diff --git "a/202602/qaEval_202602_all.json" "b/202602/qaEval_202602_all.json" new file mode 100644--- /dev/null +++ "b/202602/qaEval_202602_all.json" @@ -0,0 +1,8293 @@ +[ + { + "id": "2602.00112v1", + "paper_link": "http://arxiv.org/abs/2602.00112v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "proposition", + "content": "\\label{Prop1}\nFor every torse-forming vector field $V$ that satisfies (\\ref{TFVF}), we have\n\\begin{equation}\\label{Theta}\n\t\t\\theta ={\\rm d}\\rho - f {\\rm e}^{-2 \\rho} \\omega.\n\t\\end{equation}", + "start_pos": 5584, + "end_pos": 5812, + "label": "Prop1" + }, + "ref_dict": { + "TFVF": "\\begin{equation}\\label{TFVF}\n\t\t\\nabla_X V = fX + \\theta(X)V,\n\t\\end{equation}", + "Eq110": "\\begin{eqnarray}\\label{Eq110}\ng(\\nabla_X V, V)= X g(V,V)-g(V , \\nabla_X V) &\\Leftrightarrow & g(\\nabla_X V, V) = \\frac{1}{2} X({\\rm e}^{2\\rho}) \\notag\\\\\n&\\Leftrightarrow & g(\\nabla_X V, V) = {\\rm e}^{2\\rho} X(\\rho).\n\\end{eqnarray}", + "Eq100": "\\begin{eqnarray}\\label{Eq100}\ng(\\nabla_X V, V) &=& f g(X,V) + \\theta(X)g(V,V) \\notag\\\\\n&=& f \\omega(X) + {\\rm e}^{2\\rho} \\theta(X).\n\\end{eqnarray}" + }, + "pre_theorem_intro_text_len": 3383, + "pre_theorem_intro_text": "It is well known that generalizing mathematical concepts is a prerequisite for most researchers. It is also often an important means of developing mathematics and answering many questions that have occupied the minds of researchers in many fields. For example, the notion of a Ricci soliton is a natural generalization of an Einstein metric; at the same time, it is a stationary solution of a famous PDE for Riemannian metrics, known under the name of the Ricci flow equation. Without forgetting that Ricci soliton itself has had many generalizations made recently in many works.\n\nGenerally, a mathematical concept is initially defined by a simple expression that satisfies the need to include it. Then questions begin: What if we replace the constant with a function? What if we add a differential one-form? What happens if we add a certain term? ... And so, little by little, the simple expression begins to take on longer and more complex forms. \n\nIn principle, if these generalizations fail to answer any outstanding mathematical problems or fail to impact other concepts, then they can be considered an intellectual luxury. \n\nIn this paper we present a step in the reverse direction of generalization; as a subject of study, we will discuss the concept of the torse-forming vector field as a generalization of both the torqued vector field, the concircular vector field, the concurrent vector field, the recurrent vector field, and the parallel vector field.\n\n Torse-forming vector fields demand special attention due to their applications not only in relativity and cosmology but in theory of submanifolds also. On a Riemannian manifold $(M^n, g)$, a the torse-forming vector field is a vector field $V$ that satisfies\n\\begin{equation}\\label{TFVF}\n\t\t\\nabla_X V = fX + \\theta(X)V,\n\t\\end{equation}\nfor any vector field $X$ on $M$ where $\\nabla$ is the Levi-Civita connection of $g$, $\\theta$ is a one-form and $f$ is a smooth function on $(M^n, g)$. The 1-form $\\theta$ is called the generating form and the function $f$ is called the conformal scalar. If $\\omega$ is the corresponding $1$-form of $V$ , i.e., $\\omega(X) = g(X,V)$ then for all $X, Y$ vector fields on $M$\n\\begin{equation}\\label{TFVF2}\n (\\nabla_X \\omega)Y = g(\\nabla_X V , Y)= fg(X,Y)+ \\theta(X)\\omega(Y).\n\t\\end{equation}\n Some special types of torse-forming vector fields have been considered in various studies. A torse-forming vector field $V$ is called: \n\\begin{itemize}\n\\item[1)] proper torse-forming if $f \\neq 0$ and the 1-form $\\theta$ is nowhere zero on a dense open subset of $M$.\n\\item[2)] torqued vector field if $V$ satisfying (\\ref{TFVF}) with $\\theta(V)=0$ (see \\cite{CBY2, CBY3}).\n\\item[3)] concircular vector field if $\\theta$ is identically zero \\cite{YK2} .\n\\item[4)] concurrent vector field if $\\theta = 0$ and $f=1$.\n\\item[5)] reccurent vector field if $\\theta \\neq 0$ and $f=0$.\n\\item[6)] geodesic vector field if $\\nabla_V V=0$.\n\\item[7)] parallel vector field if $\\theta = 0$ and $f=0$.\n\\end{itemize}\n\n\\textbf{Agreement:} Through the rest of this paper, $(M,g)$ always denotes a Riemannian manifold. $V$ denotes a proper torse-forming vector field on $M$ that satisfies (\\ref{TFVF}) with $\\Vert V \\Vert = {\\rm e}^{\\rho}$, and $\\omega $ the $1$-form corresponding to $V$, i.e., for all $X$ vector field on $M$,\n$$ \\omega(X)=g(V , X)\\qquad and \\qquad \\omega(V)={\\rm e}^{2\\rho}.$$", + "context": "It is well known that generalizing mathematical concepts is a prerequisite for most researchers. It is also often an important means of developing mathematics and answering many questions that have occupied the minds of researchers in many fields. For example, the notion of a Ricci soliton is a natural generalization of an Einstein metric; at the same time, it is a stationary solution of a famous PDE for Riemannian metrics, known under the name of the Ricci flow equation. Without forgetting that Ricci soliton itself has had many generalizations made recently in many works.\n\nGenerally, a mathematical concept is initially defined by a simple expression that satisfies the need to include it. Then questions begin: What if we replace the constant with a function? What if we add a differential one-form? What happens if we add a certain term? ... And so, little by little, the simple expression begins to take on longer and more complex forms.\n\nIn this paper we present a step in the reverse direction of generalization; as a subject of study, we will discuss the concept of the torse-forming vector field as a generalization of both the torqued vector field, the concircular vector field, the concurrent vector field, the recurrent vector field, and the parallel vector field.\n\nTorse-forming vector fields demand special attention due to their applications not only in relativity and cosmology but in theory of submanifolds also. On a Riemannian manifold $(M^n, g)$, a the torse-forming vector field is a vector field $V$ that satisfies\n\\begin{equation}\\label{TFVF}\n \\nabla_X V = fX + \\theta(X)V,\n \\end{equation}\nfor any vector field $X$ on $M$ where $\\nabla$ is the Levi-Civita connection of $g$, $\\theta$ is a one-form and $f$ is a smooth function on $(M^n, g)$. The 1-form $\\theta$ is called the generating form and the function $f$ is called the conformal scalar. If $\\omega$ is the corresponding $1$-form of $V$ , i.e., $\\omega(X) = g(X,V)$ then for all $X, Y$ vector fields on $M$\n\\begin{equation}\\label{TFVF2}\n (\\nabla_X \\omega)Y = g(\\nabla_X V , Y)= fg(X,Y)+ \\theta(X)\\omega(Y).\n \\end{equation}\n Some special types of torse-forming vector fields have been considered in various studies. A torse-forming vector field $V$ is called: \n\\begin{itemize}\n\\item[1)] proper torse-forming if $f \\neq 0$ and the 1-form $\\theta$ is nowhere zero on a dense open subset of $M$.\n\\item[2)] torqued vector field if $V$ satisfying (\\ref{TFVF}) with $\\theta(V)=0$ (see \\cite{CBY2, CBY3}).\n\\item[3)] concircular vector field if $\\theta$ is identically zero \\cite{YK2} .\n\\item[4)] concurrent vector field if $\\theta = 0$ and $f=1$.\n\\item[5)] reccurent vector field if $\\theta \\neq 0$ and $f=0$.\n\\item[6)] geodesic vector field if $\\nabla_V V=0$.\n\\item[7)] parallel vector field if $\\theta = 0$ and $f=0$.\n\\end{itemize}\n\n\\textbf{Agreement:} Through the rest of this paper, $(M,g)$ always denotes a Riemannian manifold. $V$ denotes a proper torse-forming vector field on $M$ that satisfies (\\ref{TFVF}) with $\\Vert V \\Vert = {\\rm e}^{\\rho}$, and $\\omega $ the $1$-form corresponding to $V$, i.e., for all $X$ vector field on $M$,\n$$ \\omega(X)=g(V , X)\\qquad and \\qquad \\omega(V)={\\rm e}^{2\\rho}.$$\n\n\\begin{equation}\\label{TFVF}\n\t\t\\nabla_X V = fX + \\theta(X)V,\n\t\\end{equation}", + "full_context": "It is well known that generalizing mathematical concepts is a prerequisite for most researchers. It is also often an important means of developing mathematics and answering many questions that have occupied the minds of researchers in many fields. For example, the notion of a Ricci soliton is a natural generalization of an Einstein metric; at the same time, it is a stationary solution of a famous PDE for Riemannian metrics, known under the name of the Ricci flow equation. Without forgetting that Ricci soliton itself has had many generalizations made recently in many works.\n\nGenerally, a mathematical concept is initially defined by a simple expression that satisfies the need to include it. Then questions begin: What if we replace the constant with a function? What if we add a differential one-form? What happens if we add a certain term? ... And so, little by little, the simple expression begins to take on longer and more complex forms.\n\nIn this paper we present a step in the reverse direction of generalization; as a subject of study, we will discuss the concept of the torse-forming vector field as a generalization of both the torqued vector field, the concircular vector field, the concurrent vector field, the recurrent vector field, and the parallel vector field.\n\nTorse-forming vector fields demand special attention due to their applications not only in relativity and cosmology but in theory of submanifolds also. On a Riemannian manifold $(M^n, g)$, a the torse-forming vector field is a vector field $V$ that satisfies\n\\begin{equation}\\label{TFVF}\n \\nabla_X V = fX + \\theta(X)V,\n \\end{equation}\nfor any vector field $X$ on $M$ where $\\nabla$ is the Levi-Civita connection of $g$, $\\theta$ is a one-form and $f$ is a smooth function on $(M^n, g)$. The 1-form $\\theta$ is called the generating form and the function $f$ is called the conformal scalar. If $\\omega$ is the corresponding $1$-form of $V$ , i.e., $\\omega(X) = g(X,V)$ then for all $X, Y$ vector fields on $M$\n\\begin{equation}\\label{TFVF2}\n (\\nabla_X \\omega)Y = g(\\nabla_X V , Y)= fg(X,Y)+ \\theta(X)\\omega(Y).\n \\end{equation}\n Some special types of torse-forming vector fields have been considered in various studies. A torse-forming vector field $V$ is called: \n\\begin{itemize}\n\\item[1)] proper torse-forming if $f \\neq 0$ and the 1-form $\\theta$ is nowhere zero on a dense open subset of $M$.\n\\item[2)] torqued vector field if $V$ satisfying (\\ref{TFVF}) with $\\theta(V)=0$ (see \\cite{CBY2, CBY3}).\n\\item[3)] concircular vector field if $\\theta$ is identically zero \\cite{YK2} .\n\\item[4)] concurrent vector field if $\\theta = 0$ and $f=1$.\n\\item[5)] reccurent vector field if $\\theta \\neq 0$ and $f=0$.\n\\item[6)] geodesic vector field if $\\nabla_V V=0$.\n\\item[7)] parallel vector field if $\\theta = 0$ and $f=0$.\n\\end{itemize}\n\n\\textbf{Agreement:} Through the rest of this paper, $(M,g)$ always denotes a Riemannian manifold. $V$ denotes a proper torse-forming vector field on $M$ that satisfies (\\ref{TFVF}) with $\\Vert V \\Vert = {\\rm e}^{\\rho}$, and $\\omega $ the $1$-form corresponding to $V$, i.e., for all $X$ vector field on $M$,\n$$ \\omega(X)=g(V , X)\\qquad and \\qquad \\omega(V)={\\rm e}^{2\\rho}.$$\n\n\\begin{equation}\\label{TFVF}\n\t\t\\nabla_X V = fX + \\theta(X)V,\n\t\\end{equation}\n\nFor a conformal transformation, it is well known that the Levi-Civita connection $\\nabla$ and $\\tilde{\\nabla}$ associated with the metrics $g$ and $\\tilde{g}$ respectively are connected by\n\\begin{eqnarray}\\label{TransfConf}\n\\tilde{\\nabla}_{X}Y= \\nabla_{X}Y + X(\\sigma)Y + Y(\\sigma)X - g(X,Y){\\rm grad}\\sigma.\n \\end{eqnarray}\nBy using (\\ref{TFVF}) and Proposition \\ref{Prop1}, we have\n\\begin{eqnarray}\\label{TildNabV}\n\\tilde{\\nabla}_{X}V &=& \\nabla_{X}V + X(\\sigma)V + V(\\sigma)X - g(X,V){\\rm grad}\\sigma \\notag\\\\\n&=& f X + \\big( X(\\rho) - f {\\rm e}^{-2\\rho} \\omega(X) \\big)V + X(\\sigma)V + V(\\sigma)X - \\omega(X){\\rm grad}\\sigma \\notag\\\\\n&=& \\big(f + V(\\sigma) \\big) X + \\big( X(\\rho) + X(\\sigma) \\big)V - \\omega(X) \\big( f {\\rm e}^{-2\\rho} V +{\\rm grad}\\sigma \\big).\n\\end{eqnarray}\nIt is clear that $V$ is a torse-formming vector field on $(M,\\tilde{g})$ if and only if\n$${\\rm grad}\\sigma = -f {\\rm e}^{-2\\rho} V.$$\nAlso, under this condition, we get $ f + V(\\sigma)=0$ and (\\ref{TildNabV}) gives\n$$\\tilde{\\nabla}_{X}V = \\theta(X)V.$$\nConsequently, $V$ is recurrent with respect to $\\tilde{g}$. Hence, we conclude the following:\n\\begin{theorem}\\label{Th1}\nLet $V$ be a proper torse-formming vector field on $(M,g)$ which satisfying (\\ref{TFVF}). $V$ is recurrent with respect to $\\tilde{g}={\\rm e}^{2\\sigma}g$ where $\\sigma \\in \\mathcal{C}^{\\infty}(M)$ if and only if ${\\rm grad}\\sigma = -f {\\rm e}^{-2\\rho} V$.\n\\end{theorem}\n\\begin{example}\nIt is convenient to have some sort of hyperspherical coordinates on the unit sphere $\\mathbb{S}^3$ in analogy to the usual spherical coordinates on $\\mathbb{S}^2$. One such choice is to use $(x_1, x_2,x_3)$ where\n\\begin{equation}\\label{parametrisationS3}\n \\left\\{\n \\begin{array}{lll}\nx_1 = \\cos x_1,\\\\\nx_2 = \\sin x_1 \\cos x_2,\\\\\nx_3 = \\sin x_1 \\sin x_2 \\cos x_3,\\\\\nx_4 = \\sin x_1 \\sin x_2 \\sin x_3,\n \\end{array}\n \\right.\n\\end{equation}\nwhere $x_1$ and $x_2$ run over the range $0$ to $\\pi$, and $x_3$ runs over $0$ to $2 \\pi$.\n\nOne can define the Riemannian metric $g$ in these coordinates as follows:\n$$ g = d x_1^2 + \\sin^2 x_1 ( dx_2^2 + \\sin^2 x_2 \\;d x_3^2).$$\nwhere\n$$\\left\\lbrace e_1 = \\frac{\\partial}{\\partial x_1},\\quad e_2 = \\frac{1}{\\sin x_1}\\frac{\\partial}{\\partial x_2},\\quad\ne_3=\\frac{1}{\\sin x_1 \\sin x_2}\\frac{\\partial}{\\partial x_3}\\right\\rbrace ,$$\nis an orthonormal basis for $g$. By Kozsul's formula, the covariant derivatives of the basis elements are as follows:\n$$\\begin{tabular}{lll}\n$ \\nabla_{e_1} e_1 = 0,$ & $\\nabla_{e_1} e_2 =0,$ & $\\nabla_{e_1} e_3 = 0$, \\\\ \n$\\nabla_{e_2} e_1= \\cot x_1 e_2,$ & $\\nabla_{e_2} e_2= -\\cot x_1 e_1,$ & $\\nabla_{e_2} e_3 =0,$ \\\\ \n$\\nabla_{e_3} e_1= \\cot x_1 e_3,$& $\\nabla_{e_3} e_2 = \\frac{\\cot x_2 }{\\sin x_1} e_3,$ & $\\nabla_{e_3} e_3 = -\\cot x_1 e_1 - \\frac{\\cot x_2 }{\\sin x_1} e_2.$ \\\\ \n\\end{tabular} \n$$\nEasily, we can see that for $ V = h(x_3) \\sin x_1 e_1$ we have \n\\begin{equation}\\label{NabV1}\n \\left\\{\n \\begin{array}{lll}\n\\nabla_{e_1} V =h(x_3) \\cos x_1 e_1 ,\\\\\n\\nabla_{e_2} V = h(x_3) \\cos x_1 e_2,\\\\\n\\nabla_{e_3} V =\\frac{h'}{\\sin x_2 } e_1 + h(x_3) \\cos x_1 e_3,\n \\end{array}\n \\right.\n\\end{equation}\nwhere $h' = \\frac{\\partial h}{\\partial x_3}$. Then, for $\\theta = \\frac{h'}{h} dx_3$ we get\n$$\\nabla_{e_i} V = h(x_3) \\cos x_1 e_i + \\theta(e_i) V,$$\nhence, according to (\\ref{TFVF}), $V$ is a proper torse-forming vector field on $(\\mathbb{S}^3, g)$ with $f =h(x_3) \\cos x_1$ and $\\theta = \\frac{h'}{h} dx_3$.\n\n\\begin{proposition}\n Let $\\nabla$ and $\\hat{\\nabla} $ denote the Riemannian connections of $ g$ and $\\hat{g}$ respectively. Then, for all $X,Y $ vector fields on $M$, we have the relation:\n\\begin{eqnarray}\\label{NablaHat}\n\\hat{\\nabla}_{X}Y &=& \\nabla_X Y + X(\\sigma)Y + Y(\\sigma)X \\notag\\\\\n&-& g(X,Y){\\rm grad} \\sigma - \\omega(X)\\omega(Y){\\rm grad}(\\sigma +\\rho)\\notag\\\\\n&+& \\frac{1}{1+{\\rm e}^{2\\rho}} \\Big( \\big(f + V(\\sigma)\\big) g(X,Y) +X(\\rho) \\omega(Y) + \\omega(X)Y(\\rho)\\notag\\\\\n&&\\qquad\\qquad +\\big(V(\\rho)+V(\\sigma)-f {\\rm e}^{-2\\rho}\\big)\\omega(X)\\omega(Y)\\Big)V.\n \\end{eqnarray}\n\\end{proposition}\n\\begin{proof}\nUsing (\\ref{TransfConf}), we get\n\\begin{eqnarray}\\label{TransfDConf}\n\\hat{\\nabla}_{X}Y= \\overline{\\nabla}_{X}Y + X(\\sigma)Y + Y(\\sigma)X - \\overline{g}(X,Y)\\overline{{\\rm grad}}\\sigma.\n \\end{eqnarray}\nFor all $X$ vector field on $M$, we compute\n\\begin{eqnarray}\\label{Eq300}\n\\overline{g}(\\overline{{\\rm grad}}\\sigma, X) = X(\\sigma)\n &=& g({\\rm grad} \\sigma, X) \\notag\\\\\n&=& \\overline{g}( {\\rm grad} \\sigma, X) - V(\\sigma)\\omega(X) \\notag\\\\\n&=& \\overline{g}\\Big( {\\rm grad} \\sigma -\\frac{ V(\\sigma)}{1+ {\\rm e}^{2\\rho}} V,X\\Big),\n\\end{eqnarray} \nthen,\n\\begin{equation}\\label{Eq310}\n\\overline{{\\rm grad}}\\sigma = {\\rm grad} \\sigma -\\frac{ V(\\sigma)}{1+ {\\rm e}^{2\\rho}} V.\n\\end{equation}\nNow, by substituting (\\ref{NablaOverline}) and (\\ref{Eq310}) in (\\ref{TransfDConf}) taking into account $\\overline{g} = g + \\omega \\otimes \\omega$ we get our formula.\n\\end{proof}\nLet's replace $Y$ by $V$ in (\\ref{NablaHat}) and using (\\ref{TFVF}), we get\n\\begin{eqnarray}\\label{NablaHat2}\n\\hat{\\nabla}_{X}V &=& \\big(f + V(\\sigma)\\big) X + \\theta(X)V + X(\\sigma)V \\notag\\\\\n&-& (1+{\\rm e}^{2 \\rho})\\omega(X){\\rm grad} \\sigma - {\\rm e}^{2 \\rho}\\omega(X){\\rm grad}(\\rho)\\notag\\\\\n&+& \\frac{1}{1+{\\rm e}^{2\\rho}} \\Big( {\\rm e}^{2 \\rho}X(\\rho) +(1+{\\rm e}^{2 \\rho})\\big(V(\\rho)+V(\\sigma)\\big)\\omega(X)\\Big)V.\n \\end{eqnarray}\n Reorganize this equation using (\\ref{Theta}), we find\n\n\\begin{eqnarray}\\label{Eq400}\n\\hat{\\nabla}_{X}V &=& \\big(f + V(\\sigma)\\big) X \\notag\\\\\n&&+ \\Big( X(\\sigma) + \\big(V(\\rho)+V(\\sigma)- f {\\rm e}^{-2 \\rho}\\big)\\omega(X) + \\frac{1+2{\\rm e}^{2 \\rho}}{1+{\\rm e}^{2\\rho}} X(\\rho) \\Big)V \\notag\\\\\n&&- (1+{\\rm e}^{2 \\rho})\\omega(X){\\rm grad} \\sigma - {\\rm e}^{2 \\rho}\\omega(X){\\rm grad}(\\rho).\n \\end{eqnarray}\nFor $V$ to be a torse forming vector field with respect to $\\hat{g}$, it is sufficient for it to be\n\\begin{equation}\\label{Eq410}\n {\\rm grad} \\sigma= - \\frac{{\\rm e}^{2 \\rho}}{1+{\\rm e}^{2\\rho}}{\\rm grad}\\rho.\n\\end{equation}\nFrom this equation, we get\n\\begin{equation}\\label{Eq420}\nX(\\sigma) = - \\frac{{\\rm e}^{2 \\rho}}{1+{\\rm e}^{2\\rho}}X(\\rho)\\qquad and \\qquad V(\\sigma) = - \\frac{{\\rm e}^{2 \\rho}}{1+{\\rm e}^{2\\rho}}V(\\rho).\n\\end{equation}\nBy (\\ref{Eq410}) and (\\ref{Eq420}), (\\ref{Eq400}) becomes\n\\begin{equation}\\label{430}\n\\hat{\\nabla}_{X}V = \\Big(f - \\frac{{\\rm e}^{2 \\rho}}{1+{\\rm e}^{2\\rho}} \\Big) X + \\Big( X(\\rho) + \\Big( \\frac{V(\\rho)}{1+{\\rm e}^{2\\rho}} - f {\\rm e}^{-2 \\rho}\\Big)\\omega(X) \\Big)V.\n \\end{equation}\nHence, $V$ is a torse-forming with respect to $\\hat{g}$. Therfore,\nwe can extract two important cases:\\\\\n(1):\\quad For $f = \\frac{{\\rm e}^{2 \\rho}}{1+{\\rm e}^{2\\rho}}$ we get\n$ \\hat{\\nabla}_{X}V = X(\\rho)V$ this means that $V$ is reccurent on $(M, \\hat{g})$.\\\\\n(2):\\quad For $ X(\\rho) = \\Big( f {\\rm e}^{-2 \\rho} -\\frac{V(\\rho)}{1+{\\rm e}^{2\\rho}}\\Big)\\omega(X)$, by taking $X=V$ we get $ f =\\frac{1+2{\\rm e}^{2\\rho}}{1+{\\rm e}^{2\\rho}}V(\\rho)$ then we obtain $ \\hat{\\nabla}_{X}V = V(\\rho)X$.\nBy combining the arguments of the above discussion, we obtain the following theorem:\n\\begin{theorem}\nLet $V$ be a proper torse-forming vector field on $(M,g)$ with $V^{\\flat}=\\omega$ and $\\omega(V)={\\rm e}^{2\\rho}$. If\n\\begin{equation}\\label{Eq450}\n {\\rm grad} \\sigma= - \\frac{{\\rm e}^{2 \\rho}}{1+{\\rm e}^{2\\rho}}{\\rm grad}\\rho.\n\\end{equation}\nthen $V$ is a torse-forming with respect to $\\hat{g}$. Moreover,\\\\\n\\textbf{(i)}\\quad if $f = \\frac{{\\rm e}^{2 \\rho}}{1+{\\rm e}^{2\\rho}}$ then\n $V$ is reccurent vector field on $(M, \\hat{g})$.\\\\\n\\textbf{(ii)}\\quad if $ {\\rm d}\\rho = \\Big( f {\\rm e}^{-2 \\rho} -\\frac{V(\\rho)}{1+{\\rm e}^{2\\rho}}\\Big)\\omega$ then $V$ is a concircular vector field on $(M, \\hat{g})$.", + "post_theorem_intro_text_len": 643, + "post_theorem_intro_text": "\\begin{proof}\nUsing (\\ref{TFVF}), for all $X$ vector field on $M$, we have\n\\begin{eqnarray}\\label{Eq100}\ng(\\nabla_X V, V) &=& f g(X,V) + \\theta(X)g(V,V) \\notag\\\\\n&=& f \\omega(X) + {\\rm e}^{2\\rho} \\theta(X).\n\\end{eqnarray}\nOn the other hand, we have\n\\begin{eqnarray}\\label{Eq110}\ng(\\nabla_X V, V)= X g(V,V)-g(V , \\nabla_X V) &\\Leftrightarrow & g(\\nabla_X V, V) = \\frac{1}{2} X({\\rm e}^{2\\rho}) \\notag\\\\\n&\\Leftrightarrow & g(\\nabla_X V, V) = {\\rm e}^{2\\rho} X(\\rho).\n\\end{eqnarray}\nThen, from (\\ref{Eq100}) and (\\ref{Eq110}) we get\n$$ \\theta(X) = X(\\rho) - f {\\rm e}^{-2\\rho} \\omega(X).$$\nThis is what is required to be proven,\n\n\\end{proof}", + "sketch": "From (\\ref{TFVF}), compute for any vector field $X$:\n\\begin{eqnarray}\n g(\\nabla_X V, V) &=& f\\, g(X,V)+\\theta(X)g(V,V)= f\\,\\omega(X)+\\mathrm{e}^{2\\rho}\\theta(X). \\tag{\\ref{Eq100}}\n\\end{eqnarray}\nOn the other hand, using $g(\\nabla_X V,V)=\\tfrac12 X(g(V,V))$ and $g(V,V)=\\mathrm{e}^{2\\rho}$ gives\n\\begin{eqnarray}\n g(\\nabla_X V, V)=\\frac12 X(\\mathrm{e}^{2\\rho})=\\mathrm{e}^{2\\rho}X(\\rho). \\tag{\\ref{Eq110}}\n\\end{eqnarray}\nEquating (\\ref{Eq100}) and (\\ref{Eq110}) yields\n$$\\theta(X)=X(\\rho)-f\\,\\mathrm{e}^{-2\\rho}\\omega(X),$$\ni.e. $\\theta=\\mathrm{d}\\rho-f\\,\\mathrm{e}^{-2\\rho}\\omega$, which is (\\ref{Theta}).", + "expanded_sketch": "From\n\\begin{equation}\\label{TFVF}\n\t\t\\nabla_X V = fX + \\theta(X)V,\n\t\\end{equation}\ncompute for any vector field $X$:\n\\begin{eqnarray}\\label{Eq100}\ng(\\nabla_X V, V) &=& f g(X,V) + \\theta(X)g(V,V) \\\\notag\\\\\n&=& f \\omega(X) + {\\rm e}^{2\\rho} \\theta(X).\n\\end{eqnarray}\nOn the other hand, using $g(\\nabla_X V,V)=\\tfrac12 X(g(V,V))$ and $g(V,V)=\\mathrm{e}^{2\\rho}$ gives\n\\begin{eqnarray}\\label{Eq110}\ng(\\nabla_X V, V)= X g(V,V)-g(V , \\nabla_X V) &\\Leftrightarrow & g(\\nabla_X V, V) = \\\\frac{1}{2} X({\\rm e}^{2\\rho}) \\\\notag\\\\\n&\\Leftrightarrow & g(\\nabla_X V, V) = {\\rm e}^{2\\rho} X(\\rho).\n\\end{eqnarray}\nEquating the two displayed identities yields\n$$\\theta(X)=X(\\rho)-f\\,\\mathrm{e}^{-2\\rho}\\omega(X),$$\ni.e. $\\theta=\\mathrm{d}\\rho-f\\,\\mathrm{e}^{-2\\rho}\\omega$, which is (\\ref{Theta}).", + "expanded_theorem": "\\label{Prop1}\nFor every torse-forming vector field $V$ that satisfies\n\\begin{equation}\\label{TFVF}\n\t\t\\nabla_X V = fX + \\theta(X)V,\n\t\\end{equation}\nwe have\n\\begin{equation}\\label{Theta}\n\t\t\\theta ={\\rm d}\\rho - f {\\rm e}^{-2 \\rho} \\omega.\n\t\\end{equation}", + "theorem_type": [ + "Universal" + ], + "mcq": { + "question": "Let \\((M,g)\\) be a Riemannian manifold, and let \\(V\\) be a torse-forming vector field, meaning that there exist a smooth function \\(f\\) and a 1-form \\(\\theta\\) such that for every vector field \\(X\\), \\(\\nabla_X V = fX + \\theta(X)V\\). Let \\(\\omega = V^{\\flat}\\) be the metric dual 1-form of \\(V\\), so \\(\\omega(X)=g(V,X)\\), and write \\(\\|V\\|=e^{\\rho}\\) (equivalently, \\(\\omega(V)=e^{2\\rho}\\)). Which statement holds for every such vector field \\(V\\)?", + "correct_choice": { + "label": "A", + "text": "The generating 1-form satisfies \\(\\theta = d\\rho - f e^{-2\\rho}\\,\\omega\\); equivalently, for every vector field \\(X\\), \\(\\theta(X)=X(\\rho)-f e^{-2\\rho}\\omega(X)\\)." + }, + "choices": [ + { + "label": "B", + "text": "The generating 1-form satisfies \\(\\theta = d\\rho + f e^{-2\\rho}\\,\\omega\\); equivalently, for every vector field \\(X\\), \\(\\theta(X)=X(\\rho)+f e^{-2\\rho}\\omega(X)\\)." + }, + { + "label": "C", + "text": "For every vector field \\(X\\) orthogonal to \\(V\\) (that is, \\(\\omega(X)=0\\)), one has \\(\\theta(X)=X(\\rho)\\)." + }, + { + "label": "D", + "text": "The generating 1-form satisfies \\(\\theta = d\\rho - f e^{-\\rho}\\,\\omega\\); equivalently, for every vector field \\(X\\), \\(\\theta(X)=X(\\rho)-f e^{-\\rho}\\omega(X)\\)." + }, + { + "label": "E", + "text": "There exists a smooth function \\(\\lambda\\) on \\(M\\), depending only on \\(\\rho\\), such that \\(\\theta = d\\rho - \\lambda\\,\\omega\\); equivalently, for every vector field \\(X\\), \\(\\theta(X)=X(\\rho)-\\lambda\\omega(X)\\)." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "trace_identity", + "tampered_component": "sign from equating the two expressions for g(\\nabla_XV,V)", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "trace_identity", + "tampered_component": "dropped the full formula by restricting to the case \\omega(X)=0", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "trace_identity", + "tampered_component": "power of e^\\rho arising from dividing by g(V,V)=e^{2\\rho}", + "template_used": "boundary_range" + }, + { + "label": "E", + "sketch_hook_type": "trace_identity", + "tampered_component": "specific dependence of the coefficient on f and e^{-2\\rho}", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives the definition of a torse-forming vector field and relevant notation, but it does not state or strongly telegraph the target identity. The correct formula for \\(\\theta\\) must still be derived." + }, + "TAS": { + "score": 2, + "justification": "This is not a mere restatement of the definition in the stem. The correct choice is a derived consequence relating \\(\\theta\\), \\(\\rho\\), \\(f\\), and \\(\\omega\\), and the options present genuinely competing conclusions." + }, + "GPS": { + "score": 1, + "justification": "The item requires a short but real derivation: compare \\(X(g(V,V))\\) with \\(2g(\\nabla_XV,V)\\) and solve for \\(\\theta(X)\\). This is more than recall, but the reasoning is fairly direct rather than deeply generative." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically targeted: sign error (B), wrong exponent (D), a weaker-but-true statement (C), and an overgeneralized existential form (E). These reflect realistic failure modes." + }, + "total_score": 7, + "overall_assessment": "A strong MCQ: it avoids answer leakage, is not tautological, and uses well-designed distractors. The reasoning required is meaningful but relatively short, so the generative demand is moderate rather than high." + } + }, + { + "id": "2602.01138v1", + "paper_link": "http://arxiv.org/abs/2602.01138v1", + "theorems_cnt": 3, + "theorem": { + "env_name": "theorem", + "content": "\\label{propagation of chaos}\nThe assumptions of Lemma \\ref{weak solution} still hold. Let $\\{X_{N,i}^\\varepsilon\\}_{1\\le i\\le N}$ and $\\{\\overline{X}_i^\\varepsilon\\}_{1\\le i\\le N}$ be the solutions to systems \\eqref{sde} and \\eqref{mfs}, respectively. Then \nfor any $m,N\\in \\mathbb{N}^+$ and parameters $0<\\theta<\\frac{1}{2},\\quad 0<\\alpha<\\frac{\\theta}{2},\\quad m>\\frac{\\theta+1}{1-2\\theta}$, there exist a constant $C(m,T)>0$ and a parameter $\\gamma,\\eta>0$ satisfying the bounds\n\\begin{align}\n&0<\\gamma<\\min\\big\\{\\frac{\\alpha}{3},\\frac{-2\\alpha +m(1-2\\theta)-1}{4m+4}\\big\\},\\label{gamma} \\\\\n&0<\\eta\\le\\min\\{\\theta-2\\alpha, -(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}\\label{eta}\n\\end{align}\nsuch that for all $0\\le t\\le T$,\n\\begin{align}\\label{Pe}\n\\mathbb{P}\\big(\\max_{i=1,\\cdots,N}\\big|(X_{N,i}^\\varepsilon - \\overline{X}_i^\\varepsilon)(t)\\big|> N^{-\\alpha}\\big)\\le C(m,T)N^{-\\eta}, \n\\end{align} \nwhere the cut-off parameter satisfies\n$\\varepsilon\\sim N^{-\\gamma} $.", + "start_pos": 15136, + "end_pos": 16132, + "label": "propagation of chaos" + }, + "ref_dict": { + "sde": "\\begin{align}\\label{sde}\n\\begin{cases}\ndX_{N,i}^\\varepsilon(t) = \\Big(2\\exp\\Big(-\\frac{1}{N}\\sum_{j=1}^N\\Phi^\\varepsilon(X_{N,i}^\\varepsilon(t) - X_{N,j}^\\varepsilon(t))\\Big)+ 2\\Big)^{1/2}dB_i(t),\\\\\nX_{N,i}^\\varepsilon(0) =\\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}", + "mfs": "\\begin{align}\\label{mfs}\n\\begin{cases}\nd\\overline{X}_i^\\varepsilon(t) = \\big(2\\exp\\big(-\\Phi^\\varepsilon * u^\\varepsilon( \\overline{X}_i^\\varepsilon,t)\\big)+2\\big)^{1/2} dB_i(t),\\\\\n\\overline{X}_i^\\varepsilon(0) = \\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}", + "pde": "\\begin{cases}\\label{pde}\n\\partial_t u = \\Delta(e^{-v}u+ u), \\qquad &x\\in\\R^2,~ t>0,\\\\\n-\\Delta v + v =\\chi u, \\qquad &x\\in\\R^2,~ t>0,\\\\\nu(x,0) = u_0(x), \\qquad &x\\in\\R^2,\n\\end{cases}", + "Pe": "\\begin{align}\\label{Pe}\n\\mathbb{P}\\big(\\max_{i=1,\\cdots,N}\\big|(X_{N,i}^\\varepsilon - \\overline{X}_i^\\varepsilon)(t)\\big|> N^{-\\alpha}\\big)\\le C(m,T)N^{-\\eta}, \n\\end{align}", + "propagation of chaos": "\\begin{theorem}\\label{propagation of chaos}\nThe assumptions of Lemma \\ref{weak solution} still hold. Let $\\{X_{N,i}^\\varepsilon\\}_{1\\le i\\le N}$ and $\\{\\overline{X}_i^\\varepsilon\\}_{1\\le i\\le N}$ be the solutions to systems \\eqref{sde} and \\eqref{mfs}, respectively. Then \nfor any $m,N\\in \\mathbb{N}^+$ and parameters $0<\\theta<\\frac{1}{2},\\quad 0<\\alpha<\\frac{\\theta}{2},\\quad m>\\frac{\\theta+1}{1-2\\theta}$, there exist a constant $C(m,T)>0$ and a parameter $\\gamma,\\eta>0$ satisfying the bounds\n\\begin{align}\n&0<\\gamma<\\min\\big\\{\\frac{\\alpha}{3},\\frac{-2\\alpha +m(1-2\\theta)-1}{4m+4}\\big\\},\\label{gamma} \\\\\n&0<\\eta\\le\\min\\{\\theta-2\\alpha, -(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}\\label{eta}\n\\end{align}\nsuch that for all $0\\le t\\le T$,\n\\begin{align}\\label{Pe}\n\\mathbb{P}\\big(\\max_{i=1,\\cdots,N}\\big|(X_{N,i}^\\varepsilon - \\overline{X}_i^\\varepsilon)(t)\\big|> N^{-\\alpha}\\big)\\le C(m,T)N^{-\\eta}, \n\\end{align} \nwhere the cut-off parameter satisfies\n$\\varepsilon\\sim N^{-\\gamma} $.\n\\end{theorem}", + "pde1": "\\begin{cases}\\label{pde1}\n\\partial_t u = \\Delta(\\gamma(v)u),\\\\\n\\tau\\partial_t v-\\Delta v + v = u,\n\\end{cases}", + "weak solution": "\\begin{lemma}\\label{weak solution}\n\\textup{(Existence of weak solution, \\cite[Theorem 2, Lemma 16, and Lemma 18]{BOL2026113712})}.\nLet $u_0\\ge0$ be a initial probability density that satisfies\n\\begin{eqnarray*}\n&u_0\\log u_0\\in L^1(\\R^2), \\quad \\nabla\\log u_0\\in W^{1,q}(\\R^2)~(q>2),\\\\\n&u_0\\in L^1(\\R^2,|x|^2dx)\\cap L^p(\\R^2)~(1\\le p\\le \\infty). \n\\end{eqnarray*}\nFurthermore, assume $\\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo-Nirenberg inequality: $\\|\\omega\\|_{L^4(\\R^2)}^4\\le c_*\\|\\omega\\|_{L^2(\\R^2)}^2\\|\\nabla\\omega\\|_{L^2(\\R^2)}^2$. Then for any $T>0$ and $t\\in[0,T]$, the problem \\eqref{rpde} possesses weak solutions $(u^\\varepsilon, v^\\varepsilon)$ in $\\R^2\\times(0,T)$, and there exists a time $T^*\\in(0,T)$ such that \n\\begin{eqnarray}\n&\\|u^\\varepsilon\\|_{{L^2(0,T;H^1(\\R^2))}\\cap L^\\infty(0,T;L^p(\\R^2))}\\le C,\\label{uniformweaksolution}\\\\\n&\\||x|^2u^\\varepsilon\\|_{L^\\infty(0, T; L^1(\\R^2))}\\le C,\\quad\\|\\nabla\\log u^\\varepsilon\\|_{L^\\infty(0, T^*; W^{1,q}(\\R^2))}\\le C.\\label{uniformweaksolution1}\n\\end{eqnarray}\nwhere $C$ is a constant independent of $\\varepsilon$. \n\\end{lemma}", + "Ps111": "\\begin{align}\\label{Ps111}\n\\|u_{N,r}^{\\varepsilon}(t) - u^{\\varepsilon\\otimes r}(t)\\|_{L^\\infty(0,T^*;L^1(\\R^{2r}))} \\le C(r,m,T)\\varepsilon^{\\beta}, \n \\end{align}", + "rpde": "\\begin{cases}\\label{rpde}\n\\partial_t u^\\varepsilon = \\Delta(e^{-v^\\varepsilon}u^\\varepsilon+ u^\\varepsilon),\\qquad &x\\in\\R^2,~ t>0,\\\\\n-\\Delta v^\\varepsilon + v^\\varepsilon =\\chi u^\\varepsilon * j^\\varepsilon,\\qquad &x\\in\\R^2,~ t>0,\\\\\nu^\\varepsilon(x,0) = u_0*j^\\varepsilon(x), \\qquad &x\\in\\R^2.\n\\end{cases}" + }, + "pre_theorem_intro_text_len": 12702, + "pre_theorem_intro_text": "In this work, we provide a rigorous derivation of the two-dimensional Keller–Segel-type system with signal-dependent sensitivity expressed as\n\\begin{align}\n\\begin{cases}\\label{pde}\n\\partial_t u = \\Delta(e^{-v}u+ u), \\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\n-\\Delta v + v =\\chi u, \\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\nu(x,0) = u_0(x), \\qquad &x\\in\\mathbb{R}^2,\n\\end{cases}\n\\end{align}\nwhere $u(x,t)$ denotes cell density and $v(x,t)$ represents the concentration of the chemical signal. The coefficient $\\chi>0$ quantifies the signal-dependent strength of chemotactic sensitivity. \n\nThe formulation of the model is grounded in a robust biological rationale. This model describes the formation of stripe patterns through a self-trapping mechanism. Referring\nto \\cite{fu2012stripe}, we know that this process, which has been extensively studied using synthetic biological experimental methods, involves Escherichia coli secreting the signaling molecule acyl-homoserine lactone (AHL). At low concentrations of AHL, the bacteria exhibit high motility, characterized by random motion driven by typical swimming and tumbling behavior with little external interference. However, as the AHL concentration increases, the behavior of the bacterial population changes significantly, eventually leading to a macroscopically static state.\nThe partial differential equation (PDE) \\eqref{pde} is derived from the general signal-dependent Keller–Segel model proposed \n\\begin{align}\n\\begin{cases}\\label{pde1}\n\\partial_t u = \\Delta(\\gamma(v)u),\\\\\n\\tau\\partial_t v-\\Delta v + v = u,\n\\end{cases}\n\\end{align}\nwhere $\\tau$ takes the value of $0$ or $1$. Here, $\\gamma(v)$ represents the signal-dependent motility, which satisfies $\\gamma'(v)\\le 0$. \nA substantial body of research has been devoted to the existence of solutions for the model \\eqref{pde1}, particularly in cases where the signal-dependent function $\\gamma(v)$ is subject to different conditions.\nIn 2017, under the assumptions of upper and lower bounds for both $\\gamma(v)$ and its derivative $\\gamma'(v)$, Tao and Winkler \\cite{tao2017effects} proved the existence of global classical solutions for the two-dimensional case, while demonstrating that the system admits only weak solutions in higher dimensions. Fujie and Jiang \\cite{fujie2020global} established that when $\\gamma(v)$ satisfies the following conditions:\n\\begin{equation*}\n0<\\gamma(v)\\in C^3[0,\\infty),\\quad \\gamma'(v)\\leq 0 \\text{ on } [0,\\infty),\\quad \\lim_{v\\to\\infty}\\gamma(v)=0,\n\\end{equation*}\nglobal classical solutions exist for arbitrary initial data. Their subsequent work \\cite{fujie2021boundedness} showed that if the function $\\gamma(v)$ satisfies the asymptotic condition $\\lim_{v\\to\\infty} e^{\\alpha v}\\gamma(v) = +\\infty$ for any $\\alpha > 0$, then there exists a globally bounded classical solution.\nFor the case $\\gamma(v) = e^{-\\alpha v}$ ($\\alpha > 0$), the authors in \\cite{fujie2020global,fujie2021comparison,jin2020critical} derived a critical mass threshold: solutions remain uniformly bounded when the initial cell mass is below this critical value. In contrast, studies in \\cite{burger2021delayed, fujie2020global,fujie2021comparison} indicate that solutions blow up as time approaches infinity when the initial cell mass exceeds the critical mass. Nevertheless, it is impossible to blow-up in finite time, which is a difference from the Keller-Segel model. \n\nIn their seminal 2025 work on the model \\eqref{pde} (with $\\gamma(v)=e^{-v}+1$), the reference \\cite{BOL2026113712}\nrigorously established the well-posedness of solutions to the PDE \\eqref{pde} and gave error estimates between \\eqref{pde} and its regularized version. The discussion is already so comprehensive that we can cite it directly. They also conducted an in-depth study in which the convergences for the stochastic particle systems of \\eqref{pde} inspired us to refine their methodologies. \nOur focus lies particularly on the stochastic particle equations and the mean field limit corresponding to \\eqref{pde}. By introducing the concept of stopping time (achieving convergence in probability in Section 2), we obtained results with faster convergence rates than those in \\cite{BOL2026113712}. \nIn Section 3, we establish the propagation of chaos in strong senses for stochastic differential equations (SDEs), (where we apply the relative entropy method and derive a higher convergence rate), which build a foundational bridge between the macroscopic and microscopic systems.\nIn the following, we propose that the SDE corresponding to \\eqref{pde} for $N\\in\\mathbb{N}$ interacting particles $\\{X_{N,i}^\\varepsilon(\\cdot)\\}_{1\\le i\\le N}$ in $\\mathbb{R}^2$ reads as\n\\begin{align}\\label{sde}\n\\begin{cases}\ndX_{N,i}^\\varepsilon(t) = \\Big(2\\exp\\Big(-\\frac{1}{N}\\sum_{j=1}^N\\Phi^\\varepsilon(X_{N,i}^\\varepsilon(t) - X_{N,j}^\\varepsilon(t))\\Big)+ 2\\Big)^{1/2}dB_i(t),\\\\\nX_{N,i}^\\varepsilon(0) =\\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}\nwhere $0<\\varepsilon<1$. Here, we consider a filtered probability space defined by $(\\Omega,\\mathcal{F}, (\\mathcal{F})_{t\\ge0}, \\mathbb{P})$ and introduce $\\{B_i\\}_{1\\le i\\le N}$, a collection of independent $\\mathcal{F}_t$-Brownian motions. The initial data $\\zeta_1,\\zeta_2,\\cdots,\\zeta_N$ are assumed to be random variables independent and identically distributed (i.i.d.) with the common probability density function $u_0$. The potential $\\Phi^\\varepsilon$ is given by\n$$\\Phi^\\varepsilon:=\\Phi*j^\\varepsilon,\\quad \\Phi:=\\chi \\tilde{\\Phi},\\quad j^\\varepsilon(x):=\\frac{1}{\\varepsilon^2}j(\\frac{x}{\\varepsilon}),$$\nwhere $\\chi>0$ describes the strength rate of the signals in the model \\eqref{pde} and $\\tilde{\\Phi}$ is the Yukawa potential \\cite{lieb2001analysis}, which is defined for any $\\mu > 0$, by \n\\begin{align*}\n&\\tilde{\\Phi}(x) = \\int_0^\\infty(4\\pi t)^{-1}\\exp\\Big\\{-\\frac{|x|^2}{4t}-\\mu^2t\\Big\\}dt.\n\\end{align*}\nBased on \\cite[Appendix B]{li2023optimal}, the following properties are observed\n\\begin{align}\\label{Phi}\n&\\|\\tilde{\\Phi}\\|_{L^p(\\mathbb{R}^2)}<\\infty,~\\|\\nabla\\tilde{\\Phi}\\|_{L^q(\\mathbb{R}^2)}<\\infty,\\qquad p\\in[1,\\infty),~q\\in[1,2).\n\\end{align}\nThrough calculation, the following results are obtained that for any constant $C$ independent of $\\varepsilon$,\n\\begin{align}\n&\\|\\Phi^\\varepsilon\\|_{W^{1,1}(\\mathbb{R}^2)}\\le\\|\\Phi\\|_{W^{1,1}(\\mathbb{R}^2)}\\|j^\\varepsilon\\|_{L^1(\\mathbb{R}^2)}\\le C,\\label{D1phi}\\\\\n&\\|\\Phi^\\varepsilon\\|_{W^{1,\\infty}(\\mathbb{R}^2)}\\le\\|\\Phi\\|_{W^{1,1}(\\mathbb{R}^2)}\\|j^\\varepsilon\\|_{L^\\infty(\\mathbb{R}^2)}\\le \\frac{C}{\\varepsilon^2},\\label{phi}\\\\\n&\\|D^2\\Phi^\\varepsilon\\|_{L^\\infty(\\mathbb{R}^2)}\\le \\|\\nabla\\Phi\\|_{L^1(\\mathbb{R}^2)}\\|\\nabla j^\\varepsilon\\|_{L^\\infty(\\mathbb{R}^2)}\\le \\frac{C}{\\varepsilon^3}\\label{D2phi}.\n\\end{align}\nThe rigorous derivation of PDEs from stochastic particle systems represents a fundamental challenge in mathematical physics and applied analysis. \nA particularly influential framework for such derivations is the theory of moderately interacting particles, introduced in the pioneering works of Oelschl\\\"ager \\cite{oelschlager1985law, oelschlager1987fluctuation, oelschlager1989derivation},\nwhere the Law of Large Numbers for interacting diffusions was established, together with central-limit-type fluctuation results.\nThis approach, characterized by a specific scaling of the interaction potential that preserves nonlocality while weakening pointwise strength, has since been extended to various biological and physical models. For example, Stevens \\cite{stevens2000derivation} derived the Keller–Segel chemotaxis model from a moderately interacting system, overcoming the lack of ellipticity, a common assumption in earlier mean field limits, by introducing novel analytical techniques. \n\nIn recent years, significant progress has been made in handling singular interactions, such as Coulomb and Riesz potentials, which arise naturally in models of collective behavior. Lazarovici and Pickl \\cite{lazarovici2017mean} analyzed a particle system incorporating a regularized potential and random initial conditions. Serfaty \\cite{serfaty2020mean} developed the modulated energy method to treat Coulomb-type flows, providing a powerful tool for the analysis of the mean field without confinement. \nSubsequent studies on related models employing diverse regularization strategies have further expanded this line of inquiry, as documented in \\cite{boers2016mean,bolley2011stochastic,chen2017mean,chen2020combined}.\nCurrently, the relative entropy method, advanced by Jabin and Wang \\cite{jabin2017mean} and further applied by Bresch et al.\\cite{bresch2023mean} and Chen et al.\\cite{chen2025quantitative,chen2025mean}, has enabled quantitative estimates of convergence for singular attractive kernels. These techniques have been instrumental in the establishment of various modes of convergence, including convergence in the Wasserstein distance (Carrillo et al.\\cite{carrillo2019propagation}) and convergence in probability Pickl et al. \\cite{lazarovici2017mean,huang2020mean,huang2017error}, strong $L^1$ convergence (Chen et al. \\cite{chen2025quantitative} and \\cite{olivera2020quantitative,olivera2023quantitative}). More recently, the moderate interaction framework has been instrumental in the derivation of cross-diffusion models; see, for example, \\cite{carrillo2024interacting,chen2021rigorous, chen2019rigorous,li2024convergence}.\n\nTo study the relationship between the macroscopic system \\eqref{pde} and its microscopic system \\eqref{sde}, \nwe need to introduce an intermediate particle system (Mean-Field equation). A key object in the mean-field limit is the empirical measure, a random probability measure defined as\n$$\\mu^\\varepsilon_N(t)=\\frac{1}{N}\\sum_{i=1}^N\\delta_{X_{N,i}^\\varepsilon(t)},\\quad t>0,$$\nwhere $\\delta$ is the Dirac delta distribution. The work \\cite{olivera2021quantitative} shows that $\\mu^\\varepsilon_N(t)$ converges to the following PDE solution $u^\\varepsilon$. For fixed $\\varepsilon>0$, the particle system \\eqref{sde} propagates chaos in the many-particle limit $N\\to\\infty$ towards the non-linear SDE system:\n\\begin{align}\\label{mfs}\n\\begin{cases}\nd\\overline{X}_i^\\varepsilon(t) = \\big(2\\exp\\big(-\\Phi^\\varepsilon * u^\\varepsilon( \\overline{X}_i^\\varepsilon,t)\\big)+2\\big)^{1/2} dB_i(t),\\\\\n\\overline{X}_i^\\varepsilon(0) = \\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}\nwhere\n$u^\\varepsilon(\\cdot,t)$ is the density function of i.i.d. random processes\n$\\overline{X}_1^\\varepsilon(t),\\cdots, \\overline{X}_N^\\varepsilon(t)$. \nAnd the initial data $\\{\\zeta_i\\}_{i=1}^N$\nare subject to the same conditions as in \\eqref{sde}. Using It\\^{o}'s formula, the density function $u^\\varepsilon$ satisfies the so-called intermediate nonlocal problem, namely,\n\\begin{align}\n\\begin{cases}\\label{rpde}\n\\partial_t u^\\varepsilon = \\Delta(e^{-v^\\varepsilon}u^\\varepsilon+ u^\\varepsilon),\\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\n-\\Delta v^\\varepsilon + v^\\varepsilon =\\chi u^\\varepsilon * j^\\varepsilon,\\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\nu^\\varepsilon(x,0) = u_0*j^\\varepsilon(x), \\qquad &x\\in\\mathbb{R}^2.\n\\end{cases}\n\\end{align}\n\nIn this paper, our argument relies on the Lemma \\ref{weak solution} from \\cite{BOL2026113712}, which establishes the existence and uniqueness of the weak solution $u^\\varepsilon$ to the above system \\eqref{rpde} and provides the $L^\\infty$ estimate for $\\nabla\\log u^\\varepsilon$.\n\\begin{lemma}\\label{weak solution}\n\\textup{(Existence of weak solution, \\cite[Theorem 2, Lemma 16, and Lemma 18]{BOL2026113712})}.\nLet $u_0\\ge0$ be a initial probability density that satisfies\n\\begin{eqnarray*}\n&u_0\\log u_0\\in L^1(\\mathbb{R}^2), \\quad \\nabla\\log u_0\\in W^{1,q}(\\mathbb{R}^2)~(q>2),\\\\\n&u_0\\in L^1(\\mathbb{R}^2,|x|^2dx)\\cap L^p(\\mathbb{R}^2)~(1\\le p\\le \\infty). \n\\end{eqnarray*}\nFurthermore, assume $\\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo-Nirenberg inequality: $\\|\\omega\\|_{L^4(\\mathbb{R}^2)}^4\\le c_*\\|\\omega\\|_{L^2(\\mathbb{R}^2)}^2\\|\\nabla\\omega\\|_{L^2(\\mathbb{R}^2)}^2$. Then for any $T>0$ and $t\\in[0,T]$, the problem \\eqref{rpde} possesses weak solutions $(u^\\varepsilon, v^\\varepsilon)$ in $\\mathbb{R}^2\\times(0,T)$, and there exists a time $T^*\\in(0,T)$ such that \n\\begin{eqnarray}\n&\\|u^\\varepsilon\\|_{{L^2(0,T;H^1(\\mathbb{R}^2))}\\cap L^\\infty(0,T;L^p(\\mathbb{R}^2))}\\le C,\\label{uniformweaksolution}\\\\\n&\\||x|^2u^\\varepsilon\\|_{L^\\infty(0, T; L^1(\\mathbb{R}^2))}\\le C,\\quad\\|\\nabla\\log u^\\varepsilon\\|_{L^\\infty(0, T^*; W^{1,q}(\\mathbb{R}^2))}\\le C.\\label{uniformweaksolution1}\n\\end{eqnarray}\nwhere $C$ is a constant independent of $\\varepsilon$. \n\\end{lemma}\n\nUsing the above lemma, we obtain our first main result: convergence in probability of the mean-field limit under algebraic scaling.", + "context": "In this work, we provide a rigorous derivation of the two-dimensional Keller–Segel-type system with signal-dependent sensitivity expressed as\n\\begin{align}\n\\begin{cases}\\label{pde}\n\\partial_t u = \\Delta(e^{-v}u+ u), \\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\n-\\Delta v + v =\\chi u, \\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\nu(x,0) = u_0(x), \\qquad &x\\in\\mathbb{R}^2,\n\\end{cases}\n\\end{align}\nwhere $u(x,t)$ denotes cell density and $v(x,t)$ represents the concentration of the chemical signal. The coefficient $\\chi>0$ quantifies the signal-dependent strength of chemotactic sensitivity.\n\nTo study the relationship between the macroscopic system \\eqref{pde} and its microscopic system \\eqref{sde}, \nwe need to introduce an intermediate particle system (Mean-Field equation). A key object in the mean-field limit is the empirical measure, a random probability measure defined as\n$$\\mu^\\varepsilon_N(t)=\\frac{1}{N}\\sum_{i=1}^N\\delta_{X_{N,i}^\\varepsilon(t)},\\quad t>0,$$\nwhere $\\delta$ is the Dirac delta distribution. The work \\cite{olivera2021quantitative} shows that $\\mu^\\varepsilon_N(t)$ converges to the following PDE solution $u^\\varepsilon$. For fixed $\\varepsilon>0$, the particle system \\eqref{sde} propagates chaos in the many-particle limit $N\\to\\infty$ towards the non-linear SDE system:\n\\begin{align}\\label{mfs}\n\\begin{cases}\nd\\overline{X}_i^\\varepsilon(t) = \\big(2\\exp\\big(-\\Phi^\\varepsilon * u^\\varepsilon( \\overline{X}_i^\\varepsilon,t)\\big)+2\\big)^{1/2} dB_i(t),\\\\\n\\overline{X}_i^\\varepsilon(0) = \\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}\nwhere\n$u^\\varepsilon(\\cdot,t)$ is the density function of i.i.d. random processes\n$\\overline{X}_1^\\varepsilon(t),\\cdots, \\overline{X}_N^\\varepsilon(t)$. \nAnd the initial data $\\{\\zeta_i\\}_{i=1}^N$\nare subject to the same conditions as in \\eqref{sde}. Using It\\^{o}'s formula, the density function $u^\\varepsilon$ satisfies the so-called intermediate nonlocal problem, namely,\n\\begin{align}\n\\begin{cases}\\label{rpde}\n\\partial_t u^\\varepsilon = \\Delta(e^{-v^\\varepsilon}u^\\varepsilon+ u^\\varepsilon),\\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\n-\\Delta v^\\varepsilon + v^\\varepsilon =\\chi u^\\varepsilon * j^\\varepsilon,\\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\nu^\\varepsilon(x,0) = u_0*j^\\varepsilon(x), \\qquad &x\\in\\mathbb{R}^2.\n\\end{cases}\n\\end{align}\n\nIn this paper, our argument relies on the Lemma \\ref{weak solution} from \\cite{BOL2026113712}, which establishes the existence and uniqueness of the weak solution $u^\\varepsilon$ to the above system \\eqref{rpde} and provides the $L^\\infty$ estimate for $\\nabla\\log u^\\varepsilon$.\n\\begin{lemma}\\label{weak solution}\n\\textup{(Existence of weak solution, \\cite[Theorem 2, Lemma 16, and Lemma 18]{BOL2026113712})}.\nLet $u_0\\ge0$ be a initial probability density that satisfies\n\\begin{eqnarray*}\n&u_0\\log u_0\\in L^1(\\mathbb{R}^2), \\quad \\nabla\\log u_0\\in W^{1,q}(\\mathbb{R}^2)~(q>2),\\\\\n&u_0\\in L^1(\\mathbb{R}^2,|x|^2dx)\\cap L^p(\\mathbb{R}^2)~(1\\le p\\le \\infty). \n\\end{eqnarray*}\nFurthermore, assume $\\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo-Nirenberg inequality: $\\|\\omega\\|_{L^4(\\mathbb{R}^2)}^4\\le c_*\\|\\omega\\|_{L^2(\\mathbb{R}^2)}^2\\|\\nabla\\omega\\|_{L^2(\\mathbb{R}^2)}^2$. Then for any $T>0$ and $t\\in[0,T]$, the problem \\eqref{rpde} possesses weak solutions $(u^\\varepsilon, v^\\varepsilon)$ in $\\mathbb{R}^2\\times(0,T)$, and there exists a time $T^*\\in(0,T)$ such that \n\\begin{eqnarray}\n&\\|u^\\varepsilon\\|_{{L^2(0,T;H^1(\\mathbb{R}^2))}\\cap L^\\infty(0,T;L^p(\\mathbb{R}^2))}\\le C,\\label{uniformweaksolution}\\\\\n&\\||x|^2u^\\varepsilon\\|_{L^\\infty(0, T; L^1(\\mathbb{R}^2))}\\le C,\\quad\\|\\nabla\\log u^\\varepsilon\\|_{L^\\infty(0, T^*; W^{1,q}(\\mathbb{R}^2))}\\le C.\\label{uniformweaksolution1}\n\\end{eqnarray}\nwhere $C$ is a constant independent of $\\varepsilon$. \n\\end{lemma}\n\nUsing the above lemma, we obtain our first main result: convergence in probability of the mean-field limit under algebraic scaling.", + "full_context": "In this work, we provide a rigorous derivation of the two-dimensional Keller–Segel-type system with signal-dependent sensitivity expressed as\n\\begin{align}\n\\begin{cases}\\label{pde}\n\\partial_t u = \\Delta(e^{-v}u+ u), \\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\n-\\Delta v + v =\\chi u, \\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\nu(x,0) = u_0(x), \\qquad &x\\in\\mathbb{R}^2,\n\\end{cases}\n\\end{align}\nwhere $u(x,t)$ denotes cell density and $v(x,t)$ represents the concentration of the chemical signal. The coefficient $\\chi>0$ quantifies the signal-dependent strength of chemotactic sensitivity.\n\nTo study the relationship between the macroscopic system \\eqref{pde} and its microscopic system \\eqref{sde}, \nwe need to introduce an intermediate particle system (Mean-Field equation). A key object in the mean-field limit is the empirical measure, a random probability measure defined as\n$$\\mu^\\varepsilon_N(t)=\\frac{1}{N}\\sum_{i=1}^N\\delta_{X_{N,i}^\\varepsilon(t)},\\quad t>0,$$\nwhere $\\delta$ is the Dirac delta distribution. The work \\cite{olivera2021quantitative} shows that $\\mu^\\varepsilon_N(t)$ converges to the following PDE solution $u^\\varepsilon$. For fixed $\\varepsilon>0$, the particle system \\eqref{sde} propagates chaos in the many-particle limit $N\\to\\infty$ towards the non-linear SDE system:\n\\begin{align}\\label{mfs}\n\\begin{cases}\nd\\overline{X}_i^\\varepsilon(t) = \\big(2\\exp\\big(-\\Phi^\\varepsilon * u^\\varepsilon( \\overline{X}_i^\\varepsilon,t)\\big)+2\\big)^{1/2} dB_i(t),\\\\\n\\overline{X}_i^\\varepsilon(0) = \\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}\nwhere\n$u^\\varepsilon(\\cdot,t)$ is the density function of i.i.d. random processes\n$\\overline{X}_1^\\varepsilon(t),\\cdots, \\overline{X}_N^\\varepsilon(t)$. \nAnd the initial data $\\{\\zeta_i\\}_{i=1}^N$\nare subject to the same conditions as in \\eqref{sde}. Using It\\^{o}'s formula, the density function $u^\\varepsilon$ satisfies the so-called intermediate nonlocal problem, namely,\n\\begin{align}\n\\begin{cases}\\label{rpde}\n\\partial_t u^\\varepsilon = \\Delta(e^{-v^\\varepsilon}u^\\varepsilon+ u^\\varepsilon),\\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\n-\\Delta v^\\varepsilon + v^\\varepsilon =\\chi u^\\varepsilon * j^\\varepsilon,\\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\nu^\\varepsilon(x,0) = u_0*j^\\varepsilon(x), \\qquad &x\\in\\mathbb{R}^2.\n\\end{cases}\n\\end{align}\n\nIn this paper, our argument relies on the Lemma \\ref{weak solution} from \\cite{BOL2026113712}, which establishes the existence and uniqueness of the weak solution $u^\\varepsilon$ to the above system \\eqref{rpde} and provides the $L^\\infty$ estimate for $\\nabla\\log u^\\varepsilon$.\n\\begin{lemma}\\label{weak solution}\n\\textup{(Existence of weak solution, \\cite[Theorem 2, Lemma 16, and Lemma 18]{BOL2026113712})}.\nLet $u_0\\ge0$ be a initial probability density that satisfies\n\\begin{eqnarray*}\n&u_0\\log u_0\\in L^1(\\mathbb{R}^2), \\quad \\nabla\\log u_0\\in W^{1,q}(\\mathbb{R}^2)~(q>2),\\\\\n&u_0\\in L^1(\\mathbb{R}^2,|x|^2dx)\\cap L^p(\\mathbb{R}^2)~(1\\le p\\le \\infty). \n\\end{eqnarray*}\nFurthermore, assume $\\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo-Nirenberg inequality: $\\|\\omega\\|_{L^4(\\mathbb{R}^2)}^4\\le c_*\\|\\omega\\|_{L^2(\\mathbb{R}^2)}^2\\|\\nabla\\omega\\|_{L^2(\\mathbb{R}^2)}^2$. Then for any $T>0$ and $t\\in[0,T]$, the problem \\eqref{rpde} possesses weak solutions $(u^\\varepsilon, v^\\varepsilon)$ in $\\mathbb{R}^2\\times(0,T)$, and there exists a time $T^*\\in(0,T)$ such that \n\\begin{eqnarray}\n&\\|u^\\varepsilon\\|_{{L^2(0,T;H^1(\\mathbb{R}^2))}\\cap L^\\infty(0,T;L^p(\\mathbb{R}^2))}\\le C,\\label{uniformweaksolution}\\\\\n&\\||x|^2u^\\varepsilon\\|_{L^\\infty(0, T; L^1(\\mathbb{R}^2))}\\le C,\\quad\\|\\nabla\\log u^\\varepsilon\\|_{L^\\infty(0, T^*; W^{1,q}(\\mathbb{R}^2))}\\le C.\\label{uniformweaksolution1}\n\\end{eqnarray}\nwhere $C$ is a constant independent of $\\varepsilon$. \n\\end{lemma}\n\nUsing the above lemma, we obtain our first main result: convergence in probability of the mean-field limit under algebraic scaling.\n\nUsing the above lemma, we obtain our first main result: convergence in probability of the mean-field limit under algebraic scaling.\n\nThe proof of Theorem \\ref{propagation of chaos}, given in Section 2, follows the approach of \\cite{lazarovici2017mean}, adopting a superior algebraic scaling with respect to the maximum norm of the trajectory, which \\cite{BOL2026113712} only achieved results under logaritheoremic scaling. \nSince the diffusion coefficients in the stochastic models \\eqref{sde} and \\eqref{mfs} given the nonlinear dependence of the diffusion coefficients on inter-particle interactions, the use of the Burkholder-Davis-Gundy inequality becomes indispensable. The key ingredient of our proof is the introduction of stopping times, a technique that facilitates the application of the Law of Large Numbers.\n\nTherefore, plugging \\eqref{I12} and \\eqref{I11} into \\eqref{EI1i} gives\n\\begin{align}\\label{I1i}\n&\\E\\Big(N^{2\\alpha }\\int_0^{t\\land\\tau_\\alpha}\\max_{i=1,\\cdots,N}|I_{1,i}(s)|^2 ds\\Big)\\nonumber\\\\\n\\le\\ & C(T) (1+\\varepsilon^{-6}N^{-2\\alpha })\\int_0^{t\\land\\tau_\\alpha}\\E(S_\\alpha^2(s))ds + C(m, T)\\varepsilon^{-4m-4}N^{-m+1}.\n\\end{align}\nSubstituting \\eqref{I2i} and \\eqref{I1i} into \\eqref{ES} yields the inequality\n\\begin{align}\\label{initial ineq}\n\\E(S_\\alpha^2(t))\n\\le\\ & C(T) (1+\\varepsilon^{-6}N^{-2\\alpha })\\int_0^{t\\land\\tau_\\alpha}\\E(S_\\alpha^2(s))ds \\nonumber\\\\\n&+ C(m, T)(\\varepsilon^{-4m-4}N^{-m+1} +N^{(2\\alpha-\\theta)} + \\varepsilon^{-4m-4}N^{2\\alpha +m(2\\theta -1)+1}).\n\\end{align}\nWe now show that the terms multiplying the constants decay as a negative power of $N$. Recall the scaling $\\varepsilon\\sim N^{-\\gamma},\\gamma>0$. To bound the error terms, we require a positive exponent\n$\\eta>0$ such that the following bounds hold\n\\begin{align*}\n\\varepsilon^{-6}N^{-2\\alpha } = N^{6\\gamma -2\\alpha}\\le N^{0}&\\Longleftrightarrow\\gamma\\le\\frac{\\alpha}{3},\\\\\nN^{2\\alpha-\\theta}\\le N^{-\\eta}&\\Longleftrightarrow\\eta\\le\\theta-2\\alpha, \\\\\n\\varepsilon^{-4m-4}N^{-m+1} = N^{(4m+4)\\gamma -m+1}\\le N^{-\\eta}\n&\\Longleftrightarrow\\eta\\le-(4m+4)\\gamma+m-1,\\\\\n\\varepsilon^{-4m-4}N^{2\\alpha +m(2\\theta -1)+1}=N^{(4m+4)\\gamma+2\\alpha +m(2\\theta -1)+1}\\le N^{-\\eta}&\\Longleftrightarrow\\eta\\le-(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1.\n\\end{align*}\nThe condition $\\eta>0$ implies that $\\theta$, $\\alpha$, $m$, and $\\gamma$ need to be selected to meet the following relationships\n\\begin{align*}\n0<\\theta<\\frac{1}{2},\\quad 0<\\alpha<\\frac{\\theta}{2},\\quad m>\\frac{\\theta+1}{1-2\\theta},\\quad 0<\\gamma<\\frac{-2\\alpha+m(1-2\\theta)-1}{4m+4}. \n\\end{align*}\nThus choose $\\gamma$ satisfying\n$$\n0<\\gamma<\\min\\big\\{\\frac{\\alpha}{3},\\frac{-2\\alpha +m(1-2\\theta)-1}{4m+4}\\big\\},\n$$\nand $\\eta$ satisfying \n$$\n0<\\eta\\le\\min\\{\\theta-2\\alpha, -(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}.\n$$ \nThen the inequality \\eqref{initial ineq} simplifies the estimate\n\\begin{align}\\label{Sak}\n\\E(S_\\alpha^2(t))\n\\le\\ & C\\int_0^{t\\land\\tau_\\alpha}\\E(S_\\alpha^2(s))ds + C(m, T)N^{-\\eta}.\n\\end{align}\nUsing Gr\\\"{o}nwall's inequality for \\eqref{Sak}, we obtain for any $t\\in[0,T]$\n\\begin{align} \\label{Egronwall}\n\\E(S_\\alpha^2(t))\\le C(m,T)N^{-\\eta} (1+CTe^{CT}). \n\\end{align}\nBy inserting \\eqref{MK} into \\eqref{Egronwall}, we derive \n\\begin{align}\n\\mathbb{P}\\big(\\max_{i=1,\\cdots,N}\\big|(X_{N,i}^\\varepsilon - \\overline{X}_i^\\varepsilon)(t)\\big|> N^{-\\alpha}\\big)\\le C(m,T)N^{-\\eta} (1+CTe^{CT}).\n\\end{align}\nThe proof of this theorem is completed.\n\n\\setcounter{theorem}{1}\n\\begin{theorem}\\label{Propagation of chaos in the strong sense}\\textup{(Propagation of chaos in the strong sense).} \nUnder the assumptions of Lemma \\ref{weak solution}, let \n$u_{N}^\\varepsilon(t,x_1,\\cdots,x_N)$ and $u^{\\varepsilon\\otimes N}(t,x_1,\\cdots,x_N)$ be the solutions of systems \\eqref{lpde1} and \\eqref{lpde2}, respectively. For any \n$r\\in\\mathbb{N^+}$, denote by $u_{N,r}^\\varepsilon(t,x_1,\\cdots,x_r)$ and $u^{\\varepsilon\\otimes r}(t,x_1,\\cdots,x_r)$ their corresponding \n$r$-th marginal distributions. \nThen for parameters $0<\\theta<1/2$, $0 <\\alpha<\\frac{\\theta}{2}$, and $m\\in \\mathbb{N}^+$ satisfying $m \\ge \\frac{1+\\theta}{1-2\\theta}$, there exist a time $T^*\\in(0,T)$, a constant $C(m,r,T)>0$, and a parameter $\\beta>0$ satisfying the bound $1<\\beta\\le\\min\\{\\frac{2\\alpha}{\\gamma}-6,\\frac{\\eta}{\\gamma}-4\\}$ such that \n\\begin{align}\\label{Ps}\n\\|u_{N,r}^{\\varepsilon}(t) - u^{\\varepsilon\\otimes r}(t)\\|_{L^\\infty(0,T^*;L^1(\\R^{2r}))} \\le C(r,m,T)\\varepsilon^{\\beta}, \n \\end{align}\nwhere the cut-off parameter satisfies\n$\\varepsilon\\sim N^{-\\gamma} $,\nand the parameter $\\gamma$ defined by\n$$0<\\gamma<\\min\\{\\frac{2\\alpha}{7},\\frac{-2\\alpha + m(1-2\\theta)-1}{4m+4}\\},$$\nand the parameter $\\eta$ satisfies\n$$5\\gamma<\\eta<\\min\\{\\theta-2\\alpha,-(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}.$$ \n\\end{theorem}\n\nPlugging \\eqref{J1} and \\eqref{J2} into \\eqref{K21} produces the following\n\\begin{align}\nK_2\\leq &-\\int_{\\R^{2N}}\\frac{1}{N}\\sum_{i=1}^Nu_N^\\varepsilon\\Big[\\exp\\big(-\\frac{1}{N}\\sum_{j=1}^N\\Phi^\\varepsilon(x_i - x_j)\\big) \\Big|\\nabla_{x_i}\\log\\frac{u_N^\\varepsilon}{u^{\\varepsilon\\otimes N}}\\Big|^2\\Big]d_{x_1}\n\\cdots d_{x_N}\\nonumber\\\\\n&+ C(m, T)(N^{-2\\alpha}\\varepsilon^{-6} + (N^{-1} + N^{-\\eta})\\varepsilon^{-4}).\\label{K_2}\n\\end{align}\nCombining \\eqref{K1} and \\eqref{H1}, we derive\n\\begin{align}\n\\frac{d}{dt}\\mathcal{H}(u_N^\\varepsilon|u^{\\varepsilon\\otimes N})\n\\leq& -\\frac{1}{2}\\int_{\\R^{2N}}\\frac{1}{N}\\sum_{i=1}^Nu_N^\\varepsilon\\Big|\\nabla_{x_i}\\log\\frac{u_N^\\varepsilon}{u^{\\varepsilon\\otimes N}}\\Big|^2d_{x_1}\\cdots d_{x_N}\\nonumber\\\\\n&-\\frac{1}{2}\\int_{\\R^{2N}}\\frac{1}{N}\\sum_{i=1}^Nu_N^\\varepsilon\\Big[\\exp\\big(-\\frac{1}{N}\\sum_{j=1}^N\\Phi^\\varepsilon(x_i - x_j)\\big) \\Big|\\nabla_{x_i}\\log\\frac{u_N^\\varepsilon}{u^{\\varepsilon\\otimes N}}\\Big|^2\\Big]d_{x_1}\n\\cdots d_{x_N}\\nonumber\\\\\n& + C(m, T)(N^{-2\\alpha}\\varepsilon^{-6} + (N^{-1} + N^{-\\eta})\\varepsilon^{-4}).\\label{entropy}\n\\end{align}\nNoticing that the initial relative entropy is zero, it can be obtained from (\\ref{entropy}) that for any $00$ such that \n\\begin{align}\nN^{-2\\alpha}\\varepsilon^{-6} + (N^{-1} + N^{- \\eta})\\varepsilon^{-4}=\\varepsilon^\\frac{2\\alpha}{\\gamma}\\varepsilon^{-6}+(\\varepsilon^\\frac{1}{\\gamma}+\\varepsilon^\\frac{\\eta}{\\gamma})\\varepsilon^{-4}\\le \\varepsilon^\\beta, \n\\end{align}\nwhere $\\beta$ satisfies\n$$1<\\beta\\le\\min\\{\\frac{2\\alpha}{\\gamma}-6,\\frac{\\eta}{\\gamma}-4\\}.$$ \nHere we have used $\\eta<1$.\nThis further requires that $0<\\gamma<\\frac{2\\alpha}{7}$, $\\eta>5\\gamma$. Building on the assumptions in \\eqref{eta1} and \\eqref{gamma1}, we assume\n\\begin{align*}\n&0<\\gamma<\\min\\{\\frac{2\\alpha}{7},\\frac{-2\\alpha + m(1-2\\theta)-1}{4m+4}\\},\\\\\n&5\\gamma<\\eta<\\min\\{\\theta-2\\alpha,-(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}.\n\\end{align*} \nThen we obtain that for $01.\n\\end{align}\nThus, we complete the proof of propagation of chaos. \n\\end{proof}", + "post_theorem_intro_text_len": 5262, + "post_theorem_intro_text": "\\begin{remark}\nTo see the value of $\\gamma$ and $\\eta$ more clearly, we provide a special case. When $m\\in\\mathbb{N}^+$, $ m\\ge\\frac{1+\\theta}{1-2\\theta}$ and $0<\\alpha<\\frac{\\theta}{2}<\\frac{1}{2}$, we take $\\alpha=0.1$, $\\theta=0.3$ and $m=4$ such that\n$0<\\gamma<\\min\\{\\frac{\\alpha}{3},\\frac{-2\\alpha +m(1-2\\theta)-1}{2(2+2m)}\\}=\\min\\{0.033,0.02\\}=0.02$. we may assume that $\\gamma=0.019$, then $0<\\eta\\le\\min\\{\\theta-2\\alpha,-2\\gamma(2+2m)-2\\alpha -m(2\\theta -1)-1\\}=\\min\\{0.1, 0.02\\}=0.02$. \n\\end{remark}\n\nThe proof of Theorem \\ref{propagation of chaos}, given in Section 2, follows the approach of \\cite{lazarovici2017mean}, adopting a superior algebraic scaling with respect to the maximum norm of the trajectory, which \\cite{BOL2026113712} only achieved results under logaritheoremic scaling. \nSince the diffusion coefficients in the stochastic models \\eqref{sde} and \\eqref{mfs} given the nonlinear dependence of the diffusion coefficients on inter-particle interactions, the use of the Burkholder-Davis-Gundy inequality becomes indispensable. The key ingredient of our proof is the introduction of stopping times, a technique that facilitates the application of the Law of Large Numbers. \n\nBased on the uniform $L^\\infty(0,T^*;H^2(\\mathbb{R}^2))$ bound for $\\nabla\\log u_\\varepsilon$ in Lemma \\ref{weak solution} and the propagation of chaos established in Theorem \\ref{propagation of chaos}, our next main result addresses the strong $L^1$ convergence for the propagation of chaos. The proof is mainly presented in Section 3, employing the relative entropy method \\cite{jabin2018quantitative} as recently refined in \\cite{chen2025quantitative}.\n\n\\begin{theorem}\\label{Propagation of chaos in the strong sense}\\textup{(Propagation of chaos in the strong sense).} \nUnder the assumptions of Theorem \\ref{propagation of chaos}, let $r\\in \\mathbb{N^+}$, $u_{N,r}^\\varepsilon(t, x_1,\\cdots, x_r)$ be the $r$-th marginal density of the joint density $u_N^\\varepsilon(t, x_1,\\cdots, x_N)$ of $\\{X_{N,i}^\\varepsilon\\}_{1\\le i\\le N}$, and $u^{\\varepsilon \\otimes r}(t, x_1,\\cdots, x_r)$ be the tensor product of the solutions $u^{\\varepsilon}$ to the model \\eqref{rpde}. Then for parameters $0<\\theta<1/2$, $0 <\\alpha<\\frac{\\theta}{2}$, and $m\\in \\mathbb{N}^+$ satisfying $m \\ge \\frac{1+\\theta}{1-2\\theta}$, there exist a time $T^*\\in(0,T)$, a constant $C(m,r,T)>0$, and a parameter $\\beta>0$ satisfying the bound $1<\\beta\\le\\min\\{\\frac{2\\alpha}{\\gamma}-6,\\frac{\\eta}{\\gamma}-4\\}$ such that \n\\begin{align}\\label{Ps111}\n\\|u_{N,r}^{\\varepsilon}(t) - u^{\\varepsilon\\otimes r}(t)\\|_{L^\\infty(0,T^*;L^1(\\mathbb{R}^{2r}))} \\le C(r,m,T)\\varepsilon^{\\beta}, \n \\end{align}\nwhere the cut-off parameter satisfies\n$\\varepsilon\\sim N^{-\\gamma} $,\nand the parameter $\\gamma$ defined by\n$$0<\\gamma<\\min\\{\\frac{2\\alpha}{7},\\frac{-2\\alpha + m(1-2\\theta)-1}{4m+4}\\},$$\nand the parameter $\\eta$ satisfies\n$$5\\gamma<\\eta<\\min\\{\\theta-2\\alpha,-(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}.$$ \n\\end{theorem}\n\\begin{remark}\nTo illustrate the feasible range of parameters, we give an explicit example. Choose $\\theta = 0.4,\\ \\alpha = 0.1$ and $m\\ge \\frac{1+\\theta}{1-2\\theta}=7$ such that $m=7$.\nThen we obtain\n$$0<\\gamma<\\min\\{\\frac{2\\alpha}{7},\\frac{-2\\alpha + m(1-2\\theta)-1}{4m+4}\\}=\\min\\{0.029,0.006\\}=0.006.$$\nTaking $\\gamma=0.0025$ gives\n$$0.0125=5\\gamma<\\eta<\\min\\{\\theta-2\\alpha,-(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}=\\min\\{0.2,0.12\\}=0.12.$$\nso we pick $\\eta=0.03$. Finally, $$1<\\beta\\le\\min\\{\\frac{2\\alpha}{\\gamma}-6,\\frac{\\eta}{\\gamma}-4\\}=\\min\\{74,8\\}=8$$\nwe take $\\beta = 8$.\nWith these choices, the propagation of chaos estimate becomes\n\\begin{align*}\n\\max_{i=1,\\cdots,N}\\mathbb{E}\\big(\\sup_{t\\in[0,T]}\\big|(X_{N,i}^\\varepsilon - \\overline{X}_i^\\varepsilon)(t)\\big|^2\\big)\\le C\\varepsilon^8, \n\\end{align*} \n\\end{remark}\nWe comment here that the convergence of the mean-field limit have already been discussed in \\cite{BOL2026113712}. It is therefore especially gratifying that our results accelerate the convergence rates for the solutions to the SDE \\eqref{sde} and \\eqref{mfs} across different norms. Specifically, Chen et al. in \\cite[Theorem 3]{BOL2026113712} presented as\n\\begin{align*}\n\\max_{i=1,\\cdots,N}\\mathbb{E}\\big(\\sup_{t\\in[0,T]}\\big|(X_{N,i}^\\varepsilon - \\overline{X}_i^\\varepsilon)(t)\\big|^2\\big)\\le C\\varepsilon^2, \n\\end{align*} \nwhere the cut-off parameter $\\varepsilon=(\\lambda\\log N)^{-\\frac{1}{4}}$. However, in \\eqref{Pe}, we prove algebraic convergence in contrast to logarithmic convergence established in \\cite{BOL2026113712}. This represents an improvement in the convergence rate. Furthermore, in \\cite[Theorem 5]{BOL2026113712}, they obtained\n\\begin{align*}\n\\|u_{N,r}^{\\varepsilon}(t) - u^{\\varepsilon\\otimes r}(t)\\|_{L^\\infty(0,T^*;L^1(\\mathbb{R}^{2r}))}^2 \\le C(r,m,T)\\varepsilon.\n\\end{align*}\nBy contrast, our estimate in \\eqref{Ps111} yields a faster convergence rate than that bound.\n\nThe article is organized as follows. In section 2, we establish the propagation of chaos, which corresponds to the convergence in probability of solutions to the stochastic differential equations \\eqref{sde} and \\eqref{mfs}. Section 3 further derive quantitative propagation of chaos result in the strong sense by applying the relative entropy method.", + "sketch": "The proof of Theorem~\\ref{propagation of chaos} (given in Section 2) “follows the approach of \\cite{lazarovici2017mean},” and “adopt[s] a superior algebraic scaling with respect to the maximum norm of the trajectory” (contrasting with \\cite{BOL2026113712}, which “only achieved results under logarithmic scaling”). Because “the diffusion coefficients … [have] the nonlinear dependence … on inter-particle interactions,” the argument says “the use of the Burkholder-Davis-Gundy inequality becomes indispensable.” The “key ingredient” is “the introduction of stopping times,” which “facilitates the application of the Law of Large Numbers.”", + "expanded_sketch": "The proof (given later) “follows the approach of Lazarovici and Pickl, \\emph{A mean field limit for the Vlasov--Poisson system} (2017),” and “adopt[s] a superior algebraic scaling with respect to the maximum norm of the trajectory” (contrasting with \\cite{BOL2026113712}, which “only achieved results under logarithmic scaling”). Because “the diffusion coefficients … [have] the nonlinear dependence … on inter-particle interactions,” the argument says “the use of the Burkholder-Davis-Gundy inequality becomes indispensable.” The “key ingredient” is “the introduction of stopping times,” which “facilitates the application of the Law of Large Numbers.”", + "expanded_theorem": "\\label{propagation of chaos}\nThe assumptions of the following lemma still hold.\n\\begin{lemma}\\label{weak solution}\n\\textup{(Existence of weak solution, \\cite[Theorem 2, Lemma 16, and Lemma 18]{BOL2026113712})}.\nLet $u_0\\ge0$ be a initial probability density that satisfies\n\\begin{eqnarray*}\n&u_0\\log u_0\\in L^1(\\R^2), \\quad \\nabla\\log u_0\\in W^{1,q}(\\R^2)~(q>2),\\\\\n&u_0\\in L^1(\\R^2,|x|^2dx)\\cap L^p(\\R^2)~(1\\le p\\le \\infty). \n\\end{eqnarray*}\nFurthermore, assume $\\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo-Nirenberg inequality: $\\|\\omega\\|_{L^4(\\R^2)}^4\\le c_*\\|\\omega\\|_{L^2(\\R^2)}^2\\|\\nabla\\omega\\|_{L^2(\\R^2)}^2$. Then for any $T>0$ and $t\\in[0,T]$, the problem \\eqref{rpde} possesses weak solutions $(u^\\varepsilon, v^\\varepsilon)$ in $\\R^2\\times(0,T)$, and there exists a time $T^*\\in(0,T)$ such that \n\\begin{eqnarray}\n&\\|u^\\varepsilon\\|_{{L^2(0,T;H^1(\\R^2))}\\cap L^\\infty(0,T;L^p(\\R^2))}\\le C,\\label{uniformweaksolution}\\\\\n&\\||x|^2u^\\varepsilon\\|_{L^\\infty(0, T; L^1(\\R^2))}\\le C,\\quad\\|\\nabla\\log u^\\varepsilon\\|_{L^\\infty(0, T^*; W^{1,q}(\\R^2))}\\le C.\\label{uniformweaksolution1}\n\\end{eqnarray}\nwhere $C$ is a constant independent of $\\varepsilon$. \n\\end{lemma}\nLet $\\{X_{N,i}^\\varepsilon\\}_{1\\le i\\le N}$ and $\\{\\overline{X}_i^\\varepsilon\\}_{1\\le i\\le N}$ be the solutions to systems\n\\begin{align}\\label{sde}\n\\begin{cases}\ndX_{N,i}^\\varepsilon(t) = \\Big(2\\exp\\Big(-\\frac{1}{N}\\sum_{j=1}^N\\Phi^\\varepsilon(X_{N,i}^\\varepsilon(t) - X_{N,j}^\\varepsilon(t))\\Big)+ 2\\Big)^{1/2}dB_i(t),\\\\\nX_{N,i}^\\varepsilon(0) =\\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}\nand\n\\begin{align}\\label{mfs}\n\\begin{cases}\nd\\overline{X}_i^\\varepsilon(t) = \\big(2\\exp\\big(-\\Phi^\\varepsilon * u^\\varepsilon( \\overline{X}_i^\\varepsilon,t)\\big)+2\\big)^{1/2} dB_i(t),\\\\\n\\overline{X}_i^\\varepsilon(0) = \\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}\nrespectively. Then \nfor any $m,N\\in \\mathbb{N}^+$ and parameters $0<\\theta<\\frac{1}{2},\\quad 0<\\alpha<\\frac{\\theta}{2},\\quad m>\\frac{\\theta+1}{1-2\\theta}$, there exist a constant $C(m,T)>0$ and a parameter $\\gamma,\\eta>0$ satisfying the bounds\n\\begin{align}\n&0<\\gamma<\\min\\big\\{\\frac{\\alpha}{3},\\frac{-2\\alpha +m(1-2\\theta)-1}{4m+4}\\big\\},\\label{gamma} \\\\\n&0<\\eta\\le\\min\\{\\theta-2\\alpha, -(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}\\label{eta}\n\\end{align}\nsuch that for all $0\\le t\\le T$,\n\\begin{align}\\label{Pe}\n\\mathbb{P}\\big(\\max_{i=1,\\cdots,N}\\big|(X_{N,i}^\\varepsilon - \\overline{X}_i^\\varepsilon)(t)\\big|> N^{-\\alpha}\\big)\\le C(m,T)N^{-\\eta}, \n\\end{align} \nwhere the cut-off parameter satisfies\n$\\varepsilon\\sim N^{-\\gamma} $.", + "theorem_type": [ + "Existential–Universal", + "Inequality or Bound" + ], + "mcq": { + "question": "Let $u_0\\ge 0$ be an initial probability density on $\\mathbb{R}^2$ such that\n$$u_0\\log u_0\\in L^1(\\mathbb{R}^2),\\qquad \\nabla\\log u_0\\in W^{1,q}(\\mathbb{R}^2)\\ \\text{for some }q>2,$$\nand\n$$u_0\\in L^1(\\mathbb{R}^2,|x|^2dx)\\cap L^p(\\mathbb{R}^2)\\qquad (1\\le p\\le \\infty).$$\nAssume also that $\\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo--Nirenberg inequality\n$$\\|\\omega\\|_{L^4(\\mathbb{R}^2)}^4\\le c_*\\|\\omega\\|_{L^2(\\mathbb{R}^2)}^2\\|\\nabla\\omega\\|_{L^2(\\mathbb{R}^2)}^2.$$ \nLet $(u^\\varepsilon,v^\\varepsilon)$ be the weak solution of the intermediate nonlocal problem\n$$\\partial_t u^\\varepsilon=\\Delta(e^{-v^\\varepsilon}u^\\varepsilon+u^\\varepsilon),\\qquad -\\Delta v^\\varepsilon+v^\\varepsilon=\\chi \\,u^\\varepsilon * j^\\varepsilon,\\qquad u^\\varepsilon(\\cdot,0)=u_0*j^\\varepsilon,$$\non $\\mathbb{R}^2\\times(0,T)$, where $*$ denotes convolution.\nFor $1\\le i\\le N$, let $X_{N,i}^\\varepsilon$ and $\\overline X_i^\\varepsilon$ solve, with the same Brownian motions $B_i$ and the same initial data $\\zeta_i$,\n$$dX_{N,i}^\\varepsilon(t)=\\Big(2\\exp\\Big(-\\frac1N\\sum_{j=1}^N\\Phi^\\varepsilon(X_{N,i}^\\varepsilon(t)-X_{N,j}^\\varepsilon(t))\\Big)+2\\Big)^{1/2}dB_i(t),$$\n$$d\\overline X_i^\\varepsilon(t)=\\Big(2\\exp\\big(-\\Phi^\\varepsilon*u^\\varepsilon(\\overline X_i^\\varepsilon,t)\\big)+2\\Big)^{1/2}dB_i(t).$$\nFix $m,N\\in\\mathbb N^+$ and parameters $0<\\theta<\\tfrac12$, $0<\\alpha<\\tfrac\\theta2$, and $m>\\frac{\\theta+1}{1-2\\theta}$. Under these assumptions, which quantitative estimate holds for the probability that the interacting particle system and the mean-field system differ by more than $N^{-\\alpha}$ in maximum norm at time $t$?", + "correct_choice": { + "label": "A", + "text": "There exist a constant $C(m,T)>0$ and parameters $\\gamma,\\eta>0$ such that\n$$0<\\gamma<\\min\\Big\\{\\frac\\alpha3,\\frac{-2\\alpha+m(1-2\\theta)-1}{4m+4}\\Big\\},$$\n$$0<\\eta\\le \\min\\Big\\{\\theta-2\\alpha,\n-(4m+4)\\gamma-2\\alpha+m(1-2\\theta)-1\\Big\\},$$\nand, if the cutoff parameter satisfies $\\varepsilon\\sim N^{-\\gamma}$, then for every $t\\in[0,T]$,\n$$\\mathbb P\\Big(\\max_{i=1,\\dots,N}|X_{N,i}^\\varepsilon(t)-\\overline X_i^\\varepsilon(t)|>N^{-\\alpha}\\Big)\\le C(m,T)N^{-\\eta}.$$" + }, + "choices": [ + { + "label": "B", + "text": "There exist a constant $C(m,T)>0$ and parameters $\\gamma,\\eta>0$ such that\n$$0<\\gamma<\\min\\Big\\{\\frac\\alpha3,\\frac{-2\\alpha+m(1-2\\theta)-1}{4m+4}\\Big\\},$$\n$$0<\\eta\\le \\min\\Big\\{\\theta-2\\alpha,\n-(4m+4)\\gamma-2\\alpha+m(1-2\\theta)-1\\Big\\},$$\nand, if the cutoff parameter satisfies $\\varepsilon\\sim N^{-\\gamma}$, then for every $t\\in[0,T]$,\n$$\\mathbb P\\Big(\\max_{i=1,\\dots,N}|X_{N,i}^\\varepsilon(t)-\\overline X_i^\\varepsilon(t)|>N^{-\\alpha}\\Big)\\le C(m,T)N^{-2\\eta}.$$" + }, + { + "label": "C", + "text": "There exist a constant $C(m,T)>0$ and parameters $\\gamma,\\eta>0$ such that\n$$0<\\gamma<\\min\\Big\\{\\frac\\alpha3,\\frac{-2\\alpha+m(1-2\\theta)-1}{4m+4}\\Big\\},$$\n$$0<\\eta\\le \\min\\Big\\{\\theta-2\\alpha,\n-(4m+4)\\gamma-2\\alpha+m(1-2\\theta)-1\\Big\\},$$\nand, if the cutoff parameter satisfies $\\varepsilon\\sim N^{-\\gamma}$, then for every $t\\in[0,T]$,\n$$\\mathbb P\\Big(\\max_{i=1,\\dots,N}|X_{N,i}^\\varepsilon(t)-\\overline X_i^\\varepsilon(t)|>N^{-\\alpha}\\Big)\\le C(m,T).$$" + }, + { + "label": "D", + "text": "For every choice of parameters $\\gamma,\\eta>0$ satisfying\n$$0<\\gamma<\\min\\Big\\{\\frac\\alpha3,\\frac{-2\\alpha+m(1-2\\theta)-1}{4m+4}\\Big\\},$$\n$$0<\\eta\\le \\min\\Big\\{\\theta-2\\alpha,\n-(4m+4)\\gamma-2\\alpha+m(1-2\\theta)-1\\Big\\},$$\nthere exists a constant $C(m,T)>0$ such that, if the cutoff parameter satisfies $\\varepsilon\\sim N^{-\\gamma}$, then for every $t\\in[0,T]$,\n$$\\mathbb P\\Big(\\max_{i=1,\\dots,N}|X_{N,i}^\\varepsilon(t)-\\overline X_i^\\varepsilon(t)|>N^{-\\alpha}\\Big)\\le C(m,T)N^{-\\eta},$$\nwith the same constant $C(m,T)$ working uniformly for all such admissible pairs $(\\gamma,\\eta)$." + }, + { + "label": "E", + "text": "There exist a constant $C(m,T)>0$ and parameters $\\gamma,\\eta>0$ such that\n$$0<\\gamma\\le\\min\\Big\\{\\frac\\alpha3,\\frac{-2\\alpha+m(1-2\\theta)-1}{4m+4}\\Big\\},$$\n$$0<\\eta\\le \\min\\Big\\{\\theta-2\\alpha,\n-(4m+4)\\gamma-2\\alpha+m(1-2\\theta)-1\\Big\\},$$\nand, if the cutoff parameter satisfies $\\varepsilon\\sim N^{-\\gamma}$, then for every $t\\in[0,T]$,\n$$\\mathbb P\\Big(\\max_{i=1,\\dots,N}|X_{N,i}^\\varepsilon(t)-\\overline X_i^\\varepsilon(t)|>N^{-\\alpha}\\Big)\\le C(m,T)N^{-\\eta}.$$" + } + ], + "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": "algebraic decay rate from stopping-time/BDG argument", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped the quantitative decay factor $N^{-\\eta}$", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "dependence of the constant on the admissible pair $(\\gamma,\\eta)$", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "boundary_range", + "tampered_component": "strict upper bound on $\\gamma$ replaced by an attained endpoint", + "template_used": "boundary_range" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not reveal the correct option directly. It gives hypotheses and asks which conclusion holds, without explicitly signaling the exact probability estimate or the precise quantifier structure." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recall question: the correct choice is essentially the exact theorem conclusion under the listed assumptions. It is not a pure restatement in the stem, but it mainly tests recognition of the formal statement rather than deriving a new consequence." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish subtle changes in rates, strict vs. non-strict bounds, pointwise-in-time vs. supremum-in-time estimates, and explicit rate vs. mere convergence. However, this is more precision/recall of a known result than substantial generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong: they are mathematically close to the target statement and differ in realistic ways that reflect common failure modes, such as weakening the conclusion, changing the rate variable, altering endpoint admissibility, or strengthening time uniformity improperly." + }, + "total_score": 6, + "overall_assessment": "A technically strong MCQ with little answer leakage and high-quality distractors, but it leans more toward theorem-statement recognition than genuine generative reasoning." + } + }, + { + "id": "2602.01571v1", + "paper_link": "http://arxiv.org/abs/2602.01571v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "thm", + "content": "\\label{theorem:main}\n Let $f$ be a cuspidal modular form of weight $k$ on $\\SL_2(\\BZ)$. For any integers $l\\geq2$ and $d\\geq1$ and $dl>4$, one has\n \\[\\sum_{n\\leq x} \\lambda^l_{\\Sym^d f}(n)=xP_{d,l}(\\log x)+O\\left(x^{\\theta_{d,l}+\\epsilon}\\right).\\]\n Here $P_{d,l}$ is a polynomial of degree $K_{0,d,l}-1$, where $K_{i,d,l}$ denote the Kostka number defined in sections \\ref{section:Weylmod} and \\ref{section:Kostka number}. The exponent $\\theta_{d, l}$ is given by\n $$\\theta_{d,l}=1-1/\\left((d+1)^l/2-4K_{0,d,l}/21-K_{1,d,l}/3-5K_{2,d,l}/14\\right).