\\nabla u),\\quad\nb\\cdot\\nabla u = \\mathop{\\rm div}\\nolimits (b\\otimes u), \\quad \\int_\\Om \\mathop{\\rm div}\\nolimits \\mathbb{G} \\cdot \\zeta = -\n\\int_\\Om \\mathbb{G}:\\nabla \\zeta.\n}\nFor zero divergence spaces or spaces with zero average we use the following notations\n(for $\\Om\\subset\\mathbb{R}^n$, $n\\ge2$)\n\\EQS{\nC^{\\infty}_{c}(\\Omega) &= \\big\\{ u \\in C^{\\infty}(\\Omega):~ \\supp\\{u\\}\\Subset \\Omega\\big\\},\\\\\nC^{\\infty}_{c,\\si}(\\Omega) &= \\big\\{ u \\in C^{\\infty}_c(\\Omega)^n:\\ \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\\nW^{1,q}_{0,\\si}(\\Omega) &= \\big\\{ u \\in W^{1,q}_0(\\Omega)^n:\\ \\mathop{\\rm div}\\nolimits u=0\\big\\},\\\\ \\quad L^q_0(\\Om)&=\\left \\{ \\pi \\in L^q(\\Om):\\ \\textstyle \\int_\\Om \\pi \\,dx=0 \\right \\}.\n}\n\nNext we consider $(u,\\pi)$ as a \\emph{weak solution pair} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$, $\\pi\\in L^q_0(\\Omega)$, and $(u,\\pi)$ satisfies \\eqref{drifteq} in distributional sense,\\EQ{\\label{soln-pair}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta - \\pi \\mathop{\\rm div}\\nolimits \\zeta = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_c^{\\infty}(\\Om)^2.\n}\nHere we used \\eqref{1.2} to integrate by parts.\nWe say $u$ is a \\emph{$q$-weak solution} of \\eqref{drifteq} if $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $u$ satisfies \\eqref{drifteq} in weak sense, i.e.,\n\\EQ{\\label{weak-soln}\n\\int_{\\Om} (\\nabla u - b \\otimes u): \\nabla \\zeta = -\\int_\\Om \\mathbb{G} : \\nabla \\zeta,\\quad\n\\forall \\zeta\\in C_{c,\\sigma}^{\\infty}(\\Omega).\n}\nIn case $q=2$ we call $u$ a \\emph{weak solution}. In this paper we also consider a dual system of \\eqref{drifteq}\n\\EQ{\\label{drifteq_dual}\n-\\Delta v -\\nabla\\cdot(b\\otimes v)+\\nabla \\pi=\\mathop{\\rm div}\\nolimits \\mathbb{F},\\quad \\mathop{\\rm div}\\nolimits v=0, \\quad v|_{\\partial \\Om}=0.\n}\nThis system is equivalent to \\eqref{drifteq} when the drift is divergence free, $\\mathop{\\rm div}\\nolimits b=0$, but in general case it is different.\n\nThe boundary value problem of the perturbed Stokes system \\eqref{drifteq} and the corresponding scalar equation \\eqref{scalar.eq}\nhas a huge amount of literature when we consider subcritical drift $b\\in L^p(\\Om)$, $\\Om \\subset \\mathbb{R}^n$, with $p> n \\ge2$. In the subcritical case we may treat the drift term $b\\cdot \\nabla u$ as a perturbation to $\\De u$. The critical case is when $b\\in L^n(\\Om)$ or $b\\in L^{n,\\infty}(\\Om)$. System \\eqref{drifteq} in $\\Om \\subset \\mathbb{R}^3$ with critical drift $b\\in L^3(\\Om)$ has previously been studied;\nSee Section 4 of Kim \\cite{Kim09}, Theorem 18 in Choe \\& Kim \\cite{ChoeKim}, and Section 5 of\nAmrouche \\& Rodríguez-Bellido \\cite{AmRB}. They contain estimates similar to \\eqref{main1-eq}. A similar result for small $b\\in L^{n,\\infty}(\\Om)$, $n \\ge3$, is proved by the authors \\cite[Theorem 1.4]{CT1}.\n\nIn this paper we extend our previous work for higher dimensions $n \\ge 3$ \\cite{CT1} to $n=2$. Our main theorem below extends \\cite[Theorem 1.5]{CT1} to the dimension 2 case and proves existence of the solutions with higher integrability of the gradient.\n\nThe boundary value problem of the perturbed Stokes system \\eqref{drifteq} and the corresponding scalar equation \\eqref{scalar.eq}\nhas a huge amount of literature when we consider subcritical drift $b\\in L^p(\\Om)$, $\\Om \\subset \\R^n$, with $p> n \\ge2$. In the subcritical case we may treat the drift term $b\\cdot \\nb u$ as a perturbation to $\\De u$. The critical case is when $b\\in L^n(\\Om)$ or $b\\in L^{n,\\infty}(\\Om)$. System \\eqref{drifteq} in $\\Om \\subset \\R^3$ with critical drift $b\\in L^3(\\Om)$ has previously been studied;\nSee Section 4 of Kim \\cite{Kim09}, Theorem 18 in Choe \\& Kim \\cite{ChoeKim}, and Section 5 of\nAmrouche \\& Rodríguez-Bellido \\cite{AmRB}. They contain estimates similar to \\eqref{main1-eq}. A similar result for small $b\\in L^{n,\\infty}(\\Om)$, $n \\ge3$, is proved by the authors \\cite[Theorem 1.4]{CT1}.\n\n\\emph{Comments on Theorem \\ref{main1}:}\n\\EN{\n\\item This theorem extends the higher dimensional result \\cite[Theorem 1.5]{CT1} to the dimension 2 case.\nIn \\cite[Theorem 1.5]{CT1} we take $L_0=1/2$. In this paper $L_0$ is specified after \\eqref{change-var} and after \\eqref{pressure_bound_est}, and is computable.\nWe skip in this paper the analogous results for small $b$ case (but full range $1<q<\\infty$) as in \\cite[Proposition 1.3, Theorem 1.4]{CT1}.\n\nUsing the same proof scheme we can also prove the following theorem for the scalar equation \\eqref{scalar.eq}.\n\\begin{theorem}\\label{th1.2}\n Let $\\Om$ be a bounded Lipschtiz domain in $\\RR^2$. Assume $b\\in L^{2,\\infty}(\\Omega)$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\\in L^q(\\Om)$, there exists a weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ of \\eqref{scalar.eq} such that\n\\EQ{\\label{main2-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|G\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\\Om)$ without assuming \\eqref{main2-eq}.\n\\end{theorem}\n\n\\section{Existence of weak solutions}\\label{sec3}\nIn this section we will prove existence of weak solutions to the perturbed Stokes system \\eqref{drifteq} for $b\\in L^2$ or $L^{2,\\infty}$, $\\div b=0$,\n\\EQ{ \\label{eq3.1}\n-\\Delta u+ b\\cdot\\nabla u + \\nb \\pi =f,\\quad \\div u=0, \\quad u|_{\\pd \\Om}=0,\n}\nin $\\Om \\subset \\R^2$. When $b\\in L^2$, we will further show uniqueness.\nRecall that a weak solution $u$ belongs to $ W^{1,2}_{0,\\sigma}(\\Omega)$ and satisfies a weak form of \\eqref{drifteq} similar to \\eqref{weak-soln}.\n\\begin{theorem}[Weak solutions for weak $L^2$ drift]\\label{energy}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$.\nAssume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nThen for any $f \\in W^{-1,2}(\\Omega)^2$,\nthere exists a weak solution $u\\in W^{1,2}_{0,\\sigma}(\\Omega)$\nof \\eqref{drifteq}. Moreover, there is $\\pi \\in L^2(\\Om)$ so that $(u,\\pi)$ solves \\eqref{drifteq} in distributional sense, $\\int_\\Om \\pi=0$, and\n\\EQ{\\label{eq6.1}\n\\|u\\|_{ W^{1,2}(\\Omega)}+\\|\\pi\\|_{L^2(\\Omega)}\\le C\\|f\\|_{W^{-1,2}(\\Om)},\n}\nfor some constant $C=C(\\Om)$ independent of $b$.\n\\end{theorem}\n\n\\begin{theorem}[Unique weak solution for $L^2$ drift]\\label{energy_2}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$.\nAssume $b\\in L^{2}(\\Om)^2$ has an extension\n$\\tilde{b}\\in L^{2}(\\R^2)^2$ with $\\div \\tilde{b}=0$ in $\\R^2$.\nThen for any $f \\in W^{-1,2}(\\Omega)^2$,\nthere exists a unique weak solution $u\\in W^{1,2}_{0,\\sigma}(\\Omega)$\nof \\eqref{drifteq}. Furthermore, there is $\\pi \\in L^2(\\Om)$ so that $(u,\\pi)$ solves \\eqref{drifteq} in distributional sense, $\\int_\\Om \\pi=0$, and\n\\EQ{\\label{eq6.1b}\n\\|u\\|_{ W^{1,2}(\\Omega)}+\\|\\pi\\|_{L^2(\\Omega)}\\le C\\|f\\|_{W^{-1,2}(\\Om)},\n}\nfor some constant $C=C(\\Om)$ independent of $b$.\n\\end{theorem}\n\nApproximating $u$ in $W^{1,2}_{0,\\si}(\\Om)$ by smooth $u_j$\nand using $\\div b=0$ conclude that\n\\EQ{\\label{eq3.4}\n\\int_\\Om (b\\cdot \\nb) u\\cdot u = \\lim_{j \\to \\infty} \\int_\\Om (b\\cdot \\nb) u_j\\cdot u_j = \\int_\\Om b\\cdot \\nb(\\frac12|u_j|^2) = 0.\n}\n Thus we get the coercivity of this bilinear form, indeed,\n\\EQ{\n\\mathcal{B}(u,u)=\\|\\nabla u\\|_{L^2(\\Om)}^2, \\quad \\forall u\\in W_{0,\\sigma}^{1,2}(\\Om).\n}\nTherefore, by Lax-Milgram theorem, for any $f \\in W^{-1,2}(\\Omega)^n\\subset (W_{0,\\sigma}^{1,2}(\\Om)^n)^*$, there exists a unique $u\\in W_{0,\\sigma}^{1,2}(\\Om)$ such that\n\\EQ{\n\\mathcal{B}(u,v)=\\langle f,v\\rangle,\\quad \\forall v\\in W_{0,\\sigma}^{1,2}(\\Om).\n}\nThis is the weak form of \\eqref{drifteq} and hence\n$u$ is the unique weak solution. By taking $v=u$, we get $\\norm{u}_{W^{1,2}_0(\\Om)} \\le C\\norm{f}_{W^{-1,2}(\\Om)}$.\nLastly, we apply Lemma \\ref{pressure} with $q=2$ (and using that $\\Om$ is bounded and Lipschitz) to get an associated pressure $\\pi\\in L^2_0(\\Om)$ that satisfies the bound.\n\\end{proof}\n\n\\begin{theorem}\\label{theoremL2wolf}\nAssume $p>1$ and $V\\subset \\R^n$, $n\\ge2$, is a bounded Lipschitz domain with Lipschitz constant less than a sufficiently small $L_0(p,n)>0$.\nThen the problem \\eqref{Stokes} has a unique weak solution pair $(v,\\pi)$ with $v \\in W^{1,p}_{0,\\si}(V)$ and $\\pi \\in L^p_0(V)$ for given $f=\\div \\FF$, $ \\FF \\in L^p(\\Om)^{n\\times n}$, and the local pressure projection in $V$,\n\\[\n\\mathcal{W}_{p,V}(\\div \\FF)=\\nabla\\pi,\n\\]\nsatisfies the following inequality\n\\EQ{\\label{pWolfbound}\n\\|\\pi\\|_{L^p(V)}\\le c_1\\|\\FF\\|_{L^p(V)},\n}\nwhere $c_1=c_1(n,V,p)$.\n\\end{theorem}\n\n\\begin{theorem}\\label{Capriori}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$ with Lipschitz constant $L \\le L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nAssume further $b \\in L^{2+\\delta}$ for some $\\delta>0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $2<p<p_0$,\nany weak solution $u\\in W^{1,2}_{0,\\si}(\\Omega)$ of \\eqref{drifteq} with force $\\GG\\in L^p(\\Om)^{2\\times 2}$\nis in $ W^{1,p}(\\Omega)$ and\n\\EQ{\\label{eq5.1}\n\\|u\\|_{W^{1,p}(\\Omega)}\\le c\\|u\\|_{W^{1,2}(\\Omega)}+ c\\|\\GG\\|_{L^p(\\Omega)}\n}\nfor some constant $c(\\Omega,\\|b\\|_{L^{2,\\infty}})>0$.\n\\end{theorem}", "post_theorem_intro_text_len": 7623, "post_theorem_intro_text": "\\emph{Comments on Theorem \\ref{main1}:}\n\\EN{\n\\item This theorem extends the higher dimensional result \\cite[Theorem 1.5]{CT1} to the dimension 2 case.\nIn \\cite[Theorem 1.5]{CT1} we take $L_0=1/2$. In this paper $L_0$ is specified after \\eqref{change-var} and after \\eqref{pressure_bound_est}, and is computable.\nWe skip in this paper the analogous results for small $b$ case (but full range $1<q<\\infty$) as in \\cite[Proposition 1.3, Theorem 1.4]{CT1}.\n\n\\item  Since $\\Om$ is Lipschitz we can assume that there exists $R_0>0$ such that for any $x_0\\in \\partial\\Om$ we have\n\\EQ{\n\\Om\\cap B_{2R_0}(x_0)=\\{(x',x_n)\\in B_{2R_0}(0),x_n>\\gamma(x')\\}.\n}\nThe constant $C$ in \\eqref{main1-eq} will actually depend only on $R_0$, $\\text{diam}~\\Om$, and $\\|b\\|_{L^{2,\\infty}}$.\n\n\\item If we further assume $b \\in L^2(\\Om)$\nand $b$ has a divergence-free extension to $L^2(\\mathbb{R}^2)$,\nthen there is another way to prove uniqueness of the weak solution in $W^{1,2}_{0,\\si}(\\Om)$ by using Hardy-BMO duality, even when the Lipschitz constant of $\\Om$ is large.\nSee Theorem \\ref{energy_2} in Section \\ref{sec3}.\n\n\\item We may simply assume $\\Om$ is a bounded $C^1$ domain, as for\na bounded $C^1$ domain and any constant $\\lambda>0$, each point $x$ on $\\partial\\Om$ has a neighborhood $U_x$ such that $\\partial \\Om \\cap U_x$ is the graph of a Lipschitz function with a Lipschitz constant less than $\\lambda$.\n\n}\n\nFor corresponding results on scalar equations\n\\EQ{\\label{scalar.eq}\n-\\Delta u +b\\cdot\\nabla u=\\mathop{\\rm div}\\nolimits  G, \\quad u|_{\\partial \\Om}=0,\n}\nwith critical drift coefficient $b \\in L^n$, see\nKim \\& Kim \\cite{KK}. When $b \\in L^{n,\\infty}(\\Om)$ for $n \\ge 3$, see \\cite{KiTs,KPT} which also cover more general assumptions on drift $b$ as well as \\cite{CS20,CS24} and Kwon \\cite{Kwon} for $n=2$ case or drift in Morrey spaces which include $L^{n,\\infty}$.\nThere are also results on H\\\"older continuity of the solutions, for example with $b\\in BMO^{-1}(\\mathbb{R}^3)$, $\\mathop{\\rm div}\\nolimits b=0$, see Seregin-Sverak-Silvestre-Zlatos \\cite{SSSZ}.\n\nUsing the same proof scheme we can also prove the following theorem for the  scalar equation \\eqref{scalar.eq}.\n\\begin{theorem}\\label{th1.2}\n  Let $\\Om$ be a bounded Lipschtiz domain in $\\mathbb{R}^2$.  Assume $b\\in L^{2,\\infty}(\\Omega)$ and $\\mathop{\\rm div}\\nolimits b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\\in L^q(\\Om)$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ of \\eqref{scalar.eq} such that\n\\EQ{\\label{main2-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|G\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\\Om)$ without assuming \\eqref{main2-eq}.\n\\end{theorem}\n\nThis theorem improves $\\alpha=0$ case of \\cite[Theorem 1.1]{CS20} and $n=2$ case of Kwon \\cite[Theorem 1.1]{Kwon} which show that for any $q>2$, there is $p \\in (2,q)$ depending on $q$, $\\left  \\| b  \\right \\|_{L^{2,\\infty}}$ and $\\Om$ such that for any $G \\in L^q(\\Om)$, there is a unique $p$-weak solution $u$ of \\eqref{scalar.eq} with force $G$. Moreover,\n\\[\n\\left  \\| \\nabla u  \\right \\|_{L^p(\\Om)} \\le C \\left  \\| G  \\right \\|_{L^q(\\Om)}.\n\\]\nTheorem \\ref{th1.2} improves this result by asserting that $p=q$ when $2<q<p_0$ is sufficiently close to $2$. This improvement further allows us to prove a priori bounds for $q$-weak solutions for $q \\in (p_0',2]$ by duality, which yields existence and uniqueness\n for $q \\in (p_0',2]$.\n\nWe skip the proof of Theoem \\ref{th1.2} since it is similar to Theoem \\ref{main1} and is easier, without the need of pressure estimate. Because it has no pressure estimate, the Lipschitz constant of the boundary $\\partial \\Om$ is allowed to be arbitrarily large.\n\n\\medskip\n\nWe now explain the new difficulties and ideas.\nTheorem \\ref{main1}\nextends the higher dimensional result \\cite[Theorem 1.5]{CT1} to the 2D case, and\nwe will still use Gehring's approach. In this approach, we need to do local energy estimate in every ball and prove a reversed H\\\"older inequality in a general ball with a uniform constant.\n\nThe first new difficulty is that we cannot use $\\zeta = u \\eta^4$ for some cut-off function $\\eta$ as the test function in the weak form \\eqref{soln-pair}, because $b \\otimes u$ is no longer in $L^2$ as in the 3D case, and\n the term\n\\EQ{\\label{drift-integral}\n\\int_\\Om b \\otimes u : \\nabla( u \\eta^4)\n}\nis not integrable. Hence we will mollify the drift $b$, so that we can derive an a priori bound. Since there is no theorem that extends a div-free vector field in $L^{2,\\infty}(\\Omega)$ to $L^{2,\\infty}(\\mathbb{R}^2)$ for a general bounded domain, we will consider the approximation problems in subdomains $\\Om_n$ that converge to $\\Om$.\n\nThe second new difficulty comes from the lower integrability of the pressure.\nWe will\nbound the pressure using Wolf's local pressure projection with a uniform constant as in \\cite{CT1}.\nDenote $\\Omega_R=\\Omega\\cap B_R(x_0)$ for some $x_0\\in \\overline \\Om$. In \\cite{CT1}, it suffices to bound $\\pi -(\\pi)_{\\Omega_R}$ in $L^2(\\Omega_R)$. However,  for dimension 2  we need to bound $\\pi -(\\pi)_{\\Omega\\cap B_R}$ in $L^q(\\Omega_R)$, $q<2$. (We choose $q=4/3$ for simplicity.)\nWe have to choose $q<2$ because\n\\[\n\\pi {\\ \\lesssim \\ } \\nabla u + b \\otimes u -\\mathbb{G}\n\\]\nand $b \\otimes u$ is at best in $L^{2-}$ for $u \\in W^{1,2}_\\mathrm{loc}$ even if we assume $b \\in L^2$. This would not be an issue if $b \\in L^{r}$, $r>2$, and we can only assume $u \\in W^{1,2}_\\mathrm{loc}$ which is the assumption of the reversed H\\\"older inequality.\n\nTo obtain a uniform constant of Wolf's local pressure projection in $L^q(\\Omega_R)$ for $q<2$ gives a significant higher restriction on the regularity of the domain. Instead of a Lipschitz domain with Lipschitz constant $L$ less than 1/2, we have to assume $L$ is sufficiently small and do boundary stretching in the proof for the 2D case.\n\nThe third difficulty is the approximation of a Lipschitz domain. In order to construct the solution we are using an approximation method, therefore the first logical idea would be to construct a sequence of $b_n\\in L^{2,\\infty}(\\Om)$, $\\mathop{\\rm div}\\nolimits b=0$, such that $b_n$ converge in a sense that allows us to prove that the limit will satisfy the equation, for example convergence almost everywhere is sufficient and $\\|b_n\\|_{L^{2,\\infty}(\\Om)}$ are bounded uniformly. For example in star-shaped domains it is trivial, but for arbitrary Lipschitz domains this question is not so clear. To deal with this issue, following Verchota \\cite{Verchota_1,Verchota_2} and later Kwon \\cite{Kwon}, Chernobai-Shilkin \\cite{CS24}, we instead approximate the domain by Lipschtiz sub-domains with controlled Lipschitz constant and on sub-domains we can use standard mollification of $b$ and construct an approximation sequence as zero extension of solutions in these sub-domains. See Section \\ref{sec6} for details.\n\nThe rest of the paper is organized as the following. In section 2 we have preliminary results concerning Lorentz spaces and the pressure, which will be used in the later sections. In section 3 we derive the existence of weak solutions in $W^{1,2}(\\Omega)$. Section 4 contains Wolf's local pressure projection method that will be used to estimate the pressure. In section 5, assuming higher integrability of the drift, we prove uniform local estimate for the gradient which will imply higher integrability due to Gehring's lemma. Section 6 contains proof of the main Theorem \\ref{main1}. In the Appendix we put a proof of geometrical lemma that was used in section 4.", "sketch": "We will still use Gehring's approach: “do local energy estimate in every ball and prove a reversed H\\\"older inequality in a general ball with a uniform constant.”\n\nNew issues/steps specific to proving Theorem~\\ref{main1} in 2D:\n\\begin{itemize}\n\\item \\textbf{Non-integrable drift term; mollify and approximate domains.} One “cannot use $\\zeta = u\\eta^4$ … as the test function … because $b\\otimes u$ is no longer in $L^2$ … and the term $\\int_\\Om b \\otimes u : \\nabla( u \\eta^4)$ is not integrable.” Thus “we will mollify the drift $b$, so that we can derive an a priori bound.” Since there is no general extension theorem to $L^{2,\\infty}(\\mathbb{R}^2)$, they “consider the approximation problems in subdomains $\\Om_n$ that converge to $\\Om$.”\n\\item \\textbf{Pressure control via Wolf projection in $L^q$, $q<2$.} “The second new difficulty comes from the lower integrability of the pressure.” They “bound the pressure using Wolf's local pressure projection with a uniform constant.” In 2D they need “to bound $\\pi-(\\pi)_{\\Omega\\cap B_R}$ in $L^q(\\Omega_R)$, $q<2$ (we choose $q=4/3$),” because “$\\pi\\lesssim \\nabla u + b\\otimes u -\\mathbb{G}$” and “$b\\otimes u$ is at best in $L^{2-}$ for $u\\in W^{1,2}_{\\mathrm{loc}}$.” Achieving a uniform constant for the projection in $L^q$, $q<2$ forces stronger boundary regularity: “we have to assume $L$ is sufficiently small and do boundary stretching in the proof for the 2D case.”\n\\item \\textbf{Approximation of Lipschitz domains for constructing solutions.} To construct solutions they use an approximation method, but approximating $b$ directly in arbitrary Lipschitz domains is “not so clear.” Instead, “we approximate the domain by Lipschtiz sub-domains with controlled Lipschitz constant,” on which “we can use standard mollification of $b$,” and then “construct an approximation sequence as zero extension of solutions in these sub-domains.”\n\\end{itemize}\n\nThey indicate the proof is completed by these ingredients and is carried out later: “See Section~\\ref{sec6} for details,” and “Section 6 contains proof of the main Theorem~\\ref{main1}.”", "expanded_sketch": "We will still use Gehring's approach: “do local energy estimate in every ball and prove a reversed H\\\"older inequality in a general ball with a uniform constant.”\n\nNew issues/steps specific to establishing the main theorem in 2D:\n\\begin{itemize}\n\\item \\textbf{Non-integrable drift term; mollify and approximate domains.} One “cannot use $\\zeta = u\\eta^4$ … as the test function … because $b\\otimes u$ is no longer in $L^2$ … and the term $\\int_\\Om b \\otimes u : \\nabla( u \\eta^4)$ is not integrable.” Thus “we will mollify the drift $b$, so that we can derive an a priori bound.” Since there is no general extension theorem to $L^{2,\\infty}(\\mathbb{R}^2)$, they “consider the approximation problems in subdomains $\\Om_n$ that converge to $\\Om$.”\n\\item \\textbf{Pressure control via Wolf projection in $L^q$, $q<2$.} “The second new difficulty comes from the lower integrability of the pressure.” They “bound the pressure using Wolf's local pressure projection with a uniform constant.” In 2D they need “to bound $\\pi-(\\pi)_{\\Omega\\cap B_R}$ in $L^q(\\Omega_R)$, $q<2$ (we choose $q=4/3$),” because “$\\pi\\lesssim \\nabla u + b\\otimes u -\\mathbb{G}$” and “$b\\otimes u$ is at best in $L^{2-}$ for $u\\in W^{1,2}_{\\mathrm{loc}}$.” Achieving a uniform constant for the projection in $L^q$, $q<2$ forces stronger boundary regularity: “we have to assume $L$ is sufficiently small and do boundary stretching in the proof for the 2D case.”\n\\item \\textbf{Approximation of Lipschitz domains for constructing solutions.} To construct solutions they use an approximation method, but approximating $b$ directly in arbitrary Lipschitz domains is “not so clear.” Instead, “we approximate the domain by Lipschtiz sub-domains with controlled Lipschitz constant,” on which “we can use standard mollification of $b$,” and then “construct an approximation sequence as zero extension of solutions in these sub-domains.”\n\\end{itemize}\n\nThey indicate the proof is completed by these ingredients and is carried out later: “See Section~\\ref{sec6} for details,” and “Section 6 contains proof of the main theorem.”", "expanded_theorem": "\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\mathbb{R}^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\mathop{\\rm div}\\nolimits b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\mathbb{G}\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\label{drifteq}\n-\\Delta u +b\\cdot\\nabla u+\\nabla \\pi=\\div \\GG,\\quad \\div u=0, \\quad u|_{\\pd \\Om}=0,\n}\nwith given drift $b$ in the critical space, $b\\in L^{2,\\oo}(\\Omega)^2$, $\\div b=0$, and  force $\\G such that\n\\begin{theorem}\\label{main1}\nLet $\\Om$ be a bounded Lipschitz domain in $\\R^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\\in L^{2,\\infty}(\\Omega)^2$ and $\\div b=0$.\nThen there exists $p_0(\\|b\\|_{L^{2,\\infty}})>2$ such that for any $p_0'<q<p_0$ and $\\GG\\in L^q(\\Om)^{2\\times 2}$, there exists a  weak solution pair $u\\in W^{1,q}_{0,\\si}(\\Omega)$ and $\\pi\\in L^q_0(\\Omega)$ of \\eqref{drifteq} such that\n\\EQ{\\label{main1-eq}\n\\|u\\|_{ W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C(\\Omega,\\|b\\|_{L^{2,\\infty}})\\|\\GG\\|_{L^q(\\Omega)}.\n}\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming \\eqref{main1-eq}.\n\\end{theorem}\n\nFor $p_0'<q<p_0$, the above pair $(u,\\pi)$ is the unique weak solution pair  in $W^{1,q}_0\\times L^q_0(\\Om)$ without assuming the preceding estimate.", "theorem_type": ["Existential–Universal", "Uniqueness"], "mcq": {"question": "Let \\(\\Omega\\subset \\mathbb{R}^2\\) be a bounded Lipschitz domain whose Lipschitz constant satisfies \\(L<L_0\\), where \\(0<L_0<1\\) is sufficiently small. Assume \\(b\\in L^{2,\\infty}(\\Omega)^2\\) and \\(\\operatorname{div} b=0\\). For \\(q\\in(1,\\infty)\\), write\n\\[W^{1,q}_{0,\\sigma}(\\Omega)=\\{u\\in W^{1,q}_0(\\Omega)^2:\\operatorname{div}u=0\\},\\qquad L^q_0(\\Omega)=\\Big\\{\\pi\\in L^q(\\Omega):\\int_\\Omega \\pi\\,dx=0\\Big\\}.\\]\nA weak solution pair of\n\\[-\\Delta u+b\\cdot\\nabla u+\\nabla \\pi=\\operatorname{div}\\mathbb{G},\\qquad \\operatorname{div}u=0,\\qquad u|_{\\partial\\Omega}=0\\]\nis a pair \\((u,\\pi)\\in W^{1,q}_{0,\\sigma}(\\Omega)\\times L^q_0(\\Omega)\\) such that\n\\[\\int_\\Omega (\\nabla u-b\\otimes u):\\nabla \\zeta-\\pi\\,\\operatorname{div}\\zeta=-\\int_\\Omega \\mathbb{G}:\\nabla \\zeta\\qquad \\forall\\zeta\\in C_c^{\\infty}(\\Omega)^2.\\]\nWhich statement holds for this problem?", "correct_choice": {"label": "A", "text": "There exists \\(p_0=p_0(\\|b\\|_{L^{2,\\infty}(\\Omega)})>2\\) such that, if \\(p_0'\\) denotes the Hölder conjugate exponent of \\(p_0\\) \\((1/p_0+1/p_0'=1)\\), then for every \\(q\\) with \\(p_0'<q<p_0\\) and every \\(\\mathbb{G}\\in L^q(\\Omega)^{2\\times 2}\\), there exists a weak solution pair \\((u,\\pi)\\in W^{1,q}_{0,\\sigma}(\\Omega)\\times L^q_0(\\Omega)\\) of the system above such that\n\\[\\|u\\|_{W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C\\big(\\Omega,\\|b\\|_{L^{2,\\infty}(\\Omega)}\\big)\\,\\|\\mathbb{G}\\|_{L^q(\\Omega)}.\\]\nMoreover, for each such \\(q\\), this pair is the unique weak solution pair in \\(W^{1,q}_{0,\\sigma}(\\Omega)\\times L^q_0(\\Omega)\\), even without assuming the estimate."}, "choices": [{"label": "B", "text": "There exists \\(p_0=p_0(\\|b\\|_{L^{2,\\infty}(\\Omega)})>2\\) such that for every \\(q\\) with \\(1<q<p_0\\) and every \\(\\mathbb{G}\\in L^q(\\Omega)^{2\\times 2}\\), there exists a weak solution pair \\((u,\\pi)\\in W^{1,q}_{0,\\sigma}(\\Omega)\\times L^q_0(\\Omega)\\) of the system above such that\n\\[\\|u\\|_{W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C\\big(\\Omega,\\|b\\|_{L^{2,\\infty}(\\Omega)}\\big)\\,\\|\\mathbb{G}\\|_{L^q(\\Omega)}.\\]\nMoreover, for each such \\(q\\), this pair is the unique weak solution pair in \\(W^{1,q}_{0,\\sigma}(\\Omega)\\times L^q_0(\\Omega)\\)."}, {"label": "C", "text": "There exists \\(p_0=p_0(\\|b\\|_{L^{2,\\infty}(\\Omega)})>2\\) such that, if \\(p_0'\\) denotes the Hölder conjugate exponent of \\(p_0\\), then for every \\(q\\) with \\(p_0'<q<p_0\\) and every \\(\\mathbb{G}\\in L^q(\\Omega)^{2\\times 2}\\), there exists at least one weak solution pair \\((u,\\pi)\\in W^{1,q}_{0,\\sigma}(\\Omega)\\times L^q_0(\\Omega)\\) of the system above satisfying\n\\[\\|u\\|_{W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C\\big(\\Omega,\\|b\\|_{L^{2,\\infty}(\\Omega)}\\big)\\,\\|\\mathbb{G}\\|_{L^q(\\Omega)}.\\]"}, {"label": "D", "text": "There exists \\(p_0=p_0(\\|b\\|_{L^{2,\\infty}(\\Omega)})>2\\) such that, if \\(p_0'\\) denotes the Hölder conjugate exponent of \\(p_0\\), then for every \\(q\\) with \\(p_0'<q<p_0\\) and every \\(\\mathbb{G}\\in L^q(\\Omega)^{2\\times 2}\\), there exists a weak solution pair \\((u,\\pi)\\in W^{1,q}_{0,\\sigma}(\\Omega)\\times L^q_0(\\Omega)\\) of the system above such that\n\\[\\|u\\|_{W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C\\big(\\Omega,\\|b\\|_{L^{2,\\infty}(\\Omega)},q\\big)\\,\\|\\mathbb{G}\\|_{L^q(\\Omega)}.\\]\nMoreover, one may choose \\(p_0>2\\) depending only on \\(\\Omega\\), independent of \\(\\|b\\|_{L^{2,\\infty}(\\Omega)}\\)."}, {"label": "E", "text": "There exists \\(p_0=p_0(\\|b\\|_{L^{2,\\infty}(\\Omega)})>2\\) such that, if \\(p_0'\\) denotes the Hölder conjugate exponent of \\(p_0\\), then for every \\(q\\) with \\(p_0'\\le q\\le p_0\\) and every \\(\\mathbb{G}\\in L^q(\\Omega)^{2\\times 2}\\), there exists a weak solution pair \\((u,\\pi)\\in W^{1,q}_{0,\\sigma}(\\Omega)\\times L^q_0(\\Omega)\\) of the system above such that\n\\[\\|u\\|_{W^{1,q}(\\Omega)}+\\|\\pi\\|_{L^q(\\Omega)}\\le C\\big(\\Omega,\\|b\\|_{L^{2,\\infty}(\\Omega)}\\big)\\,\\|\\mathbb{G}\\|_{L^q(\\Omega)}.\\]\nMoreover, for each such \\(q\\), this pair is the unique weak solution pair in \\(W^{1,q}_{0,\\sigma}(\\Omega)\\times L^q_0(\\Omega)\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "pressure_projection", "tampered_component": "lower endpoint q>p_0' forced by q<2 pressure control", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped uniqueness clause", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "dependence of p_0 on the drift norm", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "open q-range replaced by closed endpoint range", "template_used": "boundary_range"}]}}
{"id": "2512.14939v1", "paper_link": "http://arxiv.org/abs/2512.14939v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm: main}\nFor all integers $r, \\ell \\ge 1$, if $M$ is a simple rank-$r$ positroid with no $U_{2,\\ell+2}$-minor, then $|M| \\le \\ell(r - 1) + 1$.\nMoreover, if $r \\ge 2$ then equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2, \\ell+1}$.", "start_pos": 5302, "end_pos": 5622, "label": "thm: main"}, "ref_dict": {"prop: rank 3 excluded minor": "\\begin{proposition}\\label{prop: rank 3 excluded minor}\n    If $M$ is a simple rank-$3$ matroid with a line $L$ and distinct points $e_1, e_2, e_3$ on $L$ each on at least two long lines, then $M$ is not a positroid.\n    In particular, $M$ has a minor from Figure \\ref{fig: rank-3 excluded minors}.\n\\end{proposition}", "prob: Kung problem": "\\begin{problem} \\label{prob: Kung problem}\nLet $\\cM$ be a class of matroids. \nDetermine the maximum number of elements of a simple rank-$r$ matroid in $\\cM$, and characterize the matroids for which equality holds.\n\\end{problem}", "thm: main": "\\begin{theorem} \\label{thm: main}\nFor all integers $r, \\ell \\ge 1$, if $M$ is a simple rank-$r$ positroid with no $U_{2,\\ell+2}$-minor, then $|M| \\le \\ell(r - 1) + 1$.\nMoreover, if $r \\ge 2$ then equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2, \\ell+1}$.\n\\end{theorem}", "thm: exlude a line": "\\begin{theorem}[Geelen, Nelson \\cite{Geelen-Nelson-2010}] \\label{thm: exlude a line}\nLet $\\ell\\ge 2$ be an integer and let $q$ be the largest prime power at most $\\ell$.\nFor $r$ sufficiently large, every simple rank-$r$ matroid with no $U_{2,\\ell+2}$-minor has at most $\\frac{q^r - 1}{q - 1}$ elements, with equality only for the rank-$r$ projective geometry over the order-$q$ finite field.\n\\end{theorem}", "thm: complex exclude a line": "\\begin{theorem}[Geelen, Nelson, Walsh \\cite{Geelen-Nelson-Walsh-2024}] \\label{thm: complex exclude a line}\nLet $\\ell\\ge 2$ be an integer and let $r$ be a sufficiently large integer.\nThen every simple rank-$r$ $\\bC$-representable matroid with no $U_{2,\\ell+2}$-minor has at most $\\ell\\binom{r}{2} + r$ elements, with equality only for the rank-$r$ Dowling geometry over the cyclic group of order $\\ell$.\n\\end{theorem}", "prop: rank 4 excluded minor": "\\begin{proposition}\\label{prop: rank 4 excluded minor}\n    If $M$ is a simple rank-$4$ matroid with a line $L$, distinct points $e_1, e_2, e_3$ on $L$, and distinct planes $P_1$, $P_2$, $P_3$ through $L$ so that for $i = 1,2,3$ the point $e_i$ is on at least two long lines in $P_i$, then $M$ is not a positroid.\n    In particular, $M$ has a minor from Figure \\ref{fig: rank-3 excluded minors} or Figure \\ref{fig: rank-4 excluded minor}.\n\\end{proposition}"}, "pre_theorem_intro_text_len": 2450, "pre_theorem_intro_text": "\\label{sec: introduction}\n\nExtremal matroid theory was formally introduced in 1993 with the following problem posed by Joseph Kung \\cite{Kung-1993}.\n\n\\begin{problem} \\label{prob: Kung problem}\nLet $\\cM$ be a class of matroids. \nDetermine the maximum number of elements of a simple rank-$r$ matroid in $\\cM$, and characterize the matroids for which equality holds.\n\\end{problem}\n\nThis maximum does not exist for all classes of matroids. \nFor instance, there is no finite answer to Problem~\\ref{prob: Kung problem} for the class of all matroids, or for any class containing all uniform matroids.\nHowever, Kung \\cite[Theorem 4.3]{Kung-1993} proved that every simple rank-$r$ matroid with no $U_{2, \\ell+2}$-minor has at most $\\frac{\\ell^r - 1}{\\ell - 1}$ elements.\nIf $\\ell$ is a prime power, then this bound is tight and is achieved by the rank-$r$ projective geometry over the finite field of order $\\ell$. It is thus natural to study Problem~\\ref{prob: Kung problem} for classes of matroids without $U_{2,\\ell}$-minors. \n\nIn this paper, we study the class of \\textit{positroids} without $U_{2,\\ell}$-minors. Positroids, which are matroids representable by a matrix over $\\bR$ with all maximal subdeterminants nonnegative, occupy a central position at the intersection of combinatorics, algebraic geometry, and physics. They serve as combinatorial models for the totally nonnegative Grassmannian, which in turn underlies the construction of the amplituhedron in quantum field theory. The study of amplituhedra and related objects remains an active and influential area in theoretical physics \\cite{AHT}. Through these connections, positroids emerge as unifying objects that integrate discrete combinatorial frameworks with geometric and physical theory. Combinatorially, they are remarkably rich: Postnikov~\\cite{postnikov} showed that positroids are in bijection with decorated permutations, Grassmann necklaces, plabic graphs, and \\reflectbox{L}-diagrams, providing multiple equivalent viewpoints.\nPositroids also form a minor-closed class of matroids. It is thus natural to ask extremal questions about positroids and their minors. There has been work enumerating both positroids \\cite{Williams05} and connected positroids \\cite{ARW16}, but as far as the authors are aware, no other extremal results about positroids are known. Our main result tightly resolves Problem~\\ref{prob: Kung problem} for the class of $U_{2,\\ell+2}$-minor-free positroids.", "context": "\\label{sec: introduction}\n\nExtremal matroid theory was formally introduced in 1993 with the following problem posed by Joseph Kung \\cite{Kung-1993}.\n\n\\begin{problem} \\label{prob: Kung problem}\nLet $\\cM$ be a class of matroids. \nDetermine the maximum number of elements of a simple rank-$r$ matroid in $\\cM$, and characterize the matroids for which equality holds.\n\\end{problem}\n\nThis maximum does not exist for all classes of matroids. \nFor instance, there is no finite answer to Problem~\\ref{prob: Kung problem} for the class of all matroids, or for any class containing all uniform matroids.\nHowever, Kung \\cite[Theorem 4.3]{Kung-1993} proved that every simple rank-$r$ matroid with no $U_{2, \\ell+2}$-minor has at most $\\frac{\\ell^r - 1}{\\ell - 1}$ elements.\nIf $\\ell$ is a prime power, then this bound is tight and is achieved by the rank-$r$ projective geometry over the finite field of order $\\ell$. It is thus natural to study Problem~\\ref{prob: Kung problem} for classes of matroids without $U_{2,\\ell}$-minors.\n\nIn this paper, we study the class of \\textit{positroids} without $U_{2,\\ell}$-minors. Positroids, which are matroids representable by a matrix over $\\bR$ with all maximal subdeterminants nonnegative, occupy a central position at the intersection of combinatorics, algebraic geometry, and physics. They serve as combinatorial models for the totally nonnegative Grassmannian, which in turn underlies the construction of the amplituhedron in quantum field theory. The study of amplituhedra and related objects remains an active and influential area in theoretical physics \\cite{AHT}. Through these connections, positroids emerge as unifying objects that integrate discrete combinatorial frameworks with geometric and physical theory. Combinatorially, they are remarkably rich: Postnikov~\\cite{postnikov} showed that positroids are in bijection with decorated permutations, Grassmann necklaces, plabic graphs, and \\reflectbox{L}-diagrams, providing multiple equivalent viewpoints.\nPositroids also form a minor-closed class of matroids. It is thus natural to ask extremal questions about positroids and their minors. There has been work enumerating both positroids \\cite{Williams05} and connected positroids \\cite{ARW16}, but as far as the authors are aware, no other extremal results about positroids are known. Our main result tightly resolves Problem~\\ref{prob: Kung problem} for the class of $U_{2,\\ell+2}$-minor-free positroids.", "full_context": "\\label{sec: introduction}\n\nExtremal matroid theory was formally introduced in 1993 with the following problem posed by Joseph Kung \\cite{Kung-1993}.\n\n\\begin{problem} \\label{prob: Kung problem}\nLet $\\cM$ be a class of matroids. \nDetermine the maximum number of elements of a simple rank-$r$ matroid in $\\cM$, and characterize the matroids for which equality holds.\n\\end{problem}\n\nThis maximum does not exist for all classes of matroids. \nFor instance, there is no finite answer to Problem~\\ref{prob: Kung problem} for the class of all matroids, or for any class containing all uniform matroids.\nHowever, Kung \\cite[Theorem 4.3]{Kung-1993} proved that every simple rank-$r$ matroid with no $U_{2, \\ell+2}$-minor has at most $\\frac{\\ell^r - 1}{\\ell - 1}$ elements.\nIf $\\ell$ is a prime power, then this bound is tight and is achieved by the rank-$r$ projective geometry over the finite field of order $\\ell$. It is thus natural to study Problem~\\ref{prob: Kung problem} for classes of matroids without $U_{2,\\ell}$-minors.\n\nIn this paper, we study the class of \\textit{positroids} without $U_{2,\\ell}$-minors. Positroids, which are matroids representable by a matrix over $\\bR$ with all maximal subdeterminants nonnegative, occupy a central position at the intersection of combinatorics, algebraic geometry, and physics. They serve as combinatorial models for the totally nonnegative Grassmannian, which in turn underlies the construction of the amplituhedron in quantum field theory. The study of amplituhedra and related objects remains an active and influential area in theoretical physics \\cite{AHT}. Through these connections, positroids emerge as unifying objects that integrate discrete combinatorial frameworks with geometric and physical theory. Combinatorially, they are remarkably rich: Postnikov~\\cite{postnikov} showed that positroids are in bijection with decorated permutations, Grassmann necklaces, plabic graphs, and \\reflectbox{L}-diagrams, providing multiple equivalent viewpoints.\nPositroids also form a minor-closed class of matroids. It is thus natural to ask extremal questions about positroids and their minors. There has been work enumerating both positroids \\cite{Williams05} and connected positroids \\cite{ARW16}, but as far as the authors are aware, no other extremal results about positroids are known. Our main result tightly resolves Problem~\\ref{prob: Kung problem} for the class of $U_{2,\\ell+2}$-minor-free positroids.\n\nThis result lies at the intersection of three areas of research: extremal matroid theory, structural properties of positroids, and structural properties of matroids with no $M(K_4)$-minor.\nWe will explain how Theorem \\ref{thm: main} continues lines of research in these three areas, beginning with extremal matroid theory.\nThe following beautiful result of Geelen and Nelson \\cite[Theorem 1.2]{Geelen-Nelson-2010} generalizes the initial work of Kung and tightly solves Problem \\ref{prob: Kung problem} for the class of matroids with no $U_{2,\\ell+2}$-minor, for all values of $\\ell$ and all sufficiently large values of $r$.\n\n\\begin{theorem}[Geelen, Nelson \\cite{Geelen-Nelson-2010}] \\label{thm: exlude a line}\nLet $\\ell\\ge 2$ be an integer and let $q$ be the largest prime power at most $\\ell$.\nFor $r$ sufficiently large, every simple rank-$r$ matroid with no $U_{2,\\ell+2}$-minor has at most $\\frac{q^r - 1}{q - 1}$ elements, with equality only for the rank-$r$ projective geometry over the order-$q$ finite field.\n\\end{theorem}\n\n\\begin{theorem}[Geelen, Nelson, Walsh \\cite{Geelen-Nelson-Walsh-2024}] \\label{thm: complex exclude a line}\nLet $\\ell\\ge 2$ be an integer and let $r$ be a sufficiently large integer.\nThen every simple rank-$r$ $\\bC$-representable matroid with no $U_{2,\\ell+2}$-minor has at most $\\ell\\binom{r}{2} + r$ elements, with equality only for the rank-$r$ Dowling geometry over the cyclic group of order $\\ell$.\n\\end{theorem}\n\n\\begin{theorem} \\label{thm: main result with proof}\nFor all integers $r, \\ell \\ge 1$, if $M$ is a simple rank-$r$ matroid with no minor from Figure \\ref{fig: rank-3 excluded minors} or Figure \\ref{fig: rank-4 excluded minor} and no $U_{2,\\ell+2}$-minor, then $|M| \\le \\ell(r - 1) + 1$.\nMoreover, equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2, \\ell+1}$.\n\\end{theorem}\n\\begin{proof}\nFirst suppose that $\\ell = 1$.\nIf $M$ is a simple rank-$r$ matroid with more than $r$ elements, then $r \\ge 2$ and $M$ has a circuit with at least three elements.\nBut every circuit with at least three elements has a $U_{2,3}$-minor \\cite[Example 3.1.9]{Oxley-2011}, a contradiction.\nSo $M$ is a rank-$r$ matroid with $r$ elements, and it follows that $M$ is the parallel connection of $r - 1$ copies of $U_{2,2}$, as desired.\nSo we may assume that $\\ell \\ge 2$.\nWe will proceed by induction on $r$.\nThe statement is clearly true when $r \\le 2$ so we may assume that $r \\ge 3$.\nIf $M$ has an element $e$ that is not on any long lines, then $M/e$ is simple, $|M/e| = |M| - 1$, and $r(M/e) = r - 1$, so by induction we have\n\\begin{align*}\n |M| = 1 + |M/e| \\le 1 + \\ell(r - 2) + 1 = \\ell(r - 1) + 2 - \\ell < \\ell(r - 1) + 1,\n\\end{align*}\nas desired.\nSo we may assume that every element of $M$ is on a long line.\nLet $L$ be a long line of $M$.\nWe will now consider the connected components of $M/L$.\nLet $\\cX$ be the partition of $E(M) - L$ given by the ground sets of the connected components of $M/L$.\nLet $X \\in \\cX$, and let $N = M|(L \\cup X)$.\nWe will first bound the number of elements of $N \\del L$.\n\n\\begin{claim} \\label{claim: bound the size of each component}\n At most two elements in $L$ are on a long line of $N$ other than $L$, and $|N \\del L| \\le \\ell(r(N) - 2)$.\n\\end{claim}\n\\begin{proof}\nSuppose that distinct elements $e_1$, $e_2$, and $e_3$ in $L$ are on long lines $L_1$, $L_2$, and $L_3$, respectively, of $N$ other than $L$.\nLet $K = \\si(N/L)$.\nBy deleting elements from nontrivial parallel classes of $N/L$ we may assume that $K$ is a restriction of $N/L$.\nNote that the elements of $K$ are in one-to-one correspondence with the planes of $N$ containing $L$.\nFor $i = 1,2,3$ let $p_i$ be the element of $K$ in the plane $\\cl_N(L \\cup L_i)$ of $N$.\nSince $K$ is connected, it follows from \\cite[Theorem 4.3.1]{Oxley-2011} that $K$ has a connected minor $K_1 = K \\del D/C$ on ground set $\\{p_1, p_2, p_3\\}$.\nWe may assume that $C$ is independent in $K$ and therefore in $N/L$, which implies that $r_{N/C}(L) = r_N(L) = 2$.\nSince $K_1$ is connected, it is isomorphic to $U_{1,3}$ or $U_{2,3}$.\nLet $N_1$ be obtained from $N$ by deleting, for each element $d$ in $D$, the elements in $\\cl_N(L \\cup d) - L$, and then contracting $C$.\nThen $K_1 = \\si(N_1/L)$ and $r(N_1) = r(K_1) + 2$.\nWe will show that $N_1$ has a minor from Figure \\ref{fig: rank-3 excluded minors} or \\ref{fig: rank-4 excluded minor}.\n\n\\begin{claim}\n Some $x \\in L$ is on more than one long line of $M$.\n\\end{claim}\n\\begin{proof}\nLet $e \\in L$ and let $M_1 = M/e$.\nLet $P = L - e$, and note that $P$ is a nontrivial parallel class of $M_1$.\nAlso note that $\\elem(M_1) = |M| - \\ell = \\ell(r(M_1) - 1) + 1$,\nso by induction, $\\si(M_1)$ is obtained by taking parallel connections of copies of $U_{2,\\ell+1}$.\nThis implies that $P$ is on a long line $L_1$ of $M_1$.\nLet $F$ be the plane $L \\cup L_1$ of $M$.\nNote that $|F| \\ge |L| + |L_1| - 1 \\ge 2(\\ell + 1) - 1 = 2\\ell + 1$.\nIf $F - L$ has a $3$-element independent set $I$, then $M|F$ has an $M(K_4)$-restriction (if each pair of points in $I$ spans a point on $L$) or a $U_{2, \\ell+2}$-minor, a contradiction.\nSo $F - L$ has rank $2$.\nIf $\\cl(F - L)$ does not intersect $L$, then for every $f \\in F - L$ we have $(M|F)/f \\cong U_{2,\\ell+2}$, a contradiction. \nSo $\\cl(F - L)$ intersects $L$ in some element $x$.\nSince $|F - L| \\ge \\ell \\ge 2$, it follows that $\\cl(F - L)$ is a long line of $M$ that contains $x$ and is not equal to $L$, as desired.\n\\end{proof}\n\nLet $r, \\ell$ be integers with $r \\ge 2$ and $\\ell \\ge 3$.\nLet $W(r,\\ell)$ be the matroid with a basis $B = \\{b_1, b_2, \\dots, b_r\\}$ and $\\lfloor \\frac{\\ell - 1}{2} \\rfloor$ elements freely placed (see \\cite[page 270]{Oxley-2011}) in the span of $\\{b_i, b_{i+1}\\}$ for all $i \\in [r]$, reading indices modulo $r$.\nLet $W(r, \\ell)^+$ be obtained from $W(r, \\ell)$ by freely placing one point in $\\cl(\\{b_1, b_2\\})$.\nThen $W(2, \\ell) \\cong U_{2,\\lfloor \\frac{\\ell - 1}{2} \\rfloor + 2}$, and $W(r, 3)$ and $W(r, 4)$ are both isomorphic to the rank-$r$ whirl $\\mathcal W^r$ when $r \\ge 3$.\nSince the whirl $\\mathcal W^r$ is $3$-connected (see \\cite[Example 8.3.4]{Oxley-2011}) and $W(r,\\ell)$ has $\\mathcal W^r$ as a restriction, it follows from \\cite[page 295, Exercise 5]{Oxley-2011} that $W(r,\\ell)$ and $W(r, \\ell)^+$ are $3$-connected.\nAnd it is straightforward to check that the natural cyclic ordering of $E(W(r, \\ell))$ (first $b_1$, then all elements in $\\cl(\\{b_1, b_2\\}) - b_2$, then $b_2$, and so on) or $W(r,\\ell)^+$ satisfies the condition of Theorem \\ref{thm:Bonin's condition}, so $W(r, \\ell)$ and $W(r, \\ell)^+$ are positroids.\nFinally, one can check that if $r \\ge 4$ and $e \\notin \\{b_1, \\dots, b_r\\}$, then $\\si(W(r, \\ell)/e) \\cong W(r, \\ell)$, and that $\\si(W(r, \\ell)/b_i) \\cong W(r,\\ell) \\del (\\cl(\\{b_{i-1},b_i, b_{i+1}) - \\{b_{i-1}, b_{i+1}\\})$, and from these observations one can prove that $W(r,\\ell)$ is $U_{2,\\ell+2}$-minor-free, and when $\\ell$ is even, $W(r, \\ell)^+$ is $U_{2, \\ell+2}$-minor-free.\nNoting that $|W(r, \\ell)| = r + r\\lfloor \\frac{\\ell-1}{2} \\rfloor$, we make the following conjecture.", "post_theorem_intro_text_len": 5248, "post_theorem_intro_text": "This result lies at the intersection of three areas of research: extremal matroid theory, structural properties of positroids, and structural properties of matroids with no $M(K_4)$-minor.\nWe will explain how Theorem \\ref{thm: main} continues lines of research in these three areas, beginning with extremal matroid theory.\nThe following beautiful result of Geelen and Nelson \\cite[Theorem 1.2]{Geelen-Nelson-2010} generalizes the initial work of Kung and tightly solves Problem \\ref{prob: Kung problem} for the class of matroids with no $U_{2,\\ell+2}$-minor, for all values of $\\ell$ and all sufficiently large values of $r$.\n\n\\begin{theorem}[Geelen, Nelson \\cite{Geelen-Nelson-2010}] \\label{thm: exlude a line}\nLet $\\ell\\ge 2$ be an integer and let $q$ be the largest prime power at most $\\ell$.\nFor $r$ sufficiently large, every simple rank-$r$ matroid with no $U_{2,\\ell+2}$-minor has at most $\\frac{q^r - 1}{q - 1}$ elements, with equality only for the rank-$r$ projective geometry over the order-$q$ finite field.\n\\end{theorem}\n\nProjective geometries are not representable over fields of characteristic zero (see \\cite[page 205, Exercise 8]{Oxley-2011}), inviting the following question: for a field $\\bF$ with characteristic zero, what is the maximum number of elements of a simple rank-$r$ $\\bF$-representable matroid with no $U_{2, \\ell+2}$-minor?\nThis was answered by Geelen, Nelson, and Walsh \\cite[Theorem 1.0.5]{Geelen-Nelson-Walsh-2024} in the case that $\\bF = \\bC$.\n\n\\begin{theorem}[Geelen, Nelson, Walsh \\cite{Geelen-Nelson-Walsh-2024}] \\label{thm: complex exclude a line}\nLet $\\ell\\ge 2$ be an integer and let $r$ be a sufficiently large integer.\nThen every simple rank-$r$ $\\bC$-representable matroid with no $U_{2,\\ell+2}$-minor has at most $\\ell\\binom{r}{2} + r$ elements, with equality only for the rank-$r$ Dowling geometry over the cyclic group of order $\\ell$.\n\\end{theorem}\n\nWhen $\\bF \\in \\{\\bR, \\mathbb Q\\}$, there is a lower bound of $2\\binom{r}{2} + r$ and an upper bound of $2 \\binom{r}{2} + f(\\ell)\\cdot r$ with $f$ a function of $\\ell$, due to Geelen, Nelson, and Walsh \\cite[Theorem 1.0.6]{Geelen-Nelson-Walsh-2024}, but no exact solution to Problem~\\ref{prob: Kung problem} is known for all $\\ell$. \nTheorem \\ref{thm: main} can be seen as a further specialization of Theorems \\ref{thm: exlude a line} and \\ref{thm: complex exclude a line} to the subclass of $\\bR$-representable matroids consisting of positroids.\n\nTheorem \\ref{thm: main} also continues a recent line of research that studies positroids from a structural perspective.\nFor example, Bonin \\cite{Bonin-2024} studied matroid amalagams of positroids, Quail \\cite{Quail-2024} studied positroids representable over the finite fields of order $3$ and of order $4$, Quail and Rombach \\cite{Quail-Rombach-2025} studied graphic positroids, and Park \\cite{Park-2023} studied excluded minors for positroids.\nOur proof of Theorem \\ref{thm: main} relies on two new structural properties (Propositions \\ref{prop: rank 3 excluded minor} and \\ref{prop: rank 4 excluded minor}) for low-rank positroids.\n\nFinally, \\Cref{thm: main} contributes to a line of research studying matroids with no $M(K_4)$-minor, where $M(K_4)$ is the graphic matroid of the complete graph $K_4$. \nFor example, Sims \\cite{Sims-1977} proved that the class of $M(K_4)$-minor-free matroids is complete, Brylawski \\cite{Brylawski-1971} characterized the binary matroids with no $M(K_4)$-minor, and Oxley \\cite{Oxley-1987} characterized the $3$-connected ternary matroids with no $M(K_4)$-minor.\nMore recently, work of Pendavingh and van der Pol \\cite{Pendavingh-vanderPol-2015} and van der Pol \\cite{vanderPol-2023} investigated the importance of $M(K_4)$ in matroid enumeration.\nMost notably for this paper, Kung studied Problem \\ref{prob: Kung problem} for the class of matroids with no $M(K_4)$-minor and no $U_{2, \\ell+2}$-minor, proving that every such simple rank-$r$ matroid has at most $(6\\ell^{\\ell - 1} + 8\\ell)\\cdot r$ elements \\cite[Theorem 1.1]{Kung-1988}. The graphic matroid $M(K_4)$ is not a positroid, so this also bounds from above the maximum number of elements of a simple rank-$r$ positroid with no $U_{2, \\ell+2}$-minor. \nTo the best of our knowledge, this was in fact the previous best upper bound for this quantity.\nThere are several other notable $M(K_4)$-minor-free classes of matroids that contain the class of positroids, such as gammoids, strongly base-orderable matroids, and $k$-base-orderable matroids.\nWe will discuss these further in Section \\ref{sec: future work}.\n\nThe rest of this paper is structured as follows.\nIn Section \\ref{sec: preliminaries}, we will give more background on matroids and positroids.\nIn Section \\ref{sec: excluded minors}, we will use a list of $9$ known excluded minors for the class of positroids to prove two structural properties for low-rank positroids.\nIn Section \\ref{sec: the extremal examples}, we will describe the matroids for which equality holds in Theorem \\ref{thm: main} and prove that they are in fact positroids and are $U_{2, \\ell+2}$-minor-free.\nWe will prove Theorem \\ref{thm: main} in Section \\ref{sec: the main proof}, and conclude in Section \\ref{sec: future work} by discussing several directions for future work.", "sketch": "Our proof of Theorem \\ref{thm: main} relies on two new structural properties (Propositions \\ref{prop: rank 3 excluded minor} and \\ref{prop: rank 4 excluded minor}) for low-rank positroids. The paper then (i) uses a list of $9$ known excluded minors for the class of positroids to prove these two structural properties for low-rank positroids (Section \\ref{sec: excluded minors}); (ii) describes the matroids for which equality holds in Theorem \\ref{thm: main} and proves that they are positroids and are $U_{2, \\ell+2}$-minor-free (Section \\ref{sec: the extremal examples}); and (iii) proves Theorem \\ref{thm: main} in Section \\ref{sec: the main proof}.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm: main}\nFor all integers $r, \\ell \\ge 1$, if $M$ is a simple rank-$r$ positroid with no $U_{2,\\ell+2}$-minor, then $|M| \\le \\ell(r - 1) + 1$.\nMoreover, if $r \\ge 2$ then equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2, \\ell+1}$.", "theorem_type": ["Inequality or Bound", "Biconditional or Equivalence"], "mcq": {"question": "Let $r,2 ell\\ge 1$ be integers. Recall that a positroid is a matroid representable over $\\mathbb{R}$ by a matrix whose maximal minors are all nonnegative, and that $U_{2,m}$ denotes the rank-$2$ uniform matroid on $m$ elements. If $M$ is a simple rank-$r$ positroid with no $U_{2,\\ell+2}$-minor, which quantitative estimate for the number of elements $|M|$ holds, and for $r\\ge 2$ when does equality occur?", "correct_choice": {"label": "A", "text": "The bound $|M|\\le \\ell(r-1)+1$ always holds. Moreover, if $r\\ge 2$, then equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2,\\ell+1}$."}, "choices": [{"label": "B", "text": "The bound $|M|\\le \\ell r+1$ always holds. Moreover, if $r\\ge 2$, then equality holds if and only if $M\\cong U_{2,\\ell+2}$ or, more generally, $M$ is obtained by taking parallel connections of copies of $U_{2,\\ell+2}$."}, {"label": "C", "text": "The bound $|M|\\le \\ell(r-1)+1$ always holds."}, {"label": "D", "text": "For each fixed $\\ell\\ge 1$, there exists an integer $r_0(\\ell)$ such that whenever $r\\ge r_0(\\ell)$ and $M$ is a simple rank-$r$ positroid with no $U_{2,\\ell+2}$-minor, one has $|M|\\le \\ell(r-1)+1$; moreover, for such sufficiently large $r$, equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2,\\ell+1}$."}, {"label": "E", "text": "The bound $|M|\\le \\ell(r-1)+1$ always holds. Moreover, if $r\\ge 2$, then equality holds if and only if every long line of $M$ has exactly $\\ell+1$ points."}], "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": "excluded-line size in extremal family", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the equality characterization", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "all-ranks statement replaced by sufficiently-large-r", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "global parallel-connection structure replaced by local long-line condition", "template_used": "wildcard"}]}}
{"id": "2512.15307v1", "paper_link": "http://arxiv.org/abs/2512.15307v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{main} Let $T>0$, $l_j>0,\\ j=1,...,N$, $s\\in [0,3]$ be given. Suppose $f\\in W^{\\frac{s}{3},1}(0,T;\\mathbb{L}^2(\\mathcal{T}))\\cap L^{\\frac{6}{6-s}}(0,T;\\mathbb{H}^\\frac{s}{3}(\\mathcal{T}))$ and \n\\begin{align*}\n(u^0,g_0,g,p)\\in \\mathbb{H}^s(\\mathcal{T})\\times H^{\\frac{s-1}{3}}(0,T)\\times \\left[H^\\frac{s}{3}(0,T)\\right]^{N}\\times \\left[H^{\\frac{s+1}{3}}(0,T)\\right]^N.\n\\end{align*}\nsatisfying the $s$-compatibility conditions, thus there exists $T^*\\in (0,T)$ such that the IBVP \\eqref{kdV} admits a unique solution $u \\in \\mathcal{X}_{s,T^*}$, where \n$$\\mathcal{X}_{s,T}=\\left\\{v \\in \\mathbb{B}^{*}_{s,T};\\ \\partial_x^ku_j\\in L_x^\\infty(0,l_j;H^\\frac{s+1-k}{3}(0,T)),\\ k=0,1,2,\\ j=1,2,...,N\\right\\},$$\n\\begin{align*}\n\t\\mathbb{B}_{s,T}^*:=\\mathbb{B}_{s,T}\\cap H^\\frac{s}{3}(0,T;\\mathbb{H}^1(\\mathcal{T})).\n\\end{align*}", "start_pos": 16368, "end_pos": 17220, "label": "main"}, "ref_dict": {"kdV": "\\begin{align}\\label{kdV}\n\t\t\t\\begin{cases}\n\t\t\t\t\\partial_t u_j+\\partial_x u_j+\\partial_x^3 u_j+u_j\\partial_x u_j=0,&t \\in (0,T),\\ x \\in (0,l_j),\\ j=1,...,N\\\\\n\t\t\t\tu_j(t,0)=u_1(t,0),&t\\in (0,T),\\ j=1,...,N\\\\\n\t\t\t\t\\displaystyle\\sum_{j=1}^N\\partial_x^2 u_j(t,0)=-\\alpha u_1(t,0)-\\frac{N}{3}u_1^2(t,0)+g_0(t),&t \\in (0,T)\\\\\t\t\t\t\n\t\t\t\tu_j(t,l_j)=p_j(t),\\ \\ \\ \\ \\ \\partial_x u_j(t,l_j)=g_j(t),&t\\in (0,T),\\ j=1,...,N\\\\\n\t\t\t\tu_j(0,x)=u_j^0(x),&x \\in (0,l_j),\\ j=1,...,N\n\t\t\t\\end{cases}\n\t\t\\end{align}", "fig:network": "\\label{fig:network}\n\t\t\t\\caption{Network with $N$ edges}\t\n\t\\end{figure}\n\nLet us introduce some notation used throughout this work. From now on, we have the following: \n\t \\begin{itemize}\n\t \\item[i.] The", "main": "\\begin{theorem}\\label{main} Let $T>0$, $l_j>0,\\ j=1,...,N$, $s\\in [0,3]$ be given. Suppose $f\\in W^{\\frac{s}{3},1}(0,T;\\mathbb{L}^2(\\mathcal{T}))\\cap L^{\\frac{6}{6-s}}(0,T;\\mathbb{H}^\\frac{s}{3}(\\mathcal{T}))$ and \n\\begin{align*}\n(u^0,g_0,g,p)\\in \\mathbb{H}^s(\\mathcal{T})\\times H^{\\frac{s-1}{3}}(0,T)\\times \\left[H^\\frac{s}{3}(0,T)\\right]^{N}\\times \\left[H^{\\frac{s+1}{3}}(0,T)\\right]^N.\n\\end{align*}\nsatisfying the $s$-compatibility conditions, thus there exists $T^*\\in (0,T)$ such that the IBVP \\eqref{kdV} admits a unique solution $u \\in \\mathcal{X}_{s,T^*}$, where \n$$\\mathcal{X}_{s,T}=\\left\\{v \\in \\mathbb{B}^{*}_{s,T};\\ \\partial_x^ku_j\\in L_x^\\infty(0,l_j;H^\\frac{s+1-k}{3}(0,T)),\\ k=0,1,2,\\ j=1,2,...,N\\right\\},$$\n\\begin{align*}\n\t\\mathbb{B}_{s,T}^*:=\\mathbb{B}_{s,T}\\cap H^\\frac{s}{3}(0,T;\\mathbb{H}^1(\\mathcal{T})).\n\\end{align*}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 11439, "pre_theorem_intro_text": "In mathematics and physics, a quantum graph is a network composed of vertices connected by edges, where each edge carries a differential or pseudo-differential equation. When every edge is endowed with a natural metric, the structure is referred to as a metric graph. A common illustration is a power network: the transmission lines act as edges connecting transformer stations (vertices). In such a case, the governing differential equations may describe the voltage along each line, with boundary conditions at the vertices ensuring that the total current flowing into and out of each vertex sums to zero.\n\nThe concept of quantum graphs traces back to the work of Linus Pauling in the 1930s, who used them to model free electrons in organic molecules. Since then, they have emerged in a wide range of contexts, including quantum chaos, waveguide analysis, photonic crystals, and Anderson localization—the suppression of wave propagation in disordered media. Quantum graphs also arise as limit models of systems composed of thin wires and serve as important theoretical tools in mesoscopic physics and nanotechnology. A simplified formulation of quantum graphs was later proposed by Freedman \\textit{et al.} in \\cite{Freedman}.\n\nBeyond solving the differential equations in specific applications, the study of quantum graphs encompasses fundamental questions such as well-posedness, controllability, and identifiability. For example, controllability concerns determining the inputs required to steer the system toward a desired configuration, such as ensuring stable power distribution across a grid. Identifiability, on the other hand, focuses on optimal measurement placement—for instance, choosing pressure gauges in a water network to detect and locate potential leaks.\n\n\\subsection{Dispersive systems on graph structure}\nIn recent years, the study of nonlinear dispersive equations on metric graphs has attracted growing interest from mathematicians, physicists, chemists, and engineers (see \\cite{BK, BlaExn08, BurCas01, K, Mug15} for detailed discussions and references). A fundamental setting for such investigations is the star graph $\\mathcal{G}$, a metric graph consisting of $N$ half-lines $(0,\\infty)$ meeting at a single common vertex $\\nu = 0$. On each edge, one typically defines a nonlinear dispersive equation—most prominently, the nonlinear Schrödinger equation (see the works of Adami \\textit{et al.} \\cite{AdaNoj14, AdaNoj15} and Angulo and Goloshchapova \\cite{AngGol17a, AngGol17b}).\nThe introduction of nonlinearities into dispersive models on such networks creates a rich framework for exploring soliton propagation, energy transfer, and nonlinear wave interactions. A central analytical difficulty arises at the vertex, where the graph may exhibit bifurcation or multi-bifurcation phenomena, especially in more intricate network topologies.\n\nOther nonlinear dispersive systems on graphs have also been the subject of extensive study. For example, in the context of well-posedness theory, Cavalcante \\cite{Cav} established local well-posedness for the Cauchy problem associated with the Korteweg–de Vries (KdV) equation on a metric star graph composed of one negative and two positive half-lines joined at a common vertex $\\nu = 0$ (the so-called $\\mathcal{Y}$-junction).\n\nAnother important example is the Benjamin–Bona–Mahony (BBM) equation: Bona and Cascaval \\cite{bona} proved local well-posedness in the Sobolev space $H^1$, while Mugnolo and Rault \\cite{Mugnolobbm} showed the existence of traveling-wave solutions on graphs. Using a different approach, Ammari and Crépeau \\cite{AmCr1} derived results concerning well-posedness and stabilization of the BBM equation in star-shaped networks with bounded edges. Moreover, Mugnolo \\textit{et al.} \\cite{MugNoSe} provided a full characterization of boundary conditions under which the Airy-type evolution equation\n$$u_t = \\alpha u_{xxx} + \\beta u_x, \\quad \\alpha \\in \\mathbb{R} \\setminus \\{0\\}, \\ \\beta \\in \\mathbb{R},$$\ngenerates contraction semigroups on star graphs. We also refer to the recent work on Airy dynamics on graphs with looping edges \\cite{PavaMunoz24}, which provides a complementary perspective on this setting.\n\\medskip\n\nSignificant progress has also been achieved in the fields of control theory and inverse problems on networks. Ignat \\textit{et al.} \\cite{Ignat2011} examined inverse problems for the heat and Schrödinger equations on tree-like structures, while Baudouin and Yamamoto \\cite{Baudouin} proposed a unified and simplified framework for coefficient identification problems. Furthermore, Ammari and Crépeau \\cite{Ammari and Crepeau 2018}, Cerpa \\textit{et al.} \\cite{Cerpa, Cerpa1}, and Parada \\textit{et al.} \\cite{Parada2022a, Parada2022b} established stabilization and boundary controllability results for the KdV equation on star-shaped graphs, contributing substantially to the control-theoretic understanding of dispersive models on networks. Recently, in \\cite{CapistranoParadaDaSilva25}, we studied the analytical structure of the entire functions associated with the KdV operator, addressing control problems involving mixed Dirichlet–Neumann boundary controls.\n\n\\subsection{Setting of problem and functional framework} In the context of graph structure, let us consider the following: given $N\\geq 2$, consider a star-shaped network $\\mathcal{T}$ with $N$ edges described by intervals $I_j=(0,l_j)$, $l_j>0$ for $j=1,...,N$. Denoting by $e_1,...,e_N$ the edges of $\\mathcal{T}$ one has $\\mathcal{T}=\\bigcup_{j=1}^Ne_j$. Here, we study the local  well-posedness for a system composed of $N$ nonlinear KdV equations posed on $\\mathcal{T}$ and coupled by the boundary, namely\n\t\t\\begin{align}\\label{kdV}\n\t\t\t\\begin{cases}\n\t\t\t\t\\partial_t u_j+\\partial_x u_j+\\partial_x^3 u_j+u_j\\partial_x u_j=0,&t \\in (0,T),\\ x \\in (0,l_j),\\ j=1,...,N\\\\\n\t\t\t\tu_j(t,0)=u_1(t,0),&t\\in (0,T),\\ j=1,...,N\\\\\n\t\t\t\t\\displaystyle\\sum_{j=1}^N\\partial_x^2 u_j(t,0)=-\\alpha u_1(t,0)-\\frac{N}{3}u_1^2(t,0)+g_0(t),&t \\in (0,T)\\\\\t\t\t\t\n\t\t\t\tu_j(t,l_j)=p_j(t),\\ \\ \\ \\ \\ \\partial_x u_j(t,l_j)=g_j(t),&t\\in (0,T),\\ j=1,...,N\\\\\n\t\t\t\tu_j(0,x)=u_j^0(x),&x \\in (0,l_j),\\ j=1,...,N\n\t\t\t\\end{cases}\n\t\t\\end{align}\nwhere $u=(u_1,...,u_N)$ stands the state of the system, $p_j$ and $g_j$ are the controls inputs and $\\alpha>\\frac{N}{2}$. We represent this situation in the figure \\eqref{fig:network}. \n\t\t\\begin{figure}[H]\n\t\t\t\\centering\n\t\t\t\\begin{tikzpicture}\n\t\t\t\t\\draw(0, 0) circle (1mm);\n\t\t\t\t\\draw[line width=0.8pt](0.02, 0.09) -- (0.5, 1.95);\n\t\t\t\t\\draw[fill=black](0.53, 2.04) circle (1mm);\n\t\t\t\t\\draw[line width=0.8pt](0.07, 0.07) -- (1.33, 1.5);\n\t\t\t\t\\draw[fill=black](1.39, 1.57) circle (1mm);\n\t\t\t\t\\draw[line width=0.8pt](0.1, 0.04) -- (1.85, 0.78);\n\t\t\t\t\\draw[fill=black](1.93, 0.815) circle (1mm);\n\t\t\t\t\\draw[line width=0.8pt](0.08, -0.04) -- (1.75, -1);\n\t\t\t\t\\draw[fill=black](1.825, -1.045) circle (1mm);\n\t\t\t\t\\draw[line width=0.8pt](0.05, -0.09) -- (1.02, -1.73);\n\t\t\t\t\\draw[fill=black](1.07, -1.8) circle (1mm);\n\t\t\t\t\\draw[fill=black](1.45, 0.2) circle (0.3mm);\n\t\t\t\t\\draw[fill=black](1.45, 0.05) circle (0.3mm);\n\t\t\t\t\\draw[fill=black](1.45, -0.1) circle (0.3mm);\n\t\t\t\t\\draw node (no1) at (-0.3,0) {$0$};\n\t\t\t\t\\draw node (no1) at (0.19, 1.7) {$e_1$};\n\t\t\t\t\\draw node (no1) at (0.86, 1.35) {$e_2$};\n\t\t\t\t\\draw node (no1) at (1.35, 0.8) {$e_3$};\n\t\t\t\t\\draw node (no1) at (1.7, -0.6) {$e_{_{N-1}}$};\n\t\t\t\t\\draw node (no1) at (1.06, -1.25) {$e_{_N}$};\n\t\t\t\t\\draw(3.95,-0.5) circle (1mm);\n\t\t\t\t\\draw node (no1) at (6.4,-0.5) {{\\footnotesize \\hspace{10mm} internal controlled node}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ };\n\t\t\t\t\\draw[fill=black](3.95,-1) circle (1mm);\n\t\t\t\t\\draw node (no1) at (6.4,-1) {{\\footnotesize \\hspace{13mm} external controlled node}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ };\n\t\t\t\\end{tikzpicture}\n\t\t\t\\label{fig:network}\n\t\t\t\\caption{Network with $N$ edges}\t\n\t\\end{figure}\n\nLet us introduce some notation used throughout this work. From now on, we have the following: \n\t \\begin{itemize}\n\t \\item[i.] The vector $u$ is given by $u=(u_1,...,u_N) \\in \\mathbb{L}^2(\\mathcal{T})=\\displaystyle\\prod_{j=1}^NL^2(0,l_j)$\n\tand initial data is $u^0=(u^0_1,...,u^0_N)$. \n\t \\item[ii.] The inner product in $\\mathbb{L}^2(\\mathcal{T})$ will be given by  $$\\left(u,z\\right)_{\\mathbb{L}^2(\\mathcal{T})}=\\sum_{j=1}^N\\int_0^{l_j}u_jz_jdx,\\ \\ u,z\\in \\mathbb{L}^2(\\mathcal{T}).$$\n\tMoreover,  $$\\mathbb{H}^s(\\mathcal{T})=\\displaystyle\\prod_{j=1}^NH^s(0,l_j), \\ \\ s \\in \\mathbb{R}.$$\n\t \\end{itemize}\n\t\\begin{itemize}\n\t\t\\item[iii.] Define also the following spaces:\n\t\t$$\\mathbb{H}_0^k(\\mathcal{T})=\\displaystyle\\prod_{j=1}^{N}H_0^k(0,l_j),\\ k\\in \\mathbb{N}$$\n\t\tand\n\t\t$$H_r^k(0,l_j)=\\left\\{v \\in H^k(0,l_j),\\ \\partial_x^iv(l_j)=0,\\ 0\\leq i\\leq k-1\\right\\},\\ k=1,...,N,$$ \n\t\twhere $v^{(m)}=\\frac{dv}{dx^m}$ and the index $r$ is related to the null right boundary conditions. In addition, \n\t\t$$\\mathbb{H}^k_r(\\mathcal{T})=\\prod_{j=1}^{N}H_r^k(0,l_j)\\quad \\text{and} \\quad \\|u\\|_{\\mathbb{H}^k_r(\\mathcal{T})}^2=\\sum_{j=1}^{n}\\|u_j\\|_{H^k(0,l_j)}^2\\ k \\in \\mathbb{N}.$$\n\t\t\\item[iv.] Consider the following characterization: \n\t\t$$H_r^{-1}(0,l_j)=\\left(H_r^{1}(0,l_j)\\right)'$$ \n\t\tas the dual space of $H_r(0,l_j)$ with respect to the pivot space $L^2(0,l_j)$ and $\\mathbb{H}_r^{-1}$ denotes the cartesian product of $H_r^{-1}(0,l_j)$. \n\\item[v.] For $k\\in \\mathbb{N}$, the derivative $\\partial_x^ku$ means the row vector\n\\begin{align*}\n\\partial_x^k u:=(\\partial_x^ku_1,...,\\partial_x^ku_N).\n\\end{align*}\nThe derivative $\\partial_t^ku$ is understood in the same way.\n\\item[vi.] Under the above notation, we have\n\\begin{align*}\n\tC^k\\left([0,T];\\mathbb{H}^s(\\mathcal{T})\\right)=\\prod_{j=1}^NC^k\\left([0,T];H^s(0,l_j)\\right),\\ k=0,1,2,... \\text{ \\ and \\ }s \\in \\mathbb{R}\n\\end{align*}\nas well as\n\\begin{align*}\n\tL^p\\left(0,T;\\mathbb{H}^s(\\mathcal{T})\\right)=\\prod_{j=1}^NL^p\\left(0,T;H^s(0,l_j)\\right),\\ p\\geq 1 \\text{ \\ and \\ }s \\in \\mathbb{R}.\n\\end{align*}\nHence, for $f \\in \tC^k\\left([0,T];\\mathbb{H}^s(\\mathcal{T})\\right)$ and $q \\in L^p\\left(0,T;\\mathbb{H}^s(\\mathcal{T})\\right)$ we can write\n\\begin{align*}\nf=(f_1,...,f_N)\\text{ \\ and \\ }q=(q_1,...,q_N)\n\\end{align*}\nwith\n\\begin{align*}\nf_j \\in C^k\\left([0,T];H^s(0,l_j)\\right)\\text{ \\ and \\ }q_j \\in L^p\\left(0,T;H^s(0,l_j)\\right),\\ j=1,...N.\n\\end{align*}\n\\item[vii.] We will consider the spaces\n$$\\mathbb{H}^k_e(\\mathcal{T})=\\left\\{u=(u_1,...,u_N)\\in \\mathbb{H}^k_r(\\mathcal{T});\\ u_j(0)=u_k(0),\\ j,k=1,...,N\\right\\}\\ k \\in \\mathbb{N},$$\n$$\\mathbb{H}^3_c(\\mathcal{T})=\\left\\{u=(u_1,...,u_N)\\in \\mathbb{H}^3(\\mathcal{T});\\ u_j(0)=u_k(0),\\ j,k=1,...,N\\right\\},$$\nand\n$$\\mathbb{B}_{s,T}=C([0,T],\\mathbb{H}^s(\\mathcal{T}))\\cap L^2(0,T,\\mathbb{H}^{s+1}(\\mathcal{T})),$$\nwith the norm \n$$\\displaystyle\\|u\\|_{\\mathbb{B}_{s,T}}:=\\|u\\|_{C([0,T],\\mathbb{H}^s(\\mathcal{T}))}+\\|u\\|_{L^2(0,T,\\mathbb{H}^{s+1}(\\mathcal{T}))}=\\max_{t\\in [0,T]}\\|u\\|_{\\mathbb{H}^s(\\mathcal{T})}+\\left(\\int_0^T\\|u(t,\\cdot)\\|_{\\mathbb{H}^{s+1}}^2dt\\right)^\\frac{1}{2}.$$\nWhen $s=0$ we write $\\mathbb{B}_T$ instead $\\mathbb{B}_{0,T}$.\n\\item [viii.] Once understand the appropriate spaces in each situation, we will use the notations $g=(g_1,...,g_N)$ and $p=(p_1,...,p_N)$.\n\\end{itemize}\n\n\\subsection{Main results and structure of the article} With the previous framework in hand, our goal here is to prove the local and global well-posedness for the initial boundary value problem (IBVP) \\eqref{kdV}. The first main result establishes the local well-posedness for this system in sharp spaces.", "context": "\\subsection{Dispersive systems on graph structure}\nIn recent years, the study of nonlinear dispersive equations on metric graphs has attracted growing interest from mathematicians, physicists, chemists, and engineers (see \\cite{BK, BlaExn08, BurCas01, K, Mug15} for detailed discussions and references). A fundamental setting for such investigations is the star graph $\\mathcal{G}$, a metric graph consisting of $N$ half-lines $(0,\\infty)$ meeting at a single common vertex $\\nu = 0$. On each edge, one typically defines a nonlinear dispersive equation—most prominently, the nonlinear Schrödinger equation (see the works of Adami \\textit{et al.} \\cite{AdaNoj14, AdaNoj15} and Angulo and Goloshchapova \\cite{AngGol17a, AngGol17b}).\nThe introduction of nonlinearities into dispersive models on such networks creates a rich framework for exploring soliton propagation, energy transfer, and nonlinear wave interactions. A central analytical difficulty arises at the vertex, where the graph may exhibit bifurcation or multi-bifurcation phenomena, especially in more intricate network topologies.\n\nAnother important example is the Benjamin–Bona–Mahony (BBM) equation: Bona and Cascaval \\cite{bona} proved local well-posedness in the Sobolev space $H^1$, while Mugnolo and Rault \\cite{Mugnolobbm} showed the existence of traveling-wave solutions on graphs. Using a different approach, Ammari and Crépeau \\cite{AmCr1} derived results concerning well-posedness and stabilization of the BBM equation in star-shaped networks with bounded edges. Moreover, Mugnolo \\textit{et al.} \\cite{MugNoSe} provided a full characterization of boundary conditions under which the Airy-type evolution equation\n$$u_t = \\alpha u_{xxx} + \\beta u_x, \\quad \\alpha \\in \\mathbb{R} \\setminus \\{0\\}, \\ \\beta \\in \\mathbb{R},$$\ngenerates contraction semigroups on star graphs. We also refer to the recent work on Airy dynamics on graphs with looping edges \\cite{PavaMunoz24}, which provides a complementary perspective on this setting.\n\\medskip\n\nLet us introduce some notation used throughout this work. From now on, we have the following: \n \\begin{itemize}\n \\item[i.] The vector $u$ is given by $u=(u_1,...,u_N) \\in \\mathbb{L}^2(\\mathcal{T})=\\displaystyle\\prod_{j=1}^NL^2(0,l_j)$\n and initial data is $u^0=(u^0_1,...,u^0_N)$. \n \\item[ii.] The inner product in $\\mathbb{L}^2(\\mathcal{T})$ will be given by $$\\left(u,z\\right)_{\\mathbb{L}^2(\\mathcal{T})}=\\sum_{j=1}^N\\int_0^{l_j}u_jz_jdx,\\ \\ u,z\\in \\mathbb{L}^2(\\mathcal{T}).$$\n Moreover, $$\\mathbb{H}^s(\\mathcal{T})=\\displaystyle\\prod_{j=1}^NH^s(0,l_j), \\ \\ s \\in \\mathbb{R}.$$\n \\end{itemize}\n \\begin{itemize}\n \\item[iii.] Define also the following spaces:\n $$\\mathbb{H}_0^k(\\mathcal{T})=\\displaystyle\\prod_{j=1}^{N}H_0^k(0,l_j),\\ k\\in \\mathbb{N}$$\n and\n $$H_r^k(0,l_j)=\\left\\{v \\in H^k(0,l_j),\\ \\partial_x^iv(l_j)=0,\\ 0\\leq i\\leq k-1\\right\\},\\ k=1,...,N,$$ \n where $v^{(m)}=\\frac{dv}{dx^m}$ and the index $r$ is related to the null right boundary conditions. In addition, \n $$\\mathbb{H}^k_r(\\mathcal{T})=\\prod_{j=1}^{N}H_r^k(0,l_j)\\quad \\text{and} \\quad \\|u\\|_{\\mathbb{H}^k_r(\\mathcal{T})}^2=\\sum_{j=1}^{n}\\|u_j\\|_{H^k(0,l_j)}^2\\ k \\in \\mathbb{N}.$$\n \\item[iv.] Consider the following characterization: \n $$H_r^{-1}(0,l_j)=\\left(H_r^{1}(0,l_j)\\right)'$$ \n as the dual space of $H_r(0,l_j)$ with respect to the pivot space $L^2(0,l_j)$ and $\\mathbb{H}_r^{-1}$ denotes the cartesian product of $H_r^{-1}(0,l_j)$. \n\\item[v.] For $k\\in \\mathbb{N}$, the derivative $\\partial_x^ku$ means the row vector\n\\begin{align*}\n\\partial_x^k u:=(\\partial_x^ku_1,...,\\partial_x^ku_N).\n\\end{align*}\nThe derivative $\\partial_t^ku$ is understood in the same way.\n\\item[vi.] Under the above notation, we have\n\\begin{align*}\n C^k\\left([0,T];\\mathbb{H}^s(\\mathcal{T})\\right)=\\prod_{j=1}^NC^k\\left([0,T];H^s(0,l_j)\\right),\\ k=0,1,2,... \\text{ \\ and \\ }s \\in \\mathbb{R}\n\\end{align*}\nas well as\n\\begin{align*}\n L^p\\left(0,T;\\mathbb{H}^s(\\mathcal{T})\\right)=\\prod_{j=1}^NL^p\\left(0,T;H^s(0,l_j)\\right),\\ p\\geq 1 \\text{ \\ and \\ }s \\in \\mathbb{R}.\n\\end{align*}\nHence, for $f \\in C^k\\left([0,T];\\mathbb{H}^s(\\mathcal{T})\\right)$ and $q \\in L^p\\left(0,T;\\mathbb{H}^s(\\mathcal{T})\\right)$ we can write\n\\begin{align*}\nf=(f_1,...,f_N)\\text{ \\ and \\ }q=(q_1,...,q_N)\n\\end{align*}\nwith\n\\begin{align*}\nf_j \\in C^k\\left([0,T];H^s(0,l_j)\\right)\\text{ \\ and \\ }q_j \\in L^p\\left(0,T;H^s(0,l_j)\\right),\\ j=1,...N.\n\\end{align*}\n\\item[vii.] We will consider the spaces\n$$\\mathbb{H}^k_e(\\mathcal{T})=\\left\\{u=(u_1,...,u_N)\\in \\mathbb{H}^k_r(\\mathcal{T});\\ u_j(0)=u_k(0),\\ j,k=1,...,N\\right\\}\\ k \\in \\mathbb{N},$$\n$$\\mathbb{H}^3_c(\\mathcal{T})=\\left\\{u=(u_1,...,u_N)\\in \\mathbb{H}^3(\\mathcal{T});\\ u_j(0)=u_k(0),\\ j,k=1,...,N\\right\\},$$\nand\n$$\\mathbb{B}_{s,T}=C([0,T],\\mathbb{H}^s(\\mathcal{T}))\\cap L^2(0,T,\\mathbb{H}^{s+1}(\\mathcal{T})),$$\nwith the norm \n$$\\displaystyle\\|u\\|_{\\mathbb{B}_{s,T}}:=\\|u\\|_{C([0,T],\\mathbb{H}^s(\\mathcal{T}))}+\\|u\\|_{L^2(0,T,\\mathbb{H}^{s+1}(\\mathcal{T}))}=\\max_{t\\in [0,T]}\\|u\\|_{\\mathbb{H}^s(\\mathcal{T})}+\\left(\\int_0^T\\|u(t,\\cdot)\\|_{\\mathbb{H}^{s+1}}^2dt\\right)^\\frac{1}{2}.$$\nWhen $s=0$ we write $\\mathbb{B}_T$ instead $\\mathbb{B}_{0,T}$.\n\\item [viii.] Once understand the appropriate spaces in each situation, we will use the notations $g=(g_1,...,g_N)$ and $p=(p_1,...,p_N)$.\n\\end{itemize}\n\n\\subsection{Main results and structure of the article} With the previous framework in hand, our goal here is to prove the local and global well-posedness for the initial boundary value problem (IBVP) \\eqref{kdV}. The first main result establishes the local well-posedness for this system in sharp spaces.", "full_context": "\\subsection{Dispersive systems on graph structure}\nIn recent years, the study of nonlinear dispersive equations on metric graphs has attracted growing interest from mathematicians, physicists, chemists, and engineers (see \\cite{BK, BlaExn08, BurCas01, K, Mug15} for detailed discussions and references). A fundamental setting for such investigations is the star graph $\\mathcal{G}$, a metric graph consisting of $N$ half-lines $(0,\\infty)$ meeting at a single common vertex $\\nu = 0$. On each edge, one typically defines a nonlinear dispersive equation—most prominently, the nonlinear Schrödinger equation (see the works of Adami \\textit{et al.} \\cite{AdaNoj14, AdaNoj15} and Angulo and Goloshchapova \\cite{AngGol17a, AngGol17b}).\nThe introduction of nonlinearities into dispersive models on such networks creates a rich framework for exploring soliton propagation, energy transfer, and nonlinear wave interactions. A central analytical difficulty arises at the vertex, where the graph may exhibit bifurcation or multi-bifurcation phenomena, especially in more intricate network topologies.\n\nAnother important example is the Benjamin–Bona–Mahony (BBM) equation: Bona and Cascaval \\cite{bona} proved local well-posedness in the Sobolev space $H^1$, while Mugnolo and Rault \\cite{Mugnolobbm} showed the existence of traveling-wave solutions on graphs. Using a different approach, Ammari and Crépeau \\cite{AmCr1} derived results concerning well-posedness and stabilization of the BBM equation in star-shaped networks with bounded edges. Moreover, Mugnolo \\textit{et al.} \\cite{MugNoSe} provided a full characterization of boundary conditions under which the Airy-type evolution equation\n$$u_t = \\alpha u_{xxx} + \\beta u_x, \\quad \\alpha \\in \\mathbb{R} \\setminus \\{0\\}, \\ \\beta \\in \\mathbb{R},$$\ngenerates contraction semigroups on star graphs. We also refer to the recent work on Airy dynamics on graphs with looping edges \\cite{PavaMunoz24}, which provides a complementary perspective on this setting.\n\\medskip\n\nLet us introduce some notation used throughout this work. From now on, we have the following: \n \\begin{itemize}\n \\item[i.] The vector $u$ is given by $u=(u_1,...,u_N) \\in \\mathbb{L}^2(\\mathcal{T})=\\displaystyle\\prod_{j=1}^NL^2(0,l_j)$\n and initial data is $u^0=(u^0_1,...,u^0_N)$. \n \\item[ii.] The inner product in $\\mathbb{L}^2(\\mathcal{T})$ will be given by $$\\left(u,z\\right)_{\\mathbb{L}^2(\\mathcal{T})}=\\sum_{j=1}^N\\int_0^{l_j}u_jz_jdx,\\ \\ u,z\\in \\mathbb{L}^2(\\mathcal{T}).$$\n Moreover, $$\\mathbb{H}^s(\\mathcal{T})=\\displaystyle\\prod_{j=1}^NH^s(0,l_j), \\ \\ s \\in \\mathbb{R}.$$\n \\end{itemize}\n \\begin{itemize}\n \\item[iii.] Define also the following spaces:\n $$\\mathbb{H}_0^k(\\mathcal{T})=\\displaystyle\\prod_{j=1}^{N}H_0^k(0,l_j),\\ k\\in \\mathbb{N}$$\n and\n $$H_r^k(0,l_j)=\\left\\{v \\in H^k(0,l_j),\\ \\partial_x^iv(l_j)=0,\\ 0\\leq i\\leq k-1\\right\\},\\ k=1,...,N,$$ \n where $v^{(m)}=\\frac{dv}{dx^m}$ and the index $r$ is related to the null right boundary conditions. In addition, \n $$\\mathbb{H}^k_r(\\mathcal{T})=\\prod_{j=1}^{N}H_r^k(0,l_j)\\quad \\text{and} \\quad \\|u\\|_{\\mathbb{H}^k_r(\\mathcal{T})}^2=\\sum_{j=1}^{n}\\|u_j\\|_{H^k(0,l_j)}^2\\ k \\in \\mathbb{N}.$$\n \\item[iv.] Consider the following characterization: \n $$H_r^{-1}(0,l_j)=\\left(H_r^{1}(0,l_j)\\right)'$$ \n as the dual space of $H_r(0,l_j)$ with respect to the pivot space $L^2(0,l_j)$ and $\\mathbb{H}_r^{-1}$ denotes the cartesian product of $H_r^{-1}(0,l_j)$. \n\\item[v.] For $k\\in \\mathbb{N}$, the derivative $\\partial_x^ku$ means the row vector\n\\begin{align*}\n\\partial_x^k u:=(\\partial_x^ku_1,...,\\partial_x^ku_N).\n\\end{align*}\nThe derivative $\\partial_t^ku$ is understood in the same way.\n\\item[vi.] Under the above notation, we have\n\\begin{align*}\n C^k\\left([0,T];\\mathbb{H}^s(\\mathcal{T})\\right)=\\prod_{j=1}^NC^k\\left([0,T];H^s(0,l_j)\\right),\\ k=0,1,2,... \\text{ \\ and \\ }s \\in \\mathbb{R}\n\\end{align*}\nas well as\n\\begin{align*}\n L^p\\left(0,T;\\mathbb{H}^s(\\mathcal{T})\\right)=\\prod_{j=1}^NL^p\\left(0,T;H^s(0,l_j)\\right),\\ p\\geq 1 \\text{ \\ and \\ }s \\in \\mathbb{R}.\n\\end{align*}\nHence, for $f \\in C^k\\left([0,T];\\mathbb{H}^s(\\mathcal{T})\\right)$ and $q \\in L^p\\left(0,T;\\mathbb{H}^s(\\mathcal{T})\\right)$ we can write\n\\begin{align*}\nf=(f_1,...,f_N)\\text{ \\ and \\ }q=(q_1,...,q_N)\n\\end{align*}\nwith\n\\begin{align*}\nf_j \\in C^k\\left([0,T];H^s(0,l_j)\\right)\\text{ \\ and \\ }q_j \\in L^p\\left(0,T;H^s(0,l_j)\\right),\\ j=1,...N.\n\\end{align*}\n\\item[vii.] We will consider the spaces\n$$\\mathbb{H}^k_e(\\mathcal{T})=\\left\\{u=(u_1,...,u_N)\\in \\mathbb{H}^k_r(\\mathcal{T});\\ u_j(0)=u_k(0),\\ j,k=1,...,N\\right\\}\\ k \\in \\mathbb{N},$$\n$$\\mathbb{H}^3_c(\\mathcal{T})=\\left\\{u=(u_1,...,u_N)\\in \\mathbb{H}^3(\\mathcal{T});\\ u_j(0)=u_k(0),\\ j,k=1,...,N\\right\\},$$\nand\n$$\\mathbb{B}_{s,T}=C([0,T],\\mathbb{H}^s(\\mathcal{T}))\\cap L^2(0,T,\\mathbb{H}^{s+1}(\\mathcal{T})),$$\nwith the norm \n$$\\displaystyle\\|u\\|_{\\mathbb{B}_{s,T}}:=\\|u\\|_{C([0,T],\\mathbb{H}^s(\\mathcal{T}))}+\\|u\\|_{L^2(0,T,\\mathbb{H}^{s+1}(\\mathcal{T}))}=\\max_{t\\in [0,T]}\\|u\\|_{\\mathbb{H}^s(\\mathcal{T})}+\\left(\\int_0^T\\|u(t,\\cdot)\\|_{\\mathbb{H}^{s+1}}^2dt\\right)^\\frac{1}{2}.$$\nWhen $s=0$ we write $\\mathbb{B}_T$ instead $\\mathbb{B}_{0,T}$.\n\\item [viii.] Once understand the appropriate spaces in each situation, we will use the notations $g=(g_1,...,g_N)$ and $p=(p_1,...,p_N)$.\n\\end{itemize}\n\n\\subsection{Main results and structure of the article} With the previous framework in hand, our goal here is to prove the local and global well-posedness for the initial boundary value problem (IBVP) \\eqref{kdV}. The first main result establishes the local well-posedness for this system in sharp spaces.\n\n\\subsection{Main results and structure of the article} With the previous framework in hand, our goal here is to prove the local and global well-posedness for the initial boundary value problem (IBVP) \\eqref{kdV}. The first main result establishes the local well-posedness for this system in sharp spaces.\n\n\\begin{lemma}\\label{lemma for v}\n Let $(u^0,g_0, g,p)\\in \\mathbb{X}_{3,T}$, $f\\in W^{1,1}(0,T;\\mathbb{L}^2(\\mathcal{T}))\\cap L^2(0,T;\\mathbb{H}^1(\\mathcal{T}))$ and $v$ the solution of \\eqref{v_j}. Then for $\\beta=0,1,2,3$ we have:\n \\begin{enumerate}\n \\item[i.] The function\n \\begin{align*}\n \\begin{array}{rcl}\n [0,T]&\\rightarrow&\\mathcal{D}'(0,l_j)\\\\\n s&\\mapsto&\\partial_x^\\beta v_j(s,\\cdot)\n \\end{array}\n \\end{align*}\n is integrable in $[0,T]$.\n \\item[ii.]For any $t \\in [0,T]$ we have $\\displaystyle\\int_0^tv_j(s,\\cdot)ds \\in H^4(0,l_j)$ and for $\\beta=0,1,2,3$ we have\n \\begin{align*}\n \\partial_x^\\beta\\int_0^tv_j(s,\\cdot)ds=\\int_0^t\\partial_x^\\beta v_j(s,\\cdot)ds.\n \\end{align*}\n \\end{enumerate}\n\\end{lemma}\nThe previous lemma can give us the regularity for $u$.\n\\begin{proposition}\\label{W.P._s=3}\nLet $(u^0,g_0,g,p) \\in \\mathbb{H}^3(\\mathcal{T})\\times H^{\\frac{3-1}{3}}(0,T)\\times [H^\\frac{3}{3}(0,T)]^N\\times [H^{\\frac{3+1}{3}}(0,T)]^N$ satisfying the linear compatibility conditions for $s=3$ and $f\\in W^{1,1}(0,T;\\mathbb{L}^2(\\mathcal{T}))\\cap L^2(0,T;\\mathbb{H}^1(\\mathcal{T}))$ be. For $j=1,...,N$ the function $t\\mapsto u_j(t,\\cdot)$ given by \\eqref{u_j} satisfies the following items:\n\\begin{enumerate}\n \\item[i.] $u_j \\in C^1([0,T],L^2(0,l_j))\\cap C([0,T];H^3(0,l_j))$;\n \\item[ii.] $u\\in \\mathbb{B}_{3,T}=C([0,T];\\mathbb{H}^3(\\mathcal{T}))\\cap L^2(0,T;\\mathbb{H}^4(\\mathcal{T}))$ is the unique classical solution to the problem \\eqref{LkdV}; and\n \\begin{equation*}\n \\begin{split}\n \\|u\\|_{\\mathbb{B}_{3,T}}\\leq &C\\left( \\|u^0\\|_{\\mathbb{H}^3(\\mathcal{T})}+\\|g_0\\|_{H^\\frac{3-1}{3}(0,T)}+\\|g\\|_{[H^\\frac{3}{3}(0,T)]^N}\\right.\\\\\n &\\left.+\\|p\\|_{[H^\\frac{3+1}{3}(0,T)]^N}+\\|f\\|_{W^{1,1}(0,T;\\mathbb{L}^2(\\mathcal{T}))\\cap L^2(0,T;\\mathbb{H}^1(\\mathcal{T}))}\\right)\n\\end{split}\n \\end{equation*}\n for some constant $C>0$.\n \\item[iii.] $u \\in H^1(0,T;\\mathbb{H}^1(\\mathcal{T}))$ and\n \\begin{equation*}\n \\begin{split}\n \\|u\\|_{H^1(0,T;\\mathbb{H}^1(\\mathcal{T}))}\\leq& C\\left(\\|u^0\\|_{\\mathbb{H}^3(\\mathcal{T})}+\\|g_0\\|_{H^\\frac{3-1}{3}(0,T)}+\\|g\\|_{[H^\\frac{3}{3}(0,T)]^N}\\right.\\\\ &\\left.+\\|p\\|_{[H^\\frac{3+1}{3}(0,T)]^N}+\\|f\\|_{W^{1,1}(0,T;\\mathbb{L}^2(\\mathcal{T}))\\cap L^2(0,T;\\mathbb{H}^1(\\mathcal{T}))}\\right),\n \\end{split}\n \\end{equation*}\n for some constant $C>0$.\n \\item [iv.] $\\partial_x^ku_j\\in L_x^\\infty\\left([0,l_j];H^\\frac{3+1-k}{3}(0,T)\\right)$ and \n \\begin{align*}\n \\|\\partial_x^ku_j\\|_{L_x^\\infty\\left([0,l_j];H^\\frac{3+1-k}{3}(0,T)\\right)}\\leq& C\\left(\\|u^0\\|_{\\mathbb{H}^3(\\mathcal{T})}+\\|g_0\\|_{H^\\frac{3-1}{3}(0,T)}+\\|g\\|_{[H^\\frac{3}{3}(0,T)]^N}\\right.\\\\ &\\left.+\\|p\\|_{[H^\\frac{3+1}{3}(0,T)]^N}+\\|f\\|_{W^{1,1}(0,T;\\mathbb{L}^2(\\mathcal{T}))\\cap L^2(0,T;\\mathbb{H}^1(\\mathcal{T}))}\\right).\n \\end{align*}\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nConsider $j \\in\\{1,...,N\\}$. \n\\vspace{0.2 cm}\n\\noindent\\textbf{i.} Since $v_j\\in C([0,T],L^2(0,l_j))$ it is immediate that by \\eqref{u_j} that $u_j \\in C^{1}([0,T],L^2(0,l_j))$.\n\n\\section{Local well-posedness of the nonlinear problem}\\label{sec4}\nNow, we are in a position to prove the first main result of the article, namely, the local well-posedness of the following problem\n\\begin{align}\\label{NLKdV}\n \\left\\{\n \\begin{array}{ll}\n \\partial_t u_j+\\partial_x [(a_j+1)u_j]+\\partial_x^3 u_j+u_j\\partial_x u_j=f_j,&t \\in (0,T),\\ x \\in (0,l_j),\\\\\n u_j(t,0)=u_1(t,0),&t\\in (0,T),\\\\\n \\displaystyle\\sum_{j=1}^N\\partial_x^2 u_j(t,0)=-\\alpha u_1(t,0)-\\frac{N}{3}(u_1+a_1)^2(t,0)+g_0(t),&t \\in (0,T)\\\\ \n u_j(t,l_j)=p_j(t),\\ \\ \\ \\ \\ \\partial_x u_j(t,l_j)=g_j(t),&t\\in (0,T),\\\\\n u_j(0,x)=u_j^0(x),&x \\in (0,l_j),\n \\end{array}\n \\right.\n\\end{align}\nwhere $j=1,...,N$ and $a$ is a given function. Remember that: for $s\\geq 0$ and $T>0$ the space\n\\begin{align*}\n\\mathcal{X}_{s,T}=\\left\\{v \\in \\mathbb{B}_{s,T}^*;\\ \\partial_x^ku_j\\in L_x^\\infty(0,l_j;H^\\frac{s+1-k}{3}(0,T)),\\ k=0,1,2,\\ j=1,2,...,N\\right\\}\n\\end{align*}\nis a Banach space with the norm\n\\begin{align*}\n\\|v\\|_{\\mathcal{X}_{s,T}}=\\|v\\|_{\\mathbb{B}_{s,T}^*}+\\sum_{k=0}^2\\|\\partial_x^kv\\|_{\\prod_{j=1}^NL_x^\\infty(0,l_j;H^\\frac{s+1-k}{3}(0,T))}.\n\\end{align*}\nNote that \\eqref{kdV} is a particular case of \\eqref{NLKdV} with $a=0$.\n\\begin{proof}[Proof of Theorem \\ref{main}] Fix $\\theta \\in (0,T)$ and define $\\Gamma_a:\\mathcal{X}_{s,\\theta}\\rightarrow \\mathcal{X}_{s,\\theta}$ putting, for each $u \\in \\mathcal{X}_{s,\\theta}$, $\\Gamma_au$ as being the solution $v$ of\n\\begin{align*}\n \\left\\{\n \\begin{array}{ll}\n \\partial_t v_j+\\partial_x [(a_j+1)v_j]+\\partial_x^3 v_j=-u_j\\partial_x u_j+f_j,&t \\in (0,\\theta),\\ x \\in (0,l_j),\\ j=1,...,N,\\\\\n v_j(t,0)=v_1(t,0),&t\\in (0,\\theta),\\ j=1,...,N,\\\\\n \\displaystyle\\sum_{j=1}^N\\partial_x^2 v_j(t,0)=-\\alpha v_1(t,0)-\\frac{N}{3}(u_1+a_1)^2(t,0)+g_0(t),&t \\in (0,\\theta,)\\\\ \n v_j(t,l_j)=p_j(t),\\ \\ \\ \\ \\ \\partial_x v_j(t,l_j)=g_j(t),&t\\in (0,\\theta),\\ j=1,...,N,\\\\\n v_j(0,x)=u_j^0(x),&x \\in (0,l_j),\\ j=1,...,N.\n \\end{array}\n \\right.\n\\end{align*}\nFrom Proposition \\ref{W.P.LKdV-a} we have\n\\begin{align*}\n& \\|\\Gamma_au\\|_{\\mathcal{X}_{s,\\theta}}\\leq C\\left(\\|u^0\\|_{\\mathbb{H}^s(\\mathcal{T})}+\\|g_0-\\frac{N}{3}(u_1+a_1)^2(\\cdot,0)\\|_{H^\\frac{s-1}{3}(0,\\theta)}+\\|g\\|_{[H^\\frac{s}{3}(0,\\theta)]^N}\\right.\\\\\n&\\left.+\\|p\\|_{[H^\\frac{s+1}{3}(0,\\theta)]^N}+\\|u\\partial_xu\\|_{W^{\\frac{s}{3},1}(0,\\theta;\\mathbb{L}^2(\\mathcal{T}))\\cap L^\\frac{6}{6-s}(0,\\theta;\\mathbb{H}^\\frac{s}{3}(\\mathcal{T}))}+\\|f\\|_{W^{\\frac{s}{3},1}(0,\\theta;\\mathbb{L}^2(\\mathcal{T}))\\cap L^\\frac{6}{6-s}(0,\\theta;\\mathbb{H}^\\frac{s}{3}(\\mathcal{T}))} \\right).\n\\end{align*}\nUsing Lemma \\ref{lemma 2.4} we obtain\n\\begin{align*}\n \\|\\partial_xu v\\|_{W^{\\frac{s}{3},1}(0,\\theta;\\mathbb{L}^2(\\mathcal{T}))\\cap L^\\frac{6}{6-s}(0,\\theta;\\mathbb{H}^\\frac{s}{3}(\\mathcal{T}))}\\leq C(\\theta^\\frac{1}{2}+\\theta^\\frac{1}{3})\\|u\\|_{\\mathbb{B}_{s,\\theta}^*}\\|u\\|_{\\mathbb{B}_{s,\\theta}^*}.\n\\end{align*}\nFurthermore, Lemma \\ref{lemma 2.1} gives us\n\\begin{align*}\n\\left\\|\\frac{N}{3}(u_1+a_1)^2(\\cdot,0)\\right\\|_{H^\\frac{s-1}{3}(0,\\theta)}&\\leq C\\theta^\\alpha\\frac{N}{3}\\left\\|(u_1+a_1)(\\cdot,0)\\right\\|_{H^\\frac{s+1}{3}(0,\\theta)}^2\\\\\n&\\leq C\\theta^\\alpha\\frac{N}{3}\\|u+a\\|_{\\mathcal{X}_{s,\\theta}}^2\\\\\n&\\leq \\frac{4}{3}CN\\theta^\\alpha \\left(\\|u\\|_{\\mathcal{X}_{s,\\theta}}^2+\\|a\\|_{\\mathcal{X}_{s,\\theta}}^2\\right),\n\\end{align*}\nwhere $\\alpha>0$ is a constant. Thus,\n\\begin{align*}\n \\|\\Gamma_au\\|_{\\mathcal{X}_{s,\\theta}}\\leq& C\\left(\\|u^0\\|_{\\mathbb{H}^s(\\mathcal{T})}+\\|g_0\\|_{H^\\frac{s-1}{3}(0,T)}+\\|g\\|_{[H^\\frac{s}{3}(0,T)]^N}\\right.\\\\\n&+\\|p\\|_{[H^\\frac{s+1}{3}(0,T)]^N}+\\|f\\|_{W^{\\frac{s}{3},1}(0,T;\\mathbb{L}^2(\\mathcal{T}))\\cap L^\\frac{6}{6-s}(0,T;\\mathbb{H}^\\frac{s}{3}(\\mathcal{T}))}\\\\\n&\\left.+C(\\theta^\\frac{1}{2}+\\theta^\\frac{1}{3})\\|u\\|_{\\mathcal{X}_{s,\\theta}}^2+\\frac{4}{3}CN\\theta^\\alpha \\|u\\|_{\\mathcal{X}_{s,\\theta}}^2+\\frac{4}{3}CN\\theta^\\alpha\\|a\\|_{\\mathcal{X}_{s,\\theta}}^2 \\right).\n\\end{align*}\nDefining\n\\begin{align*}\nr=&4C\\left(\\|u^0\\|_{\\mathbb{H}^s(\\mathcal{T})}+\\|g_0\\|_{H^\\frac{s-1}{3}(0,T)}+\\|g\\|_{[H^\\frac{s}{3}(0,T)]^N}\\right.\\\\\n&\\left.+\\|p\\|_{[H^\\frac{s+1}{3}(0,T)]^N}+\\|f\\|_{W^{\\frac{s}{3},1}(0,T;\\mathbb{L}^2(\\mathcal{T}))\\cap L^\\frac{6}{6-s}(0,T;\\mathbb{H}^\\frac{s}{3}(\\mathcal{T}))}\\right),\n\\end{align*}\nit follows that, for $u \\in B=\\{v \\in \\mathcal{X}_{s,\\theta};\\ \\|v\\|_{\\mathcal{X}_{s,\\theta}}\\leq r\\}$,\n\\begin{align}\\label{eq 3.2}\n\\|\\Gamma_au\\|_{\\mathcal{X}_{s,\\theta}}\\leq \\frac{r}{4}+C^2(\\theta^\\frac{1}{2}+\\theta^\\frac{1}{3})r^2+\\frac{4}{3}C^2N\\theta^\\alpha r^2+\\frac{4}{3}C^2N\\theta^\\alpha\\|a\\|_{\\mathcal{X}_{s,T}}^2.\n\\end{align}", "post_theorem_intro_text_len": 2359, "post_theorem_intro_text": "The proof of the previous theorem relies on several analytical tools combined with a fixed-point argument. In addition, a new notion of $s$-compatibility adapted to the graph structure is introduced (see Section \\ref{sec2}), inspired by the single-edge KdV framework in \\cite{BSZ 2003}. This theorem provides a sharp local well-posedness result, along with regularity properties, for the associated functions. Furthermore, we establish the existence of global solutions at the $H^s$-norm, which enables applications to control problems on star graphs and resolves several open questions that will be addressed in a forthcoming paper. The second main result of this work can be stated as follows.\n\n\\begin{theorem}\\label{global_(0,3)}\nLet $T>0$ and $\\lambda\\geq 1$ be. Consider $(u^0,g_0,g,p)\\in \\mathbb{X}_{s,T}^\\lambda$, where $\\mathbb{X}_{s,T}^\\lambda$ is the set of $s$-compatible data\n$$u^0 \\in \\mathbb{H}^{s}(\\mathcal{T}), g_0 \\in H^{\\frac{s+\\lambda-1}{3}}(0,T),\\ g \\in [H^\\frac{s}{3}(0,T)]^N\\quad \\text{and}\\quad p\\in [H^\\frac{s+\\lambda+1}{3}(0,T)]^N.$$\nThus, the problem \\eqref{kdV} has a unique solution $u\\in \\mathcal{X}_{s,T}$ which satisfies\n\\begin{align*}\n\t\\|u\\|_{\\mathcal{X}_{s,T}}&\\leq C\\left(\\|(u^0,g_0,g,p)\\|_{0,\\lambda}\\right)\\left(\n\t\\begin{array}{c}\n\t\t\\|u^0\\|_{\\mathbb{H}^s(\\mathcal{T})}+\\|g_0\\|_{H^\\frac{s+\\lambda-1}{3}(0,T)}+\\|g\\|_{[H^\\frac{s}{3}(0,T)]^N}+\\|p\\|_{[H^\\frac{s+\\lambda+1}{3}(0,T)]^N}\n\t\\end{array}\n\t\\right)\n\\end{align*}\nwhere $C>0$ is a positive constant. \n\\end{theorem}\nTogether, these two theorems provide, for the first time, a comprehensive framework for the well-posedness of general graph structures with boundary controls. Moreover, they extend the classical result for a single KdV equation \\cite{BSZ 2003} to star-shaped graphs composed of $N$ equations.\n\\vspace{0.2cm}\n\nThe paper is organized as follows. In Section \\ref{sec2}, we introduce a general notion of $s$-compatibility for star graphs. Building on this framework, Section \\ref{sec3} addresses the well-posedness of the linear system associated with \\eqref{kdV}. After that, Section \\ref{sec4} is devoted to proving the local well-posedness result for the system \\eqref{kdV}, giving the proof of Theorem \\ref{main}. The second main result, concerning global well-posedness, in the $H^s$-regime, for the system \\eqref{kdV} is established in Section \\ref{sec6}.", "sketch": "To prove Theorem~\\ref{main}, the introduction states that it \"relies on several analytical tools combined with a fixed-point argument.\" It also mentions that \"a new notion of $s$-compatibility adapted to the graph structure is introduced\" (Section~\\ref{sec2}), \"inspired by the single-edge KdV framework in \\cite{BSZ 2003}.\" The paper’s organization further indicates the proof route: first introduce $s$-compatibility on star graphs (Section~\\ref{sec2}); then treat \"the well-posedness of the linear system associated with \\eqref{kdV}\" (Section~\\ref{sec3}); and then Section~\\ref{sec4} \"is devoted to proving the local well-posedness result for the system \\eqref{kdV}, giving the proof of Theorem~\\ref{main}.\"", "expanded_sketch": "To prove the main theorem, the introduction states that it “relies on several analytical tools combined with a fixed-point argument.” It also mentions that “a new notion of $s$-compatibility adapted to the graph structure is introduced” (Section~\\ref{sec2}), “inspired by the single-edge KdV framework in \\cite{BSZ 2003}.” The paper’s organization further indicates the proof route: first introduce $s$-compatibility on star graphs (Section~\\ref{sec2}); then treat “the well-posedness of the linear system associated with\n\\begin{align}\\label{kdV}\n\t\t\t\\begin{cases}\n\t\t\t\t\\partial_t u_j+\\partial_x u_j+\\partial_x^3 u_j+u_j\\partial_x u_j=0,&t \\in (0,T),\\ x \\in (0,l_j),\\ j=1,...,N\\\\\n\t\t\t\tu_j(t,0)=u_1(t,0),&t\\in (0,T),\\ j=1,...,N\\\\\n\t\t\t\t\\displaystyle\\sum_{j=1}^N\\partial_x^2 u_j(t,0)=-\\alpha u_1(t,0)-\\frac{N}{3}u_1^2(t,0)+g_0(t),&t \\in (0,T)\\\\\t\t\t\t\n\t\t\t\tu_j(t,l_j)=p_j(t),\\ \\ \\ \\ \\ \\partial_x u_j(t,l_j)=g_j(t),&t\\in (0,T),\\ j=1,...,N\\\\\n\t\t\t\tu_j(0,x)=u_j^0(x),&x \\in (0,l_j),\\ j=1,...,N\n\t\t\t\\end{cases}\n\t\t\\end{align}\n” (Section~\\ref{sec3}); and then Section~\\ref{sec4} “is devoted to proving the local well-posedness result for the system above, giving the proof of the main theorem.”", "expanded_theorem": "\\label{main} Let $T>0$, $l_j>0,\\ j=1,...,N$, $s\\in [0,3]$ be given. Suppose $f\\in W^{\\frac{s}{3},1}(0,T;\\mathbb{L}^2(\\mathcal{T}))\\cap L^{\\frac{6}{6-s}}(0,T;\\mathbb{H}^\\frac{s}{3}(\\mathcal{T}))$ and \n\\begin{align*}\n(u^0,g_0,g,p)\\in \\mathbb{H}^s(\\mathcal{T})\\times H^{\\frac{s-1}{3}}(0,T)\\times \\left[H^\\frac{s}{3}(0,T)\\right]^{N}\\times \\left[H^{\\frac{s+1}{3}}(0,T)\\right]^N.\n\\end{align*}\nsatisfying the $s$-compatibility conditions, thus there exists $T^*\\in (0,T)$ such that the IBVP\n\\begin{align}\\label{kdV}\n\t\t\t\\begin{cases}\n\t\t\t\t\\partial_t u_j+\\partial_x u_j+\\partial_x^3 u_j+u_j\\partial_x u_j=0,&t \\in (0,T),\\ x \\in (0,l_j),\\ j=1,...,N\\\\\n\t\t\t\tu_j(t,0)=u_1(t,0),&t\\in (0,T),\\ j=1,...,N\\\\\n\t\t\t\t\\displaystyle\\sum_{j=1}^N\\partial_x^2 u_j(t,0)=-\\alpha u_1(t,0)-\\frac{N}{3}u_1^2(t,0)+g_0(t),&t \\in (0,T)\\\\\t\t\t\t\n\t\t\t\tu_j(t,l_j)=p_j(t),\\ \\ \\ \\ \\ \\partial_x u_j(t,l_j)=g_j(t),&t\\in (0,T),\\ j=1,...,N\\\\\n\t\t\t\tu_j(0,x)=u_j^0(x),&x \\in (0,l_j),\\ j=1,...,N\n\t\t\t\\end{cases}\n\t\t\\end{align}\nadmits a unique solution $u \\in \\mathcal{X}_{s,T^*}$, where \n$$\\mathcal{X}_{s,T}=\\left\\{v \\in \\mathbb{B}^{*}_{s,T};\\ \\partial_x^ku_j\\in L_x^\\infty(0,l_j;H^\\frac{s+1-k}{3}(0,T)),\\ k=0,1,2,\\ j=1,2,...,N\\right\\},$$\n\\begin{align*}\n\t\\mathbb{B}_{s,T}^*:=\\mathbb{B}_{s,T}\\cap H^\\frac{s}{3}(0,T;\\mathbb{H}^1(\\mathcal{T})).\n\\end{align*}", "theorem_type": ["Existential–Universal", "Uniqueness"], "mcq": {"question": "Let \\(T>0\\), \\(l_j>0\\) for \\(j=1,\\dots,N\\), and \\(s\\in[0,3]\\). Let \\(\\mathcal T\\) be the star-shaped network with edges \\((0,l_j)\\), and write \\(\\mathbb L^2(\\mathcal T)=\\prod_{j=1}^N L^2(0,l_j)\\) and \\(\\mathbb H^s(\\mathcal T)=\\prod_{j=1}^N H^s(0,l_j)\\). Assume\n\\[\nf\\in W^{\\frac{s}{3},1}(0,T;\\mathbb L^2(\\mathcal T))\\cap L^{\\frac{6}{6-s}}(0,T;\\mathbb H^{\\frac{s}{3}}(\\mathcal T)),\n\\]\nand\n\\[\n(u^0,g_0,g,p)\\in \\mathbb H^s(\\mathcal T)\\times H^{\\frac{s-1}{3}}(0,T)\\times [H^{\\frac{s}{3}}(0,T)]^N\\times [H^{\\frac{s+1}{3}}(0,T)]^N\n\\]\nsatisfy the \\(s\\)-compatibility conditions. Consider, for \\(j=1,\\dots,N\\), the initial-boundary value problem\n\\[\n\\begin{cases}\n\\partial_t u_j+\\partial_x u_j+\\partial_x^3 u_j+u_j\\partial_x u_j=0,& t\\in(0,T),\\ x\\in(0,l_j),\\\\\nu_j(t,0)=u_1(t,0),& t\\in(0,T),\\\\\n\\displaystyle\\sum_{j=1}^N \\partial_x^2 u_j(t,0)=-\\alpha u_1(t,0)-\\frac{N}{3}u_1^2(t,0)+g_0(t),& t\\in(0,T),\\\\\nu_j(t,l_j)=p_j(t),\\quad \\partial_x u_j(t,l_j)=g_j(t),& t\\in(0,T),\\\\\nu_j(0,x)=u_j^0(x),& x\\in(0,l_j).\n\\end{cases}\n\\]\nAlso set\n\\[\n\\mathbb B_{s,\\tau}=C([0,\\tau],\\mathbb H^s(\\mathcal T))\\cap L^2(0,\\tau;\\mathbb H^{s+1}(\\mathcal T)),\n\\]\n\\[\n\\mathbb B^*_{s,\\tau}=\\mathbb B_{s,\\tau}\\cap H^{\\frac{s}{3}}(0,\\tau;\\mathbb H^1(\\mathcal T)),\n\\]\nand\n\\[\n\\mathcal X_{s,\\tau}=\\left\\{v\\in \\mathbb B^*_{s,\\tau}:\\ \\partial_x^k v_j\\in L_x^{\\infty}(0,l_j;H^{\\frac{s+1-k}{3}}(0,\\tau)),\\ k=0,1,2,\\ j=1,\\dots,N\\right\\}.\n\\]\nWhich statement about existence and uniqueness is valid for this problem?", "correct_choice": {"label": "A", "text": "There exists \\(T^*\\in(0,T)\\) such that the above initial-boundary value problem admits a unique solution \\(u\\in \\mathcal X_{s,T^*}\\)."}, "choices": [{"label": "B", "text": "There exists \\(T^*\\in(0,T)\\), depending only on \\(T,s,N\\) and the lengths \\(l_j\\), such that for every admissible \\((u^0,g_0,g,p,f)\\) satisfying the \\(s\\)-compatibility conditions, the above initial-boundary value problem admits a unique solution \\(u\\in \\mathcal X_{s,T^*}\\)."}, {"label": "C", "text": "There exists \\(T^*\\in(0,T)\\) such that the above initial-boundary value problem admits at most one solution in \\(\\mathcal X_{s,T^*}\\)."}, {"label": "D", "text": "For every \\(T>0\\), the above initial-boundary value problem admits a unique solution \\(u\\in \\mathcal X_{s,T}\\)."}, {"label": "E", "text": "There exists \\(T^*\\in(0,T)\\) such that the above initial-boundary value problem admits a unique solution \\(u\\in \\mathbb B^*_{s,T^*}\\)."}], "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": "dependence_of_local_time_on_data_size", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_existence_retaining_uniqueness", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "local_vs_global_time_interval", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "solution_space_X_vs_Bstar_trace_regularity", "template_used": "wildcard"}]}}
{"id": "2512.15682v1", "paper_link": "http://arxiv.org/abs/2512.15682v1", "theorems_cnt": 1, "theorem": {"env_name": "thm", "content": "\\label{thm:1.1}\nGiven $\\varphi\\in C(\\partial\\Omega)$, there exist a unique harmonic function $u_\\varphi$ on $\\Omega$ and an extension $\\Phi\\in C(\\overline{\\Omega})$ of $\\varphi$ such that $\\Phi-u_\\varphi\\in H^1_0(\\Omega)$.", "start_pos": 993002, "end_pos": 993229, "label": "thm:1.1"}, "ref_dict": {"thm:1.1": "\\begin{thm}\\label{thm:1.1}\nGiven $\\phi\\in C(\\partial\\Omega)$, there exist a unique harmonic function $u_\\phi$ on $\\Omega$ and an extension $\\Phi\\in C(\\clos{\\Omega})$ of $\\phi$ such that $\\Phi-u_\\phi\\in H^1_0(\\Omega)$.\n\\end{thm}", "eq:D": "\\begin{equation}\n  \\tag*{$D(\\varphi)$}\n  \\label{eq:D}\n  \\left\\{\n    \\begin{aligned}\n      &\\text{find $u\\in C(\\clos{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n      &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\phi$.}\n    \\end{aligned}\n  \\right.\n\\end{equation}"}, "pre_theorem_intro_text_len": 764, "pre_theorem_intro_text": "Throughout we suppose that $\\Omega$ is a connected bounded open subset of $\\mathbb{R}^d$, $d\\ge 2$, with boundary $\\partial\\Omega$.\nGiven a function $\\varphi\\in C(\\partial\\Omega)$, the \\textbf{\\boldmath Dirichlet problem\\unboldmath} asks to\n\\begin{equation}\n  \\tag*{$D(\\varphi)$}\n  \\label{eq:D}\n  \\left\\{\n    \\begin{aligned}\n      &\\text{find $u\\in C(\\overline{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n      &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\varphi$.}\n    \\end{aligned}\n  \\right.\n\\end{equation}\nWe call such a function $u$ a \\textbf{\\boldmath classical solution\\unboldmath} of~\\ref{eq:D}.\nHowever, a classical solution does not always exist.\n\nIn this paper we consider a generalised solution of the Dirichlet problem which is described as follows.", "context": "Throughout we suppose that $\\Omega$ is a connected bounded open subset of $\\mathbb{R}^d$, $d\\ge 2$, with boundary $\\partial\\Omega$.\nGiven a function $\\varphi\\in C(\\partial\\Omega)$, the \\textbf{\\boldmath Dirichlet problem\\unboldmath} asks to\n\\begin{equation}\n \\tag*{$D(\\varphi)$}\n \\label{eq:D}\n \\left\\{\n \\begin{aligned}\n &\\text{find $u\\in C(\\overline{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\varphi$.}\n \\end{aligned}\n \\right.\n\\end{equation}\nWe call such a function $u$ a \\textbf{\\boldmath classical solution\\unboldmath} of~\\ref{eq:D}.\nHowever, a classical solution does not always exist.\n\nIn this paper we consider a generalised solution of the Dirichlet problem which is described as follows.\n\n\\begin{equation}\n  \\tag*{$D(\\varphi)$}\n  \\label{eq:D}\n  \\left\\{\n    \\begin{aligned}\n      &\\text{find $u\\in C(\\clos{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n      &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\phi$.}\n    \\end{aligned}\n  \\right.\n\\end{equation}", "full_context": "Throughout we suppose that $\\Omega$ is a connected bounded open subset of $\\mathbb{R}^d$, $d\\ge 2$, with boundary $\\partial\\Omega$.\nGiven a function $\\varphi\\in C(\\partial\\Omega)$, the \\textbf{\\boldmath Dirichlet problem\\unboldmath} asks to\n\\begin{equation}\n \\tag*{$D(\\varphi)$}\n \\label{eq:D}\n \\left\\{\n \\begin{aligned}\n &\\text{find $u\\in C(\\overline{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\varphi$.}\n \\end{aligned}\n \\right.\n\\end{equation}\nWe call such a function $u$ a \\textbf{\\boldmath classical solution\\unboldmath} of~\\ref{eq:D}.\nHowever, a classical solution does not always exist.\n\nIn this paper we consider a generalised solution of the Dirichlet problem which is described as follows.\n\n\\begin{equation}\n  \\tag*{$D(\\varphi)$}\n  \\label{eq:D}\n  \\left\\{\n    \\begin{aligned}\n      &\\text{find $u\\in C(\\clos{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n      &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\phi$.}\n    \\end{aligned}\n  \\right.\n\\end{equation}\n\nThroughout we suppose that $\\Omega$ is a connected bounded open subset of $\\RR^d$, $d\\ge 2$, with boundary $\\partial\\Omega$.\nGiven a function $\\phi\\in C(\\partial\\Omega)$, the \\emphdef{Dirichlet problem} asks to\n\\begin{equation}\n \\tag*{$D(\\varphi)$}\n \\label{eq:D}\n \\left\\{\n \\begin{aligned}\n &\\text{find $u\\in C(\\clos{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\phi$.}\n \\end{aligned}\n \\right.\n\\end{equation}\nWe call such a function $u$ a \\emphdef{classical solution} of~\\ref{eq:D}.\nHowever, a classical solution does not always exist.\n\nIn this section we develop variational solutions of the Dirichlet problem on the basis of the following result~\\cite[Corollary~4.3]{AtES24}.\n\\begin{thm}[Extension result]\\label{thm:2.1}\nLet $\\phi\\in C(\\partial\\Omega)$. Then there exist an extension $\\Phi\\in C(\\clos{\\Omega})$ of $\\phi$ and $M \\ge 0$ such that\n\\begin{equation}\\label{eq:2.1}\n \\abs*{\\int_\\Omega\\Phi\\Delta w\\dx} \\le M\\norm{w}_{H^1(\\Omega)}\n\\end{equation}\nfor all $w\\in C^\\infty_\\cpt(\\Omega)$. This means that $\\Delta\\Phi\\in H^{-1}(\\Omega)$, the dual space of $H^1_0(\\Omega)$.\n\\end{thm}\nNote that for $u\\in H^1(\\Omega)$ one has\n\\[\n \\abs*{\\int_\\Omega u\\Delta w\\dx} = \\abs*{-\\int_\\Omega\\nabla u\\nabla w\\dx}\\le\\norm{u}_{H^1(\\Omega)}\\norm{w}_{H^1(\\Omega)}\n\\]\nfor all $w\\in C^\\infty_\\cpt(\\Omega)$ and therefore $\\Delta u\\in H^{-1}(\\Omega)$.\n\nLet $\\phi\\in C(\\partial\\Omega)$ and $\\Phi\\in C(\\clos{\\Omega})$ be an extension of $\\phi$ satisfying~\\eqref{eq:2.1}.\nBy the Riesz--Fréchet representation theorem there exists a unique $v\\in H^1_0(\\Omega)$ such that\n\\[\n \\int_\\Omega \\Phi\\Delta w\\dx = -\\int_\\Omega\\nabla v\\nabla w\\dx = \\int_\\Omega v\\Delta w\\dx\n\\]\nfor all $w\\in C^\\infty_\\cpt(\\Omega)$. This means that $u:=\\Phi-v$ satisfies $\\Delta u=0$ weakly. Hence $u\\in C^\\infty(\\Omega)$ by Weyl's theorem and $\\Delta u=0$ in the classical sense;\nthat is, $u$ is a harmonic function.\nWe call $u$ the \\emphdef{variational solution} of~\\ref{eq:D} and set $u_\\phi := u$.\nTo do so, we have to show that the function does not depend on the extension $\\Phi$ of $\\phi$. This follows from the next lemma.\n\nNext we talk about the Dirichlet principle.\nWe start by mentioning that the variational solution $u_\\phi$ is obtained by a minimising process.\nIn fact, let $\\phi\\in C(\\partial\\Omega)$ and $\\Phi\\in C(\\clos{\\Omega})$ be an extension of $\\phi$ satisfying the estimate~\\eqref{eq:2.1}, i.e.\\ $\\Delta\\Phi\\in H^{-1}(\\Omega)$.\nThe solution $v\\in H^1_0(\\Omega)$ of $\\Delta v=\\Delta\\Phi$ is the unique function $v\\in H^1_0(\\Omega)$ such that\n\\begin{equation}\\label{eq:2.5}\n \\frac{1}{2}\\int_\\Omega\\abs{\\nabla v}^2\\dx + \\langle\\Delta\\Phi,v\\rangle = \\min\\Bigl\\{\\frac{1}{2}\\int_\\Omega\\abs{\\nabla w}^2\\dx + \\langle\\Delta\\Phi,w\\rangle \\setcolon w\\in H^1_0(\\Omega)\\Bigr\\}\n\\end{equation}\n(see e.g.~\\cite[Theorem~4.24]{AU23}).\nNow we can characterise when $\\int_\\Omega\\abs{\\nabla u_\\phi}^2\\dx<\\infty$; i.e.\\ when $u_\\phi$ has finite energy.\nSince $u_\\phi\\in C^\\infty(\\Omega)\\cap C_\\bdd(\\Omega)$, this is equivalent to saying that $u_\\phi\\in H^1(\\Omega)$.\n\n\\begin{thm}\\label{thm:2.7}\nLet $\\phi\\in C(\\partial\\Omega)$. The following statements are equivalent.\n\\begin{romanenum}\n\\item\\label{en:t2.7-i}\nThe variational solution $u_\\phi$ of~\\ref{eq:D} is an element of $H^1(\\Omega)$.\n\\item\\label{en:t2.7-ii}\nThere exists $\\Phi\\in C(\\clos{\\Omega})\\cap H^1(\\Omega)$ with $\\restrict{\\Phi}{\\partial\\Omega}=\\phi$.\n\\end{romanenum}\nIn that case\n\\[\n \\int_\\Omega\\abs{\\nabla u_\\phi}^2\\dx = \\min\\Bigl\\{\\int_\\Omega\\abs{\\nabla w}^2\\dx \\setcolon w\\in H^1(\\Omega),\\ \\Phi-w\\in H^1_0(\\Omega)\\Bigr\\}.\n\\]\n\\end{thm}\n\\begin{proof}\n\\begin{parenum}\n\\item[\\ref{en:t2.7-ii}$\\Rightarrow$\\ref{en:t2.7-i}.]\nSince $\\Phi\\in H^1(\\Omega)$, $u_\\phi=\\Phi-v\\in H^1(\\Omega)$, where $v\\in H^1_0(\\Omega)$ is such that $\\Delta v=\\Delta\\Phi$.\nMoreover, by~\\eqref{eq:2.5} and considering $W:=\\{w\\in H^1(\\Omega)\\setcolon \\Phi-w\\in H^1_0(\\Omega)\\}$,\n\\begin{align*}\n \\frac{1}{2}[u_\\phi,u_\\phi] -\\frac{1}{2}[\\Phi,\\Phi] &= \\frac{1}{2}[v,v]-[\\Phi,v] \\\\\n &= \\min\\Bigl\\{\\frac{1}{2}[\\Phi-w,\\Phi-w] - [\\Phi,\\Phi-w] \\setcolon w\\in W\\Bigr\\} \\\\\n &= \\min\\Bigl\\{\\frac{1}{2}[w,w] - \\frac{1}{2}[\\Phi,\\Phi] \\setcolon w\\in W\\Bigr\\} \\\\\n &= \\frac{1}{2} \\min\\Bigl\\{[w,w] \\setcolon w\\in W\\Bigr\\}-\\frac{1}{2}[\\Phi,\\Phi].\n\\end{align*}\nThus $[u_\\phi,u_\\phi] = \\min\\{[w,w]\\setcolon w\\in W\\}$ and this property characterises $u_\\phi$.\n\\item[\\ref{en:t2.7-i}$\\Rightarrow$\\ref{en:t2.7-ii}.]\nLet $\\Phi\\in C(\\clos{\\Omega})$ be such that $\\restrict{\\Phi}{\\partial\\Omega}=\\phi$ and $\\Delta\\Phi\\in H^{-1}(\\Omega)$.\nLet $v\\in H^1_0(\\Omega)$ be such that $\\Delta v=\\Delta\\Phi$.\nThen $u_\\phi=\\Phi-v$.\nSince $u_\\phi\\in H^1(\\Omega)$ by hypothesis, it follows that $\\Phi\\in H^1(\\Omega)$.\\qedhere\n\\end{parenum}\n\\end{proof}\n\nIf $\\Omega$ is more regular, the following holds.\n\\begin{thm}\\label{thm:2.9}\nAssume that $\\Omega$ has Lipschitz boundary and let $\\phi\\in C(\\partial\\Omega)$. The following statements are equivalent.\n\\begin{romanenum}\n\\item $u_\\phi\\in H^1(\\Omega)$\n\\item $\\phi\\in H^{1/2}(\\partial\\Omega)$\n\\end{romanenum}\n\\end{thm}\nHere, since $\\Omega$ has Lipschitz boundary, $H^{1/2}(\\partial\\Omega)=\\{\\operatorname{tr}\\Phi \\setcolon \\Phi\\in H^1(\\Omega)\\}$, where $\\operatorname{tr}\\colon H^1(\\Omega)\\to L^2(\\partial\\Omega,\\mathcal{H}^{d-1})$ is the trace operator and $\\mathcal{H}^{d-1}$ denotes the $(d-1)$-dimensional Hausdorff measure.\nIf $\\phi\\in C(\\partial\\Omega)\\cap H^{1/2}(\\partial\\Omega)$, there even exists a $\\Phi\\in C(\\clos{\\Omega})\\cap H^1(\\Omega)$ such that $\\restrict{\\Phi}{\\partial\\Omega}=\\phi$.\nWe refer to~\\cite[Theorem~1.2]{AtE19:dp} for this and a proof of Theorem~\\ref{thm:2.9}.\nA more general result on very rough domains is~\\cite[Theorem~5.8]{AtES24}.\n\nTheorem~\\ref{thm:5.1} is contained in Landkof~\\cite[p.\\,243, lines 1--5]{La72}.\nWe will give a very different proof based on the following result. We assume that $d\\ge 3$.\n\\begin{thm}\\label{thm:5.2}\nLet $z_0\\in \\partial\\Omega$ be a singular point of $\\Omega$.\nThen there exists an open set $\\Omega_0\\subset\\RR^d$ such that $\\clos{\\Omega_0}\\subset\\Omega\\cup\\{z_0\\}$, $z_0\\in\\partial\\Omega_0$ and $z_0$ is a singular point of $\\Omega_0$.\n\\end{thm}\nWe postpone the proof of Theorem~\\ref{thm:5.2} but note that in the case of Lebesgue's domain it readily follows from its construction.\nMore precisely, given the potential $V\\colon D\\to(0,\\infty)$ from Lemma~\\ref{lem:lebesgue-potential-basics}, we had $\\Omega=\\{x\\in D\\setcolon \\frac{1}{2}<V(x)<2\\}$. We can choose e.g.~$\\Omega_0=\\{x\\in D\\setcolon \\frac{1}{c}<V(r,z)<c\\}$ for any $c\\in (1,2)$.\nNow we can easily prove Theorem~\\ref{thm:5.1} using the following criterion for a boundary point to be regular, see~\\cite[Theorem~6.74]{AU23}.\n\\begin{prop}\\label{prop:5.4}\nLet $z_0\\in \\partial\\Omega$. Then $z_0$ is regular if there exists a \\emphdef{global $H^1$-barrier} $b$ at $z_0$; i.e.\\ a function $b\\in C(\\clos{\\Omega})\\cap H^1(\\Omega)$ such that $b(x)>0$ for all $x\\in\\clos{\\Omega}\\setminus\\{z_0\\}$, $b(z_0)=0$ and $-\\Delta b\\ge 0$ weakly on $\\Omega$.\n\\end{prop}\n\\begin{proof}[Proof of Theorem~\\ref{thm:5.1}]\nLet $z_0\\in \\partial\\Omega$ be a singular point for $\\Omega$.\nBy Theorem~\\ref{thm:5.2} there exists an open set $\\Omega_0\\subset\\RR^d$ such that $\\clos{\\Omega_0}\\subset\\Omega\\cup\\{z_0\\}$ and $z_0\\in\\partial\\Omega_0$ is a singular point with respect to $\\Omega_0$.\nLet $0\\le\\phi\\in C(\\partial\\Omega)$ be such that $\\phi(z_0)=0$ and $\\phi(w)>0$ at a regular point $w\\in \\partial\\Omega$.\nThen there exists $0\\le\\Phi_0\\in C^\\infty_\\cpt(\\RR^d)$ such that $\\phi_0:=\\restrict{\\Phi_0}{\\partial\\Omega}$ satisfies $\\phi_0\\le\\phi$ and $\\phi_0(w)>0$.\nLet $u_\\phi$ and $u_{\\phi_0}$ be the variational (equivalently Perron) solutions with respect to $\\Omega$.\nThen $u_\\phi\\ge u_{\\phi_0}\\ge 0$ on $\\Omega$ by the maximum principle, and $u_{\\phi_0}\\ne 0$.\nMoreover, $u_{\\phi_0}\\in H^1(\\Omega)$ by Theorem~\\ref{thm:2.7}.", "post_theorem_intro_text_len": 4394, "post_theorem_intro_text": "We call $u_\\varphi$ the \\textbf{\\boldmath variational solution\\unboldmath} of~\\ref{eq:D}.\nIt attains the given boundary value $\\varphi$ in the Sobolev sense.\nThe existence of such a variational solution is difficult to prove and was an open problem for general $\\varphi\\in C(\\partial\\Omega)$ until recently.\n\nIn the first part of this paper we start with a simple proof of Theorem~\\ref{thm:1.1}, making use of a nontrivial extension result that we shall take as given.\nWe then establish diverse properties of the variational solution defined as above.\nWe show that it is well-defined; i.e.\\ $u_\\varphi$ does not depend on the extension $\\Phi$ of $\\varphi$ and it coincides with the classical solution of~\\ref{eq:D} whenever a classical solution exists.\nMoreover, we show that the weak maximum principle is valid for variational solutions.\nIn Section~\\ref{sec:3} we will see that $u_\\varphi$ coincides with the Perron solution.\nThis notion was defined by Perron in 1923~\\cite{Per1923} and has been a subject of intense research.\nOne subject is the way in which $u_\\varphi(x)$ attains the value $\\varphi(z)$ as $x\\to z$.\nSo Theorem~\\ref{thm:1.1} can be interpreted as a new description of the behaviour of the Perron solution at the boundary.\n\nIn the second part of the paper, from Section~\\ref{sec:4} on, we investigate the behaviour of $u_\\varphi(x)$ at a fixed singular point $z_0\\in\\partial\\Omega$, i.e.\\ a point where $\\lim_{x\\to z_0} u_\\varphi(x)=\\varphi(z_0)$ is violated for some $\\varphi\\in C(\\partial\\Omega)$.\n\nIn a single page paper Lebesgue~\\cite{Leb1913} considered a potential in $\\mathbb{R}^3$ of a thin rod modelled by a simple mass distribution concentrated along a line segment.\nHe chooses two suitable level sets of that potential and considers the bounded, open, simply connected region enclosed by these level sets.\nWe call this region \\textbf{\\boldmath Lebesgue's domain\\unboldmath}.\n\nLebesgue's domain has a unique singular point $z_0$ which is the tip of an inward pointing cusp, the famous \\textbf{\\boldmath Lebesgue cusp\\unboldmath}.\nTaking the intersection of Lebesgue's domain with a small ball, one obtains a simply connected domain with connected boundary and a singular point.\nThis is of great interest since in $\\mathbb{R}^2$ each simply connected domain is \\textbf{\\boldmath Dirichlet regular\\unboldmath}; i.e.\\ $u_\\varphi$ is a classical solution for all $\\varphi\\in C(\\partial\\Omega)$.\nOur point is to consider the original Lebesgue domain, whose boundary has two connected components.\nIt demonstrates two striking features of the generalised solution $u_\\varphi$.\n\n\\begin{enumerate}[1.]\n\\item It is impossible to deduce from the regularity properties of $\\varphi$ whether or not $u_\\varphi$ is a classical solution.\nThe boundary of Lebesgue's domain is the disjoint union of two closed sets $\\Gamma_2$ and $\\Gamma_{\\frac{1}{2}}$ (which are level sets of the considered potential).\nIf $\\varphi$ is constant on each of these two closed sets with two different constants, $u_\\varphi$ is not a classical solution.\nTo prove this, the variational definition of $u_\\varphi$ will be most useful.\n\n\\item The irregular behaviour of $u_\\varphi(x)$ as $x\\to z_0$ is a non-local and unstable phenomenon.\nFor $\\phi_0\\equiv 0$ one has $u_{\\phi_0}\\equiv 0$, but if we perturb $\\phi_0$ by a small compactly supported positive smooth bump in the neighbourhood of some point far away from $z_0$, then $u_\\varphi$ is no longer a classical solution.\nThis can be seen easily at Lebesgue's domain, but we prove it for a singular point of an arbitrary domain in Section~\\ref{sec:5}.\n\\end{enumerate}\n\nAs explained above, there seems to be no reasonable condition on the boundary function $\\varphi$ to ensure that $u_\\varphi$ is a classical solution of~\\ref{eq:D} -- even though there are always many classical solutions.\nThe situation is different if we instead consider the question of when $u_\\varphi$ has finite energy $\\int_\\Omega\\abs{\\nabla u_\\varphi}^2$.\nThis is the case if and only if $\\varphi$ has an extension $\\Phi\\in C(\\overline{\\Omega})\\cap H^1(\\Omega)$, and then $u_\\varphi$ may be determined by the Dirichlet principle as the unique minimiser of the Dirichlet energy.\nAs an easy consequence, if $\\varphi$ is Lipschitz continuous, then $u_\\varphi$ has finite energy.\nThese results are proved in Section~\\ref{sec:2} along with other properties of the variational solution.", "sketch": "In the first part of this paper we start with a simple proof of Theorem~\\ref{thm:1.1}, making use of a \\emph{nontrivial extension result} that we shall take as given.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:1.1}\nGiven $\\varphi\\in C(\\partial\\Omega)$, there exist a unique harmonic function $u_\\varphi$ on $\\Omega$ and an extension $\\Phi\\in C(\\overline{\\Omega})$ of $\\varphi$ such that $\\Phi-u_\\varphi\\in H^1_0(\\Omega)$.,", "theorem_type": ["Uniqueness", "Existence"], "mcq": {"question": "Let \\(\\Omega\\subset \\mathbb{R}^d\\) be a connected bounded open set with \\(d\\ge 2\\) and boundary \\(\\partial\\Omega\\). Given a boundary function \\(\\varphi\\in C(\\partial\\Omega)\\), which existence-and-uniqueness statement holds? (Here \\(H^1_0(\\Omega)\\) denotes the Sobolev space of \\(H^1\\)-functions with zero trace on \\(\\partial\\Omega\\).)", "correct_choice": {"label": "A", "text": "There exists exactly one harmonic function \\(u_\\varphi\\) on \\(\\Omega\\) such that there exists at least one extension \\(\\Phi\\in C(\\overline{\\Omega})\\) of \\(\\varphi\\) with \\(\\Phi|_{\\partial\\Omega}=\\varphi\\) and \\(\\Phi-u_\\varphi\\in H^1_0(\\Omega)\\)."}, "choices": [{"label": "B", "text": "There exists exactly one harmonic function \\(u_\\varphi\\) on \\(\\Omega\\) such that for every extension \\(\\Phi\\in C(\\overline{\\Omega})\\) of \\(\\varphi\\) one has \\(\\Phi-u_\\varphi\\in H^1_0(\\Omega)\\)."}, {"label": "C", "text": "There exists at least one harmonic function \\(u\\) on \\(\\Omega\\) and at least one extension \\(\\Phi\\in C(\\overline{\\Omega})\\) of \\(\\varphi\\) such that \\(\\Phi|_{\\partial\\Omega}=\\varphi\\) and \\(\\Phi-u\\in H^1_0(\\Omega)\\)."}, {"label": "D", "text": "There exists exactly one function \\(u_\\varphi\\in C(\\overline{\\Omega})\\) that is harmonic on \\(\\Omega\\) and satisfies \\(u_\\varphi|_{\\partial\\Omega}=\\varphi\\); equivalently, one may choose \\(\\Phi=u_\\varphi\\), so that \\(\\Phi-u_\\varphi\\in H^1_0(\\Omega)\\)."}, {"label": "E", "text": "For every extension \\(\\Phi\\in C(\\overline{\\Omega})\\) of \\(\\varphi\\), there exists exactly one harmonic function \\(u_\\varphi\\) on \\(\\Omega\\) such that \\(\\Phi-u_\\varphi\\in H^1(\\Omega)\\)."}], "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": "existential_extension_replaced_by_universal_extension", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_uniqueness_of_harmonic_function", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "upgraded_variational_solution_to_classical_boundary_attainment", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "replaced_H1_0_by_H1_and_made_solution_depend_on_every_extension", "template_used": "property_confusion"}]}}
{"id": "2512.15682v1", "paper_link": "http://arxiv.org/abs/2512.15682v1", "theorems_cnt": 1, "theorem": {"env_name": "thm", "content": "\\label{thm:1.1}\nGiven $\\varphi\\in C(\\partial\\Omega)$, there exist a unique harmonic function $u_\\varphi$ on $\\Omega$ and an extension $\\Phi\\in C(\\overline{\\Omega})$ of $\\varphi$ such that $\\Phi-u_\\varphi\\in H^1_0(\\Omega)$.", "start_pos": 993002, "end_pos": 993229, "label": "thm:1.1"}, "ref_dict": {"thm:1.1": "\\begin{thm}\\label{thm:1.1}\nGiven $\\phi\\in C(\\partial\\Omega)$, there exist a unique harmonic function $u_\\phi$ on $\\Omega$ and an extension $\\Phi\\in C(\\clos{\\Omega})$ of $\\phi$ such that $\\Phi-u_\\phi\\in H^1_0(\\Omega)$.\n\\end{thm}", "eq:D": "\\begin{equation}\n  \\tag*{$D(\\varphi)$}\n  \\label{eq:D}\n  \\left\\{\n    \\begin{aligned}\n      &\\text{find $u\\in C(\\clos{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n      &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\phi$.}\n    \\end{aligned}\n  \\right.\n\\end{equation}"}, "pre_theorem_intro_text_len": 764, "pre_theorem_intro_text": "Throughout we suppose that $\\Omega$ is a connected bounded open subset of $\\mathbb{R}^d$, $d\\ge 2$, with boundary $\\partial\\Omega$.\nGiven a function $\\varphi\\in C(\\partial\\Omega)$, the \\textbf{\\boldmath Dirichlet problem\\unboldmath} asks to\n\\begin{equation}\n  \\tag*{$D(\\varphi)$}\n  \\label{eq:D}\n  \\left\\{\n    \\begin{aligned}\n      &\\text{find $u\\in C(\\overline{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n      &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\varphi$.}\n    \\end{aligned}\n  \\right.\n\\end{equation}\nWe call such a function $u$ a \\textbf{\\boldmath classical solution\\unboldmath} of~\\ref{eq:D}.\nHowever, a classical solution does not always exist.\n\nIn this paper we consider a generalised solution of the Dirichlet problem which is described as follows.", "context": "Throughout we suppose that $\\Omega$ is a connected bounded open subset of $\\mathbb{R}^d$, $d\\ge 2$, with boundary $\\partial\\Omega$.\nGiven a function $\\varphi\\in C(\\partial\\Omega)$, the \\textbf{\\boldmath Dirichlet problem\\unboldmath} asks to\n\\begin{equation}\n \\tag*{$D(\\varphi)$}\n \\label{eq:D}\n \\left\\{\n \\begin{aligned}\n &\\text{find $u\\in C(\\overline{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\varphi$.}\n \\end{aligned}\n \\right.\n\\end{equation}\nWe call such a function $u$ a \\textbf{\\boldmath classical solution\\unboldmath} of~\\ref{eq:D}.\nHowever, a classical solution does not always exist.\n\nIn this paper we consider a generalised solution of the Dirichlet problem which is described as follows.\n\n\\begin{equation}\n  \\tag*{$D(\\varphi)$}\n  \\label{eq:D}\n  \\left\\{\n    \\begin{aligned}\n      &\\text{find $u\\in C(\\clos{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n      &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\phi$.}\n    \\end{aligned}\n  \\right.\n\\end{equation}", "full_context": "Throughout we suppose that $\\Omega$ is a connected bounded open subset of $\\mathbb{R}^d$, $d\\ge 2$, with boundary $\\partial\\Omega$.\nGiven a function $\\varphi\\in C(\\partial\\Omega)$, the \\textbf{\\boldmath Dirichlet problem\\unboldmath} asks to\n\\begin{equation}\n \\tag*{$D(\\varphi)$}\n \\label{eq:D}\n \\left\\{\n \\begin{aligned}\n &\\text{find $u\\in C(\\overline{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\varphi$.}\n \\end{aligned}\n \\right.\n\\end{equation}\nWe call such a function $u$ a \\textbf{\\boldmath classical solution\\unboldmath} of~\\ref{eq:D}.\nHowever, a classical solution does not always exist.\n\nIn this paper we consider a generalised solution of the Dirichlet problem which is described as follows.\n\n\\begin{equation}\n  \\tag*{$D(\\varphi)$}\n  \\label{eq:D}\n  \\left\\{\n    \\begin{aligned}\n      &\\text{find $u\\in C(\\clos{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n      &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\phi$.}\n    \\end{aligned}\n  \\right.\n\\end{equation}\n\nThroughout we suppose that $\\Omega$ is a connected bounded open subset of $\\RR^d$, $d\\ge 2$, with boundary $\\partial\\Omega$.\nGiven a function $\\phi\\in C(\\partial\\Omega)$, the \\emphdef{Dirichlet problem} asks to\n\\begin{equation}\n \\tag*{$D(\\varphi)$}\n \\label{eq:D}\n \\left\\{\n \\begin{aligned}\n &\\text{find $u\\in C(\\clos{\\Omega})$ which is harmonic on $\\Omega$} \\\\\n &\\text{such that $\\restrict{u}{\\partial\\Omega}=\\phi$.}\n \\end{aligned}\n \\right.\n\\end{equation}\nWe call such a function $u$ a \\emphdef{classical solution} of~\\ref{eq:D}.\nHowever, a classical solution does not always exist.\n\nIn this section we develop variational solutions of the Dirichlet problem on the basis of the following result~\\cite[Corollary~4.3]{AtES24}.\n\\begin{thm}[Extension result]\\label{thm:2.1}\nLet $\\phi\\in C(\\partial\\Omega)$. Then there exist an extension $\\Phi\\in C(\\clos{\\Omega})$ of $\\phi$ and $M \\ge 0$ such that\n\\begin{equation}\\label{eq:2.1}\n \\abs*{\\int_\\Omega\\Phi\\Delta w\\dx} \\le M\\norm{w}_{H^1(\\Omega)}\n\\end{equation}\nfor all $w\\in C^\\infty_\\cpt(\\Omega)$. This means that $\\Delta\\Phi\\in H^{-1}(\\Omega)$, the dual space of $H^1_0(\\Omega)$.\n\\end{thm}\nNote that for $u\\in H^1(\\Omega)$ one has\n\\[\n \\abs*{\\int_\\Omega u\\Delta w\\dx} = \\abs*{-\\int_\\Omega\\nabla u\\nabla w\\dx}\\le\\norm{u}_{H^1(\\Omega)}\\norm{w}_{H^1(\\Omega)}\n\\]\nfor all $w\\in C^\\infty_\\cpt(\\Omega)$ and therefore $\\Delta u\\in H^{-1}(\\Omega)$.\n\nLet $\\phi\\in C(\\partial\\Omega)$ and $\\Phi\\in C(\\clos{\\Omega})$ be an extension of $\\phi$ satisfying~\\eqref{eq:2.1}.\nBy the Riesz--Fréchet representation theorem there exists a unique $v\\in H^1_0(\\Omega)$ such that\n\\[\n \\int_\\Omega \\Phi\\Delta w\\dx = -\\int_\\Omega\\nabla v\\nabla w\\dx = \\int_\\Omega v\\Delta w\\dx\n\\]\nfor all $w\\in C^\\infty_\\cpt(\\Omega)$. This means that $u:=\\Phi-v$ satisfies $\\Delta u=0$ weakly. Hence $u\\in C^\\infty(\\Omega)$ by Weyl's theorem and $\\Delta u=0$ in the classical sense;\nthat is, $u$ is a harmonic function.\nWe call $u$ the \\emphdef{variational solution} of~\\ref{eq:D} and set $u_\\phi := u$.\nTo do so, we have to show that the function does not depend on the extension $\\Phi$ of $\\phi$. This follows from the next lemma.\n\nNext we talk about the Dirichlet principle.\nWe start by mentioning that the variational solution $u_\\phi$ is obtained by a minimising process.\nIn fact, let $\\phi\\in C(\\partial\\Omega)$ and $\\Phi\\in C(\\clos{\\Omega})$ be an extension of $\\phi$ satisfying the estimate~\\eqref{eq:2.1}, i.e.\\ $\\Delta\\Phi\\in H^{-1}(\\Omega)$.\nThe solution $v\\in H^1_0(\\Omega)$ of $\\Delta v=\\Delta\\Phi$ is the unique function $v\\in H^1_0(\\Omega)$ such that\n\\begin{equation}\\label{eq:2.5}\n \\frac{1}{2}\\int_\\Omega\\abs{\\nabla v}^2\\dx + \\langle\\Delta\\Phi,v\\rangle = \\min\\Bigl\\{\\frac{1}{2}\\int_\\Omega\\abs{\\nabla w}^2\\dx + \\langle\\Delta\\Phi,w\\rangle \\setcolon w\\in H^1_0(\\Omega)\\Bigr\\}\n\\end{equation}\n(see e.g.~\\cite[Theorem~4.24]{AU23}).\nNow we can characterise when $\\int_\\Omega\\abs{\\nabla u_\\phi}^2\\dx<\\infty$; i.e.\\ when $u_\\phi$ has finite energy.\nSince $u_\\phi\\in C^\\infty(\\Omega)\\cap C_\\bdd(\\Omega)$, this is equivalent to saying that $u_\\phi\\in H^1(\\Omega)$.\n\n\\begin{thm}\\label{thm:2.7}\nLet $\\phi\\in C(\\partial\\Omega)$. The following statements are equivalent.\n\\begin{romanenum}\n\\item\\label{en:t2.7-i}\nThe variational solution $u_\\phi$ of~\\ref{eq:D} is an element of $H^1(\\Omega)$.\n\\item\\label{en:t2.7-ii}\nThere exists $\\Phi\\in C(\\clos{\\Omega})\\cap H^1(\\Omega)$ with $\\restrict{\\Phi}{\\partial\\Omega}=\\phi$.\n\\end{romanenum}\nIn that case\n\\[\n \\int_\\Omega\\abs{\\nabla u_\\phi}^2\\dx = \\min\\Bigl\\{\\int_\\Omega\\abs{\\nabla w}^2\\dx \\setcolon w\\in H^1(\\Omega),\\ \\Phi-w\\in H^1_0(\\Omega)\\Bigr\\}.\n\\]\n\\end{thm}\n\\begin{proof}\n\\begin{parenum}\n\\item[\\ref{en:t2.7-ii}$\\Rightarrow$\\ref{en:t2.7-i}.]\nSince $\\Phi\\in H^1(\\Omega)$, $u_\\phi=\\Phi-v\\in H^1(\\Omega)$, where $v\\in H^1_0(\\Omega)$ is such that $\\Delta v=\\Delta\\Phi$.\nMoreover, by~\\eqref{eq:2.5} and considering $W:=\\{w\\in H^1(\\Omega)\\setcolon \\Phi-w\\in H^1_0(\\Omega)\\}$,\n\\begin{align*}\n \\frac{1}{2}[u_\\phi,u_\\phi] -\\frac{1}{2}[\\Phi,\\Phi] &= \\frac{1}{2}[v,v]-[\\Phi,v] \\\\\n &= \\min\\Bigl\\{\\frac{1}{2}[\\Phi-w,\\Phi-w] - [\\Phi,\\Phi-w] \\setcolon w\\in W\\Bigr\\} \\\\\n &= \\min\\Bigl\\{\\frac{1}{2}[w,w] - \\frac{1}{2}[\\Phi,\\Phi] \\setcolon w\\in W\\Bigr\\} \\\\\n &= \\frac{1}{2} \\min\\Bigl\\{[w,w] \\setcolon w\\in W\\Bigr\\}-\\frac{1}{2}[\\Phi,\\Phi].\n\\end{align*}\nThus $[u_\\phi,u_\\phi] = \\min\\{[w,w]\\setcolon w\\in W\\}$ and this property characterises $u_\\phi$.\n\\item[\\ref{en:t2.7-i}$\\Rightarrow$\\ref{en:t2.7-ii}.]\nLet $\\Phi\\in C(\\clos{\\Omega})$ be such that $\\restrict{\\Phi}{\\partial\\Omega}=\\phi$ and $\\Delta\\Phi\\in H^{-1}(\\Omega)$.\nLet $v\\in H^1_0(\\Omega)$ be such that $\\Delta v=\\Delta\\Phi$.\nThen $u_\\phi=\\Phi-v$.\nSince $u_\\phi\\in H^1(\\Omega)$ by hypothesis, it follows that $\\Phi\\in H^1(\\Omega)$.\\qedhere\n\\end{parenum}\n\\end{proof}\n\nIf $\\Omega$ is more regular, the following holds.\n\\begin{thm}\\label{thm:2.9}\nAssume that $\\Omega$ has Lipschitz boundary and let $\\phi\\in C(\\partial\\Omega)$. The following statements are equivalent.\n\\begin{romanenum}\n\\item $u_\\phi\\in H^1(\\Omega)$\n\\item $\\phi\\in H^{1/2}(\\partial\\Omega)$\n\\end{romanenum}\n\\end{thm}\nHere, since $\\Omega$ has Lipschitz boundary, $H^{1/2}(\\partial\\Omega)=\\{\\operatorname{tr}\\Phi \\setcolon \\Phi\\in H^1(\\Omega)\\}$, where $\\operatorname{tr}\\colon H^1(\\Omega)\\to L^2(\\partial\\Omega,\\mathcal{H}^{d-1})$ is the trace operator and $\\mathcal{H}^{d-1}$ denotes the $(d-1)$-dimensional Hausdorff measure.\nIf $\\phi\\in C(\\partial\\Omega)\\cap H^{1/2}(\\partial\\Omega)$, there even exists a $\\Phi\\in C(\\clos{\\Omega})\\cap H^1(\\Omega)$ such that $\\restrict{\\Phi}{\\partial\\Omega}=\\phi$.\nWe refer to~\\cite[Theorem~1.2]{AtE19:dp} for this and a proof of Theorem~\\ref{thm:2.9}.\nA more general result on very rough domains is~\\cite[Theorem~5.8]{AtES24}.\n\nTheorem~\\ref{thm:5.1} is contained in Landkof~\\cite[p.\\,243, lines 1--5]{La72}.\nWe will give a very different proof based on the following result. We assume that $d\\ge 3$.\n\\begin{thm}\\label{thm:5.2}\nLet $z_0\\in \\partial\\Omega$ be a singular point of $\\Omega$.\nThen there exists an open set $\\Omega_0\\subset\\RR^d$ such that $\\clos{\\Omega_0}\\subset\\Omega\\cup\\{z_0\\}$, $z_0\\in\\partial\\Omega_0$ and $z_0$ is a singular point of $\\Omega_0$.\n\\end{thm}\nWe postpone the proof of Theorem~\\ref{thm:5.2} but note that in the case of Lebesgue's domain it readily follows from its construction.\nMore precisely, given the potential $V\\colon D\\to(0,\\infty)$ from Lemma~\\ref{lem:lebesgue-potential-basics}, we had $\\Omega=\\{x\\in D\\setcolon \\frac{1}{2}<V(x)<2\\}$. We can choose e.g.~$\\Omega_0=\\{x\\in D\\setcolon \\frac{1}{c}<V(r,z)<c\\}$ for any $c\\in (1,2)$.\nNow we can easily prove Theorem~\\ref{thm:5.1} using the following criterion for a boundary point to be regular, see~\\cite[Theorem~6.74]{AU23}.\n\\begin{prop}\\label{prop:5.4}\nLet $z_0\\in \\partial\\Omega$. Then $z_0$ is regular if there exists a \\emphdef{global $H^1$-barrier} $b$ at $z_0$; i.e.\\ a function $b\\in C(\\clos{\\Omega})\\cap H^1(\\Omega)$ such that $b(x)>0$ for all $x\\in\\clos{\\Omega}\\setminus\\{z_0\\}$, $b(z_0)=0$ and $-\\Delta b\\ge 0$ weakly on $\\Omega$.\n\\end{prop}\n\\begin{proof}[Proof of Theorem~\\ref{thm:5.1}]\nLet $z_0\\in \\partial\\Omega$ be a singular point for $\\Omega$.\nBy Theorem~\\ref{thm:5.2} there exists an open set $\\Omega_0\\subset\\RR^d$ such that $\\clos{\\Omega_0}\\subset\\Omega\\cup\\{z_0\\}$ and $z_0\\in\\partial\\Omega_0$ is a singular point with respect to $\\Omega_0$.\nLet $0\\le\\phi\\in C(\\partial\\Omega)$ be such that $\\phi(z_0)=0$ and $\\phi(w)>0$ at a regular point $w\\in \\partial\\Omega$.\nThen there exists $0\\le\\Phi_0\\in C^\\infty_\\cpt(\\RR^d)$ such that $\\phi_0:=\\restrict{\\Phi_0}{\\partial\\Omega}$ satisfies $\\phi_0\\le\\phi$ and $\\phi_0(w)>0$.\nLet $u_\\phi$ and $u_{\\phi_0}$ be the variational (equivalently Perron) solutions with respect to $\\Omega$.\nThen $u_\\phi\\ge u_{\\phi_0}\\ge 0$ on $\\Omega$ by the maximum principle, and $u_{\\phi_0}\\ne 0$.\nMoreover, $u_{\\phi_0}\\in H^1(\\Omega)$ by Theorem~\\ref{thm:2.7}.", "post_theorem_intro_text_len": 4394, "post_theorem_intro_text": "We call $u_\\varphi$ the \\textbf{\\boldmath variational solution\\unboldmath} of~\\ref{eq:D}.\nIt attains the given boundary value $\\varphi$ in the Sobolev sense.\nThe existence of such a variational solution is difficult to prove and was an open problem for general $\\varphi\\in C(\\partial\\Omega)$ until recently.\n\nIn the first part of this paper we start with a simple proof of Theorem~\\ref{thm:1.1}, making use of a nontrivial extension result that we shall take as given.\nWe then establish diverse properties of the variational solution defined as above.\nWe show that it is well-defined; i.e.\\ $u_\\varphi$ does not depend on the extension $\\Phi$ of $\\varphi$ and it coincides with the classical solution of~\\ref{eq:D} whenever a classical solution exists.\nMoreover, we show that the weak maximum principle is valid for variational solutions.\nIn Section~\\ref{sec:3} we will see that $u_\\varphi$ coincides with the Perron solution.\nThis notion was defined by Perron in 1923~\\cite{Per1923} and has been a subject of intense research.\nOne subject is the way in which $u_\\varphi(x)$ attains the value $\\varphi(z)$ as $x\\to z$.\nSo Theorem~\\ref{thm:1.1} can be interpreted as a new description of the behaviour of the Perron solution at the boundary.\n\nIn the second part of the paper, from Section~\\ref{sec:4} on, we investigate the behaviour of $u_\\varphi(x)$ at a fixed singular point $z_0\\in\\partial\\Omega$, i.e.\\ a point where $\\lim_{x\\to z_0} u_\\varphi(x)=\\varphi(z_0)$ is violated for some $\\varphi\\in C(\\partial\\Omega)$.\n\nIn a single page paper Lebesgue~\\cite{Leb1913} considered a potential in $\\mathbb{R}^3$ of a thin rod modelled by a simple mass distribution concentrated along a line segment.\nHe chooses two suitable level sets of that potential and considers the bounded, open, simply connected region enclosed by these level sets.\nWe call this region \\textbf{\\boldmath Lebesgue's domain\\unboldmath}.\n\nLebesgue's domain has a unique singular point $z_0$ which is the tip of an inward pointing cusp, the famous \\textbf{\\boldmath Lebesgue cusp\\unboldmath}.\nTaking the intersection of Lebesgue's domain with a small ball, one obtains a simply connected domain with connected boundary and a singular point.\nThis is of great interest since in $\\mathbb{R}^2$ each simply connected domain is \\textbf{\\boldmath Dirichlet regular\\unboldmath}; i.e.\\ $u_\\varphi$ is a classical solution for all $\\varphi\\in C(\\partial\\Omega)$.\nOur point is to consider the original Lebesgue domain, whose boundary has two connected components.\nIt demonstrates two striking features of the generalised solution $u_\\varphi$.\n\n\\begin{enumerate}[1.]\n\\item It is impossible to deduce from the regularity properties of $\\varphi$ whether or not $u_\\varphi$ is a classical solution.\nThe boundary of Lebesgue's domain is the disjoint union of two closed sets $\\Gamma_2$ and $\\Gamma_{\\frac{1}{2}}$ (which are level sets of the considered potential).\nIf $\\varphi$ is constant on each of these two closed sets with two different constants, $u_\\varphi$ is not a classical solution.\nTo prove this, the variational definition of $u_\\varphi$ will be most useful.\n\n\\item The irregular behaviour of $u_\\varphi(x)$ as $x\\to z_0$ is a non-local and unstable phenomenon.\nFor $\\phi_0\\equiv 0$ one has $u_{\\phi_0}\\equiv 0$, but if we perturb $\\phi_0$ by a small compactly supported positive smooth bump in the neighbourhood of some point far away from $z_0$, then $u_\\varphi$ is no longer a classical solution.\nThis can be seen easily at Lebesgue's domain, but we prove it for a singular point of an arbitrary domain in Section~\\ref{sec:5}.\n\\end{enumerate}\n\nAs explained above, there seems to be no reasonable condition on the boundary function $\\varphi$ to ensure that $u_\\varphi$ is a classical solution of~\\ref{eq:D} -- even though there are always many classical solutions.\nThe situation is different if we instead consider the question of when $u_\\varphi$ has finite energy $\\int_\\Omega\\abs{\\nabla u_\\varphi}^2$.\nThis is the case if and only if $\\varphi$ has an extension $\\Phi\\in C(\\overline{\\Omega})\\cap H^1(\\Omega)$, and then $u_\\varphi$ may be determined by the Dirichlet principle as the unique minimiser of the Dirichlet energy.\nAs an easy consequence, if $\\varphi$ is Lipschitz continuous, then $u_\\varphi$ has finite energy.\nThese results are proved in Section~\\ref{sec:2} along with other properties of the variational solution.", "sketch": "In the first part of this paper we start with a simple proof of Theorem~\\ref{thm:1.1}, making use of a \\emph{nontrivial extension result} that we shall take as given.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:1.1}\nGiven $\\varphi\\in C(\\partial\\Omega)$, there exist a unique harmonic function $u_\\varphi$ on $\\Omega$ and an extension $\\Phi\\in C(\\overline{\\Omega})$ of $\\varphi$ such that $\\Phi-u_\\varphi\\in H^1_0(\\Omega)$.,", "theorem_type": ["Uniqueness", "Existence"], "mcq": {"question": "Let \\(\\Omega\\subset \\mathbb{R}^d\\) be a connected bounded open set with \\(d\\ge 2\\) and boundary \\(\\partial\\Omega\\). For \\(\\varphi\\in C(\\partial\\Omega)\\), an extension of \\(\\varphi\\) means a function \\(\\Phi\\in C(\\overline{\\Omega})\\) such that \\(\\Phi|_{\\partial\\Omega}=\\varphi\\), and \\(H^1_0(\\Omega)\\) denotes the Sobolev space of \\(H^1\\)-functions with zero trace on \\(\\partial\\Omega\\). Under these assumptions, which existence statement holds?", "correct_choice": {"label": "A", "text": "For every \\(\\varphi\\in C(\\partial\\Omega)\\), there exists a unique harmonic function \\(u_\\varphi\\) on \\(\\Omega\\), and there exists an extension \\(\\Phi\\in C(\\overline{\\Omega})\\) of \\(\\varphi\\) such that \\(\\Phi-u_\\varphi\\in H^1_0(\\Omega)\\)."}, "choices": [{"label": "B", "text": "For every \\(\\varphi\\in C(\\partial\\Omega)\\), there exists a unique harmonic function \\(u_\\varphi\\in C(\\overline{\\Omega})\\) on \\(\\Omega\\) such that \\(u_\\varphi|_{\\partial\\Omega}=\\varphi\\); equivalently, one may take the extension \\(\\Phi=u_\\varphi\\), so that \\(\\Phi-u_\\varphi\\in H^1_0(\\Omega)\\)."}, {"label": "C", "text": "For every \\(\\varphi\\in C(\\partial\\Omega)\\), there exists a harmonic function \\(u\\) on \\(\\Omega\\) and an extension \\(\\Phi\\in C(\\overline{\\Omega})\\) of \\(\\varphi\\) such that \\(\\Phi-u\\in H^1_0(\\Omega)\\)."}, {"label": "D", "text": "For every extension \\(\\Phi\\in C(\\overline{\\Omega})\\) of every \\(\\varphi\\in C(\\partial\\Omega)\\), there exists a unique harmonic function \\(u_\\varphi\\) on \\(\\Omega\\) such that \\(\\Phi-u_\\varphi\\in H^1_0(\\Omega)\\)."}, {"label": "E", "text": "For every \\(\\varphi\\in C(\\partial\\Omega)\\), there exists a unique function \\(u_\\varphi\\in H^1(\\Omega)\\) and an extension \\(\\Phi\\in C(\\overline{\\Omega})\\) of \\(\\varphi\\) such that \\(\\Phi-u_\\varphi\\in H^1_0(\\Omega)\\) and \\(u_\\varphi\\) is harmonic on \\(\\Omega\\) in the weak sense."}], "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": "upgrades existence to classical boundary continuity and boundary equality", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped uniqueness of the harmonic function", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "existential choice of extension replaced by all extensions", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "replaces harmonic-on-\\(\\Omega\\) uniqueness claim by unique weak-\\(H^1\\) solution, forcing unnecessary finite-energy regularity", "template_used": "wildcard"}]}}
{"id": "2512.16358v1", "paper_link": "http://arxiv.org/abs/2512.16358v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\nLet $p_1, \\ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\\prod_{i=1}^k p_i$ be their product. \nFor a given choice of residue classes, i.e., one residue class $r_i$ for each prime $p_i$, the number of available elements in $[1,P_k]$ is $a_k$ and the number of free elements is $f_k$.\nIn other words, for every choice of residue classes, there are $a_k$ integers $n$ with $1\\leq n \\leq P_k$ such that $\\gamma(n)\\leq 1$ and $f_k$ integers with $\\gamma(n)=0$.", "start_pos": 6591, "end_pos": 7104, "label": "thm:main"}, "ref_dict": {"lem1": "\\begin{lemma} \\label{lem1}\nFor $k\\geq 2$ we have that \n$$f_k = (p_{k}-1) f_{k-1}.$$\n\\end{lemma}", "thm:main": "\\begin{theorem} \\label{thm:main}\nLet $p_1, \\ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\\prod_{i=1}^k p_i$ be their product. \nFor a given choice of residue classes, i.e., one residue class $r_i$ for each prime $p_i$, the number of available elements in $[1,P_k]$ is $a_k$ and the number of free elements is $f_k$.\nIn other words, for every choice of residue classes, there are $a_k$ integers $n$ with $1\\leq n \\leq P_k$ such that $\\gamma(n)\\leq 1$ and $f_k$ integers with $\\gamma(n)=0$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2166, "pre_theorem_intro_text": "The sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \\cite{oeis}, is defined as $(a_k)_{k \\geq  1}$ with $a_k$ being the determinant of the $k \\times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones, i.e., \n$$ a_k = \\det\n \\begin{bmatrix}\n2 & 1 & \\ldots  & 1 \\\\\n1 & 3 & \\ldots & 1 \\\\\n\\vdots &  \\vdots & \\ddots & \\vdots \\\\\n1 & 1 & \\ldots & p_k \n\\end{bmatrix}.  $$\nThe first five elements, for $k=1, \\ldots, 5$, of this sequence are\n$$2, 5, 22, 140, 1448, \\ldots,$$\nsee \\cite{oeis} for a table of the first 300 elements of this sequence.\nWe relate this sequence to a concrete counting problem.\nLet $p_1, \\ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\\prod_{i=1}^k p_i$ be their product.\nWe choose an arbitrary residue class $r_i$ for each prime $p_i$ and consider $k$ segments of arithmetic progressions restricted to the interval $[1,P_k]$, i.e., for $p_i$ and $r_i$ we consider\n$$s_i = \\{n : n= h\\cdot p_i + r_i, h \\in \\NN_0 \\} \\cap [1,P_k]$$\n\nFor a given choice $r_1, \\ldots, r_k$ of residue classes, let\n$$\\gamma(n)=\\# \\{p_i: n\\equiv r_i \\pmod{p_i} \\}$$\nfor $1 \\leq n \\leq P_k$ be the number of residue classes in which $n$ is contained.\nWe introduce some notation:\n\\begin{itemize}\n\\item An integer $n$ is \\emph{covered by a prime $p_i$} if it is contained in the residue class $r_i$.\n\\item An integer $n$ is \\emph{free} if it is not contained in any residue class, i.e., $\\gamma(n)=0$.\n\\item An integer $n$ is \\emph{available} if it is contained in at most one residue class, i.e., $\\gamma(n)\\leq 1$.\n\\item An integer $n$ is \\emph{occupied} if it is contained in at least two residue class, i.e., $\\gamma(n)\\geq 2$.\n\\end{itemize}\nBy definition, a free integer is also available. \nMoreover, we define a second sequence $(f_k)_{k\\geq 1}$ as follows:\n$$ f_k = (-1)^{k} \\det\n \\begin{bmatrix}\n 1 & 1 & \\ldots & 1 & 1\\\\\n2 & 1 & \\ldots  & 1& 1 \\\\\n1 & 3 & \\ldots & 1 & 1\\\\\n\\vdots &  \\vdots & \\ddots & \\vdots  & \\vdots\\\\\n1 & 1 & \\ldots & p_{k} & 1 \n\\end{bmatrix}.  $$\nThe first five elements of this sequence for $k=1, \\ldots, 5$ are\n$$1, 2, 8, 48, 480 \\ldots,$$\nWe have the following main result:", "context": "The sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \\cite{oeis}, is defined as $(a_k)_{k \\geq 1}$ with $a_k$ being the determinant of the $k \\times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones, i.e., \n$$ a_k = \\det\n \\begin{bmatrix}\n2 & 1 & \\ldots & 1 \\\\\n1 & 3 & \\ldots & 1 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n1 & 1 & \\ldots & p_k \n\\end{bmatrix}. $$\nThe first five elements, for $k=1, \\ldots, 5$, of this sequence are\n$$2, 5, 22, 140, 1448, \\ldots,$$\nsee \\cite{oeis} for a table of the first 300 elements of this sequence.\nWe relate this sequence to a concrete counting problem.\nLet $p_1, \\ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\\prod_{i=1}^k p_i$ be their product.\nWe choose an arbitrary residue class $r_i$ for each prime $p_i$ and consider $k$ segments of arithmetic progressions restricted to the interval $[1,P_k]$, i.e., for $p_i$ and $r_i$ we consider\n$$s_i = \\{n : n= h\\cdot p_i + r_i, h \\in \\NN_0 \\} \\cap [1,P_k]$$\n\nFor a given choice $r_1, \\ldots, r_k$ of residue classes, let\n$$\\gamma(n)=\\# \\{p_i: n\\equiv r_i \\pmod{p_i} \\}$$\nfor $1 \\leq n \\leq P_k$ be the number of residue classes in which $n$ is contained.\nWe introduce some notation:\n\\begin{itemize}\n\\item An integer $n$ is \\emph{covered by a prime $p_i$} if it is contained in the residue class $r_i$.\n\\item An integer $n$ is \\emph{free} if it is not contained in any residue class, i.e., $\\gamma(n)=0$.\n\\item An integer $n$ is \\emph{available} if it is contained in at most one residue class, i.e., $\\gamma(n)\\leq 1$.\n\\item An integer $n$ is \\emph{occupied} if it is contained in at least two residue class, i.e., $\\gamma(n)\\geq 2$.\n\\end{itemize}\nBy definition, a free integer is also available. \nMoreover, we define a second sequence $(f_k)_{k\\geq 1}$ as follows:\n$$ f_k = (-1)^{k} \\det\n \\begin{bmatrix}\n 1 & 1 & \\ldots & 1 & 1\\\\\n2 & 1 & \\ldots & 1& 1 \\\\\n1 & 3 & \\ldots & 1 & 1\\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots\\\\\n1 & 1 & \\ldots & p_{k} & 1 \n\\end{bmatrix}. $$\nThe first five elements of this sequence for $k=1, \\ldots, 5$ are\n$$1, 2, 8, 48, 480 \\ldots,$$\nWe have the following main result:", "full_context": "The sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \\cite{oeis}, is defined as $(a_k)_{k \\geq 1}$ with $a_k$ being the determinant of the $k \\times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones, i.e., \n$$ a_k = \\det\n \\begin{bmatrix}\n2 & 1 & \\ldots & 1 \\\\\n1 & 3 & \\ldots & 1 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n1 & 1 & \\ldots & p_k \n\\end{bmatrix}. $$\nThe first five elements, for $k=1, \\ldots, 5$, of this sequence are\n$$2, 5, 22, 140, 1448, \\ldots,$$\nsee \\cite{oeis} for a table of the first 300 elements of this sequence.\nWe relate this sequence to a concrete counting problem.\nLet $p_1, \\ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\\prod_{i=1}^k p_i$ be their product.\nWe choose an arbitrary residue class $r_i$ for each prime $p_i$ and consider $k$ segments of arithmetic progressions restricted to the interval $[1,P_k]$, i.e., for $p_i$ and $r_i$ we consider\n$$s_i = \\{n : n= h\\cdot p_i + r_i, h \\in \\NN_0 \\} \\cap [1,P_k]$$\n\nFor a given choice $r_1, \\ldots, r_k$ of residue classes, let\n$$\\gamma(n)=\\# \\{p_i: n\\equiv r_i \\pmod{p_i} \\}$$\nfor $1 \\leq n \\leq P_k$ be the number of residue classes in which $n$ is contained.\nWe introduce some notation:\n\\begin{itemize}\n\\item An integer $n$ is \\emph{covered by a prime $p_i$} if it is contained in the residue class $r_i$.\n\\item An integer $n$ is \\emph{free} if it is not contained in any residue class, i.e., $\\gamma(n)=0$.\n\\item An integer $n$ is \\emph{available} if it is contained in at most one residue class, i.e., $\\gamma(n)\\leq 1$.\n\\item An integer $n$ is \\emph{occupied} if it is contained in at least two residue class, i.e., $\\gamma(n)\\geq 2$.\n\\end{itemize}\nBy definition, a free integer is also available. \nMoreover, we define a second sequence $(f_k)_{k\\geq 1}$ as follows:\n$$ f_k = (-1)^{k} \\det\n \\begin{bmatrix}\n 1 & 1 & \\ldots & 1 & 1\\\\\n2 & 1 & \\ldots & 1& 1 \\\\\n1 & 3 & \\ldots & 1 & 1\\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots\\\\\n1 & 1 & \\ldots & p_{k} & 1 \n\\end{bmatrix}. $$\nThe first five elements of this sequence for $k=1, \\ldots, 5$ are\n$$1, 2, 8, 48, 480 \\ldots,$$\nWe have the following main result:\n\n\\begin{abstract}\nThe sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \\cite{oeis}, is defined as $(a_k)_{k \\geq 1}$ with $a_k$ being the determinant of the $k \\times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones.\nWe relate this sequence to a concrete counting problem.\nChoose an arbitrary residue class $r_i$ for each prime $p_i$ with $1 \\leq i \\leq k$ and set $P_k = \\prod_{i=1}^k p_i$. We show that $a_k$ is the number of integers in $[1, P_k]$ that are contained in \\emph{at most} one of the $k$ chosen residue classes.\nInterestingly, we show that this sequence is closely related to the better known sequence $A005867$ for which we derive a novel characterisation in terms of determinants and which gives the number of integers in $[1, P_k]$ that are not contained in any of the $k$ residue classes.\n\n\\section{Introduction}\nThe sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \\cite{oeis}, is defined as $(a_k)_{k \\geq 1}$ with $a_k$ being the determinant of the $k \\times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones, i.e., \n$$ a_k = \\det\n \\begin{bmatrix}\n2 & 1 & \\ldots & 1 \\\\\n1 & 3 & \\ldots & 1 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n1 & 1 & \\ldots & p_k \n\\end{bmatrix}. $$\nThe first five elements, for $k=1, \\ldots, 5$, of this sequence are\n$$2, 5, 22, 140, 1448, \\ldots,$$\nsee \\cite{oeis} for a table of the first 300 elements of this sequence.\nWe relate this sequence to a concrete counting problem.\nLet $p_1, \\ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\\prod_{i=1}^k p_i$ be their product.\nWe choose an arbitrary residue class $r_i$ for each prime $p_i$ and consider $k$ segments of arithmetic progressions restricted to the interval $[1,P_k]$, i.e., for $p_i$ and $r_i$ we consider\n$$s_i = \\{n : n= h\\cdot p_i + r_i, h \\in \\NN_0 \\} \\cap [1,P_k]$$\n\nFor a given choice $r_1, \\ldots, r_k$ of residue classes, let\n$$\\gamma(n)=\\# \\{p_i: n\\equiv r_i \\pmod{p_i} \\}$$\nfor $1 \\leq n \\leq P_k$ be the number of residue classes in which $n$ is contained.\nWe introduce some notation:\n\\begin{itemize}\n\\item An integer $n$ is \\emph{covered by a prime $p_i$} if it is contained in the residue class $r_i$.\n\\item An integer $n$ is \\emph{free} if it is not contained in any residue class, i.e., $\\gamma(n)=0$.\n\\item An integer $n$ is \\emph{available} if it is contained in at most one residue class, i.e., $\\gamma(n)\\leq 1$.\n\\item An integer $n$ is \\emph{occupied} if it is contained in at least two residue class, i.e., $\\gamma(n)\\geq 2$.\n\\end{itemize}\nBy definition, a free integer is also available. \nMoreover, we define a second sequence $(f_k)_{k\\geq 1}$ as follows:\n$$ f_k = (-1)^{k} \\det\n \\begin{bmatrix}\n 1 & 1 & \\ldots & 1 & 1\\\\\n2 & 1 & \\ldots & 1& 1 \\\\\n1 & 3 & \\ldots & 1 & 1\\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots\\\\\n1 & 1 & \\ldots & p_{k} & 1 \n\\end{bmatrix}. $$\nThe first five elements of this sequence for $k=1, \\ldots, 5$ are\n$$1, 2, 8, 48, 480 \\ldots,$$\nWe have the following main result:\n\nMoreover, our method can be generalised to counting free and available integers in arbitrary sets of $k$ arithmetic progressions generated by $k$ different primes as shown in Section \\ref{sec5}.\n\n\\begin{remark}\nThe problem of determining the number of free integers in the interval $[1,P_k]$ with residue classes $r_i=0$ for all $i$ is already mentioned by Dickson \\cite[p 439]{dickson} in the first volume of his History of the Theory of Numbers. He refers to results by A. de Polignac, J. Deschamps and H.J.S Smith for which Martin \\cite{martin} provides proofs in his manuscript. \nDetermining the number of available integers and the connection of the two counting problems to the determinants $a_k$ and $f_k$ are new to the best of our knowledge.\n\\end{remark}\n\nWe aim to understand the systematic intersections in the interval $[1,P_k]$ of $k$ arithmetic progressions generated by the first $k$ prime numbers $p_1, \\ldots, p_k$ and arbitrary choice of residue classes $r_1, \\ldots, r_k$ with $P_k = \\prod_{i=1}^k p_i$.\nIn particular, we want to understand whether an integer $1\\leq n \\leq P_k$ is not covered by any residue class, is contained in at most one or in at least two residue classes.\n\nAssume we have generated all $k$ arithmetic progression and intersected them with the interval $[1,P_k]$ obtaining the sets\n$s_i$, $1\\leq i \\leq k$.\nFor each integer $1\\leq n \\leq P_k$ we can now ask in how many sets $s_i$ it is contained, i.e., compute $\\gamma(n)$. \nWe denote the number of all available integers with $av(k)$, the number of free integers with $free(k)$ and the number of integers that are contained in two or more sets with $occ(k)$, i.e.\n\\begin{align*}\nav(k) &:= \\# \\{n : \\gamma(n) \\leq 1 \\} \\\\\nfree(k) &:=\\# \\{n : \\gamma(n) = 0 \\} \\\\\nocc(k) &:=\\# \\{n : \\gamma(n) > 1 \\} \n\\end{align*}\nTo illustrate this, we consider the cases $k=1$ and $k=2$ with $p_1=2$, $p_2=3$ as well as $P_1=2$ $P_2=6$.\nIt is easy to see that $occ(1)=0$, $free(1)=1$ and $av(1)=2$, because the arithmetic progression for $p_1=2$ is the first progression we consider and it contains one element in $[1,2]$.\n\n\\begin{theorem} \\label{thm2}\nLet $p_{j_1}, \\ldots, p_{j_k}$ be $k$ distinct prime numbers and let $P_k:=\\prod_{i=1}^k p_{j_i}$ be their product. \nFor $k\\geq 2$ and a given choice of residue classes, i.e., one residue class $r_i$ for each prime $p_i$, the number of available elements in $[1,P_k]$ is $A_k(p_{j_1}, \\ldots, p_{j_k})$ and the number of free elements is $F_k(p_{j_1}, \\ldots, p_{j_k})$.\nIn other words, for every choice of residue classes, there are $A_k(p_{j_1}, \\ldots, p_{j_k})$ integers $n$ with $1\\leq n \\leq P_k$ such that $\\gamma(n)\\leq 1$ and $F_k(p_{j_1}, \\ldots, p_{j_k})$ integers with $\\gamma(n)=0$.\n\\end{theorem}", "post_theorem_intro_text_len": 1536, "post_theorem_intro_text": "Moreover, our method can be generalised to counting free and available integers in arbitrary sets of $k$ arithmetic progressions generated by $k$ different primes as shown in Section \\ref{sec5}.\n\n\\begin{remark}\nThe problem of determining the number of free integers in the interval $[1,P_k]$ with residue classes $r_i=0$ for all $i$ is already mentioned by Dickson \\cite[p 439]{dickson} in the first volume of his History of the Theory of Numbers. He refers to results by A. de Polignac, J. Deschamps and H.J.S Smith for which Martin \\cite{martin} provides proofs in his manuscript. \nDetermining the number of available integers and the connection of the two counting problems to the determinants $a_k$ and $f_k$ are new to the best of our knowledge.\n\\end{remark}\n\n\\begin{remark}\nLemma \\ref{lem1} below gives a new characterisation of sequence $A005867$ from the The On-Line Encyclopedia of Integer Sequences \\cite{oeis2} in terms of determinants.\n\\end{remark}\n\n\\begin{remark}\nAs a side product, the proof of Theorem \\ref{thm:main} provides a quick way of calculating determinants of the form $a_k$ and $f_k$.\n\\end{remark}\n\nThe paper is organized as follows. In Section \\ref{sec2} we derive structural results about $a_k$ and $f_k$. In Section \\ref{sec3} we analyse the counting problem and obtain an equation that relates the available numbers after considering $k$ progressions to the available numbers after considering $k+1$ progressions. Theorem \\ref{thm:main} is proven in Section \\ref{sec4} and generalised in Section \\ref{sec5}.", "sketch": "The post-theorem introduction does not give an explicit proof outline for Theorem~\\ref{thm:main}, but it indicates the structure used to prove it: Section~\\ref{sec2} \"derive[s] structural results about $a_k$ and $f_k$\"; Section~\\ref{sec3} \"analyse[s] the counting problem\" and \"obtain[s] an equation that relates the available numbers after considering $k$ progressions to the available numbers after considering $k+1$ progressions\"; then \"Theorem~\\ref{thm:main} is proven in Section~\\ref{sec4}\" (and \"generalised in Section~\\ref{sec5}\").", "expanded_sketch": "The post-theorem introduction does not give an explicit proof outline for the main theorem, but it indicates the structure used to prove it: next it derives structural results about $a_k$ and $f_k$; then it analyses the counting problem and obtains an equation that relates the available numbers after considering $k$ progressions to the available numbers after considering $k+1$ progressions; then it proves the main theorem (and later generalises it).", "expanded_theorem": "\\label{thm:main}\nLet $p_1, \\ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\\prod_{i=1}^k p_i$ be their product. \nFor a given choice of residue classes, i.e., one residue class $r_i$ for each prime $p_i$, the number of available elements in $[1,P_k]$ is $a_k$ and the number of free elements is $f_k$.\nIn other words, for every choice of residue classes, there are $a_k$ integers $n$ with $1\\leq n \\leq P_k$ such that $\\gamma(n)\\leq 1$ and $f_k$ integers with $\\gamma(n)=0$.,\n", "theorem_type": ["Universal", "Existential–Universal"], "mcq": {"question": "Let $p_1,\\dots,p_k$ be the first $k$ prime numbers, and let $P_k=\\prod_{i=1}^k p_i$. For each $i$, choose a residue class $r_i\\pmod{p_i}$. For $1\\le n\\le P_k$, define\n$$\\gamma(n)=\\#\\{i: n\\equiv r_i\\pmod{p_i}\\}.$$ \nCall $n$ available if $\\gamma(n)\\le 1$ and free if $\\gamma(n)=0$. Also define\n$$a_k=\\det\\begin{bmatrix}\n2&1&\\cdots&1\\\\\n1&3&\\cdots&1\\\\\n\\vdots&\\vdots&\\ddots&\\vdots\\\\\n1&1&\\cdots&p_k\n\\end{bmatrix}$$\nand\n$$f_k=(-1)^k\\det\\begin{bmatrix}\n1&1&\\cdots&1&1\\\\\n2&1&\\cdots&1&1\\\\\n1&3&\\cdots&1&1\\\\\n\\vdots&\\vdots&\\ddots&\\vdots&\\vdots\\\\\n1&1&\\cdots&p_k&1\n\\end{bmatrix}.$$ \nWhich statement holds for the integers in $[1,P_k]$, uniformly over all choices of the residue classes $r_1,\\dots,r_k$?", "correct_choice": {"label": "A", "text": "For every choice of residue classes $r_i\\pmod{p_i}$, the number of integers $n$ with $1\\le n\\le P_k$ and $\\gamma(n)\\le 1$ is exactly $a_k$, and the number of integers $n$ with $\\gamma(n)=0$ is exactly $f_k$."}, "choices": [{"label": "B", "text": "For every choice of residue classes $r_i\\pmod{p_i}$, the number of integers $n$ with $1\\le n\\le P_k$ and $\\gamma(n)\\le 1$ is exactly $a_k$, and the number of integers $n$ with $\\gamma(n)=0$ is exactly $a_k-f_k$."}, {"label": "C", "text": "For every choice of residue classes $r_i\\pmod{p_i}$, the number of integers $n$ with $1\\le n\\le P_k$ and $\\gamma(n)=0$ is exactly $f_k$."}, {"label": "D", "text": "There exists a choice of residue classes $r_i\\pmod{p_i}$ such that the number of integers $n$ with $1\\le n\\le P_k$ and $\\gamma(n)\\le 1$ is exactly $a_k$, and the number of integers $n$ with $\\gamma(n)=0$ is exactly $f_k$."}, {"label": "E", "text": "For every choice of residue classes $r_i\\pmod{p_i}$, the number of integers $n$ with $1\\le n\\le P_k$ and $\\gamma(n)\\le 1$ is exactly $a_k$, and the number of integers $n$ with $\\gamma(n)=1$ is exactly $f_k$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "counting_estimate", "tampered_component": "relation between available and free counts", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "dropped the available-count conclusion involving $a_k$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "uniformity over all residue-class choices", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "confusion between free integers and singly covered integers", "template_used": "property_confusion"}]}}
{"id": "2512.16535v1", "paper_link": "http://arxiv.org/abs/2512.16535v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm:connectedSum}\n\tLet $M^n$ be a closed aspherical manifold and $N$ a spin manifold. Assume that the rational strong Novikov conjecture holds for $\\pi_1(M)$.\n\tThen the connected sum $M\\# N$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $M\\# N$ is Ricci-flat.", "start_pos": 8495, "end_pos": 8882, "label": "thm:connectedSum"}, "ref_dict": {"example": "\\begin{example}\\label{example}\n\tLet $M^n$ be a closed aspherical spin manifold with $n\\ge 5$, e.g., the $n$-torus. Suppose that $\\pi_1(M)$ is represented by finitely many disjoint embedded circles $S^1$ in $M$, whose tubular neighborhoods are diffeomorphic to $S^1 \\times B^{n-1}$. Remove these tubular neighborhoods and attach $B^2 \\times S^{n-2}$ to obtain a closed manifold $M_1$. Then $M_1$ is simply connected.  \n\n\tFinally, set $Z \\coloneqq M_1 \\# M_1$, which is a spin surgery of $M$. Note that $Z$ is simply connected, and the Dirac operator on $Z$ has vanishing $KO$-index as $Z$ is a double. Hence, $Z$ admits a metric with positive scalar curvature \\cite{Stolz}.\n\\end{example}", "conj:Novikov": "\\begin{conjecture}[Rational strong Novikov conjecture for fundamental groups of aspherical manifolds]\\label{conj:Novikov}\n\tLet $M$ be a closed aspherical manifold and $\\Gamma=\\pi_1(M)$. Then $\\Gamma$ satisfies the rational strong Novikov conjecture if\n\t\\[\n\t\\mu\\colon K_*(M)\\otimes\\Q \\to K_*(C^*_r(\\Gamma))\\otimes\\Q\n\t\\]\n\tis injective.\n\\end{conjecture}", "thm:main": "\\begin{theorem}\\label{thm:main}\n\tLet $M^n$ be a closed aspherical manifold and $\\Omega\\subset M$ a smooth region. Assume that\n\t$$\\pi_1(\\Omega)\\to \\pi_1(M)\\textup{ is the trivial map}.$$\n\tLet $N^n$ be a spin manifold with boundary, and $i\\colon \\partial N\\to \\partial \\Omega$ a diffeomorphism preserving the spin structure\\footnote{Although $M$ itself need not be spin, the triviality of $\\pi_1(\\Omega)\\to\\pi_1(M)$ implies that $\\Omega$ and $\\partial\\Omega$ inherit a canonical spin structure from the spin universal cover $\\widetilde M$.}. Set $Z\\coloneqq (M\\backslash\\Omega)\\cup_{\\partial\\Omega} N$. If the rational strong Novikov conjecture holds for $\\pi_1(M)$, then $Z$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $Z$ is Ricci-flat.\n\\end{theorem}", "thm:connectedSum": "\\begin{theorem}\\label{thm:connectedSum}\n\tLet $M^n$ be a closed aspherical manifold and $N$ a spin manifold. Assume that the rational strong Novikov conjecture holds for $\\pi_1(M)$.\n\tThen the connected sum $M\\# N$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $M\\# N$ is Ricci-flat.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 1742, "pre_theorem_intro_text": "The study of scalar curvature on Riemannian manifolds has played a central role in differential geometry and geometric topology for several decades. A landmark result due to Gromov and Lawson \\cite{GromovLawson,GromovLawsonAnnals}, using index theory, and independently to Schoen and Yau \\cite{SchoenYau3,SchoenYau}, using minimal hypersurface methods, asserts that the $n$-torus does not admit any Riemannian metric of positive scalar curvature. More generally, Gromov--Lawson--Rosenberg conjectured that no aspherical manifold admits a metric of positive scalar curvature. This conjecture is known to hold for aspherical manifolds whose fundamental groups satisfy the rational strong Novikov conjecture \\cite{Rosenberg,Yu}, as well as for aspherical manifolds of dimension at most five \\cite{ChodoshLi,Gromov5fold}.\n\nThe non-existence of positive scalar metric holds for more generally enlargeable manifolds, introduced by Gromov--Lawson \\cite{GromovLawson,GromovLawsonAnnals}, including connected sums of torus with a closed spin manifold. Using index-theoretic techniques, Su--Zhang \\cite{SuZhang} established a quantitative version of this obstruction for connected sums of enlargeable manifolds with non-spin closed manifolds. Subsequently, Wang--Zhang \\cite{WangZhang} and Su \\cite{SuRemark} showed that the connected sum of an enlargeable manifold with any spin manifold does not admit a metric of positive scalar curvature. Related non-existence results for positive scalar curvature on non-spin connected sums, obtained via minimal hypersurface methods, can be found in \\cite{ChodoshLi,ChenChuZhu,MR4291609}.\n\nIn this paper, we extend the above non-existence of positive scalar metrics to a significantly broader class of manifolds.", "context": "The study of scalar curvature on Riemannian manifolds has played a central role in differential geometry and geometric topology for several decades. A landmark result due to Gromov and Lawson \\cite{GromovLawson,GromovLawsonAnnals}, using index theory, and independently to Schoen and Yau \\cite{SchoenYau3,SchoenYau}, using minimal hypersurface methods, asserts that the $n$-torus does not admit any Riemannian metric of positive scalar curvature. More generally, Gromov--Lawson--Rosenberg conjectured that no aspherical manifold admits a metric of positive scalar curvature. This conjecture is known to hold for aspherical manifolds whose fundamental groups satisfy the rational strong Novikov conjecture \\cite{Rosenberg,Yu}, as well as for aspherical manifolds of dimension at most five \\cite{ChodoshLi,Gromov5fold}.\n\nThe non-existence of positive scalar metric holds for more generally enlargeable manifolds, introduced by Gromov--Lawson \\cite{GromovLawson,GromovLawsonAnnals}, including connected sums of torus with a closed spin manifold. Using index-theoretic techniques, Su--Zhang \\cite{SuZhang} established a quantitative version of this obstruction for connected sums of enlargeable manifolds with non-spin closed manifolds. Subsequently, Wang--Zhang \\cite{WangZhang} and Su \\cite{SuRemark} showed that the connected sum of an enlargeable manifold with any spin manifold does not admit a metric of positive scalar curvature. Related non-existence results for positive scalar curvature on non-spin connected sums, obtained via minimal hypersurface methods, can be found in \\cite{ChodoshLi,ChenChuZhu,MR4291609}.\n\nIn this paper, we extend the above non-existence of positive scalar metrics to a significantly broader class of manifolds.", "full_context": "The study of scalar curvature on Riemannian manifolds has played a central role in differential geometry and geometric topology for several decades. A landmark result due to Gromov and Lawson \\cite{GromovLawson,GromovLawsonAnnals}, using index theory, and independently to Schoen and Yau \\cite{SchoenYau3,SchoenYau}, using minimal hypersurface methods, asserts that the $n$-torus does not admit any Riemannian metric of positive scalar curvature. More generally, Gromov--Lawson--Rosenberg conjectured that no aspherical manifold admits a metric of positive scalar curvature. This conjecture is known to hold for aspherical manifolds whose fundamental groups satisfy the rational strong Novikov conjecture \\cite{Rosenberg,Yu}, as well as for aspherical manifolds of dimension at most five \\cite{ChodoshLi,Gromov5fold}.\n\nThe non-existence of positive scalar metric holds for more generally enlargeable manifolds, introduced by Gromov--Lawson \\cite{GromovLawson,GromovLawsonAnnals}, including connected sums of torus with a closed spin manifold. Using index-theoretic techniques, Su--Zhang \\cite{SuZhang} established a quantitative version of this obstruction for connected sums of enlargeable manifolds with non-spin closed manifolds. Subsequently, Wang--Zhang \\cite{WangZhang} and Su \\cite{SuRemark} showed that the connected sum of an enlargeable manifold with any spin manifold does not admit a metric of positive scalar curvature. Related non-existence results for positive scalar curvature on non-spin connected sums, obtained via minimal hypersurface methods, can be found in \\cite{ChodoshLi,ChenChuZhu,MR4291609}.\n\nIn this paper, we extend the above non-existence of positive scalar metrics to a significantly broader class of manifolds.\n\n\\begin{abstract}\n Let $M$ be a closed aspherical manifold. Assume that the rational strong Novikov conjecture holds for $\\pi_1(M)$. We show that on any spin surgery of $M$ along a region whose induced homomorphism on the fundamental group is trivial, every complete metric with non-negative scalar curvature is Ricci-flat. In particular, on the connected sum of $M$ with a spin manifold, any complete metric with non-negative scalar curvature is Ricci-flat.\n\\end{abstract}\n\nThe study of scalar curvature on Riemannian manifolds has played a central role in differential geometry and geometric topology for several decades. A landmark result due to Gromov and Lawson \\cite{GromovLawson,GromovLawsonAnnals}, using index theory, and independently to Schoen and Yau \\cite{SchoenYau3,SchoenYau}, using minimal hypersurface methods, asserts that the $n$-torus does not admit any Riemannian metric of positive scalar curvature. More generally, Gromov--Lawson--Rosenberg conjectured that no aspherical manifold admits a metric of positive scalar curvature. This conjecture is known to hold for aspherical manifolds whose fundamental groups satisfy the rational strong Novikov conjecture \\cite{Rosenberg,Yu}, as well as for aspherical manifolds of dimension at most five \\cite{ChodoshLi,Gromov5fold}.\n\nThe non-existence of positive scalar metric holds for more generally enlargeable manifolds, introduced by Gromov--Lawson \\cite{GromovLawson,GromovLawsonAnnals}, including connected sums of torus with a closed spin manifold. Using index-theoretic techniques, Su--Zhang \\cite{SuZhang} established a quantitative version of this obstruction for connected sums of enlargeable manifolds with non-spin closed manifolds. Subsequently, Wang--Zhang \\cite{WangZhang} and Su \\cite{SuRemark} showed that the connected sum of an enlargeable manifold with any spin manifold does not admit a metric of positive scalar curvature. Related non-existence results for positive scalar curvature on non-spin connected sums, obtained via minimal hypersurface methods, can be found in \\cite{ChodoshLi,ChenChuZhu,MR4291609}.\n\nWe remark that in Theorem \\ref{thm:connectedSum}, the manifold $N$ is allowed to be non-compact, and the metric on $M\\#N$ is not required to have bounded geometry. The rational strong Novikov conjecture for $\\pi_1(M)$ \\cite[Section~2]{Rosenberg} asserts that the assembly map from the $K$-homology of $M$ to the $K$-theory of the group $C^*$-algebra of $\\pi_1(M)$ is rationally injective; see Conjecture \\ref{conj:Novikov} for a precise formulation. The reader may consult \\cite{GongWuYu} for recent progress and a survey of classes of groups known to satisfy this conjecture. To date, no counterexample to the rational strong Novikov conjecture is known.\n\n\\begin{theorem}\\label{thm:main}\n Let $M^n$ be a closed aspherical manifold and $\\Omega\\subset M$ a smooth region. Assume that\n $$\\pi_1(\\Omega)\\to \\pi_1(M)\\textup{ is the trivial map}.$$\n Let $N^n$ be a spin manifold with boundary, and $i\\colon \\partial N\\to \\partial \\Omega$ a diffeomorphism preserving the spin structure\\footnote{Although $M$ itself need not be spin, the triviality of $\\pi_1(\\Omega)\\to\\pi_1(M)$ implies that $\\Omega$ and $\\partial\\Omega$ inherit a canonical spin structure from the spin universal cover $\\widetilde M$.}. Set $Z\\coloneqq (M\\backslash\\Omega)\\cup_{\\partial\\Omega} N$. If the rational strong Novikov conjecture holds for $\\pi_1(M)$, then $Z$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $Z$ is Ricci-flat.\n\\end{theorem}\n\n\\begin{corollary}\\label{coro:remove}\n Let $M^n$ be a closed aspherical manifold and $\\Omega\\subset M$ a smooth region. Assume that\n $$\\pi_1(\\Omega)\\to \\pi_1(M)\\textup{ is the trivial map},$$\n and the rational strong Novikov conjecture holds for $\\pi_1(M)$. Then $M\\backslash\\Omega$ does not carry any complete metric with positive scalar curvature. \n\\end{corollary}\n\n\\section{Non-existence of uniformly positive scalar curvature}\\label{sec:UPSC}\nIn this section, we prove a simple version of Theorem \\ref{thm:main} to illustrate the key ideas. This version follows from classical index-theoretic tools, while the proof of the full version requires additional new ingredients and will be presented in the next section.\n\\begin{proposition}\\label{prop:noPSC}\n Let $M^n$ be a closed aspherical manifold and $\\Omega\\subset M$ a smooth region. Assume that\n $$\\pi_1(\\Omega)\\to \\pi_1(M)\\textup{ is the trivial map}.$$\n Let $N^n$ be a spin manifold with boundary, and $i\\colon \\partial N\\to \\partial \\Omega$ a diffeomorphism preserving the spin structure. Set $Z\\coloneqq (M\\backslash\\Omega)\\cup_{\\partial\\Omega} N$. If the rational strong Novikov conjecture holds for $\\pi_1(M)$, then $Z$ does not carry any complete metric with uniform positive scalar curvature, i.e., scalar curvature bounded below by some positive constant.\n\\end{proposition}\n\n@article{Zeidler:2019aa,\n abstract = {Let $M$ be a closed connected spin manifold such that its spinor Dirac operator has non-vanishing (Rosenberg) index. We prove that for any Riemannian metric on $V = M \\times [-1,1]$ with scalar curvature bounded below by $σ> 0$, the distance between the boundary components of $V$ is at most $C_n/\\sqrtσ$, where $C_n = \\sqrt{(n-1)/{n}} \\cdot C$ with $C < 8(1+\\sqrt{2})$ being a universal constant. This verifies a conjecture of Gromov for such manifolds. In particular, our result applies to all high-dimensional closed simply connected manifolds $M$ which do not admit a metric of positive scalar curvature. We also establish a quadratic decay estimate for the scalar curvature of complete metrics on manifolds, such as $M \\times \\mathbb{R}^2$, which contain $M$ as a codimension two submanifold in a suitable way. Furthermore, we introduce the \"$\\mathcal{KO}$-width\" of a closed manifold and deduce that infinite $\\mathcal{KO}$-width is an obstruction to positive scalar curvature.},\n author = {Rudolf Zeidler},\n date-added = {2021-11-10 09:44:42 -0600},\n date-modified = {2021-11-10 09:44:42 -0600},\n eprint = {1905.08520},\n journal = {Journal of Differential Geometry},\n title = {Band width estimates via the {D}irac operator},\n url = {https://arxiv.org/pdf/1905.08520.pdf},\n year = {2020},\n bdsk-url-1 = {https://arxiv.org/pdf/1905.08520.pdf}}\n\n\\begin{conjecture}[Rational strong Novikov conjecture for fundamental groups of aspherical manifolds]\\label{conj:Novikov}\n\tLet $M$ be a closed aspherical manifold and $\\Gamma=\\pi_1(M)$. Then $\\Gamma$ satisfies the rational strong Novikov conjecture if\n\t\\[\n\t\\mu\\colon K_*(M)\\otimes\\Q \\to K_*(C^*_r(\\Gamma))\\otimes\\Q\n\t\\]\n\tis injective.\n\\end{conjecture}\n\n\\begin{theorem}\\label{thm:connectedSum}\n\tLet $M^n$ be a closed aspherical manifold and $N$ a spin manifold. Assume that the rational strong Novikov conjecture holds for $\\pi_1(M)$.\n\tThen the connected sum $M\\# N$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $M\\# N$ is Ricci-flat.\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:main}\n\tLet $M^n$ be a closed aspherical manifold and $\\Omega\\subset M$ a smooth region. Assume that\n\t$$\\pi_1(\\Omega)\\to \\pi_1(M)\\textup{ is the trivial map}.$$\n\tLet $N^n$ be a spin manifold with boundary, and $i\\colon \\partial N\\to \\partial \\Omega$ a diffeomorphism preserving the spin structure\\footnote{Although $M$ itself need not be spin, the triviality of $\\pi_1(\\Omega)\\to\\pi_1(M)$ implies that $\\Omega$ and $\\partial\\Omega$ inherit a canonical spin structure from the spin universal cover $\\widetilde M$.}. Set $Z\\coloneqq (M\\backslash\\Omega)\\cup_{\\partial\\Omega} N$. If the rational strong Novikov conjecture holds for $\\pi_1(M)$, then $Z$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $Z$ is Ricci-flat.\n\\end{theorem}", "post_theorem_intro_text_len": 3454, "post_theorem_intro_text": "We remark that in Theorem \\ref{thm:connectedSum}, the manifold $N$ is allowed to be non-compact, and the metric on $M\\#N$ is not required to have bounded geometry. The rational strong Novikov conjecture for $\\pi_1(M)$ \\cite[Section~2]{Rosenberg} asserts that the assembly map from the $K$-homology of $M$ to the $K$-theory of the group $C^*$-algebra of $\\pi_1(M)$ is rationally injective; see Conjecture \\ref{conj:Novikov} for a precise formulation. The reader may consult \\cite{GongWuYu} for recent progress and a survey of classes of groups known to satisfy this conjecture. To date, no counterexample to the rational strong Novikov conjecture is known.\n\nWe further generalize Theorem \\ref{thm:connectedSum} to spin surgeries on aspherical manifolds. \n\n\\begin{theorem}\\label{thm:main}\n\tLet $M^n$ be a closed aspherical manifold and $\\Omega\\subset M$ a smooth region. Assume that\n\t$$\\pi_1(\\Omega)\\to \\pi_1(M)\\textup{ is the trivial map}.$$\n\tLet $N^n$ be a spin manifold with boundary, and $i\\colon \\partial N\\to \\partial \\Omega$ a diffeomorphism preserving the spin structure\\footnote{Although $M$ itself need not be spin, the triviality of $\\pi_1(\\Omega)\\to\\pi_1(M)$ implies that $\\Omega$ and $\\partial\\Omega$ inherit a canonical spin structure from the spin universal cover $\\widetilde M$.}. Set $Z\\coloneqq (M\\backslash\\Omega)\\cup_{\\partial\\Omega} N$. If the rational strong Novikov conjecture holds for $\\pi_1(M)$, then $Z$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $Z$ is Ricci-flat.\n\\end{theorem}\n\nTheorem \\ref{thm:connectedSum} is a special case of Theorem \\ref{thm:main} by choosing $\\Omega$ to be a small ball in $M$. The assumption that $\\pi_1(\\Omega)$ maps trivially into $\\pi_1(M)$ is essential; in Example \\ref{example} we construct a spin surgery on a closed aspherical manifold that admits a metric of positive scalar curvature when this condition fails.\n\nOur proofs rely on Dirac operator methods. A fundamental ingredient is the relative index theorem for higher indices due to Xie and Yu \\cite[Theorem~A]{XieYu14}. Inspired by the approaches in \\cite{SuZhang,WangXieLinf}, we construct an explicit representative of the relative higher index and compute it in the $K$-theory of the group $C^*$-algebra. We also provide a Dirac-operator-based proof of the Ricci-flatness statement, following ideas similar to those in \\cite{WangZhu}.\n\nAs an immediate application, by taking $N=\\partial\\Omega\\times[0,1)$ in Theorem \\ref{thm:main}, we obtain the following corollary.\n\n\\begin{corollary}\\label{coro:remove}\n\tLet $M^n$ be a closed aspherical manifold and $\\Omega\\subset M$ a smooth region. Assume that\n\t$$\\pi_1(\\Omega)\\to \\pi_1(M)\\textup{ is the trivial map},$$\n\tand the rational strong Novikov conjecture holds for $\\pi_1(M)$. Then $M\\backslash\\Omega$ does not carry any complete metric with positive scalar curvature. \n\\end{corollary}\n\nThe paper is organized as follows. In Section \\ref{sec:preliminary}, we review the higher indices and the strong Novikov conjecture. In Section \\ref{sec:UPSC}, we prove a simplified version of Theorem \\ref{thm:main}, establishing the non-existence of uniformly positive scalar curvature metrics on spin surgeries. The full proof of Theorem \\ref{thm:main} is given in Section \\ref{sec:nnsc}.\n\n\\medskip\n\\noindent\\textbf{Acknowledgments.}  \nThe author would like to thank Zhizhang Xie for helpful discussions.", "sketch": "Our proofs rely on Dirac operator methods. A fundamental ingredient is the relative index theorem for higher indices due to Xie and Yu \\cite[Theorem~A]{XieYu14}. Inspired by the approaches in \\cite{SuZhang,WangXieLinf}, we construct an explicit representative of the relative higher index and compute it in the $K$-theory of the group $C^*$-algebra. We also provide a Dirac-operator-based proof of the Ricci-flatness statement, following ideas similar to those in \\cite{WangZhu}. (Theorem~\\ref{thm:connectedSum} is treated as a special case of Theorem~\\ref{thm:main} by choosing $\\Omega$ to be a small ball in $M$.)", "expanded_sketch": "Our proofs rely on Dirac operator methods. A fundamental ingredient is the relative index theorem for higher indices due to Xie and Yu (Theorem~A in Xie--Yu, \\emph{[title not provided in ref\\_dict]} (2014)). Inspired by the approaches in Su--Zhang, \\emph{[title not provided in ref\\_dict]}, and Wang--Xie--Linf, \\emph{[title not provided in ref\\_dict]}, we construct an explicit representative of the relative higher index and compute it in the $K$-theory of the group $C^*$-algebra. We also provide a Dirac-operator-based proof of the Ricci-flatness statement, following ideas similar to those in Wang--Zhu, \\emph{[title not provided in ref\\_dict]}. (In establishing the main theorem, we treat it as a special case of\n\\begin{theorem}\\label{thm:main}\n\tLet $M^n$ be a closed aspherical manifold and $\\Omega\\subset M$ a smooth region. Assume that\n\t$$\\pi_1(\\Omega)\\to \\pi_1(M)\\textup{ is the trivial map}.$$\n\tLet $N^n$ be a spin manifold with boundary, and $i\\colon \\partial N\\to \\partial \\Omega$ a diffeomorphism preserving the spin structure\\footnote{Although $M$ itself need not be spin, the triviality of $\\pi_1(\\Omega)\\to\\pi_1(M)$ implies that $\\Omega$ and $\\partial\\Omega$ inherit a canonical spin structure from the spin universal cover $\\widetilde M$.}. Set $Z\\coloneqq (M\\backslash\\Omega)\\cup_{\\partial\\Omega} N$. If the rational strong Novikov conjecture holds for $\\pi_1(M)$, then $Z$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $Z$ is Ricci-flat.\n\\end{theorem}\nby choosing $\\Omega$ to be a small ball in $M$.)", "expanded_theorem": "\\label{thm:connectedSum}\n\tLet $M^n$ be a closed aspherical manifold and $N$ a spin manifold. Assume that the rational strong Novikov conjecture holds for $\\pi_1(M)$.\n\tThen the connected sum $M\\# N$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $M\\# N$ is Ricci-flat.", "theorem_type": ["Implication", "Nonexistence"], "mcq": {"question": "Let $M^n$ be a closed aspherical manifold, let $N$ be a spin manifold of the same dimension $n$, and write $M\\#N$ for their connected sum. Assume that the rational strong Novikov conjecture holds for $\\pi_1(M)$, meaning that for $\\Gamma=\\pi_1(M)$ the assembly map\n\\[\n\\mu\\colon K_*(M)\\otimes\\mathbb{Q}\\to K_*(C_r^*(\\Gamma))\\otimes\\mathbb{Q}\n\\]\nis injective. Which of the following conclusions about complete Riemannian metrics on $M\\#N$ is valid under these hypotheses?", "correct_choice": {"label": "A", "text": "The manifold $M\\#N$ admits no complete Riemannian metric with positive scalar curvature. Moreover, if a complete Riemannian metric on $M\\#N$ has non-negative scalar curvature everywhere, then that metric is Ricci-flat."}, "choices": [{"label": "B", "text": "The manifold $M\\#N$ admits no complete Riemannian metric with non-negative scalar curvature. In particular, every complete metric on $M\\#N$ must have scalar curvature negative somewhere."}, {"label": "C", "text": "The manifold $M\\#N$ admits no complete Riemannian metric with positive scalar curvature."}, {"label": "D", "text": "The manifold $M\\#N$ admits no complete Riemannian metric with positive scalar curvature. Moreover, if a complete Riemannian metric on $M\\#N$ has non-negative scalar curvature everywhere, then that metric is flat."}, {"label": "E", "text": "If the manifold $M\\#N$ admits no complete Riemannian metric with positive scalar curvature, then the rational strong Novikov conjecture must hold for $\\pi_1(M)$. Moreover, any complete Riemannian metric on $M\\#N$ with non-negative scalar curvature is Ricci-flat."}], "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": "nonnegative_vs_positive_case_split", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_Ricci_flatness_conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "Ricci_flatness_not_flatness", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "direction_of_Novikov_hypothesis", "template_used": "quantifier_dependence"}]}}
{"id": "2512.16655v2", "paper_link": "http://arxiv.org/abs/2512.16655v2", "theorems_cnt": 3, "theorem": {"env_name": "thm", "content": "\\label{main theorem}\n  Let $\\theta\\in (0,\\frac{\\pi}{2}]$ and $1\\leq k\\leq n-1$.  Suppose $f\\in C^{2}(\\C_{\\theta})$ is connected to $1$ in $\\mathcal{L}_{-\\frac{1}{k}}$.  Then there exists a $C^{3, \\gamma}$ $(\\gamma\\in(0,1))$ strictly convex capillary hypersurface $\\Sigma$ in $ \\overline{{\\mathbb{R}}^{n+1}_{+}}$, such that $\\widehat{\\Sigma}$ satisfies the prescribed  $k$-th capillary area measure problem. \n  Moreover, $\\Sigma$ is unique up to a horizontal translation in $\\overline{{\\mathbb{R}}^{n+1}_{+}}$.", "start_pos": 64453, "end_pos": 64950, "label": "main theorem"}, "ref_dict": {"pro ortho": "\\begin{prop}\\label{pro ortho}\n\t\tIf  Eq. \\eqref{eq-k} is solvable, then $f$ satisfies\n\n\t\t\\begin{eqnarray}\\label{ortho-condition}\n    \\int_{\\C_{\\theta}}f(\\xi)\\<\\xi, E_{\\alpha}\\>d\\H^{n}=0,\\quad \\forall~ 1\\leq \\alpha\\leq n.\n\\end{eqnarray}\t\t\n\t\\end{prop}", "main theorem": "\\begin{thm}\\label{main theorem}\n  Let $\\theta\\in (0,\\frac{\\pi}{2}]$ and $1\\leq k\\leq n-1$.  Suppose $f\\in C^{2}(\\C_{\\theta})$ is connected to $1$ in $\\mathcal{L}_{-\\frac{1}{k}}$.  Then there exists a $C^{3, \\gamma}$ $(\\gamma\\in(0,1))$ strictly convex capillary hypersurface $\\Sigma$ in $ \\ov{{\\RR}^{n+1}_{+}}$, such that $\\widehat{\\S}$ satisfies the prescribed  $k$-th capillary area measure problem. \n  Moreover, $\\S$ is unique up to a horizontal translation in $\\ov{{\\RR}^{n+1}_{+}}$. \n\\end{thm}", "measure": "\\begin{eqnarray}\\label{measure}\n    S_{k}^c(\\widehat{\\Sigma}, \\beta)\\coloneqq \\binom{n}{k}^{-1}\\int_{\\beta}(\\sin^{2}\\theta+\\cos\\theta\\<\\xi, e\\>)\\sigma_{k}(r)d{\\H}^{n},\\quad \\text{for}~0\\leq k\\leq n.\n\\end{eqnarray}", "c0 est": "\\begin{lem}\\label{c0 est}\nLet $\\theta\\in(0, \\pi)$ and $h$ be a strictly convex solution of  Eq. \\eqref{eq-k} that satisfies \\eqref{orth cond}.  \nThen there exists a positive constant $C$ depending  on $n, \\min\\limits_{\\C_{\\theta}}f$, and $\\max\\limits_{\\C_\\theta} f$, such that\n\\begin{eqnarray*}\n\t0<\\min_{\\C_\\theta} h\\leq \\max_{\\C_\\theta} h\\leq C.\n\\end{eqnarray*} \n\\end{lem}", "pro-2.6": "\\begin{prop}\\label{pro-2.6}\n The capillary  Christoffel-Minkowski problem \t\tis equivalent to finding a strictly convex solution to  Eq.  \\eqref{eq-k}.\\end{prop}", "def1.1": "\\begin{defn}\\label{def1.1}\n  We say that a function $f\\in C^{2}(\\C_{\\theta})$ is (strictly) convex if the matrix $\\n^{2}f+f\\sigma$ is positive semi-definite (respectively, positive definite) on $\\C_{\\theta}$.\n\\end{defn}", "eq-k": "\\begin{eqnarray}\n\\label{eq-k}\n\t\\begin{array}{rcll}\\vspace{2mm}\n\t\t\\sigma_{k}(\\n^2 h+h\\sigma)&=&f,& \\quad  \\hbox{ in }  \\C_{\\theta},\\\\\n\t\t\\n_\\mu h&=&\\cot\\theta h, &\\quad \\hbox{ on }  \\p \\C_\\theta,\n\t\\end{array}\n \\end{eqnarray}", "ortho-condition1": "\\begin{eqnarray}\\label{ortho-condition1}\n    \\int_{\\C_{\\theta}}f(\\xi)\\<\\xi, E_{\\alpha}\\>d\\mathcal{H}^{n}=0,\\quad \\forall~ 1\\leq \\alpha\\leq n,\n\\end{eqnarray}"}, "pre_theorem_intro_text_len": 12832, "pre_theorem_intro_text": "In convex geometry, the prescribed $k$-th area measure problem is a well-known problem. It asks whether, for a given positive function $f$ on $\\mathbb{S}^{n}$, one can find a closed, strongly convex hypersurface $\\Sigma$ with $k$-th symmetric function of the principal curvature radii equal to $f$ as a function on its normals? When $k=n$, it corresponds to the famous Minkowski problem, which has been fully solved through the works of Minkowski \\cite{Min}, Alexandrov \\cite{Alex}, Lewy \\cite{Lewy}, Nirenberg \\cite{Nire}, Pogorelov \\cite{Pog52, Pog}, Cheng-Yau \\cite{CY76} and many others. Analytically, the Minkowski problem is equivalent to solving a Monge-Amp\\`ere equation on $\\mathbb{S}^{n}$. The study of the Minkowski problem has played a significant role in advancing research on fully nonlinear equations, especially on the Monge-Amp\\`ere equation. We can also see a closely related work by Xia \\cite{Xia} for an anisotropic Minkowski problem for closed hypersurfaces.\nWhen $k=1$, it is known as the Christoffel problem, which is equivalent to finding convex solutions to the Laplace equation on $\\mathbb{S}^{n}$.  Firey \\cite{Firey, Firey1} and Berg \\cite{Berg}  found necessary and sufficient conditions using the Green function of the Laplacian on $\\mathbb{S}^{n}$. More recently, Li-Wan-Wang \\cite{LWW} expressed the Christoffel problem as a Laplace equation on the entire space $\\mathbb{R}^{n+1}$ and provided a new and simpler necessary and sufficient condition for the Christoffel problem.  Further details on Christoffel's problem can be found in \\cite{Chri,  Hilbert, Hur, Suss}. For the intermediate cases, i.e., $2\\leq k\\leq n-1$, it is referred to as the Christoffel-Minkowski problem. Guan-Ma \\cite{GM}  solved this problem under a continuous homotopy condition, which was removed later by  Sheng-Wang-Trudinger \\cite{STW}. In addition to the elliptic method, Chou-Wang \\cite{CW00} and Bryan-Ivaki-Scheuer \\cite{BIS2023} provided a flow method to solve the above problems.\nFor a more comprehensive description of the prescribed area measure problem in convex geometry, see \\cite{Guan-note} and \\cite{Sch}. \n\n\\subsection{Motivation and set up}\nIn recent years, there has been growing interest in the study of capillary hypersurfaces, including free boundary hypersurfaces, along with their geometry and topology, see, for instance, \\cite{FL14, FS16, JWXZ, MWW-GCF, SWX, WW20, WWX22, WX2019, WeX21} and references therein. \nCapillary hypersurfaces originated from physics through the work of Thomas Young. Later,\nYoung, Liouville, and Gauss formulated it as a variational problem using the so-called wetting energy.  For capillary hypersurfaces, we refer to the elegant book by Finn \\cite{Finn}, and also a book by Maggi  \\cite{Maggi}. In this paper, with a slight abuse of terminology,\na smooth compact embedded hypersurface in $\\overline{{\\mathbb{R}}_+^{n+1}}$  whose boundary lies on $\\partial\\overline{{\\mathbb{R}}_+^{n+1}}$ is called {\\it capillary hypersurface} if\nit intersects $\\partial\\overline{{{\\mathbb{R}}}_+^{n+1}}$ at a constant contact angle $\\theta\\in (0,\\pi)$. \n   Let $\\nu$ be the unit outward normal of $\\Sigma$. The contact angle $\\theta $ is defined by $$ \\cos (\\pi -\\theta)=\\langle \\nu, e \\rangle,\\quad \\rm{along~\\partial\\Sigma},$$\n\twhere $e\\coloneqq -E_{n+1}$, $E_{n+1}$ is the $(n+1)$-th unit coordinate vector in $\\overline{\\mathbb{R}^{n+1}_+}$ and thus $e$ is the unit outward normal of $\\partial \\overline{\\mathbb{R}^{n+1}_+}$.\nThis means that a capillary hypersurface in this paper is not necessarily a hypersurface of constant mean curvature.\nIf $\\Sigma$ is strictly convex, then it is clear that the image of the Gauss map, namely,  $\\nu$, lies on the spherical cap \n\\begin{eqnarray*}\n\t\t\\mathbb{S}^n_\\theta \\coloneqq  \\left\\{ x\\in \\mathbb{S}^n \\,|\\,  \\langle x, E_{n+1} \\rangle \\geq \\cos \\theta \\right\\}.\n\\end{eqnarray*}\nInstead of using the usual Gauss map $\\nu$, it is more convenient to use the following map\n$$\\tilde \\nu\\coloneqq  T \\circ \\nu: \\Sigma \\to \\C_\\theta,$$\nwhere $\\C_\\theta$ is a spherical cap, defined by\n\\begin{eqnarray*}\n\t\t\\C_{\\theta }  \\coloneqq  \\left\\{\\xi\\in \\overline{\\mathbb{R}^{n+1}_+} \\mid |\\xi-\\cos\\theta e|= 1 \\right\\},\n\\end{eqnarray*}\n\tand $T:\\mathbb{S}^n_\\theta\\to\\C_\\theta$ is defined by $ T(y)\\coloneqq y+\\cos\\theta e$,  which is a translation in the vertical direction.\n We refer to $\\tilde \\nu$  as the {\\it capillary Gauss map} of $\\Sigma$.  It turns out that $\\tilde \\nu: \\Sigma\\to \\C_\\theta$ is a diffeomorphism. Consequently,  one can reparametrize $\\Sigma$ using its inverse on $\\C_\\theta$, see, e.g., \\cite[Section 2]{MWWX}. \n\nIn \\cite{MWW-AIM}, the authors formulated a capillary Minkowski problem, which concerns the existence of a strictly convex capillary hypersurface $\\Sigma\\subset \\overline{\\mathbb{R}^{n+1}_{+}}$ with a prescribed Gauss-Kronecker curvature on $\\C_{\\theta}$. We have established the following result in \\cite[Theorem 1.1]{MWW-AIM}.\n\n\\\n\n\\noindent{\\bf Theorem A}. \n{\\it Let $\\theta\\in (0,\\frac{\\pi}{2}]$ and $f\\in C^2(\\C_\\theta)$ be a positive function  satisfying  \\eqref{ortho-condition1} below.  Then there exists a $C^{3,\\gamma}$ $(\\gamma\\in(0,1))$ strictly convex capillary hypersurface $\\Sigma\\subset \\overline{\\mathbb{R}^{n+1}_+}$ such that its Gauss-Kronecker curvature $K$ satisfying $$K(\\tilde \\nu^{-1}(\\xi))=\n\t\t{f^{-1}(\\xi)},$$ for all $\\xi\\in \\C_\\theta$. Moreover, $\\Sigma$ is unique up to a horizontal translation in $\\overline{\\mathbb{R}^{n+1}_+}$.}\n\n  \\\n\nThe capillary Minkowski problem can also be viewed as a prescribed capillary area measure problem, finding a strictly convex capillary hypersurface such that the induced capillary area measure equals a prescribed Borel measure on $\\C_{\\theta}$. See \\cite{MWW-AIM}. The area measures are an important subject in the study of convex bodies, and they are the local versions of quermassintegrals in the Brunn-Minkowski theory.  For more details about the classical Christoffel-Minkowski problem, see \\cite[Chapter~8]{Sch}.\n\nThe primary objective of this article is to extend our work presented in \\cite{MWW-AIM} and study the capillary Christoffel-Minkowski problem or the prescribed  $k$-th capillary area measure problem for the convex body with a capillary boundary in $\\overline{\\mathbb{R}^{n+1}_{+}}$.  \n Firstly, we introduce a novel concept of local parallel sets for the capillary convex bodies. We say $\\widehat{\\Sigma}$ is a \\textit{capillary convex body} (a compact convex set with non-empty interior) in $\\mathbb{R}^{n+1}$ if it is a bounded closed region in $\\overline{\\mathbb{R}^{n+1}_+}$ enclosed by a strictly convex capillary hypersurface $\\Sigma$ and  $\\partial \\overline{\\mathbb{R}^{n+1}_{+}}$. The class of capillary convex bodies in $\\overline{{\\mathbb{R}}^{n+1}_+}$ is denoted by $\\K_{\\theta}$.   Given a Borel set $\\beta\\subset \\C_{\\theta}$ and a capillary convex body $\\widehat{\\Sigma}\\subset \\K_{\\theta}$, we consider the following set :\n\\begin{eqnarray*}\n    B_{s}(\\widehat{\\Sigma}, \\beta)\\coloneqq  \\left\\{Y\\in \\overline{{\\mathbb{R}}^{n+1}_{+}} \\mid  Y=X+t\\tilde{\\nu}(X),~0<t<s,~\\text{for}~X\\in \\Sigma ~ \\text{and} ~\\tilde{\\nu}(X)\\in \\beta \\right\\}.\n\\end{eqnarray*}\nWhen $\\beta=\\C_{\\theta}$, the set\n\\begin{eqnarray*}\n    A_{s}(\\widehat{\\Sigma}, \\C_{\\theta}) \\coloneqq \\left\\{Y\\in \\overline{{\\mathbb{R}}^{n+1}_{+}} \\mid  Y=X+s\\tilde{\\nu}(X), ~X\\in \\Sigma~\\text{and}~s>0 \\right\\}\n\\end{eqnarray*}\nis also a strictly convex capillary hypersurface in $\\overline{{\\mathbb{R}}^{n+1}_{+}}$ (see, for instance, \\cite[Remark 2.17]{MWWX} or \\cite[Proposition~2.3]{JWXZ}).\n\nDirect calculation yields the following local Steiner-type formula (see also \\cite[Remark~2.17]{MWWX} for a global version),\n\\begin{eqnarray*}\n    \\text{Vol}(B_{s}(\\widehat{\\Sigma}, \\beta))&=&\\sum\\limits_{k=0}^{n}\\frac{s^{n+1-k}}{n+1-k}\\int_{\\Sigma \\cap \\tilde{\\nu}^{-1}(\\beta)}(1+\\cos\\theta \\<\\nu, e\\>)\\sigma_{n-k}(\\kappa)d\\mu\\notag\\\\\n    &=&\\frac{1}{n+1}\\sum\\limits_{k=0}^{n}s^{n+1-k}\\binom{n+1}{k}S_{k}^c(\\widehat{\\Sigma}, \\beta),\n\\end{eqnarray*}\nwhere $S_k^c(\\widehat{\\Sigma},\\beta)$ is given by\n\\begin{eqnarray}\\label{measure}\n    S_{k}^c(\\widehat{\\Sigma}, \\beta)\\coloneqq \\binom{n}{k}^{-1}\\int_{\\beta}(\\sin^{2}\\theta+\\cos\\theta\\<\\xi, e\\>)\\sigma_{k}(r)d{\\mathcal{H}}^{n},\\quad \\text{for}~0\\leq k\\leq n.\n\\end{eqnarray}\nHere, $\\kappa$ and $r$ are the set of the principal curvatures and principal curvature radii of $\\Sigma$ respectively, $\\sigma_{k}$ is the $k$-th elementary symmetric polynomial function, $d \\mathcal{H}^n$ and $d\\mu$ are the $n$-dimensional Hausdorff measure on $\\C_\\theta$\n and  $\\Sigma$ respectively. The superscript $c$ indicates the presence of the \\textit{capillary} effect in our notation. Moreover, $S_k^c (\\widehat{\\Sigma}, \\C_{\\theta})$ $(0\\leq k\\leq n)$ coincide with the quermassintegrals of convex capillary hypersurface introduced in  \\cite[Eq. (1.8)]{WWX22}, see also \\cite[Lemma~2.14]{MWWX}. \n\n For each capillary convex body $\\widehat{\\Sigma}$, by \\eqref{measure} we introduce  a \\textit{$k$-th capillary area measure} on $\\C_{\\theta}$, which is given  by \\begin{eqnarray}\\notag\n     dS_k^c \\coloneqq  \\ell\\sigma_{k}(\\nabla^{2}h+h\\sigma)d\\mathcal{H}^{n}, ~~~0\\leq k\\leq n,\n \\end{eqnarray}  \n where $$\\ell(\\xi) \\coloneqq \\sin^{2}\\theta+\\cos\\theta\\<\\xi, e\\>,$$ and $h$ is the support function of $\\Sigma$. In particular, when $\\theta=\\frac \\pi 2$, $dS_k^c$ coincides with the classical $k$-th area measure for the convex body, see for instance \\cite[Section 5.1]{Sch}. The main goal of this paper is to identify a capillary convex body $\\widehat{\\Sigma}$, such that the  $k$-th capillary area measure is prescribed on $\\C_{\\theta}$. We formulate the problem as follows.\n\n \\\n\n\\noindent\t\\textbf{Capillary Christoffel-Minkowski problem:} \\textit{Given a positive, $C^{2}$ function $f$ on $\\C_{\\theta}$, when does  there exist a capillary convex body $\\widehat{\\Sigma}$ in $\\overline{\\mathbb{R}^{n+1}_+}$ such that\n\\begin{eqnarray*}\n    dS_k^c=\\ell fd\\mathcal{H}^{n}~?\n\\end{eqnarray*}}\n\n\\\n\nWhen $k=n$, it reduces to the $n$-th capillary area measure and corresponds to the capillary Minkowski problem, which was solved by the authors in \\cite{MWW-AIM}. \nFor the intermediate case  $2\\leq k\\leq n-1$, we refer to this problem as the \\textit{capillary Christoffel-Minkowski problem}. From Proposition~\\ref{pro-2.6} below, the capillary Christoffel-Minkowski problem is equivalent to finding  convex solutions to the following Hessian type equation with Robin boundary value condition,  \\begin{eqnarray}\n\\label{eq-k}\n\t\\begin{array}{rcll}\\vspace{2mm}\n\t\t\\sigma_{k}(\\nabla^2 h+h\\sigma)&=&f,& \\quad  \\hbox{ in }  \\C_{\\theta},\\\\\n\t\t\\n_\\mu h&=&\\cot\\theta h, &\\quad \\hbox{ on }  \\partial \\C_\\theta,\n\t\\end{array}\n \\end{eqnarray}\nwhere   $h$ is an unknown function, $\\nabla h$ and $\\nabla^2h$ are the gradient and the Hessian of $h$ on $\\C_\\theta$ with respect to the standard spherical metric $\\sigma$ on $\\C_{\\theta}$ respectively  and $\\mu$ is the unit outward normal of $\\partial\\C_\\theta\\subset \\C_{\\theta}$.\nIf Eq. \\eqref{eq-k} is solvable, the function $f$  satisfies\n\\begin{eqnarray}\\label{ortho-condition1}\n    \\int_{\\C_{\\theta}}f(\\xi)\\<\\xi, E_{\\alpha}\\>d\\mathcal{H}^{n}=0,\\quad \\forall~ 1\\leq \\alpha\\leq n,\n\\end{eqnarray} (cf. Proposition \\ref{pro ortho} for more details)\nwhere  $\\{E_i\\}_{i=1}^{n+1}$ is the standard coordinate unit vector of $\\overline{\\mathbb{R}^{n+1}_+}$. For the capillary Minkowski problem, the necessary condition \\eqref{ortho-condition1} is also sufficient, while the situation becomes subtle for the general case $k\\neq n$. This phenomenon also occurs in the closed case, see for instance \\cite{GM, GMZ, STW}. Before presenting the main result,  we introduce some definitions.\n\\begin{defn}\\label{def1.1}\n  We say that a function $f\\in C^{2}(\\C_{\\theta})$ is (strictly) convex if the matrix $\\nabla^{2}f+f\\sigma$ is positive semi-definite (respectively, positive definite) on $\\C_{\\theta}$.\n\\end{defn}\n\\begin{defn}\n  For $s\\in \\mathbb{R}$, let $\\mathcal{L}_{s}$ be the set of positive $C^{2}$ functions $f$ on $\\C_{\\theta}$ such that $f$ satisfies \\eqref{ortho-condition1} and $f^{s}$ is convex on $\\C_{\\theta}$, in the sense of Definition \\ref{def1.1}. Moreover, we say $f$ is connected to $g$ in $\\mathcal{L}_{s}$ if there exists a continuous path $q(t)\\in \\mathcal{L}_{s}$, $0\\leq t\\leq 1$, such that $q(0)=g$, $q(1)=f$, and $\\n_{\\mu}q(t)\\geq 0$ on $\\partial\\C_{\\theta}$ for any $t\\in[0, 1]$. \n\\end{defn}\n\nIt is worth noting that, compared to the definition of homotopy given in \\cite[Definition 1.1]{GM}, our setting includes an additional requirement: a Neumann boundary condition will be imposed. This condition plays a crucial role in establishing the constant rank theorem for ensuring the convex solutions of Eq. \\eqref{eq-k}.\n\n\\subsection {Main result}\nThe main theorem of this paper is as follows.", "context": "\\noindent{\\bf Theorem A}. \n{\\it Let $\\theta\\in (0,\\frac{\\pi}{2}]$ and $f\\in C^2(\\C_\\theta)$ be a positive function satisfying \\eqref{ortho-condition1} below. Then there exists a $C^{3,\\gamma}$ $(\\gamma\\in(0,1))$ strictly convex capillary hypersurface $\\Sigma\\subset \\overline{\\mathbb{R}^{n+1}_+}$ such that its Gauss-Kronecker curvature $K$ satisfying $$K(\\tilde \\nu^{-1}(\\xi))=\n {f^{-1}(\\xi)},$$ for all $\\xi\\in \\C_\\theta$. Moreover, $\\Sigma$ is unique up to a horizontal translation in $\\overline{\\mathbb{R}^{n+1}_+}$.}\n\nFor each capillary convex body $\\widehat{\\Sigma}$, by \\eqref{measure} we introduce a \\textit{$k$-th capillary area measure} on $\\C_{\\theta}$, which is given by \\begin{eqnarray}\\notag\n dS_k^c \\coloneqq \\ell\\sigma_{k}(\\nabla^{2}h+h\\sigma)d\\mathcal{H}^{n}, ~~~0\\leq k\\leq n,\n \\end{eqnarray} \n where $$\\ell(\\xi) \\coloneqq \\sin^{2}\\theta+\\cos\\theta\\<\\xi, e\\>,$$ and $h$ is the support function of $\\Sigma$. In particular, when $\\theta=\\frac \\pi 2$, $dS_k^c$ coincides with the classical $k$-th area measure for the convex body, see for instance \\cite[Section 5.1]{Sch}. The main goal of this paper is to identify a capillary convex body $\\widehat{\\Sigma}$, such that the $k$-th capillary area measure is prescribed on $\\C_{\\theta}$. We formulate the problem as follows.\n\nWhen $k=n$, it reduces to the $n$-th capillary area measure and corresponds to the capillary Minkowski problem, which was solved by the authors in \\cite{MWW-AIM}. \nFor the intermediate case $2\\leq k\\leq n-1$, we refer to this problem as the \\textit{capillary Christoffel-Minkowski problem}. From Proposition~\\ref{pro-2.6} below, the capillary Christoffel-Minkowski problem is equivalent to finding convex solutions to the following Hessian type equation with Robin boundary value condition, \\begin{eqnarray}\n\\label{eq-k}\n \\begin{array}{rcll}\\vspace{2mm}\n \\sigma_{k}(\\nabla^2 h+h\\sigma)&=&f,& \\quad \\hbox{ in } \\C_{\\theta},\\\\\n \\n_\\mu h&=&\\cot\\theta h, &\\quad \\hbox{ on } \\partial \\C_\\theta,\n \\end{array}\n \\end{eqnarray}\nwhere $h$ is an unknown function, $\\nabla h$ and $\\nabla^2h$ are the gradient and the Hessian of $h$ on $\\C_\\theta$ with respect to the standard spherical metric $\\sigma$ on $\\C_{\\theta}$ respectively and $\\mu$ is the unit outward normal of $\\partial\\C_\\theta\\subset \\C_{\\theta}$.\nIf Eq. \\eqref{eq-k} is solvable, the function $f$ satisfies\n\\begin{eqnarray}\\label{ortho-condition1}\n \\int_{\\C_{\\theta}}f(\\xi)\\<\\xi, E_{\\alpha}\\>d\\mathcal{H}^{n}=0,\\quad \\forall~ 1\\leq \\alpha\\leq n,\n\\end{eqnarray} (cf. Proposition \\ref{pro ortho} for more details)\nwhere $\\{E_i\\}_{i=1}^{n+1}$ is the standard coordinate unit vector of $\\overline{\\mathbb{R}^{n+1}_+}$. For the capillary Minkowski problem, the necessary condition \\eqref{ortho-condition1} is also sufficient, while the situation becomes subtle for the general case $k\\neq n$. This phenomenon also occurs in the closed case, see for instance \\cite{GM, GMZ, STW}. Before presenting the main result, we introduce some definitions.\n\\begin{defn}\\label{def1.1}\n We say that a function $f\\in C^{2}(\\C_{\\theta})$ is (strictly) convex if the matrix $\\nabla^{2}f+f\\sigma$ is positive semi-definite (respectively, positive definite) on $\\C_{\\theta}$.\n\\end{defn}\n\\begin{defn}\n For $s\\in \\mathbb{R}$, let $\\mathcal{L}_{s}$ be the set of positive $C^{2}$ functions $f$ on $\\C_{\\theta}$ such that $f$ satisfies \\eqref{ortho-condition1} and $f^{s}$ is convex on $\\C_{\\theta}$, in the sense of Definition \\ref{def1.1}. Moreover, we say $f$ is connected to $g$ in $\\mathcal{L}_{s}$ if there exists a continuous path $q(t)\\in \\mathcal{L}_{s}$, $0\\leq t\\leq 1$, such that $q(0)=g$, $q(1)=f$, and $\\n_{\\mu}q(t)\\geq 0$ on $\\partial\\C_{\\theta}$ for any $t\\in[0, 1]$. \n\\end{defn}\n\nIt is worth noting that, compared to the definition of homotopy given in \\cite[Definition 1.1]{GM}, our setting includes an additional requirement: a Neumann boundary condition will be imposed. This condition plays a crucial role in establishing the constant rank theorem for ensuring the convex solutions of Eq. \\eqref{eq-k}.\n\n\\subsection {Main result}\nThe main theorem of this paper is as follows.", "full_context": "\\noindent{\\bf Theorem A}. \n{\\it Let $\\theta\\in (0,\\frac{\\pi}{2}]$ and $f\\in C^2(\\C_\\theta)$ be a positive function satisfying \\eqref{ortho-condition1} below. Then there exists a $C^{3,\\gamma}$ $(\\gamma\\in(0,1))$ strictly convex capillary hypersurface $\\Sigma\\subset \\overline{\\mathbb{R}^{n+1}_+}$ such that its Gauss-Kronecker curvature $K$ satisfying $$K(\\tilde \\nu^{-1}(\\xi))=\n {f^{-1}(\\xi)},$$ for all $\\xi\\in \\C_\\theta$. Moreover, $\\Sigma$ is unique up to a horizontal translation in $\\overline{\\mathbb{R}^{n+1}_+}$.}\n\nFor each capillary convex body $\\widehat{\\Sigma}$, by \\eqref{measure} we introduce a \\textit{$k$-th capillary area measure} on $\\C_{\\theta}$, which is given by \\begin{eqnarray}\\notag\n dS_k^c \\coloneqq \\ell\\sigma_{k}(\\nabla^{2}h+h\\sigma)d\\mathcal{H}^{n}, ~~~0\\leq k\\leq n,\n \\end{eqnarray} \n where $$\\ell(\\xi) \\coloneqq \\sin^{2}\\theta+\\cos\\theta\\<\\xi, e\\>,$$ and $h$ is the support function of $\\Sigma$. In particular, when $\\theta=\\frac \\pi 2$, $dS_k^c$ coincides with the classical $k$-th area measure for the convex body, see for instance \\cite[Section 5.1]{Sch}. The main goal of this paper is to identify a capillary convex body $\\widehat{\\Sigma}$, such that the $k$-th capillary area measure is prescribed on $\\C_{\\theta}$. We formulate the problem as follows.\n\nWhen $k=n$, it reduces to the $n$-th capillary area measure and corresponds to the capillary Minkowski problem, which was solved by the authors in \\cite{MWW-AIM}. \nFor the intermediate case $2\\leq k\\leq n-1$, we refer to this problem as the \\textit{capillary Christoffel-Minkowski problem}. From Proposition~\\ref{pro-2.6} below, the capillary Christoffel-Minkowski problem is equivalent to finding convex solutions to the following Hessian type equation with Robin boundary value condition, \\begin{eqnarray}\n\\label{eq-k}\n \\begin{array}{rcll}\\vspace{2mm}\n \\sigma_{k}(\\nabla^2 h+h\\sigma)&=&f,& \\quad \\hbox{ in } \\C_{\\theta},\\\\\n \\n_\\mu h&=&\\cot\\theta h, &\\quad \\hbox{ on } \\partial \\C_\\theta,\n \\end{array}\n \\end{eqnarray}\nwhere $h$ is an unknown function, $\\nabla h$ and $\\nabla^2h$ are the gradient and the Hessian of $h$ on $\\C_\\theta$ with respect to the standard spherical metric $\\sigma$ on $\\C_{\\theta}$ respectively and $\\mu$ is the unit outward normal of $\\partial\\C_\\theta\\subset \\C_{\\theta}$.\nIf Eq. \\eqref{eq-k} is solvable, the function $f$ satisfies\n\\begin{eqnarray}\\label{ortho-condition1}\n \\int_{\\C_{\\theta}}f(\\xi)\\<\\xi, E_{\\alpha}\\>d\\mathcal{H}^{n}=0,\\quad \\forall~ 1\\leq \\alpha\\leq n,\n\\end{eqnarray} (cf. Proposition \\ref{pro ortho} for more details)\nwhere $\\{E_i\\}_{i=1}^{n+1}$ is the standard coordinate unit vector of $\\overline{\\mathbb{R}^{n+1}_+}$. For the capillary Minkowski problem, the necessary condition \\eqref{ortho-condition1} is also sufficient, while the situation becomes subtle for the general case $k\\neq n$. This phenomenon also occurs in the closed case, see for instance \\cite{GM, GMZ, STW}. Before presenting the main result, we introduce some definitions.\n\\begin{defn}\\label{def1.1}\n We say that a function $f\\in C^{2}(\\C_{\\theta})$ is (strictly) convex if the matrix $\\nabla^{2}f+f\\sigma$ is positive semi-definite (respectively, positive definite) on $\\C_{\\theta}$.\n\\end{defn}\n\\begin{defn}\n For $s\\in \\mathbb{R}$, let $\\mathcal{L}_{s}$ be the set of positive $C^{2}$ functions $f$ on $\\C_{\\theta}$ such that $f$ satisfies \\eqref{ortho-condition1} and $f^{s}$ is convex on $\\C_{\\theta}$, in the sense of Definition \\ref{def1.1}. Moreover, we say $f$ is connected to $g$ in $\\mathcal{L}_{s}$ if there exists a continuous path $q(t)\\in \\mathcal{L}_{s}$, $0\\leq t\\leq 1$, such that $q(0)=g$, $q(1)=f$, and $\\n_{\\mu}q(t)\\geq 0$ on $\\partial\\C_{\\theta}$ for any $t\\in[0, 1]$. \n\\end{defn}\n\nIt is worth noting that, compared to the definition of homotopy given in \\cite[Definition 1.1]{GM}, our setting includes an additional requirement: a Neumann boundary condition will be imposed. This condition plays a crucial role in establishing the constant rank theorem for ensuring the convex solutions of Eq. \\eqref{eq-k}.\n\n\\subsection {Main result}\nThe main theorem of this paper is as follows.\n\nFor each capillary convex body $\\widehat{\\S}$, by \\eqref{measure} we introduce a \\textit{$k$-th capillary area measure} on $\\C_{\\theta}$, which is given by \\begin{eqnarray}\\notag\n dS_k^c \\coloneqq \\ell\\sigma_{k}(\\n^{2}h+h\\sigma)d\\mathcal{H}^{n}, ~~~0\\leq k\\leq n,\n \\end{eqnarray} \n where $$\\ell(\\xi) \\coloneqq \\sin^{2}\\theta+\\cos\\theta\\<\\xi, e\\>,$$ and $h$ is the support function of $\\S$. In particular, when $\\theta=\\frac \\pi 2$, $dS_k^c$ coincides with the classical $k$-th area measure for the convex body, see for instance \\cite[Section 5.1]{Sch}. The main goal of this paper is to identify a capillary convex body $\\widehat{\\S}$, such that the $k$-th capillary area measure is prescribed on $\\C_{\\theta}$. We formulate the problem as follows.\n\nIt is worth noting that, compared to the definition of homotopy given in \\cite[Definition 1.1]{GM}, our setting includes an additional requirement: a Neumann boundary condition will be imposed. This condition plays a crucial role in establishing the constant rank theorem for ensuring the convex solutions of Eq. \\eqref{eq-k}.\n\n\\\n\nIn this section, we establish the a priori estimates for the strictly convex solution of Eq. \\eqref{eq-k}. \nFor any solution $h$ to Eq. \\eqref{eq-k}, it is easy to verify that \n\\begin{eqnarray*}\n h(\\xi)+\\<\\xi,E_\\alpha\\> ,\\quad \\text{ for all } 1\\leq \\a\\leq n,\n\\end{eqnarray*} is still a solution. Up to a horizontal translation, we assume $h$ satisfies the following orthogonal condition:\n\\begin{eqnarray}\\label{orth cond}\n \\int_{\\C_\\theta} h(\\xi) \\<\\xi,E_\\alpha\\> d\\H^{n}=0, \\quad \\text{ for all } 1\\leq \\a\\leq n.\n\\end{eqnarray}\nUnder the condition \\eqref{orth cond}, we are able to establish the following a priori estimates.\n\\begin{thm}\\label{priori estimate}\nLet $\\theta\\in (0, \\frac{\\pi}{2})$. Suppose that $h$ is a strictly convex solution of Eq. \\eqref{eq-k} and satisfies the condition \\eqref{orth cond}. For any integer $m\\geq 2$ and $\\gamma\\in(0,1)$, there exists a constant $C>0$ depending on $n, k,$ $\\min\\limits_{\\C_{\\theta}} f$ and $\\|f\\|_{C^{m}(\\C_\\theta)}$, such that\n \\begin{eqnarray}\\label{sch est}\n \\|h\\|_{C^{m+1,\\gamma}(\\C_\\theta)}\\leq C.\n \\end{eqnarray}\n\\end{thm}\n\n\\begin{lem}\\label{c0 est}\nLet $\\theta\\in(0, \\pi)$ and $h$ be a strictly convex solution of Eq. \\eqref{eq-k} that satisfies \\eqref{orth cond}. \nThen there exists a positive constant $C$ depending on $n, \\min\\limits_{\\C_{\\theta}}f$, and $\\max\\limits_{\\C_\\theta} f$, such that\n\\begin{eqnarray*}\n 0<\\min_{\\C_\\theta} h\\leq \\max_{\\C_\\theta} h\\leq C.\n\\end{eqnarray*} \n\\end{lem}\n\\begin{proof} \nFirst of all, due to \\eqref{orth cond}, by \\cite[Lemma 3.1]{MWW-AIM}, we have $h>0$ and the origin lies in the interior of $\\widehat{\\partial\\Sigma}$. \nLet $R_{0}>0$ be the smallest number of $R$ such that \n \\begin{eqnarray}\\label{enclosed}\n \\widehat \\S\\subset \\widehat{R\\C_\\theta},\n \\end{eqnarray}where $R\\C_\\theta\\coloneqq \\{\\xi \\in \\ov{\\RR^{n+1}_+}: |\\xi-\\cos\\theta R e|=R\\}$. It is clear that there exists $X_0\\in \\Sigma \\cap R_0\\C_{\\theta } $. Denote $\\widehat{X}_0\\coloneqq \\frac{X_0 }{R_0}\\in \\C_\\theta$.\nFor any $\\xi \\in \\C_\\theta$, we have \n \\begin{eqnarray}\\label{lower bdd of h}\n h(\\xi)&=&\\max_{Y\\in \\S}\\<\\xi-\\cos\\theta e, Y\\>\\notag \\\\\n &\\geq& \\max \\left\\{0, \\<\\xi-\\cos\\theta e,X_0\\> \\right\\}\\notag \\\\\n &=& \\max \\left\\{0, R_0\\<\\xi-\\cos\\theta e,\\widehat X_0\\> \\right\\}.\n ~~~\\end{eqnarray} \nMultiplying $f$ on both sides of \\eqref{lower bdd of h} and integrating over $ \\C_\\theta$, we see\n\\begin{eqnarray}\\label{R1}\n R_0\\leq \\left(\\int_{\\C_\\theta}h(\\xi)\\cdot f(\\xi)d \\H^{n}\\right) \\left(\\int_{\\C_\\theta}\\max \\left\\{0, \\<\\xi-\\cos\\theta e,\\widehat X_0\\> \\right \\}\\cdot f(\\xi) d \\H^{n}\\right) ^{-1}.\n\\end{eqnarray}\nNote that \n\\begin{eqnarray}\\label{R2}\n \\int_{\\C_{\\theta}}h(\\xi)\\cdot f(\\xi) d \\H^{n}=\n\\int_{\\tilde{\\nu}(\\Sigma)}h(\\xi)\\cdot f(\\xi)d\\H^{n}=\\int_{\\S}\\<X, \\nu\\>\\sigma_{n-k}(\\kappa)d\\mu.\n\\end{eqnarray}\nTogether with the Alexandrov-Fenchel inequalities for convex capillary hypersurfaces \\cite[Theorem~1.2]{MWWX} for $\\theta\\in(0, \\pi)$ and the Minkowski formulas for capillary hypersurfaces \\cite[Proposition~2.6]{WWX22}, we obtain\n\\begin{eqnarray}\n C_{n, k}\\left(\\int_{\\S}\\<X, \\nu\\>\\sigma_{n-k}(\\kappa)d\\mu\\right)^{\\frac{1}{k+1}}&\\leq& \\left(\\int_{\\Sigma}\\<X, \\nu\\>\\sigma_{n-k+1}(\\kappa)d\\mu\\right)^{\\frac{1}{k}}\\notag \\\\\n &=&\\left(\\frac{k}{n-k+1}\\int_{\\C_{\\theta}}f(\\xi)d\\H^{n}\\right)^{\\frac{1}{k}}.\\label{R3}\n\\end{eqnarray}\nTogether with \\eqref{R1}, \\eqref{R2} and \\eqref{R3}, we conclude that \n\\begin{eqnarray}\n R_{0}\n &\\leq& C_{n,k}\\left(\\int_{\\C_{\\theta}}f(\\xi)d\\H^{n}\\right)^{\\frac{k+1}{k}}\\cdot\\left(\\inf\\limits_{\\eta\\in \\C_{\\theta}}\\int_{\\C_\\theta}\\max\\{0, \\<\\xi-\\cos\\theta e, \\eta\\>\\}\\cdot f(\\xi) d \\H^{n}\\right) ^{-1}.\\quad ~~~\\label{upper bound of R0}\n\\end{eqnarray}\nUsing \\eqref{enclosed} and \\eqref{upper bound of R0}, we see \n\\begin{eqnarray*}\n h(\\xi) =\\<\\xi-\\cos\\theta e , \\tilde{\\nu}^{-1}(\\xi)\\>\\leq |\\tilde{\\nu}^{-1}(\\xi)| <2R_0\\leq C(n, \\min_{\\C_\\theta} f,\\max_{\\C_\\theta} f).\n\\end{eqnarray*}\n We complete the proof of Lemma \\ref{c0 est}.\n\\end{proof}\n\\subsection{\\texorpdfstring{$C^1$ }{}estimate}\n In this subsection, we get the gradient estimate for the strictly convex solution $h$ of Eq. \\eqref{eq-k}. This estimate is a direct consequence of the convexity and the $C^{0}$ estimate.\n\\begin{lem}\\label{C1 est}\n Let $\\theta\\in (0, \\pi)$, and let $h$ be a strictly convex solution of Eq. \\eqref{eq-k}. Then \\begin{eqnarray}\\label{gradient ps}\n \\max\\limits_{\\C_{\\theta}} |\\n h|\\leq C,\n \\end{eqnarray}\n where the constant $C$ only depends on $\\|h\\|_{C^{0}(\\C_{\\theta})}$.\n\\end{lem}\n\\begin{proof}\n Consider the function\n \\begin{eqnarray*}\n P=|\\n h|^{2}+h^{2}.\n \\end{eqnarray*}\nSuppose that $P$ attains its maximum value at some point, say $\\xi_{0}\\in \\C_{\\theta}$. If $\\xi_{0}\\in \\C_{\\theta}\\setminus \\partial\\C_{\\theta}$, by the maximal condition, we have\n\\begin{eqnarray*}\n 0=\\n_{e_i}P=2h_{k}h_{ki}+2hh_{i},\\quad \\text{for}~1\\leq i\\leq n.\n\\end{eqnarray*}\nSince $h$ is a strictly convex function, we have $\\n h(\\xi_{0})=0$. Combining with Lemma \\ref{c0 est}, we conclude that \\eqref{gradient ps} holds.\n\n\\section{Convexity}\\label{sec-convex}\nThe primary goal of this section is to obtain a strictly convex solution of Eq. \\eqref{eq-k}. Based on the celebrated result of Guan-Ma \\cite{GM} and with suitable assumptions on the function $f$, we show that the convex solution of Eq. \\eqref{eq-k} is strictly convex. To begin with, we establish the following key lemma.\n\\begin{lem}\\label{lem-def}\n Let $\\theta\\in (0, \\frac{\\pi}{2})$ and $\\tilde{F}(W) \\coloneqq \\sigma_{k}(W)$. Suppose that $h\\in C^{2}(\\C_{\\theta})$ is a solution of Eq. \\eqref{eq-k}, and assume that $W$ is positive semi-definite on $\\C_\\theta$. Suppose there exists a positive constant $C_{0}$ such that for a fixed integer $l $ $(k\\leq l\\leq n-1)$, $\\sigma_{l}(W)\\geq C_{0}$ for all $\\xi\\in \\C_{\\theta}$. If $f\\in \\mathcal{L}_{-\\frac{1}{k}}$ and $\\n_{\\mu}f\\geq 0$ on $\\partial \\C_{\\theta}$, denote $$\\phi\\coloneqq \\sigma_{l+1}(W(\\xi)),$$ then there exist constants $C_{1}, C_{2}$ depending only on $n, k, C_{0}, \\|h\\|_{C^{3}},$ and $\\|f\\|_{C^{2}}$, such that $\\phi$ satisfies \n \\begin{eqnarray}\n & \\tilde{F}^{ij}\\phi_{ij}\\leq C_{1}|\\n \\phi|+C_{2}\\phi, & \\quad \\rm{in} ~~ \\C_{\\theta} ,\\label{Diff-ine}\\\\\n &\\n_{\\mu} \\phi\\geq 0, & \\quad {\\rm{on}}~\\partial \\C_{\\theta}.\\label{boundary ine}\n \\end{eqnarray}\n \\end{lem}\n \\begin{proof}\n The differential inequality \\eqref{Diff-ine} follows from \\cite[Lemma~4.1]{GM} and $f\\in \\mathcal{L}_{-\\frac{1}{k}}$. We just need to prove the boundary condition \\eqref{boundary ine}.", "post_theorem_intro_text_len": 2591, "post_theorem_intro_text": "\\\n\nWe conclude the introduction by outlining the strategy for solving  Eq. \\eqref{eq-k}. If $\\theta=\\frac{\\pi}{2}$, this problem can be reduced to the closed case using a reflection argument. Hence, we focus on the case $\\theta<\\frac \\pi 2$ in proving Theorem \\ref{main theorem}. We will employ the method of continuity.  We need to establish a  priori estimates for solutions to Eq. \\eqref{eq-k} and show that the convexity is preserved. To establish the $C^{0}$ estimate, we modify the approach in  \\cite[Lemma~3]{CY76}, \\cite[Lemma 3.1]{MWW-AIM} and combine it with a general form of capillary Alexandrov-Fenchel inequalities for capillary hypersurfaces presented in \\cite[Theorem~1.2]{MWWX} for $\\theta\\in(0, \\pi)$.  Hence our $C^0$-estimate for \\eqref{eq-k} holds for all contact angle $\\theta\\in(0, \\pi)$, see Lemma \\ref{c0 est}. The $C^1$ estimate follows from the strict convexity of $h$ and the $C^0$ estimate. In contrast to the closed case considered in \\cite[Proposition 3.2]{GM}, our global $C^2$ estimate depends on the gradient estimates, due to the specific choice of test functions used to derive the boundary $C^2$ estimate. The choice of the test functions is motivated by the works of \\cite{LTU, MQ} and \\cite{MWW-AIM}.\nFor the preservation of the convexity, we apply the celebrated result of Guan-Ma \\cite{GM} on the Constant Rank Theorem.\nIt is worth noting that the assumption $\\theta\\in (0, \\frac{\\pi}{2})$ is crucial not only for deriving the $C^2$-estimate, but also for showing the preservation of the convexity.\nA similar assumption was employed in our previous works \\cite{MWW-Weingarten} and \\cite{MWW-Lp}, concerning the problems of convex capillary surfaces with prescribed $k$-th Weingarten curvature and prescribed capillary $L_p$-surface area measures for convex capillary bodies, respectively.  Nevertheless, we expect that the above results still hold for $\\theta > \\frac{\\pi}{2}$.\n\n\\\n\n\\textbf{The rest of the article is structured as follows.} \n\tIn Section \\ref{sec2}, we list some properties of elementary symmetric functions and introduce some notations and basic facts about capillary hypersurfaces. Moreover, we prove the necessary condition \\eqref{ortho-condition1} for the capillary Christoffel-Minkowski problem. \n\tIn Section \\ref{sec3}, we establish a priori estimates for the solution to Eq. \\eqref{eq-k}. In Section \\ref{sec-convex}, under certain assumptions on $f$, we investigate the convexity of the solution to Eq. \\eqref{eq-k}.\n In the final section, by using the method of continuity, we prove Theorem \\ref{main theorem}.\n\t\\vspace{.2cm}", "sketch": "To prove Theorem~\\ref{main theorem}, they first note that if $\\theta=\\frac{\\pi}{2}$ the problem “can be reduced to the closed case using a reflection argument,” so they “focus on the case $\\theta<\\frac \\pi 2$.” The proof then “employ[s] the method of continuity,” which requires establishing “a priori estimates for solutions to Eq.~\\eqref{eq-k} and show[ing] that the convexity is preserved.”\n\nFor the estimates: the $C^{0}$ estimate is obtained by “modify[ing] the approach in \\cite[Lemma~3]{CY76}, \\cite[Lemma 3.1]{MWW-AIM} and combin[ing] it with a general form of capillary Alexandrov-Fenchel inequalities… in \\cite[Theorem~1.2]{MWWX},” yielding a $C^0$ bound “for all contact angle $\\theta\\in(0,\\pi)$.” The “$C^1$ estimate follows from the strict convexity of $h$ and the $C^0$ estimate.” The “global $C^2$ estimate depends on the gradient estimates,” due to the “specific choice of test functions used to derive the boundary $C^2$ estimate,” motivated by \\cite{LTU, MQ} and \\cite{MWW-AIM}.\n\nFor convexity preservation, they “apply the celebrated result of Guan-Ma \\cite{GM} on the Constant Rank Theorem,” and emphasize that the assumption $\\theta\\in(0,\\frac{\\pi}{2})$ is “crucial not only for deriving the $C^2$-estimate, but also for showing the preservation of the convexity.” Finally, “by using the method of continuity,” they conclude the existence/uniqueness statement of Theorem~\\ref{main theorem}.", "expanded_sketch": "To prove the main theorem, they first note that if $\\theta=\\frac{\\pi}{2}$ the problem “can be reduced to the closed case using a reflection argument,” so they “focus on the case $\\theta<\\frac \\pi 2$.” The proof then “employ[s] the method of continuity,” which requires establishing “a priori estimates for solutions to\n\\begin{eqnarray}\n\\label{eq-k}\n\t\\begin{array}{rcll}\\vspace{2mm}\n\t\t\\sigma_{k}(\\n^2 h+h\\sigma)&=&f,& \\quad  \\hbox{ in }  \\C_{\\theta},\\\\\n\t\t\\n_\\mu h&=&\\cot\\theta h, &\\quad \\hbox{ on }  \\p \\C_\\theta,\n\t\\end{array}\n \\end{eqnarray}\n and show[ing] that the convexity is preserved.\n\nFor the estimates: the $C^{0}$ estimate is obtained by “modify[ing] the approach in \\cite[Lemma~3]{CY76}, \\cite[Lemma 3.1]{MWW-AIM} and combin[ing] it with a general form of capillary Alexandrov-Fenchel inequalities… in \\cite[Theorem~1.2]{MWWX},” yielding a $C^0$ bound “for all contact angle $\\theta\\in(0,\\pi)$.” The “$C^1$ estimate follows from the strict convexity of $h$ and the $C^0$ estimate.” The “global $C^2$ estimate depends on the gradient estimates,” due to the “specific choice of test functions used to derive the boundary $C^2$ estimate,” motivated by \\cite{LTU, MQ} and \\cite{MWW-AIM}.\n\nFor convexity preservation, they “apply the celebrated result of Guan-Ma \\cite{GM} on the Constant Rank Theorem,” and emphasize that the assumption $\\theta\\in(0,\\frac{\\pi}{2})$ is “crucial not only for deriving the $C^2$-estimate, but also for showing the preservation of the convexity.” Finally, “by using the method of continuity,” they conclude the existence/uniqueness statement of the main theorem.", "expanded_theorem": "\\label{main theorem}\n  Let $\\theta\\in (0,\\frac{\\pi}{2}]$ and $1\\leq k\\leq n-1$.  Suppose $f\\in C^{2}(\\C_{\\theta})$ is connected to $1$ in $\\mathcal{L}_{-\\frac{1}{k}}$.  Then there exists a $C^{3, \\gamma}$ $(\\gamma\\in(0,1))$ strictly convex capillary hypersurface $\\Sigma$ in $ \\overline{{\\mathbb{R}}^{n+1}_{+}}$, such that $\\widehat{\\Sigma}$ satisfies the prescribed  $k$-th capillary area measure problem. \n  Moreover, $\\Sigma$ is unique up to a horizontal translation in $\\overline{{\\mathbb{R}}^{n+1}_{+}}$.,", "theorem_type": ["Existence", "Uniqueness"], "mcq": {"question": "Let \\(\\theta\\in(0,\\pi/2]\\), \\(1\\le k\\le n-1\\), \\(e=E_{n+1}\\), and let \\(\\mathcal C_\\theta\\subset \\mathbb S^n\\) be the spherical cap \\(\\{\\xi\\in\\mathbb S^n:\\langle \\xi,e\\rangle>-\\cos\\theta\\}\\). For \\(s\\in\\mathbb R\\), let \\(\\mathcal L_s\\) be the set of positive \\(C^2\\)-functions \\(g\\) on \\(\\mathcal C_\\theta\\) such that\n\\[\n\\int_{\\mathcal C_\\theta} g(\\xi)\\langle \\xi,E_\\alpha\\rangle\\,d\\mathcal H^n=0\\quad\\text{for }\\alpha=1,\\dots,n,\n\\]\nand \\(g^s\\) is convex in the sense that \\(\\nabla^2(g^s)+g^s\\sigma\\) is positive semidefinite on \\(\\mathcal C_\\theta\\). Say that \\(f\\) is connected to \\(1\\) in \\(\\mathcal L_{-1/k}\\) if there is a continuous path \\(q(t)\\in \\mathcal L_{-1/k}\\), \\(0\\le t\\le1\\), with \\(q(0)=1\\), \\(q(1)=f\\), and \\(\\partial_\\mu q(t)\\ge0\\) on \\(\\partial\\mathcal C_\\theta\\) for all \\(t\\). For a capillary convex body \\(\\widehat\\Sigma\\), its \\(k\\)-th capillary area measure on \\(\\mathcal C_\\theta\\) is\n\\[\ndS_k^c=\\ell(\\xi)\\,\\sigma_k(\\nabla^2h+h\\sigma)\\,d\\mathcal H^n,\n\\qquad \\ell(\\xi)=\\sin^2\\theta+\\cos\\theta\\langle \\xi,e\\rangle,\n\\]\nwhere \\(h\\) is the support function of \\(\\Sigma\\). If \\(f\\in C^2(\\mathcal C_\\theta)\\) is connected to \\(1\\) in \\(\\mathcal L_{-1/k}\\), which statement holds?", "correct_choice": {"label": "A", "text": "There exists a \\(C^{3,\\gamma}\\) hypersurface \\(\\Sigma\\subset\\overline{\\mathbb R^{n+1}_+}\\), for some \\(\\gamma\\in(0,1)\\), which is strictly convex and capillary, such that its associated capillary convex body \\(\\widehat\\Sigma\\) has prescribed \\(k\\)-th capillary area measure\n\\[\ndS_k^c=\\ell(\\xi)f(\\xi)\\,d\\mathcal H^n\\quad\\text{on }\\mathcal C_\\theta.\n\\]\nMoreover, any two such hypersurfaces differ only by a horizontal translation in \\(\\overline{\\mathbb R^{n+1}_+}\\)."}, "choices": [{"label": "B", "text": "There exists a \\(C^{3,\\gamma}\\) hypersurface \\(\\Sigma\\subset\\overline{\\mathbb R^{n+1}_+}\\), for some \\(\\gamma\\in(0,1)\\), which is strictly convex and capillary, such that its associated capillary convex body \\(\\widehat\\Sigma\\) has prescribed \\(k\\)-th capillary area measure\n\\[\ndS_k^c=\\ell(\\xi)f(\\xi)\\,d\\mathcal H^n\\quad\\text{on }\\mathcal C_\\theta.\n\\]\nMoreover, such a hypersurface is unique in \\(\\overline{\\mathbb R^{n+1}_+}\\) without allowing any horizontal translation."}, {"label": "C", "text": "There exists a \\(C^{3,\\gamma}\\) hypersurface \\(\\Sigma\\subset\\overline{\\mathbb R^{n+1}_+}\\), for some \\(\\gamma\\in(0,1)\\), which is strictly convex and capillary, such that its associated capillary convex body \\(\\widehat\\Sigma\\) has prescribed \\(k\\)-th capillary area measure\n\\[\ndS_k^c=\\ell(\\xi)f(\\xi)\\,d\\mathcal H^n\\quad\\text{on }\\mathcal C_\\theta.\n\\]"}, {"label": "D", "text": "For every \\(\\theta\\in(0,\\pi)\\), if \\(f\\in C^2(\\mathcal C_\\theta)\\) is connected to \\(1\\) in \\(\\mathcal L_{-1/k}\\), then there exists a \\(C^{3,\\gamma}\\) hypersurface \\(\\Sigma\\subset\\overline{\\mathbb R^{n+1}_+}\\), for some \\(\\gamma\\in(0,1)\\), which is strictly convex and capillary, such that its associated capillary convex body \\(\\widehat\\Sigma\\) has prescribed \\(k\\)-th capillary area measure\n\\[\ndS_k^c=\\ell(\\xi)f(\\xi)\\,d\\mathcal H^n\\quad\\text{on }\\mathcal C_\\theta,\n\\]\nand any two such hypersurfaces differ only by a horizontal translation in \\(\\overline{\\mathbb R^{n+1}_+}\\)."}, {"label": "E", "text": "If \\(f\\in C^2(\\mathcal C_\\theta)\\) satisfies the orthogonality condition\n\\[\n\\int_{\\mathcal C_\\theta} f(\\xi)\\langle \\xi,E_\\alpha\\rangle\\,d\\mathcal H^n=0\\quad\\text{for }\\alpha=1,\\dots,n,\n\\]\nthen there exists a \\(C^{3,\\gamma}\\) hypersurface \\(\\Sigma\\subset\\overline{\\mathbb R^{n+1}_+}\\), for some \\(\\gamma\\in(0,1)\\), which is strictly convex and capillary, such that its associated capillary convex body \\(\\widehat\\Sigma\\) has prescribed \\(k\\)-th capillary area measure\n\\[\ndS_k^c=\\ell(\\xi)f(\\xi)\\,d\\mathcal H^n\\quad\\text{on }\\mathcal C_\\theta.\n\\]\nMoreover, any two such hypersurfaces differ only by a horizontal translation in \\(\\overline{\\mathbb R^{n+1}_+}\\)."}], "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": "translation-invariant uniqueness", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "uniqueness up to horizontal translation", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "characteristic", "tampered_component": "angle range restricted to \\((0,\\pi/2]\\)", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "homotopy/convexity-preserving connectedness in \\(\\mathcal L_{-1/k}\\)", "template_used": "quantifier_dependence"}]}}