$$", + "start_pos": 19116, + "end_pos": 19700, + "label": "theorem:main" + }, + "ref_dict": { + "theorem:main": "\\begin{thm}\\label{theorem:main}\n Let $f$ be a cuspidal modular form of weight $k$ on $\\SL_2(\\BZ)$. For any integers $l\\geq2$ and $d\\geq1$ and $dl>4$, one has\n \\[\\sum_{n\\leq x} \\lambda^l_{\\Sym^d f}(n)=xP_{d,l}(\\log x)+O\\left(x^{\\theta_{d,l}+\\epsilon}\\right).\\]\n Here $P_{d,l}$ is a polynomial of degree $K_{0,d,l}-1$, where $K_{i,d,l}$ denote the Kostka number defined in sections \\ref{section:Weylmod} and \\ref{section:Kostka number}. The exponent $\\theta_{d, l}$ is given by\n $$\\theta_{d,l}=1-1/\\left((d+1)^l/2-4K_{0,d,l}/21-K_{1,d,l}/3-5K_{2,d,l}/14\\right).$$\n\\end{thm}", + "equation:introduction4": "\\begin{equation}\\label{equation:introduction4}\n \\sum_{n\\leq x}\\lambda^2_{\\Sym^d f}(n)=c_dx+O\\left(x^{\\tilde{\\theta}_d+\\epsilon}\\right),\n\\end{equation}", + "equation:introduction3": "\\begin{equation} \\label{equation:introduction3}\n \\sum_{n\\leq x}\\lambda^l_{\\Sym^2 f}(n)=xP_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right)\n\\end{equation}", + "proposition:dirichletseriescomp": "\\begin{prop}\\label{proposition:dirichletseriescomp}\nThe following identities hold:\n\\begin{align*}\n &D\\left(f^{\\otimes l},s\\right)=L\\left(f^{\\otimes l},s\\right)U_{f,l}(s),D\\left(\\left(\\Sym^d f\\right)^{\\otimes l},s\\right)=L\\left(\\left(\\Sym^df\\right)^{\\otimes l},s\\right)U_{\\Sym^d f,l}(s),\\\\ \n &D\\left(f^{\\otimes l},\\chi,s\\right)=L\\left(f^{\\otimes l},\\chi,s\\right)U_{f,l,\\chi}(s),D\\left(\\left(\\Sym^d f\\right)^{\\otimes l},\\chi,s\\right)=L\\left(\\left(\\Sym^df\\right)^{\\otimes l},\\chi,s\\right)U_{\\Sym^d f,l,\\chi}(s).\n\\end{align*}\n where each $U_*(s)$ is an Euler product that is absolutely convergent for $\\Re(s)>1/2$ and uniformly convergent in the region $\\Re(s)>1/2+\\epsilon$ for any $\\epsilon>0$.\n\\end{prop}", + "equation:introduction2": "\\begin{equation}\\label{equation:introduction2}\n \\sum_{n_1^2+n_2^2\\leq x}\\lambda_f^l(n_1^2+n_2^2)=xQ'_l(\\log x)+O\\left(x^{{\\theta}'_l+\\epsilon}\\right),\n\\end{equation}", + "equation:introduction1": "\\begin{equation}\\label{equation:introduction1}\n \\sum_{n\\leq x} \\lambda^l_f(n)=xQ_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right),\n\\end{equation}", + "corollary:Kostka": "\\begin{cor}\\label{corollary:Kostka}\n \\begin{equation}\n K_{i,d,l}=\\begin{cases}\n 0\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\text{if $2\\nmid dl-i$}\\\\\n \\sum\\limits_{j=0}^{\\lfloor(dl-i)/(2d+2)\\rfloor}(-1)^j\\binom{l}{j}\\binom\n {(dl-i)/2-j(d+1)+l-2}{l-2}\n \\ \\text{otherwise}\n \\end{cases}.\n \\end{equation}\n\\end{cor}" + }, + "pre_theorem_intro_text_len": 4707, + "pre_theorem_intro_text": "Let $f$ be a cuspidal eigenform of weight $k$ on $\\SL_2(\\BZ)$, and let\n\\[f(z)=\\sum_{n=1}^\\infty \\lambda_f(n)n^{\\frac{k-1}{2}}q^n\\]\nbe its $q$-expansion. Estimates of $\\lambda_f(n)$ and their average behavior are fundamental problems in number theory. \n Estimating the sum\n$\\sum\\limits_{n\\leq x}\\lambda_f(n),$ has a long history, with contributions from\n \\cite{kloosterman1927asymptotische}, \\cite{rankin1983sums}, \\cite{weil1948some}, \\cite{wu2009power}, among others. The sum of higher power moments \n$\\sum\\limits_{n\\leq x}\\lambda_f^l(n)$\nhas also attracted considerable attention. If $l=2$, Rankin\\cite{rankin1939contributions} and Selberg\\cite{selberg1940bemerkungen} already studied this problem and recently Huang's work \\cite{huang2021rankin} provides improved estimates with\n\\[\\sum_{n\\leq x}\\lambda_f^2(n)=Cx+O\\left(x^{\\frac{3}{5}-\\frac{1}{560}+\\epsilon}\\right).\\]\n\nFor $l\\geq3$, symmetric power $L$-functions of modular forms are required. Kim and Kim \\& Shahidi \\cite{kim2003functoriality}, \\cite{kim2002cuspidality},\\cite{kim2002functorial} showed that $\\Sym^l f$ is automorphic for $l\\leq 4$. Using these results, the estimates for $l\\leq 8$ are derived in \\cite{lau2011integral}, \\cite{lu2009average} and \\cite{lu2011higher}. Later Zhai\\cite{zhai2013average} considered the sum over sums of two squares\n\\[\\sum_{n_1^2+n_2^2\\leq x}\\lambda_f^l(n_1^2+n_2^2),\\]\nand provided estimates for $l\\leq 8$.\n\nRecently, Newton \\& Thorne \\cite{newton2021symmetric1},\\cite{newton2021symmetric2} proved the automorphy and cuspidality of $\\Sym^l f$ for all $l\\geq1$. Using this celebrated work, Xu\\cite{xu2022general} derived a general formula for $\\sum\\limits_{n\\leq x}\\lambda_f^l(n)$ and $\\sum\\limits_{n_1^2+n_2^2\\leq x}\\lambda_f^l(n_1^2+n_2^2)$ for every $l\\geq1$. This result was later refined by Liu\\cite{liu2023asymptotic} as\n\\begin{equation}\\label{equation:introduction1}\n \\sum_{n\\leq x} \\lambda^l_f(n)=xQ_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right),\n\\end{equation}\n\\begin{equation}\\label{equation:introduction2}\n \\sum_{n_1^2+n_2^2\\leq x}\\lambda_f^l(n_1^2+n_2^2)=xQ'_l(\\log x)+O\\left(x^{{\\theta}'_l+\\epsilon}\\right),\n\\end{equation}\nwhere $Q_l$ and $Q'_l$ are polynomials of degree $\\binom{l}{m}-\\binom{l}{m-1}-1$ if $l=2m$ and equals 0 if $l$ is odd. Here $\\theta_l, {\\theta}'_l$ are explicit constants less than 1. \n\nAnother direction of research concerns the average behavior of Fourier coefficients of symmetric powers of $f$. Several results have been obtained in this direction, but they only focus on some special cases. For example, in \\cite{fomenko2006identities} and \\cite{fomenko2008mean}, Fomenko proved asymptotic formulas for \n\\[\\sum_{n\\leq x}\\lambda_{\\Sym^2 f}(n)\\ \\text{and}\\ \\sum_{n\\leq x}\\lambda^2_{\\Sym^2 f}(n).\\]\nIn \\cite{he2019integral} and \\cite{luo2021asymptotics}, the summations \\[\\sum_{n\\leq x}\\lambda^l_{\\Sym^2 f}(n)\\ \\text{and}\\ \\sum_{n\\leq x}\\lambda^2_{\\Sym^d f}(n)\\]\nwith $d\\geq 2$ and $l\\leq 8$ were studied. Later in \\cite{liu2023average}, Liu obtained the following results with better error terms:\n\\begin{equation} \\label{equation:introduction3}\n \\sum_{n\\leq x}\\lambda^l_{\\Sym^2 f}(n)=xP_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right)\n\\end{equation}\nfor $2\\leq l\\leq 8$, and\n\\begin{equation}\\label{equation:introduction4}\n \\sum_{n\\leq x}\\lambda^2_{\\Sym^d f}(n)=c_dx+O\\left(x^{\\tilde{\\theta}_d+\\epsilon}\\right),\n\\end{equation}\nfor any $d\\geq 2$.\nHere $P_l$ are polynomials with explicitly computed degrees, $c_d$ is a constant and $\\theta_l, \\tilde{\\theta}_d$ are positive constants less than 1.\n\nExtending these results to arbitrary integers $d\\geq1$ and $l\\geq1$ presents significant challenges, as the associated Dirichlet series become increasingly complex. In this paper, we address this problem by studying the underlying Galois representations and applying combinatorial results concerning their decompositions. \nOur work generalizes the average formulas (\\ref{equation:introduction1}),(\\ref{equation:introduction2}),(\\ref{equation:introduction3}),(\\ref{equation:introduction4}) in the following aspects.\n\\begin{itemize}\n \\item We obtain asymptotic formulas with improved error terms.\n \\item We establish a universal asymptotic formula for $\\sum\\limits_{n\\leq x}\\lambda^l_{\\Sym^d f}(n)$ valid for any $d\\geq 1$ and $l\\geq 1$,which generalizes (\\ref{equation:introduction3}) and (\\ref{equation:introduction4}).\n \\item We prove an average formula for sums over values of a binary quadratic form, namely $ \\sum\\limits_{Q(n_1,n_2)\\leq x}\\lambda^l_{\\Sym^d f}(Q(n_1,n_2))$ for any definite binary quadratic form $Q$ and any $d,l\\geq1$.\n\\end{itemize}\n\\subsection{Main theorems}\nWe now state our main results for general $d$ and $l$.", + "context": "Let $f$ be a cuspidal eigenform of weight $k$ on $\\SL_2(\\BZ)$, and let\n\\[f(z)=\\sum_{n=1}^\\infty \\lambda_f(n)n^{\\frac{k-1}{2}}q^n\\]\nbe its $q$-expansion. Estimates of $\\lambda_f(n)$ and their average behavior are fundamental problems in number theory. \n Estimating the sum\n$\\sum\\limits_{n\\leq x}\\lambda_f(n),$ has a long history, with contributions from\n \\cite{kloosterman1927asymptotische}, \\cite{rankin1983sums}, \\cite{weil1948some}, \\cite{wu2009power}, among others. The sum of higher power moments \n$\\sum\\limits_{n\\leq x}\\lambda_f^l(n)$\nhas also attracted considerable attention. If $l=2$, Rankin\\cite{rankin1939contributions} and Selberg\\cite{selberg1940bemerkungen} already studied this problem and recently Huang's work \\cite{huang2021rankin} provides improved estimates with\n\\[\\sum_{n\\leq x}\\lambda_f^2(n)=Cx+O\\left(x^{\\frac{3}{5}-\\frac{1}{560}+\\epsilon}\\right).\\]\n\nFor $l\\geq3$, symmetric power $L$-functions of modular forms are required. Kim and Kim \\& Shahidi \\cite{kim2003functoriality}, \\cite{kim2002cuspidality},\\cite{kim2002functorial} showed that $\\Sym^l f$ is automorphic for $l\\leq 4$. Using these results, the estimates for $l\\leq 8$ are derived in \\cite{lau2011integral}, \\cite{lu2009average} and \\cite{lu2011higher}. Later Zhai\\cite{zhai2013average} considered the sum over sums of two squares\n\\[\\sum_{n_1^2+n_2^2\\leq x}\\lambda_f^l(n_1^2+n_2^2),\\]\nand provided estimates for $l\\leq 8$.\n\nRecently, Newton \\& Thorne \\cite{newton2021symmetric1},\\cite{newton2021symmetric2} proved the automorphy and cuspidality of $\\Sym^l f$ for all $l\\geq1$. Using this celebrated work, Xu\\cite{xu2022general} derived a general formula for $\\sum\\limits_{n\\leq x}\\lambda_f^l(n)$ and $\\sum\\limits_{n_1^2+n_2^2\\leq x}\\lambda_f^l(n_1^2+n_2^2)$ for every $l\\geq1$. This result was later refined by Liu\\cite{liu2023asymptotic} as\n\\begin{equation}\\label{equation:introduction1}\n \\sum_{n\\leq x} \\lambda^l_f(n)=xQ_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right),\n\\end{equation}\n\\begin{equation}\\label{equation:introduction2}\n \\sum_{n_1^2+n_2^2\\leq x}\\lambda_f^l(n_1^2+n_2^2)=xQ'_l(\\log x)+O\\left(x^{{\\theta}'_l+\\epsilon}\\right),\n\\end{equation}\nwhere $Q_l$ and $Q'_l$ are polynomials of degree $\\binom{l}{m}-\\binom{l}{m-1}-1$ if $l=2m$ and equals 0 if $l$ is odd. Here $\\theta_l, {\\theta}'_l$ are explicit constants less than 1.\n\nAnother direction of research concerns the average behavior of Fourier coefficients of symmetric powers of $f$. Several results have been obtained in this direction, but they only focus on some special cases. For example, in \\cite{fomenko2006identities} and \\cite{fomenko2008mean}, Fomenko proved asymptotic formulas for \n\\[\\sum_{n\\leq x}\\lambda_{\\Sym^2 f}(n)\\ \\text{and}\\ \\sum_{n\\leq x}\\lambda^2_{\\Sym^2 f}(n).\\]\nIn \\cite{he2019integral} and \\cite{luo2021asymptotics}, the summations \\[\\sum_{n\\leq x}\\lambda^l_{\\Sym^2 f}(n)\\ \\text{and}\\ \\sum_{n\\leq x}\\lambda^2_{\\Sym^d f}(n)\\]\nwith $d\\geq 2$ and $l\\leq 8$ were studied. Later in \\cite{liu2023average}, Liu obtained the following results with better error terms:\n\\begin{equation} \\label{equation:introduction3}\n \\sum_{n\\leq x}\\lambda^l_{\\Sym^2 f}(n)=xP_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right)\n\\end{equation}\nfor $2\\leq l\\leq 8$, and\n\\begin{equation}\\label{equation:introduction4}\n \\sum_{n\\leq x}\\lambda^2_{\\Sym^d f}(n)=c_dx+O\\left(x^{\\tilde{\\theta}_d+\\epsilon}\\right),\n\\end{equation}\nfor any $d\\geq 2$.\nHere $P_l$ are polynomials with explicitly computed degrees, $c_d$ is a constant and $\\theta_l, \\tilde{\\theta}_d$ are positive constants less than 1.\n\nExtending these results to arbitrary integers $d\\geq1$ and $l\\geq1$ presents significant challenges, as the associated Dirichlet series become increasingly complex. In this paper, we address this problem by studying the underlying Galois representations and applying combinatorial results concerning their decompositions. \nOur work generalizes the average formulas (\\ref{equation:introduction1}),(\\ref{equation:introduction2}),(\\ref{equation:introduction3}),(\\ref{equation:introduction4}) in the following aspects.\n\\begin{itemize}\n \\item We obtain asymptotic formulas with improved error terms.\n \\item We establish a universal asymptotic formula for $\\sum\\limits_{n\\leq x}\\lambda^l_{\\Sym^d f}(n)$ valid for any $d\\geq 1$ and $l\\geq 1$,which generalizes (\\ref{equation:introduction3}) and (\\ref{equation:introduction4}).\n \\item We prove an average formula for sums over values of a binary quadratic form, namely $ \\sum\\limits_{Q(n_1,n_2)\\leq x}\\lambda^l_{\\Sym^d f}(Q(n_1,n_2))$ for any definite binary quadratic form $Q$ and any $d,l\\geq1$.\n\\end{itemize}\n\\subsection{Main theorems}\nWe now state our main results for general $d$ and $l$.\n\n\\begin{equation}\\label{equation:introduction1}\n \\sum_{n\\leq x} \\lambda^l_f(n)=xQ_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right),\n\\end{equation}\n\n\\begin{equation} \\label{equation:introduction3}\n \\sum_{n\\leq x}\\lambda^l_{\\Sym^2 f}(n)=xP_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right)\n\\end{equation}\n\n\\begin{equation}\\label{equation:introduction4}\n \\sum_{n\\leq x}\\lambda^2_{\\Sym^d f}(n)=c_dx+O\\left(x^{\\tilde{\\theta}_d+\\epsilon}\\right),\n\\end{equation}", + "full_context": "Let $f$ be a cuspidal eigenform of weight $k$ on $\\SL_2(\\BZ)$, and let\n\\[f(z)=\\sum_{n=1}^\\infty \\lambda_f(n)n^{\\frac{k-1}{2}}q^n\\]\nbe its $q$-expansion. Estimates of $\\lambda_f(n)$ and their average behavior are fundamental problems in number theory. \n Estimating the sum\n$\\sum\\limits_{n\\leq x}\\lambda_f(n),$ has a long history, with contributions from\n \\cite{kloosterman1927asymptotische}, \\cite{rankin1983sums}, \\cite{weil1948some}, \\cite{wu2009power}, among others. The sum of higher power moments \n$\\sum\\limits_{n\\leq x}\\lambda_f^l(n)$\nhas also attracted considerable attention. If $l=2$, Rankin\\cite{rankin1939contributions} and Selberg\\cite{selberg1940bemerkungen} already studied this problem and recently Huang's work \\cite{huang2021rankin} provides improved estimates with\n\\[\\sum_{n\\leq x}\\lambda_f^2(n)=Cx+O\\left(x^{\\frac{3}{5}-\\frac{1}{560}+\\epsilon}\\right).\\]\n\nFor $l\\geq3$, symmetric power $L$-functions of modular forms are required. Kim and Kim \\& Shahidi \\cite{kim2003functoriality}, \\cite{kim2002cuspidality},\\cite{kim2002functorial} showed that $\\Sym^l f$ is automorphic for $l\\leq 4$. Using these results, the estimates for $l\\leq 8$ are derived in \\cite{lau2011integral}, \\cite{lu2009average} and \\cite{lu2011higher}. Later Zhai\\cite{zhai2013average} considered the sum over sums of two squares\n\\[\\sum_{n_1^2+n_2^2\\leq x}\\lambda_f^l(n_1^2+n_2^2),\\]\nand provided estimates for $l\\leq 8$.\n\nRecently, Newton \\& Thorne \\cite{newton2021symmetric1},\\cite{newton2021symmetric2} proved the automorphy and cuspidality of $\\Sym^l f$ for all $l\\geq1$. Using this celebrated work, Xu\\cite{xu2022general} derived a general formula for $\\sum\\limits_{n\\leq x}\\lambda_f^l(n)$ and $\\sum\\limits_{n_1^2+n_2^2\\leq x}\\lambda_f^l(n_1^2+n_2^2)$ for every $l\\geq1$. This result was later refined by Liu\\cite{liu2023asymptotic} as\n\\begin{equation}\\label{equation:introduction1}\n \\sum_{n\\leq x} \\lambda^l_f(n)=xQ_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right),\n\\end{equation}\n\\begin{equation}\\label{equation:introduction2}\n \\sum_{n_1^2+n_2^2\\leq x}\\lambda_f^l(n_1^2+n_2^2)=xQ'_l(\\log x)+O\\left(x^{{\\theta}'_l+\\epsilon}\\right),\n\\end{equation}\nwhere $Q_l$ and $Q'_l$ are polynomials of degree $\\binom{l}{m}-\\binom{l}{m-1}-1$ if $l=2m$ and equals 0 if $l$ is odd. Here $\\theta_l, {\\theta}'_l$ are explicit constants less than 1.\n\nAnother direction of research concerns the average behavior of Fourier coefficients of symmetric powers of $f$. Several results have been obtained in this direction, but they only focus on some special cases. For example, in \\cite{fomenko2006identities} and \\cite{fomenko2008mean}, Fomenko proved asymptotic formulas for \n\\[\\sum_{n\\leq x}\\lambda_{\\Sym^2 f}(n)\\ \\text{and}\\ \\sum_{n\\leq x}\\lambda^2_{\\Sym^2 f}(n).\\]\nIn \\cite{he2019integral} and \\cite{luo2021asymptotics}, the summations \\[\\sum_{n\\leq x}\\lambda^l_{\\Sym^2 f}(n)\\ \\text{and}\\ \\sum_{n\\leq x}\\lambda^2_{\\Sym^d f}(n)\\]\nwith $d\\geq 2$ and $l\\leq 8$ were studied. Later in \\cite{liu2023average}, Liu obtained the following results with better error terms:\n\\begin{equation} \\label{equation:introduction3}\n \\sum_{n\\leq x}\\lambda^l_{\\Sym^2 f}(n)=xP_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right)\n\\end{equation}\nfor $2\\leq l\\leq 8$, and\n\\begin{equation}\\label{equation:introduction4}\n \\sum_{n\\leq x}\\lambda^2_{\\Sym^d f}(n)=c_dx+O\\left(x^{\\tilde{\\theta}_d+\\epsilon}\\right),\n\\end{equation}\nfor any $d\\geq 2$.\nHere $P_l$ are polynomials with explicitly computed degrees, $c_d$ is a constant and $\\theta_l, \\tilde{\\theta}_d$ are positive constants less than 1.\n\nExtending these results to arbitrary integers $d\\geq1$ and $l\\geq1$ presents significant challenges, as the associated Dirichlet series become increasingly complex. In this paper, we address this problem by studying the underlying Galois representations and applying combinatorial results concerning their decompositions. \nOur work generalizes the average formulas (\\ref{equation:introduction1}),(\\ref{equation:introduction2}),(\\ref{equation:introduction3}),(\\ref{equation:introduction4}) in the following aspects.\n\\begin{itemize}\n \\item We obtain asymptotic formulas with improved error terms.\n \\item We establish a universal asymptotic formula for $\\sum\\limits_{n\\leq x}\\lambda^l_{\\Sym^d f}(n)$ valid for any $d\\geq 1$ and $l\\geq 1$,which generalizes (\\ref{equation:introduction3}) and (\\ref{equation:introduction4}).\n \\item We prove an average formula for sums over values of a binary quadratic form, namely $ \\sum\\limits_{Q(n_1,n_2)\\leq x}\\lambda^l_{\\Sym^d f}(Q(n_1,n_2))$ for any definite binary quadratic form $Q$ and any $d,l\\geq1$.\n\\end{itemize}\n\\subsection{Main theorems}\nWe now state our main results for general $d$ and $l$.\n\n\\begin{equation}\\label{equation:introduction1}\n \\sum_{n\\leq x} \\lambda^l_f(n)=xQ_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right),\n\\end{equation}\n\n\\begin{equation} \\label{equation:introduction3}\n \\sum_{n\\leq x}\\lambda^l_{\\Sym^2 f}(n)=xP_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right)\n\\end{equation}\n\n\\begin{equation}\\label{equation:introduction4}\n \\sum_{n\\leq x}\\lambda^2_{\\Sym^d f}(n)=c_dx+O\\left(x^{\\tilde{\\theta}_d+\\epsilon}\\right),\n\\end{equation}\n\nThe estimation in the general case is new. And our estimates in the error terms are better than those in the work of \\cite{liu2023average} for small $d$ and $l$. We list the cases with $d=2,3\\leq l\\leq8$ and $l=2,3\\leq d\\leq8$ in the following tables.\n\nWe also have the following theorem for summation over binary quadratic forms.\n\\begin{thm}\\label{thm:binary}\n Let $f$ be a cuspidal modular form of weight $k$ on $\\SL_2(\\BZ)$. For any integers $l\\geq2,d\\geq1$ with $dl>4$ and any binary quadratic form $Q(n_1,n_2)$, one has\n \\[\\sum_{Q(n_1,n_2)\\leq x} \\lambda^l_{\\Sym^d f}(Q(n_1,n_2))=xP_{d,l,Q}(\\log x)+O\\left(x^{\\theta_{d,l,Q}+\\epsilon}\\right).\\]\n Here $P_{d,l,Q}$ is a polynomial of degree $K_{0,d,l}-1$. If the class number of $Q$ is not 1, then \n $$\\theta_{d,l,Q}=1-\\frac{3}{3(d+1)^l-K_{0,d,l}};$$ if the class number is 1, then \n $$\\theta_{d,l,Q}=1-1/\\left((d+1)^l-8K_{0,d,l}/21-2K_{1,d,l}/3-5K_{2,d,l}/7\\right).$$\n\\end{thm}\n\nWe now provide an alternative explicit description of these Kostka numbers. Let $K_{i,d,l}=K_{\\mu,\\lambda}$ (as defined above). By Proposition \\ref{prop:decomposition}, $K_{i,d,l}$ is the multiplicity of $\\Sym^i V$ in $\\left(\\Sym^d V\\right)^{\\otimes l}$ for a two dimensional representation $V$. We extend the definition of $K_{i,d,l}$ to all $i\\in\\BZ$ by setting $K_{i,d,l}=0$ for $i\\notin [0,dl]$, or when $i$ and $dl$ have different parity. From Proposition \\ref{prop:decomposition} (4), we derive a recursive formula for $K_{i,d,l}$:\n\\[K_{i,d,l}=K_{i-d,d,l-1}+K_{i-d+2,d,l-1}+\\cdots+K_{i+d,d,l-1}.\\]\nOn the other hand, consider the polynomial:\n\\[(1+x+x^2+\\cdots+x^d)^{l}=\\sum_{j=-\\infty}^{+\\infty}C_{j,d,l}x^j.\\]\nDefine $A_{i,d,l}=C_{\\lfloor\\frac{dl-i}{2}\\rfloor,d,l}-C_{\\lfloor\\frac{dl-i-1}{2}\\rfloor,d,l}.$\n\\begin{prop} We have \n $ K_{i,d,l}=A_{i,d,l}.$\n \\end{prop}\n\\begin{proof}\nWe only need to show that $A_{i,d,l}$ and $K_{i,d,l}$ satisfy the same recursive formula and boundary conditions.\n First, the coefficients $C_{i,d,l}$ satisfy the recursion:\n \\[C_{i,d,l}=C_{i-d,d,l-1}+C_{i-d+1,d,l-1}+\\cdots+C_{i,d,l-1}.\\]\n Thus\n \\begin{align*}\n A_{i,d,l}=&C_{\\lfloor\\frac{dl-i}{2}\\rfloor,d,l}-C_{\\lfloor\\frac{dl-i-1}{2}\\rfloor,d,l}\\\\\n =&\\sum_{t=j-d}^jC_{t,d,l-1}-\\sum_{t=j'-d}^{j'}C_{t,d,l-1}.\n \\end{align*}\n where $j=\\lfloor\\frac{dl-i}{2}\\rfloor, j'=\\lfloor\\frac{dl-i-1}{2}\\rfloor.$\n On the other hand,\n \\begin{align*}\n \\sum_{t=0}^dA_{i-d+2t,d,l-1}&=\\sum_{t=0}^d\\left(C_{\\lfloor\\frac{dl-d-(i-d+2t)}{2}\\rfloor,d,l-1}-C_{\\lfloor\\frac{dl-d-(i-d+2t)-1}{2}\\rfloor,d,l-1}\\right)\\\\\n &=\\sum_{t=0}^d\\left(C_{\\lfloor\\frac{dl-i-2t)}{2}\\rfloor,d,l-1}-C_{\\lfloor\\frac{dl-i-2t-1}{2}\\rfloor,d,l-1}\\right)\n \\end{align*}\n By telescoping, this sum equals $A_{i,d,l}$.\n Thus\n \\[A_{i,d,l}=A_{i-d,d,l-1}+A_{i-d+2,d,l-1}+\\cdots+A_{i+d,d,l-1}.\\]\n For the boundary condition $l=1$:\n \\begin{itemize}\n \\item $K_{d,d,1}=1$ and $K_{i,d,1}=0$ for $i\\neq d$\n \\item $A_{d,d,1}=C_{\\lfloor \\frac{d-d}{2}\\rfloor,d,1}-C_{\\lfloor \\frac{d-d-1}{2}\\rfloor,d,1}=C_{0,d,1}-C_{-1,d,1}=1-0=1$ and $A_{i,d,1}=0$ for $i\\neq d$.\n \\end{itemize}\n Thus $A_{i,d,d}$ and $K_{i,d,d}$ satisfy the same recursion and boundary conditions, so $A_{i,d,l}=K_{i,d,l}$.\n\\end{proof}\nFrom this proposition, the Kostka number $K_{i,d,l}$ can be expressed in terms of binomial coefficients. This gives the following formula.\n\\begin{cor}\\label{corollary:Kostka}\n \\begin{equation}\n K_{i,d,l}=\\begin{cases}\n 0\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\text{if $2\\nmid dl-i$}\\\\\n \\sum\\limits_{j=0}^{\\lfloor(dl-i)/(2d+2)\\rfloor}(-1)^j\\binom{l}{j}\\binom\n {(dl-i)/2-j(d+1)+l-2}{l-2}\n \\ \\text{otherwise}\n \\end{cases}.\n \\end{equation}\n\\end{cor}\n\\begin{proof}\nSince\n \\[(1+x+x^2+\\cdots+x^d)^{l}=(1-x^{d+1})^{l}(1-x)^{-l},\\]\n the coefficients $K_{i,d,l}=A_{i,d,l}$ can be computed as\n\\begin{align*}\n A_{i,d,l}=&\\sum_{j=0}^{\\lfloor(dl-i)/(2d+2)\\rfloor}(-1)^j\\binom{l}{j}\\binom{\\lfloor(dl-i)/2\\rfloor-j(d+1)+l-1}{l-1}\\\\\n &-\\sum_{j=0}^{\\lfloor(dl-i-1)/(2d+2)\\rfloor}(-1)^j\\begin{pmatrix}\n l\\\\j\n \\end{pmatrix}\\begin{pmatrix}\n \\lfloor(dl-i-1)/2\\rfloor-j(d+1)+l-1\\\\l-1\n \\end{pmatrix}.\n\\end{align*}\n\nNow let $f$ be a cuspidal eigenform on $\\SL_2(\\BZ)$. For any integer $d\\geq0$, the symmetric power lifting $\\Sym^d f$ is an automorphic form on $\\GL_{d+1}$. For a character $\\chi$ of $\\fh$, we define the associated $L$-function as follows. If $\\chi$ is non-trivial,\nlet\n\\[L\\left(\\Sym^d f, \\chi,s\\right):=\\sum_{n=1}^\\infty\\frac{\\lambda_{\\Sym^df}(n)a_\\chi(n)}{n^s}\\]\nbe the Rankin-Selberg $L$-function associated to $\\Sym^d f$ and $f_\\chi$. If $\\chi$ is trivial, define $$L\\left(\\Sym^d f, \\chi,s\\right)=L\\left(\\Sym^df,s\\right)L\\left(\\Sym^df,\\chi_F,s\\right).$$ Similarly, we can define the $L$-functions $L\\left(f^{\\otimes l},\\chi,s\\right)$ and $L\\left(\\left(\\Sym^d f\\right)^{\\otimes l},\\chi,s\\right)$ by incorporating the coefficients $a_\\chi(n)$. Leveraging the tensor product decompositions from previous sections, we obtain the following factorizations:\n \\begin{equation}\n L\\left(f^{\\otimes l},\\chi,s\\right)=\\prod_{i=0}^{[l/2]}L\\left(\\Sym^{l-2i}f,\\chi,s\\right)^{c_{l,i}},\n \\end{equation}\n \\begin{equation}\\label{equation:chilfactor}\n L\\left(\\left(\\Sym^d f\\right)^{\\otimes l},\\chi,s\\right)=\\prod_{i=0}^{dl}L\\left(\\Sym^i f,\\chi,s\\right)^{K_{i,d,l}}. \n \\end{equation}\n\\section{The estimations via Perron's formula}\n\\subsection{The Dirichlet series}\nLet $f$ be a cuspidal eigenform of weight $k$ on $\\SL_2(\\BZ)$, let $Q$ be a binary quadratic form and let $\\chi$ be a character of $\\fh$. We define the following Dirichlet series, which are central to our analysis:\n\\begin{align*}\n&D\\left(f^{\\otimes l},s\\right):=\\sum_{n=1}^\\infty \\frac{\\lambda_f^l(n)}{n^s},&D&\\left(\\left(\\Sym^df\\right)^{\\otimes l},s\\right):=\\sum_{n=1}^\\infty \\frac{\\lambda_{\\Sym^df}^l(n)}{n^s},\\\\ \n&D\\left(f^{\\otimes l},Q,s\\right):=\\sum_{n=1}^\\infty \\frac{\\lambda_f^l(n)r(n,Q)}{n^s},&D&\\left(\\left(\\Sym^d f\\right)^{\\otimes l},Q,s\\right):=\\sum_{n=1}^\\infty \\frac{\\lambda_{\\Sym^d f}^l(n)r(n,Q)}{n^s},\\\\\n&D\\left(f^{\\otimes l},\\chi,s\\right):=\\sum_{n=1}^\\infty \\frac{\\lambda_f^l(n)a_{\\chi}(n)}{n^s},&D&\\left(\\left(\\Sym^d f\\right)^{\\otimes l},\\chi,s\\right):=\\sum_{n=1}^\\infty \\frac{\\lambda_{\\Sym^d f}^l(n)a_\\chi(n)}{n^s}.\n\\end{align*}\nFrom the expression for $r(n,Q)$ given in (\\ref{equation:binarydecom}), we immediately obtain the relations\n\\begin{align*}\n D\\left(f^{\\otimes l},Q,s\\right)&=\\frac{w_D}{h(D)}\\sum_{\\chi}\\overline{\\chi(\\fa_Q)}D\\left(f^{\\otimes l},\\chi,s\\right),\\\\ \n D\\left(\\left(\\Sym^d f\\right)^{\\otimes l},Q,s\\right)&=\\frac{w_D}{h(D)}\\sum_{\\chi}\\overline{\\chi(\\fa_Q)}D\\left(\\left(\\Sym^d f\\right)^{\\otimes l},\\chi,s\\right).\n\\end{align*}\nThe connection between these Dirichlet series and the complete $L$-functions is provided by the following proposition, which is an analogue of Theorem \\ref{theorem:l-functions} for the twisted case.\n\nThe parameter $T$ is now chosen to balance the error terms.\n We first consider the case where the class number of $Q$ is one. In this case, only the trivial character contributes. Choosing\n$$T=x^{1/\\left((d+1)^l-\\frac{8}{21}K_{0,d,l}-\\frac{2}{3}K_{1,d,l}-\\frac{5}{7}K_{2,d,l}\\right)},$$\nwe obtain the asymptotic formula\n\\[\\sum_{n\\leq x}\\lambda^l_{\\Sym^df}(n)a_\\chi(n)=xP_{0,d,l}(\\log(x))+O\\left(x^{1-1/\\left((d+1)^l-\\frac{8}{21}K_{0,d,l}-\\frac{2}{3}K_{1,d,l}-\\frac{5}{7}K_{2,d,l}\\right)+\\epsilon}\\right).\\]\nNext assume the class number is greater than 1. The main term comes only from the trivial character, but the error terms include contributions from all characters. Choosing\n $$T=x^{\\frac{3}{3(d+1)^l-K_{0,d,l}}},$$ we obtain\n\\[\\sum_{n\\leq x}\\lambda^l_{\\Sym^df}(n)a_\\chi(n)=xP_{0,d,l}(\\log(x))+O\\left(x^{1-\\frac{3}{3(d+1)^l-K_{0,d,l}}+\\epsilon}\\right).\\]\nSumming over all characters $\\chi$, and using (\\ref{equation:binarydecom}) completes the proof of Theorem \\ref{thm:binary}. \n\\bibliography{bib/ref}\n\\bibliographystyle{plain}", + "post_theorem_intro_text_len": 5762, + "post_theorem_intro_text": "In Corollary \\ref{corollary:Kostka}, we provide an explicit formula for the Kostka number $K_{i,d,l}$, which determines the degrees of the main terms and the exponents in the error terms. The formula is \n\\begin{align*} K_{i,d,l}=\\begin{cases}\n 0\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\text{if $2\\nmid dl-i$}\\\\\n \\sum\\limits_{j=0}^{\\lfloor(dl-i)/(2d+2)\\rfloor}(-1)^j\\binom{l}{j}\\binom\n {(dl-i)/2-j(d+1)+l-2}{l-2}\n \\ \\text{otherwise}\n \\end{cases}.\n \\end{align*}\nNote that if $dl$ is odd, then $K_{0,d,l}=0$ and $P_{d,l}$ is 0.\n\nThe estimation in the general case is new. And our estimates in the error terms are better than those in the work of \\cite{liu2023average} for small $d$ and $l$. We list the cases with $d=2,3\\leq l\\leq8$ and $l=2,3\\leq d\\leq8$ in the following tables.\n\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\n$d=2, l=$ & Exponents in \\cite{liu2023average} & $\\theta_{d, l}$ \\\\\n\\hline\n3 & 0.9193... & 0.918287938\n \\\\\n\\hline\n4 & 0.9737... & 0.973534972\n \\\\\n\\hline\n5 & 0.99136... & 0.991304348\n\\\\\n\\hline\n6 & 0.99714... & 0.99713291\n \\\\\n\\hline\n7 & 0.9990558... & 0.999051362\n \\\\\n\\hline\n8 & 0.9996868... & 0.999685565\n \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n$l=2, d=$ & Exponents in \\cite{liu2023average} & $\\theta_{d, l}$ \\\\\n\\hline\n3 & 0.866... & 0.865814696\n \\\\\n\\hline\n4 & 0.9166... & 0.916334661\n \\\\\n\\hline\n5 & 0.9428... & 0.942701228\n \\\\\n\\hline\n6 & 0.9583... & 0.958250497\n \\\\\n\\hline\n7 & $0.96824\\ldots$ &0.968205905\n \\\\\n\\hline\n8 & 0.97499... & 0.974970203\n \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\nWe also have the following theorem for summation over binary quadratic forms.\n\\begin{thm}\\label{thm:binary}\n Let $f$ be a cuspidal modular form of weight $k$ on $\\SL_2(\\BZ)$. For any integers $l\\geq2,d\\geq1$ with $dl>4$ and any binary quadratic form $Q(n_1,n_2)$, one has\n \\[\\sum_{Q(n_1,n_2)\\leq x} \\lambda^l_{\\Sym^d f}(Q(n_1,n_2))=xP_{d,l,Q}(\\log x)+O\\left(x^{\\theta_{d,l,Q}+\\epsilon}\\right).\\]\n Here $P_{d,l,Q}$ is a polynomial of degree $K_{0,d,l}-1$. If the class number of $Q$ is not 1, then \n $$\\theta_{d,l,Q}=1-\\frac{3}{3(d+1)^l-K_{0,d,l}};$$ if the class number is 1, then \n $$\\theta_{d,l,Q}=1-1/\\left((d+1)^l-8K_{0,d,l}/21-2K_{1,d,l}/3-5K_{2,d,l}/7\\right).$$\n\\end{thm}\n\n\\subsection{Proof Sketch }\nThe proofs of our theorems follow a framework that can be viewed as an $\\ell$-adic analogue of the methods for Artin representations developed in \\cite{Yang2024second} , \\cite{Yang2024first} and \\cite{Yang2026higher}.\n\nThe starting point is Perron's formula, which translates the estimation of the sum $\\sum\\limits_{n\\leq x}\\lambda^l_{\\Sym^d f}(n)$ into the problem of evaluating a contour integral of the associated Dirichlet series $\\sum\\limits_{n=1}^\\infty\\frac{\\lambda^l_{\\Sym^d f}(n)}{n^s}$. This Dirichlet series is intimately related via its Euler product to the $L$-function $L\\left(\\left(\\Sym^d f\\right)^{\\otimes l},s\\right)$, which corresponds to the $\\ell$-adic representation $\\Sym^d(\\rho_f)^{\\otimes l}$ (see Proposition \\ref{proposition:dirichletseriescomp}).\n\nA key insight is the decomposition of this high-dimensional tensor representation into a direct sum of irreducible constituents, which are shown to be of the form $\\Sym^i \\rho_f$. Consequently, the $L$-function factors into a product of symmetric power $L$-functions:\n$$L\\left(\\left(\\Sym^d f\\right)^{\\otimes l},s\\right)=\\prod_{i=0}^{ld}L\\left(\\Sym^i f,s\\right)^{K_{i,d,l}}$$\nThe automorphy and entirety of these symmetric power $L$-functions, established by Newton and Thorne \\cite{newton2021symmetric1},\\cite{newton2021symmetric2}, are crucial here. The final asymptotic formulas are then derived by combining this factorization with established subconvexity bounds for the involved $L$-functions.\n\nFor sums over values of a binary quadratic form $Q$, the approach is similar but involves additional ingredients. The generating Dirichlet series becomes $\\sum\\limits_{n=1}^\\infty\\frac{\\lambda^l_{\\Sym^d f}(n)r(n,Q)}{n^s}$, where $r(n,Q)$ counts representations of $n$ by $Q$. Let $F=\\BQ(\\sqrt{D})$ be the corresponding imaginary quadratic field. When the class number $h(D)=1$, the series $\\sum\\limits_{n=1}^\\infty \\frac{r(n,Q)}{n^s}$ is essentially the Dedekind zeta function $\\zeta_F(s)$. In general, it can be expressed as a finite sum of Hecke $L$-functions associated to characters of the class group of $F$. Each such $L$-function corresponds to a modular form $f_\\chi$\n of weight $1$. Thus, the analysis reduces to studying Rankin-Selberg type $L$-functions of the form $L\\left(\\left(\\Sym^d f\\right)^{\\otimes l}\\otimes\\rho_{f_\\chi},s\\right)$ which again admit a decomposition into products of simpler $L$-functions. The theorem is proved by applying subconvexity estimates to these components.\n\n\\begin{remark}\n While this paper has been submitted and review, we noticed that recently Venkatasubbareddy submitted a paper\\cite{arXiv:2601.17079} working on a similar problem and obtained similar results as our Theorem \\ref{theorem:main}. But our work is different from \\cite{arXiv:2601.17079} in the following aspects. First we use a representation point of view to deal with the Dirichlet series, this allows us to connect our work with combination. Such ideas are similar with our work \\cite{Yang2024first} and \\cite{Yang2026higher}. They can be used to deal with more complicated cases and play a key role in the following work \\cite{Yang2026second}. Next we also consider average results over binary quadratic forms, which are not considered in \\cite{arXiv:2601.17079}. Finally, the estimation for the error terms in the two works are different.\n\n\\end{remark}", + "sketch": "The post-theorem introduction contains a \\subsection{Proof Sketch} explaining the strategy for Theorem~\\ref{theorem:main}:\n\n- **Perron's formula / contour integral reduction.** The “starting point is Perron's formula,” which converts estimating \\(\\sum_{n\\le x}\\lambda^l_{\\Sym^d f}(n)\\) into evaluating “a contour integral of the associated Dirichlet series” \\(\\sum_{n\\ge1}\\lambda^l_{\\Sym^d f}(n)n^{-s}\\).\n\n- **Relate the Dirichlet series to an \\(L\\)-function via Euler product.** This Dirichlet series is “intimately related via its Euler product” to \\(L((\\Sym^d f)^{\\otimes l},s)\\), corresponding to the \\(\\ell\\)-adic representation \\(\\Sym^d(\\rho_f)^{\\otimes l}\\) (cf. Proposition~\\ref{proposition:dirichletseriescomp}).\n\n- **Decompose the tensor representation and factor the \\(L\\)-function.** A “key insight” is decomposing the “high-dimensional tensor representation into a direct sum of irreducible constituents,” which are “of the form \\(\\Sym^i\\rho_f\\).” Hence\n \\[\n L\\bigl((\\Sym^d f)^{\\otimes l},s\\bigr)=\\prod_{i=0}^{ld} L(\\Sym^i f,s)^{K_{i,d,l}}.\n \\]\n\n- **Use automorphy/entireness and subconvexity to get the asymptotic.** The “automorphy and entirety of these symmetric power \\(L\\)-functions,” due to Newton--Thorne, are “crucial,” and the “final asymptotic formulas are then derived by combining this factorization with established subconvexity bounds for the involved \\(L\\)-functions.”\n\n(An additional sketch is given for the binary quadratic form theorem \\(\\ref{thm:binary}\\): one studies \\(\\sum \\lambda^l_{\\Sym^d f}(n)r(n,Q)n^{-s}\\), expresses \\(\\sum r(n,Q)n^{-s}\\) via \\(\\zeta_F(s)\\) when \\(h(D)=1\\) or as a finite sum of Hecke \\(L\\)-functions in general, reducing to Rankin--Selberg type \\(L\\)-functions \\(L((\\Sym^d f)^{\\otimes l}\\otimes \\rho_{f_\\chi},s)\\), and then applies subconvexity to their decomposed components.)", + "expanded_sketch": "No expanded sketch found.", + "expanded_theorem": "\\label{theorem:main}\n Let $f$ be a cuspidal modular form of weight $k$ on $\\SL_2(\\BZ)$. For any integers $l\\geq2$ and $d\\geq1$ and $dl>4$, one has\n \\[\\sum_{n\\leq x} \\lambda^l_{\\Sym^d f}(n)=xP_{d,l}(\\log x)+O\\left(x^{\\theta_{d,l}+\\epsilon}\\right).\\]\n Here $P_{d,l}$ is a polynomial of degree $K_{0,d,l}-1$, where $K_{i,d,l}$ denote the Kostka number defined in sections \\ref{section:Weylmod} and \\ref{section:Kostka number}. The exponent $\\theta_{d, l}$ is given by\n $$\\theta_{d,l}=1-1/\\left((d+1)^l/2-4K_{0,d,l}/21-K_{1,d,l}/3-5K_{2,d,l}/14\\right).$$,", + "theorem_type": [ + "Asymptotic or Limit", + "Universal" + ], + "mcq": { + "question": "Let $f$ be a cuspidal modular form of weight $k$ on $\\SL_2(\\mathbb{Z})$. For each $d\\ge 1$, write the $d$th symmetric-power $L$-function in the form\n\\[\nL(\\Sym^d f,s)=\\sum_{n=1}^\\infty \\lambda_{\\Sym^d f}(n)n^{-s}.\n\\]\nLet $K_{i,d,l}$ denote the Kostka numbers attached to $(i,d,l)$, equivalently the multiplicities/exponents appearing in the factorization of the $l$-fold tensor power of $\\Sym^d$, such as\n\\[\nL\\big((\\Sym^d f)^{\\otimes l},s\\big)=\\prod_{i=0}^{dl}L(\\Sym^i f,s)^{K_{i,d,l}}.\n\\]\nFor integers $l\\ge 2$ and $d\\ge 1$ with $dl>4$, which statement holds for every such $f$, $d$, and $l$?", + "correct_choice": { + "label": "A", + "text": "For every $\\varepsilon>0$,\n\\[\n\\sum_{n\\le x}\\lambda_{\\Sym^d f}(n)^l=xP_{d,l}(\\log x)+O\\bigl(x^{\\theta_{d,l}+\\varepsilon}\\bigr)\n\\]\nas $x\\to\\infty$, where $P_{d,l}$ is a polynomial of degree $K_{0,d,l}-1$ and\n\\[\n\\theta_{d,l}=1-\\frac{1}{(d+1)^l/2-4K_{0,d,l}/21-K_{1,d,l}/3-5K_{2,d,l}/14}.\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "For every $\\varepsilon>0$,\n\\[\n\\sum_{n\\le x}\\lambda_{\\Sym^d f}(n)^l=xP_{d,l}(\\log x)+O\\bigl(x^{\\theta_{d,l}+\\varepsilon}\\bigr)\n\\]\nas $x\\to\\infty$, where $P_{d,l}$ is a polynomial of degree $K_{0,d,l}$ and\n\\[\n\\theta_{d,l}=1-\\frac{1}{(d+1)^l/2-4K_{0,d,l}/21-K_{1,d,l}/3-5K_{2,d,l}/14}.\n\\]" + }, + { + "label": "C", + "text": "For every $\\varepsilon>0$,\n\\[\n\\sum_{n\\le x}\\lambda_{\\Sym^d f}(n)^l=xP_{d,l}(\\log x)+O\\bigl(x^{\\theta_{d,l}+\\varepsilon}\\bigr)\n\\]\nas $x\\to\\infty$ for some polynomial $P_{d,l}$, where\n\\[\n\\theta_{d,l}=1-\\frac{1}{(d+1)^l/2-4K_{0,d,l}/21-K_{1,d,l}/3-5K_{2,d,l}/14}.\n\\]" + }, + { + "label": "D", + "text": "There exists $\\varepsilon>0$ such that for every cuspidal modular form $f$ of weight $k$ on $\\SL_2(\\mathbb{Z})$ and every integers $l\\ge 2$, $d\\ge 1$ with $dl>4$,\n\\[\n\\sum_{n\\le x}\\lambda_{\\Sym^d f}(n)^l=xP_{d,l}(\\log x)+O\\bigl(x^{\\theta_{d,l}+\\varepsilon}\\bigr)\n\\]\nas $x\\to\\infty$, where $P_{d,l}$ is a polynomial of degree $K_{0,d,l}-1$ and\n\\[\n\\theta_{d,l}=1-\\frac{1}{(d+1)^l/2-4K_{0,d,l}/21-K_{1,d,l}/3-5K_{2,d,l}/14}.\n\\]" + }, + { + "label": "E", + "text": "For every $\\varepsilon>0$,\n\\[\n\\sum_{n\\le x}\\lambda_{\\Sym^d f}(n)^l=xP_{d,l}(\\log x)+O\\bigl(x^{\\theta_{d,l}+\\varepsilon}\\bigr)\n\\]\nas $x\\to\\infty$, where $P_{d,l}$ is a polynomial of degree $K_{0,d,l}-1$ and\n\\[\n\\theta_{d,l}=1-\\frac{1}{(d+1)^l-4K_{0,d,l}/21-K_{1,d,l}/3-5K_{2,d,l}/14}.\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": "degree_of_main_polynomial", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "explicit_degree_K0dl_minus_1", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "quantifier_order_on_epsilon", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "counting_estimate", + "tampered_component": "denominator_factor_one_half_in_theta", + "template_used": "stronger_trap" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not reveal the correct option explicitly or by an obvious cue. It sets up notation and asks for the valid theorem-level conclusion, but the exact degree, quantifier structure, and exponent formula must be identified from the choices." + }, + "TAS": { + "score": 1, + "justification": "The item is essentially a theorem-identification question: the correct answer is a precise asymptotic statement very close to a known result. It is not a pure verbatim restatement because the options introduce meaningful variants, but it still mainly tests recognition of the exact theorem statement." + }, + "GPS": { + "score": 1, + "justification": "There is some reasoning pressure in distinguishing the strongest valid statement from a weaker true one and from subtle false variants involving quantifiers, polynomial degree, and the exponent denominator. However, the problem relies much more on recall/discrimination than on generating a conclusion from first principles." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically targeted: one changes the polynomial degree by 1, one gives a weaker true statement, one alters quantifier order, and one perturbs the key denominator. These reflect realistic theorem-recall and logic mistakes." + }, + "total_score": 6, + "overall_assessment": "A solid but theorem-recall-heavy MCQ: little answer leakage and strong distractors, but only moderate generative reasoning since success depends largely on recognizing the exact statement rather than deriving it." + } + }, + { + "id": "2602.02723v1", + "paper_link": "http://arxiv.org/abs/2602.02723v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "theorem", + "content": "\\label{maintheo}Let $(M, g)$ be a Lorentzian manifold of dimension at least $3$ and let $\\bf P$ be its pseudo-group of local conformal transformations, that is the collection \n of all conformal diffeomorphisms between open subsets of $M$. We assume $M$ locally conformally homogeneous, i.e.~$M$ is an orbit of $\\bf P$. Then one of the following possibilities holds:\n\n \\begin{enumerate}\n\n \\item $\\bf P$ preserves a Lorentzian metric in the $g$-conformal class.\n\n \\item $(M, g)$ is conformally flat. \n\n \\item $(M, g)$ is locally conformally equivalent to a plane wave that is isometrically locally homogeneous. More precisely, there exists a homogeneous plane wave $X$, such that \n $(M, [g])$ is modeled on $(\\Conf(X), X)$.\n \\end{enumerate}", + "start_pos": 11292, + "end_pos": 12061, + "label": "maintheo" + }, + "ref_dict": { + "Penrose-prop": "\\begin{proposition}\\label{Penrose-prop}\nLet $g$ be a Lorentzian metric, and let $\\gamma$ be a null geodesic. Let $g_\\sigma := e^\\sigma g$ be a conformal change of $g$ (up to reparametrization, $\\gamma$ is also a null geodesic of $g_\\sigma$). Let $\\mathrm{PL}_g$ (resp.\\ $\\mathrm{PL}_{g_\\sigma}$) denote the Penrose limit of $g$ (resp.\\ of $g_\\sigma$) along $\\gamma$. Then there exists a conformal diffeomorphism\n$ \\mathrm{PL}_g \\to \\mathrm{PL}_{g_\\sigma}.$\n\\end{proposition}", + "maintheo": "\\begin{theorem}\\label{maintheo}Let $(M, g)$ be a Lorentzian manifold of dimension at least $3$ and let $\\bf P$ be its pseudo-group of local conformal transformations, that is the collection \n of all conformal diffeomorphisms between open subsets of $M$. We assume $M$ locally conformally homogeneous, i.e.~$M$ is an orbit of $\\bf P$. Then one of the following possibilities holds:\n\n \\begin{enumerate}\n\n \\item $\\bf P$ preserves a Lorentzian metric in the $g$-conformal class.\n\n \\item $(M, g)$ is conformally flat. \n\n \\item $(M, g)$ is locally conformally equivalent to a plane wave that is isometrically locally homogeneous. More precisely, there exists a homogeneous plane wave $X$, such that \n $(M, [g])$ is modeled on $(\\Conf(X), X)$.\n \\end{enumerate}\n \\end{theorem}", + "Gromov's Theory": "\\label{Gromov's Theory}\n \\addtocontents{toc}{\\setcounter{tocdepth}{1}}\n\nFor references, see the foundational work of Gromov~\\cite{Gro}, the lecture notes by D'Ambra and Gromov~\\cite{DG}, and those by" + }, + "pre_theorem_intro_text_len": 2569, + "pre_theorem_intro_text": "A fundamental problem in conformal geometry is to classify conformal manifolds with \\textit{essential} conformal transformations, that is, global conformal transformations that do not preserve a metric in the conformal class. For Riemannian conformal manifolds the situation is very rigid: up to conformal equivalence only the round sphere and Euclidean space admit essential conformal transformations. This was conjectured by Lichnerowicz in the 1960's and proved in a series of papers by Ferrand and Obata (\\cite{Lelong-Ferrand71}, \\cite{Obata71}, \\cite{Ferrand96}, with contributions in \\cite{Alekseevskii72}). \n\nFor indefinite metrics, the situation is far more flexible: there exist pseudo-Riemannian metrics that are not conformally flat yet admit essential conformal transformations, both on non-compact manifolds and, as shown in \\cite{frances12}, on compact manifolds in signatures other than the Lorentzian one. Non-compact examples in Lorentzian signature consist of plane waves. These are manifolds for which the metric in dimension $n+2$ is locally given by\n\\begin{equation}\\label{pw}\ng=2\\,\\mathrm{d} t\\, \\mathrm{d} v + \\boldsymbol{x}^\\top Q(t) \\boldsymbol{x}\\ \\mathrm{d} t^2 +\\mathrm{d}\\boldsymbol{x}^\\top\\mathrm{d}\\boldsymbol{x},\n\\end{equation}\nwhere $\\boldsymbol{x}=(x^1, \\ldots , x^n)^\\top\\in \\mathbb{R}^n$, {$Q(t)$ is} a $t$-dependent symmetric $n\\times n$-matrix. If $Q$ is not a scalar matrix, this metric is not conformally flat.\nPlane waves for which this metric is defined on all of $\\mathbb{R}^{n+2}$, for example, admit non-isometric homotheties with fixed points. Therefore these homotheties are essential (for this implication see for example \\cite[Proposition 2.5]{LeistnerTeisseire22}). \n\n\\bigskip\n\nRecently \nan important rigidity result for essential Lorentzian conformal structures in the \\textit{conformally homogeneous} setting has been obtained by Alekseevsky and Galaev\n in \\cite{AlekseevskyGalaev25}. They showed that if the manifold is simply connected and admits a transitive group of global conformal transformations that is essential, then the manifold is either conformally flat or the conformal class contains a plane wave metric that is isometrically homogeneous and geodesically complete. In the present article we will generalize this result in the following sense: we will drop the assumption that the manifold is simply connected and we will only assume that the manifold is \\textit{locally conformally homogeneous}, i.e.~the local conformal transformations act transitively. We will prove the following result.", + "context": "A fundamental problem in conformal geometry is to classify conformal manifolds with \\textit{essential} conformal transformations, that is, global conformal transformations that do not preserve a metric in the conformal class. For Riemannian conformal manifolds the situation is very rigid: up to conformal equivalence only the round sphere and Euclidean space admit essential conformal transformations. This was conjectured by Lichnerowicz in the 1960's and proved in a series of papers by Ferrand and Obata (\\cite{Lelong-Ferrand71}, \\cite{Obata71}, \\cite{Ferrand96}, with contributions in \\cite{Alekseevskii72}).\n\nFor indefinite metrics, the situation is far more flexible: there exist pseudo-Riemannian metrics that are not conformally flat yet admit essential conformal transformations, both on non-compact manifolds and, as shown in \\cite{frances12}, on compact manifolds in signatures other than the Lorentzian one. Non-compact examples in Lorentzian signature consist of plane waves. These are manifolds for which the metric in dimension $n+2$ is locally given by\n\\begin{equation}\\label{pw}\ng=2\\,\\mathrm{d} t\\, \\mathrm{d} v + \\boldsymbol{x}^\\top Q(t) \\boldsymbol{x}\\ \\mathrm{d} t^2 +\\mathrm{d}\\boldsymbol{x}^\\top\\mathrm{d}\\boldsymbol{x},\n\\end{equation}\nwhere $\\boldsymbol{x}=(x^1, \\ldots , x^n)^\\top\\in \\mathbb{R}^n$, {$Q(t)$ is} a $t$-dependent symmetric $n\\times n$-matrix. If $Q$ is not a scalar matrix, this metric is not conformally flat.\nPlane waves for which this metric is defined on all of $\\mathbb{R}^{n+2}$, for example, admit non-isometric homotheties with fixed points. Therefore these homotheties are essential (for this implication see for example \\cite[Proposition 2.5]{LeistnerTeisseire22}).\n\n\\bigskip\n\nRecently \nan important rigidity result for essential Lorentzian conformal structures in the \\textit{conformally homogeneous} setting has been obtained by Alekseevsky and Galaev\n in \\cite{AlekseevskyGalaev25}. They showed that if the manifold is simply connected and admits a transitive group of global conformal transformations that is essential, then the manifold is either conformally flat or the conformal class contains a plane wave metric that is isometrically homogeneous and geodesically complete. In the present article we will generalize this result in the following sense: we will drop the assumption that the manifold is simply connected and we will only assume that the manifold is \\textit{locally conformally homogeneous}, i.e.~the local conformal transformations act transitively. We will prove the following result.", + "full_context": "A fundamental problem in conformal geometry is to classify conformal manifolds with \\textit{essential} conformal transformations, that is, global conformal transformations that do not preserve a metric in the conformal class. For Riemannian conformal manifolds the situation is very rigid: up to conformal equivalence only the round sphere and Euclidean space admit essential conformal transformations. This was conjectured by Lichnerowicz in the 1960's and proved in a series of papers by Ferrand and Obata (\\cite{Lelong-Ferrand71}, \\cite{Obata71}, \\cite{Ferrand96}, with contributions in \\cite{Alekseevskii72}).\n\nFor indefinite metrics, the situation is far more flexible: there exist pseudo-Riemannian metrics that are not conformally flat yet admit essential conformal transformations, both on non-compact manifolds and, as shown in \\cite{frances12}, on compact manifolds in signatures other than the Lorentzian one. Non-compact examples in Lorentzian signature consist of plane waves. These are manifolds for which the metric in dimension $n+2$ is locally given by\n\\begin{equation}\\label{pw}\ng=2\\,\\mathrm{d} t\\, \\mathrm{d} v + \\boldsymbol{x}^\\top Q(t) \\boldsymbol{x}\\ \\mathrm{d} t^2 +\\mathrm{d}\\boldsymbol{x}^\\top\\mathrm{d}\\boldsymbol{x},\n\\end{equation}\nwhere $\\boldsymbol{x}=(x^1, \\ldots , x^n)^\\top\\in \\mathbb{R}^n$, {$Q(t)$ is} a $t$-dependent symmetric $n\\times n$-matrix. If $Q$ is not a scalar matrix, this metric is not conformally flat.\nPlane waves for which this metric is defined on all of $\\mathbb{R}^{n+2}$, for example, admit non-isometric homotheties with fixed points. Therefore these homotheties are essential (for this implication see for example \\cite[Proposition 2.5]{LeistnerTeisseire22}).\n\n\\bigskip\n\nRecently \nan important rigidity result for essential Lorentzian conformal structures in the \\textit{conformally homogeneous} setting has been obtained by Alekseevsky and Galaev\n in \\cite{AlekseevskyGalaev25}. They showed that if the manifold is simply connected and admits a transitive group of global conformal transformations that is essential, then the manifold is either conformally flat or the conformal class contains a plane wave metric that is isometrically homogeneous and geodesically complete. In the present article we will generalize this result in the following sense: we will drop the assumption that the manifold is simply connected and we will only assume that the manifold is \\textit{locally conformally homogeneous}, i.e.~the local conformal transformations act transitively. We will prove the following result.\n\nRecently \nan important rigidity result for essential Lorentzian conformal structures in the \\textit{conformally homogeneous} setting has been obtained by Alekseevsky and Galaev\n in \\cite{AlekseevskyGalaev25}. They showed that if the manifold is simply connected and admits a transitive group of global conformal transformations that is essential, then the manifold is either conformally flat or the conformal class contains a plane wave metric that is isometrically homogeneous and geodesically complete. In the present article we will generalize this result in the following sense: we will drop the assumption that the manifold is simply connected and we will only assume that the manifold is \\textit{locally conformally homogeneous}, i.e.~the local conformal transformations act transitively. We will prove the following result.\n\nWhen the pseudo-group $\\mathbf{P}$ does not preserve any metric in the $g$-conformal class, we say that $(M,g)$ is \\textit{weakly essential}, as introduced in Section \\ref{Section: terminology}. This is a weaker condition than being essential.\n Our proof starts off similarly as in \\cite{AlekseevskyGalaev25} by noting that weak essentiality yields the existence of an element in the isotropy group that is not contained in $\\SO(T_pM)$.\nIn \\cite{AlekseevskyGalaev25}, simple connectedness and global conformal homogeneity ensure that the isotropy group is connected, which in turn provides a specific element in its Lie algebra (that does not lie in the isometry algebra). In our setting, however, we cannot directly conclude the existence of such an element. Therefore, we adopt a different approach. We work only with the local isotropy of local conformal transformations, but we use powerful tools from Gromov's theory of rigid transformations (see Appendix \\ref{Gromov's Theory}) and from algebraic groups, which allow us to reduce the problem to the Lie algebra level (see Section \\ref{Section: isotropy}), and in particular to obtain a hyperbolic element in the isotropy algebra that is not in $\\so(T_pM)$.\nSince we assume only local homogeneity, we can pass to the universal cover, where weak essentiality is preserved. Moreover, on a simply connected real-analytic manifold, local conformal vector fields extend to global ones. We may therefore work with the Lie algebra of conformal vector fields on the universal cover. Loosely speaking, by weakening the assumption of homogeneity to local homogeneity, we are able to overcome the assumption of simple connectedness. \nThen, in Section \\ref{Section: Heis algebra}, we use the decomposition of the Lie algebra of conformal vector fields induced by the hyperbolic element to derive the existence of a codimension-one lightlike foliation of \\textit{Heisenberg type}, i.e., with a transitive action of a Heisenberg algebra. \nOur approach, together with the new characterization of plane waves obtained in Section \\ref{Section: Determining a plane wave by Killing fields}, not only leads to the generalization in Theorem \\ref{maintheo}, but also provides a more conceptual approach and simplifies the proof in \\cite{AlekseevskyGalaev25} for the simply connected globally homogeneous case.\n\nIn the last section, we consider the consequence of our result for {\\bf Penrose limits} of locally conformally homogeneous Lorentzian manifolds. \nThe Penrose limit is a famous construction in gravitational physics, which shows that any Lorentzian manifold admits a plane wave spacetime as a limit, this limit being taken along any lightlike geodesic (see \\cite{Blau-PL} for instance). \nAs shown in Theorem \\ref{maintheo}, a weakly essential locally conformally homogeneous Lorentzian manifold of dimension $\\geq 3$ is either conformally flat or locally conformal to a homogeneous plane wave.\nA natural question is then whether this plane wave coincides with the Penrose limit of the Lorentzian manifold along some lightlike geodesic. A priori, the Penrose limit is a plane wave associated with a metric, rather than with a conformal structure.\nThe underlying question is then whether two conformally related metrics have conformally equivalent Penrose limits along some lightlike geodesic (note that conformally related metrics have the same unparameterized lightlike geodesics). It turns out that the answer is affirmative, so the Penrose limit is a conformal invariant, as shown in Proposition~\\ref{Penrose-prop}.\nAs a consequence, we obtain that the plane wave metric in the conformal class of a weakly essential locally conformally homogeneous Lorentzian manifold $(M,g)$ (as in item (3) of Theorem~\\ref{maintheo}) coincides with the Penrose limit of $(M,g)$ along some null geodesic.\n\\begin{corollary} \nLet $(M, g)$ be a non-conformally flat Lorentzian manifold of dimension at least $3$ which is locally conformally homogeneous. Assume that $(M,g)$ is weakly essential. Then, $(M,g)$ is locally conformally equivalent to its Penrose limit along some null geodesic. \n\\end{corollary}\nThis raises the question of whether a direct proof of this fact can be found, which we plan to investigate in future work.\n\n\\begin{proposition}\\label{distributionprop}\nLet $\\g$ be the algebra of conformal vector fields of $M$, and assume that $\\g$ acts transitively on $M$. \nFix a point \n$p\\in M$, and let $\\g_p$ be the isotropy subalgebra at $p$, with image $\\h_p$ under the isotropy representation $\\rho_p$. \nLet $V_p$ be a subspace of $T_pM$ that is invariant under $\\h_p \\subset \\co(T_pM)$. Then\n\\begin{enumerate}\n\\item There is a vector distribution $V$ with $V_p$ as fiber at $p$ that is invariant under $\\g$, i.e. $\\d\\psi_p(V_p)=V_{\\psi(p)}$ for all local conformal transformations $\\psi \\in \\mathbf{G}$ acting locally on $M$.\n\\item If there is a Lie algebra $\\l\\subset \\g$ with stabilizer algebra $\\l_p$ such that the isomorphism $T_pM\\simeq \\g/\\g_p$ restricts to an isomorphism $V_p\\simeq \\l/\\l_p$, then $V$ is involutive with fiber at $\\psi(p)$ spanned by $\\Ad_\\psi (\\l)|_{\\psi(p)}$, for $\\psi\\in \\mathbf{G}$ acting locally on $M$. \nIn particular, if $\\l$ is an ideal of $\\g$, then $V_q$ is spanned by $\\l|_q$ for each $q\\in M$.\n\\item If $\\mathbf{L}$ is the Lie group corresponding to $\\l$, then the leaves through $q=\\psi(p)$ are given \nby the orbits of $q$ under $\\psi \\mathbf L \\psi^{-1}$. \nIn particular, if $\\l$ is an ideal, then the integral manifold through $q$ is the orbit of $q$ under $\\mathbf{L}$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proposition}\\label{Prop. existence of foliation}\nLet $(M,g)$ be a Lorentzian manifold of dimension $n+2$, with a transitive Lie algebra $\\g$ of conformal vector fields, and stabilizer algebra $\\h$ at $p\\in M$.\nAssume that \n$\\h \\subset \\g^0 \\oplus \\g^{\\pm 1}$ and $\\g^0=\\R B \\oplus \\h^{00}$ with $B=\\mathrm{diag}(\\alpha+1,\\alpha-1,\\alpha,\\ldots, \\alpha)$. \nThen there is a metric $\\hat g$ in the conformal class of $g$ and \nan algebra of Killing vector fields $\\g_1$ for $\\hat g$ that acts locally transitively at $p$. Moreover, \n\\begin{enumerate}\n \\item If $\\alpha \\notin \\left\\{\\frac{1}{2}, 1, 2\\right\\}$, then $(M,g)$ is conformally flat.\n\n\\begin{corollary}\\label{Penrose-cor}\nLet $(M, g)$ be a non-conformally flat Lorentzian manifold of dimension at least $3$ which is locally conformally homogeneous. Assume that $(M,g)$ is weakly essential. Then, $(M,g)$ \n is locally conformally equivalent to its Penrose limit along some null geodesic. \n\\end{corollary}\nNote that the proof of Theorem~\\ref{maintheo} reveals more details than those stated in this corollary. The assumption implies that there is a null line bundle that is invariant under the local conformal vector fields, which is spanned by the parallel vector field of the plane wave, and that $(M,g)$ is locally conformally equivalent to the Penrose limit along all null geodesics that are \\textit{transversal} to this null line bundle.", + "post_theorem_intro_text_len": 4292, + "post_theorem_intro_text": "When the pseudo-group $\\mathbf{P}$ does not preserve any metric in the $g$-conformal class, we say that $(M,g)$ is \\textit{weakly essential}, as introduced in Section \\ref{Section: terminology}. This is a weaker condition than being essential.\n Our proof starts off similarly as in \\cite{AlekseevskyGalaev25} by noting that weak essentiality yields the existence of an element in the isotropy group that is not contained in $\\mathbf{SO}(T_pM)$.\nIn \\cite{AlekseevskyGalaev25}, simple connectedness and global conformal homogeneity ensure that the isotropy group is connected, which in turn provides a specific element in its Lie algebra (that does not lie in the isometry algebra). In our setting, however, we cannot directly conclude the existence of such an element. Therefore, we adopt a different approach. We work only with the local isotropy of local conformal transformations, but we use powerful tools from Gromov's theory of rigid transformations (see Appendix \\ref{Gromov's Theory}) and from algebraic groups, which allow us to reduce the problem to the Lie algebra level (see Section \\ref{Section: isotropy}), and in particular to obtain a hyperbolic element in the isotropy algebra that is not in $\\mathfrak{so}(T_pM)$.\nSince we assume only local homogeneity, we can pass to the universal cover, where weak essentiality is preserved. Moreover, on a simply connected real-analytic manifold, local conformal vector fields extend to global ones. We may therefore work with the Lie algebra of conformal vector fields on the universal cover. Loosely speaking, by weakening the assumption of homogeneity to local homogeneity, we are able to overcome the assumption of simple connectedness. \nThen, in Section \\ref{Section: Heis algebra}, we use the decomposition of the Lie algebra of conformal vector fields induced by the hyperbolic element to derive the existence of a codimension-one lightlike foliation of \\textit{Heisenberg type}, i.e., with a transitive action of a Heisenberg algebra. \nOur approach, together with the new characterization of plane waves obtained in Section \\ref{Section: Determining a plane wave by Killing fields}, not only leads to the generalization in Theorem \\ref{maintheo}, but also provides a more conceptual approach and simplifies the proof in \\cite{AlekseevskyGalaev25} for the simply connected globally homogeneous case.\n\n \\bigskip\n\nIn the last section, we consider the consequence of our result for {\\bf Penrose limits} of locally conformally homogeneous Lorentzian manifolds. \nThe Penrose limit is a famous construction in gravitational physics, which shows that any Lorentzian manifold admits a plane wave spacetime as a limit, this limit being taken along any lightlike geodesic (see \\cite{Blau-PL} for instance). \nAs shown in Theorem \\ref{maintheo}, a weakly essential locally conformally homogeneous Lorentzian manifold of dimension $\\geq 3$ is either conformally flat or locally conformal to a homogeneous plane wave.\nA natural question is then whether this plane wave coincides with the Penrose limit of the Lorentzian manifold along some lightlike geodesic. A priori, the Penrose limit is a plane wave associated with a metric, rather than with a conformal structure.\nThe underlying question is then whether two conformally related metrics have conformally equivalent Penrose limits along some lightlike geodesic (note that conformally related metrics have the same unparameterized lightlike geodesics). It turns out that the answer is affirmative, so the Penrose limit is a conformal invariant, as shown in Proposition~\\ref{Penrose-prop}.\nAs a consequence, we obtain that the plane wave metric in the conformal class of a weakly essential locally conformally homogeneous Lorentzian manifold $(M,g)$ (as in item (3) of Theorem~\\ref{maintheo}) coincides with the Penrose limit of $(M,g)$ along some null geodesic.\n\\begin{corollary} \nLet $(M, g)$ be a non-conformally flat Lorentzian manifold of dimension at least $3$ which is locally conformally homogeneous. Assume that $(M,g)$ is weakly essential. Then, $(M,g)$ is locally conformally equivalent to its Penrose limit along some null geodesic. \n\\end{corollary}\nThis raises the question of whether a direct proof of this fact can be found, which we plan to investigate in future work.", + "sketch": "To prove Theorem~\\ref{maintheo}, the argument is described as follows.\n\n- Define the weakly essential case: when the pseudo-group $\\mathbf{P}$ does not preserve any metric in the $g$-conformal class.\n\n- As in \\cite{AlekseevskyGalaev25}, start by using weak essentiality to get “the existence of an element in the isotropy group that is not contained in $\\mathbf{SO}(T_pM)$.” The earlier approach relied on connectedness of the isotropy group to “provide a specific element in its Lie algebra,” but “in our setting … we cannot directly conclude the existence of such an element.”\n\n- Use a different approach: “work only with the local isotropy of local conformal transformations,” and apply “powerful tools from Gromov's theory of rigid transformations … and from algebraic groups,” which “allow us to reduce the problem to the Lie algebra level … and in particular to obtain a hyperbolic element in the isotropy algebra that is not in $\\mathfrak{so}(T_pM)$.”\n\n- Because only local homogeneity is assumed, “pass to the universal cover, where weak essentiality is preserved.” On a “simply connected real-analytic manifold, local conformal vector fields extend to global ones,” so one can “work with the Lie algebra of conformal vector fields on the universal cover.”\n\n- Then “use the decomposition of the Lie algebra of conformal vector fields induced by the hyperbolic element to derive the existence of a codimension-one lightlike foliation of \\textit{Heisenberg type}, i.e., with a transitive action of a Heisenberg algebra.”\n\n- Finally, combine this with “the new characterization of plane waves obtained in Section~\\ref{Section: Determining a plane wave by Killing fields},” which “leads to the generalization in Theorem~\\ref{maintheo}.”", + "expanded_sketch": "To prove the main theorem, the argument is described as follows.\n\n- Define the weakly essential case: when the pseudo-group $\\mathbf{P}$ does not preserve any metric in the $g$-conformal class.\n\n- As in \\cite{AlekseevskyGalaev25}, start by using weak essentiality to get “the existence of an element in the isotropy group that is not contained in $\\mathbf{SO}(T_pM)$.” The earlier approach relied on connectedness of the isotropy group to “provide a specific element in its Lie algebra,” but “in our setting … we cannot directly conclude the existence of such an element.”\n\n- Use a different approach: “work only with the local isotropy of local conformal transformations,” and apply “powerful tools from Gromov's theory of rigid transformations … and from algebraic groups,” which “allow us to reduce the problem to the Lie algebra level … and in particular to obtain a hyperbolic element in the isotropy algebra that is not in $\\mathfrak{so}(T_pM)$.”\n\n- Because only local homogeneity is assumed, “pass to the universal cover, where weak essentiality is preserved.” On a “simply connected real-analytic manifold, local conformal vector fields extend to global ones,” so one can “work with the Lie algebra of conformal vector fields on the universal cover.”\n\n- Then “use the decomposition of the Lie algebra of conformal vector fields induced by the hyperbolic element to derive the existence of a codimension-one lightlike foliation of \\textit{Heisenberg type}, i.e., with a transitive action of a Heisenberg algebra.”\n\n- Finally, combine this with “the new characterization of plane waves obtained later,” which “leads to the generalization in the main theorem.”", + "expanded_theorem": "\\label{maintheo}Let $(M, g)$ be a Lorentzian manifold of dimension at least $3$ and let $\\bf P$ be its pseudo-group of local conformal transformations, that is the collection \n of all conformal diffeomorphisms between open subsets of $M$. We assume $M$ locally conformally homogeneous, i.e.~$M$ is an orbit of $\\bf P$. Then one of the following possibilities holds:\n\n \\begin{enumerate}\n\n \\item $\\bf P$ preserves a Lorentzian metric in the $g$-conformal class.\n\n \\item $(M, g)$ is conformally flat. \n\n \\item $(M, g)$ is locally conformally equivalent to a plane wave that is isometrically locally homogeneous. More precisely, there exists a homogeneous plane wave $X$, such that \n $(M, [g])$ is modeled on $(\\Conf(X), X)$.\n \\end{enumerate}", + "theorem_type": [ + "Classification or Bijection", + "Universal" + ], + "mcq": { + "question": "Which statement holds for every Lorentzian manifold $(M,g)$ of dimension at least $3$ whose pseudo-group $\\mathbf P$ of local conformal transformations acts transitively on $M$? Here $\\mathbf P$ means all conformal diffeomorphisms between open subsets of $M$, and “locally conformally homogeneous” means that $M$ is a single orbit of $\\mathbf P$. A plane wave is a Lorentzian manifold locally given in dimension $n+2$ by\n\\[\ng=2\\,dt\\,dv+\\boldsymbol{x}^\\top Q(t)\\boldsymbol{x}\\,dt^2+d\\boldsymbol{x}^\\top d\\boldsymbol{x},\n\\]\nwhere $\\boldsymbol{x}\\in\\mathbb R^n$ and $Q(t)$ is a symmetric $n\\times n$ matrix; “homogeneous plane wave” means such a plane wave is isometrically locally homogeneous.", + "correct_choice": { + "label": "A", + "text": "At least one of the following holds: (1) $\\mathbf P$ preserves a Lorentzian metric in the conformal class $[g]$; or (2) $(M,g)$ is conformally flat; or (3) $(M,g)$ is locally conformally equivalent to an isometrically locally homogeneous plane wave. More precisely in case (3), there exists a homogeneous plane wave $X$ such that $(M,[g])$ is modeled on $(\\operatorname{Conf}(X),X)$." + }, + "choices": [ + { + "label": "B", + "text": "Exactly one of the following holds: (1) $\\mathbf P$ preserves a Lorentzian metric in the conformal class $[g]$; or (2) $(M,g)$ is conformally flat; or (3) $(M,g)$ is locally conformally equivalent to an isometrically locally homogeneous plane wave. More precisely in case (3), there exists a homogeneous plane wave $X$ such that $(M,[g])$ is modeled on $(\\operatorname{Conf}(X),X)$." + }, + { + "label": "C", + "text": "At least one of the following holds: (1) $\\mathbf P$ preserves a Lorentzian metric in the conformal class $[g]$; or (2) $(M,g)$ is conformally flat; or (3) $(M,g)$ is locally conformally equivalent to a plane wave." + }, + { + "label": "D", + "text": "At least one of the following holds: (1) after passing to the universal cover of $M$, the lifted pseudo-group $\\mathbf P$ preserves a Lorentzian metric in the lifted conformal class; or (2) $(M,g)$ is conformally flat; or (3) after passing to the universal cover, $(M,g)$ is locally conformally equivalent to an isometrically locally homogeneous plane wave. More precisely in case (3), there exists a homogeneous plane wave $X$ such that the universal cover of $(M,[g])$ is modeled on $(\\operatorname{Conf}(X),X)$." + }, + { + "label": "E", + "text": "If $\\mathbf P$ does not preserve any Lorentzian metric in the conformal class $[g]$, then $(M,g)$ is locally conformally equivalent to an isometrically locally homogeneous plane wave. More precisely, there exists a homogeneous plane wave $X$ such that $(M,[g])$ is modeled on $(\\operatorname{Conf}(X),X)$." + } + ], + "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": "nonexclusive trichotomy", + "template_used": "stronger_trap" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped homogeneous/modeling refinement in the plane-wave alternative", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "conclusion only on universal cover instead of on M itself", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "case_split", + "tampered_component": "omits conformally flat weakly essential branch", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not reveal the classification itself or uniquely signal choice A. It only sets up the setting and asks for the complete classification." + }, + "TAS": { + "score": 1, + "justification": "This is essentially a theorem-classification recall item: the correct option states the full trichotomy. However, it is not a pure tautology because the distractors are nearby variants that force attention to precise qualifiers such as local vs global, exactness, and isometric vs conformal homogeneity." + }, + "GPS": { + "score": 1, + "justification": "The item requires moderate reasoning to distinguish subtle mathematical differences among the options, but it mainly tests precise theorem recognition rather than genuinely generating a conclusion from given premises." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically meaningful: they exploit common failure modes such as weakening the statement, strengthening it improperly, confusing local and global conclusions, and conflating conformal with isometric homogeneity." + }, + "total_score": 6, + "overall_assessment": "A solid, high-precision theorem-recognition MCQ with strong distractors and little answer leakage, though it leans more toward exact recall than deep generative reasoning." + } + }, + { + "id": "2602.03027v2", + "paper_link": "http://arxiv.org/abs/2602.03027v2", + "theorems_cnt": 1, + "theorem": { + "env_name": "conjecture", + "content": "\\label{conj:main}\nThe constant $8/\\pi^2$ admits the following GCF representation:\n\\begin{equation}\\label{eq:conjecture}\n\\frac{8}{\\pi^2} = 1 - \\cfrac{2 \\cdot 1^4 - 1^3}{7 - \\cfrac{2 \\cdot 2^4 - 2^3}{19 - \\cfrac{2 \\cdot 3^4 - 3^3}{37 - \\dots}}}\n\\end{equation}\nwhere the partial denominators are $b_n = 3n^2+3n+1$ and partial numerators are $a_n = -(2n^4-n^3)$.", + "start_pos": 3657, + "end_pos": 4050, + "label": "conj:main" + }, + "ref_dict": { + "conj:main": "\\begin{conjecture}\\label{conj:main}\nThe constant $8/\\pi^2$ admits the following GCF representation:\n\\begin{equation}\\label{eq:conjecture}\n\\frac{8}{\\pi^2} = 1 - \\cfrac{2 \\cdot 1^4 - 1^3}{7 - \\cfrac{2 \\cdot 2^4 - 2^3}{19 - \\cfrac{2 \\cdot 3^4 - 3^3}{37 - \\dots}}}\n\\end{equation}\nwhere the partial denominators are $b_n = 3n^2+3n+1$ and partial numerators are $a_n = -(2n^4-n^3)$.\n\\end{conjecture}" + }, + "pre_theorem_intro_text_len": 1017, + "pre_theorem_intro_text": "The algorithmic generation of conjectures on fundamental constants, exemplified by the Ramanujan Machine \\cite{Raayoni2021}, has introduced new identities involving generalized continued fractions (GCFs). \nHistorically, these identities trace their lineage back to Srinivasa Ramanujan’s profound constructions in his \\textit{Notebooks}, particularly his series for $1/\\pi$. While Ramanujan’s classical identities typically involve quadratic or low-degree polynomial patterns, the conjectures surfaced by the Ramanujan Machine often exhibit a higher structural complexity. One such conjecture involves the constant $8/\\pi^2$, characterized by a polynomial architecture with quartic partial numerators. \nThis identity poses a significant challenge to classical limits due to its quartic degree, which exceeds the polynomial complexity of most historically known GCFs.\nIn this work, we provide a rigorous proof of this identity using the framework of operator factorization for variable-coefficient difference equations.", + "context": "The algorithmic generation of conjectures on fundamental constants, exemplified by the Ramanujan Machine \\cite{Raayoni2021}, has introduced new identities involving generalized continued fractions (GCFs). \nHistorically, these identities trace their lineage back to Srinivasa Ramanujan’s profound constructions in his \\textit{Notebooks}, particularly his series for $1/\\pi$. While Ramanujan’s classical identities typically involve quadratic or low-degree polynomial patterns, the conjectures surfaced by the Ramanujan Machine often exhibit a higher structural complexity. One such conjecture involves the constant $8/\\pi^2$, characterized by a polynomial architecture with quartic partial numerators. \nThis identity poses a significant challenge to classical limits due to its quartic degree, which exceeds the polynomial complexity of most historically known GCFs.\nIn this work, we provide a rigorous proof of this identity using the framework of operator factorization for variable-coefficient difference equations.", + "full_context": "The algorithmic generation of conjectures on fundamental constants, exemplified by the Ramanujan Machine \\cite{Raayoni2021}, has introduced new identities involving generalized continued fractions (GCFs). \nHistorically, these identities trace their lineage back to Srinivasa Ramanujan’s profound constructions in his \\textit{Notebooks}, particularly his series for $1/\\pi$. While Ramanujan’s classical identities typically involve quadratic or low-degree polynomial patterns, the conjectures surfaced by the Ramanujan Machine often exhibit a higher structural complexity. One such conjecture involves the constant $8/\\pi^2$, characterized by a polynomial architecture with quartic partial numerators. \nThis identity poses a significant challenge to classical limits due to its quartic degree, which exceeds the polynomial complexity of most historically known GCFs.\nIn this work, we provide a rigorous proof of this identity using the framework of operator factorization for variable-coefficient difference equations.\n\n\\section{Introduction}\nThe algorithmic generation of conjectures on fundamental constants, exemplified by the Ramanujan Machine \\cite{Raayoni2021}, has introduced new identities involving generalized continued fractions (GCFs). \nHistorically, these identities trace their lineage back to Srinivasa Ramanujan’s profound constructions in his \\textit{Notebooks}, particularly his series for $1/\\pi$. While Ramanujan’s classical identities typically involve quadratic or low-degree polynomial patterns, the conjectures surfaced by the Ramanujan Machine often exhibit a higher structural complexity. One such conjecture involves the constant $8/\\pi^2$, characterized by a polynomial architecture with quartic partial numerators. \nThis identity poses a significant challenge to classical limits due to its quartic degree, which exceeds the polynomial complexity of most historically known GCFs.\nIn this work, we provide a rigorous proof of this identity using the framework of operator factorization for variable-coefficient difference equations.\n\nConjecture~\\eqref{conj:main} presents a notable structural complexity where the complexity of the coefficients (quartic in $n$) seemingly contradicts the simplicity of the limit. In this paper, we provide a formal analytic regularization of this conjecture. By constructing a discrete Green's function analogue through operator factorization, we resolve the recurrence into a summation series, thereby proving the identity.\n\n\\section{Preliminaries}\nThe evaluation of GCFs of the form \n\\begin{equation}\nx = b_0 + \\K_{n=1}^{\\infty} \\left( \\frac{a_n}{b_n} \\right) = b_0 + \\cfrac{a_1}{b_1 + \\cfrac{a_2}{b_2 + \\dots}}\n\\end{equation}\nis fundamentally governed by the relationship between its partial coefficients and its convergents $A_n/B_n$.\n\n\\begin{proposition}\\label{prop:factor}\nFor the conjectured coefficients $b_n = 3n(n+1)+1$ and $a_n = -(2n-1)n^3$, the recurrence admits an exact reduction over $\\mathbb{N}^{+}$ with the auxiliary sequences:\n\\begin{equation}\nc_n = n^2, \\quad d_n = n(2n-1).