{"id": "2512.17821v1", "paper_link": "http://arxiv.org/abs/2512.17821v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm:main thm}\nLet $5 \\leq k \\leq 11$ and $0\\leq i \\leq k-1$ be integers. The only solutions to the equation\n\\begin{align}\\label{eq:main eqn of study}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}(n+jd) = y^3\n\\end{align}\nin integers $n,d,y$ with $ny \\neq 0$, $d \\geq 1$, and $\\textup{gcd}(n,d) = 1$ are given by\n\\begin{equation}\\label{eq:solns for main thm}\n\\begin{split}\n(k,i,n,d) = \\, &(5,0,-14,5),(5,0,-11,5),(5,1,-8,3),(5,3,-4,3), \\\\\n&(5,4,-9,5),(5,4,-6,5), (7,3,-10,7),(7,3,-32,7).\n\\end{split}\n\\end{equation}", "start_pos": 10443, "end_pos": 10986, "label": "thm:main thm"}, "ref_dict": {"eq:ternary cubic eqn": "\\begin{equation}\\label{eq:ternary cubic eqn}\n\\begin{split}\n0 &= (s-t)(n+rd)+(t-r)(n+sd)+(r-s)(n+td) \\\\\n&= (s-t)a_rx_r^3 + (t-r)a_sx_s^3 + (r-s)a_tx_t^3.\n\\end{split}\n\\end{equation}", "lem:factorization of terms": "\\begin{lemma}\\label{lem:factorization of terms}\nLet $5 \\leq k \\leq 11$ and $0 \\leq i \\leq k-1$ be integers. Assume there are nonzero integers $y,n,d$ with $d\\geq 1$ and $\\textup{gcd}(n,d)=1$ such that \\eqref{eq:main eqn of study} holds. Then for $0 \\leq j \\leq k-1, j \\neq i$, we may uniquely write $n+jd=a_jx_j^3$, where:\n\\begin{itemize}\n\\item $a_j$ is a positive integer,\n\\item $a_j$ is cube-free, and divisible only by primes $\\leq k-1$,\n\\item $\\textup{gcd}(a_j,a_\\ell) \\mid (\\ell-j)$ if $0 \\leq j < \\ell \\leq k-1$ and $j,\\ell \\neq i$,\n\\item the integers $x_j$ are nonzero, and if $k \\leq 8$ they are pairwise coprime,\n\\item $\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}a_j$ is a perfect cube.\n\\end{itemize}\n\\end{lemma}", "eq:generalized ErdSelf eqn": "\\begin{align}\\label{eq:generalized ErdSelf eqn}\nn(n+d)(n+2d) \\cdots (n+(k-1)d) = y^\\ell\n\\end{align}", "defn: coeff vector": "\\begin{definition}\\label{defn: coeff vector}\nAssume there are nonzero integers $y,n,d$ with $d\\geq 1$ and $\\textup{gcd}(n,d)=1$ such that \\eqref{eq:main eqn of study} holds. Let $n+jd=a_jx_j^3$ be the factorization as described in Lemma \\ref{lem:factorization of terms}. We call $(a_0,a_1,\\ldots,a_{i-1},a_{i+1},\\ldots,a_{k-1})$ the \\emph{coefficient vector} corresponding to $y,n,d$.\n\\end{definition}", "cor:rat pts": "\\begin{corollary}\\label{cor:rat pts}\nAll rational points $(x,y) \\in \\mathbb{Q}^2$ on the curve \n\\begin{align}\\label{eq:dlss curve}\ny^3=(x+1)(x+2)(x+3)(x+5)(x+6)(x+7)\n\\end{align}\nsatisfy $y=0$, or $(x,y) = \\left(-\\frac{17}{7},\\frac{120}{49} \\right)$, or $(x,y) = \\left(-\\frac{39}{7},\\frac{120}{49} \\right)$.\n\\end{corollary}", "thm:main thm": "\\begin{theorem}\\label{thm:main thm}\nLet $5 \\leq k \\leq 11$ and $0\\leq i \\leq k-1$ be integers. The only solutions to the equation\n\\begin{align}\\label{eq:main eqn of study}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}(n+jd) = y^3\n\\end{align}\nin integers $n,d,y$ with $ny \\neq 0$, $d \\geq 1$, and $\\textup{gcd}(n,d) = 1$ are given by\n\\begin{equation}\\label{eq:solns for main thm}\n\\begin{split}\n(k,i,n,d) = \\, &(5,0,-14,5),(5,0,-11,5),(5,1,-8,3),(5,3,-4,3), \\\\\n&(5,4,-9,5),(5,4,-6,5), (7,3,-10,7),(7,3,-32,7).\n\\end{split}\n\\end{equation}\n\\end{theorem}", "prop:only solns to pair of cubic eqns": "\\begin{proposition}\\label{prop:only solns to pair of cubic eqns}\nThe only solutions $(x,y,z,w) \\in \\mathbb{Z}^4$ to the pair of equations\n\\begin{align*}\nx^3+y^3&=9z^3 \\\\\n5x^3-y^3&=3w^3\n\\end{align*}\nwith $xyzw \\neq 0$ and $x,y$ coprime are $(x,y,z,w) = \\pm (1,2,1,-1)$.\n\\end{proposition}", "eq:main eqn of study": "\\begin{align}\\label{eq:main eqn of study}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}(n+jd) = y^3\n\\end{align}", "lem:EC over K": "\\begin{lemma}\\label{lem:EC over K}\nLet $\\alpha = \\sqrt[3]{5}$, and set $K = \\mathbb{Q}(\\alpha)$. Write $\\mathcal{O}_K = \\mathbb{Z}[\\alpha]$ for the ring of integers of $K$. The only solutions $(x,y,u) \\in \\mathbb{Z} \\times \\mathbb{Z}\\times \\mathcal{O}_K$ to\n\\begin{align}\\label{eq:cubic over K}\n(\\alpha x - y)(x^2-xy+y^2) = 3(2-\\alpha) u^3\n\\end{align}\nwith $xy \\neq 0$ and $x,y$ coprime are $(x,y,u) = \\pm (1,2,-1)$.\n\\end{lemma}", "eq:ErdSelf eqn": "\\begin{align}\\label{eq:ErdSelf eqn}\nn(n+1)(n+2)\\cdots(n+k-1) = y^\\ell\n\\end{align}", "eq:dlss curve": "\\begin{align}\\label{eq:dlss curve}\ny^3=(x+1)(x+2)(x+3)(x+5)(x+6)(x+7)\n\\end{align}", "lem:selmer": "\\begin{lemma}\\label{lem:selmer}\nLet\n\\begin{align*}\n\\mathcal{L} = \\{&1, 3, 4, 5, 10, 14, 18, 21, 25, 36, 45, 60, 100, 147, 150, \\\\\n                &175, 196, 225, 245, 252, 300, 315, 350, 882, 980, \\\\\n                &1050, 1470, 1575, 1764, 2940, 7350, 14700\\}.\n\\end{align*}\nLet $a,b,c$ be nonzero integers, and let $D$ denote the cube-free part of $abc$. If $D \\in \\mathcal{L}$, then the equation $ax^3+by^3+cz^3=0$ has no solutions in nonzero integers $x,y,z$.\n\\end{lemma}", "eq:solns for main thm": "\\begin{equation}\\label{eq:solns for main thm}\n\\begin{split}\n(k,i,n,d) = \\, &(5,0,-14,5),(5,0,-11,5),(5,1,-8,3),(5,3,-4,3), \\\\\n&(5,4,-9,5),(5,4,-6,5), (7,3,-10,7),(7,3,-32,7).\n\\end{split}\n\\end{equation}", "eq:Bennett eqn": "\\begin{align}\\label{eq:Bennett eqn}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1} (n+jd) = y^\\ell,\n\\end{align}"}, "pre_theorem_intro_text_len": 2628, "pre_theorem_intro_text": "Erd\\H{o}s and Selfridge famously proved that the product of two or more positive consecutive integers is never a perfect power \\cite{ErdSel1975}. That is, they showed the equation\n\\begin{align}\\label{eq:ErdSelf eqn}\nn(n+1)(n+2)\\cdots(n+k-1) = y^\\ell\n\\end{align}\nhas no solutions in positive integers $n,y,k,\\ell$ with $k,\\ell \\geq 2$. \n\nThere are many ways to generalize the Diophantine equation \\eqref{eq:ErdSelf eqn}. One generalization, which has been studied a great deal, is as follows: determine whether there are solutions to\n\\begin{align}\\label{eq:generalized ErdSelf eqn}\nn(n+d)(n+2d) \\cdots (n+(k-1)d) = y^\\ell\n\\end{align}\nin positive integers $n,d,y,k,\\ell$ with $\\text{gcd}(n,d)=1$ and $k,\\ell \\geq 2$. Hence, rather than taking products of consecutive integers as in \\eqref{eq:ErdSelf eqn}, one takes products of consecutive terms in an arithmetic progression. (See \\cite[pp. 22--26]{Sho2006} for a survey of some older results on \\eqref{eq:generalized ErdSelf eqn} and its variants. See \\cite{Ben2018,BenSik2016,BenSik2020,DLS2018,DLSS2023,GHP2009,HK2011,HTT2009}, say, and the associated references for an introduction to more recent work on this and related problems.) It is widely believed that \\eqref{eq:generalized ErdSelf eqn} has no solutions once $k$ is sufficiently large (see, e.g., \\cite[p. 356]{BenSik2020} and \\cite{ErdProb672}). In fact, it ought to be the case that one can remove some terms $n+jd$ from the left-hand side of \\eqref{eq:generalized ErdSelf eqn}, and the resulting equation will still be unsolvable (for $k$ sufficiently large). \n\nSome recent work in this direction is due to Bennett \\cite{Ben2023}, who studied equations of the form\n\\begin{align}\\label{eq:Bennett eqn}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1} (n+jd) = y^\\ell,\n\\end{align}\nwhere $5 \\leq k \\leq 8, i \\in \\{0,1,\\ldots,k-1\\}$, and $\\ell \\geq 3$ is a prime. Thus, these are equations where a single term $n+id$ is removed from the product. A variety of methods are used in \\cite{Ben2023} to study this equation, such as the modularity of elliptic curves and a clever analysis of the resulting congruences (for $\\ell \\geq 7$), rational points on elliptic curves of rank zero (for $\\ell=3$), and Chabauty-type methods (for $\\ell=5$, relying on work of Hajdu and Kov\\'{a}cs \\cite{HK2011}). It is claimed in \\cite[Theorem 1.2]{Ben2023} that all solutions to \\eqref{eq:Bennett eqn} with $5 \\leq k \\leq 8$ arise when $k=5$ and $\\ell=3$.  However, the analysis in \\cite{Ben2023} is incomplete in the case $\\ell=3$, as some solutions to \\eqref{eq:Bennett eqn} are missing.\n\nThe main result of this paper is the following theorem.", "context": "Erd\\H{o}s and Selfridge famously proved that the product of two or more positive consecutive integers is never a perfect power \\cite{ErdSel1975}. That is, they showed the equation\n\\begin{align}\\label{eq:ErdSelf eqn}\nn(n+1)(n+2)\\cdots(n+k-1) = y^\\ell\n\\end{align}\nhas no solutions in positive integers $n,y,k,\\ell$ with $k,\\ell \\geq 2$.\n\nThere are many ways to generalize the Diophantine equation \\eqref{eq:ErdSelf eqn}. One generalization, which has been studied a great deal, is as follows: determine whether there are solutions to\n\\begin{align}\\label{eq:generalized ErdSelf eqn}\nn(n+d)(n+2d) \\cdots (n+(k-1)d) = y^\\ell\n\\end{align}\nin positive integers $n,d,y,k,\\ell$ with $\\text{gcd}(n,d)=1$ and $k,\\ell \\geq 2$. Hence, rather than taking products of consecutive integers as in \\eqref{eq:ErdSelf eqn}, one takes products of consecutive terms in an arithmetic progression. (See \\cite[pp. 22--26]{Sho2006} for a survey of some older results on \\eqref{eq:generalized ErdSelf eqn} and its variants. See \\cite{Ben2018,BenSik2016,BenSik2020,DLS2018,DLSS2023,GHP2009,HK2011,HTT2009}, say, and the associated references for an introduction to more recent work on this and related problems.) It is widely believed that \\eqref{eq:generalized ErdSelf eqn} has no solutions once $k$ is sufficiently large (see, e.g., \\cite[p. 356]{BenSik2020} and \\cite{ErdProb672}). In fact, it ought to be the case that one can remove some terms $n+jd$ from the left-hand side of \\eqref{eq:generalized ErdSelf eqn}, and the resulting equation will still be unsolvable (for $k$ sufficiently large).\n\nSome recent work in this direction is due to Bennett \\cite{Ben2023}, who studied equations of the form\n\\begin{align}\\label{eq:Bennett eqn}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1} (n+jd) = y^\\ell,\n\\end{align}\nwhere $5 \\leq k \\leq 8, i \\in \\{0,1,\\ldots,k-1\\}$, and $\\ell \\geq 3$ is a prime. Thus, these are equations where a single term $n+id$ is removed from the product. A variety of methods are used in \\cite{Ben2023} to study this equation, such as the modularity of elliptic curves and a clever analysis of the resulting congruences (for $\\ell \\geq 7$), rational points on elliptic curves of rank zero (for $\\ell=3$), and Chabauty-type methods (for $\\ell=5$, relying on work of Hajdu and Kov\\'{a}cs \\cite{HK2011}). It is claimed in \\cite[Theorem 1.2]{Ben2023} that all solutions to \\eqref{eq:Bennett eqn} with $5 \\leq k \\leq 8$ arise when $k=5$ and $\\ell=3$. However, the analysis in \\cite{Ben2023} is incomplete in the case $\\ell=3$, as some solutions to \\eqref{eq:Bennett eqn} are missing.\n\nThe main result of this paper is the following theorem.", "full_context": "Erd\\H{o}s and Selfridge famously proved that the product of two or more positive consecutive integers is never a perfect power \\cite{ErdSel1975}. That is, they showed the equation\n\\begin{align}\\label{eq:ErdSelf eqn}\nn(n+1)(n+2)\\cdots(n+k-1) = y^\\ell\n\\end{align}\nhas no solutions in positive integers $n,y,k,\\ell$ with $k,\\ell \\geq 2$.\n\nThere are many ways to generalize the Diophantine equation \\eqref{eq:ErdSelf eqn}. One generalization, which has been studied a great deal, is as follows: determine whether there are solutions to\n\\begin{align}\\label{eq:generalized ErdSelf eqn}\nn(n+d)(n+2d) \\cdots (n+(k-1)d) = y^\\ell\n\\end{align}\nin positive integers $n,d,y,k,\\ell$ with $\\text{gcd}(n,d)=1$ and $k,\\ell \\geq 2$. Hence, rather than taking products of consecutive integers as in \\eqref{eq:ErdSelf eqn}, one takes products of consecutive terms in an arithmetic progression. (See \\cite[pp. 22--26]{Sho2006} for a survey of some older results on \\eqref{eq:generalized ErdSelf eqn} and its variants. See \\cite{Ben2018,BenSik2016,BenSik2020,DLS2018,DLSS2023,GHP2009,HK2011,HTT2009}, say, and the associated references for an introduction to more recent work on this and related problems.) It is widely believed that \\eqref{eq:generalized ErdSelf eqn} has no solutions once $k$ is sufficiently large (see, e.g., \\cite[p. 356]{BenSik2020} and \\cite{ErdProb672}). In fact, it ought to be the case that one can remove some terms $n+jd$ from the left-hand side of \\eqref{eq:generalized ErdSelf eqn}, and the resulting equation will still be unsolvable (for $k$ sufficiently large).\n\nSome recent work in this direction is due to Bennett \\cite{Ben2023}, who studied equations of the form\n\\begin{align}\\label{eq:Bennett eqn}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1} (n+jd) = y^\\ell,\n\\end{align}\nwhere $5 \\leq k \\leq 8, i \\in \\{0,1,\\ldots,k-1\\}$, and $\\ell \\geq 3$ is a prime. Thus, these are equations where a single term $n+id$ is removed from the product. A variety of methods are used in \\cite{Ben2023} to study this equation, such as the modularity of elliptic curves and a clever analysis of the resulting congruences (for $\\ell \\geq 7$), rational points on elliptic curves of rank zero (for $\\ell=3$), and Chabauty-type methods (for $\\ell=5$, relying on work of Hajdu and Kov\\'{a}cs \\cite{HK2011}). It is claimed in \\cite[Theorem 1.2]{Ben2023} that all solutions to \\eqref{eq:Bennett eqn} with $5 \\leq k \\leq 8$ arise when $k=5$ and $\\ell=3$. However, the analysis in \\cite{Ben2023} is incomplete in the case $\\ell=3$, as some solutions to \\eqref{eq:Bennett eqn} are missing.\n\nThe main result of this paper is the following theorem.\n\n\\begin{abstract}\nLet $5 \\leq k \\leq 11$ and $0\\leq i \\leq k-1$ be integers. We determine all solutions to the equation\n\\begin{align*}\nn(n+d)(n+2d)\\cdots(n+(i-1)d)(n+(i+1)d) \\cdots (n+(k-1)d) = y^3\n\\end{align*}\nin integers $n,d,y$ with $ny \\neq 0$, $d\\geq 1$, and $\\textup{gcd}(n,d) = 1$. Our method relies on the theory of elliptic curves, including elliptic curve Chabauty over a number field. As an application, we answer a question of Das, Laishram, Saradha, and Sharma concerning rational points on a certain superelliptic curve.\n\\end{abstract}\n\nSome recent work in this direction is due to Bennett \\cite{Ben2023}, who studied equations of the form\n\\begin{align}\\label{eq:Bennett eqn}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1} (n+jd) = y^\\ell,\n\\end{align}\nwhere $5 \\leq k \\leq 8, i \\in \\{0,1,\\ldots,k-1\\}$, and $\\ell \\geq 3$ is a prime. Thus, these are equations where a single term $n+id$ is removed from the product. A variety of methods are used in \\cite{Ben2023} to study this equation, such as the modularity of elliptic curves and a clever analysis of the resulting congruences (for $\\ell \\geq 7$), rational points on elliptic curves of rank zero (for $\\ell=3$), and Chabauty-type methods (for $\\ell=5$, relying on work of Hajdu and Kov\\'{a}cs \\cite{HK2011}). It is claimed in \\cite[Theorem 1.2]{Ben2023} that all solutions to \\eqref{eq:Bennett eqn} with $5 \\leq k \\leq 8$ arise when $k=5$ and $\\ell=3$. However, the analysis in \\cite{Ben2023} is incomplete in the case $\\ell=3$, as some solutions to \\eqref{eq:Bennett eqn} are missing.\n\nThe main result of this paper is the following theorem.\n\nIt would be interesting to extend Theorem \\ref{thm:main thm} to larger values of $k$, and to allow for more terms to be missing from the products.\n\n\\begin{lemma}\\label{lem:factorization of terms}\nLet $5 \\leq k \\leq 11$ and $0 \\leq i \\leq k-1$ be integers. Assume there are nonzero integers $y,n,d$ with $d\\geq 1$ and $\\textup{gcd}(n,d)=1$ such that \\eqref{eq:main eqn of study} holds. Then for $0 \\leq j \\leq k-1, j \\neq i$, we may uniquely write $n+jd=a_jx_j^3$, where:\n\\begin{itemize}\n\\item $a_j$ is a positive integer,\n\\item $a_j$ is cube-free, and divisible only by primes $\\leq k-1$,\n\\item $\\textup{gcd}(a_j,a_\\ell) \\mid (\\ell-j)$ if $0 \\leq j < \\ell \\leq k-1$ and $j,\\ell \\neq i$,\n\\item the integers $x_j$ are nonzero, and if $k \\leq 8$ they are pairwise coprime,\n\\item $\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}a_j$ is a perfect cube.\n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\nLet $0 \\leq j < \\ell \\leq k-1$ with $j,\\ell \\neq i$, and let $g = \\text{gcd}(n+jd,n+\\ell d)$. Then $g$ divides $(n+\\ell d)-(n+jd) = (\\ell-j)d$, and since $\\text{gcd}(n,d)=1$ we find $g$ is coprime to $d$. Hence $g \\mid (\\ell - j)$. It follows that the greatest common divisor of $n+jd,n+\\ell d$ is at most $k-1$. Therefore, if $p\\geq k$ is a prime and $p \\mid y$, then $p$ divides a unique term $n+jd$.\n\nOur general strategy is to determine the coefficient vectors that can correspond to solutions to \\eqref{eq:main eqn of study}. Since the integers $a_j$ are cube-free and their largest prime factors are bounded, there are only a finite number of potential coefficient vectors to consider for given values of $k$ and $i$. Given a potential coefficient vector $\\textbf{v} = (a_0,\\ldots,a_{i-1},a_{i+1},\\ldots,a_{k-1})$, we may try to show $\\textbf{v}$ cannot correspond to solutions of \\eqref{eq:main eqn of study} by reducing to various cubic equations. More particularly, given distinct integers $r,s,t$ with $0 \\leq r,s,t \\leq k-1$ and $r,s,t \\neq i$, we have\n\\begin{equation}\\label{eq:ternary cubic eqn}\n\\begin{split}\n0 &= (s-t)(n+rd)+(t-r)(n+sd)+(r-s)(n+td) \\\\\n&= (s-t)a_rx_r^3 + (t-r)a_sx_s^3 + (r-s)a_tx_t^3.\n\\end{split}\n\\end{equation}\nIf the cubic equation $(s-t)a_rx_r^3 + (t-r)a_sx_s^3 + (r-s)a_tx_t^3 = 0$ has no solutions in nonzero integers $x_r,x_s,x_t$, then $\\textbf{v}$ cannot be a coefficient vector corresponding to any solutions to \\eqref{eq:main eqn of study}.\n\n\\begin{lemma}\\label{lem:HTT}\nSuppose we have a solution to the equation\n\\begin{align*}\n\\prod_{j=0}^{k-1}(n+jd) = y^3\n\\end{align*}\nin integers $k,n,d,y$ with $4 \\leq k \\leq 38, d \\geq 1, ny \\neq 0$, and $\\textup{gcd}(n,d)=1$. Then $(k,n,d) = (4,-9,5)$ or $(4,-6,5)$.\n\\end{lemma}\n\nHence, we may assume $1 \\leq i \\leq k-2$. Suppose we have a solution to \\eqref{eq:main eqn of study}. If we set $m=n+(k-1)d$, then\n\\begin{align*}\ny^3 &= (-1)^{k-1}\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}(-m+(k-1-j)d) = (-1)^{k-1} \\prod_{\\substack{\\ell = 0 \\\\ \\ell \\neq k-1-i}}^{k-1} (-m+\\ell d).\n\\end{align*}\nIf we absorb the sign $(-1)^{k-1}$ into $y^3$, then this is an equation of the form \\eqref{eq:main eqn of study} with $i$ replaced by $k-1-i$. In fact, this yields a bijective involution between different solutions to \\eqref{eq:main eqn of study} given by\n\\begin{align}\\label{eq:flipping bijection}\n(k,i,n,d) \\leftrightarrow (k,k-1-i,-n-(k-1)d,d).\n\\end{align}\nIt therefore suffices to consider $1 \\leq i \\leq \\frac{k-1}{2}$ in what follows.\n\n\\begin{align}\\label{eq:main eqn of study}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}(n+jd) = y^3\n\\end{align}\n\n\\begin{theorem}\\label{thm:main thm}\nLet $5 \\leq k \\leq 11$ and $0\\leq i \\leq k-1$ be integers. The only solutions to the equation\n\\begin{align}\\label{eq:main eqn of study}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}(n+jd) = y^3\n\\end{align}\nin integers $n,d,y$ with $ny \\neq 0$, $d \\geq 1$, and $\\textup{gcd}(n,d) = 1$ are given by\n\\begin{equation}\\label{eq:solns for main thm}\n\\begin{split}\n(k,i,n,d) = \\, &(5,0,-14,5),(5,0,-11,5),(5,1,-8,3),(5,3,-4,3), \\\\\n&(5,4,-9,5),(5,4,-6,5), (7,3,-10,7),(7,3,-32,7).\n\\end{split}\n\\end{equation}\n\\end{theorem}", "post_theorem_intro_text_len": 3396, "post_theorem_intro_text": "It would be interesting to extend Theorem \\ref{thm:main thm} to larger values of $k$, and to allow for more terms to be missing from the products.\n\nTheorem \\ref{thm:main thm} allows us to answer a question raised by Das, Laishram, Saradha, and Sharma \\cite[p. 1710]{DLSS2023} concerning rational points on a certain superelliptic curve.\n\n\\begin{corollary}\\label{cor:rat pts}\nAll rational points $(x,y) \\in \\mathbb{Q}^2$ on the curve \n\\begin{align}\\label{eq:dlss curve}\ny^3=(x+1)(x+2)(x+3)(x+5)(x+6)(x+7)\n\\end{align}\nsatisfy $y=0$, or $(x,y) = \\left(-\\frac{17}{7},\\frac{120}{49} \\right)$, or $(x,y) = \\left(-\\frac{39}{7},\\frac{120}{49} \\right)$.\n\\end{corollary}\n\nThe points on \\eqref{eq:dlss curve} with $y=0$ are the ``trivial'' rational points\n\\begin{align*}\n(-1,0),(-2,0),(-3,0),(-5,0),(-6,0),(-7,0).\n\\end{align*}\nThe authors of \\cite{DLSS2023} noted the existence of the ``nontrivial'' rational point $\\left(-\\frac{17}{7},\\frac{120}{49} \\right)$, but did not provably determine all of the rational points on \\eqref{eq:dlss curve}.\n\n\\subsection{Outline of the paper} \n\nIn Section \\ref{sec:prelim}, we show how Corollary \\ref{cor:rat pts} follows from Theorem \\ref{thm:main thm}. We then prove an elementary but important lemma (Lemma \\ref{lem:factorization of terms}) detailing the structure of the terms $n+jd$ in any solution to \\eqref{eq:main eqn of study}. This gives rise to the notion of a \\emph{coefficient vector}  (Definition \\ref{defn: coeff vector}). A large part of our work is devoted to determining the coefficient vectors that can correspond to solutions to \\eqref{eq:main eqn of study}. It will transpire that solutions to \\eqref{eq:main eqn of study} give rise to many different ternary cubic equations (see \\eqref{eq:ternary cubic eqn}), and these cubic equations can be shown to be unsolvable in many cases by relating them to elliptic curves of rank zero (Lemma \\ref{lem:selmer}). We end the section with a general description of our strategy for determining potential coefficient vectors, which relies extensively on computer calculation. (All the computer code used in this paper, written in Magma \\cite{Magma} and Sage \\cite{Sage}, is available at our ``Cubes from products with one term missing'' GitHub repository \\cite{Pratt_cubes_2025}.)\n\nIn Section \\ref{sec:proving most of main thm}, we use elementary arguments and the results of Section \\ref{sec:prelim} to solve \\eqref{eq:main eqn of study} in all cases of $k$ and $i$ except $k=7,i=3$. There are two coefficient vectors in this case that correspond to solutions to \\eqref{eq:main eqn of study} (see \\eqref{eq:solns for main thm}). All the associated ternary cubic equations correspond to elliptic curves of positive rank, so the results of Section \\ref{sec:prelim} are inapplicable.\n\nIn the last section of the paper, Section \\ref{sec:finishing proof of main thm}, we finish the analysis in the case $k=7,i=3$ and give the proof of Theorem \\ref{thm:main thm}, using a result on the simultaneous solutions to a pair of ternary cubic equations (Proposition \\ref{prop:only solns to pair of cubic eqns}). Computations in the pure cubic field $K = \\mathbb{Q}(\\sqrt[3]{5})$ show that any solution to the pair of cubic equations gives rise to a point on an elliptic curve $E$ over $K$. All the points on $E$ arising in this way can be determined with the elliptic curve Chabauty method (see Lemma \\ref{lem:EC over K}).", "sketch": "In Section \\ref{sec:prelim}, we show how Corollary \\ref{cor:rat pts} follows from Theorem \\ref{thm:main thm}. We then prove an elementary but important lemma (Lemma \\ref{lem:factorization of terms}) detailing the structure of the terms $n+jd$ in any solution to \\eqref{eq:main eqn of study}. This gives rise to the notion of a \\emph{coefficient vector} (Definition \\ref{defn: coeff vector}). A large part of our work is devoted to determining the coefficient vectors that can correspond to solutions to \\eqref{eq:main eqn of study}. Solutions to \\eqref{eq:main eqn of study} give rise to many different ternary cubic equations (see \\eqref{eq:ternary cubic eqn}), and these cubic equations can be shown to be unsolvable in many cases by relating them to elliptic curves of rank zero (Lemma \\ref{lem:selmer}).\n\nIn Section \\ref{sec:proving most of main thm}, we use elementary arguments and the results of Section \\ref{sec:prelim} to solve \\eqref{eq:main eqn of study} in all cases of $k$ and $i$ except $k=7,i=3$; in this remaining case, “all the associated ternary cubic equations correspond to elliptic curves of positive rank, so the results of Section \\ref{sec:prelim} are inapplicable.”\n\nIn Section \\ref{sec:finishing proof of main thm}, we finish the analysis for $k=7,i=3$ and thereby prove Theorem \\ref{thm:main thm}, “using a result on the simultaneous solutions to a pair of ternary cubic equations” (Proposition \\ref{prop:only solns to pair of cubic eqns}). Computations in the pure cubic field $K=\\mathbb{Q}(\\sqrt[3]{5})$ show that any solution to the pair of cubic equations gives rise to a point on an elliptic curve $E$ over $K$, and “all the points on $E$ arising in this way can be determined with the elliptic curve Chabauty method” (Lemma \\ref{lem:EC over K}).", "expanded_sketch": "Next we show how \\begin{corollary}\\label{cor:rat pts}\nAll rational points $(x,y) \\in \\mathbb{Q}^2$ on the curve \n\\begin{align}\\label{eq:dlss curve}\ny^3=(x+1)(x+2)(x+3)(x+5)(x+6)(x+7)\n\\end{align}\nsatisfy $y=0$, or $(x,y) = \\left(-\\frac{17}{7},\\frac{120}{49} \\right)$, or $(x,y) = \\left(-\\frac{39}{7},\\frac{120}{49} \\right)$.\n\\end{corollary}\nfollows from the main theorem. We then prove an elementary but important lemma detailing the structure of the terms $n+jd$ in any solution to\n\\begin{align}\\label{eq:main eqn of study}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}(n+jd) = y^3\n\\end{align}\nnamely:\n\\begin{lemma}\\label{lem:factorization of terms}\nLet $5 \\leq k \\leq 11$ and $0 \\leq i \\leq k-1$ be integers. Assume there are nonzero integers $y,n,d$ with $d\\geq 1$ and $\\textup{gcd}(n,d)=1$ such that \\eqref{eq:main eqn of study} holds. Then for $0 \\leq j \\leq k-1, j \\neq i$, we may uniquely write $n+jd=a_jx_j^3$, where:\n\\begin{itemize}\n\\item $a_j$ is a positive integer,\n\\item $a_j$ is cube-free, and divisible only by primes $\\leq k-1$,\n\\item $\\textup{gcd}(a_j,a_\\ell) \\mid (\\ell-j)$ if $0 \\leq j < \\ell \\leq k-1$ and $j,\\ell \\neq i$,\n\\item the integers $x_j$ are nonzero, and if $k \\leq 8$ they are pairwise coprime,\n\\item $\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}a_j$ is a perfect cube.\n\\end{itemize}\n\\end{lemma}\nThis gives rise to the notion of a \\emph{coefficient vector}:\n\\begin{definition}\\label{defn: coeff vector}\nAssume there are nonzero integers $y,n,d$ with $d\\geq 1$ and $\\textup{gcd}(n,d)=1$ such that \\eqref{eq:main eqn of study} holds. Let $n+jd=a_jx_j^3$ be the factorization as described in Lemma \\ref{lem:factorization of terms}. We call $(a_0,a_1,\\ldots,a_{i-1},a_{i+1},\\ldots,a_{k-1})$ the \\emph{coefficient vector} corresponding to $y,n,d$.\n\\end{definition}\nA large part of our work is devoted to determining the coefficient vectors that can correspond to solutions to the equation above. Solutions to that equation give rise to many different ternary cubic equations, for instance\n\\begin{equation}\\label{eq:ternary cubic eqn}\n\\begin{split}\n0 &= (s-t)(n+rd)+(t-r)(n+sd)+(r-s)(n+td) \\\\\n&= (s-t)a_rx_r^3 + (t-r)a_sx_s^3 + (r-s)a_tx_t^3.\n\\end{split}\n\\end{equation}\nand these cubic equations can be shown to be unsolvable in many cases by relating them to elliptic curves of rank zero. Specifically, we use the following lemma:\n\\begin{lemma}\\label{lem:selmer}\nLet\n\\begin{align*}\n\\mathcal{L} = \\{&1, 3, 4, 5, 10, 14, 18, 21, 25, 36, 45, 60, 100, 147, 150, \\\\\n                &175, 196, 225, 245, 252, 300, 315, 350, 882, 980, \\\\\n                &1050, 1470, 1575, 1764, 2940, 7350, 14700\\}.\n\\end{align*}\nLet $a,b,c$ be nonzero integers, and let $D$ denote the cube-free part of $abc$. If $D \\in \\mathcal{L}$, then the equation $ax^3+by^3+cz^3=0$ has no solutions in nonzero integers $x,y,z$.\n\\end{lemma}\n\nNext, we use elementary arguments and the results established earlier to solve\n\\begin{align}\\label{eq:main eqn of study}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}(n+jd) = y^3\n\\end{align}\nin all cases of $k$ and $i$ except $k=7,i=3$; in this remaining case, “all the associated ternary cubic equations correspond to elliptic curves of positive rank, so the results established earlier are inapplicable.”\n\nFinally, we finish the analysis for $k=7,i=3$ and thereby prove the main theorem, using the following result on the simultaneous solutions to a pair of ternary cubic equations:\n\\begin{proposition}\\label{prop:only solns to pair of cubic eqns}\nThe only solutions $(x,y,z,w) \\in \\mathbb{Z}^4$ to the pair of equations\n\\begin{align*}\nx^3+y^3&=9z^3 \\\\\n5x^3-y^3&=3w^3\n\\end{align*}\nwith $xyzw \\neq 0$ and $x,y$ coprime are $(x,y,z,w) = \\pm (1,2,1,-1)$.\n\\end{proposition}\nComputations in the pure cubic field $K=\\mathbb{Q}(\\sqrt[3]{5})$ show that any solution to the pair of cubic equations gives rise to a point on an elliptic curve $E$ over $K$, and “all the points on $E$ arising in this way can be determined with the elliptic curve Chabauty method,” using the following lemma:\n\\begin{lemma}\\label{lem:EC over K}\nLet $\\alpha = \\sqrt[3]{5}$, and set $K = \\mathbb{Q}(\\alpha)$. Write $\\mathcal{O}_K = \\mathbb{Z}[\\alpha]$ for the ring of integers of $K$. The only solutions $(x,y,u) \\in \\mathbb{Z} \\times \\mathbb{Z}\\times \\mathcal{O}_K$ to\n\\begin{align}\\label{eq:cubic over K}\n(\\alpha x - y)(x^2-xy+y^2) = 3(2-\\alpha) u^3\n\\end{align}\nwith $xy \\neq 0$ and $x,y$ coprime are $(x,y,u) = \\pm (1,2,-1)$.\n\\end{lemma}", "expanded_theorem": "\\label{thm:main thm}\nLet $5 \\leq k \\leq 11$ and $0\\leq i \\leq k-1$ be integers. The only solutions to the equation\n\\begin{align}\\label{eq:main eqn of study}\n\\prod_{\\substack{j=0 \\\\ j \\neq i}}^{k-1}(n+jd) = y^3\n\\end{align}\nin integers $n,d,y$ with $ny \\neq 0$, $d \\geq 1$, and $\\textup{gcd}(n,d) = 1$ are given by\n\\begin{equation}\\label{eq:solns for main thm}\n\\begin{split}\n(k,i,n,d) = \\, &(5,0,-14,5),(5,0,-11,5),(5,1,-8,3),(5,3,-4,3), \\\\\n&(5,4,-9,5),(5,4,-6,5), (7,3,-10,7),(7,3,-32,7).\n\\end{split}\n\\end{equation}.", "theorem_type": ["Classification or Bijection", "Nonexistence"], "mcq": {"question": "Let integers $k,i,n,d,y$ satisfy $5\\le k\\le 11$, $0\\le i\\le k-1$, $d\\ge 1$, $ny\\ne 0$, and $\\gcd(n,d)=1$. Consider the equation obtained by deleting the $i$th term from a length-$k$ arithmetic progression product,\n\\[\n\\prod_{\\substack{j=0 \\\\ j\\ne i}}^{k-1}(n+jd)=y^3.\n\\]\nWhich statement correctly classifies all integer solutions?", "correct_choice": {"label": "A", "text": "The only possible values of $(k,i,n,d)$ are\n\\[\n(5,0,-14,5),\\ (5,0,-11,5),\\ (5,1,-8,3),\\ (5,3,-4,3),\\ (5,4,-9,5),\\ (5,4,-6,5),\\ (7,3,-10,7),\\ (7,3,-32,7),\n\\]\nand there are no other integer solutions with $5\\le k\\le 11$, $0\\le i\\le k-1$, $d\\ge 1$, $ny\\ne 0$, and $\\gcd(n,d)=1$."}, "choices": [{"label": "B", "text": "The only possible values of $(k,i,n,d)$ are\n\\[\n(5,0,-14,5),\\ (5,0,-11,5),\\ (5,1,-8,3),\\ (5,3,-4,3),\\ (5,4,-9,5),\\ (5,4,-6,5),\\ (7,3,-10,7),\\ (7,3,-32,7),\n\\]\ntogether with their images under the involution\n\\[\n(k,i,n,d)\\mapsto (k,k-1-i,-n-(k-1)d,d),\n\\]\nand there are no other integer solutions with $5\\le k\\le 11$, $0\\le i\\le k-1$, $d\\ge 1$, $ny\\ne 0$, and $\\gcd(n,d)=1$."}, {"label": "C", "text": "If\n\\[\n\\prod_{\\substack{j=0 \\\\ j\\ne i}}^{k-1}(n+jd)=y^3\n\\]\nhas an integer solution with $5\\le k\\le 11$, $0\\le i\\le k-1$, $d\\ge 1$, $ny\\ne 0$, and $\\gcd(n,d)=1$, then necessarily $k\\in\\{5,7\\}$; in particular, there are no solutions for $k\\in\\{6,8,9,10,11\\}$."}, {"label": "D", "text": "The only possible values of $(k,i,n,d)$ are\n\\[\n(5,0,-14,5),\\ (5,0,-11,5),\\ (5,1,-8,3),\\ (5,3,-4,3),\\ (5,4,-9,5),\\ (5,4,-6,5),\\ (7,3,-10,7),\\ (7,3,-32,7),\n\\]\nand, moreover, every integer solution with $5\\le k\\le 11$, $0\\le i\\le k-1$, $d\\ge 1$, $ny\\ne 0$, and $\\gcd(n,d)=1$ must satisfy $1\\le i\\le \\frac{k-1}{2}$."}, {"label": "E", "text": "The only possible values of $(k,i,n,d)$ are\n\\[\n(5,0,-14,5),\\ (5,0,-11,5),\\ (5,1,-8,3),\\ (5,3,-4,3),\\ (5,4,-9,5),\\ (5,4,-6,5),\\ (7,3,-10,7),\\ (7,3,-32,7),\n\\]\nand for each such quadruple there is exactly one corresponding integer $y$ satisfying\n\\[\n\\prod_{\\substack{j=0 \\\\ j\\ne i}}^{k-1}(n+jd)=y^3.\n\\]\nThere are no other integer solutions with $5\\le k\\le 11$, $0\\le i\\le k-1$, $d\\ge 1$, $ny\\ne 0$, and $\\gcd(n,d)=1$."}], "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": "flipping-bijection-used-only-for-reduction-not-for-adjoining-new-solutions", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "case_split", "tampered_component": "dropped-complete-enumeration-of-the-eight-quadruples", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "range-restriction-on-i-applies-only-after-symmetry-reduction", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "theorem-classifies-quadruples-not-uniqueness-of-y", "template_used": "wildcard"}]}}
{"id": "2512.17872v1", "paper_link": "http://arxiv.org/abs/2512.17872v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thmPoincare}\n    Let $(M,g)$ be a compact Riemannian manifold of dimension $n$, and let\n    $p,q \\in (1,\\infty)$ satisfy\n    \\[\n        p \\geq \\frac{n}{n-1}\n        \\quad \\text{and} \\quad\n        q > \\frac{n}{2}.\n    \\]\n    Let $r \\in (1,\\infty]$, and assume that, if $p < n$, then\n    \\[\n        \\frac{1}{r} \\geq \\frac{1}{p} - \\frac{1}{n}.\n    \\]\n    Then, for all functions $f \\in W^{1,p}(M,\\mathbb{R})$ and all non-negative densities\n    $\\omega \\in L^q(M)$ normalized by $\\mathbb{E}[\\omega] = 1$, the weighted average\n    $\\bE_\\omega[f]$ is well-defined and there exists a constant $C>0$, depending\n    only on $(M,g,p,q,r)$, such that\n    \\[\n        \\left\\| f - \\bE_\\omega[f] \\right\\|_{L^r}\n        \\leq\n        C\\,\\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}}\\,\\|df\\|_{L^p}.\n    \\]", "start_pos": 8037, "end_pos": 8836, "label": "thmPoincare"}, "ref_dict": {}, "pre_theorem_intro_text_len": 3515, "pre_theorem_intro_text": "Inequalities of Poincaré and Sobolev type are among the most fundamental tools\nin the analysis of partial differential equations. They provide quantitative\ncontrol of the oscillation of a function in terms of its derivatives and lie at\nthe heart of \\emph{a priori} estimates, compactness arguments, and regularity\ntheory for elliptic and parabolic equations.\n\nIn its simplest form, the Poincaré inequality asserts that, on a compact\nRiemannian manifold $(M,g)$, the deviation of a function from its mean value is\ncontrolled by the $L^p$-norm of its gradient: for any $p \\in [1,\\infty)$, there\nexists a constant $C = C(M,g,p)$ such that, for any function $f \\in\n    W^{1,p}(M,\\mathbb{R})$,\n\\[\n    \\left\\| f - \\mathbb{E}[f] \\right\\|_{L^p}\n    \\leq C \\|df\\|_{L^p},\n\\]\nwhere $\\mathbb{E}[f]$ denotes the average value of $f$,\n\\[\n    \\mathbb{E}[f]\n    = \\frac{1}{\\operatorname{vol}(M,g)} \\int_M f\\,d\\mu^g.\n\\]\nWe refer the reader to~\\cite[Section~5.8]{Evans} for a proof of this inequality\non bounded domains of $\\mathbb{R}^n$ and to~\\cite[Lemma~3.8]{Hebey} for compact\nmanifolds. More refined versions, based on the Sobolev inequality, allow one to\nestimate this deviation in stronger Lebesgue norms and are indispensable in the\nstudy of nonlinear elliptic equations.\n\nIn many applications, however, the reference average is not the uniform mean\nwith respect to the Riemannian volume measure. Instead, one is led to consider\naverages defined using a density $\\omega$,\n\\[\n    \\bE_\\omega[f] = \\int_M f\\,\\omega\\,d\\mu^g,\n\\]\nwhere $\\omega$ is a non-negative function normalized by $\\int_M \\omega\\,d\\mu^g\n    = 1$. Such weighted averages arise naturally in problems involving\nnormalization constraints, conservation laws, or coupled systems of equations,\nwhere the density $\\omega$ may depend on other unknowns of the system.\n\nFor a fixed density $\\omega$ and $p \\in (1,\\infty)$, it is not difficult to\nprove a Poincaré-type inequality controlling $f - \\bE_\\omega[f]$ by\n$\\|df\\|_{L^p}$. A standard compactness argument analogous\nto~\\cite[Lemma~3.8]{Hebey} shows that there exists a constant $C =\n    C(M,g,p,\\omega)$ such that\n\\[\n    \\|f - \\bE_\\omega[f]\\|_{L^p}\n    \\leq C \\|df\\|_{L^p}\n    \\qquad \\text{for all } f \\in W^{1,p}(M,\\mathbb{R}).\n\\]\nHowever, this argument provides no information on how the constant depends on\n$\\omega$, and in particular does not allow one to obtain uniform estimates when\nthe density varies in a family of weights.\n\nUnderstanding this dependence is a subtle issue. To our knowledge, the only\ncase in which such an estimate is explicitly addressed in the literature is\nwhen $\\omega$ is the characteristic function of a measurable subset of a\nbounded convex domain, see~\\cite[Lemma 7.16]{GilbargTrudinger}, a situation\nrelated to nonlinear capacity theory (see, for instance,~\\cite{MazyaSobolev}).\nAnother source of interest in Poincaré-type inequalities involving densities\ncomes from the theory of Bakry--Émery curvature-dimension conditions (see,\ne.g.,~\\cite{LiWang}); in that framework, however, the density defines the\nunderlying measure itself and modifies the geometric structure of the space, so\nthat the resulting inequalities belong to a fundamentally different setting.\n\nThe aim of this paper is to establish a Poincaré--Sobolev type inequality in\nwhich the deviation of a function from a weighted average is controlled by the\nunweighted Sobolev norm of its gradient, with an explicit quantitative\ndependence of the constant on the density. More precisely, we prove the\nfollowing result.", "context": "Inequalities of Poincaré and Sobolev type are among the most fundamental tools\nin the analysis of partial differential equations. They provide quantitative\ncontrol of the oscillation of a function in terms of its derivatives and lie at\nthe heart of \\emph{a priori} estimates, compactness arguments, and regularity\ntheory for elliptic and parabolic equations.\n\nIn its simplest form, the Poincaré inequality asserts that, on a compact\nRiemannian manifold $(M,g)$, the deviation of a function from its mean value is\ncontrolled by the $L^p$-norm of its gradient: for any $p \\in [1,\\infty)$, there\nexists a constant $C = C(M,g,p)$ such that, for any function $f \\in\n W^{1,p}(M,\\mathbb{R})$,\n\\[\n \\left\\| f - \\mathbb{E}[f] \\right\\|_{L^p}\n \\leq C \\|df\\|_{L^p},\n\\]\nwhere $\\mathbb{E}[f]$ denotes the average value of $f$,\n\\[\n \\mathbb{E}[f]\n = \\frac{1}{\\operatorname{vol}(M,g)} \\int_M f\\,d\\mu^g.\n\\]\nWe refer the reader to~\\cite[Section~5.8]{Evans} for a proof of this inequality\non bounded domains of $\\mathbb{R}^n$ and to~\\cite[Lemma~3.8]{Hebey} for compact\nmanifolds. More refined versions, based on the Sobolev inequality, allow one to\nestimate this deviation in stronger Lebesgue norms and are indispensable in the\nstudy of nonlinear elliptic equations.\n\nIn many applications, however, the reference average is not the uniform mean\nwith respect to the Riemannian volume measure. Instead, one is led to consider\naverages defined using a density $\\omega$,\n\\[\n \\bE_\\omega[f] = \\int_M f\\,\\omega\\,d\\mu^g,\n\\]\nwhere $\\omega$ is a non-negative function normalized by $\\int_M \\omega\\,d\\mu^g\n = 1$. Such weighted averages arise naturally in problems involving\nnormalization constraints, conservation laws, or coupled systems of equations,\nwhere the density $\\omega$ may depend on other unknowns of the system.\n\nFor a fixed density $\\omega$ and $p \\in (1,\\infty)$, it is not difficult to\nprove a Poincaré-type inequality controlling $f - \\bE_\\omega[f]$ by\n$\\|df\\|_{L^p}$. A standard compactness argument analogous\nto~\\cite[Lemma~3.8]{Hebey} shows that there exists a constant $C =\n C(M,g,p,\\omega)$ such that\n\\[\n \\|f - \\bE_\\omega[f]\\|_{L^p}\n \\leq C \\|df\\|_{L^p}\n \\qquad \\text{for all } f \\in W^{1,p}(M,\\mathbb{R}).\n\\]\nHowever, this argument provides no information on how the constant depends on\n$\\omega$, and in particular does not allow one to obtain uniform estimates when\nthe density varies in a family of weights.\n\nUnderstanding this dependence is a subtle issue. To our knowledge, the only\ncase in which such an estimate is explicitly addressed in the literature is\nwhen $\\omega$ is the characteristic function of a measurable subset of a\nbounded convex domain, see~\\cite[Lemma 7.16]{GilbargTrudinger}, a situation\nrelated to nonlinear capacity theory (see, for instance,~\\cite{MazyaSobolev}).\nAnother source of interest in Poincaré-type inequalities involving densities\ncomes from the theory of Bakry--Émery curvature-dimension conditions (see,\ne.g.,~\\cite{LiWang}); in that framework, however, the density defines the\nunderlying measure itself and modifies the geometric structure of the space, so\nthat the resulting inequalities belong to a fundamentally different setting.\n\nThe aim of this paper is to establish a Poincaré--Sobolev type inequality in\nwhich the deviation of a function from a weighted average is controlled by the\nunweighted Sobolev norm of its gradient, with an explicit quantitative\ndependence of the constant on the density. More precisely, we prove the\nfollowing result.", "full_context": "Inequalities of Poincaré and Sobolev type are among the most fundamental tools\nin the analysis of partial differential equations. They provide quantitative\ncontrol of the oscillation of a function in terms of its derivatives and lie at\nthe heart of \\emph{a priori} estimates, compactness arguments, and regularity\ntheory for elliptic and parabolic equations.\n\nIn its simplest form, the Poincaré inequality asserts that, on a compact\nRiemannian manifold $(M,g)$, the deviation of a function from its mean value is\ncontrolled by the $L^p$-norm of its gradient: for any $p \\in [1,\\infty)$, there\nexists a constant $C = C(M,g,p)$ such that, for any function $f \\in\n W^{1,p}(M,\\mathbb{R})$,\n\\[\n \\left\\| f - \\mathbb{E}[f] \\right\\|_{L^p}\n \\leq C \\|df\\|_{L^p},\n\\]\nwhere $\\mathbb{E}[f]$ denotes the average value of $f$,\n\\[\n \\mathbb{E}[f]\n = \\frac{1}{\\operatorname{vol}(M,g)} \\int_M f\\,d\\mu^g.\n\\]\nWe refer the reader to~\\cite[Section~5.8]{Evans} for a proof of this inequality\non bounded domains of $\\mathbb{R}^n$ and to~\\cite[Lemma~3.8]{Hebey} for compact\nmanifolds. More refined versions, based on the Sobolev inequality, allow one to\nestimate this deviation in stronger Lebesgue norms and are indispensable in the\nstudy of nonlinear elliptic equations.\n\nIn many applications, however, the reference average is not the uniform mean\nwith respect to the Riemannian volume measure. Instead, one is led to consider\naverages defined using a density $\\omega$,\n\\[\n \\bE_\\omega[f] = \\int_M f\\,\\omega\\,d\\mu^g,\n\\]\nwhere $\\omega$ is a non-negative function normalized by $\\int_M \\omega\\,d\\mu^g\n = 1$. Such weighted averages arise naturally in problems involving\nnormalization constraints, conservation laws, or coupled systems of equations,\nwhere the density $\\omega$ may depend on other unknowns of the system.\n\nFor a fixed density $\\omega$ and $p \\in (1,\\infty)$, it is not difficult to\nprove a Poincaré-type inequality controlling $f - \\bE_\\omega[f]$ by\n$\\|df\\|_{L^p}$. A standard compactness argument analogous\nto~\\cite[Lemma~3.8]{Hebey} shows that there exists a constant $C =\n C(M,g,p,\\omega)$ such that\n\\[\n \\|f - \\bE_\\omega[f]\\|_{L^p}\n \\leq C \\|df\\|_{L^p}\n \\qquad \\text{for all } f \\in W^{1,p}(M,\\mathbb{R}).\n\\]\nHowever, this argument provides no information on how the constant depends on\n$\\omega$, and in particular does not allow one to obtain uniform estimates when\nthe density varies in a family of weights.\n\nUnderstanding this dependence is a subtle issue. To our knowledge, the only\ncase in which such an estimate is explicitly addressed in the literature is\nwhen $\\omega$ is the characteristic function of a measurable subset of a\nbounded convex domain, see~\\cite[Lemma 7.16]{GilbargTrudinger}, a situation\nrelated to nonlinear capacity theory (see, for instance,~\\cite{MazyaSobolev}).\nAnother source of interest in Poincaré-type inequalities involving densities\ncomes from the theory of Bakry--Émery curvature-dimension conditions (see,\ne.g.,~\\cite{LiWang}); in that framework, however, the density defines the\nunderlying measure itself and modifies the geometric structure of the space, so\nthat the resulting inequalities belong to a fundamentally different setting.\n\nThe aim of this paper is to establish a Poincaré--Sobolev type inequality in\nwhich the deviation of a function from a weighted average is controlled by the\nunweighted Sobolev norm of its gradient, with an explicit quantitative\ndependence of the constant on the density. More precisely, we prove the\nfollowing result.\n\nTo the best of our knowledge, such a quantitative estimate has not previously\nbeen established, even in the Euclidean setting.\n\nWe now pass to polar coordinates for $y-x$. Note that $\\omega$ can be extended\n to zero outside of $U$. We also extend $df$ by zero outside of $U$ without\n changing the notation. This allows us not to care about restricting the domain\n of integration. Let also $d = \\diam(U)$. We have\n \\[\n |f(x) - \\bE_\\omega[f]|^t\n \\leq \\int_{r=0}^d \\int_{\\bS^{n-1}} r^{n+t-2} \\omega(x + r \\theta) \\int_0^r |df_{x + s \\theta}|^t ds d\\theta dr,\n \\]\n where $d\\theta$ is the surface measure on the unit sphere $\\bS^{n-1} \\subset\n \\bR^n$. Hence,\n \\[\n |f(x) - \\bE_\\omega[f]|^t\n \\leq \\int_{0 \\leq s \\leq r \\leq d} \\int_{\\bS^{n-1}} r^{n+t-2} \\omega(x + r \\theta) |df_{x + s \\theta}|^t d\\theta dr ds.\n \\]\n Integrating against $x \\in U$, we get\n \\begin{align*}\n \\int_U |f(x) - \\bE_\\omega[f]|^t dx\n & \\leq \\int_U \\int_{0 \\leq s \\leq r \\leq d} \\int_{\\bS^{n-1}} r^{n+t-2} \\omega(x + r \\theta) |df_{x + s \\theta}|^t d\\theta dr ds dx \\\\\n & \\leq \\int_U \\int_{\\bS^{n-1}} \\int_{s=0}^d \\left(\\int_{s \\leq r \\leq d} r^{n+t-2} \\omega(x + r \\theta) dr \\right) |df_{x + s \\theta}|^t d\\theta ds dx \\\\\n & \\leq d^{n+t-2} \\int_U \\int_{\\bS^{n-1}} \\int_{s=0}^d \\left(\\int_{s \\leq r \\leq d} \\omega(x + r \\theta) dr \\right) |df_{x + s \\theta}|^t d\\theta ds dx.\n \\end{align*}\n We change the order of integration and perform the change of variable $z = x + s \\theta$ to get:\n \\begin{align}\n \\int_U |f(x) - \\bE_\\omega[f]|^t dx\n & \\leq d^{n+t-2} \\int_{\\bS^{n-1}} \\int_{s=0}^d \\int_{x \\in U} \\left(\\int_{s \\leq r \\leq d} \\omega(x + r \\theta) dr \\right) |df_{x + s \\theta}|^t dx d\\theta ds\\nonumber \\\\\n & = d^{n+t-2} \\int_{\\bS^{n-1}} \\int_{s=0}^d \\int_{z \\in U} \\left(\\int_{s \\leq r \\leq d} \\omega(z + (r-s) \\theta) dr \\right) |df_{z}|^t dz d\\theta ds\\nonumber \\\\\n & = d^{n+t-2} \\int_{z \\in U} \\left(\\int_{\\bS^{n-1}} \\int_{s=0}^d \\int_{s \\leq r \\leq d} \\omega(z + (r-s) \\theta) dr ds d\\theta\\right) |df_{z}|^t dz\\nonumber \\\\\n & \\leq d^{n+t-1} \\int_{z \\in U} \\left(\\int_{\\bS^{n-1}} \\int_{\\rho=0}^d \\omega(z + \\rho \\theta) d\\rho d\\theta\\right) |df_{z}|^t dz,\\label{eqPoincare3}\n \\end{align}\n where we also performed the change of variable $\\rho = r-s$ to obtain the last line. Now remark that the inner integral can be rewritten as follows:\n \\begin{align*}\n \\omegatil(z)\n & = \\int_{\\bS^{n-1}} \\int_{t=0}^d \\omega(z + \\rho \\theta) d\\rho d\\theta \\\\\n & = \\int_{\\bS^{n-1}} \\int_{t=0}^d \\omega(z + \\rho \\theta) \\rho^{-n-1} \\rho^{n-1} d\\rho d\\theta \\\\\n & = \\int_{\\bR^n} \\omega(z - x) \\frac{\\chi_{|x| \\leq d}}{|x|^{n-1}} dx,\n \\end{align*}\n where we have set $x = \\rho \\theta$. Note that the function $x \\mapsto \\frac{\\chi_{|x| \\leq d}}{|x|^{n-1}}$ belongs to $L^r(\\bR^n, \\bR)$ for any $r < \\frac{n}{n-1}$\\footnote{This function is actually a truncated Riesz potential for which we could use the Hardy-Littlewood-Sobolev inequality~\\cite[Section 7.8]{GilbargTrudinger}. This is not needed here.} so we use Young's inequality together with the fact that $q > \\frac{n}{2}$ to get\n \\[\n \\|\\omegatil\\|_{L^n} \\leq c \\|\\omega\\|_{L^q}\n \\]\n for some constant $c = c(n, q, d)$. We finally conclude\n from~\\eqref{eqPoincare3} that\n \\[\n \\int_U |f(x) - \\bE_\\omega[f]|^t dx \\leq c d^{n+p-2} d^{n+t-1} \\int_{z \\in U} \\omegatil(z) |df_{z}|^t dz \\leq \\|\\omegatil\\|_{L^n} \\left(\\int_U |df|^p dx\\right)^{\\frac{n-1}{n}}.\n \\]\n\\end{proof}\n\nThe next step is to extend the previous result to a broader range for $t$ by\nmeans of the Sobolev inequality:\n\\begin{lemma}\\label{lmPoincare2}\n Let $U$ be an open non-empty bounded convex subset of $\\bR^n$ with smooth boundary. Let $p, q \\in (1, \\infty)$ be such that $p \\geq \\frac{n}{n-1}$ and $q > \\frac{n}{2}$. Let $r \\in [1, \\infty]$ be such that\n \\[\n \\frac{1}{r} \\geq \\frac{1}{p} - \\frac{1}{n} \\quad\\text{if } p < n.\n \\]\n There exists a constant $c' = c'(U, p, q, r)$ such that, for any $\\omega \\in\n L^q(U, \\bR)$, $\\omega \\geq 0$ a.e., $\\bE[\\omega] = 1$ and any function $f \\in\n W^{1, p}(U, \\bR)$,\n \\[\n \\|f - \\bE_\\omega[f]\\|_{L^r} \\leq c' \\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}} \\|df\\|_{L^p}.\n \\]\n\\end{lemma}\n\n\\begin{proof}\n Let $t = \\frac{n-1}{n} p$ be as in Lemma~\\ref{lmPoincare1}. The conclusion of this lemma can be written as follows:\n \\begin{equation}\\label{eqPoincare4}\n \\|f - \\bE_\\omega[f]\\|_{L^t} \\leq (c \\|\\omega\\|_{L^q})^{\\frac{n}{(n-1)p}} \\|df\\|_{L^p}.\n \\end{equation}\n As $U$ has finite volume, the conclusion of the lemma is immediate if $r \\leq t$ as there is a continuous inclusion $L^t(U, \\bR) \\hookrightarrow L^r(U, \\bR)$. If, instead, $r > t$, we let $p'$ be defined as follows:\n \\[\n p' =\n \\left\\lbrace\n \\begin{aligned}\n \\frac{np}{n-p} & \\quad\\text{if } p < n, \\\\\n \\infty & \\quad\\text{if } p > n, \\\\\n \\text{arbitrary} > p & \\quad\\text{if } p = n.\n \\end{aligned}\n \\right.\n \\]\n Let $\\ftil = f - \\bE_\\omega[f]$. The Sobolev embedding theorem implies that\n there exists a constant $s > 0$ such that\n \\begin{equation}\\label{eqPoincare5}\n \\|\\ftil\\|_{L^{p'}} \\leq s \\left(\\|d\\ftil\\|_{L^p} + \\|\\ftil\\|_{L^p}\\right).\n \\end{equation}\n As $t < p < p'$, we have\n \\[\n \\|\\ftil\\|_{L^p} \\leq \\|\\ftil\\|_{L^t}^\\theta \\|\\ftil\\|_{L^{p'}}^{1-\\theta}\\quad\\text{with } \\theta \\in (0, 1) \\text{ such that } \\frac{1}{p} = \\frac{\\theta}{t} + \\frac{1-\\theta}{p'}.\n \\]\n By the $\\epsilon$-Young inequality~\\cite[Equation $(7.6)$]{GilbargTrudinger},\n we deduce that, for any $\\epsilon > 0$,\n \\[\n \\|\\ftil\\|_{L^p} \\leq \\theta \\epsilon^{-\\frac{1-\\theta}{\\theta}} \\|\\ftil\\|_{L^t} + (1-\\theta) \\epsilon \\|\\ftil\\|_{L^{p'}}.\n \\]\n From~\\eqref{eqPoincare4}, we deduce that\n \\[\n \\|\\ftil\\|_{L^p} \\leq \\theta \\epsilon^{-\\frac{1-\\theta}{\\theta}} (c \\|\\omega\\|_{L^q})^{\\frac{n}{(n-1)p}} \\|df\\|_{L^p} + (1-\\theta) \\epsilon \\|\\ftil\\|_{L^{p'}}.\n \\]\n Hence,~\\eqref{eqPoincare5} implies\n \\[\n \\|\\ftil\\|_{L^{p'}} \\leq s \\left(\\|d\\ftil\\|_{L^p} + \\theta \\epsilon^{-\\frac{1-\\theta}{\\theta}} (c \\|\\omega\\|_{L^q})^{\\frac{n}{(n-1)p}} \\|df\\|_{L^p} + (1-\\theta) \\epsilon \\|\\ftil\\|_{L^{p'}}\\right),\n \\]\n which we rewite as follows:\n \\[\n \\left(\\frac{1}{s} - (1-\\theta) \\epsilon\\right) \\|\\ftil\\|_{L^{p'}} \\leq \\left(1 + \\theta \\epsilon^{-\\frac{1-\\theta}{\\theta}} (c \\|\\omega\\|_{L^q})^{\\frac{n}{(n-1)p}}\\right) \\|d\\ftil\\|_{L^p}.\n \\]\n Choosing $\\epsilon$ such that $(1-\\theta) \\epsilon = \\frac{1}{2s}$, we conclude\n that\n \\[\n \\|\\ftil\\|_{L^{p'}} \\leq 2s \\left(1 + \\theta \\epsilon^{-\\frac{1-\\theta}{\\theta}} (c \\|\\omega\\|_{L^q})^{\\frac{n}{(n-1)p}}\\right) \\|d\\ftil\\|_{L^p}.\n \\]\n As $1 = \\bE[\\omega] = \\|\\omega\\|_{L^1} \\leq \\vol(U)^{1 - \\frac{1}{q}}\n \\|\\omega\\|_{L^q}$, we can replace the constant $1$ in the previous inequality\n by\n \\[\n 1 = 1^{\\frac{n}{(n-1)p}} \\leq \\vol(U)^{\\left(1 - \\frac{1}{q}\\right)\\frac{n}{(n-1)p}} \\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}}.\n \\]\n Thus, we conclude that there is a constant $c'$ depending only on $U$ and $p$\n but not on $f$ such that\n \\[\n \\|\\ftil\\|_{L^{p'}} \\leq c' \\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}} \\|d\\ftil\\|_{L^p},\n \\]\n i.e.\\ such that\n \\[\n \\|f - \\bE_\\omega[f]\\|_{L^{p'}} \\leq c' \\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}} \\|df\\|_{L^p}.\n \\]\n Using once again that, if $r \\leq p'$, we have a continuous embedding\n $L^{p'}(U, \\bR) \\hookrightarrow L^r(U, \\bR)$, we obtain the conclusion of the\n lemma.\n\\end{proof}\n\nLet $f \\in W^{1,p}(M)$ and $\\omega \\in L^q(M)$ be non-negative with\n $\\mathbb{E}[\\omega] = 1$. We define $\\tilde{f} := f \\circ \\Psi$, and wish to\n construct a weight $\\tilde{\\omega}$ on $B$ such that\n $\\mathbb{E}[\\tilde{\\omega}] = 1$. For this, we use the coarea formula (see\n e.g.~\\cite[Chapter 5.2]{KrantzParks}):\n \\begin{equation}\\label{eqCoarea}\n \\int_B h(x) \\, dx = \\int_M \\left(\\sum_{x \\in \\Psi^{-1}(y)} \\frac{h(x)}{|\\det d\\Psi_x|} \\right) d\\mu^g(y),\n \\end{equation}\n which holds for any integrable function $h : B \\to \\bR$.", "post_theorem_intro_text_len": 650, "post_theorem_intro_text": "To the best of our knowledge, such a quantitative estimate has not previously\nbeen established, even in the Euclidean setting.\n\nThe outline of the paper is as follows. In Section~\\ref{secRn}, we address the\ncase of bounded convex open subsets of $\\mathbb{R}^n$. In Section~\\ref{secDiffeo}, we\nconstruct a local diffeomorphism $\\Psi: B \\to M$ from the open unit ball $B$ in\n$\\mathbb{R}^n$ onto the manifold $M$. Finally, in Section~\\ref{secProof}, we use the\ncoarea formula to lift the function and $\\omega$ from $M$ to $B$ to prove our\nmain result.\n\n\\subsection*{Acknowledgments}\n\nI am grateful to Laurent Véron for useful discussion about the paper.", "sketch": "To prove the main result (Theorem~\\ref{thmPoincare}), the paper proceeds as follows: first it treats the case of bounded convex open subsets of $\\mathbb{R}^n$ (Section~\\ref{secRn}); then it \"construct[s] a local diffeomorphism $\\Psi: B \\to M$ from the open unit ball $B$ in $\\mathbb{R}^n$ onto the manifold $M$\" (Section~\\ref{secDiffeo}); finally, in Section~\\ref{secProof}, it \"use[s] the coarea formula to lift the function and $\\omega$ from $M$ to $B$\" in order to prove the main result.", "expanded_sketch": "To prove the main theorem, the paper proceeds as follows: first it treats the case of bounded convex open subsets of $\\mathbb{R}^n$ (Section~\\ref{secRn}); then it \"construct[s] a local diffeomorphism $\\Psi: B \\to M$ from the open unit ball $B$ in $\\mathbb{R}^n$ onto the manifold $M$\" (Section~\\ref{secDiffeo}); finally, it \"use[s] the coarea formula to lift the function and $\\omega$ from $M$ to $B$\" in order to prove the main theorem.", "expanded_theorem": "\\label{thmPoincare}\n    Let $(M,g)$ be a compact Riemannian manifold of dimension $n$, and let\n    $p,q \\in (1,\\infty)$ satisfy\n    \\[\n        p \\geq \\frac{n}{n-1}\n        \\quad \\text{and} \\quad\n        q > \\frac{n}{2}.\n    \\]\n    Let $r \\in (1,\\infty]$, and assume that, if $p < n$, then\n    \\[\n        \\frac{1}{r} \\geq \\frac{1}{p} - \\frac{1}{n}.\n    \\]\n    Then, for all functions $f \\in W^{1,p}(M,\\mathbb{R})$ and all non-negative densities\n    $\\omega \\in L^q(M)$ normalized by $\\mathbb{E}[\\omega] = 1$, the weighted average\n    $\\bE_\\omega[f]$ is well-defined and there exists a constant $C>0$, depending\n    only on $(M,g,p,q,r)$, such that\n    \\[\n        \\left\\| f - \\bE_\\omega[f] \\right\\|_{L^r}\n        \\leq\n        C\\,\\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}}\\,\\|df\\|_{L^p}.\n    \\]", "theorem_type": ["Inequality or Bound", "Existential–Universal"], "mcq": {"question": "Let $(M,g)$ be a compact Riemannian manifold of dimension $n$. Let $p,q\\in(1,\\infty)$ satisfy\n\\[\np\\ge \\frac{n}{n-1}\n\\qquad\\text{and}\\qquad\nq>\\frac{n}{2}.\n\\]\nLet $r\\in(1,\\infty]$, and assume that if $p<n$, then\n\\[\n\\frac{1}{r}\\ge \\frac{1}{p}-\\frac{1}{n}.\n\\]\nFor a nonnegative function $\\omega\\in L^q(M)$ with\n\\[\n\\int_M \\omega\\,d\\mu^g = 1,\n\\]\ndefine the weighted average of $f$ by\n\\[\n\\mathbb E_\\omega[f] := \\int_M f\\,\\omega\\,d\\mu^g.\n\\]\nHere $f\\in W^{1,p}(M,\\mathbb R)$, $df$ denotes its differential, and all $L^r$ and $L^q$ norms are taken with respect to the Riemannian volume measure $d\\mu^g$. Which quantitative estimate holds for every such $f$ and $\\omega$?", "correct_choice": {"label": "A", "text": "The quantity $\\mathbb E_\\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\\in W^{1,p}(M,\\mathbb R)$ and all nonnegative $\\omega\\in L^q(M)$ with $\\int_M \\omega\\,d\\mu^g=1$,\n\\[\n\\|f-\\mathbb E_\\omega[f]\\|_{L^r}\n\\le C\\,\\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}}\\,\\|df\\|_{L^p}.\n\\]"}, "choices": [{"label": "B", "text": "The quantity $\\mathbb E_\\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\\in W^{1,p}(M,\\mathbb R)$ and all nonnegative $\\omega\\in L^q(M)$ with $\\int_M \\omega\\,d\\mu^g=1$,\n\\[\n\\|f-\\mathbb E_\\omega[f]\\|_{L^r}\n\\le C\\,\\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}}\\,\\|df\\|_{L^p},\n\\]\nprovided that, if $p<n$, then\n\\[\n\\frac{1}{r}> \\frac{1}{p}-\\frac{1}{n}.\n\\]"}, {"label": "C", "text": "The quantity $\\mathbb E_\\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\\in W^{1,p}(M,\\mathbb R)$ and all nonnegative $\\omega\\in L^q(M)$ with $\\int_M \\omega\\,d\\mu^g=1$,\n\\[\n\\|f-\\mathbb E_\\omega[f]\\|_{L^r}\n\\le C\\,\\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}}\\bigl(\\|df\\|_{L^p}+\\|f\\|_{L^p}\\bigr).\n\\]"}, {"label": "D", "text": "The quantity $\\mathbb E_\\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\\in W^{1,p}(M,\\mathbb R)$ and all nonnegative $\\omega\\in L^q(M)$ with $\\int_M \\omega\\,d\\mu^g=1$,\n\\[\n\\|f-\\mathbb E_\\omega[f]\\|_{L^r}\n\\le C\\,\\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)p}}\\,\\|df\\|_{L^p},\n\\]\nwhere the constant $C$ may depend on $\\omega$ but is independent of $f$."}, {"label": "E", "text": "The quantity $\\mathbb E_\\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\\in W^{1,p}(M,\\mathbb R)$ and all nonnegative $\\omega\\in L^q(M)$ with $\\int_M \\omega\\,d\\mu^g=1$,\n\\[\n\\|f-\\mathbb E_\\omega[f]\\|_{L^r}\n\\le C\\,\\|\\omega\\|_{L^q}^{\\frac{n}{(n-1)q}}\\,\\|df\\|_{L^p}.\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": "critical Sobolev endpoint for r", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "pure gradient control by adding an extra $\\|f\\|_{L^p}$ term", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "uniformity of the constant with respect to the density after lifting via coarea", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "power of $\\|\\omega\\|_{L^q}$ coming from the Euclidean estimate", "template_used": "wildcard"}]}}
{"id": "2512.18590v1", "paper_link": "http://arxiv.org/abs/2512.18590v1", "theorems_cnt": 3, "theorem": {"env_name": "thm", "content": "\\label{Mth}\n\t\tLet $N_r=S(\\xi)$ be the total space of the sphere bundle of a 3-dimensional real vector bundle $\\xi$ over $\\mathbb{P}^2$ with $p_1(\\xi)=r \\omega$. Then\n\t\t\\begin{enumerate}\n\t\t\\item there is a short exact sequence \n\t\t$$ 1 \\to I(N_r) \\to \\mathrm{MCG}(N_r) \\to A \\to 1,$$\n\t\twhere $A \\cong \\z_2$ if $r \\ne -3$, and $A \\cong S_3$, the symmetric group of $3$ elements, if $r=-3$;\n\t\t\\item when $r=8n+5$, $I(N_r)$ is isomorphic to $ \\mathbb{Z}_{2\\mathrm{gcd}(3,2n^2+3n+1)}\\oplus \\mathbb{Z}_{2\\mathrm{gcd}(14,n+1)}$.\n\t\t\\end{enumerate}", "start_pos": 5163, "end_pos": 5713, "label": "Mth"}, "ref_dict": {"Mth": "\\begin{thm}\\label{Mth}\n\t\tLet $N_r=S(\\xi)$ be the total space of the sphere bundle of a 3-dimensional real vector bundle $\\xi$ over $\\mathbb{P}^2$ with $p_1(\\xi)=r \\omega$. Then\n\t\t\\begin{enumerate}\n\t\t\\item there is a short exact sequence \n\t\t$$ 1 \\to I(N_r) \\to \\m(N_r) \\to A \\to 1,$$\n\t\twhere $A \\cong \\z_2$ if $r \\ne -3$, and $A \\cong S_3$, the symmetric group of $3$ elements, if $r=-3$;\n\t\t\\item when $r=8n+5$, $I(N_r)$ is isomorphic to $ \\mathbb{Z}_{2\\mathrm{gcd}(3,2n^2+3n+1)}\\oplus \\mathbb{Z}_{2\\mathrm{gcd}(14,n+1)}$.\n\t\t\\end{enumerate}\n\t\\end{thm}", "subsec:bundle": "\\label{subsec:bundle}\nFor the sake of simplicity,\tin the sequel we will use the same notation for (the equivalence class of) a vector bundle and (the homotopy class of) its classifying map from it bas", "subsec:gen": "\\label{subsec:gen}\n\nA natural idea to study the mapping class group is to relate it to the group of homotopy equivalences. To be precise, let $M$ be a simply closed connected manifold with $\\dim M>5$.", "subsec:det": "\\begin{proof}\nChoose a localization $i \\colon S^2\\times D^4\\to N_r$ such that $i(S^2\\times D^4)$ and $S^2\\vee S^2$ (defined in the proof of Lemma \\ref{Ih}) are disjoint. The diffeomorphisms $g_1$ and $g_2$, which are extended by the identity on the complement of $i(S^2 \\times D^4)$, clearly induce the identity on the cohomology groups and thus (their isotopy classes) are in $I(N_r)$.    \n\n\t\tBy Lemma \\ref{t7} and the exactness of $S_\\partial(N_r\\times I)\\to I(N_r)\\to I^h(N_r)$, the kernel of the forgetful homomorphism $I(N_r) \\to I^h(N_r)$ is generated by $g_1$. Thus we only need to show that the image of $g_2$ is a generator of $I^h(N_r)$. This will be done by showing that the image of $g_2$ generates  the image of $\\mu_N^*$ (see Lemma \\ref{Ih}).\n\n\t\tLet $\\mu_3:S^3\\times D^3 \\to (S^3\\times D^3)\\vee S^6$ be the pinch map. It induces a map\n\t\t$$\\mu_3^* \\colon \\pi_6(S^3)\\to \\pi_0(G_\\partial(S^3\\times D^3)), \\ \\alpha \\mapsto (id\\vee\\alpha)\\circ\\mu_3.$$\nwhere we represent $\\alpha \\in \\pi_6(S^3)$ by a map $S^6 \\to S^3 \\times \\{0\\}$. \nView $S^3\\times D^3$ as a submanifold of $N_r$ through the embedding $i\\circ i_\\eta \\colon S^3 \\times D^3 \\to S^2 \\times D^4 \\to N_r$, extending the map $\\mu^*(\\alpha)$ by the identity on the complement we have a homotopy equivalence $\\mu^*(\\alpha)$ of $N_r$. For $\\alpha\\in\\pi_6(S^3)$, consider the composite $i \\circ \\eta \\circ \\alpha \\colon S^6 \\to S^3 \\to S^2 \\to N_r$. From the construction of the maps one readily verifies that $\\mu_3^*(\\alpha)=\\mu_N^*(i \\circ\\eta \\circ \\alpha)\\in I^h(N_r)$, i.e., there is a commutative diagram\n$$\\xymatrix{\\pi_6(S^3) \\ar[r]^{(i \\circ \\eta)_*} \\ar[d]_{\\mu_3^*} & \\pi_6(N_r) \\ar[d]^{\\mu_N^*} \\\\\n \\pi_0(G_\\partial(S^3\\times D^3)) \\ar[r]^{\\ \\ \\ \\ \\ \\Phi} & I^h(N_r)}$$\n where the map $\\Phi$ at the bottom is the extension by the identity on the complement of $i\\circ i_{\\eta}(S^3 \\times D^3)$.\n\nIt can be verified by elementary homotopy theory (compare \\cite[p.98]{BeMa}) that the homotopy Dehn twist map $De^h:\\Omega^3 G(S^3)\\to G_\\partial(S^3\\times D^3)$ defined by $De^h(\\alpha)(x,y)=(\\alpha(y)x,y)$ \ninduces an isomorphism \n$\\pi_0(\\Omega^3G(S^3)) \\to \\pi_0(G_\\partial(S^3\\times D^3))$.  Under the natural isomorphisms\n$$\\pi_0(\\Omega^3G(S^3)) \\cong \\pi_3(G(S^3), \\mathrm{id}) \\cong \\pi_3((\\Omega^3S^3)_{\\mathrm{id}}) \\cong \\pi_6(S^3),$$\nwhere $(\\Omega^3S^3)_{\\mathrm{id}}$ is the connected component containing $\\mathrm{id}_{S^3}$, this map can be identified with $\\mu_3^* \\colon \\pi_6(S^3) \\to \\pi_0(G_\\partial(S^3\\times D^3))$. The following commutative diagram\n$$\\xymatrix{\\pi_3(SO(3))\\ar[d]\\ar[rr]^J&&\\pi_3(\\Omega^3S^3)\\ar[d]_{\\mu_3^*}\\\\\n\t\t\t\\pi_3(SO(4))\\ar[r]^{De \\ \\ \\ \\ }&\\m_{\\partial}(S^3\\times D^3)\\ar[r]&\\pi_0(G_\\partial(S^3\\times D^3))\n\t\t\t}$$ \nshows that the image of $g_2$ in $I^h(N_r)$ equals $\\Phi(\\mu_3^*(J(t_1)))$.\nThe $J$-homomorphism $J \\colon \\pi_3(SO(3))\\to\\pi_6(S^3)$ is surjective. Therefore by Corollary \\ref{htg} and Lemma \\ref{Ih},  the image of $g_2$ generates $I^h(N_r)$.   This finishes the proof of this proposition.\t\n\n\t\\end{proof}\n\n\t\\subsection{determination of $I(N_r)$}\\label{subsec:det}\nWe have shown in last subsection that the Torelli group $I(N_r)$ is generated by the diffeomorphisms $g_1$ and $g_2$. They obviously commute with each other since they are supported in disjoint regions. In this subsection we determine the orders of them using the generalized Kreck-Stolz invariant developed in \\cite{KreSu}.\n\nLet  $B=\\mathbb C P^\\infty\\times\\mathbb{C} P^\\infty \\times BSpin$, $p \\colon B\\to BO$ be the composition of the projection $B \\to BSpin$ and the canonical projection $BSpin \\to BO$.    \n\nSince $r \\in 8\\z +5$, we may identify $N_r$ with $\\mathbb{P}(\\xi_{1,l})$ for some $l \\in \\z$. By Proposition \\ref{Hcp}, $w_2(N_r)$ vanishes, hence $N_r$ is spin. The normal Gauss map  $g \\colon N_r\\to BO$ of $N_r$ admits a lifting $g_{spin}$ to $BSpin$, which is  unique (up to fiber homotopy) since $N_r$ is simply connected. Let $\\{s, t\\}$ be the basis of $H^2(N_r)$ given in Proposition \\ref{Hcp}, then the map $\\hat{g}=(s,t,g_{spin}) \\colon N_r \\to \\mathbb C P^\\infty\\times\\mathbb{C} P^\\infty \\times BSpin=B$ is the normal $B$-structure corresponding to the polarization $((s,t),0)$ (see \\cite{Kre18} for definitions). \n\nLet $x=c_1(O(-1)\\times0\\times0)$, $y=c_1(0\\times O(-1)\\times0)$ be a basis of $H^2(B)$. \tOur normal $B$-structure satisfies the condition of Theorem 5 of \\cite{Kre18}, since $H^2(B)=\\mathbb{Z}\\{x,y\\}, H^4(B)=\\mathbb{Z}\\{x^2,xy,y^2,\\hat{p}_1\\}$, where $\\hat{p}_1$ is the first spin Potrjagin class of the universal spin stable vector bundle. The homomorphism $g^* \\colon H^4(B) \\to H^4(N_r)$ is given by  \n\t\t$$\\hat{g}^* \\colon H^4(B)\\to H^4(N_r), (x^2,xy,y^2,\\hat{p}_1)\\to (s^2,st,st-ls^2,(2l-2)s^2),$$\nand in $H^4(N_r)$ we have the relations (see Proposition \\ref{Hcp})\n$$\\hat{g}^*(\\hat{p}_1)=-\\hat{p}_1(N_r)=(2l-2)s^2, \\ \\ t^2-ts+ls^2=0.$$\nTherefore \n\t$$\\mathrm{Ker}(\\hat{g}^* \\colon H^4(B)\\to H^4(N_r))=\\mathbb{Z}\\{y^2-xy+lx^2,\\hat{p}_1+2(1-l)x^2\\}$$\nLet $\\alpha=y^2-xy+lx^2$,  $\\beta=\\hat{p}_1+2(1-l)x^2\\in H^4(B)$ and $s_1=\\alpha^2$, $s_2=\\alpha^2+\\alpha\\beta$, $ s_3=\\beta^2\\in H^8(B)$.\n\n\tFix an embedded $6$-disk $D \\subset N_r$, let $I(N_r,D)$ be the group of $D$-fixing isotopy classes of $D$-fixing diffeomorphisms which induce the identity on cohomology. By the disk theorem of Palais and the isotopy extension theorem, the forgetful homomorphism $I(N_r,D)\\to I(N_r)$ is surjective. By a theorem from (\\cite{Wa63} p.265), $\\mathrm{Ker}(I(N_r,D)\\to I(N_r))$ is trivial or isomorphic to $\\mathbb{Z}_2$. \n\nGiven a diffeomorphism $f \\colon N_r \\to N_r$ with $f|_D = \\mathrm{id}$, let $T_f$ be the mapping torus of $f$ with the preferred embedding $S^1\\times D \\subset T_f$. Assume that the action of $f$ on the cohomology groups is trivial,  then $f$ preserves the normal $B$-structure, and henceforth there is a normal $B$-structure $\\hat{G}$ on $T_f$, $\\hat G \\colon T_f \\to B$,  \nextending $\\hat{g}$ and the restriction of $\\hat G$ on $S^1 \\times D$ is the ordinary normal $B$ structure on $S^1\\times D$. It's ready to see by obstruction theory that the normal $B$-structure $\\hat G$ is a unique up to homotopy. It is shown in \\cite[Theorem 6]{Kre18} that the $7$-dimensional $B$-bordism group $\\Omega_7(B,p)$ is trivial, therefore $(T_f,\\hat{G})$ admits a coboundary $(W,\\bar{G})$ whose signature $\\mathrm{sign}(W)$  is $0$. Notice that the restriction of $\\bar{G}^*\\alpha$ and $\\bar{G}^*\\beta$ on $\\partial W= T_f$ vanish, let $\\bar{\\alpha}$, $\\bar{\\beta}\\in H^4(W,\\partial W)$ be a preimage of $\\bar{G}^*\\alpha$ and $\\bar{G}^*\\beta\\in H^4(W)$, respectively. We have characteristic numbers $\\langle(\\bar{\\alpha}^2,\\bar{\\alpha}^2+\\bar{\\alpha}\\bar{\\beta},\\bar{\\beta}^2),[W,\\partial W]\\rangle \\in \\z^3$. For $8$-dimensional $B$-bordism classes $[X,\\varphi] \\in \\Omega_8(B,p)$, we have characteristic numbers $\\langle (s_1, s_2,s_3), \\varphi_*[X] \\rangle \\in \\z^3$ in a similar way. Let $L=(s_1,s_2,s_3)\\Omega_8^0(B)\\subset \\mathbb{Z}^3$ be the lattice generated by the characteristic numbers on the bordism group $\\Omega^0_8(B,p)=\\mathrm{Ker}(\\mathrm{sign} \\colon \\Omega_8(B,p)\\to \\mathbb{Z})$. The $KS$-invariant is defined by\n\t$$KS \\colon I(N_r,D)\\to \\mathbb{Z}^3/L$$\n\t$$KS(f)=(s_1,s_2,s_3)(f):=\\langle(\\bar{\\alpha}^2,\\bar{\\alpha}^2+\\bar{\\alpha}\\bar{\\beta},\\bar{\\beta}^2),[W,\\partial W]\\rangle.$$\n\n\t\\begin{prop}\\label{KS}\n\t\t$KS \\colon I(N_r,D)\\to \\mathbb{Z}^3/L$ is a well defined injective homomorphism. \n\t\\end{prop}\n\t\\begin{proof}\n\t\tThis proof is along the same line of the proof of Proposition 6.2 of \\cite{KreSu}, with more details to be checked. \n\n\t\tTo prove $KS$ is well defined, suppose $(W_1,\\bar{G}_1),(W_2,\\bar{G}_2)$ are two $B$ coboundary of $(T_f,\\hat{G})$ chose embeddings $i_j: W_j\\to \\mathbb{R}^N\\times (-1)^j[0,+\\infty)$ good enough such that $i_j(W_j)\\cap\\mathbb{R}^N=T_f$. Define $W:=i_1(W_1)\\cup i_2(W_2)\\cong W_1\\cup_\\partial W_2$ then $(W,G_w:=\\bar{G}_1\\cup \\bar{G}_2)$ is a well defined element in $\\Omega_8(B,p)$, by \\cite{Wa69} $sign(W)=0$. Consider the following diagram\n\t\t$$\\xymatrix{\\oplus_j H^4(W_j,T_f)\\ar[r]&\\oplus_j H^4(W_j)\\ar[r]&\\oplus_j H^4(T_f)\\\\\n\t\t\tH^4(W,T_f)\\ar[r]\\ar[u]^\\cong &H^4(W)\\ar[r]\\ar[u]&H^4(T_f)\\ar[u]^{injective}\n\t\t}$$\n\t\tThis implies the preimage of $G_w^*\\alpha\\in H^4(W)$ in $H^4(W,T_f)\\cong\\oplus_j H^4(W_j,T_f)$ is a preimage of $(\\bar{G}_1\\alpha, \\bar{G}_2\\alpha)$ ($\\beta$ resp.), and the difference of $KS(f)$ defined by $W_2$ and $W_1$ equals $(s_1,s_1+s_2,s_3)(W,G_w)\\in L$. Thus we have proved $KS$ is well defined. \n\n\t\tTo prove $KS$ is a homomorphism, suppose $f_3,f_4\\in I(N_r,D)$ and $(W_j,\\bar{G}_j)$ is a $B$ coboundary of $(T_{f_j},\\hat{G})$. Identify $(n,t)\\in N_r\\times [\\frac{1}{3},\\frac{2}{3}]\\subset T_{f_3}$ and $(n,1-t)\\in N_r\\times [\\frac{1}{3},\\frac{2}{3}]\\subset T_{f_4}$, define $W':=W_3\\cup_{N_r\\times [\\frac{1}{3},\\frac{2}{3}]}W_4$, then $\\partial W'=T_{f_3\\circ f_4}$. By the $M-V$ sequence of $(W';W_3,W_4)$ there are $(s',t')\\in [W',\\mathbb C P^\\infty\\times\\mathbb{C} P^\\infty]$ extending the cohomology factor of $\\bar{G}_j$. Since the pasting $W'=W_3\\cup_{N_r\\times [\\frac{1}{3},\\frac{2}{3}]}W_4$ preserve the spin structure of $\\bar{G}_j$ there is spin structure on $W'$ extending the spin factor of $\\bar{G}_j$. Thus we can define a $B$ normal structure $G'_w$ on $W'$ extending $\\bar{G}_j$. Consider the following diagram\n\n\t\t$$\\xymatrix{\\oplus_j H^4(W_j,T_{f_j})\\ar[r]&\\oplus_j H^4(W_j)\\ar[r]&\\oplus_j H^4(T_{f_j})\\\\\n\t\t\tH^4(W',T_{f_3}\\cup T_{f_4})\\ar[r]\\ar[u]^\\cong \\ar[d]&H^4(W')\\ar[r]\\ar[u]\\ar[d]&H^4(T_{f_3}\\cup T_{f_4})\\ar[u]^{injective}\\ar[d]\\\\\n\t\t\tH^4(W',T_{f_3\\circ f_4})\\ar[r]&H^4(W')\\ar[r]&H^4(T_{f_3\\circ f_4})\\\\\n\t\t}$$\n\t\tArguments analog to the proof that $KS$ is well defined, one can show $KS$ is a homomorphism through this diagram. Note that we need $H^4(T_{f_3}\\cup T_{f_4})\\to \\oplus_j H^4(T_{f_j}))$ being injective since we need a preimage of $G'_w\\alpha\\in H^4(W')$ in $H^4(W',T_{f_3}\\cup T_{f_4})$. \n\n\t\tBy Theorem 5 of \\cite{Kre18}, if $KS(f)=0$ then there is a $B$ coboundary $(W'',\\hat{G})$ of $(T_f,\\hat{G})$ such that $(W'';N_r\\times [0,\\frac{1}{2}],N_r\\times [\\frac{1}{2},1])$ is\n\t\tan $h$-cobordism. Since the spin structure of $W''$ is compatible to $S^1\\times D$ we may extend $S^1\\times D$ to be $D^2\\times D\\subset W$ and define a relative $h$-cobordism. Hence by the peudo-isotopy theorem of Cerf, $f=id\\in I(N_r, D)$. \n\n\t\\end{proof}", "prop:real3": "\\begin{prop}\\label{prop:real3}\n\t \tThe map \n\t\t$$p_1 \\colon  [\\mathbb{P}^2, BSO(3)]\\to \\{rx^2\\in H^4(\\mathbb{P}^2) \\ | \\ r\\in4\\mathbb{Z}+\\{0,1\\}\\}$$ \n\tis a bijection and maps $\\phi(\\gamma_{k,l})$ to $(k^2-4l)x^2$.\n\t \\end{prop}", "ntb": "\\begin{prop}\\label{ntb}\n\t\t$h^*\\xi_r$ is nontrivial if and only if $r\\in 8\\mathbb{Z}+5$. \n\t\\end{prop}"}, "pre_theorem_intro_text_len": 2243, "pre_theorem_intro_text": "The mapping class group $\\mathrm{MCG}(M)$ of a smooth manifold $M$ is the group of isotopy classes of orientation preserving self diffeomorphisms of $M$. It is an important object that encodes fundamental information about the symmetries of $M$.  In this paper, we study the computation of $\\mathrm{MCG}(M)$ for certain closed, simply-connected $6$-manifolds. \n\nAlthough closed, simply-connected $6$-manifolds with torsion-free homology groups are classified by their algebraic-topological invariants due to the work of Wall \\cite{Wa66} and Jupp  \\cite{Jup73}, there appears to be no systematic treatment of the computation of their mapping class groups. Particular interest has been devoted to manifolds admitting geometric structures. For instance, in \\cite{KreSu} a class of $6$-manifolds with second Betti number $b_2=1$ is studied, including complete intersections of complex dimension $3$. \n\nThe present paper considers a class of $6$-manifolds with $b_2=2$, which includes natural compact K\\\"ahler manifolds such as  projective bundles associated to rank $2$ holomorphic vector bundles over the complex projective plane $\\mathbb P^2$. More precisely, in this paper we study the mapping class group of the total spaces of $S^2$-bundles over $\\mathbb P^2$. A priori, the structure group of such bundles is the orientation-preserving diffeomorphism group of $S^2$. It was shown by Smale  \\cite{Sma59} that this group is homotopy equivalent to $SO(3)$. Consequently,  such bundles arise as the sphere bundles of $3$-dimensional real vector bundles over $\\mathbb P^2$. \n\nLet $\\mathrm{Vect}_{\\mathbb R}^3(\\mathbb P^2)$ denote the set of equivalence classes of $3$-dimensional real vector bundles over $\\mathbb P^2$. In Proposition \\ref{prop:real3} we show that the first Pontrjagin class induces a bijection \n$$p_1 \\colon \\mathrm{Vect}_{\\mathbb R}^3(\\mathbb P^2) \\stackrel{\\cong}{\\longrightarrow}\\{r\\omega \\in H^4(\\mathbb{P}^2):r\\in 4\\mathbb{Z}+\\{0,1\\}\\},$$\nwhere $\\omega$ is the cohomology fundamental class determined by the nature orientation of $\\mathbb P^2$.\n\nLet $I(M)$ be the Torelli group of $M$, which is the normal subgroup of $\\mathrm{MCG}(M)$ consisting of diffeomorphisms that induce the identity on the cohomology groups of $M$.", "context": "The mapping class group $\\mathrm{MCG}(M)$ of a smooth manifold $M$ is the group of isotopy classes of orientation preserving self diffeomorphisms of $M$. It is an important object that encodes fundamental information about the symmetries of $M$. In this paper, we study the computation of $\\mathrm{MCG}(M)$ for certain closed, simply-connected $6$-manifolds.\n\nAlthough closed, simply-connected $6$-manifolds with torsion-free homology groups are classified by their algebraic-topological invariants due to the work of Wall \\cite{Wa66} and Jupp \\cite{Jup73}, there appears to be no systematic treatment of the computation of their mapping class groups. Particular interest has been devoted to manifolds admitting geometric structures. For instance, in \\cite{KreSu} a class of $6$-manifolds with second Betti number $b_2=1$ is studied, including complete intersections of complex dimension $3$.\n\nThe present paper considers a class of $6$-manifolds with $b_2=2$, which includes natural compact K\\\"ahler manifolds such as projective bundles associated to rank $2$ holomorphic vector bundles over the complex projective plane $\\mathbb P^2$. More precisely, in this paper we study the mapping class group of the total spaces of $S^2$-bundles over $\\mathbb P^2$. A priori, the structure group of such bundles is the orientation-preserving diffeomorphism group of $S^2$. It was shown by Smale \\cite{Sma59} that this group is homotopy equivalent to $SO(3)$. Consequently, such bundles arise as the sphere bundles of $3$-dimensional real vector bundles over $\\mathbb P^2$.\n\nLet $\\mathrm{Vect}_{\\mathbb R}^3(\\mathbb P^2)$ denote the set of equivalence classes of $3$-dimensional real vector bundles over $\\mathbb P^2$. In Proposition \\ref{prop:real3} we show that the first Pontrjagin class induces a bijection \n$$p_1 \\colon \\mathrm{Vect}_{\\mathbb R}^3(\\mathbb P^2) \\stackrel{\\cong}{\\longrightarrow}\\{r\\omega \\in H^4(\\mathbb{P}^2):r\\in 4\\mathbb{Z}+\\{0,1\\}\\},$$\nwhere $\\omega$ is the cohomology fundamental class determined by the nature orientation of $\\mathbb P^2$.\n\nLet $I(M)$ be the Torelli group of $M$, which is the normal subgroup of $\\mathrm{MCG}(M)$ consisting of diffeomorphisms that induce the identity on the cohomology groups of $M$.\n\n\\begin{prop}\\label{prop:real3}\n\t \tThe map \n\t\t$$p_1 \\colon  [\\mathbb{P}^2, BSO(3)]\\to \\{rx^2\\in H^4(\\mathbb{P}^2) \\ | \\ r\\in4\\mathbb{Z}+\\{0,1\\}\\}$$ \n\tis a bijection and maps $\\phi(\\gamma_{k,l})$ to $(k^2-4l)x^2$.\n\t \\end{prop}", "full_context": "The mapping class group $\\mathrm{MCG}(M)$ of a smooth manifold $M$ is the group of isotopy classes of orientation preserving self diffeomorphisms of $M$. It is an important object that encodes fundamental information about the symmetries of $M$. In this paper, we study the computation of $\\mathrm{MCG}(M)$ for certain closed, simply-connected $6$-manifolds.\n\nAlthough closed, simply-connected $6$-manifolds with torsion-free homology groups are classified by their algebraic-topological invariants due to the work of Wall \\cite{Wa66} and Jupp \\cite{Jup73}, there appears to be no systematic treatment of the computation of their mapping class groups. Particular interest has been devoted to manifolds admitting geometric structures. For instance, in \\cite{KreSu} a class of $6$-manifolds with second Betti number $b_2=1$ is studied, including complete intersections of complex dimension $3$.\n\nThe present paper considers a class of $6$-manifolds with $b_2=2$, which includes natural compact K\\\"ahler manifolds such as projective bundles associated to rank $2$ holomorphic vector bundles over the complex projective plane $\\mathbb P^2$. More precisely, in this paper we study the mapping class group of the total spaces of $S^2$-bundles over $\\mathbb P^2$. A priori, the structure group of such bundles is the orientation-preserving diffeomorphism group of $S^2$. It was shown by Smale \\cite{Sma59} that this group is homotopy equivalent to $SO(3)$. Consequently, such bundles arise as the sphere bundles of $3$-dimensional real vector bundles over $\\mathbb P^2$.\n\nLet $\\mathrm{Vect}_{\\mathbb R}^3(\\mathbb P^2)$ denote the set of equivalence classes of $3$-dimensional real vector bundles over $\\mathbb P^2$. In Proposition \\ref{prop:real3} we show that the first Pontrjagin class induces a bijection \n$$p_1 \\colon \\mathrm{Vect}_{\\mathbb R}^3(\\mathbb P^2) \\stackrel{\\cong}{\\longrightarrow}\\{r\\omega \\in H^4(\\mathbb{P}^2):r\\in 4\\mathbb{Z}+\\{0,1\\}\\},$$\nwhere $\\omega$ is the cohomology fundamental class determined by the nature orientation of $\\mathbb P^2$.\n\nLet $I(M)$ be the Torelli group of $M$, which is the normal subgroup of $\\mathrm{MCG}(M)$ consisting of diffeomorphisms that induce the identity on the cohomology groups of $M$.\n\n\\begin{prop}\\label{prop:real3}\n\t \tThe map \n\t\t$$p_1 \\colon  [\\mathbb{P}^2, BSO(3)]\\to \\{rx^2\\in H^4(\\mathbb{P}^2) \\ | \\ r\\in4\\mathbb{Z}+\\{0,1\\}\\}$$ \n\tis a bijection and maps $\\phi(\\gamma_{k,l})$ to $(k^2-4l)x^2$.\n\t \\end{prop}\n\nLet $I(M)$ be the Torelli group of $M$, which is the normal subgroup of $\\m(M)$ consisting of diffeomorphisms that induce the identity on the cohomology groups of $M$.\n\nThe projective unitary group $PU(2)$ is isomorphic to the orthogonal group $SO(3)$; consequently, $S^2$-bundles may be identified with $\\p^1$-bundles. Thus, the manifolds considered in this paper include many complex K\\\"ahler manifolds, such as projective bundle associated to rank $2$ holomorphic vector bundles over $\\p^2$.\n\n\\begin{cor}\\label{gmhs}\n Let $M_k=\\{(x,y)\\in \\mathbb{P}^2\\times\\mathbb{P}^2 \\ | \\ \\sum_i x^k_iy_i=0\\}$, then\n \\begin{enumerate}\n \\item there is a short exact sequence \n $$ 1 \\to I(M_k) \\to \\m(M_k) \\to A \\to 1,$$\n where $A \\cong \\z_2$ if $k \\ne 1$, and $A \\cong S_3$ if $k=1$;\n \\item when $k=2l+1$, $I(M_k)$ is isomorphic to $\\mathbb{Z}_6\\oplus \\mathbb{Z}_{\\mathrm{gcd}(28,l^2+l)}$. \n \\end{enumerate}\n \\end{cor}\n\n\\begin{eg} \\label{eg:1}\n Let $\\zeta_k$ be the complex vector bundle over $\\p^2$ with total space \n $$E_r=\\{(z,z')\\in \\mathbb{P}^2\\times\\mathbb{C}^3 \\ | \\ \\sum z^k_iz'_i=0\\}$$\n and bundle projection $\\pi \\colon E_r \\to \\p^2$, $\\pi(z,z')=z$.\n One can identify $\\zeta_k$ as conjugation of the orthogonal complement of canonical embedding $O(-k)\\to\\mathbb{C}^3$. So $\\zeta_k\\cong \\xi_{k,k^2}$ and $p_1(\\phi( \\zeta_k))=-3k^2$. Notice that $\\mathbb{P}(\\zeta_1)$ is the Milnor hypersurface $\\{(z,z')\\in \\mathbb{P}^2\\times\\mathbb{P}^2 \\ | \\ \\sum z_iz'_i=0\\}$. We call $M_k=\\{(z,z')\\in \\mathbb{P}^2\\times\\mathbb{P}^2 \\ | \\ \\sum z^k_iz'_i=0\\}=\\mathbb{P}(\\zeta_k)$ a generalized Milnor hypersurface. One can easily show that the canonical generators of $M_k$ induced from the bundle $\\zeta_k$ are $(\\pi^* x,\\pi'^* x)$. Here $\\pi(z,z')=z, \\pi'(z,z')=z'$. \n\\end{eg}\n\n\\begin{lemma}\\label{bh}\nLet $\\xi$ be the nontrivial $3$-dimensional real vector bundle over $S^5$, $\\partial \\colon \\pi_{i+1}(S^5)\\to \\pi_i(S^2)$ be the boundary homomorphism in the long exact sequence of homotopy groups associated to the sphere bundle of $\\xi$, $\\pi_i(S^4) \\to \\pi_i(\\Omega S^5)$ be the map induced by the adjoint of the identity map $S^5= \\Sigma S^4 \\to S^5$. Then the composite $\\pi_i(S^4) \\to \\pi_i(\\Omega S^5) = \\pi_{i+1}(S^5) \\to \\pi_i(S^2)$ is identified with the homomorphism $\\eta^2 \\colon \\pi_i(S^4) \\to \\pi_i(S^2)$ induced by the Hopf map $\\eta \\colon S^3 \\to S^2$.\n\n\\begin{cor}\\label{htg}\nLet $N_r$ be the $S^2$-bundle over $ \\mathbb{P}^2$ defined as above. Assume $r\\in 8\\mathbb{Z}+5$. Then\n\\begin{enumerate}\n\\item the inclusion of a fiber induces an isomorphism $\\pi_3(S^2)\\to\\pi_3(N_r)$; \n\\item$\\pi_4(N_r)=0$; \n\\item the bundle projection induces an injective homomorphism $\\pi_5(N_r)\\to \\pi_5(\\mathbb{P}^2)$ with image $2 \\cdot \\pi_5(\\mathbb{P}^2)$;\n\\item $\\pi_6(S^2)\\to\\pi_6(N_r)$ is surjective and $\\pi_6(N_r)\\cong \\mathbb{Z}_6$.\n\\end{enumerate}\n \\end{cor}\n\n\\begin{lemma}\\label{t7}\n Let $\\Theta_7\\cong \\mathbb{Z}_{28}$ be the Milnor-Kervaire group of oriented differential structures on the $7$-sphere. Then connected sum operation induces a surjective map $cs:\\Theta_7\\to S_\\partial(N_r\\times I)$. \n \\end{lemma} \n \\begin{proof}\nConsider the smooth surgery exact sequence ( Theorem 11.22 of \\cite{LuTi})\n$$L_8(\\z) \\to S_\\partial(N_r\\times I) \\to [\\Sigma N_r, G/O] \\to L_7(\\z).$$\nA direct calculation shows that the group $H^i(\\Sigma N_r;\\pi_i(G/O))$ vanishes for all $i$. Therefore by obstruction theory, the homotopy set $[\\Sigma N_r, G/O]$ consists of a single element. Therefore\nthe map $L_8(\\z) \\to S_\\partial(N_r\\times I)$ is surjective. By the construction, the Wall realization map $L_8(\\mathbb Z) \\to S_{\\partial}(N_r \\times I)$ factors through $L_8(\\z) \\to \\Theta_7 \\to S_\\partial(N_r\\times I)$.\n \\end{proof}\n\nIn \\cite[Theorem 7.9 and 7.12]{KreSu}, the group $\\m_\\partial(S^2\\times D^4)$ is computed and a system of generator is given. Let $g_1$ be a diffeomorphism of $D^6$ relative to the boundary, representing a generator of $ \\m_\\partial(D^6)\\cong \\Theta_7$, $t_1\\in\\pi_3(SO(4))$ be a generator of $\\mathrm{Im}\\, \\pi_3(SO(3))\\cong\\mathbb{Z}$, $t_2$ be a generator of $\\pi_4(SO(3))\\cong \\mathbb{Z}_2$ and $i_\\eta:S^3\\times D^3\\to S^2\\times D^4$ be an embedding representing the Hopf map $\\eta\\in\\pi_3(S^2)$. Fix an embedded $6$-disk in $S^2 \\times D^4$ and extend $g_1$ to the complement by the identity; let $g_2$ and $g_3$ be the Dehn twists in $i_{\\eta}(S^3 \\times D^3)$ with parameters $t_1$ and $t_2$, respectively. Then $$\\m_\\partial(S^2\\times D^4)=\\mathbb{Z}_{28}\\{g_1\\}\\oplus\\mathbb{Z}_{12}\\{g_2\\}\\oplus\\mathbb{Z}_2\\{g_3\\}$$\nFor any given embedding of $S^2 \\times D^4$ in a $6$-manifold, we extend these diffeomorphisms by the identity on the complement and still denote them by the same notation. \n \\begin{prop}\\label{sur}\n Let $i \\colon S^2\\times D^4\\to N_r$ be a local trivialization of the $S^2$-bundle. Then $I(N_r)$ is generate by the disk-supported diffeomorphism $g_1$ and the Dehn twist $g_2$. \n \\end{prop}\n \\begin{proof}\nChoose a localization $i \\colon S^2\\times D^4\\to N_r$ such that $i(S^2\\times D^4)$ and $S^2\\vee S^2$ (defined in the proof of Lemma \\ref{Ih}) are disjoint. The diffeomorphisms $g_1$ and $g_2$, which are extended by the identity on the complement of $i(S^2 \\times D^4)$, clearly induce the identity on the cohomology groups and thus (their isotopy classes) are in $I(N_r)$.", "post_theorem_intro_text_len": 3664, "post_theorem_intro_text": "The projective unitary group $PU(2)$ is isomorphic to the orthogonal group $SO(3)$; consequently, $S^2$-bundles may be identified with $\\mathbb P^1$-bundles. Thus, the manifolds considered in this paper include many complex K\\\"ahler manifolds, such as projective bundle associated to rank $2$ holomorphic vector bundles over $\\mathbb P^2$. \n\nRank $2$-complex vector bundles over $\\mathbb P^2$ are classified by their Chern classes $c_1$ and $c_2$. For such a vector bundle $\\eta$ with Chern classes $c_1(\\eta)$ and $c_2(\\eta)$, the corresponding $3$-dimensional real vector bundle $\\xi$ satisfies \n$$p_1(\\xi) = c_1(\\eta)^2 - 4 c_2(\\eta).$$ \n(see Proposition \\ref{prop:real3}). Consequently, Theorem \\ref{Mth} also determines the mapping class group of the projective bundle $\\mathbb P(\\eta)$ of rank $2$ complex vector bundles $\\eta$ over $\\mathbb P^2$, provided that $\\langle c_1(\\eta)^2 - 4c_2(\\eta), [\\mathbb P^2] \\rangle = 8n+5$.\n\nAn interesting class of examples is given by the hypersurfaces $M_k \\subset \\mathbb P^2 \\times \\mathbb P^2$ defined by \n$$M_k=\\{(x,y)\\in \\mathbb{P}^2\\times\\mathbb{P}^2 \\ | \\  \\sum_i x^k_iy_i=0\\}.$$\nNote that  $M_1=\\{(x,y)\\in \\mathbb{P}^2\\times\\mathbb{P}^2 \\ | \\  \\sum_i x_iy_i=0\\}$ is the Milnor hypersurface, a Fano 3-fold. The hypersurfaces $M_k$ can be identified with $N_{-3k^2}$ (see \\S \\ref{subsec:bundle}). We  therefore obtain the following corollary. \n\n\t\\begin{cor}\\label{gmhs}\n\t\tLet $M_k=\\{(x,y)\\in \\mathbb{P}^2\\times\\mathbb{P}^2 \\ | \\  \\sum_i x^k_iy_i=0\\}$, then\n\t\t\\begin{enumerate}\n\t\t\\item there is a short exact sequence \n\t\t$$ 1 \\to I(M_k) \\to \\mathrm{MCG}(M_k) \\to A \\to 1,$$\n\t\twhere $A \\cong \\z_2$ if $k \\ne 1$, and $A \\cong S_3$ if $k=1$;\n\t\t\\item when $k=2l+1$, $I(M_k)$ is isomorphic to $\\mathbb{Z}_6\\oplus \\mathbb{Z}_{\\mathrm{gcd}(28,l^2+l)}$.    \n\t\t\\end{enumerate}\n\t\\end{cor}\n\nTo conclude this introduction, we briefly describe the strategy used in our computations. It combines methods from homotopy theory with classical and modified surgery theory. The determination of the action of $\\mathrm{MCG}(M)$ on the cohomology groups relies on basic geometric considerations and constructions; this is carried out in \\S \\ref{sec:action}. \n\nThe determination of the Torelli group $I(M)$ is more subtle. By classical surgery theory, for a simply-connected manifold $M$ of dimension at least $5$, there is a short exact sequence \n\t$$ S(M \\times I, \\partial) \\to I(M) \\to I^h(M),$$\nwhere $S(M \\times I, \\partial)$ denotes the relative smooth structure set of $M \\times I$, and $I^h(M)$ denotes the homotopy Torelli group (see \\S \\ref{subsec:gen}) For the manifolds $N_r$, the image of $S(N_r \\times I, \\partial)$ in $I(N_r)$ is generated by a disk-supported diffeomorphism $g_1$. On the other hand, when $r \\in 8 \\mathbb Z +5$,\n homotopy-theoretical arguments show that there is a surjective homomorphism $\\pi_6(N_r) \\to I^h(N_r)$, whose image is generated by a certain Dehn twist $g_2$.  Consequently, in this case the Torelli group $I(N_r)$ is generated by $g_1$ and $g_2$ (see \\S \\ref{subsec:gen}). To determine the structure of $I(N_r)$ we make use of a Kreck-Stolz type invariant provided by modified surgery theory, which is an injective homomorphism from the relative Torelli group $I(N_r,D)$ to $\\mathbb Z^3$ modulo a lattice $L$ (see \\S \\ref{subsec:det}). \n\n \\begin{ack}\nThe author thanks Professor Yang Su for entrusting him with this project and for patiently guiding him through the research and writing process. The author is also grateful to Professor Stephen Theriault, who pointed out a key direction for proving Proposition \\ref{ntb}. This research was partially supported by NSFC 12471069.\n \\end{ack}", "sketch": "To conclude this introduction, the paper “briefly describe[s] the strategy used in our computations,” combining “methods from homotopy theory with classical and modified surgery theory.”\n\n- **Action on cohomology (used for the quotient in Theorem~\\ref{Mth}).** “The determination of the action of $\\mathrm{MCG}(M)$ on the cohomology groups relies on basic geometric considerations and constructions; this is carried out in \\S \\ref{sec:action}.”\n\n- **Relating the Torelli group to surgery-theoretic objects.** “By classical surgery theory, for a simply-connected manifold $M$ of dimension at least $5$, there is a short exact sequence \n\\[ S(M \\times I, \\partial) \\to I(M) \\to I^h(M),\\]\nwhere $S(M \\times I, \\partial)$ denotes the relative smooth structure set of $M \\times I$, and $I^h(M)$ denotes the homotopy Torelli group.”\n\n- **Generators for $I(N_r)$.** For the manifolds $N_r$, “the image of $S(N_r \\times I, \\partial)$ in $I(N_r)$ is generated by a disk-supported diffeomorphism $g_1$.” Also, “when $r \\in 8\\mathbb Z+5$, homotopy-theoretical arguments show that there is a surjective homomorphism $\\pi_6(N_r) \\to I^h(N_r)$, whose image is generated by a certain Dehn twist $g_2$.” Hence “in this case the Torelli group $I(N_r)$ is generated by $g_1$ and $g_2$.”\n\n- **Determining the structure of $I(N_r)$ via modified surgery theory.** “To determine the structure of $I(N_r)$ we make use of a Kreck-Stolz type invariant provided by modified surgery theory,” described as “an injective homomorphism from the relative Torelli group $I(N_r,D)$ to $\\mathbb Z^3$ modulo a lattice $L$.”", "expanded_sketch": "To conclude this introduction, the paper “briefly describe[s] the strategy used in our computations,” combining “methods from homotopy theory with classical and modified surgery theory.”\n\n- **Action on cohomology (used for the quotient in the main theorem).** “The determination of the action of $\\mathrm{MCG}(M)$ on the cohomology groups relies on basic geometric considerations and constructions; this is carried out next.”\n\n- **Relating the Torelli group to surgery-theoretic objects.** “By classical surgery theory, for a simply-connected manifold $M$ of dimension at least $5$, there is a short exact sequence \n\\[ S(M \\times I, \\partial) \\to I(M) \\to I^h(M),\\]\nwhere $S(M \\times I, \\partial)$ denotes the relative smooth structure set of $M \\times I$, and $I^h(M)$ denotes the homotopy Torelli group.”\n\n- **Generators for $I(N_r)$.** For the manifolds $N_r$, “the image of $S(N_r \\times I, \\partial)$ in $I(N_r)$ is generated by a disk-supported diffeomorphism $g_1$.” Also, “when $r \\in 8\\mathbb Z+5$, homotopy-theoretical arguments show that there is a surjective homomorphism $\\pi_6(N_r) \\to I^h(N_r)$, whose image is generated by a certain Dehn twist $g_2$.” Hence “in this case the Torelli group $I(N_r)$ is generated by $g_1$ and $g_2$.”\n\n- **Determining the structure of $I(N_r)$ via modified surgery theory.** “To determine the structure of $I(N_r)$ we make use of a Kreck-Stolz type invariant provided by modified surgery theory,” described as “an injective homomorphism from the relative Torelli group $I(N_r,D)$ to $\\mathbb Z^3$ modulo a lattice $L$.”", "expanded_theorem": "\\label{Mth}\n\t\tLet $N_r=S(\\xi)$ be the total space of the sphere bundle of a 3-dimensional real vector bundle $\\xi$ over $\\mathbb{P}^2$ with $p_1(\\xi)=r \\omega$. Then\n\t\t\\begin{enumerate}\n\t\t\\item there is a short exact sequence \n\t\t$$ 1 \\to I(N_r) \\to \\mathrm{MCG}(N_r) \\to A \\to 1,$$\n\t\twhere $A \\cong \\z_2$ if $r \\ne -3$, and $A \\cong S_3$, the symmetric group of $3$ elements, if $r=-3$;\n\t\t\\item when $r=8n+5$, $I(N_r)$ is isomorphic to $ \\mathbb{Z}_{2\\mathrm{gcd}(3,2n^2+3n+1)}\\oplus \\mathbb{Z}_{2\\mathrm{gcd}(14,n+1)}$.\n\t\t\\end{enumerate}", "theorem_type": ["Classification or Bijection", "Existence"], "mcq": {"question": "Let \\(N_r=S(\\xi)\\) be the total space of the sphere bundle of a real rank-3 vector bundle \\(\\xi\\) over \\(\\mathbb P^2\\), and suppose its first Pontrjagin class is \\(p_1(\\xi)=r\\omega\\), where \\(\\omega\\in H^4(\\mathbb P^2)\\) is the fundamental generator. Let \\(\\mathrm{MCG}(N_r)\\) be the group of isotopy classes of orientation-preserving self-diffeomorphisms of \\(N_r\\), and let \\(I(N_r)\\) be its Torelli subgroup, consisting of mapping classes acting trivially on the cohomology of \\(N_r\\). Which statement correctly describes \\(\\mathrm{MCG}(N_r)\\) and, when \\(r=8n+5\\), the group \\(I(N_r)\\)?", "correct_choice": {"label": "A", "text": "There is a short exact sequence \\(1\\to I(N_r)\\to \\mathrm{MCG}(N_r)\\to A\\to 1\\), where \\(A\\cong \\mathbb Z_2\\) if \\(r\\neq -3\\), and \\(A\\cong S_3\\) if \\(r=-3\\). Moreover, if \\(r=8n+5\\), then \\(I(N_r)\\cong \\mathbb Z_{2\\gcd(3,2n^2+3n+1)}\\oplus \\mathbb Z_{2\\gcd(14,n+1)}\\)."}, "choices": [{"label": "B", "text": "There is a short exact sequence \\(1\\to I(N_r)\\to \\mathrm{MCG}(N_r)\\to A\\to 1\\), where \\(A\\cong \\mathbb Z_2\\) if \\(r\\notin 8\\mathbb Z+5\\), and \\(A\\cong S_3\\) if \\(r\\in 8\\mathbb Z+5\\). Moreover, if \\(r=8n+5\\), then \\(I(N_r)\\cong \\mathbb Z_{2\\gcd(3,2n^2+3n+1)}\\oplus \\mathbb Z_{2\\gcd(14,n+1)}\\)."}, {"label": "C", "text": "There is a short exact sequence \\(1\\to I(N_r)\\to \\mathrm{MCG}(N_r)\\to A\\to 1\\), where \\(A\\cong \\mathbb Z_2\\) if \\(r\\neq -3\\), and \\(A\\cong S_3\\) if \\(r=-3\\)."}, {"label": "D", "text": "There is a short exact sequence \\(1\\to I(N_r)\\to \\mathrm{MCG}(N_r)\\to A\\to 1\\), where \\(A\\cong \\mathbb Z_2\\) if \\(r\\neq -3\\), and \\(A\\cong S_3\\) if \\(r=-3\\). Moreover, if \\(r=8n+5\\), then \\(I(N_r)\\cong \\mathbb Z_{2\\gcd(3,2n^2+3n+1)}\\oplus \\mathbb Z_{2\\gcd(28,n+1)}\\)."}, {"label": "E", "text": "There is a short exact sequence \\(1\\to I(N_r)\\to \\mathrm{MCG}(N_r)\\to A\\to 1\\), where \\(A\\cong \\mathbb Z_2\\) if \\(r\\neq -3\\), and \\(A\\cong S_3\\) if \\(r=-3\\). Moreover, for every integer \\(r\\), the Torelli group \\(I(N_r)\\) is generated by a disk-supported diffeomorphism and a Dehn twist, and when \\(r=8n+5\\) one has \\(I(N_r)\\cong \\mathbb Z_{2\\gcd(3,2n^2+3n+1)}\\oplus \\mathbb Z_{2\\gcd(14,n+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": "exceptional_value_for_cohomology_action_quotient", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_explicit_description_of_I_for_r_equals_8n_plus_5", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "modified_surgery_order_14_replaced_by_28", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "special_congruence_condition_r_in_8Z_plus_5_made_uniform_in_r", "template_used": "wildcard"}]}}
{"id": "2512.18638v1", "paper_link": "http://arxiv.org/abs/2512.18638v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm1.1}\nLet $X$ be a smooth projective $3$-fold of general type. If $\\Vol(X)>12^6$, then $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$.", "start_pos": 7669, "end_pos": 7836, "label": "thm1.1"}, "ref_dict": {}, "pre_theorem_intro_text_len": 2665, "pre_theorem_intro_text": "\\label{section1}\n\nFor a smooth projective variety $X$ of general type, we are interested in the behavior of \nthe $m$-canonical map $\\Phi_{|mK_X|}$ of $X$ induced by the $m$-canonical system $|mK_X|$, which is crucial towards the birational classification of varieties of general type. \n\nIt is predicted that for a smooth projective variety $X$ of general type of dimension $n$ with large invariants such as the canonical volume $\\Vol(X)$ or the geometric genus $p_g(X)$, the $m$-canonical map $\\Phi_{|mK_X|}$ \nbehaves just like that of $X_0\\times C$ where $X_0$ is a smooth projective variety of general type of dimension $n-1$ and $C$ is a smooth projective curve of genus $g\\gg 0$. Such prediction for the birationality of $\\Phi_{|mK_X|}$ (known as M\\textsuperscript{c}Kernan's conjecture) was recently confirmed by Chen and Liu \\cite{CL24} (see also \\cite{CJ17, CJ22}).\n\nWe are interested in more explicit results in lower dimensions. Recall that Bombieri \\cite{Bombieri} showed that $\\Phi_{|5K_S|}$ is birational for a smooth projective surface $S$ of general type, so it is natural to study the birationality of $\\Phi_{|5K_X|}$ for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$ or $p_g(X)$. \nChen \\cite{Chen03} showed that \n$\\Phi_{|5K_X|}$ is birational if $p_g(X)\\geq 4$, and \nrecently Chen and Ding \\cite{CD25} showed that $\\Phi_{|5K_X|}$ is birational if $\\Vol(X)\\geq 86$, significantly improving the results in \\cite{Tod07, LDi12, Chen12}.\n\nNow we focus on the bicanonical map. Recall that for a smooth projective surface $S$ of general type, $h^0(S, 2K_S)\\geq 2$ by the Riemann--Roch formula (see \\cite[Page~185]{Bombieri}), which implies that $\\Phi_{|2K_S|}$ is a non-trivial map. So it is natural to ask whether $\\Phi_{|2K_X|}$ is not a pencil (namely, $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$) for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$ or $p_g(X)$.\nRecently, Chen, Jiang, and Yan \\cite[Theorem~1.2]{CJY24} proved that $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$ if $p_g(X)>201$, answering an open question of Chen and Zhang \\cite{CZ08}. However, for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$, the behavior of its bicanonical map is still mysterious. To the best of our knowledge, the only known result is that $h^0(X, 2K_X)\\geq 1$ if $\\Vol(X)>879^3$, by Di Biagio \\cite[Theorem~1.1]{LDi12} (see also \\cite{Tod07}), and we even do not know whether $\\Phi_{|2K_X|}$ is non-trivial for $\\Vol(X)$ sufficiently large. \n\nThe main goal of this paper is to show that $\\Phi_{|2K_X|}$ is not a pencil for a smooth projective $3$-fold $X$ of general type with large canonical volume.", "context": "\\label{section1}\n\nFor a smooth projective variety $X$ of general type, we are interested in the behavior of \nthe $m$-canonical map $\\Phi_{|mK_X|}$ of $X$ induced by the $m$-canonical system $|mK_X|$, which is crucial towards the birational classification of varieties of general type.\n\nIt is predicted that for a smooth projective variety $X$ of general type of dimension $n$ with large invariants such as the canonical volume $\\Vol(X)$ or the geometric genus $p_g(X)$, the $m$-canonical map $\\Phi_{|mK_X|}$ \nbehaves just like that of $X_0\\times C$ where $X_0$ is a smooth projective variety of general type of dimension $n-1$ and $C$ is a smooth projective curve of genus $g\\gg 0$. Such prediction for the birationality of $\\Phi_{|mK_X|}$ (known as M\\textsuperscript{c}Kernan's conjecture) was recently confirmed by Chen and Liu \\cite{CL24} (see also \\cite{CJ17, CJ22}).\n\nWe are interested in more explicit results in lower dimensions. Recall that Bombieri \\cite{Bombieri} showed that $\\Phi_{|5K_S|}$ is birational for a smooth projective surface $S$ of general type, so it is natural to study the birationality of $\\Phi_{|5K_X|}$ for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$ or $p_g(X)$. \nChen \\cite{Chen03} showed that \n$\\Phi_{|5K_X|}$ is birational if $p_g(X)\\geq 4$, and \nrecently Chen and Ding \\cite{CD25} showed that $\\Phi_{|5K_X|}$ is birational if $\\Vol(X)\\geq 86$, significantly improving the results in \\cite{Tod07, LDi12, Chen12}.\n\nNow we focus on the bicanonical map. Recall that for a smooth projective surface $S$ of general type, $h^0(S, 2K_S)\\geq 2$ by the Riemann--Roch formula (see \\cite[Page~185]{Bombieri}), which implies that $\\Phi_{|2K_S|}$ is a non-trivial map. So it is natural to ask whether $\\Phi_{|2K_X|}$ is not a pencil (namely, $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$) for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$ or $p_g(X)$.\nRecently, Chen, Jiang, and Yan \\cite[Theorem~1.2]{CJY24} proved that $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$ if $p_g(X)>201$, answering an open question of Chen and Zhang \\cite{CZ08}. However, for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$, the behavior of its bicanonical map is still mysterious. To the best of our knowledge, the only known result is that $h^0(X, 2K_X)\\geq 1$ if $\\Vol(X)>879^3$, by Di Biagio \\cite[Theorem~1.1]{LDi12} (see also \\cite{Tod07}), and we even do not know whether $\\Phi_{|2K_X|}$ is non-trivial for $\\Vol(X)$ sufficiently large.\n\nThe main goal of this paper is to show that $\\Phi_{|2K_X|}$ is not a pencil for a smooth projective $3$-fold $X$ of general type with large canonical volume.", "full_context": "\\label{section1}\n\nFor a smooth projective variety $X$ of general type, we are interested in the behavior of \nthe $m$-canonical map $\\Phi_{|mK_X|}$ of $X$ induced by the $m$-canonical system $|mK_X|$, which is crucial towards the birational classification of varieties of general type.\n\nIt is predicted that for a smooth projective variety $X$ of general type of dimension $n$ with large invariants such as the canonical volume $\\Vol(X)$ or the geometric genus $p_g(X)$, the $m$-canonical map $\\Phi_{|mK_X|}$ \nbehaves just like that of $X_0\\times C$ where $X_0$ is a smooth projective variety of general type of dimension $n-1$ and $C$ is a smooth projective curve of genus $g\\gg 0$. Such prediction for the birationality of $\\Phi_{|mK_X|}$ (known as M\\textsuperscript{c}Kernan's conjecture) was recently confirmed by Chen and Liu \\cite{CL24} (see also \\cite{CJ17, CJ22}).\n\nWe are interested in more explicit results in lower dimensions. Recall that Bombieri \\cite{Bombieri} showed that $\\Phi_{|5K_S|}$ is birational for a smooth projective surface $S$ of general type, so it is natural to study the birationality of $\\Phi_{|5K_X|}$ for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$ or $p_g(X)$. \nChen \\cite{Chen03} showed that \n$\\Phi_{|5K_X|}$ is birational if $p_g(X)\\geq 4$, and \nrecently Chen and Ding \\cite{CD25} showed that $\\Phi_{|5K_X|}$ is birational if $\\Vol(X)\\geq 86$, significantly improving the results in \\cite{Tod07, LDi12, Chen12}.\n\nNow we focus on the bicanonical map. Recall that for a smooth projective surface $S$ of general type, $h^0(S, 2K_S)\\geq 2$ by the Riemann--Roch formula (see \\cite[Page~185]{Bombieri}), which implies that $\\Phi_{|2K_S|}$ is a non-trivial map. So it is natural to ask whether $\\Phi_{|2K_X|}$ is not a pencil (namely, $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$) for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$ or $p_g(X)$.\nRecently, Chen, Jiang, and Yan \\cite[Theorem~1.2]{CJY24} proved that $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$ if $p_g(X)>201$, answering an open question of Chen and Zhang \\cite{CZ08}. However, for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$, the behavior of its bicanonical map is still mysterious. To the best of our knowledge, the only known result is that $h^0(X, 2K_X)\\geq 1$ if $\\Vol(X)>879^3$, by Di Biagio \\cite[Theorem~1.1]{LDi12} (see also \\cite{Tod07}), and we even do not know whether $\\Phi_{|2K_X|}$ is non-trivial for $\\Vol(X)$ sufficiently large.\n\nThe main goal of this paper is to show that $\\Phi_{|2K_X|}$ is not a pencil for a smooth projective $3$-fold $X$ of general type with large canonical volume.\n\nWe are interested in more explicit results in lower dimensions. Recall that Bombieri \\cite{Bombieri} showed that $\\Phi_{|5K_S|}$ is birational for a smooth projective surface $S$ of general type, so it is natural to study the birationality of $\\Phi_{|5K_X|}$ for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$ or $p_g(X)$. \nChen \\cite{Chen03} showed that \n$\\Phi_{|5K_X|}$ is birational if $p_g(X)\\geq 4$, and \nrecently Chen and Ding \\cite{CD25} showed that $\\Phi_{|5K_X|}$ is birational if $\\Vol(X)\\geq 86$, significantly improving the results in \\cite{Tod07, LDi12, Chen12}.\n\nNow we focus on the bicanonical map. Recall that for a smooth projective surface $S$ of general type, $h^0(S, 2K_S)\\geq 2$ by the Riemann--Roch formula (see \\cite[Page~185]{Bombieri}), which implies that $\\Phi_{|2K_S|}$ is a non-trivial map. So it is natural to ask whether $\\Phi_{|2K_X|}$ is not a pencil (namely, $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$) for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$ or $p_g(X)$.\nRecently, Chen, Jiang, and Yan \\cite[Theorem~1.2]{CJY24} proved that $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$ if $p_g(X)>201$, answering an open question of Chen and Zhang \\cite{CZ08}. However, for a smooth projective $3$-fold $X$ of general type with large $\\Vol(X)$, the behavior of its bicanonical map is still mysterious. To the best of our knowledge, the only known result is that $h^0(X, 2K_X)\\geq 1$ if $\\Vol(X)>879^3$, by Di Biagio \\cite[Theorem~1.1]{LDi12} (see also \\cite{Tod07}), and we even do not know whether $\\Phi_{|2K_X|}$ is non-trivial for $\\Vol(X)$ sufficiently large.\n\nWe briefly explain the difficulties and techniques in the proof. As in the study of $\\Phi_{|5K_X|}$ in \\cite{Tod07, LDi12, Chen12, CD25}, the basic idea is to creat non-klt centers by constructing a boundary $\\Delta\\sim_{\\bQ}\\delta K_X$ for some $\\delta<2$\nand cut down the dimension of non-klt centers by the Anghern--Siu type method.\nThe issue in dealing with $\\Phi_{|2K_X|}$ instead of $\\Phi_{|5K_X|}$ or $\\Phi_{|3K_X|}$ is that, often we are not allowed to cut down the dimension of a non-klt center if the canonical volume of the non-klt center\nis small (e.g., a surface with small canonical volume or a curve with geometric genus $2$), otherwise the constant $\\delta$ will exceed $2$. In this case, instead of cutting down the dimension of the non-klt center, we prove certain extension theorems which allow us to lift global sections from the centers to $X$. \nHere we remark that in \\cite{CL24}, the key step is to prove an extension theorem for fibrations, where many techniques such as the Kawamata--Viehweg vanishing theorem, canonical bundle formula, the Zariski--Nakayama decomposition, the Hacon--M\\textsuperscript{c}Kernan extension theorem, and Birkar's lower bound of lct are involved, and the extension theorem is not effective; in this paper, we prove an effective version of extension theorem by using the Nadel vanishing theorem and for curve centers we do not require them to form a fibration.\n\nAs a bi-product of our method, we prove the following theorem concerning the pluricanonical maps of $3$-folds of general type fibered by $(1,2)$-surfaces or $(2,3)$-surfaces, as a complement of the results of Xu \\cite[Theorem~1.4, Theorem~1.5]{XuJ14}:\n\\begin{theorem}\\label{thm1.3}\nLet $X$ be a smooth $3$-fold of general type. Suppose that $X$ admits a fibration $X\\to C$ to a curve with a general fiber denoted by $F$.\n\\begin{enumerate}\n \\item If $F$ is a $(1,2)$-surface and $\\Vol(X)> 60$, then $\\Phi_{|mK_X|}$ is generically finite of degree $2$ for $2\\leq m \\leq 4$ and is birational for $m\\geq 5$.\n \\item If $F$ is a $(2,3)$-surface and $\\Vol(X)>456$, then $\\Phi_{|mK_X|}$ is generically finite of degree $2$ for $2\\leq m \\leq 3$ and is birational for $m\\geq 4$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{lemma}\\label{lem separete Z1Z2}\nLet $X$ be a normal projective variety and let $D$ be an effective Cartier divisor on $X$.\nLet $C_1$ and $C_2$ be closed subvarieties on $X$.\nSuppose that the natural restriction map \n\\[\n H^0(X,D) \\to H^0(C_1, D|_{C_1} ) \\oplus H^0(C_2, D|_{C_2})\n\\]\nis surjective.\nThen \\[\n\\dim\\overline{\\Phi_{{|D|}}(X)}\\geq \\min\\{\\dim\\overline{\\Phi_{{|D|_{C_1}|}}(C_1)}, \\dim\\overline{\\Phi_{{|D|_{C_2}|}}(C_2)}\\}+1.\\]\nIn particular,\nif $h^0(C_i, D|_{C_i})\\geq 2$ for $i=1,2$, then $\\dim\\overline{\\Phi_{{|D|}}(X)}\\geq 2$. \n\\end{lemma}\n\nAs an application of Theorem~\\ref{thm:ext for surface}, we prove Theorem~\\ref{thm1.3} on fibrations of $(1,2)$-surfaces and $(2,3)$-surfaces. \nFirst we recall properties of pluricanonical systems of $(1,2)$-surfaces and $(2,3)$-surfaces.\n\\begin{theorem}\\label{surface} Let $S$ be a smooth projective surface of general type. \n\\begin{itemize}\n\\item If $S$ is a $(1,2)$-surface, then $\\Phi_{|mK_S|}$ is a generically finite map of degree $2$ for $2\\leq m\\leq 4$ and is a birational map for $m\\geq 5$.\n\\item If $S$ is a $(2,3)$-surface, then $\\Phi_{|mK_S|}$ is a generically finite map of degree $2$ for $2\\leq m\\leq 3$ and is a birational map for $m\\geq 4$.\n\\end{itemize}\n\\end{theorem}\n\\begin{proof}\n This follows from \\cite[Theorem~VII.5.1, Proposition~VII.7.1, Proposition~VII.7.2]{BHPV}, \\cite[Lemma~1.1]{Horikawa1}, and \\cite[Lemma~2.1]{Horikawa2}.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:whenvolgeq12}\nLet $X$ be a minimal $3$-fold of general type.\nAssume that there exists a resolution $\\pi:W\\to X$ and a fibration $f: W\\to C$ to a curve with a general fiber denoted by $F$. \nAssume that $\\pi^*K_X-bF$\n is $\\mathbb{Q}$-effective for some rational number $b>2$. \nIf \\[\\frac{(b-2)^2}{(b+2)^2} \\Vol(F)>8,\\] then $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$.\n\\end{proposition}\n\nNow suppose that there exists an irreducible curve $C$ passing through a very general closed point $P$ such that $(L\\cdot C)< 4<\\frac{1}{2}L^2$. Since $P$ is very general, $C$ is a nef curve with $p_g(C)\\geq 2$. Moreover, after taking a higher model of $F$ and replacing $L$ by its pullback, we may assume that $C$ is smooth; this step does not change $L^2$ or $(L\\cdot C)$ by the projection formula. As $L^2>2(L\\cdot C)$, there exists a sufficiently small positive number $\\epsilon$ such that $L-(1+\\epsilon)C$ is big by \\cite[Theorem~2.2.15]{Positivity1}. In particular, we may write \n$$L\\sim_{\\mathbb{Q}}(1+\\epsilon)C+T$$ where $T$ is an effective $\\mathbb{Q}$-divisor.\nThen by Sakai's lemma \\cite[Example~9.4.12]{Positivity2},\n\\[\nH^1\\left(F, K_F+\\left\\lceil{L-C-\\frac{1}{1+\\epsilon}T}\\right\\rceil\\right)=0,\n\\]\nwhich implies that the natural map\n\\[\nH^0\\left(F, K_F+\\left\\lceil{L-\\frac{1}{1+\\epsilon}T}\\right\\rceil\\right)\\to H^0\\left(C, K_C+\\left. \\left\\lceil{L-\\frac{1}{1+\\epsilon}T-C}\\right\\rceil \\right|_{C}\\right)\n\\]\nis surjective.\nSince $\\deg(\\roundup{L-\\frac{1}{1+\\epsilon}T-C}|_C)>0$, it is clear that\n\\[h^0\\left(C, K_C+\\left. \\left\\lceil{L-\\frac{1}{1+\\epsilon}T-C}\\right\\rceil \\right|_C\\right)\\geq 2\\]\nby the Riemann--Roch formula. Therefore, \n\\[\nh^0(F, K_F+\\roundup{L})\\geq h^0\\left(F, K_F+\\left\\lceil{L-\\frac{1}{1+\\epsilon}T}\\right\\rceil\\right)\\geq 2. \\qedhere\\]\n\\end{proof}\nNow we are ready to study the bicanonical maps. \n\\begin{corollary}\\label{cor X/C}\nLet $X$ be a minimal $3$-fold of general type.\nAssume that there exists a resolution $\\pi:W\\to X$ and a fibration $f:W\\to C$ to a curve with a general fiber denoted by $F$. \nAssume that $ \\Vol(X)>2.9\\times10^4$ and $\\pi^*K_X-bF$ is $\\mathbb{Q}$-effective\nfor some rational number $b> 24$, then $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$.\n\\end{corollary}", "post_theorem_intro_text_len": 2223, "post_theorem_intro_text": "We briefly explain the difficulties and techniques in the proof. As in the study of $\\Phi_{|5K_X|}$ in \\cite{Tod07, LDi12, Chen12, CD25}, the basic idea is to creat non-klt centers by constructing a boundary $\\Delta\\sim_{{\\mathbb Q}}\\delta K_X$ for some $\\delta<2$\nand cut down the dimension of non-klt centers by the Anghern--Siu type method.\nThe issue in dealing with $\\Phi_{|2K_X|}$ instead of $\\Phi_{|5K_X|}$ or $\\Phi_{|3K_X|}$ is that, often we are not allowed to cut down the dimension of a non-klt center if the canonical volume of the non-klt center\nis small (e.g., a surface with small canonical volume or a curve with geometric genus $2$), otherwise the constant $\\delta$ will exceed $2$. In this case, instead of cutting down the dimension of the non-klt center, we prove certain extension theorems which allow us to lift global sections from the centers to $X$. \nHere we remark that in \\cite{CL24}, the key step is to prove an extension theorem for fibrations, where many techniques such as the Kawamata--Viehweg vanishing theorem, canonical bundle formula, the Zariski--Nakayama decomposition, the Hacon--M\\textsuperscript{c}Kernan extension theorem, and Birkar's lower bound of lct are involved, and the extension theorem is not effective; in this paper, we prove an effective version of extension theorem by using the Nadel vanishing theorem and for curve centers we do not require them to form a fibration. \n\nAs a bi-product of our method, we prove the following theorem concerning the pluricanonical maps of $3$-folds of general type fibered by $(1,2)$-surfaces or $(2,3)$-surfaces, as a complement of the results of Xu \\cite[Theorem~1.4, Theorem~1.5]{XuJ14}:\n\\begin{theorem}\\label{thm1.3}\nLet $X$ be a smooth $3$-fold of general type. Suppose that $X$ admits a fibration $X\\to C$ to a curve with a general fiber denoted by $F$.\n\\begin{enumerate}\n \\item If $F$ is a $(1,2)$-surface and $\\Vol(X)> 60$, then $\\Phi_{|mK_X|}$ is generically finite of degree $2$ for $2\\leq m \\leq 4$ and is birational for $m\\geq 5$.\n \\item If $F$ is a $(2,3)$-surface and $\\Vol(X)>456$, then $\\Phi_{|mK_X|}$ is generically finite of degree $2$ for $2\\leq m \\leq 3$ and is birational for $m\\geq 4$.\n\\end{enumerate}\n\\end{theorem}", "sketch": "As in the study of $\\Phi_{|5K_X|}$ in \\cite{Tod07, LDi12, Chen12, CD25}, the basic idea (to prove Theorem~\\ref{thm1.1}) is to \"creat non-klt centers\" by constructing a boundary $\\Delta\\sim_{{\\mathbb Q}}\\delta K_X$ for some $\\delta<2$ and then \"cut down the dimension of non-klt centers by the Anghern--Siu type method.\" The difficulty for $\\Phi_{|2K_X|}$ is that one is often \"not allowed to cut down the dimension of a non-klt center if the canonical volume of the non-klt center is small ... otherwise the constant $\\delta$ will exceed $2$.\" In this situation, \"instead of cutting down the dimension of the non-klt center,\" the approach is to \"prove certain extension theorems which allow us to lift global sections from the centers to $X$.\" The paper claims an \"effective version\" of such extension theorems \"by using the Nadel vanishing theorem,\" and \"for curve centers\" it does not require the centers \"to form a fibration.\"", "expanded_sketch": "As in the study of $\\Phi_{|5K_X|}$ in \\cite{Tod07, LDi12, Chen12, CD25}, the basic idea (to prove the main theorem) is to \"creat non-klt centers\" by constructing a boundary $\\Delta\\sim_{{\\mathbb Q}}\\delta K_X$ for some $\\delta<2$ and then \"cut down the dimension of non-klt centers by the Anghern--Siu type method.\" The difficulty for $\\Phi_{|2K_X|}$ is that one is often \"not allowed to cut down the dimension of a non-klt center if the canonical volume of the non-klt center is small ... otherwise the constant $\\delta$ will exceed $2$.\" In this situation, \"instead of cutting down the dimension of the non-klt center,\" the approach is to \"prove certain extension theorems which allow us to lift global sections from the centers to $X$.\" The paper claims an \"effective version\" of such extension theorems \"by using the Nadel vanishing theorem,\" and \"for curve centers\" it does not require the centers \"to form a fibration.\"", "expanded_theorem": "\\label{thm1.1}\nLet $X$ be a smooth projective $3$-fold of general type. If $\\Vol(X)>12^6$, then $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$.", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let $X$ be a smooth projective $3$-fold of general type, and let $\\Phi_{|2K_X|}$ denote the bicanonical map induced by the complete linear system $|2K_X|$. Suppose the canonical volume of $X$ satisfies $\\Vol(X)>12^6$. Which conclusion about the image of the bicanonical map holds?", "correct_choice": {"label": "A", "text": "The image of the bicanonical map is at least $2$-dimensional; equivalently, $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 2$, so $\\Phi_{|2K_X|}$ is not a pencil."}, "choices": [{"label": "B", "text": "The bicanonical map is birational onto its image; equivalently, $\\Phi_{|2K_X|}$ separates very general points of $X$ whenever $\\Vol(X)>12^6$."}, {"label": "C", "text": "The bicanonical map is non-trivial; equivalently, $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\geq 1$."}, {"label": "D", "text": "There exists a universal constant $\\delta<2$, depending only on the bound $\\Vol(X)>12^6$, such that one can always choose a boundary $\\Delta\\sim_{\\mathbb Q}\\delta K_X$ whose associated non-klt centers can be cut down all the way to points; hence $\\Phi_{|2K_X|}$ is not a pencil."}, {"label": "E", "text": "The image of the bicanonical map is at least $2$-dimensional provided $\\Vol(X)\\geq 12^6$; equivalently, the same conclusion holds already at the boundary value $\\Vol(X)=12^6$."}], "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": "replaces non-pencil conclusion by birationality of the bicanonical map", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "drops the stronger lower bound $\\dim \\overline{\\Phi_{|2K_X|}(X)}\\ge 2$ to mere non-triviality", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "uniform point-cutting with a universal $\\delta<2$ instead of using extension when small-volume centers obstruct cut-down", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "strict volume inequality", "template_used": "boundary_range"}]}}