\n\\end{equation}\n\\end{proposition}\n\n\\subsubsection{Series Representation of the Convergent Ratio}\nThe $n$-th convergent of the continued fraction, $x_n = A_n/B_n$, is formulated by the ratio of \\eqref{eq:Bn_sum_final} to \\eqref{eq:An_product_final}. \nThe common product terms cancel exactly, yielding:\n\\begin{equation}\\label{eq:xn_final}\nx_n = \\frac{A_n}{B_n} = \\left( \\sum_{k=0}^{n} u_k \\right)^{-1}, \\quad \\text{where } u_k = \\frac{\\prod_{j=1}^{k} c_j}{\\prod_{j=1}^{k+1} d_j}.\n\\end{equation}\nThis fundamental equivalence maps the convergence of the polynomial continued fraction to the partial sums of an infinite series. The proof of the conjecture~\\eqref{conj:main} is thereby reduced to the asymptotic evaluation of the sum $S = \\sum_{k=0}^{\\infty} u_k$.\n\nThe convergence of the GCF is predicated on the existence of the limit of the convergents $\\{x_n\\}$. According to~\\eqref{eq:xn_final}, the value of the GCF, denoted by $\\mathcal{K}$, is mapped to the reciprocal of an infinite series:\n\\begin{equation}\\label{eq:limit_mapping}\n\\mathcal{K} = \\lim_{n \\to \\infty} \\frac{A_n}{B_n} = \\left( \\sum_{k=0}^{\\infty} t_k \\right)^{-1}, \\quad \\text{where } t_k = \\frac{\\prod_{j=1}^{k} c_j}{\\prod_{j=1}^{k+1} d_j}.\n\\end{equation}\n\n\\begin{theorem}[Pincherle's Theorem~\\cite{Gautschi1967}]\nThe generalized continued fraction $b_0 + \\K_{n=1}^\\infty (a_n / b_n)$ converges to a finite value if and only if the associated second-order linear difference equation \n\\begin{equation}\\label{eq:rec_pincherle}\n y_n = b_n y_{n-1} + a_n y_{n-2}\n\\end{equation}\nadmits a minimal solution $\\{f_n\\}$. A non-trivial solution $\\{f_n\\}$ is minimal if, for any solution $\\{g_n\\}$ linearly independent of $\\{f_n\\}$, the following asymptotic condition is satisfied:\n\\begin{equation}\n \\lim_{n \\to \\infty} \\frac{f_n}{g_n} = 0.\n\\end{equation}\nIf such a solution exists, the GCF converges to $f_0/f_{-1}$.\n\\end{theorem}\n\nCrucially, from the perspective of Pincherle’s Theorem, the fact that $S = \\pi^2/8$ is a \\textbf{finite constant} ensures that $B_n$ grows at an asymptotically faster rate than $A_n$ in the sense of the solution hierarchy. Any solution not proportional to $\\{A_n\\}$ will eventually be dominated by the summation-induced growth. Thus, the existence of this convergent series confirms $\\{A_n\\}$ as the minimal solution, and the GCF converges to the reciprocal of the sum:\n\\begin{equation}\n \\mathcal{K} = S^{-1} = \\left( \\sum_{m=1}^{\\infty} \\frac{2^m}{m^2 \\binom{2m}{m}} \\right)^{-1} = \\frac{8}{\\pi^2}.\n\\end{equation}\nThe proof is complete.\n\\end{proof}\n\n\\begin{conjecture}\\label{conj:main}\nThe constant $8/\\pi^2$ admits the following GCF representation:\n\\begin{equation}\\label{eq:conjecture}\n\\frac{8}{\\pi^2} = 1 - \\cfrac{2 \\cdot 1^4 - 1^3}{7 - \\cfrac{2 \\cdot 2^4 - 2^3}{19 - \\cfrac{2 \\cdot 3^4 - 3^3}{37 - \\dots}}}\n\\end{equation}\nwhere the partial denominators are $b_n = 3n^2+3n+1$ and partial numerators are $a_n = -(2n^4-n^3)$.\n\\end{conjecture}", + "post_theorem_intro_text_len": 423, + "post_theorem_intro_text": "Conjecture~\\eqref{conj:main} presents a notable structural complexity where the complexity of the coefficients (quartic in $n$) seemingly contradicts the simplicity of the limit. In this paper, we provide a formal analytic regularization of this conjecture. By constructing a discrete Green's function analogue through operator factorization, we resolve the recurrence into a summation series, thereby proving the identity.", + "sketch": "To prove Conjecture~\\eqref{conj:main}, the paper \"provide[s] a formal analytic regularization\" of the continued fraction. The argument proceeds by \"constructing a discrete Green's function analogue through operator factorization,\" and then using this construction to \"resolve the recurrence into a summation series,\" which \"thereby prov[es] the identity.\"", + "expanded_sketch": "To prove the main conjecture, the paper \"provide[s] a formal analytic regularization\" of the continued fraction. The argument proceeds by \"constructing a discrete Green's function analogue through operator factorization,\" and then using this construction to \"resolve the recurrence into a summation series,\" which \"thereby prov[es] the identity.\"", + "expanded_theorem": "\\label{conj:main}\nThe constant $8/\\pi^2$ admits the following GCF representation:\n\\begin{equation}\\label{eq:conjecture}\n\\frac{8}{\\pi^2} = 1 - \\cfrac{2 \\cdot 1^4 - 1^3}{7 - \\cfrac{2 \\cdot 2^4 - 2^3}{19 - \\cfrac{2 \\cdot 3^4 - 3^3}{37 - \\dots}}}\n\\end{equation}\nwhere the partial denominators are $b_n = 3n^2+3n+1$ and partial numerators are $a_n = -(2n^4-n^3)$.,", + "theorem_type": [ + "Existence", + "Algorithmic or Constructive" + ], + "mcq": { + "question": "Let\n\\[\n1+\\K_{n=1}^{\\infty}\\left(\\frac{a_n}{b_n}\\right)\n\\]\ndenote a generalized continued fraction, where\n\\[\na_n=-(2n^4-n^3)=-(2n-1)n^3,\\qquad b_n=3n^2+3n+1=3n(n+1)+1\\quad (n\\ge 1).\n\\]\nEquivalently, this is\n\\[\n1- \\cfrac{2\\cdot 1^4-1^3}{7-\\cfrac{2\\cdot 2^4-2^3}{19-\\cfrac{2\\cdot 3^4-3^3}{37-\\ddots}}}.\n\\]\nWhich explicit value does this generalized continued fraction represent?", + "correct_choice": { + "label": "A", + "text": "It equals \\(\\dfrac{8}{\\pi^2}\\); that is,\n\\[\n\\frac{8}{\\pi^2}=1- \\cfrac{2\\cdot 1^4-1^3}{7-\\cfrac{2\\cdot 2^4-2^3}{19-\\cfrac{2\\cdot 3^4-3^3}{37-\\ddots}}}.\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "It equals \\(\\dfrac{4}{\\pi^2}\\); that is,\n\\[\n\\frac{4}{\\pi^2}=1- \\cfrac{2\\cdot 1^4-1^3}{7-\\cfrac{2\\cdot 2^4-2^3}{19-\\cfrac{2\\cdot 3^4-3^3}{37-\\ddots}}}.\n\\]" + }, + { + "label": "C", + "text": "It equals the reciprocal of a finite positive constant; equivalently, the generalized continued fraction converges to some real number \\(L>0\\).\n\\[\n1- \\cfrac{2\\cdot 1^4-1^3}{7-\\cfrac{2\\cdot 2^4-2^3}{19-\\cfrac{2\\cdot 3^4-3^3}{37-\\ddots}}}=L.\n\\]" + }, + { + "label": "D", + "text": "It equals \\(\\dfrac{8}{\\pi}\\); that is,\n\\[\n\\frac{8}{\\pi}=1- \\cfrac{2\\cdot 1^4-1^3}{7-\\cfrac{2\\cdot 2^4-2^3}{19-\\cfrac{2\\cdot 3^4-3^3}{37-\\ddots}}}.\n\\]" + }, + { + "label": "E", + "text": "It equals \\(\\left(\\sum_{m=1}^{\\infty}\\dfrac{2^m}{m^2\\binom{2m}{m}}\\right)^{-1}\\), and this reciprocal sum is \\(\\dfrac{6}{\\pi^2}\\); that is,\n\\[\n1- \\cfrac{2\\cdot 1^4-1^3}{7-\\cfrac{2\\cdot 2^4-2^3}{19-\\cfrac{2\\cdot 3^4-3^3}{37-\\ddots}}}=\\left(\\sum_{m=1}^{\\infty}\\frac{2^m}{m^2\\binom{2m}{m}}\\right)^{-1}=\\frac{6}{\\pi^2}.\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": "final_constant_after_series_regularization", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "finiteness", + "tampered_component": "explicit_identification_with_8_over_pi_squared", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "square_on_pi_from_summation_evaluation", + "template_used": "property_confusion" + }, + { + "label": "E", + "sketch_hook_type": "geometric_construction", + "tampered_component": "exact_evaluation_of_resolved_series", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives only the continued fraction data and asks for its value; it does not state or strongly hint at 8/pi^2. There is no explicit answer leakage." + }, + "TAS": { + "score": 2, + "justification": "The item is not a direct restatement of a conclusion already embedded in the stem. It asks the solver to identify an explicit evaluation among competing alternatives." + }, + "GPS": { + "score": 2, + "justification": "Determining the exact constant represented by this nontrivial generalized continued fraction requires substantial mathematical recognition or derivation, so the correct answer is not obvious from surface features alone." + }, + "DQS": { + "score": 1, + "justification": "Several distractors are plausible mathematical confusions (4/pi^2, 8/pi, and a related-looking reciprocal sum), but choice C is a weaker true statement rather than a genuinely false distractor, which weakens single-answer integrity." + }, + "total_score": 7, + "overall_assessment": "A strong, non-leaky, reasoning-heavy MCQ, but its distractor set is weakened by the inclusion of a true-but-weaker option." + } + }, + { + "id": "2602.03179v1", + "paper_link": "http://arxiv.org/abs/2602.03179v1", + "theorems_cnt": 3, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm-finite.interp}\n The following three statements are equivalent.\n \\begin{enumerate}\n \\item There exists a valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\dots,r$.\n \\item There exists a quasi-monomial valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\dots,r$.\n \\item The equality $\\vbI(\\a_1\\cdots\\a_r)=\\sum_{j=1}^r b_j$ holds.\n \\end{enumerate}", + "start_pos": 5782, + "end_pos": 6192, + "label": "thm-finite.interp" + }, + "ref_dict": { + "thm-infinite.interp": "\\begin{theorem}\\label{thm-infinite.interp}\n Additionally assume $R$ is local with the unique maximal ideal $\\m$. Let $(\\a_j)_{j\\in\\bbZ_+}$ be a countable sequence of nonzero ideals in $R$ such that $\\sqrt{\\sum_{j=1}^{\\infty} \\a_j}=\\m$, and let $(b_j)_{j\\in\\bbZ_+}$ be a sequence of positive real numbers. Then the following statements are equivalent:\n \\begin{enumerate}\n \\item There exists a valuation $v$ on $R$ centered at $\\m$ with $A(v)<\\infty$ such that $v(\\a_j)=b_j$ for all $j\\ge 1$;\n \\item For every $r\\ge 1$,\n \\[\\vb_{I_{\\blt}^{(r)}}\\big(\\a_1\\cdots\\a_r\\big)=\\sum_{j=1}^r b_j,\\]\n and it holds that\n \\[\\sup_{r\\ge 1}\\sup_{k\\in\\bbN}\\Big(\\lct^{\\a_1^k\\cdots\\a_r^k}\\big( I^{(r)}_{\\blt}\\big)-k\\sum_{j=1}^r b_j\\Big)<\\infty.\\]\n \\end{enumerate}\n Here $I^{(r)}_{\\blt}=\\sum_{j=1}^{r}\\big(\\frac{1}{b_j}\\cdot \\a_j\\big)_{\\blt}$.\n\\end{theorem}", + "subsec-gr.fil": "\\begin{enumerate}\n \\item There exists a valuation $v$ on $R$ centered at $\\m$ with $A(v)<\\infty$ such that $v(\\a_j)=b_j$ for all $j\\ge 1$;\n \\item For every $r\\ge 1$,\n \\[\\vb_{I_{\\blt}^{(r)}}\\big(\\a_1\\cdots\\a_r\\big)=\\sum_{j=1}^r b_j,\\]\n and it holds that\n \\[\\sup_{r\\ge 1}\\sup_{k\\in\\bbN}\\Big(\\lct^{\\a_1^k\\cdots\\a_r^k}\\big( I^{(r)}_{\\blt}\\big)-k\\sum_{j=1}^r b_j\\Big)<\\infty.\\]\n \\end{enumerate}\n Here $I^{(r)}_{\\blt}=\\sum_{j=1}^{r}\\big(\\frac{1}{b_j}\\cdot \\a_j\\big)_{\\blt}$.\n\\end{theorem}\n\n\\vspace{.1in} {\\bf Organization}. This paper is organized as follows. In \\Cref{sec-r.o}, we recall the notion of asymptotic Samuel function and quickly prove the implication (1) $\\Rightarrow$ (3) in \\Cref{thm-finite.interp}. In \\Cref{sec-f.v.i}, we prove \\Cref{thm-finite.interp} by studying the asymptotic behavior of jumping numbers and an extremal problem on a cone $\\QM(Y,D)$ of quasi-monomial valuations. In \\Cref{sec-i.v.i}, we prove \\Cref{thm-infinite.interp} using valuation approximation, a strategy that more closely parallels the approach in \\cite{BGMY25}. Finally, we demonstrate how \\Cref{thm-finite.interp} recovers a result in \\cite{BGMY25} (which deals with the complex analytic case) and yields an analytic characterization, derived from \\Cref{thm-infinite.interp}, for the existence of infinite valuative interpolation.\n\n\\vspace{.1in} {\\em Acknowledgements}. The first-named author completed this work during a visit to the School of Mathematical Sciences at Peking University and is grateful for its hospitality and support. The second-named author was supported by National Key R\\&D Program of China 2021YFA1003100 and NSFC-12425101. The third-named author was supported by NSFC-12401099 and the Talent Fund of Beijing Jiaotong University 2024-004. The fourth-named author was supported by NSFC-12501106.\n\n\\section{Valuation and asymptotic Samuel function}\\label{sec-r.o}\n\nIn this section, $R$ is a noetherian integral domain.\n\n\\subsection{Valuation}\\label{subsec-val}\n\nRecall that a function $v\\colon R^*=R\\setminus\\{0\\} \\to \\bbR_{\\ge 0}$ is a (real) \\emph{valuation} if\n\\[v(xy)=v(x)+v(y) \\quad \\& \\quad v(x+y)\\ge \\min\\{v(x),v(y)\\},\\]\nfor all $x,y\\in R^*$. Set $v(0)=\\infty$. A valuation $v$ on $R$ can be extended uniquely to a valuation $v\\colon K\\to \\bbR\\cup\\{\\infty\\}$ on the field of fractions $K$ of $R$ by setting $v(x/y)=v(x)-v(y)$. Say $v\\le w$ for two valuations $v,w$ on $R$ if $v(x)\\le w(x)$ for all $x\\in R^*$. For any nonzero ideal $\\a$ of $R$, define $v(\\a)=\\inf_{x\\in \\a}v(x)$. Then it is easy to check: for nonzero ideals $\\a,\\b$ in $R$,\n\\[v(\\a\\b)=v(\\a)+v(\\b) \\quad \\& \\quad v(\\a+\\b)= \\min\\{v(\\a),v(\\b)\\}.\\]\nIn particular, we can see if the ideals $\\a_1,\\ldots,\\a_r$ in $R$ satisfies $\\sum_{j=1}^r\\a_j=R$, then\n\\[0=v(R)=v\\big(\\sum \\a_j\\big)=\\min v(\\a_j),\\]\nwhich implies $v(\\a_j)=0$ for some $j\\in\\{1,\\ldots,r\\}$.\n\n\\subsection{Graded sequence and filtration of ideals}\\label{subsec-gr.fil}\n\nA \\emph{graded sequence} of ideals $\\a_{\\blt}=(\\a_m)_{m\\in\\bbZ_{>0}}$ is a sequence of ideals in $R$ that satisfies $\\a_p\\cdot \\a_p\\subseteq \\a_{p+q}$ for all $p,q\\ge 1$. Put $\\a_0=R$. We always assume $\\a_m\\neq (0)$ for some $m\\ge 1$. For example, $v$ is a nontrivial valuation on $R$, then $\\a_{\\blt}^v$ is a graded sequence of ideals in $R$ defined by $\\a_{m}^v=\\{x\\in R \\mid v(x)\\ge m\\}$. \n\nNow let $v$ be a valuation on $R$, and $\\a_{\\blt}$ a graded sequence of ideals in $R$. Then $v(\\a_{p+q})\\le v(\\a_p)+v(\\a_q)$ for all $p,q\\ge 1$, which implies the limit\n\\[v(\\a_{\\blt})\\coloneqq \\lim_{m\\to \\infty, \\, \\a_m\\neq (0)}\\frac{v(\\a_m)}{m}=\\inf_{m\\ge 1}\\frac{v(\\a_m)}{m}\\]\nexists by Fekete's lemma. In particular, $v(\\a_{\\blt}^v)=1$ for any nontrivial valuation $v$ on $R$. A graded sequence $\\a_{\\blt}=\\{\\a_m\\}_{m\\in\\bbN}$ of ideals in $R$ is called a \\emph{filtration} if additionally it satisfies $\\a_m\\supseteq \\a_{m+1}$ for every $m\\in\\bbN$.\n\n\\subsection{Asymptotic Samuel function}\n\nThe classical asymptotic Samuel function (\\cite{Sa52}) has been generalized to the filtration case in \\cite{CP24}. Let us recall the constructions and some basic properties. Let $J_{\\blt}$ be a filtration of ideals in $R$. For every ideal $\\a$ in $R$, the \\emph{order} of $\\a$ with respect to $J_{\\blt}$ is defined by\n\\[\\vJ\\big(\\a\\big)\\coloneqq \\sup\\,\\{m\\in\\bbN \\mid \\a\\subseteq J_m\\}\\in\\bbN\\cup\\{\\infty\\}.\\]\nThen we can see $\\vJ(\\a^{p+q})\\ge \\vJ(\\a^p)+\\vJ(\\a^q)$ for every $p,q\\in\\bbN$. Therefore, it follows that\n\\[\\vb(\\a;J_{\\blt})\\coloneqq \\lim_{m\\to\\infty}\\frac{\\vJ(\\a^m)}{m}=\\sup_{m\\ge 1}\\frac{\\vJ(\\a^m)}{m}\\in \\bbR_{\\ge 0}\\cup \\{\\infty\\}\\]\nis well-defined by Fekete's lemma, called the \\emph{asymptotic Samuel function} of $J_{\\blt}$ (cf. \\cite[Theorem 3.4]{CP24}). We also denote $\\vb(\\a;J_{\\blt})\\coloneqq \\vbJ(\\a)$ in the present paper.\n\n\\begin{example}\\label{ex-asf}\n If $J_{\\blt}=\\{J^m\\}_{m\\ge 0}$ for a nonzero ideal $J$ in $R$, then $\\vb(\\a;J_{\\blt})=\\ol{\\nu}_{J}(\\a)$, which is the classical asymptotic Samuel function (cf. \\cite{Sa52,HS06,LJT09}).\n\\end{example}\n\n\\begin{example}\n If we take $J_{\\blt}=\\a_{\\blt}^v$ the filtration of valuation ideals associated to some valuation $v$ on $R$, then clearly $\\vb(\\a;J_{\\blt})=v(\\a)$.\n\\end{example}\n\nThe following lemma shows some basis properties of the asymptotic Samuel function, which are actually known in \\cite{CP24}. The proof below follows from the strategy of \\cite[Theorem 2]{Sa52}.\n\n\\begin{lemma}[{see \\cite[Proposition 3.11]{CP24}}]\\label{lem-mu.subadditivity}\n Let $J_{\\blt}$ be a filtration of ideals in $R$. Then\n \\begin{enumerate}\n \\item $\\vb(\\a;J_{\\blt})\\ge \\vb(\\b;J_{\\blt})$, for ideals $\\a\\subseteq\\b$ in $R$;\n \\item $\\vb(\\a^k;J_{\\blt})=k\\cdot\\vb(\\a;J_{\\blt})$, where $\\a$ is an ideal in $R$ and $k\\in\\bbZ_+$;\n \\item $\\vb(\\a+\\b;J_{\\blt})= \\min\\{\\vb(\\a;J_{\\blt}), \\vb(\\b;J_{\\blt})\\}$, where $\\a,\\b$ are ideals in $R$;\n \\item $\\vb(\\a\\b;J_{\\blt})\\ge \\vb(\\a;J_{\\blt}) +\\vb(\\b;J_{\\blt})$, where $\\a,\\b$ are ideals in $R$.\n \\end{enumerate}", + "subsec-val": "\\begin{enumerate}\n \\item There exists a valuation $v$ on $R$ centered at $\\m$ with $A(v)<\\infty$ such that $v(\\a_j)=b_j$ for all $j\\ge 1$;\n \\item For every $r\\ge 1$,\n \\[\\vb_{I_{\\blt}^{(r)}}\\big(\\a_1\\cdots\\a_r\\big)=\\sum_{j=1}^r b_j,\\]\n and it holds that\n \\[\\sup_{r\\ge 1}\\sup_{k\\in\\bbN}\\Big(\\lct^{\\a_1^k\\cdots\\a_r^k}\\big( I^{(r)}_{\\blt}\\big)-k\\sum_{j=1}^r b_j\\Big)<\\infty.\\]\n \\end{enumerate}\n Here $I^{(r)}_{\\blt}=\\sum_{j=1}^{r}\\big(\\frac{1}{b_j}\\cdot \\a_j\\big)_{\\blt}$.\n\\end{theorem}\n\n\\vspace{.1in} {\\bf Organization}. This paper is organized as follows. In \\Cref{sec-r.o}, we recall the notion of asymptotic Samuel function and quickly prove the implication (1) $\\Rightarrow$ (3) in \\Cref{thm-finite.interp}. In \\Cref{sec-f.v.i}, we prove \\Cref{thm-finite.interp} by studying the asymptotic behavior of jumping numbers and an extremal problem on a cone $\\QM(Y,D)$ of quasi-monomial valuations. In \\Cref{sec-i.v.i}, we prove \\Cref{thm-infinite.interp} using valuation approximation, a strategy that more closely parallels the approach in \\cite{BGMY25}. Finally, we demonstrate how \\Cref{thm-finite.interp} recovers a result in \\cite{BGMY25} (which deals with the complex analytic case) and yields an analytic characterization, derived from \\Cref{thm-infinite.interp}, for the existence of infinite valuative interpolation.\n\n\\vspace{.1in} {\\em Acknowledgements}. The first-named author completed this work during a visit to the School of Mathematical Sciences at Peking University and is grateful for its hospitality and support. The second-named author was supported by National Key R\\&D Program of China 2021YFA1003100 and NSFC-12425101. The third-named author was supported by NSFC-12401099 and the Talent Fund of Beijing Jiaotong University 2024-004. The fourth-named author was supported by NSFC-12501106.\n\n\\section{Valuation and asymptotic Samuel function}\\label{sec-r.o}\n\nIn this section, $R$ is a noetherian integral domain.\n\n\\subsection{Valuation}\\label{subsec-val}\n\nRecall that a function $v\\colon R^*=R\\setminus\\{0\\} \\to \\bbR_{\\ge 0}$ is a (real) \\emph{valuation} if\n\\[v(xy)=v(x)+v(y) \\quad \\& \\quad v(x+y)\\ge \\min\\{v(x),v(y)\\},\\]\nfor all $x,y\\in R^*$. Set $v(0)=\\infty$. A valuation $v$ on $R$ can be extended uniquely to a valuation $v\\colon K\\to \\bbR\\cup\\{\\infty\\}$ on the field of fractions $K$ of $R$ by setting $v(x/y)=v(x)-v(y)$. Say $v\\le w$ for two valuations $v,w$ on $R$ if $v(x)\\le w(x)$ for all $x\\in R^*$. For any nonzero ideal $\\a$ of $R$, define $v(\\a)=\\inf_{x\\in \\a}v(x)$. Then it is easy to check: for nonzero ideals $\\a,\\b$ in $R$,\n\\[v(\\a\\b)=v(\\a)+v(\\b) \\quad \\& \\quad v(\\a+\\b)= \\min\\{v(\\a),v(\\b)\\}.\\]\nIn particular, we can see if the ideals $\\a_1,\\ldots,\\a_r$ in $R$ satisfies $\\sum_{j=1}^r\\a_j=R$, then\n\\[0=v(R)=v\\big(\\sum \\a_j\\big)=\\min v(\\a_j),\\]\nwhich implies $v(\\a_j)=0$ for some $j\\in\\{1,\\ldots,r\\}$.\n\n\\subsection{Graded sequence and filtration of ideals}\\label{subsec-gr.fil}\n\nA \\emph{graded sequence} of ideals $\\a_{\\blt}=(\\a_m)_{m\\in\\bbZ_{>0}}$ is a sequence of ideals in $R$ that satisfies $\\a_p\\cdot \\a_p\\subseteq \\a_{p+q}$ for all $p,q\\ge 1$. Put $\\a_0=R$. We always assume $\\a_m\\neq (0)$ for some $m\\ge 1$. For example, $v$ is a nontrivial valuation on $R$, then $\\a_{\\blt}^v$ is a graded sequence of ideals in $R$ defined by $\\a_{m}^v=\\{x\\in R \\mid v(x)\\ge m\\}$. \n\nNow let $v$ be a valuation on $R$, and $\\a_{\\blt}$ a graded sequence of ideals in $R$. Then $v(\\a_{p+q})\\le v(\\a_p)+v(\\a_q)$ for all $p,q\\ge 1$, which implies the limit\n\\[v(\\a_{\\blt})\\coloneqq \\lim_{m\\to \\infty, \\, \\a_m\\neq (0)}\\frac{v(\\a_m)}{m}=\\inf_{m\\ge 1}\\frac{v(\\a_m)}{m}\\]\nexists by Fekete's lemma. In particular, $v(\\a_{\\blt}^v)=1$ for any nontrivial valuation $v$ on $R$. A graded sequence $\\a_{\\blt}=\\{\\a_m\\}_{m\\in\\bbN}$ of ideals in $R$ is called a \\emph{filtration} if additionally it satisfies $\\a_m\\supseteq \\a_{m+1}$ for every $m\\in\\bbN$.\n\n\\subsection{Asymptotic Samuel function}\n\nThe classical asymptotic Samuel function (\\cite{Sa52}) has been generalized to the filtration case in \\cite{CP24}. Let us recall the constructions and some basic properties. Let $J_{\\blt}$ be a filtration of ideals in $R$. For every ideal $\\a$ in $R$, the \\emph{order} of $\\a$ with respect to $J_{\\blt}$ is defined by\n\\[\\vJ\\big(\\a\\big)\\coloneqq \\sup\\,\\{m\\in\\bbN \\mid \\a\\subseteq J_m\\}\\in\\bbN\\cup\\{\\infty\\}.\\]\nThen we can see $\\vJ(\\a^{p+q})\\ge \\vJ(\\a^p)+\\vJ(\\a^q)$ for every $p,q\\in\\bbN$. Therefore, it follows that\n\\[\\vb(\\a;J_{\\blt})\\coloneqq \\lim_{m\\to\\infty}\\frac{\\vJ(\\a^m)}{m}=\\sup_{m\\ge 1}\\frac{\\vJ(\\a^m)}{m}\\in \\bbR_{\\ge 0}\\cup \\{\\infty\\}\\]\nis well-defined by Fekete's lemma, called the \\emph{asymptotic Samuel function} of $J_{\\blt}$ (cf. \\cite[Theorem 3.4]{CP24}). We also denote $\\vb(\\a;J_{\\blt})\\coloneqq \\vbJ(\\a)$ in the present paper.\n\n\\begin{example}\\label{ex-asf}\n If $J_{\\blt}=\\{J^m\\}_{m\\ge 0}$ for a nonzero ideal $J$ in $R$, then $\\vb(\\a;J_{\\blt})=\\ol{\\nu}_{J}(\\a)$, which is the classical asymptotic Samuel function (cf. \\cite{Sa52,HS06,LJT09}).\n\\end{example}\n\n\\begin{example}\n If we take $J_{\\blt}=\\a_{\\blt}^v$ the filtration of valuation ideals associated to some valuation $v$ on $R$, then clearly $\\vb(\\a;J_{\\blt})=v(\\a)$.\n\\end{example}\n\nThe following lemma shows some basis properties of the asymptotic Samuel function, which are actually known in \\cite{CP24}. The proof below follows from the strategy of \\cite[Theorem 2]{Sa52}.\n\n\\begin{lemma}[{see \\cite[Proposition 3.11]{CP24}}]\\label{lem-mu.subadditivity}\n Let $J_{\\blt}$ be a filtration of ideals in $R$. Then\n \\begin{enumerate}\n \\item $\\vb(\\a;J_{\\blt})\\ge \\vb(\\b;J_{\\blt})$, for ideals $\\a\\subseteq\\b$ in $R$;\n \\item $\\vb(\\a^k;J_{\\blt})=k\\cdot\\vb(\\a;J_{\\blt})$, where $\\a$ is an ideal in $R$ and $k\\in\\bbZ_+$;\n \\item $\\vb(\\a+\\b;J_{\\blt})= \\min\\{\\vb(\\a;J_{\\blt}), \\vb(\\b;J_{\\blt})\\}$, where $\\a,\\b$ are ideals in $R$;\n \\item $\\vb(\\a\\b;J_{\\blt})\\ge \\vb(\\a;J_{\\blt}) +\\vb(\\b;J_{\\blt})$, where $\\a,\\b$ are ideals in $R$.\n \\end{enumerate}", + "thm-finite.interp": "\\begin{theorem}\\label{thm-finite.interp}\n The following three statements are equivalent.\n \\begin{enumerate}\n \\item There exists a valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\dots,r$.\n \\item There exists a quasi-monomial valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\dots,r$.\n \\item The equality $\\vbI(\\a_1\\cdots\\a_r)=\\sum_{j=1}^r b_j$ holds.\n \\end{enumerate}\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 2845, + "pre_theorem_intro_text": "Let $R$ be an excellent regular domain of equicharacteristic $0$ and of dimension $n$.\n\nLet $\\a_1,\\ldots,\\a_r$ be $r$ nonzero ideals of $R$, and let $b_1,\\ldots,b_r\\in\\mathbb{R}_{\\ge 0}$. We consider the following valuative interpolation problem which was considered by the authors in the complex analytic setting (e.g. \\cite[Theorem 1.2]{BGMY25} deals with the case where $R=\\O_{\\mathbb{C}^n,o}$ the ring of holomorphic germs):\n\n\\begin{question}\nCan one give a necessary and sufficient condition for the existence of a valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\ldots, r$?\n\\end{question}\n\nWe introduce some notations for explaining our results in the present paper. Suppose $\\mathfrak{a}$ is an ideal on $X$, and $\\lambda\\in\\bbR_{\\ge 0}\\cup\\{\\infty\\}$. Denote by $(\\lambda\\cdot\\mathfrak{a})_{\\bullet}=\\big((\\lambda\\cdot\\mathfrak{a})_m\\big)_{m\\in\\mathbb{N}}$ the \\emph{filtration} (cf. \\Cref{subsec-gr.fil}) of ideals such that each term:\n\\begin{equation*}\n (\\lambda\\cdot\\mathfrak{a})_m=\n \\begin{cases}\n R & \\lambda=0 \\\\\n \\mathfrak{a}^{\\lceil \\lambda m\\rceil} & 0<\\lambda<\\infty \\\\\n 0 & \\lambda=\\infty \\\\\n \\end{cases}\n \\quad \\quad m\\in\\mathbb{N}.\n\\end{equation*}\n\nIf $(\\b_1)_{\\bullet},\\ldots,(\\b_s)_{\\bullet}$ are filtrations of ideals in $R$, then the summation $\\mathfrak{c}_{\\bullet}\\coloneqq \\sum(\\b_i)_{\\bullet}$ of these filtrations is a filtration defined as:\n\\begin{equation*}\n \\mathfrak{c}_{m}=\\sum_{\\substack{m_1+\\cdots+m_s=m \\\\ m_i\\in\\mathbb{N}}}(\\b_1)_{m_1}\\cdots(\\b_s)_{m_s}, \\quad m\\in\\mathbb{N}.\n\\end{equation*}\n\nNow we return to the valuative interpolation problem. For the ideals $\\a_1,\\ldots,\\a_r$ and non-negative real numbers $b_1,\\ldots,b_r$, if all of $b_j$ are zero, then we can directly choose the trivial valuation on $R$, so we only consider the case where at least one of them is nonzero. Hence, may assume $\\sum_{b_j>0}\\a_j\\neq R$, otherwise none of valuations $v$ on $R$ can satisfy $v(\\a_j)=b_j$ for all $j$ (cf. \\Cref{subsec-val}).\n\nConsider the filtration\n\\[I_{\\bullet}\\coloneqq \\Big(\\frac{1}{b_1}\\cdot \\a_{1}\\Big)_{\\bullet}+\\cdots+\\Big(\\frac{1}{b_r}\\cdot \\a_{r}\\Big)_{\\bullet}.\\]\nFollowing Cutkosky and Peaharaj \\cite[Definition 3.5]{CP24}, the \\emph{asymptotic Samuel function} associated with this filtration is defined as follows. For any ideal $\\mathfrak{a}$ of $R$, let\n\\[\\vbI(\\mathfrak{a})\\coloneqq \\lim_{k\\to\\infty}\\frac{\\vI\\big(\\mathfrak{a}^k\\big)}{k}\\in \\bbR_{\\ge 0}\\cup\\{\\infty\\},\\]\nwhere $\\vI\\big(\\mathfrak{a}\\big)\\coloneqq \\sup\\,\\{m\\in\\mathbb{N} \\mid \\mathfrak{a}\\subseteq I_m\\}$ denotes the \\emph{order function} of $I_{\\bullet}$. This construction generalizes the classical asymptotic Samuel function \\cite{Sa52} and serves as an algebraic analogue of the relative type introduced in complex analytic settings \\cite{Ras06} (cf. \\Cref{sec-c.a.c}).", + "context": "Let $\\a_1,\\ldots,\\a_r$ be $r$ nonzero ideals of $R$, and let $b_1,\\ldots,b_r\\in\\mathbb{R}_{\\ge 0}$. We consider the following valuative interpolation problem which was considered by the authors in the complex analytic setting (e.g. \\cite[Theorem 1.2]{BGMY25} deals with the case where $R=\\O_{\\mathbb{C}^n,o}$ the ring of holomorphic germs):\n\n\\begin{question}\nCan one give a necessary and sufficient condition for the existence of a valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\ldots, r$?\n\\end{question}\n\nWe introduce some notations for explaining our results in the present paper. Suppose $\\mathfrak{a}$ is an ideal on $X$, and $\\lambda\\in\\bbR_{\\ge 0}\\cup\\{\\infty\\}$. Denote by $(\\lambda\\cdot\\mathfrak{a})_{\\bullet}=\\big((\\lambda\\cdot\\mathfrak{a})_m\\big)_{m\\in\\mathbb{N}}$ the \\emph{filtration} (cf. \\Cref{subsec-gr.fil}) of ideals such that each term:\n\\begin{equation*}\n (\\lambda\\cdot\\mathfrak{a})_m=\n \\begin{cases}\n R & \\lambda=0 \\\\\n \\mathfrak{a}^{\\lceil \\lambda m\\rceil} & 0<\\lambda<\\infty \\\\\n 0 & \\lambda=\\infty \\\\\n \\end{cases}\n \\quad \\quad m\\in\\mathbb{N}.\n\\end{equation*}\n\nIf $(\\b_1)_{\\bullet},\\ldots,(\\b_s)_{\\bullet}$ are filtrations of ideals in $R$, then the summation $\\mathfrak{c}_{\\bullet}\\coloneqq \\sum(\\b_i)_{\\bullet}$ of these filtrations is a filtration defined as:\n\\begin{equation*}\n \\mathfrak{c}_{m}=\\sum_{\\substack{m_1+\\cdots+m_s=m \\\\ m_i\\in\\mathbb{N}}}(\\b_1)_{m_1}\\cdots(\\b_s)_{m_s}, \\quad m\\in\\mathbb{N}.\n\\end{equation*}\n\nNow we return to the valuative interpolation problem. For the ideals $\\a_1,\\ldots,\\a_r$ and non-negative real numbers $b_1,\\ldots,b_r$, if all of $b_j$ are zero, then we can directly choose the trivial valuation on $R$, so we only consider the case where at least one of them is nonzero. Hence, may assume $\\sum_{b_j>0}\\a_j\\neq R$, otherwise none of valuations $v$ on $R$ can satisfy $v(\\a_j)=b_j$ for all $j$ (cf. \\Cref{subsec-val}).\n\nConsider the filtration\n\\[I_{\\bullet}\\coloneqq \\Big(\\frac{1}{b_1}\\cdot \\a_{1}\\Big)_{\\bullet}+\\cdots+\\Big(\\frac{1}{b_r}\\cdot \\a_{r}\\Big)_{\\bullet}.\\]\nFollowing Cutkosky and Peaharaj \\cite[Definition 3.5]{CP24}, the \\emph{asymptotic Samuel function} associated with this filtration is defined as follows. For any ideal $\\mathfrak{a}$ of $R$, let\n\\[\\vbI(\\mathfrak{a})\\coloneqq \\lim_{k\\to\\infty}\\frac{\\vI\\big(\\mathfrak{a}^k\\big)}{k}\\in \\bbR_{\\ge 0}\\cup\\{\\infty\\},\\]\nwhere $\\vI\\big(\\mathfrak{a}\\big)\\coloneqq \\sup\\,\\{m\\in\\mathbb{N} \\mid \\mathfrak{a}\\subseteq I_m\\}$ denotes the \\emph{order function} of $I_{\\bullet}$. This construction generalizes the classical asymptotic Samuel function \\cite{Sa52} and serves as an algebraic analogue of the relative type introduced in complex analytic settings \\cite{Ras06} (cf. \\Cref{sec-c.a.c}).", + "full_context": "Let $\\a_1,\\ldots,\\a_r$ be $r$ nonzero ideals of $R$, and let $b_1,\\ldots,b_r\\in\\mathbb{R}_{\\ge 0}$. We consider the following valuative interpolation problem which was considered by the authors in the complex analytic setting (e.g. \\cite[Theorem 1.2]{BGMY25} deals with the case where $R=\\O_{\\mathbb{C}^n,o}$ the ring of holomorphic germs):\n\n\\begin{question}\nCan one give a necessary and sufficient condition for the existence of a valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\ldots, r$?\n\\end{question}\n\nWe introduce some notations for explaining our results in the present paper. Suppose $\\mathfrak{a}$ is an ideal on $X$, and $\\lambda\\in\\bbR_{\\ge 0}\\cup\\{\\infty\\}$. Denote by $(\\lambda\\cdot\\mathfrak{a})_{\\bullet}=\\big((\\lambda\\cdot\\mathfrak{a})_m\\big)_{m\\in\\mathbb{N}}$ the \\emph{filtration} (cf. \\Cref{subsec-gr.fil}) of ideals such that each term:\n\\begin{equation*}\n (\\lambda\\cdot\\mathfrak{a})_m=\n \\begin{cases}\n R & \\lambda=0 \\\\\n \\mathfrak{a}^{\\lceil \\lambda m\\rceil} & 0<\\lambda<\\infty \\\\\n 0 & \\lambda=\\infty \\\\\n \\end{cases}\n \\quad \\quad m\\in\\mathbb{N}.\n\\end{equation*}\n\nIf $(\\b_1)_{\\bullet},\\ldots,(\\b_s)_{\\bullet}$ are filtrations of ideals in $R$, then the summation $\\mathfrak{c}_{\\bullet}\\coloneqq \\sum(\\b_i)_{\\bullet}$ of these filtrations is a filtration defined as:\n\\begin{equation*}\n \\mathfrak{c}_{m}=\\sum_{\\substack{m_1+\\cdots+m_s=m \\\\ m_i\\in\\mathbb{N}}}(\\b_1)_{m_1}\\cdots(\\b_s)_{m_s}, \\quad m\\in\\mathbb{N}.\n\\end{equation*}\n\nNow we return to the valuative interpolation problem. For the ideals $\\a_1,\\ldots,\\a_r$ and non-negative real numbers $b_1,\\ldots,b_r$, if all of $b_j$ are zero, then we can directly choose the trivial valuation on $R$, so we only consider the case where at least one of them is nonzero. Hence, may assume $\\sum_{b_j>0}\\a_j\\neq R$, otherwise none of valuations $v$ on $R$ can satisfy $v(\\a_j)=b_j$ for all $j$ (cf. \\Cref{subsec-val}).\n\nConsider the filtration\n\\[I_{\\bullet}\\coloneqq \\Big(\\frac{1}{b_1}\\cdot \\a_{1}\\Big)_{\\bullet}+\\cdots+\\Big(\\frac{1}{b_r}\\cdot \\a_{r}\\Big)_{\\bullet}.\\]\nFollowing Cutkosky and Peaharaj \\cite[Definition 3.5]{CP24}, the \\emph{asymptotic Samuel function} associated with this filtration is defined as follows. For any ideal $\\mathfrak{a}$ of $R$, let\n\\[\\vbI(\\mathfrak{a})\\coloneqq \\lim_{k\\to\\infty}\\frac{\\vI\\big(\\mathfrak{a}^k\\big)}{k}\\in \\bbR_{\\ge 0}\\cup\\{\\infty\\},\\]\nwhere $\\vI\\big(\\mathfrak{a}\\big)\\coloneqq \\sup\\,\\{m\\in\\mathbb{N} \\mid \\mathfrak{a}\\subseteq I_m\\}$ denotes the \\emph{order function} of $I_{\\bullet}$. This construction generalizes the classical asymptotic Samuel function \\cite{Sa52} and serves as an algebraic analogue of the relative type introduced in complex analytic settings \\cite{Ras06} (cf. \\Cref{sec-c.a.c}).\n\nConsider the filtration\n\\[I_{\\blt}\\coloneqq \\Big(\\frac{1}{b_1}\\cdot \\a_{1}\\Big)_{\\blt}+\\cdots+\\Big(\\frac{1}{b_r}\\cdot \\a_{r}\\Big)_{\\blt}.\\]\nFollowing Cutkosky and Peaharaj \\cite[Definition 3.5]{CP24}, the \\emph{asymptotic Samuel function} associated with this filtration is defined as follows. For any ideal $\\a$ of $R$, let\n\\[\\vbI(\\a)\\coloneqq \\lim_{k\\to\\infty}\\frac{\\vI\\big(\\a^k\\big)}{k}\\in \\bbR_{\\ge 0}\\cup\\{\\infty\\},\\]\nwhere $\\vI\\big(\\a\\big)\\coloneqq \\sup\\,\\{m\\in\\bbN \\mid \\a\\subseteq I_m\\}$ denotes the \\emph{order function} of $I_{\\blt}$. This construction generalizes the classical asymptotic Samuel function \\cite{Sa52} and serves as an algebraic analogue of the relative type introduced in complex analytic settings \\cite{Ras06} (cf. \\Cref{sec-c.a.c}).\n\nIn fact, the equivalence of the first two statements, along with the equivalence of the existence of \\emph{Zhou valuations} (cf. \\cite{BGMY24,BGZ25}) satisfying the same interpolation conditions, has been proved in \\cite[Appendix A.1]{BGMY25}.\n\n\\begin{theorem}\\label{thm-infinite.interp}\n Additionally assume $R$ is local with the unique maximal ideal $\\m$. Let $(\\a_j)_{j\\in\\bbZ_+}$ be a countable sequence of nonzero ideals in $R$ such that $\\sqrt{\\sum_{j=1}^{\\infty} \\a_j}=\\m$, and let $(b_j)_{j\\in\\bbZ_+}$ be a sequence of positive real numbers. Then the following statements are equivalent:\n \\begin{enumerate}\n \\item There exists a valuation $v$ on $R$ centered at $\\m$ with $A(v)<\\infty$ such that $v(\\a_j)=b_j$ for all $j\\ge 1$;\n \\item For every $r\\ge 1$,\n \\[\\vb_{I_{\\blt}^{(r)}}\\big(\\a_1\\cdots\\a_r\\big)=\\sum_{j=1}^r b_j,\\]\n and it holds that\n \\[\\sup_{r\\ge 1}\\sup_{k\\in\\bbN}\\Big(\\lct^{\\a_1^k\\cdots\\a_r^k}\\big( I^{(r)}_{\\blt}\\big)-k\\sum_{j=1}^r b_j\\Big)<\\infty.\\]\n \\end{enumerate}\n Here $I^{(r)}_{\\blt}=\\sum_{j=1}^{r}\\big(\\frac{1}{b_j}\\cdot \\a_j\\big)_{\\blt}$.\n\\end{theorem}\n\n\\begin{proposition}\\label{prop-v(aj)=bj.implies.mu=bj-1}\n If $v$ is valuation on $R$ such that $v(\\a_j)\\ge b_j$ for $j=1,\\ldots, r$, then $\\vb(\\a_1\\cdots\\a_r;I_{\\blt})\\le \\sum_{j=1}^r v(\\a_j)$. Moreover, if $v(\\a_j) = b_j$ for all $j$, then $\\vb(\\a_1\\cdots\\a_r;I_{\\blt})=\\sum_{j=1}^r b_j$.\n\\end{proposition}\n\n\\begin{lemma}\\label{lem-finite.val.interp.exist.QM}\n Let $\\a_0$ be a nonzero ideal of $R$ with $\\vb(\\a_0;I_{\\blt})>0$. Suppose $(Y,D)$ is a log smooth pair over $X$ which gives a log resolution of $\\a_1\\cdots\\a_r$. Then there exists a quasi-monomial valuation $v\\in \\QM(Y,D)$ such that\n \\[\\vb(\\a_0;I_{\\blt})=\\frac{v(\\a_0)}{v(I_{\\blt})}.\\]\n\\end{lemma}\n\n\\begin{proposition}\\label{prop-val.interp.gr.seq}\n Assume $\\ord_{\\xi}(J_{\\blt})>0$ and $\\m_{\\xi}^p\\subseteq J_{q}$ for some $p,q\\in\\bbZ_+$. Let $\\a$ be a nonzero ideal of $R$ with $\\a\\subseteq\\m_{\\xi}$. Then the following two statements are equivalent:\n \\begin{enumerate}\n \\item There exists $v\\in\\Val_{X,\\xi}^{<\\infty}$ such that $v(J_{\\blt})=1$ and $v(\\a)=\\rho(\\a;J_{\\blt})$;\n \\item It holds that\n \\[M\\coloneqq \\sup_{k\\in\\bbN}\\Big(\\lct(\\a^k;J_{\\blt})-\\rho(\\a;J_{\\blt})\\cdot k\\Big)<\\infty.\\]\n \\end{enumerate} \n Moreover, the valuation $v$ in (1) can be chosen with $A(v)\\le M$ and $v(\\m_{\\xi})\\ge q/p$ if (2) holds.\n\\end{proposition}\n\n\\begin{lemma}\\label{lem-multiply.change.to.plus}\n Let $J_{\\blt}$ be a filtration of nonzero ideals in $R$ with $\\lct(J_{\\blt})<\\infty$, and let $\\a_1,\\ldots,\\a_r$ be nonzero ideals in $R$. Set $\\a=\\a_1\\cdots\\a_r$. Suppose there is a valuation $v\\in\\Val_X^{<\\infty}$ such that $v(\\a)=\\rho(\\a;J_{\\blt})$ and $v(J_{\\blt})=1$. Then $v(\\a_j)=\\rho(\\a_j;J_{\\blt})$ for $j=1,\\ldots,r$ if and only if $\\rho(\\a;J_{\\blt})=\\sum_{j=1}^r \\rho(\\a_j;J_{\\blt})$.\n\\end{lemma}\n\n\\begin{proposition}\\label{prop-gr.seq.infinite.compute}\n Let $J_{\\blt}$ be a filtration of ideals in $R$ such that $\\ord_{\\xi}(J_{\\blt})>0$ and $\\m_{\\xi}^p\\subseteq J_{q}$ for some $p,q\\in\\bbZ_+$. Assume $(\\a_j)_{j\\in\\bbZ_+}$ is a sequence of nonzero ideals of $R$ with $\\a_j\\subseteq\\m_{\\xi}$ for all $j\\ge 1$. Then the following two statements are equivalent:\n \\begin{enumerate}\n \\item There exists $v\\in\\Val_{X,\\xi}^{<\\infty}$ such that $v(J_{\\blt})=1$ and $v(\\a_j)=\\rho(\\a_j;J_{\\blt})$ for all $j\\ge 1$;\n \\item For every $r\\in\\bbZ_+$,\n \\[\\rho(\\a_1\\cdots\\a_r;J_{\\blt})=\\sum_{j=1}^r\\rho(\\a_j;J_{\\blt}),\\]\n and it holds that\n \\[\\sup_{r\\ge 1}\\sup_{k\\in\\bbN}\\Big(\\lct(\\a_1^k\\cdots\\a_r^k; J_{\\blt})-\\rho(\\a_1\\cdots\\a_r;J_{\\blt})\\cdot k\\Big)<\\infty.\\]\n \\end{enumerate}\n\\end{proposition}", + "post_theorem_intro_text_len": 2651, + "post_theorem_intro_text": "In fact, the equivalence of the first two statements, along with the equivalence of the existence of \\emph{Zhou valuations} (cf. \\cite{BGMY24,BGZ25}) satisfying the same interpolation conditions, has been proved in \\cite[Appendix A.1]{BGMY25}.\n\nIn addition, for the infinite valuative interpolation problem, we apply the method of valuation approximation to prove the following result in the present paper.\n\n\\begin{theorem}\\label{thm-infinite.interp}\n Additionally assume $R$ is local with the unique maximal ideal $\\mathfrak{m}$. Let $(\\a_j)_{j\\in\\bbZ_+}$ be a countable sequence of nonzero ideals in $R$ such that $\\sqrt{\\sum_{j=1}^{\\infty} \\a_j}=\\mathfrak{m}$, and let $(b_j)_{j\\in\\bbZ_+}$ be a sequence of positive real numbers. Then the following statements are equivalent:\n \\begin{enumerate}\n \\item There exists a valuation $v$ on $R$ centered at $\\mathfrak{m}$ with $A(v)<\\infty$ such that $v(\\a_j)=b_j$ for all $j\\ge 1$;\n \\item For every $r\\ge 1$,\n \\[\\vb_{I_{\\bullet}^{(r)}}\\big(\\a_1\\cdots\\a_r\\big)=\\sum_{j=1}^r b_j,\\]\n and it holds that\n \\[\\sup_{r\\ge 1}\\sup_{k\\in\\mathbb{N}}\\Big(\\lct^{\\a_1^k\\cdots\\a_r^k}\\big( I^{(r)}_{\\bullet}\\big)-k\\sum_{j=1}^r b_j\\Big)<\\infty.\\]\n \\end{enumerate}\n Here $I^{(r)}_{\\bullet}=\\sum_{j=1}^{r}\\big(\\frac{1}{b_j}\\cdot \\a_j\\big)_{\\bullet}$.\n\\end{theorem}\n\n\\vspace{.1in} {\\bf Organization}. This paper is organized as follows. In \\Cref{sec-r.o}, we recall the notion of asymptotic Samuel function and quickly prove the implication (1) $\\Rightarrow$ (3) in \\Cref{thm-finite.interp}. In \\Cref{sec-f.v.i}, we prove \\Cref{thm-finite.interp} by studying the asymptotic behavior of jumping numbers and an extremal problem on a cone $\\QM(Y,D)$ of quasi-monomial valuations. In \\Cref{sec-i.v.i}, we prove \\Cref{thm-infinite.interp} using valuation approximation, a strategy that more closely parallels the approach in \\cite{BGMY25}. Finally, we demonstrate how \\Cref{thm-finite.interp} recovers a result in \\cite{BGMY25} (which deals with the complex analytic case) and yields an analytic characterization, derived from \\Cref{thm-infinite.interp}, for the existence of infinite valuative interpolation.\n\n\\vspace{.1in} {\\em Acknowledgements}. The first-named author completed this work during a visit to the School of Mathematical Sciences at Peking University and is grateful for its hospitality and support. The second-named author was supported by National Key R\\&D Program of China 2021YFA1003100 and NSFC-12425101. The third-named author was supported by NSFC-12401099 and the Talent Fund of Beijing Jiaotong University 2024-004. The fourth-named author was supported by NSFC-12501106.", + "sketch": "The post-theorem introduction only indicates where proofs appear and what methods are used: it says they “quickly prove the implication (1) $\\Rightarrow$ (3) in \\Cref{thm-finite.interp}” in \\Cref{sec-r.o}; they “prove \\Cref{thm-finite.interp} by studying the asymptotic behavior of jumping numbers and an extremal problem on a cone $\\QM(Y,D)$ of quasi-monomial valuations” in \\Cref{sec-f.v.i}; and they “prove \\Cref{thm-infinite.interp} using valuation approximation.” No further proof outline for \\Cref{thm-finite.interp} is given beyond this organizational summary.", + "expanded_sketch": "The post-theorem introduction only indicates where proofs appear and what methods are used: it says they “quickly prove the implication (1) $\\Rightarrow$ (3)” of the main theorem later; they “prove the main theorem by studying the asymptotic behavior of jumping numbers and an extremal problem on a cone $\\QM(Y,D)$ of quasi-monomial valuations” later; and they “prove the following theorem using valuation approximation. \\begin{theorem}\\label{thm-infinite.interp}\n Additionally assume $R$ is local with the unique maximal ideal $\\m$. Let $(\\a_j)_{j\\in\\bbZ_+}$ be a countable sequence of nonzero ideals in $R$ such that $\\sqrt{\\sum_{j=1}^{\\infty} \\a_j}=\\m$, and let $(b_j)_{j\\in\\bbZ_+}$ be a sequence of positive real numbers. Then the following statements are equivalent:\n \\begin{enumerate}\n \\item There exists a valuation $v$ on $R$ centered at $\\m$ with $A(v)<\\infty$ such that $v(\\a_j)=b_j$ for all $j\\ge 1$;\n \\item For every $r\\ge 1$,\n \\[\\vb_{I_{\\blt}^{(r)}}\\big(\\a_1\\cdots\\a_r\\big)=\\sum_{j=1}^r b_j,\\]\n and it holds that\n \\[\\sup_{r\\ge 1}\\sup_{k\\in\\bbN}\\Big(\\lct^{\\a_1^k\\cdots\\a_r^k}\\big( I^{(r)}_{\\blt}\\big)-k\\sum_{j=1}^r b_j\\Big)<\\infty.\\]\n \\end{enumerate}\n Here $I^{(r)}_{\\blt}=\\sum_{j=1}^{r}\\big(\\frac{1}{b_j}\\cdot \\a_j\\big)_{\\blt}$.\n\\end{theorem} No further proof outline for the main theorem is given beyond this organizational summary.", + "expanded_theorem": "\\label{thm-finite.interp}\nThe following three statements are equivalent.\n\\begin{enumerate}\n\\item There exists a valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\dots,r$.\n\\item There exists a quasi-monomial valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\dots,r$.\n\\item The equality $\\vbI(\\a_1\\cdots\\a_r)=\\sum_{j=1}^r b_j$ holds.\n\\end{enumerate}", + "theorem_type": [ + "Biconditional or Equivalence", + "Existence" + ], + "mcq": { + "question": "Let \\(\\mathfrak a_1,\\ldots,\\mathfrak a_r\\) be nonzero ideals of a ring \\(R\\), and let \\(b_1,\\ldots,b_r\\in \\mathbb R_{\\ge 0}\\). For \\(\\lambda\\in \\mathbb R_{\\ge 0}\\cup\\{\\infty\\}\\) and an ideal \\(\\mathfrak a\\), define the filtration \\((\\lambda\\cdot \\mathfrak a)_\\bullet=((\\lambda\\cdot \\mathfrak a)_m)_{m\\in\\mathbb N}\\) by\n\\[\n(\\lambda\\cdot \\mathfrak a)_m=\n\\begin{cases}\nR,& \\lambda=0,\\\\\n\\mathfrak a^{\\lceil \\lambda m\\rceil},& 0<\\lambda<\\infty,\\\\\n0,& \\lambda=\\infty.\n\\end{cases}\n\\]\nIf \\((\\mathfrak b_1)_\\bullet,\\ldots,(\\mathfrak b_s)_\\bullet\\) are filtrations, their sum is the filtration \\(\\mathfrak c_\\bullet\\) with\n\\[\n\\mathfrak c_m=\\sum_{m_1+\\cdots+m_s=m}(\\mathfrak b_1)_{m_1}\\cdots(\\mathfrak b_s)_{m_s}.\n\\]\nSet\n\\[\nI_\\bullet:=\\Big(\\frac1{b_1}\\cdot \\mathfrak a_1\\Big)_\\bullet+\\cdots+\\Big(\\frac1{b_r}\\cdot \\mathfrak a_r\\Big)_\\bullet,\n\\]\nwhere \\(1/0\\) is interpreted as \\(\\infty\\). For any ideal \\(\\mathfrak a\\subseteq R\\), define the order function\n\\[\n\\nu_I(\\mathfrak a):=\\sup\\{m\\in\\mathbb N\\mid \\mathfrak a\\subseteq I_m\\},\n\\]\nand the associated asymptotic Samuel function\n\\[\n\\bar\\nu_I(\\mathfrak a):=\\lim_{k\\to\\infty}\\frac{\\nu_I(\\mathfrak a^k)}{k}.\n\\]\nWhich statement holds for these data?", + "correct_choice": { + "label": "A", + "text": "The following three statements are equivalent: (i) there exists a valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); (ii) there exists a quasi-monomial valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); and (iii) the equality \\(\\bar\\nu_I(\\mathfrak a_1\\cdots \\mathfrak a_r)=\\sum_{j=1}^r b_j\\) holds." + }, + "choices": [ + { + "label": "B", + "text": "The following three statements are equivalent: (i) there exists a valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); (ii) there exists a quasi-monomial valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); and (iii) for every nonempty subset \\(J\\subseteq\\{1,\\dots,r\\}\\), one has \\(\\bar\\nu_I\\big(\\prod_{j\\in J}\\mathfrak a_j\\big)=\\sum_{j\\in J} b_j\\)." + }, + { + "label": "C", + "text": "If there exists a valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\), then \\(\\bar\\nu_I(\\mathfrak a_1\\cdots \\mathfrak a_r)=\\sum_{j=1}^r b_j\\)." + }, + { + "label": "D", + "text": "The following three statements are equivalent: (i) there exists a valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); (ii) there exists a quasi-monomial valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); and (iii) for every ideal \\(\\mathfrak a\\subseteq R\\), one has \\(\\bar\\nu_I(\\mathfrak a)=\\inf_{v}\\frac{v(\\mathfrak a)}{v(I_\\bullet)}\\), where the infimum runs over all valuations \\(v\\) on \\(R\\) with \\(v(I_\\bullet)>0\\)." + }, + { + "label": "E", + "text": "The following three statements are equivalent: (i) there exists a valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); (ii) there exists a quasi-monomial valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); and (iii) the inequality \\(\\bar\\nu_I(\\mathfrak a_1\\cdots \\mathfrak a_r)\\ge \\sum_{j=1}^r b_j\\) holds." + } + ], + "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": "single-product equality upgraded to all partial products", + "template_used": "stronger_trap" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped converse directions and quasi-monomial equivalence, retaining only (i) => (iii)", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "geometric_construction", + "tampered_component": "extremal quasi-monomial valuation existence replaced by a global valuation-formula for every ideal", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "equality condition weakened to one-sided inequality", + "template_used": "property_confusion" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem provides only definitions and setup; it does not explicitly state the equivalence in the correct option or otherwise single out choice A. There are no obvious lexical cues that reveal the answer." + }, + "TAS": { + "score": 1, + "justification": "The item is still fairly close to theorem recognition: the correct option is essentially the precise theorem statement in this notation. However, the presence of several nearby variants means it is not a pure verbatim restatement." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish exact equivalence from weaker, stronger, or misdirected alternatives (equality vs inequality, full equivalence vs one implication, single product vs all subsets). But the task mainly tests recognition of the exact theorem rather than substantial generative problem solving." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target realistic failure modes: overstrengthening the conclusion, weakening to a one-way implication, replacing equality by inequality, or swapping in an overly broad valuation formula. They are distinct and nontrivial." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with no answer leakage and strong distractors, but it leans more toward precise theorem recall than genuine generative reasoning." + } + }, + { + "id": "2602.03768v1", + "paper_link": "http://arxiv.org/abs/2602.03768v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "thm", + "content": "\\label{thm;global}\nFor $(u_0,v_0)\\in L_+^1(\\mathbb R^2) \\times L^1_+(\\mathbb R^2)\\cap \\dot{H}^1(\\mathbb R^2)$,\nlet $(u,v)$ be the solution to \\eqref{eqn;KS} on $(0,T)\\times\\mathbb R^2$.\nSuppose that $\\|u_0\\|_1=8\\pi$.\nThen, the solution to \\eqref{eqn;KS} exists globally in time.", + "start_pos": 14650, + "end_pos": 14921, + "label": "thm;global" + }, + "ref_dict": { + "thm;global": "\\begin{thm}\\label{thm;global}\nFor $(u_0,v_0)\\in L_+^1(\\R^2) \\times L^1_+(\\R^2)\\cap \\dot{H}^1(\\R^2)$,\nlet $(u,v)$ be the solution to \\eqref{eqn;KS} on $(0,T)\\times\\R^2$.\nSuppose that $\\|u_0\\|_1=8\\pi$.\nThen, the solution to \\eqref{eqn;KS} exists globally in time.\n\\end{thm}", + "eqn;KS": "\\begin{aligned}\n&\\pt_t u =\\Delta u- \\N\\cd \\left( u \\N v \\right),\n& t>0,\\,~ &x\\in\\R^2,\n\\\\\n&\\pt_t v = \\Delta v - \\lambda v +u,\n&t>0,\\, ~&x\\in\\R^2,\n\\\\\n&(u,v)(0,x)=(u_{0},v_0)(x),\n&\\, &x\\in\\R^2\n\\end{aligned}\n\\right.\n\\label{eqn;KS}\n\\end{equation}\nwith a constant $\\lambda\\ge0$, where $u_0,v_0\\ge 0$ on $\\R^2$ and\n$u_0,v_0\\not\\equiv0$.\nThe Keller--Segel system~\\eqref{eqn;KS} is a fundamental mathematical model\nof chemotaxis, describing chemotactic aggregation in the\ncellular slime mold {\\it Dictyostelium discoideum} during its life cycle \\cite{KeSe,Pa}.\nIn this model, $u=u(t,x)$ and $v=v(t,x)$ denote the densities of cells and\nthe chemoattractant, respectively.\nThe parameter $\\lambda$ represents the degradation rate of the chemical.\nCells migrate toward regions of higher concentrations of a chemical substance\nsecreted by the cells themselves.\nFrom a biological viewpoint, it is natural to assume that the initial data\nare nonnegative functions.\nFrom a mathematical viewpoint, cell aggregation is interpreted as the\nblowup of solutions at $t=T$ in the sense that\n$\\lim_{t\\to T}\\|u(t)\\|_\\infty=\\infty.$\n\nOne of the central mathematical features of such systems is the phenomenon of critical\nmass.\nFor positive sufficiently regular solutions $(u,v)$ to~\\eqref{eqn;KS}, the first\ncomponent $u$ satisfies the mass conservation law \n$$\\|u(t)\\|_1=\\|u_0\\|_1\\quad \\text{for}~~~t>0.$$\nMoreover, in two space dimensions, the global behavior of solutions is governed\nby the size of the initial mass of $u$.\nIn particular, there exists a threshold value such that solutions with initial\nmass below this threshold exist globally in time, whereas solutions with\ninitial mass above the threshold may blow up in finite time.\nThis critical-mass phenomenon has attracted considerable attention in the\nliterature.\n\nThe aim of this paper is to establish the global behavior of solutions to\n\\eqref{eqn;KS} whose initial mass is exactly equal to this threshold value.\n\n\\vspace{3mm}\nThe second equation in \\eqref{eqn;KS} takes into account that cells are producing the chemoattractant themselves \nwhile this is diffusing into the environment. \nSince the chemoattractant attains its equilibrium on a time scale much quicker\nthan that of the cells, the simplified parabolic-elliptic system has also been\ninvestigated \\cite{JaLu,Na95}:\n\\begin{align*}\n\\left\\{\n\\begin{aligned}\n\\partial_t u &= \\Delta u - \\nabla\\cdot (u\\nabla v),\n& t>0,\\ & x\\in\\R^2,\\\\\n0 &= \\Delta v - \\lambda v + u,\n& t>0,\\ & x\\in\\R^2,\n\\\\\n&u(0,x)=u_{0}(x),\n&\\, &x\\in\\R^2.\n\\end{aligned}", + "prop;LWP": "\\begin{prop}[local-in-time solution]\\label{prop;LWP}\n\tLet $4/3 0$ in $(0,T)\\times\\R^2$ and\n\t\\begin{equation*}\n\t\\|u(t)\\|_1=\\|u_0\\|_1, \\qquad t\\in [0,T).\t\n\t\\end{equation*} \n\\end{prop}", + "eqn;PKS": "\\begin{aligned}\n\\partial_t u &= \\Delta u - \\nabla\\cdot (u\\nabla v),\n& t>0,\\ & x\\in\\R^2,\\\\\n0 &= \\Delta v - \\lambda v + u,\n& t>0,\\ & x\\in\\R^2,\n\\\\\n&u(0,x)=u_{0}(x),\n&\\, &x\\in\\R^2.\n\\end{aligned}\n\\right.\n\\eqntag\n\\label{eqn;PKS}\n\\end{align*}\nThe system \\eqref{eqn;PKS} is also related to models of gravitational interaction\nof particles \\cite{Bi-Na94,Wo}.\nIn~\\eqref{eqn;PKS}, the second equation can be written as\n$v = (-\\Delta+\\lambda)^{-1}u.$\nAs a consequence, the system~\\eqref{eqn;PKS} can be reduced to a single equation\nfor~$u$.\nOwing to this elliptic structure, the parabolic-elliptic system~\\eqref{eqn;PKS} is more amenable to analysis than the fully parabolic system~\\eqref{eqn;KS}, and has therefore been extensively studied and is now well understood in many aspects.\nIn particular, numerous works have been devoted to the critical-mass\nphenomenon.\nIndeed, the solution with $\\|u_0\\|_1\\le 8\\pi$ exists globally in time~\\cite{BiKaLaNa06,Bl-Ca-Ca,Bl-Ca-Ma,BlDoPe,DoPe,Lo-Na-Ya1,Lo-Na-Ya2,Na11,NaOg11,NaOg16,Na-Se,Wei},\nwhile the solution may blow up in finite time if $\\|u_0\\|_1>8\\pi$~\\cite{DoPe,Ko-Su,Ku-Og03,Wei}.\nFor the corresponding Cauchy--Neumann problem in bounded domains,\nsupplemented with homogeneous Neumann boundary conditions, see \\cite{BiKaLaNa06disc,Bi-Na94,Ga-Za98,Na95,Na01} for instance.\n\n\\vspace{3mm}\nUnlike the parabolic-elliptic system \\eqref{eqn;PKS}, the fully parabolic system\n\\eqref{eqn;KS} is a strongly coupled parabolic system, and many approaches\ndeveloped for \\eqref{eqn;PKS} are no longer applicable.\nThe global existence of solutions with sub-critical mass\n$\\|u_0\\|_{1}<8\\pi$ has nevertheless been established by combining\nLyapunov functionals with the Trudinger--Moser type inequality and its optimal\nconstant.\nMore precisely, the following results are known:\n\\begin{enumerate}\n\t\\item[(i)]\n\tIf $\\|u_0\\|_{1}<8\\pi$, then the corresponding solution to\n\t\\eqref{eqn;KS} exists globally in time \\cite{CaCo08,Mi13,NaOg11}.\n\n\t\\item[(ii)]\n\tIf $\\|u_0\\|_{1}>8\\pi$ and $(u_0,v_0)$ is radially symmetric, then there\n\texists a radially symmetric solution that blows up in finite time\n\t\\cite{Mi20,Mi20-SIAM}.\n\n\t\\item[(iii)]\n\tIf $\\|u_0\\|_{1}=8\\pi$ and $(u_0,v_0)$ is either radially symmetric or\n\tsatisfies the additional moment condition\n\t$u_0\\ln(1+|x|^2)\\in L^1(\\R^2)$, then the corresponding solution exists globally\n\tin time.\n\tIn contrast, for general initial data, the global behavior in the critical case\n\tremains delicate, and the solution either exists globally or blows up on the plane \\cite{CaCo08,Mi13}.\n\\end{enumerate}\nAs for positive forward self-similar solutions to \\eqref{eqn;KS}, refer to \\cite{BiCoDo}.\nFor the corresponding Cauchy--Neumann problem in bounded domains, we refer to\n\\cite{Bi98,He-Ve,HoWa,Na-Se-Yo}, for instance.\n\nAs mentioned above, although several partial results are available, no complete global existence result has been obtained\nfor solutions with critical mass and general initial data without any\nsymmetry or moment assumptions.\nOne of the main difficulties lies in controlling the behavior of solutions at~$|x|\\to\\infty$.\nMoreover, in the critical mass case, no global existence result is currently\nknown for the corresponding Cauchy--Neumann problem to~\\eqref{eqn;KS} in bounded domains.\n\nIn this paper, we establish the global existence of solutions with critical mass\nfor general initial data $(u_0,v_0)$.\nTo this end, we first recall the definition of solutions.\n\n\\vspace{2mm}\n\\begin{def*}\nLet $4/31\\}\\cap \\{\\supp\\phi\\} }|f|\\dx\\right)\\left(\\int_{ \\{|f|>1\\} }\\frac{|\\nabla f|^2}{1+|f|} \\phi\\dx\\right)\n\t\\\\\n\t&+4\\left(\\int_{\\R^2}|f \\nabla \\phi^{\\frac12}|\\dx\\right)^2+4\\left(\\int_{\\R^2}|f|\\phi\\dx\\right)\n\t\\end{align*}\n\tand for any $\\ep>0$\n\t\\begin{align*}\n\t\\int_{\\R^2}|f|^3\\phi\\dx\n\t\\le\\,&\\ep \\left(\\int_{ \\{\\supp\\phi\\} }(1+|f|)\\ln(1+|f|)\\dx\\right)\\left(\\int_{\\R^2 }|\\nabla f|^2\\phi\\dx\\right)\n\t\\\\\n\t&+C\\left(\\int_{\\R^2}|f^{\\frac32} \\nabla \\phi^{\\frac12}|\\dx\\right)^2+C_\\ep\\left(\\int_{\\R^2}|f|\\phi\\dx\\right),\n\t\\end{align*}\n\twhere the constant $C_\\ep\\to \\infty$ as $\\ep\\to0$.\n\\end{lem}\nThe proof of Lemma \\ref{lem;L2andL3} can be found in \\cite[Lemma 2.2]{NaOg16}.\n\n\\begin{lem}\\label{lem;Lp_Lq_heat}\n\tFor $1 \\le q \\le p \\le \\infty$,\n\tlet $f\\in L^q (\\R^n)$ and let $\\al$ be a multi-index.\n\tThen it follows that\n\t\\begin{equation*}\n\t\\| \\pt_x^{\\al} e^{t\\Del} f \\|_p \n\t\\le C t^{-\\frac{n}{2} \\left(\\frac{1}{q} - \\frac{1}{p} \\right) - \\frac{|\\al|}{2}} \\|f\\|_q\n\t\\end{equation*}\n\tfor all $t>0$.\n\\end{lem}\n\nThe proof is immediately obtained by use of Young's inequality and the \nconvolution expression of the heat evolution\nby the heat kernel, see for instance \\cite{GiGiSa}.\n\\begin{lem}\\label{lem;v-Lpbound}\nFor $v_0\\in L^1(\\R^2)\\cap \\dot H^1(\\R^2)$ and $f\\in L^\\infty \\left(0,\\infty; L^1(\\R^2) \\right)$,\nlet $v$ be the solution to $\\partial_t v =\\Delta v -\\lambda v +f$ on $(0,\\infty)\\times\\R^2$ with the initial data $v_0$ and $\\lambda\\ge0$. Then, for any $1\\le p<\\infty$, \n\\begin{align*}\n\\| v(t)\\|_p\n\\le\\,&\n\\left\\{\n\\begin{aligned}\n&\\|v_0\\|_p+\\frac1p\\|f\\|_{L^\\infty\\left(0,\\infty; L^1(\\R^2)\\right)} t^{\\frac1p} &\\text{if $\\lambda=0$},\n\\\\\n&\\|v_0\\|_p+\\lambda^{-\\frac1p} \\|f\\|_{L^\\infty\\left(0,\\infty; L^1(\\R^2)\\right)} \\Gamma\\left(\\frac1p\\right)&\\text{if $\\lambda>0$}.\n\\end{aligned}", + "eqn;afford": "\\begin{pr}{Proposition \\ref{prop;LWP}}\nBy virtue of Lemmas~\\ref{lem;fixed-point} and~\\ref{lem;continuous-depend},\nwe have established the existence of a unique mild solution to~\\eqref{eqn;KS} defined on a maximal time interval $[0,T)$ with~$T\\le\\infty$. As for the regularity of the solutions, we use the standard iteration argument with respect to the derivative. Define $|\\N|^\\al f(x):=\\F^{-1}[ |\\xi|^\\al \\F f(\\xi)\\,](x)$ for $x\\in\\R^2$ and $\\al>0$, where $\\F$ denotes the Fourier transform. Let $\\tau\\in (0,T)$ and $t\\in (0,\\tau)$. Recalling that, for $4/30$, $u(t)\\in L^1(\\R^2)\\cap W^{s,p}(\\R^2)$,\n$v(t)\\in \\dot{H}^{1}(\\R^2)\\cap W^{s,q}(\\R^2)$\nfor $t_00$ and some $C(\\tau,\\alpha)>0$ (cf.~\\cite{CaCo08,Mi13}). Hence, if $\\|u_0\\|<8\\pi$, then all terms on the left hand side of \\eqref{eqn;afford} are positive by choosing $\\al>0$ sufficiently small which depends on $\\|u_0\\|_1$, and Gronwall's inequality implies that\n$\\nabla v \\in L^2((0,T)\\times \\R^2)$, \nso that we also obtain the bound for the modified entropy $\\int_{\\R^2}(1+u)\\ln (1+u)\\dx$. \nNevertheless, for the critical mass $\\|u_0\\|_1=8\\pi$, this necessarily leads to $\\alpha=0$, as a result, \n\\begin{align*}\n\\int_0^t \\D_m(s)\\,\\d s\\le\\,&\\L_m(0)+C(\\tau)+\\frac14\\int_0^t\\|\\nabla v\\|_2^2\\,\\d s,\n\\eqntag\n\\label{eqn;lack-lyapunov}\n\\end{align*}\nwhich is useless and fails to obtain even the estimates for $\\D_m(t)$ due to the lack of the regularity of solutions.\nTherefore, we introduce a reconstructed Lyapunov functional $\\F_m(t)$ so as to show regularity estimates corresponding to the dissipative terms. Let $\\F_m(t)$ be the functional for solutions to \\eqref{eqn;KS} defined as\n\\begin{align*}\n\\F_m(t):=\\L_m(t)\n+\\int_{\\R^2}\\ln(1+u)\\,\\mathrm{d}x-\\int_{\\R^2}v\\,\\d x.\n\\eqntag\n\\label{eqn;modifiedLyapunov}\n\\end{align*}\nThen, the following functional differential inequality holds true:\n\\begin{prop}\\label{prop;energy-est}\nLet $(u,v)$ be the solution to \\eqref{eqn;KS}. Then, the functional $\\F_m(t)$ defined in~\\eqref{eqn;modifiedLyapunov} satisfies the following identity:\n\t\\begin{align*}\n\t\\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2\n\t\\le\\,&\n\t\\lambda\\int_{\\R^2}v\\,\\d x.\n\t\\end{align*}\n\tTherefore, if $\\lambda=0$ then\n\t\\begin{align*}\n\t\\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2 \\le\\,0.\n\t\\end{align*}\n\tIf $\\lambda>0$ then \n\t\\begin{align*}\n\t\\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2 \\le\\,\\|v_0\\|_1+\\|u_0\\|_1.\n\t\\end{align*}\n\\end{prop}\n\\vspace{5mm}\nIt is worth emphasizing that $\\F_m(t)$ is non-increasing in time when $\\lambda=0$ as well as\nunlike the identity \\eqref{eqn;usualmodifiedfunctiona} based on the usual modified functional $\\mathcal{L}_m(t)$, the error term appearing on the right-hand side of Proposition~\\ref{prop;energy-est} can be easily controlled by initial data.\nThis allows us to show the regularity estimates for the dissipative term associated with $\\F_m(t)$, see Proposition~\\ref{prop;bound-energy} below.\n\\begin{lem}\\label{lem;ModifiedLF}\n\tSuppose assumptions as in Theorem \\ref{thm;global}. Then,\n\\begin{align*}\n\\frac{\\d}{\\d t}\\F_m(t)+ \\widetilde{\\D}(t)=-\\int_{\\R^2}\\partial_t v \\frac{u}{1+u}\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\,\\mathrm{d}x+\\lambda\\int_{\\R^2}\\frac{v}{1+u}\\,\\d x,\n\\end{align*}\nwhere\n$\\F_m(t)$ is the modified functional defined in \\eqref{eqn;modifiedLyapunov} and the dissipative term $\\widetilde{\\D}(t)$ is given by\n\\begin{align*}\n\\widetilde{\\D}(t):=\\int_{\\R^2}u\\left|\\nabla \\left(\\ln(1+u)-v\\right)\\right|^2\\,\\d x+\\|\\partial_t v\\|_2^2.\n\\end{align*}\n\\end{lem}\n\n\\begin{pr}{Lemma \\ref{lem;ModifiedLF}}\nThe following computations are already well-known:\n\\begin{align*}\n\\frac{\\d}{\\d t} \\int_{\\R^2}(1+u)\\ln (1+u)\\,\\d x=\\,&\n-\\int_{\\R^2}(1+u)|\\nabla\\ln(1+u)|^2\\,\\d x\n+\\int_{\\R^2}u\\nabla\\ln(1+u)\\cdot\\nabla v\\,\\d x\n\\end{align*}\nas well as\n\\begin{align*}\n-\\frac{\\d}{\\d t}\\int_{\\R^2}uv\\,\\d x=\\,&\\int_{\\R^2}(1+u)\\nabla\\ln(1+u)\\cdot\\nabla v\\,\\d x-\\int_{\\R^2}u|\\nabla v|^2\\,\\d x\n\\\\\n&-\\|\\partial_tv\\|_2^2-\\frac12\\frac{\\d}{\\d t}\\|\\nabla v\\|_2^2-\\frac\\lambda2\\frac{\\d}{\\d t}\\|v\\|_2^2,\n\\end{align*}\nso that\n\\begin{align*}\n&\\frac{\\d}{\\d t}\\left[\\int_{\\R^2}(1+u)\\ln(1+u)\\,\\d x-\\int_{\\R^2}uv\\,\\d x+\\frac12\\left(\\|\\nabla v\\|_2^2+\\lambda\\|v\\|_2^2\\right)\\right]\n\\\\\n&+\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\|\\partial_tv\\|_2^2\n\\\\\n=\\,&-\\|\\nabla\\ln(1+u)\\|_2^2+\\int_{\\R^2}\\nabla\\ln(1+u)\\cdot\\nabla v\\,\\d x.\n\\end{align*}\nNext,\n\\begin{align*}\n\\frac{\\d}{\\d t}\\int_{\\R^2}\\ln(1+u)\\,\\d x=\\,&\\int_{\\R^2}\\frac{1}{1+u} \\left[\\Delta u -\\nabla\\cdot( u\\nabla v)\\right]\\,\\mathrm{d}x\n\\\\\n=\\,&-\\int_{\\R^2}\\nabla\\left(\\frac{1}{1+u}\\right)\\cdot \\nabla u\\,\\d x+\\int_{\\R^2}u\\nabla\\left(\\frac{1}{1+u}\\right)\\cdot \\nabla v\\,\\d x\n\\\\\n=\\,&\\int_{\\R^2}|\\nabla \\ln(1+u)|^2\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\nabla\\ln(1+u)\\cdot\\nabla v\\,\\d x\n\\\\\n=\\,&\\int_{\\R^2}|\\nabla \\ln(1+u)|^2\\,\\d x-\\int_{\\R^2}\\nabla\\ln(1+u)\\cdot\\nabla v\\,\\d x\n\\\\\n&+\\int_{\\R^2}\\frac{1}{1+u}\\nabla\\ln(1+u)\\cdot\\nabla v \\,\\d x.\n\\end{align*}\nHence, combining the above computations implies that\n\\begin{align*}\n&\\frac{\\d}{\\d t}\\left[\\int_{\\R^2}(1+u)\\ln(1+u)\\,\\d x-\\int_{\\R^2}uv\\,\\d x+\\frac12\\left(\\|\\nabla v\\|_2^2+\\lambda\\|v\\|_2^2\\right)+\\int_{\\R^2}\\ln(1+u)\\,\\d x\\right]\n\\\\\n&+\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\|\\partial_tv\\|_2^2\n\\\\\n=\\,&\\int_{\\R^2}\\frac{1}{(1+u)^2}\\nabla u\\cdot\\nabla v \\,\\d x.\n\\end{align*}\nSince\n\\begin{align*}\n\\int_{\\R^2}\\frac{1}{(1+u)^2}\\nabla u\\cdot\\nabla v \\,\\d x=\\,&-\\int_{\\R^2}\\nabla\\left(\\frac{1}{1+u}\\right)\\cdot \\nabla v\\,\\d x\n\\\\\n=\\,&\\int_{\\R^2}\\frac{1}{1+u}\\Delta v\\,\\d x\n\\\\\n=\\,&\\int_{\\R^2}\\frac{\\partial_t v}{1+u}\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\,\\d x+\\lambda\\int_{\\R^2}\\frac{v}{1+u}\\,\\d x\n\\\\\n=\\,&\\frac{\\d}{\\d t}\\int_{\\R^2}v\\,\\d x-\\int_{\\R^2}\\partial_t v \\frac{u}{1+u}\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\,\\d x+\\lambda\\int_{\\R^2}\\frac{v}{1+u}\\,\\d x,\n\\end{align*}\nwe end up with\n\\begin{align*}\n&\\frac{\\d}{\\d t}\\F_m(t)+\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\|\\partial_tv\\|_2^2\n\\\\\n=\\,&-\\int_{\\R^2}\\partial_t v \\frac{u}{1+u}\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\,\\d x+\\lambda\\int_{\\R^2}\\frac{v}{1+u}\\,\\d x,\n\\end{align*}\nas desired.\n\\end{pr}" + }, + "pre_theorem_intro_text_len": 8115, + "pre_theorem_intro_text": "We study the Cauchy problem for the parabolic-parabolic Keller--Segel system in~$\\mathbb R^2$\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n&\\pt_t u =\\Delta u- \\nabla\\cdot \\left( u \\nabla v \\right),\n& t>0,\\,~ &x\\in\\mathbb R^2,\n\\\\\n&\\pt_t v = \\Delta v - \\lambda v +u,\n&t>0,\\, ~&x\\in\\mathbb R^2,\n\\\\\n&(u,v)(0,x)=(u_{0},v_0)(x),\n&\\, &x\\in\\mathbb R^2\n\\end{aligned}\n\\right.\n\\label{eqn;KS}\n\\end{equation}\nwith a constant $\\lambda\\ge0$, where $u_0,v_0\\ge 0$ on $\\mathbb R^2$ and\n$u_0,v_0\\not\\equiv0$.\nThe Keller--Segel system~\\eqref{eqn;KS} is a fundamental mathematical model\nof chemotaxis, describing chemotactic aggregation in the\ncellular slime mold {\\it Dictyostelium discoideum} during its life cycle \\cite{KeSe,Pa}.\nIn this model, $u=u(t,x)$ and $v=v(t,x)$ denote the densities of cells and\nthe chemoattractant, respectively.\nThe parameter $\\lambda$ represents the degradation rate of the chemical.\nCells migrate toward regions of higher concentrations of a chemical substance\nsecreted by the cells themselves.\nFrom a biological viewpoint, it is natural to assume that the initial data\nare nonnegative functions.\nFrom a mathematical viewpoint, cell aggregation is interpreted as the\nblowup of solutions at $t=T$ in the sense that\n$\\lim_{t\\to T}\\|u(t)\\|_\\infty=\\infty.$\n\nOne of the central mathematical features of such systems is the phenomenon of critical\nmass.\nFor positive sufficiently regular solutions $(u,v)$ to~\\eqref{eqn;KS}, the first\ncomponent $u$ satisfies the mass conservation law \n$$\\|u(t)\\|_1=\\|u_0\\|_1\\quad \\text{for}~~~t>0.$$\nMoreover, in two space dimensions, the global behavior of solutions is governed\nby the size of the initial mass of $u$.\nIn particular, there exists a threshold value such that solutions with initial\nmass below this threshold exist globally in time, whereas solutions with\ninitial mass above the threshold may blow up in finite time.\nThis critical-mass phenomenon has attracted considerable attention in the\nliterature.\n\nThe aim of this paper is to establish the global behavior of solutions to\n\\eqref{eqn;KS} whose initial mass is exactly equal to this threshold value.\n\n\\vspace{3mm}\nThe second equation in \\eqref{eqn;KS} takes into account that cells are producing the chemoattractant themselves \nwhile this is diffusing into the environment. \nSince the chemoattractant attains its equilibrium on a time scale much quicker\nthan that of the cells, the simplified parabolic-elliptic system has also been\ninvestigated \\cite{JaLu,Na95}:\n\\begin{align*}\n\\left\\{\n\\begin{aligned}\n\\partial_t u &= \\Delta u - \\nabla\\cdot (u\\nabla v),\n& t>0,\\ & x\\in\\mathbb R^2,\\\\\n0 &= \\Delta v - \\lambda v + u,\n& t>0,\\ & x\\in\\mathbb R^2,\n\\\\\n&u(0,x)=u_{0}(x),\n&\\, &x\\in\\mathbb R^2.\n\\end{aligned}\n\\right.\n\\addtocounter{equation}{1}\\tag{\\theequation}\n\\label{eqn;PKS}\n\\end{align*}\nThe system \\eqref{eqn;PKS} is also related to models of gravitational interaction\nof particles \\cite{Bi-Na94,Wo}.\nIn~\\eqref{eqn;PKS}, the second equation can be written as\n$v = (-\\Delta+\\lambda)^{-1}u.$\nAs a consequence, the system~\\eqref{eqn;PKS} can be reduced to a single equation\nfor~$u$.\nOwing to this elliptic structure, the parabolic-elliptic system~\\eqref{eqn;PKS} is more amenable to analysis than the fully parabolic system~\\eqref{eqn;KS}, and has therefore been extensively studied and is now well understood in many aspects.\nIn particular, numerous works have been devoted to the critical-mass\nphenomenon.\nIndeed, the solution with $\\|u_0\\|_1\\le 8\\pi$ exists globally in time~\\cite{BiKaLaNa06,Bl-Ca-Ca,Bl-Ca-Ma,BlDoPe,DoPe,Lo-Na-Ya1,Lo-Na-Ya2,Na11,NaOg11,NaOg16,Na-Se,Wei},\nwhile the solution may blow up in finite time if $\\|u_0\\|_1>8\\pi$~\\cite{DoPe,Ko-Su,Ku-Og03,Wei}.\nFor the corresponding Cauchy--Neumann problem in bounded domains,\nsupplemented with homogeneous Neumann boundary conditions, see \\cite{BiKaLaNa06disc,Bi-Na94,Ga-Za98,Na95,Na01} for instance.\n\n\\vspace{3mm}\nUnlike the parabolic-elliptic system \\eqref{eqn;PKS}, the fully parabolic system\n\\eqref{eqn;KS} is a strongly coupled parabolic system, and many approaches\ndeveloped for \\eqref{eqn;PKS} are no longer applicable.\nThe global existence of solutions with sub-critical mass\n$\\|u_0\\|_{1}<8\\pi$ has nevertheless been established by combining\nLyapunov functionals with the Trudinger--Moser type inequality and its optimal\nconstant.\nMore precisely, the following results are known:\n\\begin{enumerate}\n\t\\item[(i)]\n\tIf $\\|u_0\\|_{1}<8\\pi$, then the corresponding solution to\n\t\\eqref{eqn;KS} exists globally in time \\cite{CaCo08,Mi13,NaOg11}.\n\n\t\\item[(ii)]\n\tIf $\\|u_0\\|_{1}>8\\pi$ and $(u_0,v_0)$ is radially symmetric, then there\n\texists a radially symmetric solution that blows up in finite time\n\t\\cite{Mi20,Mi20-SIAM}.\n\n\t\\item[(iii)]\n\tIf $\\|u_0\\|_{1}=8\\pi$ and $(u_0,v_0)$ is either radially symmetric or\n\tsatisfies the additional moment condition\n\t$u_0\\ln(1+|x|^2)\\in L^1(\\mathbb R^2)$, then the corresponding solution exists globally\n\tin time.\n\tIn contrast, for general initial data, the global behavior in the critical case\n\tremains delicate, and the solution either exists globally or blows up on the plane \\cite{CaCo08,Mi13}.\n\\end{enumerate}\nAs for positive forward self-similar solutions to \\eqref{eqn;KS}, refer to \\cite{BiCoDo}.\nFor the corresponding Cauchy--Neumann problem in bounded domains, we refer to\n\\cite{Bi98,He-Ve,HoWa,Na-Se-Yo}, for instance.\n\nAs mentioned above, although several partial results are available, no complete global existence result has been obtained\nfor solutions with critical mass and general initial data without any\nsymmetry or moment assumptions.\nOne of the main difficulties lies in controlling the behavior of solutions at~$|x|\\to\\infty$.\nMoreover, in the critical mass case, no global existence result is currently\nknown for the corresponding Cauchy--Neumann problem to~\\eqref{eqn;KS} in bounded domains.\n\nIn this paper, we establish the global existence of solutions with critical mass\nfor general initial data $(u_0,v_0)$.\nTo this end, we first recall the definition of solutions.\n\n\\vspace{2mm}\n\\begin{def*}\nLet $4/30,\\ & x\\in\\mathbb R^2,\\\\\n0 &= \\Delta v - \\lambda v + u,\n& t>0,\\ & x\\in\\mathbb R^2,\n\\\\\n&u(0,x)=u_{0}(x),\n&\\, &x\\in\\mathbb R^2.\n\\end{aligned}\n\\right.\n\\addtocounter{equation}{1}\\tag{\\theequation}\n\\label{eqn;PKS}\n\\end{align*}\nThe system \\eqref{eqn;PKS} is also related to models of gravitational interaction\nof particles \\cite{Bi-Na94,Wo}.\nIn~\\eqref{eqn;PKS}, the second equation can be written as\n$v = (-\\Delta+\\lambda)^{-1}u.$\nAs a consequence, the system~\\eqref{eqn;PKS} can be reduced to a single equation\nfor~$u$.\nOwing to this elliptic structure, the parabolic-elliptic system~\\eqref{eqn;PKS} is more amenable to analysis than the fully parabolic system~\\eqref{eqn;KS}, and has therefore been extensively studied and is now well understood in many aspects.\nIn particular, numerous works have been devoted to the critical-mass\nphenomenon.\nIndeed, the solution with $\\|u_0\\|_1\\le 8\\pi$ exists globally in time~\\cite{BiKaLaNa06,Bl-Ca-Ca,Bl-Ca-Ma,BlDoPe,DoPe,Lo-Na-Ya1,Lo-Na-Ya2,Na11,NaOg11,NaOg16,Na-Se,Wei},\nwhile the solution may blow up in finite time if $\\|u_0\\|_1>8\\pi$~\\cite{DoPe,Ko-Su,Ku-Og03,Wei}.\nFor the corresponding Cauchy--Neumann problem in bounded domains,\nsupplemented with homogeneous Neumann boundary conditions, see \\cite{BiKaLaNa06disc,Bi-Na94,Ga-Za98,Na95,Na01} for instance.\n\n\\vspace{3mm}\nUnlike the parabolic-elliptic system \\eqref{eqn;PKS}, the fully parabolic system\n\\eqref{eqn;KS} is a strongly coupled parabolic system, and many approaches\ndeveloped for \\eqref{eqn;PKS} are no longer applicable.\nThe global existence of solutions with sub-critical mass\n$\\|u_0\\|_{1}<8\\pi$ has nevertheless been established by combining\nLyapunov functionals with the Trudinger--Moser type inequality and its optimal\nconstant.\nMore precisely, the following results are known:\n\\begin{enumerate}\n \\item[(i)]\n If $\\|u_0\\|_{1}<8\\pi$, then the corresponding solution to\n \\eqref{eqn;KS} exists globally in time \\cite{CaCo08,Mi13,NaOg11}.\n\n\\item[(iii)]\n If $\\|u_0\\|_{1}=8\\pi$ and $(u_0,v_0)$ is either radially symmetric or\n satisfies the additional moment condition\n $u_0\\ln(1+|x|^2)\\in L^1(\\mathbb R^2)$, then the corresponding solution exists globally\n in time.\n In contrast, for general initial data, the global behavior in the critical case\n remains delicate, and the solution either exists globally or blows up on the plane \\cite{CaCo08,Mi13}.\n\\end{enumerate}\nAs for positive forward self-similar solutions to \\eqref{eqn;KS}, refer to \\cite{BiCoDo}.\nFor the corresponding Cauchy--Neumann problem in bounded domains, we refer to\n\\cite{Bi98,He-Ve,HoWa,Na-Se-Yo}, for instance.\n\n\\vspace{2mm}\n\\begin{def*}\nLet $4/30,\\ & x\\in\\mathbb R^2,\\\\\n0 &= \\Delta v - \\lambda v + u,\n& t>0,\\ & x\\in\\mathbb R^2,\n\\\\\n&u(0,x)=u_{0}(x),\n&\\, &x\\in\\mathbb R^2.\n\\end{aligned}\n\\right.\n\\addtocounter{equation}{1}\\tag{\\theequation}\n\\label{eqn;PKS}\n\\end{align*}\nThe system \\eqref{eqn;PKS} is also related to models of gravitational interaction\nof particles \\cite{Bi-Na94,Wo}.\nIn~\\eqref{eqn;PKS}, the second equation can be written as\n$v = (-\\Delta+\\lambda)^{-1}u.$\nAs a consequence, the system~\\eqref{eqn;PKS} can be reduced to a single equation\nfor~$u$.\nOwing to this elliptic structure, the parabolic-elliptic system~\\eqref{eqn;PKS} is more amenable to analysis than the fully parabolic system~\\eqref{eqn;KS}, and has therefore been extensively studied and is now well understood in many aspects.\nIn particular, numerous works have been devoted to the critical-mass\nphenomenon.\nIndeed, the solution with $\\|u_0\\|_1\\le 8\\pi$ exists globally in time~\\cite{BiKaLaNa06,Bl-Ca-Ca,Bl-Ca-Ma,BlDoPe,DoPe,Lo-Na-Ya1,Lo-Na-Ya2,Na11,NaOg11,NaOg16,Na-Se,Wei},\nwhile the solution may blow up in finite time if $\\|u_0\\|_1>8\\pi$~\\cite{DoPe,Ko-Su,Ku-Og03,Wei}.\nFor the corresponding Cauchy--Neumann problem in bounded domains,\nsupplemented with homogeneous Neumann boundary conditions, see \\cite{BiKaLaNa06disc,Bi-Na94,Ga-Za98,Na95,Na01} for instance.\n\n\\vspace{3mm}\nUnlike the parabolic-elliptic system \\eqref{eqn;PKS}, the fully parabolic system\n\\eqref{eqn;KS} is a strongly coupled parabolic system, and many approaches\ndeveloped for \\eqref{eqn;PKS} are no longer applicable.\nThe global existence of solutions with sub-critical mass\n$\\|u_0\\|_{1}<8\\pi$ has nevertheless been established by combining\nLyapunov functionals with the Trudinger--Moser type inequality and its optimal\nconstant.\nMore precisely, the following results are known:\n\\begin{enumerate}\n \\item[(i)]\n If $\\|u_0\\|_{1}<8\\pi$, then the corresponding solution to\n \\eqref{eqn;KS} exists globally in time \\cite{CaCo08,Mi13,NaOg11}.\n\n\\item[(iii)]\n If $\\|u_0\\|_{1}=8\\pi$ and $(u_0,v_0)$ is either radially symmetric or\n satisfies the additional moment condition\n $u_0\\ln(1+|x|^2)\\in L^1(\\mathbb R^2)$, then the corresponding solution exists globally\n in time.\n In contrast, for general initial data, the global behavior in the critical case\n remains delicate, and the solution either exists globally or blows up on the plane \\cite{CaCo08,Mi13}.\n\\end{enumerate}\nAs for positive forward self-similar solutions to \\eqref{eqn;KS}, refer to \\cite{BiCoDo}.\nFor the corresponding Cauchy--Neumann problem in bounded domains, we refer to\n\\cite{Bi98,He-Ve,HoWa,Na-Se-Yo}, for instance.\n\n\\vspace{2mm}\n\\begin{def*}\nLet $4/30$ and some $C(\\tau,\\alpha)>0$ (cf.~\\cite{CaCo08,Mi13}). Hence, if $\\|u_0\\|<8\\pi$, then all terms on the left hand side of \\eqref{eqn;afford} are positive by choosing $\\al>0$ sufficiently small which depends on $\\|u_0\\|_1$, and Gronwall's inequality implies that\n$\\nabla v \\in L^2((0,T)\\times \\R^2)$, \nso that we also obtain the bound for the modified entropy $\\int_{\\R^2}(1+u)\\ln (1+u)\\dx$. \nNevertheless, for the critical mass $\\|u_0\\|_1=8\\pi$, this necessarily leads to $\\alpha=0$, as a result, \n\\begin{align*}\n\\int_0^t \\D_m(s)\\,\\d s\\le\\,&\\L_m(0)+C(\\tau)+\\frac14\\int_0^t\\|\\nabla v\\|_2^2\\,\\d s,\n\\eqntag\n\\label{eqn;lack-lyapunov}\n\\end{align*}\nwhich is useless and fails to obtain even the estimates for $\\D_m(t)$ due to the lack of the regularity of solutions.\nTherefore, we introduce a reconstructed Lyapunov functional $\\F_m(t)$ so as to show regularity estimates corresponding to the dissipative terms. Let $\\F_m(t)$ be the functional for solutions to \\eqref{eqn;KS} defined as\n\\begin{align*}\n\\F_m(t):=\\L_m(t)\n+\\int_{\\R^2}\\ln(1+u)\\,\\mathrm{d}x-\\int_{\\R^2}v\\,\\d x.\n\\eqntag\n\\label{eqn;modifiedLyapunov}\n\\end{align*}\nThen, the following functional differential inequality holds true:\n\\begin{prop}\\label{prop;energy-est}\nLet $(u,v)$ be the solution to \\eqref{eqn;KS}. Then, the functional $\\F_m(t)$ defined in~\\eqref{eqn;modifiedLyapunov} satisfies the following identity:\n \\begin{align*}\n \\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2\n \\le\\,&\n \\lambda\\int_{\\R^2}v\\,\\d x.\n \\end{align*}\n Therefore, if $\\lambda=0$ then\n \\begin{align*}\n \\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2 \\le\\,0.\n \\end{align*}\n If $\\lambda>0$ then \n \\begin{align*}\n \\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2 \\le\\,\\|v_0\\|_1+\\|u_0\\|_1.\n \\end{align*}\n\\end{prop}\n\\vspace{5mm}\nIt is worth emphasizing that $\\F_m(t)$ is non-increasing in time when $\\lambda=0$ as well as\nunlike the identity \\eqref{eqn;usualmodifiedfunctiona} based on the usual modified functional $\\mathcal{L}_m(t)$, the error term appearing on the right-hand side of Proposition~\\ref{prop;energy-est} can be easily controlled by initial data.\nThis allows us to show the regularity estimates for the dissipative term associated with $\\F_m(t)$, see Proposition~\\ref{prop;bound-energy} below.\n\\begin{lem}\\label{lem;ModifiedLF}\n Suppose assumptions as in Theorem \\ref{thm;global}. Then,\n\\begin{align*}\n\\frac{\\d}{\\d t}\\F_m(t)+ \\widetilde{\\D}(t)=-\\int_{\\R^2}\\partial_t v \\frac{u}{1+u}\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\,\\mathrm{d}x+\\lambda\\int_{\\R^2}\\frac{v}{1+u}\\,\\d x,\n\\end{align*}\nwhere\n$\\F_m(t)$ is the modified functional defined in \\eqref{eqn;modifiedLyapunov} and the dissipative term $\\widetilde{\\D}(t)$ is given by\n\\begin{align*}\n\\widetilde{\\D}(t):=\\int_{\\R^2}u\\left|\\nabla \\left(\\ln(1+u)-v\\right)\\right|^2\\,\\d x+\\|\\partial_t v\\|_2^2.