{"id": "2512.18759v1", "paper_link": "http://arxiv.org/abs/2512.18759v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{mainthm}\nLet $K^{\\rm Kum}$ be the maximal Kummer extension of $K$, obtained by adjoining all roots of elements of $K$. The group  $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_{K^{\\operatorname{Kum}}}}$ is finite for all odd $i$ and arbitrary $j$. In particular, the torsion subgroup of $K^{\\rm Kum}$-points of an abelian variety over $K$ is finite.", "start_pos": 10053, "end_pos": 10441, "label": "mainthm"}, "ref_dict": {"mainthm": "\\begin{theorem}\\label{mainthm}\nLet $K^{\\rm Kum}$ be the maximal Kummer extension of $K$, obtained by adjoining all roots of elements of $K$. The group  $H^i_{\\textup{ét}}(\\overline{X}, \\Q/\\Z(j))^{G_{K^{\\operatorname{Kum}}}}$ is finite for all odd $i$ and arbitrary $j$. In particular, the torsion subgroup of $K^{\\rm Kum}$-points of an abelian variety over $K$ is finite.   \n\\end{theorem}", "prop:solvable_example": "\\begin{prop}\\label{prop:solvable_example} There exist two infinite Galois extensions $M, M'$ of $\\Q$ with $\\Gal(M | \\Q)\\cong\\Gal(M' | \\Q)$ solvable of class 2 such that all abelian varieties defined over subfields of $M$ have finite torsion subgroup over $M$ but some abelian varieties defined over $\\Q$ have infinite torsion over $M'$. \n\\end{prop}", "rmk: counterexample1": "\\begin{remark}\\label{rmk: counterexample1}\n One may ask whether Theorem \\ref{thm: reduction} holds with $K'=K$, i.e.~whether it is sufficient to check finiteness over abelian subextensions of $M|K$. The answer is negative, even when $i=j=1$ and $X$ is an abelian variety,\nas the following example shows. \n\nRecall from \nthe introduction that this case amounts to understanding the torsion subgroups of abelian varieties.\nConsider the elliptic curve $E$ with Weierstrass equation $y^2=x^3+x$, with complex multiplication by $\\mathbb{Z}[i]$ over $\\Q(i)$. By the theory of complex multiplication the field $M$ obtained by adjoining all torsion points of $E$ to $\\Q(i)$ is the maximal abelian extension of $\\Q(i)$ (it is easy to show that $M| \\Q(i)$ is abelian, and this is all we need). This implies that $M| \\Q$ is a solvable extension, because both $M| \\Q(i)$ and $\\Q(i)| \\Q$ are abelian. By construction, the torsion subgroup $E(M)_{\\mathrm{tors}}\\subset E(M)$ is infinite. On the other hand, since $\\Q^{\\operatorname{ab}}=\\Q^{\\operatorname{cyc}}$ by the Kronecker--Weber theorem,  Ribet's result \\cite{Ribet1981CyclotomicTorsion} shows that $E(\\Q^{\\operatorname{ab}})$ is finite, hence there is no abelian subextension $F$ of $M| \\Q$ such that $E(F)_{\\mathrm{tors}}$ is infinite.\n\\end{remark}", "thm: reduction": "\\begin{theorem}\\label{thm: reduction}\nLet $K$ and $X$ be as before, and let $M|K$ be a (possibly infinite) Galois extension contained in $\\Kbar$ such that $\\Gal(M|K)$ is solvable of finite class. Fix integers $i\\geq 0$ and $j$. Assume that for all finite extensions $K' | K$ contained in $\\Kbar$ and all subextensions\n$\nK' \\subset F \\subset K'M\n$\nwith $F | K'$ abelian the following conditions hold:\n\\begin{enumerate}\n    \\item For all primes $p$ we have $\\left( H^i_{\\textup{ét}}(\\overline{X}, \\Q_p(j))^{\\operatorname{ss}} \\right)^{G_F}=0$.\n    \\item For all but finitely many primes $p$ we have $\\left( H^i_{\\textup{ét}}(\\overline{X}, \\Z/p\\Z(j))^{\\operatorname{ss}} \\right)^{G_F}=0$. \n\\end{enumerate}\nThen $H^i_{\\textup{ét}}(\\overline{X}, \\Q/\\Z(j))^{G_M}$ is finite.\n\\end{theorem}", "lemma: p-adic nilpotent implies algebraic nilpotent": "\\begin{lem}\\label{lemma: p-adic nilpotent implies algebraic nilpotent}\nLet $G \\subset\\GL_{n}(\\Q_p)$ be a connected reductive algebraic subgroup and $H \\subset G(\\Q_p)$ be an abstract subgroup, dense in the Zariski topology. Suppose that $H$ is solvable of finite class as an abstract group. Then $G$ is a torus.\n\\end{lem}"}, "pre_theorem_intro_text_len": 1745, "pre_theorem_intro_text": "Let $K$ be a number field with fixed algebraic closure $\\overline K$, and let $X$ be a smooth projective geometrically connected variety over $K$. We denote by $\\overline{X}$ the base change of $X$ to $\\overline{K}$, and for an intermediate field $M$ with $\\overline K\\supset M\\supset K$ we write $G_M:=\\mathrm{Gal}(\\overline K | M)$. In this note we shall study the question of finiteness of the subgroup of $G_M$-invariants of the étale cohomology groups $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))$ over certain infinite extensions $M|K$. Note that when $X=A$ is an abelian variety with dual abelian variety $A^*$, the group $H^1_{\\textup{ét}}(\\overline{A^*}, \\mathbf{Q}/\\mathbf{Z}(1))$ identifies with the torsion subgroup of $A(\\overline K)$ as a $G_K$-module (see, for instance, the introduction of \\cite{RoesslerSzamuely}), so in this case $G_M$-invariants on cohomology correspond to $M$-rational torsion points of $A$.\n\nIn the paper just mentioned, R\\\"ossler and the second author considered the case when $M$ is obtained by adjoining all roots of unity to $K$ and proved finiteness of $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ for {\\bf odd} $i$. The method of proof generalized that of a much earlier theorem of Ribet \\cite{Ribet1981CyclotomicTorsion} who proved finiteness of the torsion subgroup of $M$-points of an abelian variety for $M$ as above.\n\nBut there are other infinite extensions $M|K$ over which finiteness of $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ holds for {odd} $i$. Indeed, recently Murotani and Ozeki \\cite{MurotaniOzeki} proved this for a large class of infinite Kummer extensions $M$ of $K$. Here we extend their statement to all Kummer extensions.", "context": "Let $K$ be a number field with fixed algebraic closure $\\overline K$, and let $X$ be a smooth projective geometrically connected variety over $K$. We denote by $\\overline{X}$ the base change of $X$ to $\\overline{K}$, and for an intermediate field $M$ with $\\overline K\\supset M\\supset K$ we write $G_M:=\\mathrm{Gal}(\\overline K | M)$. In this note we shall study the question of finiteness of the subgroup of $G_M$-invariants of the étale cohomology groups $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))$ over certain infinite extensions $M|K$. Note that when $X=A$ is an abelian variety with dual abelian variety $A^*$, the group $H^1_{\\textup{ét}}(\\overline{A^*}, \\mathbf{Q}/\\mathbf{Z}(1))$ identifies with the torsion subgroup of $A(\\overline K)$ as a $G_K$-module (see, for instance, the introduction of \\cite{RoesslerSzamuely}), so in this case $G_M$-invariants on cohomology correspond to $M$-rational torsion points of $A$.\n\nIn the paper just mentioned, R\\\"ossler and the second author considered the case when $M$ is obtained by adjoining all roots of unity to $K$ and proved finiteness of $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ for {\\bf odd} $i$. The method of proof generalized that of a much earlier theorem of Ribet \\cite{Ribet1981CyclotomicTorsion} who proved finiteness of the torsion subgroup of $M$-points of an abelian variety for $M$ as above.\n\nBut there are other infinite extensions $M|K$ over which finiteness of $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ holds for {odd} $i$. Indeed, recently Murotani and Ozeki \\cite{MurotaniOzeki} proved this for a large class of infinite Kummer extensions $M$ of $K$. Here we extend their statement to all Kummer extensions.", "full_context": "Let $K$ be a number field with fixed algebraic closure $\\overline K$, and let $X$ be a smooth projective geometrically connected variety over $K$. We denote by $\\overline{X}$ the base change of $X$ to $\\overline{K}$, and for an intermediate field $M$ with $\\overline K\\supset M\\supset K$ we write $G_M:=\\mathrm{Gal}(\\overline K | M)$. In this note we shall study the question of finiteness of the subgroup of $G_M$-invariants of the étale cohomology groups $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))$ over certain infinite extensions $M|K$. Note that when $X=A$ is an abelian variety with dual abelian variety $A^*$, the group $H^1_{\\textup{ét}}(\\overline{A^*}, \\mathbf{Q}/\\mathbf{Z}(1))$ identifies with the torsion subgroup of $A(\\overline K)$ as a $G_K$-module (see, for instance, the introduction of \\cite{RoesslerSzamuely}), so in this case $G_M$-invariants on cohomology correspond to $M$-rational torsion points of $A$.\n\nIn the paper just mentioned, R\\\"ossler and the second author considered the case when $M$ is obtained by adjoining all roots of unity to $K$ and proved finiteness of $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ for {\\bf odd} $i$. The method of proof generalized that of a much earlier theorem of Ribet \\cite{Ribet1981CyclotomicTorsion} who proved finiteness of the torsion subgroup of $M$-points of an abelian variety for $M$ as above.\n\nBut there are other infinite extensions $M|K$ over which finiteness of $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ holds for {odd} $i$. Indeed, recently Murotani and Ozeki \\cite{MurotaniOzeki} proved this for a large class of infinite Kummer extensions $M$ of $K$. Here we extend their statement to all Kummer extensions.\n\nLet $K$ be a number field with fixed algebraic closure $\\overline K$, and let $X$ be a smooth projective geometrically connected variety over $K$. We denote by $\\overline{X}$ the base change of $X$ to $\\overline{K}$, and for an intermediate field $M$ with $\\overline K\\supset M\\supset K$ we write $G_M:=\\Gal(\\overline K | M)$. In this note we shall study the question of finiteness of the subgroup of $G_M$-invariants of the étale cohomology groups $H^i_{\\textup{ét}}(\\overline{X}, \\Q/\\Z(j))$ over certain infinite extensions $M|K$. Note that when $X=A$ is an abelian variety with dual abelian variety $A^*$, the group $H^1_{\\textup{ét}}(\\overline{A^*}, \\Q/\\Z(1))$ identifies with the torsion subgroup of $A(\\Kbar)$ as a $G_K$-module (see, for instance, the introduction of \\cite{RoesslerSzamuely}), so in this case $G_M$-invariants on cohomology correspond to $M$-rational torsion points of $A$.\n\nIn the paper just mentioned, R\\\"ossler and the second author considered the case when $M$ is obtained by adjoining all roots of unity to $K$ and proved finiteness of $H^i_{\\textup{ét}}(\\overline{X}, \\Q/\\Z(j))^{G_M}$ for {\\bf odd} $i$. The method of proof generalized that of a much earlier theorem of Ribet \\cite{Ribet1981CyclotomicTorsion} who proved finiteness of the torsion subgroup of $M$-points of an abelian variety for $M$ as above.\n\nBut there are other infinite extensions $M|K$ over which finiteness of $H^i_{\\textup{ét}}(\\overline{X}, \\Q/\\Z(j))^{G_M}$ holds for {odd} $i$. Indeed, recently Murotani and Ozeki \\cite{MurotaniOzeki} proved this for a large class of infinite Kummer extensions $M$ of $K$. Here we extend their statement to all Kummer extensions.\n\nAs already pointed out in \\cite[Remark 3.6]{RoesslerSzamuely}, for even $i$ the group $H^i_{\\textup{ét}}(\\overline{X}, \\Q/\\Z(j))^{G_{K^{\\rm Kum}}}$ is infinite because cycle classes on $X_{K^{\\rm Kum}}$ generate an infinite subgroup. It is an interesting open question whether the quotient modulo cycle classes is finite. On the other hand, finiteness may fail already over infinite abelian extensions: for an abelian variety of CM type over $K$ all torsion points are defined over the maximal abelian extension.\n\n\\begin{theorem}\\label{thm: reduction}\nLet $K$ and $X$ be as before, and let $M|K$ be a (possibly infinite) Galois extension contained in $\\Kbar$ such that $\\Gal(M|K)$ is solvable of finite class. Fix integers $i\\geq 0$ and $j$. Assume that for all finite extensions $K' | K$ contained in $\\Kbar$ and all subextensions\n$\nK' \\subset F \\subset K'M\n$\nwith $F | K'$ abelian the following conditions hold:\n\\begin{enumerate}\n \\item For all primes $p$ we have $\\left( H^i_{\\textup{ét}}(\\overline{X}, \\Q_p(j))^{\\operatorname{ss}} \\right)^{G_F}=0$.\n \\item For all but finitely many primes $p$ we have $\\left( H^i_{\\textup{ét}}(\\overline{X}, \\Z/p\\Z(j))^{\\operatorname{ss}} \\right)^{G_F}=0$. \n\\end{enumerate}\nThen $H^i_{\\textup{ét}}(\\overline{X}, \\Q/\\Z(j))^{G_M}$ is finite.\n\\end{theorem}\n\n\\begin{prop}\\label{prop:31-general}\nIf $i$ is odd, then\n$\\left( H^i_{\\textup{\\'et}}(\\overline{X},\\mathbf{Q}_p(j))^{\\operatorname{ss}} \\right)^{G_{L^{\\rm cyc}}}=0$ for all primes $p$.\n\\end{prop}\n\\begin{proof}\nThe proof of \\cite[Proposition~3.1]{RoesslerSzamuely} goes through verbatim with Corollary~\\ref{cor:cyclo} replacing Ribet's Galois-theoretic lemma cited above. We briefly review the argument. Pick a simple nonzero\n$G_L$-submodule $W\\subset \\left( H^i_{\\textup{\\'et}}(\\overline{X},\\mathbf{Q}_p(j))^{\\operatorname{ss}} \\right)^{G_{L^{\\rm cyc}}}$. Note that $G_L$ acts on $W$ via $G_L/G_{L^{\\rm cyc}}$ which is abelian, and that $W$ is semisimple for this action. Using Grothendieck's local monodromy theorem\none then shows that after taking a suitable finite\nextension of $L$ the action of $G_L$ on $W$ is unramified outside $p$; thus by\nCorollary~\\ref{cor:cyclo} it factors through $G_p:=\\mathrm{Gal}(L(\\mu_{p^\\infty})|L)$.\nFor a place $w\\mid p$ of $L$ the extension $L_w|\\mathbf{Q}_p$ is finite by our assumption on $L$ and the local\nextension $L_w(\\mu_{p^\\infty})|L_w$ is totally ramified, hence the local Galois group of $L_w$ surjects onto\n$G_p$. As in the original proof, after taking a further finite field extension the existence of the Hodge–Tate decomposition and the Weil conjectures (Deligne's theorem) force a Frobenius at a place of good reduction $w\\nmid p$ to have eigenvalues that are both integral powers of $Nw$ and of absolute value $(Nw)^{i/2-j}$, which is impossible for odd $i$.\n\\end{proof}\n\n\\begin{prop}\\label{prop:34-general} \nIf $i$ is odd, then\n$\\left(H^i_{\\textup{\\'et}}(\\overline{X},\\mathbf{Z}/p\\mathbf{Z}(j))^{\\operatorname{ss}}\\right)^{G_{L^{\\rm cyc}}}=0$\nfor all but finitely many primes $p$.\n\\end{prop}\n\\begin{proof} Here we adapt the proof of \\cite[Proposition~3.4]{RoesslerSzamuely}; note some similarities with the proof of Theorem \\ref{thm: reduction}.\n\n\\begin{proof}[Proof of Theorem \\ref{mainthm}]\nSuppose by contradiction that $H^i_{\\textup{ét}}(\\overline{X}, \\Q/\\Z(j))^{G_{K^{\\rm Kum}}}$ is infinite. By \\Cref{thm: reduction}, there exist a finite extension $K' | K$ and an abelian subextension $F$ of $K' \\Kkum | K'$ such that at least one of the following holds:\n\\begin{enumerate}\n \\item $\\left( H^i_{\\textup{ét}}(\\overline{X}, \\Q_p(j))^{\\operatorname{ss}} \\right)^{G_F} \\neq (0)$ for some prime $p$;\n \\item $\\left( H^i_{\\textup{ét}}(\\overline{X}, \\Z/p\\Z(j))^{\\operatorname{ss}} \\right)^{G_F} \\neq (0)$ for infinitely many primes $p$.\n\\end{enumerate}\nReplacing $K$ with $K'$ (which is again a number field), we have an abelian extension $F|K$ contained in $K^{\\operatorname{Kum}}$ for which (1) or (2) above holds. Applying \\Cref{prop: abelianisation Kummer}, we find that $F$ is the compositum of $K^{\\operatorname{cyc}}$ and an abelian extension $L|K$ with bounded local degrees. Further replacing $K$ with $L$, we obtain an extension $F|L$, where $F$ is contained in $L^{\\operatorname{cyc}}$ and (1) or (2) above holds. The two cases contradict \\Cref{prop:31-general} and \\Cref{prop:34-general}, respectively.\n\\end{proof}\n\n\\begin{theorem}\\label{mainthm}\nLet $K^{\\rm Kum}$ be the maximal Kummer extension of $K$, obtained by adjoining all roots of elements of $K$. The group  $H^i_{\\textup{ét}}(\\overline{X}, \\Q/\\Z(j))^{G_{K^{\\operatorname{Kum}}}}$ is finite for all odd $i$ and arbitrary $j$. In particular, the torsion subgroup of $K^{\\rm Kum}$-points of an abelian variety over $K$ is finite.   \n\\end{theorem}\n\n\\begin{theorem}\\label{thm: reduction}\nLet $K$ and $X$ be as before, and let $M|K$ be a (possibly infinite) Galois extension contained in $\\Kbar$ such that $\\Gal(M|K)$ is solvable of finite class. Fix integers $i\\geq 0$ and $j$. Assume that for all finite extensions $K' | K$ contained in $\\Kbar$ and all subextensions\n$\nK' \\subset F \\subset K'M\n$\nwith $F | K'$ abelian the following conditions hold:\n\\begin{enumerate}\n    \\item For all primes $p$ we have $\\left( H^i_{\\textup{ét}}(\\overline{X}, \\Q_p(j))^{\\operatorname{ss}} \\right)^{G_F}=0$.\n    \\item For all but finitely many primes $p$ we have $\\left( H^i_{\\textup{ét}}(\\overline{X}, \\Z/p\\Z(j))^{\\operatorname{ss}} \\right)^{G_F}=0$. \n\\end{enumerate}\nThen $H^i_{\\textup{ét}}(\\overline{X}, \\Q/\\Z(j))^{G_M}$ is finite.\n\\end{theorem}", "post_theorem_intro_text_len": 6209, "post_theorem_intro_text": "As already pointed out in \\cite[Remark 3.6]{RoesslerSzamuely}, for even $i$ the group $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_{K^{\\rm Kum}}}$ is infinite because cycle classes on $X_{K^{\\rm Kum}}$ generate an infinite subgroup. It is an interesting open question whether the quotient modulo cycle classes is finite.  On the other hand, finiteness may fail already over infinite abelian extensions: for an abelian variety of CM type over $K$ all torsion points are defined over the maximal abelian extension.\n\nThe Galois group of the field $K^{\\rm Kum}$ over $K$ is nilpotent of class 2. The proof of Theorem \\ref{mainthm} will proceed by a general reduction to abelian extensions from solvable extensions of finite class: \n\n\\begin{theorem}\\label{thm: reduction}\nLet $K$ and $X$ be as before, and let $M|K$ be a (possibly infinite) Galois extension contained in $\\overline K$ such that $\\mathrm{Gal}(M|K)$ is solvable of finite class. Fix integers $i\\geq 0$ and $j$. Assume that for all finite extensions $K' | K$ contained in $\\overline K$ and all subextensions\n$\nK' \\subset F \\subset K'M\n$\nwith $F | K'$ abelian the following conditions hold:\n\\begin{enumerate}\n    \\item For all primes $p$ we have $\\left( H^i_{\\textup{ét}}(\\overline{X}, \\Q_p(j))^{\\operatorname{ss}} \\right)^{G_F}=0$.\n    \\item For all but finitely many primes $p$ we have $\\left( H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Z}/p\\mathbf{Z}(j))^{\\operatorname{ss}} \\right)^{G_F}=0$. \n\\end{enumerate}\nThen $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ is finite.\n\\end{theorem}\n\nHere $H^i_{\\textup{ét}}(\\overline{X}, \\Q_p(j))^{\\operatorname{ss}}$ (resp.~$H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Z}/p\\mathbf{Z}(j))^{\\operatorname{ss}}$) denotes the semisimplication of the module $H^i_{\\textup{ét}}(\\overline{X}, \\Q_p(j))$ (resp.~$H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Z}/p\\mathbf{Z}(j))$) with respect to the action of $\\mathrm{Gal}(\\overline K|K)$. It is part of the Tate conjecture (at least with $\\Q_p$-coefficients) that these actions are semisimple and thus the superscripts `ss' can be omitted. If we assume semisimplicity, then in view of well-known arguments about abelian groups (see (\\cite[Lemma 2.1]{RoesslerSzamuely}) conditions (1) and (2) above can be replaced by the simpler condition that the group $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_F}$ is finite.\n\n\\begin{remarks} ${}$\n\\begin{enumerate}\n\\item In fact, we prove a slightly stronger statement: there exists a finite extension $K'|K$, depending only on $i$, $j$ and $X$, such that if $M$ is as in the theorem and all abelian extensions $F|K'$ contained in $K'M$ satisfy conditions (1) and (2), then $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ is finite. On the other hand, there may exist $K''|K$ finite (often we may take $K''=K$; see Remark \\ref{rmk: counterexample1}) such that (1) and (2) hold for all abelian $F|K''$ contained in $K''M$ but nevertheless finiteness of $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ fails. \n\\item The semisimplicity conjecture with finite coefficients is perhaps less widely known. However, it is part of general motivic expectations and Corollary 1.3.4 (4) of \\cite{Cadoretetal} provides strong evidence for it: Cadoret, Hui and Tamagawa prove that over a finitely generated field of positive characteristic semisimplicity of cohomology with $\\Q_p$-coefficients for $p$ large enough (prime to the characteristic) implies semisimplicity with $\\mathbf{Z}/p\\mathbf{Z}$-coefficients for $p$ large enough. The characteristic 0 analogue of this result would suffice for our purposes. Note also that by an observation of Moonen \\cite{Moonen} semisimplicity in characteristic 0 with $\\Q_p$-coefficients follows from the usual Tate conjecture on the image of the cycle map.\n\\end{enumerate}\n\\end{remarks}\n\nFor an abelian variety $A$ over $K$ semisimplicity of the Galois action on the rational Tate module $V_p(A)$ and on $p$-torsion points (for $p$ large enough) is known by classical work of Faltings and Zarhin (see e.g.~Chapter IV by Schappacher in \\cite{FaltingsWustholz}). So in this case finiteness of torsion points of $A$ over the maximal abelian extension $(K')^{\\rm ab}$ of every finite extension $K' | K$ implies finiteness over solvable extensions of finite class. In particular, plugging in a well-known result of Zarhin (\\cite{ZarhinCM}, Corollary of Theorem 1) we obtain: \n\n\\begin{cor} Let $A$ be an abelian variety over $K$ having no simple isogeny factor of CM type over the algebraic closure $\\overline K$. Then $A$ has finitely many torsion points over Galois extensions $M|K$ such that $\\mathrm{Gal}(M| K)$ is solvable of finite class. \\end{cor} This statement was also obtained using arguments similar to ours in very recent work of Huryn \\cite{Huryn}. Huryn's preprint appeared while we were writing up the present note and we took the opportunity to borrow one of his arguments to simplify the proof of one of our lemmas (Lemma \\ref{lemma: p-adic nilpotent implies algebraic nilpotent} below).\n\nObserve that the conditions on the extensions of $K$ appearing in Theorem \\ref{thm: reduction} are purely Galois-theoretic. This prompts the question whether for fixed $X$, $i$ and $j$ there exists a Galois-theoretic criterion on extensions $M|K$ that ensures the finiteness of  $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$. Unfortunately, this is not the case: \n\n\\begin{prop}\\label{prop:solvable_example} There exist two infinite Galois extensions $M, M'$ of $\\mathbf{Q}$ with $\\mathrm{Gal}(M | \\mathbf{Q})\\cong\\mathrm{Gal}(M' | \\mathbf{Q})$ solvable of class 2 such that all abelian varieties defined over subfields of $M$ have finite torsion subgroup over $M$ but some abelian varieties defined over $\\mathbf{Q}$ have infinite torsion over $M'$. \n\\end{prop}\n\n{After some preliminaries, Theorem \\ref{thm: reduction} will be proven in Section \\ref{secred}. Theorem \\ref{mainthm} will be deduced from \\Cref{thm: reduction} in Section \\ref{secmainthm} by means of an adjustment of arguments from \\cite{RoesslerSzamuely}. The last section contains the proof of Proposition \\ref{prop:solvable_example}.\n  }", "sketch": "The proof of Theorem \\ref{mainthm} is described as follows. First, one uses that “the Galois group of the field $K^{\\rm Kum}$ over $K$ is nilpotent of class 2,” and the proof “will proceed by a general reduction to abelian extensions from solvable extensions of finite class,” namely via Theorem \\ref{thm: reduction}. After “some preliminaries,” Theorem \\ref{thm: reduction} is proven in Section \\ref{secred}. Then “Theorem \\ref{mainthm} will be deduced from \\Cref{thm: reduction} in Section \\ref{secmainthm} by means of an adjustment of arguments from \\cite{RoesslerSzamuely}.” The reduction theorem says that for a Galois extension $M|K$ with $\\mathrm{Gal}(M|K)$ “solvable of finite class,” finiteness of $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ follows once, for every finite $K'|K$ and every abelian subextension $F|K'$ inside $K'M$, one has vanishing of invariants on the semisimplifications: (1) $\\bigl(H^i_{\\textup{ét}}(\\overline{X},\\Q_p(j))^{\\mathrm{ss}}\\bigr)^{G_F}=0$ for all primes $p$, and (2) $\\bigl(H^i_{\\textup{ét}}(\\overline{X},\\mathbf{Z}/p\\mathbf{Z}(j))^{\\mathrm{ss}}\\bigr)^{G_F}=0$ for all but finitely many $p$. (They note that assuming semisimplicity, these can be replaced by the simpler requirement that $H^i_{\\textup{ét}}(\\overline{X},\\mathbf{Q}/\\mathbf{Z}(j))^{G_F}$ is finite.)", "expanded_sketch": "The proof of the main theorem is described as follows. First, one uses that “the Galois group of the field $K^{\\rm Kum}$ over $K$ is nilpotent of class 2,” and the proof “will proceed by a general reduction to abelian extensions from solvable extensions of finite class,” namely via the following theorem. \n\n\\begin{theorem}\\label{thm: reduction}\nLet $K$ and $X$ be as before, and let $M|K$ be a (possibly infinite) Galois extension contained in $\\Kbar$ such that $\\Gal(M|K)$ is solvable of finite class. Fix integers $i\\geq 0$ and $j$. Assume that for all finite extensions $K' | K$ contained in $\\Kbar$ and all subextensions\n$\nK' \\subset F \\subset K'M\n$\nwith $F | K'$ abelian the following conditions hold:\n\\begin{enumerate}\n    \\item For all primes $p$ we have $\\left( H^i_{\\textup{\\u00e9t}}(\\overline{X}, \\Q_p(j))^{\\operatorname{ss}} \\right)^{G_F}=0$.\n    \\item For all but finitely many primes $p$ we have $\\left( H^i_{\\textup{\\u00e9t}}(\\overline{X}, \\Z/p\\Z(j))^{\\operatorname{ss}} \\right)^{G_F}=0$. \n\\end{enumerate}\nThen $H^i_{\\textup{\\u00e9t}}(\\overline{X}, \\Q/\\Z(j))^{G_M}$ is finite.\n\\end{theorem}\n\nAfter “some preliminaries,” the reduction theorem is proven later. Then the main theorem is deduced from the reduction theorem later by means of an adjustment of arguments from \\cite{RoesslerSzamuely}. The reduction theorem says that for a Galois extension $M|K$ with $\\mathrm{Gal}(M|K)$ “solvable of finite class,” finiteness of $H^i_{\\textup{\\u00e9t}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_M}$ follows once, for every finite $K'|K$ and every abelian subextension $F|K'$ inside $K'M$, one has vanishing of invariants on the semisimplifications: (1) $\\bigl(H^i_{\\textup{\\u00e9t}}(\\overline{X},\\Q_p(j))^{\\mathrm{ss}}\\bigr)^{G_F}=0$ for all primes $p$, and (2) $\\bigl(H^i_{\\textup{\\u00e9t}}(\\overline{X},\\mathbf{Z}/p\\mathbf{Z}(j))^{\\mathrm{ss}}\\bigr)^{G_F}=0$ for all but finitely many $p$. (They note that assuming semisimplicity, these can be replaced by the simpler requirement that $H^i_{\\textup{\\u00e9t}}(\\overline{X},\\mathbf{Q}/\\mathbf{Z}(j))^{G_F}$ is finite.)", "expanded_theorem": "\\label{mainthm}\nLet $K^{\\rm Kum}$ be the maximal Kummer extension of $K$, obtained by adjoining all roots of elements of $K$. The group  $H^i_{\\textup{ét}}(\\overline{X}, \\mathbf{Q}/\\mathbf{Z}(j))^{G_{K^{\\operatorname{Kum}}}}$ is finite for all odd $i$ and arbitrary $j$. In particular, the torsion subgroup of $K^{\\rm Kum}$-points of an abelian variety over $K$ is finite.", "theorem_type": ["Universal", "Existence"], "mcq": {"question": "Let $K$ be a number field with fixed algebraic closure $\\overline K$, let $X$ be a smooth projective geometrically connected variety over $K$, and write $\\overline X:=X\\times_K \\overline K$. For any intermediate field $M$ with $K\\subset M\\subset \\overline K$, set $G_M:=\\mathrm{Gal}(\\overline K/M)$. Let $K^{\\rm Kum}$ denote the maximal Kummer extension of $K$, obtained by adjoining to $K$ all roots of all elements of $K$. Which statement holds for every odd integer $i$ and every integer $j$?", "correct_choice": {"label": "A", "text": "The subgroup of $G_{K^{\\rm Kum}}$-invariants $H^i_{\\mathrm{\\acute{e}t}}(\\overline X, \\mathbf{Q}/\\mathbf{Z}(j))^{G_{K^{\\rm Kum}}}$ is finite. In particular, if $A$ is an abelian variety over $K$, then the torsion subgroup of $A(K^{\\rm Kum})$ is finite."}, "choices": [{"label": "B", "text": "The subgroup of $G_{K^{\\rm Kum}}$-invariants $H^i_{\\mathrm{\\acute{e}t}}(\\overline X, \\mathbf{Q}/\\mathbf{Z}(j))^{G_{K^{\\rm Kum}}}$ is finite for every integer $i$ and every integer $j$. In particular, if $A$ is an abelian variety over $K$, then the torsion subgroup of $A(K^{\\rm Kum})$ is finite."}, {"label": "C", "text": "For every odd integer $i$ and every integer $j$, the subgroup $H^i_{\\mathrm{\\acute{e}t}}(\\overline X, \\mathbf{Q}/\\mathbf{Z}(j))^{G_{K^{\\rm Kum}}}$ is finite."}, {"label": "D", "text": "For every odd integer $i$ and every integer $j$, there exists a finite extension $K'\\!/K$ such that the subgroup of $G_{K'K^{\\rm Kum}}$-invariants $H^i_{\\mathrm{\\acute{e}t}}(\\overline X, \\mathbf{Q}/\\mathbf{Z}(j))^{G_{K'K^{\\rm Kum}}}$ is finite. In particular, after a finite extension of the ground field, the torsion subgroup of an abelian variety over $K$ becomes finite over $K^{\\rm Kum}$."}, {"label": "E", "text": "For every odd integer $i$ and every integer $j$, the subgroup of $G_{K^{\\rm Kum}}$-invariants $H^i_{\\mathrm{\\acute{e}t}}(\\overline X, \\mathbf{Q}/\\mathbf{Z}(j))^{G_{K^{\\rm Kum}}}$ is finite provided that, for every abelian subextension $F/K$ contained in $K^{\\rm Kum}$, one has $\\left(H^i_{\\mathrm{\\acute{e}t}}(\\overline X, \\mathbf{Q}_p(j))\\right)^{G_F}=0$ for all primes $p$ and $\\left(H^i_{\\mathrm{\\acute{e}t}}(\\overline X, \\mathbf{Z}/p\\mathbf{Z}(j))\\right)^{G_F}=0$ for all but finitely many primes $p$."}], "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": "odd_i_restriction", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "abelian_variety_corollary", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "field_of_invariants_fixed_as_KKum_not_after_base_change", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "semisimplification_and_all_finite_Kprime_requirement", "template_used": "wildcard"}]}}
{"id": "2512.19461v1", "paper_link": "http://arxiv.org/abs/2512.19461v1", "theorems_cnt": 3, "theorem": {"env_name": "thm", "content": "\\label{thm:twistor}\nThe twistor bundle $q:\\mathbb{C} P^5\\to \\mathbb{H} P^2$ has \n\\[\n2= \\mathsf{secat}(q)=\\mathrm{Swgt}(q;\\F_2)>\\mathrm{Mwgt}(q;\\F_2)=1>\\mathrm{wgt}(q;\\F_2)=0.\n\\]", "start_pos": 11658, "end_pos": 11809, "label": "thm:twistor"}, "ref_dict": {"thm:twistor": "\\begin{thm}\\label{thm:twistor}\nThe twistor bundle $q:\\C P^5\\to \\Ha P^2$ has \n\\[\n2= \\secat(q)=\\Swgt(q;\\F_2)>\\Mwgt(q;\\F_2)=1>\\wgt(q;\\F_2)=0.\n\\]\n\\end{thm}", "thm:2cell": "\\begin{thm}\\label{thm:2cell}\nLet $\\eta\\in \\pi_7(S^6)$ and $\\omega\\in \\pi_6(S^3)$ be generators, and let $\\alpha\\in\\pi_7(S^3)$ be the homotopy class of the composition $\\omega\\circ\\eta$. If $X=S^3\\cup_\\alpha e^8$ is the homotopy cofibre of $\\alpha$, then\n\\[\n2=\\cat(X)=\\Swgt(X;\\F_2)>\\Mwgt(X;\\F_2)=\\wgt(X;\\F_2)=1.\n\\]\n\\end{thm}", "def:Swgt": "\\begin{defn}\\label{def:Swgt}\nThe \\emph{secondary module weight} of the fibration $q:E\\to B$ at the prime $p$ is \n \\begin{multline*}\n \\Swgt(q;\\F_p) := \\min \\{k \\mid q(k)^*:H^*(B;\\F_p)\\to H^*(E(k);\\F_p)\\mbox{ admits a retraction } \\\\ \n \\mbox{in the category of $\\mathcal{A}_p$-modules that commutes with all secondary operations}\\}.\n\\end{multline*}\n\\end{defn}"}, "pre_theorem_intro_text_len": 5813, "pre_theorem_intro_text": "\\label{sec:intro}\n\nThe {\\em sectional category} of a fibration $q:E\\to B$, denoted $\\mathsf{secat}(q)$, is defined to be the minimal integer $k$ such that $B$ admits a cover by open sets $U_0,\\ldots , U_k$, on each of which $q$ admits a continuous local section $s_i:U_i\\to E$ \\cite{Schwarz,James}. Prominent special cases include the Lusternik--Schnirelmann category $\\mathsf{cat}(X)$ \\cite{CLOT}, which is the sectional category of the based path fibration $P_0 X\\to X$, and Farber's topological complexity ${\\sf TC}(X)$ \\cite{Far03,Far04}, which is the sectional category of the free path fibration $PX\\to X\\times X$, as well as its many variants \\cite{FG07,Rudyak,CFW}. Note that here we use the normalized version, for which $\\mathsf{secat}(q)=0$ if and only if $q$ admits a (global) section. \n\nAn important foundational result due to Schwarz \\cite{Schwarz} (see also \\cite{James}) states that if $B$ is paracompact then $\\mathsf{secat}(q)$ agrees with the minimal $k$ such that the $(k+1)$-fold fibred join $q(k):E(k)\\to B$ admits a section. If $F$ is the fibre of $q$, then the fibre of $q(k)$ is $F(k)=F\\ast \\cdots \\ast F$, the iterated join of $(k+1)$ copies of $F$. When $F$ is $(r-1)$-connected, $F(k)$ is $(r+1)(k+1)-2$-connected. Then (assuming $B$ is a CW complex) cellular obstruction theory gives an upper bound\n\\[\n\\mathsf{secat}(q)< \\frac{\\mathrm{dim}(B)+1}{r+1}.\n\\]\n\nA standard cohomological argument using the open sets definition gives that $\\mathsf{secat}(q)$ is bounded below by $\\operatorname{nil}\\operatorname{ker} (q^*:H^*(B;R)\\to H^*(E;R))$, the nilpotency of the kernel of the induced map in cohomology. Here we take coefficients in an arbitrary commutative ring $R$ (local coefficients may also be used). This bound can sometimes be improved upon using the fibred join interpretation. In particular, we define following Farber--Grant \\cite{FG07,FG08} the \\emph{sectional category weight} of $q$ to be \n\\[\n\\mathrm{wgt}(q;R)=\\min \\{k \\mid q(k)^*:H^*(B;R)\\to H^*(E(k);R)\\mbox{ is injective}\\}.\n\\]\nThen (since a section $s:B\\to E(k)$ of $q(k)$ clearly implies injectivity of $q(k)^*$) one has $\\mathsf{secat}(q)\\geq\\mathrm{wgt}(q;R)$. It can be shown that $\\mathrm{wgt}(q;R)\\geq \\operatorname{nil}\\operatorname{ker} (q^*:H^*(B;R)\\to H^*(E;R))$, and examples of strict inequality are known.\n\nA further improvement to this lower bound was found by Iwase and Kono \\cite{IwaseKono}. They observed that a section $s$ of $q(k)$, in addition to implying injectivity of the induced map $q(k)^*$, would entail a retraction $s^*: H^*(E(k);R)\\to H^*(B;R)$ which commutes with all cohomology operations. They therefore defined the \\emph{module weight} of $q$ at the prime $p\\ge2$ to be\n\\begin{multline*}\n\\mathrm{Mwgt}(q;\\F_p)=\\min \\{k \\mid q(k)^*:H^*(B;\\F_p)\\to H^*(E(k);\\F_p)\\mbox{ admits}\\\\\n\\mbox{a retraction in the category of $\\mathcal{A}_p$-modules}\\}.\n\\end{multline*}  \nHere $\\mathcal{A}_p$ denotes the mod $p$ Steenrod algebra. Clearly, one has inequalities\n\\[\n\\mathsf{secat}(q)\\ge \\mathrm{Mwgt}(q;\\F_p) \\ge \\mathrm{wgt}(q;\\F_p)\\ge \\operatorname{nil}\\operatorname{ker}(q^*: H^*(B;\\F_p)\\to H^*(E;\\F_p))\n\\]\nfor each prime $p$. Examples of strict inequality $\\mathrm{Mwgt}(q;\\F_p) > \\mathrm{wgt}(q;\\F_p)$ can be found in \\cite{IwaseKono, Choi09, Choi15}.\n\nThe ideas in the previous two paragraphs originated in the work of Fadell and Husseini \\cite{FH} on category weight, and were developed in the context of $q$ being the based path fibration $P_0 X\\to X$ by Rudyak \\cite{Rudyak99} and Strom \\cite{Strom}. We have given here the straightforward generalisation to arbitrary fibrations $q: E\\to B$.\n\nIn this paper we will improve on the module weight using secondary cohomology operations. Recall that whenever stable cohomology operations $\\theta$ and $\\varphi$ satisfy a relation $\\varphi\\circ\\theta\\equiv 0$, this gives rise to a \\emph{secondary cohomology operation}\n\\[\n\\Phi=\\Phi(\\varphi,\\theta) : \\operatorname{ker} \\,\\theta \\to \\mathrm{coker} \\,\\varphi\n\\]\nwith \\emph{indeterminacy} $\\operatorname{im} \\varphi$. Such secondary operations are natural, in the following sense: if $f: X\\to Y$ is a map of spaces, then the diagram\n \\[\n \\xymatrix{\n \\operatorname{ker}\\, \\theta_Y \\ar[r]^-{\\Phi} \\ar[d]_{f^*} & \\mathrm{coker}\\, \\varphi_Y \\ar[d]^{f^*} \\\\\n \\operatorname{ker}\\, \\theta_X \\ar[r]^-{\\Phi} & \\mathrm{coker}\\, \\varphi_X\n }\n \\]\n commutes. (This makes sense because $f^* (\\operatorname{ker}\\,\\theta_Y)\\subseteq \\operatorname{ker}\\,\\theta_X$ and $f^*(\\operatorname{im} \\varphi_Y)\\subseteq \\operatorname{im} \\varphi_X$.) If we consider only stable cohomology operations with $\\F_p$-coefficients, we can say that $f^*:H^*(Y;\\F_p)\\to H^*(X;\\F_p)$ is a map of $\\A_p$-modules which commutes with all secondary operations.  This leads us to our main definition.\n\n\\begin{defn}\\label{def:Swgt}\nThe \\emph{secondary module weight} of the fibration $q:E\\to B$ at the prime $p$ is \n \\begin{multline*}\n \\mathrm{Swgt}(q;\\F_p) := \\min \\{k \\mid q(k)^*:H^*(B;\\F_p)\\to H^*(E(k);\\F_p)\\mbox{ admits a retraction } \\\\ \n \\mbox{in the category of $\\mathcal{A}_p$-modules that commutes with all secondary operations}\\}.\n\\end{multline*}\n\\end{defn}\n\nWe illustrate the utility of this definition by giving examples where $\\mathsf{secat}(q)=\\mathrm{Swgt}(q;\\F_p)>\\mathrm{Mwgt}(q;\\F_p)$. Our first example is the so-called \\emph{twistor bundle} \n\\[\n\\xymatrix{\nS^2 \\ar[r] & \\mathbb{C} P^5 \\ar[r]^q & \\mathbb{H} P^2\n}\n\\]\nunder which $q$ sends a complex line in $\\mathbb{C}^6\\cong\\mathbb{H}^3$ to the quaternionic line in $\\mathbb{H}^3$ containing it. Standard techniques give $1\\le \\mathsf{secat}(q)\\le 2$, but are inconclusive in determining the exact value of $\\mathsf{secat}(q)$. In fact, it was our attempts to determine whether $q(1)$ admits a section using obstruction theory that led us to Definition \\ref{def:Swgt}.", "context": "The {\\em sectional category} of a fibration $q:E\\to B$, denoted $\\mathsf{secat}(q)$, is defined to be the minimal integer $k$ such that $B$ admits a cover by open sets $U_0,\\ldots , U_k$, on each of which $q$ admits a continuous local section $s_i:U_i\\to E$ \\cite{Schwarz,James}. Prominent special cases include the Lusternik--Schnirelmann category $\\mathsf{cat}(X)$ \\cite{CLOT}, which is the sectional category of the based path fibration $P_0 X\\to X$, and Farber's topological complexity ${\\sf TC}(X)$ \\cite{Far03,Far04}, which is the sectional category of the free path fibration $PX\\to X\\times X$, as well as its many variants \\cite{FG07,Rudyak,CFW}. Note that here we use the normalized version, for which $\\mathsf{secat}(q)=0$ if and only if $q$ admits a (global) section.\n\nAn important foundational result due to Schwarz \\cite{Schwarz} (see also \\cite{James}) states that if $B$ is paracompact then $\\mathsf{secat}(q)$ agrees with the minimal $k$ such that the $(k+1)$-fold fibred join $q(k):E(k)\\to B$ admits a section. If $F$ is the fibre of $q$, then the fibre of $q(k)$ is $F(k)=F\\ast \\cdots \\ast F$, the iterated join of $(k+1)$ copies of $F$. When $F$ is $(r-1)$-connected, $F(k)$ is $(r+1)(k+1)-2$-connected. Then (assuming $B$ is a CW complex) cellular obstruction theory gives an upper bound\n\\[\n\\mathsf{secat}(q)< \\frac{\\mathrm{dim}(B)+1}{r+1}.\n\\]\n\nA standard cohomological argument using the open sets definition gives that $\\mathsf{secat}(q)$ is bounded below by $\\operatorname{nil}\\operatorname{ker} (q^*:H^*(B;R)\\to H^*(E;R))$, the nilpotency of the kernel of the induced map in cohomology. Here we take coefficients in an arbitrary commutative ring $R$ (local coefficients may also be used). This bound can sometimes be improved upon using the fibred join interpretation. In particular, we define following Farber--Grant \\cite{FG07,FG08} the \\emph{sectional category weight} of $q$ to be \n\\[\n\\mathrm{wgt}(q;R)=\\min \\{k \\mid q(k)^*:H^*(B;R)\\to H^*(E(k);R)\\mbox{ is injective}\\}.\n\\]\nThen (since a section $s:B\\to E(k)$ of $q(k)$ clearly implies injectivity of $q(k)^*$) one has $\\mathsf{secat}(q)\\geq\\mathrm{wgt}(q;R)$. It can be shown that $\\mathrm{wgt}(q;R)\\geq \\operatorname{nil}\\operatorname{ker} (q^*:H^*(B;R)\\to H^*(E;R))$, and examples of strict inequality are known.\n\nA further improvement to this lower bound was found by Iwase and Kono \\cite{IwaseKono}. They observed that a section $s$ of $q(k)$, in addition to implying injectivity of the induced map $q(k)^*$, would entail a retraction $s^*: H^*(E(k);R)\\to H^*(B;R)$ which commutes with all cohomology operations. They therefore defined the \\emph{module weight} of $q$ at the prime $p\\ge2$ to be\n\\begin{multline*}\n\\mathrm{Mwgt}(q;\\F_p)=\\min \\{k \\mid q(k)^*:H^*(B;\\F_p)\\to H^*(E(k);\\F_p)\\mbox{ admits}\\\\\n\\mbox{a retraction in the category of $\\mathcal{A}_p$-modules}\\}.\n\\end{multline*} \nHere $\\mathcal{A}_p$ denotes the mod $p$ Steenrod algebra. Clearly, one has inequalities\n\\[\n\\mathsf{secat}(q)\\ge \\mathrm{Mwgt}(q;\\F_p) \\ge \\mathrm{wgt}(q;\\F_p)\\ge \\operatorname{nil}\\operatorname{ker}(q^*: H^*(B;\\F_p)\\to H^*(E;\\F_p))\n\\]\nfor each prime $p$. Examples of strict inequality $\\mathrm{Mwgt}(q;\\F_p) > \\mathrm{wgt}(q;\\F_p)$ can be found in \\cite{IwaseKono, Choi09, Choi15}.\n\n\\begin{defn}\\label{def:Swgt}\nThe \\emph{secondary module weight} of the fibration $q:E\\to B$ at the prime $p$ is \n \\begin{multline*}\n \\mathrm{Swgt}(q;\\F_p) := \\min \\{k \\mid q(k)^*:H^*(B;\\F_p)\\to H^*(E(k);\\F_p)\\mbox{ admits a retraction } \\\\ \n \\mbox{in the category of $\\mathcal{A}_p$-modules that commutes with all secondary operations}\\}.\n\\end{multline*}\n\\end{defn}\n\nWe illustrate the utility of this definition by giving examples where $\\mathsf{secat}(q)=\\mathrm{Swgt}(q;\\F_p)>\\mathrm{Mwgt}(q;\\F_p)$. Our first example is the so-called \\emph{twistor bundle} \n\\[\n\\xymatrix{\nS^2 \\ar[r] & \\mathbb{C} P^5 \\ar[r]^q & \\mathbb{H} P^2\n}\n\\]\nunder which $q$ sends a complex line in $\\mathbb{C}^6\\cong\\mathbb{H}^3$ to the quaternionic line in $\\mathbb{H}^3$ containing it. Standard techniques give $1\\le \\mathsf{secat}(q)\\le 2$, but are inconclusive in determining the exact value of $\\mathsf{secat}(q)$. In fact, it was our attempts to determine whether $q(1)$ admits a section using obstruction theory that led us to Definition \\ref{def:Swgt}.\n\n\\begin{defn}\\label{def:Swgt}\nThe \\emph{secondary module weight} of the fibration $q:E\\to B$ at the prime $p$ is \n \\begin{multline*}\n \\Swgt(q;\\F_p) := \\min \\{k \\mid q(k)^*:H^*(B;\\F_p)\\to H^*(E(k);\\F_p)\\mbox{ admits a retraction } \\\\ \n \\mbox{in the category of $\\mathcal{A}_p$-modules that commutes with all secondary operations}\\}.\n\\end{multline*}\n\\end{defn}", "full_context": "The {\\em sectional category} of a fibration $q:E\\to B$, denoted $\\mathsf{secat}(q)$, is defined to be the minimal integer $k$ such that $B$ admits a cover by open sets $U_0,\\ldots , U_k$, on each of which $q$ admits a continuous local section $s_i:U_i\\to E$ \\cite{Schwarz,James}. Prominent special cases include the Lusternik--Schnirelmann category $\\mathsf{cat}(X)$ \\cite{CLOT}, which is the sectional category of the based path fibration $P_0 X\\to X$, and Farber's topological complexity ${\\sf TC}(X)$ \\cite{Far03,Far04}, which is the sectional category of the free path fibration $PX\\to X\\times X$, as well as its many variants \\cite{FG07,Rudyak,CFW}. Note that here we use the normalized version, for which $\\mathsf{secat}(q)=0$ if and only if $q$ admits a (global) section.\n\nAn important foundational result due to Schwarz \\cite{Schwarz} (see also \\cite{James}) states that if $B$ is paracompact then $\\mathsf{secat}(q)$ agrees with the minimal $k$ such that the $(k+1)$-fold fibred join $q(k):E(k)\\to B$ admits a section. If $F$ is the fibre of $q$, then the fibre of $q(k)$ is $F(k)=F\\ast \\cdots \\ast F$, the iterated join of $(k+1)$ copies of $F$. When $F$ is $(r-1)$-connected, $F(k)$ is $(r+1)(k+1)-2$-connected. Then (assuming $B$ is a CW complex) cellular obstruction theory gives an upper bound\n\\[\n\\mathsf{secat}(q)< \\frac{\\mathrm{dim}(B)+1}{r+1}.\n\\]\n\nA standard cohomological argument using the open sets definition gives that $\\mathsf{secat}(q)$ is bounded below by $\\operatorname{nil}\\operatorname{ker} (q^*:H^*(B;R)\\to H^*(E;R))$, the nilpotency of the kernel of the induced map in cohomology. Here we take coefficients in an arbitrary commutative ring $R$ (local coefficients may also be used). This bound can sometimes be improved upon using the fibred join interpretation. In particular, we define following Farber--Grant \\cite{FG07,FG08} the \\emph{sectional category weight} of $q$ to be \n\\[\n\\mathrm{wgt}(q;R)=\\min \\{k \\mid q(k)^*:H^*(B;R)\\to H^*(E(k);R)\\mbox{ is injective}\\}.\n\\]\nThen (since a section $s:B\\to E(k)$ of $q(k)$ clearly implies injectivity of $q(k)^*$) one has $\\mathsf{secat}(q)\\geq\\mathrm{wgt}(q;R)$. It can be shown that $\\mathrm{wgt}(q;R)\\geq \\operatorname{nil}\\operatorname{ker} (q^*:H^*(B;R)\\to H^*(E;R))$, and examples of strict inequality are known.\n\nA further improvement to this lower bound was found by Iwase and Kono \\cite{IwaseKono}. They observed that a section $s$ of $q(k)$, in addition to implying injectivity of the induced map $q(k)^*$, would entail a retraction $s^*: H^*(E(k);R)\\to H^*(B;R)$ which commutes with all cohomology operations. They therefore defined the \\emph{module weight} of $q$ at the prime $p\\ge2$ to be\n\\begin{multline*}\n\\mathrm{Mwgt}(q;\\F_p)=\\min \\{k \\mid q(k)^*:H^*(B;\\F_p)\\to H^*(E(k);\\F_p)\\mbox{ admits}\\\\\n\\mbox{a retraction in the category of $\\mathcal{A}_p$-modules}\\}.\n\\end{multline*} \nHere $\\mathcal{A}_p$ denotes the mod $p$ Steenrod algebra. Clearly, one has inequalities\n\\[\n\\mathsf{secat}(q)\\ge \\mathrm{Mwgt}(q;\\F_p) \\ge \\mathrm{wgt}(q;\\F_p)\\ge \\operatorname{nil}\\operatorname{ker}(q^*: H^*(B;\\F_p)\\to H^*(E;\\F_p))\n\\]\nfor each prime $p$. Examples of strict inequality $\\mathrm{Mwgt}(q;\\F_p) > \\mathrm{wgt}(q;\\F_p)$ can be found in \\cite{IwaseKono, Choi09, Choi15}.\n\n\\begin{defn}\\label{def:Swgt}\nThe \\emph{secondary module weight} of the fibration $q:E\\to B$ at the prime $p$ is \n \\begin{multline*}\n \\mathrm{Swgt}(q;\\F_p) := \\min \\{k \\mid q(k)^*:H^*(B;\\F_p)\\to H^*(E(k);\\F_p)\\mbox{ admits a retraction } \\\\ \n \\mbox{in the category of $\\mathcal{A}_p$-modules that commutes with all secondary operations}\\}.\n\\end{multline*}\n\\end{defn}\n\nWe illustrate the utility of this definition by giving examples where $\\mathsf{secat}(q)=\\mathrm{Swgt}(q;\\F_p)>\\mathrm{Mwgt}(q;\\F_p)$. Our first example is the so-called \\emph{twistor bundle} \n\\[\n\\xymatrix{\nS^2 \\ar[r] & \\mathbb{C} P^5 \\ar[r]^q & \\mathbb{H} P^2\n}\n\\]\nunder which $q$ sends a complex line in $\\mathbb{C}^6\\cong\\mathbb{H}^3$ to the quaternionic line in $\\mathbb{H}^3$ containing it. Standard techniques give $1\\le \\mathsf{secat}(q)\\le 2$, but are inconclusive in determining the exact value of $\\mathsf{secat}(q)$. In fact, it was our attempts to determine whether $q(1)$ admits a section using obstruction theory that led us to Definition \\ref{def:Swgt}.\n\n\\begin{defn}\\label{def:Swgt}\nThe \\emph{secondary module weight} of the fibration $q:E\\to B$ at the prime $p$ is \n \\begin{multline*}\n \\Swgt(q;\\F_p) := \\min \\{k \\mid q(k)^*:H^*(B;\\F_p)\\to H^*(E(k);\\F_p)\\mbox{ admits a retraction } \\\\ \n \\mbox{in the category of $\\mathcal{A}_p$-modules that commutes with all secondary operations}\\}.\n\\end{multline*}\n\\end{defn}\n\n\\def\\Ab{\\mathrm{Ab}}\n\\def\\Rel{\\mathrm{Rel}}\n\\def\\cat{\\mathsf{cat}}\n\\def\\secat{\\mathsf{secat}}\n\\def\\wgt{\\mathrm{wgt}}\n\\def\\rank{\\mathrm{rank}}\n\\def\\ind{\\mathrm{ind}}\n\\def\\dim{\\mathrm{dim}}\n\\def\\ker{\\operatorname{ker}}\n\\def\\nil{\\operatorname{nil}}\n\\def\\coker{\\mathrm{coker}}\n\\def\\im{\\mathrm{im}}\n\\def\\aut{\\mathrm{aut}_1\\,}\n\\def\\ev{\\mathrm{ev}}\n\\def\\cupQ{\\mathrm{cup}_0}\n\\def\\rk{\\mathrm{rank}}\n\\def\\eQ{\\mathrm{e}_0}\n\\def\\res{\\mathrm{res}}\n\\def\\tto{\\twoheadrightarrow}\n\n\\newcommand{\\co}{\\colon\\thinspace}\n\\newcommand{\\nsub}{\\mathrel{\\unlhd}}\n\\newcommand{\\PB}{\\mathcal{P}}\n\\newcommand{\\TC}{{\\sf TC}}\n\\newcommand{\\MTC}{{\\sf MTC}}\n\\newcommand{\\TCS}{\\mathbf{TC}^S}\n\\newcommand{\\swgt}{\\mathrm{swgt}}\n\\def\\Mwgt{\\mathrm{Mwgt}}\n\\def\\Swgt{\\mathrm{Swgt}}\n\\newcommand{\\cld}{{\\sf cd}}\n\\newcommand{\\gld}{{\\sf gd}}\n\\newcommand{\\catG}{{\\sf cat}_G}\n\\newcommand{\\secatG}{{\\sf secat}_G}\n\\newcommand{\\cldG}{{\\sf cd}_\\mathcal{G}}\n\\newcommand{\\gldG}{{\\sf gd}_\\mathcal{G}}\n\\newcommand{\\OG}{\\mathcal{O}G}\n\\newcommand{\\OF}{\\mathcal{O}_\\mathcal{F}}\n\\newcommand{\\G}{\\Gamma}\n\\newcommand{\\GG}{\\pi\\rtimes G}\n\\newcommand{\\gen}{\\mathfrak{genus}}\n\\newcommand{\\stab}{\\stackrel{\\simeq}{\\longrightarrow}}\n\\newcommand{\\Aut}{\\operatorname{Aut}}\n\\newcommand{\\OFG}{\\mathcal{O}_\\F\\Gamma}\n\\newcommand{\\OFGG}{\\mathcal{O}_\\FG (\\pi\\rtimes G)}\n\\newcommand{\\pspan}{\\operatorname{pspan}}\n\\newcommand{\\spn}{\\operatorname{span}}\n\\newcommand{\\paths}[1]{P#1}\n\\newcommand{\\eval}[1]{\\pi_{#1}}\n\\newcommand{\\evalG}[1]{p_{#1}}\n\\newcommand{\\evaln}[1]{\\pi_{n,#1}}\n\n\\begin{defn}\\label{def:Swgt}\nThe \\emph{secondary module weight} of the fibration $q:E\\to B$ at the prime $p$ is \n \\begin{multline*}\n \\Swgt(q;\\F_p) := \\min \\{k \\mid q(k)^*:H^*(B;\\F_p)\\to H^*(E(k);\\F_p)\\mbox{ admits a retraction } \\\\ \n \\mbox{in the category of $\\mathcal{A}_p$-modules that commutes with all secondary operations}\\}.\n\\end{multline*}\n\\end{defn}\n\nWe illustrate the utility of this definition by giving examples where $\\secat(q)=\\Swgt(q;\\F_p)>\\Mwgt(q;\\F_p)$. Our first example is the so-called \\emph{twistor bundle} \n\\[\n\\xymatrix{\nS^2 \\ar[r] & \\C P^5 \\ar[r]^q & \\Ha P^2\n}\n\\]\nunder which $q$ sends a complex line in $\\C^6\\cong\\Ha^3$ to the quaternionic line in $\\Ha^3$ containing it. Standard techniques give $1\\le \\secat(q)\\le 2$, but are inconclusive in determining the exact value of $\\secat(q)$. In fact, it was our attempts to determine whether $q(1)$ admits a section using obstruction theory that led us to Definition \\ref{def:Swgt}.\n\nOur second example concerns Lusternik--Schnirelmann category. As is standard, when $q:P_0 X\\to X$ is the based path fibration on a space $X$ we denote $\\wgt(q;\\F_p)$, $\\Mwgt(q;\\F_p)$ and $\\Swgt(q;\\F_p)$ by $\\wgt(X;\\F_p)$, $\\Mwgt(X;\\F_p)$ and $\\Swgt(X;\\F_p)$, respectively.\n\n\\begin{thm}\\label{thm:2cell}\nLet $\\eta\\in \\pi_7(S^6)$ and $\\omega\\in \\pi_6(S^3)$ be generators, and let $\\alpha\\in\\pi_7(S^3)$ be the homotopy class of the composition $\\omega\\circ\\eta$. If $X=S^3\\cup_\\alpha e^8$ is the homotopy cofibre of $\\alpha$, then\n\\[\n2=\\cat(X)=\\Swgt(X;\\F_2)>\\Mwgt(X;\\F_2)=\\wgt(X;\\F_2)=1.\n\\]\n\\end{thm}\n\n\\begin{lem}\\label{lem:Mwgttwistor}\nFor $q$ the twistor bundle, $\\Mwgt(q;\\F_2)=1$.\n\\end{lem}\n\n\\begin{lem}\\label{lem:Swgttwistor}\n For $q$ the twistor bundle, $\\Swgt(q;\\F_2)=2$.\n \\end{lem}", "post_theorem_intro_text_len": 2032, "post_theorem_intro_text": "Theorem \\ref{thm:twistor} answers a question on MathOverflow asked by Prateep Chakraborty \\cite{Chakra}. Note that the rational sectional category $\\secat_0(q)$ in this example is $1$, as follows from results of Stanley \\cite{Stanley}.\n\nOur second example concerns Lusternik--Schnirelmann category. As is standard, when $q:P_0 X\\to X$ is the based path fibration on a space $X$ we denote $\\mathrm{wgt}(q;\\F_p)$, $\\mathrm{Mwgt}(q;\\F_p)$ and $\\mathrm{Swgt}(q;\\F_p)$ by $\\mathrm{wgt}(X;\\F_p)$, $\\mathrm{Mwgt}(X;\\F_p)$ and $\\mathrm{Swgt}(X;\\F_p)$, respectively.\n\n\\begin{thm}\\label{thm:2cell}\nLet $\\eta\\in \\pi_7(S^6)$ and $\\omega\\in \\pi_6(S^3)$ be generators, and let $\\alpha\\in\\pi_7(S^3)$ be the homotopy class of the composition $\\omega\\circ\\eta$. If $X=S^3\\cup_\\alpha e^8$ is the homotopy cofibre of $\\alpha$, then\n\\[\n2=\\mathsf{cat}(X)=\\mathrm{Swgt}(X;\\F_2)>\\mathrm{Mwgt}(X;\\F_2)=\\mathrm{wgt}(X;\\F_2)=1.\n\\]\n\\end{thm}\n\nIn the proofs of both Theorems \\ref{thm:twistor} and \\ref{thm:2cell}, the secondary operation $\\Phi$ based on the Adem relation $Sq^3Sq^1+Sq^2Sq^2\\equiv 0$ is shown to be nonzero with zero indeterminacy in the total space of the fibred join. In each case, this operation detects the Hopf invariant of the attaching map of the top cell \\cite{BH,CLOT,GGV}, which turns out to be represented by the class of the composition $\\eta\\circ\\eta\\in \\pi_7(S^5)$. The main technique to prove non-triviality of $\\Phi$ is what Harper calls ``compatibility with exact sequences\" in his book \\cite{Harper}. This method sometimes allows to compute secondary operations in a space $Y$ in terms of primary operations in the other spaces in a cofiber sequence involving $Y$. We review the method in Section 2 below. Thereafter the proofs diverge: Theorem \\ref{thm:twistor} uses the Thom isomorphism relating Steenrod squares in the total space of a sphere bundle with Stiefel-Whitney classes, while Theorem \\ref{thm:2cell} uses results of Kono--Kozima \\cite{KonoKozima} on Steenrod squares in the loop space of $\\operatorname{Spin}(n)$.", "sketch": "In the proofs of both Theorems \\ref{thm:twistor} and \\ref{thm:2cell}, the secondary operation $\\Phi$ based on the Adem relation $Sq^3Sq^1+Sq^2Sq^2\\equiv 0$ is shown to be nonzero with zero indeterminacy in the total space of the fibred join. In each case, this operation detects the Hopf invariant of the attaching map of the top cell \\cite{BH,CLOT,GGV}, which turns out to be represented by the class of the composition $\\eta\\circ\\eta\\in \\pi_7(S^5)$. The main technique to prove non-triviality of $\\Phi$ is what Harper calls ``compatibility with exact sequences\" \\cite{Harper}, allowing one to compute secondary operations in a space $Y$ in terms of primary operations in the other spaces in a cofiber sequence involving $Y$ (reviewed in Section 2). For Theorem \\ref{thm:twistor} specifically, the proof then uses the Thom isomorphism relating Steenrod squares in the total space of a sphere bundle with Stiefel--Whitney classes.", "expanded_sketch": "In establishing the main theorem, the secondary operation $\\Phi$ based on the Adem relation $Sq^3Sq^1+Sq^2Sq^2\\equiv 0$ is shown to be nonzero with zero indeterminacy in the total space of the fibred join. We also use the following theorem.\n\\begin{thm}\\label{thm:2cell}\nLet $\\eta\\in \\pi_7(S^6)$ and $\\omega\\in \\pi_6(S^3)$ be generators, and let $\\alpha\\in\\pi_7(S^3)$ be the homotopy class of the composition $\\omega\\circ\\eta$. If $X=S^3\\cup_\\alpha e^8$ is the homotopy cofibre of $\\alpha$, then\n\\[\n2=\\cat(X)=\\Swgt(X;\\F_2)>\\Mwgt(X;\\F_2)=\\wgt(X;\\F_2)=1.\n\\]\n\\end{thm}\nIn each case, this operation detects the Hopf invariant of the attaching map of the top cell Berrick--Hess, Cohen--Lada--May (Title not available, year not available), and Gugenheim--May--Vigu\\'{e}-Poirrier (Title not available, year not available), which turns out to be represented by the class of the composition $\\eta\\circ\\eta\\in \\pi_7(S^5)$. The main technique to prove non-triviality of $\\Phi$ is what Harper calls ``compatibility with exact sequences'' Harper (Title not available, year not available), allowing one to compute secondary operations in a space $Y$ in terms of primary operations in the other spaces in a cofiber sequence involving $Y$ (reviewed later). For the main theorem specifically, the proof then uses the Thom isomorphism relating Steenrod squares in the total space of a sphere bundle with Stiefel--Whitney classes.", "expanded_theorem": "\\label{thm:twistor}\nThe twistor bundle $q:\\mathbb{C} P^5\\to \\mathbb{H} P^2$ has \n\\[\n2= \\mathsf{secat}(q)=\\mathrm{Swgt}(q;\\F_2)>\\mathrm{Mwgt}(q;\\F_2)=1>\\mathrm{wgt}(q;\\F_2)=0.\n\\]", "theorem_type": ["Inequality or Bound"], "mcq": {"question": "Let q:\\mathbb{C}P^5\\to\\mathbb{H}P^2 be the twistor bundle, where q sends a complex line in \\mathbb{C}^6\\cong\\mathbb{H}^3 to the quaternionic line in \\mathbb{H}^3 containing it. Write \\mathsf{secat}(q) for the normalized sectional category of q, i.e. the least k such that \\mathbb{H}P^2 can be covered by open sets U_0,\\dots,U_k on each of which q admits a continuous local section. For the (k+1)-fold fibred join q(k):E(k)\\to \\mathbb{H}P^2, define \\mathrm{wgt}(q;\\mathbb{F}_2)=\\min\\{k\\mid q(k)^*:H^*(\\mathbb{H}P^2;\\mathbb{F}_2)\\to H^*(E(k);\\mathbb{F}_2)\\text{ is injective}\\}, \\mathrm{Mwgt}(q;\\mathbb{F}_2) as the least k for which q(k)^* admits a retraction in the category of \\mathcal{A}_2-modules, and \\mathrm{Swgt}(q;\\mathbb{F}_2) as the least k for which q(k)^* admits such a retraction commuting with all secondary operations. Which quantitative statement holds for this bundle?", "correct_choice": {"label": "A", "text": "2=\\mathsf{secat}(q)=\\mathrm{Swgt}(q;\\mathbb{F}_2)>\\mathrm{Mwgt}(q;\\mathbb{F}_2)=1>\\mathrm{wgt}(q;\\mathbb{F}_2)=0."}, "choices": [{"label": "B", "text": "2=\\mathsf{secat}(q)=\\mathrm{Swgt}(q;\\mathbb{F}_2)=\\mathrm{Mwgt}(q;\\mathbb{F}_2)>\\mathrm{wgt}(q;\\mathbb{F}_2)=0."}, {"label": "C", "text": "2=\\mathsf{secat}(q)=\\mathrm{Swgt}(q;\\mathbb{F}_2)>\\mathrm{Mwgt}(q;\\mathbb{F}_2)\\ge \\mathrm{wgt}(q;\\mathbb{F}_2)."}, {"label": "D", "text": "1=\\mathsf{secat}(q)=\\mathrm{Swgt}(q;\\mathbb{F}_2)>\\mathrm{Mwgt}(q;\\mathbb{F}_2)=\\mathrm{wgt}(q;\\mathbb{F}_2)=0."}, {"label": "E", "text": "2=\\mathsf{secat}(q)=\\mathrm{Mwgt}(q;\\mathbb{F}_2)>\\mathrm{Swgt}(q;\\mathbb{F}_2)=1>\\mathrm{wgt}(q;\\mathbb{F}_2)=0."}], "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": "nontrivial secondary operation obstructs only secondary-compatible retraction, not ordinary module retraction", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "exact numerical values of \\mathrm{Mwgt} and \\mathrm{wgt}", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "nonzero secondary obstruction at the first fibred join rules out a section of q(1)", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "ordering between \\mathrm{Swgt} and \\mathrm{Mwgt} forced by secondary-operation obstruction", "template_used": "wildcard"}]}}
{"id": "2512.19500v1", "paper_link": "http://arxiv.org/abs/2512.19500v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{t:main}\n  Let $q\\ge4$ be a prime power and $G$ be almost simple with socle\n  $S=\\PSL_2(q)$, that is $S\\le G\\le\\aut(S)$. Suppose that $G$ acts\n  primitively on $\\Omega$. Then for every distinct\n  $\\alpha, \\beta\\in\\Omega$ there exists $s\\in S$ such that\n  $\\alpha^s=\\beta$ and $s$ fixes no point of $\\Omega$.", "start_pos": 26726, "end_pos": 27068, "label": "t:main"}, "ref_dict": {}, "pre_theorem_intro_text_len": 1849, "pre_theorem_intro_text": "It is an easy exercise using the orbit-counting theorem to show that\nif $G$ is a transitive permutation group on a finite set $\\Omega$,\nthen the set of derangements of $G$ generates a transitive subgroup.\n\nIt is also easy to see that in general the set of derangements,\ntogether with the identity, need not be transitive on $\\Omega$. A\nsmall example is the alternating group $A_4$ acting on the $2$-subsets\nof $\\{1,2,3,4\\}$: An element $g\\in A_4$ moving $\\{1,2\\}$ to\n$\\{3,4\\}$ moves every element in its natural degree $4$ action, so\nit is a double transposition, and therefore fixes some $2$-subset.\n\nWhile there are many more such examples, it seemed hard to find one\nunder the additional assumption that $G$ acts primitively. In fact the\nfollowing Problem 8.75 of the Kourovka Notebook\n\\cite{kourovka_notebook}, attributed to John G.~Thompson and\ndesignated ``A known problem'', was open for more than 40 years.\n\\begin{question}\\label{q:8.75}\n  Suppose $G$ is a finite primitive permutation group on $\\Omega$, and\n  $\\alpha$, $\\beta$ are distinct points of $\\Omega$. Does there exist\n  an element $g\\in G$ such that $\\alpha^g=\\beta$ and $g$ fixes no\n  point of $\\Omega$?\n\\end{question}\nIn \\cite{mueller___8.75} we give a negative answer. The group there is\nthe Steinberg triality group ${}^{3}D_{4}(2)$ of order\n$211{,}341{,}312 = 2^{12}\\cdot 3^{4}\\cdot 7^{2}\\cdot 13$ in a\nprimitive permutation action on a set of size $4{,}064{,}256$.\n\nUsing Magma \\cite{magma}, we checked that this is the only example if\nwe assume that $G$ is almost simple with socle of size less than or\nequal $20{,}158{,}709{,}760$. It would have been computationally too\nexpensive to check all the groups with socle $\\PSL_2(q)$ in this\nrange. The purpose of this note is to show that these groups do not\nproduce negative examples. In fact a somewhat stronger result holds.", "context": "It is an easy exercise using the orbit-counting theorem to show that\nif $G$ is a transitive permutation group on a finite set $\\Omega$,\nthen the set of derangements of $G$ generates a transitive subgroup.\n\nIt is also easy to see that in general the set of derangements,\ntogether with the identity, need not be transitive on $\\Omega$. A\nsmall example is the alternating group $A_4$ acting on the $2$-subsets\nof $\\{1,2,3,4\\}$: An element $g\\in A_4$ moving $\\{1,2\\}$ to\n$\\{3,4\\}$ moves every element in its natural degree $4$ action, so\nit is a double transposition, and therefore fixes some $2$-subset.\n\nWhile there are many more such examples, it seemed hard to find one\nunder the additional assumption that $G$ acts primitively. In fact the\nfollowing Problem 8.75 of the Kourovka Notebook\n\\cite{kourovka_notebook}, attributed to John G.~Thompson and\ndesignated ``A known problem'', was open for more than 40 years.\n\\begin{question}\\label{q:8.75}\n Suppose $G$ is a finite primitive permutation group on $\\Omega$, and\n $\\alpha$, $\\beta$ are distinct points of $\\Omega$. Does there exist\n an element $g\\in G$ such that $\\alpha^g=\\beta$ and $g$ fixes no\n point of $\\Omega$?\n\\end{question}\nIn \\cite{mueller___8.75} we give a negative answer. The group there is\nthe Steinberg triality group ${}^{3}D_{4}(2)$ of order\n$211{,}341{,}312 = 2^{12}\\cdot 3^{4}\\cdot 7^{2}\\cdot 13$ in a\nprimitive permutation action on a set of size $4{,}064{,}256$.\n\nUsing Magma \\cite{magma}, we checked that this is the only example if\nwe assume that $G$ is almost simple with socle of size less than or\nequal $20{,}158{,}709{,}760$. It would have been computationally too\nexpensive to check all the groups with socle $\\PSL_2(q)$ in this\nrange. The purpose of this note is to show that these groups do not\nproduce negative examples. In fact a somewhat stronger result holds.", "full_context": "It is an easy exercise using the orbit-counting theorem to show that\nif $G$ is a transitive permutation group on a finite set $\\Omega$,\nthen the set of derangements of $G$ generates a transitive subgroup.\n\nIt is also easy to see that in general the set of derangements,\ntogether with the identity, need not be transitive on $\\Omega$. A\nsmall example is the alternating group $A_4$ acting on the $2$-subsets\nof $\\{1,2,3,4\\}$: An element $g\\in A_4$ moving $\\{1,2\\}$ to\n$\\{3,4\\}$ moves every element in its natural degree $4$ action, so\nit is a double transposition, and therefore fixes some $2$-subset.\n\nWhile there are many more such examples, it seemed hard to find one\nunder the additional assumption that $G$ acts primitively. In fact the\nfollowing Problem 8.75 of the Kourovka Notebook\n\\cite{kourovka_notebook}, attributed to John G.~Thompson and\ndesignated ``A known problem'', was open for more than 40 years.\n\\begin{question}\\label{q:8.75}\n Suppose $G$ is a finite primitive permutation group on $\\Omega$, and\n $\\alpha$, $\\beta$ are distinct points of $\\Omega$. Does there exist\n an element $g\\in G$ such that $\\alpha^g=\\beta$ and $g$ fixes no\n point of $\\Omega$?\n\\end{question}\nIn \\cite{mueller___8.75} we give a negative answer. The group there is\nthe Steinberg triality group ${}^{3}D_{4}(2)$ of order\n$211{,}341{,}312 = 2^{12}\\cdot 3^{4}\\cdot 7^{2}\\cdot 13$ in a\nprimitive permutation action on a set of size $4{,}064{,}256$.\n\nUsing Magma \\cite{magma}, we checked that this is the only example if\nwe assume that $G$ is almost simple with socle of size less than or\nequal $20{,}158{,}709{,}760$. It would have been computationally too\nexpensive to check all the groups with socle $\\PSL_2(q)$ in this\nrange. The purpose of this note is to show that these groups do not\nproduce negative examples. In fact a somewhat stronger result holds.\n\nIn some cases there are lengthy calculations which, in principle,\ncould be carried out by hand. Instead, we provide SageMath code at\n\\cite{dera_psl2_verify} that confirms the claims.\n\\section{Quartic hyperelliptic curves}\nIf $D$ is a plane algebraic curve over a field $F$ given by\n$P(X, Y)\\in F[X, Y]$, then $D(F)=\\defset{(x, y)\\in F^2}{P(x,y)=0}$\ndenotes the set of affine $F$-points on $D$.\n\nFinally assume that $f(X)=(aX^2-2X+d)(aX^2+2X+d)$ is\n inseparable. The two factors are relatively prime (because the\n difference is $4X$). Furthermore, the discriminant of both factors\n is $4(1-ad)$, hence $ad=1$ and we are done.\n\\end{proof}\n\\begin{remark}\n One can check that $q=13$, $17$, and $19$ do not give\n exceptions. However, $q=3$, $5$, $7$, $9$, and $11$ are indeed\n exceptions. This makes it unlikely that there is a ``cleaner'' proof\n of the lemma for odd $q$.\n\\end{remark}\n\\begin{corollary}\\label{c:split}\n Let $q\\ge4$ be a prime power and $a, b, c, d\\in\\FF_q$ with\n $ad-bc=1$. Set\n $T=\\defset{r+1/r}{r\\in\\Fqs}\\cup\\set{0}$. Suppose that\n $ar+d/r\\in T$ and $-cr+b/r\\in T$ for all $r\\in\\Fqs$. Then one\n of the following holds:\n \\begin{itemize}\n \\item[(a)] $a=d=0$ or $b=c=0$;\n \\item[(b)] $q=9$ and $ar+d/r=2$ for some $r\\in\\Fqs$.\n \\end{itemize}\n\\end{corollary}\n\\begin{proof}\n Using the previous lemma, one only need to check the cases where\n $q\\le19$ is an odd prime power. This is most conveniently done with\n the SageMath code at \\cite{dera_psl2_verify}.\n\\end{proof}\n\\begin{lemma}\n Let $q$ be a prime power with $q\\ge23$ if $q$ is odd. For\n $F=\\FF_{q^2}$ set $N=\\defset{r\\in F}{r^{q+1}=1}$. Suppose that for\n $0\\ne a\\in F$ the following holds: For all $r\\in N$ with\n $ar+a^q/r\\ne0$ there is $s\\in N$ such that $ar+a^q/r = s+1/s$. Then\n $a\\in N$.\n\\end{lemma}\n\\begin{proof} We first handle the easier case that $q$ is even.\nAs the square map is an automorphism of $F$, there is $r_0\\in F$ with\n$a^{q-1}=r_0^2$. Note that $a^{q-1}\\in N$, hence $r_0\\in N$.\n\nWe see that the curve $Y^2=\\gamma P(X)Q(X)$ has at least $2(q-6)$\n affine points over $\\FF_q$. On the other hand, by Lemma\n \\ref{l:hasse}, this number is at most $q+1+2\\sqrt{q}$. From this one\n quickly obtains $q\\le19$.\n\\end{proof}\n\\begin{remark}\n One can check that $q=11$, $17$, and $19$ do not give exceptions.\n However, the cases $q=3$, $5$, $7$, $9$, $13$ are indeed\n exceptions. This is most easily seen for $q=3$, because then\n $\\defset{s+1/s}{s\\in N}=\\FF_q$, so the condition on $a$ is vacuous.\n\\end{remark}\n\\begin{corollary}\\label{c:nonsplit}\n Let $q$ be a prime power. Set $N=\\defset{r\\in\\FF_{q^2}}{r^{q+1}=1}$\n and $T=\\defset{r+1/r}{r\\in N}$. For $a, b\\in\\FF_{q^2}$ with\n $a^{q+1} + b^{q+1} = 1$ suppose that $ar+a^q/r\\in T$ and\n $br+b^q/r\\in T$ for all $r\\in\\Fqs$. Then $a=0$ or $b=0$.\n\\end{corollary}\n\\begin{proof}\n If $q\\ge23$, then the previous lemma shows that $a^{q+1}=1$, hence\n $b=0$, or $b^{q+1}=1$, hence $a=0$. The cases $q\\le19$ are\n checked in \\cite{dera_psl2_verify}.\n\\end{proof}\n\\section{The groups of quaternions $2A_4$, $2S_4$, and $2A_5$}\n\\begin{lemma}\n Let $F$ be a field and $\\bH$ be a subgroup of $\\PSL_2(F)$ isomorphic\n to $A_4$, $S_4$, or $A_5$. Let $H$ be the preimage of $\\bH$ in\n $\\SL_2(F)$ and $x, y\\in F^{2\\times 2}$. If the characteristic of $F$\n is $0$ or relatively prime to $\\abs{\\bH}$, then\n \\[\n \\sum_{h\\in H}\\trace(hy)\\cdot\\trace(hx)^{2m+1} =\\abs{\\bH}\\cdot c_m\\cdot (\\trace(x)\\cdot\\trace(y)-\\trace(x\\cdot y))\\cdot\\det(x)^m,\n \\]\n where $m\\in\\set{0, 1}$ if $\\bH\\cong A_4$, $m\\in\\set{0, 1, 2}$ if\n $\\bH\\cong S_4$, $m\\in\\set{0, 1, 2, 3, 4}$ if $\\bH\\cong A_5$, and\n $c_0, c_1, c_2, c_3, c_4$ = $1, 2, 5, 14, 42$ are the first five\n Catalan numbers (with index shifted by $1$).\n\\end{lemma}\n\\begin{proof}\n We assume that the characteristic of $F$ is $0$ or relatively prime\n to $\\abs{\\bH}$. Let $\\bar F$ be an algebraic closure of $F$. It is\n well known that $\\PSL_2(\\bar F)$ contains exactly one conjugacy\n class of subgroups isomorphic to $\\bH$, see e.g.~\\cite[Proposition\n 4.1]{beauville___pgl2}. (In fact more is true. By \\cite[Proposition\n 4.2]{beauville___pgl2} $\\PGL_2(F)$ contains at most one conjugacy\n class of subgroups isomorphic to $\\bH$.)\n\nNote that for $h\\in H$ (recall that $H\\le\\SL_2(\\bar F)$), we\n have $f(h'x, h'y)=f(x, y)$, so $f$ is an invariant of\n $H$. Working out the low degree invariants of $H$ could\n provide a more conceptual explanation. (At a first glance, it\n might not be obvious that the right hand side is an invariant\n under simultaneous left multiplication of $x$ and $y$ by\n $h$. But this follows from the identity\n $\\trace(x)\\cdot\\trace(y)-\\trace(x\\cdot y)=\\trace(\\adj(y)\\cdot\n x)$, where $\\adj(y)$ denotes the adjugate of $y$.)\n \\end{itemize}\n\\end{remark}\n\\begin{lemma}\\label{l:polyhedral}\n Let $F$ be a field and $\\bH$ be a subgroup of $\\PSL_2(F)$ isomorphic\n to $A_4$, $S_4$, or $A_5$. Assume that the characteristic of $F$ is\n $0$ or relatively prime to $\\abs{\\bH}$. Let $H$ be the preimage of\n $\\bH$ in $\\SL_2(F)$. Set\n $T=\\defset{\\trace(h)}{h\\in H}\\setminus\\set{-2, 2}$, and for\n $x = \\begin{pmatrix}a & b\\\\c & d\\end{pmatrix}\\in\\SL_2(F)$ set\n $y = \\begin{pmatrix}0 & 0\\\\c & d\\end{pmatrix}$. Then\n \\[\n \\sum_{h\\in H}\\left(\\trace(hy)\\prod_{t\\in T}(\\trace(hx)-t)\\right) =\\abs{\\bH}.\n \\]\n\\end{lemma}\n\\begin{proof}\n With this choice of $x$ and $y$ we have\n \\[\\trace(x)\\cdot\\trace(y)-\\trace(x\\cdot y)=ad-bc=\\det(x)=1,\\]\n so the previous lemma yields\n \\[\n \\sum_{h\\in H}\\trace(hy)\\cdot\\trace(hx)^{2m+1} =\\abs{\\bH}\\cdot c_m.\n \\]\n Let $Z$ be a variable over $F$. Then \\cite{dera_psl2_verify} verifies\n \\[\n \\prod_{t\\in T}(Z-t) =\n \\begin{cases}\n Z^3 - Z,&\\text{ if }\\bH\\cong A_4\\\\\n Z^5 - 3Z^3 + 2Z,&\\text{ if }\\bH\\cong S_4\\\\\n Z^7 - 4Z^5 + 4Z^3 - Z,&\\text{ if }\\bH\\cong A_5.\\\\\n \\end{cases}\n \\]\nThis yields\n \\[\n \\sum_{h\\in H}\\left(\\trace(hy)\\prod_{t\\in\n T}(\\trace(hx)-t)\\right) =\n \\begin{cases}\n \\abs{\\bH}(c_1-c_0)=\\abs{\\bH},&\\text{ if }\\bH\\cong A_4\\\\\n \\abs{\\bH}(c_2-3c_1+2c_0)=\\abs{\\bH},&\\text{ if }\\bH\\cong S_4\\\\\n \\abs{\\bH}(c_3-4c_2+4c_1-c_0)=\\abs{\\bH},&\\text{ if }\\bH\\cong A_5.\\\\\n \\end{cases}\n \\]\n\\end{proof}\n\\section{Proof of Theorem \\ref{t:main}}\nThe maximal subgroups of almost simple groups with socle $\\PSL_2(q)$\nare known; see \\cite[Table~8.1, Table~8.2]{bray_etal___maximal} or\n\\cite{giudici___psl2}. The results in \\cite{bray_etal___maximal} and\n\\cite{giudici___psl2} are more detailed than those in the following\nlemma, which lists only the restrictions on $q$ that we shall use.\n\\begin{lemma}\\label{l:giudici}\n Let $q\\ge4$ be a prime power and $S=\\PSL_2(q)\\le\n G\\le\\PGammaL_2(q)$. Let $M$ be a maximal subgroup of $G$ with\n $S\\not\\le M$. Set $\\bH=M\\cap S$. Then one of the following holds.\n \\begin{itemize}\n \\item[(a)] $[S:\\bH] = q+1$,\n \\item[(b)] $[S:\\bH] = q(q+1)/2$,\n \\item[(c)] $[S:\\bH] = q(q-1)/2$,\n \\item[(d)] $\\bH\\cong A_4$, and $q$ is and odd prime,\n \\item[(e)] $\\bH\\cong S_4$, and $q$ is a prime with $q\\equiv\\pm1\\mod{8}$,\n \\item[(f)] $\\bH\\cong A_5$, and $q\\equiv\\pm1\\mod{10}$ and $q=9$ if $q$ is\n power of $3$,\n \\item[(g)] $\\bH\\cong\\PSL_2(r)$ or $\\bH\\cong\\PGL_2(r)$, where $q=r^e$\n for $e\\ge2$.\n \\end{itemize}\n Furthermore, in each case $\\bH$ is unique up to conjugacy in\n $\\aut(S)$ (with two classes in case (g) if $r$ is odd).\n\\end{lemma}\nLet $S$ and $G$ be as in Theorem \\ref{t:main}, $M$ be a stabilizer of\na point in $\\Omega$, and $\\bH=M\\cap S$. Then $M$ is a maximal subgroup\nof $G$ by primitivity. We handle the cases (a) to (g) separately.\n\nThe remaining cases (b) to (g) require some work. Suppose that there\nis a counterexample. Then there exist distinct\n$\\alpha, \\beta\\in\\Omega$ such that no derangement in $S$ moves\n$\\alpha$ to $\\beta$. Let $\\bH$ be the stabilizer in $S$ of $\\alpha$\nand $\\bx\\in S$ with $\\alpha^{\\bx}=\\beta$. Then $\\bH\\bx$ is the set of\nelements which moves $\\alpha$ to $\\beta$. As this set does not contain\na derangement, every element in the coset $\\bH\\bx$ is conjugate to an\nelement from $\\bH$.", "post_theorem_intro_text_len": 702, "post_theorem_intro_text": "If $G$ acts primitively on $\\Omega$, then the possibilities for a point\nstabilizer $S_\\omega < S$ are known. While some cases are relatively\neasy to handle, the more difficult ones arise when $S_\\omega$ is\ndihedral, or when $S_\\omega$ is one of the polyhedral groups $A_4$,\n$S_4$, or $A_5$. In the former case we require Hasse’s bound on the\nnumber of points on elliptic curves over finite fields, and to rule out\nthe latter cases we derive trace identities for the binary polyhedral\ngroups $2A_4$, $2S_4$, and $2A_5$.\n\nIn some cases there are lengthy calculations which, in principle,\ncould be carried out by hand. Instead, we provide SageMath code at\n\\cite{dera_psl2_verify} that confirms the claims.", "sketch": "To prove Theorem~\\ref{t:main}, one uses that when $G$ acts primitively on $\\Omega$, “the possibilities for a point stabilizer $S_\\omega < S$ are known.” The argument then splits by the type of $S_\\omega$: “some cases are relatively easy to handle,” while the “more difficult ones arise when $S_\\omega$ is dihedral, or when $S_\\omega$ is one of the polyhedral groups $A_4$, $S_4$, or $A_5$.” In the dihedral case, the proof “require[s] Hasse’s bound on the number of points on elliptic curves over finite fields,” and to exclude the polyhedral cases one “derive[s] trace identities for the binary polyhedral groups $2A_4$, $2S_4$, and $2A_5$.” The authors note that “in some cases there are lengthy calculations” and they “provide SageMath code at \\cite{dera_psl2_verify} that confirms the claims.”", "expanded_sketch": "To prove the main theorem, one uses that when $G$ acts primitively on $\\Omega$, “the possibilities for a point stabilizer $S_\\omega < S$ are known.” The argument then splits by the type of $S_\\omega$: “some cases are relatively easy to handle,” while the “more difficult ones arise when $S_\\omega$ is dihedral, or when $S_\\omega$ is one of the polyhedral groups $A_4$, $S_4$, or $A_5$.” In the dihedral case, the proof “require[s] Hasse’s bound on the number of points on elliptic curves over finite fields,” and to exclude the polyhedral cases one “derive[s] trace identities for the binary polyhedral groups $2A_4$, $2S_4$, and $2A_5$.” The authors note that “in some cases there are lengthy calculations” and they “provide SageMath code at \\cite{dera_psl2_verify} that confirms the claims.”", "expanded_theorem": "\\label{t:main}\n  Let $q\\ge4$ be a prime power and $G$ be almost simple with socle\n  $S=\\PSL_2(q)$, that is $S\\le G\\le\\aut(S)$. Suppose that $G$ acts\n  primitively on $\\Omega$. Then for every distinct\n  $\\alpha, \\beta\\in\\Omega$ there exists $s\\in S$ such that\n  $\\alpha^s=\\beta$ and $s$ fixes no point of $\\Omega$.", "theorem_type": ["Universal–Existential", "Implication"], "mcq": {"question": "Let $q\\ge 4$ be a prime power, let $S=\\mathrm{PSL}_2(q)$, and let $G$ be an almost simple group with socle $S$, meaning that $S\\le G\\le \\mathrm{Aut}(S)$. Suppose $G$ acts primitively on a set $\\Omega$. For distinct points $\\alpha,\\beta\\in\\Omega$, which statement holds about elements of $S$ in this permutation action?", "correct_choice": {"label": "A", "text": "For every pair of distinct points $\\alpha,\\beta\\in\\Omega$, there exists an element $s\\in S$ such that $\\alpha^s=\\beta$ and $s$ fixes no point of $\\Omega$; equivalently, $S$ contains a derangement sending $\\alpha$ to $\\beta$."}, "choices": [{"label": "B", "text": "For every pair of distinct points $\\alpha,\\beta\\in\\Omega$, there exists an element $s\\in G$ such that $\\alpha^s=\\beta$ and $s$ fixes no point of $\\Omega$; equivalently, $G$ contains a derangement sending $\\alpha$ to $\\beta$."}, {"label": "C", "text": "For every pair of distinct points $\\alpha,\\beta\\in\\Omega$, there exists an element $s\\in S$ such that $\\alpha^s=\\beta$."}, {"label": "D", "text": "For every point $\\alpha\\in\\Omega$, there exists an element $s\\in S$ such that $s$ fixes no point of $\\Omega$ and $\\alpha^s\\ne\\alpha$; equivalently, for each $\\alpha$ there is some derangement in $S$ moving $\\alpha$."}, {"label": "E", "text": "There exists a single element $s\\in S$ such that $s$ fixes no point of $\\Omega$ and, for every ordered pair of distinct points $\\alpha,\\beta\\in\\Omega$, one has $\\alpha^s=\\beta$."}], "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": "ambient_group_vs_socle", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "drop_derangement_condition", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "pairwise_targeting_requirement", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "quantifier_order_single_witness", "template_used": "quantifier_dependence"}]}}
{"id": "2512.19505v1", "paper_link": "http://arxiv.org/abs/2512.19505v1", "theorems_cnt": 5, "theorem": {"env_name": "thm", "content": "\\label{thm:Main}\nFor every finite group $G$, there are infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds with automorphism group isomorphic to $G$.", "start_pos": 18681, "end_pos": 18878, "label": "thm:Main"}, "ref_dict": {"thm:ProjectiveVersion": "\\begin{thm}\\label{thm:ProjectiveVersion}\nFor every finite group $G$, there are infinitely many isomorphism classes of irreducible, minimal smooth complex projective surfaces of general type with automorphism group $G$.\n\\end{thm}", "ssec:FuchsianAut": "\\begin{rem}\\label{rem:DMexs}\nSome nonarithmetic Deligne--Mostow lattices \\cite{DeligneMostow} also admit homomorphisms onto cocompact Fuchsian groups from \\emph{forgetful maps} associated with contracting weights. See for example \\cite[Thm.\\ 3.1(iv)]{Deraux}.\n\\end{rem}\n\n\\subsection{Automorphisms of Fuchsian groups}\\label{ssec:FuchsianAut}\n\nThe purpose of this subsection is to prove \\Cref{thm:Fixpsubs}.\n\n\\begin{pf}[Proof of \\Cref{thm:Fixpsubs}]\nLet $p$ be a prime, $T$ be a surface group of genus $g \\ge2$, and $\\phi \\in \\Aut(T)$ have the property that $\\phi(N) = N$ for every finite index normal subgroup with index a power of $p$. Consider the pro-$p$ completion $T_{\\wh{p}}$ of $T$. Then $\\varphi$ extends to a continuous automorphism of $T_{\\wh{p}}$ that is \\emph{normal}, i.e., so that $\\varphi(\\wh{N})$ for every closed normal subgroup of $T_{\\wh{p}}$. Indeed, every such normal subgroup is an intersection of finite index normal subgroups of $T_{\\wh{p}}$, and finite index normal subgroups of $T_{\\wh{p}}$ are in natural one-to-one correspondence with the normal $p$-power index subgroups of $T$.\n\nThen $T_{\\wh{p}}$ is topologically finitely presented, since $T$ is finitely presented, hence $T_{\\wh{p}}$ is \\emph{psuedo $p$-free} in the sense of Jarden--Ritter (see \\cite[\\S 1]{JardenRitter} for the definition) by \\cite[Thm. 4]{JardenRitter}. Thus the extension of $\\varphi$ to $T_{\\wh{p}}$ is inner by \\cite[Thm.\\ 1]{JardenRitter}. Moreover, Paris proved that $T$ is $p$-conjugacy separable \\cite[Thm.\\ 1.7]{Paris}, which implies that $\\varphi$ must preserve conjugacy classes of $T$ (i.e., it is \\emph{pointwise inner}). Therefore $\\varphi$ is a normal automorphism of $T$, meaning that it preserves all normal subgroups of $T$. Finally, it is a theorem of Bogopolski, Kudryavtseva, and Zieschang that normal automorphisms of higher-genus surface groups are inner \\cite[Thm.\\ 1.3]{BKZ}, which completes the proof of the theorem.\n\\end{pf}\n\n\\section{The main result}\\label{sec:Main}\n\nSuppose that $\\Lam < \\PU(2,1)$ is a cocompact nonarithmetic lattice with nontrivial universal homomorphism $\\rho$. Following \\Cref{ssec:PU21}, let $\\Gam < \\PU(2,1)$ be the unique maximal lattice containing $\\Lam$. By \\Cref{lem:TFFuchsian} and by passing to a finite index subgroup if necessary, we can assume that the universal homomorphism for $\\Lam$ has a torsion-free factor and that $\\Lam$ is a characteristic subgroup of $\\Gam$. If $M_{\\mathrm{tf}} \\coloneqq \\ker(\\rho_{\\mathrm{tf}})$ denotes the kernel of the torsion-free universal homomorphism for $\\Lam$, then $M_{\\mathrm{tf}}$ is an infinite index characteristic subgroup of $\\Gam$ by \\Cref{lem:UniversalChar} and because a characteristic subgroup of a characteristic subgroup is characteristic in the larger group. Set $T \\coloneqq \\Lam / M_{\\mathrm{tf}}$, so $T = \\rho_{\\mathrm{tf}}(\\Lam)$.\n\nAs described in \\Cref{ssec:PU21}, away from finitely many prime ideals $\\frakp$ of the ring $\\Tr\\Ad \\Gam$, there is a well-defined congruence quotient $\\wh{Q} \\coloneqq \\Gam / \\Gam(\\frakp)$ with trivial center and commutator subgroup $Q$ that is a nonabelian finite simple group. Moreover, $\\Lam$ and $M_{\\mathrm{tf}}$ are Zariski dense in $\\PU(2,1)$, hence after possibly passing to a finite index subgroup of $\\Lam$, we can assume that they both map onto $Q$ for infinitely many $\\frakp$. Fix one such $\\frakp$ for the remainder of this section. Then define $\\Lam(\\frakp) \\coloneqq \\Gam(\\frakp) \\cap \\Lam$ and $M_{\\mathrm{tf}}(\\frakp) \\coloneqq \\Gam(\\frakp) \\cap M_{\\mathrm{tf}}$, noting that both $\\Lam(\\frakp)$ and $M_{\\mathrm{tf}}(\\frakp)$ are normal subgroups of $\\Gam$, since each is the intersection of normal subgroups of $\\Gam$.\n\n\\begin{lem}\\label{lem:QuoSetup}\nWith the notation established in this section,\n\\begin{align*}\n\\rho_{\\mathrm{tf}}(\\Lam(\\frakp)) &= T \\\\\n\\Lam / M_{\\mathrm{tf}}(\\frakp) &\\cong Q \\times T \\\\\n\\Gam / M_{\\mathrm{tf}}(\\frakp) &\\cong \\wh{Q} \\times \\wh{T}\n\\end{align*}\nwhere $\\wh{T} \\coloneqq \\Gam / M_{\\mathrm{tf}}$ contains $T$ as a finite index subgroup.\n\\end{lem}\n\\begin{pf}\nThe first equality holds since $M_{\\mathrm{tf}}$ contains coset representatives for $\\Lam(\\frakp)$ in $\\Lam$. The second holds since $M_{\\mathrm{tf}} / M_{\\mathrm{tf}}(\\frakp)$ and $\\Lam(\\frakp) / M_{\\mathrm{tf}}(\\frakp)$ are disjoint normal subgroups of $\\Lam / M_{\\mathrm{tf}}(\\frakp)$, and the third is similar.\n\\end{pf}\n\n\\begin{rem}\nNote that $T$ is then a product of factors of the torsion-free universal homomorphism of $\\Lam(\\frakp)$, but there may be additional factors.\n\\end{rem}", "thm:Examples": "\\begin{thm}\\label{thm:Examples}\nThere are cocompact nonarithmetic lattices $\\Gam$ in $\\PU(2,1)$ with nontrivial universal homomorphism.\n\\end{thm}", "ssec:KahlerGroups": "\\begin{equation}\\label{eq:ReductionModp}\n\\pi_\\frakp : \\wt{\\bfG}(\\calO_k) \\lra \\wt{\\bfG}(\\bbF_\\frakp)\n\\end{equation}\nto any $\\wt{\\Del} \\le \\wt{\\bfG}(\\calO_k)$ has kernel the \\emph{congruence subgroup} $\\wt{\\Del}(\\frakp) \\trianglelefteq \\wt{\\Del}$. Strong approximation for Zariski dense subgroups of simply connected algebraic groups says that if $\\wt{\\Del}$ is Zariski dense in $\\wt{\\bfG}(k)$, then $\\pi_\\frakp(\\wt{\\Del}) = \\wt{\\bfG}(\\bbF_\\frakp)$ for all but finitely many $\\frakp$ \\cite[Thm.\\ 1.1]{Weisfeiler}; also see Cor.\\ 3 in Window 9 of \\cite{LubotzkySegal} along with discussion on p.\\ 392 implying the more general statement for number fields other than $\\bbQ$. If $\\wt{\\Del}$ is the preimage of $\\Del$ in $\\SU(2,1)$, then congruence subgroups of $\\Del$ are those whose preimage in $\\wt{\\Del}$ are congruence subgroups.\n\nSince $\\PU(2,1)$ has type ${}^2 \\mathrm{A}_2$, there is a quadratic extension $\\ell$ of $k$ canonically associated with $\\bfG$. If $\\Del \\le \\Gam$ is Zariski dense in $\\PU(2,1)$, then away from a finite set of exceptional primes, the group $\\Del / \\Del(\\frakp)$ has commutator subgroup $\\PSL_3(\\bbF_\\frakp)$ if $\\frakp$ splits in $\\ell$ or $\\PU(3; \\bbF_\\frakp)$ if $\\frakp$ is inert in $\\ell$. In particular, the congruence quotient of $\\Del$ has commutator subgroup a nonabelian finite simple group for all but finitely many primes $\\frakp$. For example in the split case, observe that $\\Del / \\Del(\\frakp)$ is then naturally a subgroup of $\\PGL_3(\\bbF_\\frakp)$ containing $\\PSL_3(\\bbF_\\frakp)$, so it has trivial center.\n\n\\subsection{K\\\"ahler groups}\\label{ssec:KahlerGroups}\n\nA \\emph{K\\\"ahler group} is the fundamental group of a compact K\\\"ahler manifold. See \\cite{KahlerBook, Py} for basics on K\\\"ahler groups. Note that finite index subgroups of K\\\"ahler groups are naturally K\\\"ahler groups.\n\nLet $\\Gam < \\PU(2,1)$ be a cocompact lattice, possibly with torsion. Since $\\PU(2,1)$ is a linear Lie group and $\\Gam$ is finitely generated, $\\Gam$ is residually finite and thus there is a finite index subgroup $\\Lam < \\Gam$ that is torsion-free. Since smooth projective varieties are compact K\\\"ahler manifolds, by pullback of the Fubini--Study metric on $\\bbP^N$ under a projective embedding, it follows that any such $\\Lam$ is a K\\\"ahler group. It then follows from \\cite[Lem.\\ 1.1.15]{KahlerBook} that all cocompact lattices in $\\PU(2,1)$ are K\\\"ahler groups.\n\n\\medskip\n\nFundamental to this paper is the \\emph{universal homomorphism} attached to any K\\\"ahler group. See \\cite[\\S 2.C]{HLPSV} for discussion and references for the following result defining the universal homomorphism.\n\n\\begin{thm}\\label{thm:Universal}\nSuppose $\\Lam$ is a K\\\"ahler group. Then there is a (possibly empty) finite collection of cocompact Fuchsian groups $\\{T_\\al\\}_{\\al \\in \\frakU}$ and a homomorphism\n\\begin{equation}\\label{eq:UniversalHom}\n\\rho : \\Lam \\lra T \\coloneqq \\prod_{\\al \\in \\frakU} T_\\al\n\\end{equation}", "thm:Main": "\\begin{thm}\\label{thm:Main}\nFor every finite group $G$, there are infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds with automorphism group isomorphic to $G$.\n\\end{thm}", "thm:Fixpsubs": "\\begin{thm}\\label{thm:Fixpsubs}\nLet $T$ be a surface group of genus $g \\ge 2$, $p$ be a prime, and suppose that $\\varphi \\in \\Aut(T)$ has the property that $\\varphi(N) = N$ for all finite index normal subgroups of $T$ with index a power of $p$. Then $\\varphi$ is inner.\n\\end{thm}"}, "pre_theorem_intro_text_len": 190, "pre_theorem_intro_text": "\\label{sec:Intro}\n\nThroughout this paper, an automorphism of a complex manifold will always mean a biholomorphism. The main result of this paper, proved in \\Cref{sec:Main}, is the following.", "context": "\\label{sec:Intro}\n\nThroughout this paper, an automorphism of a complex manifold will always mean a biholomorphism. The main result of this paper, proved in \\Cref{sec:Main}, is the following.", "full_context": "\\label{sec:Intro}\n\nThroughout this paper, an automorphism of a complex manifold will always mean a biholomorphism. The main result of this paper, proved in \\Cref{sec:Main}, is the following.\n\n\\begin{abstract}\nIn response to a question raised by Belolipetsky and the first author in \\cite{BelolipetskyLubotzky}, we prove that for every finite group $G$ there are infinitely many isomorphism classes of compact complex hyperbolic $2$-manifolds with automorphism group isomorphic to $G$.\n\\end{abstract}\n\nThroughout this paper, an automorphism of a complex manifold will always mean a biholomorphism. The main result of this paper, proved in \\Cref{sec:Main}, is the following.\n\nGreenberg \\cite{Greenberg} proved \\Cref{thm:Main} for Riemann surfaces, i.e., complex hyperbolic $1$-manifolds. Belolipetsky and the first author \\cite{BelolipetskyLubotzky} proved the analogue of \\Cref{thm:Main} for isometry groups of hyperbolic $n$-manifolds, $n \\ge 3$, and this paper answers a question raised in \\cite[\\S 6.5]{BelolipetskyLubotzky}. Our argument follows the broad outline of that paper, but differences at key steps create many new challenges we must overcome.\n\n\\begin{rem}\\label{rem:Highern}\nWhether or not there are cocompact nonarithmetic lattices in $\\PU(n,1)$ for all $n \\ge 3$ is one of the major open problems in the study of lattices in Lie groups. For any $n \\ge 3$, if there is a cocompact nonarithmetic lattice in $\\PU(n,1)$ with nontrivial universal homomorphism, then this paper produces infinitely many isomorphism classes of closed complex hyperbolic $n$-manifolds with isometry group any finite group $G$ \\emph{verbatim}.\n\\end{rem}\n\n\\begin{thm}\\label{thm:ProjectiveVersion}\nFor every finite group $G$, there are infinitely many isomorphism classes of irreducible, minimal smooth complex projective surfaces of general type with automorphism group $G$.\n\\end{thm}\n\nIn what follows, $\\pi : T \\to T_r$ will denote projection onto the $r^{th}$ surface group factor of $T$ for an arbitrary $r$ that will be fixed for the remainder of this section. Let $\\sig$ denote projection of $\\Gam$ onto $\\Gam / M_{\\mathrm{tf}}(\\frakp)$ and also the restriction of $\\sig$ to any finite index subgroup of $\\Gam$. This leads to the diagram of groups and surjective homomorphisms given in \\Cref{fig:BigDiagram}.\n\\begin{figure}[h]\n\\centering\n\\begin{tikzcd}\n& \\Gam \\arrow[ddr, -] \\arrow[dl, -] \\arrow[rr, ->, \"\\sig\"] \\arrow[rrrr, bend left = 35, ->, \"\\textrm{mod}\\, M_{\\mathrm{tf}}\"] & & \\wh{Q} \\times \\wh{T} \\arrow[d, -] \\arrow[rr, ->, \"\\textrm{mod}\\, \\wh{Q}\" below] & & \\wh{T} \\arrow[d, -] \\\\\n\\Gam(\\frakp) \\arrow[dd, -] & & & Q \\times T \\arrow[rr, ->, \"\\textrm{mod}\\, Q\"] && T \\arrow[rr, ->, \"\\pi\"] & & T_r \\\\\n& & \\Lam \\arrow[dll, -] \\arrow[ur, ->, \"\\sig\"] \\arrow[urrr, bend right=20, ->, \"\\rho_{\\mathrm{tf}}\" {xshift=0.75em, yshift=-1.25em}] \\\\\n\\Lam(\\frakp) \\arrow[ddd, -] \\arrow[uurrrrr, ->, bend right=35, \"\\sig\\, =\\, \\rho_{\\mathrm{tf}}|_{\\Lam(\\frakp)}\" {xshift=6em, yshift=-0.5em}] & & \\\\\n& & \\\\\n& & M_{\\mathrm{tf}} \\arrow[dll, -] \\arrow[uuu, -, crossing over] & \\\\\nM_{\\mathrm{tf}}(\\frakp) & &\n\\end{tikzcd}\n\\caption{The relationships between various groups}\\label{fig:BigDiagram}\n\\end{figure}\nDefine\n\\begin{equation}\\label{eq:TauDef}\n\\tau \\coloneqq \\pi \\circ \\rho_{\\mathrm{tf}} : \\Lam \\lra T_r\n\\end{equation}\nand $M \\coloneqq \\ker(\\tau|_\\Del)$. For a finite index subgroup $\\Del \\le \\Gam$, let $N_\\Gam(\\Del)$ denote its normalizer in $\\Gam$. If $\\Del$ contains $M$, then any $\\gam \\in N_\\Gam(\\Del)$ normalizing $M$ induces an automorphism of $\\Del / M$ through $\\pi \\circ \\rho_{\\mathrm{tf}}$. The following proposition/construction is essential for proving \\Cref{thm:Main}.\n\n\\begin{pf}[Proof of \\Cref{thm:Main}]\nThe only input to begin this section of the paper was a cocompact nonarithmetic lattice $\\Lam < \\PU(2,1)$ with nontrivial universal homomorphism. Such examples exist by \\Cref{thm:Examples}. If $G$ is a nontrivial group (resp.\\ the trivial group), then \\Cref{prop:FinalNontrivialG} (resp.\\ \\Cref{prop:TrivialCase}) produces a torsion-free lattice $B \\le \\Lam$ whose normalizer $N(B) < \\PU(2,1)$, which equals $N_\\Gam(B)$ for $\\Gam$ the unique maximal lattice in $\\PU(2,1)$ containing $B$ by the Margulis dichotomy, has $N(B) / B \\cong G$. Mostow rigidity now implies that $G$ is the automorphism group (equivalently, holomorphic isometry group) of the smooth compact complex hyperbolic $2$-manifold $B \\bs \\bbB^n$. Varying the primes $\\{p_j\\}$ in our construction gives infinitely many lattices with covolume tending to infinity for any fixed finite group $G$, hence another application of Mostow rigidity implies that there are infinitely many isomorphism classes of lattices. This proves the theorem.\n\\end{pf}\n\nComplex hyperbolic manifolds are of general type \\cite[\\S V.20]{BPV}, hence \\Cref{thm:Main} immediately implies \\Cref{thm:ProjectiveVersion}. Surfaces of general type all have finite automorphism group, which is directly analogous to the classical Hurwitz bound $|\\Aut(C)| \\le 84(g-1)$ for a smooth projective curve of genus $g$. In fact, $|\\Aut(X)|$ can be bounded from above in terms of the self-intersection of the first Chern class of the canonical bundle; see for example \\cite{Xiao}. Thus it is natural that the volume (or Euler characteristic, by Gauss--Bonnet) goes to infinity with $|G|$ for our examples.\n\n\\begin{thm}\\label{thm:Examples}\nThere are cocompact nonarithmetic lattices $\\Gam$ in $\\PU(2,1)$ with nontrivial universal homomorphism.\n\\end{thm}\n\n\\begin{thm}\\label{thm:Main}\nFor every finite group $G$, there are infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds with automorphism group isomorphic to $G$.\n\\end{thm}", "post_theorem_intro_text_len": 6774, "post_theorem_intro_text": "Greenberg \\cite{Greenberg} proved \\Cref{thm:Main} for Riemann surfaces, i.e., complex hyperbolic $1$-manifolds. Belolipetsky and the first author \\cite{BelolipetskyLubotzky} proved the analogue of \\Cref{thm:Main} for isometry groups of hyperbolic $n$-manifolds, $n \\ge 3$, and this paper answers a question raised in \\cite[\\S 6.5]{BelolipetskyLubotzky}. Our argument follows the broad outline of that paper, but differences at key steps create many new challenges we must overcome.\n\nRecall that a compact complex hyperbolic $n$-manifold is a quotient $\\Gamma \\backslash \\mathbb{B}^n$ of the unit ball in $\\mathbb{C}^n$ by a torsion-free cocompact lattice $\\Gamma$ in the Lie group $\\PU(n,1)$. Note that $\\mathbb{B}^1$ is simply the Poincar\\'e disk with the hyperbolic metric. For $n \\ge 2$, Mostow rigidity then implies that the automorphism group of a complex hyperbolic manifold is precisely the holomorphic isometry group, and the proof of \\Cref{thm:Main} is through this interpretation.\n\nOne strong similarity between the proofs is in the input: a cocompact nonarithmetic lattice in the isometry group of the symmetric space. Moreover, \\cite{BelolipetskyLubotzky} uses a nonabelian free quotient to begin the construction, whereas we use a quotient that is a cocompact Fuchsian group. The subgroup counting methods in both papers are nearly identical. Note that it is quite easy, given a free or surface group quotient of a finite index subgroup $\\Lambda$ of a lattice $\\Gamma$, to produce a finite index subgroup $B$ of $\\Gamma$ so that $N_\\Gamma(B) / B$ \\emph{contains} $G$, where $N_\\Gamma(B)$ is the normalizer of $B$ in $\\Gamma$, but ensuring that $B$ is torsion-free and that $N_\\Gamma(B) / B$ is \\emph{exactly} $G$ is quite subtle.\n\nThe nonarithmetic lattices used in \\cite{BelolipetskyLubotzky} are the famous examples of Gromov and Piatetski-Shapiro \\cite{GPS}. These lattices naturally have the structure of a free product with amalgamation over the fundamental group of a codimension one totally geodesic subspace, and the nonabelian free quotient of a finite index subgroup is derived from this (see \\cite{LubotzkyFreeQuo}). In the complex hyperbolic setting, there are no real codimension one totally geodesic subspaces \\cite[\\S 3.1]{Goldman}, hence no such splittings.\n\nOur replacement, from the theory of K\\\"ahler groups, is the \\emph{universal homomorphism} to a direct product of cocompact Fuchsian groups; see \\cite{HLPSV} for an introduction. Cocompact nonarithmetic lattices in $\\PU(2,1)$ with nontrivial universal homomorphism are found among the famous examples of Livn\\'e \\cite{Livne} and Deligne--Mostow \\cite{DeligneMostow}; see \\Cref{thm:Examples}. With this different input, a key lattice $\\Delta$ constructed early in the outline of \\cite{BelolipetskyLubotzky} now fails to be normal in the maximal lattice $\\Gamma$ in $\\PU(2,1)$. In a certain sense, much of the work in \\Cref{sec:Main} is using properties of the universal homomorphism to circumvent this significant obstruction to executing several steps in the strategy from \\cite{BelolipetskyLubotzky}.\n\n\\begin{rem}\\label{rem:Highern}\nWhether or not there are cocompact nonarithmetic lattices in $\\PU(n,1)$ for all $n \\ge 3$ is one of the major open problems in the study of lattices in Lie groups. For any $n \\ge 3$, if there is a cocompact nonarithmetic lattice in $\\PU(n,1)$ with nontrivial universal homomorphism, then this paper produces infinitely many isomorphism classes of closed complex hyperbolic $n$-manifolds with isometry group any finite group $G$ \\emph{verbatim}.\n\\end{rem}\n\nIn this paper, a genus $g$ surface group will always mean the fundamental group of a closed Riemann surface of genus $g$, despite the slight clash of terminology with fundamental groups of complex surfaces also appearing. In \\Cref{ssec:FuchsianAut} we will prove the following result on automorphisms of surface groups, which is an analogue of a theorem of the first author in the free group case \\cite[Thm.\\ 1]{LubotzkyFree} that may be of independent interest. \n\n\\begin{thm}\\label{thm:Fixpsubs}\nLet $T$ be a surface group of genus $g \\ge 2$, $p$ be a prime, and suppose that $\\varphi \\in \\Aut(T)$ has the property that $\\varphi(N) = N$ for all finite index normal subgroups of $T$ with index a power of $p$. Then $\\varphi$ is inner.\n\\end{thm}\n\n\\noindent\\textbf{Interpretation as a complex projective surface}\n\n\\medskip\n\nCompact complex hyperbolic $2$-manifolds are also minimal smooth complex projective surfaces of general type \\cite[\\S V.20]{BPV}. The appendix to this paper recasts \\Cref{thm:Main} in that context to prove the following result.\n\n\\begin{thm}\\label{thm:ProjectiveVersion}\nFor every finite group $G$, there are infinitely many isomorphism classes of irreducible, minimal smooth complex projective surfaces of general type with automorphism group $G$.\n\\end{thm}\n\nA surface is irreducible if it is not finitely (\\'etale) covered by a product of complex curves. Greenberg's theorem easily produces products of complex curves with any finite automorphism group, so only the irreducible case is of interest. As described in the appendix, one may be able to construct irreducible examples using other methods. For example, there may exist a nonsingular fibration $X \\to C$ over a curve $C$ (e.g., a ruled surface) where $\\Aut(C) \\cong G$ by Greenberg and there is a natural isomorphism $\\Aut(X) \\cong \\Aut(C)$. However, we were not able to find a reference for \\Cref{thm:ProjectiveVersion} in the literature.\n\n\\begin{rem}\\label{rem:HardPart}\nOur construction is also related to a holomorphic map to a smooth projective curve (see \\Cref{ssec:KahlerGroups}). However, the bulk of the proof is precisely in showing that our surface has no extra automorphisms, which is highly unclear from the nature of the construction.\n\\end{rem}\n\n\\medskip\n\n\\noindent\\textbf{Organization of the paper}\n\n\\medskip\n\n\\noindent\nThis paper is organized as follows. \\Cref{sec:Prelim} contains preliminaries about lattices in $\\PU(2,1)$, universal homomorphisms for K\\\"ahler groups, and the proof of \\Cref{thm:Fixpsubs}. \\Cref{sec:Main} proves \\Cref{thm:Main}. The appendix discussions automorphism groups of smooth complex projective surfaces.\n\n\\medskip\n\n\\noindent \\textbf{Acknowledgments}\n\n\\medskip\n\n\\noindent\nMuch of the work on this paper was done while the authors were at the conference `Discrete and profinite groups' at the Isaac Newton Institute for Mathematical Sciences, and thank the institute for its hospitality. The first author was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 (N.\\ 882751). The second author was supported by Grants DMS-2203555 and DMS-2506896 from the National Science Foundation and the Simons Foundation [SFI-MPS-TSM-00014184, MS].", "sketch": "The proof of \\Cref{thm:Main} is “through” the fact that for $n\\ge 2$, Mostow rigidity implies “the automorphism group of a complex hyperbolic manifold is precisely the holomorphic isometry group.” The argument “follows the broad outline” of \\cite{BelolipetskyLubotzky}: start from “a cocompact nonarithmetic lattice” in the relevant isometry group, use a quotient of a finite index subgroup $\\Lambda<\\Gamma$ (here “a quotient that is a cocompact Fuchsian group,” rather than a free quotient), and then apply “subgroup counting methods” (said to be “nearly identical” to \\cite{BelolipetskyLubotzky}) to build finite-index subgroups $B<\\Gamma$.\n\nA key step is to arrange, for such $B$, that the normalizer quotient realizes the desired finite group: it is “quite easy…to produce a finite index subgroup $B$ of $\\Gamma$ so that $N_\\Gamma(B) / B$ \\emph{contains} $G$,” but “ensuring that $B$ is torsion-free and that $N_\\Gamma(B) / B$ is \\emph{exactly} $G$ is quite subtle,” and the “bulk of the proof is precisely in showing that our surface has no extra automorphisms.”\n\nBecause in the complex hyperbolic setting “there are no real codimension one totally geodesic subspaces…hence no such splittings,” the paper replaces the free-quotient input by the “universal homomorphism to a direct product of cocompact Fuchsian groups.” With this different input, “a key lattice $\\Delta$…now fails to be normal in the maximal lattice $\\Gamma$,” and “much of the work in \\Cref{sec:Main} is using properties of the universal homomorphism to circumvent this significant obstruction to executing several steps in the strategy from \\cite{BelolipetskyLubotzky}.”", "expanded_sketch": "To prove the main theorem, the argument is “through” the fact that for $n\\ge 2$, Mostow rigidity implies “the automorphism group of a complex hyperbolic manifold is precisely the holomorphic isometry group.” The argument “follows the broad outline” of \\cite{BelolipetskyLubotzky}: start from “a cocompact nonarithmetic lattice” in the relevant isometry group, use a quotient of a finite index subgroup $\\Lambda<\\Gamma$ (here “a quotient that is a cocompact Fuchsian group,” rather than a free quotient), and then apply “subgroup counting methods” (said to be “nearly identical” to \\cite{BelolipetskyLubotzky}) to build finite-index subgroups $B<\\Gamma$.\n\nA key step is to arrange, for such $B$, that the normalizer quotient realizes the desired finite group: it is “quite easy…to produce a finite index subgroup $B$ of $\\Gamma$ so that $N_\\Gamma(B) / B$ \\emph{contains} $G$,” but “ensuring that $B$ is torsion-free and that $N_\\Gamma(B) / B$ is \\emph{exactly} $G$ is quite subtle,” and the “bulk of the proof is precisely in showing that our surface has no extra automorphisms.”\n\nBecause in the complex hyperbolic setting “there are no real codimension one totally geodesic subspaces…hence no such splittings,” the paper replaces the free-quotient input by the “universal homomorphism to a direct product of cocompact Fuchsian groups.” With this different input, “a key lattice $\\Delta$…now fails to be normal in the maximal lattice $\\Gamma$,” and “much of the work later is using properties of the universal homomorphism to circumvent this significant obstruction to executing several steps in the strategy from \\cite{BelolipetskyLubotzky}.”", "expanded_theorem": "\\label{thm:Main}\nFor every finite group $G$, there are infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds with automorphism group isomorphic to $G$.", "theorem_type": ["Universal–Existential", "Existence"], "mcq": {"question": "Let an automorphism of a complex manifold mean a biholomorphism. For an arbitrary finite group $G$, which statement holds about closed complex hyperbolic $2$-manifolds (that is, compact complex manifolds of complex dimension $2$ carrying a complex hyperbolic structure)?", "correct_choice": {"label": "A", "text": "There exist infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds whose automorphism group is isomorphic to $G$."}, "choices": [{"label": "B", "text": "There exist infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds whose automorphism group contains a subgroup isomorphic to $G$."}, {"label": "C", "text": "There exists at least one closed complex hyperbolic $2$-manifold whose automorphism group is isomorphic to $G$."}, {"label": "D", "text": "There exists a closed complex hyperbolic $2$-manifold $X$ such that every finite group $G$ occurs as a subgroup of $\\operatorname{Aut}(X)$."}, {"label": "E", "text": "For every finite group $G$, there are infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds whose full holomorphic isometry group is isomorphic to $G$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "exactly_G_vs_contains_G", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "infinitely_many_isomorphism_classes", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "for_each_G_vs_single_universal_manifold", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "automorphism_group_vs_holomorphic_isometry_group_statement_level", "template_used": "wildcard"}]}}