\n\\end{align*}\n\\end{lem}\n\n\\vspace{5mm}\n\\begin{pr}{Theorem \\ref{thm;global}}\nLet $R$ be taken as in Lemma \\ref{lem;L^2-exterior}.\nBy Lemma \\ref{lem;LlogL-bound-exterior},\n\\begin{align*}\n\\int_{ |x|>2R } (1+u) \\ln(1+u) \\dx\\le\\, C(t_0,\\tau,R).\n\\end{align*}\nOn the other hand, Lemma \\ref{prop;interior-bdd-entropy} gives\n\\begin{align*}\n\\int_{ \\R^2}(u\\ln u)\\psi_R^2\\dx\\le\\, C(t_0,\\tau,R)\n\\end{align*}\nfor $t\\in[t_0,\\tau]\\cap(0,T)$.\nNow, according to \\cite[Lemma 2.3]{NaOg16},\n\\begin{align*}\n\\int_{ \\Omega} (1+u)\\ln(1+u)\\dx\\le\\,2\\int_{ \\Omega} u |\\ln u | \\dx+(2\\ln 2)\\int_{ \\Omega} u\\dx,\n\\end{align*}\nwhere $\\Omega$ is a measurable set in $\\R^2$,\n so that since $\\supp\\psi_R\\subset \\{x; |x|\\le 16 R\\}$\n \\begin{align*}\n \\int_{ |x|<8R }(1+u)\\ln(1+u)\\dx\\le\\,&2\\int_{ |x|<8R }u|\\ln u| \\dx+(2\\ln 2)\\|u_0\\|_1\n \\\\\n \\le\\,\\,&2\\int_{ \\R^2 }u|\\ln u| \\psi_R^2\\dx+(2\\ln 2)\\|u_0\\|_1\n \\\\\n =\\,&2\\int_{ \\R^2 }(u\\ln u) \\psi_R^2\\dx-4\\int_{ \\R^2 }(u\\ln u)_-\\psi_R^2\\dx+(2\\ln 2)\\|u_0\\|_1\n \\\\\n \\le\\,&C(t_0,\\tau,R) +\\frac4e|B_{16R}(0)|\n \\end{align*}\n for $t\\in[t_0,\\tau]\\cap(0,T)$, where we use $x\\ln x \\ge -1/e$ for $x\\ge0$.\n This implies that\n \\begin{align*}\n&\\int_{ \\R^2 }(1+u)\\ln(1+u)\\dx\n\\\\=\\,&\\int_{ |x|>2R } (1+u) \\ln(1+u) \\dx\n+\\int_{ |x|\\le 2R } (1+u) \\ln(1+u) \\dx\n\\\\\n\\le\\,&\\int_{ |x|>2R } (1+u) \\ln(1+u) \\dx+\\int_{ |x|< 8R } (1+u) \\ln(1+u) \\dx\n\\\\\n\\le\\,&C(t_0,\\tau,R) \n \\end{align*}\n for $t\\in[t_0,\\tau]\\cap(0,T)$.\n Hence, we have along with Proposition \\ref{prop;bound-energy}\n \\begin{align*}\n\\int_{ \\R^2 }(1+u) \\ln (1+u)\\dx+\\int_{t_0}^t \\|\\partial_tv\\|_2^2\\,\\d s\\le\\, C(t_0,\\tau,R)\n \\eqntag\n \\label{eqn;uniform-ulogu}\n \\end{align*}\n for $t\\in[t_0,\\tau]\\cap(0,T)$.\nBy the parabolic regularity argument, we obtain a uniform $L^2$-bound for~$u$\non $[t_0,\\tau]\\cap(0,T)$ from \\eqref{eqn;uniform-ulogu} (cf. \\cite{Mi13,Na-Se-Yo}), which yields further regularity estimates.\nIn particular, the solution~$(u,v)$ cannot blow up in finite time.\nTherefore, the solution to \\eqref{eqn;KS} exists globally in time.\n\\end{pr}", + "post_theorem_intro_text_len": 2221, + "post_theorem_intro_text": "\\vspace{2mm}\n\n\\begin{rem}\nIn Theorem~\\ref{thm;global}, the initial data are only required to satisfy\n\\[\n(u_0,v_0)\\in L_+^1(\\mathbb R^2)\\times\\bigl(L_+^1(\\mathbb R^2)\\cap \\dot H^1(\\mathbb R^2)\\bigr)\\quad \\text{with}~~\\|u_0\\|_1=8\\pi.\n\\]\nIn particular, no additional symmetry or moment assumptions are imposed.\n\\end{rem}\n\\vspace{2mm}\nAs mentioned above, the global existence of solutions with critical mass\nwithout any symmetry or moment assumptions remains delicate.\nIn~\\cite[Theorem~1.2]{Mi13}, it is shown that, for general initial data with\ncritical mass, solutions either exist globally in time or blow up on the plane,\nby means of a contradiction argument.\nIndeed, although in the sub-critical case regularity estimates can be obtained\nby combining (modified) Lyapunov functionals with the Trudinger--Moser\ninequality, such estimates are no longer directly available in the critical case due to the lack of regularity of solutions,\nsee~\\eqref{eqn;afford} below.\nThis difficulty prevents the direct extension of classical entropy methods.\n\nTo overcome this difficulty, we introduce a reconstructed Lyapunov functional.\nThe main novelty of this work lies in the construction of a refined Lyapunov\nfunctional specifically adapted to the whole space setting, which allows us\nto control the behavior of solutions at $|x|\\to\\infty$ and to derive\nregularity estimates even in the critical mass regime, without\nimposing any symmetry or moment assumptions on the initial data.\nAs a consequence, we establish global-in-time existence for general initial\ndata at the critical mass, a result that was previously out of reach by existing\nmethods, see Subsection~\\ref{sect;modified-Lyapunov} for details.\nMoreover, the present approach is expected to be applicable to a broad class\nof chemotaxis systems in the whole space setting.\n\n\\vspace{3mm}\nThe remainder of this paper is organized as follows.\nIn Section~\\ref{sect;preliminary}, we recall several preliminary lemmas needed\nto prove the main result.\nSection~\\ref{sect;LWP} is devoted to the local well-posedness of~\\eqref{eqn;KS}, based on the above definition of solutions.\nFinally, in Section~\\ref{sect;global}, we present the proof of\nTheorem~\\ref{thm;global}.\n\n\\vspace{5mm}", + "sketch": "The post-theorem introduction does not give a step-by-step proof, but it outlines the strategy for proving Theorem~\\ref{thm;global}: classical entropy/Lyapunov approaches for sub-critical mass (using “(modified) Lyapunov functionals with the Trudinger--Moser inequality”) “are no longer directly available in the critical case due to the lack of regularity of solutions,” which “prevents the direct extension of classical entropy methods.” To overcome this, the authors “introduce a reconstructed Lyapunov functional,” whose “main novelty…lies in the construction of a refined Lyapunov functional specifically adapted to the whole space setting,” allowing them “to control the behavior of solutions at $|x|\\to\\infty$ and to derive regularity estimates even in the critical mass regime, without imposing any symmetry or moment assumptions on the initial data.” With these estimates, they “establish global-in-time existence for general initial data at the critical mass.” The paper’s structure supporting the proof is: preliminaries (Section~\\ref{sect;preliminary}), local well-posedness (Section~\\ref{sect;LWP}), then the global argument proving Theorem~\\ref{thm;global} (Section~\\ref{sect;global}).", + "expanded_sketch": "The post-theorem introduction does not give a step-by-step proof, but it outlines the strategy for proving the main theorem: classical entropy/Lyapunov approaches for sub-critical mass (using “(modified) Lyapunov functionals with the Trudinger--Moser inequality”) “are no longer directly available in the critical case due to the lack of regularity of solutions,” which “prevents the direct extension of classical entropy methods.” To overcome this, the authors “introduce a reconstructed Lyapunov functional,” whose “main novelty…lies in the construction of a refined Lyapunov functional specifically adapted to the whole space setting,” allowing them “to control the behavior of solutions at $|x|\\to\\infty$ and to derive regularity estimates even in the critical mass regime, without imposing any symmetry or moment assumptions on the initial data.” With these estimates, they “establish global-in-time existence for general initial data at the critical mass.” The paper’s structure supporting the proof is: preliminaries (proved next), local well-posedness (proved after that), then the global argument establishing the main theorem (proved later).", + "expanded_theorem": "\\label{thm;global}\nFor $(u_0,v_0)\\in L_+^1(\\mathbb R^2) \\times L^1_+(\\mathbb R^2)\\cap \\dot{H}^1(\\mathbb R^2)$,\nlet $(u,v)$ be the solution to \n\\begin{aligned}\n&\\pt_t u =\\Delta u- \\N\\cd \\left( u \\N v \\right),\n& t>0,\\,~ &x\\in\\R^2,\n\\\\\n&\\pt_t v = \\Delta v - \\lambda v +u,\n&t>0,\\, ~&x\\in\\R^2,\n\\\\\n&(u,v)(0,x)=(u_{0},v_0)(x),\n&\\, &x\\in\\R^2\n\\end{aligned}\n\\right.\n\\label{eqn;KS}\n\\end{equation}\nwith a constant $\\lambda\\ge0$, where $u_0,v_0\\ge 0$ on $\\R^2$ and\n$u_0,v_0\\not\\equiv0$\non $(0,T)\\times\\mathbb R^2$.\nSuppose that $\\|u_0\\|_1=8\\pi$.\nThen, in establishing the main theorem, the solution to the system above exists globally in time.", + "theorem_type": [ + "Implication", + "Existence" + ], + "mcq": { + "question": "Consider the two-dimensional fully parabolic Keller--Segel system\n\\[\n\\begin{cases}\n\\partial_t u = \\Delta u-\\nabla\\!\\cdot\\!\\big(u\\nabla v\\big), & t>0,\\ x\\in\\mathbb R^2,\\\\\n\\partial_t v = \\Delta v-\\lambda v+u, & t>0,\\ x\\in\\mathbb R^2,\\\\\n(u,v)(0,x)=(u_0,v_0)(x), & x\\in\\mathbb R^2,\n\\end{cases}\n\\]\nwhere \\(\\lambda\\ge 0\\). Assume\n\\(u_0\\in L_+^1(\\mathbb R^2)\\) and \\(v_0\\in L_+^1(\\mathbb R^2)\\cap \\dot H^1(\\mathbb R^2)\\), with\n\\[\nL_+^1(\\mathbb R^2):=\\{f\\in L^1(\\mathbb R^2): f\\ge 0\\ \\text{and}\\ f\\not\\equiv 0\\},\n\\]\nso in particular \\(u_0,v_0\\) are nonnegative and nontrivial, and suppose that the initial cell mass satisfies\n\\[\n\\|u_0\\|_{L^1(\\mathbb R^2)}=8\\pi.\n\\]\nWhich of the following conclusions about the corresponding solution \\((u,v)\\) holds?", + "correct_choice": { + "label": "A", + "text": "The solution \\((u,v)\\) exists globally in time; equivalently, it can be continued for all \\(t\\ge 0\\) and does not blow up in finite time." + }, + "choices": [ + { + "label": "B", + "text": "The solution \\((u,v)\\) exists globally in time provided, in addition, that the initial data are either radially symmetric or satisfy a finite logarithmic moment condition such as \\(u_0\\ln(1+|x|^2)\\in L^1(\\mathbb R^2)\\); without one of these extra assumptions, finite-time blow-up may occur at mass \\(8\\pi\\)." + }, + { + "label": "C", + "text": "The corresponding solution \\((u,v)\\) has a local-in-time mild solution on some interval \\([0,T)\\) for a positive maximal existence time \\(T>0\\)." + }, + { + "label": "D", + "text": "For every such initial datum with \\(\\|u_0\\|_{L^1(\\mathbb R^2)}=8\\pi\\), the solution \\((u,v)\\) exists globally in time and moreover remains uniformly bounded in \\(L^\\infty(\\mathbb R^2)\\times \\dot W^{1,\\infty}(\\mathbb R^2)\\) for all \\(t\\ge 0\\)." + }, + { + "label": "E", + "text": "There exists a time \\(T=T(u_0,v_0)>0\\) such that the solution \\((u,v)\\) exists on \\([0,T)\\), and if it is global then this conclusion requires constants in the a priori estimates to depend on additional decay of the data at spatial infinity; at critical mass \\(8\\pi\\), global existence is therefore not guaranteed for arbitrary \\(u_0\\in L_+^1(\\mathbb R^2)\\) and \\(v_0\\in L_+^1(\\mathbb R^2)\\cap \\dot H^1(\\mathbb R^2)\\)." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "regularity", + "tampered_component": "removal_of_symmetry_or_moment_hypotheses", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "regularity", + "tampered_component": "dropped_global_in_time_and_no_blowup_conclusion", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "upgrade_from_global_existence_to_uniform_Linfty_control", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "geometric_construction", + "tampered_component": "whole_space_tail_control_for_arbitrary_data", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly reveal the correct conclusion. It states the PDE, hypotheses, and critical-mass condition, but the answer still depends on knowing or reasoning about the exact existence theorem." + }, + "TAS": { + "score": 1, + "justification": "The item is close to theorem recall: it essentially asks which existence conclusion matches the stated hypotheses. However, it is not a pure restatement because the options include weaker, stronger, and conditionally modified alternatives." + }, + "GPS": { + "score": 1, + "justification": "The question requires some discrimination among nearby claims (local vs global existence, extra decay assumptions, stronger boundedness conclusions), but it mainly tests recognition of the correct theorem rather than substantial generative reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target realistic confusions: requiring extra symmetry/decay assumptions, settling for only local existence, or overstrengthening the conclusion to uniform L^infty bounds." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with good distractors and no major answer leakage, though it leans more toward recall of a specific result than deep generative reasoning." + } + }, + { + "id": "2602.03768v1", + "paper_link": "http://arxiv.org/abs/2602.03768v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "thm", + "content": "\\label{thm;global}\nFor $(u_0,v_0)\\in L_+^1(\\mathbb R^2) \\times L^1_+(\\mathbb R^2)\\cap \\dot{H}^1(\\mathbb R^2)$,\nlet $(u,v)$ be the solution to \\eqref{eqn;KS} on $(0,T)\\times\\mathbb R^2$.\nSuppose that $\\|u_0\\|_1=8\\pi$.\nThen, the solution to \\eqref{eqn;KS} exists globally in time.", + "start_pos": 14650, + "end_pos": 14921, + "label": "thm;global" + }, + "ref_dict": { + "thm;global": "\\begin{thm}\\label{thm;global}\nFor $(u_0,v_0)\\in L_+^1(\\R^2) \\times L^1_+(\\R^2)\\cap \\dot{H}^1(\\R^2)$,\nlet $(u,v)$ be the solution to \\eqref{eqn;KS} on $(0,T)\\times\\R^2$.\nSuppose that $\\|u_0\\|_1=8\\pi$.\nThen, the solution to \\eqref{eqn;KS} exists globally in time.\n\\end{thm}", + "eqn;KS": "\\begin{aligned}\n&\\pt_t u =\\Delta u- \\N\\cd \\left( u \\N v \\right),\n& t>0,\\,~ &x\\in\\R^2,\n\\\\\n&\\pt_t v = \\Delta v - \\lambda v +u,\n&t>0,\\, ~&x\\in\\R^2,\n\\\\\n&(u,v)(0,x)=(u_{0},v_0)(x),\n&\\, &x\\in\\R^2\n\\end{aligned}\n\\right.\n\\label{eqn;KS}\n\\end{equation}\nwith a constant $\\lambda\\ge0$, where $u_0,v_0\\ge 0$ on $\\R^2$ and\n$u_0,v_0\\not\\equiv0$.\nThe Keller--Segel system~\\eqref{eqn;KS} is a fundamental mathematical model\nof chemotaxis, describing chemotactic aggregation in the\ncellular slime mold {\\it Dictyostelium discoideum} during its life cycle \\cite{KeSe,Pa}.\nIn this model, $u=u(t,x)$ and $v=v(t,x)$ denote the densities of cells and\nthe chemoattractant, respectively.\nThe parameter $\\lambda$ represents the degradation rate of the chemical.\nCells migrate toward regions of higher concentrations of a chemical substance\nsecreted by the cells themselves.\nFrom a biological viewpoint, it is natural to assume that the initial data\nare nonnegative functions.\nFrom a mathematical viewpoint, cell aggregation is interpreted as the\nblowup of solutions at $t=T$ in the sense that\n$\\lim_{t\\to T}\\|u(t)\\|_\\infty=\\infty.$\n\nOne of the central mathematical features of such systems is the phenomenon of critical\nmass.\nFor positive sufficiently regular solutions $(u,v)$ to~\\eqref{eqn;KS}, the first\ncomponent $u$ satisfies the mass conservation law \n$$\\|u(t)\\|_1=\\|u_0\\|_1\\quad \\text{for}~~~t>0.$$\nMoreover, in two space dimensions, the global behavior of solutions is governed\nby the size of the initial mass of $u$.\nIn particular, there exists a threshold value such that solutions with initial\nmass below this threshold exist globally in time, whereas solutions with\ninitial mass above the threshold may blow up in finite time.\nThis critical-mass phenomenon has attracted considerable attention in the\nliterature.\n\nThe aim of this paper is to establish the global behavior of solutions to\n\\eqref{eqn;KS} whose initial mass is exactly equal to this threshold value.\n\n\\vspace{3mm}\nThe second equation in \\eqref{eqn;KS} takes into account that cells are producing the chemoattractant themselves \nwhile this is diffusing into the environment. \nSince the chemoattractant attains its equilibrium on a time scale much quicker\nthan that of the cells, the simplified parabolic-elliptic system has also been\ninvestigated \\cite{JaLu,Na95}:\n\\begin{align*}\n\\left\\{\n\\begin{aligned}\n\\partial_t u &= \\Delta u - \\nabla\\cdot (u\\nabla v),\n& t>0,\\ & x\\in\\R^2,\\\\\n0 &= \\Delta v - \\lambda v + u,\n& t>0,\\ & x\\in\\R^2,\n\\\\\n&u(0,x)=u_{0}(x),\n&\\, &x\\in\\R^2.\n\\end{aligned}", + "prop;LWP": "\\begin{prop}[local-in-time solution]\\label{prop;LWP}\n\tLet $4/3 0$ in $(0,T)\\times\\R^2$ and\n\t\\begin{equation*}\n\t\\|u(t)\\|_1=\\|u_0\\|_1, \\qquad t\\in [0,T).\t\n\t\\end{equation*} \n\\end{prop}", + "eqn;PKS": "\\begin{aligned}\n\\partial_t u &= \\Delta u - \\nabla\\cdot (u\\nabla v),\n& t>0,\\ & x\\in\\R^2,\\\\\n0 &= \\Delta v - \\lambda v + u,\n& t>0,\\ & x\\in\\R^2,\n\\\\\n&u(0,x)=u_{0}(x),\n&\\, &x\\in\\R^2.\n\\end{aligned}\n\\right.\n\\eqntag\n\\label{eqn;PKS}\n\\end{align*}\nThe system \\eqref{eqn;PKS} is also related to models of gravitational interaction\nof particles \\cite{Bi-Na94,Wo}.\nIn~\\eqref{eqn;PKS}, the second equation can be written as\n$v = (-\\Delta+\\lambda)^{-1}u.$\nAs a consequence, the system~\\eqref{eqn;PKS} can be reduced to a single equation\nfor~$u$.\nOwing to this elliptic structure, the parabolic-elliptic system~\\eqref{eqn;PKS} is more amenable to analysis than the fully parabolic system~\\eqref{eqn;KS}, and has therefore been extensively studied and is now well understood in many aspects.\nIn particular, numerous works have been devoted to the critical-mass\nphenomenon.\nIndeed, the solution with $\\|u_0\\|_1\\le 8\\pi$ exists globally in time~\\cite{BiKaLaNa06,Bl-Ca-Ca,Bl-Ca-Ma,BlDoPe,DoPe,Lo-Na-Ya1,Lo-Na-Ya2,Na11,NaOg11,NaOg16,Na-Se,Wei},\nwhile the solution may blow up in finite time if $\\|u_0\\|_1>8\\pi$~\\cite{DoPe,Ko-Su,Ku-Og03,Wei}.\nFor the corresponding Cauchy--Neumann problem in bounded domains,\nsupplemented with homogeneous Neumann boundary conditions, see \\cite{BiKaLaNa06disc,Bi-Na94,Ga-Za98,Na95,Na01} for instance.\n\n\\vspace{3mm}\nUnlike the parabolic-elliptic system \\eqref{eqn;PKS}, the fully parabolic system\n\\eqref{eqn;KS} is a strongly coupled parabolic system, and many approaches\ndeveloped for \\eqref{eqn;PKS} are no longer applicable.\nThe global existence of solutions with sub-critical mass\n$\\|u_0\\|_{1}<8\\pi$ has nevertheless been established by combining\nLyapunov functionals with the Trudinger--Moser type inequality and its optimal\nconstant.\nMore precisely, the following results are known:\n\\begin{enumerate}\n\t\\item[(i)]\n\tIf $\\|u_0\\|_{1}<8\\pi$, then the corresponding solution to\n\t\\eqref{eqn;KS} exists globally in time \\cite{CaCo08,Mi13,NaOg11}.\n\n\t\\item[(ii)]\n\tIf $\\|u_0\\|_{1}>8\\pi$ and $(u_0,v_0)$ is radially symmetric, then there\n\texists a radially symmetric solution that blows up in finite time\n\t\\cite{Mi20,Mi20-SIAM}.\n\n\t\\item[(iii)]\n\tIf $\\|u_0\\|_{1}=8\\pi$ and $(u_0,v_0)$ is either radially symmetric or\n\tsatisfies the additional moment condition\n\t$u_0\\ln(1+|x|^2)\\in L^1(\\R^2)$, then the corresponding solution exists globally\n\tin time.\n\tIn contrast, for general initial data, the global behavior in the critical case\n\tremains delicate, and the solution either exists globally or blows up on the plane \\cite{CaCo08,Mi13}.\n\\end{enumerate}\nAs for positive forward self-similar solutions to \\eqref{eqn;KS}, refer to \\cite{BiCoDo}.\nFor the corresponding Cauchy--Neumann problem in bounded domains, we refer to\n\\cite{Bi98,He-Ve,HoWa,Na-Se-Yo}, for instance.\n\nAs mentioned above, although several partial results are available, no complete global existence result has been obtained\nfor solutions with critical mass and general initial data without any\nsymmetry or moment assumptions.\nOne of the main difficulties lies in controlling the behavior of solutions at~$|x|\\to\\infty$.\nMoreover, in the critical mass case, no global existence result is currently\nknown for the corresponding Cauchy--Neumann problem to~\\eqref{eqn;KS} in bounded domains.\n\nIn this paper, we establish the global existence of solutions with critical mass\nfor general initial data $(u_0,v_0)$.\nTo this end, we first recall the definition of solutions.\n\n\\vspace{2mm}\n\\begin{def*}\nLet $4/31\\}\\cap \\{\\supp\\phi\\} }|f|\\dx\\right)\\left(\\int_{ \\{|f|>1\\} }\\frac{|\\nabla f|^2}{1+|f|} \\phi\\dx\\right)\n\t\\\\\n\t&+4\\left(\\int_{\\R^2}|f \\nabla \\phi^{\\frac12}|\\dx\\right)^2+4\\left(\\int_{\\R^2}|f|\\phi\\dx\\right)\n\t\\end{align*}\n\tand for any $\\ep>0$\n\t\\begin{align*}\n\t\\int_{\\R^2}|f|^3\\phi\\dx\n\t\\le\\,&\\ep \\left(\\int_{ \\{\\supp\\phi\\} }(1+|f|)\\ln(1+|f|)\\dx\\right)\\left(\\int_{\\R^2 }|\\nabla f|^2\\phi\\dx\\right)\n\t\\\\\n\t&+C\\left(\\int_{\\R^2}|f^{\\frac32} \\nabla \\phi^{\\frac12}|\\dx\\right)^2+C_\\ep\\left(\\int_{\\R^2}|f|\\phi\\dx\\right),\n\t\\end{align*}\n\twhere the constant $C_\\ep\\to \\infty$ as $\\ep\\to0$.\n\\end{lem}\nThe proof of Lemma \\ref{lem;L2andL3} can be found in \\cite[Lemma 2.2]{NaOg16}.\n\n\\begin{lem}\\label{lem;Lp_Lq_heat}\n\tFor $1 \\le q \\le p \\le \\infty$,\n\tlet $f\\in L^q (\\R^n)$ and let $\\al$ be a multi-index.\n\tThen it follows that\n\t\\begin{equation*}\n\t\\| \\pt_x^{\\al} e^{t\\Del} f \\|_p \n\t\\le C t^{-\\frac{n}{2} \\left(\\frac{1}{q} - \\frac{1}{p} \\right) - \\frac{|\\al|}{2}} \\|f\\|_q\n\t\\end{equation*}\n\tfor all $t>0$.\n\\end{lem}\n\nThe proof is immediately obtained by use of Young's inequality and the \nconvolution expression of the heat evolution\nby the heat kernel, see for instance \\cite{GiGiSa}.\n\\begin{lem}\\label{lem;v-Lpbound}\nFor $v_0\\in L^1(\\R^2)\\cap \\dot H^1(\\R^2)$ and $f\\in L^\\infty \\left(0,\\infty; L^1(\\R^2) \\right)$,\nlet $v$ be the solution to $\\partial_t v =\\Delta v -\\lambda v +f$ on $(0,\\infty)\\times\\R^2$ with the initial data $v_0$ and $\\lambda\\ge0$. Then, for any $1\\le p<\\infty$, \n\\begin{align*}\n\\| v(t)\\|_p\n\\le\\,&\n\\left\\{\n\\begin{aligned}\n&\\|v_0\\|_p+\\frac1p\\|f\\|_{L^\\infty\\left(0,\\infty; L^1(\\R^2)\\right)} t^{\\frac1p} &\\text{if $\\lambda=0$},\n\\\\\n&\\|v_0\\|_p+\\lambda^{-\\frac1p} \\|f\\|_{L^\\infty\\left(0,\\infty; L^1(\\R^2)\\right)} \\Gamma\\left(\\frac1p\\right)&\\text{if $\\lambda>0$}.\n\\end{aligned}", + "eqn;afford": "\\begin{pr}{Proposition \\ref{prop;LWP}}\nBy virtue of Lemmas~\\ref{lem;fixed-point} and~\\ref{lem;continuous-depend},\nwe have established the existence of a unique mild solution to~\\eqref{eqn;KS} defined on a maximal time interval $[0,T)$ with~$T\\le\\infty$. As for the regularity of the solutions, we use the standard iteration argument with respect to the derivative. Define $|\\N|^\\al f(x):=\\F^{-1}[ |\\xi|^\\al \\F f(\\xi)\\,](x)$ for $x\\in\\R^2$ and $\\al>0$, where $\\F$ denotes the Fourier transform. Let $\\tau\\in (0,T)$ and $t\\in (0,\\tau)$. Recalling that, for $4/30$, $u(t)\\in L^1(\\R^2)\\cap W^{s,p}(\\R^2)$,\n$v(t)\\in \\dot{H}^{1}(\\R^2)\\cap W^{s,q}(\\R^2)$\nfor $t_00$ and some $C(\\tau,\\alpha)>0$ (cf.~\\cite{CaCo08,Mi13}). Hence, if $\\|u_0\\|<8\\pi$, then all terms on the left hand side of \\eqref{eqn;afford} are positive by choosing $\\al>0$ sufficiently small which depends on $\\|u_0\\|_1$, and Gronwall's inequality implies that\n$\\nabla v \\in L^2((0,T)\\times \\R^2)$, \nso that we also obtain the bound for the modified entropy $\\int_{\\R^2}(1+u)\\ln (1+u)\\dx$. \nNevertheless, for the critical mass $\\|u_0\\|_1=8\\pi$, this necessarily leads to $\\alpha=0$, as a result, \n\\begin{align*}\n\\int_0^t \\D_m(s)\\,\\d s\\le\\,&\\L_m(0)+C(\\tau)+\\frac14\\int_0^t\\|\\nabla v\\|_2^2\\,\\d s,\n\\eqntag\n\\label{eqn;lack-lyapunov}\n\\end{align*}\nwhich is useless and fails to obtain even the estimates for $\\D_m(t)$ due to the lack of the regularity of solutions.\nTherefore, we introduce a reconstructed Lyapunov functional $\\F_m(t)$ so as to show regularity estimates corresponding to the dissipative terms. Let $\\F_m(t)$ be the functional for solutions to \\eqref{eqn;KS} defined as\n\\begin{align*}\n\\F_m(t):=\\L_m(t)\n+\\int_{\\R^2}\\ln(1+u)\\,\\mathrm{d}x-\\int_{\\R^2}v\\,\\d x.\n\\eqntag\n\\label{eqn;modifiedLyapunov}\n\\end{align*}\nThen, the following functional differential inequality holds true:\n\\begin{prop}\\label{prop;energy-est}\nLet $(u,v)$ be the solution to \\eqref{eqn;KS}. Then, the functional $\\F_m(t)$ defined in~\\eqref{eqn;modifiedLyapunov} satisfies the following identity:\n\t\\begin{align*}\n\t\\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2\n\t\\le\\,&\n\t\\lambda\\int_{\\R^2}v\\,\\d x.\n\t\\end{align*}\n\tTherefore, if $\\lambda=0$ then\n\t\\begin{align*}\n\t\\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2 \\le\\,0.\n\t\\end{align*}\n\tIf $\\lambda>0$ then \n\t\\begin{align*}\n\t\\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2 \\le\\,\\|v_0\\|_1+\\|u_0\\|_1.\n\t\\end{align*}\n\\end{prop}\n\\vspace{5mm}\nIt is worth emphasizing that $\\F_m(t)$ is non-increasing in time when $\\lambda=0$ as well as\nunlike the identity \\eqref{eqn;usualmodifiedfunctiona} based on the usual modified functional $\\mathcal{L}_m(t)$, the error term appearing on the right-hand side of Proposition~\\ref{prop;energy-est} can be easily controlled by initial data.\nThis allows us to show the regularity estimates for the dissipative term associated with $\\F_m(t)$, see Proposition~\\ref{prop;bound-energy} below.\n\\begin{lem}\\label{lem;ModifiedLF}\n\tSuppose assumptions as in Theorem \\ref{thm;global}. Then,\n\\begin{align*}\n\\frac{\\d}{\\d t}\\F_m(t)+ \\widetilde{\\D}(t)=-\\int_{\\R^2}\\partial_t v \\frac{u}{1+u}\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\,\\mathrm{d}x+\\lambda\\int_{\\R^2}\\frac{v}{1+u}\\,\\d x,\n\\end{align*}\nwhere\n$\\F_m(t)$ is the modified functional defined in \\eqref{eqn;modifiedLyapunov} and the dissipative term $\\widetilde{\\D}(t)$ is given by\n\\begin{align*}\n\\widetilde{\\D}(t):=\\int_{\\R^2}u\\left|\\nabla \\left(\\ln(1+u)-v\\right)\\right|^2\\,\\d x+\\|\\partial_t v\\|_2^2.\n\\end{align*}\n\\end{lem}\n\n\\begin{pr}{Lemma \\ref{lem;ModifiedLF}}\nThe following computations are already well-known:\n\\begin{align*}\n\\frac{\\d}{\\d t} \\int_{\\R^2}(1+u)\\ln (1+u)\\,\\d x=\\,&\n-\\int_{\\R^2}(1+u)|\\nabla\\ln(1+u)|^2\\,\\d x\n+\\int_{\\R^2}u\\nabla\\ln(1+u)\\cdot\\nabla v\\,\\d x\n\\end{align*}\nas well as\n\\begin{align*}\n-\\frac{\\d}{\\d t}\\int_{\\R^2}uv\\,\\d x=\\,&\\int_{\\R^2}(1+u)\\nabla\\ln(1+u)\\cdot\\nabla v\\,\\d x-\\int_{\\R^2}u|\\nabla v|^2\\,\\d x\n\\\\\n&-\\|\\partial_tv\\|_2^2-\\frac12\\frac{\\d}{\\d t}\\|\\nabla v\\|_2^2-\\frac\\lambda2\\frac{\\d}{\\d t}\\|v\\|_2^2,\n\\end{align*}\nso that\n\\begin{align*}\n&\\frac{\\d}{\\d t}\\left[\\int_{\\R^2}(1+u)\\ln(1+u)\\,\\d x-\\int_{\\R^2}uv\\,\\d x+\\frac12\\left(\\|\\nabla v\\|_2^2+\\lambda\\|v\\|_2^2\\right)\\right]\n\\\\\n&+\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\|\\partial_tv\\|_2^2\n\\\\\n=\\,&-\\|\\nabla\\ln(1+u)\\|_2^2+\\int_{\\R^2}\\nabla\\ln(1+u)\\cdot\\nabla v\\,\\d x.\n\\end{align*}\nNext,\n\\begin{align*}\n\\frac{\\d}{\\d t}\\int_{\\R^2}\\ln(1+u)\\,\\d x=\\,&\\int_{\\R^2}\\frac{1}{1+u} \\left[\\Delta u -\\nabla\\cdot( u\\nabla v)\\right]\\,\\mathrm{d}x\n\\\\\n=\\,&-\\int_{\\R^2}\\nabla\\left(\\frac{1}{1+u}\\right)\\cdot \\nabla u\\,\\d x+\\int_{\\R^2}u\\nabla\\left(\\frac{1}{1+u}\\right)\\cdot \\nabla v\\,\\d x\n\\\\\n=\\,&\\int_{\\R^2}|\\nabla \\ln(1+u)|^2\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\nabla\\ln(1+u)\\cdot\\nabla v\\,\\d x\n\\\\\n=\\,&\\int_{\\R^2}|\\nabla \\ln(1+u)|^2\\,\\d x-\\int_{\\R^2}\\nabla\\ln(1+u)\\cdot\\nabla v\\,\\d x\n\\\\\n&+\\int_{\\R^2}\\frac{1}{1+u}\\nabla\\ln(1+u)\\cdot\\nabla v \\,\\d x.\n\\end{align*}\nHence, combining the above computations implies that\n\\begin{align*}\n&\\frac{\\d}{\\d t}\\left[\\int_{\\R^2}(1+u)\\ln(1+u)\\,\\d x-\\int_{\\R^2}uv\\,\\d x+\\frac12\\left(\\|\\nabla v\\|_2^2+\\lambda\\|v\\|_2^2\\right)+\\int_{\\R^2}\\ln(1+u)\\,\\d x\\right]\n\\\\\n&+\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\|\\partial_tv\\|_2^2\n\\\\\n=\\,&\\int_{\\R^2}\\frac{1}{(1+u)^2}\\nabla u\\cdot\\nabla v \\,\\d x.\n\\end{align*}\nSince\n\\begin{align*}\n\\int_{\\R^2}\\frac{1}{(1+u)^2}\\nabla u\\cdot\\nabla v \\,\\d x=\\,&-\\int_{\\R^2}\\nabla\\left(\\frac{1}{1+u}\\right)\\cdot \\nabla v\\,\\d x\n\\\\\n=\\,&\\int_{\\R^2}\\frac{1}{1+u}\\Delta v\\,\\d x\n\\\\\n=\\,&\\int_{\\R^2}\\frac{\\partial_t v}{1+u}\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\,\\d x+\\lambda\\int_{\\R^2}\\frac{v}{1+u}\\,\\d x\n\\\\\n=\\,&\\frac{\\d}{\\d t}\\int_{\\R^2}v\\,\\d x-\\int_{\\R^2}\\partial_t v \\frac{u}{1+u}\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\,\\d x+\\lambda\\int_{\\R^2}\\frac{v}{1+u}\\,\\d x,\n\\end{align*}\nwe end up with\n\\begin{align*}\n&\\frac{\\d}{\\d t}\\F_m(t)+\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\|\\partial_tv\\|_2^2\n\\\\\n=\\,&-\\int_{\\R^2}\\partial_t v \\frac{u}{1+u}\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\,\\d x+\\lambda\\int_{\\R^2}\\frac{v}{1+u}\\,\\d x,\n\\end{align*}\nas desired.\n\\end{pr}" + }, + "pre_theorem_intro_text_len": 8115, + "pre_theorem_intro_text": "We study the Cauchy problem for the parabolic-parabolic Keller--Segel system in~$\\mathbb R^2$\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n&\\pt_t u =\\Delta u- \\nabla\\cdot \\left( u \\nabla v \\right),\n& t>0,\\,~ &x\\in\\mathbb R^2,\n\\\\\n&\\pt_t v = \\Delta v - \\lambda v +u,\n&t>0,\\, ~&x\\in\\mathbb R^2,\n\\\\\n&(u,v)(0,x)=(u_{0},v_0)(x),\n&\\, &x\\in\\mathbb R^2\n\\end{aligned}\n\\right.\n\\label{eqn;KS}\n\\end{equation}\nwith a constant $\\lambda\\ge0$, where $u_0,v_0\\ge 0$ on $\\mathbb R^2$ and\n$u_0,v_0\\not\\equiv0$.\nThe Keller--Segel system~\\eqref{eqn;KS} is a fundamental mathematical model\nof chemotaxis, describing chemotactic aggregation in the\ncellular slime mold {\\it Dictyostelium discoideum} during its life cycle \\cite{KeSe,Pa}.\nIn this model, $u=u(t,x)$ and $v=v(t,x)$ denote the densities of cells and\nthe chemoattractant, respectively.\nThe parameter $\\lambda$ represents the degradation rate of the chemical.\nCells migrate toward regions of higher concentrations of a chemical substance\nsecreted by the cells themselves.\nFrom a biological viewpoint, it is natural to assume that the initial data\nare nonnegative functions.\nFrom a mathematical viewpoint, cell aggregation is interpreted as the\nblowup of solutions at $t=T$ in the sense that\n$\\lim_{t\\to T}\\|u(t)\\|_\\infty=\\infty.$\n\nOne of the central mathematical features of such systems is the phenomenon of critical\nmass.\nFor positive sufficiently regular solutions $(u,v)$ to~\\eqref{eqn;KS}, the first\ncomponent $u$ satisfies the mass conservation law \n$$\\|u(t)\\|_1=\\|u_0\\|_1\\quad \\text{for}~~~t>0.$$\nMoreover, in two space dimensions, the global behavior of solutions is governed\nby the size of the initial mass of $u$.\nIn particular, there exists a threshold value such that solutions with initial\nmass below this threshold exist globally in time, whereas solutions with\ninitial mass above the threshold may blow up in finite time.\nThis critical-mass phenomenon has attracted considerable attention in the\nliterature.\n\nThe aim of this paper is to establish the global behavior of solutions to\n\\eqref{eqn;KS} whose initial mass is exactly equal to this threshold value.\n\n\\vspace{3mm}\nThe second equation in \\eqref{eqn;KS} takes into account that cells are producing the chemoattractant themselves \nwhile this is diffusing into the environment. \nSince the chemoattractant attains its equilibrium on a time scale much quicker\nthan that of the cells, the simplified parabolic-elliptic system has also been\ninvestigated \\cite{JaLu,Na95}:\n\\begin{align*}\n\\left\\{\n\\begin{aligned}\n\\partial_t u &= \\Delta u - \\nabla\\cdot (u\\nabla v),\n& t>0,\\ & x\\in\\mathbb R^2,\\\\\n0 &= \\Delta v - \\lambda v + u,\n& t>0,\\ & x\\in\\mathbb R^2,\n\\\\\n&u(0,x)=u_{0}(x),\n&\\, &x\\in\\mathbb R^2.\n\\end{aligned}\n\\right.\n\\addtocounter{equation}{1}\\tag{\\theequation}\n\\label{eqn;PKS}\n\\end{align*}\nThe system \\eqref{eqn;PKS} is also related to models of gravitational interaction\nof particles \\cite{Bi-Na94,Wo}.\nIn~\\eqref{eqn;PKS}, the second equation can be written as\n$v = (-\\Delta+\\lambda)^{-1}u.$\nAs a consequence, the system~\\eqref{eqn;PKS} can be reduced to a single equation\nfor~$u$.\nOwing to this elliptic structure, the parabolic-elliptic system~\\eqref{eqn;PKS} is more amenable to analysis than the fully parabolic system~\\eqref{eqn;KS}, and has therefore been extensively studied and is now well understood in many aspects.\nIn particular, numerous works have been devoted to the critical-mass\nphenomenon.\nIndeed, the solution with $\\|u_0\\|_1\\le 8\\pi$ exists globally in time~\\cite{BiKaLaNa06,Bl-Ca-Ca,Bl-Ca-Ma,BlDoPe,DoPe,Lo-Na-Ya1,Lo-Na-Ya2,Na11,NaOg11,NaOg16,Na-Se,Wei},\nwhile the solution may blow up in finite time if $\\|u_0\\|_1>8\\pi$~\\cite{DoPe,Ko-Su,Ku-Og03,Wei}.\nFor the corresponding Cauchy--Neumann problem in bounded domains,\nsupplemented with homogeneous Neumann boundary conditions, see \\cite{BiKaLaNa06disc,Bi-Na94,Ga-Za98,Na95,Na01} for instance.\n\n\\vspace{3mm}\nUnlike the parabolic-elliptic system \\eqref{eqn;PKS}, the fully parabolic system\n\\eqref{eqn;KS} is a strongly coupled parabolic system, and many approaches\ndeveloped for \\eqref{eqn;PKS} are no longer applicable.\nThe global existence of solutions with sub-critical mass\n$\\|u_0\\|_{1}<8\\pi$ has nevertheless been established by combining\nLyapunov functionals with the Trudinger--Moser type inequality and its optimal\nconstant.\nMore precisely, the following results are known:\n\\begin{enumerate}\n\t\\item[(i)]\n\tIf $\\|u_0\\|_{1}<8\\pi$, then the corresponding solution to\n\t\\eqref{eqn;KS} exists globally in time \\cite{CaCo08,Mi13,NaOg11}.\n\n\t\\item[(ii)]\n\tIf $\\|u_0\\|_{1}>8\\pi$ and $(u_0,v_0)$ is radially symmetric, then there\n\texists a radially symmetric solution that blows up in finite time\n\t\\cite{Mi20,Mi20-SIAM}.\n\n\t\\item[(iii)]\n\tIf $\\|u_0\\|_{1}=8\\pi$ and $(u_0,v_0)$ is either radially symmetric or\n\tsatisfies the additional moment condition\n\t$u_0\\ln(1+|x|^2)\\in L^1(\\mathbb R^2)$, then the corresponding solution exists globally\n\tin time.\n\tIn contrast, for general initial data, the global behavior in the critical case\n\tremains delicate, and the solution either exists globally or blows up on the plane \\cite{CaCo08,Mi13}.\n\\end{enumerate}\nAs for positive forward self-similar solutions to \\eqref{eqn;KS}, refer to \\cite{BiCoDo}.\nFor the corresponding Cauchy--Neumann problem in bounded domains, we refer to\n\\cite{Bi98,He-Ve,HoWa,Na-Se-Yo}, for instance.\n\nAs mentioned above, although several partial results are available, no complete global existence result has been obtained\nfor solutions with critical mass and general initial data without any\nsymmetry or moment assumptions.\nOne of the main difficulties lies in controlling the behavior of solutions at~$|x|\\to\\infty$.\nMoreover, in the critical mass case, no global existence result is currently\nknown for the corresponding Cauchy--Neumann problem to~\\eqref{eqn;KS} in bounded domains.\n\nIn this paper, we establish the global existence of solutions with critical mass\nfor general initial data $(u_0,v_0)$.\nTo this end, we first recall the definition of solutions.\n\n\\vspace{2mm}\n\\begin{def*}\nLet $4/30,\\ & x\\in\\mathbb R^2,\\\\\n0 &= \\Delta v - \\lambda v + u,\n& t>0,\\ & x\\in\\mathbb R^2,\n\\\\\n&u(0,x)=u_{0}(x),\n&\\, &x\\in\\mathbb R^2.\n\\end{aligned}\n\\right.\n\\addtocounter{equation}{1}\\tag{\\theequation}\n\\label{eqn;PKS}\n\\end{align*}\nThe system \\eqref{eqn;PKS} is also related to models of gravitational interaction\nof particles \\cite{Bi-Na94,Wo}.\nIn~\\eqref{eqn;PKS}, the second equation can be written as\n$v = (-\\Delta+\\lambda)^{-1}u.$\nAs a consequence, the system~\\eqref{eqn;PKS} can be reduced to a single equation\nfor~$u$.\nOwing to this elliptic structure, the parabolic-elliptic system~\\eqref{eqn;PKS} is more amenable to analysis than the fully parabolic system~\\eqref{eqn;KS}, and has therefore been extensively studied and is now well understood in many aspects.\nIn particular, numerous works have been devoted to the critical-mass\nphenomenon.\nIndeed, the solution with $\\|u_0\\|_1\\le 8\\pi$ exists globally in time~\\cite{BiKaLaNa06,Bl-Ca-Ca,Bl-Ca-Ma,BlDoPe,DoPe,Lo-Na-Ya1,Lo-Na-Ya2,Na11,NaOg11,NaOg16,Na-Se,Wei},\nwhile the solution may blow up in finite time if $\\|u_0\\|_1>8\\pi$~\\cite{DoPe,Ko-Su,Ku-Og03,Wei}.\nFor the corresponding Cauchy--Neumann problem in bounded domains,\nsupplemented with homogeneous Neumann boundary conditions, see \\cite{BiKaLaNa06disc,Bi-Na94,Ga-Za98,Na95,Na01} for instance.\n\n\\vspace{3mm}\nUnlike the parabolic-elliptic system \\eqref{eqn;PKS}, the fully parabolic system\n\\eqref{eqn;KS} is a strongly coupled parabolic system, and many approaches\ndeveloped for \\eqref{eqn;PKS} are no longer applicable.\nThe global existence of solutions with sub-critical mass\n$\\|u_0\\|_{1}<8\\pi$ has nevertheless been established by combining\nLyapunov functionals with the Trudinger--Moser type inequality and its optimal\nconstant.\nMore precisely, the following results are known:\n\\begin{enumerate}\n \\item[(i)]\n If $\\|u_0\\|_{1}<8\\pi$, then the corresponding solution to\n \\eqref{eqn;KS} exists globally in time \\cite{CaCo08,Mi13,NaOg11}.\n\n\\item[(iii)]\n If $\\|u_0\\|_{1}=8\\pi$ and $(u_0,v_0)$ is either radially symmetric or\n satisfies the additional moment condition\n $u_0\\ln(1+|x|^2)\\in L^1(\\mathbb R^2)$, then the corresponding solution exists globally\n in time.\n In contrast, for general initial data, the global behavior in the critical case\n remains delicate, and the solution either exists globally or blows up on the plane \\cite{CaCo08,Mi13}.\n\\end{enumerate}\nAs for positive forward self-similar solutions to \\eqref{eqn;KS}, refer to \\cite{BiCoDo}.\nFor the corresponding Cauchy--Neumann problem in bounded domains, we refer to\n\\cite{Bi98,He-Ve,HoWa,Na-Se-Yo}, for instance.\n\n\\vspace{2mm}\n\\begin{def*}\nLet $4/30,\\ & x\\in\\mathbb R^2,\\\\\n0 &= \\Delta v - \\lambda v + u,\n& t>0,\\ & x\\in\\mathbb R^2,\n\\\\\n&u(0,x)=u_{0}(x),\n&\\, &x\\in\\mathbb R^2.\n\\end{aligned}\n\\right.\n\\addtocounter{equation}{1}\\tag{\\theequation}\n\\label{eqn;PKS}\n\\end{align*}\nThe system \\eqref{eqn;PKS} is also related to models of gravitational interaction\nof particles \\cite{Bi-Na94,Wo}.\nIn~\\eqref{eqn;PKS}, the second equation can be written as\n$v = (-\\Delta+\\lambda)^{-1}u.$\nAs a consequence, the system~\\eqref{eqn;PKS} can be reduced to a single equation\nfor~$u$.\nOwing to this elliptic structure, the parabolic-elliptic system~\\eqref{eqn;PKS} is more amenable to analysis than the fully parabolic system~\\eqref{eqn;KS}, and has therefore been extensively studied and is now well understood in many aspects.\nIn particular, numerous works have been devoted to the critical-mass\nphenomenon.\nIndeed, the solution with $\\|u_0\\|_1\\le 8\\pi$ exists globally in time~\\cite{BiKaLaNa06,Bl-Ca-Ca,Bl-Ca-Ma,BlDoPe,DoPe,Lo-Na-Ya1,Lo-Na-Ya2,Na11,NaOg11,NaOg16,Na-Se,Wei},\nwhile the solution may blow up in finite time if $\\|u_0\\|_1>8\\pi$~\\cite{DoPe,Ko-Su,Ku-Og03,Wei}.\nFor the corresponding Cauchy--Neumann problem in bounded domains,\nsupplemented with homogeneous Neumann boundary conditions, see \\cite{BiKaLaNa06disc,Bi-Na94,Ga-Za98,Na95,Na01} for instance.\n\n\\vspace{3mm}\nUnlike the parabolic-elliptic system \\eqref{eqn;PKS}, the fully parabolic system\n\\eqref{eqn;KS} is a strongly coupled parabolic system, and many approaches\ndeveloped for \\eqref{eqn;PKS} are no longer applicable.\nThe global existence of solutions with sub-critical mass\n$\\|u_0\\|_{1}<8\\pi$ has nevertheless been established by combining\nLyapunov functionals with the Trudinger--Moser type inequality and its optimal\nconstant.\nMore precisely, the following results are known:\n\\begin{enumerate}\n \\item[(i)]\n If $\\|u_0\\|_{1}<8\\pi$, then the corresponding solution to\n \\eqref{eqn;KS} exists globally in time \\cite{CaCo08,Mi13,NaOg11}.\n\n\\item[(iii)]\n If $\\|u_0\\|_{1}=8\\pi$ and $(u_0,v_0)$ is either radially symmetric or\n satisfies the additional moment condition\n $u_0\\ln(1+|x|^2)\\in L^1(\\mathbb R^2)$, then the corresponding solution exists globally\n in time.\n In contrast, for general initial data, the global behavior in the critical case\n remains delicate, and the solution either exists globally or blows up on the plane \\cite{CaCo08,Mi13}.\n\\end{enumerate}\nAs for positive forward self-similar solutions to \\eqref{eqn;KS}, refer to \\cite{BiCoDo}.\nFor the corresponding Cauchy--Neumann problem in bounded domains, we refer to\n\\cite{Bi98,He-Ve,HoWa,Na-Se-Yo}, for instance.\n\n\\vspace{2mm}\n\\begin{def*}\nLet $4/30$ and some $C(\\tau,\\alpha)>0$ (cf.~\\cite{CaCo08,Mi13}). Hence, if $\\|u_0\\|<8\\pi$, then all terms on the left hand side of \\eqref{eqn;afford} are positive by choosing $\\al>0$ sufficiently small which depends on $\\|u_0\\|_1$, and Gronwall's inequality implies that\n$\\nabla v \\in L^2((0,T)\\times \\R^2)$, \nso that we also obtain the bound for the modified entropy $\\int_{\\R^2}(1+u)\\ln (1+u)\\dx$. \nNevertheless, for the critical mass $\\|u_0\\|_1=8\\pi$, this necessarily leads to $\\alpha=0$, as a result, \n\\begin{align*}\n\\int_0^t \\D_m(s)\\,\\d s\\le\\,&\\L_m(0)+C(\\tau)+\\frac14\\int_0^t\\|\\nabla v\\|_2^2\\,\\d s,\n\\eqntag\n\\label{eqn;lack-lyapunov}\n\\end{align*}\nwhich is useless and fails to obtain even the estimates for $\\D_m(t)$ due to the lack of the regularity of solutions.\nTherefore, we introduce a reconstructed Lyapunov functional $\\F_m(t)$ so as to show regularity estimates corresponding to the dissipative terms. Let $\\F_m(t)$ be the functional for solutions to \\eqref{eqn;KS} defined as\n\\begin{align*}\n\\F_m(t):=\\L_m(t)\n+\\int_{\\R^2}\\ln(1+u)\\,\\mathrm{d}x-\\int_{\\R^2}v\\,\\d x.\n\\eqntag\n\\label{eqn;modifiedLyapunov}\n\\end{align*}\nThen, the following functional differential inequality holds true:\n\\begin{prop}\\label{prop;energy-est}\nLet $(u,v)$ be the solution to \\eqref{eqn;KS}. Then, the functional $\\F_m(t)$ defined in~\\eqref{eqn;modifiedLyapunov} satisfies the following identity:\n \\begin{align*}\n \\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2\n \\le\\,&\n \\lambda\\int_{\\R^2}v\\,\\d x.\n \\end{align*}\n Therefore, if $\\lambda=0$ then\n \\begin{align*}\n \\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2 \\le\\,0.\n \\end{align*}\n If $\\lambda>0$ then \n \\begin{align*}\n \\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2 \\le\\,\\|v_0\\|_1+\\|u_0\\|_1.\n \\end{align*}\n\\end{prop}\n\\vspace{5mm}\nIt is worth emphasizing that $\\F_m(t)$ is non-increasing in time when $\\lambda=0$ as well as\nunlike the identity \\eqref{eqn;usualmodifiedfunctiona} based on the usual modified functional $\\mathcal{L}_m(t)$, the error term appearing on the right-hand side of Proposition~\\ref{prop;energy-est} can be easily controlled by initial data.\nThis allows us to show the regularity estimates for the dissipative term associated with $\\F_m(t)$, see Proposition~\\ref{prop;bound-energy} below.\n\\begin{lem}\\label{lem;ModifiedLF}\n Suppose assumptions as in Theorem \\ref{thm;global}. Then,\n\\begin{align*}\n\\frac{\\d}{\\d t}\\F_m(t)+ \\widetilde{\\D}(t)=-\\int_{\\R^2}\\partial_t v \\frac{u}{1+u}\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\,\\mathrm{d}x+\\lambda\\int_{\\R^2}\\frac{v}{1+u}\\,\\d x,\n\\end{align*}\nwhere\n$\\F_m(t)$ is the modified functional defined in \\eqref{eqn;modifiedLyapunov} and the dissipative term $\\widetilde{\\D}(t)$ is given by\n\\begin{align*}\n\\widetilde{\\D}(t):=\\int_{\\R^2}u\\left|\\nabla \\left(\\ln(1+u)-v\\right)\\right|^2\\,\\d x+\\|\\partial_t v\\|_2^2.\n\\end{align*}\n\\end{lem}\n\n\\vspace{5mm}\n\\begin{pr}{Theorem \\ref{thm;global}}\nLet $R$ be taken as in Lemma \\ref{lem;L^2-exterior}.\nBy Lemma \\ref{lem;LlogL-bound-exterior},\n\\begin{align*}\n\\int_{ |x|>2R } (1+u) \\ln(1+u) \\dx\\le\\, C(t_0,\\tau,R).\n\\end{align*}\nOn the other hand, Lemma \\ref{prop;interior-bdd-entropy} gives\n\\begin{align*}\n\\int_{ \\R^2}(u\\ln u)\\psi_R^2\\dx\\le\\, C(t_0,\\tau,R)\n\\end{align*}\nfor $t\\in[t_0,\\tau]\\cap(0,T)$.\nNow, according to \\cite[Lemma 2.3]{NaOg16},\n\\begin{align*}\n\\int_{ \\Omega} (1+u)\\ln(1+u)\\dx\\le\\,2\\int_{ \\Omega} u |\\ln u | \\dx+(2\\ln 2)\\int_{ \\Omega} u\\dx,\n\\end{align*}\nwhere $\\Omega$ is a measurable set in $\\R^2$,\n so that since $\\supp\\psi_R\\subset \\{x; |x|\\le 16 R\\}$\n \\begin{align*}\n \\int_{ |x|<8R }(1+u)\\ln(1+u)\\dx\\le\\,&2\\int_{ |x|<8R }u|\\ln u| \\dx+(2\\ln 2)\\|u_0\\|_1\n \\\\\n \\le\\,\\,&2\\int_{ \\R^2 }u|\\ln u| \\psi_R^2\\dx+(2\\ln 2)\\|u_0\\|_1\n \\\\\n =\\,&2\\int_{ \\R^2 }(u\\ln u) \\psi_R^2\\dx-4\\int_{ \\R^2 }(u\\ln u)_-\\psi_R^2\\dx+(2\\ln 2)\\|u_0\\|_1\n \\\\\n \\le\\,&C(t_0,\\tau,R) +\\frac4e|B_{16R}(0)|\n \\end{align*}\n for $t\\in[t_0,\\tau]\\cap(0,T)$, where we use $x\\ln x \\ge -1/e$ for $x\\ge0$.\n This implies that\n \\begin{align*}\n&\\int_{ \\R^2 }(1+u)\\ln(1+u)\\dx\n\\\\=\\,&\\int_{ |x|>2R } (1+u) \\ln(1+u) \\dx\n+\\int_{ |x|\\le 2R } (1+u) \\ln(1+u) \\dx\n\\\\\n\\le\\,&\\int_{ |x|>2R } (1+u) \\ln(1+u) \\dx+\\int_{ |x|< 8R } (1+u) \\ln(1+u) \\dx\n\\\\\n\\le\\,&C(t_0,\\tau,R) \n \\end{align*}\n for $t\\in[t_0,\\tau]\\cap(0,T)$.\n Hence, we have along with Proposition \\ref{prop;bound-energy}\n \\begin{align*}\n\\int_{ \\R^2 }(1+u) \\ln (1+u)\\dx+\\int_{t_0}^t \\|\\partial_tv\\|_2^2\\,\\d s\\le\\, C(t_0,\\tau,R)\n \\eqntag\n \\label{eqn;uniform-ulogu}\n \\end{align*}\n for $t\\in[t_0,\\tau]\\cap(0,T)$.\nBy the parabolic regularity argument, we obtain a uniform $L^2$-bound for~$u$\non $[t_0,\\tau]\\cap(0,T)$ from \\eqref{eqn;uniform-ulogu} (cf. \\cite{Mi13,Na-Se-Yo}), which yields further regularity estimates.\nIn particular, the solution~$(u,v)$ cannot blow up in finite time.\nTherefore, the solution to \\eqref{eqn;KS} exists globally in time.\n\\end{pr}", + "post_theorem_intro_text_len": 2221, + "post_theorem_intro_text": "\\vspace{2mm}\n\n\\begin{rem}\nIn Theorem~\\ref{thm;global}, the initial data are only required to satisfy\n\\[\n(u_0,v_0)\\in L_+^1(\\mathbb R^2)\\times\\bigl(L_+^1(\\mathbb R^2)\\cap \\dot H^1(\\mathbb R^2)\\bigr)\\quad \\text{with}~~\\|u_0\\|_1=8\\pi.\n\\]\nIn particular, no additional symmetry or moment assumptions are imposed.\n\\end{rem}\n\\vspace{2mm}\nAs mentioned above, the global existence of solutions with critical mass\nwithout any symmetry or moment assumptions remains delicate.\nIn~\\cite[Theorem~1.2]{Mi13}, it is shown that, for general initial data with\ncritical mass, solutions either exist globally in time or blow up on the plane,\nby means of a contradiction argument.\nIndeed, although in the sub-critical case regularity estimates can be obtained\nby combining (modified) Lyapunov functionals with the Trudinger--Moser\ninequality, such estimates are no longer directly available in the critical case due to the lack of regularity of solutions,\nsee~\\eqref{eqn;afford} below.\nThis difficulty prevents the direct extension of classical entropy methods.\n\nTo overcome this difficulty, we introduce a reconstructed Lyapunov functional.\nThe main novelty of this work lies in the construction of a refined Lyapunov\nfunctional specifically adapted to the whole space setting, which allows us\nto control the behavior of solutions at $|x|\\to\\infty$ and to derive\nregularity estimates even in the critical mass regime, without\nimposing any symmetry or moment assumptions on the initial data.\nAs a consequence, we establish global-in-time existence for general initial\ndata at the critical mass, a result that was previously out of reach by existing\nmethods, see Subsection~\\ref{sect;modified-Lyapunov} for details.\nMoreover, the present approach is expected to be applicable to a broad class\nof chemotaxis systems in the whole space setting.\n\n\\vspace{3mm}\nThe remainder of this paper is organized as follows.\nIn Section~\\ref{sect;preliminary}, we recall several preliminary lemmas needed\nto prove the main result.\nSection~\\ref{sect;LWP} is devoted to the local well-posedness of~\\eqref{eqn;KS}, based on the above definition of solutions.\nFinally, in Section~\\ref{sect;global}, we present the proof of\nTheorem~\\ref{thm;global}.\n\n\\vspace{5mm}", + "sketch": "The post-theorem introduction does not give a step-by-step proof, but it outlines the strategy for proving Theorem~\\ref{thm;global}: classical entropy/Lyapunov approaches for sub-critical mass (using “(modified) Lyapunov functionals with the Trudinger--Moser inequality”) “are no longer directly available in the critical case due to the lack of regularity of solutions,” which “prevents the direct extension of classical entropy methods.” To overcome this, the authors “introduce a reconstructed Lyapunov functional,” whose “main novelty…lies in the construction of a refined Lyapunov functional specifically adapted to the whole space setting,” allowing them “to control the behavior of solutions at $|x|\\to\\infty$ and to derive regularity estimates even in the critical mass regime, without imposing any symmetry or moment assumptions on the initial data.” With these estimates, they “establish global-in-time existence for general initial data at the critical mass.” The paper’s structure supporting the proof is: preliminaries (Section~\\ref{sect;preliminary}), local well-posedness (Section~\\ref{sect;LWP}), then the global argument proving Theorem~\\ref{thm;global} (Section~\\ref{sect;global}).", + "expanded_sketch": "The post-theorem introduction does not give a step-by-step proof, but it outlines the strategy for proving the main theorem: classical entropy/Lyapunov approaches for sub-critical mass (using “(modified) Lyapunov functionals with the Trudinger--Moser inequality”) “are no longer directly available in the critical case due to the lack of regularity of solutions,” which “prevents the direct extension of classical entropy methods.” To overcome this, the authors “introduce a reconstructed Lyapunov functional,” whose “main novelty…lies in the construction of a refined Lyapunov functional specifically adapted to the whole space setting,” allowing them “to control the behavior of solutions at $|x|\\to\\infty$ and to derive regularity estimates even in the critical mass regime, without imposing any symmetry or moment assumptions on the initial data.” With these estimates, they “establish global-in-time existence for general initial data at the critical mass.” The paper’s structure supporting the proof is: preliminaries (proved next), local well-posedness (proved after that), then the global argument establishing the main theorem (proved later).", + "expanded_theorem": "\\label{thm;global}\nFor $(u_0,v_0)\\in L_+^1(\\mathbb R^2) \\times L^1_+(\\mathbb R^2)\\cap \\dot{H}^1(\\mathbb R^2)$,\nlet $(u,v)$ be the solution to \n\\begin{aligned}\n&\\pt_t u =\\Delta u- \\N\\cd \\left( u \\N v \\right),\n& t>0,\\,~ &x\\in\\R^2,\n\\\\\n&\\pt_t v = \\Delta v - \\lambda v +u,\n&t>0,\\, ~&x\\in\\R^2,\n\\\\\n&(u,v)(0,x)=(u_{0},v_0)(x),\n&\\, &x\\in\\R^2\n\\end{aligned}\n\\right.\n\\label{eqn;KS}\n\\end{equation}\nwith a constant $\\lambda\\ge0$, where $u_0,v_0\\ge 0$ on $\\R^2$ and\n$u_0,v_0\\not\\equiv0$\non $(0,T)\\times\\mathbb R^2$.\nSuppose that $\\|u_0\\|_1=8\\pi$.\nThen, in establishing the main theorem, the solution to the system above exists globally in time.", + "theorem_type": [ + "Implication", + "Existence" + ], + "mcq": { + "question": "Consider the two-dimensional fully parabolic Keller--Segel system\n\\[\n\\begin{cases}\n\\partial_t u = \\Delta u-\\nabla\\!\\cdot\\!\\big(u\\nabla v\\big), & t>0,\\ x\\in\\mathbb R^2,\\\\\n\\partial_t v = \\Delta v-\\lambda v+u, & t>0,\\ x\\in\\mathbb R^2,\\\\\n(u,v)(0,x)=(u_0,v_0)(x), & x\\in\\mathbb R^2,\n\\end{cases}\n\\]\nwhere \\(\\lambda\\ge 0\\). Assume\n\\(u_0\\in L_+^1(\\mathbb R^2)\\) and \\(v_0\\in L_+^1(\\mathbb R^2)\\cap \\dot H^1(\\mathbb R^2)\\), with\n\\[\nL_+^1(\\mathbb R^2):=\\{f\\in L^1(\\mathbb R^2): f\\ge 0\\ \\text{and}\\ f\\not\\equiv 0\\},\n\\]\nso in particular \\(u_0,v_0\\) are nonnegative and nontrivial, and suppose that the initial cell mass satisfies\n\\[\n\\|u_0\\|_{L^1(\\mathbb R^2)}=8\\pi.\n\\]\nWhich of the following conclusions about the corresponding solution \\((u,v)\\) holds?", + "correct_choice": { + "label": "A", + "text": "The solution \\((u,v)\\) exists globally in time; equivalently, it can be continued for all \\(t\\ge 0\\) and does not blow up in finite time." + }, + "choices": [ + { + "label": "B", + "text": "The solution \\((u,v)\\) exists globally in time provided, in addition, that the initial data are either radially symmetric or satisfy a finite logarithmic moment condition such as \\(u_0\\ln(1+|x|^2)\\in L^1(\\mathbb R^2)\\); without one of these extra assumptions, finite-time blow-up may occur at mass \\(8\\pi\\)." + }, + { + "label": "C", + "text": "The corresponding solution \\((u,v)\\) has a local-in-time mild solution on some interval \\([0,T)\\) for a positive maximal existence time \\(T>0\\)." + }, + { + "label": "D", + "text": "For every such initial datum with \\(\\|u_0\\|_{L^1(\\mathbb R^2)}=8\\pi\\), the solution \\((u,v)\\) exists globally in time and moreover remains uniformly bounded in \\(L^\\infty(\\mathbb R^2)\\times \\dot W^{1,\\infty}(\\mathbb R^2)\\) for all \\(t\\ge 0\\)." + }, + { + "label": "E", + "text": "There exists a time \\(T=T(u_0,v_0)>0\\) such that the solution \\((u,v)\\) exists on \\([0,T)\\), and if it is global then this conclusion requires constants in the a priori estimates to depend on additional decay of the data at spatial infinity; at critical mass \\(8\\pi\\), global existence is therefore not guaranteed for arbitrary \\(u_0\\in L_+^1(\\mathbb R^2)\\) and \\(v_0\\in L_+^1(\\mathbb R^2)\\cap \\dot H^1(\\mathbb R^2)\\)." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "regularity", + "tampered_component": "removal_of_symmetry_or_moment_hypotheses", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "regularity", + "tampered_component": "dropped_global_in_time_and_no_blowup_conclusion", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "upgrade_from_global_existence_to_uniform_Linfty_control", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "geometric_construction", + "tampered_component": "whole_space_tail_control_for_arbitrary_data", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem states hypotheses and asks for the resulting conclusion, but it does not explicitly reveal that global existence is the correct outcome. The critical mass 8π is mathematically relevant, not answer leakage." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recall question: under the stated assumptions, choose the theorem’s conclusion. It is not a pure verbatim restatement because the options include weaker, stronger, and hypothesis-altered alternatives." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to reject plausible variants such as needing extra symmetry/moment assumptions, accepting only local existence, or inferring uniform L∞ bounds. However, the main task is still recognition of the correct theorem-level conclusion rather than substantial derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and distinct: one imports extra assumptions from related results, one gives a weaker true local statement, one overstates regularity, and one casts doubt on critical-mass global existence. These align with realistic mathematical confusions." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with strong distractors and no real answer leakage, but it is primarily a theorem-identification item rather than a deeply generative reasoning question." + } + }, + { + "id": "2602.03774v1", + "paper_link": "http://arxiv.org/abs/2602.03774v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm-convergence}\n Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \\geq 2$ vertices. Given $c>0$, let $p= cn^{-1/\\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$. There exists a constant $m(F,c)$ such that\n \\begin{align*}\n \\frac{{\\rm MinM}(\\mathbb{G}(n,p),F)}{n}\n \\to m(F,c),\n \\end{align*}\n\\whp~ as $n\\to\\infty$.", + "start_pos": 10979, + "end_pos": 11342, + "label": "thm-convergence" + }, + "ref_dict": { + "thm-convergence": "\\begin{theorem}\\label{thm-convergence}\n Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \\geq 2$ vertices. Given $c>0$, let $p= cn^{-1/\\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$. There exists a constant $m(F,c)$ such that\n \\begin{align*}\n \\frac{{\\rm MinM}(\\G(n,p),F)}{n}\n \\to m(F,c),\n \\end{align*}\n\\whp~ as $n\\to\\infty$.\n\\end{theorem}", + "thm-limit": "\\begin{theorem}\\label{thm-limit}\n Under the assumption of \\Cref{thm-convergence},\n\\begin{align*}\n m(F,c)=\\kappa(F,c,s,r)+\n (1+o_r(1))\\sqrt{(2\\log 2)\\kappa(F,c,s,r)}\n +o_{c}(c^{s/2}).\n \\end{align*}\n\\end{theorem}", + "prop:coupling": "\\begin{proposition}[{\\cite[Theorem 1.6]{burghart2024sharp}}]\\label{thm-bur-1.6}\\label{prop:coupling}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. There exist constants $\\delta, \\varepsilon > 0$ such that the following holds. Fix any two sequences $p,q$ satisfying $p \\leq n^{-1/\\rd_1(F) + \\varepsilon}$ and $q \\leq (1 - n^{-\\delta})p^{s}$. Whp as $n\\to\\infty$ there exists a permutation $\\tau: [n] \\to [n]$ such that \n for every $F(v_1, \\ldots, v_r) \\in E(\\Hg_F(n,q))$ also $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subset \\G(n,p)$.\n\\end{proposition}" + }, + "pre_theorem_intro_text_len": 5796, + "pre_theorem_intro_text": "The study of monochromatic subgraphs in graph bipartitions is a widely investigated problem in extremal combinatorics and Ramsey theory. \nA classical result by Goodman~\\cite{g1959} states that in any bipartition of the complete graph $K_n$, the number of monochromatic triangles admits a sharp lower bound\n\\[\nT(n) = \n\\begin{cases}\n \\dfrac{u(u-1)(u-2)}{3}, & \\text{if } n = 2u; \\\\\n \\dfrac{2u(u-1)(4u+1)}{3}, & \\text{if } n = 4u+1; \\\\\n \\dfrac{2u(u+1)(4u-1)}{3}, & \\text{if } n = 4u+3.\n\\end{cases}\n\\]\n\\ignore{\n\\zhu{The general case was studied by Paul Erdős, who proved that the minimum number of monochromatic $K_r$ in any bipartition of $K_n$ is at most $\\binom{n}{r}/2^{\\binom{r}{2}-1}$ and conjectured that this bound is asymptotically tight for large $n$ \\cite{erdHos1962number}. This conjecture was later disproved by Thomason \\cite{thomason1989disproof}. In 2012, Conlon proved that any bipartition of $K_n$ contains at least $n^r / C^{(1+o(1))r^2}$ monochromatic $K_r$ subgraphs, where $C \\approx 2.18$ \\cite{conlon2012ramsey}. This phenomenon, whereby a positive fraction of all $r$-cliques are monochromatic, also appears in other deterministic contexts \\cite{frankl1988quantitative}. In general, it can often be characterized by the principle that monochromatic substructures are evenly distributed among configurations that force their occurrence \\cite{csv}.}\n\\dg{I am not sure we need such a detailed summary about deterministic setup. It is somewhat off-topic}\n}\n\nWhile these deterministic settings have attracted considerable attention and yielded several intriguing results, \nthe same question in the context of random graphs,\nsuch as Erdős–Rényi random graphs, remains far less understood, even for the special case of triangles. In this paper, we consider this question of finding the smallest number of monochromatic subgraphs in bipartitions of random graphs. We do so in a general setting, \nwhere the subgraph is only assumed to be\nstrictly balanced.\n\nTo set the stage, let $F$ be a graph on $r$ vertices. An $F$-graph $H$ is defined as any pair $(V=V(H), E=E(H))$, where $V$ is a set of vertices, and \n$E$ is a collection of copies $e$ of $F$ called hyperedges, where for each $e$, the vertex set of $e$ is subset of $V$.\n A random \\( F \\)-graph \\( \\Hg_F(n, q) \\) is an \\( F \\)-graph with vertex set \\( V=[n]=\\cbc{1,2,\\ldots,n} \\), where each of the\n\\begin{align}\\label{eq:N-trials}\n\\binom{n}{r} \\cdot \\frac{r!}{{\\rm aut}(F)}\n\\end{align}\npotential copies of \\( F \\) on vertices in \\( [n] \\) is included in the edge set $E$ independently with probability \\( q \\). Here ${\\rm aut}(H)$ denotes the set of automorphisms of a graph $H$. In the special case\nwhen $F$ is $K_2$ (two nodes\nconnected by an edge), we use a more common notation $\\mathbb{G}(n,q)$.\n\nGiven a simple graph $G=(V,E)$ (so that $F$ is just $K_2$) and given its bipartition \\( V=V_1 \\cup V_2, V_1\\cap V_2=\\emptyset \\), a cut, or more specifically an $F$-cut associated with this bipartition is defined as the set of subgraphs $e$ of $G$ which are isomorphic to $F$, and whose vertex sets have a non-empty intersection with \\( V_1 \\) and \\( V_2 \\). That is, each hyperedge of the cut is neither entirely contained in \\( V_1 \\), nor is it entirely contained in \\( V_2 \\). The \\emph{cut value} associated with a cut is \nthe cardinality of this set.\n The \\emph{max-cut value}\n denoted by ${\\rm MaxC}(G,F)$\n is defined as the largest such value taken over all possible bipartitions of \\(V\\). Thinking of $V_1$ and $V_2$ as being associated with two distinct colors, the cut associated with the bipartition is simply the set of all non-monochromatic subgraphs of $G$ isomorphic to $F$\n associated with the coloring scheme $V_1,V_2$. Conversely, every $F$-subgraph of $G$ not participating in the cut is monochromatic by definition. The minimum\n number of monochromatic $F$-subgraphs is denoted by ${\\rm MinM}(G,F)$. Naturally, ${\\rm MinM}(G,F)+{\\rm MaxC}(G,F)$ is the total number of $F$-subgraphs of $G$ as this is the case for every bipartition.\n\nIn combinatorial optimization and theoretical computer science, there is a long-standing interest in understanding the minimum number of monochromatic subgraphs achievable by varying over all bipartitions of graphs, both in the special case of max-cut values associated with $F=K_2$, and beyond ~\\cite{ckpsty2013,dms2017,dl2018,sen2018optimization,shabanov2021maximum}. This is the question we address in our paper.\n\n\\relax\n\nLet $F$ be a graph. We say $F$ is a strictly 1-balanced graph if $\\rd_1(F')<{\\rm d}(F)$ for any subgraph $F'$ of $F$ such that $F'\\neq F$. Here\n \\begin{align*}\n \\rd_1(G)=\\frac{\\abs{E(G)}}{\\abs{V(G )}-1}.\n \\end{align*}\nThere are many examples of strictly 1-balanced graphs, including cycles and complete graphs. However, any disconnected graph is not strictly 1-balanced, and neither is a tree-graph.\n\nRecently a great progress was achieved in understanding the relationship between $\\mathbb{G}(n,p)$ and $\\Hg_F(n,q)$\nwith $q$ judiciously chosen\nas $q=p^{|E(F)|}$. In particular, as shown in \\cite{burghart2024sharp} (see Proposition~\\ref{prop:coupling} below), when $p$ is at most $n^{-1/d_1(F)}$, the graphs\n$\\mathbb{G}(n,p)$ and $\\mathbb{H}(n,q)$ can be coupled in such a way that the number of $F$-hyperedges in $\\mathbb{H}$ nearly matches the number of copies of $F$ naturally occurring in $\\mathbb{G}(n,p)$. As a result a difficult problem of studying minimal number of monochromatic $F$-subgraphs of $\\mathbb{G}(n,p)$ can be reduced to simpler version, one defined on $\\mathbb{H}$ where the occurances of $F$-s are independent by design. This is the insight we use to obtain several asymptotic results on max-cut values for general graphs $F$ in $\\mathbb{G}(n,p)$. \n\nOur first result is as follows.", + "context": "The study of monochromatic subgraphs in graph bipartitions is a widely investigated problem in extremal combinatorics and Ramsey theory. \nA classical result by Goodman~\\cite{g1959} states that in any bipartition of the complete graph $K_n$, the number of monochromatic triangles admits a sharp lower bound\n\\[\nT(n) = \n\\begin{cases}\n \\dfrac{u(u-1)(u-2)}{3}, & \\text{if } n = 2u; \\\\\n \\dfrac{2u(u-1)(4u+1)}{3}, & \\text{if } n = 4u+1; \\\\\n \\dfrac{2u(u+1)(4u-1)}{3}, & \\text{if } n = 4u+3.\n\\end{cases}\n\\]\n\\ignore{\n\\zhu{The general case was studied by Paul Erdős, who proved that the minimum number of monochromatic $K_r$ in any bipartition of $K_n$ is at most $\\binom{n}{r}/2^{\\binom{r}{2}-1}$ and conjectured that this bound is asymptotically tight for large $n$ \\cite{erdHos1962number}. This conjecture was later disproved by Thomason \\cite{thomason1989disproof}. In 2012, Conlon proved that any bipartition of $K_n$ contains at least $n^r / C^{(1+o(1))r^2}$ monochromatic $K_r$ subgraphs, where $C \\approx 2.18$ \\cite{conlon2012ramsey}. This phenomenon, whereby a positive fraction of all $r$-cliques are monochromatic, also appears in other deterministic contexts \\cite{frankl1988quantitative}. In general, it can often be characterized by the principle that monochromatic substructures are evenly distributed among configurations that force their occurrence \\cite{csv}.}\n\\dg{I am not sure we need such a detailed summary about deterministic setup. It is somewhat off-topic}\n}\n\nTo set the stage, let $F$ be a graph on $r$ vertices. An $F$-graph $H$ is defined as any pair $(V=V(H), E=E(H))$, where $V$ is a set of vertices, and \n$E$ is a collection of copies $e$ of $F$ called hyperedges, where for each $e$, the vertex set of $e$ is subset of $V$.\n A random \\( F \\)-graph \\( \\Hg_F(n, q) \\) is an \\( F \\)-graph with vertex set \\( V=[n]=\\cbc{1,2,\\ldots,n} \\), where each of the\n\\begin{align}\\label{eq:N-trials}\n\\binom{n}{r} \\cdot \\frac{r!}{{\\rm aut}(F)}\n\\end{align}\npotential copies of \\( F \\) on vertices in \\( [n] \\) is included in the edge set $E$ independently with probability \\( q \\). Here ${\\rm aut}(H)$ denotes the set of automorphisms of a graph $H$. In the special case\nwhen $F$ is $K_2$ (two nodes\nconnected by an edge), we use a more common notation $\\mathbb{G}(n,q)$.\n\nGiven a simple graph $G=(V,E)$ (so that $F$ is just $K_2$) and given its bipartition \\( V=V_1 \\cup V_2, V_1\\cap V_2=\\emptyset \\), a cut, or more specifically an $F$-cut associated with this bipartition is defined as the set of subgraphs $e$ of $G$ which are isomorphic to $F$, and whose vertex sets have a non-empty intersection with \\( V_1 \\) and \\( V_2 \\). That is, each hyperedge of the cut is neither entirely contained in \\( V_1 \\), nor is it entirely contained in \\( V_2 \\). The \\emph{cut value} associated with a cut is \nthe cardinality of this set.\n The \\emph{max-cut value}\n denoted by ${\\rm MaxC}(G,F)$\n is defined as the largest such value taken over all possible bipartitions of \\(V\\). Thinking of $V_1$ and $V_2$ as being associated with two distinct colors, the cut associated with the bipartition is simply the set of all non-monochromatic subgraphs of $G$ isomorphic to $F$\n associated with the coloring scheme $V_1,V_2$. Conversely, every $F$-subgraph of $G$ not participating in the cut is monochromatic by definition. The minimum\n number of monochromatic $F$-subgraphs is denoted by ${\\rm MinM}(G,F)$. Naturally, ${\\rm MinM}(G,F)+{\\rm MaxC}(G,F)$ is the total number of $F$-subgraphs of $G$ as this is the case for every bipartition.\n\nLet $F$ be a graph. We say $F$ is a strictly 1-balanced graph if $\\rd_1(F')<{\\rm d}(F)$ for any subgraph $F'$ of $F$ such that $F'\\neq F$. Here\n \\begin{align*}\n \\rd_1(G)=\\frac{\\abs{E(G)}}{\\abs{V(G )}-1}.\n \\end{align*}\nThere are many examples of strictly 1-balanced graphs, including cycles and complete graphs. However, any disconnected graph is not strictly 1-balanced, and neither is a tree-graph.\n\nRecently a great progress was achieved in understanding the relationship between $\\mathbb{G}(n,p)$ and $\\Hg_F(n,q)$\nwith $q$ judiciously chosen\nas $q=p^{|E(F)|}$. In particular, as shown in \\cite{burghart2024sharp} (see Proposition~\\ref{prop:coupling} below), when $p$ is at most $n^{-1/d_1(F)}$, the graphs\n$\\mathbb{G}(n,p)$ and $\\mathbb{H}(n,q)$ can be coupled in such a way that the number of $F$-hyperedges in $\\mathbb{H}$ nearly matches the number of copies of $F$ naturally occurring in $\\mathbb{G}(n,p)$. As a result a difficult problem of studying minimal number of monochromatic $F$-subgraphs of $\\mathbb{G}(n,p)$ can be reduced to simpler version, one defined on $\\mathbb{H}$ where the occurances of $F$-s are independent by design. This is the insight we use to obtain several asymptotic results on max-cut values for general graphs $F$ in $\\mathbb{G}(n,p)$.\n\nOur first result is as follows.\n\n\\begin{proposition}[{\\cite[Theorem 1.6]{burghart2024sharp}}]\\label{thm-bur-1.6}\\label{prop:coupling}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. There exist constants $\\delta, \\varepsilon > 0$ such that the following holds. Fix any two sequences $p,q$ satisfying $p \\leq n^{-1/\\rd_1(F) + \\varepsilon}$ and $q \\leq (1 - n^{-\\delta})p^{s}$. Whp as $n\\to\\infty$ there exists a permutation $\\tau: [n] \\to [n]$ such that \n for every $F(v_1, \\ldots, v_r) \\in E(\\Hg_F(n,q))$ also $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subset \\G(n,p)$.\n\\end{proposition}", + "full_context": "The study of monochromatic subgraphs in graph bipartitions is a widely investigated problem in extremal combinatorics and Ramsey theory. \nA classical result by Goodman~\\cite{g1959} states that in any bipartition of the complete graph $K_n$, the number of monochromatic triangles admits a sharp lower bound\n\\[\nT(n) = \n\\begin{cases}\n \\dfrac{u(u-1)(u-2)}{3}, & \\text{if } n = 2u; \\\\\n \\dfrac{2u(u-1)(4u+1)}{3}, & \\text{if } n = 4u+1; \\\\\n \\dfrac{2u(u+1)(4u-1)}{3}, & \\text{if } n = 4u+3.\n\\end{cases}\n\\]\n\\ignore{\n\\zhu{The general case was studied by Paul Erdős, who proved that the minimum number of monochromatic $K_r$ in any bipartition of $K_n$ is at most $\\binom{n}{r}/2^{\\binom{r}{2}-1}$ and conjectured that this bound is asymptotically tight for large $n$ \\cite{erdHos1962number}. This conjecture was later disproved by Thomason \\cite{thomason1989disproof}. In 2012, Conlon proved that any bipartition of $K_n$ contains at least $n^r / C^{(1+o(1))r^2}$ monochromatic $K_r$ subgraphs, where $C \\approx 2.18$ \\cite{conlon2012ramsey}. This phenomenon, whereby a positive fraction of all $r$-cliques are monochromatic, also appears in other deterministic contexts \\cite{frankl1988quantitative}. In general, it can often be characterized by the principle that monochromatic substructures are evenly distributed among configurations that force their occurrence \\cite{csv}.}\n\\dg{I am not sure we need such a detailed summary about deterministic setup. It is somewhat off-topic}\n}\n\nTo set the stage, let $F$ be a graph on $r$ vertices. An $F$-graph $H$ is defined as any pair $(V=V(H), E=E(H))$, where $V$ is a set of vertices, and \n$E$ is a collection of copies $e$ of $F$ called hyperedges, where for each $e$, the vertex set of $e$ is subset of $V$.\n A random \\( F \\)-graph \\( \\Hg_F(n, q) \\) is an \\( F \\)-graph with vertex set \\( V=[n]=\\cbc{1,2,\\ldots,n} \\), where each of the\n\\begin{align}\\label{eq:N-trials}\n\\binom{n}{r} \\cdot \\frac{r!}{{\\rm aut}(F)}\n\\end{align}\npotential copies of \\( F \\) on vertices in \\( [n] \\) is included in the edge set $E$ independently with probability \\( q \\). Here ${\\rm aut}(H)$ denotes the set of automorphisms of a graph $H$. In the special case\nwhen $F$ is $K_2$ (two nodes\nconnected by an edge), we use a more common notation $\\mathbb{G}(n,q)$.\n\nGiven a simple graph $G=(V,E)$ (so that $F$ is just $K_2$) and given its bipartition \\( V=V_1 \\cup V_2, V_1\\cap V_2=\\emptyset \\), a cut, or more specifically an $F$-cut associated with this bipartition is defined as the set of subgraphs $e$ of $G$ which are isomorphic to $F$, and whose vertex sets have a non-empty intersection with \\( V_1 \\) and \\( V_2 \\). That is, each hyperedge of the cut is neither entirely contained in \\( V_1 \\), nor is it entirely contained in \\( V_2 \\). The \\emph{cut value} associated with a cut is \nthe cardinality of this set.\n The \\emph{max-cut value}\n denoted by ${\\rm MaxC}(G,F)$\n is defined as the largest such value taken over all possible bipartitions of \\(V\\). Thinking of $V_1$ and $V_2$ as being associated with two distinct colors, the cut associated with the bipartition is simply the set of all non-monochromatic subgraphs of $G$ isomorphic to $F$\n associated with the coloring scheme $V_1,V_2$. Conversely, every $F$-subgraph of $G$ not participating in the cut is monochromatic by definition. The minimum\n number of monochromatic $F$-subgraphs is denoted by ${\\rm MinM}(G,F)$. Naturally, ${\\rm MinM}(G,F)+{\\rm MaxC}(G,F)$ is the total number of $F$-subgraphs of $G$ as this is the case for every bipartition.\n\nLet $F$ be a graph. We say $F$ is a strictly 1-balanced graph if $\\rd_1(F')<{\\rm d}(F)$ for any subgraph $F'$ of $F$ such that $F'\\neq F$. Here\n \\begin{align*}\n \\rd_1(G)=\\frac{\\abs{E(G)}}{\\abs{V(G )}-1}.\n \\end{align*}\nThere are many examples of strictly 1-balanced graphs, including cycles and complete graphs. However, any disconnected graph is not strictly 1-balanced, and neither is a tree-graph.\n\nRecently a great progress was achieved in understanding the relationship between $\\mathbb{G}(n,p)$ and $\\Hg_F(n,q)$\nwith $q$ judiciously chosen\nas $q=p^{|E(F)|}$. In particular, as shown in \\cite{burghart2024sharp} (see Proposition~\\ref{prop:coupling} below), when $p$ is at most $n^{-1/d_1(F)}$, the graphs\n$\\mathbb{G}(n,p)$ and $\\mathbb{H}(n,q)$ can be coupled in such a way that the number of $F$-hyperedges in $\\mathbb{H}$ nearly matches the number of copies of $F$ naturally occurring in $\\mathbb{G}(n,p)$. As a result a difficult problem of studying minimal number of monochromatic $F$-subgraphs of $\\mathbb{G}(n,p)$ can be reduced to simpler version, one defined on $\\mathbb{H}$ where the occurances of $F$-s are independent by design. This is the insight we use to obtain several asymptotic results on max-cut values for general graphs $F$ in $\\mathbb{G}(n,p)$.\n\nOur first result is as follows.\n\n\\begin{proposition}[{\\cite[Theorem 1.6]{burghart2024sharp}}]\\label{thm-bur-1.6}\\label{prop:coupling}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. There exist constants $\\delta, \\varepsilon > 0$ such that the following holds. Fix any two sequences $p,q$ satisfying $p \\leq n^{-1/\\rd_1(F) + \\varepsilon}$ and $q \\leq (1 - n^{-\\delta})p^{s}$. Whp as $n\\to\\infty$ there exists a permutation $\\tau: [n] \\to [n]$ such that \n for every $F(v_1, \\ldots, v_r) \\in E(\\Hg_F(n,q))$ also $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subset \\G(n,p)$.\n\\end{proposition}\n\nOur first result is as follows.\n\n\\ignore{\n\\yd{Maybe we should specify that $m(F,p)$ is a deterministic scalar.}\\dg{good point. Done}\n}\n\nThe following key result from \\cite{burghart2024sharp} helps us understand the ordered copies of a strictly 1-balanced graph $F$ in $\\G(n,p)$, and establishes a connection to hyperedges in the random $F$-graph $\\Hg_F(n,q)$:\n\\begin{proposition}[{\\cite[Theorem 1.6]{burghart2024sharp}}]\\label{thm-bur-1.6}\\label{prop:coupling}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. There exist constants $\\delta, \\varepsilon > 0$ such that the following holds. Fix any two sequences $p,q$ satisfying $p \\leq n^{-1/\\rd_1(F) + \\varepsilon}$ and $q \\leq (1 - n^{-\\delta})p^{s}$. Whp as $n\\to\\infty$ there exists a permutation $\\tau: [n] \\to [n]$ such that \n for every $F(v_1, \\ldots, v_r) \\in E(\\Hg_F(n,q))$ also $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subset \\G(n,p)$.\n\\end{proposition}\n\\ignore{\n\\yd{I think the order should be \"whp as $n\\to\\infty$, here exists a permutation $\\tau: [n] \\to [n]$\". The permutations aren't fixed beforehand.}\n\\dg{done}\n\\yd{Can't $\\epsilon, \\delta$ be arbitrarily small?}\n\\dg{We don't want them to be small. In fact the larger is $\\epsilon$ the stronger the result. $\\epsilon$ controls how high above $n^{-{1\\over d_1(F)}}$ we can go and theorem still be valid. Makes sense?}\n\\yd{}\n}\nIn the above $A\\subset B$ means $A$ is a subgraph of $B$ (not necessarily induced one). \n\\Cref{thm-bur-1.6} implies that if we choose $q = (1 - n^{-\\delta})p^{s}$, each $F$-hyperedge in $\\Hg_F(n,q)$ corresponds to an ordered copy of $F$ in $\\G(n,p)$. If we instead choose $q = p^s$, then, on the one hand, the number of $F$-hyperedges in $\\Hg_F(n,q)$ does not increase too much, while on the other hand, it is close to the expected number of ordered copies of $F$ in $\\G(n,p)$. In particular,\nwe claim the following.\n\n\\begin{corollary}\\label{cor-erg-hg}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. Fix $c > 0$, and let $p = c n^{-1/\\rd_1(F)}$, $q = p^s = c^s n^{1-r}$. Whp as $n\\to\\infty$, there exists a permutation\n$\\tau:[n]\\to [n]$ such that the number of distinct ordered copies $F(v_1, \\ldots, v_r)$ for which exactly one of the events $F(v_1, \\ldots, v_r) \\in E(\\Hg_F(n,q))$ or $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subset \\G(n,p)$ holds, is $o(n)$.\n\\end{corollary}\n\\ignore{\n\\yd{Again the existence of permutation is whp}\n\\dg{done}\n}\nThe proof crucially uses the fact that\nfor strictly 1-balanced graph that when $p=O(n^{-{1/d_1(F)}})$, the number of pairs of distinct ordered copies of $F$ in $\\G(n,p)$ which share at least one edge is small. \n\\ignore{\n\\yd{remove \"ensures that\"}\\dg{done}\n}\n\nThroughout this paper, we use $G(n,p)$ to denote the Erdős–Rényi random graph with vertex set $[n]$ and edge probability $p$.\nRecent progress in the study of random \\( F \\)-graphs enables us to extend this relation to more complex substructures, by analyzing the minimum number of monochromatic \\( F \\)-subgraphs over all bipartitions of Erdős–Rényi random graphs, where \\( F \\) is a strictly 1-balanced graph~\\cite{burghart2024sharp}. \n\\begin{theorem}\\label{thm-convergence}\n Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \\geq 2$ vertices. For $c>0$, let $p= cn^{-1/\\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$. Let $\\Min(F,n,p)$ be the minimum number of monochromatic \\( F \\)-subgraphs over all bipartitions of the Erdős–Rényi random graph $G(n,p)$. Let $\\Max(F,n,q)$ be the size of max-cut in the random $F$-graph $R_F(n,q)$. Then, there exists $m(F,p)$ such that\n \\begin{align*}\n \\frac{\\Min(F,n,p)}{n}\\xlongrightarrow{\\text{p}}m(F,p),\n \\end{align*}\n and\n \\begin{align*}\n \\frac{\\Max(F,n,q)}{n}\\xlongrightarrow{\\text{p}}\\frac{c^s}{\\aut(F)}-m(F,p).