{"id": "2512.20498v1", "paper_link": "http://arxiv.org/abs/2512.20498v1", "theorems_cnt": 3, "theorem": {"env_name": "teorema", "content": "\\label{thm:main}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Then $\\mathcal{S}_\\mu^k$ is contained in a countable union of $k$-dimensional Lipschitz graphs. Moreover, the approximate tangent plane to $\\mathcal{S}_\\mu^k$ at $x$ coincides with $D_x$ for $\\mathcal{H}^{k}$-a.e.\\ $x \\in \\mathcal{S}_\\mu^k$.", "start_pos": 13260, "end_pos": 13630, "label": "thm:main"}, "ref_dict": {"thm:main": "\\begin{teorema}\\label{thm:main}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Then $\\mathcal{S}_\\mu^k$ is contained in a countable union of $k$-dimensional Lipschitz graphs. Moreover, the approximate tangent plane to $\\mathcal{S}_\\mu^k$ at $x$ coincides with $D_x$ for $\\mathcal{H}^{k}$-a.e.\\ $x \\in \\mathcal{S}_\\mu^k$.\n\\end{teorema}", "lmm:0-hom": "\\begin{lemma}\\label{lmm:0-hom}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and let $\\mu_x \\neq 0$ be the unique blow-up of $\\mu$ at $x \\in \\mathcal{S}_\\mu$. Then there exists $\\alpha=\\alpha(x)\\geq 0$ such that $\\mu_x$ and $|\\mu_x|$ are $(\\alpha-n)$-homogeneous,\\footnote{According to this definition, the measure induced by a $p$-homogenous locally integrable function $f$, with respect to the Lebesgue measure, is $p$-homogeneous as well.} namely\n\\begin{equation}\\label{eq:hom_blow-up}\n(\\tau_{0,\\lambda})_\\# \\mu_x=\\lambda^\\alpha \\mu_x,\n\\quad\n(\\tau_{0,\\lambda})_\\# |\\mu_x|=\\lambda^\\alpha |\\mu_x|\n\\qquad\n\\forall \\lambda>0.\n\\end{equation}\nFor $\\mu_x$, and more generally for every $(\\alpha-n)$-homogeneous Radon measure, the set\n\\begin{equation}\nC_x\\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,\\lambda})_\\# \\mu_x=\\lambda^\\alpha (\\tau_{y,1})_\\# \\mu_x \\,\\,\\,\\,\\forall \\lambda>0 \\right \\}\n\\end{equation}\ncoincides with the invariant subspace $D_x$ of $\\mu_x$, that is\n\\begin{equation}\nD_x \\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,1})_\\# \\mu_x= \\mu_x \\right \\}.\n\\end{equation}\nMoreover $D_x$ is a linear subspace of $\\mathbb{R}^n$ and, provided $\\mu_x \\neq 0$, it holds $\\dim D_x \\leq \\alpha$.\n\\end{lemma}", "eq:def_s_mu_k": "\\begin{equation}\\label{eq:def_s_mu_k}\n\\mathcal{S}_\\mu = \\bigcup_{k=1}^n \\mathcal{S}_\\mu^k,\n\\qquad\\text{where} \\qquad\n\\mathcal{S}_\\mu^k \\coloneqq \\left \\{x \\in \\mathcal{S}_\\mu \\colon \\dim D_x=k \\right \\}.\n\\end{equation}", "eq:def-unique_blowup": "\\begin{equation}\\label{eq:def-unique_blowup}\n\\mu_{x,r_j} \\coloneqq\n\\frac{1}{|\\mu|(B_{r_j}(x))} (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\mu_x,\n\\end{equation}", "cor:rectif_criterion": "\\begin{corollario}\\label{cor:rectif_criterion}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Let us assume that, for $|\\mu|$-a.e.\\ $x \\in \\mathbb{R}^n$, there exists a Radon measure $\\nu_x$ whose invariant linear subspace $V_x$ is $k$-dimensional and with the property that, for every sequence $r_j \\to 0^+$ there are a subsequence, not relabeled, and a constant $c\\neq 0$ such that\n\\begin{equation}\n\\frac{1}{r_j^k}  (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\nu_x.\n\\end{equation}\nThen $\\mu$ is $k$-rectifiable: there exists a signed function $\\theta \\in L^1_{\\mathrm{loc}}(\\mathcal{S}_\\mu^k,\\mathcal{H}^k)$ such that \\mbox{$\\mu= \\theta \\mathcal{H}^k \\llcorner \\mathcal{S}_\\mu^k$,} where $\\mathcal{S}_\\mu^k$ is defined in \\eqref{eq:def_s_mu_k}.\n\\end{corollario}"}, "pre_theorem_intro_text_len": 3739, "pre_theorem_intro_text": "The notion of blow-up of a geometric or analytic object, i.e.\\ the limit of its enlargements around a point as the zooming factor goes to infinity, is of fundamental importance in many areas of Mathematics. The blow-up of a function or a manifold provides a simplified, but somewhat essential, description of the local behavior of the object ``at infinitesimal scale\".\nFor this and other reasons, the classification of the blow-ups of various mathematical items has become a central theme in geometric and functional analysis.\n\nFor a Radon measure $\\mu$ in $\\mathbb{R}^n$, a natural definition of blow-up at a point $x \\in \\mathbb{R}^n$ is the one in the sense of Preiss, \\cite{preiss87}: the limit in the weak*-topology of Radon measures (in duality with $C_c(\\mathbb{R}^n)$) of the rescalings\n\\begin{equation}\\label{eq:rescalings}\nc_j (\\tau_{x,r_j})_\\# \\mu,\n\\end{equation}\nfor some sequences $c_j>0$, $r_j \\to 0$, where $\\tau_{x,r}(y)\\coloneqq \\frac{y-x}{r}$ and ${}_\\#$ is the push-forward of Radon measures.\n\nWe say that $\\mu$ has unique blow-up at $x \\in \\operatorname{supp} \\mu$ if there exists a Radon measure $\\mu_x$ with either $|\\mu_x|(B_1(0)) = 1$ or $\\mu_x = 0$ and such that, for every sequence $r_j \\to 0^+$, there are a subsequence, not relabeled, and a constant $c\\neq 0$ for which\n\\begin{equation}\\label{eq:def-unique_blowup}\n\\mu_{x,r_j} \\coloneqq\n\\frac{1}{|\\mu|(B_{r_j}(x))} (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\mu_x,\n\\end{equation}\nwhere $|\\mu|$ is the total variation measure of $\\mu$ and $\\overset{\\ast}{\\rightharpoonup}$ denotes the weak*-convergence of Radon measures.\nWe will denote by $\\mathcal{S}_\\mu$ the set of points where $\\mu$ has unique blow-up.\n\nMattila studied positive measures having a unique blow-up at almost every point: a way to rephrase his main result \\cite[Theorem 3.2]{mattila_unique05} is that,\nfor $\\mu$-a.e.\\ $x \\in \\mathcal{S}_\\mu$, the blow-up of $\\mu$ at $x$ is a multiple of the $k$-dimensional Hausdorff measure $\\mathcal{H}^k$ on a $k$-plane.\nAlthough this describes the blow-ups of $\\mu$ at $\\mu$-a.e.\\ point in $\\mathcal{S}_\\mu$, it can be interesting to study the behavior of $\\mu$ in the remaining part of $\\mathcal{S}_\\mu$.\n\nThis problem has already received some attention: in \\cite{delnin21} it is proved that, for a given $u \\in L^1_{\\text{loc}}(\\mathbb{R}^n)$, the set $\\Sigma_u$ of points $x$ where $\\frac{1}{r^n}u \\left (\\frac{\\cdot-x}{r} \\right )$ converges, in the $L^1_{\\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs. \n\\begin{itemize}\n\\item The results in \\cite{delnin21} rely on the fact that the measure $\\mu$ is induced by a $L^1_{\\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\\mathrm{loc}}$-topology, while in $\\mathcal{S}_\\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\\mathcal{S}_\\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\\Sigma_u$.\nIs this set rectifiable as well? What can be said if $\\mu$ is a general Radon measure?\n\n\\item It is easily seen (Lemma \\ref{lmm:0-hom}) that the unique blow-up $\\mu_x$ of $\\mu$ at a point in $x \\in \\mathcal{S}_\\mu$ has an \\emph{invariant linear subspace} $D_x$ of $\\mathbb{R}^n$, namely $(\\tau_{y,1})_\\# \\mu_x = \\mu_x$ for $y \\in D_x$. Therefore\n\\begin{equation}\\label{eq:def_s_mu_k}\n\\mathcal{S}_\\mu = \\bigcup_{k=1}^n \\mathcal{S}_\\mu^k,\n\\qquad\\text{where} \\qquad\n\\mathcal{S}_\\mu^k \\coloneqq \\left \\{x \\in \\mathcal{S}_\\mu \\colon \\dim D_x=k \\right \\}.\n\\end{equation}\nDo these $\\mathcal{S}_\\mu^k$ enjoy finer rectifiability properties?\n\\end{itemize}\n\nThe main result of this note provides an answer to these questions.", "context": "For a Radon measure $\\mu$ in $\\mathbb{R}^n$, a natural definition of blow-up at a point $x \\in \\mathbb{R}^n$ is the one in the sense of Preiss, \\cite{preiss87}: the limit in the weak*-topology of Radon measures (in duality with $C_c(\\mathbb{R}^n)$) of the rescalings\n\\begin{equation}\\label{eq:rescalings}\nc_j (\\tau_{x,r_j})_\\# \\mu,\n\\end{equation}\nfor some sequences $c_j>0$, $r_j \\to 0$, where $\\tau_{x,r}(y)\\coloneqq \\frac{y-x}{r}$ and ${}_\\#$ is the push-forward of Radon measures.\n\nWe say that $\\mu$ has unique blow-up at $x \\in \\operatorname{supp} \\mu$ if there exists a Radon measure $\\mu_x$ with either $|\\mu_x|(B_1(0)) = 1$ or $\\mu_x = 0$ and such that, for every sequence $r_j \\to 0^+$, there are a subsequence, not relabeled, and a constant $c\\neq 0$ for which\n\\begin{equation}\\label{eq:def-unique_blowup}\n\\mu_{x,r_j} \\coloneqq\n\\frac{1}{|\\mu|(B_{r_j}(x))} (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\mu_x,\n\\end{equation}\nwhere $|\\mu|$ is the total variation measure of $\\mu$ and $\\overset{\\ast}{\\rightharpoonup}$ denotes the weak*-convergence of Radon measures.\nWe will denote by $\\mathcal{S}_\\mu$ the set of points where $\\mu$ has unique blow-up.\n\nMattila studied positive measures having a unique blow-up at almost every point: a way to rephrase his main result \\cite[Theorem 3.2]{mattila_unique05} is that,\nfor $\\mu$-a.e.\\ $x \\in \\mathcal{S}_\\mu$, the blow-up of $\\mu$ at $x$ is a multiple of the $k$-dimensional Hausdorff measure $\\mathcal{H}^k$ on a $k$-plane.\nAlthough this describes the blow-ups of $\\mu$ at $\\mu$-a.e.\\ point in $\\mathcal{S}_\\mu$, it can be interesting to study the behavior of $\\mu$ in the remaining part of $\\mathcal{S}_\\mu$.\n\nThis problem has already received some attention: in \\cite{delnin21} it is proved that, for a given $u \\in L^1_{\\text{loc}}(\\mathbb{R}^n)$, the set $\\Sigma_u$ of points $x$ where $\\frac{1}{r^n}u \\left (\\frac{\\cdot-x}{r} \\right )$ converges, in the $L^1_{\\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs. \n\\begin{itemize}\n\\item The results in \\cite{delnin21} rely on the fact that the measure $\\mu$ is induced by a $L^1_{\\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\\mathrm{loc}}$-topology, while in $\\mathcal{S}_\\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\\mathcal{S}_\\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\\Sigma_u$.\nIs this set rectifiable as well? What can be said if $\\mu$ is a general Radon measure?\n\n\\item It is easily seen (Lemma \\ref{lmm:0-hom}) that the unique blow-up $\\mu_x$ of $\\mu$ at a point in $x \\in \\mathcal{S}_\\mu$ has an \\emph{invariant linear subspace} $D_x$ of $\\mathbb{R}^n$, namely $(\\tau_{y,1})_\\# \\mu_x = \\mu_x$ for $y \\in D_x$. Therefore\n\\begin{equation}\\label{eq:def_s_mu_k}\n\\mathcal{S}_\\mu = \\bigcup_{k=1}^n \\mathcal{S}_\\mu^k,\n\\qquad\\text{where} \\qquad\n\\mathcal{S}_\\mu^k \\coloneqq \\left \\{x \\in \\mathcal{S}_\\mu \\colon \\dim D_x=k \\right \\}.\n\\end{equation}\nDo these $\\mathcal{S}_\\mu^k$ enjoy finer rectifiability properties?\n\\end{itemize}\n\nThe main result of this note provides an answer to these questions.\n\n\\begin{lemma}\\label{lmm:0-hom}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and let $\\mu_x \\neq 0$ be the unique blow-up of $\\mu$ at $x \\in \\mathcal{S}_\\mu$. Then there exists $\\alpha=\\alpha(x)\\geq 0$ such that $\\mu_x$ and $|\\mu_x|$ are $(\\alpha-n)$-homogeneous,\\footnote{According to this definition, the measure induced by a $p$-homogenous locally integrable function $f$, with respect to the Lebesgue measure, is $p$-homogeneous as well.} namely\n\\begin{equation}\\label{eq:hom_blow-up}\n(\\tau_{0,\\lambda})_\\# \\mu_x=\\lambda^\\alpha \\mu_x,\n\\quad\n(\\tau_{0,\\lambda})_\\# |\\mu_x|=\\lambda^\\alpha |\\mu_x|\n\\qquad\n\\forall \\lambda>0.\n\\end{equation}\nFor $\\mu_x$, and more generally for every $(\\alpha-n)$-homogeneous Radon measure, the set\n\\begin{equation}\nC_x\\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,\\lambda})_\\# \\mu_x=\\lambda^\\alpha (\\tau_{y,1})_\\# \\mu_x \\,\\,\\,\\,\\forall \\lambda>0 \\right \\}\n\\end{equation}\ncoincides with the invariant subspace $D_x$ of $\\mu_x$, that is\n\\begin{equation}\nD_x \\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,1})_\\# \\mu_x= \\mu_x \\right \\}.\n\\end{equation}\nMoreover $D_x$ is a linear subspace of $\\mathbb{R}^n$ and, provided $\\mu_x \\neq 0$, it holds $\\dim D_x \\leq \\alpha$.\n\\end{lemma}", "full_context": "For a Radon measure $\\mu$ in $\\mathbb{R}^n$, a natural definition of blow-up at a point $x \\in \\mathbb{R}^n$ is the one in the sense of Preiss, \\cite{preiss87}: the limit in the weak*-topology of Radon measures (in duality with $C_c(\\mathbb{R}^n)$) of the rescalings\n\\begin{equation}\\label{eq:rescalings}\nc_j (\\tau_{x,r_j})_\\# \\mu,\n\\end{equation}\nfor some sequences $c_j>0$, $r_j \\to 0$, where $\\tau_{x,r}(y)\\coloneqq \\frac{y-x}{r}$ and ${}_\\#$ is the push-forward of Radon measures.\n\nWe say that $\\mu$ has unique blow-up at $x \\in \\operatorname{supp} \\mu$ if there exists a Radon measure $\\mu_x$ with either $|\\mu_x|(B_1(0)) = 1$ or $\\mu_x = 0$ and such that, for every sequence $r_j \\to 0^+$, there are a subsequence, not relabeled, and a constant $c\\neq 0$ for which\n\\begin{equation}\\label{eq:def-unique_blowup}\n\\mu_{x,r_j} \\coloneqq\n\\frac{1}{|\\mu|(B_{r_j}(x))} (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\mu_x,\n\\end{equation}\nwhere $|\\mu|$ is the total variation measure of $\\mu$ and $\\overset{\\ast}{\\rightharpoonup}$ denotes the weak*-convergence of Radon measures.\nWe will denote by $\\mathcal{S}_\\mu$ the set of points where $\\mu$ has unique blow-up.\n\nMattila studied positive measures having a unique blow-up at almost every point: a way to rephrase his main result \\cite[Theorem 3.2]{mattila_unique05} is that,\nfor $\\mu$-a.e.\\ $x \\in \\mathcal{S}_\\mu$, the blow-up of $\\mu$ at $x$ is a multiple of the $k$-dimensional Hausdorff measure $\\mathcal{H}^k$ on a $k$-plane.\nAlthough this describes the blow-ups of $\\mu$ at $\\mu$-a.e.\\ point in $\\mathcal{S}_\\mu$, it can be interesting to study the behavior of $\\mu$ in the remaining part of $\\mathcal{S}_\\mu$.\n\nThis problem has already received some attention: in \\cite{delnin21} it is proved that, for a given $u \\in L^1_{\\text{loc}}(\\mathbb{R}^n)$, the set $\\Sigma_u$ of points $x$ where $\\frac{1}{r^n}u \\left (\\frac{\\cdot-x}{r} \\right )$ converges, in the $L^1_{\\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs. \n\\begin{itemize}\n\\item The results in \\cite{delnin21} rely on the fact that the measure $\\mu$ is induced by a $L^1_{\\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\\mathrm{loc}}$-topology, while in $\\mathcal{S}_\\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\\mathcal{S}_\\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\\Sigma_u$.\nIs this set rectifiable as well? What can be said if $\\mu$ is a general Radon measure?\n\n\\item It is easily seen (Lemma \\ref{lmm:0-hom}) that the unique blow-up $\\mu_x$ of $\\mu$ at a point in $x \\in \\mathcal{S}_\\mu$ has an \\emph{invariant linear subspace} $D_x$ of $\\mathbb{R}^n$, namely $(\\tau_{y,1})_\\# \\mu_x = \\mu_x$ for $y \\in D_x$. Therefore\n\\begin{equation}\\label{eq:def_s_mu_k}\n\\mathcal{S}_\\mu = \\bigcup_{k=1}^n \\mathcal{S}_\\mu^k,\n\\qquad\\text{where} \\qquad\n\\mathcal{S}_\\mu^k \\coloneqq \\left \\{x \\in \\mathcal{S}_\\mu \\colon \\dim D_x=k \\right \\}.\n\\end{equation}\nDo these $\\mathcal{S}_\\mu^k$ enjoy finer rectifiability properties?\n\\end{itemize}\n\nThe main result of this note provides an answer to these questions.\n\n\\begin{lemma}\\label{lmm:0-hom}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and let $\\mu_x \\neq 0$ be the unique blow-up of $\\mu$ at $x \\in \\mathcal{S}_\\mu$. Then there exists $\\alpha=\\alpha(x)\\geq 0$ such that $\\mu_x$ and $|\\mu_x|$ are $(\\alpha-n)$-homogeneous,\\footnote{According to this definition, the measure induced by a $p$-homogenous locally integrable function $f$, with respect to the Lebesgue measure, is $p$-homogeneous as well.} namely\n\\begin{equation}\\label{eq:hom_blow-up}\n(\\tau_{0,\\lambda})_\\# \\mu_x=\\lambda^\\alpha \\mu_x,\n\\quad\n(\\tau_{0,\\lambda})_\\# |\\mu_x|=\\lambda^\\alpha |\\mu_x|\n\\qquad\n\\forall \\lambda>0.\n\\end{equation}\nFor $\\mu_x$, and more generally for every $(\\alpha-n)$-homogeneous Radon measure, the set\n\\begin{equation}\nC_x\\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,\\lambda})_\\# \\mu_x=\\lambda^\\alpha (\\tau_{y,1})_\\# \\mu_x \\,\\,\\,\\,\\forall \\lambda>0 \\right \\}\n\\end{equation}\ncoincides with the invariant subspace $D_x$ of $\\mu_x$, that is\n\\begin{equation}\nD_x \\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,1})_\\# \\mu_x= \\mu_x \\right \\}.\n\\end{equation}\nMoreover $D_x$ is a linear subspace of $\\mathbb{R}^n$ and, provided $\\mu_x \\neq 0$, it holds $\\dim D_x \\leq \\alpha$.\n\\end{lemma}\n\nThis problem has already received some attention: in \\cite{delnin21} it is proved that, for a given $u \\in L^1_{\\text{loc}}(\\mathbb{R}^n)$, the set $\\Sigma_u$ of points $x$ where $\\frac{1}{r^n}u \\left (\\frac{\\cdot-x}{r} \\right )$ converges, in the $L^1_{\\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs. \n\\begin{itemize}\n\\item The results in \\cite{delnin21} rely on the fact that the measure $\\mu$ is induced by a $L^1_{\\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\\mathrm{loc}}$-topology, while in $\\mathcal{S}_\\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\\mathcal{S}_\\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\\Sigma_u$.\nIs this set rectifiable as well? What can be said if $\\mu$ is a general Radon measure?\n\nThe main result of this note provides an answer to these questions.\n\nThe idea of the proof is that, for every $x \\in \\mathcal{S}_\\mu^k$, there is a radius $r>0$ for which the points in $\\mathcal{S}_\\mu^k \\cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$. Otherwise, by contradiction, one can find a sequence of such points $y_j \\notin x+C$ with $r_j=|x-y_j| \\to 0$ and such that the measures $c_j \\mu_{y_j}$ and $\\mu_{y_j,r_j}$ are as close as we want. Now, on one hand $c_j\\mu_{y_j}$ converges to a measure $\\nu$ whose invariant subspace is close to $D_x$; on the other hand $\\mu_{y_j,r_j}$, being translations of $\\mu_{x,r_j}$ with respect to points outside $C$, converge to a translation of $\\mu_x$ whose invariant subspace is not contained in $C$, contradicting the closeness of $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$.\n\n\\begin{corollario}\\label{cor:rectif_criterion}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Let us assume that, for $|\\mu|$-a.e.\\ $x \\in \\mathbb{R}^n$, there exists a Radon measure $\\nu_x$ whose invariant linear subspace $V_x$ is $k$-dimensional and with the property that, for every sequence $r_j \\to 0^+$ there are a subsequence, not relabeled, and a constant $c\\neq 0$ such that\n\\begin{equation}\n\\frac{1}{r_j^k} (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\nu_x.\n\\end{equation}\nThen $\\mu$ is $k$-rectifiable: there exists a signed function $\\theta \\in L^1_{\\mathrm{loc}}(\\mathcal{S}_\\mu^k,\\mathcal{H}^k)$ such that \\mbox{$\\mu= \\theta \\mathcal{H}^k \\llcorner \\mathcal{S}_\\mu^k$,} where $\\mathcal{S}_\\mu^k$ is defined in \\eqref{eq:def_s_mu_k}.\n\\end{corollario}\n\n\\begin{corollario}\\label{cor:almost_everywhere_rect}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Then, for $|\\mu|$-a.e.\\ $x \\in \\mathcal{S}_\\mu^k$, the unique blow-up of $\\mu$ at $x$ is a multiple of $\\mathcal{H}^k\\llcorner D_x$.\n\\end{corollario}\n\n\\begin{lemma}\\label{lmm:0-hom}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and let $\\mu_x \\neq 0$ be the unique blow-up of $\\mu$ at $x \\in \\mathcal{S}_\\mu$. Then there exists $\\alpha=\\alpha(x)\\geq 0$ such that $\\mu_x$ and $|\\mu_x|$ are $(\\alpha-n)$-homogeneous,\\footnote{According to this definition, the measure induced by a $p$-homogenous locally integrable function $f$, with respect to the Lebesgue measure, is $p$-homogeneous as well.} namely\n\\begin{equation}\\label{eq:hom_blow-up}\n(\\tau_{0,\\lambda})_\\# \\mu_x=\\lambda^\\alpha \\mu_x,\n\\quad\n(\\tau_{0,\\lambda})_\\# |\\mu_x|=\\lambda^\\alpha |\\mu_x|\n\\qquad\n\\forall \\lambda>0.\n\\end{equation}\nFor $\\mu_x$, and more generally for every $(\\alpha-n)$-homogeneous Radon measure, the set\n\\begin{equation}\nC_x\\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,\\lambda})_\\# \\mu_x=\\lambda^\\alpha (\\tau_{y,1})_\\# \\mu_x \\,\\,\\,\\,\\forall \\lambda>0 \\right \\}\n\\end{equation}\ncoincides with the invariant subspace $D_x$ of $\\mu_x$, that is\n\\begin{equation}\nD_x \\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,1})_\\# \\mu_x= \\mu_x \\right \\}.\n\\end{equation}\nMoreover $D_x$ is a linear subspace of $\\mathbb{R}^n$ and, provided $\\mu_x \\neq 0$, it holds $\\dim D_x \\leq \\alpha$.\n\\end{lemma}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main}]\nThe statement is trivial for $k=n$, thus we assume $k \\in \\{0,1,\\dots,n-1\\}$ be fixed throughout the proof.\nLet us moreover fix $\\gamma \\in \\left (0,\\frac{1}{16}\\right )$. It is possible to choose a finite number $m$ of $k$-dimensional linear subspaces $\\{V_\\ell\\}_{\\ell=1}^m$ such that every $k$-dimensional linear subspace $V$ is contained in $\\overline{C(0,V_\\ell,\\gamma)}$ for some $\\ell \\in \\{1,\\dots,m\\}$. For every $\\ell \\in \\{1,\\dots,m\\}$ and every $i \\in \\mathbb{N}$, we define the set\n\\begin{equation}\nE_{\\ell,i} \\coloneqq \\left \\{ x \\in \\mathcal{S}^k \\colon D_x \\subset \\overline{C(0,V_\\ell,\\gamma)},\\,\\,\\, \\alpha(x) \\leq i,\\,\\,\\, \\eta(x)\\geq \\frac{1}{i},\\,\\,\\, d(c\\mu_{x},\\omega)\\geq \\frac{1}{i} \\,\\,\\,\\forall |c| \\in \\left [\\frac{1}{i},1 \\right ],\\, \\forall\\omega \\in \\operatorname{Hom}_{k+1}^i \\right \\},\n\\end{equation}\nwhere $D_x$ and $\\alpha(x)$ are respectively the invariant subspace and the homogeneity exponent of $\\mu_x$ given by Lemma \\ref{lmm:0-hom}, $\\eta=\\eta(x)$ is given by Lemma \\ref{lmm:lower_bound_mass_blow-up}, while $d$ and $\\operatorname{Hom}_{k+1}^i$ are respectively the distance and the set of $(\\alpha-n)$-homogeneous measures described in Section \\ref{sec:notations}.\nSince $\\mathcal{S}^k= \\bigcup_{i \\in \\mathbb{N}}\\bigcup_{\\ell=1}^m E_{i,\\ell}$, it is enough to prove that each $E_{i,\\ell}$ can be covered by a countable union of Lipschitz graphs.\n\nLet us fix $i \\in \\mathbb{N}$ and $\\ell \\in \\{1,\\dots,m\\}$. \nThen fix the value $\\varepsilon$ given by Lemma \\ref{lmm:too_close} for $\\beta=i$, $\\delta=\\frac{1}{i}$, $M=i \\cdot 2^\\alpha$ and, for every $\\rho>0$, define the set\n\\begin{equation}\nE_{i,\\ell,\\rho} \\coloneqq\n\\left \\{x \\in E_{i,\\ell} \\colon \\inf_{|c| \\in [\\eta,1]}d(\\mu_{x,r},c\\mu_x)< \\varepsilon \\,\\,\\,\\forall r \\in (0,\\rho) \\right \\}.\n\\end{equation}\nIf $x \\in E_{i,\\ell}$, then it belongs to $E_{i,\\ell,\\rho}$ for some $\\rho>0$, because the existence of a sequence $r_j \\to 0$ such that $d(\\mu_{x,r_j},c\\mu_x) \\geq \\varepsilon$ for every $j \\in \\mathbb{N}$ and every $|c| \\in [\\eta,1]$ would contradict $x \\in \\mathcal{S}$. Therefore\n\\begin{equation}\nE_{i,\\ell} = \\bigcup_{\\rho \\in \\mathbb{Q} \\cap (0,+\\infty)} E_{i,\\ell,\\rho}\n\\end{equation}\nand it suffice to prove that, for a fixed $\\rho>0$, the set $E_{i,\\ell,\\rho}$ can be covered with a countable union of Lipschitz graphs.\nIn order to do so, we claim that, for every $x \\in E_{i,\\ell,\\rho}$, there exists $r>0$ such that\n\\begin{equation}\\label{eq:outside_cones}\nE_{i,\\ell,\\rho} \\cap B_{r}(x) \\setminus C(x,D_x,3\\gamma)= \\{x\\}.\n\\end{equation}\nLet us fix $x\\in E_{i,\\ell,\\rho}$, let $\\mu_x$ be $(\\alpha-n)$-homogeneous and let us assume, toward a contradiction, that for every $j \\in \\mathbb{N}$ there exists \n\\begin{equation}\\label{eq:y_j_final}\ny_j \\in E_{i,\\ell,\\rho} \\cap \\overline{B_{\\frac{r_j}{2}}(x)} \\setminus\\Big( B_{\\frac{r_j}{4}}(x) \\cup C(x,D_x,3\\gamma) \\Big),\n\\end{equation}\nfor some $r_j \\leq \\rho$ such that $r_j \\downarrow 0$. This in particular implies the existence of constants $c_j$ with $|c_j| \\in \\left [\\frac{1}{i},1 \\right ]$ such that\n\\begin{equation}\\label{eq:distance_muj}\nd( c_j\\mu_{y_j}, \\mu_{y_j,r_j})< \\varepsilon\n\\qquad\n\\forall j \\in \\mathbb{N}.\n\\end{equation}\nUp to selecting a non relabeled subsequence and without loss of generality, we can assume $c_j>0$ and that $\\mu_{x,r_j} \\overset{\\ast}{\\rightharpoonup} c\\mu_x$ for some $c \\in \\left [\\frac{1}{i},1 \\right ]$.\nWe now analyze the weak*-limit (up to a suitable subsequence) of the two sequences of measures in \\eqref{eq:distance_muj}.", "post_theorem_intro_text_len": 5662, "post_theorem_intro_text": "The idea of the proof is that, for every $x \\in \\mathcal{S}_\\mu^k$, there is a radius $r>0$ for which the points in $\\mathcal{S}_\\mu^k \\cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$. Otherwise, by contradiction, one can find a sequence of such points $y_j \\notin x+C$ with $r_j=|x-y_j| \\to 0$ and such that the measures $c_j \\mu_{y_j}$ and $\\mu_{y_j,r_j}$ are as close as we want. Now, on one hand $c_j\\mu_{y_j}$ converges to a measure $\\nu$ whose invariant subspace is close to $D_x$; on the other hand $\\mu_{y_j,r_j}$, being translations of $\\mu_{x,r_j}$ with respect to points outside $C$, converge to a translation of $\\mu_x$ whose invariant subspace is not contained in $C$, contradicting the closeness of $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$.\n\n\\medskip\n\nA first outcome of Theorem \\ref{thm:main} is a rectifiability criterion for signed Radon measures in terms of their blow-ups which, to the best of our knowledge, is new.\n\n\\begin{corollario}\\label{cor:rectif_criterion}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Let us assume that, for $|\\mu|$-a.e.\\ $x \\in \\mathbb{R}^n$, there exists a Radon measure $\\nu_x$ whose invariant linear subspace $V_x$ is $k$-dimensional and with the property that, for every sequence $r_j \\to 0^+$ there are a subsequence, not relabeled, and a constant $c\\neq 0$ such that\n\\begin{equation}\n\\frac{1}{r_j^k}  (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\nu_x.\n\\end{equation}\nThen $\\mu$ is $k$-rectifiable: there exists a signed function $\\theta \\in L^1_{\\mathrm{loc}}(\\mathcal{S}_\\mu^k,\\mathcal{H}^k)$ such that \\mbox{$\\mu= \\theta \\mathcal{H}^k \\llcorner \\mathcal{S}_\\mu^k$,} where $\\mathcal{S}_\\mu^k$ is defined in \\eqref{eq:def_s_mu_k}.\n\\end{corollario}\n\nThe above result extends to signed Radon measures the well-known criterion for positive ones, see for instance \\cite[Theorem 11.8]{simonlectures83}, where one usually sets $\\nu_x=\\theta_0 \\mathcal{H}^k \\llcorner V_x$ for some $k$-dimensional linear subspace $V_x$ and assumes the existence of the limit $r^{-k}(\\tau_{x,r_j})_\\# \\mu \\overset{\\ast}{\\rightharpoonup} \\theta_0 \\mathcal{H}^k \\llcorner V_x$ as $r \\to 0^+$.\n\nWe point out that Corollary \\ref{cor:rectif_criterion} cannot be trivially deduced from the corresponding result for positive measures, since the fact that $\\mu$ has a certain blow-up at a point does not provide any suitable information on the blow-ups of $|\\mu|$ at the same point, not even their uniqueness; nor does it appear that the classical proof of \\cite[Theorem 11.8]{simonlectures83} can be readily adapted to the case of signed measures.\n\n\\medskip\n\nA further consequence of Theorem \\ref{thm:main} is the following extension of \\cite[Theorem 3.2]{mattila_unique05} to signed Radon measures. It is based on the intuitive idea that, since each $\\mathcal{S}_{\\mu}^k$ is $k$-rectifiable, it is reasonable to expect that its subset of points where the blow-up of $\\mu$ is more diffused than a $k$-dimensional measure is $|\\mu|$-negligible.\n\n\\begin{corollario}\\label{cor:almost_everywhere_rect}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Then, for $|\\mu|$-a.e.\\ $x \\in \\mathcal{S}_\\mu^k$, the unique blow-up of $\\mu$ at $x$ is a multiple of $\\mathcal{H}^k\\llcorner D_x$.\n\\end{corollario}\n\nWe conclude this introduction with a comment on the definition of unique blow-up at a point.\nAs already said above, we say that $\\mu_x$ is the unique blow-up of $\\mu$ at $x \\in \\operatorname{supp}\\mu$ if, for every sequence $r_j \\to 0$, \\emph{there exists} a subsequence for which \\eqref{eq:def-unique_blowup} holds true. This differs from the definition used in \\cite{mattila_unique05} for positive Radon measures, where one does not require the convergence of a subsequence for every choice of $\\{r_j\\}_j$: according to \\cite{mattila_unique05}, one says that  $\\mu$ has unique blow-up at $x \\in \\mathbb{R}$ if there exists $\\nu$ such that any non-trivial weak*-limit (in case it exists) of $c_j (\\tau_{x,r_j})_\\# \\mu$, where $c_j>0$ and $r_j \\to 0^+$, is of the form $c\\nu$ for some $c>0$. \n\n\\cite[Lemma 2.5]{mattila_unique05} shows, provided $\\mu$ is positive, that this condition actually implies $\\mu_{x,r} \\overset{\\ast}{\\rightharpoonup} c\\mu_x$ as $r \\to 0$ and, subsequently, our definition (they are actually equivalent).\nHowever, this does not hold in the case of signed Radon measures: let us consider two sequences of positive real numbers $a_j, \\rho_j$ such that\n\\begin{equation}\n\\lim_{j \\to +\\infty} \\frac{1}{a_j} \\sum_{i=j+1}^{+\\infty} a_i = 0,\n\\qquad\n\\lim_{j \\to +\\infty} \\frac{\\rho_{j+1}}{\\rho_j}=0.\n\\end{equation}\nThen the Radon measure on $\\mathbb{R}$ defined as\n\\begin{equation}\n\\mu \\coloneqq \\sum_{j \\in \\mathbb{N}} a_j\\left ( \\delta_{2\\rho_j + \\rho_j^2} - \\delta_{2\\rho_j - \\rho_j^2} \\right )\n\\end{equation}\nhas the null measure as unique blow-up at $x=0$ in the sense of \\cite{mattila_unique05}, but $\\mu_{0,\\rho_j} =\\frac{1}{|\\mu|(B_{\\rho_j}(0))} (\\tau_{0,\\rho_j})_\\# \\mu $ has no converging subsequences because $|\\mu_{x,\\rho_j}|\\big(B_2(0)\\big) \\to +\\infty$ as $j\\to +\\infty$.\n\n\\subsection*{Acknowledgments}\nThe author would like to thank Andrea Marchese, Paolo Bonicatto and Gian Paolo Leonardi for inspiring conversations. He has been supportend by PRIN project 2022PJ9EFL ``Geometric Measure Theory: Structure of Singular Measures, Regularity Theory and Applications in the Calculus of Variations\" CUP:E53D23005860006 and  partially supported by INDAM-GNAMPA. Part of the work has been carried out during the thematic program on ``Free boundary problems\" at ESI (Wien), which the author wishes to thank.", "sketch": "The post-theorem introduction gives the following proof idea for Theorem~\\ref{thm:main}.\n\n1. **Cone containment near each point.** For every $x\\in\\mathcal{S}_\\mu^k$, there exists $r>0$ such that the points in $\\mathcal{S}_\\mu^k\\cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$.\n\n2. **Contradiction setup if cone containment fails.** Otherwise, one can find a sequence $y_j\\notin x+C$ with $r_j=|x-y_j|\\to 0$ such that the measures $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$ are “as close as we want.”\n\n3. **Two limiting descriptions of the same asymptotically close measures.**\n   - On the one hand, $c_j\\mu_{y_j}$ converges to a measure $\\nu$ whose invariant subspace is close to $D_x$.\n   - On the other hand, $\\mu_{y_j,r_j}$ are translations of $\\mu_{x,r_j}$ “with respect to points outside $C$,” hence they converge to a translation of $\\mu_x$ whose invariant subspace is not contained in $C$.\n\n4. **Contradiction.** This “contradict[s] the closeness of $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$,” forcing the desired cone property; this yields the Lipschitz-graph covering and the identification of the approximate tangent plane with $D_x$ for $\\mathcal{H}^k$-a.e. $x\\in\\mathcal{S}_\\mu^k$.", "expanded_sketch": "The post-theorem introduction gives the following proof idea for the main theorem.\n\n1. **Cone containment near each point.** For every $x\\in\\mathcal{S}_\\mu^k$, there exists $r>0$ such that the points in $\\mathcal{S}_\\mu^k\\cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$.\n\n2. **Contradiction setup if cone containment fails.** Otherwise, one can find a sequence $y_j\\notin x+C$ with $r_j=|x-y_j|\\to 0$ such that the measures $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$ are “as close as we want.”\n\n3. **Two limiting descriptions of the same asymptotically close measures.**\n   - On the one hand, $c_j\\mu_{y_j}$ converges to a measure $\\nu$ whose invariant subspace is close to $D_x$.\n   - On the other hand, $\\mu_{y_j,r_j}$ are translations of $\\mu_{x,r_j}$ “with respect to points outside $C$,” hence they converge to a translation of $\\mu_x$ whose invariant subspace is not contained in $C$.\n\n4. **Contradiction.** This “contradict[s] the closeness of $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$,” forcing the desired cone property; this yields the Lipschitz-graph covering and the identification of the approximate tangent plane with $D_x$ for $\\mathcal{H}^k$-a.e. $x\\in\\mathcal{S}_\\mu^k$.", "expanded_theorem": "\\label{thm:main}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Then $\\mathcal{S}_\\mu^k$ is contained in a countable union of $k$-dimensional Lipschitz graphs. Moreover, the approximate tangent plane to $\\mathcal{S}_\\mu^k$ at $x$ coincides with $D_x$ for $\\mathcal{H}^{k}$-a.e.\\ $x \\in \\mathcal{S}_\\mu^k$.,", "theorem_type": ["Universal", "Implication"], "mcq": {"question": "Let \\(\\mu\\) be a signed Radon measure on \\(\\mathbb{R}^n\\). For \\(x\\in \\operatorname{supp}\\mu\\), say that \\(\\mu\\) has a unique blow-up at \\(x\\) if there exists a Radon measure \\(\\mu_x\\), with either \\(|\\mu_x|(B_1(0))=1\\) or \\(\\mu_x=0\\), such that for every sequence \\(r_j\\to 0^+\\) there are a subsequence and a constant \\(c\\neq 0\\) for which\n\\[\n\\mu_{x,r_j}:=\\frac{1}{|\\mu|(B_{r_j}(x))}(\\tau_{x,r_j})_\\#\\mu \\overset{*}{\\rightharpoonup} c\\mu_x,\n\\qquad \\tau_{x,r}(y)=\\frac{y-x}{r}.\n\\]\nLet \\(\\mathcal S_\\mu\\) be the set of points where \\(\\mu\\) has a unique blow-up. For \\(x\\in \\mathcal S_\\mu\\), let\n\\[\nD_x:=\\{y\\in\\mathbb R^n:(\\tau_{y,1})_\\#\\mu_x=\\mu_x\\}\n\\]\nbe the invariant linear subspace of the blow-up, and for \\(k\\in\\{0,1,\\dots,n\\}\\) define\n\\[\n\\mathcal S_\\mu^k:=\\{x\\in\\mathcal S_\\mu:\\dim D_x=k\\}.\n\\]\nWhich statement holds for every signed Radon measure \\(\\mu\\) on \\(\\mathbb R^n\\) and every \\(k\\in\\{0,1,\\dots,n\\}\\)?", "correct_choice": {"label": "A", "text": "The set \\(\\mathcal S_\\mu^k\\) is contained in a countable union of \\(k\\)-dimensional Lipschitz graphs, and for \\(\\mathcal H^k\\)-almost every \\(x\\in \\mathcal S_\\mu^k\\), the approximate tangent plane to \\(\\mathcal S_\\mu^k\\) at \\(x\\) is exactly \\(D_x\\)."}, "choices": [{"label": "B", "text": "The set \\(\\mathcal S_\\mu^k\\) is contained in a countable union of \\((k+1)\\)-dimensional Lipschitz graphs, and for \\(\\mathcal H^{k+1}\\)-almost every \\(x\\in \\mathcal S_\\mu^k\\), the approximate tangent plane to \\(\\mathcal S_\\mu^k\\) at \\(x\\) contains \\(D_x\\)."}, {"label": "C", "text": "The set \\(\\mathcal S_\\mu^k\\) is contained in a countable union of \\(k\\)-dimensional Lipschitz graphs."}, {"label": "D", "text": "The set \\(\\mathcal S_\\mu^k\\) is contained in a countable union of \\(k\\)-dimensional Lipschitz graphs, and for every \\(x\\in \\mathcal S_\\mu^k\\), the approximate tangent plane to \\(\\mathcal S_\\mu^k\\) at \\(x\\) exists and is exactly \\(D_x\\)."}, {"label": "E", "text": "The set \\(\\mathcal S_\\mu^k\\) is contained in a finite union of \\(k\\)-dimensional Lipschitz graphs, and for \\(\\mathcal H^k\\)-almost every \\(x\\in \\mathcal S_\\mu^k\\), the approximate tangent plane to \\(\\mathcal S_\\mu^k\\) at \\(x\\) is a \\(k\\)-plane parallel to \\(D_x\\)."}], "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": "dimension_from_cone_containment", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "geometric_construction", "tampered_component": "dropped_a.e._tangent_plane_identification", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "a.e._pointwise_tangent_identification_upgraded_to_every_point", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "countable_cover_replaced_by_finite_cover_and_exact_plane_identification_relaxed_to_parallelism", "template_used": "wildcard"}]}}
{"id": "2512.20498v1", "paper_link": "http://arxiv.org/abs/2512.20498v1", "theorems_cnt": 3, "theorem": {"env_name": "teorema", "content": "\\label{thm:main}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Then $\\mathcal{S}_\\mu^k$ is contained in a countable union of $k$-dimensional Lipschitz graphs. Moreover, the approximate tangent plane to $\\mathcal{S}_\\mu^k$ at $x$ coincides with $D_x$ for $\\mathcal{H}^{k}$-a.e.\\ $x \\in \\mathcal{S}_\\mu^k$.", "start_pos": 13260, "end_pos": 13630, "label": "thm:main"}, "ref_dict": {"thm:main": "\\begin{teorema}\\label{thm:main}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Then $\\mathcal{S}_\\mu^k$ is contained in a countable union of $k$-dimensional Lipschitz graphs. Moreover, the approximate tangent plane to $\\mathcal{S}_\\mu^k$ at $x$ coincides with $D_x$ for $\\mathcal{H}^{k}$-a.e.\\ $x \\in \\mathcal{S}_\\mu^k$.\n\\end{teorema}", "lmm:0-hom": "\\begin{lemma}\\label{lmm:0-hom}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and let $\\mu_x \\neq 0$ be the unique blow-up of $\\mu$ at $x \\in \\mathcal{S}_\\mu$. Then there exists $\\alpha=\\alpha(x)\\geq 0$ such that $\\mu_x$ and $|\\mu_x|$ are $(\\alpha-n)$-homogeneous,\\footnote{According to this definition, the measure induced by a $p$-homogenous locally integrable function $f$, with respect to the Lebesgue measure, is $p$-homogeneous as well.} namely\n\\begin{equation}\\label{eq:hom_blow-up}\n(\\tau_{0,\\lambda})_\\# \\mu_x=\\lambda^\\alpha \\mu_x,\n\\quad\n(\\tau_{0,\\lambda})_\\# |\\mu_x|=\\lambda^\\alpha |\\mu_x|\n\\qquad\n\\forall \\lambda>0.\n\\end{equation}\nFor $\\mu_x$, and more generally for every $(\\alpha-n)$-homogeneous Radon measure, the set\n\\begin{equation}\nC_x\\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,\\lambda})_\\# \\mu_x=\\lambda^\\alpha (\\tau_{y,1})_\\# \\mu_x \\,\\,\\,\\,\\forall \\lambda>0 \\right \\}\n\\end{equation}\ncoincides with the invariant subspace $D_x$ of $\\mu_x$, that is\n\\begin{equation}\nD_x \\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,1})_\\# \\mu_x= \\mu_x \\right \\}.\n\\end{equation}\nMoreover $D_x$ is a linear subspace of $\\mathbb{R}^n$ and, provided $\\mu_x \\neq 0$, it holds $\\dim D_x \\leq \\alpha$.\n\\end{lemma}", "eq:def_s_mu_k": "\\begin{equation}\\label{eq:def_s_mu_k}\n\\mathcal{S}_\\mu = \\bigcup_{k=1}^n \\mathcal{S}_\\mu^k,\n\\qquad\\text{where} \\qquad\n\\mathcal{S}_\\mu^k \\coloneqq \\left \\{x \\in \\mathcal{S}_\\mu \\colon \\dim D_x=k \\right \\}.\n\\end{equation}", "eq:def-unique_blowup": "\\begin{equation}\\label{eq:def-unique_blowup}\n\\mu_{x,r_j} \\coloneqq\n\\frac{1}{|\\mu|(B_{r_j}(x))} (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\mu_x,\n\\end{equation}", "cor:rectif_criterion": "\\begin{corollario}\\label{cor:rectif_criterion}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Let us assume that, for $|\\mu|$-a.e.\\ $x \\in \\mathbb{R}^n$, there exists a Radon measure $\\nu_x$ whose invariant linear subspace $V_x$ is $k$-dimensional and with the property that, for every sequence $r_j \\to 0^+$ there are a subsequence, not relabeled, and a constant $c\\neq 0$ such that\n\\begin{equation}\n\\frac{1}{r_j^k}  (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\nu_x.\n\\end{equation}\nThen $\\mu$ is $k$-rectifiable: there exists a signed function $\\theta \\in L^1_{\\mathrm{loc}}(\\mathcal{S}_\\mu^k,\\mathcal{H}^k)$ such that \\mbox{$\\mu= \\theta \\mathcal{H}^k \\llcorner \\mathcal{S}_\\mu^k$,} where $\\mathcal{S}_\\mu^k$ is defined in \\eqref{eq:def_s_mu_k}.\n\\end{corollario}"}, "pre_theorem_intro_text_len": 3739, "pre_theorem_intro_text": "The notion of blow-up of a geometric or analytic object, i.e.\\ the limit of its enlargements around a point as the zooming factor goes to infinity, is of fundamental importance in many areas of Mathematics. The blow-up of a function or a manifold provides a simplified, but somewhat essential, description of the local behavior of the object ``at infinitesimal scale\".\nFor this and other reasons, the classification of the blow-ups of various mathematical items has become a central theme in geometric and functional analysis.\n\nFor a Radon measure $\\mu$ in $\\mathbb{R}^n$, a natural definition of blow-up at a point $x \\in \\mathbb{R}^n$ is the one in the sense of Preiss, \\cite{preiss87}: the limit in the weak*-topology of Radon measures (in duality with $C_c(\\mathbb{R}^n)$) of the rescalings\n\\begin{equation}\\label{eq:rescalings}\nc_j (\\tau_{x,r_j})_\\# \\mu,\n\\end{equation}\nfor some sequences $c_j>0$, $r_j \\to 0$, where $\\tau_{x,r}(y)\\coloneqq \\frac{y-x}{r}$ and ${}_\\#$ is the push-forward of Radon measures.\n\nWe say that $\\mu$ has unique blow-up at $x \\in \\operatorname{supp} \\mu$ if there exists a Radon measure $\\mu_x$ with either $|\\mu_x|(B_1(0)) = 1$ or $\\mu_x = 0$ and such that, for every sequence $r_j \\to 0^+$, there are a subsequence, not relabeled, and a constant $c\\neq 0$ for which\n\\begin{equation}\\label{eq:def-unique_blowup}\n\\mu_{x,r_j} \\coloneqq\n\\frac{1}{|\\mu|(B_{r_j}(x))} (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\mu_x,\n\\end{equation}\nwhere $|\\mu|$ is the total variation measure of $\\mu$ and $\\overset{\\ast}{\\rightharpoonup}$ denotes the weak*-convergence of Radon measures.\nWe will denote by $\\mathcal{S}_\\mu$ the set of points where $\\mu$ has unique blow-up.\n\nMattila studied positive measures having a unique blow-up at almost every point: a way to rephrase his main result \\cite[Theorem 3.2]{mattila_unique05} is that,\nfor $\\mu$-a.e.\\ $x \\in \\mathcal{S}_\\mu$, the blow-up of $\\mu$ at $x$ is a multiple of the $k$-dimensional Hausdorff measure $\\mathcal{H}^k$ on a $k$-plane.\nAlthough this describes the blow-ups of $\\mu$ at $\\mu$-a.e.\\ point in $\\mathcal{S}_\\mu$, it can be interesting to study the behavior of $\\mu$ in the remaining part of $\\mathcal{S}_\\mu$.\n\nThis problem has already received some attention: in \\cite{delnin21} it is proved that, for a given $u \\in L^1_{\\text{loc}}(\\mathbb{R}^n)$, the set $\\Sigma_u$ of points $x$ where $\\frac{1}{r^n}u \\left (\\frac{\\cdot-x}{r} \\right )$ converges, in the $L^1_{\\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs. \n\\begin{itemize}\n\\item The results in \\cite{delnin21} rely on the fact that the measure $\\mu$ is induced by a $L^1_{\\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\\mathrm{loc}}$-topology, while in $\\mathcal{S}_\\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\\mathcal{S}_\\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\\Sigma_u$.\nIs this set rectifiable as well? What can be said if $\\mu$ is a general Radon measure?\n\n\\item It is easily seen (Lemma \\ref{lmm:0-hom}) that the unique blow-up $\\mu_x$ of $\\mu$ at a point in $x \\in \\mathcal{S}_\\mu$ has an \\emph{invariant linear subspace} $D_x$ of $\\mathbb{R}^n$, namely $(\\tau_{y,1})_\\# \\mu_x = \\mu_x$ for $y \\in D_x$. Therefore\n\\begin{equation}\\label{eq:def_s_mu_k}\n\\mathcal{S}_\\mu = \\bigcup_{k=1}^n \\mathcal{S}_\\mu^k,\n\\qquad\\text{where} \\qquad\n\\mathcal{S}_\\mu^k \\coloneqq \\left \\{x \\in \\mathcal{S}_\\mu \\colon \\dim D_x=k \\right \\}.\n\\end{equation}\nDo these $\\mathcal{S}_\\mu^k$ enjoy finer rectifiability properties?\n\\end{itemize}\n\nThe main result of this note provides an answer to these questions.", "context": "For a Radon measure $\\mu$ in $\\mathbb{R}^n$, a natural definition of blow-up at a point $x \\in \\mathbb{R}^n$ is the one in the sense of Preiss, \\cite{preiss87}: the limit in the weak*-topology of Radon measures (in duality with $C_c(\\mathbb{R}^n)$) of the rescalings\n\\begin{equation}\\label{eq:rescalings}\nc_j (\\tau_{x,r_j})_\\# \\mu,\n\\end{equation}\nfor some sequences $c_j>0$, $r_j \\to 0$, where $\\tau_{x,r}(y)\\coloneqq \\frac{y-x}{r}$ and ${}_\\#$ is the push-forward of Radon measures.\n\nWe say that $\\mu$ has unique blow-up at $x \\in \\operatorname{supp} \\mu$ if there exists a Radon measure $\\mu_x$ with either $|\\mu_x|(B_1(0)) = 1$ or $\\mu_x = 0$ and such that, for every sequence $r_j \\to 0^+$, there are a subsequence, not relabeled, and a constant $c\\neq 0$ for which\n\\begin{equation}\\label{eq:def-unique_blowup}\n\\mu_{x,r_j} \\coloneqq\n\\frac{1}{|\\mu|(B_{r_j}(x))} (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\mu_x,\n\\end{equation}\nwhere $|\\mu|$ is the total variation measure of $\\mu$ and $\\overset{\\ast}{\\rightharpoonup}$ denotes the weak*-convergence of Radon measures.\nWe will denote by $\\mathcal{S}_\\mu$ the set of points where $\\mu$ has unique blow-up.\n\nMattila studied positive measures having a unique blow-up at almost every point: a way to rephrase his main result \\cite[Theorem 3.2]{mattila_unique05} is that,\nfor $\\mu$-a.e.\\ $x \\in \\mathcal{S}_\\mu$, the blow-up of $\\mu$ at $x$ is a multiple of the $k$-dimensional Hausdorff measure $\\mathcal{H}^k$ on a $k$-plane.\nAlthough this describes the blow-ups of $\\mu$ at $\\mu$-a.e.\\ point in $\\mathcal{S}_\\mu$, it can be interesting to study the behavior of $\\mu$ in the remaining part of $\\mathcal{S}_\\mu$.\n\nThis problem has already received some attention: in \\cite{delnin21} it is proved that, for a given $u \\in L^1_{\\text{loc}}(\\mathbb{R}^n)$, the set $\\Sigma_u$ of points $x$ where $\\frac{1}{r^n}u \\left (\\frac{\\cdot-x}{r} \\right )$ converges, in the $L^1_{\\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs. \n\\begin{itemize}\n\\item The results in \\cite{delnin21} rely on the fact that the measure $\\mu$ is induced by a $L^1_{\\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\\mathrm{loc}}$-topology, while in $\\mathcal{S}_\\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\\mathcal{S}_\\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\\Sigma_u$.\nIs this set rectifiable as well? What can be said if $\\mu$ is a general Radon measure?\n\n\\item It is easily seen (Lemma \\ref{lmm:0-hom}) that the unique blow-up $\\mu_x$ of $\\mu$ at a point in $x \\in \\mathcal{S}_\\mu$ has an \\emph{invariant linear subspace} $D_x$ of $\\mathbb{R}^n$, namely $(\\tau_{y,1})_\\# \\mu_x = \\mu_x$ for $y \\in D_x$. Therefore\n\\begin{equation}\\label{eq:def_s_mu_k}\n\\mathcal{S}_\\mu = \\bigcup_{k=1}^n \\mathcal{S}_\\mu^k,\n\\qquad\\text{where} \\qquad\n\\mathcal{S}_\\mu^k \\coloneqq \\left \\{x \\in \\mathcal{S}_\\mu \\colon \\dim D_x=k \\right \\}.\n\\end{equation}\nDo these $\\mathcal{S}_\\mu^k$ enjoy finer rectifiability properties?\n\\end{itemize}\n\nThe main result of this note provides an answer to these questions.\n\n\\begin{lemma}\\label{lmm:0-hom}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and let $\\mu_x \\neq 0$ be the unique blow-up of $\\mu$ at $x \\in \\mathcal{S}_\\mu$. Then there exists $\\alpha=\\alpha(x)\\geq 0$ such that $\\mu_x$ and $|\\mu_x|$ are $(\\alpha-n)$-homogeneous,\\footnote{According to this definition, the measure induced by a $p$-homogenous locally integrable function $f$, with respect to the Lebesgue measure, is $p$-homogeneous as well.} namely\n\\begin{equation}\\label{eq:hom_blow-up}\n(\\tau_{0,\\lambda})_\\# \\mu_x=\\lambda^\\alpha \\mu_x,\n\\quad\n(\\tau_{0,\\lambda})_\\# |\\mu_x|=\\lambda^\\alpha |\\mu_x|\n\\qquad\n\\forall \\lambda>0.\n\\end{equation}\nFor $\\mu_x$, and more generally for every $(\\alpha-n)$-homogeneous Radon measure, the set\n\\begin{equation}\nC_x\\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,\\lambda})_\\# \\mu_x=\\lambda^\\alpha (\\tau_{y,1})_\\# \\mu_x \\,\\,\\,\\,\\forall \\lambda>0 \\right \\}\n\\end{equation}\ncoincides with the invariant subspace $D_x$ of $\\mu_x$, that is\n\\begin{equation}\nD_x \\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,1})_\\# \\mu_x= \\mu_x \\right \\}.\n\\end{equation}\nMoreover $D_x$ is a linear subspace of $\\mathbb{R}^n$ and, provided $\\mu_x \\neq 0$, it holds $\\dim D_x \\leq \\alpha$.\n\\end{lemma}", "full_context": "For a Radon measure $\\mu$ in $\\mathbb{R}^n$, a natural definition of blow-up at a point $x \\in \\mathbb{R}^n$ is the one in the sense of Preiss, \\cite{preiss87}: the limit in the weak*-topology of Radon measures (in duality with $C_c(\\mathbb{R}^n)$) of the rescalings\n\\begin{equation}\\label{eq:rescalings}\nc_j (\\tau_{x,r_j})_\\# \\mu,\n\\end{equation}\nfor some sequences $c_j>0$, $r_j \\to 0$, where $\\tau_{x,r}(y)\\coloneqq \\frac{y-x}{r}$ and ${}_\\#$ is the push-forward of Radon measures.\n\nWe say that $\\mu$ has unique blow-up at $x \\in \\operatorname{supp} \\mu$ if there exists a Radon measure $\\mu_x$ with either $|\\mu_x|(B_1(0)) = 1$ or $\\mu_x = 0$ and such that, for every sequence $r_j \\to 0^+$, there are a subsequence, not relabeled, and a constant $c\\neq 0$ for which\n\\begin{equation}\\label{eq:def-unique_blowup}\n\\mu_{x,r_j} \\coloneqq\n\\frac{1}{|\\mu|(B_{r_j}(x))} (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\mu_x,\n\\end{equation}\nwhere $|\\mu|$ is the total variation measure of $\\mu$ and $\\overset{\\ast}{\\rightharpoonup}$ denotes the weak*-convergence of Radon measures.\nWe will denote by $\\mathcal{S}_\\mu$ the set of points where $\\mu$ has unique blow-up.\n\nMattila studied positive measures having a unique blow-up at almost every point: a way to rephrase his main result \\cite[Theorem 3.2]{mattila_unique05} is that,\nfor $\\mu$-a.e.\\ $x \\in \\mathcal{S}_\\mu$, the blow-up of $\\mu$ at $x$ is a multiple of the $k$-dimensional Hausdorff measure $\\mathcal{H}^k$ on a $k$-plane.\nAlthough this describes the blow-ups of $\\mu$ at $\\mu$-a.e.\\ point in $\\mathcal{S}_\\mu$, it can be interesting to study the behavior of $\\mu$ in the remaining part of $\\mathcal{S}_\\mu$.\n\nThis problem has already received some attention: in \\cite{delnin21} it is proved that, for a given $u \\in L^1_{\\text{loc}}(\\mathbb{R}^n)$, the set $\\Sigma_u$ of points $x$ where $\\frac{1}{r^n}u \\left (\\frac{\\cdot-x}{r} \\right )$ converges, in the $L^1_{\\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs. \n\\begin{itemize}\n\\item The results in \\cite{delnin21} rely on the fact that the measure $\\mu$ is induced by a $L^1_{\\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\\mathrm{loc}}$-topology, while in $\\mathcal{S}_\\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\\mathcal{S}_\\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\\Sigma_u$.\nIs this set rectifiable as well? What can be said if $\\mu$ is a general Radon measure?\n\n\\item It is easily seen (Lemma \\ref{lmm:0-hom}) that the unique blow-up $\\mu_x$ of $\\mu$ at a point in $x \\in \\mathcal{S}_\\mu$ has an \\emph{invariant linear subspace} $D_x$ of $\\mathbb{R}^n$, namely $(\\tau_{y,1})_\\# \\mu_x = \\mu_x$ for $y \\in D_x$. Therefore\n\\begin{equation}\\label{eq:def_s_mu_k}\n\\mathcal{S}_\\mu = \\bigcup_{k=1}^n \\mathcal{S}_\\mu^k,\n\\qquad\\text{where} \\qquad\n\\mathcal{S}_\\mu^k \\coloneqq \\left \\{x \\in \\mathcal{S}_\\mu \\colon \\dim D_x=k \\right \\}.\n\\end{equation}\nDo these $\\mathcal{S}_\\mu^k$ enjoy finer rectifiability properties?\n\\end{itemize}\n\nThe main result of this note provides an answer to these questions.\n\n\\begin{lemma}\\label{lmm:0-hom}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and let $\\mu_x \\neq 0$ be the unique blow-up of $\\mu$ at $x \\in \\mathcal{S}_\\mu$. Then there exists $\\alpha=\\alpha(x)\\geq 0$ such that $\\mu_x$ and $|\\mu_x|$ are $(\\alpha-n)$-homogeneous,\\footnote{According to this definition, the measure induced by a $p$-homogenous locally integrable function $f$, with respect to the Lebesgue measure, is $p$-homogeneous as well.} namely\n\\begin{equation}\\label{eq:hom_blow-up}\n(\\tau_{0,\\lambda})_\\# \\mu_x=\\lambda^\\alpha \\mu_x,\n\\quad\n(\\tau_{0,\\lambda})_\\# |\\mu_x|=\\lambda^\\alpha |\\mu_x|\n\\qquad\n\\forall \\lambda>0.\n\\end{equation}\nFor $\\mu_x$, and more generally for every $(\\alpha-n)$-homogeneous Radon measure, the set\n\\begin{equation}\nC_x\\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,\\lambda})_\\# \\mu_x=\\lambda^\\alpha (\\tau_{y,1})_\\# \\mu_x \\,\\,\\,\\,\\forall \\lambda>0 \\right \\}\n\\end{equation}\ncoincides with the invariant subspace $D_x$ of $\\mu_x$, that is\n\\begin{equation}\nD_x \\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,1})_\\# \\mu_x= \\mu_x \\right \\}.\n\\end{equation}\nMoreover $D_x$ is a linear subspace of $\\mathbb{R}^n$ and, provided $\\mu_x \\neq 0$, it holds $\\dim D_x \\leq \\alpha$.\n\\end{lemma}\n\nThis problem has already received some attention: in \\cite{delnin21} it is proved that, for a given $u \\in L^1_{\\text{loc}}(\\mathbb{R}^n)$, the set $\\Sigma_u$ of points $x$ where $\\frac{1}{r^n}u \\left (\\frac{\\cdot-x}{r} \\right )$ converges, in the $L^1_{\\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs. \n\\begin{itemize}\n\\item The results in \\cite{delnin21} rely on the fact that the measure $\\mu$ is induced by a $L^1_{\\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\\mathrm{loc}}$-topology, while in $\\mathcal{S}_\\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\\mathcal{S}_\\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\\Sigma_u$.\nIs this set rectifiable as well? What can be said if $\\mu$ is a general Radon measure?\n\nThe main result of this note provides an answer to these questions.\n\nThe idea of the proof is that, for every $x \\in \\mathcal{S}_\\mu^k$, there is a radius $r>0$ for which the points in $\\mathcal{S}_\\mu^k \\cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$. Otherwise, by contradiction, one can find a sequence of such points $y_j \\notin x+C$ with $r_j=|x-y_j| \\to 0$ and such that the measures $c_j \\mu_{y_j}$ and $\\mu_{y_j,r_j}$ are as close as we want. Now, on one hand $c_j\\mu_{y_j}$ converges to a measure $\\nu$ whose invariant subspace is close to $D_x$; on the other hand $\\mu_{y_j,r_j}$, being translations of $\\mu_{x,r_j}$ with respect to points outside $C$, converge to a translation of $\\mu_x$ whose invariant subspace is not contained in $C$, contradicting the closeness of $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$.\n\n\\begin{corollario}\\label{cor:rectif_criterion}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Let us assume that, for $|\\mu|$-a.e.\\ $x \\in \\mathbb{R}^n$, there exists a Radon measure $\\nu_x$ whose invariant linear subspace $V_x$ is $k$-dimensional and with the property that, for every sequence $r_j \\to 0^+$ there are a subsequence, not relabeled, and a constant $c\\neq 0$ such that\n\\begin{equation}\n\\frac{1}{r_j^k} (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\nu_x.\n\\end{equation}\nThen $\\mu$ is $k$-rectifiable: there exists a signed function $\\theta \\in L^1_{\\mathrm{loc}}(\\mathcal{S}_\\mu^k,\\mathcal{H}^k)$ such that \\mbox{$\\mu= \\theta \\mathcal{H}^k \\llcorner \\mathcal{S}_\\mu^k$,} where $\\mathcal{S}_\\mu^k$ is defined in \\eqref{eq:def_s_mu_k}.\n\\end{corollario}\n\n\\begin{corollario}\\label{cor:almost_everywhere_rect}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Then, for $|\\mu|$-a.e.\\ $x \\in \\mathcal{S}_\\mu^k$, the unique blow-up of $\\mu$ at $x$ is a multiple of $\\mathcal{H}^k\\llcorner D_x$.\n\\end{corollario}\n\n\\begin{lemma}\\label{lmm:0-hom}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and let $\\mu_x \\neq 0$ be the unique blow-up of $\\mu$ at $x \\in \\mathcal{S}_\\mu$. Then there exists $\\alpha=\\alpha(x)\\geq 0$ such that $\\mu_x$ and $|\\mu_x|$ are $(\\alpha-n)$-homogeneous,\\footnote{According to this definition, the measure induced by a $p$-homogenous locally integrable function $f$, with respect to the Lebesgue measure, is $p$-homogeneous as well.} namely\n\\begin{equation}\\label{eq:hom_blow-up}\n(\\tau_{0,\\lambda})_\\# \\mu_x=\\lambda^\\alpha \\mu_x,\n\\quad\n(\\tau_{0,\\lambda})_\\# |\\mu_x|=\\lambda^\\alpha |\\mu_x|\n\\qquad\n\\forall \\lambda>0.\n\\end{equation}\nFor $\\mu_x$, and more generally for every $(\\alpha-n)$-homogeneous Radon measure, the set\n\\begin{equation}\nC_x\\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,\\lambda})_\\# \\mu_x=\\lambda^\\alpha (\\tau_{y,1})_\\# \\mu_x \\,\\,\\,\\,\\forall \\lambda>0 \\right \\}\n\\end{equation}\ncoincides with the invariant subspace $D_x$ of $\\mu_x$, that is\n\\begin{equation}\nD_x \\coloneqq \\left \\{ y \\in \\mathbb{R}^n \\colon (\\tau_{y,1})_\\# \\mu_x= \\mu_x \\right \\}.\n\\end{equation}\nMoreover $D_x$ is a linear subspace of $\\mathbb{R}^n$ and, provided $\\mu_x \\neq 0$, it holds $\\dim D_x \\leq \\alpha$.\n\\end{lemma}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main}]\nThe statement is trivial for $k=n$, thus we assume $k \\in \\{0,1,\\dots,n-1\\}$ be fixed throughout the proof.\nLet us moreover fix $\\gamma \\in \\left (0,\\frac{1}{16}\\right )$. It is possible to choose a finite number $m$ of $k$-dimensional linear subspaces $\\{V_\\ell\\}_{\\ell=1}^m$ such that every $k$-dimensional linear subspace $V$ is contained in $\\overline{C(0,V_\\ell,\\gamma)}$ for some $\\ell \\in \\{1,\\dots,m\\}$. For every $\\ell \\in \\{1,\\dots,m\\}$ and every $i \\in \\mathbb{N}$, we define the set\n\\begin{equation}\nE_{\\ell,i} \\coloneqq \\left \\{ x \\in \\mathcal{S}^k \\colon D_x \\subset \\overline{C(0,V_\\ell,\\gamma)},\\,\\,\\, \\alpha(x) \\leq i,\\,\\,\\, \\eta(x)\\geq \\frac{1}{i},\\,\\,\\, d(c\\mu_{x},\\omega)\\geq \\frac{1}{i} \\,\\,\\,\\forall |c| \\in \\left [\\frac{1}{i},1 \\right ],\\, \\forall\\omega \\in \\operatorname{Hom}_{k+1}^i \\right \\},\n\\end{equation}\nwhere $D_x$ and $\\alpha(x)$ are respectively the invariant subspace and the homogeneity exponent of $\\mu_x$ given by Lemma \\ref{lmm:0-hom}, $\\eta=\\eta(x)$ is given by Lemma \\ref{lmm:lower_bound_mass_blow-up}, while $d$ and $\\operatorname{Hom}_{k+1}^i$ are respectively the distance and the set of $(\\alpha-n)$-homogeneous measures described in Section \\ref{sec:notations}.\nSince $\\mathcal{S}^k= \\bigcup_{i \\in \\mathbb{N}}\\bigcup_{\\ell=1}^m E_{i,\\ell}$, it is enough to prove that each $E_{i,\\ell}$ can be covered by a countable union of Lipschitz graphs.\n\nLet us fix $i \\in \\mathbb{N}$ and $\\ell \\in \\{1,\\dots,m\\}$. \nThen fix the value $\\varepsilon$ given by Lemma \\ref{lmm:too_close} for $\\beta=i$, $\\delta=\\frac{1}{i}$, $M=i \\cdot 2^\\alpha$ and, for every $\\rho>0$, define the set\n\\begin{equation}\nE_{i,\\ell,\\rho} \\coloneqq\n\\left \\{x \\in E_{i,\\ell} \\colon \\inf_{|c| \\in [\\eta,1]}d(\\mu_{x,r},c\\mu_x)< \\varepsilon \\,\\,\\,\\forall r \\in (0,\\rho) \\right \\}.\n\\end{equation}\nIf $x \\in E_{i,\\ell}$, then it belongs to $E_{i,\\ell,\\rho}$ for some $\\rho>0$, because the existence of a sequence $r_j \\to 0$ such that $d(\\mu_{x,r_j},c\\mu_x) \\geq \\varepsilon$ for every $j \\in \\mathbb{N}$ and every $|c| \\in [\\eta,1]$ would contradict $x \\in \\mathcal{S}$. Therefore\n\\begin{equation}\nE_{i,\\ell} = \\bigcup_{\\rho \\in \\mathbb{Q} \\cap (0,+\\infty)} E_{i,\\ell,\\rho}\n\\end{equation}\nand it suffice to prove that, for a fixed $\\rho>0$, the set $E_{i,\\ell,\\rho}$ can be covered with a countable union of Lipschitz graphs.\nIn order to do so, we claim that, for every $x \\in E_{i,\\ell,\\rho}$, there exists $r>0$ such that\n\\begin{equation}\\label{eq:outside_cones}\nE_{i,\\ell,\\rho} \\cap B_{r}(x) \\setminus C(x,D_x,3\\gamma)= \\{x\\}.\n\\end{equation}\nLet us fix $x\\in E_{i,\\ell,\\rho}$, let $\\mu_x$ be $(\\alpha-n)$-homogeneous and let us assume, toward a contradiction, that for every $j \\in \\mathbb{N}$ there exists \n\\begin{equation}\\label{eq:y_j_final}\ny_j \\in E_{i,\\ell,\\rho} \\cap \\overline{B_{\\frac{r_j}{2}}(x)} \\setminus\\Big( B_{\\frac{r_j}{4}}(x) \\cup C(x,D_x,3\\gamma) \\Big),\n\\end{equation}\nfor some $r_j \\leq \\rho$ such that $r_j \\downarrow 0$. This in particular implies the existence of constants $c_j$ with $|c_j| \\in \\left [\\frac{1}{i},1 \\right ]$ such that\n\\begin{equation}\\label{eq:distance_muj}\nd( c_j\\mu_{y_j}, \\mu_{y_j,r_j})< \\varepsilon\n\\qquad\n\\forall j \\in \\mathbb{N}.\n\\end{equation}\nUp to selecting a non relabeled subsequence and without loss of generality, we can assume $c_j>0$ and that $\\mu_{x,r_j} \\overset{\\ast}{\\rightharpoonup} c\\mu_x$ for some $c \\in \\left [\\frac{1}{i},1 \\right ]$.\nWe now analyze the weak*-limit (up to a suitable subsequence) of the two sequences of measures in \\eqref{eq:distance_muj}.", "post_theorem_intro_text_len": 5662, "post_theorem_intro_text": "The idea of the proof is that, for every $x \\in \\mathcal{S}_\\mu^k$, there is a radius $r>0$ for which the points in $\\mathcal{S}_\\mu^k \\cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$. Otherwise, by contradiction, one can find a sequence of such points $y_j \\notin x+C$ with $r_j=|x-y_j| \\to 0$ and such that the measures $c_j \\mu_{y_j}$ and $\\mu_{y_j,r_j}$ are as close as we want. Now, on one hand $c_j\\mu_{y_j}$ converges to a measure $\\nu$ whose invariant subspace is close to $D_x$; on the other hand $\\mu_{y_j,r_j}$, being translations of $\\mu_{x,r_j}$ with respect to points outside $C$, converge to a translation of $\\mu_x$ whose invariant subspace is not contained in $C$, contradicting the closeness of $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$.\n\n\\medskip\n\nA first outcome of Theorem \\ref{thm:main} is a rectifiability criterion for signed Radon measures in terms of their blow-ups which, to the best of our knowledge, is new.\n\n\\begin{corollario}\\label{cor:rectif_criterion}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Let us assume that, for $|\\mu|$-a.e.\\ $x \\in \\mathbb{R}^n$, there exists a Radon measure $\\nu_x$ whose invariant linear subspace $V_x$ is $k$-dimensional and with the property that, for every sequence $r_j \\to 0^+$ there are a subsequence, not relabeled, and a constant $c\\neq 0$ such that\n\\begin{equation}\n\\frac{1}{r_j^k}  (\\tau_{x,r_j})_\\# \\mu\n\\overset{\\ast}{\\rightharpoonup} c \\nu_x.\n\\end{equation}\nThen $\\mu$ is $k$-rectifiable: there exists a signed function $\\theta \\in L^1_{\\mathrm{loc}}(\\mathcal{S}_\\mu^k,\\mathcal{H}^k)$ such that \\mbox{$\\mu= \\theta \\mathcal{H}^k \\llcorner \\mathcal{S}_\\mu^k$,} where $\\mathcal{S}_\\mu^k$ is defined in \\eqref{eq:def_s_mu_k}.\n\\end{corollario}\n\nThe above result extends to signed Radon measures the well-known criterion for positive ones, see for instance \\cite[Theorem 11.8]{simonlectures83}, where one usually sets $\\nu_x=\\theta_0 \\mathcal{H}^k \\llcorner V_x$ for some $k$-dimensional linear subspace $V_x$ and assumes the existence of the limit $r^{-k}(\\tau_{x,r_j})_\\# \\mu \\overset{\\ast}{\\rightharpoonup} \\theta_0 \\mathcal{H}^k \\llcorner V_x$ as $r \\to 0^+$.\n\nWe point out that Corollary \\ref{cor:rectif_criterion} cannot be trivially deduced from the corresponding result for positive measures, since the fact that $\\mu$ has a certain blow-up at a point does not provide any suitable information on the blow-ups of $|\\mu|$ at the same point, not even their uniqueness; nor does it appear that the classical proof of \\cite[Theorem 11.8]{simonlectures83} can be readily adapted to the case of signed measures.\n\n\\medskip\n\nA further consequence of Theorem \\ref{thm:main} is the following extension of \\cite[Theorem 3.2]{mattila_unique05} to signed Radon measures. It is based on the intuitive idea that, since each $\\mathcal{S}_{\\mu}^k$ is $k$-rectifiable, it is reasonable to expect that its subset of points where the blow-up of $\\mu$ is more diffused than a $k$-dimensional measure is $|\\mu|$-negligible.\n\n\\begin{corollario}\\label{cor:almost_everywhere_rect}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Then, for $|\\mu|$-a.e.\\ $x \\in \\mathcal{S}_\\mu^k$, the unique blow-up of $\\mu$ at $x$ is a multiple of $\\mathcal{H}^k\\llcorner D_x$.\n\\end{corollario}\n\nWe conclude this introduction with a comment on the definition of unique blow-up at a point.\nAs already said above, we say that $\\mu_x$ is the unique blow-up of $\\mu$ at $x \\in \\operatorname{supp}\\mu$ if, for every sequence $r_j \\to 0$, \\emph{there exists} a subsequence for which \\eqref{eq:def-unique_blowup} holds true. This differs from the definition used in \\cite{mattila_unique05} for positive Radon measures, where one does not require the convergence of a subsequence for every choice of $\\{r_j\\}_j$: according to \\cite{mattila_unique05}, one says that  $\\mu$ has unique blow-up at $x \\in \\mathbb{R}$ if there exists $\\nu$ such that any non-trivial weak*-limit (in case it exists) of $c_j (\\tau_{x,r_j})_\\# \\mu$, where $c_j>0$ and $r_j \\to 0^+$, is of the form $c\\nu$ for some $c>0$. \n\n\\cite[Lemma 2.5]{mattila_unique05} shows, provided $\\mu$ is positive, that this condition actually implies $\\mu_{x,r} \\overset{\\ast}{\\rightharpoonup} c\\mu_x$ as $r \\to 0$ and, subsequently, our definition (they are actually equivalent).\nHowever, this does not hold in the case of signed Radon measures: let us consider two sequences of positive real numbers $a_j, \\rho_j$ such that\n\\begin{equation}\n\\lim_{j \\to +\\infty} \\frac{1}{a_j} \\sum_{i=j+1}^{+\\infty} a_i = 0,\n\\qquad\n\\lim_{j \\to +\\infty} \\frac{\\rho_{j+1}}{\\rho_j}=0.\n\\end{equation}\nThen the Radon measure on $\\mathbb{R}$ defined as\n\\begin{equation}\n\\mu \\coloneqq \\sum_{j \\in \\mathbb{N}} a_j\\left ( \\delta_{2\\rho_j + \\rho_j^2} - \\delta_{2\\rho_j - \\rho_j^2} \\right )\n\\end{equation}\nhas the null measure as unique blow-up at $x=0$ in the sense of \\cite{mattila_unique05}, but $\\mu_{0,\\rho_j} =\\frac{1}{|\\mu|(B_{\\rho_j}(0))} (\\tau_{0,\\rho_j})_\\# \\mu $ has no converging subsequences because $|\\mu_{x,\\rho_j}|\\big(B_2(0)\\big) \\to +\\infty$ as $j\\to +\\infty$.\n\n\\subsection*{Acknowledgments}\nThe author would like to thank Andrea Marchese, Paolo Bonicatto and Gian Paolo Leonardi for inspiring conversations. He has been supportend by PRIN project 2022PJ9EFL ``Geometric Measure Theory: Structure of Singular Measures, Regularity Theory and Applications in the Calculus of Variations\" CUP:E53D23005860006 and  partially supported by INDAM-GNAMPA. Part of the work has been carried out during the thematic program on ``Free boundary problems\" at ESI (Wien), which the author wishes to thank.", "sketch": "The post-theorem introduction gives the following proof idea for Theorem~\\ref{thm:main}.\n\n1. **Cone containment near each point.** For every $x\\in\\mathcal{S}_\\mu^k$, there exists $r>0$ such that the points in $\\mathcal{S}_\\mu^k\\cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$.\n\n2. **Contradiction setup if cone containment fails.** Otherwise, one can find a sequence $y_j\\notin x+C$ with $r_j=|x-y_j|\\to 0$ such that the measures $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$ are “as close as we want.”\n\n3. **Two limiting descriptions of the same asymptotically close measures.**\n   - On the one hand, $c_j\\mu_{y_j}$ converges to a measure $\\nu$ whose invariant subspace is close to $D_x$.\n   - On the other hand, $\\mu_{y_j,r_j}$ are translations of $\\mu_{x,r_j}$ “with respect to points outside $C$,” hence they converge to a translation of $\\mu_x$ whose invariant subspace is not contained in $C$.\n\n4. **Contradiction.** This “contradict[s] the closeness of $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$,” forcing the desired cone property; this yields the Lipschitz-graph covering and the identification of the approximate tangent plane with $D_x$ for $\\mathcal{H}^k$-a.e. $x\\in\\mathcal{S}_\\mu^k$.", "expanded_sketch": "The post-theorem introduction gives the following proof idea for the main theorem.\n\n1. **Cone containment near each point.** For every $x\\in\\mathcal{S}_\\mu^k$, there exists $r>0$ such that the points in $\\mathcal{S}_\\mu^k\\cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$.\n\n2. **Contradiction setup if cone containment fails.** Otherwise, one can find a sequence $y_j\\notin x+C$ with $r_j=|x-y_j|\\to 0$ such that the measures $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$ are “as close as we want.”\n\n3. **Two limiting descriptions of the same asymptotically close measures.**\n   - On the one hand, $c_j\\mu_{y_j}$ converges to a measure $\\nu$ whose invariant subspace is close to $D_x$.\n   - On the other hand, $\\mu_{y_j,r_j}$ are translations of $\\mu_{x,r_j}$ “with respect to points outside $C$,” hence they converge to a translation of $\\mu_x$ whose invariant subspace is not contained in $C$.\n\n4. **Contradiction.** This “contradict[s] the closeness of $c_j\\mu_{y_j}$ and $\\mu_{y_j,r_j}$,” forcing the desired cone property; this yields the Lipschitz-graph covering and the identification of the approximate tangent plane with $D_x$ for $\\mathcal{H}^k$-a.e. $x\\in\\mathcal{S}_\\mu^k$.", "expanded_theorem": "\\label{thm:main}\nLet $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$ and $k \\in \\{0,1,\\dots,n\\}$. Then $\\mathcal{S}_\\mu^k$ is contained in a countable union of $k$-dimensional Lipschitz graphs. Moreover, the approximate tangent plane to $\\mathcal{S}_\\mu^k$ at $x$ coincides with $D_x$ for $\\mathcal{H}^{k}$-a.e.\\ $x \\in \\mathcal{S}_\\mu^k$.,", "theorem_type": ["Universal", "Implication"], "mcq": {"question": "Let $\\mu$ be a signed Radon measure on $\\mathbb{R}^n$, and for $x\\in \\operatorname{supp}\\mu$ and $r>0$ define the rescaled measure\n\\[\n\\mu_{x,r}:=\\frac{1}{|\\mu|(B_r(x))}(\\tau_{x,r})_\\#\\mu,\n\\qquad \\tau_{x,r}(y):=\\frac{y-x}{r}.\n\\]\nSay that $\\mu$ has a unique blow-up at $x$ if there exists a Radon measure $\\mu_x$ with either $|\\mu_x|(B_1(0))=1$ or $\\mu_x=0$ such that, for every sequence $r_j\\to 0^+$, one can extract a subsequence and find a constant $c\\neq 0$ for which\n\\[\n\\mu_{x,r_j}\\overset{*}{\\rightharpoonup} c\\mu_x\n\\]\nin the weak-* topology of Radon measures. Let $\\mathcal S_\\mu$ be the set of points where $\\mu$ has unique blow-up. For $x\\in \\mathcal S_\\mu$, define the invariant subspace of the blow-up by\n\\[\nD_x:=\\{y\\in \\mathbb R^n:(\\tau_{y,1})_\\#\\mu_x=\\mu_x\\},\n\\]\nand, for $k\\in\\{0,1,\\dots,n\\}$, set\n\\[\n\\mathcal S_\\mu^k:=\\{x\\in \\mathcal S_\\mu:\\dim D_x=k\\}.\n\\]\nUnder these assumptions, which statement about $\\mathcal S_\\mu^k$ holds?", "correct_choice": {"label": "A", "text": "The set $\\mathcal S_\\mu^k$ is contained in a countable union of $k$-dimensional Lipschitz graphs. Moreover, for $\\mathcal H^k$-almost every $x\\in \\mathcal S_\\mu^k$, the approximate tangent plane to $\\mathcal S_\\mu^k$ at $x$ exists and coincides with $D_x$."}, "choices": [{"label": "B", "text": "The set $\\mathcal S_\\mu^k$ is contained in a countable union of $(k+1)$-dimensional Lipschitz graphs. Moreover, for $\\mathcal H^{k+1}$-almost every $x\\in \\mathcal S_\\mu^k$, the approximate tangent plane to $\\mathcal S_\\mu^k$ at $x$ exists and contains $D_x$."}, {"label": "C", "text": "The set $\\mathcal S_\\mu^k$ is countably $\\mathcal H^k$-rectifiable; in particular, it is contained in a countable union of Lipschitz images of subsets of $\\mathbb R^k$."}, {"label": "D", "text": "The set $\\mathcal S_\\mu^k$ is contained in a countable union of $k$-dimensional Lipschitz graphs. Moreover, for every $x\\in \\mathcal S_\\mu^k$, the approximate tangent plane to $\\mathcal S_\\mu^k$ at $x$ exists and coincides with $D_x$."}, {"label": "E", "text": "The set $\\mathcal S_\\mu^k$ is contained in a finite union of $k$-dimensional Lipschitz graphs. Moreover, for $\\mathcal H^k$-almost every $x\\in \\mathcal S_\\mu^k$, the approximate tangent plane to $\\mathcal S_\\mu^k$ at $x$ exists and is parallel to $D_x$."}], "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": "dimension_of_cone_and_tangent_plane", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "a.e._identification_of_approximate_tangent_plane_with_D_x", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "almost_everywhere_vs_everywhere_tangent_identification", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "countable_union_and_exact_plane_identification", "template_used": "wildcard"}]}}
{"id": "2512.20751v1", "paper_link": "http://arxiv.org/abs/2512.20751v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{Teo1}\n\tAssume that the potential $W$ satisfies \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$}. Then, for every $\\varepsilon > 0$ and $a_0 > 0$, there exists $\\delta > 0$ such that for all $a \\in (0, a_0]$, and any solution $u_a$ of \\eqref{Equation} with initial data satisfying\n\t$$\n\t\\|u_a(0)-u_*\\| + \\|\\dot{u}_a(0)\\| < \\delta,\n\t$$\n\tthe following hold:\n\t\\begin{itemize}\n\t\t\\item[\\emph{(a)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le \\varepsilon$;\n\t\t\\item[\\emph{(b)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tIf, additionally, \\hyperlink{W2}{$(W_2)$} and \\hyperlink{W3}{$(W_3)$} hold with $\\delta = \\lambda = +\\infty$, then there exists $R > 0$ such that, for all $a \\in (0, a_0]$, and any solution $u_a$ of \\eqref{Equation},\n\t\\begin{itemize}\n\t\t\\item[\\emph{(c)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le R$;\n\t\t\\item[\\emph{(d)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and  $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tMoreover, under the additional assumption \\hyperlink{W5}{$(W_5)$}, for each $a >0$ there exist $\\gamma(a) > 0$ and $C(a)>0$ such that the corresponding solution $u_a$ satisfies the exponential decay estimate\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C(a) e^{-\\gamma(a) t}, \\quad \\forall\\, t \\ge 0.\n\t\\end{equation*}\n\tIn particular, if $0<a_1<a_2$ and $a\\in[a_1,a_2]$, then there exist $C_*>0$ and $\\gamma_*>0$, depending only on $a_1,a_2$ such that\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C_* e^{-\\gamma_* t},\\quad \\forall\\, t \\ge 0,\\ \\forall\\, a\\in[a_1,a_2].\n\t\\end{equation*}", "start_pos": 1552652, "end_pos": 1554414, "label": "Teo1"}, "ref_dict": {"Equation": "\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}", "Teo1": "\\begin{theorem}\\label{Teo1}\n\tAssume that the potential $W$ satisfies \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$}. Then, for every $\\varepsilon > 0$ and $a_0 > 0$, there exists $\\delta > 0$ such that for all $a \\in (0, a_0]$, and any solution $u_a$ of \\eqref{Equation} with initial data satisfying\n\t$$\n\t\\|u_a(0)-u_*\\| + \\|\\dot{u}_a(0)\\| < \\delta,\n\t$$\n\tthe following hold:\n\t\\begin{itemize}\n\t\t\\item[\\emph{(a)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le \\varepsilon$;\n\t\t\\item[\\emph{(b)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tIf, additionally, \\hyperlink{W2}{$(W_2)$} and \\hyperlink{W3}{$(W_3)$} hold with $\\delta = \\lambda = +\\infty$, then there exists $R > 0$ such that, for all $a \\in (0, a_0]$, and any solution $u_a$ of \\eqref{Equation},\n\t\\begin{itemize}\n\t\t\\item[\\emph{(c)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le R$;\n\t\t\\item[\\emph{(d)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and  $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tMoreover, under the additional assumption \\hyperlink{W5}{$(W_5)$}, for each $a >0$ there exist $\\gamma(a) > 0$ and $C(a)>0$ such that the corresponding solution $u_a$ satisfies the exponential decay estimate\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C(a) e^{-\\gamma(a) t}, \\quad \\forall\\, t \\ge 0.\n\t\\end{equation*}\n\tIn particular, if $0<a_1<a_2$ and $a\\in[a_1,a_2]$, then there exist $C_*>0$ and $\\gamma_*>0$, depending only on $a_1,a_2$ such that\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C_* e^{-\\gamma_* t},\\quad \\forall\\, t \\ge 0,\\ \\forall\\, a\\in[a_1,a_2].\n\t\\end{equation*}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 6996, "pre_theorem_intro_text": "Second-order dissipative systems of the form\n\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\narise naturally in mechanics, optimization, and nonlinear dynamics. A classical example is the \\emph{heavy-ball method with friction}, in which a particle moves on the graph of a potential $W$ under the action of inertia and a linear damping force with coefficient $a>0$. The heavy-ball method was introduced by Polyak in 1964 to speed up the convergence of iterative schemes \\cite{Polyak}, and was formalized by Attouch, Goudou, and Redont \\cite{AttouchGoudouRedont}, who analyzed the frictional equation to explore the minima of potential $W$. Beyond heavy-ball method with friction, the prototype equation \\eqref{Equation} also models the motion of a \\emph{damped simple pendulum}.  \nLet $u=\\theta$ be the pendulum angle, take $W(\\theta)=\\frac{g}{\\ell}(1-\\cos\\theta)$, then\n$$\n\\ddot{\\theta}(t)+ a\\,\\dot{\\theta}(t)+\\dfrac{g}{\\ell}\\sin\\theta(t)=0,\n$$\nwhich is the standard damped pendulum equation. This modeling is classical in nonlinear dynamics and the term $a\\theta'$ represents the damping force, with $a > 0$. Depending on the initial energy, this system exhibits oscillatory motions, convergence to stable equilibria, and heteroclinic trajectories connecting different critical points of $W$. A comprehensive analysis of the pendulum dynamics can be found in \\cite{Mawhin}. In fact, beyond these classical examples, a number of other mechanical and computational systems reduce to the general dissipative structure \\eqref{Equation}, which reinforces the broad applicability and theoretical interest of the problem.\n\nSecond-order equations with constant damping also appear in the study of generalized gradient systems. Recent works have achieved significant progress in the asymptotic analysis of such systems. Jendoubi and collaborators showed that an equation of the type \\eqref{Equation} can be interpreted as a quasi-gradient system via small deformations of the total energy. They proved that the desingularizing function in the Kurdyka-Łojasiewicz inequality satisfies $\\varphi(t) \\ge c\\sqrt{t}$ when the potential $W$ is definable and of class $C^2$ \\cite{Jendoubi1}. This result implies that every trajectory of a quasi-gradient system either converges to a critical point or its norm diverges to infinity. Another relevant development concerns Newton-type systems with {\\it Hessian-driven damping}, combining the continuous Newton method with the heavy-ball system. Boţ and Csetnek in \\cite{BotCsetnek} observed that such systems are second-order in both time and space, due to the presence of the acceleration term and the Hessian, and can be seen as a mixture of Newton’s method with the heavy-ball system. This hybrid approach inherits advantages from both methods, as shown in the optimization context by Alvarez et al. \\cite{Alvarez}. We also mention that several recent works have focused on nonhomogeneous variants of \\eqref{Equation} and on models where the damping parameter depends on time, such as \\cite{1,2,3,4,6,7}.\n\nThe term $a\\dot{u}$ in \\eqref{Equation} represents a damping force, while $\\nabla W(u)$ acts as a restoring force derived from the potential energy $W(u)$. In this work we focus on the second-order dissipative systems \\eqref{Equation}, where $a > 0$ is a damping coefficient, $u\\colon \\mathbb{R} \\to \\mathbb{R}^N$ is the unknown trajectory, and $W\\colon \\mathbb{R}^N \\to \\mathbb{R}$ is a potential. Throughout the paper we impose the following assumptions on $W$:\n\\begin{itemize} \n\t\\item[\\hypertarget{W1}{($W_1$)}] $W\\in C^2(\\mathbb{R}^N; \\mathbb{R})$.\n\n\t\\item[\\hypertarget{W2}{($W_2$)}] There exists $\\delta > 0$ and $u_* \\in \\mathbb{R}^{N}$ such that\n\t$$\n\tW(u_*) = 0 \\quad \\text{and} \\quad W(u) > 0 \\quad \\text{for all} \\quad u \\in B_\\delta(u_*) \\setminus \\{u_*\\}.\n\t$$\n\n\t\\item[\\hypertarget{W3}{($W_3$)}] There exists $\\lambda > 0$ such that\n\t$$\n\t\\langle \\nabla W(u), u - u_* \\rangle > 0 \\quad \\text{for all} \\quad u \\in B_\\lambda(u_*) \\setminus \\{u_*\\}.\n\t$$\n\n\t\\item[\\hypertarget{W4}{($W_4$)}] $W$ is coercive, that is,\n\t$$\n\t\\lim_{\\|u\\| \\to +\\infty} W(u) = +\\infty.\n\t$$\n\n\t\\item[\\hypertarget{W5}{($W_5$)}] There exist constants $\\alpha, \\beta, \\mu > 0$ such that, for every $x \\in B_\\lambda(u_*)$,\n\t$$\n\t\\alpha\\|x - u_*\\|^2 \\le W(x) \\le \\beta\\|x - u_*\\|^2 \\quad \\text{and} \\quad \\langle \\nabla W(x), x - u_* \\rangle \\ge \\mu\\|x - u_*\\|^2.\n\t$$\n\\end{itemize}\n\nA simple example satisfying \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_5)$} with $u_*=0$ is the harmonic potential\n$$\nW(u)=\\frac{1}{2}\\|u\\|^2.\n$$\nIn this case, \\eqref{Equation} reduces to the classical damped harmonic oscillator. This model describes a {\\it mass-spring-damper system}, where $a$ represents the viscous damping and $u$ is the Hookean restoring force. Solutions exhibit decaying oscillations and converge to the equilibrium $u=0$. Such systems appear  not only in mechanics but also in electrical circuits and other dissipative models driven by an energy potential, see \\cite{Edwards,Sunday1,Sunday2}. Beyond the quadratic case, the class of potentials satisfying assumptions \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$} is broad and includes many models from mechanics, nonlinear dynamics, and phase-transition theory. A general and important family is obtained by taking  \n$$\nW(u)=\\Phi(u-u_*),\n$$\nwhere $\\Phi\\colon\\mathbb{R}^N\\to\\mathbb{R}$ is a convex, positive function with a global minimum at the origin. Such potentials arise naturally in the study of quasilinear Allen-Cahn type equations, see, for instance, \\cite{Isneri}. In particular, coercive polynomial potentials, such as $W(u)=\\|u-u_*\\|^p$, with $p\\ge 2$, and the exponential potential  \n$$\nW(u)=\\frac12\\big(e^{\\|u-u_*\\|^2}-1\\big),\n$$\nwhich is convex and coercive, fits perfectly into this model. Another classical example is the Ginzburg-Landau potential  \n$$\nW(u)=\\frac14\\left(\\|u\\|^2-1\\right)^2,\n$$\nwhose set of global minima is the unit sphere $\\{u\\in\\mathbb{R}^N : \\|u\\| = 1\\}$. This potential play a central role in the theory of phase transitions and superconductivity, see \\cite{Allen,5}. A further relevant example, frequently appearing in the study of dissipative hyperbolic PDEs, is the quartic double-well potential  \n$$\nW(u)=c_1\\|u\\|^{4}-c_2\\|u\\|^{2}+c_3,\\quad c_1,c_2,c_3>0,\n$$\nwhose structure is extensively used in nonlinear wave equations with damping. Such models constitute a natural infinite-dimensional counterpart of \\eqref{Equation}, see, for instance, \\cite{8}. These examples illustrate that the hypotheses imposed on $W$ encompass both linear and nonlinear restoring forces, as in the damped mass-spring system, as well as potentials arising in gradient flows, optimization, and models of phase separation.\n\nWith these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \\eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.", "context": "Second-order dissipative systems of the form\n\\begin{equation}\\label{Equation}\n \\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\narise naturally in mechanics, optimization, and nonlinear dynamics. A classical example is the \\emph{heavy-ball method with friction}, in which a particle moves on the graph of a potential $W$ under the action of inertia and a linear damping force with coefficient $a>0$. The heavy-ball method was introduced by Polyak in 1964 to speed up the convergence of iterative schemes \\cite{Polyak}, and was formalized by Attouch, Goudou, and Redont \\cite{AttouchGoudouRedont}, who analyzed the frictional equation to explore the minima of potential $W$. Beyond heavy-ball method with friction, the prototype equation \\eqref{Equation} also models the motion of a \\emph{damped simple pendulum}. \nLet $u=\\theta$ be the pendulum angle, take $W(\\theta)=\\frac{g}{\\ell}(1-\\cos\\theta)$, then\n$$\n\\ddot{\\theta}(t)+ a\\,\\dot{\\theta}(t)+\\dfrac{g}{\\ell}\\sin\\theta(t)=0,\n$$\nwhich is the standard damped pendulum equation. This modeling is classical in nonlinear dynamics and the term $a\\theta'$ represents the damping force, with $a > 0$. Depending on the initial energy, this system exhibits oscillatory motions, convergence to stable equilibria, and heteroclinic trajectories connecting different critical points of $W$. A comprehensive analysis of the pendulum dynamics can be found in \\cite{Mawhin}. In fact, beyond these classical examples, a number of other mechanical and computational systems reduce to the general dissipative structure \\eqref{Equation}, which reinforces the broad applicability and theoretical interest of the problem.\n\nThe term $a\\dot{u}$ in \\eqref{Equation} represents a damping force, while $\\nabla W(u)$ acts as a restoring force derived from the potential energy $W(u)$. In this work we focus on the second-order dissipative systems \\eqref{Equation}, where $a > 0$ is a damping coefficient, $u\\colon \\mathbb{R} \\to \\mathbb{R}^N$ is the unknown trajectory, and $W\\colon \\mathbb{R}^N \\to \\mathbb{R}$ is a potential. Throughout the paper we impose the following assumptions on $W$:\n\\begin{itemize} \n \\item[\\hypertarget{W1}{($W_1$)}] $W\\in C^2(\\mathbb{R}^N; \\mathbb{R})$.\n\n\\item[\\hypertarget{W2}{($W_2$)}] There exists $\\delta > 0$ and $u_* \\in \\mathbb{R}^{N}$ such that\n $$\n W(u_*) = 0 \\quad \\text{and} \\quad W(u) > 0 \\quad \\text{for all} \\quad u \\in B_\\delta(u_*) \\setminus \\{u_*\\}.\n $$\n\n\\item[\\hypertarget{W5}{($W_5$)}] There exist constants $\\alpha, \\beta, \\mu > 0$ such that, for every $x \\in B_\\lambda(u_*)$,\n $$\n \\alpha\\|x - u_*\\|^2 \\le W(x) \\le \\beta\\|x - u_*\\|^2 \\quad \\text{and} \\quad \\langle \\nabla W(x), x - u_* \\rangle \\ge \\mu\\|x - u_*\\|^2.\n $$\n\\end{itemize}\n\nA simple example satisfying \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_5)$} with $u_*=0$ is the harmonic potential\n$$\nW(u)=\\frac{1}{2}\\|u\\|^2.\n$$\nIn this case, \\eqref{Equation} reduces to the classical damped harmonic oscillator. This model describes a {\\it mass-spring-damper system}, where $a$ represents the viscous damping and $u$ is the Hookean restoring force. Solutions exhibit decaying oscillations and converge to the equilibrium $u=0$. Such systems appear not only in mechanics but also in electrical circuits and other dissipative models driven by an energy potential, see \\cite{Edwards,Sunday1,Sunday2}. Beyond the quadratic case, the class of potentials satisfying assumptions \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$} is broad and includes many models from mechanics, nonlinear dynamics, and phase-transition theory. A general and important family is obtained by taking \n$$\nW(u)=\\Phi(u-u_*),\n$$\nwhere $\\Phi\\colon\\mathbb{R}^N\\to\\mathbb{R}$ is a convex, positive function with a global minimum at the origin. Such potentials arise naturally in the study of quasilinear Allen-Cahn type equations, see, for instance, \\cite{Isneri}. In particular, coercive polynomial potentials, such as $W(u)=\\|u-u_*\\|^p$, with $p\\ge 2$, and the exponential potential \n$$\nW(u)=\\frac12\\big(e^{\\|u-u_*\\|^2}-1\\big),\n$$\nwhich is convex and coercive, fits perfectly into this model. Another classical example is the Ginzburg-Landau potential \n$$\nW(u)=\\frac14\\left(\\|u\\|^2-1\\right)^2,\n$$\nwhose set of global minima is the unit sphere $\\{u\\in\\mathbb{R}^N : \\|u\\| = 1\\}$. This potential play a central role in the theory of phase transitions and superconductivity, see \\cite{Allen,5}. A further relevant example, frequently appearing in the study of dissipative hyperbolic PDEs, is the quartic double-well potential \n$$\nW(u)=c_1\\|u\\|^{4}-c_2\\|u\\|^{2}+c_3,\\quad c_1,c_2,c_3>0,\n$$\nwhose structure is extensively used in nonlinear wave equations with damping. Such models constitute a natural infinite-dimensional counterpart of \\eqref{Equation}, see, for instance, \\cite{8}. These examples illustrate that the hypotheses imposed on $W$ encompass both linear and nonlinear restoring forces, as in the damped mass-spring system, as well as potentials arising in gradient flows, optimization, and models of phase separation.\n\nWith these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \\eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.\n\n\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}", "full_context": "Second-order dissipative systems of the form\n\\begin{equation}\\label{Equation}\n \\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\narise naturally in mechanics, optimization, and nonlinear dynamics. A classical example is the \\emph{heavy-ball method with friction}, in which a particle moves on the graph of a potential $W$ under the action of inertia and a linear damping force with coefficient $a>0$. The heavy-ball method was introduced by Polyak in 1964 to speed up the convergence of iterative schemes \\cite{Polyak}, and was formalized by Attouch, Goudou, and Redont \\cite{AttouchGoudouRedont}, who analyzed the frictional equation to explore the minima of potential $W$. Beyond heavy-ball method with friction, the prototype equation \\eqref{Equation} also models the motion of a \\emph{damped simple pendulum}. \nLet $u=\\theta$ be the pendulum angle, take $W(\\theta)=\\frac{g}{\\ell}(1-\\cos\\theta)$, then\n$$\n\\ddot{\\theta}(t)+ a\\,\\dot{\\theta}(t)+\\dfrac{g}{\\ell}\\sin\\theta(t)=0,\n$$\nwhich is the standard damped pendulum equation. This modeling is classical in nonlinear dynamics and the term $a\\theta'$ represents the damping force, with $a > 0$. Depending on the initial energy, this system exhibits oscillatory motions, convergence to stable equilibria, and heteroclinic trajectories connecting different critical points of $W$. A comprehensive analysis of the pendulum dynamics can be found in \\cite{Mawhin}. In fact, beyond these classical examples, a number of other mechanical and computational systems reduce to the general dissipative structure \\eqref{Equation}, which reinforces the broad applicability and theoretical interest of the problem.\n\nThe term $a\\dot{u}$ in \\eqref{Equation} represents a damping force, while $\\nabla W(u)$ acts as a restoring force derived from the potential energy $W(u)$. In this work we focus on the second-order dissipative systems \\eqref{Equation}, where $a > 0$ is a damping coefficient, $u\\colon \\mathbb{R} \\to \\mathbb{R}^N$ is the unknown trajectory, and $W\\colon \\mathbb{R}^N \\to \\mathbb{R}$ is a potential. Throughout the paper we impose the following assumptions on $W$:\n\\begin{itemize} \n \\item[\\hypertarget{W1}{($W_1$)}] $W\\in C^2(\\mathbb{R}^N; \\mathbb{R})$.\n\n\\item[\\hypertarget{W2}{($W_2$)}] There exists $\\delta > 0$ and $u_* \\in \\mathbb{R}^{N}$ such that\n $$\n W(u_*) = 0 \\quad \\text{and} \\quad W(u) > 0 \\quad \\text{for all} \\quad u \\in B_\\delta(u_*) \\setminus \\{u_*\\}.\n $$\n\n\\item[\\hypertarget{W5}{($W_5$)}] There exist constants $\\alpha, \\beta, \\mu > 0$ such that, for every $x \\in B_\\lambda(u_*)$,\n $$\n \\alpha\\|x - u_*\\|^2 \\le W(x) \\le \\beta\\|x - u_*\\|^2 \\quad \\text{and} \\quad \\langle \\nabla W(x), x - u_* \\rangle \\ge \\mu\\|x - u_*\\|^2.\n $$\n\\end{itemize}\n\nA simple example satisfying \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_5)$} with $u_*=0$ is the harmonic potential\n$$\nW(u)=\\frac{1}{2}\\|u\\|^2.\n$$\nIn this case, \\eqref{Equation} reduces to the classical damped harmonic oscillator. This model describes a {\\it mass-spring-damper system}, where $a$ represents the viscous damping and $u$ is the Hookean restoring force. Solutions exhibit decaying oscillations and converge to the equilibrium $u=0$. Such systems appear not only in mechanics but also in electrical circuits and other dissipative models driven by an energy potential, see \\cite{Edwards,Sunday1,Sunday2}. Beyond the quadratic case, the class of potentials satisfying assumptions \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$} is broad and includes many models from mechanics, nonlinear dynamics, and phase-transition theory. A general and important family is obtained by taking \n$$\nW(u)=\\Phi(u-u_*),\n$$\nwhere $\\Phi\\colon\\mathbb{R}^N\\to\\mathbb{R}$ is a convex, positive function with a global minimum at the origin. Such potentials arise naturally in the study of quasilinear Allen-Cahn type equations, see, for instance, \\cite{Isneri}. In particular, coercive polynomial potentials, such as $W(u)=\\|u-u_*\\|^p$, with $p\\ge 2$, and the exponential potential \n$$\nW(u)=\\frac12\\big(e^{\\|u-u_*\\|^2}-1\\big),\n$$\nwhich is convex and coercive, fits perfectly into this model. Another classical example is the Ginzburg-Landau potential \n$$\nW(u)=\\frac14\\left(\\|u\\|^2-1\\right)^2,\n$$\nwhose set of global minima is the unit sphere $\\{u\\in\\mathbb{R}^N : \\|u\\| = 1\\}$. This potential play a central role in the theory of phase transitions and superconductivity, see \\cite{Allen,5}. A further relevant example, frequently appearing in the study of dissipative hyperbolic PDEs, is the quartic double-well potential \n$$\nW(u)=c_1\\|u\\|^{4}-c_2\\|u\\|^{2}+c_3,\\quad c_1,c_2,c_3>0,\n$$\nwhose structure is extensively used in nonlinear wave equations with damping. Such models constitute a natural infinite-dimensional counterpart of \\eqref{Equation}, see, for instance, \\cite{8}. These examples illustrate that the hypotheses imposed on $W$ encompass both linear and nonlinear restoring forces, as in the damped mass-spring system, as well as potentials arising in gradient flows, optimization, and models of phase separation.\n\nWith these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \\eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.\n\n\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\n\nWith these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \\eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.\n\nIn this section we discuss the dynamics of the conservative system\n\\begin{equation}\\label{EqConservative}\n \\ddot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\nwhich arises as the limit of \\eqref{Equation} when the damping parameter $a$ tends to zero. Throughout this section, we continue to assume that $W$ satisfies \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$} and, without loss of generality, that the global minimum of $W$ is located at $u_*=0$. In contrast with the dissipative\ncase, the conservative dynamics do not generate asymptotic stability, since no decay mechanism is present. To analyze the qualitative behavior of \\eqref{EqConservative}, it is convenient to rewrite it as a first-order system in $\\mathbb{R}^{2N}$ by setting $x=u$ and $y=\\dot{u}$:\n\\begin{equation}\\label{S0}\n \\begin{cases}\n \\dot{x} = y,\\\\[2mm]\n \\dot{y} = -\\nabla W(x).\n \\end{cases}\n\\end{equation}\nThe natural energy associated with \\eqref{EqConservative},\n$$\nE(x,y) := \\frac12\\|y\\|^2 + W(x), \\quad (x,y)\\in B_\\eta(0)\\times \\mathbb{R}^N,\n$$\nwhere $\\eta=\\min\\{\\delta,\\lambda\\}$ with $\\delta$ and $\\lambda$ given in \\hyperlink{W2}{$(W_2)$}-\\hyperlink{W3}{$(W_3)$}, acts as a Lyapunov function for the equilibrium $(0,0)$. Indeed, if $(x(t),y(t))$ is a solution of \\eqref{S0}, then deriving \n$E(x(t),y(t))$ along the trajectory yields\n$$\n\\frac{d}{dt}E(x(t),y(t))= \\langle y(t),\\dot{y}(t)\\rangle + \\langle \\nabla W(x(t)),\\dot{x}(t)\\rangle.\n$$\nUsing $\\dot{x}=y$ and $\\dot{y}=-\\nabla W(x)$, we immediately obtain\n$$\n\\frac{d}{dt}E(x(t),y(t))= -\\langle y(t),\\nabla W(x(t))\\rangle+ \\langle \\nabla W(x(t)),y(t)\\rangle= 0.\n$$\nThus, if $u(t)$ is any solution of \\eqref{EqConservative}, then the energy is conserved in time, that is,\n\\begin{equation}\\label{EnergyConservation}\n \\frac12\\|\\dot{u}(t)\\|^2 + W(u(t)) = \\tau,\\quad \\forall\\, t\\in\\mathbb{R},\n\\end{equation}\nfor some $\\tau\\ge 0$ determined by the initial condition. From the Lyapunov Criterion, conservation of energy implies that \n$(0,0)$ is stable: for every $\\varepsilon>0$ there exists \n$\\delta>0$ such that\n$$\n\\|u(0)\\| + \\|\\dot{u}(0)\\| < \\delta\\quad\\Longrightarrow\\quad\\sup_{t\\ge 0} \\big( \\|u(t)\\| + \\|\\dot{u}(t)\\| \\big) \\le \\varepsilon.\n$$\nHowever, stability cannot be asymptotic. Indeed, if a nontrivial solution $(u(t),\\dot{u}(t))$ were to converge to $(0,0)$ as $t\\to+\\infty$, then \\eqref{EnergyConservation} would force $\\tau=0$, which implies that $u(t)\\equiv 0$ identically. Hence the only trajectory converging to the equilibrium is the constant solution.\n\nWe now turn to the relation between the conservative regime and the dissipative dynamics studied in Section \\ref{Sec2}. Assume that $W$ satisfies \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W5}{$(W_5)$}. The conservative system \\eqref{EqConservative} arises as the limit of the dissipative equation \\eqref{Equation} when the damping parameter $a$ tends to zero. In the dissipative regime, Section \\ref{Sec2} established that solutions are asymptotically stable and even converge exponentially fast to the equilibrium with decay rate $\\gamma(a)>0$ depending on $a$. A natural question is whether this behavior persists, in some sense, in the limit as $a\\to 0^{+}$. The answer is negative: the limit dynamics is not asymptotically stable, and the passage $a\\to 0^{+}$ is singular. This phenomenon can be described rigorously through a perturbation argument. Let $(a_n)$ be a sequence of positive numbers with $a_n \\to 0$, and let $u_n$ be the solution of the perturbed problem\n\\begin{equation}\\label{EqDamped-n}\n \\ddot u_n(t) + a_n \\dot u_n(t) + \\nabla W(u_n(t)) = 0,\n\\end{equation}\nwith fixed initial data $(u_n(0),\\dot u_n(0)) = (z_0,z_1)$, where $(z_0,z_1)\\neq(0,0)$ and $\\|(z_0,z_1)\\|<\\sigma$. Here $\\sigma>0$ depends on a prescribed $\\varepsilon\\in(0,\\delta)$, with $\\delta>0$ given in \\hyperlink{W2}{\\((W_2)\\)}, and is provided by the uniform stability result of Lemma \\ref{L5}. In particular, Lemma \\ref{L5} ensures that\n$$\n\\lim_{t\\to+\\infty} (u_n(t),\\dot u_n(t)) = (0,0)\\quad\\text{and}\\quad\\|u_n(t)\\| + \\|\\dot u_n(t)\\| \\le \\varepsilon \\quad\\text{for all } t\\ge 0\\text{ and }n\\in\\mathbb{N}.\n$$\nMoreover, by Lemma \\ref{L7}, each $u_n$ satisfies the exponential decay estimate\n\\begin{equation}\\label{Exp}\n \\|u_n(t)\\| + \\|\\dot u_n(t)\\|\\le C e^{-\\gamma(a_n) t}, \\qquad t\\ge 0,\n\\end{equation}\nwhere $C>0$ is a constant independent of $a_n$. A compactness argument shows that, up to a subsequence, $u_n$ converges in $C^2_{\\mathrm{loc}}([0,+\\infty))$ to a limit\nfunction $u_0$, which solves the conservative system\n$$\n\\ddot u_0 + \\nabla W(u_0)=0\n$$\nwith initial data $u_0(0)=z_0$ and $\\dot u_0(0)=z_1$, and satisfying\n\\begin{equation*}\n \\sup_{t\\ge 0}\\left(\\|u_0(t)\\|+\\|\\dot{u}_0(t)\\|\\right) \\leq \\epsilon.\n\\end{equation*}\nAlthough each $u_n$ converges asymptotically to the equilibrium, the limit solution $u_0$ does not. Indeed, suppose by contradiction that\n$$\n\\lim_{t\\to+\\infty} u_0(t)=0\\quad\\text{and}\\quad\n\\lim_{t\\to+\\infty} \\dot u_0(t)=0.\n$$\nSince $W$ is continuous and $W(0)=0$, it follows that $W(u_0(t))\\to 0$ and $\\|\\dot u_0(t)\\|\\to 0$. Inserting these limits into the conservation law \\eqref{EnergyConservation}, we obtain $\\tau=0$. But $\\tau=0$ implies\n$$\n\\dot u_0(t)\\equiv 0\\quad\\text{and}\\quad W(u_0(t))\\equiv 0,\n$$\nso that $u_0(t)= 0$ for all $t\\in\\mathbb{R}$. This contradicts the fact that $(u_0(0),\\dot u_0(0))\\neq(0,0)$. Therefore,\n$$\n\\lim_{t \\to +\\infty} (u_0(t),\\dot{u}_0(t)) \\neq (0,0).\n$$\nThis argument shows that although each damped solution converges to the equilibrium, the limit solution does not. Equivalently, the exponential decay rate $\\gamma(a)$ obtained for $a>0$ must necessarily satisfy $\\gamma(a)\\to 0$ as $a\\to 0^+$, and no uniform decay estimate can hold on an interval of the form $(0,a_0]$. The transition from $a>0$ to $a=0$ represents a loss of asymptotic stability and a qualitative change in the global dynamics.\n\n\\begin{lemma}\\label{asd}\n Let $B\\subset \\mathbb R^{2N}$ be any bounded set containing the initial data $z_0$.\n Then there exists a bounded set $K\\subset \\mathbb{R}^{2N}$ such that \n $$\n T(t) B \\subset K,\\quad \\forall\\, t\\ge 0.\n $$\n In other words, $K$ serves as a absorbing set for the family of semigroup $\\{T(t)\\}_{t\\ge 0}$.\n\\end{lemma}\n\\begin{proof}\nFix $a>0$ and let $z_0=(x_0,y_0)\\in B$. Since $B$ is bounded in $\\mathbb{R}^{2N}$ and $W$ is continuous by \\hyperlink{W1}{$(W_1)$}, there exists a $R_B>0$ such that\n$$\n\\sup_{z_0\\in B} E(z_0)\\le R_B.\n$$\nTherefore, by the monotonicity property of the energy in \\eqref{decreasing-property}, we obtain\n$$\nW(x_a(t))\\leq R_B\\quad\\text{and}\\quad\\|y_a(t)\\|^2\\leq 2R_B\\quad\\forall \\, t\\geq 0,\n$$\nwhere $x_a$ and $y_a$ denote the components of the trajectory $\\varphi_a(t;z_0)$. Next, by the coercivity assumption \\hyperlink{W4}{$(W_4)$}, the sublevel set\n$$\n\\mathcal{S}_{B} \\coloneqq \\{ x \\in \\mathbb{R}^{N}:W(x)\\le R_B\\}\n$$\nis bounded. Consequently, we may define the positive real number\n$$\nC_B \\coloneqq \\sup \\left\\{\\|x\\|^{2} : x \\in \\mathcal{S}_{B}\\right\\} < \\infty.\n$$\nIn this way, we have \n$$\n\\Vert T(t)z_0 \\Vert_{\\ast}^2 = \\Vert (x_a(t), y_a(t))\\Vert_{\\ast}^2 = \\Vert x_a(t) \\Vert^2 + \\Vert y_a(t) \\Vert^2 \\leq C_B + 2R_B,\\quad \\forall\\, t\\ge 0, \n$$\nNow, defining\n$$\nK\\coloneqq\\Big\\{(x,y)\\in\\mathbb{R}^{2N}:\\|(x,y)\\|_{\\ast}^{2}\\le \\sqrt{C_B + 2R_B}\\Big\\}\n$$\nit follows the inclusion $T(t)B \\subset K$ for all $t\\geq 0$, which finishes our proof.\n\\end{proof}", "post_theorem_intro_text_len": 6953, "post_theorem_intro_text": "The Theorem \\ref{Teo1} asserts that the equilibrium $(u_*,0)$ of the system \\eqref{Equation} is uniformly asymptotically stable with respect to the parameter $a\\in(0,a_0]$: every solution starting sufficiently close to $(u_*,0)$ remains in an arbitrarily small neighborhood of the equilibrium and converges to it as $t\\to +\\infty$, with bounds that are uniform in $a$. The fundamental point of our study is the construction of a strict Lyapunov functional for the first-order formulation of the dissipative system\n\\begin{equation}\\label{S-intro}\n\t\\begin{cases}\n\t\t\\dot{x}=y,\\\\[1mm]\n\t\t\\dot{y}=-a\\,y-\\nabla W(x),\n\t\\end{cases}\n\\end{equation}\nwhich encodes the full dynamics of \\eqref{Equation} in the phase space $\\mathbb{R}^{2N}$. To this end, we introduce the Lyapunov functional\n\\begin{equation}\n\tV_a(x,y):=\\frac{1}{2}\\|y\\|^2+2W(x)+\\frac{1}{2}\\|y+a(x-u_*)\\|^2,\n\\end{equation}\nwhich can be interpreted as a smooth perturbation of the mechanical energy\n$$\nE(x,y):=\\frac12\\|y\\|^2+W(x).\n$$\nThe functional $V_a$ allows us to obtain stability estimates that are uniform with respect to $a$. The assumptions \\hyperlink{W1}{($W_1$)}-\\hyperlink{W4}{($W_4$)} play a crucial role in ensuring that $V_a$ is strictly decreasing along non-trivial trajectories. The condition \\hyperlink{W5}{($W_5$)} is only required to derive exponential decay of solutions, this decay cannot be uniform as $a\\to 0^{+}$, since the corresponding rate $\\gamma(a)$ necessarily satisfies $\\gamma(a)\\to 0$. Furthermore, the introduction of the Lyapunov functional $V_a$ therefore provides a robust framework for establishing the uniform asymptotic stability of \\eqref{Equation}. Furthermore, we believe that $V_a$ highlights a mechanism that can be extended to non-homogeneous cases, to time-dampened systems, and to more general configurations such as trajectories over Hilbert space.\n\nWe emphasize that the conclusions of Theorem \\ref{Teo1} are not restricted to a specific equilibrium. In fact, the result applies to any global minimizer $u_*$ of the potential $W$ satisfying assumptions \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$}, yielding uniform asymptotic stability of the corresponding equilibrium $(u_*,0)$. Moreover, when the additional condition \\hyperlink{W5}{$(W_5)$} holds, the convergence toward $(u_*,0)$ is exponential. In this sense, Theorem \\ref{Teo1} provides a robust stability result around arbitrary admissible minima of the potential.\n\nWhen $a=0$, the equation \\eqref{Equation} reduces to the conservative system\n$$\n\\ddot u(t) + \\nabla W(u(t))=0,\n$$\ncorresponding to the motion of a particle on $W$ without friction, so that mechanical energy is preserved. Here, $u(t)$ is the position of the particle, $\\ddot{u}(t)$ its acceleration, and $\\nabla W(u)$ the force derived from $W$. Under assumptions \\hyperlink{W1}{($W_1$)}-\\hyperlink{W4}{($W_4$)}, and in particular the coercivity of $W$, every solution remains globally bounded in $\\mathbb{R}^{N}$, that is, \n$$\n\\sup_{t\\in\\mathbb{R}}\\left(\\|u(t)\\|+\\|\\dot u(t)\\|\\right)<\\infty.\n$$\nThus blow-up cannot occur and no trajectory can converge asymptotically to the equilibrium. In this Hamiltonian-type regime, the combination of confinement and energy conservation leads to the typical behaviors of conservative dynamics: periodic orbits and transition solutions (heteroclinic or homoclinic). Hence, the case $a=0$ produces a qualitative change in the dynamics, yielding globally bounded but non-convergent trajectories.\n\nIn addition to the study of the asymptotic stability of \\eqref{Equation}, two additional components complement our study. In Section \\ref{Sec4}, we revisit the dynamics from the point of view of nonlinear semigroup theory, aiming for a comprehensive qualitative description of the dissipative flux generated by the first-order system \\eqref{Equation}. This approach shows, in particular, the existence of a global attractor $\\mathcal{A}$ for the corresponding semigroup, which is a compact, invariant subset of the phase space that attracts all bounded sets in forward time. Moreover, $\\mathcal{A}$ contains the entire set of equilibria of the system, that is, \n$$\\{(x,0)\\in\\mathbb{R}^{2N}:\\nabla W (x)=0\\}\\subset \\mathcal{A}.\n$$ \nFor readers interested in the general theory of global attractors and nonlinear semigroups, we refer to the classical monographs \\cite{Livro-Alexandre,Hale,Temam}. Finally, Section \\ref{Sec5} presents numerical simulations for some cases of \\eqref{Equation} involving quadratic, exponential, and double-well potentials implemented in a computational environment based on \\emph{R}. The simulations rely on a functional-programming framework supported by \\texttt{tidyverse} (data manipulation), \\texttt{deSolve} (time-integration of differential equations), and \\texttt{ggplot2} (phase-plane visualization and graphical output). This computational component has a dual purpose: to geometrically illustrate the results developed in the previous sections and to reveal characteristics of the dynamics that remain inaccessible to our methods. For instance, when the potential $W$ possesses multiple global minima, such as $W(u)=(u^2-1)^2/4$, numerical evidence suggests that local maxima separating successive minima act as unstable equilibria. In the conservative regime $a=0$, simulations indicate that trajectories in the phase plane $\\mathbb{R}^{2N}$ form closed periodic orbits. Furthermore, when $W(u)=u^2/2$, equation \\eqref{Equation} is linear and exhibits the classical critical damping threshold at $a=2$. Our numerical experiments indicate that the same phenomenon persists in nonlinear settings: for instance, when $W(u)=(e^{u^{2}}-1)/2$, simulations suggest that $a=2$ again separates oscillatory (underdamped) dynamics from non-oscillatory (overdamped) decay. This numerical evidence points to a broader conjectural picture: nonlinear potentials with a single global minimizer may still inherit a critical damping regime analogous to the linear case. We have included complete executable \\emph{R} scripts at the end of Section \\ref{Sec5}. The reader can inspect, modify, or compile them to visualize trajectories, replicate phase plane geometry, and experiment with potentials or additional parameter regimes. Therefore, numerical simulation serves as a gateway to more in-depth mathematical and computational investigations.\n\nThe plan of the paper is as follows. Section \\ref{Sec2} contains the proof of Theorem \\ref{Teo1}, based on the construction of a strict Lyapunov functional adapted to the dissipative system. In Section \\ref{Sec3} we briefly discuss the conservative case $a=0$ and the singular nature of the limit as $a\\to 0^{+}$. Section \\ref{Sec4} examines the global dynamics of nonlinear semigroup theory, providing a complementary viewpoint to the Lyapunov analysis of Section \\ref{Sec2}. Finally, Section \\ref{Sec5} presents numerical simulations for representative potentials, including quadratic, exponential, and double-well examples.", "sketch": "To prove Theorem~\\ref{Teo1} the key idea is the “construction of a strict Lyapunov functional for the first-order formulation” of \\eqref{Equation} written as the phase-space system \\eqref{S-intro}. The argument introduces\n\\[\nV_a(x,y):=\\tfrac12\\|y\\|^2+2W(x)+\\tfrac12\\|y+a(x-u_*)\\|^2,\n\\]\nviewed as a “smooth perturbation of the mechanical energy” \\(E(x,y)=\\frac12\\|y\\|^2+W(x)\\). The functional \\(V_a\\) is used to obtain “stability estimates that are uniform with respect to \\(a\\)”. The assumptions \\hyperlink{W1}{\\((W_1)\\)}–\\hyperlink{W4}{\\((W_4)\\)} are used to ensure that \\(V_a\\) is “strictly decreasing along non-trivial trajectories,” which yields uniform asymptotic stability (solutions starting sufficiently close remain close and converge as \\(t\\to+\\infty\\)). Under the additional condition \\hyperlink{W5}{\\((W_5)\\)}, the same Lyapunov framework is used to “derive exponential decay of solutions,” with a non-uniform rate as \\(a\\to0^+\\) (the text notes “\\(\\gamma(a)\\to0\\)”).", "expanded_sketch": "To prove the main theorem, the key idea is the “construction of a strict Lyapunov functional for the first-order formulation” of\n\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\nwritten as the phase-space system \\eqref{S-intro}. The argument introduces\n\\[\nV_a(x,y):=\\tfrac12\\|y\\|^2+2W(x)+\\tfrac12\\|y+a(x-u_*)\\|^2,\n\\]\nviewed as a “smooth perturbation of the mechanical energy” \\(E(x,y)=\\frac12\\|y\\|^2+W(x)\\). The functional \\(V_a\\) is used to obtain “stability estimates that are uniform with respect to \\(a\\)”. The assumptions \\hyperlink{W1}{\\((W_1)\\)}–\\hyperlink{W4}{\\((W_4)\\)} are used to ensure that \\(V_a\\) is “strictly decreasing along non-trivial trajectories,” which yields uniform asymptotic stability (solutions starting sufficiently close remain close and converge as \\(t\\to+\\infty\\)). Under the additional condition \\hyperlink{W5}{\\((W_5)\\)}, the same Lyapunov framework is used to “derive exponential decay of solutions,” with a non-uniform rate as \\(a\\to0^+\\) (the text notes “\\(\\gamma(a)\\to0\\)”).", "expanded_theorem": "\\label{Teo1}\n\tAssume that the potential $W$ satisfies \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$}. Then, for every $\\varepsilon > 0$ and $a_0 > 0$, there exists $\\delta > 0$ such that for all $a \\in (0, a_0]$, and any solution $u_a$ of\n\t\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\n\twith initial data satisfying\n\t$$\n\t\\|u_a(0)-u_*\\| + \\|\\dot{u}_a(0)\\| < \\delta,\n\t$$\n\tthe following hold:\n\t\\begin{itemize}\n\t\t\\item[\\emph{(a)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le \\varepsilon$;\n\t\t\\item[\\emph{(b)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tIf, additionally, \\hyperlink{W2}{$(W_2)$} and \\hyperlink{W3}{$(W_3)$} hold with $\\delta = \\lambda = +\\infty$, then there exists $R > 0$ such that, for all $a \\in (0, a_0]$, and any solution $u_a$ of the equation above,\n\t\\begin{itemize}\n\t\t\\item[\\emph{(c)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le R$;\n\t\t\\item[\\emph{(d)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and  $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tMoreover, under the additional assumption \\hyperlink{W5}{$(W_5)$}, for each $a >0$ there exist $\\gamma(a) > 0$ and $C(a)>0$ such that the corresponding solution $u_a$ satisfies the exponential decay estimate\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C(a) e^{-\\gamma(a) t}, \\quad \\forall\\, t \\ge 0.\n\t\\end{equation*}\n\tIn particular, if $0<a_1<a_2$ and $a\\in[a_1,a_2]$, then there exist $C_*>0$ and $\\gamma_*>0$, depending only on $a_1,a_2$ such that\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C_* e^{-\\gamma_* t},\\quad \\forall\\, t \\ge 0,\\ \\forall\\, a\\in[a_1,a_2].\n\t\\end{equation*}", "theorem_type": ["Universal–Existential", "Asymptotic or Limit"], "mcq": {"question": "Let $u_a:\\mathbb{R}\\to\\mathbb{R}^N$ solve the damped second-order system\n$$\n\\ddot u(t)+a\\dot u(t)+\\nabla W(u(t))=0,\n$$\nwhere $a>0$ and $W:\\mathbb{R}^N\\to\\mathbb{R}$ is a potential. Assume $W\\in C^2(\\mathbb{R}^N)$, there exist $u_*\\in\\mathbb{R}^N$ and radii $\\delta,\\lambda>0$ such that $W(u_*)=0$, $W(x)>0$ for every $x\\in B_\\delta(u_*)\\setminus\\{u_*\\}$, $\\langle \\nabla W(x),x-u_*\\rangle>0$ for every $x\\in B_\\lambda(u_*)\\setminus\\{u_*\\}$, and $W$ is coercive. In addition, when needed, assume the stronger local estimate\n$$\n\\alpha\\|x-u_*\\|^2\\le W(x)\\le \\beta\\|x-u_*\\|^2,\n\\qquad\n\\langle \\nabla W(x),x-u_*\\rangle\\ge \\mu\\|x-u_*\\|^2\n$$\nfor all $x\\in B_\\lambda(u_*)$, for some $\\alpha,\\beta,\\mu>0$. Which statement holds about the stability and decay of solutions?", "correct_choice": {"label": "A", "text": "For every $\\varepsilon>0$ and $a_0>0$, there exists $\\rho>0$ such that for every $a\\in(0,a_0]$, any solution $u_a$ with\n$$\n\\|u_a(0)-u_*\\|+\\|\\dot u_a(0)\\|<\\rho\n$$\nsatisfies\n$$\n\\sup_{t\\ge0}\\bigl(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\bigr)\\le \\varepsilon,\n$$\nand\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nIf, moreover, the positivity condition $W(x)>0$ for $x\\neq u_*$ and the directional monotonicity condition $\\langle \\nabla W(x),x-u_*\\rangle>0$ for $x\\neq u_*$ hold globally (that is, with $\\delta=\\lambda=+\\infty$), then there exists $R>0$ such that for every $a\\in(0,a_0]$ and every solution $u_a$,\n$$\n\\sup_{t\\ge0}\\bigl(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\bigr)\\le R,\n$$\nand again\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nMoreover, under the additional quadratic/monotonicity assumption above, for each $a>0$ there exist constants $\\gamma(a)>0$ and $C(a)>0$ such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C(a)e^{-\\gamma(a)t}\\qquad\\forall t\\ge0.\n$$\nIn particular, if $0<a_1<a_2$ and $a\\in[a_1,a_2]$, then there exist constants $C_*>0$ and $\\gamma_*>0$, depending only on $a_1,a_2$, such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C_*e^{-\\gamma_* t}\\qquad\\forall t\\ge0,\\ \\forall a\\in[a_1,a_2].\n$$"}, "choices": [{"label": "B", "text": "For every $\\varepsilon>0$ and $a_0>0$, there exists $\\rho>0$ such that for every $a\\in(0,a_0]$, any solution $u_a$ with\n$$\n\\|u_a(0)-u_*\\|+\\|\\dot u_a(0)\\|<\\rho\n$$\nsatisfies\n$$\n\\sup_{t\\ge0}\\bigl(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\bigr)\\le \\varepsilon,\n$$\nand\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nIf, moreover, the positivity condition $W(x)>0$ for $x\\neq u_*$ and the directional monotonicity condition $\\langle \\nabla W(x),x-u_*\\rangle>0$ for $x\\neq u_*$ hold globally, then there exist constants $R>0$, $\\gamma_*>0$, and $C_*>0$, depending only on $a_0$, such that for every $a\\in(0,a_0]$ and every solution $u_a$,\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C_*e^{-\\gamma_* t}\\qquad\\forall t\\ge0.\n$$"}, {"label": "C", "text": "For every $\\varepsilon>0$ and $a_0>0$, there exists $\\rho>0$ such that for every $a\\in(0,a_0]$, any solution $u_a$ with\n$$\n\\|u_a(0)-u_*\\|+\\|\\dot u_a(0)\\|<\\rho\n$$\nsatisfies\n$$\n\\sup_{t\\ge0}\\bigl(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\bigr)\\le \\varepsilon,\n$$\nand\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nMoreover, under the additional quadratic/monotonicity assumption above, for each fixed $a>0$ there exist constants $\\gamma(a)>0$ and $C(a)>0$ such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C(a)e^{-\\gamma(a)t}\\qquad\\forall t\\ge0.\n$$"}, {"label": "D", "text": "For every $\\varepsilon>0$ there exists $\\rho>0$ such that for every $a>0$, any solution $u_a$ with\n$$\n\\|u_a(0)-u_*\\|+\\|\\dot u_a(0)\\|<\\rho\n$$\nsatisfies\n$$\n\\sup_{t\\ge0}\\bigl(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\bigr)\\le \\varepsilon,\n$$\nand\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nMoreover, under the additional quadratic/monotonicity assumption above, there exist constants $\\gamma_*>0$ and $C_*>0$, independent of $a>0$, such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C_*e^{-\\gamma_* t}\\qquad\\forall t\\ge0,\\ \\forall a>0.\n$$"}, {"label": "E", "text": "For every $\\varepsilon>0$ and $a_0>0$, there exists $\\rho>0$ such that for every $a\\in(0,a_0]$, any solution $u_a$ with\n$$\n\\|u_a(0)-u_*\\|+\\|\\dot u_a(0)\\|<\\rho\n$$\nsatisfies\n$$\n\\sup_{t\\ge0}\\bigl(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\bigr)\\le \\varepsilon.\n$$\nIf, moreover, the positivity condition $W(x)>0$ for $x\\neq u_*$ and the directional monotonicity condition $\\langle \\nabla W(x),x-u_*\\rangle>0$ for $x\\neq u_*$ hold globally, then there exists $R>0$ such that for every $a\\in(0,a_0]$ and every solution $u_a$,\n$$\n\\sup_{t\\ge0}\\bigl(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\bigr)\\le R,\n$$\nand every bounded trajectory converges to some equilibrium point of $W$ as $t\\to+\\infty$.\n$$\n$$\nMoreover, under the additional quadratic/monotonicity assumption above, for each $a>0$ there exist constants $\\gamma(a)>0$ and $C(a)>0$ such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C(a)e^{-\\gamma(a)t}\\qquad\\forall t\\ge0.\n$$"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "extra global exponential estimate without (W5)", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped the global-positivity/global-monotonicity conclusion and the uniform-in-$[a_1,a_2]$ exponential constants", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "uniformity in $a$ down to $a=0$ for stability radius and exponential rate", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "limit identified only as some equilibrium rather than the distinguished $u_*$ under global assumptions", "template_used": "wildcard"}]}}