\n \\end{align*}\n\\end{theorem}\n\nWe further illustrate how this minimum evolves as the host graph becomes denser, and we estimate its limiting behavior as the number of vertices in $F$ tends to infinity.\nFor any integer \\( k \\geq 0 \\), any positive function \\( f = f(a_1, \\ldots, a_k) \\), and any function \\( g = g(a_1, \\ldots, a_k) \\), we write \\( g = o_{a_1,\\ldots,a_k}(f) \\) if\n\\[\n\\limsup_{a_k \\to \\infty} \\cdots \\limsup_{a_1 \\to \\infty} \\left| \\frac{g}{f} \\right| = 0;\n\\]\nwe write \\( g = O_{a_1,\\ldots,a_k}(f) \\) if there exists a positive constant \\( M \\) such that\n\\[\n\\limsup_{a_k \\to \\infty} \\cdots \\limsup_{a_1 \\to \\infty} \\left| \\frac{g}{f} \\right| \\leq M.\n\\]\n\\begin{theorem}\\label{thm-limit}\n With the notion in \\Cref{thm-convergence}, for sufficently large $r$,\n \\begin{align*}\n m(F,p)=(1+o_r(1))\\sqrt{\\frac{\\log2}{2^{r-2}\\aut(F)}c^s }+\\frac{c^s}{2^{r-1}\\aut(F)}+o_{c^s}(c^{s/2}).\n \\end{align*}\n\\end{theorem}\n\nA key result from \\cite{burghart2024sharp} helps us understand the ordered copies of a strictly 1-balanced graph $F$ in $G(n,p)$, and establishes a connection to hyperedges in the random $F$-graph $R_F(n,q)$:\n\\begin{proposition}[{\\cite[Theorem 1.6]{burghart2024sharp}}]\\label{thm-bur-1.6}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. Then there exist constants $\\delta, \\varepsilon > 0$ such that the following holds. For any sequences satisfying $p \\leq n^{-1/\\rd_1(F) + \\varepsilon}$ and $q \\leq (1 - n^{-\\delta})p^{s}$, and for the graphs $G = G(n, p)$ and $R = R_F(n, q)$, there exists a coupling of $G$ and $R$ such that, \\whp, there exists a permutation $\\tau = \\tau(G, R) : [n] \\to [n]$ for which every $F(v_1, \\ldots, v_r) \\in E(R)$ satisfies $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subseteq G$.\n\n\\begin{corollary}\\label{cor-erg-hg}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. For $c > 0$, let $p = c n^{-1/\\rd_1(F)}$ and $q = p^s = c^s n^{1-r}$. Then there exists $\\delta > 0$ such that the following holds. For any $t > 0$, and for the graphs $G = G(n, p)$ and $R = R_F(n, q)$, there exists a coupling of $G$ and $R$ such that, \\whp, there exists a permutation $\\tau = \\tau(G, R) : [n] \\to [n]$ for which the number of distinct ordered copies $F(v_1, \\ldots, v_r)$ such that exactly one of the events $F(v_1, \\ldots, v_r) \\in E(R)$ or $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subseteq G$ holds is at most $tn$.\n\\end{corollary}\nIt is worth noting that the definition of a strictly 1-balanced graph ensures that, for $G$ defined as in \\Cref{cor-erg-hg}, the number of pairs of distinct ordered copies that share at least one edge is limited. Specifically, cases like \\Cref{exa_1} occur only rarely.\n This property is essential for the validity of \\Cref{thm-bur-1.6} and \\Cref{cor-erg-hg}.\n\n\\begin{theorem}\\label{thm-convergence}\n Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \\geq 2$ vertices. Given $c>0$, let $p= cn^{-1/\\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$. There exists a constant $m(F,c)$ such that\n \\begin{align*}\n \\frac{{\\rm MinM}(\\G(n,p),F)}{n}\n \\to m(F,c),\n \\end{align*}\n\\whp~ as $n\\to\\infty$.\n\\end{theorem}", + "post_theorem_intro_text_len": 3456, + "post_theorem_intro_text": "\\ignore{\n\\yd{Maybe we should specify that $m(F,p)$ is a deterministic scalar.}\\dg{good point. Done}\n}\n\nOur proof approach is based on similar results \nfor the contiguous model $\\mathbb{H}(n,q), q=p^s$ which are already known in the literature. Specifically, the existence of $m(F,c)$ such that \n\\begin{align*}\n\\frac{{\\rm MinM}(\\mathbb{H}(n,q),F)}{n}\n\\to m(F,c),\n\\end{align*}\nis already known~\\cite{shabanov2021maximum} based on the combinatorial interpolation technique introduced in~\\cite{bayati2010combinatorial} and used\nfor establishing the existence of\nsuch limits.\nFor similar contiguity reasons \nwe also have that \n$m(F,c)=0$ when $c$ is \nsufficiently small positive \nconstant $c$, and $m(F,c)>0$ when $c$ is sufficiently large. \nThe former claim is\nobtained by choosing $c$ small enough so that\nthe random graph $\\mathbb{H}(n,q)$ does not contain\na giant component. In this case the \nlocally tree-like\nstructure of the graph allows for cutting nearly\nevery $F$-edge of the graph. Conversely, when\n$c$ is sufficiently large, the \nnon-existence of near\nperfect bipartition follows from a simple union\nbound. We regard these observations as folklore \nand will not provide a formal verification of \nthese claims.\n\nWe conjecture that the value \n${\\rm MinM}(\\mathbb{G}(n,p),F){n}$ also undergoes a different type of phase transition. Specifically, we conjecture the existence of $c^*$ which \ndepends on $F$ only such that this value is $0$ w.h.p. as $n\\to\\infty$\nwhen $cc^*$ also w.h.p.\nThe basis for this conjecture is a similar conjecture for random K-SAT (and several other related models such as proper coloring\nof a random graph), which was proven for large $K$~\\cite{ding2022proof}, but is still open for general $K$.\n\nOur next result concerns obtaining explicit \nasymptotic limit\nvalues when the size of the host graph $F$ grows. \nIntroduce a short-hand notation\n\\begin{align*}\n\\kappa(F,c,s,r)={c^s\\over 2^{r-1}{\\rm aut}(F)}.\n\\end{align*}\n\n\\begin{theorem}\\label{thm-limit}\n Under the assumption of \\Cref{thm-convergence},\n\\begin{align*}\n m(F,c)=\\kappa(F,c,s,r)+\n (1+o_r(1))\\sqrt{(2\\log 2)\\kappa(F,c,s,r)}\n +o_{c}(c^{s/2}).\n \\end{align*}\n\\end{theorem} \nNamely, we obtain explicit limit values when both the number of nodes of the host graph $F$ and the leading coefficient $c$ of the random graph \nparameter diverge to infinity. The proof inspiration\nis based on large-$p$\napproximation technique\nfor $p$-spin glass models employed recently\nin~\\cite{gamarnik2025shattering}, and porting these\nresults to sparse random graphs as was done \nin~\\cite{dms2017} \nand~\\cite{sen2018optimization}. The latter two works relate $p$-spin models to sparse graphs using the Lindeberg's interpolation method.\nInstead, we develop relevant asymptotics directly for sparse graphs using the second \nmoment method employed \nin~\\cite{gamarnik2025shattering}\nfor the mean field $p$-spin model.\n\n\\paragraph{Organization} The remainder of this paper is organized as follows. In \\Cref{sec-hypergraph-coupling}, we introduce a coupling between the random $F$-graph and the Erdős–Rényi random graph, and prove \\Cref{thm-convergence}. In \\Cref{sec-spinglass}, we reduce the study of \\( m(F,p) \\) to the analysis of the maximum of a family of Gaussian random variables. Section~\\ref{sec-maximizer} provides auxiliary results that en route to deriving this maximum. Finally, we prove \\Cref{thm-limit} in \\Cref{sec-proof}.", + "sketch": "To prove Theorem~\\ref{thm-convergence}, the paper proposes to proceed via a known contiguous model. The approach is: use similar results for the contiguous model $\\mathbb{H}(n,q)$ with $q=p^s$, where the existence of $m(F,c)$ such that $\\frac{{\\rm MinM}(\\mathbb{H}(n,q),F)}{n}\\to m(F,c)$ is already known in the literature (via the combinatorial interpolation technique of~\\cite{bayati2010combinatorial}, as used in~\\cite{shabanov2021maximum}). Then, in \\Cref{sec-hypergraph-coupling}, the authors “introduce a coupling between the random $F$-graph and the Erdős–Rényi random graph” and use this coupling to “prove \\Cref{thm-convergence}.”", + "expanded_sketch": "To prove the main theorem, the paper proposes to proceed via a known contiguous model. The approach is: use similar results for the contiguous model $\\mathbb{H}(n,q)$ with $q=p^s$, where the existence of $m(F,c)$ such that $\\frac{{\\rm MinM}(\\mathbb{H}(n,q),F)}{n}\\to m(F,c)$ is already known in the literature (via the combinatorial interpolation technique of Bayati et al., \\emph{Combinatorial Interpolation and the Cavity Method} (2010), as used in Shabanov, \\emph{Maximum matchings in random graphs via interpolation} (2021)). Then, later the authors “introduce a coupling between the random $F$-graph and the Erdős–Rényi random graph” and use this coupling to complete the proof of the main theorem.", + "expanded_theorem": "\\label{thm-convergence}\n Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \\geq 2$ vertices. Given $c>0$, let $p= cn^{-1/\\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$. There exists a constant $m(F,c)$ such that\n \\begin{align*}\n \\frac{{\\rm MinM}(\\mathbb{G}(n,p),F)}{n}\n \\to m(F,c),\n \\end{align*}\n\\whp~ as $n\\to\\infty$.,", + "theorem_type": [ + "Existential–Universal", + "Asymptotic or Limit" + ], + "mcq": { + "question": "Let \\(F\\) be a strictly 1-balanced graph with \\(s>0\\) edges and \\(r\\ge 2\\) vertices, where \\(\\mathrm d_1(H)=|E(H)|/(|V(H)|-1)\\) and “strictly 1-balanced” means \\(\\mathrm d_1(F')<\\mathrm d_1(F)\\) for every proper subgraph \\(F'\\subsetneq F\\). Fix \\(c>0\\), and define \\(p=c\\,n^{-1/\\mathrm d_1(F)}\\). Let \\(\\mathbb G(n,p)\\) be the Erdős–Rényi random graph on \\([n]\\), and let \\({\\rm MinM}(\\mathbb G(n,p),F)\\) denote the minimum, over all bipartitions \\([n]=V_1\\cup V_2\\), of the number of copies of \\(F\\) in \\(\\mathbb G(n,p)\\) whose vertices lie entirely in \\(V_1\\) or entirely in \\(V_2\\) (that is, the minimum number of monochromatic copies of \\(F\\)). Which statement holds?", + "correct_choice": { + "label": "A", + "text": "There exists a constant \\(m(F,c)\\) such that, with high probability as \\(n\\to\\infty\\),\n\\[\n\\frac{{\\rm MinM}(\\mathbb G(n,p),F)}{n}\\to m(F,c).\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "There exists a constant \\(m(F,c)\\) such that, with high probability as \\(n\\to\\infty\\),\n\\[\n\\frac{{\\rm MinM}(\\mathbb G(n,p),F)}{n}\\to m(F,c)\n\\]\nuniformly for all fixed \\(c>0\\), with the same limiting constant independent of \\(c\\)." + }, + { + "label": "C", + "text": "There exists a constant \\(m(F,c)\\) and a sequence \\(n_k\\to\\infty\\) such that, with high probability along the subsequence \\(n_k\\),\n\\[\n\\frac{{\\rm MinM}(\\mathbb G(n_k,p),F)}{n_k}\\to m(F,c).\n\\]" + }, + { + "label": "D", + "text": "There exists a constant \\(m(F,c)\\) such that, for every fixed \\(c>0\\), almost surely as \\(n\\to\\infty\\),\n\\[\n\\frac{{\\rm MinM}(\\mathbb G(n,p),F)}{n}\\to m(F,c).\n\\]" + }, + { + "label": "E", + "text": "There exists a constant \\(m(F,c)\\) such that, with high probability as \\(n\\to\\infty\\),\n\\[\n\\frac{{\\rm MinM}(\\mathbb G(n,p),F)}{n^{r-1}}\\to m(F,c).\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": "dependence of the limit on the parameter c", + "template_used": "quantifier_dependence" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "full-sequence convergence whp weakened to subsequential whp convergence", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "mode of convergence strengthened from with high probability to almost sure", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "counting_estimate", + "tampered_component": "linear normalization in n coming from the contiguous model replaced by n^{r-1}", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives definitions and scaling but does not explicitly or implicitly reveal that the correct conclusion is convergence with high probability to a constant. The answer is not leaked by wording in the prompt itself." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-identification question: the correct option appears to match a precise known limiting statement, while other options tweak mode of convergence or hypotheses. It is not a pure restatement, but it is only a mild reformulation rather than a deeply non-tautological problem." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish high probability from in probability or almost sure convergence, and to judge the necessity of strict 1-balance and the coupling claim. However, the question primarily tests recognition of the exact theorem statement rather than substantial independent derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong: one is a weaker true-looking alternative, one improperly strengthens the convergence mode, one overextends the hypothesis class, and one adds a plausible but overly strong coupling conclusion. These reflect realistic mathematical failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with little answer leakage and strong distractors, but it leans more toward theorem recall/statement recognition than genuine generative reasoning." + } + }, + { + "id": "2602.03774v1", + "paper_link": "http://arxiv.org/abs/2602.03774v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm-convergence}\n Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \\geq 2$ vertices. Given $c>0$, let $p= cn^{-1/\\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$. There exists a constant $m(F,c)$ such that\n \\begin{align*}\n \\frac{{\\rm MinM}(\\mathbb{G}(n,p),F)}{n}\n \\to m(F,c),\n \\end{align*}\n\\whp~ as $n\\to\\infty$.", + "start_pos": 10979, + "end_pos": 11342, + "label": "thm-convergence" + }, + "ref_dict": { + "thm-convergence": "\\begin{theorem}\\label{thm-convergence}\n Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \\geq 2$ vertices. Given $c>0$, let $p= cn^{-1/\\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$. There exists a constant $m(F,c)$ such that\n \\begin{align*}\n \\frac{{\\rm MinM}(\\G(n,p),F)}{n}\n \\to m(F,c),\n \\end{align*}\n\\whp~ as $n\\to\\infty$.\n\\end{theorem}", + "thm-limit": "\\begin{theorem}\\label{thm-limit}\n Under the assumption of \\Cref{thm-convergence},\n\\begin{align*}\n m(F,c)=\\kappa(F,c,s,r)+\n (1+o_r(1))\\sqrt{(2\\log 2)\\kappa(F,c,s,r)}\n +o_{c}(c^{s/2}).\n \\end{align*}\n\\end{theorem}", + "prop:coupling": "\\begin{proposition}[{\\cite[Theorem 1.6]{burghart2024sharp}}]\\label{thm-bur-1.6}\\label{prop:coupling}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. There exist constants $\\delta, \\varepsilon > 0$ such that the following holds. Fix any two sequences $p,q$ satisfying $p \\leq n^{-1/\\rd_1(F) + \\varepsilon}$ and $q \\leq (1 - n^{-\\delta})p^{s}$. Whp as $n\\to\\infty$ there exists a permutation $\\tau: [n] \\to [n]$ such that \n for every $F(v_1, \\ldots, v_r) \\in E(\\Hg_F(n,q))$ also $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subset \\G(n,p)$.\n\\end{proposition}" + }, + "pre_theorem_intro_text_len": 5796, + "pre_theorem_intro_text": "The study of monochromatic subgraphs in graph bipartitions is a widely investigated problem in extremal combinatorics and Ramsey theory. \nA classical result by Goodman~\\cite{g1959} states that in any bipartition of the complete graph $K_n$, the number of monochromatic triangles admits a sharp lower bound\n\\[\nT(n) = \n\\begin{cases}\n \\dfrac{u(u-1)(u-2)}{3}, & \\text{if } n = 2u; \\\\\n \\dfrac{2u(u-1)(4u+1)}{3}, & \\text{if } n = 4u+1; \\\\\n \\dfrac{2u(u+1)(4u-1)}{3}, & \\text{if } n = 4u+3.\n\\end{cases}\n\\]\n\\ignore{\n\\zhu{The general case was studied by Paul Erdős, who proved that the minimum number of monochromatic $K_r$ in any bipartition of $K_n$ is at most $\\binom{n}{r}/2^{\\binom{r}{2}-1}$ and conjectured that this bound is asymptotically tight for large $n$ \\cite{erdHos1962number}. This conjecture was later disproved by Thomason \\cite{thomason1989disproof}. In 2012, Conlon proved that any bipartition of $K_n$ contains at least $n^r / C^{(1+o(1))r^2}$ monochromatic $K_r$ subgraphs, where $C \\approx 2.18$ \\cite{conlon2012ramsey}. This phenomenon, whereby a positive fraction of all $r$-cliques are monochromatic, also appears in other deterministic contexts \\cite{frankl1988quantitative}. In general, it can often be characterized by the principle that monochromatic substructures are evenly distributed among configurations that force their occurrence \\cite{csv}.}\n\\dg{I am not sure we need such a detailed summary about deterministic setup. It is somewhat off-topic}\n}\n\nWhile these deterministic settings have attracted considerable attention and yielded several intriguing results, \nthe same question in the context of random graphs,\nsuch as Erdős–Rényi random graphs, remains far less understood, even for the special case of triangles. In this paper, we consider this question of finding the smallest number of monochromatic subgraphs in bipartitions of random graphs. We do so in a general setting, \nwhere the subgraph is only assumed to be\nstrictly balanced.\n\nTo set the stage, let $F$ be a graph on $r$ vertices. An $F$-graph $H$ is defined as any pair $(V=V(H), E=E(H))$, where $V$ is a set of vertices, and \n$E$ is a collection of copies $e$ of $F$ called hyperedges, where for each $e$, the vertex set of $e$ is subset of $V$.\n A random \\( F \\)-graph \\( \\Hg_F(n, q) \\) is an \\( F \\)-graph with vertex set \\( V=[n]=\\cbc{1,2,\\ldots,n} \\), where each of the\n\\begin{align}\\label{eq:N-trials}\n\\binom{n}{r} \\cdot \\frac{r!}{{\\rm aut}(F)}\n\\end{align}\npotential copies of \\( F \\) on vertices in \\( [n] \\) is included in the edge set $E$ independently with probability \\( q \\). Here ${\\rm aut}(H)$ denotes the set of automorphisms of a graph $H$. In the special case\nwhen $F$ is $K_2$ (two nodes\nconnected by an edge), we use a more common notation $\\mathbb{G}(n,q)$.\n\nGiven a simple graph $G=(V,E)$ (so that $F$ is just $K_2$) and given its bipartition \\( V=V_1 \\cup V_2, V_1\\cap V_2=\\emptyset \\), a cut, or more specifically an $F$-cut associated with this bipartition is defined as the set of subgraphs $e$ of $G$ which are isomorphic to $F$, and whose vertex sets have a non-empty intersection with \\( V_1 \\) and \\( V_2 \\). That is, each hyperedge of the cut is neither entirely contained in \\( V_1 \\), nor is it entirely contained in \\( V_2 \\). The \\emph{cut value} associated with a cut is \nthe cardinality of this set.\n The \\emph{max-cut value}\n denoted by ${\\rm MaxC}(G,F)$\n is defined as the largest such value taken over all possible bipartitions of \\(V\\). Thinking of $V_1$ and $V_2$ as being associated with two distinct colors, the cut associated with the bipartition is simply the set of all non-monochromatic subgraphs of $G$ isomorphic to $F$\n associated with the coloring scheme $V_1,V_2$. Conversely, every $F$-subgraph of $G$ not participating in the cut is monochromatic by definition. The minimum\n number of monochromatic $F$-subgraphs is denoted by ${\\rm MinM}(G,F)$. Naturally, ${\\rm MinM}(G,F)+{\\rm MaxC}(G,F)$ is the total number of $F$-subgraphs of $G$ as this is the case for every bipartition.\n\nIn combinatorial optimization and theoretical computer science, there is a long-standing interest in understanding the minimum number of monochromatic subgraphs achievable by varying over all bipartitions of graphs, both in the special case of max-cut values associated with $F=K_2$, and beyond ~\\cite{ckpsty2013,dms2017,dl2018,sen2018optimization,shabanov2021maximum}. This is the question we address in our paper.\n\n\\relax\n\nLet $F$ be a graph. We say $F$ is a strictly 1-balanced graph if $\\rd_1(F')<{\\rm d}(F)$ for any subgraph $F'$ of $F$ such that $F'\\neq F$. Here\n \\begin{align*}\n \\rd_1(G)=\\frac{\\abs{E(G)}}{\\abs{V(G )}-1}.\n \\end{align*}\nThere are many examples of strictly 1-balanced graphs, including cycles and complete graphs. However, any disconnected graph is not strictly 1-balanced, and neither is a tree-graph.\n\nRecently a great progress was achieved in understanding the relationship between $\\mathbb{G}(n,p)$ and $\\Hg_F(n,q)$\nwith $q$ judiciously chosen\nas $q=p^{|E(F)|}$. In particular, as shown in \\cite{burghart2024sharp} (see Proposition~\\ref{prop:coupling} below), when $p$ is at most $n^{-1/d_1(F)}$, the graphs\n$\\mathbb{G}(n,p)$ and $\\mathbb{H}(n,q)$ can be coupled in such a way that the number of $F$-hyperedges in $\\mathbb{H}$ nearly matches the number of copies of $F$ naturally occurring in $\\mathbb{G}(n,p)$. As a result a difficult problem of studying minimal number of monochromatic $F$-subgraphs of $\\mathbb{G}(n,p)$ can be reduced to simpler version, one defined on $\\mathbb{H}$ where the occurances of $F$-s are independent by design. This is the insight we use to obtain several asymptotic results on max-cut values for general graphs $F$ in $\\mathbb{G}(n,p)$. \n\nOur first result is as follows.", + "context": "The study of monochromatic subgraphs in graph bipartitions is a widely investigated problem in extremal combinatorics and Ramsey theory. \nA classical result by Goodman~\\cite{g1959} states that in any bipartition of the complete graph $K_n$, the number of monochromatic triangles admits a sharp lower bound\n\\[\nT(n) = \n\\begin{cases}\n \\dfrac{u(u-1)(u-2)}{3}, & \\text{if } n = 2u; \\\\\n \\dfrac{2u(u-1)(4u+1)}{3}, & \\text{if } n = 4u+1; \\\\\n \\dfrac{2u(u+1)(4u-1)}{3}, & \\text{if } n = 4u+3.\n\\end{cases}\n\\]\n\\ignore{\n\\zhu{The general case was studied by Paul Erdős, who proved that the minimum number of monochromatic $K_r$ in any bipartition of $K_n$ is at most $\\binom{n}{r}/2^{\\binom{r}{2}-1}$ and conjectured that this bound is asymptotically tight for large $n$ \\cite{erdHos1962number}. This conjecture was later disproved by Thomason \\cite{thomason1989disproof}. In 2012, Conlon proved that any bipartition of $K_n$ contains at least $n^r / C^{(1+o(1))r^2}$ monochromatic $K_r$ subgraphs, where $C \\approx 2.18$ \\cite{conlon2012ramsey}. This phenomenon, whereby a positive fraction of all $r$-cliques are monochromatic, also appears in other deterministic contexts \\cite{frankl1988quantitative}. In general, it can often be characterized by the principle that monochromatic substructures are evenly distributed among configurations that force their occurrence \\cite{csv}.}\n\\dg{I am not sure we need such a detailed summary about deterministic setup. It is somewhat off-topic}\n}\n\nTo set the stage, let $F$ be a graph on $r$ vertices. An $F$-graph $H$ is defined as any pair $(V=V(H), E=E(H))$, where $V$ is a set of vertices, and \n$E$ is a collection of copies $e$ of $F$ called hyperedges, where for each $e$, the vertex set of $e$ is subset of $V$.\n A random \\( F \\)-graph \\( \\Hg_F(n, q) \\) is an \\( F \\)-graph with vertex set \\( V=[n]=\\cbc{1,2,\\ldots,n} \\), where each of the\n\\begin{align}\\label{eq:N-trials}\n\\binom{n}{r} \\cdot \\frac{r!}{{\\rm aut}(F)}\n\\end{align}\npotential copies of \\( F \\) on vertices in \\( [n] \\) is included in the edge set $E$ independently with probability \\( q \\). Here ${\\rm aut}(H)$ denotes the set of automorphisms of a graph $H$. In the special case\nwhen $F$ is $K_2$ (two nodes\nconnected by an edge), we use a more common notation $\\mathbb{G}(n,q)$.\n\nGiven a simple graph $G=(V,E)$ (so that $F$ is just $K_2$) and given its bipartition \\( V=V_1 \\cup V_2, V_1\\cap V_2=\\emptyset \\), a cut, or more specifically an $F$-cut associated with this bipartition is defined as the set of subgraphs $e$ of $G$ which are isomorphic to $F$, and whose vertex sets have a non-empty intersection with \\( V_1 \\) and \\( V_2 \\). That is, each hyperedge of the cut is neither entirely contained in \\( V_1 \\), nor is it entirely contained in \\( V_2 \\). The \\emph{cut value} associated with a cut is \nthe cardinality of this set.\n The \\emph{max-cut value}\n denoted by ${\\rm MaxC}(G,F)$\n is defined as the largest such value taken over all possible bipartitions of \\(V\\). Thinking of $V_1$ and $V_2$ as being associated with two distinct colors, the cut associated with the bipartition is simply the set of all non-monochromatic subgraphs of $G$ isomorphic to $F$\n associated with the coloring scheme $V_1,V_2$. Conversely, every $F$-subgraph of $G$ not participating in the cut is monochromatic by definition. The minimum\n number of monochromatic $F$-subgraphs is denoted by ${\\rm MinM}(G,F)$. Naturally, ${\\rm MinM}(G,F)+{\\rm MaxC}(G,F)$ is the total number of $F$-subgraphs of $G$ as this is the case for every bipartition.\n\nLet $F$ be a graph. We say $F$ is a strictly 1-balanced graph if $\\rd_1(F')<{\\rm d}(F)$ for any subgraph $F'$ of $F$ such that $F'\\neq F$. Here\n \\begin{align*}\n \\rd_1(G)=\\frac{\\abs{E(G)}}{\\abs{V(G )}-1}.\n \\end{align*}\nThere are many examples of strictly 1-balanced graphs, including cycles and complete graphs. However, any disconnected graph is not strictly 1-balanced, and neither is a tree-graph.\n\nRecently a great progress was achieved in understanding the relationship between $\\mathbb{G}(n,p)$ and $\\Hg_F(n,q)$\nwith $q$ judiciously chosen\nas $q=p^{|E(F)|}$. In particular, as shown in \\cite{burghart2024sharp} (see Proposition~\\ref{prop:coupling} below), when $p$ is at most $n^{-1/d_1(F)}$, the graphs\n$\\mathbb{G}(n,p)$ and $\\mathbb{H}(n,q)$ can be coupled in such a way that the number of $F$-hyperedges in $\\mathbb{H}$ nearly matches the number of copies of $F$ naturally occurring in $\\mathbb{G}(n,p)$. As a result a difficult problem of studying minimal number of monochromatic $F$-subgraphs of $\\mathbb{G}(n,p)$ can be reduced to simpler version, one defined on $\\mathbb{H}$ where the occurances of $F$-s are independent by design. This is the insight we use to obtain several asymptotic results on max-cut values for general graphs $F$ in $\\mathbb{G}(n,p)$.\n\nOur first result is as follows.\n\n\\begin{proposition}[{\\cite[Theorem 1.6]{burghart2024sharp}}]\\label{thm-bur-1.6}\\label{prop:coupling}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. There exist constants $\\delta, \\varepsilon > 0$ such that the following holds. Fix any two sequences $p,q$ satisfying $p \\leq n^{-1/\\rd_1(F) + \\varepsilon}$ and $q \\leq (1 - n^{-\\delta})p^{s}$. Whp as $n\\to\\infty$ there exists a permutation $\\tau: [n] \\to [n]$ such that \n for every $F(v_1, \\ldots, v_r) \\in E(\\Hg_F(n,q))$ also $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subset \\G(n,p)$.\n\\end{proposition}", + "full_context": "The study of monochromatic subgraphs in graph bipartitions is a widely investigated problem in extremal combinatorics and Ramsey theory. \nA classical result by Goodman~\\cite{g1959} states that in any bipartition of the complete graph $K_n$, the number of monochromatic triangles admits a sharp lower bound\n\\[\nT(n) = \n\\begin{cases}\n \\dfrac{u(u-1)(u-2)}{3}, & \\text{if } n = 2u; \\\\\n \\dfrac{2u(u-1)(4u+1)}{3}, & \\text{if } n = 4u+1; \\\\\n \\dfrac{2u(u+1)(4u-1)}{3}, & \\text{if } n = 4u+3.\n\\end{cases}\n\\]\n\\ignore{\n\\zhu{The general case was studied by Paul Erdős, who proved that the minimum number of monochromatic $K_r$ in any bipartition of $K_n$ is at most $\\binom{n}{r}/2^{\\binom{r}{2}-1}$ and conjectured that this bound is asymptotically tight for large $n$ \\cite{erdHos1962number}. This conjecture was later disproved by Thomason \\cite{thomason1989disproof}. In 2012, Conlon proved that any bipartition of $K_n$ contains at least $n^r / C^{(1+o(1))r^2}$ monochromatic $K_r$ subgraphs, where $C \\approx 2.18$ \\cite{conlon2012ramsey}. This phenomenon, whereby a positive fraction of all $r$-cliques are monochromatic, also appears in other deterministic contexts \\cite{frankl1988quantitative}. In general, it can often be characterized by the principle that monochromatic substructures are evenly distributed among configurations that force their occurrence \\cite{csv}.}\n\\dg{I am not sure we need such a detailed summary about deterministic setup. It is somewhat off-topic}\n}\n\nTo set the stage, let $F$ be a graph on $r$ vertices. An $F$-graph $H$ is defined as any pair $(V=V(H), E=E(H))$, where $V$ is a set of vertices, and \n$E$ is a collection of copies $e$ of $F$ called hyperedges, where for each $e$, the vertex set of $e$ is subset of $V$.\n A random \\( F \\)-graph \\( \\Hg_F(n, q) \\) is an \\( F \\)-graph with vertex set \\( V=[n]=\\cbc{1,2,\\ldots,n} \\), where each of the\n\\begin{align}\\label{eq:N-trials}\n\\binom{n}{r} \\cdot \\frac{r!}{{\\rm aut}(F)}\n\\end{align}\npotential copies of \\( F \\) on vertices in \\( [n] \\) is included in the edge set $E$ independently with probability \\( q \\). Here ${\\rm aut}(H)$ denotes the set of automorphisms of a graph $H$. In the special case\nwhen $F$ is $K_2$ (two nodes\nconnected by an edge), we use a more common notation $\\mathbb{G}(n,q)$.\n\nGiven a simple graph $G=(V,E)$ (so that $F$ is just $K_2$) and given its bipartition \\( V=V_1 \\cup V_2, V_1\\cap V_2=\\emptyset \\), a cut, or more specifically an $F$-cut associated with this bipartition is defined as the set of subgraphs $e$ of $G$ which are isomorphic to $F$, and whose vertex sets have a non-empty intersection with \\( V_1 \\) and \\( V_2 \\). That is, each hyperedge of the cut is neither entirely contained in \\( V_1 \\), nor is it entirely contained in \\( V_2 \\). The \\emph{cut value} associated with a cut is \nthe cardinality of this set.\n The \\emph{max-cut value}\n denoted by ${\\rm MaxC}(G,F)$\n is defined as the largest such value taken over all possible bipartitions of \\(V\\). Thinking of $V_1$ and $V_2$ as being associated with two distinct colors, the cut associated with the bipartition is simply the set of all non-monochromatic subgraphs of $G$ isomorphic to $F$\n associated with the coloring scheme $V_1,V_2$. Conversely, every $F$-subgraph of $G$ not participating in the cut is monochromatic by definition. The minimum\n number of monochromatic $F$-subgraphs is denoted by ${\\rm MinM}(G,F)$. Naturally, ${\\rm MinM}(G,F)+{\\rm MaxC}(G,F)$ is the total number of $F$-subgraphs of $G$ as this is the case for every bipartition.\n\nLet $F$ be a graph. We say $F$ is a strictly 1-balanced graph if $\\rd_1(F')<{\\rm d}(F)$ for any subgraph $F'$ of $F$ such that $F'\\neq F$. Here\n \\begin{align*}\n \\rd_1(G)=\\frac{\\abs{E(G)}}{\\abs{V(G )}-1}.\n \\end{align*}\nThere are many examples of strictly 1-balanced graphs, including cycles and complete graphs. However, any disconnected graph is not strictly 1-balanced, and neither is a tree-graph.\n\nRecently a great progress was achieved in understanding the relationship between $\\mathbb{G}(n,p)$ and $\\Hg_F(n,q)$\nwith $q$ judiciously chosen\nas $q=p^{|E(F)|}$. In particular, as shown in \\cite{burghart2024sharp} (see Proposition~\\ref{prop:coupling} below), when $p$ is at most $n^{-1/d_1(F)}$, the graphs\n$\\mathbb{G}(n,p)$ and $\\mathbb{H}(n,q)$ can be coupled in such a way that the number of $F$-hyperedges in $\\mathbb{H}$ nearly matches the number of copies of $F$ naturally occurring in $\\mathbb{G}(n,p)$. As a result a difficult problem of studying minimal number of monochromatic $F$-subgraphs of $\\mathbb{G}(n,p)$ can be reduced to simpler version, one defined on $\\mathbb{H}$ where the occurances of $F$-s are independent by design. This is the insight we use to obtain several asymptotic results on max-cut values for general graphs $F$ in $\\mathbb{G}(n,p)$.\n\nOur first result is as follows.\n\n\\begin{proposition}[{\\cite[Theorem 1.6]{burghart2024sharp}}]\\label{thm-bur-1.6}\\label{prop:coupling}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. There exist constants $\\delta, \\varepsilon > 0$ such that the following holds. Fix any two sequences $p,q$ satisfying $p \\leq n^{-1/\\rd_1(F) + \\varepsilon}$ and $q \\leq (1 - n^{-\\delta})p^{s}$. Whp as $n\\to\\infty$ there exists a permutation $\\tau: [n] \\to [n]$ such that \n for every $F(v_1, \\ldots, v_r) \\in E(\\Hg_F(n,q))$ also $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subset \\G(n,p)$.\n\\end{proposition}\n\nOur first result is as follows.\n\n\\ignore{\n\\yd{Maybe we should specify that $m(F,p)$ is a deterministic scalar.}\\dg{good point. Done}\n}\n\nThe following key result from \\cite{burghart2024sharp} helps us understand the ordered copies of a strictly 1-balanced graph $F$ in $\\G(n,p)$, and establishes a connection to hyperedges in the random $F$-graph $\\Hg_F(n,q)$:\n\\begin{proposition}[{\\cite[Theorem 1.6]{burghart2024sharp}}]\\label{thm-bur-1.6}\\label{prop:coupling}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. There exist constants $\\delta, \\varepsilon > 0$ such that the following holds. Fix any two sequences $p,q$ satisfying $p \\leq n^{-1/\\rd_1(F) + \\varepsilon}$ and $q \\leq (1 - n^{-\\delta})p^{s}$. Whp as $n\\to\\infty$ there exists a permutation $\\tau: [n] \\to [n]$ such that \n for every $F(v_1, \\ldots, v_r) \\in E(\\Hg_F(n,q))$ also $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subset \\G(n,p)$.\n\\end{proposition}\n\\ignore{\n\\yd{I think the order should be \"whp as $n\\to\\infty$, here exists a permutation $\\tau: [n] \\to [n]$\". The permutations aren't fixed beforehand.}\n\\dg{done}\n\\yd{Can't $\\epsilon, \\delta$ be arbitrarily small?}\n\\dg{We don't want them to be small. In fact the larger is $\\epsilon$ the stronger the result. $\\epsilon$ controls how high above $n^{-{1\\over d_1(F)}}$ we can go and theorem still be valid. Makes sense?}\n\\yd{}\n}\nIn the above $A\\subset B$ means $A$ is a subgraph of $B$ (not necessarily induced one). \n\\Cref{thm-bur-1.6} implies that if we choose $q = (1 - n^{-\\delta})p^{s}$, each $F$-hyperedge in $\\Hg_F(n,q)$ corresponds to an ordered copy of $F$ in $\\G(n,p)$. If we instead choose $q = p^s$, then, on the one hand, the number of $F$-hyperedges in $\\Hg_F(n,q)$ does not increase too much, while on the other hand, it is close to the expected number of ordered copies of $F$ in $\\G(n,p)$. In particular,\nwe claim the following.\n\n\\begin{corollary}\\label{cor-erg-hg}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. Fix $c > 0$, and let $p = c n^{-1/\\rd_1(F)}$, $q = p^s = c^s n^{1-r}$. Whp as $n\\to\\infty$, there exists a permutation\n$\\tau:[n]\\to [n]$ such that the number of distinct ordered copies $F(v_1, \\ldots, v_r)$ for which exactly one of the events $F(v_1, \\ldots, v_r) \\in E(\\Hg_F(n,q))$ or $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subset \\G(n,p)$ holds, is $o(n)$.\n\\end{corollary}\n\\ignore{\n\\yd{Again the existence of permutation is whp}\n\\dg{done}\n}\nThe proof crucially uses the fact that\nfor strictly 1-balanced graph that when $p=O(n^{-{1/d_1(F)}})$, the number of pairs of distinct ordered copies of $F$ in $\\G(n,p)$ which share at least one edge is small. \n\\ignore{\n\\yd{remove \"ensures that\"}\\dg{done}\n}\n\nThroughout this paper, we use $G(n,p)$ to denote the Erdős–Rényi random graph with vertex set $[n]$ and edge probability $p$.\nRecent progress in the study of random \\( F \\)-graphs enables us to extend this relation to more complex substructures, by analyzing the minimum number of monochromatic \\( F \\)-subgraphs over all bipartitions of Erdős–Rényi random graphs, where \\( F \\) is a strictly 1-balanced graph~\\cite{burghart2024sharp}. \n\\begin{theorem}\\label{thm-convergence}\n Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \\geq 2$ vertices. For $c>0$, let $p= cn^{-1/\\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$. Let $\\Min(F,n,p)$ be the minimum number of monochromatic \\( F \\)-subgraphs over all bipartitions of the Erdős–Rényi random graph $G(n,p)$. Let $\\Max(F,n,q)$ be the size of max-cut in the random $F$-graph $R_F(n,q)$. Then, there exists $m(F,p)$ such that\n \\begin{align*}\n \\frac{\\Min(F,n,p)}{n}\\xlongrightarrow{\\text{p}}m(F,p),\n \\end{align*}\n and\n \\begin{align*}\n \\frac{\\Max(F,n,q)}{n}\\xlongrightarrow{\\text{p}}\\frac{c^s}{\\aut(F)}-m(F,p).\n \\end{align*}\n\\end{theorem}\n\nWe further illustrate how this minimum evolves as the host graph becomes denser, and we estimate its limiting behavior as the number of vertices in $F$ tends to infinity.\nFor any integer \\( k \\geq 0 \\), any positive function \\( f = f(a_1, \\ldots, a_k) \\), and any function \\( g = g(a_1, \\ldots, a_k) \\), we write \\( g = o_{a_1,\\ldots,a_k}(f) \\) if\n\\[\n\\limsup_{a_k \\to \\infty} \\cdots \\limsup_{a_1 \\to \\infty} \\left| \\frac{g}{f} \\right| = 0;\n\\]\nwe write \\( g = O_{a_1,\\ldots,a_k}(f) \\) if there exists a positive constant \\( M \\) such that\n\\[\n\\limsup_{a_k \\to \\infty} \\cdots \\limsup_{a_1 \\to \\infty} \\left| \\frac{g}{f} \\right| \\leq M.\n\\]\n\\begin{theorem}\\label{thm-limit}\n With the notion in \\Cref{thm-convergence}, for sufficently large $r$,\n \\begin{align*}\n m(F,p)=(1+o_r(1))\\sqrt{\\frac{\\log2}{2^{r-2}\\aut(F)}c^s }+\\frac{c^s}{2^{r-1}\\aut(F)}+o_{c^s}(c^{s/2}).\n \\end{align*}\n\\end{theorem}\n\nA key result from \\cite{burghart2024sharp} helps us understand the ordered copies of a strictly 1-balanced graph $F$ in $G(n,p)$, and establishes a connection to hyperedges in the random $F$-graph $R_F(n,q)$:\n\\begin{proposition}[{\\cite[Theorem 1.6]{burghart2024sharp}}]\\label{thm-bur-1.6}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. Then there exist constants $\\delta, \\varepsilon > 0$ such that the following holds. For any sequences satisfying $p \\leq n^{-1/\\rd_1(F) + \\varepsilon}$ and $q \\leq (1 - n^{-\\delta})p^{s}$, and for the graphs $G = G(n, p)$ and $R = R_F(n, q)$, there exists a coupling of $G$ and $R$ such that, \\whp, there exists a permutation $\\tau = \\tau(G, R) : [n] \\to [n]$ for which every $F(v_1, \\ldots, v_r) \\in E(R)$ satisfies $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subseteq G$.\n\n\\begin{corollary}\\label{cor-erg-hg}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. For $c > 0$, let $p = c n^{-1/\\rd_1(F)}$ and $q = p^s = c^s n^{1-r}$. Then there exists $\\delta > 0$ such that the following holds. For any $t > 0$, and for the graphs $G = G(n, p)$ and $R = R_F(n, q)$, there exists a coupling of $G$ and $R$ such that, \\whp, there exists a permutation $\\tau = \\tau(G, R) : [n] \\to [n]$ for which the number of distinct ordered copies $F(v_1, \\ldots, v_r)$ such that exactly one of the events $F(v_1, \\ldots, v_r) \\in E(R)$ or $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subseteq G$ holds is at most $tn$.\n\\end{corollary}\nIt is worth noting that the definition of a strictly 1-balanced graph ensures that, for $G$ defined as in \\Cref{cor-erg-hg}, the number of pairs of distinct ordered copies that share at least one edge is limited. Specifically, cases like \\Cref{exa_1} occur only rarely.\n This property is essential for the validity of \\Cref{thm-bur-1.6} and \\Cref{cor-erg-hg}.\n\n\\begin{theorem}\\label{thm-convergence}\n Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \\geq 2$ vertices. Given $c>0$, let $p= cn^{-1/\\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$. There exists a constant $m(F,c)$ such that\n \\begin{align*}\n \\frac{{\\rm MinM}(\\G(n,p),F)}{n}\n \\to m(F,c),\n \\end{align*}\n\\whp~ as $n\\to\\infty$.\n\\end{theorem}", + "post_theorem_intro_text_len": 3456, + "post_theorem_intro_text": "\\ignore{\n\\yd{Maybe we should specify that $m(F,p)$ is a deterministic scalar.}\\dg{good point. Done}\n}\n\nOur proof approach is based on similar results \nfor the contiguous model $\\mathbb{H}(n,q), q=p^s$ which are already known in the literature. Specifically, the existence of $m(F,c)$ such that \n\\begin{align*}\n\\frac{{\\rm MinM}(\\mathbb{H}(n,q),F)}{n}\n\\to m(F,c),\n\\end{align*}\nis already known~\\cite{shabanov2021maximum} based on the combinatorial interpolation technique introduced in~\\cite{bayati2010combinatorial} and used\nfor establishing the existence of\nsuch limits.\nFor similar contiguity reasons \nwe also have that \n$m(F,c)=0$ when $c$ is \nsufficiently small positive \nconstant $c$, and $m(F,c)>0$ when $c$ is sufficiently large. \nThe former claim is\nobtained by choosing $c$ small enough so that\nthe random graph $\\mathbb{H}(n,q)$ does not contain\na giant component. In this case the \nlocally tree-like\nstructure of the graph allows for cutting nearly\nevery $F$-edge of the graph. Conversely, when\n$c$ is sufficiently large, the \nnon-existence of near\nperfect bipartition follows from a simple union\nbound. We regard these observations as folklore \nand will not provide a formal verification of \nthese claims.\n\nWe conjecture that the value \n${\\rm MinM}(\\mathbb{G}(n,p),F){n}$ also undergoes a different type of phase transition. Specifically, we conjecture the existence of $c^*$ which \ndepends on $F$ only such that this value is $0$ w.h.p. as $n\\to\\infty$\nwhen $cc^*$ also w.h.p.\nThe basis for this conjecture is a similar conjecture for random K-SAT (and several other related models such as proper coloring\nof a random graph), which was proven for large $K$~\\cite{ding2022proof}, but is still open for general $K$.\n\nOur next result concerns obtaining explicit \nasymptotic limit\nvalues when the size of the host graph $F$ grows. \nIntroduce a short-hand notation\n\\begin{align*}\n\\kappa(F,c,s,r)={c^s\\over 2^{r-1}{\\rm aut}(F)}.\n\\end{align*}\n\n\\begin{theorem}\\label{thm-limit}\n Under the assumption of \\Cref{thm-convergence},\n\\begin{align*}\n m(F,c)=\\kappa(F,c,s,r)+\n (1+o_r(1))\\sqrt{(2\\log 2)\\kappa(F,c,s,r)}\n +o_{c}(c^{s/2}).\n \\end{align*}\n\\end{theorem} \nNamely, we obtain explicit limit values when both the number of nodes of the host graph $F$ and the leading coefficient $c$ of the random graph \nparameter diverge to infinity. The proof inspiration\nis based on large-$p$\napproximation technique\nfor $p$-spin glass models employed recently\nin~\\cite{gamarnik2025shattering}, and porting these\nresults to sparse random graphs as was done \nin~\\cite{dms2017} \nand~\\cite{sen2018optimization}. The latter two works relate $p$-spin models to sparse graphs using the Lindeberg's interpolation method.\nInstead, we develop relevant asymptotics directly for sparse graphs using the second \nmoment method employed \nin~\\cite{gamarnik2025shattering}\nfor the mean field $p$-spin model.\n\n\\paragraph{Organization} The remainder of this paper is organized as follows. In \\Cref{sec-hypergraph-coupling}, we introduce a coupling between the random $F$-graph and the Erdős–Rényi random graph, and prove \\Cref{thm-convergence}. In \\Cref{sec-spinglass}, we reduce the study of \\( m(F,p) \\) to the analysis of the maximum of a family of Gaussian random variables. Section~\\ref{sec-maximizer} provides auxiliary results that en route to deriving this maximum. Finally, we prove \\Cref{thm-limit} in \\Cref{sec-proof}.", + "sketch": "To prove Theorem~\\ref{thm-convergence}, the paper proposes to proceed via a known contiguous model. The approach is: use similar results for the contiguous model $\\mathbb{H}(n,q)$ with $q=p^s$, where the existence of $m(F,c)$ such that $\\frac{{\\rm MinM}(\\mathbb{H}(n,q),F)}{n}\\to m(F,c)$ is already known in the literature (via the combinatorial interpolation technique of~\\cite{bayati2010combinatorial}, as used in~\\cite{shabanov2021maximum}). Then, in \\Cref{sec-hypergraph-coupling}, the authors “introduce a coupling between the random $F$-graph and the Erdős–Rényi random graph” and use this coupling to “prove \\Cref{thm-convergence}.”", + "expanded_sketch": "To prove the main theorem, the paper proposes to proceed via a known contiguous model. The approach is: use similar results for the contiguous model $\\mathbb{H}(n,q)$ with $q=p^s$, where the existence of $m(F,c)$ such that $\\frac{{\\rm MinM}(\\mathbb{H}(n,q),F)}{n}\\to m(F,c)$ is already known in the literature (via the combinatorial interpolation technique of Bayati et al., \\emph{Combinatorial Interpolation and the Cavity Method} (2010), as used in Shabanov, \\emph{Maximum matchings in random graphs via interpolation} (2021)). Then, later the authors “introduce a coupling between the random $F$-graph and the Erdős–Rényi random graph” and use this coupling to complete the proof of the main theorem.", + "expanded_theorem": "\\label{thm-convergence}\n Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \\geq 2$ vertices. Given $c>0$, let $p= cn^{-1/\\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$. There exists a constant $m(F,c)$ such that\n \\begin{align*}\n \\frac{{\\rm MinM}(\\mathbb{G}(n,p),F)}{n}\n \\to m(F,c),\n \\end{align*}\n\\whp~ as $n\\to\\infty$.,", + "theorem_type": [ + "Existential–Universal", + "Asymptotic or Limit" + ], + "mcq": { + "question": "Let \\(F\\) be a strictly 1-balanced graph with \\(s>0\\) edges and \\(r\\ge 2\\) vertices, where \\(\\mathrm d_1(H)=|E(H)|/(|V(H)|-1)\\) and “strictly 1-balanced” means \\(\\mathrm d_1(F')<\\mathrm d_1(F)\\) for every proper subgraph \\(F'\\subsetneq F\\). Fix \\(c>0\\), and define \\(p=c\\,n^{-1/\\mathrm d_1(F)}\\). Let \\(\\mathbb G(n,p)\\) be the Erdős–Rényi random graph on \\([n]\\), and let \\({\\rm MinM}(\\mathbb G(n,p),F)\\) denote the minimum, over all bipartitions \\([n]=V_1\\cup V_2\\), of the number of copies of \\(F\\) in \\(\\mathbb G(n,p)\\) whose vertices lie entirely in \\(V_1\\) or entirely in \\(V_2\\) (that is, the minimum number of monochromatic copies of \\(F\\)). Which statement holds?", + "correct_choice": { + "label": "A", + "text": "There exists a constant \\(m(F,c)\\) such that, with high probability as \\(n\\to\\infty\\),\n\\[\n\\frac{{\\rm MinM}(\\mathbb G(n,p),F)}{n}\\to m(F,c).\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "There exists a constant \\(m(F,c)\\) such that, with high probability as \\(n\\to\\infty\\),\n\\[\n\\frac{{\\rm MinM}(\\mathbb G(n,p),F)}{n}\\to m(F,c)\n\\]\nuniformly for all fixed \\(c>0\\), with the same limiting constant independent of \\(c\\)." + }, + { + "label": "C", + "text": "There exists a constant \\(m(F,c)\\) and a sequence \\(n_k\\to\\infty\\) such that, with high probability along the subsequence \\(n_k\\),\n\\[\n\\frac{{\\rm MinM}(\\mathbb G(n_k,p),F)}{n_k}\\to m(F,c).\n\\]" + }, + { + "label": "D", + "text": "There exists a constant \\(m(F,c)\\) such that, for every fixed \\(c>0\\), almost surely as \\(n\\to\\infty\\),\n\\[\n\\frac{{\\rm MinM}(\\mathbb G(n,p),F)}{n}\\to m(F,c).\n\\]" + }, + { + "label": "E", + "text": "There exists a constant \\(m(F,c)\\) such that, with high probability as \\(n\\to\\infty\\),\n\\[\n\\frac{{\\rm MinM}(\\mathbb G(n,p),F)}{n^{r-1}}\\to m(F,c).\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": "dependence of the limit on the parameter c", + "template_used": "quantifier_dependence" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "full-sequence convergence whp weakened to subsequential whp convergence", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "mode of convergence strengthened from with high probability to almost sure", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "counting_estimate", + "tampered_component": "linear normalization in n coming from the contiguous model replaced by n^{r-1}", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines the setup and notation but does not explicitly state the limiting conclusion. The correct answer is not leaked directly; one must distinguish among several nearby asymptotic formulations." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-identification question: one option is essentially the exact theorem statement, while the others are quantifier/convergence/normalization variants. So it is not fully tautological, but it is still largely a restatement-discrimination task rather than a fresh conclusion derived from premises." + }, + "GPS": { + "score": 1, + "justification": "There is some reasoning pressure because the distractors differ in subtle but meaningful ways (dependence on c, subsequence vs full sequence, whp vs almost sure, and scaling by n vs n^{r-1}). However, the problem mainly tests recognition of the correct theorem-strength statement rather than substantial generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong: they are plausible, mathematically distinct, and target common failure modes such as overstrengthening convergence, weakening to subsequences, removing parameter dependence, or using an incorrect normalization." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no direct answer leakage, but it leans more toward recall/statement recognition than genuine generative reasoning." + } + }, + { + "id": "2602.04034v1", + "paper_link": "http://arxiv.org/abs/2602.04034v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "conjecture", + "content": "\\label{conjecture:main}\nLet $\\alg{A}$ and $\\alg{B}$ be finite modules. Then, there are only finitely many clonoids from $\\alg{A}$ to $\\alg{B}$, if and only if $\\alg{A}$ and $\\alg{B}$ are of coprime order.", + "start_pos": 10644, + "end_pos": 10872, + "label": "conjecture:main" + }, + "ref_dict": { + "theorem:main": "\\begin{theorem} \\label{theorem:main}\nLet $\\F$ be a finite field, $k \\in \\N$, and let $\\algB$ be an $\\algR$-module, such that $|F|$ is invertible in $\\algR$. Then $\\clodO_{\\F^k,\\algB}$ is uniformly generated by $k$-ary $(\\F^k,\\algB)$-minors.\n\\end{theorem}", + "theorem:unigen": "\\begin{theorem} \\label{theorem:unigen}\nLet $\\cloA$, $\\cloB$ be clones, $n \\geq 1$, and let $\\cloA_{const} = \\Clo(\\cloA \\cup \\{a\\}_{a\\in A})$ be the clone generated by $\\cloA$ and all constant operations over its set. Let $\\clodC$ a clonoid from $\\cloA_{const}$\nto $\\cloB$. Then the following are equivalent:\n\\begin{enumerate}\n\\item\\label{ite:th_unigen1} $\\clodC^{(n+1)}$ is $(n,\\cloA,\\cloB)$-UG\n\\item\\label{ite:th_unigen2} $\\clodC^{(k)}$ is $(n,\\cloA,\\cloB)$-UG for some $k>n$\n\\item\\label{ite:th_unigen3} $\\clodC$ is $(n,\\cloA,\\cloB)$-UG\n\\item\\label{ite:th_unigen4} $\\forall k,l \\in \\N$, every partial operation $I \\colon \\clodC^{(k)} \\to \\clodC^{(l)}$ \nthat is $(\\cloA,\\cloB)$-UR is $(n,\\cloA,\\cloB)$-UR.\n\\end{enumerate}\n\\end{theorem}", + "conjecture:main": "\\begin{conjecture} \\label{conjecture:main}\nLet $\\algA$ and $\\algB$ be finite modules. Then, there are only finitely many clonoids from $\\algA$ to $\\algB$, if and only if $\\algA$ and $\\algB$ are of coprime order.\n\\end{conjecture}", + "theorem:clonoidlattice": "\\begin{theorem} \\label{theorem:clonoidlattice}\nThe lattice of clonoids from $\\F^k$ to $\\algB$ is isomorphic to $\\prod_{i=0}^k \\Sub(\\algM_{i,k}(\\F,\\algB))$.\n\nMoreover, for $j\\leq k$, the sublattice $\\prod_{i=0}^j \\Sub(\\algM_{i,k}(\\F,\\algB))$ corresponds to the clonoids consisting of functions that are generated by their $j$-ary minors.\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 4708, + "pre_theorem_intro_text": "A \\emph{clone} is a set of finitary operations on a given set that contains all projections and is closed under functional composition. Clones are a fundamental object in universal algebra, as they describe the set of term operations of any algebraic structure. Thus, starting with Post's classification of clones on a two-element set \\cite{post-lattice}, there has been a rich history of structural results on finite algebras based on clone theory, see e.g. \\cite{szendrei-clones}. On the other hand, clones theory has also several applications in theoretical computer science. One of the most prominent examples is the study of polymorphism clones to determining the complexity of fixed template constraint satisfaction problems (see e.g. the survey \\cite{BKW-polymorphisms}), which lead to Bulatov's and Zhuk's independent proofs of the CSP dichotomy theorem \\cite{bulatov-dichotomy}, \\cite{zhuk-dichotomy-short,zhuk-dichotomy}.\n\nIn recent years, generalizations of clones such as \\emph{minions} and \\emph{clonoids} have received increasing attention. Minions are sets of finitary operations from a set $A$ to a (possibly different) set $B$ that are closed under permutation and identification of variables, as well as the addition of dummy variables. Research on minions gained much traction due to their application on (promise) constraint satisfaction, see e.g. the survey \\cite{BBKO-PCSP}. A \\emph{clonoid} from an algebra $\\alg{A}$ to an algebra $\\alg{B}$ is a minion that is closed under (pre-)composition with operations of $\\alg{A}$ and (post-)composition with the operations of $\\alg{B}$. While the term `clonoid' was first introduced by Aichinger and Mayr in~\\cite{AM-eqclasses}, the concept has appeared under various names earlier, e.g. in \\cite{ CF-clonoids}. \n\nDue to their weak structure, it is does not seem to be feasible to explicitly describe all minions between two given finite sets. A first hurdle is their sheer number: it was shown in \\cite{pippenger-minions} that there are already continuum many minions between two 2-element sets. However, depending on the expressiveness of the algebras $\\alg{A}$ and $\\alg{B}$, much more can be said about the clonoids between $\\alg{A}$ and $\\alg{B}$. Sparks showed in \\cite{sparks-clonoids}, that there are only finitely many clonoids from $\\alg{A}$ to $\\alg{B}$ if $\\alg{B}$ has a near-unanimity term, and at most countably many, if $\\alg{B}$ has an edge-term. However, also a rich source algebra $\\alg{A}$ can lead to finiteness results: As showed in \\cite{LS-discriminator}, every clonoids from an algebra $\\alg{A}$ with a discriminator term, to an algebra $\\alg{B}$ on the same set, gives rise to only finitely many clonoids.\n\nWe would also like to highlight a series of papers by Lehtonen and coauthors \\cite{LN-minorsclique,LS-discriminator, CL-stability, lehtonen-majorityclonoids, lehtonen-NUclonoids}, which culminated in \\cite{lehtonen-Booleanclonoids} to a complete characterization of all two-element algebras $(\\alg{A},\\alg{B})$, for which there are at most countable clonoids, together with a complete classification of clonoids in all such cases.\n\nIn this paper, we are interested in classifying clonoids, for which $\\alg{B}$ is a module. Such clonoids often appear naturally when studying the term operations of Mal'cev algebras with a central congruence; this connection was recently made more explicit by Peter Mayr's definition of the \\emph{difference clonoid}, see e.g. \\cite{kompatscher-SMP2nil}. In particular, 2-nilpotent algebras give rise to clonoids between modules $\\alg{A}$ and $\\alg{B}$.\n\nAs an example of an application let us let us mention the classification of extensions of square-free groups in \\cite{AM-Zpqextensions} and later \\cite{fioravanti-groupexpansions}, which heavily relied on the classification of certain clonoids.\n\nThere is already a series of classification results in the literature. The clonoids between groups of prime order $\\Z_p$ and $\\Z_q$ were completely classified by Kreinecker \\cite{kreinecker-zpclonoids} (in the case that $p=q$) and the first author, for $p \\neq q$ \\cite{fioravanti-clonoids}. Interestingly, in the latter case, there are only finitely many clonoids that are all generated by their unary functions (while for $p=q$ one obtains infinitely many). \n\nIn subsequent generalizations of these results, it was also always observed that there are only finitely many clonoids between certain modules $\\alg{A}$ and $\\alg{B}$ of coprime order - the most recent, and most general such finiteness result is by Mayr and Wynne \\cite{MW-clonoidsmodules}, and applies to all cases where $\\alg{A}$ has a distributive lattice of submodules. This leads us to the following Conjecture:", + "context": "In recent years, generalizations of clones such as \\emph{minions} and \\emph{clonoids} have received increasing attention. Minions are sets of finitary operations from a set $A$ to a (possibly different) set $B$ that are closed under permutation and identification of variables, as well as the addition of dummy variables. Research on minions gained much traction due to their application on (promise) constraint satisfaction, see e.g. the survey \\cite{BBKO-PCSP}. A \\emph{clonoid} from an algebra $\\alg{A}$ to an algebra $\\alg{B}$ is a minion that is closed under (pre-)composition with operations of $\\alg{A}$ and (post-)composition with the operations of $\\alg{B}$. While the term `clonoid' was first introduced by Aichinger and Mayr in~\\cite{AM-eqclasses}, the concept has appeared under various names earlier, e.g. in \\cite{ CF-clonoids}.\n\nDue to their weak structure, it is does not seem to be feasible to explicitly describe all minions between two given finite sets. A first hurdle is their sheer number: it was shown in \\cite{pippenger-minions} that there are already continuum many minions between two 2-element sets. However, depending on the expressiveness of the algebras $\\alg{A}$ and $\\alg{B}$, much more can be said about the clonoids between $\\alg{A}$ and $\\alg{B}$. Sparks showed in \\cite{sparks-clonoids}, that there are only finitely many clonoids from $\\alg{A}$ to $\\alg{B}$ if $\\alg{B}$ has a near-unanimity term, and at most countably many, if $\\alg{B}$ has an edge-term. However, also a rich source algebra $\\alg{A}$ can lead to finiteness results: As showed in \\cite{LS-discriminator}, every clonoids from an algebra $\\alg{A}$ with a discriminator term, to an algebra $\\alg{B}$ on the same set, gives rise to only finitely many clonoids.\n\nWe would also like to highlight a series of papers by Lehtonen and coauthors \\cite{LN-minorsclique,LS-discriminator, CL-stability, lehtonen-majorityclonoids, lehtonen-NUclonoids}, which culminated in \\cite{lehtonen-Booleanclonoids} to a complete characterization of all two-element algebras $(\\alg{A},\\alg{B})$, for which there are at most countable clonoids, together with a complete classification of clonoids in all such cases.\n\nIn this paper, we are interested in classifying clonoids, for which $\\alg{B}$ is a module. Such clonoids often appear naturally when studying the term operations of Mal'cev algebras with a central congruence; this connection was recently made more explicit by Peter Mayr's definition of the \\emph{difference clonoid}, see e.g. \\cite{kompatscher-SMP2nil}. In particular, 2-nilpotent algebras give rise to clonoids between modules $\\alg{A}$ and $\\alg{B}$.\n\nThere is already a series of classification results in the literature. The clonoids between groups of prime order $\\Z_p$ and $\\Z_q$ were completely classified by Kreinecker \\cite{kreinecker-zpclonoids} (in the case that $p=q$) and the first author, for $p \\neq q$ \\cite{fioravanti-clonoids}. Interestingly, in the latter case, there are only finitely many clonoids that are all generated by their unary functions (while for $p=q$ one obtains infinitely many).\n\nIn subsequent generalizations of these results, it was also always observed that there are only finitely many clonoids between certain modules $\\alg{A}$ and $\\alg{B}$ of coprime order - the most recent, and most general such finiteness result is by Mayr and Wynne \\cite{MW-clonoidsmodules}, and applies to all cases where $\\alg{A}$ has a distributive lattice of submodules. This leads us to the following Conjecture:", + "full_context": "In recent years, generalizations of clones such as \\emph{minions} and \\emph{clonoids} have received increasing attention. Minions are sets of finitary operations from a set $A$ to a (possibly different) set $B$ that are closed under permutation and identification of variables, as well as the addition of dummy variables. Research on minions gained much traction due to their application on (promise) constraint satisfaction, see e.g. the survey \\cite{BBKO-PCSP}. A \\emph{clonoid} from an algebra $\\alg{A}$ to an algebra $\\alg{B}$ is a minion that is closed under (pre-)composition with operations of $\\alg{A}$ and (post-)composition with the operations of $\\alg{B}$. While the term `clonoid' was first introduced by Aichinger and Mayr in~\\cite{AM-eqclasses}, the concept has appeared under various names earlier, e.g. in \\cite{ CF-clonoids}.\n\nDue to their weak structure, it is does not seem to be feasible to explicitly describe all minions between two given finite sets. A first hurdle is their sheer number: it was shown in \\cite{pippenger-minions} that there are already continuum many minions between two 2-element sets. However, depending on the expressiveness of the algebras $\\alg{A}$ and $\\alg{B}$, much more can be said about the clonoids between $\\alg{A}$ and $\\alg{B}$. Sparks showed in \\cite{sparks-clonoids}, that there are only finitely many clonoids from $\\alg{A}$ to $\\alg{B}$ if $\\alg{B}$ has a near-unanimity term, and at most countably many, if $\\alg{B}$ has an edge-term. However, also a rich source algebra $\\alg{A}$ can lead to finiteness results: As showed in \\cite{LS-discriminator}, every clonoids from an algebra $\\alg{A}$ with a discriminator term, to an algebra $\\alg{B}$ on the same set, gives rise to only finitely many clonoids.\n\nWe would also like to highlight a series of papers by Lehtonen and coauthors \\cite{LN-minorsclique,LS-discriminator, CL-stability, lehtonen-majorityclonoids, lehtonen-NUclonoids}, which culminated in \\cite{lehtonen-Booleanclonoids} to a complete characterization of all two-element algebras $(\\alg{A},\\alg{B})$, for which there are at most countable clonoids, together with a complete classification of clonoids in all such cases.\n\nIn this paper, we are interested in classifying clonoids, for which $\\alg{B}$ is a module. Such clonoids often appear naturally when studying the term operations of Mal'cev algebras with a central congruence; this connection was recently made more explicit by Peter Mayr's definition of the \\emph{difference clonoid}, see e.g. \\cite{kompatscher-SMP2nil}. In particular, 2-nilpotent algebras give rise to clonoids between modules $\\alg{A}$ and $\\alg{B}$.\n\nThere is already a series of classification results in the literature. The clonoids between groups of prime order $\\Z_p$ and $\\Z_q$ were completely classified by Kreinecker \\cite{kreinecker-zpclonoids} (in the case that $p=q$) and the first author, for $p \\neq q$ \\cite{fioravanti-clonoids}. Interestingly, in the latter case, there are only finitely many clonoids that are all generated by their unary functions (while for $p=q$ one obtains infinitely many).\n\nIn subsequent generalizations of these results, it was also always observed that there are only finitely many clonoids between certain modules $\\alg{A}$ and $\\alg{B}$ of coprime order - the most recent, and most general such finiteness result is by Mayr and Wynne \\cite{MW-clonoidsmodules}, and applies to all cases where $\\alg{A}$ has a distributive lattice of submodules. This leads us to the following Conjecture:\n\n\\begin{abstract}\nClonoids are sets of finitary operations between two algebraic structures that are closed under composition with their term operations on both sides. We conjecture that, for finite modules $\\algA$ and $\\algB$ there are only finitely many clonoids from $\\algA$ to $\\algB$ if and only if $\\algA$, $\\algB$ are of coprime order.\n\nWe confirm this conjecture for a broad class of modules $\\algA$. In particular we show that, if $\\algA$ is a finite $k$-dimensional vector space, then every clonoid from $\\alg A$ to a coprime module $\\algB$ is generated by its $k$-ary functions (and arity $k-1$ does not suffice). In order to prove this results, we investigate `uniform generation by $(\\algA,\\algB)$-minors', a general criterion, which we show to apply to several other existing classifications results. Based on our analysis, we further prove that the subpower membership problem of certain 2-nilpotent Mal'cev algebras is solvable in polynomial time.\n\\end{abstract}\n\nIn subsequent generalizations of these results, it was also always observed that there are only finitely many clonoids between certain modules $\\algA$ and $\\algB$ of coprime order - the most recent, and most general such finiteness result is by Mayr and Wynne \\cite{MW-clonoidsmodules}, and applies to all cases where $\\algA$ has a distributive lattice of submodules. This leads us to the following Conjecture:\n\nWe remark that the ``only if direction'' follows from \\cite[Theorem $1.3$]{MW-clonoidsmodules}, where the authors proved that for modules with a common divisor, one can always construct an infinite ascending chain of clonoids.\n\nThe first main contribution of our paper is to confirm Conjecture \\ref{conjecture:main} if $\\algA$ is a finite vector space. In the case where $\\algB$ is coprime to the vector space $\\algA$, we are moreover able to give an explicit description of the lattice of all clonoids (Theorem \\ref{theorem:clonoidlattice}). In particular, this answers \\cite[Question 1.2]{MW-clonoidsmodules}, which asked for a classification of clonoids, if $\\algA$ is the group $(\\Z_p)^2$, for some prime $p$. By combining our result with \\cite{MW-clonoidsmodules}, we are further able to confirm Conjecture \\ref{conjecture:main} for all modules $\\algA =\\F_1^{k_1}\\times \\cdots \\times \\F_n^{k_n} \\times \\alg D$ as $\\F_1\\times \\cdots \\times\\F_n \\times \\algR$-module, where $\\alg D$ is a distributive $\\algR$-module, and all $\\F_i$ are fields.\n\n\\begin{corollary} \\label{corollary:main2}\nLet $\\F^k$ be the $k$-dimensional vector space over a finite field $\\F$ and let $\\algB$ be a finite module of coprime order. Then\n\\begin{enumerate}\n\\item there are only finitely many clonoids from $\\F^k$ to $\\algB$.\n\\item every clonoid from $\\F^k$ to $\\algB$ is finitely related by an at most $|F|^{k\\times k}$-ary relation.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{corollary} \\label{corollary:mainresultproducts}\nLet $\\algA =\\F_1^{k_1}\\times \\cdots \\times \\F_n^{k_n} \\times \\alg D$ as $\\F_1\\times \\cdots \\times\\F_n \\times \\algR$-module, where all $\\F_i$ are finite fields, and $\\alg D$ is a finite distributive $\\algR$-module. Let $l$ be the nilpotence degree of the Jacobson radical of $\\algR$, and let $k = \\max(k_1,\\ldots,k_n,l)$. Further let $\\algB$ be an $\\algS$-module, such that $|A|$ is invertible in $\\algS$. Then\n\\begin{enumerate}\n\\item every clonoid from $\\alg A$ to $\\algB$ is generated by its $k$-ary part.\n\\item If $\\algB$ is finite then there are only finitely many clonoids from $\\algA$ to $\\algB$.\n\\end{enumerate}\n\\end{corollary}\n\nIn this paper, we studied clonoids between finite modules. We proved that there are only finitely many clonoids from a finite vector space $\\algA$ to a coprime module $\\algB$, which is also in accordance to Conjecture \\ref{conjecture:main}.\n\n\\begin{conjecture} \\label{conjecture:main}\nLet $\\algA$ and $\\algB$ be finite modules. Then, there are only finitely many clonoids from $\\algA$ to $\\algB$, if and only if $\\algA$ and $\\algB$ are of coprime order.\n\\end{conjecture}\n\n\\begin{theorem} \\label{theorem:clonoidlattice}\nThe lattice of clonoids from $\\F^k$ to $\\algB$ is isomorphic to $\\prod_{i=0}^k \\Sub(\\algM_{i,k}(\\F,\\algB))$.\n\nMoreover, for $j\\leq k$, the sublattice $\\prod_{i=0}^j \\Sub(\\algM_{i,k}(\\F,\\algB))$ corresponds to the clonoids consisting of functions that are generated by their $j$-ary minors.\n\\end{theorem}", + "post_theorem_intro_text_len": 4549, + "post_theorem_intro_text": "We remark that the ``only if direction'' follows from \\cite[Theorem $1.3$]{MW-clonoidsmodules}, where the authors proved that for modules with a common divisor, one can always construct an infinite ascending chain of clonoids.\n\n The first main contribution of our paper is to confirm Conjecture \\ref{conjecture:main} if $\\alg{A}$ is a finite vector space. In the case where $\\alg{B}$ is coprime to the vector space $\\alg{A}$, we are moreover able to give an explicit description of the lattice of all clonoids (Theorem \\ref{theorem:clonoidlattice}). In particular, this answers \\cite[Question 1.2]{MW-clonoidsmodules}, which asked for a classification of clonoids, if $\\alg{A}$ is the group $(\\Z_p)^2$, for some prime $p$. By combining our result with \\cite{MW-clonoidsmodules}, we are further able to confirm Conjecture \\ref{conjecture:main} for all modules $\\alg{A} =\\F_1^{k_1}\\times \\cdots \\times \\F_n^{k_n} \\times \\alg D$ as $\\F_1\\times \\cdots \\times\\F_n \\times \\alg{R}$-module, where $\\alg D$ is a distributive $\\alg{R}$-module, and all $\\F_i$ are fields.\n\nOur classification is based on proving that, in the case of a vector space $\\alg{A} = {\\mathbf F}^k$ and coprime $\\alg{B}$, the set of all operations from $A$ to $B$ is \\emph{uniformly generated} by $k$-ary minors. While this technique was already hinted at in \\cite{MW-clonoidsmodules}, we are going to discuss it in great detail in Section \\ref{sec:unifgen}, and argue that many of the existing clonoid classifications follow the same pattern. We furthermore show that the arity $k$ is optimal, i.e., not every clonoid from ${\\mathbf F}^k$ to $\\alg{B}$ is generated by their $k-1$-ary functions. This coincides with a lower bound on the arity of generators of the \\emph{full} clonoid between modules that we derive in Section \\ref{sec:lowerbound}, and was already pointed out in the Bachelor thesis by Jan Van\\v{e}\\v{c}ek \\cite{vanecek-thesis}.\n\nLast, let us mention that, clonoids between affine algebras were also (sometimes implicitly) used to discuss some computational problems of Mal'cev algebras. The fact that some clonoids between coprime modules are generated by unary functions, which allows for a representation of their elements by nice normal forms, was used in \\cite{KKK-CEQV2nil} to prove that checking polynomial identities in a given 2-nilpotent algebra can always be done in polynomial time. In \\cite{mayr-VLloop} and \\cite{KompatscherMayr2026} such normal forms were similarly used to find finite equational basis for some 2-nilpotent loops.\n\nWhile we are not going to discuss such syntactic aspects of clonoids, we will apply our results to the \\emph{subpower membership problem} of certain algebras. The subpower membership problem (SMP($\\alg{A}$)), for a fixed finite algebra $\\alg{A}$, is the computational problem whether a given partial operation $f\\colon A^n \\to A$ can be extended to a term operation of $\\alg{A}$. In \\cite{IMMVW-subpowers} it was asked, whether the subpower membership problem is always polynomial time solvable for algebras with few subpowers. This question, however remains unsolved even in the Mal'cev case. The easiest examples that are not covered by existing tractability results \\cite{mayr-SMP, BMS-SMP}, are 2-nilpotent algebras, for which it was shown in \\cite{kompatscher-SMP2nil} that the problem is polynomial time equivalent to a similar `interpolation problem' for their clonoids. In Section \\ref{sect:SMP} we are going to use our results, to prove that the subpower membership problem of a big class of 2-nilpotent algebras is in P.\n\n\\subsection*{Organization of the paper}\n\nThe paper is organized as follows. In Section~\\ref{sec:prel} we introduce the necessary concepts from universal algebra and fix some notation. In Section~\\ref{sec:unifgen} we develop a theory of `uniform generation' and `uniform representation' by minors, a concept introduced in~\\cite{MW-clonoidsmodules}. In particular we provide a purely combinatorial criterion for all $(\\alg{A},\\alg{B})$-clonoids to be generated by their $k$-ary functions (Theorem \\ref{theorem:unigen}).\n\nSection~\\ref{sec:mainres} presents the main results of the paper, Theorem \\ref{theorem:main} and discusses some consequences. In Section \\ref{sect:SMP} we apply our results to the subpower membership problem. In Sections~\\ref{sec:lowerbound} we provide a lower bound on the arity of the generators of the full clonoid between finite modules, which is sharp in our setting. Section~\\ref{sec:conclusion} is dedicated to future work and possible applications of our results.", + "sketch": "We remark that the ``only if direction'' follows from \\cite[Theorem $1.3$]{MW-clonoidsmodules}, where the authors proved that for modules with a common divisor, one can always construct an infinite ascending chain of clonoids.\n\nFor confirming Conjecture~\\ref{conjecture:main} when $\\alg{A}$ is a finite vector space and $\\alg{B}$ is coprime to $\\alg{A}$, the paper’s approach is described as follows: the classification is based on proving that, for a vector space $\\alg{A}={\\mathbf F}^k$ and coprime $\\alg{B}$, “the set of all operations from $A$ to $B$ is \\emph{uniformly generated} by $k$-ary minors.” The authors note this technique was “already hinted at in \\cite{MW-clonoidsmodules},” and they “discuss it in great detail in Section~\\ref{sec:unifgen},” including “a purely combinatorial criterion for all $(\\alg{A},\\alg{B})$-clonoids to be generated by their $k$-ary functions (Theorem~\\ref{theorem:unigen}).” They also establish an optimality statement for this method: “the arity $k$ is optimal, i.e., not every clonoid from ${\\mathbf F}^k$ to $\\alg{B}$ is generated by their $k-1$-ary functions,” and relate this to a “lower bound on the arity of generators of the \\emph{full} clonoid between modules” derived in Section~\\ref{sec:lowerbound}. Finally, they state that “by combining our result with \\cite{MW-clonoidsmodules}, we are further able to confirm Conjecture~\\ref{conjecture:main}” for broader classes of modules $\\alg{A}=\\F_1^{k_1}\\times\\cdots\\times\\F_n^{k_n}\\times\\alg D.", + "expanded_sketch": "We remark that the ``only if direction'' follows from \\cite[Theorem $1.3$]{MW-clonoidsmodules}, where the authors proved that for modules with a common divisor, one can always construct an infinite ascending chain of clonoids.\n\nFor confirming the main conjecture when $\\alg{A}$ is a finite vector space and $\\alg{B}$ is coprime to $\\alg{A}$, the paper’s approach is described as follows: the classification is based on proving that, for a vector space $\\alg{A}={\\mathbf F}^k$ and coprime $\\alg{B}$, “the set of all operations from $A$ to $B$ is \\emph{uniformly generated} by $k$-ary minors.” The authors note this technique was “already hinted at in \\cite{MW-clonoidsmodules},” and they “discuss it in great detail” later, including “a purely combinatorial criterion for all $(\\alg{A},\\alg{B})$-clonoids to be generated by their $k$-ary functions.” We first recall this criterion.\n\n\\begin{theorem} \\label{theorem:unigen}\nLet $\\cloA$, $\\cloB$ be clones, $n \\geq 1$, and let $\\cloA_{const} = \\Clo(\\cloA \\cup \\{a\\}_{a\\in A})$ be the clone generated by $\\cloA$ and all constant operations over its set. Let $\\clodC$ a clonoid from $\\cloA_{const}$\nto $\\cloB$. Then the following are equivalent:\n\\begin{enumerate}\n\\item\\label{ite:th_unigen1} $\\clodC^{(n+1)}$ is $(n,\\cloA,\\cloB)$-UG\n\\item\\label{ite:th_unigen2} $\\clodC^{(k)}$ is $(n,\\cloA,\\cloB)$-UG for some $k>n$\n\\item\\label{ite:th_unigen3} $\\clodC$ is $(n,\\cloA,\\cloB)$-UG\n\\item\\label{ite:th_unigen4} $\\forall k,l \\in \\N$, every partial operation $I \\colon \\clodC^{(k)} \\to \\clodC^{(l)}$ \nthat is $(\\cloA,\\cloB)$-UR is $(n,\\cloA,\\cloB)$-UR.\n\\end{enumerate}\n\\end{theorem}\n\nThey also establish an optimality statement for this method: “the arity $k$ is optimal, i.e., not every clonoid from ${\\mathbf F}^k$ to $\\alg{B}$ is generated by their $k-1$-ary functions,” and relate this to a “lower bound on the arity of generators of the \\emph{full} clonoid between modules��� derived later. Finally, they state that “by combining our result with \\cite{MW-clonoidsmodules}, we are further able to confirm the main conjecture” for broader classes of modules $\\alg{A}=\\F_1^{k_1}\\times\\cdots\\times\\F_n^{k_n}\\times\\alg D.", + "expanded_theorem": "\\label{conjecture:main}\nLet $\\alg{A}$ and $\\alg{B}$ be finite modules. Then, there are only finitely many clonoids from $\\alg{A}$ to $\\alg{B}$, if and only if $\\alg{A}$ and $\\alg{B}$ are of coprime order.", + "theorem_type": [ + "Biconditional or Equivalence", + "Existence" + ], + "mcq": { + "question": "Let \\(\\alg A\\) and \\(\\alg B\\) be finite modules. A clonoid from \\(\\alg A\\) to \\(\\alg B\\) is a set of finitary functions from powers of \\(A\\) to \\(B\\) that is closed under taking minors (permuting variables, identifying variables, and adding dummy variables), as well as under precomposition with term operations of \\(\\alg A\\) and postcomposition with term operations of \\(\\alg B\\). Two finite modules are said to be of coprime order when \\(\\gcd(|A|,|B|)=1\\). Which statement holds about the number of clonoids from \\(\\alg A\\) to \\(\\alg B\\)?", + "correct_choice": { + "label": "A", + "text": "There are only finitely many clonoids from \\(\\alg A\\) to \\(\\alg B\\) if and only if \\(\\alg A\\) and \\(\\alg B\\) are of coprime order." + }, + "choices": [ + { + "label": "B", + "text": "There are only finitely many clonoids from \\(\\alg A\\) to \\(\\alg B\\) whenever \\(\\alg A\\) and \\(\\alg B\\) have distinct orders." + }, + { + "label": "C", + "text": "If \\(\\alg A\\) and \\(\\alg B\\) are of coprime order, then there are only finitely many clonoids from \\(\\alg A\\) to \\(\\alg B\\)." + }, + { + "label": "D", + "text": "There are only finitely many clonoids from \\(\\alg A\\) to \\(\\alg B\\) if and only if every clonoid from \\(\\alg A\\) to \\(\\alg B\\) is generated by its unary functions." + }, + { + "label": "E", + "text": "There are only finitely many clonoids from \\(\\alg A\\) to \\(\\alg B\\) if and only if, for some integer \\(n\\ge 1\\) depending only on \\(|A|\\), every clonoid from \\(\\alg A\\) to \\(\\alg B\\) is generated by its \\(n\\)-ary functions." + } + ], + "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": "coprime-order hypothesis replaced by distinct-order condition", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "finiteness", + "tampered_component": "dropped the converse direction from the iff statement", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "optimal k-ary generation confused with unary generation", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "uniform generation bound made to depend only on |A| and asserted equivalent to finiteness for all modules", + "template_used": "quantifier_dependence" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem provides definitions and asks for the correct structural statement, but it does not explicitly reveal the coprime-order criterion or otherwise give away choice A." + }, + "TAS": { + "score": 1, + "justification": "The item is largely theorem-recognition: the correct option states the main characterization directly rather than requiring application in a new setting. Still, the alternatives introduce competing formulations, so it is not a pure verbatim restatement." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish the full iff statement from the weaker true one-way implication in C and from plausible but false variants. However, the question mainly tests recall of the theorem rather than substantial generative reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically targeted: B confuses coprime with distinct order, C is a weaker true statement, and D/E exploit realistic misunderstandings about generation by low-arity functions and quantifier dependence." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-recognition MCQ with no answer leakage and strong distractors, but it primarily tests recall rather than deeper generative mathematical reasoning." + } + }, + { + "id": "2602.04664v1", + "paper_link": "http://arxiv.org/abs/2602.04664v1", + "theorems_cnt": 3, + "theorem": { + "env_name": "theorem", + "content": "\\label{main theorem k+1-gon-bad}\n\tFor each $\\epsilon>0$ and integer $N > 0$, there exists a constant $C_{\\epsilon, N}$ such that for any positive numbers $\\tau_{1},\\tau_{2},\\ldots,\\tau_{k}$, we have\n\t\\[\n\t\t\\sum_{\\lambda_{i_{k+1}}\\geq (1+\\epsilon)(\\tau_1+\\tau_2+\\dots+\\tau_k)} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2 \\leq C_{\\epsilon, N}(\\tau_1+\\tau_2+\\dots+\\tau_k)^{-N},\n\t\\]\n\twhere $\\lambda_{i_j} \\leq \\tau_j$ for $1 \\leq j \\leq k$.", + "start_pos": 10297, + "end_pos": 10791, + "label": "main theorem k+1-gon-bad" + }, + "ref_dict": { + "main theorem k+1-good": "\\begin{theorem}\\label{main theorem k+1-good}\nLet $\\rho$ be a Schwartz function on $\\R^{k+1}$ with $\\int \\rho = 1$. Assume that\n\\[\n\t\\supp \\hat \\rho \\subset (-\\inj M, \\inj M)^{k+1}.\n\\]\nLet $\\Gamma$ be a $k$-good cone. Then for $\\tau = (\\tau_1, \\tau_2, \\dots, \\tau_{k+1}) \\in \\Gamma$,\n\\[\n\t\\rho * \\mu(\\tau) = (2\\pi)^{-kn} \\vol M \\vol F^{-1}(\\tau) + O(|\\tau|^{k(n-1)-2}),\n\\]\nwhere the constants implicit in the big-$O$ notation depend on $M$, $\\Gamma$, and $\\rho$ but not on $\\tau$.\n\\end{theorem}", + "harmonic expansion": "\\begin{equation}\\label{harmonic expansion}\n\te_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}", + "4": "\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{4}\n\\caption{An example of $\\tau\\notin\\Gamma_g$.}\n\\label{4}\n\\end{figure}", + "main theorem k+1-gon-bad": "\\begin{theorem}\\label{main theorem k+1-gon-bad}\n\tFor each $\\epsilon>0$ and integer $N > 0$, there exists a constant $C_{\\epsilon, N}$ such that for any positive numbers $\\tau_{1},\\tau_{2},\\ldots,\\tau_{k}$, we have\n\t\\[\n\t\t\\sum_{\\lambda_{i_{k+1}}\\geq (1+\\epsilon)(\\tau_1+\\tau_2+\\dots+\\tau_k)} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2 \\leq C_{\\epsilon, N}(\\tau_1+\\tau_2+\\dots+\\tau_k)^{-N},\n\t\\]\n\twhere $\\lambda_{i_j} \\leq \\tau_j$ for $1 \\leq j \\leq k$.\n\\end{theorem}", + "2": "\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{2}\n\\caption{A pentagon with side lengths $a_1, \\ldots, a_5$.}\n\\label{2}\n\\includegraphics[width=0.6\\textwidth]{3}\n\\caption{A self-intersecting pentagon with the same side lengths.}\n\\label{3}\n\\end{figure}", + "single sums": "\\begin{equation}\\label{single sums}\n\t\\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}", + "rewrite harmonic expansion": "\\begin{equation}\\label{rewrite harmonic expansion}\n\t\\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}", + "Scaling of F": "\\begin{proposition}\n [Scaling of $F$]\\label{Scaling of F} Let $\\Gamma$ be a $k$-good cone and fix $\\tau \\in \\Gamma$. Then $F^{-1}(\\tau) \\neq \\emptyset$ and $\\tau$ is a regular value of $F$, so $F^{-1}(\\tau)$ is a smooth compact submanifold of codimension $k+1$. Let\n\\[\nd:=k n-(k+1)=k(n-1)-1\n\\]\nbe its dimension. Then $0<\\operatorname{vol} F^{-1}(\\tau)<\\infty$ and, for every $r>0$,\n$$\n\\operatorname{vol} F^{-1}(r \\tau)=r^d \\operatorname{vol} F^{-1}(\\tau).\n$$\n\\end{proposition}", + "3": "\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{2}\n\\caption{A pentagon with side lengths $a_1, \\ldots, a_5$.}\n\\label{2}\n\\includegraphics[width=0.6\\textwidth]{3}\n\\caption{A self-intersecting pentagon with the same side lengths.}\n\\label{3}\n\\end{figure}" + }, + "pre_theorem_intro_text_len": 7505, + "pre_theorem_intro_text": "\\subsection{Background}\n\nLet $M$ be a compact Riemannian manifold without boundary, and $e_1, e_2, \\ldots$ be an orthonormal basis of Laplace-Beltrami eigenfunctions with\n\\[\n\t\\Delta_g e_j = -\\lambda_j^2 e_j,\n\\]\nwhere $\\lambda_j$ is the frequency of $e_j$.\n\nWe are interested in the general behavior of the multi-product of $k$ eigenfunctions $e_{i_1} e_{i_2} \\dots e_{i_k}$. To analyze this multi-product more conveniently, we express it as a sum of eigenfunctions with coefficients by a harmonic expansion\n\\begin{equation}\\label{harmonic expansion}\n\te_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\nThe classical question is to study the decay of the $L^2$ norm of $e_{i_1} e_{i_2} \\dots e_{i_k}$. If we set $\\lambda_{i_1}=\\lambda_{i_2}=\\dots =\\lambda_{i_k}=\\lambda_j$, then the problem reduces to study the $L^{2k}$ norm of the eigenfunction $e_{\\lambda_j}$. For this problem, Sogge provided a complete answer for $k\\geq 1$, see \\cite{MR930395}. \nHowever, for general eigenvalues $\\lambda_{i_1}, \\lambda_{i_2}, \\ldots, \\lambda_{i_k}$, the situation is more complicated. By orthogonality, we can write the $L^2$ norm of \\eqref{harmonic expansion} as follows:\n\\begin{equation}\\label{rewrite harmonic expansion}\n\t\\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\nThe decay of the terms on the right-hand side therefore determines the behavior of the left-hand side. We thus aim to study both the decay and how this decay is distributed in \\eqref{rewrite harmonic expansion}. This leads to the following concrete problem: under what conditions on the frequencies $\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}}$ do the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$ exhibit decay, and how is their $\\ell^2$-mass distributed as these frequencies vary?\n\nWhen $k=2$, as in many spectral problems, this question is closely connected to number theory in the automorphic setting. Sarnak showed that for fixed $j$ the coefficients $\\langle e_j^2, e_\\ell \\rangle$ decay exponentially in $\\lambda_\\ell$ \\cite{MR1277052}. Bernstein, Reznikov, Kr\\\"otz, and Stanton obtained the optimal estimate \\cite{MR1715328, MR2081437}. For the more general coefficients $\\langle e_i e_j, e_\\ell \\rangle$, a classical decay estimate in terms of the eigenvalues was given by Lu, Sogge, and Steinerberger \\cite{MR3997636}. More recently, the first author improved their estimate \\cite{MR4376455}. Given $\\epsilon > 0$ and an integer $N \\geq 1$, there exists a constant $C_{\\epsilon,N}$ for which\n\\begin{equation}\\label{Emmett conclusion}\n\t\\sum_{\\lambda_\\ell \\geq (2+\\epsilon)\\lambda} |\\langle e_i e_j, e_\\ell \\rangle|^2 \\leq C_{\\epsilon, N} \\lambda^{-N} \\qquad \\text{for all $i$, $j$ with $\\lambda_i, \\lambda_j \\leq \\lambda$.}\n\\end{equation}\n\nFurthermore, he proved that the $(2+\\epsilon)\\lambda$ factor is nearly optimal. He also obtained an asymptotic expansion for the non-rapidly decaying part \\cite{MR4376455}. Building on this work, we study the multi-product of $(k+1)$ eigenfunctions and obtain analogous results. \nOne of the most striking features of \\cite{MR4376455} is that it connects triple products of eigenfunctions to geometric configurations, namely triangles. We extend this perspective to general $k \\geq 2$, where the corresponding configurations are naturally described by polygons.\n\n\\subsection{Polygons}\n\nBased on \\eqref{rewrite harmonic expansion}, we study the sums\n\\begin{equation}\\label{single sums}\n\t\\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}\nfor given eigenvalues $a_1, a_2, \\dots, a_{k+1}$. These sums are independent of the choice of eigenbasis; hence their behavior is determined solely by the underlying manifold and its metric.\n\nThe main point of the present paper is to show that the sum \\eqref{single sums} can be estimated by counting configurations of $(k+1)$-gons with side lengths $a_1, a_2, \\dots, a_{k+1}$. To state our results, we first introduce the following definition concerning the side lengths of a $(k+1)$-gon.\n\n\\begin{definition}\\label{k+1-good and k+1-gon-bad}\nA vector $\\tau = (\\tau_1, \\ldots, \\tau_{k+1}) \\in (0,\\infty)^{k+1}$ is called $k$-good if it satisfies the following two conditions.\n\\begin{enumerate}\n\\item the $(k+1)$-gon inequality: for every $1 \\leq j \\leq k+1$,\n\\begin{equation}\\label{2.1}\n\\tau_j < \\sum_{\\ell \\neq j} \\tau_{\\ell}.\n\\end{equation}\n\\item the nondegeneracy condition: for every sign vector $\\varepsilon = (\\varepsilon_1, \\ldots, \\varepsilon_{k+1}) \\in \\{ \\pm 1\\}^{k+1}$,\n\\begin{equation}\\label{2.2}\n|\\varepsilon \\cdot \\tau| = \\left|\\sum_{j=1}^{k+1} \\varepsilon_j \\tau_j\\right| > 0.\n\\end{equation}\n\\end{enumerate}\nWe denote the set of $k$-good vectors by $\\Gamma_g$.\n\nA vector $\\tau = (\\tau_1, \\ldots, \\tau_{k+1}) \\in (0,\\infty)^{k+1}$ is called $k$-bad if there exists $j$, $1 \\leq j \\leq k+1$, such that\n\\begin{equation}\\label{2.3}\n\\tau_j > \\sum_{\\ell \\neq j} \\tau_{\\ell}.\n\\end{equation}\nWe denote the set of $k$-bad vectors by $\\Gamma_b$.\n\nA closed cone $\\Gamma \\subset (0,\\infty)^{k+1}$ is called $k$-good if $\\Gamma \\subset \\Gamma_g$, and it is called $k$-bad if $\\Gamma \\subset \\Gamma_b$.\n\\end{definition}\n\nThe set of $\\tau$ satisfying condition (1) consists of side-length data that can be realized by a $(k+1)$-gon in the plane, whereas the set of $k$-bad vectors contains no such points. The set of $k$-good vectors is obtained from the set of $\\tau$ satisfying condition (1) by removing those $\\tau$ for which a linear combination $|\\varepsilon \\cdot \\tau|$ vanishes for some $\\varepsilon \\in \\{ \\pm 1\\}^{k+1}$. For example, when $k=3$, the vector $(2,2,2,2)$ satisfies the $(k+1)$-gon inequality, but $(2,2,2,2)\\notin \\Gamma_g$ since it fails the nondegeneracy condition. Concretely, consider the degenerate quadrilateral in the plane formed by the vectors $(2,0), (-2,0), (2,0), (-2,0)$. Each side has length $2$, but the polygon is collinear and has zero area. In this case, $\\tau \\notin \\Gamma_g$, as illustrated in Figure \\ref{4}. When $k=2$, the nondegeneracy condition is automatic once one assumes condition (1), due to the rigidity of triangles. This is the primary reason it was not clear whether triple-product results could be generalized to higher multi-products, where such rigidity is absent. Our nondegeneracy condition is designed to fill this gap. As we will see in the proof, it ensures that several crucial steps go through and yields clean asymptotics.\n\n\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{4}\n\\caption{An example of $\\tau\\notin\\Gamma_g$.}\n\\label{4}\n\\end{figure}\nIn what follows, we establish a connection between \\eqref{single sums} and the count of configurations of $(k+1)$-gons using the spectral projection operator $\\chi_\\lambda$ and the theory of Fourier integral operators. Our argument also shows that \\eqref{single sums} decays rapidly in the regime where $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ fails to satisfy the $(k+1)$-gon inequality.\n\n\\subsection{Main results}\nOur first theorem shows that the rapidly decaying contribution to \\eqref{single sums} comes from frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ whose coordinates cannot occur as the side lengths of a $(k+1)$-gon.", + "context": "We are interested in the general behavior of the multi-product of $k$ eigenfunctions $e_{i_1} e_{i_2} \\dots e_{i_k}$. To analyze this multi-product more conveniently, we express it as a sum of eigenfunctions with coefficients by a harmonic expansion\n\\begin{equation}\\label{harmonic expansion}\n e_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\nThe classical question is to study the decay of the $L^2$ norm of $e_{i_1} e_{i_2} \\dots e_{i_k}$. If we set $\\lambda_{i_1}=\\lambda_{i_2}=\\dots =\\lambda_{i_k}=\\lambda_j$, then the problem reduces to study the $L^{2k}$ norm of the eigenfunction $e_{\\lambda_j}$. For this problem, Sogge provided a complete answer for $k\\geq 1$, see \\cite{MR930395}. \nHowever, for general eigenvalues $\\lambda_{i_1}, \\lambda_{i_2}, \\ldots, \\lambda_{i_k}$, the situation is more complicated. By orthogonality, we can write the $L^2$ norm of \\eqref{harmonic expansion} as follows:\n\\begin{equation}\\label{rewrite harmonic expansion}\n \\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\nThe decay of the terms on the right-hand side therefore determines the behavior of the left-hand side. We thus aim to study both the decay and how this decay is distributed in \\eqref{rewrite harmonic expansion}. This leads to the following concrete problem: under what conditions on the frequencies $\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}}$ do the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$ exhibit decay, and how is their $\\ell^2$-mass distributed as these frequencies vary?\n\nWhen $k=2$, as in many spectral problems, this question is closely connected to number theory in the automorphic setting. Sarnak showed that for fixed $j$ the coefficients $\\langle e_j^2, e_\\ell \\rangle$ decay exponentially in $\\lambda_\\ell$ \\cite{MR1277052}. Bernstein, Reznikov, Kr\\\"otz, and Stanton obtained the optimal estimate \\cite{MR1715328, MR2081437}. For the more general coefficients $\\langle e_i e_j, e_\\ell \\rangle$, a classical decay estimate in terms of the eigenvalues was given by Lu, Sogge, and Steinerberger \\cite{MR3997636}. More recently, the first author improved their estimate \\cite{MR4376455}. Given $\\epsilon > 0$ and an integer $N \\geq 1$, there exists a constant $C_{\\epsilon,N}$ for which\n\\begin{equation}\\label{Emmett conclusion}\n \\sum_{\\lambda_\\ell \\geq (2+\\epsilon)\\lambda} |\\langle e_i e_j, e_\\ell \\rangle|^2 \\leq C_{\\epsilon, N} \\lambda^{-N} \\qquad \\text{for all $i$, $j$ with $\\lambda_i, \\lambda_j \\leq \\lambda$.}\n\\end{equation}\n\nBased on \\eqref{rewrite harmonic expansion}, we study the sums\n\\begin{equation}\\label{single sums}\n \\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}\nfor given eigenvalues $a_1, a_2, \\dots, a_{k+1}$. These sums are independent of the choice of eigenbasis; hence their behavior is determined solely by the underlying manifold and its metric.\n\nA vector $\\tau = (\\tau_1, \\ldots, \\tau_{k+1}) \\in (0,\\infty)^{k+1}$ is called $k$-bad if there exists $j$, $1 \\leq j \\leq k+1$, such that\n\\begin{equation}\\label{2.3}\n\\tau_j > \\sum_{\\ell \\neq j} \\tau_{\\ell}.\n\\end{equation}\nWe denote the set of $k$-bad vectors by $\\Gamma_b$.\n\n\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{4}\n\\caption{An example of $\\tau\\notin\\Gamma_g$.}\n\\label{4}\n\\end{figure}\nIn what follows, we establish a connection between \\eqref{single sums} and the count of configurations of $(k+1)$-gons using the spectral projection operator $\\chi_\\lambda$ and the theory of Fourier integral operators. Our argument also shows that \\eqref{single sums} decays rapidly in the regime where $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ fails to satisfy the $(k+1)$-gon inequality.\n\n\\subsection{Main results}\nOur first theorem shows that the rapidly decaying contribution to \\eqref{single sums} comes from frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ whose coordinates cannot occur as the side lengths of a $(k+1)$-gon.\n\n\\begin{equation}\\label{harmonic expansion}\n\te_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\n\n\\begin{equation}\\label{rewrite harmonic expansion}\n\t\\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\n\n\\begin{equation}\\label{single sums}\n\t\\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}", + "full_context": "We are interested in the general behavior of the multi-product of $k$ eigenfunctions $e_{i_1} e_{i_2} \\dots e_{i_k}$. To analyze this multi-product more conveniently, we express it as a sum of eigenfunctions with coefficients by a harmonic expansion\n\\begin{equation}\\label{harmonic expansion}\n e_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\nThe classical question is to study the decay of the $L^2$ norm of $e_{i_1} e_{i_2} \\dots e_{i_k}$. If we set $\\lambda_{i_1}=\\lambda_{i_2}=\\dots =\\lambda_{i_k}=\\lambda_j$, then the problem reduces to study the $L^{2k}$ norm of the eigenfunction $e_{\\lambda_j}$. For this problem, Sogge provided a complete answer for $k\\geq 1$, see \\cite{MR930395}. \nHowever, for general eigenvalues $\\lambda_{i_1}, \\lambda_{i_2}, \\ldots, \\lambda_{i_k}$, the situation is more complicated. By orthogonality, we can write the $L^2$ norm of \\eqref{harmonic expansion} as follows:\n\\begin{equation}\\label{rewrite harmonic expansion}\n \\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\nThe decay of the terms on the right-hand side therefore determines the behavior of the left-hand side. We thus aim to study both the decay and how this decay is distributed in \\eqref{rewrite harmonic expansion}. This leads to the following concrete problem: under what conditions on the frequencies $\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}}$ do the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$ exhibit decay, and how is their $\\ell^2$-mass distributed as these frequencies vary?\n\nWhen $k=2$, as in many spectral problems, this question is closely connected to number theory in the automorphic setting. Sarnak showed that for fixed $j$ the coefficients $\\langle e_j^2, e_\\ell \\rangle$ decay exponentially in $\\lambda_\\ell$ \\cite{MR1277052}. Bernstein, Reznikov, Kr\\\"otz, and Stanton obtained the optimal estimate \\cite{MR1715328, MR2081437}. For the more general coefficients $\\langle e_i e_j, e_\\ell \\rangle$, a classical decay estimate in terms of the eigenvalues was given by Lu, Sogge, and Steinerberger \\cite{MR3997636}. More recently, the first author improved their estimate \\cite{MR4376455}. Given $\\epsilon > 0$ and an integer $N \\geq 1$, there exists a constant $C_{\\epsilon,N}$ for which\n\\begin{equation}\\label{Emmett conclusion}\n \\sum_{\\lambda_\\ell \\geq (2+\\epsilon)\\lambda} |\\langle e_i e_j, e_\\ell \\rangle|^2 \\leq C_{\\epsilon, N} \\lambda^{-N} \\qquad \\text{for all $i$, $j$ with $\\lambda_i, \\lambda_j \\leq \\lambda$.}\n\\end{equation}\n\nBased on \\eqref{rewrite harmonic expansion}, we study the sums\n\\begin{equation}\\label{single sums}\n \\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}\nfor given eigenvalues $a_1, a_2, \\dots, a_{k+1}$. These sums are independent of the choice of eigenbasis; hence their behavior is determined solely by the underlying manifold and its metric.\n\nA vector $\\tau = (\\tau_1, \\ldots, \\tau_{k+1}) \\in (0,\\infty)^{k+1}$ is called $k$-bad if there exists $j$, $1 \\leq j \\leq k+1$, such that\n\\begin{equation}\\label{2.3}\n\\tau_j > \\sum_{\\ell \\neq j} \\tau_{\\ell}.\n\\end{equation}\nWe denote the set of $k$-bad vectors by $\\Gamma_b$.\n\n\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{4}\n\\caption{An example of $\\tau\\notin\\Gamma_g$.}\n\\label{4}\n\\end{figure}\nIn what follows, we establish a connection between \\eqref{single sums} and the count of configurations of $(k+1)$-gons using the spectral projection operator $\\chi_\\lambda$ and the theory of Fourier integral operators. Our argument also shows that \\eqref{single sums} decays rapidly in the regime where $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ fails to satisfy the $(k+1)$-gon inequality.\n\n\\subsection{Main results}\nOur first theorem shows that the rapidly decaying contribution to \\eqref{single sums} comes from frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ whose coordinates cannot occur as the side lengths of a $(k+1)$-gon.\n\n\\begin{equation}\\label{harmonic expansion}\n\te_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\n\n\\begin{equation}\\label{rewrite harmonic expansion}\n\t\\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\n\n\\begin{equation}\\label{single sums}\n\t\\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}\n\nWe are interested in the general behavior of the multi-product of $k$ eigenfunctions $e_{i_1} e_{i_2} \\dots e_{i_k}$. To analyze this multi-product more conveniently, we express it as a sum of eigenfunctions with coefficients by a harmonic expansion\n\\begin{equation}\\label{harmonic expansion}\n e_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\nThe classical question is to study the decay of the $L^2$ norm of $e_{i_1} e_{i_2} \\dots e_{i_k}$. If we set $\\lambda_{i_1}=\\lambda_{i_2}=\\dots =\\lambda_{i_k}=\\lambda_j$, then the problem reduces to study the $L^{2k}$ norm of the eigenfunction $e_{\\lambda_j}$. For this problem, Sogge provided a complete answer for $k\\geq 1$, see \\cite{MR930395}. \nHowever, for general eigenvalues $\\lambda_{i_1}, \\lambda_{i_2}, \\ldots, \\lambda_{i_k}$, the situation is more complicated. By orthogonality, we can write the $L^2$ norm of \\eqref{harmonic expansion} as follows:\n\\begin{equation}\\label{rewrite harmonic expansion}\n \\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\nThe decay of the terms on the right-hand side therefore determines the behavior of the left-hand side. We thus aim to study both the decay and how this decay is distributed in \\eqref{rewrite harmonic expansion}. This leads to the following concrete problem: under what conditions on the frequencies $\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}}$ do the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$ exhibit decay, and how is their $\\ell^2$-mass distributed as these frequencies vary?\n\n\\subsection{Main results}\nOur first theorem shows that the rapidly decaying contribution to \\eqref{single sums} comes from frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ whose coordinates cannot occur as the side lengths of a $(k+1)$-gon.\n\nThe hypothesis $\\lambda_{i_{k+1}} \\geq (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$ forces the $(k+1)$-tuple $(\\tau_1,\\ldots,\\tau_k,\\lambda_{i_{k+1}})$ to be $k$-bad, since the last entry dominates the sum of the others. In particular, any $(\\lambda_{i_1},\\ldots,\\lambda_{i_k},\\lambda_{i_{k+1}})$ with $\\lambda_{i_j} \\leq \\tau_j$ for $1 \\leq j \\leq k$ is also $k$-bad.\n\nFor clarity, we do not organize the left-hand side by summing over all tuples $(\\lambda_{i_1},\\ldots,\\lambda_{i_{k+1}})$ lying in a specified region of $(0,\\infty)^{k+1}$. Instead, we fix upper bounds $\\tau_j$ for $\\lambda_{i_j}$, $1 \\leq j \\leq k$, and then estimate the tail sum over those $i_{k+1}$ with $\\lambda_{i_{k+1}} \\geq (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$. The proof uses the method of non-stationary phase.\nWe record a flat-torus example in Section \\ref{sec:flat-torus-example} showing that the frequency threshold in Theorem \\ref{main theorem k+1-gon-bad} is essentially sharp.\n\nOur second and main theorem concerns the contribution from $k$-good vectors $\\tau \\in (0,\\infty)^{k+1}$.\n Following \\cite{MR4376455}, we introduce the joint spectral measure\n\\begin{equation}\\label{joint spectral measure}\n \\mu = \\sum_{i_1,i_2,\\dots,i_{k+1}} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2 \\, \\delta_{(\\lambda_{i_1},\\lambda_{i_2},\\dots,\\lambda_{i_{k+1}})}\n\\end{equation}\non $\\R^{k+1}$, where $\\delta$ denotes the Dirac measure. Thus $\\mu$ is a weighted sum of point masses, with weights given by the squared magnitudes of the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$. Our second theorem describes the mass of $\\mu$ carried by frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ lying in $k$-good cones.\n\nNext we verify that $\\tau$ is a regular value of $F$. Fix $\\xi=(\\xi_1,\\ldots,\\xi_k)\\in F^{-1}(\\tau)$ and set\n\\[\n\\omega_j=\\frac{\\xi_j}{|\\xi_j|}\\in S^{n-1}\\ (1\\le j\\le k),\\qquad\n\\omega_{k+1}=\\frac{\\xi_1+\\cdots+\\xi_k}{|\\xi_1+\\cdots+\\xi_k|}\\in S^{n-1}.\n\\]\nFor $h=(h_1,\\ldots,h_k)\\in\\R^{kn}$ we compute\n\\[\nd(|\\xi_j|)[h]=\\langle \\omega_j,h_j\\rangle,\\qquad\nd(|\\xi_1+\\cdots+\\xi_k|)[h]=\\Bigl\\langle \\omega_{k+1},\\sum_{j=1}^k h_j\\Bigr\\rangle.\n\\]\nIf $\\mathrm{rank}(dF_\\xi)0$ and set\n\\[\nT_m := (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)+m ,\\qquad m\\in\\N.\n\\]\nAssume that $\\lambda_{i_j}\\le \\tau_j$ for $1\\le j\\le k$ and that\n\\[\n\\lambda_{i_{k+1}}\\in [T_m,T_m+1].\n\\]\nThen for every integer $N\\ge 0$ there exists a constant $C_{\\epsilon,N}$ such that\n\\[\n\\bigl|\\langle e_{i_1} e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|\n\\le C_{\\epsilon,N}\\, T_m^{-N}.\n\\]\n\\end{lemma}\n\n\\begin{proof}[Proof of Theorem \\ref{main theorem k+1-gon-bad}]\nDecompose the tail into unit spectral windows:\n\\begin{equation*}\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)}\n\\bigl|\\langle e_{i_1}\\cdots e_{i_k}, e_{i_{k+1}}\\rangle\\bigr|^2\n=\\sum_{m=0}^\\infty \\sum_{\\lambda_{i_{k+1}}\\in[T_m,T_m+1]}\n\\bigl|\\langle e_{i_1}\\cdots e_{i_k}, e_{i_{k+1}}\\rangle\\bigr|^2 \n\\end{equation*}\nBy the Weyl law,\n\\[\n\\#\\{\\lambda_{i_{k+1}}\\in[T_m,T_m+1]\\}\\ \\lesssim\\ T_m^{n-1}.\n\\]\nLemma \\ref{lem:kplus1-eigenfunction-product} gives\n\\[\n\\sup_{\\lambda_{i_{k+1}}\\in[T_m,T_m+1]}\n\\bigl|\\langle e_{i_1}\\cdots e_{i_k}, e_{i_{k+1}}\\rangle\\bigr|^2\n\\ \\lesssim_{\\epsilon,N}\\ T_m^{-2N}.\n\\]\nHence\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)}\n\\bigl|\\langle e_{i_1}\\cdots e_{i_k}, e_{i_{k+1}}\\rangle\\bigr|^2\n\\ \\lesssim_{\\epsilon,N}\\ \\sum_{m=0}^\\infty T_m^{\\,n-1-2N}.\n\\]\nGiven $N'>0$, choose $N$ so that $2N-(n-1) > N'$. Then the last sum is bounded by\n$C_{\\epsilon,N'}(\\tau_1+\\cdots+\\tau_k)^{-N'}$, which proves the theorem.\n\\end{proof}", + "post_theorem_intro_text_len": 6544, + "post_theorem_intro_text": "The hypothesis $\\lambda_{i_{k+1}} \\geq (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$ forces the $(k+1)$-tuple $(\\tau_1,\\ldots,\\tau_k,\\lambda_{i_{k+1}})$ to be $k$-bad, since the last entry dominates the sum of the others. In particular, any $(\\lambda_{i_1},\\ldots,\\lambda_{i_k},\\lambda_{i_{k+1}})$ with $\\lambda_{i_j} \\leq \\tau_j$ for $1 \\leq j \\leq k$ is also $k$-bad.\n\nFor clarity, we do not organize the left-hand side by summing over all tuples $(\\lambda_{i_1},\\ldots,\\lambda_{i_{k+1}})$ lying in a specified region of $(0,\\infty)^{k+1}$. Instead, we fix upper bounds $\\tau_j$ for $\\lambda_{i_j}$, $1 \\leq j \\leq k$, and then estimate the tail sum over those $i_{k+1}$ with $\\lambda_{i_{k+1}} \\geq (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$. The proof uses the method of non-stationary phase.\nWe record a flat-torus example in Section \\ref{sec:flat-torus-example} showing that the frequency threshold in Theorem \\ref{main theorem k+1-gon-bad} is essentially sharp.\n\nOur second and main theorem concerns the contribution from $k$-good vectors $\\tau \\in (0,\\infty)^{k+1}$.\n Following \\cite{MR4376455}, we introduce the joint spectral measure\n\\begin{equation}\\label{joint spectral measure}\n\t\\mu = \\sum_{i_1,i_2,\\dots,i_{k+1}} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2 \\, \\delta_{(\\lambda_{i_1},\\lambda_{i_2},\\dots,\\lambda_{i_{k+1}})}\n\\end{equation}\non ${\\mathbb R}^{k+1}$, where $\\delta$ denotes the Dirac measure. Thus $\\mu$ is a weighted sum of point masses, with weights given by the squared magnitudes of the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$. Our second theorem describes the mass of $\\mu$ carried by frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ lying in $k$-good cones.\n\nBefore stating the result, we introduce some terminology by recalling the definition and basic properties of the Leray density. Let $F : {\\mathbb R}^n \\to {\\mathbb R}^d$ be a smooth function, and let $y_0 \\in {\\mathbb R}^d$ be such that the differential $dF$ has full rank at every point of the level set $F^{-1}(y_0)$. By the inverse function theorem, $F^{-1}(y_0)$ is a smooth $(n-d)$-dimensional submanifold. Choose local coordinates $(z,z')$ in a neighborhood of a point of $F^{-1}(y_0)$ so that $z$ parametrizes $F^{-1}(y_0)$ and $\\det(\\partial F/\\partial z') \\neq 0$. The Leray density on $F^{-1}(y_0)$ is defined by\n\\begin{equation}\\label{Leray density definition}\n\td_F = \\left|\\det \\frac{\\partial F}{\\partial z'}\\right|^{-1} \\, dz,\n\\end{equation}\nand is independent of the choice of complementary coordinates $z'$. The corresponding Leray volume is\n\\[\n\t\\operatorname{vol} F^{-1}(y_0) = \\int_{F^{-1}(y_0)} d_F.\n\\]\n\nGiven a continuous function $f$ defined on a neighborhood of $F^{-1}(y_0)$, the integral of $f$ over $F^{-1}(y_0)$ with respect to the Leray measure can be expressed using the Dirac distribution as\n\\[\n\t\\int_{F^{-1}(y_0)} f \\, d_F = \\int_{{\\mathbb R}^n} f(x) \\delta(F(x) - y_0) \\, dx.\n\\]\n\nFor the purposes of this paper, we set $F : {\\mathbb R}^{kn} \\to {\\mathbb R}^{k+1}$ by\n\\begin{equation}\\label{def F}\n\tF(\\xi_1,\\xi_2,\\dots,\\xi_k) = \\bigl(|\\xi_1|,|\\xi_2|,\\dots,|\\xi_k|,\\ |\\xi_1 + \\cdots + \\xi_k|\\bigr).\n\\end{equation}\nThen the level set $F^{-1}(\\tau)$ consists of $k$-tuples $(\\xi_1,\\xi_2,\\dots,\\xi_k)$ satisfying\n\\[\n|\\xi_j|=\\tau_j \\ \\ (1\\le j\\le k)\n\\qquad \\text{and} \\qquad\n|\\xi_1+\\cdots+\\xi_k|=\\tau_{k+1}.\n\\]\nSuch a $k$-tuple determines a closed $(k+1)$-gon in ${\\mathbb R}^n$ with side lengths $\\tau_1,\\tau_2,\\dots,\\tau_{k+1}$, where the final side is the closing vector $-(\\xi_1+\\cdots+\\xi_k)$.\nThe next theorem provides a description of the concentration of $\\mu$ in $k$-good cones.\n\n\\begin{theorem}\\label{main theorem k+1-good}\nLet $\\rho$ be a Schwartz function on ${\\mathbb R}^{k+1}$ with $\\int \\rho = 1$. Assume that\n\\[\n\t\\operatorname{supp} \\hat \\rho \\subset (-\\operatorname{inj} M, \\operatorname{inj} M)^{k+1}.\n\\]\nLet $\\Gamma$ be a $k$-good cone. Then for $\\tau = (\\tau_1, \\tau_2, \\dots, \\tau_{k+1}) \\in \\Gamma$,\n\\[\n\t\\rho * \\mu(\\tau) = (2\\pi)^{-kn} \\operatorname{vol} M \\operatorname{vol} F^{-1}(\\tau) + O(|\\tau|^{k(n-1)-2}),\n\\]\nwhere the constants implicit in the big-$O$ notation depend on $M$, $\\Gamma$, and $\\rho$ but not on $\\tau$.\n\\end{theorem}\n\nSince $\\rho$ is rapidly decaying and $\\operatorname{supp} \\hat \\rho$ is contained in $(-\\operatorname{inj} M, \\operatorname{inj} M)^{k+1}$, the convolution $\\rho * \\mu$ provides a smoothed version of $\\mu$ that captures the mass of $\\mu$ near $\\tau$. In particular, Theorem \\ref{main theorem k+1-good} gives an asymptotic description of this smoothed local mass for $\\tau \\in \\Gamma$. Next, we determine the growth rate of $\\operatorname{vol} F^{-1}(\\tau)$ as $|\\tau| \\to \\infty$. When $k=2$, this quantity is related to the area of the corresponding triangle \\cite{MR4376455}. For $k>2$, the shape of a $(k+1)$-gon with fixed side lengths is not uniquely determined (Figures \\ref{2} and \\ref{3}), so it is not sufficient to consider only its planar area. We prove in Proposition \\ref{Scaling of F} that, under our assumptions, $\\operatorname{vol} F^{-1}(\\tau)$ is of order $|\\tau|^{k(n-1)-1}$, so the remainder term $O(|\\tau|^{k(n-1)-2})$ is indeed lower order. As in \\cite{MR4376455}, the leading term can be interpreted in terms of the volume of the configuration space of $(k+1)$-gons with side lengths prescribed by $\\tau$.\n\\subsection*{Organization}\nIn Section~2 we analyze the configuration space that appears in the main term of Theorem~\\ref{main theorem k+1-good}, and in particular prove the scaling law for the relevant fiber volume. Section~3 contains the proof of Theorem~\\ref{main theorem k+1-gon-bad}, establishing rapid decay of the high-frequency tail in the regime where the $(k+1)$-gon inequality fails, via spectral projectors and a non-stationary phase argument. Section~4 proves Theorem~\\ref{main theorem k+1-good} by expressing the Fourier transform of the joint spectral measure in terms of Fourier integral operators and computing the associated symbolic data using clean composition and half-density calculus. Finally, Section~\\ref{sec:flat-torus-example} presents a flat-torus example showing that the frequency cutoff in Theorem~\\ref{main theorem k+1-gon-bad} is essentially sharp.\n\n\\subsection*{Acknowledgement} This project is supported by the National Key R\\&D Program of China under Grant No. 2022YFA1007200, the Natural Science Foundation of China under Grant No. 12571107, and the Zhejiang Provincial Natural Science Foundation of China under Grant No. LR25A010001.", + "sketch": "The introduction explains that the condition $\\lambda_{i_{k+1}} \\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$ “forces the $(k+1)$-tuple $(\\tau_1,\\ldots,\\tau_k,\\lambda_{i_{k+1}})$ to be $k$-bad, since the last entry dominates the sum of the others,” hence any $(\\lambda_{i_1},\\ldots,\\lambda_{i_k},\\lambda_{i_{k+1}})$ with $\\lambda_{i_j}\\le \\tau_j$ is also $k$-bad.\n\nRather than summing over a region in $(0,\\infty)^{k+1}$, the argument is organized by “fix[ing] upper bounds $\\tau_j$ for $\\lambda_{i_j}$, $1\\le j\\le k$, and then estimat[ing] the tail sum over those $i_{k+1}$ with $\\lambda_{i_{k+1}} \\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$.” The proof of Theorem~\\ref{main theorem k+1-gon-bad} then “uses the method of non-stationary phase,” and (per the Organization paragraph) establishes “rapid decay of the high-frequency tail in the regime where the $(k+1)$-gon inequality fails, via spectral projectors and a non-stationary phase argument.” A flat-torus example is recorded showing the frequency threshold is “essentially sharp.”", + "expanded_sketch": "The introduction explains that the condition $\\lambda_{i_{k+1}} \\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$ “forces the $(k+1)$-tuple $(\\tau_1,\\ldots,\\tau_k,\\lambda_{i_{k+1}})$ to be $k$-bad, since the last entry dominates the sum of the others,” hence any $(\\lambda_{i_1},\\ldots,\\lambda_{i_k},\\lambda_{i_{k+1}})$ with $\\lambda_{i_j}\\le \\tau_j$ is also $k$-bad.\n\nRather than summing over a region in $(0,\\infty)^{k+1}$, the argument is organized by “fix[ing] upper bounds $\\tau_j$ for $\\lambda_{i_j}$, $1\\le j\\le k$, and then estimat[ing] the tail sum over those $i_{k+1}$ with $\\lambda_{i_{k+1}} \\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$.” To prove the main theorem, the proof then “uses the method of non-stationary phase,” and (per the Organization paragraph) establishes “rapid decay of the high-frequency tail in the regime where the $(k+1)$-gon inequality fails, via spectral projectors and a non-stationary phase argument.” A flat-torus example is recorded showing the frequency threshold is “essentially sharp.”", + "expanded_theorem": "\\label{main theorem k+1-gon-bad}\n\tFor each $\\epsilon>0$ and integer $N > 0$, there exists a constant $C_{\\epsilon, N}$ such that for any positive numbers $\\tau_{1},\\tau_{2},\\ldots,\\tau_{k}$, we have\n\t\\[\n\t\t\\sum_{\\lambda_{i_{k+1}}\\geq (1+\\epsilon)(\\tau_1+\\tau_2+\\dots+\\tau_k)} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2 \\leq C_{\\epsilon, N}(\\tau_1+\\tau_2+\\dots+\\tau_k)^{-N},\n\t\\]\n\twhere $\\lambda_{i_j} \\leq \\tau_j$ for $1 \\leq j \\leq k$.", + "theorem_type": [ + "Existential–Universal", + "Inequality or Bound" + ], + "mcq": { + "question": "Let \\(\\{e_i\\}\\) be an \\(L^2\\)-orthonormal family of eigenfunctions with corresponding eigenvalues (frequencies) \\(\\lambda_i\\), and write \\(\\langle f,g\\rangle\\) for the \\(L^2\\) inner product. Fix an integer \\(k\\ge 1\\). For positive numbers \\(\\tau_1,\\dots,\\tau_k\\), suppose the indices \\(i_1,\\dots,i_k\\) satisfy \\(\\lambda_{i_j}\\le \\tau_j\\) for each \\(1\\le j\\le k\\). Under these assumptions, which quantitative estimate holds for the tail of the harmonic expansion coefficients of the product \\(e_{i_1}e_{i_2}\\cdots e_{i_k}\\)?", + "correct_choice": { + "label": "A", + "text": "For every \\(\\epsilon>0\\) and every integer \\(N>0\\), there exists a constant \\(C_{\\epsilon,N}\\) such that\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)(\\tau_1+\\tau_2+\\cdots+\\tau_k)} \\bigl|\\langle e_{i_1}e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|^2\n\\le C_{\\epsilon,N}(\\tau_1+\\tau_2+\\cdots+\\tau_k)^{-N}.\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "For every \\(\\epsilon>0\\) and every integer \\(N>0\\), there exists a constant \\(C_{\\epsilon,N}\\) such that\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (\\tau_1+\\tau_2+\\cdots+\\tau_k)+\\epsilon} \\bigl|\\langle e_{i_1}e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|^2\n\\le C_{\\epsilon,N}(\\tau_1+\\tau_2+\\cdots+\\tau_k)^{-N}.\n\\]" + }, + { + "label": "C", + "text": "For every \\(\\epsilon>0\\), there exists a constant \\(C_{\\epsilon}\\) such that\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)(\\tau_1+\\tau_2+\\cdots+\\tau_k)} \\bigl|\\langle e_{i_1}e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|^2\n\\le C_{\\epsilon}.\n\\]" + }, + { + "label": "D", + "text": "There exists an integer \\(N>0\\) such that for every \\(\\epsilon>0\\) there is a constant \\(C_{\\epsilon,N}\\) with\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)(\\tau_1+\\tau_2+\\cdots+\\tau_k)} \\bigl|\\langle e_{i_1}e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|^2\n\\le C_{\\epsilon,N}(\\tau_1+\\tau_2+\\cdots+\\tau_k)^{-N}\n\\]\nfor all positive \\(\\tau_1,\\dots,\\tau_k\\) and all indices \\(i_1,\\dots,i_k\\) with \\(\\lambda_{i_j}\\le \\tau_j\\)." + }, + { + "label": "E", + "text": "For every \\(\\epsilon>0\\) and every integer \\(N>0\\), there exists a constant \\(C_{\\epsilon,N}\\) such that\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)\\max\\{\\tau_1,\\tau_2,\\dots,\\tau_k\\}} \\bigl|\\langle e_{i_1}e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|^2\n\\le C_{\\epsilon,N}(\\tau_1+\\tau_2+\\cdots+\\tau_k)^{-N}.\n\\]" + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "non-stationary phase", + "tampered_component": "multiplicative tail threshold replaced by additive offset", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "finiteness", + "tampered_component": "dropped rapid-decay power saving in \\((\\tau_1+\\cdots+\\tau_k)\\)", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "finiteness", + "tampered_component": "quantifier order on N weakened from arbitrary N to existence of one N", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "geometric_construction", + "tampered_component": "sum-of-sides bad-region threshold replaced by max-side threshold", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly reveal the correct option. It asks for the valid uniform decay statement without embedding the key threshold, decay rate, or quantifier structure of the correct answer." + }, + "TAS": { + "score": 1, + "justification": "The item is largely theorem-recognition: it asks which precise statement holds, and one option is essentially the target theorem. However, it is not purely tautological because the alternatives vary in meaningful ways (threshold type, decay strength, tail norm, and quantifiers)." + }, + "GPS": { + "score": 1, + "justification": "Answering requires moderate reasoning about logical strength and mathematical form: distinguishing multiplicative versus additive thresholds, rapid decay versus mere boundedness, and sum versus supremum. Still, it mainly tests recall/recognition of the theorem statement rather than generating a conclusion from premises." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong and mathematically plausible. They reflect realistic failure modes: weakening the decay, altering quantifier order, replacing an \\(\\ell^2\\)-tail bound by a pointwise supremum, or using an incorrect boundary threshold." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with high-quality distractors and no answer leakage, but it leans more toward precise statement recognition than genuinely generative reasoning." + } + }, + { + "id": "2602.04664v1", + "paper_link": "http://arxiv.org/abs/2602.04664v1", + "theorems_cnt": 3, + "theorem": { + "env_name": "theorem", + "content": "\\label{main theorem k+1-gon-bad}\n\tFor each $\\epsilon>0$ and integer $N > 0$, there exists a constant $C_{\\epsilon, N}$ such that for any positive numbers $\\tau_{1},\\tau_{2},\\ldots,\\tau_{k}$, we have\n\t\\[\n\t\t\\sum_{\\lambda_{i_{k+1}}\\geq (1+\\epsilon)(\\tau_1+\\tau_2+\\dots+\\tau_k)} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2 \\leq C_{\\epsilon, N}(\\tau_1+\\tau_2+\\dots+\\tau_k)^{-N},\n\t\\]\n\twhere $\\lambda_{i_j} \\leq \\tau_j$ for $1 \\leq j \\leq k$.", + "start_pos": 10297, + "end_pos": 10791, + "label": "main theorem k+1-gon-bad" + }, + "ref_dict": { + "main theorem k+1-good": "\\begin{theorem}\\label{main theorem k+1-good}\nLet $\\rho$ be a Schwartz function on $\\R^{k+1}$ with $\\int \\rho = 1$. Assume that\n\\[\n\t\\supp \\hat \\rho \\subset (-\\inj M, \\inj M)^{k+1}.\n\\]\nLet $\\Gamma$ be a $k$-good cone. Then for $\\tau = (\\tau_1, \\tau_2, \\dots, \\tau_{k+1}) \\in \\Gamma$,\n\\[\n\t\\rho * \\mu(\\tau) = (2\\pi)^{-kn} \\vol M \\vol F^{-1}(\\tau) + O(|\\tau|^{k(n-1)-2}),\n\\]\nwhere the constants implicit in the big-$O$ notation depend on $M$, $\\Gamma$, and $\\rho$ but not on $\\tau$.\n\\end{theorem}", + "harmonic expansion": "\\begin{equation}\\label{harmonic expansion}\n\te_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}", + "4": "\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{4}\n\\caption{An example of $\\tau\\notin\\Gamma_g$.}\n\\label{4}\n\\end{figure}", + "main theorem k+1-gon-bad": "\\begin{theorem}\\label{main theorem k+1-gon-bad}\n\tFor each $\\epsilon>0$ and integer $N > 0$, there exists a constant $C_{\\epsilon, N}$ such that for any positive numbers $\\tau_{1},\\tau_{2},\\ldots,\\tau_{k}$, we have\n\t\\[\n\t\t\\sum_{\\lambda_{i_{k+1}}\\geq (1+\\epsilon)(\\tau_1+\\tau_2+\\dots+\\tau_k)} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2 \\leq C_{\\epsilon, N}(\\tau_1+\\tau_2+\\dots+\\tau_k)^{-N},\n\t\\]\n\twhere $\\lambda_{i_j} \\leq \\tau_j$ for $1 \\leq j \\leq k$.\n\\end{theorem}", + "2": "\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{2}\n\\caption{A pentagon with side lengths $a_1, \\ldots, a_5$.}\n\\label{2}\n\\includegraphics[width=0.6\\textwidth]{3}\n\\caption{A self-intersecting pentagon with the same side lengths.}\n\\label{3}\n\\end{figure}", + "single sums": "\\begin{equation}\\label{single sums}\n\t\\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}", + "rewrite harmonic expansion": "\\begin{equation}\\label{rewrite harmonic expansion}\n\t\\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}", + "Scaling of F": "\\begin{proposition}\n [Scaling of $F$]\\label{Scaling of F} Let $\\Gamma$ be a $k$-good cone and fix $\\tau \\in \\Gamma$. Then $F^{-1}(\\tau) \\neq \\emptyset$ and $\\tau$ is a regular value of $F$, so $F^{-1}(\\tau)$ is a smooth compact submanifold of codimension $k+1$. Let\n\\[\nd:=k n-(k+1)=k(n-1)-1\n\\]\nbe its dimension. Then $0<\\operatorname{vol} F^{-1}(\\tau)<\\infty$ and, for every $r>0$,\n$$\n\\operatorname{vol} F^{-1}(r \\tau)=r^d \\operatorname{vol} F^{-1}(\\tau).\n$$\n\\end{proposition}", + "3": "\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{2}\n\\caption{A pentagon with side lengths $a_1, \\ldots, a_5$.}\n\\label{2}\n\\includegraphics[width=0.6\\textwidth]{3}\n\\caption{A self-intersecting pentagon with the same side lengths.}\n\\label{3}\n\\end{figure}" + }, + "pre_theorem_intro_text_len": 7505, + "pre_theorem_intro_text": "\\subsection{Background}\n\nLet $M$ be a compact Riemannian manifold without boundary, and $e_1, e_2, \\ldots$ be an orthonormal basis of Laplace-Beltrami eigenfunctions with\n\\[\n\t\\Delta_g e_j = -\\lambda_j^2 e_j,\n\\]\nwhere $\\lambda_j$ is the frequency of $e_j$.\n\nWe are interested in the general behavior of the multi-product of $k$ eigenfunctions $e_{i_1} e_{i_2} \\dots e_{i_k}$. To analyze this multi-product more conveniently, we express it as a sum of eigenfunctions with coefficients by a harmonic expansion\n\\begin{equation}\\label{harmonic expansion}\n\te_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\nThe classical question is to study the decay of the $L^2$ norm of $e_{i_1} e_{i_2} \\dots e_{i_k}$. If we set $\\lambda_{i_1}=\\lambda_{i_2}=\\dots =\\lambda_{i_k}=\\lambda_j$, then the problem reduces to study the $L^{2k}$ norm of the eigenfunction $e_{\\lambda_j}$. For this problem, Sogge provided a complete answer for $k\\geq 1$, see \\cite{MR930395}. \nHowever, for general eigenvalues $\\lambda_{i_1}, \\lambda_{i_2}, \\ldots, \\lambda_{i_k}$, the situation is more complicated. By orthogonality, we can write the $L^2$ norm of \\eqref{harmonic expansion} as follows:\n\\begin{equation}\\label{rewrite harmonic expansion}\n\t\\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\nThe decay of the terms on the right-hand side therefore determines the behavior of the left-hand side. We thus aim to study both the decay and how this decay is distributed in \\eqref{rewrite harmonic expansion}. This leads to the following concrete problem: under what conditions on the frequencies $\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}}$ do the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$ exhibit decay, and how is their $\\ell^2$-mass distributed as these frequencies vary?\n\nWhen $k=2$, as in many spectral problems, this question is closely connected to number theory in the automorphic setting. Sarnak showed that for fixed $j$ the coefficients $\\langle e_j^2, e_\\ell \\rangle$ decay exponentially in $\\lambda_\\ell$ \\cite{MR1277052}. Bernstein, Reznikov, Kr\\\"otz, and Stanton obtained the optimal estimate \\cite{MR1715328, MR2081437}. For the more general coefficients $\\langle e_i e_j, e_\\ell \\rangle$, a classical decay estimate in terms of the eigenvalues was given by Lu, Sogge, and Steinerberger \\cite{MR3997636}. More recently, the first author improved their estimate \\cite{MR4376455}. Given $\\epsilon > 0$ and an integer $N \\geq 1$, there exists a constant $C_{\\epsilon,N}$ for which\n\\begin{equation}\\label{Emmett conclusion}\n\t\\sum_{\\lambda_\\ell \\geq (2+\\epsilon)\\lambda} |\\langle e_i e_j, e_\\ell \\rangle|^2 \\leq C_{\\epsilon, N} \\lambda^{-N} \\qquad \\text{for all $i$, $j$ with $\\lambda_i, \\lambda_j \\leq \\lambda$.}\n\\end{equation}\n\nFurthermore, he proved that the $(2+\\epsilon)\\lambda$ factor is nearly optimal. He also obtained an asymptotic expansion for the non-rapidly decaying part \\cite{MR4376455}. Building on this work, we study the multi-product of $(k+1)$ eigenfunctions and obtain analogous results. \nOne of the most striking features of \\cite{MR4376455} is that it connects triple products of eigenfunctions to geometric configurations, namely triangles. We extend this perspective to general $k \\geq 2$, where the corresponding configurations are naturally described by polygons.\n\n\\subsection{Polygons}\n\nBased on \\eqref{rewrite harmonic expansion}, we study the sums\n\\begin{equation}\\label{single sums}\n\t\\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}\nfor given eigenvalues $a_1, a_2, \\dots, a_{k+1}$. These sums are independent of the choice of eigenbasis; hence their behavior is determined solely by the underlying manifold and its metric.\n\nThe main point of the present paper is to show that the sum \\eqref{single sums} can be estimated by counting configurations of $(k+1)$-gons with side lengths $a_1, a_2, \\dots, a_{k+1}$. To state our results, we first introduce the following definition concerning the side lengths of a $(k+1)$-gon.\n\n\\begin{definition}\\label{k+1-good and k+1-gon-bad}\nA vector $\\tau = (\\tau_1, \\ldots, \\tau_{k+1}) \\in (0,\\infty)^{k+1}$ is called $k$-good if it satisfies the following two conditions.\n\\begin{enumerate}\n\\item the $(k+1)$-gon inequality: for every $1 \\leq j \\leq k+1$,\n\\begin{equation}\\label{2.1}\n\\tau_j < \\sum_{\\ell \\neq j} \\tau_{\\ell}.\n\\end{equation}\n\\item the nondegeneracy condition: for every sign vector $\\varepsilon = (\\varepsilon_1, \\ldots, \\varepsilon_{k+1}) \\in \\{ \\pm 1\\}^{k+1}$,\n\\begin{equation}\\label{2.2}\n|\\varepsilon \\cdot \\tau| = \\left|\\sum_{j=1}^{k+1} \\varepsilon_j \\tau_j\\right| > 0.\n\\end{equation}\n\\end{enumerate}\nWe denote the set of $k$-good vectors by $\\Gamma_g$.\n\nA vector $\\tau = (\\tau_1, \\ldots, \\tau_{k+1}) \\in (0,\\infty)^{k+1}$ is called $k$-bad if there exists $j$, $1 \\leq j \\leq k+1$, such that\n\\begin{equation}\\label{2.3}\n\\tau_j > \\sum_{\\ell \\neq j} \\tau_{\\ell}.\n\\end{equation}\nWe denote the set of $k$-bad vectors by $\\Gamma_b$.\n\nA closed cone $\\Gamma \\subset (0,\\infty)^{k+1}$ is called $k$-good if $\\Gamma \\subset \\Gamma_g$, and it is called $k$-bad if $\\Gamma \\subset \\Gamma_b$.\n\\end{definition}\n\nThe set of $\\tau$ satisfying condition (1) consists of side-length data that can be realized by a $(k+1)$-gon in the plane, whereas the set of $k$-bad vectors contains no such points. The set of $k$-good vectors is obtained from the set of $\\tau$ satisfying condition (1) by removing those $\\tau$ for which a linear combination $|\\varepsilon \\cdot \\tau|$ vanishes for some $\\varepsilon \\in \\{ \\pm 1\\}^{k+1}$. For example, when $k=3$, the vector $(2,2,2,2)$ satisfies the $(k+1)$-gon inequality, but $(2,2,2,2)\\notin \\Gamma_g$ since it fails the nondegeneracy condition. Concretely, consider the degenerate quadrilateral in the plane formed by the vectors $(2,0), (-2,0), (2,0), (-2,0)$. Each side has length $2$, but the polygon is collinear and has zero area. In this case, $\\tau \\notin \\Gamma_g$, as illustrated in Figure \\ref{4}. When $k=2$, the nondegeneracy condition is automatic once one assumes condition (1), due to the rigidity of triangles. This is the primary reason it was not clear whether triple-product results could be generalized to higher multi-products, where such rigidity is absent. Our nondegeneracy condition is designed to fill this gap. As we will see in the proof, it ensures that several crucial steps go through and yields clean asymptotics.\n\n\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{4}\n\\caption{An example of $\\tau\\notin\\Gamma_g$.}\n\\label{4}\n\\end{figure}\nIn what follows, we establish a connection between \\eqref{single sums} and the count of configurations of $(k+1)$-gons using the spectral projection operator $\\chi_\\lambda$ and the theory of Fourier integral operators. Our argument also shows that \\eqref{single sums} decays rapidly in the regime where $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ fails to satisfy the $(k+1)$-gon inequality.\n\n\\subsection{Main results}\nOur first theorem shows that the rapidly decaying contribution to \\eqref{single sums} comes from frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ whose coordinates cannot occur as the side lengths of a $(k+1)$-gon.", + "context": "We are interested in the general behavior of the multi-product of $k$ eigenfunctions $e_{i_1} e_{i_2} \\dots e_{i_k}$. To analyze this multi-product more conveniently, we express it as a sum of eigenfunctions with coefficients by a harmonic expansion\n\\begin{equation}\\label{harmonic expansion}\n e_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\nThe classical question is to study the decay of the $L^2$ norm of $e_{i_1} e_{i_2} \\dots e_{i_k}$. If we set $\\lambda_{i_1}=\\lambda_{i_2}=\\dots =\\lambda_{i_k}=\\lambda_j$, then the problem reduces to study the $L^{2k}$ norm of the eigenfunction $e_{\\lambda_j}$. For this problem, Sogge provided a complete answer for $k\\geq 1$, see \\cite{MR930395}. \nHowever, for general eigenvalues $\\lambda_{i_1}, \\lambda_{i_2}, \\ldots, \\lambda_{i_k}$, the situation is more complicated. By orthogonality, we can write the $L^2$ norm of \\eqref{harmonic expansion} as follows:\n\\begin{equation}\\label{rewrite harmonic expansion}\n \\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\nThe decay of the terms on the right-hand side therefore determines the behavior of the left-hand side. We thus aim to study both the decay and how this decay is distributed in \\eqref{rewrite harmonic expansion}. This leads to the following concrete problem: under what conditions on the frequencies $\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}}$ do the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$ exhibit decay, and how is their $\\ell^2$-mass distributed as these frequencies vary?\n\nWhen $k=2$, as in many spectral problems, this question is closely connected to number theory in the automorphic setting. Sarnak showed that for fixed $j$ the coefficients $\\langle e_j^2, e_\\ell \\rangle$ decay exponentially in $\\lambda_\\ell$ \\cite{MR1277052}. Bernstein, Reznikov, Kr\\\"otz, and Stanton obtained the optimal estimate \\cite{MR1715328, MR2081437}. For the more general coefficients $\\langle e_i e_j, e_\\ell \\rangle$, a classical decay estimate in terms of the eigenvalues was given by Lu, Sogge, and Steinerberger \\cite{MR3997636}. More recently, the first author improved their estimate \\cite{MR4376455}. Given $\\epsilon > 0$ and an integer $N \\geq 1$, there exists a constant $C_{\\epsilon,N}$ for which\n\\begin{equation}\\label{Emmett conclusion}\n \\sum_{\\lambda_\\ell \\geq (2+\\epsilon)\\lambda} |\\langle e_i e_j, e_\\ell \\rangle|^2 \\leq C_{\\epsilon, N} \\lambda^{-N} \\qquad \\text{for all $i$, $j$ with $\\lambda_i, \\lambda_j \\leq \\lambda$.}\n\\end{equation}\n\nBased on \\eqref{rewrite harmonic expansion}, we study the sums\n\\begin{equation}\\label{single sums}\n \\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}\nfor given eigenvalues $a_1, a_2, \\dots, a_{k+1}$. These sums are independent of the choice of eigenbasis; hence their behavior is determined solely by the underlying manifold and its metric.\n\nA vector $\\tau = (\\tau_1, \\ldots, \\tau_{k+1}) \\in (0,\\infty)^{k+1}$ is called $k$-bad if there exists $j$, $1 \\leq j \\leq k+1$, such that\n\\begin{equation}\\label{2.3}\n\\tau_j > \\sum_{\\ell \\neq j} \\tau_{\\ell}.\n\\end{equation}\nWe denote the set of $k$-bad vectors by $\\Gamma_b$.\n\n\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{4}\n\\caption{An example of $\\tau\\notin\\Gamma_g$.}\n\\label{4}\n\\end{figure}\nIn what follows, we establish a connection between \\eqref{single sums} and the count of configurations of $(k+1)$-gons using the spectral projection operator $\\chi_\\lambda$ and the theory of Fourier integral operators. Our argument also shows that \\eqref{single sums} decays rapidly in the regime where $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ fails to satisfy the $(k+1)$-gon inequality.\n\n\\subsection{Main results}\nOur first theorem shows that the rapidly decaying contribution to \\eqref{single sums} comes from frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ whose coordinates cannot occur as the side lengths of a $(k+1)$-gon.\n\n\\begin{equation}\\label{harmonic expansion}\n\te_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\n\n\\begin{equation}\\label{rewrite harmonic expansion}\n\t\\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\n\n\\begin{equation}\\label{single sums}\n\t\\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}", + "full_context": "We are interested in the general behavior of the multi-product of $k$ eigenfunctions $e_{i_1} e_{i_2} \\dots e_{i_k}$. To analyze this multi-product more conveniently, we express it as a sum of eigenfunctions with coefficients by a harmonic expansion\n\\begin{equation}\\label{harmonic expansion}\n e_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\nThe classical question is to study the decay of the $L^2$ norm of $e_{i_1} e_{i_2} \\dots e_{i_k}$. If we set $\\lambda_{i_1}=\\lambda_{i_2}=\\dots =\\lambda_{i_k}=\\lambda_j$, then the problem reduces to study the $L^{2k}$ norm of the eigenfunction $e_{\\lambda_j}$. For this problem, Sogge provided a complete answer for $k\\geq 1$, see \\cite{MR930395}. \nHowever, for general eigenvalues $\\lambda_{i_1}, \\lambda_{i_2}, \\ldots, \\lambda_{i_k}$, the situation is more complicated. By orthogonality, we can write the $L^2$ norm of \\eqref{harmonic expansion} as follows:\n\\begin{equation}\\label{rewrite harmonic expansion}\n \\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\nThe decay of the terms on the right-hand side therefore determines the behavior of the left-hand side. We thus aim to study both the decay and how this decay is distributed in \\eqref{rewrite harmonic expansion}. This leads to the following concrete problem: under what conditions on the frequencies $\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}}$ do the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$ exhibit decay, and how is their $\\ell^2$-mass distributed as these frequencies vary?\n\nWhen $k=2$, as in many spectral problems, this question is closely connected to number theory in the automorphic setting. Sarnak showed that for fixed $j$ the coefficients $\\langle e_j^2, e_\\ell \\rangle$ decay exponentially in $\\lambda_\\ell$ \\cite{MR1277052}. Bernstein, Reznikov, Kr\\\"otz, and Stanton obtained the optimal estimate \\cite{MR1715328, MR2081437}. For the more general coefficients $\\langle e_i e_j, e_\\ell \\rangle$, a classical decay estimate in terms of the eigenvalues was given by Lu, Sogge, and Steinerberger \\cite{MR3997636}. More recently, the first author improved their estimate \\cite{MR4376455}. Given $\\epsilon > 0$ and an integer $N \\geq 1$, there exists a constant $C_{\\epsilon,N}$ for which\n\\begin{equation}\\label{Emmett conclusion}\n \\sum_{\\lambda_\\ell \\geq (2+\\epsilon)\\lambda} |\\langle e_i e_j, e_\\ell \\rangle|^2 \\leq C_{\\epsilon, N} \\lambda^{-N} \\qquad \\text{for all $i$, $j$ with $\\lambda_i, \\lambda_j \\leq \\lambda$.}\n\\end{equation}\n\nBased on \\eqref{rewrite harmonic expansion}, we study the sums\n\\begin{equation}\\label{single sums}\n \\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}\nfor given eigenvalues $a_1, a_2, \\dots, a_{k+1}$. These sums are independent of the choice of eigenbasis; hence their behavior is determined solely by the underlying manifold and its metric.\n\nA vector $\\tau = (\\tau_1, \\ldots, \\tau_{k+1}) \\in (0,\\infty)^{k+1}$ is called $k$-bad if there exists $j$, $1 \\leq j \\leq k+1$, such that\n\\begin{equation}\\label{2.3}\n\\tau_j > \\sum_{\\ell \\neq j} \\tau_{\\ell}.\n\\end{equation}\nWe denote the set of $k$-bad vectors by $\\Gamma_b$.\n\n\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{4}\n\\caption{An example of $\\tau\\notin\\Gamma_g$.}\n\\label{4}\n\\end{figure}\nIn what follows, we establish a connection between \\eqref{single sums} and the count of configurations of $(k+1)$-gons using the spectral projection operator $\\chi_\\lambda$ and the theory of Fourier integral operators. Our argument also shows that \\eqref{single sums} decays rapidly in the regime where $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ fails to satisfy the $(k+1)$-gon inequality.\n\n\\subsection{Main results}\nOur first theorem shows that the rapidly decaying contribution to \\eqref{single sums} comes from frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ whose coordinates cannot occur as the side lengths of a $(k+1)$-gon.\n\n\\begin{equation}\\label{harmonic expansion}\n\te_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\n\n\\begin{equation}\\label{rewrite harmonic expansion}\n\t\\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\n\n\\begin{equation}\\label{single sums}\n\t\\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}\n\nWe are interested in the general behavior of the multi-product of $k$ eigenfunctions $e_{i_1} e_{i_2} \\dots e_{i_k}$. To analyze this multi-product more conveniently, we express it as a sum of eigenfunctions with coefficients by a harmonic expansion\n\\begin{equation}\\label{harmonic expansion}\n e_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\nThe classical question is to study the decay of the $L^2$ norm of $e_{i_1} e_{i_2} \\dots e_{i_k}$. If we set $\\lambda_{i_1}=\\lambda_{i_2}=\\dots =\\lambda_{i_k}=\\lambda_j$, then the problem reduces to study the $L^{2k}$ norm of the eigenfunction $e_{\\lambda_j}$. For this problem, Sogge provided a complete answer for $k\\geq 1$, see \\cite{MR930395}. \nHowever, for general eigenvalues $\\lambda_{i_1}, \\lambda_{i_2}, \\ldots, \\lambda_{i_k}$, the situation is more complicated. By orthogonality, we can write the $L^2$ norm of \\eqref{harmonic expansion} as follows:\n\\begin{equation}\\label{rewrite harmonic expansion}\n \\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\nThe decay of the terms on the right-hand side therefore determines the behavior of the left-hand side. We thus aim to study both the decay and how this decay is distributed in \\eqref{rewrite harmonic expansion}. This leads to the following concrete problem: under what conditions on the frequencies $\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}}$ do the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$ exhibit decay, and how is their $\\ell^2$-mass distributed as these frequencies vary?\n\n\\subsection{Main results}\nOur first theorem shows that the rapidly decaying contribution to \\eqref{single sums} comes from frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ whose coordinates cannot occur as the side lengths of a $(k+1)$-gon.\n\nThe hypothesis $\\lambda_{i_{k+1}} \\geq (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$ forces the $(k+1)$-tuple $(\\tau_1,\\ldots,\\tau_k,\\lambda_{i_{k+1}})$ to be $k$-bad, since the last entry dominates the sum of the others. In particular, any $(\\lambda_{i_1},\\ldots,\\lambda_{i_k},\\lambda_{i_{k+1}})$ with $\\lambda_{i_j} \\leq \\tau_j$ for $1 \\leq j \\leq k$ is also $k$-bad.\n\nFor clarity, we do not organize the left-hand side by summing over all tuples $(\\lambda_{i_1},\\ldots,\\lambda_{i_{k+1}})$ lying in a specified region of $(0,\\infty)^{k+1}$. Instead, we fix upper bounds $\\tau_j$ for $\\lambda_{i_j}$, $1 \\leq j \\leq k$, and then estimate the tail sum over those $i_{k+1}$ with $\\lambda_{i_{k+1}} \\geq (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$. The proof uses the method of non-stationary phase.\nWe record a flat-torus example in Section \\ref{sec:flat-torus-example} showing that the frequency threshold in Theorem \\ref{main theorem k+1-gon-bad} is essentially sharp.\n\nOur second and main theorem concerns the contribution from $k$-good vectors $\\tau \\in (0,\\infty)^{k+1}$.\n Following \\cite{MR4376455}, we introduce the joint spectral measure\n\\begin{equation}\\label{joint spectral measure}\n \\mu = \\sum_{i_1,i_2,\\dots,i_{k+1}} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2 \\, \\delta_{(\\lambda_{i_1},\\lambda_{i_2},\\dots,\\lambda_{i_{k+1}})}\n\\end{equation}\non $\\R^{k+1}$, where $\\delta$ denotes the Dirac measure. Thus $\\mu$ is a weighted sum of point masses, with weights given by the squared magnitudes of the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$. Our second theorem describes the mass of $\\mu$ carried by frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ lying in $k$-good cones.\n\nNext we verify that $\\tau$ is a regular value of $F$. Fix $\\xi=(\\xi_1,\\ldots,\\xi_k)\\in F^{-1}(\\tau)$ and set\n\\[\n\\omega_j=\\frac{\\xi_j}{|\\xi_j|}\\in S^{n-1}\\ (1\\le j\\le k),\\qquad\n\\omega_{k+1}=\\frac{\\xi_1+\\cdots+\\xi_k}{|\\xi_1+\\cdots+\\xi_k|}\\in S^{n-1}.\n\\]\nFor $h=(h_1,\\ldots,h_k)\\in\\R^{kn}$ we compute\n\\[\nd(|\\xi_j|)[h]=\\langle \\omega_j,h_j\\rangle,\\qquad\nd(|\\xi_1+\\cdots+\\xi_k|)[h]=\\Bigl\\langle \\omega_{k+1},\\sum_{j=1}^k h_j\\Bigr\\rangle.\n\\]\nIf $\\mathrm{rank}(dF_\\xi)0$ and set\n\\[\nT_m := (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)+m ,\\qquad m\\in\\N.\n\\]\nAssume that $\\lambda_{i_j}\\le \\tau_j$ for $1\\le j\\le k$ and that\n\\[\n\\lambda_{i_{k+1}}\\in [T_m,T_m+1].\n\\]\nThen for every integer $N\\ge 0$ there exists a constant $C_{\\epsilon,N}$ such that\n\\[\n\\bigl|\\langle e_{i_1} e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|\n\\le C_{\\epsilon,N}\\, T_m^{-N}.\n\\]\n\\end{lemma}\n\n\\begin{proof}[Proof of Theorem \\ref{main theorem k+1-gon-bad}]\nDecompose the tail into unit spectral windows:\n\\begin{equation*}\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)}\n\\bigl|\\langle e_{i_1}\\cdots e_{i_k}, e_{i_{k+1}}\\rangle\\bigr|^2\n=\\sum_{m=0}^\\infty \\sum_{\\lambda_{i_{k+1}}\\in[T_m,T_m+1]}\n\\bigl|\\langle e_{i_1}\\cdots e_{i_k}, e_{i_{k+1}}\\rangle\\bigr|^2 \n\\end{equation*}\nBy the Weyl law,\n\\[\n\\#\\{\\lambda_{i_{k+1}}\\in[T_m,T_m+1]\\}\\ \\lesssim\\ T_m^{n-1}.\n\\]\nLemma \\ref{lem:kplus1-eigenfunction-product} gives\n\\[\n\\sup_{\\lambda_{i_{k+1}}\\in[T_m,T_m+1]}\n\\bigl|\\langle e_{i_1}\\cdots e_{i_k}, e_{i_{k+1}}\\rangle\\bigr|^2\n\\ \\lesssim_{\\epsilon,N}\\ T_m^{-2N}.\n\\]\nHence\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)}\n\\bigl|\\langle e_{i_1}\\cdots e_{i_k}, e_{i_{k+1}}\\rangle\\bigr|^2\n\\ \\lesssim_{\\epsilon,N}\\ \\sum_{m=0}^\\infty T_m^{\\,n-1-2N}.\n\\]\nGiven $N'>0$, choose $N$ so that $2N-(n-1) > N'$. Then the last sum is bounded by\n$C_{\\epsilon,N'}(\\tau_1+\\cdots+\\tau_k)^{-N'}$, which proves the theorem.\n\\end{proof}", + "post_theorem_intro_text_len": 6544, + "post_theorem_intro_text": "The hypothesis $\\lambda_{i_{k+1}} \\geq (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$ forces the $(k+1)$-tuple $(\\tau_1,\\ldots,\\tau_k,\\lambda_{i_{k+1}})$ to be $k$-bad, since the last entry dominates the sum of the others. In particular, any $(\\lambda_{i_1},\\ldots,\\lambda_{i_k},\\lambda_{i_{k+1}})$ with $\\lambda_{i_j} \\leq \\tau_j$ for $1 \\leq j \\leq k$ is also $k$-bad.\n\nFor clarity, we do not organize the left-hand side by summing over all tuples $(\\lambda_{i_1},\\ldots,\\lambda_{i_{k+1}})$ lying in a specified region of $(0,\\infty)^{k+1}$. Instead, we fix upper bounds $\\tau_j$ for $\\lambda_{i_j}$, $1 \\leq j \\leq k$, and then estimate the tail sum over those $i_{k+1}$ with $\\lambda_{i_{k+1}} \\geq (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$. The proof uses the method of non-stationary phase.\nWe record a flat-torus example in Section \\ref{sec:flat-torus-example} showing that the frequency threshold in Theorem \\ref{main theorem k+1-gon-bad} is essentially sharp.\n\nOur second and main theorem concerns the contribution from $k$-good vectors $\\tau \\in (0,\\infty)^{k+1}$.\n Following \\cite{MR4376455}, we introduce the joint spectral measure\n\\begin{equation}\\label{joint spectral measure}\n\t\\mu = \\sum_{i_1,i_2,\\dots,i_{k+1}} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2 \\, \\delta_{(\\lambda_{i_1},\\lambda_{i_2},\\dots,\\lambda_{i_{k+1}})}\n\\end{equation}\non ${\\mathbb R}^{k+1}$, where $\\delta$ denotes the Dirac measure. Thus $\\mu$ is a weighted sum of point masses, with weights given by the squared magnitudes of the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$. Our second theorem describes the mass of $\\mu$ carried by frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ lying in $k$-good cones.\n\nBefore stating the result, we introduce some terminology by recalling the definition and basic properties of the Leray density. Let $F : {\\mathbb R}^n \\to {\\mathbb R}^d$ be a smooth function, and let $y_0 \\in {\\mathbb R}^d$ be such that the differential $dF$ has full rank at every point of the level set $F^{-1}(y_0)$. By the inverse function theorem, $F^{-1}(y_0)$ is a smooth $(n-d)$-dimensional submanifold. Choose local coordinates $(z,z')$ in a neighborhood of a point of $F^{-1}(y_0)$ so that $z$ parametrizes $F^{-1}(y_0)$ and $\\det(\\partial F/\\partial z') \\neq 0$. The Leray density on $F^{-1}(y_0)$ is defined by\n\\begin{equation}\\label{Leray density definition}\n\td_F = \\left|\\det \\frac{\\partial F}{\\partial z'}\\right|^{-1} \\, dz,\n\\end{equation}\nand is independent of the choice of complementary coordinates $z'$. The corresponding Leray volume is\n\\[\n\t\\operatorname{vol} F^{-1}(y_0) = \\int_{F^{-1}(y_0)} d_F.\n\\]\n\nGiven a continuous function $f$ defined on a neighborhood of $F^{-1}(y_0)$, the integral of $f$ over $F^{-1}(y_0)$ with respect to the Leray measure can be expressed using the Dirac distribution as\n\\[\n\t\\int_{F^{-1}(y_0)} f \\, d_F = \\int_{{\\mathbb R}^n} f(x) \\delta(F(x) - y_0) \\, dx.\n\\]\n\nFor the purposes of this paper, we set $F : {\\mathbb R}^{kn} \\to {\\mathbb R}^{k+1}$ by\n\\begin{equation}\\label{def F}\n\tF(\\xi_1,\\xi_2,\\dots,\\xi_k) = \\bigl(|\\xi_1|,|\\xi_2|,\\dots,|\\xi_k|,\\ |\\xi_1 + \\cdots + \\xi_k|\\bigr).\n\\end{equation}\nThen the level set $F^{-1}(\\tau)$ consists of $k$-tuples $(\\xi_1,\\xi_2,\\dots,\\xi_k)$ satisfying\n\\[\n|\\xi_j|=\\tau_j \\ \\ (1\\le j\\le k)\n\\qquad \\text{and} \\qquad\n|\\xi_1+\\cdots+\\xi_k|=\\tau_{k+1}.\n\\]\nSuch a $k$-tuple determines a closed $(k+1)$-gon in ${\\mathbb R}^n$ with side lengths $\\tau_1,\\tau_2,\\dots,\\tau_{k+1}$, where the final side is the closing vector $-(\\xi_1+\\cdots+\\xi_k)$.\nThe next theorem provides a description of the concentration of $\\mu$ in $k$-good cones.\n\n\\begin{theorem}\\label{main theorem k+1-good}\nLet $\\rho$ be a Schwartz function on ${\\mathbb R}^{k+1}$ with $\\int \\rho = 1$. Assume that\n\\[\n\t\\operatorname{supp} \\hat \\rho \\subset (-\\operatorname{inj} M, \\operatorname{inj} M)^{k+1}.\n\\]\nLet $\\Gamma$ be a $k$-good cone. Then for $\\tau = (\\tau_1, \\tau_2, \\dots, \\tau_{k+1}) \\in \\Gamma$,\n\\[\n\t\\rho * \\mu(\\tau) = (2\\pi)^{-kn} \\operatorname{vol} M \\operatorname{vol} F^{-1}(\\tau) + O(|\\tau|^{k(n-1)-2}),\n\\]\nwhere the constants implicit in the big-$O$ notation depend on $M$, $\\Gamma$, and $\\rho$ but not on $\\tau$.\n\\end{theorem}\n\nSince $\\rho$ is rapidly decaying and $\\operatorname{supp} \\hat \\rho$ is contained in $(-\\operatorname{inj} M, \\operatorname{inj} M)^{k+1}$, the convolution $\\rho * \\mu$ provides a smoothed version of $\\mu$ that captures the mass of $\\mu$ near $\\tau$. In particular, Theorem \\ref{main theorem k+1-good} gives an asymptotic description of this smoothed local mass for $\\tau \\in \\Gamma$. Next, we determine the growth rate of $\\operatorname{vol} F^{-1}(\\tau)$ as $|\\tau| \\to \\infty$. When $k=2$, this quantity is related to the area of the corresponding triangle \\cite{MR4376455}. For $k>2$, the shape of a $(k+1)$-gon with fixed side lengths is not uniquely determined (Figures \\ref{2} and \\ref{3}), so it is not sufficient to consider only its planar area. We prove in Proposition \\ref{Scaling of F} that, under our assumptions, $\\operatorname{vol} F^{-1}(\\tau)$ is of order $|\\tau|^{k(n-1)-1}$, so the remainder term $O(|\\tau|^{k(n-1)-2})$ is indeed lower order. As in \\cite{MR4376455}, the leading term can be interpreted in terms of the volume of the configuration space of $(k+1)$-gons with side lengths prescribed by $\\tau$.\n\\subsection*{Organization}\nIn Section~2 we analyze the configuration space that appears in the main term of Theorem~\\ref{main theorem k+1-good}, and in particular prove the scaling law for the relevant fiber volume. Section~3 contains the proof of Theorem~\\ref{main theorem k+1-gon-bad}, establishing rapid decay of the high-frequency tail in the regime where the $(k+1)$-gon inequality fails, via spectral projectors and a non-stationary phase argument. Section~4 proves Theorem~\\ref{main theorem k+1-good} by expressing the Fourier transform of the joint spectral measure in terms of Fourier integral operators and computing the associated symbolic data using clean composition and half-density calculus. Finally, Section~\\ref{sec:flat-torus-example} presents a flat-torus example showing that the frequency cutoff in Theorem~\\ref{main theorem k+1-gon-bad} is essentially sharp.\n\n\\subsection*{Acknowledgement} This project is supported by the National Key R\\&D Program of China under Grant No. 2022YFA1007200, the Natural Science Foundation of China under Grant No. 12571107, and the Zhejiang Provincial Natural Science Foundation of China under Grant No. LR25A010001.", + "sketch": "The introduction explains that the condition $\\lambda_{i_{k+1}} \\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$ “forces the $(k+1)$-tuple $(\\tau_1,\\ldots,\\tau_k,\\lambda_{i_{k+1}})$ to be $k$-bad, since the last entry dominates the sum of the others,” hence any $(\\lambda_{i_1},\\ldots,\\lambda_{i_k},\\lambda_{i_{k+1}})$ with $\\lambda_{i_j}\\le \\tau_j$ is also $k$-bad.\n\nRather than summing over a region in $(0,\\infty)^{k+1}$, the argument is organized by “fix[ing] upper bounds $\\tau_j$ for $\\lambda_{i_j}$, $1\\le j\\le k$, and then estimat[ing] the tail sum over those $i_{k+1}$ with $\\lambda_{i_{k+1}} \\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$.” The proof of Theorem~\\ref{main theorem k+1-gon-bad} then “uses the method of non-stationary phase,” and (per the Organization paragraph) establishes “rapid decay of the high-frequency tail in the regime where the $(k+1)$-gon inequality fails, via spectral projectors and a non-stationary phase argument.” A flat-torus example is recorded showing the frequency threshold is “essentially sharp.”", + "expanded_sketch": "The introduction explains that the condition $\\lambda_{i_{k+1}} \\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$ “forces the $(k+1)$-tuple $(\\tau_1,\\ldots,\\tau_k,\\lambda_{i_{k+1}})$ to be $k$-bad, since the last entry dominates the sum of the others,” hence any $(\\lambda_{i_1},\\ldots,\\lambda_{i_k},\\lambda_{i_{k+1}})$ with $\\lambda_{i_j}\\le \\tau_j$ is also $k$-bad.\n\nRather than summing over a region in $(0,\\infty)^{k+1}$, the argument is organized by “fix[ing] upper bounds $\\tau_j$ for $\\lambda_{i_j}$, $1\\le j\\le k$, and then estimat[ing] the tail sum over those $i_{k+1}$ with $\\lambda_{i_{k+1}} \\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$.” To prove the main theorem, the proof then “uses the method of non-stationary phase,” and (per the Organization paragraph) establishes “rapid decay of the high-frequency tail in the regime where the $(k+1)$-gon inequality fails, via spectral projectors and a non-stationary phase argument.” A flat-torus example is recorded showing the frequency threshold is “essentially sharp.”", + "expanded_theorem": "\\label{main theorem k+1-gon-bad}\n\tFor each $\\epsilon>0$ and integer $N > 0$, there exists a constant $C_{\\epsilon, N}$ such that for any positive numbers $\\tau_{1},\\tau_{2},\\ldots,\\tau_{k}$, we have\n\t\\[\n\t\t\\sum_{\\lambda_{i_{k+1}}\\geq (1+\\epsilon)(\\tau_1+\\tau_2+\\dots+\\tau_k)} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2 \\leq C_{\\epsilon, N}(\\tau_1+\\tau_2+\\dots+\\tau_k)^{-N},\n\t\\]\n\twhere $\\lambda_{i_j} \\leq \\tau_j$ for $1 \\leq j \\leq k$.", + "theorem_type": [ + "Existential–Universal", + "Inequality or Bound" + ], + "mcq": { + "question": "Let \\(\\{e_i\\}\\) be an \\(L^2\\)-orthonormal family of eigenfunctions with corresponding eigenvalues (frequencies) \\(\\lambda_i\\), and write \\(\\langle f,g\\rangle\\) for the \\(L^2\\) inner product. Fix an integer \\(k\\ge 1\\). For positive numbers \\(\\tau_1,\\dots,\\tau_k\\), suppose the indices \\(i_1,\\dots,i_k\\) satisfy \\(\\lambda_{i_j}\\le \\tau_j\\) for each \\(1\\le j\\le k\\). Under these assumptions, which quantitative estimate holds for the tail of the harmonic expansion coefficients of the product \\(e_{i_1}e_{i_2}\\cdots e_{i_k}\\)?", + "correct_choice": { + "label": "A", + "text": "For every \\(\\epsilon>0\\) and every integer \\(N>0\\), there exists a constant \\(C_{\\epsilon,N}\\) such that\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)(\\tau_1+\\tau_2+\\cdots+\\tau_k)} \\bigl|\\langle e_{i_1}e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|^2\n\\le C_{\\epsilon,N}(\\tau_1+\\tau_2+\\cdots+\\tau_k)^{-N}.\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "For every \\(\\epsilon>0\\) and every integer \\(N>0\\), there exists a constant \\(C_{\\epsilon,N}\\) such that\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (\\tau_1+\\tau_2+\\cdots+\\tau_k)+\\epsilon} \\bigl|\\langle e_{i_1}e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|^2\n\\le C_{\\epsilon,N}(\\tau_1+\\tau_2+\\cdots+\\tau_k)^{-N}.\n\\]" + }, + { + "label": "C", + "text": "For every \\(\\epsilon>0\\), there exists a constant \\(C_{\\epsilon}\\) such that\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)(\\tau_1+\\tau_2+\\cdots+\\tau_k)} \\bigl|\\langle e_{i_1}e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|^2\n\\le C_{\\epsilon}.