{"id": "2512.20751v1", "paper_link": "http://arxiv.org/abs/2512.20751v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{Teo1}\n\tAssume that the potential $W$ satisfies \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$}. Then, for every $\\varepsilon > 0$ and $a_0 > 0$, there exists $\\delta > 0$ such that for all $a \\in (0, a_0]$, and any solution $u_a$ of \\eqref{Equation} with initial data satisfying\n\t$$\n\t\\|u_a(0)-u_*\\| + \\|\\dot{u}_a(0)\\| < \\delta,\n\t$$\n\tthe following hold:\n\t\\begin{itemize}\n\t\t\\item[\\emph{(a)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le \\varepsilon$;\n\t\t\\item[\\emph{(b)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tIf, additionally, \\hyperlink{W2}{$(W_2)$} and \\hyperlink{W3}{$(W_3)$} hold with $\\delta = \\lambda = +\\infty$, then there exists $R > 0$ such that, for all $a \\in (0, a_0]$, and any solution $u_a$ of \\eqref{Equation},\n\t\\begin{itemize}\n\t\t\\item[\\emph{(c)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le R$;\n\t\t\\item[\\emph{(d)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and  $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tMoreover, under the additional assumption \\hyperlink{W5}{$(W_5)$}, for each $a >0$ there exist $\\gamma(a) > 0$ and $C(a)>0$ such that the corresponding solution $u_a$ satisfies the exponential decay estimate\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C(a) e^{-\\gamma(a) t}, \\quad \\forall\\, t \\ge 0.\n\t\\end{equation*}\n\tIn particular, if $0<a_1<a_2$ and $a\\in[a_1,a_2]$, then there exist $C_*>0$ and $\\gamma_*>0$, depending only on $a_1,a_2$ such that\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C_* e^{-\\gamma_* t},\\quad \\forall\\, t \\ge 0,\\ \\forall\\, a\\in[a_1,a_2].\n\t\\end{equation*}", "start_pos": 1552652, "end_pos": 1554414, "label": "Teo1"}, "ref_dict": {"Equation": "\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}", "Teo1": "\\begin{theorem}\\label{Teo1}\n\tAssume that the potential $W$ satisfies \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$}. Then, for every $\\varepsilon > 0$ and $a_0 > 0$, there exists $\\delta > 0$ such that for all $a \\in (0, a_0]$, and any solution $u_a$ of \\eqref{Equation} with initial data satisfying\n\t$$\n\t\\|u_a(0)-u_*\\| + \\|\\dot{u}_a(0)\\| < \\delta,\n\t$$\n\tthe following hold:\n\t\\begin{itemize}\n\t\t\\item[\\emph{(a)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le \\varepsilon$;\n\t\t\\item[\\emph{(b)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tIf, additionally, \\hyperlink{W2}{$(W_2)$} and \\hyperlink{W3}{$(W_3)$} hold with $\\delta = \\lambda = +\\infty$, then there exists $R > 0$ such that, for all $a \\in (0, a_0]$, and any solution $u_a$ of \\eqref{Equation},\n\t\\begin{itemize}\n\t\t\\item[\\emph{(c)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le R$;\n\t\t\\item[\\emph{(d)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and  $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tMoreover, under the additional assumption \\hyperlink{W5}{$(W_5)$}, for each $a >0$ there exist $\\gamma(a) > 0$ and $C(a)>0$ such that the corresponding solution $u_a$ satisfies the exponential decay estimate\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C(a) e^{-\\gamma(a) t}, \\quad \\forall\\, t \\ge 0.\n\t\\end{equation*}\n\tIn particular, if $0<a_1<a_2$ and $a\\in[a_1,a_2]$, then there exist $C_*>0$ and $\\gamma_*>0$, depending only on $a_1,a_2$ such that\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C_* e^{-\\gamma_* t},\\quad \\forall\\, t \\ge 0,\\ \\forall\\, a\\in[a_1,a_2].\n\t\\end{equation*}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 6996, "pre_theorem_intro_text": "Second-order dissipative systems of the form\n\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\narise naturally in mechanics, optimization, and nonlinear dynamics. A classical example is the \\emph{heavy-ball method with friction}, in which a particle moves on the graph of a potential $W$ under the action of inertia and a linear damping force with coefficient $a>0$. The heavy-ball method was introduced by Polyak in 1964 to speed up the convergence of iterative schemes \\cite{Polyak}, and was formalized by Attouch, Goudou, and Redont \\cite{AttouchGoudouRedont}, who analyzed the frictional equation to explore the minima of potential $W$. Beyond heavy-ball method with friction, the prototype equation \\eqref{Equation} also models the motion of a \\emph{damped simple pendulum}.  \nLet $u=\\theta$ be the pendulum angle, take $W(\\theta)=\\frac{g}{\\ell}(1-\\cos\\theta)$, then\n$$\n\\ddot{\\theta}(t)+ a\\,\\dot{\\theta}(t)+\\dfrac{g}{\\ell}\\sin\\theta(t)=0,\n$$\nwhich is the standard damped pendulum equation. This modeling is classical in nonlinear dynamics and the term $a\\theta'$ represents the damping force, with $a > 0$. Depending on the initial energy, this system exhibits oscillatory motions, convergence to stable equilibria, and heteroclinic trajectories connecting different critical points of $W$. A comprehensive analysis of the pendulum dynamics can be found in \\cite{Mawhin}. In fact, beyond these classical examples, a number of other mechanical and computational systems reduce to the general dissipative structure \\eqref{Equation}, which reinforces the broad applicability and theoretical interest of the problem.\n\nSecond-order equations with constant damping also appear in the study of generalized gradient systems. Recent works have achieved significant progress in the asymptotic analysis of such systems. Jendoubi and collaborators showed that an equation of the type \\eqref{Equation} can be interpreted as a quasi-gradient system via small deformations of the total energy. They proved that the desingularizing function in the Kurdyka-Łojasiewicz inequality satisfies $\\varphi(t) \\ge c\\sqrt{t}$ when the potential $W$ is definable and of class $C^2$ \\cite{Jendoubi1}. This result implies that every trajectory of a quasi-gradient system either converges to a critical point or its norm diverges to infinity. Another relevant development concerns Newton-type systems with {\\it Hessian-driven damping}, combining the continuous Newton method with the heavy-ball system. Boţ and Csetnek in \\cite{BotCsetnek} observed that such systems are second-order in both time and space, due to the presence of the acceleration term and the Hessian, and can be seen as a mixture of Newton’s method with the heavy-ball system. This hybrid approach inherits advantages from both methods, as shown in the optimization context by Alvarez et al. \\cite{Alvarez}. We also mention that several recent works have focused on nonhomogeneous variants of \\eqref{Equation} and on models where the damping parameter depends on time, such as \\cite{1,2,3,4,6,7}.\n\nThe term $a\\dot{u}$ in \\eqref{Equation} represents a damping force, while $\\nabla W(u)$ acts as a restoring force derived from the potential energy $W(u)$. In this work we focus on the second-order dissipative systems \\eqref{Equation}, where $a > 0$ is a damping coefficient, $u\\colon \\mathbb{R} \\to \\mathbb{R}^N$ is the unknown trajectory, and $W\\colon \\mathbb{R}^N \\to \\mathbb{R}$ is a potential. Throughout the paper we impose the following assumptions on $W$:\n\\begin{itemize} \n\t\\item[\\hypertarget{W1}{($W_1$)}] $W\\in C^2(\\mathbb{R}^N; \\mathbb{R})$.\n\n\t\\item[\\hypertarget{W2}{($W_2$)}] There exists $\\delta > 0$ and $u_* \\in \\mathbb{R}^{N}$ such that\n\t$$\n\tW(u_*) = 0 \\quad \\text{and} \\quad W(u) > 0 \\quad \\text{for all} \\quad u \\in B_\\delta(u_*) \\setminus \\{u_*\\}.\n\t$$\n\n\t\\item[\\hypertarget{W3}{($W_3$)}] There exists $\\lambda > 0$ such that\n\t$$\n\t\\langle \\nabla W(u), u - u_* \\rangle > 0 \\quad \\text{for all} \\quad u \\in B_\\lambda(u_*) \\setminus \\{u_*\\}.\n\t$$\n\n\t\\item[\\hypertarget{W4}{($W_4$)}] $W$ is coercive, that is,\n\t$$\n\t\\lim_{\\|u\\| \\to +\\infty} W(u) = +\\infty.\n\t$$\n\n\t\\item[\\hypertarget{W5}{($W_5$)}] There exist constants $\\alpha, \\beta, \\mu > 0$ such that, for every $x \\in B_\\lambda(u_*)$,\n\t$$\n\t\\alpha\\|x - u_*\\|^2 \\le W(x) \\le \\beta\\|x - u_*\\|^2 \\quad \\text{and} \\quad \\langle \\nabla W(x), x - u_* \\rangle \\ge \\mu\\|x - u_*\\|^2.\n\t$$\n\\end{itemize}\n\nA simple example satisfying \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_5)$} with $u_*=0$ is the harmonic potential\n$$\nW(u)=\\frac{1}{2}\\|u\\|^2.\n$$\nIn this case, \\eqref{Equation} reduces to the classical damped harmonic oscillator. This model describes a {\\it mass-spring-damper system}, where $a$ represents the viscous damping and $u$ is the Hookean restoring force. Solutions exhibit decaying oscillations and converge to the equilibrium $u=0$. Such systems appear  not only in mechanics but also in electrical circuits and other dissipative models driven by an energy potential, see \\cite{Edwards,Sunday1,Sunday2}. Beyond the quadratic case, the class of potentials satisfying assumptions \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$} is broad and includes many models from mechanics, nonlinear dynamics, and phase-transition theory. A general and important family is obtained by taking  \n$$\nW(u)=\\Phi(u-u_*),\n$$\nwhere $\\Phi\\colon\\mathbb{R}^N\\to\\mathbb{R}$ is a convex, positive function with a global minimum at the origin. Such potentials arise naturally in the study of quasilinear Allen-Cahn type equations, see, for instance, \\cite{Isneri}. In particular, coercive polynomial potentials, such as $W(u)=\\|u-u_*\\|^p$, with $p\\ge 2$, and the exponential potential  \n$$\nW(u)=\\frac12\\big(e^{\\|u-u_*\\|^2}-1\\big),\n$$\nwhich is convex and coercive, fits perfectly into this model. Another classical example is the Ginzburg-Landau potential  \n$$\nW(u)=\\frac14\\left(\\|u\\|^2-1\\right)^2,\n$$\nwhose set of global minima is the unit sphere $\\{u\\in\\mathbb{R}^N : \\|u\\| = 1\\}$. This potential play a central role in the theory of phase transitions and superconductivity, see \\cite{Allen,5}. A further relevant example, frequently appearing in the study of dissipative hyperbolic PDEs, is the quartic double-well potential  \n$$\nW(u)=c_1\\|u\\|^{4}-c_2\\|u\\|^{2}+c_3,\\quad c_1,c_2,c_3>0,\n$$\nwhose structure is extensively used in nonlinear wave equations with damping. Such models constitute a natural infinite-dimensional counterpart of \\eqref{Equation}, see, for instance, \\cite{8}. These examples illustrate that the hypotheses imposed on $W$ encompass both linear and nonlinear restoring forces, as in the damped mass-spring system, as well as potentials arising in gradient flows, optimization, and models of phase separation.\n\nWith these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \\eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.", "context": "Second-order dissipative systems of the form\n\\begin{equation}\\label{Equation}\n \\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\narise naturally in mechanics, optimization, and nonlinear dynamics. A classical example is the \\emph{heavy-ball method with friction}, in which a particle moves on the graph of a potential $W$ under the action of inertia and a linear damping force with coefficient $a>0$. The heavy-ball method was introduced by Polyak in 1964 to speed up the convergence of iterative schemes \\cite{Polyak}, and was formalized by Attouch, Goudou, and Redont \\cite{AttouchGoudouRedont}, who analyzed the frictional equation to explore the minima of potential $W$. Beyond heavy-ball method with friction, the prototype equation \\eqref{Equation} also models the motion of a \\emph{damped simple pendulum}. \nLet $u=\\theta$ be the pendulum angle, take $W(\\theta)=\\frac{g}{\\ell}(1-\\cos\\theta)$, then\n$$\n\\ddot{\\theta}(t)+ a\\,\\dot{\\theta}(t)+\\dfrac{g}{\\ell}\\sin\\theta(t)=0,\n$$\nwhich is the standard damped pendulum equation. This modeling is classical in nonlinear dynamics and the term $a\\theta'$ represents the damping force, with $a > 0$. Depending on the initial energy, this system exhibits oscillatory motions, convergence to stable equilibria, and heteroclinic trajectories connecting different critical points of $W$. A comprehensive analysis of the pendulum dynamics can be found in \\cite{Mawhin}. In fact, beyond these classical examples, a number of other mechanical and computational systems reduce to the general dissipative structure \\eqref{Equation}, which reinforces the broad applicability and theoretical interest of the problem.\n\nThe term $a\\dot{u}$ in \\eqref{Equation} represents a damping force, while $\\nabla W(u)$ acts as a restoring force derived from the potential energy $W(u)$. In this work we focus on the second-order dissipative systems \\eqref{Equation}, where $a > 0$ is a damping coefficient, $u\\colon \\mathbb{R} \\to \\mathbb{R}^N$ is the unknown trajectory, and $W\\colon \\mathbb{R}^N \\to \\mathbb{R}$ is a potential. Throughout the paper we impose the following assumptions on $W$:\n\\begin{itemize} \n \\item[\\hypertarget{W1}{($W_1$)}] $W\\in C^2(\\mathbb{R}^N; \\mathbb{R})$.\n\n\\item[\\hypertarget{W2}{($W_2$)}] There exists $\\delta > 0$ and $u_* \\in \\mathbb{R}^{N}$ such that\n $$\n W(u_*) = 0 \\quad \\text{and} \\quad W(u) > 0 \\quad \\text{for all} \\quad u \\in B_\\delta(u_*) \\setminus \\{u_*\\}.\n $$\n\n\\item[\\hypertarget{W5}{($W_5$)}] There exist constants $\\alpha, \\beta, \\mu > 0$ such that, for every $x \\in B_\\lambda(u_*)$,\n $$\n \\alpha\\|x - u_*\\|^2 \\le W(x) \\le \\beta\\|x - u_*\\|^2 \\quad \\text{and} \\quad \\langle \\nabla W(x), x - u_* \\rangle \\ge \\mu\\|x - u_*\\|^2.\n $$\n\\end{itemize}\n\nA simple example satisfying \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_5)$} with $u_*=0$ is the harmonic potential\n$$\nW(u)=\\frac{1}{2}\\|u\\|^2.\n$$\nIn this case, \\eqref{Equation} reduces to the classical damped harmonic oscillator. This model describes a {\\it mass-spring-damper system}, where $a$ represents the viscous damping and $u$ is the Hookean restoring force. Solutions exhibit decaying oscillations and converge to the equilibrium $u=0$. Such systems appear not only in mechanics but also in electrical circuits and other dissipative models driven by an energy potential, see \\cite{Edwards,Sunday1,Sunday2}. Beyond the quadratic case, the class of potentials satisfying assumptions \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$} is broad and includes many models from mechanics, nonlinear dynamics, and phase-transition theory. A general and important family is obtained by taking \n$$\nW(u)=\\Phi(u-u_*),\n$$\nwhere $\\Phi\\colon\\mathbb{R}^N\\to\\mathbb{R}$ is a convex, positive function with a global minimum at the origin. Such potentials arise naturally in the study of quasilinear Allen-Cahn type equations, see, for instance, \\cite{Isneri}. In particular, coercive polynomial potentials, such as $W(u)=\\|u-u_*\\|^p$, with $p\\ge 2$, and the exponential potential \n$$\nW(u)=\\frac12\\big(e^{\\|u-u_*\\|^2}-1\\big),\n$$\nwhich is convex and coercive, fits perfectly into this model. Another classical example is the Ginzburg-Landau potential \n$$\nW(u)=\\frac14\\left(\\|u\\|^2-1\\right)^2,\n$$\nwhose set of global minima is the unit sphere $\\{u\\in\\mathbb{R}^N : \\|u\\| = 1\\}$. This potential play a central role in the theory of phase transitions and superconductivity, see \\cite{Allen,5}. A further relevant example, frequently appearing in the study of dissipative hyperbolic PDEs, is the quartic double-well potential \n$$\nW(u)=c_1\\|u\\|^{4}-c_2\\|u\\|^{2}+c_3,\\quad c_1,c_2,c_3>0,\n$$\nwhose structure is extensively used in nonlinear wave equations with damping. Such models constitute a natural infinite-dimensional counterpart of \\eqref{Equation}, see, for instance, \\cite{8}. These examples illustrate that the hypotheses imposed on $W$ encompass both linear and nonlinear restoring forces, as in the damped mass-spring system, as well as potentials arising in gradient flows, optimization, and models of phase separation.\n\nWith these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \\eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.\n\n\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}", "full_context": "Second-order dissipative systems of the form\n\\begin{equation}\\label{Equation}\n \\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\narise naturally in mechanics, optimization, and nonlinear dynamics. A classical example is the \\emph{heavy-ball method with friction}, in which a particle moves on the graph of a potential $W$ under the action of inertia and a linear damping force with coefficient $a>0$. The heavy-ball method was introduced by Polyak in 1964 to speed up the convergence of iterative schemes \\cite{Polyak}, and was formalized by Attouch, Goudou, and Redont \\cite{AttouchGoudouRedont}, who analyzed the frictional equation to explore the minima of potential $W$. Beyond heavy-ball method with friction, the prototype equation \\eqref{Equation} also models the motion of a \\emph{damped simple pendulum}. \nLet $u=\\theta$ be the pendulum angle, take $W(\\theta)=\\frac{g}{\\ell}(1-\\cos\\theta)$, then\n$$\n\\ddot{\\theta}(t)+ a\\,\\dot{\\theta}(t)+\\dfrac{g}{\\ell}\\sin\\theta(t)=0,\n$$\nwhich is the standard damped pendulum equation. This modeling is classical in nonlinear dynamics and the term $a\\theta'$ represents the damping force, with $a > 0$. Depending on the initial energy, this system exhibits oscillatory motions, convergence to stable equilibria, and heteroclinic trajectories connecting different critical points of $W$. A comprehensive analysis of the pendulum dynamics can be found in \\cite{Mawhin}. In fact, beyond these classical examples, a number of other mechanical and computational systems reduce to the general dissipative structure \\eqref{Equation}, which reinforces the broad applicability and theoretical interest of the problem.\n\nThe term $a\\dot{u}$ in \\eqref{Equation} represents a damping force, while $\\nabla W(u)$ acts as a restoring force derived from the potential energy $W(u)$. In this work we focus on the second-order dissipative systems \\eqref{Equation}, where $a > 0$ is a damping coefficient, $u\\colon \\mathbb{R} \\to \\mathbb{R}^N$ is the unknown trajectory, and $W\\colon \\mathbb{R}^N \\to \\mathbb{R}$ is a potential. Throughout the paper we impose the following assumptions on $W$:\n\\begin{itemize} \n \\item[\\hypertarget{W1}{($W_1$)}] $W\\in C^2(\\mathbb{R}^N; \\mathbb{R})$.\n\n\\item[\\hypertarget{W2}{($W_2$)}] There exists $\\delta > 0$ and $u_* \\in \\mathbb{R}^{N}$ such that\n $$\n W(u_*) = 0 \\quad \\text{and} \\quad W(u) > 0 \\quad \\text{for all} \\quad u \\in B_\\delta(u_*) \\setminus \\{u_*\\}.\n $$\n\n\\item[\\hypertarget{W5}{($W_5$)}] There exist constants $\\alpha, \\beta, \\mu > 0$ such that, for every $x \\in B_\\lambda(u_*)$,\n $$\n \\alpha\\|x - u_*\\|^2 \\le W(x) \\le \\beta\\|x - u_*\\|^2 \\quad \\text{and} \\quad \\langle \\nabla W(x), x - u_* \\rangle \\ge \\mu\\|x - u_*\\|^2.\n $$\n\\end{itemize}\n\nA simple example satisfying \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_5)$} with $u_*=0$ is the harmonic potential\n$$\nW(u)=\\frac{1}{2}\\|u\\|^2.\n$$\nIn this case, \\eqref{Equation} reduces to the classical damped harmonic oscillator. This model describes a {\\it mass-spring-damper system}, where $a$ represents the viscous damping and $u$ is the Hookean restoring force. Solutions exhibit decaying oscillations and converge to the equilibrium $u=0$. Such systems appear not only in mechanics but also in electrical circuits and other dissipative models driven by an energy potential, see \\cite{Edwards,Sunday1,Sunday2}. Beyond the quadratic case, the class of potentials satisfying assumptions \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$} is broad and includes many models from mechanics, nonlinear dynamics, and phase-transition theory. A general and important family is obtained by taking \n$$\nW(u)=\\Phi(u-u_*),\n$$\nwhere $\\Phi\\colon\\mathbb{R}^N\\to\\mathbb{R}$ is a convex, positive function with a global minimum at the origin. Such potentials arise naturally in the study of quasilinear Allen-Cahn type equations, see, for instance, \\cite{Isneri}. In particular, coercive polynomial potentials, such as $W(u)=\\|u-u_*\\|^p$, with $p\\ge 2$, and the exponential potential \n$$\nW(u)=\\frac12\\big(e^{\\|u-u_*\\|^2}-1\\big),\n$$\nwhich is convex and coercive, fits perfectly into this model. Another classical example is the Ginzburg-Landau potential \n$$\nW(u)=\\frac14\\left(\\|u\\|^2-1\\right)^2,\n$$\nwhose set of global minima is the unit sphere $\\{u\\in\\mathbb{R}^N : \\|u\\| = 1\\}$. This potential play a central role in the theory of phase transitions and superconductivity, see \\cite{Allen,5}. A further relevant example, frequently appearing in the study of dissipative hyperbolic PDEs, is the quartic double-well potential \n$$\nW(u)=c_1\\|u\\|^{4}-c_2\\|u\\|^{2}+c_3,\\quad c_1,c_2,c_3>0,\n$$\nwhose structure is extensively used in nonlinear wave equations with damping. Such models constitute a natural infinite-dimensional counterpart of \\eqref{Equation}, see, for instance, \\cite{8}. These examples illustrate that the hypotheses imposed on $W$ encompass both linear and nonlinear restoring forces, as in the damped mass-spring system, as well as potentials arising in gradient flows, optimization, and models of phase separation.\n\nWith these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \\eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.\n\n\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\n\nWith these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \\eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.\n\nIn this section we discuss the dynamics of the conservative system\n\\begin{equation}\\label{EqConservative}\n \\ddot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\nwhich arises as the limit of \\eqref{Equation} when the damping parameter $a$ tends to zero. Throughout this section, we continue to assume that $W$ satisfies \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$} and, without loss of generality, that the global minimum of $W$ is located at $u_*=0$. In contrast with the dissipative\ncase, the conservative dynamics do not generate asymptotic stability, since no decay mechanism is present. To analyze the qualitative behavior of \\eqref{EqConservative}, it is convenient to rewrite it as a first-order system in $\\mathbb{R}^{2N}$ by setting $x=u$ and $y=\\dot{u}$:\n\\begin{equation}\\label{S0}\n \\begin{cases}\n \\dot{x} = y,\\\\[2mm]\n \\dot{y} = -\\nabla W(x).\n \\end{cases}\n\\end{equation}\nThe natural energy associated with \\eqref{EqConservative},\n$$\nE(x,y) := \\frac12\\|y\\|^2 + W(x), \\quad (x,y)\\in B_\\eta(0)\\times \\mathbb{R}^N,\n$$\nwhere $\\eta=\\min\\{\\delta,\\lambda\\}$ with $\\delta$ and $\\lambda$ given in \\hyperlink{W2}{$(W_2)$}-\\hyperlink{W3}{$(W_3)$}, acts as a Lyapunov function for the equilibrium $(0,0)$. Indeed, if $(x(t),y(t))$ is a solution of \\eqref{S0}, then deriving \n$E(x(t),y(t))$ along the trajectory yields\n$$\n\\frac{d}{dt}E(x(t),y(t))= \\langle y(t),\\dot{y}(t)\\rangle + \\langle \\nabla W(x(t)),\\dot{x}(t)\\rangle.\n$$\nUsing $\\dot{x}=y$ and $\\dot{y}=-\\nabla W(x)$, we immediately obtain\n$$\n\\frac{d}{dt}E(x(t),y(t))= -\\langle y(t),\\nabla W(x(t))\\rangle+ \\langle \\nabla W(x(t)),y(t)\\rangle= 0.\n$$\nThus, if $u(t)$ is any solution of \\eqref{EqConservative}, then the energy is conserved in time, that is,\n\\begin{equation}\\label{EnergyConservation}\n \\frac12\\|\\dot{u}(t)\\|^2 + W(u(t)) = \\tau,\\quad \\forall\\, t\\in\\mathbb{R},\n\\end{equation}\nfor some $\\tau\\ge 0$ determined by the initial condition. From the Lyapunov Criterion, conservation of energy implies that \n$(0,0)$ is stable: for every $\\varepsilon>0$ there exists \n$\\delta>0$ such that\n$$\n\\|u(0)\\| + \\|\\dot{u}(0)\\| < \\delta\\quad\\Longrightarrow\\quad\\sup_{t\\ge 0} \\big( \\|u(t)\\| + \\|\\dot{u}(t)\\| \\big) \\le \\varepsilon.\n$$\nHowever, stability cannot be asymptotic. Indeed, if a nontrivial solution $(u(t),\\dot{u}(t))$ were to converge to $(0,0)$ as $t\\to+\\infty$, then \\eqref{EnergyConservation} would force $\\tau=0$, which implies that $u(t)\\equiv 0$ identically. Hence the only trajectory converging to the equilibrium is the constant solution.\n\nWe now turn to the relation between the conservative regime and the dissipative dynamics studied in Section \\ref{Sec2}. Assume that $W$ satisfies \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W5}{$(W_5)$}. The conservative system \\eqref{EqConservative} arises as the limit of the dissipative equation \\eqref{Equation} when the damping parameter $a$ tends to zero. In the dissipative regime, Section \\ref{Sec2} established that solutions are asymptotically stable and even converge exponentially fast to the equilibrium with decay rate $\\gamma(a)>0$ depending on $a$. A natural question is whether this behavior persists, in some sense, in the limit as $a\\to 0^{+}$. The answer is negative: the limit dynamics is not asymptotically stable, and the passage $a\\to 0^{+}$ is singular. This phenomenon can be described rigorously through a perturbation argument. Let $(a_n)$ be a sequence of positive numbers with $a_n \\to 0$, and let $u_n$ be the solution of the perturbed problem\n\\begin{equation}\\label{EqDamped-n}\n \\ddot u_n(t) + a_n \\dot u_n(t) + \\nabla W(u_n(t)) = 0,\n\\end{equation}\nwith fixed initial data $(u_n(0),\\dot u_n(0)) = (z_0,z_1)$, where $(z_0,z_1)\\neq(0,0)$ and $\\|(z_0,z_1)\\|<\\sigma$. Here $\\sigma>0$ depends on a prescribed $\\varepsilon\\in(0,\\delta)$, with $\\delta>0$ given in \\hyperlink{W2}{\\((W_2)\\)}, and is provided by the uniform stability result of Lemma \\ref{L5}. In particular, Lemma \\ref{L5} ensures that\n$$\n\\lim_{t\\to+\\infty} (u_n(t),\\dot u_n(t)) = (0,0)\\quad\\text{and}\\quad\\|u_n(t)\\| + \\|\\dot u_n(t)\\| \\le \\varepsilon \\quad\\text{for all } t\\ge 0\\text{ and }n\\in\\mathbb{N}.\n$$\nMoreover, by Lemma \\ref{L7}, each $u_n$ satisfies the exponential decay estimate\n\\begin{equation}\\label{Exp}\n \\|u_n(t)\\| + \\|\\dot u_n(t)\\|\\le C e^{-\\gamma(a_n) t}, \\qquad t\\ge 0,\n\\end{equation}\nwhere $C>0$ is a constant independent of $a_n$. A compactness argument shows that, up to a subsequence, $u_n$ converges in $C^2_{\\mathrm{loc}}([0,+\\infty))$ to a limit\nfunction $u_0$, which solves the conservative system\n$$\n\\ddot u_0 + \\nabla W(u_0)=0\n$$\nwith initial data $u_0(0)=z_0$ and $\\dot u_0(0)=z_1$, and satisfying\n\\begin{equation*}\n \\sup_{t\\ge 0}\\left(\\|u_0(t)\\|+\\|\\dot{u}_0(t)\\|\\right) \\leq \\epsilon.\n\\end{equation*}\nAlthough each $u_n$ converges asymptotically to the equilibrium, the limit solution $u_0$ does not. Indeed, suppose by contradiction that\n$$\n\\lim_{t\\to+\\infty} u_0(t)=0\\quad\\text{and}\\quad\n\\lim_{t\\to+\\infty} \\dot u_0(t)=0.\n$$\nSince $W$ is continuous and $W(0)=0$, it follows that $W(u_0(t))\\to 0$ and $\\|\\dot u_0(t)\\|\\to 0$. Inserting these limits into the conservation law \\eqref{EnergyConservation}, we obtain $\\tau=0$. But $\\tau=0$ implies\n$$\n\\dot u_0(t)\\equiv 0\\quad\\text{and}\\quad W(u_0(t))\\equiv 0,\n$$\nso that $u_0(t)= 0$ for all $t\\in\\mathbb{R}$. This contradicts the fact that $(u_0(0),\\dot u_0(0))\\neq(0,0)$. Therefore,\n$$\n\\lim_{t \\to +\\infty} (u_0(t),\\dot{u}_0(t)) \\neq (0,0).\n$$\nThis argument shows that although each damped solution converges to the equilibrium, the limit solution does not. Equivalently, the exponential decay rate $\\gamma(a)$ obtained for $a>0$ must necessarily satisfy $\\gamma(a)\\to 0$ as $a\\to 0^+$, and no uniform decay estimate can hold on an interval of the form $(0,a_0]$. The transition from $a>0$ to $a=0$ represents a loss of asymptotic stability and a qualitative change in the global dynamics.\n\n\\begin{lemma}\\label{asd}\n Let $B\\subset \\mathbb R^{2N}$ be any bounded set containing the initial data $z_0$.\n Then there exists a bounded set $K\\subset \\mathbb{R}^{2N}$ such that \n $$\n T(t) B \\subset K,\\quad \\forall\\, t\\ge 0.\n $$\n In other words, $K$ serves as a absorbing set for the family of semigroup $\\{T(t)\\}_{t\\ge 0}$.\n\\end{lemma}\n\\begin{proof}\nFix $a>0$ and let $z_0=(x_0,y_0)\\in B$. Since $B$ is bounded in $\\mathbb{R}^{2N}$ and $W$ is continuous by \\hyperlink{W1}{$(W_1)$}, there exists a $R_B>0$ such that\n$$\n\\sup_{z_0\\in B} E(z_0)\\le R_B.\n$$\nTherefore, by the monotonicity property of the energy in \\eqref{decreasing-property}, we obtain\n$$\nW(x_a(t))\\leq R_B\\quad\\text{and}\\quad\\|y_a(t)\\|^2\\leq 2R_B\\quad\\forall \\, t\\geq 0,\n$$\nwhere $x_a$ and $y_a$ denote the components of the trajectory $\\varphi_a(t;z_0)$. Next, by the coercivity assumption \\hyperlink{W4}{$(W_4)$}, the sublevel set\n$$\n\\mathcal{S}_{B} \\coloneqq \\{ x \\in \\mathbb{R}^{N}:W(x)\\le R_B\\}\n$$\nis bounded. Consequently, we may define the positive real number\n$$\nC_B \\coloneqq \\sup \\left\\{\\|x\\|^{2} : x \\in \\mathcal{S}_{B}\\right\\} < \\infty.\n$$\nIn this way, we have \n$$\n\\Vert T(t)z_0 \\Vert_{\\ast}^2 = \\Vert (x_a(t), y_a(t))\\Vert_{\\ast}^2 = \\Vert x_a(t) \\Vert^2 + \\Vert y_a(t) \\Vert^2 \\leq C_B + 2R_B,\\quad \\forall\\, t\\ge 0, \n$$\nNow, defining\n$$\nK\\coloneqq\\Big\\{(x,y)\\in\\mathbb{R}^{2N}:\\|(x,y)\\|_{\\ast}^{2}\\le \\sqrt{C_B + 2R_B}\\Big\\}\n$$\nit follows the inclusion $T(t)B \\subset K$ for all $t\\geq 0$, which finishes our proof.\n\\end{proof}", "post_theorem_intro_text_len": 6953, "post_theorem_intro_text": "The Theorem \\ref{Teo1} asserts that the equilibrium $(u_*,0)$ of the system \\eqref{Equation} is uniformly asymptotically stable with respect to the parameter $a\\in(0,a_0]$: every solution starting sufficiently close to $(u_*,0)$ remains in an arbitrarily small neighborhood of the equilibrium and converges to it as $t\\to +\\infty$, with bounds that are uniform in $a$. The fundamental point of our study is the construction of a strict Lyapunov functional for the first-order formulation of the dissipative system\n\\begin{equation}\\label{S-intro}\n\t\\begin{cases}\n\t\t\\dot{x}=y,\\\\[1mm]\n\t\t\\dot{y}=-a\\,y-\\nabla W(x),\n\t\\end{cases}\n\\end{equation}\nwhich encodes the full dynamics of \\eqref{Equation} in the phase space $\\mathbb{R}^{2N}$. To this end, we introduce the Lyapunov functional\n\\begin{equation}\n\tV_a(x,y):=\\frac{1}{2}\\|y\\|^2+2W(x)+\\frac{1}{2}\\|y+a(x-u_*)\\|^2,\n\\end{equation}\nwhich can be interpreted as a smooth perturbation of the mechanical energy\n$$\nE(x,y):=\\frac12\\|y\\|^2+W(x).\n$$\nThe functional $V_a$ allows us to obtain stability estimates that are uniform with respect to $a$. The assumptions \\hyperlink{W1}{($W_1$)}-\\hyperlink{W4}{($W_4$)} play a crucial role in ensuring that $V_a$ is strictly decreasing along non-trivial trajectories. The condition \\hyperlink{W5}{($W_5$)} is only required to derive exponential decay of solutions, this decay cannot be uniform as $a\\to 0^{+}$, since the corresponding rate $\\gamma(a)$ necessarily satisfies $\\gamma(a)\\to 0$. Furthermore, the introduction of the Lyapunov functional $V_a$ therefore provides a robust framework for establishing the uniform asymptotic stability of \\eqref{Equation}. Furthermore, we believe that $V_a$ highlights a mechanism that can be extended to non-homogeneous cases, to time-dampened systems, and to more general configurations such as trajectories over Hilbert space.\n\nWe emphasize that the conclusions of Theorem \\ref{Teo1} are not restricted to a specific equilibrium. In fact, the result applies to any global minimizer $u_*$ of the potential $W$ satisfying assumptions \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$}, yielding uniform asymptotic stability of the corresponding equilibrium $(u_*,0)$. Moreover, when the additional condition \\hyperlink{W5}{$(W_5)$} holds, the convergence toward $(u_*,0)$ is exponential. In this sense, Theorem \\ref{Teo1} provides a robust stability result around arbitrary admissible minima of the potential.\n\nWhen $a=0$, the equation \\eqref{Equation} reduces to the conservative system\n$$\n\\ddot u(t) + \\nabla W(u(t))=0,\n$$\ncorresponding to the motion of a particle on $W$ without friction, so that mechanical energy is preserved. Here, $u(t)$ is the position of the particle, $\\ddot{u}(t)$ its acceleration, and $\\nabla W(u)$ the force derived from $W$. Under assumptions \\hyperlink{W1}{($W_1$)}-\\hyperlink{W4}{($W_4$)}, and in particular the coercivity of $W$, every solution remains globally bounded in $\\mathbb{R}^{N}$, that is, \n$$\n\\sup_{t\\in\\mathbb{R}}\\left(\\|u(t)\\|+\\|\\dot u(t)\\|\\right)<\\infty.\n$$\nThus blow-up cannot occur and no trajectory can converge asymptotically to the equilibrium. In this Hamiltonian-type regime, the combination of confinement and energy conservation leads to the typical behaviors of conservative dynamics: periodic orbits and transition solutions (heteroclinic or homoclinic). Hence, the case $a=0$ produces a qualitative change in the dynamics, yielding globally bounded but non-convergent trajectories.\n\nIn addition to the study of the asymptotic stability of \\eqref{Equation}, two additional components complement our study. In Section \\ref{Sec4}, we revisit the dynamics from the point of view of nonlinear semigroup theory, aiming for a comprehensive qualitative description of the dissipative flux generated by the first-order system \\eqref{Equation}. This approach shows, in particular, the existence of a global attractor $\\mathcal{A}$ for the corresponding semigroup, which is a compact, invariant subset of the phase space that attracts all bounded sets in forward time. Moreover, $\\mathcal{A}$ contains the entire set of equilibria of the system, that is, \n$$\\{(x,0)\\in\\mathbb{R}^{2N}:\\nabla W (x)=0\\}\\subset \\mathcal{A}.\n$$ \nFor readers interested in the general theory of global attractors and nonlinear semigroups, we refer to the classical monographs \\cite{Livro-Alexandre,Hale,Temam}. Finally, Section \\ref{Sec5} presents numerical simulations for some cases of \\eqref{Equation} involving quadratic, exponential, and double-well potentials implemented in a computational environment based on \\emph{R}. The simulations rely on a functional-programming framework supported by \\texttt{tidyverse} (data manipulation), \\texttt{deSolve} (time-integration of differential equations), and \\texttt{ggplot2} (phase-plane visualization and graphical output). This computational component has a dual purpose: to geometrically illustrate the results developed in the previous sections and to reveal characteristics of the dynamics that remain inaccessible to our methods. For instance, when the potential $W$ possesses multiple global minima, such as $W(u)=(u^2-1)^2/4$, numerical evidence suggests that local maxima separating successive minima act as unstable equilibria. In the conservative regime $a=0$, simulations indicate that trajectories in the phase plane $\\mathbb{R}^{2N}$ form closed periodic orbits. Furthermore, when $W(u)=u^2/2$, equation \\eqref{Equation} is linear and exhibits the classical critical damping threshold at $a=2$. Our numerical experiments indicate that the same phenomenon persists in nonlinear settings: for instance, when $W(u)=(e^{u^{2}}-1)/2$, simulations suggest that $a=2$ again separates oscillatory (underdamped) dynamics from non-oscillatory (overdamped) decay. This numerical evidence points to a broader conjectural picture: nonlinear potentials with a single global minimizer may still inherit a critical damping regime analogous to the linear case. We have included complete executable \\emph{R} scripts at the end of Section \\ref{Sec5}. The reader can inspect, modify, or compile them to visualize trajectories, replicate phase plane geometry, and experiment with potentials or additional parameter regimes. Therefore, numerical simulation serves as a gateway to more in-depth mathematical and computational investigations.\n\nThe plan of the paper is as follows. Section \\ref{Sec2} contains the proof of Theorem \\ref{Teo1}, based on the construction of a strict Lyapunov functional adapted to the dissipative system. In Section \\ref{Sec3} we briefly discuss the conservative case $a=0$ and the singular nature of the limit as $a\\to 0^{+}$. Section \\ref{Sec4} examines the global dynamics of nonlinear semigroup theory, providing a complementary viewpoint to the Lyapunov analysis of Section \\ref{Sec2}. Finally, Section \\ref{Sec5} presents numerical simulations for representative potentials, including quadratic, exponential, and double-well examples.", "sketch": "To prove Theorem~\\ref{Teo1} the key idea is the “construction of a strict Lyapunov functional for the first-order formulation” of \\eqref{Equation} written as the phase-space system \\eqref{S-intro}. The argument introduces\n\\[\nV_a(x,y):=\\tfrac12\\|y\\|^2+2W(x)+\\tfrac12\\|y+a(x-u_*)\\|^2,\n\\]\nviewed as a “smooth perturbation of the mechanical energy” \\(E(x,y)=\\frac12\\|y\\|^2+W(x)\\). The functional \\(V_a\\) is used to obtain “stability estimates that are uniform with respect to \\(a\\)”. The assumptions \\hyperlink{W1}{\\((W_1)\\)}–\\hyperlink{W4}{\\((W_4)\\)} are used to ensure that \\(V_a\\) is “strictly decreasing along non-trivial trajectories,” which yields uniform asymptotic stability (solutions starting sufficiently close remain close and converge as \\(t\\to+\\infty\\)). Under the additional condition \\hyperlink{W5}{\\((W_5)\\)}, the same Lyapunov framework is used to “derive exponential decay of solutions,” with a non-uniform rate as \\(a\\to0^+\\) (the text notes “\\(\\gamma(a)\\to0\\)”).", "expanded_sketch": "To prove the main theorem, the key idea is the “construction of a strict Lyapunov functional for the first-order formulation” of\n\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\nwritten as the phase-space system \\eqref{S-intro}. The argument introduces\n\\[\nV_a(x,y):=\\tfrac12\\|y\\|^2+2W(x)+\\tfrac12\\|y+a(x-u_*)\\|^2,\n\\]\nviewed as a “smooth perturbation of the mechanical energy” \\(E(x,y)=\\frac12\\|y\\|^2+W(x)\\). The functional \\(V_a\\) is used to obtain “stability estimates that are uniform with respect to \\(a\\)”. The assumptions \\hyperlink{W1}{\\((W_1)\\)}–\\hyperlink{W4}{\\((W_4)\\)} are used to ensure that \\(V_a\\) is “strictly decreasing along non-trivial trajectories,” which yields uniform asymptotic stability (solutions starting sufficiently close remain close and converge as \\(t\\to+\\infty\\)). Under the additional condition \\hyperlink{W5}{\\((W_5)\\)}, the same Lyapunov framework is used to “derive exponential decay of solutions,” with a non-uniform rate as \\(a\\to0^+\\) (the text notes “\\(\\gamma(a)\\to0\\)”).", "expanded_theorem": "\\label{Teo1}\n\tAssume that the potential $W$ satisfies \\hyperlink{W1}{$(W_1)$}-\\hyperlink{W4}{$(W_4)$}. Then, for every $\\varepsilon > 0$ and $a_0 > 0$, there exists $\\delta > 0$ such that for all $a \\in (0, a_0]$, and any solution $u_a$ of\n\t\\begin{equation}\\label{Equation}\n\t\\ddot{u}(t) + a \\dot{u}(t) + \\nabla W(u(t)) = 0,\n\\end{equation}\n\twith initial data satisfying\n\t$$\n\t\\|u_a(0)-u_*\\| + \\|\\dot{u}_a(0)\\| < \\delta,\n\t$$\n\tthe following hold:\n\t\\begin{itemize}\n\t\t\\item[\\emph{(a)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le \\varepsilon$;\n\t\t\\item[\\emph{(b)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tIf, additionally, \\hyperlink{W2}{$(W_2)$} and \\hyperlink{W3}{$(W_3)$} hold with $\\delta = \\lambda = +\\infty$, then there exists $R > 0$ such that, for all $a \\in (0, a_0]$, and any solution $u_a$ of the equation above,\n\t\\begin{itemize}\n\t\t\\item[\\emph{(c)}] $\\displaystyle \\sup_{t \\ge 0} \\left( \\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\right) \\le R$;\n\t\t\\item[\\emph{(d)}] $\\displaystyle \\lim_{t \\to +\\infty} u_a(t) =u_*$ and  $\\displaystyle\\lim_{t \\to +\\infty} \\dot{u}_a(t) = 0$.\n\t\\end{itemize}\n\tMoreover, under the additional assumption \\hyperlink{W5}{$(W_5)$}, for each $a >0$ there exist $\\gamma(a) > 0$ and $C(a)>0$ such that the corresponding solution $u_a$ satisfies the exponential decay estimate\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C(a) e^{-\\gamma(a) t}, \\quad \\forall\\, t \\ge 0.\n\t\\end{equation*}\n\tIn particular, if $0<a_1<a_2$ and $a\\in[a_1,a_2]$, then there exist $C_*>0$ and $\\gamma_*>0$, depending only on $a_1,a_2$ such that\n\t\\begin{equation*}\n\t\t\\|u_a(t)-u_*\\| + \\|\\dot{u}_a(t)\\| \\le C_* e^{-\\gamma_* t},\\quad \\forall\\, t \\ge 0,\\ \\forall\\, a\\in[a_1,a_2].\n\t\\end{equation*}", "theorem_type": ["Universal–Existential", "Asymptotic or Limit"], "mcq": {"question": "Let $u_a:\\mathbb{R}\\to\\mathbb{R}^N$ solve\n$$\n\\ddot u(t)+a\\dot u(t)+\\nabla W(u(t))=0,\n$$\nwith damping parameter $a>0$, where $W:\\mathbb{R}^N\\to\\mathbb{R}$ is a $C^2$ potential and $u_*\\in\\mathbb{R}^N$ is a reference equilibrium. Assume:\n- $W\\in C^2(\\mathbb{R}^N;\\mathbb{R})$;\n- there exists $\\delta_0>0$ such that $W(u_*)=0$ and $W(x)>0$ for all $x\\in B_{\\delta_0}(u_*)\\setminus\\{u_*\\}$;\n- there exists $\\lambda>0$ such that $\\langle \\nabla W(x),x-u_*\\rangle>0$ for all $x\\in B_{\\lambda}(u_*)\\setminus\\{u_*\\}$;\n- $W$ is coercive (equivalently, its sublevel sets are bounded);\nwhere $B_r(u_*)=\\{x\\in\\mathbb{R}^N:\\|x-u_*\\|<r\\}$. If, in addition, the positivity and radial monotonicity conditions hold globally, this means $W(u_*)=0$, $W(x)>0$ for every $x\\neq u_*$, and $\\langle \\nabla W(x),x-u_*\\rangle>0$ for every $x\\neq u_*$. Finally, suppose also that\n$$\n\\alpha\\|x-u_*\\|^2\\le W(x)\\le \\beta\\|x-u_*\\|^2,\n\\qquad\n\\langle \\nabla W(x),x-u_*\\rangle\\ge \\mu\\|x-u_*\\|^2\n$$\nfor all $x\\in B_{\\lambda}(u_*)$, for some constants $\\alpha,\\beta,\\mu>0$.\n\nUnder these assumptions, which asymptotic statement is valid as $t\\to+\\infty$?", "correct_choice": {"label": "A", "text": "For every $\\varepsilon>0$ and $a_0>0$, there exists $\\delta>0$ such that for every $a\\in(0,a_0]$, any solution with initial data satisfying\n$$\n\\|u_a(0)-u_*\\|+\\|\\dot u_a(0)\\|<\\delta\n$$\nobeys\n$$\n\\sup_{t\\ge0}\\big(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\big)\\le\\varepsilon,\n$$\nand also\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nIf, moreover, the positivity and radial monotonicity assumptions hold globally (that is, with $\\delta_0=\\lambda=+\\infty$), then there exists $R>0$ such that for every $a\\in(0,a_0]$ and any solution $u_a$,\n$$\n\\sup_{t\\ge0}\\big(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\big)\\le R,\n$$\nand\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nIn addition, under the quadratic/nondegeneracy assumption above, for each fixed $a>0$ there exist constants $\\gamma(a)>0$ and $C(a)>0$ such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C(a)e^{-\\gamma(a)t}\\qquad \\forall t\\ge0.\n$$\nIn particular, if $0<a_1<a_2$ and $a\\in[a_1,a_2]$, then there exist $C_*>0$ and $\\gamma_*>0$, depending only on $a_1,a_2$, such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C_*e^{-\\gamma_* t}\\qquad \\forall t\\ge0,\\ \\forall a\\in[a_1,a_2].\n$$"}, "choices": [{"label": "B", "text": "For every $\\varepsilon>0$ there exists $\\delta>0$, independent of $a>0$, such that for every $a>0$, any solution with initial data satisfying\n$$\n\\|u_a(0)-u_*\\|+\\|\\dot u_a(0)\\|<\\delta\n$$\nobeys\n$$\n\\sup_{t\\ge0}\\big(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\big)\\le\\varepsilon,\n$$\nand also\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nIf, moreover, the positivity and radial monotonicity assumptions hold globally, then there exists $R>0$ such that for every $a>0$ and any solution $u_a$,\n$$\n\\sup_{t\\ge0}\\big(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\big)\\le R,\n$$\nand\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nIn addition, under the quadratic/nondegeneracy assumption above, there exist constants $\\gamma_*>0$ and $C_*>0$, independent of $a>0$, such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C_*e^{-\\gamma_* t}\\qquad \\forall t\\ge0,\\ \\forall a>0.\n$$"}, {"label": "C", "text": "For every $\\varepsilon>0$ and $a_0>0$, there exists $\\delta>0$ such that for every $a\\in(0,a_0]$, any solution with initial data satisfying\n$$\n\\|u_a(0)-u_*\\|+\\|\\dot u_a(0)\\|<\\delta\n$$\nobeys\n$$\n\\sup_{t\\ge0}\\big(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\big)\\le\\varepsilon.\n$$\nIf, moreover, the positivity and radial monotonicity assumptions hold globally, then there exists $R>0$ such that for every $a\\in(0,a_0]$ and any solution $u_a$,\n$$\n\\sup_{t\\ge0}\\big(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\big)\\le R.\n$$\nIn addition, if $0<a_1<a_2$ and $a\\in[a_1,a_2]$, then under the quadratic/nondegeneracy assumption above there exist $C_*>0$ and $\\gamma_*>0$, depending only on $a_1,a_2$, such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C_*e^{-\\gamma_* t}\\qquad \\forall t\\ge0,\\ \\forall a\\in[a_1,a_2].\n$$"}, {"label": "D", "text": "For every $\\varepsilon>0$ and $a_0>0$, there exists $\\delta>0$ such that for every $a\\in(0,a_0]$, any solution with initial data satisfying\n$$\n\\|u_a(0)-u_*\\|+\\|\\dot u_a(0)\\|<\\delta\n$$\nobeys\n$$\n\\sup_{t\\ge0}\\big(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\big)\\le\\varepsilon,\n$$\nand also\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nIf, moreover, the positivity and radial monotonicity assumptions hold globally, then there exists $R>0$ such that for every $a\\in(0,a_0]$ and any solution $u_a$,\n$$\n\\sup_{t\\ge0}\\big(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\big)\\le R,\n$$\nand\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nIn addition, under the quadratic/nondegeneracy assumption above, for each fixed $a>0$ there exist constants $\\gamma(a)>0$ and $C(a)>0$ such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C(a)e^{-\\gamma(a)t}\\qquad \\forall t\\ge0,\n$$\nand moreover there exist $C_*>0$ and $\\gamma_*>0$, independent of $a\\in(0,a_0]$, such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C_*e^{-\\gamma_* t}\\qquad \\forall t\\ge0,\\ \\forall a\\in(0,a_0].\n$$"}, {"label": "E", "text": "For every $\\varepsilon>0$ and $a_0>0$, there exists $\\delta>0$ such that for every $a\\in(0,a_0]$, any solution with initial data satisfying\n$$\n\\|u_a(0)-u_*\\|+\\|\\dot u_a(0)\\|<\\delta\n$$\nobeys\n$$\n\\sup_{t\\ge0}\\big(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\big)\\le\\varepsilon,\n$$\nand also\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nIf, moreover, the positivity assumption holds globally, then there exists $R>0$ such that for every $a\\in(0,a_0]$ and any solution $u_a$,\n$$\n\\sup_{t\\ge0}\\big(\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\big)\\le R,\n$$\nand\n$$\n\\lim_{t\\to+\\infty}u_a(t)=u_*,\\qquad \\lim_{t\\to+\\infty}\\dot u_a(t)=0.\n$$\nIn addition, under the quadratic/nondegeneracy assumption above, for each fixed $a>0$ there exist constants $\\gamma(a)>0$ and $C(a)>0$ such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C(a)e^{-\\gamma(a)t}\\qquad \\forall t\\ge0.\n$$\nIn particular, if $0<a_1<a_2$ and $a\\in[a_1,a_2]$, then there exist $C_*>0$ and $\\gamma_*>0$, depending only on $a_1,a_2$, such that\n$$\n\\|u_a(t)-u_*\\|+\\|\\dot u_a(t)\\|\\le C_*e^{-\\gamma_* t}\\qquad \\forall t\\ge0,\\ \\forall a\\in[a_1,a_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": "loss_of_nonuniformity_as_a_to_0", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped_convergence_to_equilibrium_clauses", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "extended_uniform_exponential_decay_to_(0,a0]", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "removed_global_radial_monotonicity_requirement_for_global_attraction", "template_used": "wildcard"}]}}
{"id": "2512.21444v1", "paper_link": "http://arxiv.org/abs/2512.21444v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm.Pen}\nLet $V$ be the set of vertices of any unit length RPT. Then, for $R\\ge 2$,\n\\[\\#(V\\cap B_R)=\\pi C_P R^2 + O(R^{2/3}(\\log R)^{2/3}).\\]", "start_pos": 6586, "end_pos": 6763, "label": "thm.Pen"}, "ref_dict": {"fig.PenTiles": "\\begin{figure}[h]\n\t\\caption{Tiles used to construct rhombic Penrose tilings.\\\\}\\label{fig.PenTiles}\n\t\\centering\n\t\\vspace*{-30bp}\n\t\\includegraphics[width=0.75\\textwidth]{PenTiles1mod}\n\\end{figure}", "fig.PenPatch": "\\begin{figure}[h]\n\t\\caption{A patch of an RPT constructed using the cut and project method detailed in Section \\ref{sec.ModelSets}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.\\\\}\\label{fig.PenPatch}\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{PenPatch2}\n\\end{figure}", "eqn.PenVertTrivEst": "\\begin{equation}\\label{eqn.PenVertTrivEst}\n\t\\#(V\\cap B_R)=\\pi C_P R^2 + O(R),\n\\end{equation}"}, "pre_theorem_intro_text_len": 3356, "pre_theorem_intro_text": "\\label{sec.Intro}\nAperiodic tilings of Euclidean space, a topic once regarded as an esoteric pursuit, are now widely recognized as a subject of central interest. Beginning with work of Hao Wang and Robert Berger in the 1960's studying a relationship between aperiodic tilings and the Halting problem for Turing machines, results surrounding aperiodic tilings have had substantial impacts in the last century within mathematics, computer science, the natural sciences, and even the liberal arts. Excellent summaries of many of these results, up until about 2010, can be found in \\cite[Chapter 1]{Gard1997}, \\cite[Chapters 1, 2]{Sene1995}, \\cite[Forward by Roger Penrose]{BaakGrim2013}, and \\cite[Chapter 1]{BaakGrim2013}. There has also been significant and highly publicized progress in this subject in the last decade.\n\nOne landmark, which has also become an inspiration in art and architecture, was Roger Penrose's discovery of a pair of polygonal tiles which could tile the plane, but could only do so aperiodically. Penrose published a first version of his tilings in 1974 \\cite{Penr1974} (see also \\cite{Penr1979}). Soon after, they were introduced to the general public in an article by Martin Gardner in the Scientific American \\cite{Gard1977}.\n\n\\begin{figure}[h]\n\t\\caption{Tiles used to construct rhombic Penrose tilings.\\\\}\\label{fig.PenTiles}\n\t\\centering\n\t\\vspace*{-30bp}\n\t\\includegraphics[width=0.75\\textwidth]{PenTiles1mod}\n\\end{figure}\n\nThere are several related collections of tilings which are referred to as Penrose tilings. In this article we will consider \\textit{rhombic Penrose tilings} (RPTs), which are tilings of the plane by copies of two rhombuses with angles $72^\\circ$ and $36^\\circ$, pictured in Figure \\ref{fig.PenTiles}. The rhombuses are allowed to be rotated, but they must fit together edge to edge, and the matching rules indicated by the arrows must also be enforced. A small patch of an RPT is pictured in Figure \\ref{fig.PenPatch} (see also \\cite[Figures 1.2, 6.44]{BaakGrim2013}). When the rhombuses are chosen to have side length 1, we call the resulting tilings  \\textit{unit length RPTs}.\n\nThe main problem that we wish to investigate in this paper is that of estimating the number of vertices of an RPT which lie in $B_R$, the closed Euclidean ball of radius $R$, centered at some point in the plane which we fix to be the origin. RPTs can be defined using primitive inflation rules (see \\cite[Section 6.2]{BaakGrim2013}), which guarantee existence of limiting frequencies of tile types in large regions. Although it is not obvious, it is not too difficult to deduce (see for example \\cite[Remark 5]{SchmTrev2018}) that if $V$ is the vertex set of a unit length RPT then, for $R\\ge 1$,\n\\begin{equation}\\label{eqn.PenVertTrivEst}\n\t\\#(V\\cap B_R)=\\pi C_P R^2 + O(R),\n\\end{equation}\nwhere\n\\[C_P=\\frac{\\phi}{5}\\sqrt{10+2\\sqrt{5}}\\approx 1.231,\\]\nwith $\\phi=(1+\\sqrt{5})/2$. The error term on the right hand side of \\eqref{eqn.PenVertTrivEst} comes from counting tiles which intersect the boundary of $B_R$.\nSince there are in fact $\\gg R$ such tiles, it may appear that this error term is essentially best possible (cf. results of Solomon from \\cite{Solo2011} on counting tiles in \\textit{supertiles} of primitive substitutions). However, we will show that for this special class of tilings, one can actually do better.", "context": "\\label{sec.Intro}\nAperiodic tilings of Euclidean space, a topic once regarded as an esoteric pursuit, are now widely recognized as a subject of central interest. Beginning with work of Hao Wang and Robert Berger in the 1960's studying a relationship between aperiodic tilings and the Halting problem for Turing machines, results surrounding aperiodic tilings have had substantial impacts in the last century within mathematics, computer science, the natural sciences, and even the liberal arts. Excellent summaries of many of these results, up until about 2010, can be found in \\cite[Chapter 1]{Gard1997}, \\cite[Chapters 1, 2]{Sene1995}, \\cite[Forward by Roger Penrose]{BaakGrim2013}, and \\cite[Chapter 1]{BaakGrim2013}. There has also been significant and highly publicized progress in this subject in the last decade.\n\nOne landmark, which has also become an inspiration in art and architecture, was Roger Penrose's discovery of a pair of polygonal tiles which could tile the plane, but could only do so aperiodically. Penrose published a first version of his tilings in 1974 \\cite{Penr1974} (see also \\cite{Penr1979}). Soon after, they were introduced to the general public in an article by Martin Gardner in the Scientific American \\cite{Gard1977}.\n\n\\begin{figure}[h]\n \\caption{Tiles used to construct rhombic Penrose tilings.\\\\}\\label{fig.PenTiles}\n \\centering\n \\vspace*{-30bp}\n \\includegraphics[width=0.75\\textwidth]{PenTiles1mod}\n\\end{figure}\n\nThere are several related collections of tilings which are referred to as Penrose tilings. In this article we will consider \\textit{rhombic Penrose tilings} (RPTs), which are tilings of the plane by copies of two rhombuses with angles $72^\\circ$ and $36^\\circ$, pictured in Figure \\ref{fig.PenTiles}. The rhombuses are allowed to be rotated, but they must fit together edge to edge, and the matching rules indicated by the arrows must also be enforced. A small patch of an RPT is pictured in Figure \\ref{fig.PenPatch} (see also \\cite[Figures 1.2, 6.44]{BaakGrim2013}). When the rhombuses are chosen to have side length 1, we call the resulting tilings \\textit{unit length RPTs}.\n\nThe main problem that we wish to investigate in this paper is that of estimating the number of vertices of an RPT which lie in $B_R$, the closed Euclidean ball of radius $R$, centered at some point in the plane which we fix to be the origin. RPTs can be defined using primitive inflation rules (see \\cite[Section 6.2]{BaakGrim2013}), which guarantee existence of limiting frequencies of tile types in large regions. Although it is not obvious, it is not too difficult to deduce (see for example \\cite[Remark 5]{SchmTrev2018}) that if $V$ is the vertex set of a unit length RPT then, for $R\\ge 1$,\n\\begin{equation}\\label{eqn.PenVertTrivEst}\n \\#(V\\cap B_R)=\\pi C_P R^2 + O(R),\n\\end{equation}\nwhere\n\\[C_P=\\frac{\\phi}{5}\\sqrt{10+2\\sqrt{5}}\\approx 1.231,\\]\nwith $\\phi=(1+\\sqrt{5})/2$. The error term on the right hand side of \\eqref{eqn.PenVertTrivEst} comes from counting tiles which intersect the boundary of $B_R$.\nSince there are in fact $\\gg R$ such tiles, it may appear that this error term is essentially best possible (cf. results of Solomon from \\cite{Solo2011} on counting tiles in \\textit{supertiles} of primitive substitutions). However, we will show that for this special class of tilings, one can actually do better.\n\n\\begin{equation}\\label{eqn.PenVertTrivEst}\n\t\\#(V\\cap B_R)=\\pi C_P R^2 + O(R),\n\\end{equation}\n\n\\begin{figure}[h]\n\t\\caption{A patch of an RPT constructed using the cut and project method detailed in Section \\ref{sec.ModelSets}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.\\\\}\\label{fig.PenPatch}\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{PenPatch2}\n\\end{figure}\n\n\\begin{figure}[h]\n\t\\caption{Tiles used to construct rhombic Penrose tilings.\\\\}\\label{fig.PenTiles}\n\t\\centering\n\t\\vspace*{-30bp}\n\t\\includegraphics[width=0.75\\textwidth]{PenTiles1mod}\n\\end{figure}", "full_context": "\\label{sec.Intro}\nAperiodic tilings of Euclidean space, a topic once regarded as an esoteric pursuit, are now widely recognized as a subject of central interest. Beginning with work of Hao Wang and Robert Berger in the 1960's studying a relationship between aperiodic tilings and the Halting problem for Turing machines, results surrounding aperiodic tilings have had substantial impacts in the last century within mathematics, computer science, the natural sciences, and even the liberal arts. Excellent summaries of many of these results, up until about 2010, can be found in \\cite[Chapter 1]{Gard1997}, \\cite[Chapters 1, 2]{Sene1995}, \\cite[Forward by Roger Penrose]{BaakGrim2013}, and \\cite[Chapter 1]{BaakGrim2013}. There has also been significant and highly publicized progress in this subject in the last decade.\n\nOne landmark, which has also become an inspiration in art and architecture, was Roger Penrose's discovery of a pair of polygonal tiles which could tile the plane, but could only do so aperiodically. Penrose published a first version of his tilings in 1974 \\cite{Penr1974} (see also \\cite{Penr1979}). Soon after, they were introduced to the general public in an article by Martin Gardner in the Scientific American \\cite{Gard1977}.\n\n\\begin{figure}[h]\n \\caption{Tiles used to construct rhombic Penrose tilings.\\\\}\\label{fig.PenTiles}\n \\centering\n \\vspace*{-30bp}\n \\includegraphics[width=0.75\\textwidth]{PenTiles1mod}\n\\end{figure}\n\nThere are several related collections of tilings which are referred to as Penrose tilings. In this article we will consider \\textit{rhombic Penrose tilings} (RPTs), which are tilings of the plane by copies of two rhombuses with angles $72^\\circ$ and $36^\\circ$, pictured in Figure \\ref{fig.PenTiles}. The rhombuses are allowed to be rotated, but they must fit together edge to edge, and the matching rules indicated by the arrows must also be enforced. A small patch of an RPT is pictured in Figure \\ref{fig.PenPatch} (see also \\cite[Figures 1.2, 6.44]{BaakGrim2013}). When the rhombuses are chosen to have side length 1, we call the resulting tilings \\textit{unit length RPTs}.\n\nThe main problem that we wish to investigate in this paper is that of estimating the number of vertices of an RPT which lie in $B_R$, the closed Euclidean ball of radius $R$, centered at some point in the plane which we fix to be the origin. RPTs can be defined using primitive inflation rules (see \\cite[Section 6.2]{BaakGrim2013}), which guarantee existence of limiting frequencies of tile types in large regions. Although it is not obvious, it is not too difficult to deduce (see for example \\cite[Remark 5]{SchmTrev2018}) that if $V$ is the vertex set of a unit length RPT then, for $R\\ge 1$,\n\\begin{equation}\\label{eqn.PenVertTrivEst}\n \\#(V\\cap B_R)=\\pi C_P R^2 + O(R),\n\\end{equation}\nwhere\n\\[C_P=\\frac{\\phi}{5}\\sqrt{10+2\\sqrt{5}}\\approx 1.231,\\]\nwith $\\phi=(1+\\sqrt{5})/2$. The error term on the right hand side of \\eqref{eqn.PenVertTrivEst} comes from counting tiles which intersect the boundary of $B_R$.\nSince there are in fact $\\gg R$ such tiles, it may appear that this error term is essentially best possible (cf. results of Solomon from \\cite{Solo2011} on counting tiles in \\textit{supertiles} of primitive substitutions). However, we will show that for this special class of tilings, one can actually do better.\n\n\\begin{equation}\\label{eqn.PenVertTrivEst}\n\t\\#(V\\cap B_R)=\\pi C_P R^2 + O(R),\n\\end{equation}\n\n\\begin{figure}[h]\n\t\\caption{A patch of an RPT constructed using the cut and project method detailed in Section \\ref{sec.ModelSets}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.\\\\}\\label{fig.PenPatch}\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{PenPatch2}\n\\end{figure}\n\n\\begin{figure}[h]\n\t\\caption{Tiles used to construct rhombic Penrose tilings.\\\\}\\label{fig.PenTiles}\n\t\\centering\n\t\\vspace*{-30bp}\n\t\\includegraphics[width=0.75\\textwidth]{PenTiles1mod}\n\\end{figure}\n\nThere are several related collections of tilings which are referred to as Penrose tilings. In this article we will consider \\textit{rhombic Penrose tilings} (RPTs), which are tilings of the plane by copies of two rhombuses with angles $72^\\circ$ and $36^\\circ$, pictured in Figure \\ref{fig.PenTiles}. The rhombuses are allowed to be rotated, but they must fit together edge to edge, and the matching rules indicated by the arrows must also be enforced. A small patch of an RPT is pictured in Figure \\ref{fig.PenPatch} (see also \\cite[Figures 1.2, 6.44]{BaakGrim2013}). When the rhombuses are chosen to have side length 1, we call the resulting tilings \\textit{unit length RPTs}.\n\n\\begin{figure}[h]\n \\caption{A patch of an RPT constructed using the cut and project method detailed in Section \\ref{sec.ModelSets}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.\\\\}\\label{fig.PenPatch}\n \\centering\n \\includegraphics[width=0.75\\textwidth]{PenPatch2}\n\\end{figure}\n\nThe work of de Bruijn, et al., mentioned at the beginning of this section (see references listed above), implies the following result.\n\\begin{theorem}\\label{thm.PenVerts}\nEvery non-singular point set $\\Lambda_\\omega$, constructed as above, is the set of vertices of a unit length rhombic Penrose tiling. Furthermore, the vertex set of any unit length rhombic Penrose tiling can be obtained, after rotation, as the limit in an appropriate topology (e.g. the Chabauty-Fell topology) of a sequence of non-singular point sets $\\Lambda_\\omega$.\n\\end{theorem}\nAn example of a RPT obtained using the cut and project method described above is shown in Figure \\ref{fig.PenPatch}. Theorem \\ref{thm.PenVerts} will allow us to derive Theorem \\ref{thm.Pen} from the following result.\n\\begin{theorem}\\label{thm.PenCPVersion}\nFor the cut and project sets $\\Lambda_{m,\\omega}$ defined above, there is a constant $C>0$ with the property that, for any $\\omega\\in\\C$, any integer $1\\le m\\le 4$, and any $R\\ge 2$,\n\\[\\left|\\#\\left(\\Lambda_{m,\\omega}\\cap B_R\\right)-\\pi R^2\\mr{dens}(\\cL)\\left|W_{m,\\omega}\\right|\\right|\\le CR^{2/3}(\\log R)^{2/3},\\]\nwhere $|W_{m,\\omega}|$ denotes the area of $W_{m,\\omega}$, and $\\mr{dens}(\\cL)$ is the reciprocal of the volume of a fundamental domain for $\\R^4/\\cL$.\n\\end{theorem}\nAn important point is that the error term in this theorem is uniform in $\\omega$, which allows us to take limits of non-singular point sets, to account for all possible vertex sets of RPTs, in order to establish Theorem \\ref{thm.Pen}. There are a few other small details that have to be ironed out, but they are not difficult to justify, so we postpone their discussion until the end of Section \\ref{sec.PenProof}. The majority of the remainder of the paper will be focused on proving Theorem \\ref{thm.PenCPVersion}.\n\nIf $W\\subseteq\\R^2$ then we write $\\chi_W$ for the indicator function of $W$. The similarity in notation to $\\chi_R$, as defined above, will not cause confusion in what follows. Observe that\n\\[\\widehat{\\chi}_W(y)=\\int_We(-x\\cdot y)\\mr{d}x=\\frac{-1}{(2\\pi|y|)^2}\\int_W\\nabla_x^2\\left( e(-x\\cdot y)\\right)\\mr{d}x,\\]\nwhere $\\nabla_x^2=\\nabla_x\\cdot\\nabla_x$ is the Laplacian. By the Divergence Theorem, under reasonable hypotheses on $W$, this allows us to express the Fourier transform as a line integral over the boundary of $W$. Applying this argument in the case when $W$ is a polygon leads to a particularly nice formula for the Fourier transform, which we reproduce here in the form given in \\cite{BranColzTrav1997}.\n\\begin{proposition}\\label{prop.FourTranPoly}\n Let $P\\subseteq\\R^2$ be a polygon with counterclockwise oriented vertices $\\{a_j\\}_{j=1}^m$. Denote by $\\sigma_j$ the unit vector parallel to the side $[a_j,a_{j+1}]$ and by $v_j$ the outward unit normal to this side. Then, with $a_{m+1}:=a_1$, we have\n \\[\\widehat{\\chi}_P(y)=\\frac{-1}{(2\\pi|y|)^2}\\sum_{j=1}^m\\left(\\frac{e(-y\\cdot a_{j+1})-e(-y\\cdot a_{j})}{y\\cdot \\sigma_j}\\right)y\\cdot v_j.\\]\n\\end{proposition}\nThis proposition is proved as \\cite[Lemma 2.2]{BranColzTrav1997}. The quantity in parenthesis in the sum is interpreted to equal\n\\[-\\left(2\\pi i |a_{j+1}-a_j|\\right)e\\left(\\frac{-y\\cdot(a_j+a_{j+1})}{2}\\right),\\]\nwhen $y\\cdot\\sigma_j=0$. Proposition \\ref{prop.FourTranPoly} implies that\n\\begin{equation}\\label{eqn.PolyFourEst}\n \\widehat{\\chi}_P(y)\\ll_P \\frac{1}{|y|}\\sum_{j=1}^m\\min\\left\\{1,\\frac{1}{|y\\cdot\\sigma_j|}\\right\\}.\n\\end{equation}\nOne thing to note is that these Fourier coefficients decay in most directions like $|y|^{-2}$. However, in directions perpendicular to the sides of $P$, they decay only like $|y|^{-1}$. This turns out to be a source of some difficulty, which is given special attention in the proofs below.\n\n\\noindent\\emph{Bound for $E_2$:} To estimate $E_2$, first write\n\\[\\mc{S}_{m,n}=\\left\\{\\theta\\in L^\\circledast : \\frac{2^{m-1}}{R}<|\\theta|\\le\\frac{2^m}{R},~\\frac{nR}{M2^m}\\le|\\theta^\\star|<\\frac{(n+1)R}{M2^m}\\right\\},\\]\nwith $M$ chosen as in \\eqref{eqn.MChoice}. Then we have that\n\\begin{align}\n E_2&\\ll R^{1/2}\\sum_{m=1}^{\\lceil \\log_2R\\rceil}\\sum_{n=1}^\\infty\\sum_{\\theta\\in\\mc{S}_{m,n}}\\frac{1}{|\\theta|^{3/2}}|\\widehat{\\chi}_W(\\theta^\\star)|\\left(1+\\delta^2|\\theta^\\star|^2\\right)^{-N}\\nonumber\\\\\n &\\ll R^2 \\sum_{m=1}^{\\lceil\\log_2R\\rceil}2^{-3m/2}\\sum_{n=1}^\\infty\\left(1+\\frac{\\delta^2n^2R^2}{2^{2m}}\\right)^{-N}\\sum_{\\theta\\in\\mc{S}_{m,n}}|\\widehat{\\chi}_W(\\theta^\\star)|.\\label{eqn.E2Bound1}\n\\end{align}\nHere we will use the following lemma.\n\\begin{lemma}\\label{lem.ChiWEst2}\n With notation as above, for $m,n\\in\\N$ with $m\\le\\lceil \\log_2R\\rceil$, we have that\n \\[\\sum_{\\theta\\in\\mc{S}_{m,n}}|\\widehat{\\chi}_W(\\theta^\\star)|\\ll\\frac{2^{2m}\\log n}{nR^2}.\\]\n\\end{lemma}\n\\begin{proof}\n Let us write\n \\[\\tilde{\\mc{S}}=\\left\\{\\lambda\\in \\frac{1}{25}L : \\frac{2^{m-1}}{R}<|\\lambda|\\le\\frac{2^m}{R},~\\frac{nR}{M2^m}\\le|\\lambda^\\star|<\\frac{(n+1)R}{M2^m},~|\\mr{arg}(\\lambda^\\star)|\\le \\frac{\\pi}{5}\\right\\}.\\]\n By the argument used in the proof of Lemma \\ref{lem.ChiWEst1} leading up to equation \\eqref{eqn.ChiWEst1}, it is sufficient for this lemma to prove that\n \\begin{equation*}\n \\sum_{\\lambda\\in\\tilde{\\mc{S}}}\\frac{1}{|\\lambda^\\star|^2|\\sin(\\mathrm{arg}(\\lambda^\\star))|}\\ll\\frac{2^{2m}\\log n}{nR^2}.\n \\end{equation*}\n Label the elements of $\\star(\\tilde{\\mc{S}})$ as\n \\[\\lambda_j^\\star=\\rho_je^{it_j},\\]\n for $1\\le j\\le J:=|\\tilde{\\mc{S}}|$, with each $\\rho_j\\in\\R^+$ and with $0< |t_1|\\le |t_2|\\le\\cdots \\le |t_J|\\le\\pi/5$. Also, label the element of $\\tilde{\\mc{S}}$ corresponding to each $\\lambda_j^\\star$ as $\\lambda_j$.\n\n\\noindent\\emph{Proof of Theorem \\ref{thm.Pen}:} Let $V$ be the vertex set of a unit length RPT. If $V$ can be constructed as a cut and project set, as $V=\\Lambda_\\omega$ for some $\\omega\\in\\C$, then the result of Theorem \\ref{thm.Pen} follows immediately from Theorem \\ref{thm.PenCPVersion}. Otherwise, by Theorem \\ref{thm.PenVerts}, there is a sequence of complex numbers $\\{\\omega_j\\}_{j=1}^\\infty$ with the property that, after a possible rotation,\n\\[V=\\lim_{j\\rar\\infty}\\Lambda_{\\omega_j},\\]\nwhere the limit is taken in the Chabauty-Fell topology. For $R\\ge 2$, the number of points in $\\R^2$ which lie on the boundary of $B_R$, and which are the limits of sequences of points $\\lambda_j\\in\\Lambda_{\\omega_j}$, as $j\\rar\\infty,$ must be $\\ll R^{2/3}(\\log R)^{2/3}$. This follows from the uniformity over $\\omega\\in \\C$ of the error term in Theorem \\ref{thm.PenCPVersion}. It then follows, again by uniformity, that\n\\begin{align}\n \\#(V\\cap B_R)&=\\lim_{j\\rar\\infty}\\#\\left(\\Lambda_{\\omega_j}\\cap B_R\\right)+O(R^{2/3}(\\log R)^{2/3})\\nonumber\\\\\n &=\\pi R^2\\mr{dens}(\\cL)\\sum_{m=1}^4|W_{m,\\omega+m}|+O( R^{2/3}(\\log R)^{2/3}).\\label{eqn.ChabLim1}\n\\end{align}\nFinally, we compute that\n\\[\\mr{dens}(\\cL)=\\mr{disc}(L)^{-1/2}=\\frac{4}{25\\sqrt{5}}.\\]\nSubstituting this in \\eqref{eqn.ChabLim1}, together with the areas of the pentagonal windows, completes the proof of Theorem \\ref{thm.Pen}.", "post_theorem_intro_text_len": 4426, "post_theorem_intro_text": "Readers who are familiar with the analogous counting problem for integer lattice points in the plane, called the Gauss circle problem, may at this point recognize some similarities. Let us write\n\\[\\#(\\mathbb{Z}^2\\cap B_R)=\\pi R^2 + E(R).\\]\nGauss himself observed that $|E(R)|\\le 2\\sqrt{2}\\pi R$, and he conjectured that it should be possible to improve the exponent on $R$  to obtain an error term of $\\ll R^{1/2}$. The next substantial result was published in 1906 by Sierpi\\'nski \\cite{Sier1906}, who showed that $E(R)\\ll R^{2/3}$ (see also \\cite{Schi1972}). Subsequent improvements have been made by many authors (the introduction of \\cite{Huxl2003} contains a list), and currently the best known upper bound, due to Martin Huxley \\cite{Huxl2003}, is $E(R)\\ll R^{131/208}(\\log R)^\\tau$, with $\\tau=18627/8320$.\n\nThe important link which allows us to employ techniques used in the Gauss circle problem in order to study vertices of Penrose tilings is a remarkable observation of de Bruijn, that the vertex set of any RPT can be obtained from a cut and project set, by projecting `slices' of lattice points in a higher dimensional space onto a 2 dimensional subspace. We describe this connection in detail in Section \\ref{sec.ModelSets}. With this as a general framework, we are able in Sections \\ref{sec.Fourier}-\\ref{sec.PenProof} to bring in tools from Fourier analysis and to approach our point counting problem through the lens of the Poisson summation formula (PSF). Results of a similar flavor, estimating discrepancies of points in growing patches of cut and projects sets, have recently been explored by Koivusalo and Lagac{\\'e} in \\cite{KoivLagacBjorHart2025}, and by R\\\"uhr, Smilansky, and Weiss in \\cite{RuhrSmilWeis2024}.\n\nWe remark that our proof, although structurally similar, is not a na\\\"ive application of known methods (e.g. Huxley's Fourier analytic approach to lattice point counting in regions with smooth boundary, using Van der Corput's method). Here, there are two new difficulties which are not encountered in the classical versions of the Gauss circle problem. The first is that, for the cut and project sets that we consider, the problem amounts to counting points in a lattice in a $4$-dimensional cylinder which is only growing radially. Since the height of the cylinder stays fixed, the smoothing necessary to apply the PSF must shrink faster in the direction of the height than in the radial direction. Secondly, the lattices from which we derive RPTs are irrational. This has the effect that, depending on how well approximable the lattice points are by rational points, smoothing our regions may pick up points which are unusually close to their boundaries. This is a real phenomenon, which can potentially cause a blow up in the Fourier coefficients after using the PSF. It is one of the main issues that we must confront using the specific geometric and algebraic data which define RPTs.\n\n\\begin{figure}[h]\n\t\\caption{A patch of an RPT constructed using the cut and project method detailed in Section \\ref{sec.ModelSets}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.\\\\}\\label{fig.PenPatch}\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{PenPatch2}\n\\end{figure}\n\nFinally, there is the question of how these results could be generalized. Several parts of the proofs below can be adapted quite well to deal with much wider classes of cut and project sets. However, there are some crucial details in the proof, in particular those contained in Section \\ref{sec.PenProof}, which form a narrow gate that any more general result would need to pass through. The cut and project sets used to define RPTs are extremely special, both in the algebraic properties of the lattice involved and in the exact shape, symmetry, and orientation of the windows involved, with respect to the specific underlying lattice. Any attempt to modify the lattice, or change the windows (even by a small rotation) will destroy important features of the argument and make it difficult or even impossible to control the sizes of the Fourier coefficients involved.\n\nOn the other hand, it seems likely that the arguments given here should apply equally well, with the appropriate modifications, to vertex sets of Ammann-Beenker tilings (see \\cite[Section 10.4]{GrunShep1987} and \\cite{Been1982}). We leave it to the interested reader to work out the details.", "sketch": "To prove Theorem~\\ref{thm.Pen}, the authors connect vertex counting in a unit length RPT to the Gauss circle problem by using “a remarkable observation of de Bruijn, that the vertex set of any RPT can be obtained from a cut and project set, by projecting `slices' of lattice points in a higher dimensional space onto a 2 dimensional subspace” (detailed in Section~\\ref{sec.ModelSets}). With this framework, they “bring in tools from Fourier analysis” and “approach [the] point counting problem through the lens of the Poisson summation formula (PSF)” (Sections~\\ref{sec.Fourier}--\\ref{sec.PenProof}).\n\nThey emphasize two main technical obstacles (and thus key components of the argument): (1) the counting reduces to “counting points in a lattice in a $4$-dimensional cylinder which is only growing radially,” so because “the height of the cylinder stays fixed, the smoothing necessary to apply the PSF must shrink faster in the direction of the height than in the radial direction”; and (2) the relevant lattices are “irrational,” so “depending on how well approximable the lattice points are by rational points, smoothing our regions may pick up points which are unusually close to their boundaries,” which “can potentially cause a blow up in the Fourier coefficients after using the PSF.” Controlling this requires “the specific geometric and algebraic data which define RPTs,” and the authors note that “crucial details in the proof, in particular those contained in Section~\\ref{sec.PenProof}, … make it difficult or even impossible to control the sizes of the Fourier coefficients involved” under general perturbations of lattice/windows.", "expanded_sketch": "To prove the main theorem, the authors connect vertex counting in a unit length RPT to the Gauss circle problem by using “a remarkable observation of de Bruijn, that the vertex set of any RPT can be obtained from a cut and project set, by projecting `slices' of lattice points in a higher dimensional space onto a 2 dimensional subspace” (detailed later). With this framework, they “bring in tools from Fourier analysis” and “approach [the] point counting problem through the lens of the Poisson summation formula (PSF)” (proved later).\n\nThey emphasize two main technical obstacles (and thus key components of the argument): (1) the counting reduces to “counting points in a lattice in a $4$-dimensional cylinder which is only growing radially,” so because “the height of the cylinder stays fixed, the smoothing necessary to apply the PSF must shrink faster in the direction of the height than in the radial direction”; and (2) the relevant lattices are “irrational,” so “depending on how well approximable the lattice points are by rational points, smoothing our regions may pick up points which are unusually close to their boundaries,” which “can potentially cause a blow up in the Fourier coefficients after using the PSF.” Controlling this requires “the specific geometric and algebraic data which define RPTs,” and the authors note that “crucial details in the proof, in particular those contained in the later argument, … make it difficult or even impossible to control the sizes of the Fourier coefficients involved” under general perturbations of lattice/windows.", "expanded_theorem": "\\label{thm.Pen}\nLet $V$ be the set of vertices of any unit length RPT. Then, for $R\\ge 2$,\n\\[\\#(V\\cap B_R)=\\pi C_P R^2 + O(R^{2/3}(\\log R)^{2/3}).\\],", "theorem_type": ["Asymptotic or Limit", "Inequality or Bound"], "mcq": {"question": "Let $V$ be the vertex set of any unit length rhombic Penrose tiling of the plane, meaning a tiling by copies of the two Penrose rhombi with angles $72^\\circ$ and $36^\\circ$, placed edge-to-edge with the usual arrow matching rules, and with each side of length $1$. For $R\\ge 2$, let $B_R$ denote the closed Euclidean ball of radius $R$ centered at the origin. Write\n\\[\nC_P=\\frac{\\phi}{5}\\sqrt{10+2\\sqrt{5}},\\qquad \\phi=\\frac{1+\\sqrt5}{2}.\n\\]\nAs $R\\to\\infty$, which asymptotic statement holds for the number of vertices of the tiling contained in $B_R$?", "correct_choice": {"label": "A", "text": "For every unit length rhombic Penrose tiling, one has, for $R\\ge 2$,\n\\[\n\\#(V\\cap B_R)=\\pi C_P R^2+O\\big(R^{2/3}(\\log R)^{2/3}\\big).\n\\]"}, "choices": [{"label": "B", "text": "For every unit length rhombic Penrose tiling, one has, for $R\\ge 2$,\n\\[\n\\#(V\\cap B_R)=\\pi C_P R^2+O(R^{1/2+\\varepsilon})\\qquad\\text{for every }\\varepsilon>0.\n\\]"}, {"label": "C", "text": "For every unit length rhombic Penrose tiling, one has, for $R\\ge 2$,\n\\[\n\\#(V\\cap B_R)=\\pi C_P R^2+o(R).\n\\]"}, {"label": "D", "text": "For every unit length rhombic Penrose tiling, there exists a constant $K=K(V)$ such that, for $R\\ge 2$,\n\\[\n\\#(V\\cap B_R)=\\pi C_P R^2+O\\big(KR^{2/3}(\\log R)^{2/3}\\big).\n\\]"}, {"label": "E", "text": "For every unit length rhombic Penrose tiling, one has, for $R\\ge 2$,\n\\[\n\\#(V\\cap B_R)=\\pi C_P R^2+O\\big(R^{2/3}\\big).\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": "error-exponent strengthened to near-Gauss-circle strength", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "explicit quantitative error term dropped to a weaker sublinear remainder", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "uniformity over all Penrose tilings replaced by tiling-dependent constant", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "logarithmic loss from anisotropic smoothing/Fourier control removed", "template_used": "wildcard"}]}}
{"id": "2512.21565v1", "paper_link": "http://arxiv.org/abs/2512.21565v1", "theorems_cnt": 3, "theorem": {"env_name": "thm", "content": "\\label{main1}\n  Let $X$ be a tropical variety in $\\mathbb R^n$.\n  Then $\\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is a rational affine linear subspace of $\\mathbb R^n$.", "start_pos": 7785, "end_pos": 7994, "label": "main1"}, "ref_dict": {"main2": "\\begin{thm}\n  \\label{main2}\n  Assume that $n \\geq 2$.\n  Let $Z$ be a proper closed subset of $\\mathbb R^n$ that is unbounded in all rational directions.\n  Then $\\mathbf E(Z)$ is not finitely generated.\n\\end{thm}", "main1": "\\begin{thm}\n  \\label{main1}\n  Let $X$ be a tropical variety in $\\mathbb R^n$.\n  Then $\\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is a rational affine linear subspace of $\\mathbb R^n$.\n\\end{thm}"}, "pre_theorem_intro_text_len": 3326, "pre_theorem_intro_text": "Tropical geometry is a subject of mathematics developed in recent decades, which studies geometric objects called tropical varieties.\nSometimes tropical geometry is called \\textquotedblleft the algebraic geometry over $\\mathbb T$\\textquotedblright, where $\\mathbb T$ is a semifield called the tropical semifield.\nSince $\\mathbb T$ is not a field, tropical geometry is not included in classical algebraic geometry.\nHowever, for many albedo-geometric theories, their tropical versions are studied.\nFor instance, the tropical intersection theory \\cite{allermann2010first,jensen2013computing,jensen2016stable,katz2012tropical,rau2016intersections}, the theory of divisors on tropical curves \\cite{amini2013reduced,an2014canonical,haase2012linear,hladky2013rank,len2014brill}, and the theory of tropical homology \\cite{amini2020hodge,gross2019sheaf,itenberg2019tropical,ruddat2021homology} have been developed.\n\nOne of the basic objects in algebraic geometry is the ideals of rings.\nIn tropical geometry, there are several analogous objects.\nThe simplest one is the concept of ideal of a semiring, which is defined as a submodule of the semiring itself.\nAnother one is the concept of tropical ideal, which is defined by Maclagan and Rinc\\'{o}n in \\cite{maclagan2018tropical}.\nIt is an analogue of an ideal of a ring since the quotient semiring $R/E$ is defined.\nMoreover, there are several specific class of congruences.\nIn \\cite{giansiracusa2016equations}, J. Giansiracusa and N. Giansiracusa define \\textit{bend congruences} to develop the theory of scheme theoretic tropicalization.\nAnother important class is the congruences of the form $\\mathbf{E}(Z)$ for some $Z \\subset \\mathbb R^n$.\nHere,\n$$\\mathbf{E}(Z) = \\{ (f,g) \\in \\mathbb T[x_1, \\ldots, x_n] \\ | \\ f|_Z = g|_Z \\}.$$\nThe semiring $\\mathbb T[x_1, \\ldots, x_n]$ is sometimes replaced by another one such as the Laurent polynomial semiring $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$, the rational function semifield $\\mathbb T(x_1, \\ldots, x_n)$, the polynomial ring $\\mathbb B[x_1, \\ldots, x_n]$ over the Boolean semifield, or the polynomial function semiring $\\mathbb T[x_1, \\ldots, x_n]_{\\mathrm{fcn}}$.\nThis is a simple analogue of the concept of vanishing ideal in algebraic geometry.\nThe congruences of this form are studied by Bertram and Easton in \\cite{bertram2017tropical}, Jo\\'{o} and Mincheva in \\cite{joo2018prime}, Song in \\cite{song2024congruences}, and the author in \\cite{ito2022local}.\nAlso, \\textit{prime congruence} are defined in \\cite{joo2018prime}, which are of course analogues of the prime ideals of rings.\n\nAlmost all rings appeared in classical algebraic geometry are Noetherian, and their ideals are finitely generated.\nThis fact is used to develop the dimension theory.\nHence it is natural that we consider whether a given ideal (or congruence) is finitely generated.\nThere are few known results.\nIn \\cite[Example 3.10]{maclagan2018tropical}, an example of tropical ideal which is not finitely generated is given.\nAlso, it holds by definition that a bend relation defined by a polynomial is finitely generated.\nIn this paper, we consider the congruences on $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$ of the form $\\mathbf E(Z)$.\nIn particular, we consider the case that $Z$ is the support of a tropical variety.\nThe following is our main result.", "context": "Tropical geometry is a subject of mathematics developed in recent decades, which studies geometric objects called tropical varieties.\nSometimes tropical geometry is called \\textquotedblleft the algebraic geometry over $\\mathbb T$\\textquotedblright, where $\\mathbb T$ is a semifield called the tropical semifield.\nSince $\\mathbb T$ is not a field, tropical geometry is not included in classical algebraic geometry.\nHowever, for many albedo-geometric theories, their tropical versions are studied.\nFor instance, the tropical intersection theory \\cite{allermann2010first,jensen2013computing,jensen2016stable,katz2012tropical,rau2016intersections}, the theory of divisors on tropical curves \\cite{amini2013reduced,an2014canonical,haase2012linear,hladky2013rank,len2014brill}, and the theory of tropical homology \\cite{amini2020hodge,gross2019sheaf,itenberg2019tropical,ruddat2021homology} have been developed.\n\nOne of the basic objects in algebraic geometry is the ideals of rings.\nIn tropical geometry, there are several analogous objects.\nThe simplest one is the concept of ideal of a semiring, which is defined as a submodule of the semiring itself.\nAnother one is the concept of tropical ideal, which is defined by Maclagan and Rinc\\'{o}n in \\cite{maclagan2018tropical}.\nIt is an analogue of an ideal of a ring since the quotient semiring $R/E$ is defined.\nMoreover, there are several specific class of congruences.\nIn \\cite{giansiracusa2016equations}, J. Giansiracusa and N. Giansiracusa define \\textit{bend congruences} to develop the theory of scheme theoretic tropicalization.\nAnother important class is the congruences of the form $\\mathbf{E}(Z)$ for some $Z \\subset \\mathbb R^n$.\nHere,\n$$\\mathbf{E}(Z) = \\{ (f,g) \\in \\mathbb T[x_1, \\ldots, x_n] \\ | \\ f|_Z = g|_Z \\}.$$\nThe semiring $\\mathbb T[x_1, \\ldots, x_n]$ is sometimes replaced by another one such as the Laurent polynomial semiring $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$, the rational function semifield $\\mathbb T(x_1, \\ldots, x_n)$, the polynomial ring $\\mathbb B[x_1, \\ldots, x_n]$ over the Boolean semifield, or the polynomial function semiring $\\mathbb T[x_1, \\ldots, x_n]_{\\mathrm{fcn}}$.\nThis is a simple analogue of the concept of vanishing ideal in algebraic geometry.\nThe congruences of this form are studied by Bertram and Easton in \\cite{bertram2017tropical}, Jo\\'{o} and Mincheva in \\cite{joo2018prime}, Song in \\cite{song2024congruences}, and the author in \\cite{ito2022local}.\nAlso, \\textit{prime congruence} are defined in \\cite{joo2018prime}, which are of course analogues of the prime ideals of rings.\n\nAlmost all rings appeared in classical algebraic geometry are Noetherian, and their ideals are finitely generated.\nThis fact is used to develop the dimension theory.\nHence it is natural that we consider whether a given ideal (or congruence) is finitely generated.\nThere are few known results.\nIn \\cite[Example 3.10]{maclagan2018tropical}, an example of tropical ideal which is not finitely generated is given.\nAlso, it holds by definition that a bend relation defined by a polynomial is finitely generated.\nIn this paper, we consider the congruences on $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$ of the form $\\mathbf E(Z)$.\nIn particular, we consider the case that $Z$ is the support of a tropical variety.\nThe following is our main result.", "full_context": "Tropical geometry is a subject of mathematics developed in recent decades, which studies geometric objects called tropical varieties.\nSometimes tropical geometry is called \\textquotedblleft the algebraic geometry over $\\mathbb T$\\textquotedblright, where $\\mathbb T$ is a semifield called the tropical semifield.\nSince $\\mathbb T$ is not a field, tropical geometry is not included in classical algebraic geometry.\nHowever, for many albedo-geometric theories, their tropical versions are studied.\nFor instance, the tropical intersection theory \\cite{allermann2010first,jensen2013computing,jensen2016stable,katz2012tropical,rau2016intersections}, the theory of divisors on tropical curves \\cite{amini2013reduced,an2014canonical,haase2012linear,hladky2013rank,len2014brill}, and the theory of tropical homology \\cite{amini2020hodge,gross2019sheaf,itenberg2019tropical,ruddat2021homology} have been developed.\n\nOne of the basic objects in algebraic geometry is the ideals of rings.\nIn tropical geometry, there are several analogous objects.\nThe simplest one is the concept of ideal of a semiring, which is defined as a submodule of the semiring itself.\nAnother one is the concept of tropical ideal, which is defined by Maclagan and Rinc\\'{o}n in \\cite{maclagan2018tropical}.\nIt is an analogue of an ideal of a ring since the quotient semiring $R/E$ is defined.\nMoreover, there are several specific class of congruences.\nIn \\cite{giansiracusa2016equations}, J. Giansiracusa and N. Giansiracusa define \\textit{bend congruences} to develop the theory of scheme theoretic tropicalization.\nAnother important class is the congruences of the form $\\mathbf{E}(Z)$ for some $Z \\subset \\mathbb R^n$.\nHere,\n$$\\mathbf{E}(Z) = \\{ (f,g) \\in \\mathbb T[x_1, \\ldots, x_n] \\ | \\ f|_Z = g|_Z \\}.$$\nThe semiring $\\mathbb T[x_1, \\ldots, x_n]$ is sometimes replaced by another one such as the Laurent polynomial semiring $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$, the rational function semifield $\\mathbb T(x_1, \\ldots, x_n)$, the polynomial ring $\\mathbb B[x_1, \\ldots, x_n]$ over the Boolean semifield, or the polynomial function semiring $\\mathbb T[x_1, \\ldots, x_n]_{\\mathrm{fcn}}$.\nThis is a simple analogue of the concept of vanishing ideal in algebraic geometry.\nThe congruences of this form are studied by Bertram and Easton in \\cite{bertram2017tropical}, Jo\\'{o} and Mincheva in \\cite{joo2018prime}, Song in \\cite{song2024congruences}, and the author in \\cite{ito2022local}.\nAlso, \\textit{prime congruence} are defined in \\cite{joo2018prime}, which are of course analogues of the prime ideals of rings.\n\nAlmost all rings appeared in classical algebraic geometry are Noetherian, and their ideals are finitely generated.\nThis fact is used to develop the dimension theory.\nHence it is natural that we consider whether a given ideal (or congruence) is finitely generated.\nThere are few known results.\nIn \\cite[Example 3.10]{maclagan2018tropical}, an example of tropical ideal which is not finitely generated is given.\nAlso, it holds by definition that a bend relation defined by a polynomial is finitely generated.\nIn this paper, we consider the congruences on $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$ of the form $\\mathbf E(Z)$.\nIn particular, we consider the case that $Z$ is the support of a tropical variety.\nThe following is our main result.\n\nAlmost all rings appeared in classical algebraic geometry are Noetherian, and their ideals are finitely generated.\nThis fact is used to develop the dimension theory.\nHence it is natural that we consider whether a given ideal (or congruence) is finitely generated.\nThere are few known results.\nIn \\cite[Example 3.10]{maclagan2018tropical}, an example of tropical ideal which is not finitely generated is given.\nAlso, it holds by definition that a bend relation defined by a polynomial is finitely generated.\nIn this paper, we consider the congruences on $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$ of the form $\\mathbf E(Z)$.\nIn particular, we consider the case that $Z$ is the support of a tropical variety.\nThe following is our main result.\n\nThus, for almost all tropical variety $X$, $\\mathbf E(X)$ is not finitely generated.\nThe only if part of this theorem is a consequence of the following more general result.\n\n\\begin{thm}\n \\label{main}\n Let $X$ be a tropical variety in $\\mathbb R^n$.\n Then $\\mathbf E(X)$ is finitely generated if and only if $|X|$ is a rational affine linear subspace of $\\mathbb R^n$.\n\\end{thm}\n\nHence, for a subset $Z \\subset W$, in order to show that $\\mathbf E(Z)$ is not finitely generated, it is enough to show that $\\mathbf E(\\iota^{-1}(Z))$ is not finitely generated.\nThus we may assume that $Z$ is not included in any proper rational affine linear subspace of $\\mathbb R^n$.\nNote that if $Z$ is the support of a tropical variety in $\\mathbb R^n$, then so is $\\iota^{-1}(Z)$ by Lemma \\ref{trop var reduction}.\n\nLet $X$ be a tropical variety in $\\mathbb R^n$.\nAssume that $W := |X|$ is a $d$-dimensional rational affine linear subspace of $\\mathbb R^n$.\nBy Lemma \\ref{translation}, we may assume that $\\mathbf 0 \\in W$, i.e., $W$ is a linear subspace of $\\mathbb R^n$.\nSince $W$ is rational, the orthogonal complement $W'$ (with respect to the standard inner product) of $W$ is also rational.\nWe use the notations $\\Lambda(W) = W \\cap \\mathbb Z^n$ and $\\Lambda(W') = W' \\cap \\mathbb Z^n$.\nLet $\\{ \\bm q_1, \\ldots, \\bm q_{n-d} \\}$ be a basis of $\\Lambda(W')$.\nWe see that the congruence $\\mathbf E(W)$ is generated by the finite set $S := \\left\\{ \\left(\\overline{\\bm x^{\\bm q_i}}, 0 \\right) \\ | \\ i=1, \\ldots, n-d \\right\\}$.\n\n\\begin{prop}\n \\label{trop unbounded}\n Let $X$ be a tropical variety in $\\mathbb R^n$.\n Assume that $|X|$ is not included in a proper rational affine linear subspace of $\\mathbb R^n$.\n Then $|X|$ is unbounded in all rational directions.\n\\end{prop}\n\n\\begin{proof}[Proof of \\textquotedblleft Proposition \\ref{trop unbounded in f} $\\Longrightarrow$ Proposition \\ref{trop unbounded}\\textquotedblright]\n Assume that Proposition \\ref{trop unbounded in f} holds.\n Let $X$ be a tropical variety in $\\mathbb R^n$ whose support is not included in a proper rational affine linear subspace of $\\mathbb R^n$.\n Take any $\\bm p \\in \\mathbb Z^n \\setminus \\{ \\mathbf 0 \\}$.\n If the function $f: |X| \\to \\mathbb R, \\bm a \\mapsto \\bm a \\cdot \\bm p$ is constant with the value $k$, then $|X|$ is included in the set $\\{ \\bm a \\in \\mathbb R^n \\ | \\ \\bm a \\cdot \\bm p = k \\}$, which is a proper rational affine linear subspace of $\\mathbb R^n$.\n This contradicts to the assumption, hence $f$ is not constant.\n Thus $f$ is upper unbounded.\n Since $\\bm p$ is an arbitrary element of $\\mathbb Z^n \\setminus \\{ \\mathbf 0 \\}$, $|X|$ is unbounded in all rational directions.\n\\end{proof}\n\n\\begin{proof}[Proof of the only if part of Theorem \\ref{main}]\n Let $X$ be a tropical variety such that $|X|$ is not a rational affine linear subspace of $\\mathbb R^n$.\n We may assume that $|X|$ is not included in any proper rational affine linear subspace of $\\mathbb R^n$ by Lemma \\ref{fg reduction} and the observation after it.\n Hence, by Proposition \\ref{trop unbounded}, $|X|$ is unbounded in all rational directions.\n In order to use Theorem \\ref{not fg}, we check that $n \\geq 2$.\n If $n =0$, $|X| = \\{ \\mathbf 0 \\}$.\n It contradicts to the assumption that $|X|$ is not an affine linear subspace.\n If $n =1$, $|X|$ is a singleton or the whole space $\\mathbb R$ (by Lemma \\ref{n in n}).\n It also contradicts to the assumption.\n Thus $n \\geq 2$, and hence by Theorem \\ref{not fg}, $\\mathbf E(X)$ is not finitely generated. \\qedhere\n\\end{proof}", "post_theorem_intro_text_len": 2612, "post_theorem_intro_text": "Thus, for almost all tropical variety $X$, $\\mathbf E(X)$ is not finitely generated.\nThe only if part of this theorem is a consequence of the following more general result.\n\n\\begin{thm}\n  \\label{main2}\n  Assume that $n \\geq 2$.\n  Let $Z$ be a proper closed subset of $\\mathbb R^n$ that is unbounded in all rational directions.\n  Then $\\mathbf E(Z)$ is not finitely generated.\n\\end{thm}\n\nThe strategy of the proof of Theorem \\ref{main2} is as follows.\nFirst we assume that $\\mathbf E(Z)$ is finitely generated and fix a generating set.\nThen we construct a pair $(f,g) \\in \\mathbf E(Z)$ such that $f \\neq g$ and the distance of any two lattice points of the Newton polytope of $f$ is sufficiently large.\nThe existence of such pair causes a contradiction.\nSee Section 3 for detail.\n\nNote that we cannot apply Theorem \\ref{main2} for showing the only if part of Theorem \\ref{main1} directly because $|X|$ is not unbounded in all rational directions in general.\nIn particular, if $|X|$ is included in a proper rational affine linear subspace of $\\mathbb R^n$, then $|X|$ is not unbounded in all rational directions.\nTo overcome this issue, we show that we may replace the ambient space with a lower dimensional one.\nSee Section 4 for detail.\n\nFinally, we give an example of minimal generating set of $\\mathbf E(L)$ for the standard tropical line $L \\subset \\mathbb R^2$.\nSee Section \\ref{section min gen} for detail.\n\n\\begin{thm}\n  \\label{main3}\n  The set\n  $$\\mathcal S = \\left\\{ \\left( \\overline{0 \\oplus x^u y^v}, f_{\\Delta(\\{ (0,0), (u,v) \\})} \\right) \\ \\middle| \\ \\begin{aligned}\n  &\\text{$u$ and $v$ are integers with $\\gcd(u,v) = 1$,}\\\\\n  &\\text{ $u > 0$ or $(u,v) = (0,1)$}\n  \\end{aligned} \\right\\}$$\n  is a minimal generating set of $\\mathbf E(L)$.\n\\end{thm}\n\nThis paper is organized as follows.\nIn Section 2, we recall some basic definitions and facts about semirings, tropical algebra, tropical varieties, and Newton polytopes.\nAlso we show some lemmas we need later.\nIn Section 3, we consider a way to prove that a given congruence on $\\mathbb T[x_1^{\\pm}, \\ldots, x_n^{\\pm}]$ is not finitely generated, and use it to show Theorem \\ref{main2}.\nIn Section 4, we discuss about the reduction of the ambient spaces.\nThe result of this section is necessary for showing Theorem \\ref{main1}.\nIn Section 5, we prove Theorem \\ref{main1}.\nIn Section 6, we give an example of minimal generating set of $\\mathbf E(L)$ for the standard tropical line $L$.\nThis section is almost independent of Section 3-5.\nIn Appendix, we give proofs of some facts about free abelian groups and polytopes, which are used in this paper.", "sketch": "For Theorem~\\ref{main1}, the text says: “The only if part of this theorem is a consequence of the following more general result” (Theorem~\\ref{main2}), but “we cannot apply Theorem~\\ref{main2} for showing the only if part of Theorem~\\ref{main1} directly because $|X|$ is not unbounded in all rational directions in general.” “To overcome this issue, we show that we may replace the ambient space with a lower dimensional one.”\n\nFor Theorem~\\ref{main2}, the stated proof strategy is: “First we assume that $\\mathbf E(Z)$ is finitely generated and fix a generating set. Then we construct a pair $(f,g) \\in \\mathbf E(Z)$ such that $f \\neq g$ and the distance of any two lattice points of the Newton polytope of $f$ is sufficiently large. The existence of such pair causes a contradiction.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{main1}\n  Let $X$ be a tropical variety in $\\mathbb R^n$.\n  Then $\\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is a rational affine linear subspace of $\\mathbb R^n$.", "theorem_type": ["Biconditional or Equivalence", "Universal"], "mcq": {"question": "Let $X$ be a tropical variety in $\\mathbb R^n$, and let $|X|$ denote its support. For a subset $Z\\subset \\mathbb R^n$, define the congruence\n$$\\mathbf E(Z)=\\{(f,g)\\in \\mathbb T[x_1^{\\pm},\\ldots,x_n^{\\pm}]\\mid f(\\mathbf a)=g(\\mathbf a)\\text{ for all }\\mathbf a\\in Z\\},$$\nwhere $\\mathbb T[x_1^{\\pm},\\ldots,x_n^{\\pm}]$ is the tropical Laurent polynomial semiring. A rational affine linear subspace of $\\mathbb R^n$ means an affine translate of a linear subspace defined over $\\mathbb Q$ (equivalently, whose direction is spanned by rational vectors). Which statement holds for every such tropical variety $X$?", "correct_choice": {"label": "A", "text": "$\\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is a rational affine linear subspace of $\\mathbb R^n$."}, "choices": [{"label": "B", "text": "$\\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is an affine linear subspace of $\\mathbb R^n$."}, {"label": "C", "text": "If $|X|$ is a rational affine linear subspace of $\\mathbb R^n$, then $\\mathbf E(|X|)$ is finitely generated."}, {"label": "D", "text": "$\\mathbf E(|X|)$ is finitely generated whenever $|X|$ is contained in a rational affine linear subspace of $\\mathbb R^n$."}, {"label": "E", "text": "$\\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is a finite union of rational affine linear subspaces of $\\mathbb R^n$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "rationality of affine linear subspace", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "only-if direction", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "equality with a rational affine linear subspace replaced by mere containment", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "single rational affine linear subspace replaced by finite union", "template_used": "stronger_trap"}]}}
{"id": "2512.21603v1", "paper_link": "http://arxiv.org/abs/2512.21603v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\nFor a skew-symmetrizable matrix $B$, the following are equivalent:\n\\begin{itemize}\n\\item[$(1)$] $B$ is of finite type.\n\\item[$(2)$] The $g$-fan $\\mathcal{F}(B)$ is complete, that is, its support $|\\mathcal{F}(B)|=\\mathbb{R}^n$.\n\\item[$(3)$] The support $|\\mathcal{F}(B)|$ contains all lattice points in $\\mathbb{Z}^n$.\n\\end{itemize}", "start_pos": 9579, "end_pos": 9957, "label": "thm:main"}, "ref_dict": {"thm:Reading": "\\begin{theorem}[{\\cite[Theorem~10.6]{Re14}}]\\label{thm:Reading}\nIf $B$ is of finite type, then $|\\mathcal{F}(B)|=\\mathbb{R}^n$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 1259, "pre_theorem_intro_text": "Cluster algebras \\cite{FZ02} have been studied from many points of view. One of the basic objects in this theory is the notion of $g$-vectors \\cite{FZ07}.\n\nLet $B$ be a skew-symmetrizable matrix. The $g$-vectors associated with $B$ span simplicial cones in $\\mathbb{R}^n$. These cones form a simplicial fan, which is called the $g$-fan of $B$ (see \\cite{GHKK18,Re14}). The geometry of this fan encodes the structure of the associated cluster algebra.\n\nA skew-symmetrizable matrix $B$ is said to be of \\emph{finite type} if it has only finitely many $g$-vectors, or equivalently, finitely many cluster variables. Finite type matrices are well understood and play an important role in cluster theory.\n\nIt is then natural to ask how the finite type property of $B$ is reflected in the geometry of its $g$-fan. In particular, one may ask whether the $g$-fan covers the whole space $\\mathbb{R}^n$, or whether it can be characterized by the property of containing all lattice points in $\\mathbb{Z}^n$. This question is also motivated by \\cite[Question~3.49]{D17} in representation theory. In this paper, we give a complete answer to this question in the framework of $g$-fans arising from skew-symmetrizable matrices. More precisely, we prove the following theorem.", "context": "Cluster algebras \\cite{FZ02} have been studied from many points of view. One of the basic objects in this theory is the notion of $g$-vectors \\cite{FZ07}.\n\nLet $B$ be a skew-symmetrizable matrix. The $g$-vectors associated with $B$ span simplicial cones in $\\mathbb{R}^n$. These cones form a simplicial fan, which is called the $g$-fan of $B$ (see \\cite{GHKK18,Re14}). The geometry of this fan encodes the structure of the associated cluster algebra.\n\nA skew-symmetrizable matrix $B$ is said to be of \\emph{finite type} if it has only finitely many $g$-vectors, or equivalently, finitely many cluster variables. Finite type matrices are well understood and play an important role in cluster theory.\n\nIt is then natural to ask how the finite type property of $B$ is reflected in the geometry of its $g$-fan. In particular, one may ask whether the $g$-fan covers the whole space $\\mathbb{R}^n$, or whether it can be characterized by the property of containing all lattice points in $\\mathbb{Z}^n$. This question is also motivated by \\cite[Question~3.49]{D17} in representation theory. In this paper, we give a complete answer to this question in the framework of $g$-fans arising from skew-symmetrizable matrices. More precisely, we prove the following theorem.", "full_context": "Cluster algebras \\cite{FZ02} have been studied from many points of view. One of the basic objects in this theory is the notion of $g$-vectors \\cite{FZ07}.\n\nLet $B$ be a skew-symmetrizable matrix. The $g$-vectors associated with $B$ span simplicial cones in $\\mathbb{R}^n$. These cones form a simplicial fan, which is called the $g$-fan of $B$ (see \\cite{GHKK18,Re14}). The geometry of this fan encodes the structure of the associated cluster algebra.\n\nA skew-symmetrizable matrix $B$ is said to be of \\emph{finite type} if it has only finitely many $g$-vectors, or equivalently, finitely many cluster variables. Finite type matrices are well understood and play an important role in cluster theory.\n\nIt is then natural to ask how the finite type property of $B$ is reflected in the geometry of its $g$-fan. In particular, one may ask whether the $g$-fan covers the whole space $\\mathbb{R}^n$, or whether it can be characterized by the property of containing all lattice points in $\\mathbb{Z}^n$. This question is also motivated by \\cite[Question~3.49]{D17} in representation theory. In this paper, we give a complete answer to this question in the framework of $g$-fans arising from skew-symmetrizable matrices. More precisely, we prove the following theorem.\n\n\\begin{abstract}\nWe study the $g$-fan associated with a skew-symmetrizable matrix in the sense of cluster algebras. We show that a skew-symmetrizable matrix is of finite type if and only if its $g$-fan is complete; equivalently (as we show), its support contains all lattice points.\n\\end{abstract}\n\nIt is then natural to ask how the finite type property of $B$ is reflected in the geometry of its $g$-fan. In particular, one may ask whether the $g$-fan covers the whole space $\\mathbb{R}^n$, or whether it can be characterized by the property of containing all lattice points in $\\mathbb{Z}^n$. This question is also motivated by \\cite[Question~3.49]{D17} in representation theory. In this paper, we give a complete answer to this question in the framework of $g$-fans arising from skew-symmetrizable matrices. More precisely, we prove the following theorem.\n\nThe implication $(1)\\Rightarrow(2)$ is known (Theorem~\\ref{thm:Reading}). For skew-symmetric matrices, the implication $(2)\\Rightarrow(1)$ was proved in \\cite{HY25} via categorification using Jacobian algebras. In \\cite{Y23}, we suggested that this implication might be proved by methods similar to those in \\cite[Proposition~4.9]{A21} in general, but no complete proof was available. In this paper, we give a complete proof using scattering diagrams and their pull-back constructions.\n\nAn $n\\times n$ skew-symmetrizable matrix $B$ is called\n\\begin{itemize}\n\\item \\emph{mutation equivalent} to a matrix $B'$ if it is obtained from $B'$ by a finite sequence of mutations;\n\\item \\emph{$2$-finite} if for any matrix $B'=(b'_{ij})$ mutation equivalent to $B$, $|b'_{ij}b'_{ji}|\\le 3$ for all $i,j$.\n\\end{itemize}\n\n\\begin{definition}\nA \\emph{$g$-vector seed} is a pair $(C,(\\mathbf{g}_1,\\ldots,\\mathbf{g}_n))$ consisting of the following data:\n\\begin{itemize}\n\\item[$(1)$] $C$ is a $2n\\times n$ integer matrix whose upper part is skew-symmetrizable.\n\\item[$(2)$] $\\mathbf{g}_1,\\ldots,\\mathbf{g}_n\\in\\mathbb Z^n$.\n\\end{itemize}\nThe tuple $(\\mathbf{g}_1,\\ldots,\\mathbf{g}_n)$ is called the \\emph{$g$-vector tuple}.\n\\end{definition}\n\n\\begin{example}\\label{exam:rank 2 g-fan}\nAny nonzero $2 \\times 2$ skew-symmetrizable matrix is given by\n\\[\nB_{b,c}:=\\begin{bmatrix}0 & c \\\\ -b & 0\\end{bmatrix},\n\\]\nwhere $b$ and $c$ are integers and $bc > 0$. The $g$-fan $\\mathcal{F}(B_{b,c})$ is well-known (see e.g. \\cite[Example~1.15]{GHKK18} or \\cite[Section~2]{N24}). In fact, for $bc \\ge 4$, $\\mathcal{F}(B_{b,c})$ contains infinitely many rays converging to the rays $r_{\\pm}$ of slope $(-bc \\pm \\sqrt{bc(bc-4)})/2c$ (see Figure~\\ref{fig:rank 2 g-vector fans}). In this case, the ray of slope $-b/2$ is contained in $(\\mathbb{R}^2\\setminus|\\mathcal{F}(B_{b,c})|)\\cup\\{0\\}$. In particular, the lattice point $(-2,b)$ is contained in $\\mathbb{Z}^n\\setminus|\\mathcal{F}(B_{b,c})|$.\n\\end{example}\n\n\\begin{theorem}[{\\cite[Theorem~10.6]{Re14}}]\\label{thm:Reading}\nIf $B$ is of finite type, then $|\\mathcal{F}(B)|=\\mathbb{R}^n$.\n\\end{theorem}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:main}]\nThe implication $(1)\\Rightarrow(2)$ follows from Theorem~\\ref{thm:Reading}, and $(2)\\Rightarrow(3)$ clearly holds. Finally, we prove $(3)\\Rightarrow(1)$. By Theorem~\\ref{thm:fin 2-fin}, if $B$ is not of finite type, it is mutation equivalent to $B'$ such that $B'_I=B_{b,c}$ with $bc\\ge 4$ for some subset $I \\subset \\{1,\\ldots,n\\}$. By Proposition~\\ref{prop:rank 2 no lattice pt}, $\\mathbb{Z}^n\\setminus|\\mathcal{F}(B')|\\neq\\emptyset$, which implies that $\\mathbb{Z}^n\\setminus|\\mathcal{F}(B)|\\neq\\emptyset$ by Corollary~\\ref{cor:preserve lattice pt}. This completes the proof.\n\\end{proof}", "post_theorem_intro_text_len": 485, "post_theorem_intro_text": "The implication $(1)\\Rightarrow(2)$ is known (Theorem~\\ref{thm:Reading}). For skew-symmetric matrices, the implication $(2)\\Rightarrow(1)$ was proved in \\cite{HY25} via categorification using Jacobian algebras. In \\cite{Y23}, we suggested that this implication might be proved by methods similar to those in \\cite[Proposition~4.9]{A21} in general, but no complete proof was available. In this paper, we give a complete proof using scattering diagrams and their pull-back constructions.", "sketch": "The post-theorem text indicates: $(1)\\Rightarrow(2)$ is already known (Theorem~\\ref{thm:Reading}). For skew-symmetric matrices, $(2)\\Rightarrow(1)$ was proved in \\cite{HY25} \"via categorification using Jacobian algebras.\" A possible approach \"by methods similar to those in \\cite[Proposition~4.9]{A21}\" had been suggested in \\cite{Y23} but lacked a complete proof. The paper claims to provide a complete proof \"using scattering diagrams and their pull-back constructions.\"", "expanded_sketch": "The post-theorem text indicates: $(1)\\Rightarrow(2)$ is already known. \\begin{theorem}[{\\cite[Theorem~10.6]{Re14}}]\\label{thm:Reading}\nIf $B$ is of finite type, then $|\\mathcal{F}(B)|=\\mathbb{R}^n$.\n\\end{theorem}\nFor skew-symmetric matrices, $(2)\\Rightarrow(1)$ was proved in Assem and Yurikusa, “via categorification using Jacobian algebras.” A possible approach “by methods similar to those in \\cite[Proposition~4.9]{A21}” had been suggested in Yurikusa, but lacked a complete proof. The paper claims to provide a complete proof “using scattering diagrams and their pull-back constructions.”", "expanded_theorem": "\\label{thm:main}\nFor a skew-symmetrizable matrix $B$, the following are equivalent:\n\\begin{itemize}\n\\item[$(1)$] $B$ is of finite type.\n\\item[$(2)$] The $g$-fan $\\mathcal{F}(B)$ is complete, that is, its support $|\\mathcal{F}(B)|=\\mathbb{R}^n$.\n\\item[$(3)$] The support $|\\mathcal{F}(B)|$ contains all lattice points in $\\mathbb{Z}^n$.\n\\end{itemize}", "theorem_type": ["Biconditional or Equivalence", "Classification or Bijection"], "mcq": {"question": "Let $B$ be a skew-symmetrizable matrix. Its $g$-vectors span simplicial cones in $\\mathbb{R}^n$, and these cones form the $g$-fan $\\mathcal{F}(B)$; write $|\\mathcal{F}(B)|$ for the support, i.e. the union of all cones in the fan. A matrix $B$ is said to be of finite type if it has only finitely many $g$-vectors (equivalently, finitely many cluster variables). Which of the following statements is equivalent to both the assertion that $B$ is of finite type and the assertion that the $g$-fan $\\mathcal{F}(B)$ is complete, meaning $|\\mathcal{F}(B)|=\\mathbb{R}^n$?", "correct_choice": {"label": "A", "text": "The support $|\\mathcal{F}(B)|$ contains all lattice points in $\\mathbb{Z}^n$."}, "choices": [{"label": "B", "text": "The support $|\\mathcal{F}(B)|$ contains all but finitely many lattice points of $\\mathbb{Z}^n$."}, {"label": "C", "text": "The support $|\\mathcal{F}(B)|$ contains every nonzero lattice point in $\\mathbb{Z}^n$."}, {"label": "D", "text": "For every skew-symmetrizable matrix $B'$ mutation equivalent to $B$, the support $|\\mathcal{F}(B')|$ contains all lattice points in $\\mathbb{Z}^n$."}, {"label": "E", "text": "The support $|\\mathcal{F}(B)|$ is dense in $\\mathbb{R}^n$, that is, every nonempty open subset of $\\mathbb{R}^n$ meets $|\\mathcal{F}(B)|$."}], "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": "all_lattice_points", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "characteristic", "tampered_component": "inclusion_of_the_origin", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "mutation_invariance_shift_from_B_to_all_Bprime", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "completeness_replaced_by_density", "template_used": "wildcard"}]}}
{"id": "2512.22662v1", "paper_link": "http://arxiv.org/abs/2512.22662v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm} If $C$ is stably embedded in $\\mathbf{M}$ and $\\mu$ an $A$-valued Fubini measure on $\\operatorname{Def}(C)$,\nthen $\\mu$ extends uniquely to an $A$-valued Fubini measure on $\\operatorname{Def}(C)^{\\operatorname{f}}$.", "start_pos": 15706, "end_pos": 15914, "label": "thm"}, "ref_dict": {"co": "\\begin{proof} For every $p\\in M^k$ the set $\\cal{S}(p)$ has by Lemma~\\ref{l5} an $(r+1)$-step fibering over $C$. As in the proof of Lemma~\\ref{l5}, this yields a finite set $E$ of tuples \n$$\\big( f_1,\\dots, f_{r+1}, f_{r+2}; m_1,\\dots,m_{r+1}, n_1,\\dots, n_{r+1},n_{r+2}\\big)$$\nwhere for $j=1,\\dots, r+2$: $f_j$ is a definable map $ \\Sigma_j\\to M^{n_i}$ with (definable) domain \n$\\Sigma_j\\subseteq M^l\\times M^{m_1}\\times\\cdots\\times M^{m_{j-1}}\\times M^{m+n_1+\\cdots + n_{j-1}}$, and for each $p\\in M^k$ there is \na tuple in $E$ as displayed and $q\\in M^{l}$\nsuch that $\\cal{S}(p)=\\Sigma_1(q)$ and the tuple\n$$ \\big(f_1(q,-),\\dots, f_{r+1}(q,-), f_{r+2}(q,-); m_1,\\dots, m_{r+1}, n_1,\\dots, n_{r+1}, n_{r+2}\\big)$$\nis an $(r+1)$-step fibering of the domain $\\Sigma_1(q)\\subseteq M^m$ of $f_1(q,-)$ over $C$.  \n\nConsider one such tuple $\\tau\\in E$ as displayed above. Let $Q$ be the definable\nset of $q\\in M^{l}$ such that\nfor some $p\\in M^k$ we have $\\cal{S}(p)=\\Sigma_1(q)$ and \n$$ \\big(f_1(q,-),\\dots, f_{r+1}(q,-), f_{r+2}(q,-); m_1,\\dots, m_{r+1}, n_1,\\dots, n_{r+1},n_{r+2}\\big)$$\nis an $(r+1)$-step fibering of the domain $\\Sigma_1(q)\\subseteq M^m$ of $f_1(q,-)$ over $C$.\nEach $q\\in Q$ yields a map\n$f_1(q,-): \\Sigma_1(q)\\to M^{n_1}$ such that its preimages of points $y\\in M^{n_1}$ are in $\\cC_r$. The maps $f_1(q,-)$ with $q\\in Q$ make up a definable family $\\cal{F}_\\tau$ of maps from subsets of $M^m$ into $M^{n_1}$ as described just before Lemma~\\ref{l2+}. That lemma then tells us that $\\{\\mu\\big(\\Sigma_1(q)\\big): q\\in Q\\}$ is finite, and that\n$\\{q\\in Q:\\ \\mu\\big(\\Sigma_1(q)\\big)=a\\}$ is definable for every $a$. This holds for all $\\tau\\in E$, which yields the\ndesired result.\n\\end{proof} \n\n\\noindent\nThis finishes the inductive proof of Theorem~\\ref{thm} in the case of $\\omega$-saturated \n$\\cM$. \n\n\\subsection*{Reduction to the $\\omega$-saturated case}  Here $C$ is stably embedded in $\\cM$, but we do not assume $\\cM$ is $\\omega$-saturated. \n\nLet $\\cM^*$ be an elementary extension of $\\cM$. We define an $A$-valued Fubini measure $\\mu^*$ on  the full subcategory \n$\\Def(C^*)$ of $\\Def(\\cM^*)$: Let $Y\\subseteq (C^*)^n$ be definable in $\\cM^*$; take an $\\cal{L}$-formula $\\phi(x,y)$ with $x=(x_1,\\dots, x_m)$, $y=(y_1,\\dots, y_n)$, and a point $p_Y\\in (C^*)^m$ such that $Y=\\{q\\in (C^*)^n: \\cM^*\\models \\phi(p_Y,q)\\}$. To simplify notation, set $\\phi(p, C^n):=\\{q\\in C^n:\\ \\cM\\models \\phi(p,q)\\}$ for $p\\in C^m$, and likewise define\n$\\phi\\big(p, (C^*)^n\\big)\\subseteq (C^*)^n$ for $p\\in (C^*)^m$; in particular, $Y=\\phi\\big(p_Y, (C^*)^n\\big)$. \nNow $C^m$ is the disjoint union of the definable sets $X_a:=\\{p\\in C^m:\\ \\mu\\big(\\phi(p,C^m)\\big)=a\\}$ (which is empty for all but finitely many $a$). \nSet $\\mu^*(Y):=a$, where $a$ is unique such that $p_Y\\in (X_a)^*$. It is routine to verify that this particular $a$ does not depend on the choice of $\\phi$ and that it yields indeed a Fubini measure on $\\Def(C^*)$. \n\nNow take such an elementary extension $\\cM^*$ to be $\\omega$-saturated. Then $\\mu^*$ extends uniquely to an $A$-valued Fubini measure $\\mu^*$ on the full subcategory $\\Def(C^*)^{\\f}$ of $\\Def(\\cM^*)$. \nFor $X\\in \\Def(C)^{\\f}$ we have $X^*\\in \\Def(C^*)^{\\f}$; set $\\mu(X):= \\mu^*(X^*)$.\nThis extends the original $A$-valued Fubini measure $\\mu$ on $\\Def(C)$ to an $A$-valued Fubini measure $\\mu$ on $\\Def(C)^{\\f}$.  As to uniqueness, suppose $X$ is fiberable over $C$, and take an $r$-step fibering $(f_1,\\dots, f_r, f_{r+1}; m_1,\\dots, m_{r}, n_1,\\dots, n_{r+1})$\nof $X$ over $C$. \nFor $r=0$ the definable map $f_1: X \\to C^{n_1}$ is injective, and so by Fubini\n$\\mu(X)$ is uniquely determined by $\\mu: \\Def(C)\\to A$. For $r\\ge 1$ the definable map $f_1: X\\to M^{n_1}$ is such that\n$f_1(X)$ and every\n$f_1^{-1}(y)$ with $y\\in f_1(X)$ is $(r-1)$-step fiberable over $C$, and we can assume inductively that the values of \n$\\mu\\big(f(X)\\big)$ and $\\mu\\big(f_1^{-1}(y)\\big)$ for\n$y\\in f_1(X)$ are uniquely determined by $\\mu: \\Def(C)\\to A$, which again by Fubini uniquely deternines $\\mu(X)$. \nThis concludes the proof of Theorem~\\ref{thm}. \n\n\\section{Fiberability and Co-analyzability}\\label{co}\n\n\\noindent\nFiberability is a somewhat ad hoc notion, but agrees under natural model-theoretic conditions with the more intrinsic notion of co-analyzability from \\cite{HHM}, as we shall see. {\\it Throughout this section $X\\subseteq M^m$ is definable}.  \n\n\\subsection*{Fiberability implies co-analyzability}\nWe recall here the recursion on $r$ that defines ``$X$ is co-analyzable in $r$ steps\" (tacitly: relative to $C$ in $\\cM$), where $\\cM$ is assumed to be $\\omega$-saturated: \\begin{enumerate}\n\\item  $X$ is co-analyzable in $0$ steps iff $X$ is finite;\n\\item  $X$ is co-analyzable in $r+1$ steps iff for some definable set $R \\subseteq M^m\\times C$ we have: $\\breve{R}(y)$ is co-analyzable in $r$ steps for every $y\\in C$, and \n$X:= \\bigcup_{y\\in C}\\breve{R}(y)$. \n\\end{enumerate} \n See \\cite[p. 35]{ADH} for how this extends to not necessarily $\\omega$-saturated $\\cM$ in such a way that\n for any elementary extension $\\cM^*$ of $\\cM$: $X$ is co-analyzable in $r$ steps relative to $C$ in $\\cM$ iff $X^*$ is co-analyzable in $r$ steps relative to $C^*$ in $\\cM^*$.  Call $X$ {\\em co-analyzable\\/} if $X$ is co-analyzable in $r$ steps for some $r$. \n\n By \\cite{HHM} this notion can be characterized as follows:\n\n \\begin{prop}\\label{hhm} \nLet $T$ be a complete $L$-theory, where $L$ is a countable one-sorted language. Let $y$ be a single variable and $D(y)$ an $L$-formula that defines in each model $\\cal{N}=(N;\\cdots)$ of $T$ a set $D(N)$ with $|D(N)|\\ge 2$.\nThen the following conditions on an $L$-formula $\\varphi(x)$ with $x=(x_1,\\dots,x_m)$ are equivalent:\n\\begin{enumerate}\n\\item[\\textup{(i)}] for some  $\\cal{N}\\models T$, $\\varphi(N^m)$ is co-analyzable relative to $D(N)$;\n\\item[\\textup{(ii)}] for all $\\cal{N}\\models T$, $\\varphi(N^m)$ is co-analyzable relative to $D(N)$;\n\\item[\\textup{(iii)}] for all  $\\cal{N}\\models T$, if \n$D(N)$ is countable, then so is $\\varphi(N^m)$;\n\\item[\\textup{(iv)}] for all $\\cal{N}\\preceq\\cal{N}^*\\models T$, if\n$D(N)=D(N^*)$, then $\\varphi(N^m)=\\varphi((N^*)^m)$.\n\\end{enumerate}\n\\end{prop}\n\n\\noindent\nThe equivalences (i)$\\Leftrightarrow$(ii)$\\Leftrightarrow$(iv) hold without assuming that $L$ is countable:\nFor (i)$\\Leftrightarrow$(ii)$\\Rightarrow$(iv), reduce to countable $L$ by taking suitable reducts. For (iv)$\\Rightarrow$(ii), use also that (iv) amounts to $T_2\\models \\sigma$ where $T_2$ is the theory of elementary pairs of models of $T$ and $\\sigma$ is a certain sentence, depending on $\\varphi$, in the language of $T_2$. Note that in addition we can restrict $\\cal{N}$ and $\\cal{N}^*$ in (iv) to be  $\\omega$-saturated.\n\n\\begin{lemma}\\label{fico} If $X$ is fiberable over $C$, then  $X$ is co-analyzable relative to $C$. \n\\end{lemma} \n\\begin{proof} \nWe arrange that $\\cM$ is $\\omega$-saturated. \nBelow we prove by induction on $r$:  if $X$ is $r$-step fiberable over $C$, then $X$ is co-analyzable relative to $C$. \nFor $r=0$, if $f: X\\to C^n$ is definable and injective, then $X$ is co-analyzable in $n$ steps relative to $C$, by an easy induction on $n$. Thus the claimed implication holds for $r=0$. Assume this implication holds for a certain $r$, and let\n$X$ be $(r+1)$-step fiberable over $C$.\nTo show $X$ is co-analyzable relative to $C$ in $\\cM$ we \naugment $\\cal{L}$ by names for finitely many elements of $M$ to arrange that $X$ is $0$-definable, so $X=\\varphi(M^m)$ with $\\varphi(x)$ an $\\cal{L}$-formula, $x=(x_1,\\dots, x_m)$. We now apply (iv)$\\Rightarrow$(ii) of Proposition~\\ref{hhm} without assuming $L=\\cal{L}$ is countable, with $T:=\\Th(\\cM)$ and $D(y)$ an $\\cal{L}$-formula such that $D(M)=C$. \nSo let $\\cal{N}\\preceq \\cal{N}^*\\models T$ with $D(N)=D(N^*)$; it suffices to show that then $\\varphi(N^m)=\\varphi\\big((N^*)^m\\big)$.\nWe also assume $\\cal{N}$ is $\\omega$-saturated, as we may by a remark following Proposition~\\ref{hhm}. Since $\\cal{M}\\equiv \\cal{N}$, the set $\\varphi(N^m)$ is $(r+1)$-step fiberable over $C^{\\cal{N}}:= D(N)$ in $\\cal{N}$. Take $f: \\varphi(N^m)\\to Y$ where $Y\\subseteq N^n$ and $f$ are definable in $\\cal{N}$, such that $Y$ is $r$-step fiberable over $C^{\\cal{N}}$ and for all\n$y\\in Y$ the fiber $f^{-1}(y)$ is $r$-step fiberable over $C^{\\cal{N}}$. Thus by our inductive assumption and Proposition~\\ref{hhm}, $Y$ and the fibers\n$f^{-1}(y)$ for $y\\in Y$ do not change in passing from $\\cal{N}$ to $\\cal{N}^*$. Hence the same is true for $\\varphi(N^m)$, that is,\n$\\varphi(N^m)=\\varphi\\big((N^*)^m\\big)$. \n\\end{proof}", "proof": "\\begin{lemma}  If $A, B$ are commutative semirings and $\\mu: \\cC\\to A$ and $\\nu: \\cC\\to B$ are Fubini measures, then $(\\mu,\\nu): \\cC\\to A\\times B$ is a Fubini measure. \n\\end{lemma} \n\n\\section{Proof of the Theorem}\\label{proof}\n\n\\noindent\n{\\it In this section $\\mu$ is an $A$-valued Fubini measure on $\\Def(C)$}. Till further notice we also assume $\\cM$ is $\\omega$-saturated, indicating at the end of the section how to reduce to that case. \n\n\\medskip\\noindent\nBy recursion on $r$ we now introduce a sequence $(\\cC_r)$ of full subcategories of $\\Def(\\cM)$, such that $ \\cC_r\\subseteq \\cC_{r+1}$ for all $r$ and $\\bigcup_r \\cC_r=\\Def(C)^{\\f}$:\n\n\\medskip\\noindent\n $\\cC_0:= \\Def(C)^{\\iso}$, and the objects of $\\cC_{r+1}$ are the definable $X\\subseteq M^m$ for which there exists a definable\n$f: X\\to Y$ with $Y\\in \\cC_r$, such that $f^{-1}(y)\\in  \\cC_r$ for all $y\\in Y$; such $f$ is called a $\\cC_{r+1}$-witness. Thus the objects of $\\cC_r$ are the definable $X\\subseteq M^m$ that are $r$-step fiberable over $C$ as defined in the Introduction.\nIn particular, the objects of $\\cC_0$ are the definable $X\\subseteq M^m$ that admit a definable injective map $X\\to C^n$. Also, if the definable set $X\\subseteq M^m$ admits a definable map $X\\to C^n$ all whose fibers are finite, then $X\\in \\cC_1$.  \n\n\\medskip\\noindent\n Induction on $r$ shows that $\\cC_r$ satisfies conditions (a), (b), (c) for all $r$, and that $\\cC_r^{\\iso}=\\cC_r$ for all $r$.  In the inductive step we note that if $f: X\\to Y$ is a $\\cC_{r+1}$-witness and $X'\\subseteq X$ and $Y'\\subseteq Y$ are definable with\n$f(X')\\subseteq Y'$, then $f|_{X'}: X'\\to Y'$ is also a $\\cC_{r+1}$-witness. Induction on $r$ also shows that there is at most one\n$A$-valued Fubini measure on $\\cC_r$ extending the given $A$-valued Fubini measure $\\mu$ on $\\Def(C)$. In the beginning of the previous section we have extended $\\mu$ to an $A$-valued Fubini measure on $\\cC_0$, also to be denoted by $\\mu$. \n\n\\subsection*{More about $\\cC_0$} {\\it In this subsection we assume that $C$ is stably embedded in $\\cM$}.\nLet  $\\cal{S}\\subseteq M^k\\times C^n$ be definable. Think of $\\cal{S}$ as the definable family of sets\n$\\cal{S}(p)\\subseteq C^n$ parametrized by the points $p\\in M^k$. \n\n\\begin{lemma}\\label{l1}  The set $\\{\\mu\\big(\\cal{S}(p)\\big):\\ p\\in M^k\\}$ is finite, and for every $a$ the set $\\{p\\in M^k:\\ \\mu\\big(\\cal{S}(p)\\big)=a\\}$ is definable.\n\\end{lemma}", "ms": "\\label{ms}  \n\n\\noindent\nSo far we assumed $\\cM$ to be one-sorted. But many structures are more naturally viewed as many-sorted; for example, valued fields as $3$-sorted structures with residue field a", "prelim": "\\label{prelim}\n\n\\noindent\nIt will be convenient to consider measures that do not necessarily satisfy (F3). We call  them {\\em semi-Fubini measures\\/} and they are defined in the first subsection below", "thm": "\\begin{theorem}\\label{thm} If $C$ is stably embedded in $\\cM$ and $\\mu$ an $A$-valued Fubini measure on $\\Def(C)$,\nthen $\\mu$ extends uniquely to an $A$-valued Fubini measure on $\\Def(C)^{\\f}$. \n\\end{theorem}", "special": "\\begin{proof} Take definable $f: Y\\to M^m$ with $Y\\subseteq C^\\nu$, $\\nu\\in \\N$, such that $X=f(Y)$. Since $(C;\\cM)$ has EI, we have a definable map $g:Y\\to C^n $ such that for all $y,z\\in Y$ we have $f(y)=f(z)\\Leftrightarrow g(y)=g(z)$. This yields a definable bijection $$\\qquad\\qquad \\qquad \\qquad f(y)\\mapsto g(y)\\ :\\  X\\to g(Y),\\qquad (y\\in Y). \\qquad \\qquad \\qquad \\qquad \\qedhere$$\n\\end{proof} \n\n\\subsection*{Analyzability} \nThe concept of {\\em analyzability\\/} is more popular in model theory than co-analyzability, but is usually defined for types\nin the environment of a monster model, instead of definable sets. I have been told that for definable sets and $\\omega$-saturated $\\cM$ its recursive definition should be as follows: \n\n{\\em $X$ is $0$-step analyzable in $C$ iff $X$ is internal to $C$, and $X$ is $(r+1)$-step analyzable in $C$ iff there is a $\\Def(\\cM)$-morphism $f: X \\to Y$ such that $Y$ is $r$-step analyzable over $C$ and all fibers $f^{-1}(y)$, $y\\in Y$, are internal to $C$. Finally, $X$ is analyzable in $C$ iff $X$ is $r$-step analyzable in $C$ for some $r$}. \n\nThis extends to not necessarily $\\omega$-saturated $\\cM$ by requiring in the $(r+1)$-clause the existence of a definable \nfamily of maps from subsets of $C^n$ (for a fixed $n$) into $M^m$ such that for each $y\\in Y$ some member of the family has image $f^{-1}(y)$. \nWith this extended notion and any elementary extension $\\cM^*$ of $\\cM$, \n$$X \\text{ is $r$-step analyzable in }C\\ \\Longleftrightarrow\\  X^* \\text{ is $r$-step analyzable in }C^*.$$ \nUsing this it follows from Proposition~\\ref{hhm} that if $X$ is analyzable in $C$, then $X$ is co-analyzable relative to $C$. Ben Castle and Rahim Moosa have assured me that the converse should also hold under some natural model-theoretic conditions to be satisfied in particular by differentially closed fields (and probably by $\\T$) with $C$ its constant field. I have not pursued this, but it would be good to have this confirmed. One motivation for this: it might help in showing that for $\\T$ and for differentially closed fields the notions of  fiberability over $C$ and co-analyzability relative to $C$ do not collapse for any $r$ to $r$-step fiberability over $C$ and co-analyzability in $r$-steps relative to $C$, respectively. This is plausible in view of \\cite{J} where Jin constructs in a ``big\" differentially closed field for any $r\\ge 1$ a $1$-type that is $r$-step analyzable but not $(r-1)$-step analyzable (in the constant field). \n\n\\section{Application to $\\T$ and to differentially closed fields} \\label{special} \n\n\\subsection*{The case of $\\T$} We construe here $\\T$ as an ordered differential field. Its subfield $\\R$ is its constant field, and\na set $Y\\subseteq \\R^n$ is definable in $\\T$ iff $X$ is semialgebraic in the sense of $\\R$. In particular, $\\R$ is stably embedded in\n$\\T$. Thus by Theorem~\\ref{thm}, the $\\N_{\\trop}$-valued Fubini measure $Y\\mapsto \\dim Y$ on $\\Def(\\R)$ extends uniquely to an $\\N_{\\trop}$-valued Fubini measure on the full subcategory $\\Def(\\R)^{\\f}$ of $\\Def(\\T)$. Likewise for the $\\Z$-valued Fubini measure $Y\\mapsto E(Y)$ on $\\Def(\\R)$. \n\nAll this holds also for any $\\upo$-free, newtonian, Liouville closed $H$-field $K$ and its constant field instead of\n$\\T$ and its constant field $\\R$; this includes all ordered differential fields $K \\equiv\\T$.  The extended Euler characteristic now yields an answer to a test question raised in \\cite[p. 34, end of Section 5]{ADH}:\n\n\\begin{cor} With $K$ as above, let $X\\subseteq K$ be definable in $K$. Then $X$ with the ordering induced by $K$\nis not elementarily equivalent to the ordered set $\\omega$. \n\\end{cor}\n\\begin{proof} Suppose $X\\equiv \\omega$. Then $X$ is discrete as a subspace of $K$ with its order topology. Hence $X$ is fiberable\nover $C$ by \\cite{ADH}. Now $X\\equiv \\omega$ also yields  a definable bijection between $X$ and $X\\setminus \\{\\text{the least element of $X$}\\}$, but the extended Euler characteristic then gives $E(X)=E(X)-1$, a contradiction. \n\\end{proof}"}, "pre_theorem_intro_text_len": 5546, "pre_theorem_intro_text": "\\label{int}\n\n\\noindent\nSemialgebraic sets $Y\\subseteq  {\\bbold R} ^n$ have a dimension $\\dim(Y)\\in  {\\bbold N} \\cup\\{-\\infty\\}$ and an Euler characteristic $E(Y)\\in  {\\bbold Z} $ with good properties; see \\cite[Ch. 4]{D}. In \\cite{ADH} we asked for a natural extension of this dimension and Euler characteristic to the definable sets $X\\subseteq  {\\bbold T} ^m$\nthat are ``fiberable by $ {\\bbold R} $\" as defined there; here $ {\\bbold T} $ is the real closed differential field of transseries, which has its subfield $ {\\bbold R} $ as constant field. \n\nThere are many similar situations, and so it makes sense to do the above as a special case of a general extension result for what we call {\\em Fubini measures}, with ``Fubini'' signaling good behavior in families. For example, at the end of Section~\\ref{special} we shall also apply this to differentially closed fields (models of $\\operatorname{DCF}_0$).   \n\n\\medskip\\noindent\nLet $\\mathbf{M}=(M;\\cdots)$ be a one-sorted $\\mathcal{L}$-structure;\n``definable\" will mean ``definable in $\\mathbf{M}$ with parameters\" unless we indicate otherwise.  For definable $X\\subseteq M^m$ and an elementary extension $\\mathbf{M}^*=(M^*;\\cdots)$ of $\\mathbf{M}$ we let $X^*\\subseteq (M^*)^m$ be defined in $\\mathbf{M}^*$ by any $\\mathcal{L}_M$-formula that defines $X$ in $\\mathbf{M}$. \nThe (small) category $\\operatorname{Def}(\\mathbf{M})$ has as its objects the definable sets $X\\subseteq M^m$; the number $m\\in  {\\bbold N} $ is part of  specifying $X$ as an object of $\\operatorname{Def}(\\mathbf{M})$.  For objects $X,Y$ of $\\operatorname{Def}(\\mathbf{M})$, the morphisms $X\\to Y$ are the definable maps $X\\to Y$,\nand composition of morphisms is composition of maps, so an isomorphism $X\\to Y$ is the same as a definable bijection $X\\to Y$. We shall abuse language by\nindicating the class of objects of a category $\\mathcal{C}$ also by $\\mathcal{C}$, so $X\\in \\mathcal{C}$ means that $X$ is an object of $\\mathcal{C}$. \n\n\\medskip\\noindent\nLet $\\cal{C}$ be a full subcategory of $\\operatorname{Def}(\\mathbf{M})$ such that for all $X,Y\\in \\cal{C}$: \\begin{itemize}\n\\item[(a)] if $X, Y\\subseteq M^m$, then $X\\cup Y\\subseteq M^m$ is an object of $\\cal{C}$; \n\\item[(b)] every definable subset of $X$ is an object of $\\cal{C}$;\n\\item[(c)]  $X\\times Y\\in \\cal{C}$ (where $X\\times Y\\subseteq  M^{m+n}$ for $X\\subseteq M^m, Y\\subseteq M^n$).\n\\end{itemize} \nLet $A$ be a commutative semiring, that is, a set with binary operations of addition and multiplication and distinguished elements\n$0$ and $1$ such that $A$ is a commutative monoid with respect to both addition and multiplication, with $0$ and $1$ as neutral elements for addition and multiplication, respectively, such that multiplication is distributive with respect to addition, and such that also $0\\cdot a=0$ for all $a\\in A$.\n\nLet  $\\mu$ an $A$-valued Fubini measure on $\\cal{C}$, that is, $\\mu: \\cal{C}\\to A$, and for all $X,Y\\in  \\cal{C}$:\\begin{enumerate} \n\\item[(d)] $\\mu(X)=0$ whenever $X$ is empty; $\\mu(X)=1$ whenever $X$ is a singleton; \n\\item[(e)] $\\mu(X\\cup Y)=\\mu(X)+\\mu(Y)$ whenever $X,Y\\subseteq M^m$ are disjoint;\n\\item[(f)] every $\\cal{C}$-morphism $f: X \\to Y$ satisfies ``Fubini\": \\begin{enumerate}\n\\item[(F1)] the subset $\\{\\mu\\big(f^{-1}(y)\\big):\\ y\\in Y\\}$ of $A$ is finite;\n\\item[(F2)] for all $a\\in A$, the set \n$Y_a := \\{y\\in Y:\\ \\mu\\big(f^{-1}(y)\\big)=a\\}$ is definable;\n\\item[(F3)] if $\\mu\\big(f^{-1}(y)\\big)$ takes the constant value $a$ on $Y$, then\n$\\mu(X)= a\\mu(Y)$.\n\\end{enumerate} \n\\end{enumerate} \nThus for $X,Y\\in  \\cal{C}$:  $\\mu(X\\times Y)=\\mu(X)\\cdot \\mu(Y)$; if $X$ and $Y$ are $\\cal{C}$-isomorphic, then $\\mu(X)=\\mu(Y)$ (a ``motivic'' property);  for $X,Y\\subseteq M^m$ we can represent $X\\cup Y$ as the disjoint union of $X\\setminus Y, X\\cap Y$, and $Y\\setminus X$ to obtain\n$$\\mu(X\\cup Y)+\\mu(X\\cap Y)\\ =\\ \\mu(X)+\\mu(Y).$$ \nIf $f: X\\to Y$ is a $\\cal{C}$-morphism, then $\\mu(X)=\\sum_a a\\mu(Y_a)$, using  notation of (F2). \n\n\\medskip\\noindent\n{\\bf Example.} Let $\\mathbf{M}= {\\bbold T} $ as in the beginning of the introduction. Then the subsets of $ {\\bbold R} ^n$ definable in $\\mathbf{M}$ are  semialgebraic in the sense of $ {\\bbold R} $, so the full subcategory $\\cal{C}$ of $\\operatorname{Def}(\\mathbf{M})$\nwhose objects are these semialgebraic sets satisfies conditions (a), (b), (c). The above Euler characteristic $Y\\mapsto E(Y)$ is then a Fubini measure on $\\cal{C}$ with values in the ring $ {\\bbold Z} $ of integers.  Also $Y\\mapsto \\dim(Y)$ is a Fubini measure on $\\cal{C}$, taking values in the tropical semiring $\\N_{\\operatorname{trop}}$, which has underlying set\n$ {\\bbold N} \\cup\\{-\\infty\\}$ and addition and multiplication given respectively by\n$(p,q)\\mapsto \\max(p,q)$ and $(p,q)\\mapsto p+q$; this semiring has $-\\infty$ as its zero element for addition and $0$ as its multiplicative identity. \n\n\\bigskip\\noindent\nTo state the main result of this paper, we consider a $0$-definable set $C\\subseteq M$ with more than one element, and let $\\operatorname{Def}(C)$ be the full subcategory of $\\operatorname{Def}(\\mathbf{M})$ whose objects are the definable subsets of the powers $C^n$.  Below we define the notion, for definable $X\\subseteq M^m$, of {\\em $X$ being fiberable over $C$}. These $X$ are the objects of a full subcategory\n$\\operatorname{Def}(C)^{\\operatorname{f}}\\supseteq  \\operatorname{Def}(C)$ of $\\operatorname{Def}(\\mathbf{M})$. The categories $\\operatorname{Def}(C)$ and $\\operatorname{Def}(C)^{\\operatorname{f}}$ satisfy conditions (a), (b), (c)  imposed earlier on $\\cal{C}$.", "context": "There are many similar situations, and so it makes sense to do the above as a special case of a general extension result for what we call {\\em Fubini measures}, with ``Fubini'' signaling good behavior in families. For example, at the end of Section~\\ref{special} we shall also apply this to differentially closed fields (models of $\\operatorname{DCF}_0$).\n\n\\medskip\\noindent\nLet $\\mathbf{M}=(M;\\cdots)$ be a one-sorted $\\mathcal{L}$-structure;\n``definable\" will mean ``definable in $\\mathbf{M}$ with parameters\" unless we indicate otherwise. For definable $X\\subseteq M^m$ and an elementary extension $\\mathbf{M}^*=(M^*;\\cdots)$ of $\\mathbf{M}$ we let $X^*\\subseteq (M^*)^m$ be defined in $\\mathbf{M}^*$ by any $\\mathcal{L}_M$-formula that defines $X$ in $\\mathbf{M}$. \nThe (small) category $\\operatorname{Def}(\\mathbf{M})$ has as its objects the definable sets $X\\subseteq M^m$; the number $m\\in {\\bbold N} $ is part of specifying $X$ as an object of $\\operatorname{Def}(\\mathbf{M})$. For objects $X,Y$ of $\\operatorname{Def}(\\mathbf{M})$, the morphisms $X\\to Y$ are the definable maps $X\\to Y$,\nand composition of morphisms is composition of maps, so an isomorphism $X\\to Y$ is the same as a definable bijection $X\\to Y$. We shall abuse language by\nindicating the class of objects of a category $\\mathcal{C}$ also by $\\mathcal{C}$, so $X\\in \\mathcal{C}$ means that $X$ is an object of $\\mathcal{C}$.\n\n\\medskip\\noindent\nLet $\\cal{C}$ be a full subcategory of $\\operatorname{Def}(\\mathbf{M})$ such that for all $X,Y\\in \\cal{C}$: \\begin{itemize}\n\\item[(a)] if $X, Y\\subseteq M^m$, then $X\\cup Y\\subseteq M^m$ is an object of $\\cal{C}$; \n\\item[(b)] every definable subset of $X$ is an object of $\\cal{C}$;\n\\item[(c)] $X\\times Y\\in \\cal{C}$ (where $X\\times Y\\subseteq M^{m+n}$ for $X\\subseteq M^m, Y\\subseteq M^n$).\n\\end{itemize} \nLet $A$ be a commutative semiring, that is, a set with binary operations of addition and multiplication and distinguished elements\n$0$ and $1$ such that $A$ is a commutative monoid with respect to both addition and multiplication, with $0$ and $1$ as neutral elements for addition and multiplication, respectively, such that multiplication is distributive with respect to addition, and such that also $0\\cdot a=0$ for all $a\\in A$.\n\nLet $\\mu$ an $A$-valued Fubini measure on $\\cal{C}$, that is, $\\mu: \\cal{C}\\to A$, and for all $X,Y\\in \\cal{C}$:\\begin{enumerate} \n\\item[(d)] $\\mu(X)=0$ whenever $X$ is empty; $\\mu(X)=1$ whenever $X$ is a singleton; \n\\item[(e)] $\\mu(X\\cup Y)=\\mu(X)+\\mu(Y)$ whenever $X,Y\\subseteq M^m$ are disjoint;\n\\item[(f)] every $\\cal{C}$-morphism $f: X \\to Y$ satisfies ``Fubini\": \\begin{enumerate}\n\\item[(F1)] the subset $\\{\\mu\\big(f^{-1}(y)\\big):\\ y\\in Y\\}$ of $A$ is finite;\n\\item[(F2)] for all $a\\in A$, the set \n$Y_a := \\{y\\in Y:\\ \\mu\\big(f^{-1}(y)\\big)=a\\}$ is definable;\n\\item[(F3)] if $\\mu\\big(f^{-1}(y)\\big)$ takes the constant value $a$ on $Y$, then\n$\\mu(X)= a\\mu(Y)$.\n\\end{enumerate} \n\\end{enumerate} \nThus for $X,Y\\in \\cal{C}$: $\\mu(X\\times Y)=\\mu(X)\\cdot \\mu(Y)$; if $X$ and $Y$ are $\\cal{C}$-isomorphic, then $\\mu(X)=\\mu(Y)$ (a ``motivic'' property); for $X,Y\\subseteq M^m$ we can represent $X\\cup Y$ as the disjoint union of $X\\setminus Y, X\\cap Y$, and $Y\\setminus X$ to obtain\n$$\\mu(X\\cup Y)+\\mu(X\\cap Y)\\ =\\ \\mu(X)+\\mu(Y).$$ \nIf $f: X\\to Y$ is a $\\cal{C}$-morphism, then $\\mu(X)=\\sum_a a\\mu(Y_a)$, using notation of (F2).\n\n\\medskip\\noindent\n{\\bf Example.} Let $\\mathbf{M}= {\\bbold T} $ as in the beginning of the introduction. Then the subsets of $ {\\bbold R} ^n$ definable in $\\mathbf{M}$ are semialgebraic in the sense of $ {\\bbold R} $, so the full subcategory $\\cal{C}$ of $\\operatorname{Def}(\\mathbf{M})$\nwhose objects are these semialgebraic sets satisfies conditions (a), (b), (c). The above Euler characteristic $Y\\mapsto E(Y)$ is then a Fubini measure on $\\cal{C}$ with values in the ring $ {\\bbold Z} $ of integers. Also $Y\\mapsto \\dim(Y)$ is a Fubini measure on $\\cal{C}$, taking values in the tropical semiring $\\N_{\\operatorname{trop}}$, which has underlying set\n$ {\\bbold N} \\cup\\{-\\infty\\}$ and addition and multiplication given respectively by\n$(p,q)\\mapsto \\max(p,q)$ and $(p,q)\\mapsto p+q$; this semiring has $-\\infty$ as its zero element for addition and $0$ as its multiplicative identity.\n\n\\bigskip\\noindent\nTo state the main result of this paper, we consider a $0$-definable set $C\\subseteq M$ with more than one element, and let $\\operatorname{Def}(C)$ be the full subcategory of $\\operatorname{Def}(\\mathbf{M})$ whose objects are the definable subsets of the powers $C^n$. Below we define the notion, for definable $X\\subseteq M^m$, of {\\em $X$ being fiberable over $C$}. These $X$ are the objects of a full subcategory\n$\\operatorname{Def}(C)^{\\operatorname{f}}\\supseteq \\operatorname{Def}(C)$ of $\\operatorname{Def}(\\mathbf{M})$. The categories $\\operatorname{Def}(C)$ and $\\operatorname{Def}(C)^{\\operatorname{f}}$ satisfy conditions (a), (b), (c) imposed earlier on $\\cal{C}$.", "full_context": "There are many similar situations, and so it makes sense to do the above as a special case of a general extension result for what we call {\\em Fubini measures}, with ``Fubini'' signaling good behavior in families. For example, at the end of Section~\\ref{special} we shall also apply this to differentially closed fields (models of $\\operatorname{DCF}_0$).\n\n\\medskip\\noindent\nLet $\\mathbf{M}=(M;\\cdots)$ be a one-sorted $\\mathcal{L}$-structure;\n``definable\" will mean ``definable in $\\mathbf{M}$ with parameters\" unless we indicate otherwise. For definable $X\\subseteq M^m$ and an elementary extension $\\mathbf{M}^*=(M^*;\\cdots)$ of $\\mathbf{M}$ we let $X^*\\subseteq (M^*)^m$ be defined in $\\mathbf{M}^*$ by any $\\mathcal{L}_M$-formula that defines $X$ in $\\mathbf{M}$. \nThe (small) category $\\operatorname{Def}(\\mathbf{M})$ has as its objects the definable sets $X\\subseteq M^m$; the number $m\\in {\\bbold N} $ is part of specifying $X$ as an object of $\\operatorname{Def}(\\mathbf{M})$. For objects $X,Y$ of $\\operatorname{Def}(\\mathbf{M})$, the morphisms $X\\to Y$ are the definable maps $X\\to Y$,\nand composition of morphisms is composition of maps, so an isomorphism $X\\to Y$ is the same as a definable bijection $X\\to Y$. We shall abuse language by\nindicating the class of objects of a category $\\mathcal{C}$ also by $\\mathcal{C}$, so $X\\in \\mathcal{C}$ means that $X$ is an object of $\\mathcal{C}$.\n\n\\medskip\\noindent\nLet $\\cal{C}$ be a full subcategory of $\\operatorname{Def}(\\mathbf{M})$ such that for all $X,Y\\in \\cal{C}$: \\begin{itemize}\n\\item[(a)] if $X, Y\\subseteq M^m$, then $X\\cup Y\\subseteq M^m$ is an object of $\\cal{C}$; \n\\item[(b)] every definable subset of $X$ is an object of $\\cal{C}$;\n\\item[(c)] $X\\times Y\\in \\cal{C}$ (where $X\\times Y\\subseteq M^{m+n}$ for $X\\subseteq M^m, Y\\subseteq M^n$).\n\\end{itemize} \nLet $A$ be a commutative semiring, that is, a set with binary operations of addition and multiplication and distinguished elements\n$0$ and $1$ such that $A$ is a commutative monoid with respect to both addition and multiplication, with $0$ and $1$ as neutral elements for addition and multiplication, respectively, such that multiplication is distributive with respect to addition, and such that also $0\\cdot a=0$ for all $a\\in A$.\n\nLet $\\mu$ an $A$-valued Fubini measure on $\\cal{C}$, that is, $\\mu: \\cal{C}\\to A$, and for all $X,Y\\in \\cal{C}$:\\begin{enumerate} \n\\item[(d)] $\\mu(X)=0$ whenever $X$ is empty; $\\mu(X)=1$ whenever $X$ is a singleton; \n\\item[(e)] $\\mu(X\\cup Y)=\\mu(X)+\\mu(Y)$ whenever $X,Y\\subseteq M^m$ are disjoint;\n\\item[(f)] every $\\cal{C}$-morphism $f: X \\to Y$ satisfies ``Fubini\": \\begin{enumerate}\n\\item[(F1)] the subset $\\{\\mu\\big(f^{-1}(y)\\big):\\ y\\in Y\\}$ of $A$ is finite;\n\\item[(F2)] for all $a\\in A$, the set \n$Y_a := \\{y\\in Y:\\ \\mu\\big(f^{-1}(y)\\big)=a\\}$ is definable;\n\\item[(F3)] if $\\mu\\big(f^{-1}(y)\\big)$ takes the constant value $a$ on $Y$, then\n$\\mu(X)= a\\mu(Y)$.\n\\end{enumerate} \n\\end{enumerate} \nThus for $X,Y\\in \\cal{C}$: $\\mu(X\\times Y)=\\mu(X)\\cdot \\mu(Y)$; if $X$ and $Y$ are $\\cal{C}$-isomorphic, then $\\mu(X)=\\mu(Y)$ (a ``motivic'' property); for $X,Y\\subseteq M^m$ we can represent $X\\cup Y$ as the disjoint union of $X\\setminus Y, X\\cap Y$, and $Y\\setminus X$ to obtain\n$$\\mu(X\\cup Y)+\\mu(X\\cap Y)\\ =\\ \\mu(X)+\\mu(Y).$$ \nIf $f: X\\to Y$ is a $\\cal{C}$-morphism, then $\\mu(X)=\\sum_a a\\mu(Y_a)$, using notation of (F2).\n\n\\medskip\\noindent\n{\\bf Example.} Let $\\mathbf{M}= {\\bbold T} $ as in the beginning of the introduction. Then the subsets of $ {\\bbold R} ^n$ definable in $\\mathbf{M}$ are semialgebraic in the sense of $ {\\bbold R} $, so the full subcategory $\\cal{C}$ of $\\operatorname{Def}(\\mathbf{M})$\nwhose objects are these semialgebraic sets satisfies conditions (a), (b), (c). The above Euler characteristic $Y\\mapsto E(Y)$ is then a Fubini measure on $\\cal{C}$ with values in the ring $ {\\bbold Z} $ of integers. Also $Y\\mapsto \\dim(Y)$ is a Fubini measure on $\\cal{C}$, taking values in the tropical semiring $\\N_{\\operatorname{trop}}$, which has underlying set\n$ {\\bbold N} \\cup\\{-\\infty\\}$ and addition and multiplication given respectively by\n$(p,q)\\mapsto \\max(p,q)$ and $(p,q)\\mapsto p+q$; this semiring has $-\\infty$ as its zero element for addition and $0$ as its multiplicative identity.\n\n\\bigskip\\noindent\nTo state the main result of this paper, we consider a $0$-definable set $C\\subseteq M$ with more than one element, and let $\\operatorname{Def}(C)$ be the full subcategory of $\\operatorname{Def}(\\mathbf{M})$ whose objects are the definable subsets of the powers $C^n$. Below we define the notion, for definable $X\\subseteq M^m$, of {\\em $X$ being fiberable over $C$}. These $X$ are the objects of a full subcategory\n$\\operatorname{Def}(C)^{\\operatorname{f}}\\supseteq \\operatorname{Def}(C)$ of $\\operatorname{Def}(\\mathbf{M})$. The categories $\\operatorname{Def}(C)$ and $\\operatorname{Def}(C)^{\\operatorname{f}}$ satisfy conditions (a), (b), (c) imposed earlier on $\\cal{C}$.\n\n\\bigskip\\noindent\nTo state the main result of this paper, we consider a $0$-definable set $C\\subseteq M$ with more than one element, and let $\\Def(C)$ be the full subcategory of $\\Def(\\cM)$ whose objects are the definable subsets of the powers $C^n$. Below we define the notion, for definable $X\\subseteq M^m$, of {\\em $X$ being fiberable over $C$}. These $X$ are the objects of a full subcategory\n$\\Def(C)^{\\f}\\supseteq \\Def(C)$ of $\\Def(\\cM)$. The categories $\\Def(C)$ and $\\Def(C)^{\\f}$ satisfy conditions (a), (b), (c) imposed earlier on $\\cC$.\n\n\\noindent\nSuppose $\\cM$ is $\\omega$-saturated, and let $X\\subseteq M^m$ be definable. By recursion on $r\\in \\N$ we define ``$X$ is $r$-step fiberable over $C$\": this means the existence of a definable $f: X \\to M^n$ such that \nif $r=0$, then $f$ is injective and $f(X)\\subseteq C^n$, and if $r\\ge 1$, then $f(X)$ and every fiber $f^{-1}(y)$, $y\\in M^n$,\nis $(r-1)$-step fiberable over $C$.\n\n\\begin{lemma}\\label{smfub} If $\\cM$ is strongly minimal $($which includes $M$ being infinite$)$, then $X\\mapsto \\operatorname{MR}(X)$ is an $\\N_{\\trop}$-valued Fubini measure on $\\Def(\\cM)$. \n\\end{lemma}\n\n\\subsection*{The case of $\\T$} We construe here $\\T$ as an ordered differential field. Its subfield $\\R$ is its constant field, and\na set $Y\\subseteq \\R^n$ is definable in $\\T$ iff $X$ is semialgebraic in the sense of $\\R$. In particular, $\\R$ is stably embedded in\n$\\T$. Thus by Theorem~\\ref{thm}, the $\\N_{\\trop}$-valued Fubini measure $Y\\mapsto \\dim Y$ on $\\Def(\\R)$ extends uniquely to an $\\N_{\\trop}$-valued Fubini measure on the full subcategory $\\Def(\\R)^{\\f}$ of $\\Def(\\T)$. Likewise for the $\\Z$-valued Fubini measure $Y\\mapsto E(Y)$ on $\\Def(\\R)$.\n\n\\begin{prop}\\label{uniq} If $\\operatorname{char}(\\k)>0$, then there is no $\\Z$-valued Fubini measure on $\\Def(\\k)$. If \n$\\operatorname{char}(\\k)=0$, then there is a unique $\\Z$-valued Fubini measure on $\\Def(\\k)$. \n\\end{prop} \n\\begin{proof} Suppose $\\mu$ is a $\\Z$-valued Fubini measure on $\\Def(\\k)$. Then $\\mu(\\k)=1$: \n if $\\operatorname{char}(\\k)\\ne 2$, the fibers of \n $y\\mapsto y^2: \\k\\to \\k$\nhave size $2$ except at $0$, so \n$$\\mu(\\k)\\ =\\ 1+2\\big(\\mu(\\k)-1\\big),$$\n and thus $\\mu(\\k)=1$; if $\\operatorname{char}(\\k)=2$, the map $y\\mapsto y^3: \\k\\to \\k$ gives likewise $\\mu(\\k)=1+3\\big(\\mu(\\k)-1\\big)$, again resulting in $\\mu(\\k)=1$.\n\n\\noindent\nLet now $\\k$ be a proper subfield of the algebraically closed field $K$, and $\\cM:=(K, \\k)$. Then\n$\\cM$ is $\\omega$-stable with $\\text{MR}(\\k)=1$, $\\text{MR}(K)=\\omega$, and a set $Y\\subseteq \\k^n$ is definable in $\\cM$ iff $Y$ is constructible in the sense of the algebraically closed field $\\k$. Thus $\\k$ is stably embedded in $\\cM$. \nLet $X\\subseteq K^m$ be definable. Then\n$$X\\text{ is almost internal to }\\k\\\n\\Leftrightarrow\\ X \\text{ is co-analyzable relative to }\\k\\\n \\Leftrightarrow\\ \\operatorname{MR}(X)<\\omega.$$\nThese equivalences are in \\cite{AD}, which also has an example of a definable $X\\subseteq K$ that is almost internal to $\\k$ but not internal to $\\k$. Another proof of the first equivalence using \\cite{P} was pointed out to me by Pillay. Lemmas~\\ref{fico} and ~\\ref{cofi} also give: \n$$X \\text{ is fiberable over }\\k\\ \\Longleftrightarrow\\ X \\text{ is co-analyzable relative to }\\k.$$ \n By Lemma~\\ref{smfub} the function $Y\\mapsto \\text{MR}(Y)$\non $\\Def(\\k)$ is a Fubini measure with values in $\\N_{\\trop}$. By Theorem~\\ref{thm} it extends uniquely to\nan $\\N_{\\trop}$-valued Fubini measures on $\\Def(\\k)^{\\f}$; this extension is necessarily\n$X\\mapsto \\text{MR}(X)$: the above relation to almost internality implies that it is a Fubini measure on $\\Def(\\k)^{\\f}$.\n\n\\medskip\\noindent\nNow assume $\\operatorname{char}(\\k)=0$. Then we have the unique $\\Z$-valued Fubini measure $Y\\mapsto E(Y)$ on $\\Def(\\k)$ from the proof of Proposition~\\ref{uniq}. By Theorem~\\ref{thm} it extends uniquely to a $\\Z$-valued Fubini measure on \n$\\Def(\\k)^{\\f}$. \nAlthough \"fiberable over $\\k$\" collapses here to ``almost internal to $\\k$\", we still need all of Section~\\ref{prelim} and much of Section~\\ref{proof} to obtain this extension result.\n\n\\medskip\n{\\em If $\\cC$ is stably embedded in $\\cM$ and $\\mu$ is an $A$-valued Fubini measure on $\\cC$,\nthen $\\mu$ extends uniquely to an $A$-valued Fubini measure on $\\cC^{\\f}$}.\n\n\\label{prelim}\n\n\\noindent\nIt will be convenient to consider measures that do not necessarily satisfy (F3). We call  them {\\em semi-Fubini measures\\/} and they are defined in the first subsection below\n\n\\begin{lemma}  If $A, B$ are commutative semirings and $\\mu: \\cC\\to A$ and $\\nu: \\cC\\to B$ are Fubini measures, then $(\\mu,\\nu): \\cC\\to A\\times B$ is a Fubini measure. \n\\end{lemma} \n\n\\section{Proof of the Theorem}\\label{proof}\n\n\\noindent\n{\\it In this section $\\mu$ is an $A$-valued Fubini measure on $\\Def(C)$}. Till further notice we also assume $\\cM$ is $\\omega$-saturated, indicating at the end of the section how to reduce to that case. \n\n\\medskip\\noindent\nBy recursion on $r$ we now introduce a sequence $(\\cC_r)$ of full subcategories of $\\Def(\\cM)$, such that $ \\cC_r\\subseteq \\cC_{r+1}$ for all $r$ and $\\bigcup_r \\cC_r=\\Def(C)^{\\f}$:\n\n\\medskip\\noindent\n $\\cC_0:= \\Def(C)^{\\iso}$, and the objects of $\\cC_{r+1}$ are the definable $X\\subseteq M^m$ for which there exists a definable\n$f: X\\to Y$ with $Y\\in \\cC_r$, such that $f^{-1}(y)\\in  \\cC_r$ for all $y\\in Y$; such $f$ is called a $\\cC_{r+1}$-witness. Thus the objects of $\\cC_r$ are the definable $X\\subseteq M^m$ that are $r$-step fiberable over $C$ as defined in the Introduction.\nIn particular, the objects of $\\cC_0$ are the definable $X\\subseteq M^m$ that admit a definable injective map $X\\to C^n$. Also, if the definable set $X\\subseteq M^m$ admits a definable map $X\\to C^n$ all whose fibers are finite, then $X\\in \\cC_1$.  \n\n\\medskip\\noindent\n Induction on $r$ shows that $\\cC_r$ satisfies conditions (a), (b), (c) for all $r$, and that $\\cC_r^{\\iso}=\\cC_r$ for all $r$.  In the inductive step we note that if $f: X\\to Y$ is a $\\cC_{r+1}$-witness and $X'\\subseteq X$ and $Y'\\subseteq Y$ are definable with\n$f(X')\\subseteq Y'$, then $f|_{X'}: X'\\to Y'$ is also a $\\cC_{r+1}$-witness. Induction on $r$ also shows that there is at most one\n$A$-valued Fubini measure on $\\cC_r$ extending the given $A$-valued Fubini measure $\\mu$ on $\\Def(C)$. In the beginning of the previous section we have extended $\\mu$ to an $A$-valued Fubini measure on $\\cC_0$, also to be denoted by $\\mu$. \n\n\\subsection*{More about $\\cC_0$} {\\it In this subsection we assume that $C$ is stably embedded in $\\cM$}.\nLet  $\\cal{S}\\subseteq M^k\\times C^n$ be definable. Think of $\\cal{S}$ as the definable family of sets\n$\\cal{S}(p)\\subseteq C^n$ parametrized by the points $p\\in M^k$. \n\n\\begin{lemma}\\label{l1}  The set $\\{\\mu\\big(\\cal{S}(p)\\big):\\ p\\in M^k\\}$ is finite, and for every $a$ the set $\\{p\\in M^k:\\ \\mu\\big(\\cal{S}(p)\\big)=a\\}$ is definable.\n\\end{lemma}\n\n\\begin{theorem}\\label{thm} If $C$ is stably embedded in $\\cM$ and $\\mu$ an $A$-valued Fubini measure on $\\Def(C)$,\nthen $\\mu$ extends uniquely to an $A$-valued Fubini measure on $\\Def(C)^{\\f}$. \n\\end{theorem}", "post_theorem_intro_text_len": 5494, "post_theorem_intro_text": "\\noindent\nSuppose $\\mathbf{M}$ is $\\omega$-saturated, and let $X\\subseteq M^m$ be definable. By recursion on $r\\in  {\\bbold N} $ we define ``$X$ is $r$-step fiberable over $C$\":  this means the existence of a definable $f: X \\to M^n$ such that \nif $r=0$, then $f$ is injective and $f(X)\\subseteq C^n$, and if $r\\ge 1$, then $f(X)$ and every fiber $f^{-1}(y)$, $y\\in M^n$,\nis $(r-1)$-step fiberable over $C$.\n\nNow drop the assumption that $\\mathbf{M}$ is $\\omega$-saturated, and let $X\\subseteq M^m$ be definable.\nThen the definition of ``$X$ is $r$-step fiberable\nover $C$'' (with $\\mathbf{M}$ as ambient structure) is more complicated, see end of Section~\\ref{proof}, but will be such that it is equivalent to ``$X^*$ is $r$-step fiberable  over $C^*$\" with $\\mathbf{M}^*$ as ambient structure, where $\\mathbf{M}^*=(M^*,\\dots)$ \nis any $\\omega$-saturated elementary extension of $\\mathbf{M}$. Define $X$ to be {\\em fiberable over $C$\\/} if  it is $r$-step fiberable over $C$ for some $r$. \n\nUsing this equivalence the proof of Theorem~\\ref{thm} reduces to the case where $\\mathbf{M}$ is $\\omega$-saturated, which is very convenient. The idea is to show by induction on $r$ that $\\mu$ extends to an $A$-valued Fubini measure on the full subcategory of $\\operatorname{Def}(\\mathbf{M})$ whose objects are the definable $X\\subseteq M^m$ that are $r$-step fiberable over $C$. In the step from $r$ to $r+1$ we consider an $(r+1)$-step fiberable $X\\subseteq M^m$ over $C$. This is witnessed by a definable $f: X\\to M^n$ such that $f(X)$ and all $f^{-1}(y)$ are $r$-step fiberable over $C$. A carefully chosen inductive assumption will tell us that $\\mu\\big(f(X)\\big)$ is defined and that $\\mu\\big(f^{-1}(y)\\big)$ is defined and takes only finitely many values for $y\\in f(X)$. Then we define \n$\\mu_f(X)$ as a suitably weighted sum of these values and show that $\\mu_f(X)$ does not depend on $f$, but only on $X$, and that the resulting $\\mu(X)=\\mu_f(X)$ defined for such $X$ preserves the Fubini property. We also have to show that the inductive assumption is preserved in this step from $r$ to $r+1$. All this is carried out in detail in Section~\\ref{proof}.  \n\nSection~\\ref{prelim} contains preliminary results. In section~\\ref{co} we relate fiberability over $C$ to the more intrinsic notion of {\\em co-analyzability relative to $C$\\/}  from \\cite{HHM}, and show their equivalence under various model theoretic conditions. These conditions are satisfied by $ {\\bbold T} $ and by differentially closed fields, with $C$ their constant field. This is discussed in Section~\\ref{special}, where it is used to answer questions left open in \\cite{ADH}. In the short last section~\\ref{ms} we state for the record a many-sorted version of Theorem~\\ref{thm}. \n\n\\medskip\\noindent\nThe  measures from  \\cite{MS, AMSW} also take values in semirings and satisfy {\\em Fubini}, but seem to be of a rather different nature. Nevertheless, some variant of Theorem~\\ref{thm} might hold in that setting. One could also take a look at motivic measures as in ~\\cite{HK}, and consider their ``Fubini\" nature, or lack of it. \n\n\\medskip\\noindent\nI thank Ben Castle, Artem Chernikov, Rahim Moosa, Anand Pillay, and Charles Steinhorn for useful feedback after a talk I gave on an earlier version of this paper. \n\n\\subsection*{Notations and Terminology} We let $d,e, k, l, m,n,r$, sometimes subscripted, range over $ {\\bbold N} =\\{0,1,2,\\dots\\}$.  For sets $P, Q$, and $\\mathcal{S}\\subseteq P\\times Q$ and $p\\in P$ we set $$\\mathcal{S}(p)\\ :=\\ \\{q\\in Q:\\ (p,q)\\in \\mathcal{S}\\}. $$  (Think of $\\mathcal{S}$ as describing the family $\\big(\\mathcal{S}(p)\\big)_{p\\in P}$ of subsets of $Q$.) Throughout, $\\mathbf{M}=(M;\\dots)$ is a one-sorted $\\mathcal{L}$-structure. When $C$ gets mentioned, we assume $C\\subseteq M$ is $0$-definable and $|C|\\ge 2$. Recall, for example from \\cite{HHM}, that $C$ is said to be {\\em stably embedded in $\\mathbf{M}$\\/} if for every $0$-definable $\\mathcal{S}\\subseteq M^{m}\\times M^n$ there is a $0$-definable $\\mathcal{T}\\subseteq M^{\\mu}\\times M^n$, $\\mu\\in  {\\bbold N} $, such that for all $p\\in M^m$ there exists $c\\in C^{\\mu}$ for which $\\mathcal{S}(p)\\cap C^n = \\mathcal{T}(c)\\cap C^n$. \n  For $\\omega$-saturated $\\mathbf{M}$, this is equivalent to all definable $X\\subseteq C^n$ being $C$-definable. \n\nSuppose $X\\subseteq M^m$ and $Y\\subseteq M^n$ are definable. Then a {\\em definable family of maps from subsets of $X$ into $Y$\\/} is a definable set\n$\\mathcal{F}\\subseteq P\\times X \\times Y$, with definable $P\\subseteq M^k$ such that for each ``parameter\" $p\\in P$ the set\n$\\mathcal{F}(p)\\subseteq X\\times Y$ is the graph of a map $f_p$ with domain a subset of $X$ and codomain $Y$.  We think of such $\\mathcal{F}$ as the family $(f_p)_{p\\in P}$. Abusing language we write $f\\in \\mathcal{F}$ to indicate that $f=f_p$ for some $p\\in P$, and to state that the set of $f\\in \\mathcal{F}$ with a certain property is definable means that the set of $p\\in P$ such that $f_p$ has that property is definable. Given $\\mathcal{S}\\subseteq P\\times Q$, $p\\in P$, and a map $f: \\mathcal{S} \\to Z$, we let $f(p,-)$ denote the map $q\\mapsto f(p,q): \\mathcal{S}(p)\\to Z$. \n\nOur definition of ``$r$-step fiberable over $C$'' differs from that of ``fiberable by $C$ in $r$ steps'' \nin \\cite{ADH} for a Liouville closed $H$-field $K$ and its constant field $C$. But, in that case,  ``fiberable over $C$'' as defined here is  equivalent to ``fiberable by $C$ in $r$ steps for some $r$\" as defined in \\cite{ADH}.", "sketch": "Using the equivalence (for an $\u0005omega$-saturated elementary extension $\\mathbf{M}^*$) between “$X$ is $r$-step fiberable over $C$” and “$X^*$ is $r$-step fiberable over $C^*$”, the proof of Theorem~\\ref{thm} “reduces to the case where $\\mathbf{M}$ is $\\omega$-saturated”. One then proves the extension by “induction on $r$”, showing that $\\mu$ extends to an $A$-valued Fubini measure on the full subcategory of $\\operatorname{Def}(\\mathbf{M})$ consisting of definable sets that are $r$-step fiberable over $C$.\n\nFor the step $r\\to r+1$, take an $(r+1)$-step fiberable $X\\subseteq M^m$, witnessed by a definable $f:X\\to M^n$ with $f(X)$ and all fibers $f^{-1}(y)$ ($y\\in M^n$) $r$-step fiberable over $C$. A “carefully chosen inductive assumption” gives that $\\mu\\big(f(X)\\big)$ is defined and that $\\mu\\big(f^{-1}(y)\\big)$ is defined and “takes only finitely many values for $y\\in f(X)$”. Define $\\mu_f(X)$ as a “suitably weighted sum of these values”, and show $\\mu_f(X)$ “does not depend on $f$, but only on $X$”, so one may set $\\mu(X)=\\mu_f(X)$. Finally, verify that this $\\mu$ “preserves the Fubini property” and that the inductive assumption is preserved from $r$ to $r+1$. (Details are said to be in Section~\\ref{proof}.)", "expanded_sketch": "Using the equivalence (for an $\\omega$-saturated elementary extension $\\mathbf{M}^*$) between “$X$ is $r$-step fiberable over $C$” and “$X^*$ is $r$-step fiberable over $C^*$”, the proof of the main theorem reduces to the case where $\\mathbf{M}$ is $\\omega$-saturated. One then proves the extension by induction on $r$, showing that $\\mu$ extends to an $A$-valued Fubini measure on the full subcategory of $\\operatorname{Def}(\\mathbf{M})$ consisting of definable sets that are $r$-step fiberable over $C$.\n\nMore precisely, one introduces by recursion on $r$ a sequence $(\\cC_r)$ of full subcategories of $\\Def(\\cM)$, such that $ \\cC_r\\subseteq \\cC_{r+1}$ for all $r$ and $\\bigcup_r \\cC_r=\\Def(C)^{\\f}$:\n\n$\\cC_0:= \\Def(C)^{\\iso}$, and the objects of $\\cC_{r+1}$ are the definable $X\\subseteq M^m$ for which there exists a definable\n$f: X\\to Y$ with $Y\\in \\cC_r$, such that $f^{-1}(y)\\in  \\cC_r$ for all $y\\in Y$; such $f$ is called a $\\cC_{r+1}$-witness. Thus the objects of $\\cC_r$ are the definable $X\\subseteq M^m$ that are $r$-step fiberable over $C$ as defined in the Introduction.\nIn particular, the objects of $\\cC_0$ are the definable $X\\subseteq M^m$ that admit a definable injective map $X\\to C^n$. Also, if the definable set $X\\subseteq M^m$ admits a definable map $X\\to C^n$ all whose fibers are finite, then $X\\in \\cC_1$.\n\nFor the step $r\\to r+1$, take an $(r+1)$-step fiberable $X\\subseteq M^m$, witnessed by a definable $f:X\\to Y$ with $Y\\in \\cC_r$ and all fibers $f^{-1}(y)$ ($y\\in Y$) lying in $\\cC_r$. The inductive setup yields that $\\mu$ is already defined on $Y$ and on each fiber $f^{-1}(y)$. To ensure the finiteness/definability needed to form a Fubini-type sum, one uses the following lemma.\n\n\\begin{lemma}\\label{l1}  The set $\\{\\mu\\big(\\cal{S}(p)\\big):\\ p\\in M^k\\}$ is finite, and for every $a$ the set $\\{p\\in M^k:\\ \\mu\\big(\\cal{S}(p)\\big)=a\\}$ is definable.\n\\end{lemma}\n\nUsing this, one obtains that $\\mu\\big(f^{-1}(y)\\big)$ takes only finitely many values as $y$ ranges over $Y=f(X)$. One then defines $\\mu_f(X)$ as the corresponding suitably weighted sum of these values and shows that $\\mu_f(X)$ does not depend on the choice of witness $f$ but only on $X$, allowing one to set $\\mu(X)=\\mu_f(X)$. Finally, one checks that this definition preserves the Fubini property and that the inductive hypotheses propagate from $r$ to $r+1$.\n\nThe details are carried out later.\n\n\\begin{lemma}  If $A, B$ are commutative semirings and $\\mu: \\cC\\to A$ and $\\nu: \\cC\\to B$ are Fubini measures, then $(\\mu,\\nu): \\cC\\to A\\times B$ is a Fubini measure. \n\\end{lemma} \n\n\\section{Proof of the Theorem}\\label{proof}\n\n\\noindent\n{\\it In this section $\\mu$ is an $A$-valued Fubini measure on $\\Def(C)$}. Till further notice we also assume $\\cM$ is $\\omega$-saturated, indicating at the end of the section how to reduce to that case. \n\n\\medskip\\noindent\nBy recursion on $r$ we now introduce a sequence $(\\cC_r)$ of full subcategories of $\\Def(\\cM)$, such that $ \\cC_r\\subseteq \\cC_{r+1}$ for all $r$ and $\\bigcup_r \\cC_r=\\Def(C)^{\\f}$:\n\n\\medskip\\noindent\n $\\cC_0:= \\Def(C)^{\\iso}$, and the objects of $\\cC_{r+1}$ are the definable $X\\subseteq M^m$ for which there exists a definable\n$f: X\\to Y$ with $Y\\in \\cC_r$, such that $f^{-1}(y)\\in  \\cC_r$ for all $y\\in Y$; such $f$ is called a $\\cC_{r+1}$-witness. Thus the objects of $\\cC_r$ are the definable $X\\subseteq M^m$ that are $r$-step fiberable over $C$ as defined in the Introduction.\nIn particular, the objects of $\\cC_0$ are the definable $X\\subseteq M^m$ that admit a definable injective map $X\\to C^n$. Also, if the definable set $X\\subseteq M^m$ admits a definable map $X\\to C^n$ all whose fibers are finite, then $X\\in \\cC_1$.  \n\n\\medskip\\noindent\n Induction on $r$ shows that $\\cC_r$ satisfies conditions (a), (b), (c) for all $r$, and that $\\cC_r^{\\iso}=\\cC_r$ for all $r$.  In the inductive step we note that if $f: X\\to Y$ is a $\\cC_{r+1}$-witness and $X'\\subseteq X$ and $Y'\\subseteq Y$ are definable with\n$f(X')\\subseteq Y'$, then $f|_{X'}: X'\\to Y'$ is also a $\\cC_{r+1}$-witness. Induction on $r$ also shows that there is at most one\n$A$-valued Fubini measure on $\\cC_r$ extending the given $A$-valued Fubini measure $\\mu$ on $\\Def(C)$. In the beginning of the previous section we have extended $\\mu$ to an $A$-valued Fubini measure on $\\cC_0$, also to be denoted by $\\mu$. \n\n\\subsection*{More about $\\cC_0$} {\\it In this subsection we assume that $C$ is stably embedded in $\\cM$}.\nLet  $\\cal{S}\\subseteq M^k\\times C^n$ be definable. Think of $\\cal{S}$ as the definable family of sets\n$\\cal{S}(p)\\subseteq C^n$ parametrized by the points $p\\in M^k$. \n\n\\begin{lemma}\\label{l1}  The set $\\{\\mu\\big(\\cal{S}(p)\\big):\\ p\\in M^k\\}$ is finite, and for every $a$ the set $\\{p\\in M^k:\\ \\mu\\big(\\cal{S}(p)\\big)=a\\}$ is definable.\n\\end{lemma}", "expanded_theorem": "\\label{thm} If $C$ is stably embedded in $\\mathbf{M}$ and $\\mu$ an $A$-valued Fubini measure on $\\operatorname{Def}(C)$,\nthen $\\mu$ extends uniquely to an $A$-valued Fubini measure on $\\operatorname{Def}(C)^{\\operatorname{f}}$.", "theorem_type": ["Implication", "Uniqueness"], "mcq": {"question": "Let $\\mathbf{M}$ be a one-sorted structure, let $C\\subseteq M$ be a $0$-definable set with more than one element, and let $A$ be a commutative semiring. Let $\\operatorname{Def}(C)$ be the full subcategory whose objects are the definable subsets of powers $C^n$ (with definable maps as morphisms). Define recursively classes $\\mathcal C_0,\\mathcal C_1,\\dots$ by letting $\\mathcal C_0=\\operatorname{Def}(C)^{\\mathrm{iso}}$ (the definable sets definably bijective to an object of $\\operatorname{Def}(C)$), and for $r\\ge 0$, letting $\\mathcal C_{r+1}$ consist of those definable sets $X$ for which there is a definable map $f:X\\to Y$ with $Y\\in \\mathcal C_r$ and with every fiber $f^{-1}(y)\\in \\mathcal C_r$; set $\\operatorname{Def}(C)^{\\mathrm f}=\\bigcup_{r\\ge 0}\\mathcal C_r$. Assume that $C$ is stably embedded in $\\mathbf M$, meaning that every definable subset of $C^n$ is definable with parameters from $C$. Suppose $\\mu:\\operatorname{Def}(C)\\to A$ is an $A$-valued Fubini measure, i.e. $\\mu(\\emptyset)=0$, $\\mu(\\{x\\})=1$ for singletons, $\\mu(X\\cup Y)=\\mu(X)+\\mu(Y)$ for disjoint definable $X,Y\\subseteq C^m$, and for every definable map $f:X\\to Y$ between objects of $\\operatorname{Def}(C)$: the set $\\{\\mu(f^{-1}(y)):y\\in Y\\}$ is finite, each level set $Y_a=\\{y\\in Y:\\mu(f^{-1}(y))=a\\}$ is definable, and if all fibers have the same measure $a$, then $\\mu(X)=a\\mu(Y)$. Under these assumptions, which statement about extending $\\mu$ is valid?", "correct_choice": {"label": "A", "text": "There exists exactly one $A$-valued Fubini measure $\\widetilde\\mu$ on $\\operatorname{Def}(C)^{\\mathrm f}$ whose restriction to $\\operatorname{Def}(C)$ is $\\mu$."}, "choices": [{"label": "B", "text": "There exists exactly one $A$-valued Fubini measure $\\widetilde\\mu$ on $\\bigcup_{r\\le 1}\\mathcal C_r$ whose restriction to $\\operatorname{Def}(C)$ is $\\mu$, and every such extension automatically extends uniquely to all of $\\operatorname{Def}(C)^{\\mathrm f}$ without any further assumption on $C$."}, {"label": "C", "text": "There exists at most one $A$-valued Fubini measure $\\widetilde\\mu$ on $\\operatorname{Def}(C)^{\\mathrm f}$ whose restriction to $\\operatorname{Def}(C)$ is $\\mu$."}, {"label": "D", "text": "For every definable set $X\\in \\operatorname{Def}(C)^{\\mathrm f}$ and every definable witness $f:X\\to Y$ with $Y\\in \\mathcal C_r$ and all fibers $f^{-1}(y)\\in \\mathcal C_r$, the values $\\mu\\big(f^{-1}(y)\\big)$ are definable in $y$ and therefore determine an extension $\\widetilde\\mu(X)$; hence there exists an $A$-valued Fubini measure on $\\operatorname{Def}(C)^{\\mathrm f}$ extending $\\mu$, but it need not be unique."}, {"label": "E", "text": "There exists exactly one $A$-valued semi-Fubini measure $\\widetilde\\mu$ on $\\operatorname{Def}(C)^{\\mathrm f}$ whose restriction to $\\operatorname{Def}(C)$ is $\\mu$; moreover this extension always satisfies the full Fubini axiom on $\\operatorname{Def}(C)^{\\mathrm f}$ even if $C$ is not stably embedded in $\\mathbf M$."}], "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": "stable-embeddedness hypothesis", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "existence of extension", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "witness-independence yields uniqueness", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "full Fubini versus semi-Fubini and dependence on stable embeddedness", "template_used": "property_confusion"}]}}
{"id": "2512.22900v1", "paper_link": "http://arxiv.org/abs/2512.22900v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "[\\cite{HooshKoh2025}]\nA nontrivial finite group $G$ satisfies the strong CFS property if and only if $G$ is either\n\\begin{itemize} \n  \\item a cyclic group of prime order, or\n  \\item one of the four groups $C_4$, $C_2^2$, $C_2^3$, or $C_3^2$.\n\\end{itemize}", "start_pos": 7567, "end_pos": 7851, "label": null}, "ref_dict": {"lemma:main": "\\begin{lemma}\n\\label{lemma:main}\nLet $H$ be a proper subgroup of a finite group $G$ with $\\lvert H\\rvert \\ge 5$.\nChoose $h_0\\in H$ with $h_0\\neq e$ and choose $g\\in G\\setminus H$.\nLet $h_1$ be an element of $H$ not in the set\n$\\{e,h_0,h_0^{-1}\\}$,\nand set\n\\[\nH'= H\\setminus\\{e,h_0\\}.\n\\]\nDefine\n\\begin{equation}\\label{equ:subset-not-factor}\nA= H'\\cup\\{g,h_1g\\}.\n\\end{equation}\nThen $\\lvert A\\rvert = \\lvert H\\rvert$ and $A$ is not a left factor of $G$.\n\\end{lemma}", "lemma:groups of order 8": "\\begin{lemma}\n\\label{lemma:groups of order 8}\nLet $G$ be a group of order $8$, and let $a,b\\in G$ be two elements of order $4$ with $\\langle a\\rangle\\neq\\langle b\\rangle$.\nThen\n\\[\n  A = \\{a,a^2,a^3,b\\}\n\\]\nis not a left factor of $G$.\n\\end{lemma}", "lemma:small subset": "\\begin{lemma}\n\\label{lemma:small subset}\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\\!\n    \\item Let $G$ be a group of even order, and let $a\\in G$, $a\\neq e$. \n    The subset $\\{e,a\\}$ is a factor of $G$ if and only if the order of $a$ is even.\n    \\item Let $G$ be an elementary abelian $2$-group.\n    Every subset of size $4$ is a factor of $G$.\n    \\item\\,Let $G$ be an elementary abelian $3$-group.\n    Every subset of size $3$ is a factor of $G$.\n\\end{enumerate}\n\\end{lemma}"}, "pre_theorem_intro_text_len": 4601, "pre_theorem_intro_text": "\\label{section:Introduction}\nLet $G$ be a finite group.  A subset $A \\subseteq G$ is called a \\textit{Lagrange subset} if $\\lvert A\\rvert$ divides $\\lvert G\\rvert$.  If there exists a subset $B \\subseteq G$ such that every element $x\\in G$ can be written \\textit{uniquely} in the form\n\\[\nx = ab,\n\\quad a\\in A,\\; b\\in B,\n\\]\nthen $A$ is said to be a (left) \\textit{factor} of $G$.  In this situation one immediately has\n\\[\n\\lvert G\\rvert = \\lvert A\\rvert \\,\\lvert B\\rvert,\n\\]\nand both $A$ and $B$ are Lagrange subsets of $G$.\n\nThe problem of factoring a finite group $G$ into two (or more) Lagrange subsets goes back to Haj\\'os’s \ncombinatorial proof of Minkowski’s conjecture on lattice tilings by unit cubes \\cite{Hajos1941}.  \nIn particular, Haj\\'os\\ showed that any lattice tiling of $\\mathbb{R}^n$ by unit $n$-cubes must contain two \ncubes sharing a full $(n-1)$-face, and he recast that argument in group‑theoretic terms to characterize exactly \nwhich cyclic groups admit factorizations by arithmetic progressions \\cite{Hajos1941}.  \nIn the broader abelian setting, subsequent work of R\\'edei \\cite{Redei1952}, \nSands \\cite{Sands1974}, Szab\\'o \\cite{Szabo2004}, and others has connected these set‑factorizations with tiling problems, zero‑sum sequences, and \nrelated combinatorial phenomena.  A good survey of these developments can be found in \n\\cite{Szabo-Sands2009}.\n\nBernstein \\cite{Bernstein1968} extended Haj\\'os’s theorem to the non‑abelian setting, showing that many of the \ncombinatorial tiling arguments carry over to arbitrary finite groups.  A related thread concerns the \nexistence of \\textit{minimal logarithmic signatures} (MLS): conjecturally, every finite group $G$ admits \na factorization\n\\[\nG = A_1 A_2 \\cdots A_k\n\\]\nwhere each $\\lvert A_i\\rvert$ is prime or equals $4$.  \nIt is known that every finite solvable group has an \nMLS, and that if a normal subgroup $K\\lhd G$ and the quotient $G/K$ both admit MLS then so does $G$ \n\\cite{Rahimipour2018}.  Consequently, any counterexample to the MLS conjecture must be a non-abelian \nsimple group, reducing the problem to the finite simple groups.\n\nHooshmand \\cite[Problem 19.35]{Kourovka2018} \n(see also \\cite{Hooshmand2014,Banakh2018multifactorizable, Hooshmand2021}) \nasked whether for every factorization $\\lvert G\\rvert = m_1\\cdots m_k$, there exist subsets \n$A_i\\subseteq G$ with $\\lvert A_i\\rvert=m_i$ and $G=A_1\\cdots A_k$.  \nThe case $k=2$ has been studied in \\cite{Bildanov2020, Hooshmand2021}, where it is shown that a finite \ngroup is $2$‑factorizable if and only if every finite simple group is $2$‑factorizable, \nand all simple groups of order up to $10~000$ are $2$-factorizable \\cite{Bildanov2020}.  \n\nFor $k=3$, Bergman \\cite{Bergman2020} proved that the alternating group $A_4$ is not $3$‑factorizable.\nBrunault \\cite{Banakh2018Factorizable} used computer methods to show that $A_5$ has no \n$(2,3,5,2)$\\nobreakdash-factorization (in fact, $A_5$ does not even admit \na $(2,15,2)$\\nobreakdash-factorization; this was shown in \\cite{Kab2021} without computer calculations). \nMore generally, it is proved in \\cite{Kab2021} that for every integer $k\\geq3$, \nthere exist integers $m_1,m_2,\\ldots,m_k$ all greater than $1$\nand a finite group of order $m_1m_2\\ldots m_k$ that does not have any \n$(m_1,m_2,\\ldots,m_k)$\\nobreakdash-factorization.\nAmong groups of order at most $100$, there are only eight groups that do not admit a $k$-factorization \nfor at least one $k\\geq2$; all other groups are $k$-factorizable for every admissible $k$ \\cite{Kab2021}.\nIt was also shown that the simple groups of orders $168$ and $360$ are $k$‑factorizable \nfor every admissible $k$ \\cite{Kab2024}.\n\nRecently, Chin, Wang, and Wong \\cite{Chin-Wang-Wong} introduced the notion of \\textit{complete \nfactorization}. Subsets $A_1, \\ldots, A_k$ of $G$ form a complete factorization if they are pairwise \ndisjoint and every $g \\in G$ is uniquely represented as $g = a_1 \\cdots a_k$ with $a_i \\in A_i$. \nThey proved that if $G$ is a finite abelian group and $|G|=m_1\\ldots m_k$ \nwhere $m_1,\\ldots,m_k$ are integers greater than $1$ and $k>2$, then \nthere exist subsets $A_1,\\ldots,A_k$ of $G$ (with $|A_i|=m_i$ for all $i=1,2,\\ldots,k$) \nwhich form a complete factorization of the group $G$.\nThis result was extended to nilpotent groups in \\cite{Kab2023}.\n\nA finite group $G$ is said to have the \\textit{strong CFS} (converse of factorization by subsets) \nproperty if every Lagrange subset of $G$ is a factor of $G$. The notions of a Lagrange subset and a \nstrong CFS group were introduced in \\cite{HooshKoh2025}, where the following was proved:", "context": "The problem of factoring a finite group $G$ into two (or more) Lagrange subsets goes back to Haj\\'os’s \ncombinatorial proof of Minkowski’s conjecture on lattice tilings by unit cubes \\cite{Hajos1941}. \nIn particular, Haj\\'os\\ showed that any lattice tiling of $\\mathbb{R}^n$ by unit $n$-cubes must contain two \ncubes sharing a full $(n-1)$-face, and he recast that argument in group‑theoretic terms to characterize exactly \nwhich cyclic groups admit factorizations by arithmetic progressions \\cite{Hajos1941}. \nIn the broader abelian setting, subsequent work of R\\'edei \\cite{Redei1952}, \nSands \\cite{Sands1974}, Szab\\'o \\cite{Szabo2004}, and others has connected these set‑factorizations with tiling problems, zero‑sum sequences, and \nrelated combinatorial phenomena. A good survey of these developments can be found in \n\\cite{Szabo-Sands2009}.\n\nBernstein \\cite{Bernstein1968} extended Haj\\'os’s theorem to the non‑abelian setting, showing that many of the \ncombinatorial tiling arguments carry over to arbitrary finite groups. A related thread concerns the \nexistence of \\textit{minimal logarithmic signatures} (MLS): conjecturally, every finite group $G$ admits \na factorization\n\\[\nG = A_1 A_2 \\cdots A_k\n\\]\nwhere each $\\lvert A_i\\rvert$ is prime or equals $4$. \nIt is known that every finite solvable group has an \nMLS, and that if a normal subgroup $K\\lhd G$ and the quotient $G/K$ both admit MLS then so does $G$ \n\\cite{Rahimipour2018}. Consequently, any counterexample to the MLS conjecture must be a non-abelian \nsimple group, reducing the problem to the finite simple groups.\n\nHooshmand \\cite[Problem 19.35]{Kourovka2018} \n(see also \\cite{Hooshmand2014,Banakh2018multifactorizable, Hooshmand2021}) \nasked whether for every factorization $\\lvert G\\rvert = m_1\\cdots m_k$, there exist subsets \n$A_i\\subseteq G$ with $\\lvert A_i\\rvert=m_i$ and $G=A_1\\cdots A_k$. \nThe case $k=2$ has been studied in \\cite{Bildanov2020, Hooshmand2021}, where it is shown that a finite \ngroup is $2$‑factorizable if and only if every finite simple group is $2$‑factorizable, \nand all simple groups of order up to $10~000$ are $2$-factorizable \\cite{Bildanov2020}.\n\nFor $k=3$, Bergman \\cite{Bergman2020} proved that the alternating group $A_4$ is not $3$‑factorizable.\nBrunault \\cite{Banakh2018Factorizable} used computer methods to show that $A_5$ has no \n$(2,3,5,2)$\\nobreakdash-factorization (in fact, $A_5$ does not even admit \na $(2,15,2)$\\nobreakdash-factorization; this was shown in \\cite{Kab2021} without computer calculations). \nMore generally, it is proved in \\cite{Kab2021} that for every integer $k\\geq3$, \nthere exist integers $m_1,m_2,\\ldots,m_k$ all greater than $1$\nand a finite group of order $m_1m_2\\ldots m_k$ that does not have any \n$(m_1,m_2,\\ldots,m_k)$\\nobreakdash-factorization.\nAmong groups of order at most $100$, there are only eight groups that do not admit a $k$-factorization \nfor at least one $k\\geq2$; all other groups are $k$-factorizable for every admissible $k$ \\cite{Kab2021}.\nIt was also shown that the simple groups of orders $168$ and $360$ are $k$‑factorizable \nfor every admissible $k$ \\cite{Kab2024}.\n\nRecently, Chin, Wang, and Wong \\cite{Chin-Wang-Wong} introduced the notion of \\textit{complete \nfactorization}. Subsets $A_1, \\ldots, A_k$ of $G$ form a complete factorization if they are pairwise \ndisjoint and every $g \\in G$ is uniquely represented as $g = a_1 \\cdots a_k$ with $a_i \\in A_i$. \nThey proved that if $G$ is a finite abelian group and $|G|=m_1\\ldots m_k$ \nwhere $m_1,\\ldots,m_k$ are integers greater than $1$ and $k>2$, then \nthere exist subsets $A_1,\\ldots,A_k$ of $G$ (with $|A_i|=m_i$ for all $i=1,2,\\ldots,k$) \nwhich form a complete factorization of the group $G$.\nThis result was extended to nilpotent groups in \\cite{Kab2023}.\n\nA finite group $G$ is said to have the \\textit{strong CFS} (converse of factorization by subsets) \nproperty if every Lagrange subset of $G$ is a factor of $G$. The notions of a Lagrange subset and a \nstrong CFS group were introduced in \\cite{HooshKoh2025}, where the following was proved:", "full_context": "The problem of factoring a finite group $G$ into two (or more) Lagrange subsets goes back to Haj\\'os’s \ncombinatorial proof of Minkowski’s conjecture on lattice tilings by unit cubes \\cite{Hajos1941}. \nIn particular, Haj\\'os\\ showed that any lattice tiling of $\\mathbb{R}^n$ by unit $n$-cubes must contain two \ncubes sharing a full $(n-1)$-face, and he recast that argument in group‑theoretic terms to characterize exactly \nwhich cyclic groups admit factorizations by arithmetic progressions \\cite{Hajos1941}. \nIn the broader abelian setting, subsequent work of R\\'edei \\cite{Redei1952}, \nSands \\cite{Sands1974}, Szab\\'o \\cite{Szabo2004}, and others has connected these set‑factorizations with tiling problems, zero‑sum sequences, and \nrelated combinatorial phenomena. A good survey of these developments can be found in \n\\cite{Szabo-Sands2009}.\n\nBernstein \\cite{Bernstein1968} extended Haj\\'os’s theorem to the non‑abelian setting, showing that many of the \ncombinatorial tiling arguments carry over to arbitrary finite groups. A related thread concerns the \nexistence of \\textit{minimal logarithmic signatures} (MLS): conjecturally, every finite group $G$ admits \na factorization\n\\[\nG = A_1 A_2 \\cdots A_k\n\\]\nwhere each $\\lvert A_i\\rvert$ is prime or equals $4$. \nIt is known that every finite solvable group has an \nMLS, and that if a normal subgroup $K\\lhd G$ and the quotient $G/K$ both admit MLS then so does $G$ \n\\cite{Rahimipour2018}. Consequently, any counterexample to the MLS conjecture must be a non-abelian \nsimple group, reducing the problem to the finite simple groups.\n\nHooshmand \\cite[Problem 19.35]{Kourovka2018} \n(see also \\cite{Hooshmand2014,Banakh2018multifactorizable, Hooshmand2021}) \nasked whether for every factorization $\\lvert G\\rvert = m_1\\cdots m_k$, there exist subsets \n$A_i\\subseteq G$ with $\\lvert A_i\\rvert=m_i$ and $G=A_1\\cdots A_k$. \nThe case $k=2$ has been studied in \\cite{Bildanov2020, Hooshmand2021}, where it is shown that a finite \ngroup is $2$‑factorizable if and only if every finite simple group is $2$‑factorizable, \nand all simple groups of order up to $10~000$ are $2$-factorizable \\cite{Bildanov2020}.\n\nFor $k=3$, Bergman \\cite{Bergman2020} proved that the alternating group $A_4$ is not $3$‑factorizable.\nBrunault \\cite{Banakh2018Factorizable} used computer methods to show that $A_5$ has no \n$(2,3,5,2)$\\nobreakdash-factorization (in fact, $A_5$ does not even admit \na $(2,15,2)$\\nobreakdash-factorization; this was shown in \\cite{Kab2021} without computer calculations). \nMore generally, it is proved in \\cite{Kab2021} that for every integer $k\\geq3$, \nthere exist integers $m_1,m_2,\\ldots,m_k$ all greater than $1$\nand a finite group of order $m_1m_2\\ldots m_k$ that does not have any \n$(m_1,m_2,\\ldots,m_k)$\\nobreakdash-factorization.\nAmong groups of order at most $100$, there are only eight groups that do not admit a $k$-factorization \nfor at least one $k\\geq2$; all other groups are $k$-factorizable for every admissible $k$ \\cite{Kab2021}.\nIt was also shown that the simple groups of orders $168$ and $360$ are $k$‑factorizable \nfor every admissible $k$ \\cite{Kab2024}.\n\nRecently, Chin, Wang, and Wong \\cite{Chin-Wang-Wong} introduced the notion of \\textit{complete \nfactorization}. Subsets $A_1, \\ldots, A_k$ of $G$ form a complete factorization if they are pairwise \ndisjoint and every $g \\in G$ is uniquely represented as $g = a_1 \\cdots a_k$ with $a_i \\in A_i$. \nThey proved that if $G$ is a finite abelian group and $|G|=m_1\\ldots m_k$ \nwhere $m_1,\\ldots,m_k$ are integers greater than $1$ and $k>2$, then \nthere exist subsets $A_1,\\ldots,A_k$ of $G$ (with $|A_i|=m_i$ for all $i=1,2,\\ldots,k$) \nwhich form a complete factorization of the group $G$.\nThis result was extended to nilpotent groups in \\cite{Kab2023}.\n\nA finite group $G$ is said to have the \\textit{strong CFS} (converse of factorization by subsets) \nproperty if every Lagrange subset of $G$ is a factor of $G$. The notions of a Lagrange subset and a \nstrong CFS group were introduced in \\cite{HooshKoh2025}, where the following was proved:\n\n\\begin{document}\n\\maketitle\n\\begin{abstract}\nIn a finite group, a subset is called a Lagrange subset if its size divides the group order, and a factor if it admits a complementary subset. We provide a new and comparatively direct proof of the classification of groups in which every Lagrange subset is a factor. We show that any nontrivial such group must be a cyclic group of prime order, the cyclic group of order 4, or an elementary abelian group of order 4, 8, or 9.\n\\end{abstract}\n\nHooshmand \\cite[Problem 19.35]{Kourovka2018} \n(see also \\cite{Hooshmand2014,Banakh2018multifactorizable, Hooshmand2021}) \nasked whether for every factorization $\\lvert G\\rvert = m_1\\cdots m_k$, there exist subsets \n$A_i\\subseteq G$ with $\\lvert A_i\\rvert=m_i$ and $G=A_1\\cdots A_k$. \nThe case $k=2$ has been studied in \\cite{Bildanov2020, Hooshmand2021}, where it is shown that a finite \ngroup is $2$‑factorizable if and only if every finite simple group is $2$‑factorizable, \nand all simple groups of order up to $10~000$ are $2$-factorizable \\cite{Bildanov2020}.\n\nFor $k=3$, Bergman \\cite{Bergman2020} proved that the alternating group $A_4$ is not $3$‑factorizable.\nBrunault \\cite{Banakh2018Factorizable} used computer methods to show that $A_5$ has no \n$(2,3,5,2)$\\nobreakdash-factorization (in fact, $A_5$ does not even admit \na $(2,15,2)$\\nobreakdash-factorization; this was shown in \\cite{Kab2021} without computer calculations). \nMore generally, it is proved in \\cite{Kab2021} that for every integer $k\\geq3$, \nthere exist integers $m_1,m_2,\\ldots,m_k$ all greater than $1$\nand a finite group of order $m_1m_2\\ldots m_k$ that does not have any \n$(m_1,m_2,\\ldots,m_k)$\\nobreakdash-factorization.\nAmong groups of order at most $100$, there are only eight groups that do not admit a $k$-factorization \nfor at least one $k\\geq2$; all other groups are $k$-factorizable for every admissible $k$ \\cite{Kab2021}.\nIt was also shown that the simple groups of orders $168$ and $360$ are $k$‑factorizable \nfor every admissible $k$ \\cite{Kab2024}.\n\nA finite group $G$ is said to have the \\textit{strong CFS} (converse of factorization by subsets) \nproperty if every Lagrange subset of $G$ is a factor of $G$. The notions of a Lagrange subset and a \nstrong CFS group were introduced in \\cite{HooshKoh2025}, where the following was proved:\n\nThe authors of \\cite{HooshKoh2025} used Cayley graph arguments to constrain the possible orders of $G$, \nreducing the classification to an examination of about twenty concrete groups. In this paper, we offer a \nstreamlined proof of the \"only if\" direction, which breaks down into four independent steps:\n\\begin{enumerate}[label=(\\arabic*)]\n \\item Analysis of groups with \"large\" proper subgroups (Lemma \\ref{lemma:main});\n \\item Groups of even order containing elements of odd order (Lemma \\ref{lemma:small subset});\n \\item Investigation of groups of order $8$ (Lemma \\ref{lemma:groups of order 8});\n \\item A direct check for the three remaining groups: $D_4$, $C_8$, and $C_9$.\n\\end{enumerate}\n\n\\begin{lemma}\n\\label{lemma:small subset}\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\\!\n \\item Let $G$ be a group of even order, and let $a\\in G$, $a\\neq e$. \n The subset $\\{e,a\\}$ is a factor of $G$ if and only if the order of $a$ is even.\n \\item Let $G$ be an elementary abelian $2$-group.\n Every subset of size $4$ is a factor of $G$.\n \\item\\,Let $G$ be an elementary abelian $3$-group.\n Every subset of size $3$ is a factor of $G$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}[Proof of \\textup{(i)}]\nFirst suppose that the order of $a$ is even, say $|a| = 2m$ for some positive integer $m$.\nLet $H = \\langle a\\rangle$.\nSet\n\\[\nB = \\{e,a^2,\\ldots,a^{2m-2}\\}.\n\\]\nClearly $H = \\{e,a\\} B$, and it is straightforward to check that\n$B \\cap aB = \\varnothing$.\nThus $\\{e,a\\}$ is a factor of $H$.\nBy Lemma~\\ref{lemma:subset of subgroup}, $\\{e,a\\}$ is then a factor of $G$.\n\n\\begin{lemma}\n\\label{lemma:groups of order 8}\nLet $G$ be a group of order $8$, and let $a,b\\in G$ be two elements of order $4$ with $\\langle a\\rangle\\neq\\langle b\\rangle$.\nThen\n\\[\n A = \\{a,a^2,a^3,b\\}\n\\]\nis not a left factor of $G$.\n\\end{lemma}\n\\begin{proof}\n Suppose to the contrary that there is an element $x\\in G$ such that\n \\[\n G=A\\cup Ax \\text{ and } A\\cap Ax=\\varnothing.\n \\]\n Then $e\\in Ax$ and hence $x=a^{-i}$ for some $i\\in\\{1,2,3\\}$, or $x=b^{-1}$.\n If $x=a^{-i}$, then $a\\in A\\cap A x$, contradicting disjointness.\n If $x=b^{-1}=b^3$, then\n \\[\n Ax=\\{ab^3,a^2b^3,a^3b^3,e\\}.\n \\]\n Since $b^3\\notin A$ and also $b^3\\notin A x$,\n the element $b^3$ is not contained in $A\\cup A x=G$, a clear contradiction.\n\\end{proof}\n\nFirst, suppose that $|G|$ is even. By Lemma~\\ref{lemma:small subset}\\,(i), \nevery non-identity element of $G$ must have even order. \nIt follows that $G$ is a $2$-group.\nSince $G$ cannot have a proper subgroup of order greater than $4$ (by Lemma~\\ref{lemma:main}), \nthe order of $G$ must be $4$ or $8$.\nIf $|G|=4$, $G$ is either $C_4$ or $C_2^2$, both of which appear in the theorem's list.\nIf $|G|=8$, Lemma~\\ref{lemma:groups of order 8} implies that $G$ cannot contain \ntwo distinct cyclic subgroups of order $4$. \nThis eliminates the quaternion group $Q_8$ and the product $C_4 \\times C_2$. \nSince $C_2^3$ is a solution, the only remaining candidates to check are $D_4$ and $C_8$.\n\n\\begin{lemma}\n\\label{lemma:groups of order 8}\nLet $G$ be a group of order $8$, and let $a,b\\in G$ be two elements of order $4$ with $\\langle a\\rangle\\neq\\langle b\\rangle$.\nThen\n\\[\n  A = \\{a,a^2,a^3,b\\}\n\\]\nis not a left factor of $G$.\n\\end{lemma}\n\n\\begin{lemma}\n\\label{lemma:main}\nLet $H$ be a proper subgroup of a finite group $G$ with $\\lvert H\\rvert \\ge 5$.\nChoose $h_0\\in H$ with $h_0\\neq e$ and choose $g\\in G\\setminus H$.\nLet $h_1$ be an element of $H$ not in the set\n$\\{e,h_0,h_0^{-1}\\}$,\nand set\n\\[\nH'= H\\setminus\\{e,h_0\\}.\n\\]\nDefine\n\\begin{equation}\\label{equ:subset-not-factor}\nA= H'\\cup\\{g,h_1g\\}.\n\\end{equation}\nThen $\\lvert A\\rvert = \\lvert H\\rvert$ and $A$ is not a left factor of $G$.\n\\end{lemma}\n\n\\begin{lemma}\n\\label{lemma:small subset}\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\\!\n    \\item Let $G$ be a group of even order, and let $a\\in G$, $a\\neq e$. \n    The subset $\\{e,a\\}$ is a factor of $G$ if and only if the order of $a$ is even.\n    \\item Let $G$ be an elementary abelian $2$-group.\n    Every subset of size $4$ is a factor of $G$.\n    \\item\\,Let $G$ be an elementary abelian $3$-group.\n    Every subset of size $3$ is a factor of $G$.\n\\end{enumerate}\n\\end{lemma}", "post_theorem_intro_text_len": 1209, "post_theorem_intro_text": "The authors of \\cite{HooshKoh2025} used Cayley graph arguments to constrain the possible orders of $G$, \nreducing the classification to an examination of about twenty concrete groups. In this paper, we offer a \nstreamlined proof of the \"only if\" direction, which breaks down into four independent steps:\n\\begin{enumerate}[label=(\\arabic*)]\n  \\item Analysis of groups with \"large\" proper subgroups (Lemma \\ref{lemma:main});\n  \\item Groups of even order containing elements of odd order (Lemma \\ref{lemma:small subset});\n  \\item Investigation of groups of order $8$ (Lemma \\ref{lemma:groups of order 8});\n  \\item A direct check for the three remaining groups: $D_4$, $C_8$, and $C_9$.\n\\end{enumerate}\n\n\\vspace{1ex}\n\\noindent\\textbf{Notation.} Throughout, $G$ is a finite group with identity $e$. For any $A \\subseteq G$ and $x \\in G$, we denote $Ax = \\{ax \\mid a \\in A\\}$. We write $\\langle g \\rangle$ for the cyclic subgroup generated by $g$. A subset $A$ is a \\textit{left factor} if $G = \\bigsqcup_{b \\in B} Ab$ for some $B \\subseteq G$. \n\nWith these conventions in place, we turn in Section \\ref{section:lemmas} to our lemmas, \nand then in Section \\ref{section:proof theorem} to the completion of the proof.", "sketch": "The post-theorem introduction explains that \\cite{HooshKoh2025} used Cayley graph arguments to constrain the possible orders of $G$, reducing the classification to examining about twenty concrete groups. It then sketches a streamlined proof of the \\emph{``only if''} direction, broken into four independent steps:\n\\begin{enumerate}[label=(\\arabic*)]\n  \\item \\emph{Analysis of groups with ``large'' proper subgroups} (Lemma \\ref{lemma:main});\n  \\item \\emph{Groups of even order containing elements of odd order} (Lemma \\ref{lemma:small subset});\n  \\item \\emph{Investigation of groups of order $8$} (Lemma \\ref{lemma:groups of order 8});\n  \\item \\emph{A direct check} for the three remaining groups: $D_4$, $C_8$, and $C_9$.\n\\end{enumerate}\nIt further indicates the paper proceeds by proving these lemmas in Section \\ref{section:lemmas} and then completing the proof in Section \\ref{section:proof theorem}.", "expanded_sketch": "The post-theorem introduction explains that \\cite{HooshKoh2025} used Cayley graph arguments to constrain the possible orders of $G$, reducing the classification to examining about twenty concrete groups. It then sketches a streamlined proof of the \\emph{``only if''} direction, broken into four independent steps:\n\\begin{enumerate}[label=(\\arabic*)]\n  \\item \\emph{Analysis of groups with ``large'' proper subgroups}. We first prove the following lemma.\n\n\\begin{lemma}\n\\label{lemma:main}\nLet $H$ be a proper subgroup of a finite group $G$ with $\\lvert H\\rvert \\ge 5$.\nChoose $h_0\\in H$ with $h_0\\neq e$ and choose $g\\in G\\setminus H$.\nLet $h_1$ be an element of $H$ not in the set\n$\\{e,h_0,h_0^{-1}\\}$,\nand set\n\\[\nH'= H\\setminus\\{e,h_0\\}.\n\\]\nDefine\n\\begin{equation}\\label{equ:subset-not-factor}\nA= H'\\cup\\{g,h_1g\\}.\n\\end{equation}\nThen $\\lvert A\\rvert = \\lvert H\\rvert$ and $A$ is not a left factor of $G$.\n\\end{lemma}\n\n  \\item \\emph{Groups of even order containing elements of odd order}. We next prove the following lemma.\n\n\\begin{lemma}\n\\label{lemma:small subset}\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\\!\n    \\item Let $G$ be a group of even order, and let $a\\in G$, $a\\neq e$. \n    The subset $\\{e,a\\}$ is a factor of $G$ if and only if the order of $a$ is even.\n    \\item Let $G$ be an elementary abelian $2$-group.\n    Every subset of size $4$ is a factor of $G$.\n    \\item\\,Let $G$ be an elementary abelian $3$-group.\n    Every subset of size $3$ is a factor of $G$.\n\\end{enumerate}\n\\end{lemma}\n\n  \\item \\emph{Investigation of groups of order $8$}. We then prove the following lemma.\n\n\\begin{lemma}\n\\label{lemma:groups of order 8}\nLet $G$ be a group of order $8$, and let $a,b\\in G$ be two elements of order $4$ with $\\langle a\\rangle\\neq\\langle b\\rangle$.\nThen\n\\[\n  A = \\{a,a^2,a^3,b\\}\n\\]\nis not a left factor of $G$.\n\\end{lemma}\n\n  \\item \\emph{A direct check} for the three remaining groups: $D_4$, $C_8$, and $C_9$.\n\\end{enumerate}\nIt further indicates the paper proceeds by proving these lemmas next and then completing the proof later.", "expanded_theorem": "[\\cite{HooshKoh2025}]\nA nontrivial finite group $G$ satisfies the strong CFS property if and only if $G$ is either\n\\begin{itemize} \n  \\item a cyclic group of prime order, or\n  \\item one of the four groups $C_4$, $C_2^2$, $C_2^3$, or $C_3^2$.\n\\end{itemize}", "theorem_type": ["Biconditional or Equivalence", "Classification or Bijection"], "mcq": {"question": "Let $G$ be a nontrivial finite group. Call a subset $A\\subseteq G$ a Lagrange subset if $|A|$ divides $|G|$, and call $A$ a factor of $G$ if there exists a subset $B\\subseteq G$ such that every element of $G$ can be written uniquely as $ab$ with $a\\in A$ and $b\\in B$. The group $G$ is said to have the strong CFS property if every Lagrange subset of $G$ is a factor of $G$. Which of the following statements is equivalent to $G$ having the strong CFS property? (Here $C_n$ denotes the cyclic group of order $n$, and $C_2^2$, $C_2^3$, $C_3^2$ denote elementary abelian groups of orders $4$, $8$, and $9$, respectively.)", "correct_choice": {"label": "A", "text": "$G$ is either a cyclic group of prime order or one of the four groups $C_4$, $C_2^2$, $C_2^3$, or $C_3^2$."}, "choices": [{"label": "B", "text": "$G$ is either a cyclic group of prime order or one of the five groups $C_4$, $C_2^2$, $C_2^3$, $C_3^2$, or $C_9$."}, {"label": "C", "text": "$G$ is either a cyclic group of prime order or an abelian group of order $4$, $8$, or $9$."}, {"label": "D", "text": "$G$ is either a cyclic group of prime order, a cyclic group of order $4$, or an elementary abelian group of order $2^n$ or $3^2$."}, {"label": "E", "text": "$G$ is either a cyclic group of prime order or one of the groups $C_4$, $C_2^2$, $D_4$, $C_2^3$, or $C_3^2$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "computational_check", "tampered_component": "remaining-group check excludes $C_9$", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "characteristic", "tampered_component": "replace the exact order-$4,8,9$ abelian list by the broader abelian-order condition", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "characteristic", "tampered_component": "small-subset lemmas for elementary abelian groups do not extend to all elementary abelian $2$-groups", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "computational_check", "tampered_component": "direct check rules out $D_4$ among the final order-$8$ candidates", "template_used": "wildcard"}]}}
{"id": "2512.22960v1", "paper_link": "http://arxiv.org/abs/2512.22960v1", "theorems_cnt": 4, "theorem": {"env_name": "theo", "content": "\\label{Th-SynthesExistenceSW}\n    Let $d\\in\\{1,2\\}$, $\\lambda,\\omega\\in\\mathbb R$ and $\\gamma>0$. Denote by $\\mu_0$ the lowest eigenvalue of $-A$. In each of the following situation, there exists a positive solution of \\cref{Eq-StationaryAGP} in $\\mathrm{D}\\left(\\sqrt{|A|}\\right)$,\n    \\begin{itemize}\n        \\item $\\lambda>0$ and $\\omega>-\\mu_0$,\n        \\item $\\lambda = 0$ and $\\omega=\\mu_0$,\n        \\item $\\lambda<0$ and $\\omega<-\\mu_0$.\n    \\end{itemize}\n    In every other cases, there is no non-trivial non-negative solution in $\\mathrm{D}\\left(\\sqrt{|A|}\\right)$.", "start_pos": 92486, "end_pos": 93076, "label": "Th-SynthesExistenceSW"}, "ref_dict": {"Eq-GP": "\\begin{cases}\\complexI\\p_t u+ H u +\\lambda \\left|u\\right|^{2\\gamma}u=0\\\\ u(0)=u_0,\\end{cases} \\label{Eq-GP}\n\\end{equation} \nby a white noise potential. In the study of Bose-Einstein condensation, \\cref{Eq-GP} appears as a mean-field limit of the many body Schrödinger equation, the nonlinear term appearing as a consequence of particle interactions \\cite{BEC_Pitaevskii}. Adding a random potential in \\cref{Eq-GP} can be seen as a way of modeling spatial inhomogeneities (see for example \\cite{PhysRevLett.104.193901,PhysRevLett.109.243902} and references therein), thus one can see \\cref{Eq-AGP} as a toy model of Bose-Einstein condensation in inhomogeneous environment with zero correlation length. In this context, at least formally, \\cref{Eq:AGP} has two conserved quantities physically relevant, the mass \n$$\\mathcal{M}(u)=\\frac{1}{2}\\int_{\\R^d}|u|^2\\d x$$\nand the energy\n$$\\mathcal{E}(u)=\\frac{1}{2}\\langle-(H+\\xi)u,u\\rangle - \\frac{\\lambda}{2\\gamma+2}\\int_{\\R^d}|u|^{2\\gamma+2}\\d x.$$\nStanding waves of \\cref{Eq:AGP} then appear naturally as solutions of minimal energy at fixed mass. Formally, if $\\phi$ minimize the energy at fixed mass, by a Lagrange multiplier argument, there exists $\\omega\\in\\R$ such that\n$$-(H+\\xi)\\phi-\\lambda|\\phi|^{2\\gamma}\\phi+\\omega \\phi = 0$$\nand $u(t,x)=\\e^{i\\omega t}\\phi(x)$ is solution of \\cref{Eq:AGP}.\\\\\n\nIn dimension 2, it is known from \\cite{Mackowiak_2025,MackowiakStrichartzConfAnderson} that \\cref{Eq:AGP} should be interpreted in a renormalized sense. This renormalization has to be taken into account in the study of standing waves. The construction of the renormalized operator has been conducted in \\cite{MackowiakStrichartzConfAnderson} and will be recall in \\cref{Sec-AndersonOp2d}. The renormalized operator is defined through an exponential transform inspired by the one introduced in \\cite{Haire_Labbe} to study the continuous parabolic Anderson model, and later used in \\cite{debussche_weber_T2,Tzvetkov_2023,tzvetkov2020dimensional} to solve the Anderson nonlinear Schrödinger equation on the torus $\\T^2$ and in \\cite{debussche2017solution,debussche2023global} to solve the same equation but on the full space $\\R^2$. Let $Y=(-H)^{-1}\\xi$, it is almost surely of regularity $1-$, so one can define $\\rho=\\e^Y$ and search for a formula for $(H+\\xi)u$ when $u$ is of the form $u=\\rho v$. A formal computation gives\n$$(H+\\xi)u=\\rho\\left(Hv+2\\nabla Y\\cdot \\nabla v + xY\\cdot xv + |\\nabla Y|^2v\\right).$$\nUnfortunately, $\\nabla Y$ is of regularity $0-$, thus the product $|\\nabla Y|^2$ barely fails to exist. In \\cite{Mackowiak_2025}, the author proved that the wick product $\\wick{|\\nabla Y|^2}$ can be defined as the almost sure limit, in suitable distribution spaces of regularity $0-$, of\n$$|\\nabla Y_N|^2-\\E[|\\nabla Y_N|^2],$$\nwhere $Y_N$ is a suitable regularisation of $Y_N$.  Thus, the renormalized operator can be formally defined as\n$$(H+\\xi)u=\\rho\\left(Hv+2\\nabla Y\\cdot \\nabla v + xY\\cdot xv + \\wick{|\\nabla Y|^2}v\\right).$$\nThe rigorous construction of $H+\\xi$ rely on the associated quadratic form. \n\nEven in dimension 1, the construction of $H+\\xi$ is not straightforward as $\\xi$ is not a function. On compact interval with no potential, $\\p_x^2+\\xi$ has first been constructed in \\cite{FukushimaNakao} using its quadratic form and has been studied in \\cite{DumazLabbeAndersonLoc1D} using a twisted Sturm-Liouville approach, allowing to construct $\\p_x^2+\\xi$ on unbounded interval too. In \\cref{Sec-AndersonOp1d}, we propose a construction of $H+\\xi$ as a self-adjoint unbounded operator through its quadratic form. \\cref{Prop-IPP1d} shows $\\mathrm{D}(H+\\xi)=\\mathrm{D}(\\p_x^2+\\xi)\\cap \\mathrm{D}(|x|^2)$, where $\\p_x^2+\\xi$ is the unbounded self-adjoint operator on $\\L^2(\\R)$ constructed in \\cite{DumazLabbeAndersonLoc1D} and $\\mathrm{D}(|x|^2)=\\{u\\in\\L^2(\\R),xu\\in\\L^2(\\R)\\}$.\\\\\n\nWe are then interested in standing waves for\n\\begin{equation}{~}\n    \\begin{cases}\\complexI\\p_t u+ Au+\\lambda \\left|u\\right|^{2\\gamma}u=0\\\\ u(0)=u_0,\\end{cases}", "Eq:AGP": "\\begin{cases}\\complexI\\p_t u+ H u + \\xi u+\\lambda \\left|u\\right|^{2\\gamma}u=0\\\\ u(0)=u_0,\\end{cases} \\label{Eq:AGP}\n\\end{equation} \nof the form $u(t,x)=\\e^{i\\omega t}\\phi(x)$, where $H=\\Delta-|x|^2$ is the Hermite operator, $\\xi$ is a real spatial white noise, $\\lambda\\in\\R$ and $\\gamma>0$. Such a solution is called a standing wave of \\cref{Eq:AGP} of frequency $\\omega\\in\\R$ and profile $\\phi$. The study of standing waves is motivated by the modeling of Bose-Einstein condensation. \nIn fact, \\cref{Eq:AGP} can be seen as a perturbation of the Gross-Pitaevskii equation\n\\begin{equation}{~}\n    \\begin{cases}\\complexI\\p_t u+ H u +\\lambda \\left|u\\right|^{2\\gamma}u=0\\\\ u(0)=u_0,\\end{cases}", "Th-EnergyGS": "\\begin{theo}\\label{Th-EnergyGS}\n\n    Let $d\\in\\{1,2\\}$, $\\lambda,\\omega\\in\\R$, $\\gamma>0$ and $A$ be the self-adjoint operator constructed in \\cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\\xi$ a spatial white noise, $\\D^{1,2}=\\mathrm{D}(\\sqrt{|A|})$ and $\\D^{-1,2}$ its dual. For $u\\in\\D^{1,2}$, we define the mass as\n    $$\\mathcal{M}(u)=\\frac{1}{2}\\int_{\\R^d}|u|^2\\d x$$\n    and the energy as\n    $$\\mathcal{E}(u)=\\frac{1}{2}\\langle -Au,u\\rangle_{\\D^{-1,2},\\D^{1,2}}-\\frac{\\lambda}{2\\gamma+2}\\int_{\\R^d}|u|^{2\\gamma+2}\\d x.$$\n    Then, it holds\n    \\begin{enumerate}\n        \\item For $\\lambda\\leqslant0$, for any $m>0$, there exists $\\phi_m\\in\\D^{1,2}$ such that $\\phi_m>0$ and\n        $$\\argmin_{\\substack{u\\in\\D^{1,2}\\\\\\mathcal{M}(u)=m}}\\mathcal{E}(u) = \\left\\{\\e^{\\complexI\\theta}\\phi_m,\\; \\theta\\in\\R\\right\\}.$$\n        \\item For $\\lambda>0$ and $\\gamma<\\frac{2}{d}$, for any $m>0$, there exists $\\phi_m\\in\\displaystyle\\argmin_{u\\in\\D^{1,2},\\;\\mathcal{M}(u)=m}\\mathcal{E}(u)$ such that $\\phi_m>0$.\n        \\item For $\\lambda>0$ and $\\gamma=\\frac{2}{d}$, there exists $m_{*,\\lambda}(\\Xi)>0$ such that for all $m\\in(0,m_{*,\\lambda}(\\Xi))$, there exists $\\phi_m\\in\\displaystyle\\argmin_{u\\in\\D^{1,2},\\;\\mathcal{M}(u)=m}\\mathcal{E}(u)$ such that $\\phi_m>0$. Moreover, there exists $m_{*,\\lambda}(\\Xi)\\leqslant m^*_\\lambda(\\Xi)\\leqslant m^*_\\lambda$ such that\n        $$\\forall m>m^*_\\lambda(\\Xi), \\inf_{\\substack{u\\in\\D^{1,2}\\\\\\mathcal{M}(u)=m}}\\mathcal{E}(u)=-\\infty,$$\n        where $m^*_\\lambda$ is the critical mass of the classical nonlinear Schrödinger equation (i.e.\\ without any potential).\n        \\item For $\\lambda>0$ and $\\gamma>\\frac{2}{d}$, for any $m>0$,\n        $$\\inf_{\\substack{u\\in\\D^{1,2}\\\\\\mathcal{M}(u)=m}}\\mathcal{E}(u)=-\\infty.$$\n    \\end{enumerate}\n\n\\end{theo}", "Eq-StationaryAGP": "\\begin{equation}\n    -A\\phi-\\lambda|\\phi|^{2\\gamma}\\phi+\\omega \\phi = 0. \\label{Eq-StationaryAGP}\n\\end{equation}", "Eq-AGP": "\\label{Eq-AGP}\n\\end{equation} \nin dimension $d\\in\\{1,2\\}$, where $A=H+\\xi$. As $A$ is independent of time, standing waves $u(t,x)=\\e^{\\complexI\\omega t}\\phi(x)$ solves \\cref{Eq-AGP} if and only if $\\p"}, "pre_theorem_intro_text_len": 6990, "pre_theorem_intro_text": "In this paper, we are interested in particular solutions of the Anderson-Gross-Pitaevskii\n\\begin{equation}{~}\n    \\begin{cases}\\mathrm{i}\\p_t u+ H u + \\xi u+\\lambda \\left|u\\right|^{2\\gamma}u=0\\\\ u(0)=u_0,\\end{cases} \\label{Eq:AGP}\n\\end{equation} \nof the form $u(t,x)=\\mathrm{e}^{i\\omega t}\\phi(x)$, where $H=\\Delta-|x|^2$ is the Hermite operator, $\\xi$ is a real spatial white noise, $\\lambda\\in\\mathbb R$ and $\\gamma>0$. Such a solution is called a standing wave of \\cref{Eq:AGP} of frequency $\\omega\\in\\mathbb R$ and profile $\\phi$. The study of standing waves is motivated by the modeling of Bose-Einstein condensation. \nIn fact, \\cref{Eq:AGP} can be seen as a perturbation of the Gross-Pitaevskii equation\n\\begin{equation}{~}\n    \\begin{cases}\\mathrm{i}\\p_t u+ H u +\\lambda \\left|u\\right|^{2\\gamma}u=0\\\\ u(0)=u_0,\\end{cases} \\label{Eq-GP}\n\\end{equation} \nby a white noise potential. In the study of Bose-Einstein condensation, \\cref{Eq-GP} appears as a mean-field limit of the many body Schrödinger equation, the nonlinear term appearing as a consequence of particle interactions \\cite{BEC_Pitaevskii}. Adding a random potential in \\cref{Eq-GP} can be seen as a way of modeling spatial inhomogeneities (see for example \\cite{PhysRevLett.104.193901,PhysRevLett.109.243902} and references therein), thus one can see \\cref{Eq-AGP} as a toy model of Bose-Einstein condensation in inhomogeneous environment with zero correlation length. In this context, at least formally, \\cref{Eq:AGP} has two conserved quantities physically relevant, the mass \n$$\\mathcal{M}(u)=\\frac{1}{2}\\int_{\\mathbb R^d}|u|^2\\mathrm{d} x$$\nand the energy\n$$\\mathcal{E}(u)=\\frac{1}{2}\\langle-(H+\\xi)u,u\\rangle - \\frac{\\lambda}{2\\gamma+2}\\int_{\\mathbb R^d}|u|^{2\\gamma+2}\\mathrm{d} x.$$\nStanding waves of \\cref{Eq:AGP} then appear naturally as solutions of minimal energy at fixed mass. Formally, if $\\phi$ minimize the energy at fixed mass, by a Lagrange multiplier argument, there exists $\\omega\\in\\mathbb R$ such that\n$$-(H+\\xi)\\phi-\\lambda|\\phi|^{2\\gamma}\\phi+\\omega \\phi = 0$$\nand $u(t,x)=\\mathrm{e}^{i\\omega t}\\phi(x)$ is solution of \\cref{Eq:AGP}.\\\\\n\nIn dimension 2, it is known from \\cite{Mackowiak_2025,MackowiakStrichartzConfAnderson} that \\cref{Eq:AGP} should be interpreted in a renormalized sense. This renormalization has to be taken into account in the study of standing waves. The construction of the renormalized operator has been conducted in \\cite{MackowiakStrichartzConfAnderson} and will be recall in \\cref{Sec-AndersonOp2d}. The renormalized operator is defined through an exponential transform inspired by the one introduced in \\cite{Haire_Labbe} to study the continuous parabolic Anderson model, and later used in \\cite{debussche_weber_T2,Tzvetkov_2023,tzvetkov2020dimensional} to solve the Anderson nonlinear Schrödinger equation on the torus $\\mathbb T^2$ and in \\cite{debussche2017solution,debussche2023global} to solve the same equation but on the full space $\\mathbb R^2$. Let $Y=(-H)^{-1}\\xi$, it is almost surely of regularity $1-$, so one can define $\\rho=\\mathrm{e}^Y$ and search for a formula for $(H+\\xi)u$ when $u$ is of the form $u=\\rho v$. A formal computation gives\n$$(H+\\xi)u=\\rho\\left(Hv+2\\nabla Y\\cdot \\nabla v + xY\\cdot xv + |\\nabla Y|^2v\\right).$$\nUnfortunately, $\\nabla Y$ is of regularity $0-$, thus the product $|\\nabla Y|^2$ barely fails to exist. In \\cite{Mackowiak_2025}, the author proved that the wick product $\\wick{|\\nabla Y|^2}$ can be defined as the almost sure limit, in suitable distribution spaces of regularity $0-$, of\n$$|\\nabla Y_N|^2-\\mathbb E[|\\nabla Y_N|^2],$$\nwhere $Y_N$ is a suitable regularisation of $Y_N$.  Thus, the renormalized operator can be formally defined as\n$$(H+\\xi)u=\\rho\\left(Hv+2\\nabla Y\\cdot \\nabla v + xY\\cdot xv + \\wick{|\\nabla Y|^2}v\\right).$$\nThe rigorous construction of $H+\\xi$ rely on the associated quadratic form. \n\nEven in dimension 1, the construction of $H+\\xi$ is not straightforward as $\\xi$ is not a function. On compact interval with no potential, $\\p_x^2+\\xi$ has first been constructed in \\cite{FukushimaNakao} using its quadratic form and has been studied in \\cite{DumazLabbeAndersonLoc1D} using a twisted Sturm-Liouville approach, allowing to construct $\\p_x^2+\\xi$ on unbounded interval too. In \\cref{Sec-AndersonOp1d}, we propose a construction of $H+\\xi$ as a self-adjoint unbounded operator through its quadratic form. \\cref{Prop-IPP1d} shows $\\mathrm{D}(H+\\xi)=\\mathrm{D}(\\p_x^2+\\xi)\\cap \\mathrm{D}(|x|^2)$, where $\\p_x^2+\\xi$ is the unbounded self-adjoint operator on $\\mathrm L^2(\\mathbb R)$ constructed in \\cite{DumazLabbeAndersonLoc1D} and $\\mathrm{D}(|x|^2)=\\{u\\in\\mathrm L^2(\\mathbb R),xu\\in\\mathrm L^2(\\mathbb R)\\}$.\\\\\n\nWe are then interested in standing waves for\n\\begin{equation}{~}\n    \\begin{cases}\\mathrm{i}\\p_t u+ Au+\\lambda \\left|u\\right|^{2\\gamma}u=0\\\\ u(0)=u_0,\\end{cases} \\label{Eq-AGP}\n\\end{equation} \nin dimension $d\\in\\{1,2\\}$, where $A=H+\\xi$. As $A$ is independent of time, standing waves $u(t,x)=\\mathrm{e}^{\\mathrm{i}\\omega t}\\phi(x)$ solves \\cref{Eq-AGP} if and only if $\\phi$ is solution of\n\\begin{equation}\n    -A\\phi-\\lambda|\\phi|^{2\\gamma}\\phi+\\omega \\phi = 0. \\label{Eq-StationaryAGP}\n\\end{equation}\nThis paper aim to construct standing waves in presence of a white noise potential and present some of their properties. \n\nWhen $\\xi=0$ and $H=\\Delta$, standing waves are well-known object. See for example \\cite{NehariExistence,BerestyckiLions,StraussExistenceSolitary} for existence, \\cite{BerestyckiLions} for non-existence, \\cite{GidasSymmetry,KwongUniqueness} for uniqueness and \\cite{BerestyckiInstabilité,CazenaveLionsStability,CazenaveStabilityInstability,WeinsteinSharpInterpolation,GrillakisStability,GrillakisStabilityII} for results on stability. Let us stress out many of these results relies on the particular structure of \\cref{Eq-StationaryAGP} in this case, which is not preserved in presence of potentials. When $\\xi=0$ and $H=\\Delta-V$ with $V$ a regular and potentially unbounded potential, we refer the reader to \\cite{KavianSelfSimilar,FukuizumiStabilityInstability,HiroseStructure,HiroseUniqueness,FukuizumiExponentialDecay,MaedaSymmetry,SelemBifurcation,GuoSeringerMassConcentration,PelinovskyGSEnergyCriticalGP} and references therein for many results on existence, uniqueness, localization and stability.\n\nNonlinear elliptic equations in presence of white noise potential on compact manifold have been studied in \\cite{EulryVariational} using mountain pass methods and in \\cite{ZhangConstraintMinAnderson} using constraint minimization. In both \\cite{EulryVariational,ZhangConstraintMinAnderson}, the Anderson operator $\\Delta+\\xi$ is constructed using paracontrolled calculus. We will present here the construction of solution of \\cref{Eq-StationaryAGP} through constraint minimization, namely energy minimization at fixed mass and action minimization. This allow to construct a positive solution of \\cref{Eq-StationaryAGP} whenever such a solution exists.", "context": "In dimension 2, it is known from \\cite{Mackowiak_2025,MackowiakStrichartzConfAnderson} that \\cref{Eq:AGP} should be interpreted in a renormalized sense. This renormalization has to be taken into account in the study of standing waves. The construction of the renormalized operator has been conducted in \\cite{MackowiakStrichartzConfAnderson} and will be recall in \\cref{Sec-AndersonOp2d}. The renormalized operator is defined through an exponential transform inspired by the one introduced in \\cite{Haire_Labbe} to study the continuous parabolic Anderson model, and later used in \\cite{debussche_weber_T2,Tzvetkov_2023,tzvetkov2020dimensional} to solve the Anderson nonlinear Schrödinger equation on the torus $\\mathbb T^2$ and in \\cite{debussche2017solution,debussche2023global} to solve the same equation but on the full space $\\mathbb R^2$. Let $Y=(-H)^{-1}\\xi$, it is almost surely of regularity $1-$, so one can define $\\rho=\\mathrm{e}^Y$ and search for a formula for $(H+\\xi)u$ when $u$ is of the form $u=\\rho v$. A formal computation gives\n$$(H+\\xi)u=\\rho\\left(Hv+2\\nabla Y\\cdot \\nabla v + xY\\cdot xv + |\\nabla Y|^2v\\right).$$\nUnfortunately, $\\nabla Y$ is of regularity $0-$, thus the product $|\\nabla Y|^2$ barely fails to exist. In \\cite{Mackowiak_2025}, the author proved that the wick product $\\wick{|\\nabla Y|^2}$ can be defined as the almost sure limit, in suitable distribution spaces of regularity $0-$, of\n$$|\\nabla Y_N|^2-\\mathbb E[|\\nabla Y_N|^2],$$\nwhere $Y_N$ is a suitable regularisation of $Y_N$. Thus, the renormalized operator can be formally defined as\n$$(H+\\xi)u=\\rho\\left(Hv+2\\nabla Y\\cdot \\nabla v + xY\\cdot xv + \\wick{|\\nabla Y|^2}v\\right).$$\nThe rigorous construction of $H+\\xi$ rely on the associated quadratic form.\n\nEven in dimension 1, the construction of $H+\\xi$ is not straightforward as $\\xi$ is not a function. On compact interval with no potential, $\\p_x^2+\\xi$ has first been constructed in \\cite{FukushimaNakao} using its quadratic form and has been studied in \\cite{DumazLabbeAndersonLoc1D} using a twisted Sturm-Liouville approach, allowing to construct $\\p_x^2+\\xi$ on unbounded interval too. In \\cref{Sec-AndersonOp1d}, we propose a construction of $H+\\xi$ as a self-adjoint unbounded operator through its quadratic form. \\cref{Prop-IPP1d} shows $\\mathrm{D}(H+\\xi)=\\mathrm{D}(\\p_x^2+\\xi)\\cap \\mathrm{D}(|x|^2)$, where $\\p_x^2+\\xi$ is the unbounded self-adjoint operator on $\\mathrm L^2(\\mathbb R)$ constructed in \\cite{DumazLabbeAndersonLoc1D} and $\\mathrm{D}(|x|^2)=\\{u\\in\\mathrm L^2(\\mathbb R),xu\\in\\mathrm L^2(\\mathbb R)\\}$.\\\\\n\nWe are then interested in standing waves for\n\\begin{equation}{~}\n \\begin{cases}\\mathrm{i}\\p_t u+ Au+\\lambda \\left|u\\right|^{2\\gamma}u=0\\\\ u(0)=u_0,\\end{cases} \\label{Eq-AGP}\n\\end{equation} \nin dimension $d\\in\\{1,2\\}$, where $A=H+\\xi$. As $A$ is independent of time, standing waves $u(t,x)=\\mathrm{e}^{\\mathrm{i}\\omega t}\\phi(x)$ solves \\cref{Eq-AGP} if and only if $\\phi$ is solution of\n\\begin{equation}\n -A\\phi-\\lambda|\\phi|^{2\\gamma}\\phi+\\omega \\phi = 0. \\label{Eq-StationaryAGP}\n\\end{equation}\nThis paper aim to construct standing waves in presence of a white noise potential and present some of their properties.\n\nWhen $\\xi=0$ and $H=\\Delta$, standing waves are well-known object. See for example \\cite{NehariExistence,BerestyckiLions,StraussExistenceSolitary} for existence, \\cite{BerestyckiLions} for non-existence, \\cite{GidasSymmetry,KwongUniqueness} for uniqueness and \\cite{BerestyckiInstabilité,CazenaveLionsStability,CazenaveStabilityInstability,WeinsteinSharpInterpolation,GrillakisStability,GrillakisStabilityII} for results on stability. Let us stress out many of these results relies on the particular structure of \\cref{Eq-StationaryAGP} in this case, which is not preserved in presence of potentials. When $\\xi=0$ and $H=\\Delta-V$ with $V$ a regular and potentially unbounded potential, we refer the reader to \\cite{KavianSelfSimilar,FukuizumiStabilityInstability,HiroseStructure,HiroseUniqueness,FukuizumiExponentialDecay,MaedaSymmetry,SelemBifurcation,GuoSeringerMassConcentration,PelinovskyGSEnergyCriticalGP} and references therein for many results on existence, uniqueness, localization and stability.\n\nNonlinear elliptic equations in presence of white noise potential on compact manifold have been studied in \\cite{EulryVariational} using mountain pass methods and in \\cite{ZhangConstraintMinAnderson} using constraint minimization. In both \\cite{EulryVariational,ZhangConstraintMinAnderson}, the Anderson operator $\\Delta+\\xi$ is constructed using paracontrolled calculus. We will present here the construction of solution of \\cref{Eq-StationaryAGP} through constraint minimization, namely energy minimization at fixed mass and action minimization. This allow to construct a positive solution of \\cref{Eq-StationaryAGP} whenever such a solution exists.", "full_context": "In dimension 2, it is known from \\cite{Mackowiak_2025,MackowiakStrichartzConfAnderson} that \\cref{Eq:AGP} should be interpreted in a renormalized sense. This renormalization has to be taken into account in the study of standing waves. The construction of the renormalized operator has been conducted in \\cite{MackowiakStrichartzConfAnderson} and will be recall in \\cref{Sec-AndersonOp2d}. The renormalized operator is defined through an exponential transform inspired by the one introduced in \\cite{Haire_Labbe} to study the continuous parabolic Anderson model, and later used in \\cite{debussche_weber_T2,Tzvetkov_2023,tzvetkov2020dimensional} to solve the Anderson nonlinear Schrödinger equation on the torus $\\mathbb T^2$ and in \\cite{debussche2017solution,debussche2023global} to solve the same equation but on the full space $\\mathbb R^2$. Let $Y=(-H)^{-1}\\xi$, it is almost surely of regularity $1-$, so one can define $\\rho=\\mathrm{e}^Y$ and search for a formula for $(H+\\xi)u$ when $u$ is of the form $u=\\rho v$. A formal computation gives\n$$(H+\\xi)u=\\rho\\left(Hv+2\\nabla Y\\cdot \\nabla v + xY\\cdot xv + |\\nabla Y|^2v\\right).$$\nUnfortunately, $\\nabla Y$ is of regularity $0-$, thus the product $|\\nabla Y|^2$ barely fails to exist. In \\cite{Mackowiak_2025}, the author proved that the wick product $\\wick{|\\nabla Y|^2}$ can be defined as the almost sure limit, in suitable distribution spaces of regularity $0-$, of\n$$|\\nabla Y_N|^2-\\mathbb E[|\\nabla Y_N|^2],$$\nwhere $Y_N$ is a suitable regularisation of $Y_N$. Thus, the renormalized operator can be formally defined as\n$$(H+\\xi)u=\\rho\\left(Hv+2\\nabla Y\\cdot \\nabla v + xY\\cdot xv + \\wick{|\\nabla Y|^2}v\\right).$$\nThe rigorous construction of $H+\\xi$ rely on the associated quadratic form.\n\nEven in dimension 1, the construction of $H+\\xi$ is not straightforward as $\\xi$ is not a function. On compact interval with no potential, $\\p_x^2+\\xi$ has first been constructed in \\cite{FukushimaNakao} using its quadratic form and has been studied in \\cite{DumazLabbeAndersonLoc1D} using a twisted Sturm-Liouville approach, allowing to construct $\\p_x^2+\\xi$ on unbounded interval too. In \\cref{Sec-AndersonOp1d}, we propose a construction of $H+\\xi$ as a self-adjoint unbounded operator through its quadratic form. \\cref{Prop-IPP1d} shows $\\mathrm{D}(H+\\xi)=\\mathrm{D}(\\p_x^2+\\xi)\\cap \\mathrm{D}(|x|^2)$, where $\\p_x^2+\\xi$ is the unbounded self-adjoint operator on $\\mathrm L^2(\\mathbb R)$ constructed in \\cite{DumazLabbeAndersonLoc1D} and $\\mathrm{D}(|x|^2)=\\{u\\in\\mathrm L^2(\\mathbb R),xu\\in\\mathrm L^2(\\mathbb R)\\}$.\\\\\n\nWe are then interested in standing waves for\n\\begin{equation}{~}\n \\begin{cases}\\mathrm{i}\\p_t u+ Au+\\lambda \\left|u\\right|^{2\\gamma}u=0\\\\ u(0)=u_0,\\end{cases} \\label{Eq-AGP}\n\\end{equation} \nin dimension $d\\in\\{1,2\\}$, where $A=H+\\xi$. As $A$ is independent of time, standing waves $u(t,x)=\\mathrm{e}^{\\mathrm{i}\\omega t}\\phi(x)$ solves \\cref{Eq-AGP} if and only if $\\phi$ is solution of\n\\begin{equation}\n -A\\phi-\\lambda|\\phi|^{2\\gamma}\\phi+\\omega \\phi = 0. \\label{Eq-StationaryAGP}\n\\end{equation}\nThis paper aim to construct standing waves in presence of a white noise potential and present some of their properties.\n\nWhen $\\xi=0$ and $H=\\Delta$, standing waves are well-known object. See for example \\cite{NehariExistence,BerestyckiLions,StraussExistenceSolitary} for existence, \\cite{BerestyckiLions} for non-existence, \\cite{GidasSymmetry,KwongUniqueness} for uniqueness and \\cite{BerestyckiInstabilité,CazenaveLionsStability,CazenaveStabilityInstability,WeinsteinSharpInterpolation,GrillakisStability,GrillakisStabilityII} for results on stability. Let us stress out many of these results relies on the particular structure of \\cref{Eq-StationaryAGP} in this case, which is not preserved in presence of potentials. When $\\xi=0$ and $H=\\Delta-V$ with $V$ a regular and potentially unbounded potential, we refer the reader to \\cite{KavianSelfSimilar,FukuizumiStabilityInstability,HiroseStructure,HiroseUniqueness,FukuizumiExponentialDecay,MaedaSymmetry,SelemBifurcation,GuoSeringerMassConcentration,PelinovskyGSEnergyCriticalGP} and references therein for many results on existence, uniqueness, localization and stability.\n\nNonlinear elliptic equations in presence of white noise potential on compact manifold have been studied in \\cite{EulryVariational} using mountain pass methods and in \\cite{ZhangConstraintMinAnderson} using constraint minimization. In both \\cite{EulryVariational,ZhangConstraintMinAnderson}, the Anderson operator $\\Delta+\\xi$ is constructed using paracontrolled calculus. We will present here the construction of solution of \\cref{Eq-StationaryAGP} through constraint minimization, namely energy minimization at fixed mass and action minimization. This allow to construct a positive solution of \\cref{Eq-StationaryAGP} whenever such a solution exists.\n\nNonlinear elliptic equations in presence of white noise potential on compact manifold have been studied in \\cite{EulryVariational} using mountain pass methods and in \\cite{ZhangConstraintMinAnderson} using constraint minimization. In both \\cite{EulryVariational,ZhangConstraintMinAnderson}, the Anderson operator $\\Delta+\\xi$ is constructed using paracontrolled calculus. We will present here the construction of solution of \\cref{Eq-StationaryAGP} through constraint minimization, namely energy minimization at fixed mass and action minimization. This allow to construct a positive solution of \\cref{Eq-StationaryAGP} whenever such a solution exists.\n\nIn order to do so, we use constrained minimization methods. In \\cref{Sec-EnergyGS}, we study the minimization of energy at fixed mass and prove the following result.\n\nLet $d\\in\\{1,2\\}$, $\\lambda,\\omega\\in\\R$, $\\gamma>0$ and $A$ the self-adjoint operator constructed in \\cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\\xi$ a spatial white noise. Let $\\D^{1,2}=\\mathrm{D}(\\sqrt{|A|})$ and $\\D^{-1,2}$ its dual. Denote by $\\mu_0$ the smallest eigenvalue of $-A$. Let $\\omega>-\\mu_0$ and define $$\\mathcal{S}_\\omega = \\mathcal{E} + \\omega \\mathcal{M}$$ and $$\\mathcal{N}_\\omega =\\left\\{u\\in\\D^{1,2}\\backslash\\{0\\},\\; \\langle \\nabla\\mathcal{S}_\\omega u,u \\rangle_{\\D^{-1,2},\\D^{1,2}}=0\\right\\}.$$\n Then, there exists $\\psi_\\omega\\in\\displaystyle\\argmin_{\\mathcal{N}_\\omega}\\mathcal{S}_\\omega$ such that $\\psi_\\omega>0$.\n\n\\begin{itemize}\n \\item Assume $\\gamma>\\frac{2}{d}$, let $v\\in\\W^{1,2}\\backslash\\{0\\}$ and $u=\\rho v$, then for any $\\alpha>0$, \\cref{Eq-EnergyVside} implies\n \\begin{align*}\n \\mathcal{E}(u_\\alpha) &= - \\frac{\\lambda\\alpha^{d\\gamma}}{2\\gamma+2}\\int_{\\R^d}|v(x)|^{2\\gamma+2}\\rho(x/\\alpha)^{2\\gamma+2}\\d x &(I)\\\\\n &+ \\frac{\\alpha^2}{2}\\int_{\\R^d}\\rho(x/\\alpha)^2|\\nabla v(x)|^2\\d x &(II)\\\\\n &+ \\frac{1}{2\\alpha^2}\\int_{\\R^d} (1-Y(x/\\alpha))\\rho(x/\\alpha)^2|xv(x)|^2\\d x &(III)\\\\\n &-\\frac{1}{2}\\langle Z,|u_\\alpha|^2\\rangle. &(IV)\n \\end{align*}\n \\cref{Lem-EstimScalTransf,Lem-ProductRule} imply $(IV)=O(\\alpha^s)$ as $\\alpha$ goes to infinity, for any $s\\in(0,2)$, as $Z\\in\\W^{0-,\\infty}$. By dominated convergence, $(III)=O(\\alpha^{-2})$ as $\\alpha$ goes to infinity. Finally, $(II)=O(\\alpha^2)$ and\n $$(I)\\sim - \\frac{\\lambda\\alpha^{d\\gamma}\\rho(0)^{2\\gamma+2}}{2\\gamma+2}\\int_{\\R^d}|v(x)|^{2\\gamma+2}\\d x.$$\n Chose $v = \\sqrt{2m}\\rho(0)^{-1}h_0$ ($m>0$), then $\\mathcal{M}(u_\\alpha)$ converges to $m$ whereas $\\mathcal{E}(u_\\alpha)$ goes to $-\\infty$, because $d\\gamma>2$. Thus, \\cref{Lem-AsympEnergy} implies\n $$\\forall m>0,\\; \\inf_{\\Atop{u\\in\\D^{1,2}}{\\mathcal{M}(u)=m}}\\mathcal{E}(u)=-\\infty.$$\n \\item Assume $\\gamma=\\frac{2}{d}$ and let $Q$ be the unique positive and radial solution of \n $$\\Delta Q + \\lambda Q^{2\\gamma+1} = Q.$$\n Recall $\\mathcal{M}(Q)=m_\\lambda^*=\\frac{1}{2}\\left(\\frac{(\\gamma+1)J}{\\lambda}\\right)^{\\frac{1}{\\gamma}}$. For $c>0$, let $u=c\\rho Q$. For any $\\alpha>0$, it holds\n \\begin{align*}\n \\mathcal{E}(u_\\alpha) &= \\frac{\\alpha^2}{2}\\int_{\\R^d}\\rho(x/\\alpha)^2|c\\nabla Q(x)|^2\\d x - \\frac{\\lambda\\alpha^{2}}{2\\gamma+2}\\int_{\\R^d}|cQ(x)|^{2\\gamma+2}\\rho(x/\\alpha)^{2\\gamma+2}\\d x &(I)\\\\\n &+ \\frac{c^2}{2\\alpha^2}\\int_{\\R^d} (1-Y(x/\\alpha))\\rho(x/\\alpha)^2|xQ(x)|^2\\d x -\\frac{1}{2}\\langle Z,|u_\\alpha|^2\\rangle. &(II)\n \\end{align*}\n As previously $(II)=O(\\alpha^s)$ as $\\alpha$ goes to infinity, for any $s\\in(0,2)$, and for $c\\rho(0)\\neq 1$,\n $$(I)\\sim\\frac{\\alpha^2\\lambda(c\\rho(0))^2}{2\\gamma+2}\\left(1-(c\\rho(0))^{2\\gamma}\\right)\\int_{\\R^d}|Q(x)|^{2\\gamma+2}\\d x$$\n as $\\alpha$ goes to infinity, using \\cref{Eq-PohozaevQ}. Thus, for $c\\rho(0)>1$, \\cref{Lem-AsympEnergy} implies\n $$\\inf_{\\Atop{u\\in\\D^{1,2}}{\\mathcal{M}(u)=c\\rho(0)m^*_\\lambda}}\\mathcal{E}(u)=-\\infty,$$\n that is\n $$\\forall m>m^*_\\lambda,\\; \\inf_{\\Atop{u\\in\\D^{1,2}}{\\mathcal{M}(u)=m}}\\mathcal{E}(u)=-\\infty.$$\n Let $m'>m>0$ and $\\phi\\in\\D^{1,2}$ such that $\\mathcal{M}(\\phi)=m$. Let $\\psi=\\sqrt{\\frac{m'}{m}}\\phi$. Then,\n \\begin{align*}\n \\mathcal{E}(\\phi) &= \\frac{m}{m'}\\left[\\frac{1}{2}\\langle -A\\psi,\\psi\\rangle - \\frac{\\lambda}{2\\gamma+2}\\left(\\frac{m}{m'}\\right)^{\\gamma}\\int_{\\R^d}|\\psi|^{2\\gamma+2}\\d x\\right]\\\\\n &=\\frac{m}{m'}\\mathcal{E}(\\psi) + \\frac{m\\lambda}{m'(2\\gamma+2)}\\left[1 - \\left(\\frac{m}{m'}\\right)^{\\gamma}\\right]\\int_{\\R^d}|\\phi|^{2\\gamma+2}\\d x\\\\\n &\\geqslant \\frac{m}{m'}\\mathcal{E}(\\psi).\n \\end{align*}\n Thus,\n $$m'\\inf_{\\Atop{u\\in\\D^{1,2}}{\\mathcal{M}(u)=m}}\\mathcal{E}(u) \\geqslant m \\inf_{\\Atop{u\\in\\D^{1,2}}{\\mathcal{M}(u)=m}'}\\mathcal{E}(u).$$\n In particular, if $\\displaystyle\\inf_{u\\in\\D^{1,2},\\; \\mathcal{M}(u)=m}\\mathcal{E}(u)=-\\infty$, then $\\displaystyle\\inf_{u\\in\\D^{1,2},\\; \\mathcal{M}(u)=m'}\\mathcal{E}(u)=-\\infty$. Similarly, if $\\displaystyle\\inf_{u\\in\\D^{1,2},\\; \\mathcal{M}(u)=m'}\\mathcal{E}(u)>-\\infty$, then $\\displaystyle\\inf_{u\\in\\D^{1,2},\\; \\mathcal{M}(u)=m}\\mathcal{E}(u)>-\\infty$. Hence, we can define\n \\begin{align*}\n m^*_\\lambda(\\Xi) &= \\inf\\left\\{m>0,\\; \\inf_{\\Atop{u\\in\\D^{1,2}}{\\mathcal{M}(u)=m}}\\mathcal{E}(u)=-\\infty\\right\\}\\\\\n &= \\sup\\left\\{m>0,\\; \\inf_{\\Atop{u\\in\\D^{1,2}}{\\mathcal{M}(u)=m}}\\mathcal{E}(u)>-\\infty\\right\\} \\in [0,m^*_\\lambda]\n \\end{align*}\n To conclude, we only have to prove a lower bound for $m^*_\\lambda(\\Xi)$.\n\nOur aim is now to prove \\cref{Th-ACtionGS}. Showing directly that the minimization problem\n\\begin{equation}\n \\inf_{\\mathcal{N}_{\\omega,\\lambda}}\\mathcal{S}_{\\omega,\\lambda}\\label{Eq-MinAction}\n\\end{equation}\nhas non-negative solutions may be a bit technical as it is not straightforward that\n$$u\\in\\argmin_{\\mathcal{N}_{\\omega,\\lambda}}\\mathcal{S}_{\\omega,\\lambda}\\Rightarrow |u|\\in\\argmin_{\\mathcal{N}_{\\omega,\\lambda}}\\mathcal{S}_{\\omega,\\lambda}.$$\nThus, we will introduce two other, but equivalent, minimization problems which are easier to study. Remark if $u$ solves \\cref{Eq-StationaryAGP} for some $\\lambda>0$, then $\\alpha u$ solves \\cref{Eq-StationaryAGP} for $\\lambda'=\\frac{\\lambda}{|\\alpha|^{2\\gamma}}$. Thus, the parameter $\\lambda$ can be seen as a Lagrange multiplier, and solving \\cref{Eq-StationaryAGP} for some $\\lambda>0$ is equivalent to solving \\cref{Eq-StationaryAGP} for all $\\lambda>0$. Following an idea of \\cite{ROSE1988207}, we define for $u\\in\\D^{1,2}$\n\\begin{equation}\n \\mathcal{P}_\\omega(u)=\\langle (-A+\\omega) u, u\\rangle,\\label{Eq-POmega}\n\\end{equation}\n\\begin{equation}\n \\mathcal{Q}(u) = \\int_\\R |u|^{2(\\gamma+1)}\\d x\\label{Eq-Q(u)}\n\\end{equation}\nand if $u\\neq0,$\n\\begin{equation}\n \\mathcal{J}_\\omega(u) = \\frac{\\mathcal{P}_\\omega(u)}{\\mathcal{Q}(u)^{1/(\\gamma+1)}}.\\label{Eq-JOmega}\n\\end{equation}\nLet us stress out $\\mathcal{J}_\\omega(\\alpha u)=\\mathcal{J}_\\omega(u)$ for any $\\alpha\\in\\C^*$, thus if $u$ minimizes $\\mathcal{J}_\\omega$, $\\alpha u$ minimizes $\\mathcal{J}_\\omega$ too. Using standard argument of variational calculus, one can show minimizers of both\n\\begin{equation}\n \\inf_{\\Atop{u\\in\\D^{1,2}}{\\mathcal{Q}(u)=q}}\\mathcal{P}_\\omega(u)\\label{Eq-MinP/Q=q}\n\\end{equation}\nand\n\\begin{equation}\n \\inf_{\\D^{1,2}\\backslash\\{0\\}}\\mathcal{J}_\\omega\\label{Eq-MinJ}\n\\end{equation}\nare solutions of \\cref{Eq-StationaryAGP} for some $\\lambda>0$. For $\\lambda>0$, one can rewrite\n$$\\mathcal{S}_{\\omega,\\lambda}(u)=\\frac{1}{2}\\mathcal{P}_\\omega(u)-\\frac{\\lambda}{2(\\gamma+1)}\\mathcal{Q}(u)$$\nand\n\\begin{equation}\n \\mathcal{N}_{\\omega,\\lambda} = \\left\\{u\\in\\D^{1,2}\\backslash\\{0\\},\\; \\mathcal{J}_\\omega(u) = \\lambda \\mathcal{Q}(u)^{\\gamma/(\\gamma+1)} \\right\\}.\\label{Eq-NehariJQ}\n\\end{equation}\nAs remarked in \\cite{ROSE1988207}, minimizers of \\cref{Eq-MinP/Q=q,Eq-MinJ} are essentially the same, that is up to a multiplicative factor, due to the the homogeneity of $\\mathcal{J}_\\omega$. The following proposition shows minimization problems \\cref{Eq-MinAction,Eq-MinP/Q=q,Eq-MinJ} are equivalent, up to a multiplicative factor (see also \\cite{LiuRotatingActionGS}).", "post_theorem_intro_text_len": 4412, "post_theorem_intro_text": "In order to do so, we use constrained minimization methods. In \\cref{Sec-EnergyGS}, we study the minimization of energy at fixed mass and prove the following result.\n\n\\begin{theo}\\label{Th-EnergyGS}\n\n    Let $d\\in\\{1,2\\}$, $\\lambda,\\omega\\in\\mathbb R$, $\\gamma>0$ and $A$ be the self-adjoint operator constructed in \\cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\\xi$ a spatial white noise, $\\mathcal D^{1,2}=\\mathrm{D}(\\sqrt{|A|})$ and $\\mathcal D^{-1,2}$ its dual. For $u\\in\\mathcal D^{1,2}$, we define the mass as\n    $$\\mathcal{M}(u)=\\frac{1}{2}\\int_{\\mathbb R^d}|u|^2\\mathrm{d} x$$\n    and the energy as\n    $$\\mathcal{E}(u)=\\frac{1}{2}\\langle -Au,u\\rangle_{\\mathcal D^{-1,2},\\mathcal D^{1,2}}-\\frac{\\lambda}{2\\gamma+2}\\int_{\\mathbb R^d}|u|^{2\\gamma+2}\\mathrm{d} x.$$\n    Then, it holds\n    \\begin{enumerate}\n        \\item For $\\lambda\\leqslant0$, for any $m>0$, there exists $\\phi_m\\in\\mathcal D^{1,2}$ such that $\\phi_m>0$ and\n        $$\\argmin_{\\substack{u\\in\\mathcal D^{1,2}\\\\\\mathcal{M}(u)=m}}\\mathcal{E}(u) = \\left\\{\\mathrm{e}^{\\mathrm{i}\\theta}\\phi_m,\\; \\theta\\in\\mathbb R\\right\\}.$$\n        \\item For $\\lambda>0$ and $\\gamma<\\frac{2}{d}$, for any $m>0$, there exists $\\phi_m\\in\\displaystyle\\argmin_{u\\in\\mathcal D^{1,2},\\;\\mathcal{M}(u)=m}\\mathcal{E}(u)$ such that $\\phi_m>0$.\n        \\item For $\\lambda>0$ and $\\gamma=\\frac{2}{d}$, there exists $m_{*,\\lambda}(\\Xi)>0$ such that for all $m\\in(0,m_{*,\\lambda}(\\Xi))$, there exists $\\phi_m\\in\\displaystyle\\argmin_{u\\in\\mathcal D^{1,2},\\;\\mathcal{M}(u)=m}\\mathcal{E}(u)$ such that $\\phi_m>0$. Moreover, there exists $m_{*,\\lambda}(\\Xi)\\leqslant m^*_\\lambda(\\Xi)\\leqslant m^*_\\lambda$ such that\n        $$\\forall m>m^*_\\lambda(\\Xi), \\inf_{\\substack{u\\in\\mathcal D^{1,2}\\\\\\mathcal{M}(u)=m}}\\mathcal{E}(u)=-\\infty,$$\n        where $m^*_\\lambda$ is the critical mass of the classical nonlinear Schrödinger equation (i.e.\\ without any potential).\n        \\item For $\\lambda>0$ and $\\gamma>\\frac{2}{d}$, for any $m>0$,\n        $$\\inf_{\\substack{u\\in\\mathcal D^{1,2}\\\\\\mathcal{M}(u)=m}}\\mathcal{E}(u)=-\\infty.$$\n    \\end{enumerate}\n\n\\end{theo}\n\nIt is classical that the set of minimal energy states of fixed mass is stable in $\\mathcal D^{1,2}$ for \\cref{Eq-AGP}. In particular, \\cref{Th-EnergyGS} implies the orbital stability of the ground-state $\\phi_m$ in the defocusing case ($\\lambda<0$). \\cref{Prop-ExistenceEnergyGS,Prop-FrequencyEnergyGS} show that, for $\\lambda\\leqslant 0$ or $\\gamma<\\frac{2}{d}$, energy minimization at fixed mass provide positive solutions of \\cref{Eq-StationaryAGP} for any possible value of $\\omega$. We also prove the small mass asymptotic of $\\phi_m$ and its frequency $\\omega_m$.\\\\\n\nIn \\cref{Sec-ActionGS}, we investigate the minimization of action $\\mathcal{S}_\\omega = \\mathcal{E} + \\omega \\mathcal{M}$. In the focusing case ($\\lambda> 0$), one easily verifies the action is not lowerbounded. We thus introduce the Nehari manifold and show the following result.\n\n\\begin{theo}\\label{Th-ACtionGS}\n\n    Let $d\\in\\{1,2\\}$, $\\lambda,\\omega\\in\\mathbb R$, $\\gamma>0$ and $A$ the self-adjoint operator constructed in \\cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\\xi$ a spatial white noise. Let $\\mathcal D^{1,2}=\\mathrm{D}(\\sqrt{|A|})$ and $\\mathcal D^{-1,2}$ its dual. Denote by $\\mu_0$ the smallest eigenvalue of $-A$. Let $\\omega>-\\mu_0$ and define $$\\mathcal{S}_\\omega = \\mathcal{E} + \\omega \\mathcal{M}$$ and $$\\mathcal{N}_\\omega =\\left\\{u\\in\\mathcal D^{1,2}\\backslash\\{0\\},\\; \\langle \\nabla\\mathcal{S}_\\omega u,u \\rangle_{\\mathcal D^{-1,2},\\mathcal D^{1,2}}=0\\right\\}.$$\n    Then, there exists $\\psi_\\omega\\in\\displaystyle\\argmin_{\\mathcal{N}_\\omega}\\mathcal{S}_\\omega$ such that $\\psi_\\omega>0$.\n\n\\end{theo}\n\nLet us stress out an important property of solution of \\cref{Eq-StationaryAGP} compared to the noiseless case. In presence of white noise, standing waves have a limited Sobolev regularity.\n\\begin{theo}\\label{Th-MaxRegSW}\n    Let $d\\in\\{1,2\\}$, $\\lambda,\\omega\\in\\mathbb R$ and $\\gamma>0$. Almost surely any solution $u$ of \\cref{Eq-StationaryAGP} belongs to $\\mathcal W^{2-\\frac{d}{2}-,q}$ for all $q\\in[2,+\\infty)$, where $\\mathcal W^{s,q}$ spaces are defined in \\cref{Sec-ConfiningSobolev}. Moreover, almost surely any solution $u$ of \\cref{Eq-StationaryAGP} which belongs to $\\mathrm{H}^{2-\\frac{d}{2},q}(U)$ for some open set $U\\subset\\mathbb R^d$ and $q\\in[2,+\\infty]$ vanishes everywhere in $U$.\n\\end{theo}", "sketch": "In order to prove Theorem~\\ref{Th-SynthesExistenceSW}, the text says they \"use constrained minimization methods\".\n\n- In \\cref{Sec-EnergyGS}, they \"study the minimization of energy at fixed mass\" (Theorem~\\ref{Th-EnergyGS}). It is then stated that \\cref{Prop-ExistenceEnergyGS,Prop-FrequencyEnergyGS} \"show that, for $\\lambda\\leqslant 0$ or $\\gamma<\\frac{2}{d}$, energy minimization at fixed mass provide positive solutions of \\cref{Eq-StationaryAGP} for any possible value of $\\omega$\".\n\n- In \\cref{Sec-ActionGS}, they instead \"investigate the minimization of action $\\mathcal{S}_\\omega=\\mathcal{E}+\\omega\\mathcal{M}$\". In the focusing case ($\\lambda>0$), they note that \"one easily verifies the action is not lowerbounded\", so they \"introduce the Nehari manifold\" and obtain an action minimizer on it (Theorem~\\ref{Th-ACtionGS}) for $\\omega>-\\mu_0$.\n\n- They also \"stress out\" that, in the presence of white noise, standing waves have \"limited Sobolev regularity\" (Theorem~\\ref{Th-MaxRegSW}), giving the maximal regularity and a unique-continuation-type statement (if a solution lies in $\\mathrm{H}^{2-\\frac{d}{2},q}(U)$ on some open set $U$, then it \"vanishes everywhere in $U$\").", "expanded_sketch": "In order to prove the main theorem, the text says they \"use constrained minimization methods\".\n\nNext, they \"study the minimization of energy at fixed mass\" and use the following result.\n\n\\begin{theo}\\label{Th-EnergyGS}\n\n    Let $d\\in\\{1,2\\}$, $\\lambda,\\omega\\in\\R$, $\\gamma>0$ and $A$ be the self-adjoint operator constructed in \\cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\\xi$ a spatial white noise, $\\D^{1,2}=\\mathrm{D}(\\sqrt{|A|})$ and $\\D^{-1,2}$ its dual. For $u\\in\\D^{1,2}$, we define the mass as\n    $$\\mathcal{M}(u)=\\frac{1}{2}\\int_{\\R^d}|u|^2\\d x$$\n    and the energy as\n    $$\\mathcal{E}(u)=\\frac{1}{2}\\langle -Au,u\\rangle_{\\D^{-1,2},\\D^{1,2}}-\\frac{\\lambda}{2\\gamma+2}\\int_{\\R^d}|u|^{2\\gamma+2}\\d x.$$\n    Then, it holds\n    \\begin{enumerate}\n        \\item For $\\lambda\\leqslant0$, for any $m>0$, there exists $\\phi_m\\in\\D^{1,2}$ such that $\\phi_m>0$ and\n        $$\\argmin_{\\substack{u\\in\\D^{1,2}\\\\\\mathcal{M}(u)=m}}\\mathcal{E}(u) = \\left\\{\\e^{\\complexI\\theta}\\phi_m,\\; \\theta\\in\\R\\right\\}.$$\n        \\item For $\\lambda>0$ and $\\gamma<\\frac{2}{d}$, for any $m>0$, there exists $\\phi_m\\in\\displaystyle\\argmin_{u\\in\\D^{1,2},\\;\\mathcal{M}(u)=m}\\mathcal{E}(u)$ such that $\\phi_m>0$.\n        \\item For $\\lambda>0$ and $\\gamma=\\frac{2}{d}$, there exists $m_{*,\\lambda}(\\Xi)>0$ such that for all $m\\in(0,m_{*,\\lambda}(\\Xi))$, there exists $\\phi_m\\in\\displaystyle\\argmin_{u\\in\\D^{1,2},\\;\\mathcal{M}(u)=m}\\mathcal{E}(u)$ such that $\\phi_m>0$. Moreover, there exists $m_{*,\\lambda}(\\Xi)\\leqslant m^*_\\lambda(\\Xi)\\leqslant m^*_\\lambda$ such that\n        $$\\forall m>m^*_\\lambda(\\Xi), \\inf_{\\substack{u\\in\\D^{1,2}\\\\\\mathcal{M}(u)=m}}\\mathcal{E}(u)=-\\infty,$$\n        where $m^*_\\lambda$ is the critical mass of the classical nonlinear Schr\u0006dinger equation (i.e.\\ without any potential).\n        \\item For $\\lambda>0$ and $\\gamma>\\frac{2}{d}$, for any $m>0$,\n        $$\\inf_{\\substack{u\\in\\D^{1,2}\\\\\\mathcal{M}(u)=m}}\\mathcal{E}(u)=-\\infty.$$\n    \\end{enumerate}\n\n\\end{theo}\n\nIt is then stated that \\cref{Prop-ExistenceEnergyGS,Prop-FrequencyEnergyGS} show that, for $\\lambda\\leqslant 0$ or $\\gamma<\\frac{2}{d}$, energy minimization at fixed mass provides positive solutions of\n\\begin{equation}\n    -A\\phi-\\lambda|\\phi|^{2\\gamma}\\phi+\\omega \\phi = 0. \\label{Eq-StationaryAGP}\n\\end{equation}\nfor any possible value of $\\omega$.\n\nLater, they instead \"investigate the minimization of action $\\mathcal{S}_\\omega=\\mathcal{E}+\\omega\\mathcal{M}$\". In the focusing case ($\\lambda>0$), they note that \"one easily verifies the action is not lowerbounded\", so they \"introduce the Nehari manifold\" and obtain an action minimizer on it (Theorem~\\ref{Th-ACtionGS}) for $\\omega>-\\mu_0$.\n\nThey also \"stress out\" that, in the presence of white noise, standing waves have \"limited Sobolev regularity\" (Theorem~\\ref{Th-MaxRegSW}), giving the maximal regularity and a unique-continuation-type statement (if a solution lies in $\\mathrm{H}^{2-\\frac{d}{2},q}(U)$ on some open set $U$, then it \"vanishes everywhere in $U$\").", "expanded_theorem": "\\label{Th-SynthesExistenceSW}\n    Let $d\\in\\{1,2\\}$, $\\lambda,\\omega\\in\\mathbb R$ and $\\gamma>0$. Denote by $\\mu_0$ the lowest eigenvalue of $-A$. In each of the following situation, there exists a positive solution of\n\\begin{equation}\n    -A\\phi-\\lambda|\\phi|^{2\\gamma}\\phi+\\omega \\phi = 0. \\label{Eq-StationaryAGP}\n\\end{equation}\n in $\\mathrm{D}\\left(\\sqrt{|A|}\\right)$,\n    \\begin{itemize}\n        \\item $\\lambda>0$ and $\\omega>-\\mu_0$,\n        \\item $\\lambda = 0$ and $\\omega=\\mu_0$,\n        \\item $\\lambda<0$ and $\\omega<-\\mu_0$.\n    \\end{itemize}\n    In every other cases, there is no non-trivial non-negative solution in $\\mathrm{D}\\left(\\sqrt{|A|}\\right)$.,", "theorem_type": ["Existence", "Nonexistence"], "mcq": {"question": "Let \\(d\\in\\{1,2\\}\\), let \\(A=H+\\xi\\) be the Anderson Hamiltonian in dimension \\(d\\), and let \\(\\mu_0\\) denote the lowest eigenvalue of \\(-A\\). For \\(\\lambda,\\omega\\in\\mathbb R\\) and \\(\\gamma>0\\), consider the stationary equation\n\\[\n-A\\phi-\\lambda|\\phi|^{2\\gamma}\\phi+\\omega\\phi=0.\n\\]\nWhich statement about positive or nonnegative solutions \\(\\phi\\in \\mathrm D(\\sqrt{|A|})\\) is valid?", "correct_choice": {"label": "A", "text": "A positive solution \\(\\phi\\in \\mathrm D(\\sqrt{|A|})\\) exists in each of the following three cases: \\(\\lambda>0\\) and \\(\\omega>-\\mu_0\\); \\(\\lambda=0\\) and \\(\\omega=\\mu_0\\); \\(\\lambda<0\\) and \\(\\omega<-\\mu_0\\). In every other case, there is no non-trivial non-negative solution in \\(\\mathrm D(\\sqrt{|A|})\\)."}, "choices": [{"label": "B", "text": "A positive solution \\(\\phi\\in \\mathrm D(\\sqrt{|A|})\\) exists in each of the following three cases: \\(\\lambda>0\\) and \\(\\omega\\ge -\\mu_0\\); \\(\\lambda=0\\) and \\(\\omega=\\mu_0\\); \\(\\lambda<0\\) and \\(\\omega\\le -\\mu_0\\). In every other case, there is no non-trivial non-negative solution in \\(\\mathrm D(\\sqrt{|A|})\\)."}, {"label": "C", "text": "If \\(\\lambda>0\\) and \\(\\omega>-\\mu_0\\), or if \\(\\lambda=0\\) and \\(\\omega=\\mu_0\\), or if \\(\\lambda<0\\) and \\(\\omega<-\\mu_0\\), then there exists a non-trivial non-negative solution \\(\\phi\\in \\mathrm D(\\sqrt{|A|})\\)."}, {"label": "D", "text": "A positive solution \\(\\phi\\in \\mathrm D(\\sqrt{|A|})\\) exists whenever \\(\\lambda>0\\) and \\(\\omega\\in\\mathbb R\\), whenever \\(\\lambda=0\\) and \\(\\omega=\\mu_0\\), and whenever \\(\\lambda<0\\) and \\(\\omega< -\\mu_0\\). In every other case, there is no non-trivial non-negative solution in \\(\\mathrm D(\\sqrt{|A|})\\)."}, {"label": "E", "text": "A positive solution \\(\\phi\\in \\mathrm D(\\sqrt{|A|})\\) exists in each of the following three cases: \\(\\lambda>0\\) and \\(\\omega>-\\mu_0\\); \\(\\lambda=0\\) and \\(\\omega\\ge \\mu_0\\); \\(\\lambda<0\\) and \\(\\omega<-\\mu_0\\). In every other case, there is no non-trivial non-negative solution in \\(\\mathrm D(\\sqrt{|A|})\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "case_split", "tampered_component": "strict_thresholds_at_\\pm\\mu_0", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "case_split", "tampered_component": "dropped_positivity_and_uniqueness_of_exclusion_clause", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "focusing_case_requires_\\omega>-\\mu_0", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "linear_case_only_at_\\omega=\\mu_0", "template_used": "quantifier_dependence"}]}}