\n\\]" + }, + { + "label": "D", + "text": "There exists an integer \\(N>0\\) such that for every \\(\\epsilon>0\\) there is a constant \\(C_{\\epsilon,N}\\) with\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)(\\tau_1+\\tau_2+\\cdots+\\tau_k)} \\bigl|\\langle e_{i_1}e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|^2\n\\le C_{\\epsilon,N}(\\tau_1+\\tau_2+\\cdots+\\tau_k)^{-N}\n\\]\nfor all positive \\(\\tau_1,\\dots,\\tau_k\\) and all indices \\(i_1,\\dots,i_k\\) with \\(\\lambda_{i_j}\\le \\tau_j\\)." + }, + { + "label": "E", + "text": "For every \\(\\epsilon>0\\) and every integer \\(N>0\\), there exists a constant \\(C_{\\epsilon,N}\\) such that\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)\\max\\{\\tau_1,\\tau_2,\\dots,\\tau_k\\}} \\bigl|\\langle e_{i_1}e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|^2\n\\le C_{\\epsilon,N}(\\tau_1+\\tau_2+\\cdots+\\tau_k)^{-N}.\n\\]" + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "non-stationary phase", + "tampered_component": "multiplicative tail threshold replaced by additive offset", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "finiteness", + "tampered_component": "dropped rapid-decay power saving in \\((\\tau_1+\\cdots+\\tau_k)\\)", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "finiteness", + "tampered_component": "quantifier order on N weakened from arbitrary N to existence of one N", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "geometric_construction", + "tampered_component": "sum-of-sides bad-region threshold replaced by max-side threshold", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not state the exact estimate or its key features such as the multiplicative threshold \u001d(1+\u001b)\u001d and rapid decay for every \u001dN\u001d. It only asks which estimate holds, so there is no explicit answer leakage." + }, + "TAS": { + "score": 1, + "justification": "The item is very close to theorem recall: it presents the full setup and asks for the correct quantitative conclusion. However, the options differ in meaningful ways (threshold scale, power decay, quantifier order), so it is not a pure verbatim restatement." + }, + "GPS": { + "score": 1, + "justification": "Selecting the correct choice requires some reasoning about which formulation is strongest and correctly quantified, but it mainly tests recognition of the theorem's exact statement rather than substantial derivation or synthesis." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically targeted: additive versus multiplicative cutoff, loss of rapid decay, weakened quantifiers, and replacing the sum by a max. These align with realistic failure modes and are clearly distinct." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and little answer leakage, though it leans more toward precise recall than deep generative reasoning." + } + }, + { + "id": "2602.05203v1", + "paper_link": "http://arxiv.org/abs/2602.05203v1", + "theorems_cnt": 5, + "theorem": { + "env_name": "theorem", + "content": "\\label{EPS} Assume that $p=\\frac{2n}{n-2k}$ and $n\\geq2k+2$.\n Then there is a positive function $f_0\\in H\\left(\\mathbb{B}^n\\right)$ which achieves the equality \n for the inequality \\eqref{BPS}. Furthermore, the extremal function $f_0$ must be radially symmetric and monotone decreasing about some point $P\\in \\mathbb{B}^n$, that is, $f_0$ is a constant on the geodesic sphere centered at $P\\in \\mathbb{B}^n$ and radially decreasing about the geodesic distance from P.", + "start_pos": 12151, + "end_pos": 12655, + "label": "EPS" + }, + "ref_dict": { + "BSobolev": "\\begin{equation}\\label{BSobolev}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV\\geq S_{n,k}\\left(\\int_{\\mathbb{B}^n}|f|^{\\frac{2n}{n-2k}}dV\\right)^{\\frac{n-2k}{n}},\n\\end{equation}", + "sublambdaBN": "\\begin{equation}\\label{sublambdaBN}\nP_{k}\\left(f\\right)-\\alpha f=|f|^{p-2}f.\n\\end{equation}", + "HHSM": "\\begin{equation}\\label{HHSM}\n\\int_{\\mathbb{R}^{n}_{+}}|\\nabla^{k}u|^2dx-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{R}^{n}_{+}}\\frac{u^2}{x_1^{2k}}dx\\geq C_{n,k,p} \\left(\\int_{\\mathbb{R}^{n}_{+}}x_1^\\gamma|u|^{p}dx\\right)^{\\frac{2}{p}}.\n\\end{equation}", + "AHSM": "\\begin{equation}\\label{AHSM}\n\\int_{\\mathbb{R}^{n}_{+}}|\\nabla^{k}u|^2dx-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{R}^{n}_{+}}\\frac{u^2}{x_1^{2k}}dx\\geq C_{n,k,\\frac{2n}{n-2k}} \\left(\\int_{\\mathbb{R}^{n}_{+}}|u|^{\\frac{2n}{n-2k}}dx\\right)^{\\frac{n-2k}{n}}.\n\\end{equation}", + "E-subHPS": "\\begin{theorem}\\label{E-subHPS}\n Assume that $22k$. Then there is a positive function $f_0\\in H\\left(\\mathbb{B}^n\\right)$ which achieves the equality in the inequality \\eqref{BPS},\n\\end{theorem}", + "Preliminaries": "\\label{Preliminaries}\n\nIn this section, we recall some facts on the hyperbolic space. There are several models of hyperbolic space, such as the Poincar\\'e half space model and the Poincar\\'e ball mod", + "EPS": "\\begin{theorem}\\label{EPS} Assume that $p=\\frac{2n}{n-2k}$ and $n\\geq2k+2$.\n Then there is a positive function $f_0\\in H\\left(\\mathbb{B}^n\\right)$ which achieves the equality \n for the inequality \\eqref{BPS}. Furthermore, the extremal function $f_0$ must be radially symmetric and monotone decreasing about some point $P\\in \\mathbb{B}^n$, that is, $f_0$ is a constant on the geodesic sphere centered at $P\\in \\mathbb{B}^n$ and radially decreasing about the geodesic distance from P. \n\\end{theorem}", + "BPS": "\\begin{equation}\\label{BPS}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f|^2dV\\geq C_{n,k,p}\\left(\\int_{\\mathbb{B}^n}|f|^{p}dV\\right)^{\\frac{2}{p}},\n\\end{equation}" + }, + "pre_theorem_intro_text_len": 7570, + "pre_theorem_intro_text": "The classical Hardy-Sobolev-Maz'ya (HSM) inequality provides a refinement of both the Sobolev and the Hardy inequalities on the upper half space $\\mathbb{R}^n_+$. It reads as follows: for $n>2k$, $22k$ and $22k$, $22k$ and $22k$, $22k$ and $22k$. Then there is a positive function $f_0\\in H\\left(\\mathbb{B}^n\\right)$ which achieves the equality in the inequality \\eqref{BPS},\n\\end{theorem}\n\n\\begin{theorem}\\label{E-subHHSM}\nAssume that $22k$. \n Then there is a positive function $f_0\\in \\widetilde{H}\\left(\\mathbb{R}^{n}_{+}\\right)$ which achieves the equality in inequality \\eqref{HHSM}.\n\n\\begin{corollary}\\label{EL}\nLet $\\alpha=\\prod\\limits_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}$. If either $2k+2\\leq n$, $p=\\frac{2n}{n-2k}$ or $2k0$ such that\n$f_m\\left(R\\right)\\leq \\left(1-R\\right)^{\\frac{n-1}{q_0}}$, where $q_0$ is chosen as $q_00$, there exists $y_k\\in \\mathbb{R}^n$ and $R_{\\epsilon}>0$ such that $\\int_{|x-y_k|\\geq R}\\rho_k(x)dx<\\epsilon$ for any $R>R_{\\epsilon}$. This is achieved by scaling cut-off function and \n$$\\int_{R_0<|x-y_k|2k$. We have the following results:\n\n\\begin{theorem}\\label{E-subHPS}\n Assume that $22k$. Then there is a positive function $f_0\\in H\\left(\\mathbb{B}^n\\right)$ which achieves the equality in the inequality \\eqref{BPS},\n\\end{theorem}\n\n\\begin{theorem}\\label{E-subHHSM}\nAssume that $22k$. \n Then there is a positive function $f_0\\in \\widetilde{H}\\left(\\mathbb{R}^{n}_{+}\\right)$ which achieves the equality in inequality \\eqref{HHSM}.\n\n\\end{theorem}\n\nThe Poincar\\'e-Sobolev inequalities are highly linked with the Brezis-Nirenberg problem \\cite{Brezis} on the entire hyperbolic space $\\mathbb{B}^n$:\n\\begin{equation}\\label{sublambdaBN}\nP_{k}\\left(f\\right)-\\alpha f=|f|^{p-2}f.\n\\end{equation}\nFor $k=1$, Mancini and Sandeep \\cite{MS} showed that the entire solution exists either when $n\\geq4$, $p=\\frac{2n}{n-2}$, $0<\\alpha\\leq\\frac{1}{4}$ or when $n\\geq3$, $12k$. Then there is a positive function $f_0\\in H\\left(\\mathbb{B}^n\\right)$ which achieves the equality in the inequality \\eqref{BPS},\n\\end{theorem}", + "Preliminaries": "\\label{Preliminaries}\n\nIn this section, we recall some facts on the hyperbolic space. There are several models of hyperbolic space, such as the Poincar\\'e half space model and the Poincar\\'e ball mod", + "EPS": "\\begin{theorem}\\label{EPS} Assume that $p=\\frac{2n}{n-2k}$ and $n\\geq2k+2$.\n Then there is a positive function $f_0\\in H\\left(\\mathbb{B}^n\\right)$ which achieves the equality \n for the inequality \\eqref{BPS}. Furthermore, the extremal function $f_0$ must be radially symmetric and monotone decreasing about some point $P\\in \\mathbb{B}^n$, that is, $f_0$ is a constant on the geodesic sphere centered at $P\\in \\mathbb{B}^n$ and radially decreasing about the geodesic distance from P. \n\\end{theorem}", + "BPS": "\\begin{equation}\\label{BPS}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f|^2dV\\geq C_{n,k,p}\\left(\\int_{\\mathbb{B}^n}|f|^{p}dV\\right)^{\\frac{2}{p}},\n\\end{equation}" + }, + "pre_theorem_intro_text_len": 7570, + "pre_theorem_intro_text": "The classical Hardy-Sobolev-Maz'ya (HSM) inequality provides a refinement of both the Sobolev and the Hardy inequalities on the upper half space $\\mathbb{R}^n_+$. It reads as follows: for $n>2k$, $22k$ and $22k$, $22k$ and $22k$, $22k$ and $22k$. Then there is a positive function $f_0\\in H\\left(\\mathbb{B}^n\\right)$ which achieves the equality in the inequality \\eqref{BPS},\n\\end{theorem}\n\n\\begin{theorem}\\label{E-subHHSM}\nAssume that $22k$. \n Then there is a positive function $f_0\\in \\widetilde{H}\\left(\\mathbb{R}^{n}_{+}\\right)$ which achieves the equality in inequality \\eqref{HHSM}.\n\n\\begin{corollary}\\label{EL}\nLet $\\alpha=\\prod\\limits_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}$. If either $2k+2\\leq n$, $p=\\frac{2n}{n-2k}$ or $2k0$ such that\n$f_m\\left(R\\right)\\leq \\left(1-R\\right)^{\\frac{n-1}{q_0}}$, where $q_0$ is chosen as $q_00$, there exists $y_k\\in \\mathbb{R}^n$ and $R_{\\epsilon}>0$ such that $\\int_{|x-y_k|\\geq R}\\rho_k(x)dx<\\epsilon$ for any $R>R_{\\epsilon}$. This is achieved by scaling cut-off function and \n$$\\int_{R_0<|x-y_k|2k$. We have the following results:\n\n\\begin{theorem}\\label{E-subHPS}\n Assume that $22k$. Then there is a positive function $f_0\\in H\\left(\\mathbb{B}^n\\right)$ which achieves the equality in the inequality \\eqref{BPS},\n\\end{theorem}\n\n\\begin{theorem}\\label{E-subHHSM}\nAssume that $22k$. \n Then there is a positive function $f_0\\in \\widetilde{H}\\left(\\mathbb{R}^{n}_{+}\\right)$ which achieves the equality in inequality \\eqref{HHSM}.\n\n\\end{theorem}\n\nThe Poincar\\'e-Sobolev inequalities are highly linked with the Brezis-Nirenberg problem \\cite{Brezis} on the entire hyperbolic space $\\mathbb{B}^n$:\n\\begin{equation}\\label{sublambdaBN}\nP_{k}\\left(f\\right)-\\alpha f=|f|^{p-2}f.\n\\end{equation}\nFor $k=1$, Mancini and Sandeep \\cite{MS} showed that the entire solution exists either when $n\\geq4$, $p=\\frac{2n}{n-2}$, $0<\\alpha\\leq\\frac{1}{4}$ or when $n\\geq3$, $10$ are integration constants, and $\\mathrm{sn}(\\,\\cdot,\\,ik)$ denotes the Jacobian elliptic sine function with modulus $i k$. \nApplying the classical Jacobi imaginary modulus transformation, the solution also admits the equivalent form \n\\[\nv = \\operatorname{arcsinh}\\Bigl( \\frac{k}{\\sqrt{1+k^2}} \\; \\mathrm{sn}\\bigl( \\sqrt{1+k^2}\\,(2u_1 + \\delta),\\; \\frac{k}{\\sqrt{1+k^2}} \\bigr) \\Bigr).\\]\n\nGeometrically, given a solution of~\\eqref{eq-ellip}, one first obtains a minimal surface $\\psi:\\mathbb{C}\\rightarrow \\mathbb{S}^3$ with induced metric $e^{2v}|dz|^2$ \nand Hopf differential $\\frac i 2 dz\\otimes dz$. \nSuch a surface should be necessarily homogeneous or of cohomogeneity one and hence belongs to the $T_{m,k,a}$ family constructed and classified by Hsiang and Lawson in \\cite{Hsiang-Lawson}. This family contains infinitely many closed minimal surfaces, whose topology is either a torus or a Klein bottle. Denote by $N$ the Gauss map of such a surface. Then the map \\emph{(}see the proof of Corollary $1$ in \\cite{Urbano}\\emph{)}\n\\[\n\\phi(z)=(V_\\psi, e^{2i u_2}):\\mathbb{C}\\rightarrow \\mathbb{S}^2\\times\\mathbb{S}^1\n\\]\ndefines a minimal surface in $\\mathbb{S}^2\\times\\mathbb{S}^1$, where\n\\[\nV_\\psi=\\frac{1}{\\sqrt{2}}(-2ie^{-2v}\\psi_{z}\\wedge\\psi_{\\overline{z}}+\\psi\\wedge N):\\mathbb{C}\\rightarrow \\mathbb{S}^2\\subset \\Lambda^2\\mathbb{R}^4. \n\\]\nApart from the obvious slices, these surfaces constitute all minimal-Willmore surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^1$. \n\\end{remark}", + "thm1": "\\begin{thm}\\label{thm1}\n Let \\( x: \\Sigma \\rightarrow \\mathbb{S}^2 \\times \\mathbb{S}^2 \\) be a minimal-Willmore surface. Then either \\( x \\) is a special complex curve given by a slice or a diagonal; or, up to an isometry, is contained in a totally geodesic submanifold $\\mathbb{S}^2 \\times \\mathbb{S}^1$, and can be described by a solution of the sinh-Gordon equation in one variable.\n\\end{thm}" + }, + "pre_theorem_intro_text_len": 2863, + "pre_theorem_intro_text": "For a closed surface $x:\\Sigma\\rightarrow (N^n,g)$ in a Riemannian manifold, the squared $L^2$-norm of the trace-free second fundamental form is a fundamental conformal invariant; that is, it is preserved under conformal changes of the metric. This functional, referred to as the conformally invariant Willmore functional \\cite{Michelat,Mondino}, will henceforth be called simply the Willmore functional. The Willmore functional is equivalent, up to a topological invariant, to the integral \n$$\\int_{\\Sigma}(|\\vec{H}|^2+K_{1212})dA,$$\nwhere $\\vec{H}$ is the mean curvature vector and $K_{1212}$ denotes the sectional curvature of $(N^n,g)$ restricted to the tangent plane of the surface. In what follows, we will denote this integral by $\\mathcal{W}$ and also refer to it as the Willmore functional. This functional exhibits notable links with other fundamental quantities, including, among others, the renormalized area functional within the AdS/CFT correspondence \\cite{Alexakis, Graham}. Surfaces that satisfy the Euler-Lagrange equation of this functional are called {\\em Willmore surfaces}. \n\nWhen the ambient space is a real space form, both the Willmore functional and Willmore surfaces have been extensively investigated. Notable advances include the resolution of the Willmore conjecture in $\\mathbb{S}^3$, originally proposed by Willmore in \\cite{Willmore} and proved by Marques and Neves in \\cite{Marques}. Regarding the construction and classification of Willmore surfaces, we refer to \\cite{Bryant, Ejiri, WD, MWW} and references therein. A basic fact is that every minimal surface in a real space form is automatically Willmore. \nWhen the ambient space is a non-space form Riemannian manifold, the Euler–Lagrange equation of the Willmore functional has been derived independently by several groups of geometers from different viewpoints; see, for example, \\cite{Hu-Li, Mondino, Pedit, Wang-Xie}. Owing to the presence of ambient curvature terms in the Euler–Lagrange equation, not every minimal surface is Willmore in a non-space form Riemannian manifold. In \\cite{Montiel-Urbano}, Montiel and Urbano proved that the only surfaces in $\\mathbb{CP}^2$ that are both minimal and Willmore are the superminimal surfaces of positive spin, i.e., complex curves and minimal Lagrangian surfaces. Recently, the last two authors of this paper have generalized this result to the self-dual K\\\"ahler-Einstein surfaces (see Proposition 2.3 in \\cite{Wang-Xie}). Note that $\\mathbb{S}^2 \\times \\mathbb{S}^2$ (equipped with the standard product metric and complex structure), being neither self-dual nor anti-self-dual, is another canonical example of a K\\\"ahler-Einstein surface. This motivates the study of which minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ are Willmore. We refer to a surface as minimal-Willmore if it is both minimal and Willmore.", + "context": "For a closed surface $x:\\Sigma\\rightarrow (N^n,g)$ in a Riemannian manifold, the squared $L^2$-norm of the trace-free second fundamental form is a fundamental conformal invariant; that is, it is preserved under conformal changes of the metric. This functional, referred to as the conformally invariant Willmore functional \\cite{Michelat,Mondino}, will henceforth be called simply the Willmore functional. The Willmore functional is equivalent, up to a topological invariant, to the integral \n$$\\int_{\\Sigma}(|\\vec{H}|^2+K_{1212})dA,$$\nwhere $\\vec{H}$ is the mean curvature vector and $K_{1212}$ denotes the sectional curvature of $(N^n,g)$ restricted to the tangent plane of the surface. In what follows, we will denote this integral by $\\mathcal{W}$ and also refer to it as the Willmore functional. This functional exhibits notable links with other fundamental quantities, including, among others, the renormalized area functional within the AdS/CFT correspondence \\cite{Alexakis, Graham}. Surfaces that satisfy the Euler-Lagrange equation of this functional are called {\\em Willmore surfaces}.\n\nWhen the ambient space is a real space form, both the Willmore functional and Willmore surfaces have been extensively investigated. Notable advances include the resolution of the Willmore conjecture in $\\mathbb{S}^3$, originally proposed by Willmore in \\cite{Willmore} and proved by Marques and Neves in \\cite{Marques}. Regarding the construction and classification of Willmore surfaces, we refer to \\cite{Bryant, Ejiri, WD, MWW} and references therein. A basic fact is that every minimal surface in a real space form is automatically Willmore. \nWhen the ambient space is a non-space form Riemannian manifold, the Euler–Lagrange equation of the Willmore functional has been derived independently by several groups of geometers from different viewpoints; see, for example, \\cite{Hu-Li, Mondino, Pedit, Wang-Xie}. Owing to the presence of ambient curvature terms in the Euler–Lagrange equation, not every minimal surface is Willmore in a non-space form Riemannian manifold. In \\cite{Montiel-Urbano}, Montiel and Urbano proved that the only surfaces in $\\mathbb{CP}^2$ that are both minimal and Willmore are the superminimal surfaces of positive spin, i.e., complex curves and minimal Lagrangian surfaces. Recently, the last two authors of this paper have generalized this result to the self-dual K\\\"ahler-Einstein surfaces (see Proposition 2.3 in \\cite{Wang-Xie}). Note that $\\mathbb{S}^2 \\times \\mathbb{S}^2$ (equipped with the standard product metric and complex structure), being neither self-dual nor anti-self-dual, is another canonical example of a K\\\"ahler-Einstein surface. This motivates the study of which minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ are Willmore. We refer to a surface as minimal-Willmore if it is both minimal and Willmore.", + "full_context": "For a closed surface $x:\\Sigma\\rightarrow (N^n,g)$ in a Riemannian manifold, the squared $L^2$-norm of the trace-free second fundamental form is a fundamental conformal invariant; that is, it is preserved under conformal changes of the metric. This functional, referred to as the conformally invariant Willmore functional \\cite{Michelat,Mondino}, will henceforth be called simply the Willmore functional. The Willmore functional is equivalent, up to a topological invariant, to the integral \n$$\\int_{\\Sigma}(|\\vec{H}|^2+K_{1212})dA,$$\nwhere $\\vec{H}$ is the mean curvature vector and $K_{1212}$ denotes the sectional curvature of $(N^n,g)$ restricted to the tangent plane of the surface. In what follows, we will denote this integral by $\\mathcal{W}$ and also refer to it as the Willmore functional. This functional exhibits notable links with other fundamental quantities, including, among others, the renormalized area functional within the AdS/CFT correspondence \\cite{Alexakis, Graham}. Surfaces that satisfy the Euler-Lagrange equation of this functional are called {\\em Willmore surfaces}.\n\nWhen the ambient space is a real space form, both the Willmore functional and Willmore surfaces have been extensively investigated. Notable advances include the resolution of the Willmore conjecture in $\\mathbb{S}^3$, originally proposed by Willmore in \\cite{Willmore} and proved by Marques and Neves in \\cite{Marques}. Regarding the construction and classification of Willmore surfaces, we refer to \\cite{Bryant, Ejiri, WD, MWW} and references therein. A basic fact is that every minimal surface in a real space form is automatically Willmore. \nWhen the ambient space is a non-space form Riemannian manifold, the Euler–Lagrange equation of the Willmore functional has been derived independently by several groups of geometers from different viewpoints; see, for example, \\cite{Hu-Li, Mondino, Pedit, Wang-Xie}. Owing to the presence of ambient curvature terms in the Euler–Lagrange equation, not every minimal surface is Willmore in a non-space form Riemannian manifold. In \\cite{Montiel-Urbano}, Montiel and Urbano proved that the only surfaces in $\\mathbb{CP}^2$ that are both minimal and Willmore are the superminimal surfaces of positive spin, i.e., complex curves and minimal Lagrangian surfaces. Recently, the last two authors of this paper have generalized this result to the self-dual K\\\"ahler-Einstein surfaces (see Proposition 2.3 in \\cite{Wang-Xie}). Note that $\\mathbb{S}^2 \\times \\mathbb{S}^2$ (equipped with the standard product metric and complex structure), being neither self-dual nor anti-self-dual, is another canonical example of a K\\\"ahler-Einstein surface. This motivates the study of which minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ are Willmore. We refer to a surface as minimal-Willmore if it is both minimal and Willmore.\n\n\\section{Introduction}\nFor a closed surface $x:\\Sigma\\rightarrow (N^n,g)$ in a Riemannian manifold, the squared $L^2$-norm of the trace-free second fundamental form is a fundamental conformal invariant; that is, it is preserved under conformal changes of the metric. This functional, referred to as the conformally invariant Willmore functional \\cite{Michelat,Mondino}, will henceforth be called simply the Willmore functional. The Willmore functional is equivalent, up to a topological invariant, to the integral \n$$\\int_{\\Sigma}(|\\vec{H}|^2+K_{1212})dA,$$\nwhere $\\vec{H}$ is the mean curvature vector and $K_{1212}$ denotes the sectional curvature of $(N^n,g)$ restricted to the tangent plane of the surface. In what follows, we will denote this integral by $\\mathcal{W}$ and also refer to it as the Willmore functional. This functional exhibits notable links with other fundamental quantities, including, among others, the renormalized area functional within the AdS/CFT correspondence \\cite{Alexakis, Graham}. Surfaces that satisfy the Euler-Lagrange equation of this functional are called {\\em Willmore surfaces}.\n\nIn their work \\cite{Urbano}, Tobarro and Urbano gave a beautiful local characterization of minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ that possess no complex points. Their description is formulated in terms of a pair of solutions $v$ and $w$ to the sinh‑Gordon equation \n$$v_{z\\bar z}+\\frac{1}{2}\\sinh (2v)=0.$$\nWithin this framework, the Willmore condition introduces a new partial differential equation linking $v$ and $w$, see \\eqref{eq-mwill}. By analyzing this coupled system, we establish the theorem above.\n\n\\begin{example}\\label{ex-slice} For any point $p\\in\\mathbb{S}^2$, the corresponding slices \n\\[\n\\mathbb{S}^2\\times\\{p\\}=\\{(x,p)\\in\\mathbb{S}^2\\times\\mathbb{S}^2 \\mid x\\in\\mathbb{S}^2\\},\n\\]\n\\[\n\\{p\\}\\times\\mathbb{S}^2=\\{(p,x)\\in\\mathbb{S}^2\\times\\mathbb{S}^2 \\mid x\\in\\mathbb{S}^2\\}.\n\\]\nare totally geodesic with the first one satisfying $C_1=C_2=1$ and the second one satisfying $C_1=-C_2=1$. That is, the slices are complex with respect to both $J_1$ and $J_2$.\n\\end{example}\n\\begin{example}\\label{ex-diagonal} The diagonal \n$$D\\triangleq\\{(x,x)\\in\\mathbb{S}^2\\times\\mathbb{S}^2 \\mid x\\in\\mathbb{S}^2\\}$$\nis totally geodesic and satisfies $C_1=1$ and $C_2=0$, which means that $D$ is complex with respect to $J_1$ and Lagrangian with respect to $J_2$.\n\\end{example}\n\\begin{example}\\label{ex-Clifford}\n The torus defined by \n $$T\\triangleq\\{(x,y)\\in\\mathbb{S}^2\\times\\mathbb{S}^2 \\mid x_1=y_1=0\\}$$ \n is totally geodesic and satisfies $C_1=C_2=0$, which means that $T$ is Lagrangian for both $J_1$ and $J_2$. This surface is known as the Clifford torus in $\\mathbb{S}^2 \\times \\mathbb{S}^2$, characterized by the flatness of both its tangent and normal bundles. \n\\end{example}\n\\begin{example} \nLet $\\Sigma=\\mathbb{C}/\\Lambda$ be a torus generated by the lattice $\\Lambda=\\{m+n\\tau \\mid m,n\\in\\mathbb{Z}\\}$, where $\\tau$ is a complex number with $\\operatorname{Im}\\tau>0$. Let $\\wp:\\Sigma\\rightarrow\\mathbb{S}^2$ be the Weierstrass $\\wp$-function with a second-order pole at the origin, and $z_0\\in\\Sigma$ be a point at which $\\wp$ is not ramified. Then $x=\\big(\\wp(z),\\wp(z-z_0)\\big):\\Sigma\\rightarrow(\\mathbb{S}^2\\times\\mathbb{S}^2)$ is a holomorphic embedding with the complex structure $J_1$. Such minimal tori are referred to as Weierstrass tori.\n\\end{example}\n\n\\begin{proposition}\\label{pro-mw}\n Let $x:\\Sigma \\rightarrow \\mathbb{S}^2\\times \\mathbb{S}^2$ be a complex curve or minimal Lagrangian surface. Then $x$ is Willmore if and only if it is one of the surfaces given in Example~\\ref{ex-slice} $\\sim$ Example~\\ref{ex-Clifford}. \n\\end{proposition}\n\\begin{proof}\nWithout loss of generality, we assume that $x$ is complex or minimal Lagrangian with respect to the complex structure $J_1$. Consequently, $C_1$ is a constant which must be either $0$ or $\\pm 1$. It follows from \\cite[Proposition 3]{Urbano} (see also \\cite{Chen}) that we only need to prove that $C_2$ is a constant.\n\nBy \\eqref{eq-C1z}, we have $f_1\\bar{\\gamma_1}=0$. Substituting this into \\eqref{eq-mW} yields $\\bar{f_1}{\\gamma_2}=0$, which, by \\eqref{eq-C2z}, implies that $C_2$ is holomorphic. Since $C_2$ is real-valued, it must be constant. \n\\end{proof}\n\\begin{theorem}\\label{thm-mw}\n Let $x:\\Sigma \\rightarrow \\mathbb{S}^2\\times \\mathbb{S}^2$ be a minimal surface without complex points. Then $x$ is Willmore if and only if up to an isometry it is a minimal-Willmore surface in $\\mathbb{S}^2\\times\\mathbb{S}^1$. \n\\end{theorem}\n\nLet $x:\\Sigma \\rightarrow \\mathbb{S}^2\\times \\mathbb{S}^2$ be a minimal immersion of a simply connected surface without complex points and $z$ be a complex coordinate such that the Hopf differential $\\Theta(z)\\triangleq\\frac{\\langle J_1 x_z, J_2 x_z\\rangle}{2}dz^2$ (which is holomorphic) satisfies $\\Theta(z)=dz^2$. \n Then the functions \n $$v\\triangleq\\frac{1}{2}\\ln \\sqrt{\\frac{(1+C_1)(1+C_2)}{(1-C_1)(1-C_2)}},~~~w\\triangleq\\frac{1}{2}\\ln \\sqrt{\\frac{(1-C_1)(1+C_2)}{(1+C_1)(1-C_2)}}$$\n satisfy the sinh-Gordon equation, i.e., \n $$v_{z\\bar z}+\\frac{1}{2}\\sinh (2v)=0,~~~w_{z\\bar z}+\\frac{1}{2}\\sinh (2w)=0.$$\n Conversely, given two solutions $v, w:\\mathbb{C}\\rightarrow \\mathbb{R}$ of the sinh-Gordon equation, one can construct a 1-parameter family of minimal immersions $X_t: \\mathbb{S}^2 \\times \\mathbb{S}^2$ by taking the following quantities as the fundamental data, \n$$\\sigma=\\frac{1}{2}\\ln\\big(4\\cosh(v+w)\\cosh(v-w)\\big),~~~ \\rho=\\left(\\ln \\sqrt{\\frac{\\cosh(v+w)}{\\cosh(v-w)}}\\right)_{\\!\\!z},~~~$$\n$$C_1=\\tanh(v-w),~~~C_2=\\tanh(v+w),$$\n$$\\gamma_1=\\sqrt{2}e^{\\frac{it}{2}}\\sqrt{\\frac{\\cosh(v+w)}{\\cosh(v-w)}},~~~\\gamma_2=\\sqrt{2}e^{\\frac{it}{2}}\\sqrt{\\frac{\\cosh(v-w)}{\\cosh(v+w)}},~~~f_1=-i \\gamma_1 (v-w)_z,~~~f_2=-i\\gamma_2(v+w)_z.$$\nGeometrically, this construction originates from the Gauss maps of two minimal surfaces in \\( \\mathbb{S}^3 \\), whose induced metrics are \\( e^{2v} |dz|^2 \\) and \\( e^{2w} |dw|^2 \\), respectively, and which share the same Hopf differential \\( \\theta(z) = \\frac{i}{2} dz^2 \\). For further details, we refer to Section~5 of \\cite{Urbano}.\n\nTherefore, we establish the claim that $p=\\pm q$, which means either $v=0$ or $w=0$. It then follows from Proposition 4 in \\cite{Urbano} that $x$ is non-full in $\\mathbb{S}^2 \\times \\mathbb{S}^2$. This means it lies in a totally geodesic hypersurface of the ambient space, which, up to an isometry, is an open subset of $\\mathbb{S}^2 \\times \\mathbb{S}^1$. \n\\end{proof}\n\\begin{remark}\\label{rk-mini}If $w=0$, then it follows from \\eqref{eq-vw} that $v$ only depends on the variable $u_1$, and satisfies \n\\begin{equation}\\label{eq-ellip}\n\\frac{d^2 v}{d u_1 ^2}=-2\\sinh(2v).\n\\end{equation}\nThe solution to this ordinary differential equation can be expressed using an elliptic function, \n\\[\nv = \\operatorname{arcsinh}\\Bigl( k \\; \\mathrm{sn}\\bigl( 2u_1 + \\delta,\\; i k \\bigr) \\Bigr),\n\\]\nwhere $\\delta$ and $k>0$ are integration constants, and $\\mathrm{sn}(\\,\\cdot,\\,ik)$ denotes the Jacobian elliptic sine function with modulus $i k$. \nApplying the classical Jacobi imaginary modulus transformation, the solution also admits the equivalent form \n\\[\nv = \\operatorname{arcsinh}\\Bigl( \\frac{k}{\\sqrt{1+k^2}} \\; \\mathrm{sn}\\bigl( \\sqrt{1+k^2}\\,(2u_1 + \\delta),\\; \\frac{k}{\\sqrt{1+k^2}} \\bigr) \\Bigr).\\]", + "post_theorem_intro_text_len": 2347, + "post_theorem_intro_text": "\\noindent In other words, when the ambient space is $\\mathbb{S}^2\\times\\mathbb{S}^2$, within the minimal class the Willmore condition turns out to be rigid: it forces special holomorphicity, or reduces the problem to a totally geodesic hypersurface. \n Geometrically, the last class of minimal surfaces in Theorem~\\ref{thm1} arises from a certain Gauss-map construction applied to a cohomogeneity {one} minimal surface in $\\mathbb{S}^3\\subset\\mathbb{R}^4$, see Remark~\\ref{rk-mini}. \n\nIn their work \\cite{Urbano}, Tobarro and Urbano gave a beautiful local characterization of minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ that possess no complex points. Their description is formulated in terms of a pair of solutions $v$ and $w$ to the sinh‑Gordon equation \n$$v_{z\\bar z}+\\frac{1}{2}\\sinh (2v)=0.$$\nWithin this framework, the Willmore condition introduces a new partial differential equation linking $v$ and $w$, see \\eqref{eq-mwill}. By analyzing this coupled system, we establish the theorem above. \n\nExcept for seeking Willmore surfaces among minimal surfaces, another natural approach is to use the product structure of $\\mathbb{S}^2\\times\\mathbb{S}^2$ to produce Willmore surfaces. \n\\begin{thm}\\label{thm2}\n All Willmore surfaces of product type in $\\mathbb{S}^2 \\times \\mathbb{S}^2$ are exhausted by products of an elastic curve in $\\mathbb{S}^2$ with a great circle. \n\\end{thm}\n\nNote that both of these two classes of examples yield Willmore surfaces that are non-linearly full in $\\mathbb{S}^2 \\times \\mathbb{S}^2$. No linearly full examples are currently known. We point out that constructing explicit examples, or even addressing the existence problem, in non-space-form Riemannian manifolds is particularly challenging; see the efforts in \\cite{Modino,Ikoma,Michelat} and the references therein. \n\nThe paper is organized as follows. {Section~\\ref{sec2} begins with the preliminaries on $\\mathbb{S}^2\\times \\mathbb{S}^2$ and on surfaces within it. In Section~\\ref{sec3}, we first derive the fundamental equations for surfaces in $\\mathbb{S}^2\\times \\mathbb{S}^2$, and subsequently compute the corresponding Willmore equation. Section~\\ref{sec4} is devoted to the classification of minimal-Willmore surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$. Finally, in Section~\\ref{sec5}, we classify Willmore surfaces of product type.", + "sketch": "Using the local characterization of Tobarro--Urbano \\cite{Urbano} for minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ with no complex points, the surface is described \"in terms of a pair of solutions $v$ and $w$ to the sinh-Gordon equation\" $$v_{z\\bar z}+\\tfrac12\\sinh(2v)=0.$$ \"Within this framework, the Willmore condition introduces a new partial differential equation linking $v$ and $w$, see \\eqref{eq-mwill}.\" The proof then proceeds by \"analyzing this coupled system\" (the sinh-Gordon equation together with the additional PDE imposed by the Willmore condition), which yields the classification stated in Theorem~\\ref{thm1}.", + "expanded_sketch": "Using the local characterization of Tobarro--Urbano \\cite{Urbano} for minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ with no complex points, the surface is described \"in terms of a pair of solutions $v$ and $w$ to the sinh-Gordon equation\" $$v_{z\\bar z}+\\tfrac12\\sinh(2v)=0.$$ \"Within this framework, the Willmore condition introduces a new partial differential equation linking $v$ and $w,\" namely\n\\begin{equation}\\label{eq-mwill}\n e^{it}\\cosh^3(v-w)(v+w)_{\\bar{z}} = \\cosh^3(v+w)(v-w)_z.\n \\end{equation}\nThe proof then proceeds by \"analyzing this coupled system\" (the sinh-Gordon equation together with the additional PDE imposed by the Willmore condition). To prove the main theorem, one shows that this analysis yields the stated classification.", + "expanded_theorem": "\\label{thm1}\n Let \\( x: \\Sigma \\rightarrow \\mathbb{S}^2 \\times \\mathbb{S}^2 \\) be a minimal-Willmore surface. Then either \\( x \\) is a special complex curve given by a slice or a diagonal; or, up to an isometry, is contained in a totally geodesic submanifold $\\mathbb{S}^2 \\times \\mathbb{S}^1$, and can be described by a solution of the sinh-Gordon equation in one variable.", + "theorem_type": [ + "Classification or Bijection", + "Universal" + ], + "mcq": { + "question": "A minimal-Willmore surface in \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\) means a surface \\(x:\\Sigma\\to \\mathbb{S}^2\\times\\mathbb{S}^2\\) that is both minimal and Willmore, where a Willmore surface satisfies the Euler-Lagrange equation of the Willmore functional \\(\\mathcal W=\\int_\\Sigma (|\\vec H|^2+K_{1212})\\,dA\\). A slice is a surface of the form \\(\\mathbb{S}^2\\times\\{p\\}\\) or \\(\\{p\\}\\times\\mathbb{S}^2\\), and a diagonal means a diagonal copy of \\(\\mathbb{S}^2\\) in \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\). Which statement holds for every minimal-Willmore surface \\(x:\\Sigma\\to \\mathbb{S}^2\\times\\mathbb{S}^2\\)?", + "correct_choice": { + "label": "A", + "text": "Either \\(x\\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\), the image of \\(x\\) is contained in a totally geodesic submanifold \\(\\mathbb{S}^2\\times\\mathbb{S}^1\\), and the surface is described by a solution of the sinh-Gordon equation depending on a single variable." + }, + "choices": [ + { + "label": "B", + "text": "Either \\(x\\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\), the image of \\(x\\) is contained in a totally geodesic submanifold \\(\\mathbb{S}^2\\times\\mathbb{S}^1\\), and the surface is described by a pair of independent solutions of the sinh-Gordon equation in two variables." + }, + { + "label": "C", + "text": "Either \\(x\\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\), the image of \\(x\\) is contained in a totally geodesic submanifold \\(\\mathbb{S}^2\\times\\mathbb{S}^1\\)." + }, + { + "label": "D", + "text": "Either \\(x\\) is a special complex curve, namely a slice or a diagonal; or the image of \\(x\\) is contained in a totally geodesic submanifold \\(\\mathbb{S}^2\\times\\mathbb{S}^1\\), and every such non-complex minimal-Willmore surface is described by a solution of the sinh-Gordon equation depending on two variables." + }, + { + "label": "E", + "text": "Either \\(x\\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\), the image of \\(x\\) is contained in a totally geodesic submanifold \\(\\mathbb{S}^2\\times\\mathbb{S}^1\\), and the surface is described by a solution of the sinh-Gordon equation, with no restriction that the solution depend on only one variable." + } + ], + "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": "collapse from coupled system to one-variable reduction", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "geometric_construction", + "tampered_component": "single-variable sinh-Gordon description", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "one-variable dependence conclusion", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "restriction to one-variable solutions", + "template_used": "stronger_trap" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem names the setting and asks for the complete classification, but it does not reveal the specific conclusion. The correct answer is not stated or strongly hinted beyond the topic itself." + }, + "TAS": { + "score": 1, + "justification": "This is still very close to a direct theorem-recall question: it asks for the complete classification of minimal-Willmore surfaces. However, the presence of subtly varied alternatives means it is not a pure verbatim restatement." + }, + "GPS": { + "score": 1, + "justification": "Selecting the correct option requires moderate reasoning/discrimination, especially distinguishing the complete statement from a weaker true version and from overgeneralized variants. Still, the task is mainly recognition of the theorem rather than genuine derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong: one is weaker but true, others are plausible overstatements or property confusions, and they target realistic mathematical mistakes about completeness, generality, and PDE structure." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-classification MCQ with excellent distractors and no answer leakage, but it remains largely a recall/recognition item rather than a strongly generative reasoning question." + } + }, + { + "id": "2602.05303v1", + "paper_link": "http://arxiv.org/abs/2602.05303v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "thm", + "content": "\\label{thm1}\n Let \\( x: \\Sigma \\rightarrow \\mathbb{S}^2 \\times \\mathbb{S}^2 \\) be a minimal-Willmore surface. Then either \\( x \\) is a special complex curve given by a slice or a diagonal; or, up to an isometry, is contained in a totally geodesic submanifold $\\mathbb{S}^2 \\times \\mathbb{S}^1$, and can be described by a solution of the sinh-Gordon equation in one variable.", + "start_pos": 4463, + "end_pos": 4860, + "label": "thm1" + }, + "ref_dict": { + "eq-mwill": "\\begin{equation}\\label{eq-mwill}\n e^{it}\\cosh^3(v-w)(v+w)_{\\bar{z}} = \\cosh^3(v+w)(v-w)_z.\n \\end{equation}", + "rk-mini": "\\begin{remark}\\label{rk-mini}If $w=0$, then it follows from \\eqref{eq-vw} that $v$ only depends on the variable $u_1$, and satisfies \n\\begin{equation}\\label{eq-ellip}\n\\frac{d^2 v}{d u_1 ^2}=-2\\sinh(2v).\n\\end{equation}\nThe solution to this ordinary differential equation can be expressed using an elliptic function, \n\\[\nv = \\operatorname{arcsinh}\\Bigl( k \\; \\mathrm{sn}\\bigl( 2u_1 + \\delta,\\; i k \\bigr) \\Bigr),\n\\]\nwhere $\\delta$ and $k>0$ are integration constants, and $\\mathrm{sn}(\\,\\cdot,\\,ik)$ denotes the Jacobian elliptic sine function with modulus $i k$. \nApplying the classical Jacobi imaginary modulus transformation, the solution also admits the equivalent form \n\\[\nv = \\operatorname{arcsinh}\\Bigl( \\frac{k}{\\sqrt{1+k^2}} \\; \\mathrm{sn}\\bigl( \\sqrt{1+k^2}\\,(2u_1 + \\delta),\\; \\frac{k}{\\sqrt{1+k^2}} \\bigr) \\Bigr).\\]\n\nGeometrically, given a solution of~\\eqref{eq-ellip}, one first obtains a minimal surface $\\psi:\\mathbb{C}\\rightarrow \\mathbb{S}^3$ with induced metric $e^{2v}|dz|^2$ \nand Hopf differential $\\frac i 2 dz\\otimes dz$. \nSuch a surface should be necessarily homogeneous or of cohomogeneity one and hence belongs to the $T_{m,k,a}$ family constructed and classified by Hsiang and Lawson in \\cite{Hsiang-Lawson}. This family contains infinitely many closed minimal surfaces, whose topology is either a torus or a Klein bottle. Denote by $N$ the Gauss map of such a surface. Then the map \\emph{(}see the proof of Corollary $1$ in \\cite{Urbano}\\emph{)}\n\\[\n\\phi(z)=(V_\\psi, e^{2i u_2}):\\mathbb{C}\\rightarrow \\mathbb{S}^2\\times\\mathbb{S}^1\n\\]\ndefines a minimal surface in $\\mathbb{S}^2\\times\\mathbb{S}^1$, where\n\\[\nV_\\psi=\\frac{1}{\\sqrt{2}}(-2ie^{-2v}\\psi_{z}\\wedge\\psi_{\\overline{z}}+\\psi\\wedge N):\\mathbb{C}\\rightarrow \\mathbb{S}^2\\subset \\Lambda^2\\mathbb{R}^4. \n\\]\nApart from the obvious slices, these surfaces constitute all minimal-Willmore surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^1$. \n\\end{remark}", + "thm1": "\\begin{thm}\\label{thm1}\n Let \\( x: \\Sigma \\rightarrow \\mathbb{S}^2 \\times \\mathbb{S}^2 \\) be a minimal-Willmore surface. Then either \\( x \\) is a special complex curve given by a slice or a diagonal; or, up to an isometry, is contained in a totally geodesic submanifold $\\mathbb{S}^2 \\times \\mathbb{S}^1$, and can be described by a solution of the sinh-Gordon equation in one variable.\n\\end{thm}" + }, + "pre_theorem_intro_text_len": 2863, + "pre_theorem_intro_text": "For a closed surface $x:\\Sigma\\rightarrow (N^n,g)$ in a Riemannian manifold, the squared $L^2$-norm of the trace-free second fundamental form is a fundamental conformal invariant; that is, it is preserved under conformal changes of the metric. This functional, referred to as the conformally invariant Willmore functional \\cite{Michelat,Mondino}, will henceforth be called simply the Willmore functional. The Willmore functional is equivalent, up to a topological invariant, to the integral \n$$\\int_{\\Sigma}(|\\vec{H}|^2+K_{1212})dA,$$\nwhere $\\vec{H}$ is the mean curvature vector and $K_{1212}$ denotes the sectional curvature of $(N^n,g)$ restricted to the tangent plane of the surface. In what follows, we will denote this integral by $\\mathcal{W}$ and also refer to it as the Willmore functional. This functional exhibits notable links with other fundamental quantities, including, among others, the renormalized area functional within the AdS/CFT correspondence \\cite{Alexakis, Graham}. Surfaces that satisfy the Euler-Lagrange equation of this functional are called {\\em Willmore surfaces}. \n\nWhen the ambient space is a real space form, both the Willmore functional and Willmore surfaces have been extensively investigated. Notable advances include the resolution of the Willmore conjecture in $\\mathbb{S}^3$, originally proposed by Willmore in \\cite{Willmore} and proved by Marques and Neves in \\cite{Marques}. Regarding the construction and classification of Willmore surfaces, we refer to \\cite{Bryant, Ejiri, WD, MWW} and references therein. A basic fact is that every minimal surface in a real space form is automatically Willmore. \nWhen the ambient space is a non-space form Riemannian manifold, the Euler–Lagrange equation of the Willmore functional has been derived independently by several groups of geometers from different viewpoints; see, for example, \\cite{Hu-Li, Mondino, Pedit, Wang-Xie}. Owing to the presence of ambient curvature terms in the Euler–Lagrange equation, not every minimal surface is Willmore in a non-space form Riemannian manifold. In \\cite{Montiel-Urbano}, Montiel and Urbano proved that the only surfaces in $\\mathbb{CP}^2$ that are both minimal and Willmore are the superminimal surfaces of positive spin, i.e., complex curves and minimal Lagrangian surfaces. Recently, the last two authors of this paper have generalized this result to the self-dual K\\\"ahler-Einstein surfaces (see Proposition 2.3 in \\cite{Wang-Xie}). Note that $\\mathbb{S}^2 \\times \\mathbb{S}^2$ (equipped with the standard product metric and complex structure), being neither self-dual nor anti-self-dual, is another canonical example of a K\\\"ahler-Einstein surface. This motivates the study of which minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ are Willmore. We refer to a surface as minimal-Willmore if it is both minimal and Willmore.", + "context": "For a closed surface $x:\\Sigma\\rightarrow (N^n,g)$ in a Riemannian manifold, the squared $L^2$-norm of the trace-free second fundamental form is a fundamental conformal invariant; that is, it is preserved under conformal changes of the metric. This functional, referred to as the conformally invariant Willmore functional \\cite{Michelat,Mondino}, will henceforth be called simply the Willmore functional. The Willmore functional is equivalent, up to a topological invariant, to the integral \n$$\\int_{\\Sigma}(|\\vec{H}|^2+K_{1212})dA,$$\nwhere $\\vec{H}$ is the mean curvature vector and $K_{1212}$ denotes the sectional curvature of $(N^n,g)$ restricted to the tangent plane of the surface. In what follows, we will denote this integral by $\\mathcal{W}$ and also refer to it as the Willmore functional. This functional exhibits notable links with other fundamental quantities, including, among others, the renormalized area functional within the AdS/CFT correspondence \\cite{Alexakis, Graham}. Surfaces that satisfy the Euler-Lagrange equation of this functional are called {\\em Willmore surfaces}.\n\nWhen the ambient space is a real space form, both the Willmore functional and Willmore surfaces have been extensively investigated. Notable advances include the resolution of the Willmore conjecture in $\\mathbb{S}^3$, originally proposed by Willmore in \\cite{Willmore} and proved by Marques and Neves in \\cite{Marques}. Regarding the construction and classification of Willmore surfaces, we refer to \\cite{Bryant, Ejiri, WD, MWW} and references therein. A basic fact is that every minimal surface in a real space form is automatically Willmore. \nWhen the ambient space is a non-space form Riemannian manifold, the Euler–Lagrange equation of the Willmore functional has been derived independently by several groups of geometers from different viewpoints; see, for example, \\cite{Hu-Li, Mondino, Pedit, Wang-Xie}. Owing to the presence of ambient curvature terms in the Euler–Lagrange equation, not every minimal surface is Willmore in a non-space form Riemannian manifold. In \\cite{Montiel-Urbano}, Montiel and Urbano proved that the only surfaces in $\\mathbb{CP}^2$ that are both minimal and Willmore are the superminimal surfaces of positive spin, i.e., complex curves and minimal Lagrangian surfaces. Recently, the last two authors of this paper have generalized this result to the self-dual K\\\"ahler-Einstein surfaces (see Proposition 2.3 in \\cite{Wang-Xie}). Note that $\\mathbb{S}^2 \\times \\mathbb{S}^2$ (equipped with the standard product metric and complex structure), being neither self-dual nor anti-self-dual, is another canonical example of a K\\\"ahler-Einstein surface. This motivates the study of which minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ are Willmore. We refer to a surface as minimal-Willmore if it is both minimal and Willmore.", + "full_context": "For a closed surface $x:\\Sigma\\rightarrow (N^n,g)$ in a Riemannian manifold, the squared $L^2$-norm of the trace-free second fundamental form is a fundamental conformal invariant; that is, it is preserved under conformal changes of the metric. This functional, referred to as the conformally invariant Willmore functional \\cite{Michelat,Mondino}, will henceforth be called simply the Willmore functional. The Willmore functional is equivalent, up to a topological invariant, to the integral \n$$\\int_{\\Sigma}(|\\vec{H}|^2+K_{1212})dA,$$\nwhere $\\vec{H}$ is the mean curvature vector and $K_{1212}$ denotes the sectional curvature of $(N^n,g)$ restricted to the tangent plane of the surface. In what follows, we will denote this integral by $\\mathcal{W}$ and also refer to it as the Willmore functional. This functional exhibits notable links with other fundamental quantities, including, among others, the renormalized area functional within the AdS/CFT correspondence \\cite{Alexakis, Graham}. Surfaces that satisfy the Euler-Lagrange equation of this functional are called {\\em Willmore surfaces}.\n\nWhen the ambient space is a real space form, both the Willmore functional and Willmore surfaces have been extensively investigated. Notable advances include the resolution of the Willmore conjecture in $\\mathbb{S}^3$, originally proposed by Willmore in \\cite{Willmore} and proved by Marques and Neves in \\cite{Marques}. Regarding the construction and classification of Willmore surfaces, we refer to \\cite{Bryant, Ejiri, WD, MWW} and references therein. A basic fact is that every minimal surface in a real space form is automatically Willmore. \nWhen the ambient space is a non-space form Riemannian manifold, the Euler–Lagrange equation of the Willmore functional has been derived independently by several groups of geometers from different viewpoints; see, for example, \\cite{Hu-Li, Mondino, Pedit, Wang-Xie}. Owing to the presence of ambient curvature terms in the Euler–Lagrange equation, not every minimal surface is Willmore in a non-space form Riemannian manifold. In \\cite{Montiel-Urbano}, Montiel and Urbano proved that the only surfaces in $\\mathbb{CP}^2$ that are both minimal and Willmore are the superminimal surfaces of positive spin, i.e., complex curves and minimal Lagrangian surfaces. Recently, the last two authors of this paper have generalized this result to the self-dual K\\\"ahler-Einstein surfaces (see Proposition 2.3 in \\cite{Wang-Xie}). Note that $\\mathbb{S}^2 \\times \\mathbb{S}^2$ (equipped with the standard product metric and complex structure), being neither self-dual nor anti-self-dual, is another canonical example of a K\\\"ahler-Einstein surface. This motivates the study of which minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ are Willmore. We refer to a surface as minimal-Willmore if it is both minimal and Willmore.\n\n\\section{Introduction}\nFor a closed surface $x:\\Sigma\\rightarrow (N^n,g)$ in a Riemannian manifold, the squared $L^2$-norm of the trace-free second fundamental form is a fundamental conformal invariant; that is, it is preserved under conformal changes of the metric. This functional, referred to as the conformally invariant Willmore functional \\cite{Michelat,Mondino}, will henceforth be called simply the Willmore functional. The Willmore functional is equivalent, up to a topological invariant, to the integral \n$$\\int_{\\Sigma}(|\\vec{H}|^2+K_{1212})dA,$$\nwhere $\\vec{H}$ is the mean curvature vector and $K_{1212}$ denotes the sectional curvature of $(N^n,g)$ restricted to the tangent plane of the surface. In what follows, we will denote this integral by $\\mathcal{W}$ and also refer to it as the Willmore functional. This functional exhibits notable links with other fundamental quantities, including, among others, the renormalized area functional within the AdS/CFT correspondence \\cite{Alexakis, Graham}. Surfaces that satisfy the Euler-Lagrange equation of this functional are called {\\em Willmore surfaces}.\n\nIn their work \\cite{Urbano}, Tobarro and Urbano gave a beautiful local characterization of minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ that possess no complex points. Their description is formulated in terms of a pair of solutions $v$ and $w$ to the sinh‑Gordon equation \n$$v_{z\\bar z}+\\frac{1}{2}\\sinh (2v)=0.$$\nWithin this framework, the Willmore condition introduces a new partial differential equation linking $v$ and $w$, see \\eqref{eq-mwill}. By analyzing this coupled system, we establish the theorem above.\n\n\\begin{example}\\label{ex-slice} For any point $p\\in\\mathbb{S}^2$, the corresponding slices \n\\[\n\\mathbb{S}^2\\times\\{p\\}=\\{(x,p)\\in\\mathbb{S}^2\\times\\mathbb{S}^2 \\mid x\\in\\mathbb{S}^2\\},\n\\]\n\\[\n\\{p\\}\\times\\mathbb{S}^2=\\{(p,x)\\in\\mathbb{S}^2\\times\\mathbb{S}^2 \\mid x\\in\\mathbb{S}^2\\}.\n\\]\nare totally geodesic with the first one satisfying $C_1=C_2=1$ and the second one satisfying $C_1=-C_2=1$. That is, the slices are complex with respect to both $J_1$ and $J_2$.\n\\end{example}\n\\begin{example}\\label{ex-diagonal} The diagonal \n$$D\\triangleq\\{(x,x)\\in\\mathbb{S}^2\\times\\mathbb{S}^2 \\mid x\\in\\mathbb{S}^2\\}$$\nis totally geodesic and satisfies $C_1=1$ and $C_2=0$, which means that $D$ is complex with respect to $J_1$ and Lagrangian with respect to $J_2$.\n\\end{example}\n\\begin{example}\\label{ex-Clifford}\n The torus defined by \n $$T\\triangleq\\{(x,y)\\in\\mathbb{S}^2\\times\\mathbb{S}^2 \\mid x_1=y_1=0\\}$$ \n is totally geodesic and satisfies $C_1=C_2=0$, which means that $T$ is Lagrangian for both $J_1$ and $J_2$. This surface is known as the Clifford torus in $\\mathbb{S}^2 \\times \\mathbb{S}^2$, characterized by the flatness of both its tangent and normal bundles. \n\\end{example}\n\\begin{example} \nLet $\\Sigma=\\mathbb{C}/\\Lambda$ be a torus generated by the lattice $\\Lambda=\\{m+n\\tau \\mid m,n\\in\\mathbb{Z}\\}$, where $\\tau$ is a complex number with $\\operatorname{Im}\\tau>0$. Let $\\wp:\\Sigma\\rightarrow\\mathbb{S}^2$ be the Weierstrass $\\wp$-function with a second-order pole at the origin, and $z_0\\in\\Sigma$ be a point at which $\\wp$ is not ramified. Then $x=\\big(\\wp(z),\\wp(z-z_0)\\big):\\Sigma\\rightarrow(\\mathbb{S}^2\\times\\mathbb{S}^2)$ is a holomorphic embedding with the complex structure $J_1$. Such minimal tori are referred to as Weierstrass tori.\n\\end{example}\n\n\\begin{proposition}\\label{pro-mw}\n Let $x:\\Sigma \\rightarrow \\mathbb{S}^2\\times \\mathbb{S}^2$ be a complex curve or minimal Lagrangian surface. Then $x$ is Willmore if and only if it is one of the surfaces given in Example~\\ref{ex-slice} $\\sim$ Example~\\ref{ex-Clifford}. \n\\end{proposition}\n\\begin{proof}\nWithout loss of generality, we assume that $x$ is complex or minimal Lagrangian with respect to the complex structure $J_1$. Consequently, $C_1$ is a constant which must be either $0$ or $\\pm 1$. It follows from \\cite[Proposition 3]{Urbano} (see also \\cite{Chen}) that we only need to prove that $C_2$ is a constant.\n\nBy \\eqref{eq-C1z}, we have $f_1\\bar{\\gamma_1}=0$. Substituting this into \\eqref{eq-mW} yields $\\bar{f_1}{\\gamma_2}=0$, which, by \\eqref{eq-C2z}, implies that $C_2$ is holomorphic. Since $C_2$ is real-valued, it must be constant. \n\\end{proof}\n\\begin{theorem}\\label{thm-mw}\n Let $x:\\Sigma \\rightarrow \\mathbb{S}^2\\times \\mathbb{S}^2$ be a minimal surface without complex points. Then $x$ is Willmore if and only if up to an isometry it is a minimal-Willmore surface in $\\mathbb{S}^2\\times\\mathbb{S}^1$. \n\\end{theorem}\n\nLet $x:\\Sigma \\rightarrow \\mathbb{S}^2\\times \\mathbb{S}^2$ be a minimal immersion of a simply connected surface without complex points and $z$ be a complex coordinate such that the Hopf differential $\\Theta(z)\\triangleq\\frac{\\langle J_1 x_z, J_2 x_z\\rangle}{2}dz^2$ (which is holomorphic) satisfies $\\Theta(z)=dz^2$. \n Then the functions \n $$v\\triangleq\\frac{1}{2}\\ln \\sqrt{\\frac{(1+C_1)(1+C_2)}{(1-C_1)(1-C_2)}},~~~w\\triangleq\\frac{1}{2}\\ln \\sqrt{\\frac{(1-C_1)(1+C_2)}{(1+C_1)(1-C_2)}}$$\n satisfy the sinh-Gordon equation, i.e., \n $$v_{z\\bar z}+\\frac{1}{2}\\sinh (2v)=0,~~~w_{z\\bar z}+\\frac{1}{2}\\sinh (2w)=0.$$\n Conversely, given two solutions $v, w:\\mathbb{C}\\rightarrow \\mathbb{R}$ of the sinh-Gordon equation, one can construct a 1-parameter family of minimal immersions $X_t: \\mathbb{S}^2 \\times \\mathbb{S}^2$ by taking the following quantities as the fundamental data, \n$$\\sigma=\\frac{1}{2}\\ln\\big(4\\cosh(v+w)\\cosh(v-w)\\big),~~~ \\rho=\\left(\\ln \\sqrt{\\frac{\\cosh(v+w)}{\\cosh(v-w)}}\\right)_{\\!\\!z},~~~$$\n$$C_1=\\tanh(v-w),~~~C_2=\\tanh(v+w),$$\n$$\\gamma_1=\\sqrt{2}e^{\\frac{it}{2}}\\sqrt{\\frac{\\cosh(v+w)}{\\cosh(v-w)}},~~~\\gamma_2=\\sqrt{2}e^{\\frac{it}{2}}\\sqrt{\\frac{\\cosh(v-w)}{\\cosh(v+w)}},~~~f_1=-i \\gamma_1 (v-w)_z,~~~f_2=-i\\gamma_2(v+w)_z.$$\nGeometrically, this construction originates from the Gauss maps of two minimal surfaces in \\( \\mathbb{S}^3 \\), whose induced metrics are \\( e^{2v} |dz|^2 \\) and \\( e^{2w} |dw|^2 \\), respectively, and which share the same Hopf differential \\( \\theta(z) = \\frac{i}{2} dz^2 \\). For further details, we refer to Section~5 of \\cite{Urbano}.\n\nTherefore, we establish the claim that $p=\\pm q$, which means either $v=0$ or $w=0$. It then follows from Proposition 4 in \\cite{Urbano} that $x$ is non-full in $\\mathbb{S}^2 \\times \\mathbb{S}^2$. This means it lies in a totally geodesic hypersurface of the ambient space, which, up to an isometry, is an open subset of $\\mathbb{S}^2 \\times \\mathbb{S}^1$. \n\\end{proof}\n\\begin{remark}\\label{rk-mini}If $w=0$, then it follows from \\eqref{eq-vw} that $v$ only depends on the variable $u_1$, and satisfies \n\\begin{equation}\\label{eq-ellip}\n\\frac{d^2 v}{d u_1 ^2}=-2\\sinh(2v).\n\\end{equation}\nThe solution to this ordinary differential equation can be expressed using an elliptic function, \n\\[\nv = \\operatorname{arcsinh}\\Bigl( k \\; \\mathrm{sn}\\bigl( 2u_1 + \\delta,\\; i k \\bigr) \\Bigr),\n\\]\nwhere $\\delta$ and $k>0$ are integration constants, and $\\mathrm{sn}(\\,\\cdot,\\,ik)$ denotes the Jacobian elliptic sine function with modulus $i k$. \nApplying the classical Jacobi imaginary modulus transformation, the solution also admits the equivalent form \n\\[\nv = \\operatorname{arcsinh}\\Bigl( \\frac{k}{\\sqrt{1+k^2}} \\; \\mathrm{sn}\\bigl( \\sqrt{1+k^2}\\,(2u_1 + \\delta),\\; \\frac{k}{\\sqrt{1+k^2}} \\bigr) \\Bigr).\\]", + "post_theorem_intro_text_len": 2347, + "post_theorem_intro_text": "\\noindent In other words, when the ambient space is $\\mathbb{S}^2\\times\\mathbb{S}^2$, within the minimal class the Willmore condition turns out to be rigid: it forces special holomorphicity, or reduces the problem to a totally geodesic hypersurface. \n Geometrically, the last class of minimal surfaces in Theorem~\\ref{thm1} arises from a certain Gauss-map construction applied to a cohomogeneity {one} minimal surface in $\\mathbb{S}^3\\subset\\mathbb{R}^4$, see Remark~\\ref{rk-mini}. \n\nIn their work \\cite{Urbano}, Tobarro and Urbano gave a beautiful local characterization of minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ that possess no complex points. Their description is formulated in terms of a pair of solutions $v$ and $w$ to the sinh‑Gordon equation \n$$v_{z\\bar z}+\\frac{1}{2}\\sinh (2v)=0.$$\nWithin this framework, the Willmore condition introduces a new partial differential equation linking $v$ and $w$, see \\eqref{eq-mwill}. By analyzing this coupled system, we establish the theorem above. \n\nExcept for seeking Willmore surfaces among minimal surfaces, another natural approach is to use the product structure of $\\mathbb{S}^2\\times\\mathbb{S}^2$ to produce Willmore surfaces. \n\\begin{thm}\\label{thm2}\n All Willmore surfaces of product type in $\\mathbb{S}^2 \\times \\mathbb{S}^2$ are exhausted by products of an elastic curve in $\\mathbb{S}^2$ with a great circle. \n\\end{thm}\n\nNote that both of these two classes of examples yield Willmore surfaces that are non-linearly full in $\\mathbb{S}^2 \\times \\mathbb{S}^2$. No linearly full examples are currently known. We point out that constructing explicit examples, or even addressing the existence problem, in non-space-form Riemannian manifolds is particularly challenging; see the efforts in \\cite{Modino,Ikoma,Michelat} and the references therein. \n\nThe paper is organized as follows. {Section~\\ref{sec2} begins with the preliminaries on $\\mathbb{S}^2\\times \\mathbb{S}^2$ and on surfaces within it. In Section~\\ref{sec3}, we first derive the fundamental equations for surfaces in $\\mathbb{S}^2\\times \\mathbb{S}^2$, and subsequently compute the corresponding Willmore equation. Section~\\ref{sec4} is devoted to the classification of minimal-Willmore surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$. Finally, in Section~\\ref{sec5}, we classify Willmore surfaces of product type.", + "sketch": "Using the local characterization of Tobarro--Urbano \\cite{Urbano} for minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ with no complex points, the surface is described \"in terms of a pair of solutions $v$ and $w$ to the sinh-Gordon equation\" $$v_{z\\bar z}+\\tfrac12\\sinh(2v)=0.$$ \"Within this framework, the Willmore condition introduces a new partial differential equation linking $v$ and $w$, see \\eqref{eq-mwill}.\" The proof then proceeds by \"analyzing this coupled system\" (the sinh-Gordon equation together with the additional PDE imposed by the Willmore condition), which yields the classification stated in Theorem~\\ref{thm1}.", + "expanded_sketch": "Using the local characterization of Tobarro--Urbano \\cite{Urbano} for minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ with no complex points, the surface is described \"in terms of a pair of solutions $v$ and $w$ to the sinh-Gordon equation\" $$v_{z\\bar z}+\\tfrac12\\sinh(2v)=0.$$ \"Within this framework, the Willmore condition introduces a new partial differential equation linking $v$ and $w,\" namely\n\\begin{equation}\\label{eq-mwill}\n e^{it}\\cosh^3(v-w)(v+w)_{\\bar{z}} = \\cosh^3(v+w)(v-w)_z.\n \\end{equation}\nThe proof then proceeds by \"analyzing this coupled system\" (the sinh-Gordon equation together with the additional PDE imposed by the Willmore condition). To prove the main theorem, one shows that this analysis yields the stated classification.", + "expanded_theorem": "\\label{thm1}\n Let \\( x: \\Sigma \\rightarrow \\mathbb{S}^2 \\times \\mathbb{S}^2 \\) be a minimal-Willmore surface. Then either \\( x \\) is a special complex curve given by a slice or a diagonal; or, up to an isometry, is contained in a totally geodesic submanifold $\\mathbb{S}^2 \\times \\mathbb{S}^1$, and can be described by a solution of the sinh-Gordon equation in one variable.", + "theorem_type": [ + "Classification or Bijection", + "Universal" + ], + "mcq": { + "question": "A minimal-Willmore surface in \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\) means a surface \\(x:\\Sigma\\to \\mathbb{S}^2\\times\\mathbb{S}^2\\) that is both minimal and Willmore, where a Willmore surface satisfies the Euler-Lagrange equation of the Willmore functional \\(\\mathcal W=\\int_\\Sigma (|\\vec H|^2+K_{1212})\\,dA\\). A slice is a surface of the form \\(\\mathbb{S}^2\\times\\{p\\}\\) or \\(\\{p\\}\\times\\mathbb{S}^2\\), and a diagonal means a diagonal copy of \\(\\mathbb{S}^2\\) in \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\). Which statement holds for every minimal-Willmore surface \\(x:\\Sigma\\to \\mathbb{S}^2\\times\\mathbb{S}^2\\)?", + "correct_choice": { + "label": "A", + "text": "Either \\(x\\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\), the image of \\(x\\) is contained in a totally geodesic submanifold \\(\\mathbb{S}^2\\times\\mathbb{S}^1\\), and the surface is described by a solution of the sinh-Gordon equation depending on a single variable." + }, + "choices": [ + { + "label": "B", + "text": "Either \\(x\\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\), the image of \\(x\\) is contained in a totally geodesic submanifold \\(\\mathbb{S}^2\\times\\mathbb{S}^1\\), and the surface is described by a pair of independent solutions of the sinh-Gordon equation in two variables." + }, + { + "label": "C", + "text": "Either \\(x\\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\), the image of \\(x\\) is contained in a totally geodesic submanifold \\(\\mathbb{S}^2\\times\\mathbb{S}^1\\)." + }, + { + "label": "D", + "text": "Either \\(x\\) is a special complex curve, namely a slice or a diagonal; or the image of \\(x\\) is contained in a totally geodesic submanifold \\(\\mathbb{S}^2\\times\\mathbb{S}^1\\), and every such non-complex minimal-Willmore surface is described by a solution of the sinh-Gordon equation depending on two variables." + }, + { + "label": "E", + "text": "Either \\(x\\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\), the image of \\(x\\) is contained in a totally geodesic submanifold \\(\\mathbb{S}^2\\times\\mathbb{S}^1\\), and the surface is described by a solution of the sinh-Gordon equation, with no restriction that the solution depend on only one variable." + } + ], + "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": "collapse from coupled system to one-variable reduction", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "geometric_construction", + "tampered_component": "single-variable sinh-Gordon description", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "one-variable dependence conclusion", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "restriction to one-variable solutions", + "template_used": "stronger_trap" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines the setting and terminology but does not reveal the classification result or explicitly hint at the one-variable sinh-Gordon reduction. The correct answer is not leaked by the wording of the question itself." + }, + "TAS": { + "score": 1, + "justification": "The item is essentially a theorem-identification question: the correct option states a precise classification theorem rather than requiring derivation from the definitions in the stem. However, it is not a pure tautology because the distractors alter meaningful logical strength and technical details." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to choose the strongest true statement over weaker or overextended variants, especially regarding the one-variable sinh-Gordon condition. Still, this mainly tests precise recall/discrimination of a known result, not substantial generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically structured: one is weaker but true, while others introduce realistic confusions about dimensional dependence or strength of the conclusion. They are distinct and align with common failure modes in reading classification theorems." + }, + "total_score": 6, + "overall_assessment": "A reasonably strong theorem-discrimination MCQ with no answer leakage and good distractors, but it leans more toward recall of a classification result than genuine generative reasoning." + } + }, + { + "id": "2602.05697v1", + "paper_link": "http://arxiv.org/abs/2602.05697v1", + "theorems_cnt": 3, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm1}\n Let $\\psi$ be a Hecke–Maass form on $X$ with spectral parameter $\\lambda\\gg1$. We have\n \\begin{equation}\\label{i2}\n \\| \\psi \\|_{L^6(X)} \\lesssim_\\varepsilon\n \\lambda^{\\frac{1}{6}-\\frac{1}{144}+\\varepsilon}.\n \\end{equation}", + "start_pos": 25651, + "end_pos": 25933, + "label": "thm1" + }, + "ref_dict": { + "eq: IS bound": "\\begin{align}\\label{eq: IS bound}\n \\|\\psi\\|_{L^\\infty(X)} \\lesssim_\\epsilon\\lambda^{\\frac{5}{12}+\\epsilon},\n\\end{align}", + "eq: HK L4": "\\begin{align}\\label{eq: HK L4}\n \\| \\psi \\|_{L^4(X)}\\lesssim_\\epsilon \\la^{\\frac{3}{152}+\\epsilon}.\n\\end{align}", + "i4": "\\begin{equation}\\label{i4}\n \\|e_\\la\\|_{L^\\infty(M)}\\lesssim_\\e \\la^{\\frac12-\\delta_2+\\e}\n \\end{equation}", + "eq: Marshall L4": "\\begin{align}\\label{eq: Marshall L4}\n \\| \\psi \\|_{L^4(X)}\\lesssim_\\epsilon \\la^{\\frac{1}{8}-\\frac{1}{56}+\\epsilon}.\n\\end{align}", + "thm1": "\\begin{theorem}\\label{thm1}\n Let $\\psi$ be a Hecke–Maass form on $X$ with spectral parameter $\\lambda\\gg1$. We have\n \\begin{equation}\\label{i2}\n \\| \\psi \\|_{L^6(X)} \\lesssim_\\epsilon\n \\lambda^{\\frac{1}{6}-\\frac{1}{144}+\\epsilon}.\n \\end{equation}\n\\end{theorem}", + "prop: bound for I_A": "\\begin{proposition}\\label{prop: bound for I_A}\n Let $\\phi\\in C^\\infty(X)$ be such that $\\|\\phi\\|_{L^{6/5}(X)}=1$. \n \\begin{enumerate}\n \\item For any $\\gamma\\in G$, $I_\\sA(\\lambda,\\phi,\\gamma)\\lesssim_\\epsilon\\lambda^{1/3+\\epsilon}$.\n \\item Fix $0<\\epsilon_0\\ll\\delta_1$. If $d(\\gamma,e)\\lesssim 1$ and $d(g_0^{-1}\\gamma g_0,A)\\geq \\lambda^{-\\delta_1+2\\epsilon_0}$, then $I_\\sA(\\lambda,\\phi,\\gamma)\\lesssim_{\\epsilon_0,N} \\lambda^{-N}$ for any $N>0$.\n \\end{enumerate}\n\\end{proposition}", + "eq: HK L6": "\\begin{align}\\label{eq: HK L6}\n \\| \\psi \\|_{L^6(X)}\\lesssim_\\epsilon \\la^{\\frac{26}{171}+\\epsilon}=\\la^{\\frac{1}{6}-\\frac{5}{342}+\\epsilon}.\n\\end{align}", + "qnusymbol": "\\begin{equation}\\label{qnusymbol}\nA_\\nu^\\theta(x,\\xi)=\\phi(x) \\, a_j^\\theta(x,\\xi) \\, \\tilde\\beta\\bigl(p(x,\\xi)/\\la\\bigr),\n\\quad \\nu =\\theta j\\in \\theta \\cdot {\\mathbb Z}^{2}.\n\\end{equation}", + "r1": "\\begin{equation}\\label{r1}\n\\|e_\\la\\|_{L^q(M)}\\lesssim \\la^{\\delta(q,n)}\\end{equation}", + "1.2": "\\begin{equation}\\label{1.2}\n(\\Delta_g+\\la^2_j )e_{\\la_j}=0, \\quad\n\\text{and } \\quad \\int_M |e_{\\la_j}(x)|^2 \\, dx=1.\n\\end{equation}", + "i3": "\\begin{equation}\\label{i3}\n \\sup_\\nu\\|A_\\nu^{\\theta_0} e_\\la\\|_{L^6(M)}\\lesssim_\\e \\la^{\\frac16-a{\\delta_1}+\\e},\n \\end{equation}", + "thm2": "\\begin{theorem}\\label{thm2}\n Let $(M,g)$ be a compact $2$-dimensional Riemannian manifold without boundary. Let $e_\\la$ be an $L^2$ normalized eigenfunction of $-\\Delta_g$ with eigenvalue $\\la^2$, satisfying \\eqref{1.2} for some $\\la\\gg1$. Set $\\theta_0=\\la^{-\\delta_1}$ with $0<\\delta_1< \\frac12$, and let $A_\\nu^{\\theta_0}$ be the pseudo-differential operator defined in \\eqref{qnusymbol}. Assume that \n\\begin{equation}\\label{i3}\n \\sup_\\nu\\|A_\\nu^{\\theta_0} e_\\la\\|_{L^6(M)}\\lesssim_\\e \\la^{\\frac16-a{\\delta_1}+\\e},\n \\end{equation}\n and \n \\begin{equation}\\label{i4}\n \\|e_\\la\\|_{L^\\infty(M)}\\lesssim_\\e \\la^{\\frac12-\\delta_2+\\e}\n \\end{equation}\nfor some constant $a>0$ and $0<\\delta_2<\\frac{1}{2}$. Then \n \\begin{equation}\\label{i5}\n \\|e_\\la\\|_{L^6(M)}\\lesssim_\\e \\la^{\\frac16-\\frac23a{\\delta_1}+\\e}+\\la^{\\frac23(-\\frac14+\\frac53\\delta_1)}+\\la^{\\frac16(1+\\delta_1-2\\delta_2)+\\e}.\n \\end{equation}\n\\end{theorem}", + "thm microlocal KN": "\\begin{theorem}\\label{thm microlocal KN}\n Let $\\psi$ be a Hecke–Maass form on $X$ with spectral parameter $\\lambda\\gg1$. Set $\\theta_0=\\la^{-\\delta_1}$ with $0<\\delta_1<\\frac12$, and let $A_\\nu^{\\theta_0}$ be the pseudo-differential operator defined in \\eqref{qnusymbol}. We have\n \\begin{equation*}\n \\sup_\\nu\\|A_\\nu^{\\theta_0} \\psi\\|_{L^6(X)}\\lesssim_\\e \\la^{\\frac16-\\frac{\\delta_1}{12}+\\e}.\n \\end{equation*}\n\\end{theorem}", + "prop: amplification inequality": "\\begin{proposition}\\label{prop: amplification inequality}\n For any $\\phi\\in C^\\infty(X)$, we have\n \\[\n \\left|\\langle \\sA\\cT\\psi,\\phi\\rangle_X\\right|^2\\leq\\sum_{m,n\\leq N} |\\alpha_m \\alpha_n | \\sum_{d|(m,n)} \\frac{d}{\\sqrt{mn}}\\sum_{\\gamma\\in R(mn/d^2)} |I_\\sA(\\la,\\phi,\\gamma)|\n \\]\n\\end{proposition}", + "eq: Marshall geodesic bound": "\\begin{align}\\label{eq: Marshall geodesic bound}\n \\sup_{\\gamma\\in\\Pi} \\| \\psi|_\\gamma \\|_{L^2(\\gamma)}\\lesssim_\\epsilon \\lambda^{\\frac{3}{14}+\\epsilon}.\n\\end{align}" + }, + "pre_theorem_intro_text_len": 9150, + "pre_theorem_intro_text": "The purpose of this paper is to obtain improved $L^q$-norm estimates for eigenfunctions on an arithmetic hyperbolic surface for the critical exponent $q=6$, which in turn also yield improved estimates for all $q>2$ by interpolation.\n\nLet $(M,g)$ be a compact $n$-dimensional Riemannian manifold without boundary, \n$\\Delta_g$ be the\nLaplace-Beltrami operator associated with the metric $g$ on $M$. We let $0=\\la_0^2<\\lambda^2_1\\le \\lambda^2_2\\le \\cdots$ denote the\neigenvalues labeled with respect to the multiplicity of \n${-\\Delta_g}$\nand $e_{\\la_j}$ the associated $L^2$-normalized eigenfunctions, that is,\n\\begin{equation}\\label{1.2}\n(\\Delta_g+\\lambda^2_j )e_{\\la_j}=0, \\quad\n\\text{and } \\quad \\int_M |e_{\\la_j}(x)|^2 \\, dx=1.\n\\end{equation}\n\nIn Sogge~\\cite{sogge881} (see also Avakumovi\\'c \\cite{Avakumovic} and Levitan \\cite{Levitan} when $p = \\infty$), the universal bound for $q\\ge2$\n\\begin{equation}\\label{r1}\n\\|e_\\lambda\\|_{L^q(M)}\\lesssim \\lambda^{\\delta(q,n)}\\end{equation}\nwas obtained. Here $\\delta(q,n)$ is given by\n\\begin{equation*}\n \\delta(q,n) = \\begin{dcases}\n \\frac{n-1}{4} - \\frac{n-1}{2q} \\;&\\text{ for }\\;2\\leq q\\leq q_c,\\\\\n \\frac{n-1}{2} - \\frac{n}{q} \\;\\;\\;\\;&\\text{ for }\\;q_c\\leq q\\leq\\infty,\n \\end{dcases}\n \\quad\\text{ and }\\quad q_c=\\frac{2(n+1)}{n-1}.\n\\end{equation*}\nThe result also applies to quasimodes that are linear combinations of eigenfunctions whose eigenvalues lie in the unit interval $[\\lambda,\\lambda+1]$. \nWhen $M$ is a compact locally symmetric space of higher rank, besides the Laplace-Beltrami operator, one can consider eigenfunctions of the full ring of differential operators on $M$. The general local bound was obtained by Sarnak \\cite{sarnak2004letter} (for $q=\\infty$) and Marshall \\cite{Mar16HigerRank} (for any $q\\ge2$ with some log losses at the corresponding critical exponents), which extends and improves the bound \\eqref{r1}. \n\nThe estimate \\eqref{r1} is sharp on the standard round sphere $\\mathbb{S}^n$ by taking $e_\\lambda$ to be the zonal functions or the highest weight spherical harmonics, which are point-focusing and geodesic-focusing eigenfunctions, respectively. Under additional geometric assumptions--such as nonpositive sectional curvature or on flat tori--one can improve \\eqref{r1}. See e.g., the work of Bourgain-Demeter \\cite[Section 2.3]{BoDe}, as well as Germain-Myerson \\cite{germain2022bounds}, and the reference therein for related results on the torus. On compact manifolds with non-positive sectional curvature,\n$(\\log\\lambda)^{-\\frac12}$ improvements to \\eqref{r1} was obtained in B\\'erard \\cite{Berard} for $q=\\infty$, and by Hassell and Tacy \\cite{HassellTacy} for $q>q_c=\\tfrac{2(n+1)}{n-1}$. The same improvement also holds, more generally, for spectral projection operators onto the shorter interval $[\\lambda,\\lambda+(\\log\\lambda)^{-1}]$. These results were further generalized under weaker dynamical assumptions in the recent work of Canzani-Galkowski \\cite{CGGrowth}. \n\nIn the other direction, for $q\\le q_c$ similar improvements were obtained in Huang--Sogge \\cite{huang2024curvature}. More generally, they proved sharp quasimode estimates for logarithmic quasimodes, which were also used to characterize compact manifolds with constant curvature. See also Blair and Sogge \\cite{blair2018concerning,SBLog} and Blair, Huang and Sogge \\cite{blair2024improved} for related earlier work in this direction.\n\n\\subsection{\\texorpdfstring{$L^q$}{Lq}-norm bounds for Hecke-Maass forms}\nThe main result of the paper concerns improvements of \\eqref{r1} for arithmetic eigenfunctions on hyperbolic surfaces. We let $\\mathbb{H}$ be the upper half plane and $X=\\Gamma\\backslash\\mathbb{H}$ an arithmetic hyperbolic surface. Here $\\Gamma\\subset \\operatorname{SL}(2,\\mathbb{R})$ is taken to be the norm one units of a maximal order of an\nindefinite quaternion division algebra over $\\mathbb{Q}$ (see Section \\ref{sec: arith hyper sur} for the precise definitions) so that $X$ is compact. Let $\\psi$ be an $L^2$-normalized Hecke–Maass form on $X$. Namely, $\\int_X|\\psi(x)|^2dx=1$ and $\\psi$ is a joint eigenfunction of the Laplace-Beltrami operator $\\Delta_g$ and Hecke operators on $X$. Let $\\lambda$ be the spectral parameter of $\\psi$, so that $\\Delta \\psi+(\\frac14+\\lambda^2)\\psi=0$. As we are considering large-eigenvalue asymptotics, we will also assume that $\\lambda$ is real and $\\lambda\\gg1$. Iwaniec and Sarnak \\cite{IS95} made a Lindel\\\"of hypothesis type conjecture that $\\|\\psi\\|_{L^\\infty(X)}\\lesssim_\\varepsilon\\lambda^{\\varepsilon}$, which is strong and widely open. The conjecture implies that $\\|\\psi\\|_{L^q(X)}\\lesssim_\\varepsilon\\lambda^{\\varepsilon}$ for any $q>2$.\n\nFor these $X$ and $\\psi$, the first power-saving result was obtained by Iwaniec and Sarnak \\cite{IS95}. They showed that \n\\begin{align}\\label{eq: IS bound}\n \\|\\psi\\|_{L^\\infty(X)} \\lesssim_\\varepsilon\\lambda^{\\frac{5}{12}+\\varepsilon},\n\\end{align}\nwhich is a $\\lambda^{\\frac1{12}}$ improvement over the uniform bound \\eqref{r1} for $n=2$ and $q=\\infty$. In two-dimensional cases, the critical exponent is $q_c=6$, by interpolation, \\eqref{eq: IS bound} implies an improvement for $ \\| \\psi \\|_{L^q(X)}$ for all $q>6$. Their proof uses the technique known as arithmetic amplification. This approach has since been used by many authors to bound $L^\\infty$-norms of Hecke-Maass forms on other groups. For instance, see \\cite{BM13,Mar14,BHM16,BM16,BP16,BHM20} for results in the spectral aspect.\n\nFor $q<6$, Marshall \\cite{Mar16} studied the related $L^2$ geodesic restriction problem for the Hecke-Maass form $\\psi$ and obtained a similar $\\lambda$-power improvement over the uniform result of Burq, G{\\'e}rard and Tzvetkov \\cite{BGTrestr}. We let $\\varPi$ denote the space of all unit-length geodesic segments on $X$. Marshall obtained the uniform bound\n\\begin{align}\\label{eq: Marshall geodesic bound}\n \\sup_{\\gamma\\in\\varPi} \\| \\psi|_\\gamma \\|_{L^2(\\gamma)}\\lesssim_\\varepsilon \\lambda^{\\frac{3}{14}+\\varepsilon}.\n\\end{align}\nHe reduced $L^2$-norm bounds for $\\psi$ along $\\gamma\\in\\varPi$ to bounds for various\nFourier coefficients along $\\gamma$.\nHis improvements over the local bounds extend the technique of arithmetic amplification developed by Iwaniec-Sarnak.\nUsing a result of Blair and Sogge \\cite[Theorem 1.1]{blair2015refined}, the geodesic restriction bound \\eqref{eq: Marshall geodesic bound} implies that\n\\begin{align}\\label{eq: Marshall L4}\n \\| \\psi \\|_{L^4(X)}\\lesssim_\\varepsilon \\lambda^{\\frac{1}{8}-\\frac{1}{56}+\\varepsilon}.\n\\end{align}\nMoreover, \\eqref{eq: Marshall L4} also implies an improvement over \\eqref{r1} for all $22$.