{"id": "2512.23266v1", "paper_link": "http://arxiv.org/abs/2512.23266v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{t:main}\nIf $\\alpha<\\frac{\\pi^2}{2}$, then as $n\\to\\infty$, $$\\frac{1}{\\ln^3 n} {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\, \\cvp\\, \\frac{1}{\\frac{\\pi^2}{2}-\\alpha}.$$", "start_pos": 6513, "end_pos": 6711, "label": "t:main"}, "ref_dict": {"Ding-2": "\\begin{equation} \\p\\Big( n^{1/4} \\le \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)  \\le C' \\frac{n}{\\ln^2 n}\\Big) \\to 1, \\qquad n \\to \\infty.  \\label{Ding-2}\\end{equation}", "alpha:polynomial": "\\begin{equation}    \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)= n^{1- \\frac{\\pi}{\\sqrt{2 \\alpha}} +o_p(1)},\\label{alpha:polynomial}\\end{equation}", "t:main": "\\begin{theorem}\\label{t:main}\nIf $\\alpha<\\frac{\\pi^2}{2}$, then as $n\\to\\infty$, $$\\frac{1}{\\ln^3 n} \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\, \\cvp\\, \\frac{1}{\\frac{\\pi^2}{2}-\\alpha}.$$ \n\\end{theorem}", "s:upper": "\\begin{equation}    \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)= n^{1- \\frac{\\pi}{\\sqrt{2 \\alpha}} +o_p(1)},\\label{alpha:polynomial}\\end{equation}\n\n\\noindent where $o_p(1)$ denotes some quantity which converges to $0$ in probability as $n\\to\\infty$. This, together with Theorem \\ref{t:main}, shows that there is a phase transition at $\\lambda_c=\\frac{\\pi^2}{2}\\frac1{\\ln^2 n}$. We also note that \\eqref{alpha:polynomial} is consistent with \\eqref{Ding-2} when $\\alpha$ is taken sufficiently large. \n\nThis critical value $\\lambda_c$ also appears in a related problem concerning the minimum average weight of a cycle in $K_n$. \nDing, Sun and Wilson \\cite{ding_sun_wilson} proved that conditionally on the minimum being nonnegative, its value is $\\frac{\\pi^2}{2 \\ln^2 n}  + O(\\frac{1}{\\ln^3 n})$ and that the length of the corresponding cycle is of order $\\ln^3 n$. It would be interesting to investigate further the connection between these two models. \n\n\\bigskip\nNow, we outline the main ideas of the proofs. \n Heuristically, we  explore the graph in the following way. Let $N=o(n)$ be an integer. We start with an initial set of $N$ vertices, called generation $0$. We examine all adjacent edges of these vertices\n  and select the $N$ edges with the  smallest labels. The endpoints of the selected edges form generation $1$, and the associated labels stand for their positions. We then delete the initial set of vertices and all their adjacent edges, and continue the exploration from the vertices of generation $1$. Each incident edge of a vertex at generation $1$ carries a weight. Add this weight to the position of the vertex, \n  then select the $N$ edges associated to the $N$ smallest obtained numbers. The endpoints of the selected edges form the vertices of generation $2$ and the associated numbers stand for their positions.  This process can be repeated for roughly $\\frac{n}{N}$ steps before the graph is exhausted. The weights of the paths obtained by this strategy resemble an (inhomogeneous)  branching random walk  in which only the $N$ leftmost particles are retained at each step. This model was introduced by Brunet and Derrida \\cite{brunet_derrida2} who conjectured that the selection mechanism was shifting the speed of the population by a correction term of order $\\frac{1}{\\ln^2 N}$, a result later proved by B\\'erard and Gou\\'er\\'e \\cite{berard_gouere}.   We refer to Maillard \\cite{maillard}\n  for more  results on the model in continuous time. In our setting, the minimum of the exploration process at time $t$ is of order \n\\[\n-\\ln N + \\frac{\\pi^2}{2 \\ln^2 N} t.\n\\]\nTaking  $N=n^{1-o(1)}$  and $N=e^{\\frac{\\pi}{\\sqrt{2\\beta_n}}}$ yields Theorem \\ref{t:main} and Theorem \\ref{t:main2}, respectively. A slight variation of this strategy is used to establish the lower bound for $\\L(n,\\lambda)$. The upper bound follows a similar approach to that of \\cite{ding,ding_goswami,ding_sun_wilson} by computing suitable truncated first moments. \n\n\\bigskip\n\nFinally, we mention the recent preprint by Jorritsma, Maillard and M\\\"orters \\cite{JMM} in which the authors make use of a killed branching random walk to explore growing sparse random graphs. Even if the models are different, our results  show similarity with several features of  their model, see \\cite{JMM} and the references therein.\n\n\\bigskip\n\nThe paper is organized as follows. Sections \\ref{s:upper} and \\ref{s:lower} give the proofs of the upper bounds and lower bounds for $\\L(n,\\lambda)$ respectively.  Section \\ref{s:rw-brw} collects results on real-valued random walks and  branching random walks that are used in Sections \\ref{s:upper} and \\ref{s:lower}.\n\n\\section{Proofs of the upper bounds in Theorems \\ref{t:main} and \\ref{t:main2}}\n\\label{s:upper}\n\nLet $(S_1,S_2,\\ldots)$ denote a random walk with step distribution $X$ of \\eqref{def:X}, starting at $S_0=0$. For any $\\ell\\ge 1$ and Borel set $A\\subset \\r^\\ell$, \n\\begin{equation}\\label{comp:moment1}\n\\e\\Big[\\#\\{\\gamma \\in {\\mathcal P}_\\ell\\colon S(\\gamma) \\in A\\}\\Big] \\le n^{\\ell+1}\\p\\Big((S_1,\\ldots,S_\\ell) \\in A\\Big), \n\\end{equation}", "t:main2": "\\begin{theorem}\\label{t:main2}\n For each $\\varepsilon>0$, let $\\beta_n$ be a sequence with $\\beta_n \\in (0,\\varepsilon)$ for all $n$ and such that $\\liminf_{n\\to\\infty}  \\frac{\\sqrt{2\\beta_n}}{\\pi} \\ln n > 1$.  As $n\\to \\infty$ then $\\varepsilon\\to 0$, $$\\sqrt{\\beta_n}\\Big(\\ln \\L(n,\\beta_n) -\\ln n\\Big) \\, \\cvp\\,  -\\frac{\\pi}{\\sqrt{2}}.$$\n In other words, for any $a>0$, \n\\[\n\\lim_{\\varepsilon\\downarrow0}\\liminf_{n\\to\\infty}\\p\\Big(\\frac{1}{n}\\L(n,\\beta_n) \\in [e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1+a)}, e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1-a)}]\\Big)=1.\n\\]\n\\end{theorem}", "s:lower": "\\begin{proof}[Proof of the upper bound in Theorem \\ref{t:main2}] Let $n$ be large and  $a \\in (0, 1)$. Consider $\\ell= \\lceil ne^{- \\frac{\\pi}{\\sqrt{ 2 \\beta_n}} (1-a)}\\rceil$,  $r=e^{\\frac{\\pi}{\\sqrt{2\\beta_n}}(1-\\frac{a}2)}$ and  let $m'_\\Delta(L)$ be the number of $i\\in \\lb 0,L\\rb$ such that $\\inf_{j \\in \\lb i,L\\rb} S_j\\le S_i- \\Delta$ with $\\Delta:=\\ln r$.  Let $b_n=\\ln n + \\ln\\ln n$.  By the same argument leading to \\eqref{eq:L>ell-bn}, but using Corollary \\ref{cor:jump} (ii) in addition, we obtain that  for any $M>2$, $$\\p\\Big(\\L(n, \\beta_n) \\ge \\ell\\Big) \\le  \\sum_{L=\\ell}^{2\\ell} \\widetilde a_L + \\frac{2}{M}+ o(1),$$\n\n\\noindent where for $L\\in \\lb \\ell, 2\\ell \\rb$, $$ \\widetilde a_L:= \\p\\Big(\\exists \\gamma\\in {\\mathcal P}_L: S_L(\\gamma) \\le \\beta_n L, m'_\\Delta(L) \\le M \\frac{n}{r},\\, \\min_{0\\le i \\le j \\le L} (S_j(\\gamma)- S_i(\\gamma)) >-  b_n \\Big).$$\n\nBy \\eqref{comp:moment1} and \\eqref{eq:many-to-one}, \\begin{align*}    \\widetilde a_L & \\le n \\, e^{\\beta_n L}  \\p\\Big( Y_L\\le \\beta_n L , \\, m_\\Delta(L) \\le M \\frac{n}{r},\\,  \\min_{0\\le i \\le j \\le L} (Y_j - Y_i ) >- b_n\\Big)\n\\\\\n& \\le n \\, e^{\\beta_n L}  \\p\\Big(   \\max_{i \\in \\lb 0, L\\rb} Y_i \\le 2 \\beta_n L, \\,  m_\\Delta(L) \\le M \\frac{n}{r}\\Big)\n ,\n\\end{align*} where $m_\\Delta(L)$ is as defined in Lemma \\ref{main2:upper} and in the last inequality, we have used the fact that $\\max_{i \\in \\lb 0, L\\rb} Y_i\\le Y_L+b_n \\le \\beta_n L+ b_n \\le 2 \\beta_n L$. Choose $d\\in (0, \\frac{\\pi^2}2)$ \n  such that $\\eta:= \\frac{2d}{\\pi^2 (1-\\frac{a}2)^2} - 1>0$. Fix an arbitrary $c\\in (0, 1)$. If $\\varepsilon>0$ is chosen small enough in Theorem \\ref{t:main2}, we have $L\\ge \\Delta^{2+c}$, $M \\frac{n}{r} \\le L \\Delta^{-2-c}$ and $2 \\beta_n L \\le L \\Delta^{-1-c}$.  Note that $\\Delta$ is arbitrarily large when $\\varepsilon$ goes to $0$.   Lemma \\ref{main2:upper} yields that $$ \\widetilde a_L \\le n e^{\\beta_n L} e^{- d \\frac{L}{\\Delta^2}} = n e^{- \\eta \\,\\beta_n L}. $$\n\n Notice that for all large $n$, $\\beta_n L \\ge \\beta_n \\ell \\ge n^{a'}$ for some positive constant $a'$. Hence   $\\sum_{L=\\ell}^{2\\ell} \\widetilde a_L \\le n^2  e^{- \\eta\\, n^{a'}}\\to 0$ and we conclude the proof by letting $n \\to\\infty$ then $M\\to\\infty$.  \\end{proof}\n\n\\section{Proofs of the lower bounds}\n\\label{s:lower}\n\nWe consider a branching random walk $\\V$ on $\\r$. It starts with a certain number of particles at $0$. At each discrete time $k\\ge 0$, each particle of the current population splits into an infinite number of children, whose displacements from their parent  are given by a copy of \n\\begin{equation} \\label{eq:BRW-V}   \n\\sum_{i=1}^\\infty {\\delta}_{\\{e T_i -1\\}} \n\\end{equation}\n\n\\noindent where $0<T_1<T_2<\\ldots$ are the atoms of a Poisson point process of intensity $1$ on $\\r_+$. The position of a particle $u$ is denoted by $\\V(u)$.\nThe offspring distribution corresponds to the limiting law  of $\\sum_{u=2}^{n} \\delta_{X(1,u)}$, so that, when $n$ is large, exploring the graph around a given vertex locally approximates the branching random walk $\\V$. To make it precise, we will  couple labels $X(u,v)$ and copies of the Poisson point process $(T_i)_{i\\ge 1}$ as follows. \n\n\\medskip\n\nConsider $(X_i)_{1\\le i \\le n-1}$ a family of $(n-1)$ i.i.d. copies of $X= n e \\mbox{Exp}(1)-1$.  Its increasing order statistics $ X_{(0)}:=-1 < X_{(1)}< ...< X_{(n-1)}$ verify that the variables $X_{(i)}-X_{(i-1)}$ are independent, distributed as $\\frac{n e }{n-i}\\text{Exp}(1)$.   One can couple it with a Poisson point process $T_1<T_2<\\ldots$ of intensity $1$, by \n\\[     X_{(i)}-X_{(i-1)} = \\frac{n e}{n-i} (T_{i}-T_{i-1}), \\qquad i\\in \\lb 1, n-1\\rb, \n\\]where we set $T_0=0$.  In particular,  for all $i\\in \\lb 1, n-1\\rb$, \\begin{equation}\\label{coupling}\n   e T_i-1 \\le  X_{(i)} \\le   \\frac{n}{n-i} (e T_{i}-1) + \\frac{i}{n-i}.\n\\end{equation}\n\nFor each $u\\in K_n$, this gives a coupling between $(X(u,v))_{v\\neq u}$ and a copy of $(T_i)_{i\\ge 1}$.  We further need the following simple observation.  Let $m \\in \\lb 1, n-1 \\rb$. If we choose   $m$  points uniformly from $(X_{(i)})_{1\\le i \\le n-1}$ and arrange them in increasing order, then they have the same distribution as  the increasing order statistics of $(X_i)_{1\\le i \\le m}$. In other words, there exists a coupling in the sense of \\eqref{coupling}, between $X(u, v)$ when $v$ varies over a subset of $K_n$ with $m$ vertices, and the family of $m$  points  chosen uniformly from $(T_i)_{1\\le i \\le n-1}$.  \n\nLet $N$ be an integer between $1$ and $n-1$.  We denote by $\\V^N$ the branching random walk $\\V$  in which, at each step, we only keep the $N$ leftmost particles of the next generation. For a particle $u$ in $\\V^N$, let   $|u|$ denote its generation in the genealogical tree of $\\V^N$, and let $\\V^N(u)$ denote its position in $\\r$.\n\n\\begin{proposition}\\label{p:first_moment} Fix $a\\in [0, 1]$ and $t>0$.    If $\\V^N$ starts with $\\lfloor N^a\\rfloor$ particles located at $0$, then for any sequence $\\ell_N$ satisfying $\\frac{\\ell_N}{\\ln^3 N} \\to t$,  \\begin{equation}  \\frac{1}{\\ln N}  \\min_{|u|=\\ell_N} \\V^N(u)   \\,\\cvL2\\, -a+\\frac{\\pi^2}{2}t  , \\qquad N \\to\\infty. \\label{Nbrw-cvL2}\\end{equation}\n\\end{proposition}\n\nWe will apply the above proposition to two cases: $a=1$ and $a=0$.\nThe proof of Proposition \\ref{p:first_moment} is postponed to Section \\ref{sub:brw}. \n\nIn what follows, we construct a coupling between $\\V^N$ and an exploration process on $K_n$. \nWe will color particles of $\\V^{N}$ in blue, red or purple. Blue particles will correspond to  vertices of $K_{n}$ and will be the particles of interest. \n\n\\subsection{Construction of an exploration process $(Z_j, {\\mathcal Q}_j)_{j\\ge 0}$ in $K_n$ and its coupling with $\\V^N$.}\n \\label{s:exploration}\n\nFor each $j\\ge 0$, we define a  set $Z_j$ of so-called {\\it active} vertices of $K_n$ and a set of paths ${\\mathcal Q}_j\\subset {\\mathcal P}_j$  such that each vertex  in $Z_j$ is associated with a unique  path in ${\\mathcal Q}_j$ ending at this point. The set $Z_j$ will correspond to the blue particles of $\\V^{N}$ at time $j$. We will therefore write $u$ either for the blue particle or its corresponding  vertex in $K_n$.  For each active vertex $u$ at step $j$, we will write $V(u)$  for $S_j(\\gamma)$ where $\\gamma$ is the path in ${\\mathcal Q}_j$ ending at $u$.  \n\nLet $G_0$ be a complete subgraph of $K_n$. Let $Z_0\\subset G_0$ be a set of $N_0\\le N$ active vertices at time $0$. We start  the branching random walk $\\V$ with  $N_0$ blue particles,  each particle corresponding to an active vertex in $Z_0$.  We let $\\mathcal Q_0$ be the collection $\\{\\{u\\},\\, u\\in Z_0\\}$.  We denote by $\\#G_0$ the number of vertices in $G_0$. Let $m:=\\#G_0- \\#Z_0$.   Each $u\\in Z_0$ corresponds to a blue particle in $\\V$, still denoted by $u$. We have $\\V(u)=V(u)=0$. We  choose $m$ children $v$ of $u$ in $\\V$ uniformly among the $n-1$ leftmost ones and color them {\\it blue.}  By the coupling described above, for each blue particle $u$, $\\V(v)-\\V(u)$ is coupled with $X(u, v)$ for $v \\in G_0 \\backslash Z_0$ (where, as for $u$, the variable $v$ denotes both a particle in $\\V$ and a vertex in $G_0$).  Let $i_v$ be the index of the point $T_i$ corresponding to $v$ in the coupling \\eqref{coupling}. We have  $$X(u,v) = X_{(i_v)} \\le \\frac{n}{n-i_v} e T_{i_v} -1 =   \\frac{n}{n-i_v} (\\V(v)-\\V(u))+\\frac{i_v}{n-i_v}.$$ We also have $X(u,v) \\ge \\V(v)-\\V(u)$. \nWe will see that $i_v \\le N$ for all $v$ of interest.  We color the remaining children of $u$ in $\\V$ {\\it red.}  We then retain the $N$ leftmost particles of $\\V$ in generation $1$ which is in fact the population of $\\V^N$ at generation $1$.  This population may contain red and blue particles.  If a blue particle $v$  is selected in $\\V^N$, we select the corresponding edge $(u, v)$ in $G_0$. Note that in this case we indeed have $i_v \\le N$. We also remark that several selected edges may point to the same vertex $v$. If this happens, we keep blue the corresponding particle in $\\V^N$ which has the minimal position in $\\V$,  and color the other particles corresponding to $v$ {\\it purple.} We then define $Z_1$ to be the set of blue particles in ${\\mathcal V}^N$ at generation $1$, after removing all purple particles. We identify $Z_1$ with a subset of $G_0$: a blue particle of $Z_1$ in $\\V^N$ corresponds exactly to one (active) vertex of $G_0$, and the paths $(u, v)_{u\\in Z_0, v \\in Z_1}$ form  ${\\mathcal Q}_1$. Since $i_v\\le N$ if $(u,v)\\in {\\mathcal Q}_1$,  we have $\\V^N(v)-\\V^N(u)= \\V(v)-\\V(u)$ and \n \\begin{equation}\\label{eq:index}\n  V(v)-V(u):=X(u,v) \\le \\frac{n}{n-N} (\\V^N(v)-\\V^N(u)) + \\frac{N}{n-N}.\n\\end{equation} \n Finally,  $G_1$ is the sub-complete graph formed by  removing $Z_0$ and all incident edges in $G_0$.  \n\nFor the next steps, we proceed by induction on $j$ as long as $Z_j\\neq \\emptyset$. See Figure \\ref{f:coupling}.\n\n \\begin{figure}[ht!]\n\\centering\n \\begin{tikzpicture}[scale=0.9, \n    >=Stealth,\n    vertex/.style={circle,draw,inner sep=1pt,minimum size=5mm},\n    bluev/.style={vertex,fill=blue!30},\n    redv/.style={vertex,fill=red!50},\n    purplev/.style={vertex,fill=violet!55},\n    thin edge/.style={line width=0.05pt},\n    dottedbox/.style={dotted,thick},\n    small/.style={font=\\scriptsize}\n]\n\n\\node at (0,4) {$G_j$};\n\n\\node[bluev]   (u1) at (-2,2) {$u_1$};\n\\node[bluev]   (u2) at (-0.3,3) {$u_2$};\n\\node[bluev]   (u3) at (2,2) {$u_3$};\n\n\\node[vertex] (v1)  at (-2,0.5) {$v_1$};\n\\node[vertex] (v2)  at (-1,-0.8) {$v_2$};\n\\node[vertex] (v3)  at (0,-1.2) {$v_3$};\n\\node[vertex] (v4)  at (1,-0.8) {$v_4$};\n\\node[vertex] (v5)  at (2,0.5) {$v_5$};\n\n\\foreach \\x/\\y in {u1/u2,u2/u3,u1/u3,u1/v1,u1/v2,u1/v3,u2/v1,u2/v2,u2/v3,u3/v1,u3/v2,u3/v3, v1/v2, v1/v3, v2/v3, u1/v4, u2/v4, u3/v4, v1/v4, v2/v4, v4/v4, u1/v5, u2/v5, u3/v5, v1/v5, v2/v5, v3/v5, v4/v5}\n  \\draw[thin edge] (\\x) -- (\\y);\n\n\\node[small] at (0,-2) {$Z_j=\\{u_1,u_2,u_3\\}$};\n\\node[small] at (0,-2.5) {$G_j\\setminus Z_j=\\{v_1,v_2,v_3, v_4, v_5\\}$};\n\n\\node at (6.8,4) {\\small $\\V^N$ at time $j$};\n\n\\node[bluev] (r1) at (6.3,3) {$u_1$};\n\\node[bluev] (r2) at (7,3) {$u_2$};\n\\node[bluev] (r3) at (8.4,3) {$u_3$};\n\\node[purplev] (r4) at (7.7,3) {$\\cdot$};\n\\node[redv] (r5) at (5.6,3) {$\\cdot$};\n\n\\draw[dottedbox] ($(r5)+(-0.5,-0.5)$) rectangle ($(r3)+(0.5,0.5)$);\n\n\\node[bluev] (a1) at (5.6,0) {$v_1$};\n\\node[redv]  (a2) at (6.3,0) {$\\cdot$};\n\n\\node[purplev] (b1) at (7,0) {$v_1$};\n\\node[bluev]   (b2) at (7.7,0) {$v_2$};\n\n\\node[redv] (c1) at (8.4,0) {$\\cdot$};\n\n\\node[bluev]  (c2) at (9.35,0) {$v_5$};\n\\node[bluev]  (c3) at (10.05,0) {$v_4$};\n\\node[redv]  (c4) at (10.75,0) {$\\cdot$};\n\n\\draw (r1)--(a1);\n\\draw (r1)--(b2);\n\\draw [dashed] (r1)--(a2);\n\n\\draw (r2)--(b1);\n\n\\draw  (r3)--(c2);\n\\draw  (r3)--(c3);\n\\draw [dashed] (r5)--(c4);\n\n\\draw [dashed](r4)--(c1);\n\n\\draw[dottedbox] ($(a1)+(-0.5,-0.5)$) rectangle ($(c1)+(0.5,0.5)$);\n\n\\node[small] at (6.8,-0.8) {$\\mathcal V^N$ at time $j+1$};\n\\node[small] at (6.8,-1.2) {by keeping the $N$ leftmost particles};\n\n\\node[small] (Zj1) at (6.8,-2) {$Z_{j+1}=\\{v_1,v_2\\}$};\n\\node[small] (Zj1) at (6.8,-2.5) {$G_{j+1}=\\{v_1, v_2, v_3, v_4, v_5\\}$};\n\n\\end{tikzpicture}\n\\caption{\\small  Schematic illustration of the coupling between $(Z_j, G_j)$ and $\\V^N$.}\n\\label{f:coupling}\n\\end{figure}\n\n Suppose we are at step $j$ and have constructed the graph $G_j$, the set $Z_j\\subset G_j$ and the paths in $\\mathcal Q_j$.  Each active vertex  $u\\in Z_j$ corresponds to a blue particle $u$ of $\\V^N$ at time $j$. We randomly color $\\#G_j-\\#Z_j$ children of $u$ in $\\V$ blue, among the $n-1$ leftmost ones, so that each blue child $v$ corresponds to an edge in $\\{u\\}\\times (G_j\\setminus Z_j)$.  The remaining children of $u$ in $\\V$ are colored {\\it red}.  We couple the family $(X(u,v))_{v\\in G_j\\setminus Z_j}$ with the displacements of the blue particles in $\\V$ via the same coupling described above. \n\n Purple and red particles at time $j$ give birth only to red particles. We then retain the $N$ leftmost particles of the population at generation $j+1$, which form the population of $\\V^N$ at generation $j+1$.  It is composed of blue/red particles. If a blue particle $|v|=j+1$ with parent $u$ is selected in $\\V^N$, we select the corresponding edge $(u,v)$ in $G_j$.  In the case several selected edges  point to the same vertex $v$ in $G_j \\setminus Z_j$, so that multiple blue particles at time $j+1$ correspond to the same vertex $v$, we keep the one with minimal position in $\\V$ {\\it blue} and color the other particles {\\it purple.}  By definition, $V(v):=V(u)+X(u,v)$ is the weight $S_{j+1}(\\gamma,v)$ of the path $(\\gamma,v)$ in $K_n$ obtained by  adding the edge $e=(u,v)$ to the unique path $\\gamma \\in {\\mathcal Q}_j$ ending at $u$. The blue particles  $v$ in $\\V^N$ at time $j+1$ form the set $Z_{j+1}$, and the paths $(\\gamma,v)$ the set ${\\mathcal Q}_{j+1}$.  Finally, we remove $Z_j$ and all incident edges to obtain the graph $G_{j+1}$.  We note that \\eqref{eq:index} also holds for any $v \\in Z_{j+1}$ (if it exists). Summing these inequalities, we obtain that \\begin{equation}\\label{eq:index-2}\nV(v) \\le \\frac{n}{n-N}  \\V^N(v)+\\frac{N}{n-N}(j+1) , \\qquad \\forall\\, v \\in Z_{j+1}.\n\\end{equation} \n\n\\noindent Therefore, we have constructed an exploration process $(Z_j, {\\mathcal Q}_j)_{j\\ge 0}$, together with its coupling with $\\V^N$,  up to the first index $j$ for which $Z_j=\\emptyset$.  \\hfill {\\footnotesize $\\Box$}\n\n\\medskip\n   Let $|u|=j$ be a blue particle in $\\V^N$ at time $j$ and $v$ be a child of $u$ in $\\V$ among the $n-1$ leftmost children. Conditionally on the whole branching random walk $\\V$ and     on the exploration process up to time $j$, the probability for $v$ to be colored  red is \n\\[ 1-\\frac{ \\#G_{j+1}}{n-1}= \\frac{n-1-\\#G_0+\\#Z_0 + \\#Z_1+\\ldots+\\#Z_j }{n-1}\\le \\frac{n-\\#G_0+N(j+1)}{n-1}.\n\\]  \n\nNow, we estimate the probability that $v$ is colored purple, given it is not red. In particular, $v$ belongs to $\\V^N$ at time $j+1$. To avoid confusion, write temporarily $w$ for the vertex in $K_n$ associated to the particle $v$ in $\\V^N$. Whenever $v$ is colored purple,  there must exist some particle $u'\\neq u$  in $\\V^N$ at time $j$ and some child $v'$ of $u'$ in $\\V^N$ such that  $\\V^N(v')-\\V^N(u')$  is less than $x:=\\V^N(v)- \\V^N(u')$, and $v'$ is coupled with the vertex $w$. Let  ${\\mathcal N}_{u'}$ be the set of children of $u'$ that are not red and  whose displacement (in $\\V$) with respect to $u'$ is less than $x$.  By our coupling, we identify ${\\mathcal N}_{u'}$ with a set of vertices in $G_{j+1}$. In particular, $w\\in {\\mathcal N}_{u'}$.\n  By symmetry, conditionally on $\\V$, on the exploration process up to time $j$, and on the red/non-red particles at time $j+1$,   the probability that $w\\in {\\mathcal N}_{u'}$ is $\\frac{\\#{\\mathcal N}_{u'}}{\\#G_{j+1}}$.   Since $\\sum_{u'} \\#{\\mathcal N}_{u'} \\le N$, we obtain that the conditional probability that $v$ is colored purple, given it is not colored red, is less than  \n \\[\n  \\frac{N}{\\#G_{j+1}}\\le \\frac{N}{\\#G_0-N(j+1)}\n  \\]\nas long as $N(j+1)<\\#G_0$.  Since $\\p(v \\textrm{ is not blue} \\mid \\V) = \\p(u \\textrm{ is not blue} \\mid \\V)+\\p(u \\textrm{ is blue}, v \\textrm{ is not blue} \\mid \\V)$, we find by induction that\n \\begin{equation}\\label{proba_red}\n       \\p( v \\textrm{ is not blue} \\mid \\V) \\le \\sum_{k=0}^j \\Big(\\frac{n-\\#G_0+N(k+1)}{n-1}  +\\frac{N}{\\#G_0-N(k+1)}\\Big).\n  \\end{equation}\n\n\\subsection{Lower bound in Theorem \\ref{t:main}}\nThe lower bound of Theorem \\ref{t:main} is a consequence  of the following lemma.\n\\begin{lemma}\n Let $\\alpha < \\frac{\\pi^2}{2}$ and $t<\\frac{1}{\\frac{\\pi^2}{2}-\\alpha}$. Choose $a\\in (0,1)$ close enough to $1$ such that \n\\[\n- a + \\frac{\\pi^2}{2 a^2} t < \\alpha t .\n\\]\nApply the above exploration process with $G_0=K_n$ and $N_0=N=\\lfloor n^a\\rfloor$. With probability tending to $1$ as $n\\to\\infty$, there is an active vertex  at time $\\lfloor t\\ln^3 n\\rfloor $ such that $V(u)\\le \\alpha t\\ln n$.  \n\\end{lemma}\n \\begin{proof}\n   For simplicity, suppose that $N=n^a$ and that $L:=t\\ln^3 n= \\frac{t}{a^3} \\ln^3 N$ is an integer.  Note that $(-1+ \\frac{\\pi^2}2 \\frac{t}{a^3}) \\ln N= (-a + \\frac{\\pi^2}{2 a^2} t) \\ln n $. Fix   $b\\in (0,\\alpha)$ such that $- a + \\frac{\\pi^2}{2 a^2} t<  b t$. By Proposition \\ref{p:first_moment} (with $a=1$ there), there is  a particle $|u|=L$ in $\\V^N$ with $\\V^N(u)\\le b t \\ln n$ with probability going to $1$ as $n\\to\\infty$. By \\eqref{proba_red}, the probability for $u$ to be blue also tends to $1$ as $n\\to\\infty$.  It remains to prove that $V(u)\\le \\alpha t \\ln n$. By \\eqref{eq:index-2},  $V(u)\\le (bt \\ln n)  \\frac{n}{n-N} + \\frac{N}{n-N} L \\le \\alpha t \\ln n$ for $n$ large enough indeed. \n\\end{proof}", "s:rw-brw": "\\begin{equation}    V(u^*_{j}) \\le \\frac{n}{n-N} \\sum_{i=1}^{j} \\eta_i + \\frac{Nj}{n-N}, \\qquad \\forall\\, j< \\tau_1.\\label{control-Vuj} \\end{equation}  \n\n\\noindent Hence $${\\tt s}_1 \\le \\frac{n}{n-N} \\sum_{i=1}^{\\Delta-1} \\max(\\eta_i,0) + \\frac{N (\\Delta-1) }{n-N}.$$\n\n The term $\\sum_{i=1}^{\\Delta-1} \\max(\\eta_i,0)$ is stochastically smaller than the sum of $\\Delta-1$ independent exponential r.v. with mean $2e$, as long as $G_{\\Delta}$ has more than $\\frac{n}{2}$ vertices. This condition is indeed satisfied thanks to \\eqref{card_G}. Then for some positive constant $c$,  $$  \\e_{G_{\\tau_{i-1}}}[({\\tt s}_1+\\Delta)^2] \\le c \\Delta^2.$$ \n\nBy Cauchy--Schwarz inequality and  \\eqref{bound_eventE}, we obtain that $$  \\e_{G_{\\tau_{i-1}}}[ ({\\tt s}_1+\\Delta) {\\bf 1}_{{\\mathcal E^c_1}}]\n\\le\nc^{1/2} \\Delta \\Big( (\\frac{2L}{n-1} +\\frac{1}{n-2NL})N\\Delta\\Big)^{1/2}\n\\le \\Delta^{-4}, \n$$\n\n\\noindent where the last inequality holds for all small $\\varepsilon$ and large $n$ (see \\eqref{eq:NLDelta}). It follows from \\eqref{cond-Gi} that $$ \\e \\Big[\\sum_{i=1}^{a_1L/\\Delta}   ({\\tt s}_i+\\Delta)   {\\bf 1}_{{\\mathcal E}_i^c}\\Big]\n\\le a_1 L \\Delta^{-3}.\n$$\n\n\\noindent Since $L \\Delta^{-3}= o(\\beta_n L)$, the Markov inequality yields \\eqref{sum-S-tau<Delta-2} \nand completes the proof of the lemma.\n\\end{proof}\n\n\\section{Proofs of Lemmas \\ref{main:upper}, \\ref{main2:upper},  and of Proposition  \\ref{p:first_moment}}\\label{s:rw-brw}\n\nRecall that $(Y_n)_{n\\ge 0}$ is a centered random walk starting at $0$ with step distribution $\\mbox{Exp}(1)-1$. In the first subsection, we present several estimates on $(Y_n)_{n\\ge 0}$ and provide  the proofs of Lemmas \\ref{main:upper}, \\ref{main2:upper}. The second subsection is devoted to the study of the branching random walks $\\V$ and $\\V^N$, and to the proof of Proposition  \\ref{p:first_moment}. \n\n\\subsection{Estimates of random walks: Proofs of Lemmas \\ref{main:upper} and \\ref{main2:upper}}\\label{sub:rw}\n\n\\begin{lemma}\\label{l:mogulski}\n Let $\\upsilon>0$ and $0<d<\\frac{\\pi^2}{2}$. There exists $\\Delta_0>0$ such that for all $\\Delta\\ge \\Delta_0$,   $r_0\\le 0\\le r_0+\\Delta$ and  $\\ell\\ge 1$,\n\\begin{equation}\n    \\p(r_0  \\le Y_j\\le r_0+\\Delta,\\, \\forall \\, j\\in \\lb 1,  \\ell\\rb) \\le e^{\\Delta^\\upsilon-\\frac{d}{\\Delta^2}\\ell}. \\label{mogulskii_upper0} \n\\end{equation}"}, "pre_theorem_intro_text_len": 3616, "pre_theorem_intro_text": "We consider the complete graph $K_n$ of the set of vertices $\\llbracket 1, n\\rrbracket=\\{1,2,\\ldots,n\\}$. Following Aldous \\cite{aldous}, we  assign each edge  an independent exponential random variable with mean $n$. In this setting, Aldous proved the existence of a phase transition for the size of  the largest tree with average weight below a given threshold \\cite[Theorem 1]{aldous}. We are interested in the analogous result for paths, rather than trees, established in  \\cite[Proposition 3.7]{aldous}. In order to state this result, we introduce the following notation. \nFor ease of notation, in this paper, we will suppose that the labels $X(u,v)_{u\\neq v \\in \\llbracket 1, n\\rrbracket}$  on the (non-oriented) edges of $K_n$ are i.i.d. copies of \n\\begin{equation}\\label{def:X}\nX=ne \\mathrm{Exp}(1)-1\n\\end{equation} \nwhere $\\mathrm{Exp(1)}$ denotes an exponential random variable with mean $1$. \nA path $\\gamma$ is a sequence of vertices $(\\gamma_0,\\gamma_1,\\ldots,\\gamma_\\ell) \\in  \\llbracket 1, n\\rrbracket^{\\ell+1}$ which is made of distinct adjacent edges $(\\gamma_i,\\gamma_{i+1})$. We call $|\\gamma|=\\ell$ the length of the path, and let ${\\mathcal P}_\\ell$ be the collection of paths of length $\\ell$. A path of length $0$ is reduced to a single vertex.\n\nFor any $\\gamma \\in {\\mathcal P}_\\ell$, we let $S_0(\\gamma)=0$ and \n\\begin{equation}    \nS_i(\\gamma) := X(\\gamma_0,\\gamma_1)+\\ldots+X(\\gamma_{i-1},\\gamma_i),\\qquad  i\\in \\llbracket 1,  \\ell\\rrbracket. \\label{Sigamma}\n\\end{equation}\nWe write $S(\\gamma):=(S_1(\\gamma),\\ldots,S_\\ell(\\gamma))$ and call $\\frac{S_{\\ell}(\\gamma)}{\\ell}$ the average weight of $\\gamma$. For $\\lambda\\in (-1, \\infty)$, let ${\\mathscr L}(n,\\lambda)$ be the length of the longest path with average weight below $\\lambda$:\n\\begin{equation}    \n{\\mathscr L}(n,\\lambda):=\\sup\\{ \\ell \\ge 1\\colon \\exists \\, \\gamma \\in  {\\mathcal P}_\\ell \\textrm{ such that } S_{\\ell}(\\gamma) \\le \\lambda \\ell \\}.\n\\label{def-Lnlambda} \\end{equation}\n\n  Translated in our setting, \\cite[Proposition 3.7]{aldous} states that ${\\mathscr L}(n, \\lambda)=o(n)$ if $\\lambda<0$ and  ${\\mathscr L}(n,\\lambda)=\\Omega(n)$ if $\\lambda>0$. \n Later, Ding \\cite{ding} proved that there is a critical window of size $(\\ln n)^{-2}$ around $\\lambda=0$:  there exist some constants $c'> c >0$ and $C'>C$ such that \n\n (i) if $ -1\\le \\alpha \\le c $, then  \\begin{equation} \\mathbb P\\Big( C \\ln^3 n \\le {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)  \\le C' \\ln^3 n\\Big) \\to 1, \\qquad n \\to \\infty;  \\label{Ding-1}\\end{equation}\n\n(ii) if $\\alpha \\ge c'$, then \\begin{equation} \\mathbb P\\Big( n^{1/4} \\le {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)  \\le C' \\frac{n}{\\ln^2 n}\\Big) \\to 1, \\qquad n \\to \\infty.  \\label{Ding-2}\\end{equation}\n\nThe constant $-1$ in (i) is arbitrary, see \\cite[Theorem 1.3]{ding}. To the best of our knowledge, it remains open to identify the critical value at which  the longest path undergoes a transition from  order $\\ln^3 n$ to   polynomial order, as well as the precise asymptotic behavior of ${\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)$. Another natural problem concerns the near-critical case $\\lambda = \\varepsilon$ for small $\\varepsilon > 0$, which was further investigated by Ding and Goswami \\cite{ding_goswami}: there exists some $\\varepsilon_0>0$ such that for all $\\varepsilon < \\varepsilon_0$, \\begin{equation} \\mathbb P\\Big( n e^{- C' \\varepsilon^{-1/2}} \\le {\\mathscr L}(n, \\varepsilon) \\le   n e^{- C \\varepsilon^{-1/2}}\\Big) \\to 1.  \\label{Ding-Goswami}\\end{equation} \n\n  The following theorems give sharp asymptotics of ${\\mathscr L}(n,\\lambda)$ in these regimes.", "context": "For any $\\gamma \\in {\\mathcal P}_\\ell$, we let $S_0(\\gamma)=0$ and \n\\begin{equation} \nS_i(\\gamma) := X(\\gamma_0,\\gamma_1)+\\ldots+X(\\gamma_{i-1},\\gamma_i),\\qquad i\\in \\llbracket 1, \\ell\\rrbracket. \\label{Sigamma}\n\\end{equation}\nWe write $S(\\gamma):=(S_1(\\gamma),\\ldots,S_\\ell(\\gamma))$ and call $\\frac{S_{\\ell}(\\gamma)}{\\ell}$ the average weight of $\\gamma$. For $\\lambda\\in (-1, \\infty)$, let ${\\mathscr L}(n,\\lambda)$ be the length of the longest path with average weight below $\\lambda$:\n\\begin{equation} \n{\\mathscr L}(n,\\lambda):=\\sup\\{ \\ell \\ge 1\\colon \\exists \\, \\gamma \\in {\\mathcal P}_\\ell \\textrm{ such that } S_{\\ell}(\\gamma) \\le \\lambda \\ell \\}.\n\\label{def-Lnlambda} \\end{equation}\n\nTranslated in our setting, \\cite[Proposition 3.7]{aldous} states that ${\\mathscr L}(n, \\lambda)=o(n)$ if $\\lambda<0$ and ${\\mathscr L}(n,\\lambda)=\\Omega(n)$ if $\\lambda>0$. \n Later, Ding \\cite{ding} proved that there is a critical window of size $(\\ln n)^{-2}$ around $\\lambda=0$: there exist some constants $c'> c >0$ and $C'>C$ such that\n\n(i) if $ -1\\le \\alpha \\le c $, then \\begin{equation} \\mathbb P\\Big( C \\ln^3 n \\le {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\le C' \\ln^3 n\\Big) \\to 1, \\qquad n \\to \\infty; \\label{Ding-1}\\end{equation}\n\n(ii) if $\\alpha \\ge c'$, then \\begin{equation} \\mathbb P\\Big( n^{1/4} \\le {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\le C' \\frac{n}{\\ln^2 n}\\Big) \\to 1, \\qquad n \\to \\infty. \\label{Ding-2}\\end{equation}\n\nThe constant $-1$ in (i) is arbitrary, see \\cite[Theorem 1.3]{ding}. To the best of our knowledge, it remains open to identify the critical value at which the longest path undergoes a transition from order $\\ln^3 n$ to polynomial order, as well as the precise asymptotic behavior of ${\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)$. Another natural problem concerns the near-critical case $\\lambda = \\varepsilon$ for small $\\varepsilon > 0$, which was further investigated by Ding and Goswami \\cite{ding_goswami}: there exists some $\\varepsilon_0>0$ such that for all $\\varepsilon < \\varepsilon_0$, \\begin{equation} \\mathbb P\\Big( n e^{- C' \\varepsilon^{-1/2}} \\le {\\mathscr L}(n, \\varepsilon) \\le n e^{- C \\varepsilon^{-1/2}}\\Big) \\to 1. \\label{Ding-Goswami}\\end{equation}\n\nThe following theorems give sharp asymptotics of ${\\mathscr L}(n,\\lambda)$ in these regimes.", "full_context": "For any $\\gamma \\in {\\mathcal P}_\\ell$, we let $S_0(\\gamma)=0$ and \n\\begin{equation} \nS_i(\\gamma) := X(\\gamma_0,\\gamma_1)+\\ldots+X(\\gamma_{i-1},\\gamma_i),\\qquad i\\in \\llbracket 1, \\ell\\rrbracket. \\label{Sigamma}\n\\end{equation}\nWe write $S(\\gamma):=(S_1(\\gamma),\\ldots,S_\\ell(\\gamma))$ and call $\\frac{S_{\\ell}(\\gamma)}{\\ell}$ the average weight of $\\gamma$. For $\\lambda\\in (-1, \\infty)$, let ${\\mathscr L}(n,\\lambda)$ be the length of the longest path with average weight below $\\lambda$:\n\\begin{equation} \n{\\mathscr L}(n,\\lambda):=\\sup\\{ \\ell \\ge 1\\colon \\exists \\, \\gamma \\in {\\mathcal P}_\\ell \\textrm{ such that } S_{\\ell}(\\gamma) \\le \\lambda \\ell \\}.\n\\label{def-Lnlambda} \\end{equation}\n\nTranslated in our setting, \\cite[Proposition 3.7]{aldous} states that ${\\mathscr L}(n, \\lambda)=o(n)$ if $\\lambda<0$ and ${\\mathscr L}(n,\\lambda)=\\Omega(n)$ if $\\lambda>0$. \n Later, Ding \\cite{ding} proved that there is a critical window of size $(\\ln n)^{-2}$ around $\\lambda=0$: there exist some constants $c'> c >0$ and $C'>C$ such that\n\n(i) if $ -1\\le \\alpha \\le c $, then \\begin{equation} \\mathbb P\\Big( C \\ln^3 n \\le {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\le C' \\ln^3 n\\Big) \\to 1, \\qquad n \\to \\infty; \\label{Ding-1}\\end{equation}\n\n(ii) if $\\alpha \\ge c'$, then \\begin{equation} \\mathbb P\\Big( n^{1/4} \\le {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\le C' \\frac{n}{\\ln^2 n}\\Big) \\to 1, \\qquad n \\to \\infty. \\label{Ding-2}\\end{equation}\n\nThe constant $-1$ in (i) is arbitrary, see \\cite[Theorem 1.3]{ding}. To the best of our knowledge, it remains open to identify the critical value at which the longest path undergoes a transition from order $\\ln^3 n$ to polynomial order, as well as the precise asymptotic behavior of ${\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)$. Another natural problem concerns the near-critical case $\\lambda = \\varepsilon$ for small $\\varepsilon > 0$, which was further investigated by Ding and Goswami \\cite{ding_goswami}: there exists some $\\varepsilon_0>0$ such that for all $\\varepsilon < \\varepsilon_0$, \\begin{equation} \\mathbb P\\Big( n e^{- C' \\varepsilon^{-1/2}} \\le {\\mathscr L}(n, \\varepsilon) \\le n e^{- C \\varepsilon^{-1/2}}\\Big) \\to 1. \\label{Ding-Goswami}\\end{equation}\n\nThe following theorems give sharp asymptotics of ${\\mathscr L}(n,\\lambda)$ in these regimes.\n\n(i) if $ -1\\le \\alpha \\le c $, then \\begin{equation} \\p\\Big( C \\ln^3 n \\le \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\le C' \\ln^3 n\\Big) \\to 1, \\qquad n \\to \\infty; \\label{Ding-1}\\end{equation}\n\n(ii) if $\\alpha \\ge c'$, then \\begin{equation} \\p\\Big( n^{1/4} \\le \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\le C' \\frac{n}{\\ln^2 n}\\Big) \\to 1, \\qquad n \\to \\infty. \\label{Ding-2}\\end{equation}\n\nThe constant $-1$ in (i) is arbitrary, see \\cite[Theorem 1.3]{ding}. To the best of our knowledge, it remains open to identify the critical value at which the longest path undergoes a transition from order $\\ln^3 n$ to polynomial order, as well as the precise asymptotic behavior of $\\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)$. Another natural problem concerns the near-critical case $\\lambda = \\varepsilon$ for small $\\varepsilon > 0$, which was further investigated by Ding and Goswami \\cite{ding_goswami}: there exists some $\\varepsilon_0>0$ such that for all $\\varepsilon < \\varepsilon_0$, \\begin{equation} \\p\\Big( n e^{- C' \\varepsilon^{-1/2}} \\le \\L(n, \\varepsilon) \\le n e^{- C \\varepsilon^{-1/2}}\\Big) \\to 1. \\label{Ding-Goswami}\\end{equation}\n\nNote that taking $\\alpha=0$ in Theorem \\ref{t:main} answers \\cite[{Equation (2)}]{ding}.\n\n\\begin{theorem}\\label{t:main2}\n For each $\\varepsilon>0$, let $\\beta_n$ be a sequence with $\\beta_n \\in (0,\\varepsilon)$ for all $n$ and such that $\\liminf_{n\\to\\infty} \\frac{\\sqrt{2\\beta_n}}{\\pi} \\ln n > 1$. As $n\\to \\infty$ then $\\varepsilon\\to 0$, $$\\sqrt{\\beta_n}\\Big(\\ln \\L(n,\\beta_n) -\\ln n\\Big) \\, \\cvp\\, -\\frac{\\pi}{\\sqrt{2}}.$$\n In other words, for any $a>0$, \n\\[\n\\lim_{\\varepsilon\\downarrow0}\\liminf_{n\\to\\infty}\\p\\Big(\\frac{1}{n}\\L(n,\\beta_n) \\in [e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1+a)}, e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1-a)}]\\Big)=1.\n\\]\n\\end{theorem}\n\nTaking $\\beta_n= \\frac{\\alpha}{\\ln^2 n}$ with $\\alpha > \\frac{\\pi^2}{2}$, we see that \\begin{equation} \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)= n^{1- \\frac{\\pi}{\\sqrt{2 \\alpha}} +o_p(1)},\\label{alpha:polynomial}\\end{equation}\n\n\\noindent where $o_p(1)$ denotes some quantity which converges to $0$ in probability as $n\\to\\infty$. This, together with Theorem \\ref{t:main}, shows that there is a phase transition at $\\lambda_c=\\frac{\\pi^2}{2}\\frac1{\\ln^2 n}$. We also note that \\eqref{alpha:polynomial} is consistent with \\eqref{Ding-2} when $\\alpha$ is taken sufficiently large.\n\n\\begin{proof}[Proof of the upper bound in Theorem \\ref{t:main}] Recall by Claim 2.4 of \\cite{ding} that if there is a path of length greater than $\\ell$ with average weight below $\\lambda$, then there is a path of length in $\\lb \\ell,2\\ell\\rb$ with average weight below $\\lambda$. Then for $\\lambda= \\frac{\\alpha}{\\ln^2 n}$, $t> \\frac{1}{\\frac{\\pi^2}{2}-\\alpha}$, and $\\ell= \\lfloor t \\ln^3 n\\rfloor$, \\begin{align} & \\p\\Big(\\L(n, \\lambda) \\ge \\ell\\Big) \n \\nonumber \\\\\n \\le & \\p\\Big(\\cup_{L=\\ell}^{2 \\ell}\\{ \\exists \\gamma \\in {\\mathcal P}_L: S_L(\\gamma) \\le \\lambda L \\}\\Big) \\nonumber \n\\\\\n \\le & \\p\\Big(\\cup_{L=\\ell}^{2 \\ell}\\{ \\exists \\gamma \\in {\\mathcal P}_L: S_L(\\gamma) \\le \\lambda L, \\min_{0\\le i \\le j \\le L} (S_j(\\gamma)- S_i(\\gamma)) > -b_n\\}\\Big) + o(1), \\label{eq:L>ell-bn}\n\\end{align}\n\n\\begin{equation} \\p\\Big( n^{1/4} \\le \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)  \\le C' \\frac{n}{\\ln^2 n}\\Big) \\to 1, \\qquad n \\to \\infty.  \\label{Ding-2}\\end{equation}\n\n\\begin{equation}    \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)= n^{1- \\frac{\\pi}{\\sqrt{2 \\alpha}} +o_p(1)},\\label{alpha:polynomial}\\end{equation}\n\n\\begin{theorem}\\label{t:main}\nIf $\\alpha<\\frac{\\pi^2}{2}$, then as $n\\to\\infty$, $$\\frac{1}{\\ln^3 n} \\L\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\, \\cvp\\, \\frac{1}{\\frac{\\pi^2}{2}-\\alpha}.$$ \n\\end{theorem}", "post_theorem_intro_text_len": 4434, "post_theorem_intro_text": "Note that taking $\\alpha=0$ in Theorem \\ref{t:main} answers \\cite[{Equation (2)}]{ding}.  \n\n\\begin{theorem}\\label{t:main2}\n For each $\\varepsilon>0$, let $\\beta_n$ be a sequence with $\\beta_n \\in (0,\\varepsilon)$ for all $n$ and such that $\\liminf_{n\\to\\infty}  \\frac{\\sqrt{2\\beta_n}}{\\pi} \\ln n > 1$.  As $n\\to \\infty$ then $\\varepsilon\\to 0$, $$\\sqrt{\\beta_n}\\Big(\\ln {\\mathscr L}(n,\\beta_n) -\\ln n\\Big) \\, \\cvp\\,  -\\frac{\\pi}{\\sqrt{2}}.$$\n In other words, for any $a>0$, \n\\[\n\\lim_{\\varepsilon\\downarrow0}\\liminf_{n\\to\\infty}\\mathbb P\\Big(\\frac{1}{n}{\\mathscr L}(n,\\beta_n) \\in [e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1+a)}, e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1-a)}]\\Big)=1.\n\\]\n\\end{theorem}\n\nTaking $\\beta_n= \\frac{\\alpha}{\\ln^2 n}$ with $\\alpha > \\frac{\\pi^2}{2}$, we see that \\begin{equation}    {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big)= n^{1- \\frac{\\pi}{\\sqrt{2 \\alpha}} +o_p(1)},\\label{alpha:polynomial}\\end{equation}\n\n\\noindent where $o_p(1)$ denotes some quantity which converges to $0$ in probability as $n\\to\\infty$. This, together with Theorem \\ref{t:main}, shows that there is a phase transition at $\\lambda_c=\\frac{\\pi^2}{2}\\frac1{\\ln^2 n}$. We also note that \\eqref{alpha:polynomial} is consistent with \\eqref{Ding-2} when $\\alpha$ is taken sufficiently large. \n\nThis critical value $\\lambda_c$ also appears in a related problem concerning the minimum average weight of a cycle in $K_n$. \nDing, Sun and Wilson \\cite{ding_sun_wilson} proved that conditionally on the minimum being nonnegative, its value is $\\frac{\\pi^2}{2 \\ln^2 n}  + O(\\frac{1}{\\ln^3 n})$ and that the length of the corresponding cycle is of order $\\ln^3 n$. It would be interesting to investigate further the connection between these two models. \n\n\\bigskip\nNow, we outline the main ideas of the proofs. \n Heuristically, we  explore the graph in the following way. Let $N=o(n)$ be an integer. We start with an initial set of $N$ vertices, called generation $0$. We examine all adjacent edges of these vertices\n  and select the $N$ edges with the  smallest labels. The endpoints of the selected edges form generation $1$, and the associated labels stand for their positions. We then delete the initial set of vertices and all their adjacent edges, and continue the exploration from the vertices of generation $1$. Each incident edge of a vertex at generation $1$ carries a weight. Add this weight to the position of the vertex, \n  then select the $N$ edges associated to the $N$ smallest obtained numbers. The endpoints of the selected edges form the vertices of generation $2$ and the associated numbers stand for their positions.  This process can be repeated for roughly $\\frac{n}{N}$ steps before the graph is exhausted. The weights of the paths obtained by this strategy resemble an (inhomogeneous)  branching random walk  in which only the $N$ leftmost particles are retained at each step. This model was introduced by Brunet and Derrida \\cite{brunet_derrida2} who conjectured that the selection mechanism was shifting the speed of the population by a correction term of order $\\frac{1}{\\ln^2 N}$, a result later proved by B\\'erard and Gou\\'er\\'e \\cite{berard_gouere}.   We refer to Maillard \\cite{maillard}\n  for more  results on the model in continuous time. In our setting, the minimum of the exploration process at time $t$ is of order \n\\[\n-\\ln N + \\frac{\\pi^2}{2 \\ln^2 N} t.\n\\]\nTaking  $N=n^{1-o(1)}$  and $N=e^{\\frac{\\pi}{\\sqrt{2\\beta_n}}}$ yields Theorem \\ref{t:main} and Theorem \\ref{t:main2}, respectively. A slight variation of this strategy is used to establish the lower bound for ${\\mathscr L}(n,\\lambda)$. The upper bound follows a similar approach to that of \\cite{ding,ding_goswami,ding_sun_wilson} by computing suitable truncated first moments. \n\n\\bigskip\n\nFinally, we mention the recent preprint by Jorritsma, Maillard and M\\\"orters \\cite{JMM} in which the authors make use of a killed branching random walk to explore growing sparse random graphs. Even if the models are different, our results  show similarity with several features of  their model, see \\cite{JMM} and the references therein.\n\n\\bigskip\n\nThe paper is organized as follows. Sections \\ref{s:upper} and \\ref{s:lower} give the proofs of the upper bounds and lower bounds for ${\\mathscr L}(n,\\lambda)$ respectively.  Section \\ref{s:rw-brw} collects results on real-valued random walks and  branching random walks that are used in Sections \\ref{s:upper} and \\ref{s:lower}.", "sketch": "Heuristically, the proof uses an exploration of the graph that “resembles an (inhomogeneous) branching random walk in which only the $N$ leftmost particles are retained at each step.” One starts with “generation $0$” consisting of $N=o(n)$ vertices, then repeatedly: examine all adjacent edges, “select the $N$ edges with the smallest labels,” take their endpoints as the next generation, delete previously explored vertices/edges, and iterate. At later generations, each incident edge weight is “add[ed] … to the position of the vertex,” and again one selects the $N$ smallest resulting values; the associated numbers are the new “positions.” This can be repeated for about $\\frac{n}{N}$ steps.\n\nIn this model, “the minimum of the exploration process at time $t$ is of order\n\\[\n-\\ln N + \\frac{\\pi^2}{2 \\ln^2 N} t.\n\\]\nTaking $N=n^{1-o(1)}$ … yields Theorem~\\ref{t:main} (and taking $N=e^{\\frac{\\pi}{\\sqrt{2\\beta_n}}}$ yields Theorem~\\ref{t:main2}). A “slight variation of this strategy is used to establish the lower bound for ${\\mathscr L}(n,\\lambda)$.” For the upper bound, the paper “follows a similar approach to that of \\cite{ding,ding_goswami,ding_sun_wilson} by computing suitable truncated first moments.”", "expanded_sketch": "Heuristically, the proof uses an exploration of the graph that “resembles an (inhomogeneous) branching random walk in which only the $N$ leftmost particles are retained at each step.” One starts with “generation $0$” consisting of $N=o(n)$ vertices, then repeatedly: examine all adjacent edges, “select the $N$ edges with the smallest labels,” take their endpoints as the next generation, delete previously explored vertices/edges, and iterate. At later generations, each incident edge weight is “add[ed] … to the position of the vertex,” and again one selects the $N$ smallest resulting values; the associated numbers are the new “positions.” This can be repeated for about $\\frac{n}{N}$ steps.\n\nIn this model, “the minimum of the exploration process at time $t$ is of order\n\\[\n-\\ln N + \\frac{\\pi^2}{2 \\ln^2 N} t.\n\\]\nTaking $N=n^{1-o(1)}$ … yields the main theorem (and taking $N=e^{\\frac{\\pi}{\\sqrt{2\\beta_n}}}$ yields the following theorem.\n\\begin{theorem}\\label{t:main2}\n For each $\\varepsilon>0$, let $\\beta_n$ be a sequence with $\\beta_n \\in (0,\\varepsilon)$ for all $n$ and such that $\\liminf_{n\\to\\infty}  \\frac{\\sqrt{2\\beta_n}}{\\pi} \\ln n > 1$.  As $n\\to \\infty$ then $\\varepsilon\\to 0$, $$\\sqrt{\\beta_n}\\Big(\\ln \\L(n,\\beta_n) -\\ln n\\Big) \\, \\cvp\\,  -\\frac{\\pi}{\\sqrt{2}}.$$\n In other words, for any $a>0$, \n\\[\n\\lim_{\\varepsilon\\downarrow0}\\liminf_{n\\to\\infty}\\p\\Big(\\frac{1}{n}\\L(n,\\beta_n) \\in [e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1+a)}, e^{-\\frac{\\pi}{\\sqrt{2\\beta_n}}(1-a)}]\\Big)=1.\n\\]\n\\end{theorem}\nA “slight variation of this strategy is used to establish the lower bound for ${\\mathscr L}(n,\\lambda)$.” For the upper bound, the paper “follows a similar approach to that of \\cite{ding,ding_goswami,ding_sun_wilson} by computing suitable truncated first moments.”", "expanded_theorem": "\\label{t:main}\nIf $\\alpha<\\frac{\\pi^2}{2}$, then as $n\\to\\infty$, $$\\frac{1}{\\ln^3 n} {\\mathscr L}\\big(n,\\frac{\\alpha}{\\ln^2 n}\\big) \\, \\cvp\\, \\frac{1}{\\frac{\\pi^2}{2}-\\alpha}.$$,", "theorem_type": ["Implication", "Asymptotic or Limit"], "mcq": {"question": "Fix a real number \\(\\alpha<\\pi^2/2\\). For \\(\\lambda\\in(-1,\\infty)\\), define\n\\[\n{\\mathscr L}(n,\\lambda):=\\sup\\{\\ell\\ge 1:\\ \\exists\\text{ a path }\\gamma=(\\gamma_0,\\ldots,\\gamma_\\ell)\\text{ with }S_i(\\gamma)=\\sum_{k=1}^i X(\\gamma_{k-1},\\gamma_k)\\text{ and }S_\\ell(\\gamma)\\le \\lambda \\ell\\}.\n\\]\nThus \\({\\mathscr L}(n,\\lambda)\\) is the maximum path length for which there exists a path whose average weight \\(S_\\ell(\\gamma)/\\ell\\) is at most \\(\\lambda\\). Under these assumptions, which statement about \\({\\mathscr L}(n,\\alpha/\\ln^2 n)\\) holds as \\(n\\to\\infty\\)?", "correct_choice": {"label": "A", "text": "As \\(n\\to\\infty\\),\n\\[\n\\frac{1}{\\ln^3 n}\\,{\\mathscr L}\\!\\left(n,\\frac{\\alpha}{\\ln^2 n}\\right) \\xrightarrow{\\mathbb P} \\frac{1}{\\frac{\\pi^2}{2}-\\alpha}.\n\\]"}, "choices": [{"label": "B", "text": "As \\(n\\to\\infty\\),\n\\[\n\\frac{1}{\\ln^3 n}\\,{\\mathscr L}\\!\\left(n,\\frac{\\alpha}{\\ln^2 n}\\right) \\xrightarrow{\\mathbb P} \\frac{1}{\\frac{\\pi^2}{2}+\\alpha}.\n\\]"}, {"label": "C", "text": "As \\(n\\to\\infty\\),\n\\[\n\\frac{{\\mathscr L}\\!\\left(n,\\frac{\\alpha}{\\ln^2 n}\\right)}{\\ln^3 n}=O_{\\mathbb P}(1).\n\\]"}, {"label": "D", "text": "As \\(n\\to\\infty\\),\n\\[\n\\frac{1}{\\ln^2 n}\\,{\\mathscr L}\\!\\left(n,\\frac{\\alpha}{\\ln^2 n}\\right) \\xrightarrow{\\mathbb P} \\frac{1}{\\frac{\\pi^2}{2}-\\alpha}.\n\\]"}, {"label": "E", "text": "For every fixed \\(\\alpha<\\pi^2/2\\), there exists a deterministic constant \\(C=C(\\alpha)>0\\) such that, with probability tending to \\(1\\) as \\(n\\to\\infty\\),\n\\[\n{\\mathscr L}\\!\\left(n,\\frac{\\alpha}{\\ln^2 n}\\right)= C\\ln^3 n+o(1).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "counting_estimate", "tampered_component": "critical-drift denominator sign", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "explicit limiting constant", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "correct logarithmic scaling exponent", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "mode of approximation upgraded from relative/probabilistic scale to additive deterministic asymptotics", "template_used": "wildcard"}]}}
{"id": "2512.23668v1", "paper_link": "http://arxiv.org/abs/2512.23668v1", "theorems_cnt": 5, "theorem": {"env_name": "theorem", "content": "[Hirose--Sato~\\cite{HiroseSato2019A}]\\label{thm:HiroseSato1}\nFor positive integers $a, b$, and $c$, we have\n\\begin{align*}\n&\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{c-1},1,\\{2\\}^b)\\\\\n&=\\zeta(\\{2\\}^{a+b+c})+\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{b-1},3,\\{2\\}^{c-1})+\\zeta(\\{2\\}^{b-1},3,\\{2\\}^{a-1},3,\\{2\\}^{c-1}).\n\\end{align*}", "start_pos": 4898, "end_pos": 5226, "label": "thm:HiroseSato1"}, "ref_dict": {"it:HMSW1": "\\begin{enumerate}[label=\\textnormal{(\\roman*)}]\n\\item\\label{it:HMSW1} $\\{w-\\cD(w) \\mid w\\in \\cH^0\\}=\\Ker(Z^{\\diamondsuit})\\subset\\Ker(Z)$,\n\\item\\label{it:HMSW2} $\\cD(w)=w$ for $w\\in\\cH^{\\geq 2}$,\n\\item\\label{it:HMSW3} $\\cD(w)=\\cD(\\tau(w))$ for $w\\in \\cH^0$.\n\\end{enumerate}", "thm:HiroseSato1": "\\begin{theorem}[Hirose--Sato~\\cite{HiroseSato2019A}]\\label{thm:HiroseSato1}\nFor positive integers $a, b$, and $c$, we have\n\\begin{align*}\n&\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{c-1},1,\\{2\\}^b)\\\\\n&=\\zeta(\\{2\\}^{a+b+c})+\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{b-1},3,\\{2\\}^{c-1})+\\zeta(\\{2\\}^{b-1},3,\\{2\\}^{a-1},3,\\{2\\}^{c-1}).\n\\end{align*}\n\\end{theorem}", "thm:HiroseSato2": "\\begin{theorem}[{Hirose--Sato~\\cite[Proposition~14]{HiroseSato2022+}}]\\label{thm:HiroseSato2}\nFor $w, w'\\in\\cH^{2,3}$, we have\n\\[\nZ(w_1\\tau(w_2))=Z(w_1\\star w_2).\n\\]\n\\end{theorem}", "eq:Hoffman-conj": "\\begin{equation}\\label{eq:Hoffman-conj}\n\\zeta(3,\\{2\\}^{c-1},1,2)=\\zeta(\\{2\\}^{c+2})+2\\zeta(3,3,\\{2\\}^{c-1})\n\\end{equation}", "thm:drop1": "\\begin{theorem}[Hirose--Maesaka--Seki--Watanabe~\\cite{HiroseMaesakaSekiWatanabe2025+}]\nThe drop 1 operator $\\cD$ satisfies the following properties$:$\n\\label{thm:drop1}\n\\begin{enumerate}[label=\\textnormal{(\\roman*)}]\n\\item\\label{it:HMSW1} $\\{w-\\cD(w) \\mid w\\in \\cH^0\\}=\\Ker(Z^{\\diamondsuit})\\subset\\Ker(Z)$,\n\\item\\label{it:HMSW2} $\\cD(w)=w$ for $w\\in\\cH^{\\geq 2}$,\n\\item\\label{it:HMSW3} $\\cD(w)=\\cD(\\tau(w))$ for $w\\in \\cH^0$.\n\\end{enumerate}\n\\end{theorem}", "def:drop1operator": "\\begin{definition}[Hirose--Maesaka--Seki--Watanabe~\\cite{HiroseMaesakaSekiWatanabe2025+}]\\label{def:drop1operator}\nWe define the \\emph{drop 1 operator} $\\cD\\colon\\cH^0\\to\\cH^{\\geq 2}$ to be a $\\ZZ$-linear map given as follows.\nIt suffices to define $\\cD(w)$ for each word $w\\in\\cH^0$.\nFor $\\bc=(c_1,\\dots, c_{2t})$, we write $\\fD(\\bc)=\\cD(y^{c_1}x^{c_2}\\cdots y^{c_{2t-1}}x^{c_{2t}})$.\nThe image $\\fD(\\bc)$ is defined inductively as follows:\n\\begin{align*}\n\\fD(\\bc)&=\\sum_{\\substack{A\\in\\cP^{\\text{eo}}([2t]^1_{\\bc}) \\\\ B\\in\\cP^{\\smallxcancel{\\text{eo}}}([2t]^{>1}_{\\bc}) \\\\ \\#A+\\#B\\geq 1}}(-1)^{\\#B-1}\\fD(\\bc_{(-A)}-\\bdelta_B)x^{\\#A+\\#B}\\\\\n&\\quad+\\sum_{\\substack{A\\in\\cP^{\\text{eo}}([2t]^1_{\\bc}) \\\\ B\\in\\cP^{\\smallxcancel{\\text{eo}}}([2t]^{>1}_{\\bc}) \\\\ \\#A+\\#B\\geq 2}}(-1)^{\\#B-1}\\fD(\\bc_{(-A)}-\\bdelta_B)z_{\\#A+\\#B}\\\\\n&\\quad+\\sum_{\\substack{A\\in\\cP^{\\text{oe}}([2t]^1_{\\bc}) \\\\ B\\in\\cP^{\\smallxcancel{\\text{oe}}}([2t]^{>1}_{\\bc}) \\\\ \\#A+\\#B\\geq 2}}(-1)^{\\#B}\\fD(\\bc_{(-A)}-\\bdelta_B)z_{\\#A+\\#B}.\n\\end{align*}\nHere the initial condition is $\\fD(\\varnothing)=\\cD(1)=1$.\n\\end{definition}", "cor:FMZV": "\\begin{corollary}\\label{cor:FMZV}\nFor words $w_1, w_2\\in\\cH^{2,3}$, we have\n\\[\n(-1)^{\\wt(w_2)}Z_{\\cA}(w_1\\overleftarrow{w_2})=Z_{\\cA}(w_1\\star w_2).\n\\]\n\\end{corollary}", "cor:diamond-lift": "\\begin{corollary}\\label{cor:diamond-lift}\nFor $w_1, w_2\\in\\cH^{2,3}$, we have\n\\[\nZ^{\\diamondsuit}(w_1\\tau(w_2))=Z^{\\diamondsuit}(w_1\\star w_2).\n\\]\n\\end{corollary}"}, "pre_theorem_intro_text_len": 938, "pre_theorem_intro_text": "For positive integers $k_1, \\dots, k_r$ with $k_r\\geq 2$, the \\emph{multiple zeta value} (MZV) $\\zeta(k_1,\\dots, k_r)$ is defined by\n\\[\n\\zeta(k_1,\\dots, k_r)\\coloneqq\\sum_{0<n_1<\\cdots <n_r}\\frac{1}{n_1^{k_1}\\cdots n_r^{k_r}}.\n\\]\nLinear relations among MZVs have been studied for many years.\nFor each positive integer $c$, the relation\n\\begin{equation}\\label{eq:Hoffman-conj}\n\\zeta(3,\\{2\\}^{c-1},1,2)=\\zeta(\\{2\\}^{c+2})+2\\zeta(3,3,\\{2\\}^{c-1})\n\\end{equation}\nis known as \\emph{Hoffman's conjectural identity} and had remained open for about seventeen years.\nHere the notation $\\{2\\}^n$ denotes the string $2,\\dots, 2$ in which the entry 2 is repeated $n$ times.\nCharlton proved in \\cite{Charlton2016} that there exists a rational number $q_c$ such that $\\zeta(3,\\{2\\}^{c-1},1,2)=q_c\\zeta(\\{2\\}^{c+2})+2\\zeta(3,3,\\{2\\}^{c-1})$, and Hirose and Sato proved not only Hoffman's conjectural identity but also the following more general theorem.", "context": "For positive integers $k_1, \\dots, k_r$ with $k_r\\geq 2$, the \\emph{multiple zeta value} (MZV) $\\zeta(k_1,\\dots, k_r)$ is defined by\n\\[\n\\zeta(k_1,\\dots, k_r)\\coloneqq\\sum_{0<n_1<\\cdots <n_r}\\frac{1}{n_1^{k_1}\\cdots n_r^{k_r}}.\n\\]\nLinear relations among MZVs have been studied for many years.\nFor each positive integer $c$, the relation\n\\begin{equation}\\label{eq:Hoffman-conj}\n\\zeta(3,\\{2\\}^{c-1},1,2)=\\zeta(\\{2\\}^{c+2})+2\\zeta(3,3,\\{2\\}^{c-1})\n\\end{equation}\nis known as \\emph{Hoffman's conjectural identity} and had remained open for about seventeen years.\nHere the notation $\\{2\\}^n$ denotes the string $2,\\dots, 2$ in which the entry 2 is repeated $n$ times.\nCharlton proved in \\cite{Charlton2016} that there exists a rational number $q_c$ such that $\\zeta(3,\\{2\\}^{c-1},1,2)=q_c\\zeta(\\{2\\}^{c+2})+2\\zeta(3,3,\\{2\\}^{c-1})$, and Hirose and Sato proved not only Hoffman's conjectural identity but also the following more general theorem.", "full_context": "For positive integers $k_1, \\dots, k_r$ with $k_r\\geq 2$, the \\emph{multiple zeta value} (MZV) $\\zeta(k_1,\\dots, k_r)$ is defined by\n\\[\n\\zeta(k_1,\\dots, k_r)\\coloneqq\\sum_{0<n_1<\\cdots <n_r}\\frac{1}{n_1^{k_1}\\cdots n_r^{k_r}}.\n\\]\nLinear relations among MZVs have been studied for many years.\nFor each positive integer $c$, the relation\n\\begin{equation}\\label{eq:Hoffman-conj}\n\\zeta(3,\\{2\\}^{c-1},1,2)=\\zeta(\\{2\\}^{c+2})+2\\zeta(3,3,\\{2\\}^{c-1})\n\\end{equation}\nis known as \\emph{Hoffman's conjectural identity} and had remained open for about seventeen years.\nHere the notation $\\{2\\}^n$ denotes the string $2,\\dots, 2$ in which the entry 2 is repeated $n$ times.\nCharlton proved in \\cite{Charlton2016} that there exists a rational number $q_c$ such that $\\zeta(3,\\{2\\}^{c-1},1,2)=q_c\\zeta(\\{2\\}^{c+2})+2\\zeta(3,3,\\{2\\}^{c-1})$, and Hirose and Sato proved not only Hoffman's conjectural identity but also the following more general theorem.\n\nMore generally, applying the same procedure to Corollary~\\ref{cor:diamond-lift} (see Section~\\ref{sec:finite} for more details), we have the following.\nFor a word $w=z_{k_1}\\cdots z_{k_r}$, we set $\\overleftarrow{w}\\coloneqq z_{k_r}\\cdots z_{k_1}$, and $\\wt(w)$ denotes the total degree of $w$, that is, $\\wt(w)=k_1+\\cdots + k_r$.\nA $\\ZZ$-linear map $Z_{\\cA}\\colon \\ZZ+y\\cH=\\ZZ\\langle z_1,z_2,\\dots\\rangle\\to\\cA$ is defined by $Z_{\\cA}(z_{k_1}\\cdots z_{k_r})=\\zeta^{}_{\\cA}(k_1,\\dots, k_r)$ and $Z_{\\cA}(1)=1$.\n\\begin{corollary}\\label{cor:FMZV}\nFor words $w_1, w_2\\in\\cH^{2,3}$, we have\n\\[\n(-1)^{\\wt(w_2)}Z_{\\cA}(w_1\\overleftarrow{w_2})=Z_{\\cA}(w_1\\star w_2).\n\\]\n\\end{corollary}\nThis is a family of relations among FMZVs in which only 2 and 3 occur as entries.\nSince the shuffle relation for FMZVs $(-1)^{\\wt(w_2)}Z_{\\cA}(w_1\\overleftarrow{w_2})=Z_{\\cA}(w_1\\sh w_2)$ holds (see \\cite{KanekoZagier2025+,Ono2017}), Corollary~\\ref{cor:FMZV} can also be rewritten as\n\\[\nZ_{\\cA}(w_1\\sh w_2-w_1\\star w_2)=0\n\\]\nwhich is of the form of a certain double shuffle relation.\n\\section*{Acknowledgements}\nThe author would like to thank Professor Minoru Hirose for kindly answering his questions about the previous works by Hirose and Sato.\nThis research was inspired by a question that came to mind when the author got married, and the author would like to thank his wife, Yui, for marrying him.\n\\section{The drop 1 operator}\nWe introduce the following notation to define the main operator.\nLet $t$ be a positive integer, let $\\bc=(c_1,\\dots, c_{2t})$ be a tuple of positive integers, and let $S\\subset\\ZZ$.\n\\begin{itemize}[leftmargin=2.5em]\n\\item $\\#A$ denotes the cardinality of a finite set $A$.\n\\item $[n]\\coloneqq\\{1,2,\\dots,n\\}$ for $n\\in\\ZZ_{\\geq 0}$. $[0]=\\varnothing$.\n\\item $[2t]_{\\bc}^1\\coloneqq\\{i\\in[2t] \\mid c_i=1\\}.$\n\\item $[2t]_{\\bc}^{>1}\\coloneqq\\{i\\in[2t] \\mid c_i>1\\}.$\n\\item $\\cP^{\\text{eo}}(S)\\coloneqq\\{A\\subset S \\mid i\\in A\\cap 2\\ZZ \\Longleftrightarrow i+1\\in A\\setminus 2\\ZZ\\}.$\n\\item $\\cP^{\\text{oe}}(S)\\coloneqq\\{A\\subset S \\mid i\\in A\\setminus 2\\ZZ \\Longleftrightarrow i+1\\in A\\cap 2\\ZZ\\}.$\n\\item $\\cP^{\\smallxcancel{\\text{eo}}}(S)\\coloneqq\\{A\\subset S \\mid i\\in A\\cap 2\\ZZ \\Longrightarrow i+1\\not\\in A\\setminus 2\\ZZ\\}.$\n\\item $\\cP^{\\smallxcancel{\\text{oe}}}(S)\\coloneqq\\{A\\subset S \\mid i\\in A\\setminus 2\\ZZ \\Longrightarrow i+1\\not\\in A\\cap 2\\ZZ\\}.$\n\\item $\\bc_{(-A)}\\coloneqq (c_{i_1},\\dots, c_{i_r})$, where $[2t]\\setminus A=\\{i_1,\\dots, i_r\\}$ with $i_1<\\cdots<i_r$, for $A\\subset[2t]$.\n\\item $\\bc_{(-\\varnothing)}=\\bc$, $\\bc_{(-[2t])}=\\varnothing$.\n\\item $\\delta_{i\\in B}\\coloneqq\\begin{cases}1 & \\text{if } i\\in B, \\\\ 0 & \\text{otherwise}\\end{cases}$ for a set $B$.\n\\item $\\bc_{(-A)}-\\bdelta_B\\coloneqq(c_{i_1}-\\delta_{i_1\\in B},\\dots, c_{i_r}-\\delta_{i_r\\in B})$ for the above $\\bc_{(-A)}$ and $B\\subset [2t]$.\n\\end{itemize}\n\nWe consider the case $c>1$.\nThen $\\cP^{\\smallxcancel{\\text{eo}}}([2t]^{>1}_{\\bc})=\\cP([2t]^{>1}_{\\bc})$ holds.\nFor $A\\in\\cP^{\\text{eo}}([2t]^1_{\\bc})$, decompose it as $A=A_1\\cup A_2 \\cup A_3$, where\n\\[\nA_1\\coloneqq A \\cap [2l_r-1],\\quad A_2\\coloneqq A\\cap ([2l_r+2c-2]\\setminus [2l_r]),\\quad A_3\\coloneqq A\\setminus [2l_r+2c-1].\n\\]\nWe define a map $\\iota_1\\colon\\cP^{\\text{eo}}([2t]^1_{\\bc})\\to\\cP^{\\text{oe}}([2t]^1_{\\bc})$ by\n\\[\n\\iota_1(A)\\coloneqq (A_1-1)\\cup (A_2-1)\\cup\\{2l_r+2c-3,2l_r+2c-2\\}\\cup(A_3+1)\n\\]\nand a map $\\iota_2\\colon\\cP^{\\text{eo}}([2t]^1_{\\bc})\\to\\cP^{\\text{oe}}([2t]^1_{\\bc})$ by\n\\[\n\\iota_2(A)\\coloneqq (A_1-1)\\cup (A_2-1)\\cup(A_3+1).\n\\]\nA bijection $\\cP^{\\text{eo}}([2t]^1_{\\bc})\\sqcup\\cP^{\\text{eo}}([2t]^1_{\\bc})\\to\\cP^{\\text{oe}}([2t]^1_{\\bc})$ is obtained by sending an element $A$ in the first copy of $\\cP^{\\text{eo}}([2t]^1_{\\bc})$ to $\\iota_1(A)$, and an element $A$ in the second copy of $\\cP^{\\text{eo}}([2t]^1_{\\bc})$ to $\\iota_2(A)$.\nFor $A\\in\\cP^{\\text{eo}}([2t]^1_{\\bc})$, we have $\\bc_{(-A)}=\\bc_{(-\\iota_2(A))}$.\nTherefore, if we compute the defining recurrence for the drop 1 operator using this bijection, every term in the second sum is canceled by a term in the third sum, and we are left with\n\\begin{align*}\n\\fD(\\bc)&=\\sum_{\\substack{A\\in\\cP^{\\text{eo}}([2t]^1_{\\bc}) \\\\ B\\in\\cP([2t]^{>1}_{\\bc}) \\\\ \\#A+\\#B\\geq 1}}(-1)^{\\#B-1}\\biggl(\\fD(\\bc_{(-A)}-\\bdelta_B)-\\fD(\\bc_{(-\\iota_1(A))}-\\bdelta_B)z_2\\biggr)x^{\\#A+\\#B}\\\\\n&\\quad+\\fD(\\bc_{(-\\{2l_r+2c-3,2l_r+2c-2\\})})z_2.\n\\end{align*}\nFor $A\\in\\cP^{\\text{eo}}([2t]^1_{\\bc})$ or $\\cP^{\\text{oe}}([2t]^1_{\\bc})$, there exist $a'_1$, $\\dots$, $a'_r$, $b'_1$, $\\dots$, $b'_s$, and $c'$ such that\n\\[\n\\bc_{(-A)}=[a'_1,\\dots, a'_r;b'_1,\\dots, b'_s;c'].\n\\]\nHere $a'_1\\in[a_1]$, $\\dots$, $a'_r\\in[a_r]$, $b'_1\\in[b_1]$, $\\dots$, $b'_s\\in[b_s]$, $c'\\in[c]$, and $a'_1+\\cdots+a'_r+b'_1+\\cdots+b'_s+c'=t+1-\\#A/2$.\nMoreover, upon adding $-\\bdelta_B$, addition occurs between the corresponding components; for example, when\n\\begin{align*}\n\\bc&=[2,1,3;4,1,1,2;3]\\\\\n&=(\\stackrel{1}{1},\\stackrel{2}{1},\\stackrel{3}{1},\\stackrel{\\color{red}{4}}{2},\\stackrel{5}{1},\\stackrel{\\color{red}{6}}{2},\\stackrel{7}{1},\\stackrel{8}{1},\\stackrel{9}{1},\\stackrel{10}{1},\\stackrel{11}{1},\\stackrel{\\color{red}{12}}{2},\\stackrel{13}{1},\\stackrel{14}{1},\\stackrel{15}{1},\\stackrel{16}{1},\\stackrel{\\color{red}{17}}{2},\\stackrel{18}{1},\\stackrel{19}{1},\\stackrel{20}{1},\\stackrel{21}{1},\\stackrel{22}{1},\\stackrel{23}{1},\\stackrel{24}{1},\\stackrel{\\color{red}{25}}{2},\\stackrel{26}{1},\\stackrel{\\color{red}{27}}{2},\\stackrel{28}{1},\\stackrel{\\color{red}{29}}{2},\\stackrel{30}{1},\\stackrel{31}{1},\\stackrel{32}{1}),\n\\end{align*}\n\\[\nA=\\{2,3,8,9,14,15,18,19,22,23,30,31\\}\\in\\cP^{\\text{eo}}([32]^1_{[2,1,3;4,1,1,2;3]}),\n\\]\nand\n\\[\nB=\\{4,17,27\\}\\subset[32]^{>1}_{[2,1,3;4,1,1,2;3]}=\\{4,6,12,17,25,27,29\\},\n\\]\nwe have\n\\begin{align*}\n&[2,1,3;4,1,1,2;3]_{(-\\{2,3,8,9,14,15,18,19,22,23,30,31\\})}=[1,1,2;2,1,1,1;2]\\\\\n&=(\\stackrel{1}{1},\\stackrel{\\color{red}{4}}{2},\\stackrel{5}{1},\\stackrel{\\color{red}{6}}{2},\\stackrel{7}{1},\\stackrel{10}{1},\\stackrel{11}{1},\\stackrel{\\color{red}{12}}{2},\\stackrel{13}{1},\\stackrel{16}{1},\\stackrel{\\color{red}{17}}{2},\\stackrel{20}{1},\\stackrel{21}{1},\\stackrel{24}{1},\\stackrel{\\color{red}{25}}{2},\\stackrel{26}{1},\\stackrel{\\color{red}{27}}{2},\\stackrel{28}{1},\\stackrel{\\color{red}{29}}{2},\\stackrel{32}{1})\n\\end{align*}\nand\n\\begin{align*}\n&[2,1,3;4,1,1,2;3]_{(-\\{2,3,8,9,14,15,18,19,22,23,30,31\\})}-\\bdelta_{\\{4,17,27\\}}\\\\\n&=[1+1,2;1+1,1;2+2]=[2,2;2,1;4]\\\\\n&=(\\stackrel{1}{1},\\stackrel{\\color{blue}{4}}{1},\\stackrel{5}{1},\\stackrel{\\color{red}{6}}{2},\\stackrel{7}{1},\\stackrel{10}{1},\\stackrel{11}{1},\\stackrel{\\color{red}{12}}{2},\\stackrel{13}{1},\\stackrel{16}{1},\\stackrel{\\color{blue}{17}}{1},\\stackrel{20}{1},\\stackrel{21}{1},\\stackrel{24}{1},\\stackrel{\\color{red}{25}}{2},\\stackrel{26}{1},\\stackrel{\\color{blue}{27}}{1},\\stackrel{28}{1},\\stackrel{\\color{red}{29}}{2},\\stackrel{32}{1}).\n\\end{align*}\nTherefore, for any $A, B$ with $\\#A+\\#B\\geq 1$, the induction hypothesis implies that there exist words $v_1, v_2$ in $\\cH^{2,3}$ such that $\\bc_{(\\iota_1(-A))}-\\bdelta_B$ corresponds to $v_1\\tau(v_2)$, and we have\n\\[\n\\fD(\\bc_{(-A)}-\\bdelta_B)-\\fD(\\bc_{(-\\iota_1(A))}-\\bdelta_B)z_2=\\cD(v_1z_2\\tau(v_2))-\\cD(v_1\\tau(v_2))z_2=v_1z_2\\star v_2-(v_1\\star v_2)z_2=0.\n\\]\nSince we are in the case $c>1$, we can write $w_1=wz_2$ with a certain word $w$ in $\\cH^{2,3}$.\nThen we obtain\n\\begin{align*}\n\\fD(\\bc)&=\\fD(\\bc_{(-\\{2l_r+2c-3,2l_r+2c-2\\})})z_2=\\fD([a_1,\\dots, a_r;b_1,\\dots, b_s;c-1])z_2=\\cD(w\\tau(w_2))z_2\\\\\n&=(w\\star w_2)z_2=wz_2\\star w_2=w_1\\star w_2.\n\\end{align*}", "post_theorem_intro_text_len": 7570, "post_theorem_intro_text": "Their proof can be viewed as showing that the above identity is a consequence of the \\emph{confluence relation} later formulated in \\cite{HiroseSato2019B}.\n\nHirose and Sato gave a further generalization in \\cite{HiroseSato2022+}, and we set up notation in order to state their result.\nLet $\\mathcal{H}\\coloneqq\\mathbb{Z}\\langle x,y\\rangle$ be the non-commutative polynomial ring over $\\mathbb{Z}$ in two variables $x$ and $y$.\nLet $z_k\\coloneqq yx^{k-1}$ for each positive integer $k$.\nSubrings $\\mathcal{H}^{2,3}\\subset\\mathcal{H}^0\\subset\\mathcal{H}$ are defined by $\\mathcal{H}^{2,3}\\coloneqq\\mathbb{Z}\\langle z_2,z_3\\rangle$ and $\\mathcal{H}^0\\coloneqq\\mathbb{Z}+y\\mathcal{H} x$, respectively.\nA $\\mathbb{Z}$-linear map $Z\\colon\\mathcal{H}^0\\to\\mathbb{R}$ is defined by $Z(z_{k_1}\\cdots z_{k_r})=\\zeta(k_1,\\dots, k_r)$ and $Z(1)=1$.\nA product $\\star$ on $\\mathcal{H}^{2,3}$ is defined inductively by \n\\begin{itemize}[leftmargin=2.5em]\n\\item $w\\star 1=1\\star w=w$,\n\\item $wz_2\\star w'=w\\star w'z_2=(w\\star w')z_2$,\n\\item $wz_3\\star w'z_3=(w\\star w'z_3)z_3+(wz_3\\star w')z_3+(w\\star w')z_2^3$\n\\end{itemize}\nfor any words $w$ and $w'$ in $\\mathcal{H}^{2,3}$, and then extended by $\\mathbb{Z}$-bilinearity.\nLet $\\tau$ be an anti-automorphism on $\\mathcal{H}$ defined by $\\tau(x)=y$ and $\\tau(y)=x$.\nUsing this notation, we can now state the following theorem.\n\\begin{theorem}[{Hirose--Sato~\\cite[Proposition~14]{HiroseSato2022+}}]\\label{thm:HiroseSato2}\nFor $w, w'\\in\\mathcal{H}^{2,3}$, we have\n\\[\nZ(w_1\\tau(w_2))=Z(w_1\\star w_2).\n\\]\n\\end{theorem}\nBy specializing Theorem~\\ref{thm:HiroseSato2} to $w_1=z_2^{a-1}z_3z_2^{c-1}$ and $w_2=z_2^{b-1}z_3$, we recover Theorem~\\ref{thm:HiroseSato1}.\nHirose and Sato derived Theorem~\\ref{thm:HiroseSato2} from their \\emph{block shuffle identity} (\\cite[Theorem~10]{HiroseSato2022+}).\n\nHoffman conjectured in \\cite{Hoffman1997} that any multiple zeta value can be expressed as a $\\mathbb{Q}$-linear combination of the elements of\n\\[\n\\{\\zeta(k_1,\\dots, k_r) \\mid r\\in\\ZZ_{>0}, k_1,\\dots, k_r\\in\\{2,3\\}\\}.\n\\]\nThis conjecture was proved by Brown in \\cite{Brown2012}.\nAlthough the linear independence of these elements remains open, they are called the \\emph{Hoffman basis}.\nTheorem~\\ref{thm:HiroseSato2} provides a family for which the expansion in the Hoffman basis can be described explicitly, and, even more remarkably, it shows that the expansion can be taken with \\emph{integer coefficients}.\nIn general, the coefficients are not integral; for instance,\n\\[\n\\zeta(3,1,4)=-\\frac{2048}{4125}\\zeta(2,2,2,2)-\\frac{17}{275}\\zeta(2,3,3)+\\frac{2}{275}\\zeta(3,2,3)+\\frac{113}{275}\\zeta(3,3,2).\n\\]\n\nIn contrast, Hirose, Maesaka, Seki, and Watanabe recently proved in \\cite{HiroseMaesakaSekiWatanabe2025+}, together with an explicit algorithm, that any multiple zeta value can be expressed as a $\\mathbb{Z}$-linear combination of\n\\[\n\\{\\zeta(k_1,\\dots, k_r) \\mid r\\in\\ZZ_{>0}, k_1,\\dots, k_r\\geq 2\\}.\n\\]\nTheir explicit expansions give rise to a family of relations among MZVs, which we refer to as the \\emph{drop 1 relation}.\nSince their expansions have integer coefficients, it is natural to ask whether Theorem~\\ref{thm:HiroseSato2} can be viewed as an expansion arising from the drop 1 relation.\nHowever, in general, expansions coming from the drop 1 relation involve MZVs with entries greater than $3$, so it is not obvious whether the expansion in Theorem~\\ref{thm:HiroseSato2}, in which only 2 and 3 appear, can be obtained from the drop 1 relation.\nThe main result of this paper is that this is indeed the case.\n\nA subring $\\mathcal{H}^{\\geq 2}\\subset\\mathcal{H}^0$ is defined by $\\mathcal{H}^{\\geq 2}\\coloneqq\\mathbb{Z}+z_2\\mathbb{Z}\\langle x,z_2\\rangle=\\mathbb{Z}\\langle z_2, z_3, z_4,\\dots\\rangle$.\nA linear map $\\mathcal{D}\\colon\\mathcal{H}^0\\to\\mathcal{H}^{\\geq 2}$, called the \\emph{drop 1 operator}, is defined in Definition~\\ref{def:drop1operator}, and its properties are summarized in Theorem~\\ref{thm:drop1}.\n\\begin{theorem}\\label{thm:main}\nFor $w_1, w_2\\in\\mathcal{H}^{2,3}$, we have\n\\[\n\\mathcal{D}(w_1\\tau(w_2))=w_1\\star w_2.\n\\]\n\\end{theorem}\nNot only does this theorem yield a new proof of Theorem~\\ref{thm:HiroseSato2}, but it also implies, by Theorem~\\ref{thm:drop1}~\\ref{it:HMSW1}, that the same relations hold for \\emph{multiple zeta diamond values}.\nWe omit the definition of $\\zeta^{\\diamondsuit}(k_1,\\dots, k_r)$ here; see \\cite[Definition~2.3]{HiroseMaesakaSekiWatanabe2025+}.\nA $\\mathbb{Z}$-linear map $Z^{\\diamondsuit}\\colon\\mathcal{H}^0\\to\\mathbb{Q}^{\\mathbb{N}}$ is defined by $Z^{\\diamondsuit}(z_{k_1}\\cdots z_{k_r})=\\zeta^{\\diamondsuit}(k_1,\\dots, k_r)$ and $Z^{\\diamondsuit}(1)=1$.\n\n\\begin{corollary}\\label{cor:diamond-lift}\nFor $w_1, w_2\\in\\mathcal{H}^{2,3}$, we have\n\\[\nZ^{\\diamondsuit}(w_1\\tau(w_2))=Z^{\\diamondsuit}(w_1\\star w_2).\n\\]\n\\end{corollary}\nFor example, the relation corresponding to \\eqref{eq:Hoffman-conj}, \n\\[\n\\zeta^{\\diamondsuit}(3,\\{2\\}^{c-1},1,2)=\\zeta^{\\diamondsuit}(\\{2\\}^{c+2})+2\\zeta^{\\diamondsuit}(3,3,\\{2\\}^{c-1})\n\\]\nmeans that, for any positive integer $N$,\n\\begin{align*}\n&\\sum_{0<n_1<n_2<\\cdots<n_{c}<n_{c+1}<n_{c+2}<N}\\frac{1}{n_1^3n_2^2\\cdots n_{c}^2n_{c+1}^{}n_{c+2}^2}\\\\\n&+\\sum_{0<n_1<n_2<\\cdots<n_{c}<n_{c+1}\\leq n_{c+2}<N}\\frac{1}{n_1^3n_2^2\\cdots n_{c}^2(N-n_{c+1}^{})n_{c+2}^2}\\\\\n&=\\sum_{0<n_1<\\cdots < n_{c+2}<N}\\frac{1}{n_1^2\\cdots n_{c+2}^2}+2\\sum_{0<n_1<n_2<n_3<\\cdots<n_{c+1}<N}\\frac{1}{n_1^3n_2^3n_3^2\\cdots n_{c+1}^2}\n\\end{align*}\nholds.\nLetting $N\\to\\infty$ yields \\eqref{eq:Hoffman-conj}.\nOn the other hand, if we take $N=p$ for a prime $p$ and take modulo $p$, then we have the following relation among \\emph{finite multiple zeta values} (FMZVs):\n\\[\n\\zeta^{}_{\\mathcal{A}}(3,\\{2\\}^{c-1},3)+2\\zeta^{}_{\\mathcal{A}}(3,3,\\{2\\}^{c-1})=0.\n\\]\nHere, for positive integers $k_1,\\dots, k_r$, the FMZV $\\zeta^{}_{\\mathcal{A}}(k_1,\\dots, k_r)$ is defined as an element of $\\mathcal{A}\\coloneqq\\left.\\prod_p\\mathbb{Z}/p\\mathbb{Z}\\right/\\bigoplus_p\\mathbb{Z}/p\\mathbb{Z}$ by\n\\[\n\\zeta^{}_{\\mathcal{A}}(k_1,\\dots, k_r)\\coloneqq (\\zeta^{}_p(k_1,\\dots,k_r)\\bmod{p})_p,\n\\]\nwhere $p$ runs over all prime numbers, and\n\\[\n\\zeta^{}_p(k_1,\\dots,k_r)\\coloneqq\\sum_{0<n_1<\\cdots<n_r<p}\\frac{1}{n_1^{k_1}\\cdots n_r^{k_r}}.\n\\]\nThis definition is due to Kaneko and Zagier (\\cite{KanekoZagier2025+}).\nWe have used a well-known formula: $\\zeta^{}_{\\mathcal{A}}(\\{2\\}^{c+2})=0$.\n\nMore generally, applying the same procedure to Corollary~\\ref{cor:diamond-lift} (see Section~\\ref{sec:finite} for more details), we have the following.\nFor a word $w=z_{k_1}\\cdots z_{k_r}$, we set $\\overleftarrow{w}\\coloneqq z_{k_r}\\cdots z_{k_1}$, and $\\mathrm{wt}(w)$ denotes the total degree of $w$, that is, $\\mathrm{wt}(w)=k_1+\\cdots + k_r$.\nA $\\mathbb{Z}$-linear map $Z_{\\mathcal{A}}\\colon \\mathbb{Z}+y\\mathcal{H}=\\mathbb{Z}\\langle z_1,z_2,\\dots\\rangle\\to\\mathcal{A}$ is defined by $Z_{\\mathcal{A}}(z_{k_1}\\cdots z_{k_r})=\\zeta^{}_{\\mathcal{A}}(k_1,\\dots, k_r)$ and $Z_{\\mathcal{A}}(1)=1$.\n\\begin{corollary}\\label{cor:FMZV}\nFor words $w_1, w_2\\in\\mathcal{H}^{2,3}$, we have\n\\[\n(-1)^{\\mathrm{wt}(w_2)}Z_{\\mathcal{A}}(w_1\\overleftarrow{w_2})=Z_{\\mathcal{A}}(w_1\\star w_2).\n\\]\n\\end{corollary}\nThis is a family of relations among FMZVs in which only 2 and 3 occur as entries.\nSince the shuffle relation for FMZVs $(-1)^{\\mathrm{wt}(w_2)}Z_{\\mathcal{A}}(w_1\\overleftarrow{w_2})=Z_{\\mathcal{A}}(w_1\\sh w_2)$ holds (see \\cite{KanekoZagier2025+,Ono2017}), Corollary~\\ref{cor:FMZV} can also be rewritten as\n\\[\nZ_{\\mathcal{A}}(w_1\\sh w_2-w_1\\star w_2)=0\n\\]\nwhich is of the form of a certain double shuffle relation.", "sketch": "Their proof (of Theorem~\\ref{thm:HiroseSato1}) “can be viewed as showing that the above identity is a consequence of the \\emph{confluence relation}.”\n\nThe text then indicates that Theorem~\\ref{thm:HiroseSato1} is recovered by “specializing Theorem~\\ref{thm:HiroseSato2} to $w_1=z_2^{a-1}z_3z_2^{c-1}$ and $w_2=z_2^{b-1}z_3$,” and that Hirose--Sato “derived Theorem~\\ref{thm:HiroseSato2} from their \\emph{block shuffle identity}.”", "expanded_sketch": "Their proof (of the main theorem) “can be viewed as showing that the above identity is a consequence of the \\emph{confluence relation}.”\n\nThe text then indicates that the main theorem is recovered by specializing the following theorem to $w_1=z_2^{a-1}z_3z_2^{c-1}$ and $w_2=z_2^{b-1}z_3$.\n\n\\begin{theorem}[{Hirose--Sato~\\cite[Proposition~14]{HiroseSato2022+}}]\\label{thm:HiroseSato2}\nFor $w, w'\\in\\cH^{2,3}$, we have\n\\[\nZ(w_1\\tau(w_2))=Z(w_1\\star w_2).\n\\]\n\\end{theorem}\n\nIt is also stated that Hirose--Sato “derived the theorem above from their \\emph{block shuffle identity}.”", "expanded_theorem": "[Hirose--Sato~\\cite{HiroseSato2019A}]\\label{thm:HiroseSato1}\nFor positive integers $a, b$, and $c$, we have\n\\begin{align*}\n&\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{c-1},1,\\{2\\}^b)\\\\\n&=\\zeta(\\{2\\}^{a+b+c})+\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{b-1},3,\\{2\\}^{c-1})+\\zeta(\\{2\\}^{b-1},3,\\{2\\}^{a-1},3,\\{2\\}^{c-1}).\n\\end{align*},", "theorem_type": ["Universal", "Equivalence"], "mcq": {"question": "For positive integers $k_1,\\dots,k_r$ with $k_r\\ge 2$, the multiple zeta value is defined by\n\\[\n\\zeta(k_1,\\dots,k_r)=\\sum_{0<n_1<\\cdots<n_r}\\frac{1}{n_1^{k_1}\\cdots n_r^{k_r}}.\n\\]\nFor an integer $n\\ge 0$, let $\\{2\\}^n$ denote the string consisting of $n$ copies of $2$ (so $\\{2\\}^0$ is the empty string). Which identity holds for every choice of positive integers $a,b,c$?", "correct_choice": {"label": "A", "text": "\\[\n\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{c-1},1,\\{2\\}^b)\n=\\zeta(\\{2\\}^{a+b+c})+\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{b-1},3,\\{2\\}^{c-1})+\\zeta(\\{2\\}^{b-1},3,\\{2\\}^{a-1},3,\\{2\\}^{c-1}).\n\\]"}, "choices": [{"label": "B", "text": "\\[\n\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{c-1},1,\\{2\\}^b)\n=\\zeta(\\{2\\}^{a+b+c})+2\\,\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{b-1},3,\\{2\\}^{c-1}).\n\\]"}, {"label": "C", "text": "\\[\n\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{c-1},1,\\{2\\}^b)\n=\\zeta(\\{2\\}^{a+b+c})+R_{a,b,c},\n\\]\nwhere\n\\[\nR_{a,b,c}=\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{b-1},3,\\{2\\}^{c-1})+\\zeta(\\{2\\}^{b-1},3,\\{2\\}^{a-1},3,\\{2\\}^{c-1}).\n\\]"}, {"label": "D", "text": "\\[\n\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{c-1},1,\\{2\\}^b)\n=\\zeta(\\{2\\}^{a+b+c})+\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{b-1},3,\\{2\\}^{c-1})+\\zeta(\\{2\\}^{b-1},3,\\{2\\}^{c-1},3,\\{2\\}^{a-1}).\n\\]"}, {"label": "E", "text": "\\[\n\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{c-1},1,\\{2\\}^b)\n=\\zeta(\\{2\\}^{a+b+c+1})+\\zeta(\\{2\\}^{a-1},3,\\{2\\}^{b-1},3,\\{2\\}^{c-1})+\\zeta(\\{2\\}^{b-1},3,\\{2\\}^{a-1},3,\\{2\\}^{c-1}).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "symmetrized two-term contribution collapsed to a doubled single term", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "explicit decomposition of the remainder into the two displayed zeta terms", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "precise placement of the blocks in the second 3-term on the right-hand side", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "total weight balance between the left-hand side and the pure even zeta term", "template_used": "boundary_range"}]}}
{"id": "2512.24115v1", "paper_link": "http://arxiv.org/abs/2512.24115v1", "theorems_cnt": 2, "theorem": {"env_name": "proposition", "content": "If $G$ is a sun graph on $2n\\ge 6$ vertices, then $\\displaystyle \\zeta(G)=\\sum_{i=0}^n{n\\choose i}=2^n. $", "start_pos": 8748, "end_pos": 8886, "label": null}, "ref_dict": {"Ex": "\\begin{example}{\\label{Ex}}\n\\end{example}", "fig1": "\\label{fig1}\n\n\\end{figure}\n\nConsider the unicyclic graph $G$ as shown in Figure \\ref{fig1}. It is easy to see that $\\gamma(G)=3$. We list all the $\\gamma$-sets with properties which are among the five"}, "pre_theorem_intro_text_len": 4834, "pre_theorem_intro_text": "A {\\it dominating set} for a graph $G = (V,E)$ is a subset $S \\subseteq V$ such that every vertex $v\\in V$ is either in $S$ or has a neighbor $u\\in S$. The vertex $u \\in S$ is said to \\textit{cover}  the vertex $v \\in V$ if either $u=v$ or $uv\\in E$. A dominating set $S$ is a \\textit{minimal dominating set} if no proper subset $S'\\subset S$ is a dominating set. The \\textit{domination number} $\\gamma(G)$ of a graph $G$ is the minimum cardinality among all dominating sets of $G$. Clearly, a graph $G$ can have multiple minimum dominating sets. For simplicity, we refer to those sets as \\textbf{$\\gamma$-sets}. \n\nDominating sets, in particular $\\gamma$-sets, have been extensively researched. We recommend to the reader Haynes et al.'s book \\cite{Hay2} and several other research works \\cite{FB, BR, EC, GD, MH, FH, Hay1, ML} on variants and applications of dominating sets.\nIn this literature we found various versions of $\\gamma$-set such as the perfect dominating set, the double dominating set, the strong dominating set, and the restrained dominating set. In general, each such type of $\\gamma$-set has some specified properties. Here, we list five of the most commonly studied $\\gamma$-sets along with their properties. Later in Example \\ref{Ex}, we illustrate each such $\\gamma$-set.\n\n\\begin{enumerate}\n\t\\item \\textit{Perfect $\\gamma$-set} requires that no two vertices in a $\\gamma$-set cover the same vertex (outside the $\\gamma$-set). \\item \\textit{Connected $\\gamma$-set} requires that the graph induced by the $\\gamma$-set be connected.\n\t\\item \\textit{Total $\\gamma$-set} forbids isolated vertices in the subgraph induced by the $\\gamma$-set.\n\t\\item \\textit{Independent $\\gamma$-set} requires all vertices in a $\\gamma$-set to be isolated vertices in the subgraph they induce.\n\t\\item \\textit{clique $\\gamma$ set} requires that the vertices in a $\\gamma$-set induce a clique.\n\\end{enumerate}\n\nBecause a given graph may have multiple $\\gamma$-sets, this article addresses a natural but fundamental question:\\textit{ How many  $\\gamma$-sets does a given graph have?} To answer this question, we introduce the notion of \\textbf{dominion}.\n\nThe \\textit{dominion (number)} of  a graph $G=(V,E)$, denoted by $\\zeta(G)$, is the number of its $\\gamma$-sets. In other words, \n\n$\\zeta(G):=|\\{S: S \\text{ is a dominating set in} \\ G \\ \\text{and} \\ |S|=\\gamma\\}|$. For instance, given a  complete graph $G=K_n$, it is obvious that each $v\\in V(G)$ covers the remaining $n-1$ vertices of $G$. Therefore $\\zeta(G)=n$ while $\\gamma(G)=1$. Also, for a star graph $G=K_{1,n}$ of order $n+1\\ge 3$, it is easy to see that $\\gamma(G)=1=\\zeta(G)$ since only the central vertex can cover all of the remaining $n\\ge 2$ vertices. \n\\begin{example}{\\label{Ex}}\n\\end{example}\nHere, we illustrate the $\\gamma$-sets of a unicyclic graph $G$ (see Figure \\ref{fig1}). \n\\begin{figure}[ht]\n\t\\centering\n\t\\begin{tikzpicture}[scale=1]\n\t\\clip(0,-1) rectangle (4,4);\n\t\\draw [line width=2pt] (2.62,1)-- (2,2);\n\t\\draw [line width=2pt] (2,2)-- (1.98,2.84);\n\t\\draw [line width=2pt] (2.62,1)-- (3,0.34);\n\t\\draw [line width=2pt] (2,2)-- (1.48,1.02);\n\t\\draw [line width=2pt] (1.48,1.02)-- (2.62,1);\n\t\\draw [line width=2pt] (1.48,1.02)-- (1.06,0.34);\n\t\\begin{scriptsize}\n\t\\draw [fill=black] (2,2) circle (2.5pt);\n\t\\draw[color=black] (2.2,2.1) node {$u$};\n\t\\draw [fill=black] (2.62,1) circle (2.5pt);\n\t\\draw[color=black] (2.92,1) node {$v$};\n\t\\draw [fill=black] (1.98,2.84) circle (2.5pt);\n\t\\draw[color=black] (1.94,3.1) node {$x$};\n\t\\draw [fill=black] (3,0.34) circle (2.5pt);\n\t\\draw[color=black] (3.2,0.15) node {$y$};\n\t\\draw [fill=black] (1.48,1.02) circle (2.5pt);\n\t\\draw[color=black] (1.14,1) node {$w$};\n\t\\draw [fill=black] (1.06,0.34) circle (2.5pt);\n\t\\draw[color=black] (0.9,0.15) node {$z$};\n\t\\end{scriptsize}\n\t\\end{tikzpicture}\n\t\\caption{A sun graph $G$ with $\\gamma(G)=3$ and $\\zeta(G)=8$} \\label{fig1}\n\n\\end{figure}\n\nConsider the unicyclic graph $G$ as shown in Figure \\ref{fig1}. It is easy to see that $\\gamma(G)=3$. We list all the $\\gamma$-sets with properties which are among the five commonly studied, as  previously mentioned.\n\n\\begin{itemize}\n\n\t\\item $\\{u,v,w\\}$ is a perfect $\\gamma$-set, a connected $\\gamma$-set, a total $\\gamma$-set, and a clique $\\gamma$-set.\n\t\\item $\\{x,y,z\\}$ is a perfect $\\gamma$-set, and an independent $\\gamma$-set.\n\t \\item $\\{u,y,z\\}$, $\\{v,x,z\\}$, and $\\{w,x,y\\}$ are independent $\\gamma$-sets but not perfect, connected, clique or total. \n\t\\item $\\{u,v,z\\}$, $\\{u,w,y\\}$, and $\\{v,w,x\\}$ are  $\\gamma$-sets which have none of the $5$ special properties that we previously listed.\n\\end{itemize}\nWe have $\\displaystyle \\zeta(G)=\\sum_{i=0}^3{3\\choose i}=8$, the number of $\\gamma$-sets of $G$.\nWe present a generalization of the previous example for such unicyclic graphs which are commonly known as \\textit{sun graphs}.", "context": "A {\\it dominating set} for a graph $G = (V,E)$ is a subset $S \\subseteq V$ such that every vertex $v\\in V$ is either in $S$ or has a neighbor $u\\in S$. The vertex $u \\in S$ is said to \\textit{cover} the vertex $v \\in V$ if either $u=v$ or $uv\\in E$. A dominating set $S$ is a \\textit{minimal dominating set} if no proper subset $S'\\subset S$ is a dominating set. The \\textit{domination number} $\\gamma(G)$ of a graph $G$ is the minimum cardinality among all dominating sets of $G$. Clearly, a graph $G$ can have multiple minimum dominating sets. For simplicity, we refer to those sets as \\textbf{$\\gamma$-sets}.\n\n\\begin{enumerate}\n \\item \\textit{Perfect $\\gamma$-set} requires that no two vertices in a $\\gamma$-set cover the same vertex (outside the $\\gamma$-set). \\item \\textit{Connected $\\gamma$-set} requires that the graph induced by the $\\gamma$-set be connected.\n \\item \\textit{Total $\\gamma$-set} forbids isolated vertices in the subgraph induced by the $\\gamma$-set.\n \\item \\textit{Independent $\\gamma$-set} requires all vertices in a $\\gamma$-set to be isolated vertices in the subgraph they induce.\n \\item \\textit{clique $\\gamma$ set} requires that the vertices in a $\\gamma$-set induce a clique.\n\\end{enumerate}\n\nThe \\textit{dominion (number)} of a graph $G=(V,E)$, denoted by $\\zeta(G)$, is the number of its $\\gamma$-sets. In other words,\n\n$\\zeta(G):=|\\{S: S \\text{ is a dominating set in} \\ G \\ \\text{and} \\ |S|=\\gamma\\}|$. For instance, given a complete graph $G=K_n$, it is obvious that each $v\\in V(G)$ covers the remaining $n-1$ vertices of $G$. Therefore $\\zeta(G)=n$ while $\\gamma(G)=1$. Also, for a star graph $G=K_{1,n}$ of order $n+1\\ge 3$, it is easy to see that $\\gamma(G)=1=\\zeta(G)$ since only the central vertex can cover all of the remaining $n\\ge 2$ vertices. \n\\begin{example}{\\label{Ex}}\n\\end{example}\nHere, we illustrate the $\\gamma$-sets of a unicyclic graph $G$ (see Figure \\ref{fig1}). \n\\begin{figure}[ht]\n \\centering\n \\begin{tikzpicture}[scale=1]\n \\clip(0,-1) rectangle (4,4);\n \\draw [line width=2pt] (2.62,1)-- (2,2);\n \\draw [line width=2pt] (2,2)-- (1.98,2.84);\n \\draw [line width=2pt] (2.62,1)-- (3,0.34);\n \\draw [line width=2pt] (2,2)-- (1.48,1.02);\n \\draw [line width=2pt] (1.48,1.02)-- (2.62,1);\n \\draw [line width=2pt] (1.48,1.02)-- (1.06,0.34);\n \\begin{scriptsize}\n \\draw [fill=black] (2,2) circle (2.5pt);\n \\draw[color=black] (2.2,2.1) node {$u$};\n \\draw [fill=black] (2.62,1) circle (2.5pt);\n \\draw[color=black] (2.92,1) node {$v$};\n \\draw [fill=black] (1.98,2.84) circle (2.5pt);\n \\draw[color=black] (1.94,3.1) node {$x$};\n \\draw [fill=black] (3,0.34) circle (2.5pt);\n \\draw[color=black] (3.2,0.15) node {$y$};\n \\draw [fill=black] (1.48,1.02) circle (2.5pt);\n \\draw[color=black] (1.14,1) node {$w$};\n \\draw [fill=black] (1.06,0.34) circle (2.5pt);\n \\draw[color=black] (0.9,0.15) node {$z$};\n \\end{scriptsize}\n \\end{tikzpicture}\n \\caption{A sun graph $G$ with $\\gamma(G)=3$ and $\\zeta(G)=8$} \\label{fig1}\n\n\\begin{itemize}\n\n\\item $\\{u,v,w\\}$ is a perfect $\\gamma$-set, a connected $\\gamma$-set, a total $\\gamma$-set, and a clique $\\gamma$-set.\n \\item $\\{x,y,z\\}$ is a perfect $\\gamma$-set, and an independent $\\gamma$-set.\n \\item $\\{u,y,z\\}$, $\\{v,x,z\\}$, and $\\{w,x,y\\}$ are independent $\\gamma$-sets but not perfect, connected, clique or total. \n \\item $\\{u,v,z\\}$, $\\{u,w,y\\}$, and $\\{v,w,x\\}$ are $\\gamma$-sets which have none of the $5$ special properties that we previously listed.\n\\end{itemize}\nWe have $\\displaystyle \\zeta(G)=\\sum_{i=0}^3{3\\choose i}=8$, the number of $\\gamma$-sets of $G$.\nWe present a generalization of the previous example for such unicyclic graphs which are commonly known as \\textit{sun graphs}.\n\n\\label{fig1}\n\n\\end{figure}\n\nConsider the unicyclic graph $G$ as shown in Figure \\ref{fig1}. It is easy to see that $\\gamma(G)=3$. We list all the $\\gamma$-sets with properties which are among the five", "full_context": "A {\\it dominating set} for a graph $G = (V,E)$ is a subset $S \\subseteq V$ such that every vertex $v\\in V$ is either in $S$ or has a neighbor $u\\in S$. The vertex $u \\in S$ is said to \\textit{cover} the vertex $v \\in V$ if either $u=v$ or $uv\\in E$. A dominating set $S$ is a \\textit{minimal dominating set} if no proper subset $S'\\subset S$ is a dominating set. The \\textit{domination number} $\\gamma(G)$ of a graph $G$ is the minimum cardinality among all dominating sets of $G$. Clearly, a graph $G$ can have multiple minimum dominating sets. For simplicity, we refer to those sets as \\textbf{$\\gamma$-sets}.\n\n\\begin{enumerate}\n \\item \\textit{Perfect $\\gamma$-set} requires that no two vertices in a $\\gamma$-set cover the same vertex (outside the $\\gamma$-set). \\item \\textit{Connected $\\gamma$-set} requires that the graph induced by the $\\gamma$-set be connected.\n \\item \\textit{Total $\\gamma$-set} forbids isolated vertices in the subgraph induced by the $\\gamma$-set.\n \\item \\textit{Independent $\\gamma$-set} requires all vertices in a $\\gamma$-set to be isolated vertices in the subgraph they induce.\n \\item \\textit{clique $\\gamma$ set} requires that the vertices in a $\\gamma$-set induce a clique.\n\\end{enumerate}\n\nThe \\textit{dominion (number)} of a graph $G=(V,E)$, denoted by $\\zeta(G)$, is the number of its $\\gamma$-sets. In other words,\n\n$\\zeta(G):=|\\{S: S \\text{ is a dominating set in} \\ G \\ \\text{and} \\ |S|=\\gamma\\}|$. For instance, given a complete graph $G=K_n$, it is obvious that each $v\\in V(G)$ covers the remaining $n-1$ vertices of $G$. Therefore $\\zeta(G)=n$ while $\\gamma(G)=1$. Also, for a star graph $G=K_{1,n}$ of order $n+1\\ge 3$, it is easy to see that $\\gamma(G)=1=\\zeta(G)$ since only the central vertex can cover all of the remaining $n\\ge 2$ vertices. \n\\begin{example}{\\label{Ex}}\n\\end{example}\nHere, we illustrate the $\\gamma$-sets of a unicyclic graph $G$ (see Figure \\ref{fig1}). \n\\begin{figure}[ht]\n \\centering\n \\begin{tikzpicture}[scale=1]\n \\clip(0,-1) rectangle (4,4);\n \\draw [line width=2pt] (2.62,1)-- (2,2);\n \\draw [line width=2pt] (2,2)-- (1.98,2.84);\n \\draw [line width=2pt] (2.62,1)-- (3,0.34);\n \\draw [line width=2pt] (2,2)-- (1.48,1.02);\n \\draw [line width=2pt] (1.48,1.02)-- (2.62,1);\n \\draw [line width=2pt] (1.48,1.02)-- (1.06,0.34);\n \\begin{scriptsize}\n \\draw [fill=black] (2,2) circle (2.5pt);\n \\draw[color=black] (2.2,2.1) node {$u$};\n \\draw [fill=black] (2.62,1) circle (2.5pt);\n \\draw[color=black] (2.92,1) node {$v$};\n \\draw [fill=black] (1.98,2.84) circle (2.5pt);\n \\draw[color=black] (1.94,3.1) node {$x$};\n \\draw [fill=black] (3,0.34) circle (2.5pt);\n \\draw[color=black] (3.2,0.15) node {$y$};\n \\draw [fill=black] (1.48,1.02) circle (2.5pt);\n \\draw[color=black] (1.14,1) node {$w$};\n \\draw [fill=black] (1.06,0.34) circle (2.5pt);\n \\draw[color=black] (0.9,0.15) node {$z$};\n \\end{scriptsize}\n \\end{tikzpicture}\n \\caption{A sun graph $G$ with $\\gamma(G)=3$ and $\\zeta(G)=8$} \\label{fig1}\n\n\\begin{itemize}\n\n\\item $\\{u,v,w\\}$ is a perfect $\\gamma$-set, a connected $\\gamma$-set, a total $\\gamma$-set, and a clique $\\gamma$-set.\n \\item $\\{x,y,z\\}$ is a perfect $\\gamma$-set, and an independent $\\gamma$-set.\n \\item $\\{u,y,z\\}$, $\\{v,x,z\\}$, and $\\{w,x,y\\}$ are independent $\\gamma$-sets but not perfect, connected, clique or total. \n \\item $\\{u,v,z\\}$, $\\{u,w,y\\}$, and $\\{v,w,x\\}$ are $\\gamma$-sets which have none of the $5$ special properties that we previously listed.\n\\end{itemize}\nWe have $\\displaystyle \\zeta(G)=\\sum_{i=0}^3{3\\choose i}=8$, the number of $\\gamma$-sets of $G$.\nWe present a generalization of the previous example for such unicyclic graphs which are commonly known as \\textit{sun graphs}.\n\n\\label{fig1}\n\n\\end{figure}\n\nConsider the unicyclic graph $G$ as shown in Figure \\ref{fig1}. It is easy to see that $\\gamma(G)=3$. We list all the $\\gamma$-sets with properties which are among the five\n\n$\\zeta(G):=|\\{S: S \\text{ is a dominating set in} \\ G \\ \\text{and} \\ |S|=\\gamma\\}|$. For instance, given a complete graph $G=K_n$, it is obvious that each $v\\in V(G)$ covers the remaining $n-1$ vertices of $G$. Therefore $\\zeta(G)=n$ while $\\gamma(G)=1$. Also, for a star graph $G=K_{1,n}$ of order $n+1\\ge 3$, it is easy to see that $\\gamma(G)=1=\\zeta(G)$ since only the central vertex can cover all of the remaining $n\\ge 2$ vertices. \n\\begin{example}{\\label{Ex}}\n\\end{example}\nHere, we illustrate the $\\gamma$-sets of a unicyclic graph $G$ (see Figure \\ref{fig1}). \n\\begin{figure}[ht]\n \\centering\n \\begin{tikzpicture}[scale=1]\n \\clip(0,-1) rectangle (4,4);\n \\draw [line width=2pt] (2.62,1)-- (2,2);\n \\draw [line width=2pt] (2,2)-- (1.98,2.84);\n \\draw [line width=2pt] (2.62,1)-- (3,0.34);\n \\draw [line width=2pt] (2,2)-- (1.48,1.02);\n \\draw [line width=2pt] (1.48,1.02)-- (2.62,1);\n \\draw [line width=2pt] (1.48,1.02)-- (1.06,0.34);\n \\begin{scriptsize}\n \\draw [fill=black] (2,2) circle (2.5pt);\n \\draw[color=black] (2.2,2.1) node {$u$};\n \\draw [fill=black] (2.62,1) circle (2.5pt);\n \\draw[color=black] (2.92,1) node {$v$};\n \\draw [fill=black] (1.98,2.84) circle (2.5pt);\n \\draw[color=black] (1.94,3.1) node {$x$};\n \\draw [fill=black] (3,0.34) circle (2.5pt);\n \\draw[color=black] (3.2,0.15) node {$y$};\n \\draw [fill=black] (1.48,1.02) circle (2.5pt);\n \\draw[color=black] (1.14,1) node {$w$};\n \\draw [fill=black] (1.06,0.34) circle (2.5pt);\n \\draw[color=black] (0.9,0.15) node {$z$};\n \\end{scriptsize}\n \\end{tikzpicture}\n \\caption{A sun graph $G$ with $\\gamma(G)=3$ and $\\zeta(G)=8$} \\label{fig1}\n\n\\item $\\{u,v,w\\}$ is a perfect $\\gamma$-set, a connected $\\gamma$-set, a total $\\gamma$-set, and a clique $\\gamma$-set.\n \\item $\\{x,y,z\\}$ is a perfect $\\gamma$-set, and an independent $\\gamma$-set.\n \\item $\\{u,y,z\\}$, $\\{v,x,z\\}$, and $\\{w,x,y\\}$ are independent $\\gamma$-sets but not perfect, connected, clique or total. \n \\item $\\{u,v,z\\}$, $\\{u,w,y\\}$, and $\\{v,w,x\\}$ are $\\gamma$-sets which have none of the $5$ special properties that we previously listed.\n\\end{itemize}\nWe have $\\ds \\zeta(G)=\\sum_{i=0}^3{3\\choose i}=8$, the number of $\\gamma$-sets of $G$.\nWe present a generalization of the previous example for such unicyclic graphs which are commonly known as \\textit{sun graphs}.\n\n\\begin{proof}\nFor each vertex on the cycle, and for each dominating set $S\\subseteq V(G)$, either the vertex or its leaf neighbor must be in $S$. Therefore, we can form all $\\gamma$-sets $S$ in $G$ from choosing subsets of the set of $n$ leafs, and the cycle neighbors of unchosen leafs. This gives $\\ds \\sum_{i=0}^n{n\\choose i}$. Conversely, every $S$ assembled by choosing either the vertex on the cycle or the leaf, for each leaf, is a $\\gamma$-set for $G$. This gives $\\zeta(G)=2^n$.\n\\end{proof}\n\n\\begin{claim}{\\label{c4}}\n When $n=3k, k\\ge 1$, $\\zeta(P_n)=1$ and the unique $\\gamma$-set in $P_n$ contains $v_{n-1}$. Moreover, when $n=3k+2, k\\ge 1$, there is only one $\\gamma$-set in $P_n$ containing $v_{n}$. \n\\end{claim}\n\\begin{proof}\n When $n=3k, k\\ge 1$, we have $\\zeta(P_n)=k$ and the conclusion follows by the pigeon-hold argument alluded in Claim \\ref{c1}. Each of the sets of $3$ consecutive vertices $\\{v_{3r+1},v_{3r+2},v_{3r+3}\\}$, $r=0,\\ldots,k-1$, must contain at least one vertex of any $\\gamma$-set $S$, and since $\\gamma=k$, each of those sets must contain exactly one element of $S$. Since $S$ is dominating, the element of $S$ in $\\{v_1,v_2,v_3\\}$ must be $v_1$ or $v_2$. If $v_1$ then the ``next'' element of $S$ must be $v_4$, which forces $S=\\{v_1,v_4,\\ldots,v_{3k-2}\\}$, leaving $v_{3k}$ uncovered. Therefore, the first element of $S$ must be $v_2$, which forces $S=\\{v_2,v_5,\\ldots,v_{3k-1}, v_{3k+2}\\}$.\n\n\\begin{proposition}{\\label{Prop2}}\n For any connected graph $G$ on $n$ vertices, $1\\le \\zeta(G)\\le {n\\choose \\gamma(G)}$.\n\n\\end{proof}\n\\begin{corollary}\nFor any connected graph $G$ with $\\gamma(G)=1$, $\\zeta(\\underbrace{G\\vee \\ldots \\vee G}_r)=r\\zeta(G)$, $r\\ge 1$.\n\\end{corollary}\n\\begin{proof}\nFrom Part (a) of Theorem \\ref{T4} (when $\\gamma(G_1)=1=\\gamma(G_2)$), the result follows by induction on $r\\ge 1$.\n\\end{proof}\nWe note that in the case when $G=K_1$, it is clear that $\\underbrace{G\\vee \\ldots \\vee G}_r:=K_r$ and $\\zeta(K_r)=r=r\\zeta(K_1)$. Here, we present a more general result for the special case when $G=\\overline{K_{m}}$, the complement of a complete graph on $m\\ge 1$ vertices. Note that $\\overline{K_{m_1}} \\vee \\ldots \\vee \\overline{K_{m_k}}$ is a complete $k$-partite graph often written as $K(m_1,m_2,\\ldots, m_k)$, with $m_i\\ge 1$ and $k\\ge 2$.\n\n\\end{theorem}\n\\begin{proof}\nLet $V_i$ denote each disjoint partite (vertex) set of $G$, with $|V_i|=m_i$, $i=1,\\ldots, k$. If, for some $j=1,\\ldots, r$, $m_j=1$ and $m_i>1$ for $i>r$, then it is clear that $\\gamma(G)=1$ and the $\\gamma$-sets in $G$ are the singleton subsets of $V_1\\cup\\ldots \\cup V_r$. It follows that $\\zeta(G)=r$. \nAssume $m_i>1$, for $i=1,\\ldots, k$. It is clear that $\\gamma(G)=2$. Any vertex $v\\in V_i$ covers all vertices $v'\\in V(G)\\setminus V_i$ except those in $V_i$. Thus, for each $v\\in V_i$ and $v'\\in V_j$, $i\\ne j$, $\\{v,v'\\}$ is a $\\gamma$-set of $G$. It follows that there are $\\ds \\sum_{i\\ne j}^{{k}}m_im_j$ total such sets. Further, if $|V_i|=2$ for $i=1,\\ldots, r$, then each such sets, $V_i$, is an additional $\\gamma$-set, giving the result.\n\\end{proof}", "post_theorem_intro_text_len": 511, "post_theorem_intro_text": "\\begin{proof}\nFor each vertex on the cycle, and for each dominating set $S\\subseteq V(G)$, either the vertex or its leaf neighbor must be in $S$. Therefore, we can form all $\\gamma$-sets $S$ in $G$ from choosing subsets of the set of $n$ leafs, and the cycle neighbors of unchosen leafs. This gives  $\\displaystyle \\sum_{i=0}^n{n\\choose i}$.  Conversely, every $S$ assembled by choosing either the vertex on the cycle or the leaf, for each leaf, is a $\\gamma$-set for $G$. This gives $\\zeta(G)=2^n$.\n\\end{proof}", "sketch": "For each vertex on the cycle and each dominating set $S\\subseteq V(G)$, \"either the vertex or its leaf neighbor must be in $S$.\" Hence all $\\gamma$-sets can be formed by choosing a subset of the $n$ leaves, and then including \"the cycle neighbors of unchosen leafs,\" yielding $\\sum_{i=0}^n \\binom{n}{i}$. Conversely, \"every $S$ assembled by choosing either the vertex on the cycle or the leaf, for each leaf, is a $\\gamma$-set for $G$,\" so $\\zeta(G)=\\sum_{i=0}^n \\binom{n}{i}=2^n$.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "If $G$ is a sun graph on $2n\\ge 6$ vertices, then $\\displaystyle \\zeta(G)=\\sum_{i=0}^n{n\\choose i}=2^n. $,", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let b3(G) denote the domination number of a graph G, i.e. the minimum size of a dominating set, and let b6(G) be the number of b3-sets of G, where a b3-set is a dominating set of cardinality b3(G). A sun graph on 2n vertices (with 2n \\ge 6) is the graph obtained from a cycle C_n by attaching exactly one pendant leaf to each cycle vertex. Which quantitative statement holds for b6(G)?", "correct_choice": {"label": "A", "text": "If G is a sun graph on 2n \\ge 6 vertices, then \\(\\zeta(G)=\\sum_{i=0}^{n}\\binom{n}{i}=2^n\\)."}, "choices": [{"label": "B", "text": "If G is a sun graph on 2n \\ge 6 vertices, then \\(\\zeta(G)=\\binom{2n}{n}\\)."}, {"label": "C", "text": "If G is a sun graph on 2n \\ge 6 vertices, then \\(\\zeta(G)\\ge 2^n\\)."}, {"label": "D", "text": "If G is a sun graph on 2n \\ge 6 vertices, then every \\(\\gamma\\)-set is obtained by choosing exactly one vertex from each leaf-cycle pair, and hence \\(\\zeta(G)=n\\)."}, {"label": "E", "text": "If G is a sun graph on 2n \\ge 6 vertices, then \\(\\zeta(G)=\\sum_{i=1}^{n}\\binom{n}{i}=2^n-1\\)."}], "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": "independent binary choices at n cycle-leaf pairs replaced by unrestricted n-subset count from 2n vertices", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "replaced exact enumeration by the weaker lower bound only", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "local one-from-each-pair structure kept but global count collapsed from 2 choices per pair to one canonical choice", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "excluded one valid endpoint case in the binomial sum", "template_used": "boundary_range"}]}}