\n\nFor these $X$ and $\\psi$, the first power-saving result was obtained by Iwaniec and Sarnak \\cite{IS95}. They showed that \n\\begin{align}\\label{eq: IS bound}\n \\|\\psi\\|_{L^\\infty(X)} \\lesssim_\\varepsilon\\lambda^{\\frac{5}{12}+\\varepsilon},\n\\end{align}\nwhich is a $\\lambda^{\\frac1{12}}$ improvement over the uniform bound \\eqref{r1} for $n=2$ and $q=\\infty$. In two-dimensional cases, the critical exponent is $q_c=6$, by interpolation, \\eqref{eq: IS bound} implies an improvement for $ \\| \\psi \\|_{L^q(X)}$ for all $q>6$. Their proof uses the technique known as arithmetic amplification. This approach has since been used by many authors to bound $L^\\infty$-norms of Hecke-Maass forms on other groups. For instance, see \\cite{BM13,Mar14,BHM16,BM16,BP16,BHM20} for results in the spectral aspect.\n\nFor $q<6$, Marshall \\cite{Mar16} studied the related $L^2$ geodesic restriction problem for the Hecke-Maass form $\\psi$ and obtained a similar $\\lambda$-power improvement over the uniform result of Burq, G{\\'e}rard and Tzvetkov \\cite{BGTrestr}. We let $\\varPi$ denote the space of all unit-length geodesic segments on $X$. Marshall obtained the uniform bound\n\\begin{align}\\label{eq: Marshall geodesic bound}\n \\sup_{\\gamma\\in\\varPi} \\| \\psi|_\\gamma \\|_{L^2(\\gamma)}\\lesssim_\\varepsilon \\lambda^{\\frac{3}{14}+\\varepsilon}.\n\\end{align}\nHe reduced $L^2$-norm bounds for $\\psi$ along $\\gamma\\in\\varPi$ to bounds for various\nFourier coefficients along $\\gamma$.\nHis improvements over the local bounds extend the technique of arithmetic amplification developed by Iwaniec-Sarnak.\nUsing a result of Blair and Sogge \\cite[Theorem 1.1]{blair2015refined}, the geodesic restriction bound \\eqref{eq: Marshall geodesic bound} implies that\n\\begin{align}\\label{eq: Marshall L4}\n \\| \\psi \\|_{L^4(X)}\\lesssim_\\varepsilon \\lambda^{\\frac{1}{8}-\\frac{1}{56}+\\varepsilon}.\n\\end{align}\nMoreover, \\eqref{eq: Marshall L4} also implies an improvement over \\eqref{r1} for all $22$.\n\nFor these $X$ and $\\psi$, the first power-saving result was obtained by Iwaniec and Sarnak \\cite{IS95}. They showed that \n\\begin{align}\\label{eq: IS bound}\n \\|\\psi\\|_{L^\\infty(X)} \\lesssim_\\varepsilon\\lambda^{\\frac{5}{12}+\\varepsilon},\n\\end{align}\nwhich is a $\\lambda^{\\frac1{12}}$ improvement over the uniform bound \\eqref{r1} for $n=2$ and $q=\\infty$. In two-dimensional cases, the critical exponent is $q_c=6$, by interpolation, \\eqref{eq: IS bound} implies an improvement for $ \\| \\psi \\|_{L^q(X)}$ for all $q>6$. Their proof uses the technique known as arithmetic amplification. This approach has since been used by many authors to bound $L^\\infty$-norms of Hecke-Maass forms on other groups. For instance, see \\cite{BM13,Mar14,BHM16,BM16,BP16,BHM20} for results in the spectral aspect.\n\nFor $q<6$, Marshall \\cite{Mar16} studied the related $L^2$ geodesic restriction problem for the Hecke-Maass form $\\psi$ and obtained a similar $\\lambda$-power improvement over the uniform result of Burq, G{\\'e}rard and Tzvetkov \\cite{BGTrestr}. We let $\\varPi$ denote the space of all unit-length geodesic segments on $X$. Marshall obtained the uniform bound\n\\begin{align}\\label{eq: Marshall geodesic bound}\n \\sup_{\\gamma\\in\\varPi} \\| \\psi|_\\gamma \\|_{L^2(\\gamma)}\\lesssim_\\varepsilon \\lambda^{\\frac{3}{14}+\\varepsilon}.\n\\end{align}\nHe reduced $L^2$-norm bounds for $\\psi$ along $\\gamma\\in\\varPi$ to bounds for various\nFourier coefficients along $\\gamma$.\nHis improvements over the local bounds extend the technique of arithmetic amplification developed by Iwaniec-Sarnak.\nUsing a result of Blair and Sogge \\cite[Theorem 1.1]{blair2015refined}, the geodesic restriction bound \\eqref{eq: Marshall geodesic bound} implies that\n\\begin{align}\\label{eq: Marshall L4}\n \\| \\psi \\|_{L^4(X)}\\lesssim_\\varepsilon \\lambda^{\\frac{1}{8}-\\frac{1}{56}+\\varepsilon}.\n\\end{align}\nMoreover, \\eqref{eq: Marshall L4} also implies an improvement over \\eqref{r1} for all $26$. Their proof uses the technique known as arithmetic amplification. This approach has since been used by many authors to bound $L^\\infty$-norms of Hecke-Maass forms on other groups. For instance, see \\cite{BM13,Mar14,BHM16,BM16,BP16,BHM20} for results in the spectral aspect.\n\nOn the noncompact quotient $\\SL(2,\\BZ)\\backslash\\BH$, for a Hecke-Maass cusp form $\\psi$ whose spectral parameter is $\\la$, both $L^\\infty$-norm bound \\eqref{eq: IS bound} and $L^4$-norm bound \\eqref{eq: HK L4} hold as well, so one still has the same $L^6$-norm bound \\eqref{eq: HK L6}. In \\cite{ki20234}, Ki showed an essentially sharp upper bound $\\|\\psi\\|_{L^4(\\SL(2,\\BZ)\\backslash\\BH)}\\lesssim_\\epsilon\\la^{\\epsilon}$, by using the\nFourier-Whittaker expansions of Hecke-Maass cusp forms. This result justifies Iwaniec-Sarnak's conjecture for $20$ and $0<\\delta_2<\\frac{1}{2}$. Then \n \\begin{equation}\\label{i5}\n \\|e_\\la\\|_{L^6(M)}\\lesssim_\\e \\la^{\\frac16-\\frac23a{\\delta_1}+\\e}+\\la^{\\frac23(-\\frac14+\\frac53\\delta_1)}+\\la^{\\frac16(1+\\delta_1-2\\delta_2)+\\e}.\n \\end{equation}\n\\end{theorem}\nFor each fixed $\\nu$, the operator $A_\\nu^{\\theta_0}$ is microlocally supported in a $\\theta_0$-neighborhood of a geodesic segment and is frequency-localized to directions near that geodesic; the precise definition will be given in Section \\ref{sec: microlocal decomposition}. The proof of Theorem~\\ref{thm2} follows a strategy similar to that of \\cite{huang2024curvature}, relying on bilinear techniques from harmonic analysis together with the improved $L^\\infty$ estimate \\eqref{i4}. We will prove Theorem~\\ref{thm2} in Section \\ref{sec: 2}. \nThe idea of using improved microlocal $L^6$-norm estimate to derive improved global $L^6$-norm estimate is inspired by the recent work \\cite{blair2024improved}.\n\n\\begin{theorem}\\label{thm microlocal KN}\n Let $\\psi$ be a Hecke–Maass form on $X$ with spectral parameter $\\lambda\\gg1$. Set $\\theta_0=\\la^{-\\delta_1}$ with $0<\\delta_1<\\frac12$, and let $A_\\nu^{\\theta_0}$ be the pseudo-differential operator defined in \\eqref{qnusymbol}. We have\n \\begin{equation*}\n \\sup_\\nu\\|A_\\nu^{\\theta_0} \\psi\\|_{L^6(X)}\\lesssim_\\e \\la^{\\frac16-\\frac{\\delta_1}{12}+\\e}.\n \\end{equation*}\n\\end{theorem}\n\nWe let $\\Delta_g$ be the Laplace-Beltrami operator on $\\BH$ and $X$, which is induced from the standard hyperbolic Riemannian metric. Let $\\psi\\in L^2(X)$ be a Hecke-Maass form that is an eigenfunction of $\\Delta_g$ and the operators $T_n$ with $(n,q)=1$. We let $\\lambda(n)$ be the Hecke eigenvalues of $\\psi$ and $\\lambda$ be its spectral parameter, so that\n\\begin{equation}\\label{def}\n \\begin{aligned}\n &T_n\\psi=\\lambda(n)\\psi,\\\\\n &\\Delta_g\\psi+(\\frac{1}{4}+\\lambda^2)\\psi=0. \n \\end{aligned}\n\\end{equation}\nWe assume that $\\| \\psi \\|_{L^2(X)}=1$ with respect to the hyperbolic volume on $X$ and assume that $\\lambda\\gg1$. Note that because $\\Delta_g$ and $T_n$ with $(n,q)=1$ are self-adjoint, we may assume that $\\psi$ is real-valued. For functions on $X$, we will also think of them as functions on $G$ that are left $\\Gamma$-invariant and right $K$-invariant.\n\n\\begin{proof}[Proof of Theorem \\ref{thm microlocal KN}]\n By Lemma \\ref{lem 3.10}, we may bound $\\| \\sA\\psi\\|_{L^6(X)}$ instead, and moreover by duality, it suffices to bound $\\langle\\sA\\psi,\\phi\\rangle_X$ uniformly for any $\\phi\\in C^\\infty(X)$ with $\\|\\phi\\|_{L^{6/5}(X)}=1$.\n By Proposition \\ref{prop: amplification inequality}, we have\n \\begin{align}\\label{eq: amplification inequality}\n \\left|\\langle \\sA\\cT\\psi,\\phi\\rangle_X\\right|^2\\leq\\sum_{m,n\\leq N} |\\alpha_m \\alpha_n | \\sum_{d|(m,n)} \\frac{d}{\\sqrt{mn}}\\sum_{\\gamma\\in R(mn/d^2)} |I_\\sA(\\la,\\phi,\\gamma )|.\n \\end{align}\n Fix $0<\\epsilon_0\\ll\\delta_1$. By assuming the support of $k_\\la$ and $\\Omega$ are sufficiently small, and by Proposition \\ref{prop: bound for I_A}, we only need to consider the terms in \\eqref{eq: amplification inequality} with $d(g_0^{-1}\\gamma g_0,e)\\leq1$ and $d(g_0^{-1}\\gamma g_0,A)\\leq\\lambda^{-\\delta_1+2\\epsilon_0}$. Lemma \\ref{lem: Hecke return} gives\n \\[\n M(g_0,n,\\lambda^{-\\delta_1 +2\\epsilon_0})\\ll_\\epsilon n^\\epsilon \\lambda^{\\epsilon_0+\\epsilon}(n\\lambda^{-\\delta_1/2}+1),\n \\]\n and so by the uniform estimate $|I_\\sA(\\la,\\phi,\\gamma )|\\lesssim_\\epsilon \\la^{1/3+\\epsilon}$ from Proposition \\ref{prop: bound for I_A} we have\n \\begin{align*}\n &\\sum_{m,n\\leq N} |\\alpha_m\\alpha_n| \\sum_{d|(m,n)} \\frac{d}{\\sqrt{mn}}\\sum_{\\gamma\\in R(mn/d^2)}\\left| I_{\\sA}(\\lambda, \\phi, \\gamma ) \\right|\\\\\n \\lesssim& \\sum_{m,n\\leq N} |\\alpha_m\\alpha_n| \\sum_{d|(m,n)} \\frac{d}{\\sqrt{mn}}\\lambda^{1/3+\\e} M(g_0,\\frac{mn}{d^2},\\lambda^{-\\delta_1+2\\epsilon_0})+O_{\\epsilon_0,A}(\\lambda^{-A})\\\\\n \\lesssim&_\\epsilon N^\\epsilon\\lambda^{\\epsilon_0+\\epsilon}\\sum_{m,n\\leq N} |\\alpha_m\\alpha_n| \\sum_{d|(m,n)} \\left(\\frac{\\sqrt{mn}}{d}\\lambda^{1/3-\\delta_1/2}+\\frac{d}{\\sqrt{mn}}\\lambda^{1/3}\\right) +O_{\\epsilon_0,A}(\\lambda^{-A}).\n \\end{align*}\n As in \\cite[p.310]{IS95}, we have\n \\begin{align*}\n \\sum_{m,n\\leq N} \\sum_{d|(m,n)}|\\alpha_m\\alpha_n|\\frac{\\sqrt{mn}}{d}\\lesssim_\\epsilon N^{1+\\epsilon} \\left( \\sum_{n\\leq N} |\\alpha_n|\\right)^2\n \\end{align*}\n and\n \\begin{align*}\n \\sum_{m,n\\leq N} \\sum_{d|(m,n)}|\\alpha_m\\alpha_n|\\frac{d}{\\sqrt{mn}}&=\\sum_{\\substack{m,n\\leq N\\\\(m,n)=1}} \\sum_{ml,nl\\leq N}\\sum_{d|l}|\\alpha_{ml}\\alpha_{nl}|\\frac{d}{l\\sqrt{mn}}\\\\\n &\\lesssim_\\epsilon N^\\epsilon\\sum_{ml,nl\\leq N}\\left( \\frac{|\\alpha_{ml}|^2}{n} + \\frac{|\\alpha_{nl}|^2}{m} \\right)\\\\\n &\\lesssim_\\epsilon N^\\epsilon \\sum_{n\\leq N} |\\alpha_n|^2.\n \\end{align*}\n Hence from \\eqref{eq: amplification inequality} we have\n \\begin{align*}\n \\left| \\langle \\sA\\cT\\psi,\\phi\\rangle_X \\right|^2 \\lesssim&_\\epsilon N^\\epsilon\\lambda^{\\epsilon_0+\\epsilon}\\left(N\\lambda^{1/3-\\delta_1/2}\\left( \\sum_{n\\leq N} |\\alpha_n|\\right)^2+\\lambda^{1/3}\\sum_{n\\leq N} |\\alpha_n|^2\\right) +O_{\\epsilon_0,A}(\\lambda^{-A}).\n \\end{align*}\n Combining this with \\eqref{eq: moment for alpha} and \\eqref{eq: lower bound for amplifier eigenvalue} and choosing $\\epsilon_0$ small gives\n \\begin{align*}\n N^{1-\\epsilon}\\left|\\langle \\sA\\psi,\\phi\\rangle_X \\right|^2\\lesssim&_\\epsilon N^\\epsilon\\lambda^{\\epsilon}\\left(N^2\\lambda^{1/3-\\delta_1/2}+N^{1/2}\\lambda^{1/3}\\right)+O_{\\epsilon,A}(\\lambda^{-A}),\n \\end{align*}\n which completes the proof by choosing $N=\\lambda^{\\delta_1/3}$.\n\\end{proof}", + "post_theorem_intro_text_len": 7172, + "post_theorem_intro_text": "\\subsection{Proof strategy}\nA main step in the proof of Theorem~\\ref{thm1} is the following.\n\\begin{theorem}\\label{thm2}\n Let $(M,g)$ be a compact $2$-dimensional Riemannian manifold without boundary. Let $e_\\lambda$ be an $L^2$ normalized eigenfunction of $-\\Delta_g$ with eigenvalue $\\lambda^2$, satisfying \\eqref{1.2} for some $\\lambda\\gg1$. Set $\\theta_0=\\lambda^{-\\delta_1}$ with $0<\\delta_1< \\frac12$, and let $A_\\nu^{\\theta_0}$ be the pseudo-differential operator defined in \\eqref{qnusymbol}. Assume that \n\\begin{equation}\\label{i3}\n \\sup_\\nu\\|A_\\nu^{\\theta_0} e_\\lambda\\|_{L^6(M)}\\lesssim_\\varepsilon \\lambda^{\\frac16-a{\\delta_1}+\\varepsilon},\n \\end{equation}\n and \n \\begin{equation}\\label{i4}\n \\|e_\\lambda\\|_{L^\\infty(M)}\\lesssim_\\varepsilon \\lambda^{\\frac12-\\delta_2+\\varepsilon}\n \\end{equation}\nfor some constant $a>0$ and $0<\\delta_2<\\frac{1}{2}$. Then \n \\begin{equation}\\label{i5}\n \\|e_\\lambda\\|_{L^6(M)}\\lesssim_\\varepsilon \\lambda^{\\frac16-\\frac23a{\\delta_1}+\\varepsilon}+\\lambda^{\\frac23(-\\frac14+\\frac53\\delta_1)}+\\lambda^{\\frac16(1+\\delta_1-2\\delta_2)+\\varepsilon}.\n \\end{equation}\n\\end{theorem}\nFor each fixed $\\nu$, the operator $A_\\nu^{\\theta_0}$ is microlocally supported in a $\\theta_0$-neighborhood of a geodesic segment and is frequency-localized to directions near that geodesic; the precise definition will be given in Section \\ref{sec: microlocal decomposition}. The proof of Theorem~\\ref{thm2} follows a strategy similar to that of \\cite{huang2024curvature}, relying on bilinear techniques from harmonic analysis together with the improved $L^\\infty$ estimate \\eqref{i4}. We will prove Theorem~\\ref{thm2} in Section \\ref{sec: 2}. \nThe idea of using improved microlocal $L^6$-norm estimate to derive improved global $L^6$-norm estimate is inspired by the recent work \\cite{blair2024improved}.\n\nNote that for a Hecke–Maass form $\\psi$ on $X$, as in Theorem \\ref{thm1}, the bound \\eqref{i4} holds with $\\delta_2=\\tfrac{1}{12}$ by Iwaniec-Sarnak's $L^\\infty$-norm bound \\eqref{eq: IS bound}. If, in addition, \\eqref{i3} holds for such $X$ and $\\psi$ with $a=\\frac1{12}$, then Theorem~\\ref{thm1} follows immediately by choosing $\\delta_1=\\frac{3}{2}\\delta_2=\\tfrac{1}{8}$.\n\n\\begin{theorem}\\label{thm microlocal KN}\n Let $\\psi$ be a Hecke–Maass form on $X$ with spectral parameter $\\lambda\\gg1$. Set $\\theta_0=\\lambda^{-\\delta_1}$ with $0<\\delta_1<\\frac12$, and let $A_\\nu^{\\theta_0}$ be the pseudo-differential operator defined in \\eqref{qnusymbol}. We have\n \\begin{equation*}\n \\sup_\\nu\\|A_\\nu^{\\theta_0} \\psi\\|_{L^6(X)}\\lesssim_\\varepsilon \\lambda^{\\frac16-\\frac{\\delta_1}{12}+\\varepsilon}.\n \\end{equation*}\n\\end{theorem}\n\nWe will prove Theorem \\ref{thm microlocal KN} in Section \\ref{sec 3}. This theorem can be viewed as the $L^6$-version of Theorem 1.1 and Lemma 3.9 in \\cite{Mar16}, where Marshall used the amplification method to bound the $L^2$-mass of $\\psi$ restricted to a geodesic segment with frequency localized. We give the amplification inequality in Proposition \\ref{prop: amplification inequality} by integrating the amplified pretrace formula against a test function under the operator $A_\\nu^{\\theta_0}$. Let $\\gamma$ be the geodesic segment centered in $A_\\nu^{\\theta_0}$.\nSimilar to usual amplification arguments, we have a counting problem and an analytic problem. The counting problem is to estimate the number of times a Hecke operator maps $\\gamma$ back close to itself, which has been studied by Marshall \\cite[Section 3.2]{Mar16}. Using the microlocal support of $A_\\nu^{\\theta_0}$, we can reduce the analytic problem to several known results from \\cite{sogge881,Mar16,Mar16HigerRank}, which will be proved in Proposition \\ref{prop: bound for I_A}.\n\n\\subsection{Generalizations}\nIn Theorem \\ref{thm1}, the arithmetic assumption that $\\psi$ is a Hecke eigenfunction is only used to invoke the bound Iwaniec-Sarnak \\cite{IS95} and to prove Theorem \\ref{thm microlocal KN}, both of which rely on arithmetic amplification. Hence, the maximal order in the theorem can be replaced with an Eichler order more generally, as in \\cite{IS95}. Moreover, we use only unramified Hecke operators, and we may assume $\\psi$ to be an eigenfunction under even fewer Hecke operators as long as the amplification argument can be adapted.\n\nThe strategy developed in this paper can also be generalized to study the $L^{q_c}$-norm problems for Hecke-Maass forms on compact locally symmetric spaces of rank one, as long as the corresponding $L^\\infty$-norm problem and microlocal $L^{q_c}$ Kakeya-Nikodym problem can be solved by the arithmetic amplification technique; we will explore this direction in future work.\nTheorem \\ref{thm2} is expected to be generalized to Laplace-Beltrami eigenfunctions on compact Riemannian manifolds of any dimension, which allows us to control the $L^{q_c}$-norms by the $L^\\infty$-norms and the microlocal Kakeya-Nikodym norms. \nFor example, when the manifold is taken to be a compact arithmetic hyperbolic 3-manifold (the critical exponent is $q_c=4$), the corresponding $L^\\infty$-norm problem for Hecke-Maass forms is expected to be solved (see \\cite{BHM16} in the cases of noncompact arithmetic\nhyperbolic 3-manifolds with cusps). The corresponding microlocal $L^4$ Kakeya-Nikodym estimate should be established in a similar way as Theorem \\ref{thm microlocal KN}, by modifying the proof for the usual $L^2$ Kakeya-Nikodym estimate in \\cite{hou2024restrictions}.\n\n\\subsection{Notation}\nThroughout the paper, the notation $A\\lesssim B$ means that there is a positive constant $C$ such that $|A| \\leq C B$, and $A\\sim B$ means that $A\\lesssim B\\lesssim A$. We also use $A =O(B)$ to mean $A\\lesssim B$. The notation $A\\gg B$ means there is a sufficiently large positive constant $C$ such that $A\\ge CB$, and similarly for $A\\ll B$.\n\nIf $f\\in L^1(\\mathbb{R}^n)$, the Fourier transform $\\hat{f}$ of $f$ in this paper is defined as\n\\begin{align*}\n \\hat{f}(\\xi)=\\int_{\\mathbb{R}^n} f(x)e^{-i\\langle x,\\xi\\rangle} dx. \n\\end{align*}\nHere $x=(x_i),\\xi=(\\xi_i)\\in\\mathbb{R}^n$ and $\\langle x,\\xi\\rangle=\\sum_i x_i\\xi_i$.\nIf $f\\in\\mathcal{S}(\\mathbb{R}^n)$ is a Schwartz function, the Fourier inversion formula reads\n\\[\n f(x) = (2\\pi)^{-n}\\int_{\\mathbb{R}^n}\\hat{f}(\\xi)e^{i\\langle x,\\xi\\rangle} d\\xi.\n\\]\nWe say a symbol $p(x,\\xi)$ is in the class $S^m_{\\rho,\\delta}$ if\\[\n|\\partial_x^\\beta\\partial^\\alpha_\\xi p(x,\\xi)|\\lesssim_{\\alpha,\\beta} \\langle\\xi\\rangle^{m-\\rho|\\alpha|+\\delta|\\beta|},\n\\]\nwhere $\\langle\\xi\\rangle=(1+|\\xi|^2)^{1/2}$ is the Japanese bracket.\nThe pseudo-differential operator $p(x,D)$ is defined by the integral\n\\[\np(x,D)f(x)=(2\\pi)^{-n}\\int p(x,\\xi)\\hat{f}(\\xi) e^{i\\langle x,\\xi\\rangle}d\\xi=(2\\pi)^{-n}\\iint p(x,\\xi)f(y) e^{i\\langle x-y,\\xi\\rangle}d\\xi dy.\n\\]\nWe say $P$ is the pseudo-differential operator with the compound symbol $p(x,y,\\xi)$ if\n\\[Pf(x)=(2\\pi)^{-n}\\iint p(x,y,\\xi)f(y) e^{i\\langle x-y,\\xi\\rangle}d\\xi dy.\\]\n\n\\subsection{Acknowledgements}\nThe authors would like to thank Simon Marshall for helpful comments. The second author was supported in part by the Simons Foundation and NSF (DMS-2452860). \n\\bigskip", + "sketch": "A stated proof strategy for Theorem~\\ref{thm1} is:\n\n- Reduce Theorem~\\ref{thm1} to Theorem~\\ref{thm2}: “A main step in the proof of Theorem~\\ref{thm1} is the following” theorem (Theorem~\\ref{thm2}), which turns (i) a microlocal $L^6$ bound of the form \\eqref{i3} for $A_\\nu^{\\theta_0}e_\\lambda$ and (ii) an improved $L^\\infty$ bound \\eqref{i4} into a global $L^6$ bound \\eqref{i5}. Here, “for each fixed $\\nu$, the operator $A_\\nu^{\\theta_0}$ is microlocally supported in a $\\theta_0$-neighborhood of a geodesic segment and is frequency-localized to directions near that geodesic.”\n\n- Establish Theorem~\\ref{thm2} by “a strategy similar to that of \\cite{huang2024curvature}, relying on bilinear techniques from harmonic analysis together with the improved $L^\\infty$ estimate \\eqref{i4}.”\n\n- For Hecke–Maass forms $\\psi$ on $X$ (as in Theorem~\\ref{thm1}), use that \\eqref{i4} holds “with $\\delta_2=\\tfrac{1}{12}$ by Iwaniec-Sarnak’s $L^\\infty$-norm bound.” Then prove the needed microlocal estimate \\eqref{i3} via Theorem~\\ref{thm microlocal KN}, which gives\n \\[\\sup_\\nu\\|A_\\nu^{\\theta_0} \\psi\\|_{L^6(X)}\\lesssim_\\varepsilon \\lambda^{\\frac16-\\frac{\\delta_1}{12}+\\varepsilon},\\]\n i.e. \\eqref{i3} “with $a=\\frac1{12}$.”\n\n- Conclude Theorem~\\ref{thm1} by parameter choice: “If, in addition, \\eqref{i3} holds for such $X$ and $\\psi$ with $a=\\frac1{12}$, then Theorem~\\ref{thm1} follows immediately by choosing $\\delta_1=\\frac{3}{2}\\delta_2=\\tfrac{1}{8}$.”\n\n- Proof approach for Theorem~\\ref{thm microlocal KN} (supplying \\eqref{i3}): it “can be viewed as the $L^6$-version” of Marshall’s microlocal restriction results; the authors “give the amplification inequality … by integrating the amplified pretrace formula against a test function under the operator $A_\\nu^{\\theta_0}$.” As in “usual amplification arguments,” there is “a counting problem” (how often a Hecke operator maps the relevant geodesic segment $\\gamma$ back close to itself) and “an analytic problem,” and “using the microlocal support of $A_\\nu^{\\theta_0}$, we can reduce the analytic problem to several known results from \\cite{sogge881,Mar16,Mar16HigerRank}.”", + "expanded_sketch": "A stated proof strategy for the main theorem is:\n\n- Reduce the main theorem to the following auxiliary theorem.\n\n\\begin{theorem}\\label{thm2}\n Let $(M,g)$ be a compact $2$-dimensional Riemannian manifold without boundary. Let $e_\\la$ be an $L^2$ normalized eigenfunction of $-\\Delta_g$ with eigenvalue $\\la^2$, satisfying \\eqref{1.2} for some $\\la\\gg1$. Set $\\theta_0=\\la^{-\\delta_1}$ with $0<\\delta_1< \\frac12$, and let $A_\\nu^{\\theta_0}$ be the pseudo-differential operator defined in \\eqref{qnusymbol}. Assume that \n\\begin{equation}\\label{i3}\n \\sup_\\nu\\|A_\\nu^{\\theta_0} e_\\la\\|_{L^6(M)}\\lesssim_\\e \\la^{\\frac16-a{\\delta_1}+\\e},\n \\end{equation}\n and \n \\begin{equation}\\label{i4}\n \\|e_\\la\\|_{L^\\infty(M)}\\lesssim_\\e \\la^{\\frac12-\\delta_2+\\e}\n \\end{equation}\nfor some constant $a>0$ and $0<\\delta_2<\\frac{1}{2}$. Then \n \\begin{equation}\\label{i5}\n \\|e_\\la\\|_{L^6(M)}\\lesssim_\\e \\la^{\\frac16-\\frac23a{\\delta_1}+\\e}+\\la^{\\frac23(-\\frac14+\\frac53\\delta_1)}+\\la^{\\frac16(1+\\delta_1-2\\delta_2)+\\e}.\n \\end{equation}\n\\end{theorem}\n\nHere, “for each fixed $\\nu$, the operator $A_\\nu^{\\theta_0}$ is microlocally supported in a $\\theta_0$-neighborhood of a geodesic segment and is frequency-localized to directions near that geodesic.”\n\n- Establish the auxiliary theorem by “a strategy similar to that of \\cite{huang2024curvature}, relying on bilinear techniques from harmonic analysis together with the improved $L^\\infty$ estimate\n\\begin{equation}\\label{i4}\n \\|e_\\la\\|_{L^\\infty(M)}\\lesssim_\\e \\la^{\\frac12-\\delta_2+\\e}\n \\end{equation}.”\n\n- For Hecke–Maass forms $\\psi$ on $X$ (as in the main theorem), use that the preceding estimate holds “with $\\delta_2=\\tfrac{1}{12}$ by Iwaniec-Sarnak’s $L^\\infty$-norm bound.” Then prove the needed microlocal estimate\n\\begin{equation}\\label{i3}\n \\sup_\\nu\\|A_\\nu^{\\theta_0} e_\\la\\|_{L^6(M)}\\lesssim_\\e \\la^{\\frac16-a{\\delta_1}+\\e},\n \\end{equation}\nvia the following theorem.\n\n\\begin{theorem}\\label{thm microlocal KN}\n Let $\\psi$ be a Hecke\u0013Maass form on $X$ with spectral parameter $\\lambda\\gg1$. Set $\\theta_0=\\la^{-\\delta_1}$ with $0<\\delta_1<\\frac12$, and let $A_\\nu^{\\theta_0}$ be the pseudo-differential operator defined in \\eqref{qnusymbol}. We have\n \\begin{equation*}\n \\sup_\\nu\\|A_\\nu^{\\theta_0} \\psi\\|_{L^6(X)}\\lesssim_\\e \\la^{\\frac16-\\frac{\\delta_1}{12}+\\e}.\n \\end{equation*}\n\\end{theorem}\n\nThis gives\n\\[\\sup_\\nu\\|A_\\nu^{\\theta_0} \\psi\\|_{L^6(X)}\\lesssim_\\varepsilon \\lambda^{\\frac16-\\frac{\\delta_1}{12}+\\varepsilon},\\]\nthat is, the microlocal bound above “with $a=\\frac1{12}$.”\n\n- Conclude the main theorem by parameter choice: “If, in addition, the microlocal bound above holds for such $X$ and $\\psi$ with $a=\\frac1{12}$, then the main theorem follows immediately by choosing $\\delta_1=\\frac{3}{2}\\delta_2=\\tfrac{1}{8}$.”\n\n- Proof approach for the preceding microlocal theorem (supplying the microlocal bound): it “can be viewed as the $L^6$-version” of Marshall’s microlocal restriction results; the authors “give the amplification inequality … by integrating the amplified pretrace formula against a test function under the operator $A_\\nu^{\\theta_0}$.” As in “usual amplification arguments,” there is “a counting problem” (how often a Hecke operator maps the relevant geodesic segment $\\gamma$ back close to itself) and “an analytic problem,” and “using the microlocal support of $A_\\nu^{\\theta_0}$, we can reduce the analytic problem to several known results from \\cite{sogge881,Mar16,Mar16HigerRank}.”", + "expanded_theorem": "\\label{thm1}\n Let $\\psi$ be a Hecke–Maass form on $X$ with spectral parameter $\\lambda\\gg1$. We have\n \\begin{equation}\\label{i2}\n \\| \\psi \\|_{L^6(X)} \\lesssim_\\varepsilon\n \\lambda^{\\frac{1}{6}-\\frac{1}{144}+\\varepsilon}.\n \\end{equation}", + "theorem_type": [ + "Inequality or Bound", + "Universal" + ], + "mcq": { + "question": "Let $X=\\Gamma\\backslash\\mathbb H$ be the compact arithmetic hyperbolic surface obtained from the upper half-plane $\\mathbb H$ by quotienting by a group $\\Gamma\\subset \\mathrm{SL}(2,\\mathbb R)$ coming from the norm-one units of a maximal order in an indefinite quaternion division algebra over $\\mathbb Q$. Let $\\psi$ be an $L^2$-normalized Hecke--Maass form on $X$, meaning that $\\int_X |\\psi(x)|^2\\,dx=1$ and $\\psi$ is a joint eigenfunction of the Laplace--Beltrami operator and the Hecke operators on $X$. If $\\psi$ has spectral parameter $\\lambda\\gg 1$, i.e. it satisfies\n\\[\n\\Delta \\psi + \\left(\\tfrac14+\\lambda^2\\right)\\psi=0,\n\\]\nwhich statement holds for every such $\\psi$?", + "correct_choice": { + "label": "A", + "text": "For every $\\varepsilon>0$, one has\n\\[\n\\|\\psi\\|_{L^6(X)} \\lesssim_{\\varepsilon} \\lambda^{\\frac16-\\frac{1}{144}+\\varepsilon},\n\\]\nthat is, there exists a constant $C_{\\varepsilon}$ such that\n\\[\n\\|\\psi\\|_{L^6(X)} \\le C_{\\varepsilon}\\,\\lambda^{\\frac16-\\frac{1}{144}+\\varepsilon}.\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "For every $\\varepsilon>0$, one has\n\\[\n\\|\\psi\\|_{L^6(X)} \\lesssim_{\\varepsilon} \\lambda^{\\frac16-\\frac{1}{72}+\\varepsilon},\n\\]\nthat is, there exists a constant $C_{\\varepsilon}$ such that\n\\[\n\\|\\psi\\|_{L^6(X)} \\le C_{\\varepsilon}\\,\\lambda^{\\frac16-\\frac{1}{72}+\\varepsilon}.\n\\]" + }, + { + "label": "C", + "text": "For every $\\varepsilon>0$, one has\n\\[\n\\|\\psi\\|_{L^6(X)} \\lesssim_{\\varepsilon} \\lambda^{\\frac16+\\varepsilon},\n\\]\nthat is, there exists a constant $C_{\\varepsilon}$ such that\n\\[\n\\|\\psi\\|_{L^6(X)} \\le C_{\\varepsilon}\\,\\lambda^{\\frac16+\\varepsilon}.\n\\]" + }, + { + "label": "D", + "text": "There exists $\\varepsilon_0>0$ such that one has\n\\[\n\\|\\psi\\|_{L^6(X)} \\lesssim \\lambda^{\\frac16-\\frac{1}{144}+\\varepsilon_0},\n\\]\nthat is, there exists a constant $C$ such that\n\\[\n\\|\\psi\\|_{L^6(X)} \\le C\\,\\lambda^{\\frac16-\\frac{1}{144}+\\varepsilon_0}.\n\\]" + }, + { + "label": "E", + "text": "For every $\\varepsilon>0$, one has\n\\[\n\\|\\psi\\|_{L^6(X)} \\lesssim_{\\varepsilon} \\lambda^{\\frac16-\\frac{1}{96}+\\varepsilon},\n\\]\nthat is, there exists a constant $C_{\\varepsilon}$ such that\n\\[\n\\|\\psi\\|_{L^6(X)} \\le C_{\\varepsilon}\\,\\lambda^{\\frac16-\\frac{1}{96}+\\varepsilon}.\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": "final exponent from parameter choice $\\delta_1=1/8$", + "template_used": "stronger_trap" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped the power-saving term $-1/144$", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "quantifier on $\\varepsilon$ and dependence of implicit constant", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "miscomputed optimization of auxiliary exponents $a=1/12$, $\\delta_2=1/12$, $\\delta_1=1/8$", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives only the geometric and spectral setup; it does not state or strongly hint at the specific $L^6$ bound or the exponent $1/6-1/144$. The correct answer is not leaked." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recall question: it asks which precise bound holds for this standard setup. Although the options differ by subtle exponent and quantifier changes, the task is still largely a reformulation of a known result rather than an independently derived conclusion." + }, + "GPS": { + "score": 1, + "justification": "There is some reasoning pressure because the choices differ in strength, sharpness, and quantifier dependence, so one must identify the strongest valid statement. However, success depends more on recalling the exact theorem than on generating a conclusion from the stem." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and well targeted: one is a weaker true-type bound, others are overly strong exponents, and one tampers with the quantifier/constant dependence. These reflect realistic failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-identification MCQ with no answer leakage and strong distractors, but it mainly tests precise recall/recognition of a known bound rather than deeper generative reasoning." + } + }, + { + "id": "2602.06332v1", + "paper_link": "http://arxiv.org/abs/2602.06332v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm}\n\tLet $s>0, \\lambda>0, d=2,3$ and $T>0$. Then the data-to-solution map $(u_0,b_0)\\mapsto(u,b)$ for the equations \\eqref{mhd} is non-uniformly continuous from a bounded subset in $H^{s}(\\mathbb{R}^d)\\times H^{s}(\\mathbb{R}^d)$ into $C\\left([0,T],H^{s}(\\mathbb{R}^d)\\right)\\times C\\left([0,T],H^{s}(\\mathbb{R}^d)\\right)$.\n\n More precisely, for any $\\gamma>0$ and arbitrary constant magnetic field ${\\bf B_0}\\in \\mathbb{R}^d$, there exists two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$ such that\n\t\\begin{itemize}\n\t\t\\item the solutions satisfy\n\\begin{align*}\n\t\t\t(u^{\\pm 1,\\lambda}, b^{\\pm 1,\\lambda} - {\\bf B_0}) \\in C([0,T], H^{s}(\\mathbb{R}^{d})) \\times C([0,T], H^{s}(\\mathbb{R}^{d}));\n\\end{align*}\n\t\t\\item the initial data satisfy\n\\begin{equation}\\label{nonuniform-initial-bound}\n\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\|u^{+1,\\lambda}(0,\\cdot)\\|_{H^{s}} + \\|b^{+1,\\lambda}(0,\\cdot) - {\\bf B_0}\\|_{H^{s}} \\leq \\gamma, \\\\[2mm]\n\t\t\t\\|u^{-1,\\lambda}(0,\\cdot)\\|_{H^{s}} + \\|b^{-1,\\lambda}(0,\\cdot) - {\\bf B_0}\\|_{H^{s}} \\leq \\gamma;\n\t\\end{array}\n\\end{equation}\n\t\t\\item and the non-uniform continuity is characterized by\n\t\t\\begin{itemize}\n\t\t\t\\item at initial time $t=0$,\n\t\t\t\\begin{align}\\label{nonuniform-initial-diff}\n\t\t\t\t\\lim_{\\lambda\\to\\infty} \\|u^{+1,\\lambda}(0,\\cdot) - u^{-1,\\lambda}(0,\\cdot)\\|_{H^{s}} = 0,\n\t\t\t\\end{align}\n\t\t\t\\item for evolution times $t>0$,\n\t\t\t\\begin{align}\\label{nonuniform-low-bound-sint}\n\t\t\t\t\\lim_{\\lambda \\to\\infty} \\|u^{+1,\\lambda}(t) - u^{-1,\\lambda}(t)\\|_{H^{s}} \\geq c\\gamma |\\sin t|,\n\t\t\t\\end{align}\n\t\t\\end{itemize}\n\t\twhere $c = c(s,d) > 0$ is a constant depending only on $s$ and $d$.\n\t\\end{itemize}", + "start_pos": 10028, + "end_pos": 11702, + "label": "thm" + }, + "ref_dict": { + "1.2": "\\begin{align}\\label{1.2}\n\t\tu^{h,\\pm1,\\lambda}(x,t)=\\nabla^\\perp \\left(\\lambda^{-\\delta-s-1}\\phi\\left(\\dfrac{x_1}{\\lambda^\\delta}\\right)\\phi\\left(\\dfrac{x_2}{\\lambda^\\delta}\\right)\\sin (\\lambda x_2\\mp t)\\right).\n\t\\end{align}", + "mhd-B": "\\begin{align}\\label{mhd-B}\n\t\\begin{cases}\n\t\t\\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b+{\\bf B_0}\\cdot\\nabla b,\\\\[1mm]\n\t\t\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u+{\\bf B_0}\\cdot\\nabla u, \\\\[1mm]\n\t\t\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\n\t\tu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n\t\\end{cases}\n\\end{align}", + "1.5": "\\begin{align}\\label{1.4}\n\tE^{\\pm1,\\lambda}&=\\partial_t u^{h,\\pm1,\\lambda}+u^{l,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{l,\\pm1,\\lambda},\\\\\\label{1.5}\n\tF^{\\pm1,\\lambda}&=u^{h,\\pm1,\\lambda}\\cdot\\nabla b^{\\pm1,\\lambda}-b^{\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda},\n\\end{align}", + "mhd": "\\begin{align}\\label{mhd}\n\t\\begin{cases}\n\t\\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b,\\\\[1mm]\n\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u, \\\\[1mm]\n\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\nu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n\t\\end{cases}\n\\end{align}", + "thm": "\\begin{theorem}\\label{thm}\n\tLet $s>0, \\lambda>0, d=2,3$ and $T>0$. Then the data-to-solution map $(u_0,b_0)\\mapsto(u,b)$ for the equations \\eqref{mhd} is non-uniformly continuous from a bounded subset in $H^{s}(\\mr^d)\\times H^{s}(\\mr^d)$ into $C\\left([0,T],H^{s}(\\mr^d)\\right)\\times C\\left([0,T],H^{s}(\\mr^d)\\right)$.\n\n More precisely, for any $\\gamma>0$ and arbitrary constant magnetic field ${\\bf B_0}\\in \\mr^d$, there exists two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$ such that\n\t\\begin{itemize}\n\t\t\\item the solutions satisfy\n\\begin{align*}\n\t\t\t(u^{\\pm 1,\\lambda}, b^{\\pm 1,\\lambda} - {\\bf B_0}) \\in C([0,T], H^{s}(\\mathbb{R}^{d})) \\times C([0,T], H^{s}(\\mathbb{R}^{d}));\n\\end{align*}\n\t\t\\item the initial data satisfy\n\\begin{equation}\\label{nonuniform-initial-bound}\n\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\|u^{+1,\\lambda}(0,\\cdot)\\|_{H^{s}} + \\|b^{+1,\\lambda}(0,\\cdot) - {\\bf B_0}\\|_{H^{s}} \\leq \\gamma, \\\\[2mm]\n\t\t\t\\|u^{-1,\\lambda}(0,\\cdot)\\|_{H^{s}} + \\|b^{-1,\\lambda}(0,\\cdot) - {\\bf B_0}\\|_{H^{s}} \\leq \\gamma;\n\t\\end{array}\n\\end{equation}\n\t\t\\item and the non-uniform continuity is characterized by\n\t\t\\begin{itemize}\n\t\t\t\\item at initial time $t=0$,\n\t\t\t\\begin{align}\\label{nonuniform-initial-diff}\n\t\t\t\t\\lim_{\\lambda\\to\\infty} \\|u^{+1,\\lambda}(0,\\cdot) - u^{-1,\\lambda}(0,\\cdot)\\|_{H^{s}} = 0,\n\t\t\t\\end{align}\n\t\t\t\\item for evolution times $t>0$,\n\t\t\t\\begin{align}\\label{nonuniform-low-bound-sint}\n\t\t\t\t\\lim_{\\lambda \\to\\infty} \\|u^{+1,\\lambda}(t) - u^{-1,\\lambda}(t)\\|_{H^{s}} \\geq c\\gamma |\\sin t|,\n\t\t\t\\end{align}\n\t\t\\end{itemize}\n\t\twhere $c = c(s,d) > 0$ is a constant depending only on $s$ and $d$.\n\t\\end{itemize}\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 7134, + "pre_theorem_intro_text": "\\,\\,\\,\\,\\,\\,\\,\\, In this paper, we consider the Cauchy problem for the incompressible MHD equations with only magnetic diffusion (hereinafter called the resistive MHD equations):\n\\begin{align}\\label{mhd}\n\t\\begin{cases}\n\t\\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b,\\\\[1mm]\n\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u, \\\\[1mm]\n\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\nu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n\t\\end{cases}\n\\end{align}\nwhere $u(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}^d$,\n$b(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}^d$,\nand $p(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}$\ndenote the velocity field, magnetic field, and pressure field, respectively,\nwith $x \\in \\mathbb{R}^d~(d=2,3)$ and $t>0$. Clearly, when $b\\equiv0$, \\eqref{mhd} reduces to the classical Euler equations.\n\nThe resistive MHD equations provide a fundamental framework for modeling key plasma phenomena where finite electrical resistivity plays a critical role, particularly in astrophysical magnetic reconnection processes that govern energy release in solar flares and magnetospheric activity, as well as in the geodynamo mechanisms responsible for generating and sustaining planetary magnetic fields \\cite{priest-2000,roberts1967introduction}.\n\n There have been a large number of mathematical studies on the well-posedness theory for the incompressible MHD equations under various assumptions on velocity dissipation and magnetic diffusion.\nPrevious results on local well-posedness can be found in \\cite{sermange-1983,jiu-2006-local,li-local-nonresistive-2017-adv,chemin-local-nonresistive-2016-adv,fefferman-local-nonressitive-2014-jfa,fefferman-local-nonressitive-2017-arma}. The pioneering work of Bardos, Sulem and Sulem \\cite{full-not-bardos} and Lin and Zhang \\cite{lin-2014-GlobalSmallSolutions} on perturbation theory near constant background magnetic fields first revealed the stabilizing role of such equilibrium configurations in the MHD equations. Physical interpretations of this stabilization effect are discussed in \\cite{background-phy1,background-phy2}. These foundational results have inspired extensive research on global well-posedness and stability of the MHD equations, as documented in \\cite{lei-bkm-2009-dcds,full-1-duvaut-lions,sermange-1983,cao-wu-1-mix,cao-wu-2-mix,full-not-bardos,lin-2014-GlobalSmallSolutions,chen-2022-3dmhd-Diophant,deng-zhang-2018-decay,panGlobalClassicalSolutions2018,ren2014global,zhang-2016-non-resistive-global-jde,xie-2024-cvpde-dio,wei--global-resistive-2020-cmr,zhou-zhu-glbal-symmetry-jmp-2018,ye-global-resistive-2022-acta} and references therein.\n\n In recent years, ill-posedness theory for the incompressible MHD equations has also been invetigated. For the non-resistive MHD equations which contain velocity dissipation but no magnetic diffusion, Chen, Nie and Ye \\cite{chen-ye-sharp-ill-nonresistive-2024-jfa} established sharp strong ill-posedness results that provide a striking contrast to the local well-posedness theory developed by Fefferman et al. \\cite{fefferman-local-nonressitive-2017-arma}. For the resistive case \\eqref{mhd}, Wu and Zhao \\cite{wu-zhao-mild-resistive-2023-IMRN} obtained mild ill-posedness results near the background magnetic field $(1,0)$ in $\\mathbb{R}^2$.\n\nIt is noted that the concept of ill-posedness manifests rather strongly in many fundamental cases. As evidenced by \\cite{chen-ye-sharp-ill-nonresistive-2024-jfa,wu-zhao-mild-resistive-2023-IMRN} and other works of PDE systems, such behavior typically occurs in critical or supercritical function spaces, while some problems remain unresolved to this day. These substantial difficulties have motivated researchers to consider relaxed notions of ill-posedness by examining weaker properties.\n For certain equations, the solution operator may exhibit non-uniform continuity properties under stronger topological frameworks, which provides meaningful insights into the refined continuity structure of solution mappings and can be seen a kind of instability of the solutions or ill-posedness the equations.\n\nIn this paper, we are concerned with the non-uniform continuity properties of the data-to-solution map for the resistive MHD equations. We begin by precisely defining the notion of non-uniform continuity as follows:\n\\begin{definition}\\label{define-nonuniform}\n\tLet $X$ be a Banach space, and consider the Cauchy problem:\n\t\\begin{align*}\n\t\t\\begin{cases}\n\t\t\t\\partial_t v = N(v), \\\\\n\t\t\tv(0) = v_0,\n\t\t\\end{cases}\n\t\\end{align*}\n\twhere $N$ is a (possibly nonlinear) differential operator. The \\textbf{data-to-solution map} $\\Phi_t \\colon X \\to X$ (for fixed $t > 0$) defined by $\\Phi_t(v_0) = v(t)$ is said to be \\textbf{non-uniformly continuous} on $X$ if the following holds:\n\n\t\\noindent\n\t\\textbf{Sequential Formulation:} \\\\\n\tFor every $t > 0$, there exists $\\epsilon_0 > 0$ and sequences $\\{v_{1,n}(0)\\}, \\{v_{2,n}(0)\\} \\subset X$ such that:\n\t\\begin{align*}\n\t\t\\lim_{n \\to \\infty} \\| v_{1,n}(0) - v_{2,n}(0) \\|_X = 0,\n\t\\end{align*}\n\tbut\n\t\\begin{align*}\n\t\t\\limsup_{n \\to \\infty} \t\\| \\Phi_t(v_{1,n}(0)) - \\Phi_t(v_{2,n}(0)) \\|_X \\geq \\epsilon_0.\n\t\\end{align*}\n\n\t\\noindent\n\t\\textbf{$\\delta$-$\\epsilon$ Formulation (Equivalent):} \\\\\n\tFor every $t > 0$, there exists $\\epsilon_0 > 0$ such that for any $\\delta > 0$, one can find initial data $v_1(0), v_2(0) \\in X$ satisfying:\n\t\\begin{align*}\n\t\t\\| v_1(0) - v_2(0) \\|_X < \\delta,\n\t\\end{align*}\n\tbut\n\t\\begin{align*}\n\t\t\\| \\Phi_t(v_1(0)) - \\Phi_t(v_2(0)) \\|_X \\geq \\epsilon_0.\n\t\\end{align*}\n\\end{definition}\n\nIn \\cite{nonuniform-2010-cmp-himonas}, Himonas and Misiołek first proved non-uniform continuity of the data-to-solution map on the incompressible Euler equations in both $H^s(\\mathbb{R}^d)$ ($s > 0$) and $H^r(\\mathbb{T}^d)$ ($r \\in \\mathbb{R}$) with $d = 2,3$, which was later extended by Li and Bourgain \\cite{nonuniform-2019-cmp-bourgain-li} through Galilean boost techniques to the endpoint case $s \\geq 0$ in $H^s(\\mathbb{R}^d)$, where they further proved the stronger property of nowhere uniform continuity.\nFor the non-resistive MHD equations with only velocity dissipation, recent work by Li, Yin, and Zhu \\cite{nonuniforem-2023-adv-li-yin-zhu} demonstrated non-uniform continuity in $H^s(\\mathbb{R}^d)$ for $s > \\frac{d}{2}$.\n\n We will prove in this paper the non-uniform continuity of the data-to-solution map for the resistive MHD equations in Sobolev spaces $H^s(\\mathbb{R}^d)$ for all $s>0$ and $d=2,3$. In comparison with the work by Li, Yin, and Zhu \\cite{nonuniforem-2023-adv-li-yin-zhu} which is for non-resistive MHD equations with only velocity dissipation and in $H^s(\\mathbb{R}^d)$ for $s > \\frac{d}{2}$, our results hold for the resistive MHD equations and in $H^s(\\mathbb{R}^d)$ for all $s>0$ and $d=2,3$. Moreover, our results permit the solution perturbation around an arbitrary constant background magnetic fields $\\mathbf{B_0} \\in \\mathbb{R}^d$, which shows that the strong magnetic field may provide the stabilization effect but no help for the uniform continuity of the data-to-solution map.\n\nOur main results can be stated as follows:", + "context": "\\,\\,\\,\\,\\,\\,\\,\\, In this paper, we consider the Cauchy problem for the incompressible MHD equations with only magnetic diffusion (hereinafter called the resistive MHD equations):\n\\begin{align}\\label{mhd}\n \\begin{cases}\n \\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b,\\\\[1mm]\n\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u, \\\\[1mm]\n\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\nu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n \\end{cases}\n\\end{align}\nwhere $u(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}^d$,\n$b(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}^d$,\nand $p(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}$\ndenote the velocity field, magnetic field, and pressure field, respectively,\nwith $x \\in \\mathbb{R}^d~(d=2,3)$ and $t>0$. Clearly, when $b\\equiv0$, \\eqref{mhd} reduces to the classical Euler equations.\n\nIn this paper, we are concerned with the non-uniform continuity properties of the data-to-solution map for the resistive MHD equations. We begin by precisely defining the notion of non-uniform continuity as follows:\n\\begin{definition}\\label{define-nonuniform}\n Let $X$ be a Banach space, and consider the Cauchy problem:\n \\begin{align*}\n \\begin{cases}\n \\partial_t v = N(v), \\\\\n v(0) = v_0,\n \\end{cases}\n \\end{align*}\n where $N$ is a (possibly nonlinear) differential operator. The \\textbf{data-to-solution map} $\\Phi_t \\colon X \\to X$ (for fixed $t > 0$) defined by $\\Phi_t(v_0) = v(t)$ is said to be \\textbf{non-uniformly continuous} on $X$ if the following holds:\n\n\\noindent\n \\textbf{Sequential Formulation:} \\\\\n For every $t > 0$, there exists $\\epsilon_0 > 0$ and sequences $\\{v_{1,n}(0)\\}, \\{v_{2,n}(0)\\} \\subset X$ such that:\n \\begin{align*}\n \\lim_{n \\to \\infty} \\| v_{1,n}(0) - v_{2,n}(0) \\|_X = 0,\n \\end{align*}\n but\n \\begin{align*}\n \\limsup_{n \\to \\infty} \\| \\Phi_t(v_{1,n}(0)) - \\Phi_t(v_{2,n}(0)) \\|_X \\geq \\epsilon_0.\n \\end{align*}\n\n\\noindent\n \\textbf{$\\delta$-$\\epsilon$ Formulation (Equivalent):} \\\\\n For every $t > 0$, there exists $\\epsilon_0 > 0$ such that for any $\\delta > 0$, one can find initial data $v_1(0), v_2(0) \\in X$ satisfying:\n \\begin{align*}\n \\| v_1(0) - v_2(0) \\|_X < \\delta,\n \\end{align*}\n but\n \\begin{align*}\n \\| \\Phi_t(v_1(0)) - \\Phi_t(v_2(0)) \\|_X \\geq \\epsilon_0.\n \\end{align*}\n\\end{definition}\n\nWe will prove in this paper the non-uniform continuity of the data-to-solution map for the resistive MHD equations in Sobolev spaces $H^s(\\mathbb{R}^d)$ for all $s>0$ and $d=2,3$. In comparison with the work by Li, Yin, and Zhu \\cite{nonuniforem-2023-adv-li-yin-zhu} which is for non-resistive MHD equations with only velocity dissipation and in $H^s(\\mathbb{R}^d)$ for $s > \\frac{d}{2}$, our results hold for the resistive MHD equations and in $H^s(\\mathbb{R}^d)$ for all $s>0$ and $d=2,3$. Moreover, our results permit the solution perturbation around an arbitrary constant background magnetic fields $\\mathbf{B_0} \\in \\mathbb{R}^d$, which shows that the strong magnetic field may provide the stabilization effect but no help for the uniform continuity of the data-to-solution map.\n\nOur main results can be stated as follows:\n\n\\begin{align}\\label{mhd}\n\t\\begin{cases}\n\t\\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b,\\\\[1mm]\n\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u, \\\\[1mm]\n\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\nu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n\t\\end{cases}\n\\end{align}", + "full_context": "\\,\\,\\,\\,\\,\\,\\,\\, In this paper, we consider the Cauchy problem for the incompressible MHD equations with only magnetic diffusion (hereinafter called the resistive MHD equations):\n\\begin{align}\\label{mhd}\n \\begin{cases}\n \\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b,\\\\[1mm]\n\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u, \\\\[1mm]\n\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\nu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n \\end{cases}\n\\end{align}\nwhere $u(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}^d$,\n$b(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}^d$,\nand $p(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}$\ndenote the velocity field, magnetic field, and pressure field, respectively,\nwith $x \\in \\mathbb{R}^d~(d=2,3)$ and $t>0$. Clearly, when $b\\equiv0$, \\eqref{mhd} reduces to the classical Euler equations.\n\nIn this paper, we are concerned with the non-uniform continuity properties of the data-to-solution map for the resistive MHD equations. We begin by precisely defining the notion of non-uniform continuity as follows:\n\\begin{definition}\\label{define-nonuniform}\n Let $X$ be a Banach space, and consider the Cauchy problem:\n \\begin{align*}\n \\begin{cases}\n \\partial_t v = N(v), \\\\\n v(0) = v_0,\n \\end{cases}\n \\end{align*}\n where $N$ is a (possibly nonlinear) differential operator. The \\textbf{data-to-solution map} $\\Phi_t \\colon X \\to X$ (for fixed $t > 0$) defined by $\\Phi_t(v_0) = v(t)$ is said to be \\textbf{non-uniformly continuous} on $X$ if the following holds:\n\n\\noindent\n \\textbf{Sequential Formulation:} \\\\\n For every $t > 0$, there exists $\\epsilon_0 > 0$ and sequences $\\{v_{1,n}(0)\\}, \\{v_{2,n}(0)\\} \\subset X$ such that:\n \\begin{align*}\n \\lim_{n \\to \\infty} \\| v_{1,n}(0) - v_{2,n}(0) \\|_X = 0,\n \\end{align*}\n but\n \\begin{align*}\n \\limsup_{n \\to \\infty} \\| \\Phi_t(v_{1,n}(0)) - \\Phi_t(v_{2,n}(0)) \\|_X \\geq \\epsilon_0.\n \\end{align*}\n\n\\noindent\n \\textbf{$\\delta$-$\\epsilon$ Formulation (Equivalent):} \\\\\n For every $t > 0$, there exists $\\epsilon_0 > 0$ such that for any $\\delta > 0$, one can find initial data $v_1(0), v_2(0) \\in X$ satisfying:\n \\begin{align*}\n \\| v_1(0) - v_2(0) \\|_X < \\delta,\n \\end{align*}\n but\n \\begin{align*}\n \\| \\Phi_t(v_1(0)) - \\Phi_t(v_2(0)) \\|_X \\geq \\epsilon_0.\n \\end{align*}\n\\end{definition}\n\nWe will prove in this paper the non-uniform continuity of the data-to-solution map for the resistive MHD equations in Sobolev spaces $H^s(\\mathbb{R}^d)$ for all $s>0$ and $d=2,3$. In comparison with the work by Li, Yin, and Zhu \\cite{nonuniforem-2023-adv-li-yin-zhu} which is for non-resistive MHD equations with only velocity dissipation and in $H^s(\\mathbb{R}^d)$ for $s > \\frac{d}{2}$, our results hold for the resistive MHD equations and in $H^s(\\mathbb{R}^d)$ for all $s>0$ and $d=2,3$. Moreover, our results permit the solution perturbation around an arbitrary constant background magnetic fields $\\mathbf{B_0} \\in \\mathbb{R}^d$, which shows that the strong magnetic field may provide the stabilization effect but no help for the uniform continuity of the data-to-solution map.\n\nOur main results can be stated as follows:\n\n\\begin{align}\\label{mhd}\n\t\\begin{cases}\n\t\\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b,\\\\[1mm]\n\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u, \\\\[1mm]\n\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\nu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n\t\\end{cases}\n\\end{align}\n\n\\section{Preliminaries}\\label{sec:2}\n\\subsection{Auxiliary Analysis Tools}\\label{sec:2.1}\nIn this subsection, we introduce two key lemmas needed later. The first one is\n\\begin{lemma}\\cite{nonuniform-2010-cmp-himonas}\\label{phi-cmp-2010}\n Let $\\sigma \\geq 0, \\delta\\geq 0, a\\in \\mr$ and $\\lambda\\gg 1$. For any Schwartz function $\\psi\\in \\mathcal{S}(\\mr)$, it holds that\n \\begin{align*}\n \\lambda^{\\delta/2}\\left\\|\\psi\\right\\|_{L^2(\\mathbb{R})}\\leq\\left\\|\\psi\\left(\\frac{\\cdot}{\\lambda^\\delta}\\right)\\right\\|_{H^\\sigma(\\mathbb{R})}\\leq\\lambda^{\\delta/2}\\left\\|\\psi\\right\\|_{H^\\sigma(\\mathbb{R})},\n \\end{align*}\nand \\begin{align}\\label{622}\n \\left\\|\\psi\\left(\\frac{\\cdot}{\\lambda^\\delta}\\right)\\cos(\\lambda\\cdot-a)\\right\\|_{H^\\sigma(\\mathbb{R})}\\simeq\\lambda^{\\sigma+\\delta/2}\\|\\psi\\|_{L^2(\\mathbb{R})}.\n\\end{align}\nMoreover, \\eqref{622} holds true if $\\cos(\\lambda \\cdot-a)$ is replaced by $\\sin(\\lambda \\cdot-a)$.\n\\end{lemma}\n\n\\subsection{Approximate Solutions Scheme}\\label{sec:3.1}\nWe begin by constructing two sequences of approximate solutions $b^{w,\\lambda}$ and $u^{w,\\lambda}$ with $w=\\pm 1$, where $b^{w,\\lambda}$ contains low frequencies only and $u^{w,\\lambda}$ contain both high and low frequencies. More precisely, we set\n\\begin{align}\\label{u-low+high}\n u^{w,\\lambda}(x,t)=u^{l,w,\\lambda}(x,t)+u^{h,w,\\lambda}(x,t).\n\\end{align}\nThe high-frequency components $u^{h,w,\\lambda}$ are constructed as\n\\begin{align}\\label{u-high}\n u^{h,w,\\lambda}(x,t)=\\begin{cases}\n \\nabla^\\perp \\phi^{h,w,\\lambda}(x,t)=(\\partial_{x_2}\\phi^{h,w,\\lambda}(x,t),-\\partial_{x_1}\\phi^{h,w,\\lambda}(x,t)), &~d=2,\\\\[3mm]\n \\left(\\nabla^\\perp \\phi^{h,w,\\lambda}(x,t),0 \\right)=(\\partial_{x_2}\\phi^{h,w,\\lambda}(x,t),-\\partial_{x_1}\\phi^{h,w,\\lambda}(x,t),0), &~d=3.\n \\end{cases}\n\\end{align}\nIn \\eqref{u-high}, the function $\\phi^{h,w,\\lambda}(x,t)$ is defined as\n\\begin{align}\\label{phi-high}\n \\phi^{h,w,\\lambda}(x,t)=\\begin{cases}\n \\lambda^{-s-\\delta-1}\\phi(\\dfrac{x_1}{\\lambda^\\delta})\\phi(\\dfrac{x_2}{\\lambda^\\delta})\\sin (\\lambda x_2-wt),&~d=2,\\\\[3mm]\n \\lambda^{-s-\\delta-1}\\phi(\\dfrac{x_1}{\\lambda^\\delta})\\phi(\\dfrac{x_2}{\\lambda^\\delta})\\sin (\\lambda x_2-wt)\\phi(x_3),&~d=3,\n \\end{cases}\n\\end{align}\nwhere $s > 0, \\lambda>0, w=\\pm 1$, the parameter $\\delta > 0$ is to be specified later and the smooth function $\\phi \\in C_c^\\infty(\\mathbb{R})$ satisfies $\\text{supp}\\,\\phi \\subset [-2,2]$ and $\\phi(x) \\equiv 1$ on $|x| < 1$.\n\nFollowing similar estimates established in \\eqref{u,b-low-t-local-bound} and \\eqref{u-high-bound}, we can obtain\n\\begin{align}\\label{4.5}\n \\|\\bar{u}^{l,w,\\lambda}(t)\\|_{H^m} + \\|\\bar{b}^{w,\\lambda}(t)\\|_{H^{m}}\n \\leq C\\lambda^{-1+\\delta},\n\\end{align}\nand\n\\begin{align}\\label{4.6}\n \\|\\bar{u}^{h,w,\\lambda}(t)\\|_{H^r}\\leq C\\lambda^{r-s},\\quad \\|\\bar{u}^{h,w,\\lambda}(t)\\|_{L^\\infty}\\leq C\\lambda^{-s-\\delta},\\quad \\|\\nabla \\bar{u}^{h,w,\\lambda}(t)\\|_{L^\\infty}\\leq C\\lambda^{-s+1-\\delta}\n\\end{align}\nfor any $0 \\leq t \\leq T\\leq 1$, where $[0,T]$ denotes the existence interval of the solution guaranteed by Lemma~\\ref{local-well-pri}.\n\\subsection{Error Estimates for the Approximate Solutions}\\label{sec:4.2}\nFollowing the approach in subsection \\ref{sec:3.2} and based on system \\eqref{mhd-B}, we construct a perturbed system as follows\n\\begin{align}\\label{4.9}\n \\begin{cases}\n \\partial_t \\bar{u}^{w,\\lambda}+\\bar{u}^{w,\\lambda} \\cdot \\nabla \\bar{u}^{w,\\lambda}+\\nabla p^{w,\\lambda}=\\bar{b}^{w,\\lambda}\\cdot \\nabla \\bar{b}^{w,\\lambda}+{\\bf B_0}\\cdot\\nabla \\bar{b}^{w,\\lambda}+E_1^{w,\\lambda},\\\\[1mm]\n \\partial_t \\bar{b}^{w,\\lambda}-\\Delta \\bar{b}^{w,\\lambda}+\\bar{u}^{w,\\lambda}\\cdot \\nabla \\bar{b}^{w,\\lambda}=\\bar{b}^{w,\\lambda}\\cdot\\nabla \\bar{u}^{w,\\lambda}+{\\bf B_0}\\cdot\\nabla \\bar{u}^{w,\\lambda}+F_1^{w,\\lambda}, \\\\[1mm]\n \\diver\\, \\bar{u}^{w,\\lambda}=\\diver\\, \\bar{b}^{w,\\lambda}=0,\n \\end{cases}\n\\end{align}\nwhere\n\\begin{align*}\n E_1^{w,\\lambda}&=\\partial_t \\bar{u}^{h,w,\\lambda}+\\bar{u}^{l,w,\\lambda}\\cdot\\nabla \\bar{u}^{h,w,\\lambda}+\\bar{u}^{h,w,\\lambda}\\cdot\\nabla \\bar{u}^{h,w,\\lambda}+\\bar{u}^{h,w,\\lambda}\\cdot\\nabla \\bar{u}^{l,w,\\lambda},\\\\\\nonumber\n F_1^{w,\\lambda}&=\\bar{u}^{h,w,\\lambda}\\cdot\\nabla \\bar{b}^{w,\\lambda}-\\bar{b}^{w,\\lambda}\\cdot\\nabla \\bar{u}^{h,w,\\lambda}-{\\bf B_0}\\cdot \\nabla \\bar{u}^{h,w,\\lambda}\\\\\n &\\define\\diver \\tilde{F}_1^{w,\\lambda}-{\\bf B_0}\\cdot \\nabla \\bar{u}^{h,w,\\lambda}.\n\\end{align*}\nThe perturbed system \\eqref{4.9} is analogous to \\eqref{mhd-low+high}. The directional derivative of the high-frequency component along the background magnetic field ${\\bf B_0} = (B_1, B_2)$ is\n\\begin{align*}\n {\\bf B_0} \\cdot \\nabla \\bar{u}^{h,w,\\lambda} = B_1\\partial_{x_1}\\bar{u}^{h,w,\\lambda} + B_2\\partial_{x_2}\\bar{u}^{h,w,\\lambda}.\n\\end{align*}\nRemarkably, we observe that the oscillatory part in $\\bar{u}^{h,w,\\lambda}$ satisfies\n\\begin{align*}\n {\\bf B_0} \\cdot \\nabla \\left[\\phi\\left(\\frac{B_1x_2 - B_2x_1}{\\lambda^\\delta}\\right) \\cos\\left(\\lambda(B_1x_2 - B_2x_1) - wt\\right)\\right] = 0,\n\\end{align*}\nwhich leads to the following simplified expression\n\\begin{align*}\n & {\\bf B_0}\\cdot \\nabla \\left(\\lambda^{-s-\\delta}\\phi\\left(\\dfrac{x_1}{\\lambda^\\delta}\\right)\n \\phi\\left(\\dfrac{B_1x_2-B_2x_1}{\\lambda^\\delta}\\right)\\cos \\left(\\lambda \\left(B_1x_2-B_2x_1\\right)-wt\\right)\\right)\\\\\n &=B_1\\lambda^{-s-2\\delta}\\phi'\\left(\\frac{x_1}{\\lambda^{\\delta}}\\right) \\phi\\left(\\dfrac{B_1x_2-B_2x_1}{\\lambda^\\delta}\\right)\\cos \\left(\\lambda \\left(B_1x_2-B_2x_1\\right)-wt\\right)\n\\end{align*}\nTherefore, following analogous arguments as \\eqref{4.3}, we can obtain\n\\begin{align}\\label{4.7}\n \\left\\| {\\bf B_0}\\cdot\\nabla \\bar{u}^{h,w,\\lambda}\\right\\|_{L^2}\\leq C\\lambda^{-s-\\delta}.\n\\end{align}\nCombining the estimates \\eqref{4.5}, \\eqref{4.6} with Lemma \\ref{E,F-error-estimate}, we deduce that\n\\begin{align}\\label{4.8}\n \\|E_1^{w,\\lambda}(t)\\|_{L^2}\\leq C\\lambda^{-\\sigma_{s,\\delta}},\\quad \\|\\tilde{F}_1^{w,\\lambda}(t)\\|_{L^2}\\leq C\\lambda^{-s-1},\n\\end{align}\nwhere \\begin{align*}\n \\sigma_{s,\\delta}=\\min\\{s+1-\\delta,2s-1+\\delta\\}.\n\\end{align*}\n\\subsection{Exact Solutions}\\label{sec:4.3}\nLet $(\\bar{u}_{w,\\lambda},\\bar{b}_{w,\\lambda})$ be the unique solution to system \\eqref{mhd-B} with the initial data $\\left(\\bar{u}^{w,\\lambda}(x,0),\\bar{b}^{w,\\lambda}(x,0)\\right)$, satisfying\n\\begin{align*}\n \\begin{cases}\n \\partial_t \\bar{u}_{w,\\lambda}+\\bar{u}_{w,\\lambda} \\cdot \\nabla \\bar{u}_{w,\\lambda}+\\nabla p_{w,\\lambda}=\\bar{b}_{w,\\lambda}\\cdot \\nabla \\bar{b}_{w,\\lambda}+{\\bf B_0}\\cdot \\nabla \\bar{b}_{w,\\lambda},\\\\[1mm]\n \\partial_t \\bar{b}_{w,\\lambda}-\\Delta \\bar{b}_{w,\\lambda}+\\bar{u}_{w,\\lambda}\\cdot \\nabla \\bar{b}_{w,\\lambda}=\\bar{b}_{w,\\lambda}\\cdot\\nabla \\bar{u}_{w,\\lambda}+{\\bf B_0}\\cdot \\nabla \\bar{u}_{w,\\lambda}, \\\\[1mm]\n \\diver\\, \\bar{u}_{w,\\lambda}=\\diver\\, \\bar{b}_{w,\\lambda}=0,\n \\end{cases}\n\\end{align*}\nIt follows from \\eqref{4.9} that\n\\begin{align*}\n \\begin{cases}\n \\partial_t \\bar{u}^{w,\\lambda}+\\bar{u}^{w,\\lambda} \\cdot \\nabla \\bar{u}^{w,\\lambda}+\\nabla p^{w,\\lambda}=\\bar{b}^{w,\\lambda}\\cdot \\nabla \\bar{b}^{w,\\lambda}+{\\bf B_0}\\cdot\\nabla \\bar{b}^{w,\\lambda}+E_1^{w,\\lambda},\\\\[1mm]\n \\partial_t \\bar{b}^{w,\\lambda}-\\Delta \\bar{b}^{w,\\lambda}+\\bar{u}^{w,\\lambda}\\cdot \\nabla \\bar{b}^{w,\\lambda}=\\bar{b}^{w,\\lambda}\\cdot\\nabla \\bar{u}^{w,\\lambda}+{\\bf B_0}\\cdot\\nabla \\bar{u}^{w,\\lambda}+\\diver \\tilde{F}_1^{w,\\lambda}-{\\bf B_0}\\cdot \\nabla \\bar{u}^{h,w,\\lambda}, \\\\[1mm]\n \\diver\\, \\bar{u}^{w,\\lambda}=\\diver\\, \\bar{b}^{w,\\lambda}=0.\n\\end{cases}\n\\end{align*}\nThe two systems share same initial conditions, which are\n\\begin{align*}\n \\bar{u}_{w,\\lambda}(x, 0)&=\\bar{u}^{w,\\lambda}(x,0)=\\nabla^\\perp\\left( \\bar{\\phi}^{l,w,\\lambda}(x)+\\bar{\\phi}^{h,w,\\lambda}(x,0)\\right),\\\\[1mm]\n \\bar{b}_{w,\\lambda}(x, 0)&=\\bar{b}^{w,\\lambda}(x,0)=\\nabla^\\perp\\bar{\\phi}^{l,w,\\lambda}(x)\n\\end{align*}", + "post_theorem_intro_text_len": 6867, + "post_theorem_intro_text": "\\begin{remark}\n\tThe stabilizing role of background magnetic fields in the MHD equations has been well studied in previous works \\cite{lin-2014-GlobalSmallSolutions,full-not-bardos}. Our analysis reveals that despite the stabilization effect of the background magnetic fields, the data-to-solution map for the resistive MHD equations \\eqref{mhd} maintains its non-uniform continuity property for arbitrary non-zero constant fields ${\\bf B_0}\\neq {\\bf 0}$.\n\\end{remark}\n\nTo simplify the notations, we write the perturbed system as follows\n\\begin{align}\\label{mhd-B}\n\t\\begin{cases}\n\t\t\\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b+{\\bf B_0}\\cdot\\nabla b,\\\\[1mm]\n\t\t\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u+{\\bf B_0}\\cdot\\nabla u, \\\\[1mm]\n\t\t\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\n\t\tu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n\t\\end{cases}\n\\end{align}\nwhere we denote $b - {\\bf B_0}$ by $b$ for convenience.\n\nWe now explain the main ideas of the proof of Theorem \\ref{thm}. It is recalled that Himonas and Misiołek \\cite{nonuniform-2010-cmp-himonas} introduced a frequency decomposition to construct approximate solutions of the two-dimensional incompressible Euler equations. Denote\n \\begin{align}\\label{1.1}\n\tu^{\\pm1,\\lambda}(x,t)=u^{l,\\pm1,\\lambda}(x,t)+u^{h,\\pm1,\\lambda}(x,t),\n\\end{align}\nwhere\n\\begin{itemize}\n\t\\item[(i)] the low-frequency components $u^{l,\\pm1,\\lambda}$ are obtained by solving the Euler equations with low-frequency initial data;\n\t\\item[(ii)] the high-frequency components $u^{h,\\pm1,\\lambda}$ are explicitly constructed by using oscillatory profiles, with frequency parameter $\\lambda$ controlling the spatial concentration,\n\t\\begin{align}\\label{1.2}\n\t\tu^{h,\\pm1,\\lambda}(x,t)=\\nabla^\\perp \\left(\\lambda^{-\\delta-s-1}\\phi\\left(\\dfrac{x_1}{\\lambda^\\delta}\\right)\\phi\\left(\\dfrac{x_2}{\\lambda^\\delta}\\right)\\sin (\\lambda x_2\\mp t)\\right).\n\t\\end{align}\nHere, $\\lambda>0, \\max\\{1-s,0\\}<\\delta < 1$ and $\\phi\\in C_c^\\infty(\\mathbb{R})$ with $\\text{supp}\\,\\phi \\subset [-2,2]$ and $\\phi(x) \\equiv 1$ on $|x| < 1$.\n\\end{itemize}\n\nIt is required to modify the construction of the approximate solutions in the presence of the magnetic field. To the resistive MHD equations, due to the diffusive nature of the magnetic field evolution (versus the purely transport equations of the Euler equations ), we restrict high-frequency perturbations to the velocity field alone and maintain low-frequency components for the magnetic field. More precisely, in the case $\\mathbf{B_0} = \\mathbf{0}$ (no background magnetic field) and $d=2$, we set\n\\begin{itemize}\n\t\\item the low-frequency pairs $(u^{l,\\pm1,\\lambda}, b^{\\pm1,\\lambda})$ by solving the resistive MHD equations \\eqref{mhd} (equivalently, system \\eqref{mhd-B} with $\\mathbf{B_0} = \\mathbf{0}$) with low-frequency initial data;\n\t\\item the high-frequency velocity components $u^{h,\\pm1,\\lambda}$ by retain their profiles as in \\eqref{1.2}.\n\\end{itemize}\n\nIn the three-dimensional case $(d = 3)$, we modify the high-frequency velocity components as\n\t\\begin{align}\\label{1.3}\n\tu^{h,\\pm1,\\lambda}(x,t)=\\begin{pmatrix}\n\t\t\\partial_{x_1}\\\\[1mm]\n\t\t-\\partial_{x_2}\\\\[1mm]\n\t\t0\n\t\\end{pmatrix} \\left(\\lambda^{-\\delta-s-1}\\phi\\left(\\dfrac{x_1}{\\lambda^\\delta}\\right)\\phi\\left(\\dfrac{x_2}{\\lambda^\\delta}\\right)\\sin (\\lambda x_2\\mp t)\\phi(x_3)\\right).\n\\end{align}\n\nThe error terms induced by the high-frequency $u^{h,\\pm1,\\lambda}$ can be written as\n\\begin{align}\\label{1.4}\n\tE^{\\pm1,\\lambda}&=\\partial_t u^{h,\\pm1,\\lambda}+u^{l,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{l,\\pm1,\\lambda},\\\\\\label{1.5}\n\tF^{\\pm1,\\lambda}&=u^{h,\\pm1,\\lambda}\\cdot\\nabla b^{\\pm1,\\lambda}-b^{\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda},\n\\end{align}\nrespectively.\n\nDirect estimation of \\eqref{1.5} will lead to uncontrolled error terms. Our alternative approach consists of three key steps:\n\\begin{enumerate}\n\t\\item {\\bf Divergence reformulation:} We express $F^{\\pm1,\\lambda} =\\text{div}\\,\\, \\tilde{F}^{\\pm1,\\lambda}$ and establish estimates for $\\tilde{F}^{\\pm1,\\lambda}$;\n\t\\item {\\bf Integration by parts:} In $L^2$ inner product computations, we transfer the divergence operator to $b^{\\pm1,\\lambda}$ via integration by parts;\n\t\\item {\\bf Diffusion cancellation:} The resulting terms are precisely canceled by exploiting the magnetic diffusion term $\\Delta b^{\\pm1,\\lambda}$.\n\\end{enumerate}\n\nFor non-zero background fields $\\mathbf{B_0}=(B_1,B_2) \\neq \\mathbf{0}$, we introduce the coordinate transformation\n\\begin{align*}\n\tx_2 \\mapsto B_1x_2 - B_2x_1,\n\\end{align*}\nwhich maintains all norm estimates while achieving exact cancellation of the dominant high-frequency linear term $\\mathbf{B_0} \\cdot \\nabla u^{h,\\pm 1,\\lambda}$ appearing in the error analysis of $F^{\\pm1,\\lambda}$.\n\nA key observation is that the directional derivative exhibits a better decay:\n\\begin{align*}\n\t\\left\\|\\mathbf{B_0} \\cdot \\nabla u^{h,\\pm 1,\\lambda}\\right\\|_{L^2}\\leq C\\lambda^{-s-\\delta} \\ll \\left\\|u^{h,\\pm 1,\\lambda}\\right\\|_{L^2}\\leq C\\lambda^{-s}, \\text{~for~}\\lambda \\to \\infty,\n\\end{align*}\nwhere $\\delta>0$ represent the improved decay rate.\n\n\t The paper is organized as follows. section \\ref{sec:2} is the preliminary analytical framework, in which subsection \\ref{sec:2.1} is about key Lemmas and subsection \\ref{sec:2.2} is on the local well-posedness theory. The core strategy of the proof is implemented in section \\ref{sec:3} (zero background field case, $\\mathbf{B_0} = \\mathbf{0}$) and section \\ref{sec:4} (non-zero background field case, $\\mathbf{B_0} \\neq \\mathbf{0}$), following the approach as follows: construction of approximate solutions in subsections \\ref{sec:3.1} and \\ref{sec:4.1}, precise estimation of approximation errors in subsections \\ref{sec:3.2} and \\ref{sec:4.2}, rigorous construction of exact solutions in subsections \\ref{sec:3.3} and \\ref{sec:4.3}, finishing the proof of Theorem in the case $\\mathbf{B_0} = \\mathbf{0}$ and $\\mathbf{B_0} \\neq \\mathbf{0}$ in subsections \\ref{sec:3.4} and \\ref{sec:4.4}, respectively.\n\n{\\bf Notations:}\n\\begin{enumerate}\n\\item {\\bf Function Spaces:} Throughout this work, let $X$ denote a Banach space equipped with norm $\\|\\cdot\\|_X$. Since all function spaces considered here are defined on $\\mathbb{R}^d$ for $d = 2,3$, we will suppress the domain $\\mathbb{R}^d$ in our notation unless otherwise specified.\n\t\\item {\\bf Differential Operators:} For $x \\in \\mathbb{R}^2$, we define the perpendicular gradient operator as $\\nabla^\\perp := (\\partial_{x_2}, -\\partial_{x_1})$.\n\t\\item {\\bf Joint Norms:} Given functions $f, g \\in X(\\mathbb{R}^d)$, we define their joint norm by\n\\begin{align*}\n\t\\|f(\\cdot),g(\\cdot)\\|_{X(\\mathbb{R}^d)}=\t\\|f(\\cdot)\\|_{X(\\mathbb{R}^d)}+\t\\|g(\\cdot)\\|_{X(\\mathbb{R}^d)}\n\\end{align*}\nadopting this concise notation for simplicity.\n\\end{enumerate}", + "sketch": "To prove Theorem~\\ref{thm}, the authors adapt the “frequency decomposition” method of Himonas–Misio\\l{}ek. They decompose the velocity as\n\\[\n u^{\\pm1,\\lambda}(x,t)=u^{l,\\pm1,\\lambda}(x,t)+u^{h,\\pm1,\\lambda}(x,t),\n\\]\nwhere (i) the low-frequency parts $u^{l,\\pm1,\\lambda}$ come from solving the (appropriate) equations with low-frequency initial data, and (ii) the high-frequency parts $u^{h,\\pm1,\\lambda}$ are “explicitly constructed by using oscillatory profiles,” with frequency parameter $\\lambda$ (in 2D, profile as in \\eqref{1.2}; in 3D, modified as in \\eqref{1.3}).\n\nFor resistive MHD, “due to the diffusive nature of the magnetic field evolution,” they “restrict high-frequency perturbations to the velocity field alone and maintain low-frequency components for the magnetic field”: the low-frequency pair $(u^{l,\\pm1,\\lambda}, b^{\\pm1,\\lambda})$ is obtained by solving the resistive MHD system with low-frequency initial data, while $u^{h,\\pm1,\\lambda}$ retains the oscillatory profile.\n\nThey then estimate the approximation errors induced by $u^{h,\\pm1,\\lambda}$, written as\n\\[\nE^{\\pm1,\\lambda}=\\partial_t u^{h,\\pm1,\\lambda}+u^{l,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{l,\\pm1,\\lambda},\n\\]\n\\[\nF^{\\pm1,\\lambda}=u^{h,\\pm1,\\lambda}\\cdot\\nabla b^{\\pm1,\\lambda}-b^{\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}.\n\\]\nSince “direct estimation of \\eqref{1.5} will lead to uncontrolled error terms,” their alternative consists of three steps: (1) “express $F^{\\pm1,\\lambda}=\\mathrm{div}\\,\\tilde F^{\\pm1,\\lambda}$” and estimate $\\tilde F^{\\pm1,\\lambda}$; (2) “transfer the divergence operator to $b^{\\pm1,\\lambda}$ via integration by parts” in $L^2$ inner products; (3) use “diffusion cancellation,” where “the resulting terms are precisely canceled by exploiting the magnetic diffusion term $\\Delta b^{\\pm1,\\lambda}$.”\n\nFor nonzero constant background fields $\\mathbf{B_0}\\neq\\mathbf{0}$, they “introduce the coordinate transformation $x_2\\mapsto B_1x_2-B_2x_1$,” which “maintains all norm estimates” and yields “exact cancellation of the dominant high-frequency linear term $\\mathbf{B_0}\\cdot\\nabla u^{h,\\pm1,\\lambda}$” in the error analysis of $F^{\\pm1,\\lambda}$. They also note the key decay bound\n\\[\n\\|\\mathbf{B_0}\\cdot\\nabla u^{h,\\pm1,\\lambda}\\|_{L^2}\\le C\\lambda^{-s-\\delta}\\ll \\|u^{h,\\pm1,\\lambda}\\|_{L^2}\\le C\\lambda^{-s}\\quad (\\lambda\\to\\infty),\n\\]\nwith $\\delta>0$ giving “improved decay.”\n\nFinally, they describe the proof strategy as: “construction of approximate solutions,” “precise estimation of approximation errors,” “rigorous construction of exact solutions,” and “finishing the proof” in the cases $\\mathbf{B_0}=\\mathbf{0}$ and $\\mathbf{B_0}\\neq\\mathbf{0}$.", + "expanded_sketch": "To prove the main theorem, the authors adapt the “frequency decomposition” method of Himonas–Misio\\l{}ek. They decompose the velocity as\n\\[\n u^{\\pm1,\\lambda}(x,t)=u^{l,\\pm1,\\lambda}(x,t)+u^{h,\\pm1,\\lambda}(x,t),\n\\]\nwhere (i) the low-frequency parts $u^{l,\\pm1,\\lambda}$ come from solving the (appropriate) equations with low-frequency initial data, and (ii) the high-frequency parts $u^{h,\\pm1,\\lambda}$ are “explicitly constructed by using oscillatory profiles,” with frequency parameter $\\lambda$ (in 2D, profile as in\n\\begin{align}\\label{1.2}\n\t\tu^{h,\\pm1,\\lambda}(x,t)=\\nabla^\\perp \\left(\\lambda^{-\\delta-s-1}\\phi\\left(\\dfrac{x_1}{\\lambda^\\delta}\\right)\\phi\\left(\\dfrac{x_2}{\\lambda^\\delta}\\right)\\sin (\\lambda x_2\\mp t)\\right).\n\t\\end{align}\n; in 3D, modified as in \\eqref{1.3}).\n\nFor resistive MHD, “due to the diffusive nature of the magnetic field evolution,” they “restrict high-frequency perturbations to the velocity field alone and maintain low-frequency components for the magnetic field”: the low-frequency pair $(u^{l,\\pm1,\\lambda}, b^{\\pm1,\\lambda})$ is obtained by solving the resistive MHD system with low-frequency initial data, while $u^{h,\\pm1,\\lambda}$ retains the oscillatory profile.\n\nThey then estimate the approximation errors induced by $u^{h,\\pm1,\\lambda}$, written as\n\\begin{align}\\label{1.4}\n\tE^{\\pm1,\\lambda}&=\\partial_t u^{h,\\pm1,\\lambda}+u^{l,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{l,\\pm1,\\lambda},\\\\\\label{1.5}\n\tF^{\\pm1,\\lambda}&=u^{h,\\pm1,\\lambda}\\cdot\\nabla b^{\\pm1,\\lambda}-b^{\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda},\n\\end{align}\nSince “direct estimation of the equation above will lead to uncontrolled error terms,” their alternative consists of three steps: (1) “express $F^{\\pm1,\\lambda}=\\mathrm{div}\\,\\tilde F^{\\pm1,\\lambda}$” and estimate $\\tilde F^{\\pm1,\\lambda}$; (2) “transfer the divergence operator to $b^{\\pm1,\\lambda}$ via integration by parts” in $L^2$ inner products; (3) use “diffusion cancellation,” where “the resulting terms are precisely canceled by exploiting the magnetic diffusion term $\\Delta b^{\\pm1,\\lambda}$.”\n\nFor nonzero constant background fields $\\mathbf{B_0}\\neq\\mathbf{0}$, they “introduce the coordinate transformation $x_2\\mapsto B_1x_2-B_2x_1$,” which “maintains all norm estimates” and yields “exact cancellation of the dominant high-frequency linear term $\\mathbf{B_0}\\cdot\\nabla u^{h,\\pm1,\\lambda}$” in the error analysis of $F^{\\pm1,\\lambda}$. They also note the key decay bound\n\\[\n\\|\\mathbf{B_0}\\cdot\\nabla u^{h,\\pm1,\\lambda}\\|_{L^2}\\le C\\lambda^{-s-\\delta}\\ll \\|u^{h,\\pm1,\\lambda}\\|_{L^2}\\le C\\lambda^{-s}\\quad (\\lambda\\to\\infty),\n\\]\nwith $\\delta>0$ giving “improved decay.”\n\nFinally, they describe the proof strategy as: “construction of approximate solutions,” “precise estimation of approximation errors,” “rigorous construction of exact solutions,” and “finishing the proof” in the cases $\\mathbf{B_0}=\\mathbf{0}$ and $\\mathbf{B_0}\\neq\\mathbf{0}$.", + "expanded_theorem": "\\label{thm}\n\tLet $s>0, \\lambda>0, d=2,3$ and $T>0$. Then the data-to-solution map $(u_0,b_0)\\mapsto(u,b)$ for the equations\n\\begin{align}\\label{mhd}\n\t\\begin{cases}\n\t\\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b,\\\\[1mm]\n\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u, \\\\[1mm]\n\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\\nu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n\t\\end{cases}\n\\end{align}\n is non-uniformly continuous from a bounded subset in $H^{s}(\\mathbb{R}^d)\\times H^{s}(\\mathbb{R}^d)$ into $C\\left([0,T],H^{s}(\\mathbb{R}^d)\\right)\\times C\\left([0,T],H^{s}(\\mathbb{R}^d)\\right)$.\n\n More precisely, for any $\\gamma>0$ and arbitrary constant magnetic field ${\\bf B_0}\\in \\mathbb{R}^d$, there exists two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$ such that\n\t\\begin{itemize}\n\t\t\\item the solutions satisfy\n\\begin{align*}\n\t\t\t(u^{\\pm 1,\\lambda}, b^{\\pm 1,\\lambda} - {\\bf B_0}) \\in C([0,T], H^{s}(\\mathbb{R}^{d})) \\times C([0,T], H^{s}(\\mathbb{R}^{d}));\n\\end{align*}\n\t\t\\item the initial data satisfy\n\\begin{equation}\\label{nonuniform-initial-bound}\n\t\\begin{array}{l}\n\t\t\t\t\t\t\\|u^{+1,\\lambda}(0,\\cdot)\\|_{H^{s}} + \\|b^{+1,\\lambda}(0,\\cdot) - {\\bf B_0}\\|_{H^{s}} \\leq \\gamma, \\\\[2mm]\n\t\t\t\\|u^{-1,\\lambda}(0,\\cdot)\\|_{H^{s}} + \\|b^{-1,\\lambda}(0,\\cdot) - {\\bf B_0}\\|_{H^{s}} \\leq \\gamma;\n\t\\end{array}\n\\end{equation}\n\t\t\\item and the non-uniform continuity is characterized by\n\t\t\\begin{itemize}\n\t\t\t\\item at initial time $t=0$,\n\t\t\t\\begin{align}\\label{nonuniform-initial-diff}\n\t\t\t\t\\lim_{\\lambda\\to\\infty} \\|u^{+1,\\lambda}(0,\\cdot) - u^{-1,\\lambda}(0,\\cdot)\\|_{H^{s}} = 0,\n\t\t\t\\end{align}\n\t\t\t\\item for evolution times $t>0$,\n\t\t\t\\begin{align}\\label{nonuniform-low-bound-sint}\n\t\t\t\t\\lim_{\\lambda \\to\\infty} \\|u^{+1,\\lambda}(t) - u^{-1,\\lambda}(t)\\|_{H^{s}} \\geq c\\gamma |\\sin t|,\n\t\t\t\\end{align}\n\t\t\\end{itemize}\n\t\twhere $c = c(s,d) > 0$ is a constant depending only on $s$ and $d$.\n\t\\end{itemize}", + "theorem_type": [ + "Existential–Universal", + "Asymptotic or Limit" + ], + "mcq": { + "question": "For the incompressible resistive MHD system on $\\mathbb{R}^d$ with $d\\in\\{2,3\\}$,\n\\[\n\\begin{cases}\n\\partial_t u+u\\cdot\\nabla u+\\nabla p=b\\cdot\\nabla b,\\\\\n\\partial_t b-\\Delta b+u\\cdot\\nabla b=b\\cdot\\nabla u,\\\\\n\\operatorname{div}u=\\operatorname{div}b=0,\\\\\nu(x,0)=u_0(x),\\quad b(x,0)=b_0(x),\n\\end{cases}\n\\]\nlet the data-to-solution map send initial data $(u_0,b_0)$ to the corresponding solution $(u,b)$. Fix $s>0$ and $T>0$, and view this map as acting from a bounded subset of $H^s(\\mathbb{R}^d)\\times H^s(\\mathbb{R}^d)$ into $C([0,T],H^s(\\mathbb{R}^d))\\times C([0,T],H^s(\\mathbb{R}^d))$. Which statement holds, including the explicit sequential form of non-uniform continuity around an arbitrary constant background magnetic field ${\\bf B}_0\\in\\mathbb{R}^d$?", + "correct_choice": { + "label": "A", + "text": "The data-to-solution map is non-uniformly continuous from a bounded subset of $H^{s}(\\mathbb{R}^d)\\times H^{s}(\\mathbb{R}^d)$ into $C([0,T],H^{s}(\\mathbb{R}^d))\\times C([0,T],H^{s}(\\mathbb{R}^d))$. More precisely, for every $\\gamma>0$ and every constant magnetic field ${\\bf B}_0\\in\\mathbb{R}^d$, there exist two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$, indexed by $\\lambda\\to\\infty$, such that\n\\[\n(u^{\\pm1,\\lambda},\\, b^{\\pm1,\\lambda}-{\\\\bf B}_0)\\in C([0,T],H^s(\\mathbb{R}^d))\\times C([0,T],H^s(\\mathbb{R}^d)),\n\\]\nwith initial bounds\n\\[\n\\|u^{+1,\\lambda}(0)\\|_{H^s}+\\|b^{+1,\\lambda}(0)-{\\bf B}_0\\|_{H^s}\\le \\gamma,\n\\qquad\n\\|u^{-1,\\lambda}(0)\\|_{H^s}+\\|b^{-1,\\lambda}(0)-{\\bf B}_0\\|_{H^s}\\le \\gamma,\n\\]\nand such that\n\\[\n\\lim_{\\lambda\\to\\infty}\\|u^{+1,\\lambda}(0)-u^{-1,\\lambda}(0)\\|_{H^s}=0,\n\\]\nwhile for every evolution time $t>0$ (in particular, $t\\in(0,T]$),\n\\[\n\\lim_{\\lambda\\to\\infty}\\|u^{+1,\\lambda}(t)-u^{-1,\\lambda}(t)\\|_{H^s}\\ge c(s,d)\\,\\gamma\\,|\\sin t|,\n\\]\nwhere $c(s,d)>0$ depends only on $s$ and $d$." + }, + "choices": [ + { + "label": "B", + "text": "The data-to-solution map is non-uniformly continuous from a bounded subset of $H^{s}(\\mathbb{R}^d)\\times H^{s}(\\mathbb{R}^d)$ into $C([0,T],H^{s}(\\mathbb{R}^d))\\times C([0,T],H^{s}(\\mathbb{R}^d))$. More precisely, for every $\\gamma>0$ and every constant magnetic field ${\\bf B}_0\\in\\mathbb{R}^d$, there exist two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$, indexed by $\\lambda\\to\\infty$, such that\n\\[\n(u^{\\pm1,\\lambda},\\, b^{\\pm1,\\lambda}-{\\bf B}_0)\\in C([0,T],H^s(\\mathbb{R}^d))\\times C([0,T],H^s(\\mathbb{R}^d)),\n\\]\nwith the same initial bounds as in the theorem, and such that\n\\[\n\\lim_{\\lambda\\to\\infty}\\Big(\\|u^{+1,\\lambda}(0)-u^{-1,\\lambda}(0)\\|_{H^s}+\\|b^{+1,\\lambda}(0)-b^{-1,\\lambda}(0)\\|_{H^s}\\Big)=0,\n\\]\nwhile for every evolution time $t>0$,\n\\[\n\\lim_{\\lambda\\to\\infty}\\Big(\\|u^{+1,\\lambda}(t)-u^{-1,\\lambda}(t)\\|_{H^s}+\\|b^{+1,\\lambda}(t)-b^{-1,\\lambda}(t)\\|_{H^s}\\Big)\\ge c(s,d)\\,\\gamma\\,|\\sin t|,\n\\]\nwhere $c(s,d)>0$ depends only on $s$ and $d$." + }, + { + "label": "C", + "text": "The data-to-solution map is non-uniformly continuous from a bounded subset of $H^{s}(\\mathbb{R}^d)\\times H^{s}(\\mathbb{R}^d)$ into $C([0,T],H^{s}(\\mathbb{R}^d))\\times C([0,T],H^{s}(\\mathbb{R}^d))$. More precisely, for every $\\gamma>0$ and every constant magnetic field ${\\bf B}_0\\in\\mathbb{R}^d$, there exist two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$, indexed by $\\lambda\\to\\infty$, such that\n\\[\n(u^{\\pm1,\\lambda},\\, b^{\\pm1,\\lambda}-{\\bf B}_0)\\in C([0,T],H^s(\\mathbb{R}^d))\\times C([0,T],H^s(\\mathbb{R}^d)),\n\\]\nwith initial bounds\n\\[\n\\|u^{+1,\\lambda}(0)\\|_{H^s}+\\|b^{+1,\\lambda}(0)-{\\bf B}_0\\|_{H^s}\\le \\gamma,\n\\qquad\n\\|u^{-1,\\lambda}(0)\\|_{H^s}+\\|b^{-1,\\lambda}(0)-{\\bf B}_0\\|_{H^s}\\le \\gamma,\n\\]\nand such that\n\\[\n\\lim_{\\lambda\\to\\infty}\\|u^{+1,\\lambda}(0)-u^{-1,\\lambda}(0)\\|_{H^s}=0.\n\\]\nIn particular, the data-to-solution map fails to be uniformly continuous on that bounded subset." + }, + { + "label": "D", + "text": "The data-to-solution map is non-uniformly continuous from a bounded subset of $H^{s}(\\mathbb{R}^d)\\times H^{s}(\\mathbb{R}^d)$ into $C([0,T],H^{s}(\\mathbb{R}^d))\\times C([0,T],H^{s}(\\mathbb{R}^d))$. More precisely, there exists a constant $c=c(s,d)>0$ such that for every $\\gamma>0$ one can choose two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$ with the property that for every constant magnetic field ${\\bf B}_0\\in\\mathbb{R}^d$,\n\\[\n(u^{\\pm1,\\lambda},\\, b^{\\pm1,\\lambda}-{\\bf B}_0)\\in C([0,T],H^s(\\mathbb{R}^d))\\times C([0,T],H^s(\\mathbb{R}^d)),\n\\]\nwith the same initial bounds, and such that\n\\[\n\\lim_{\\lambda\\to\\infty}\\|u^{+1,\\lambda}(0)-u^{-1,\\lambda}(0)\\|_{H^s}=0,\n\\]\nwhile for every evolution time $t>0$,\n\\[\n\\lim_{\\lambda\\to\\infty}\\|u^{+1,\\lambda}(t)-u^{-1,\\lambda}(t)\\|_{H^s}\\ge c\\,\\gamma\\,|\\sin t|.\n\\]" + }, + { + "label": "E", + "text": "The data-to-solution map is non-uniformly continuous from a bounded subset of $H^{s}(\\mathbb{R}^d)\\times H^{s}(\\mathbb{R}^d)$ into $C([0,T],H^{s}(\\mathbb{R}^d))\\times C([0,T],H^{s}(\\mathbb{R}^d))$. More precisely, for every $\\gamma>0$ and every constant magnetic field ${\\bf B}_0\\in\\mathbb{R}^d$, there exist two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$, indexed by $\\lambda\\to\\infty$, such that\n\\[\n(u^{\\pm1,\\lambda},\\, b^{\\pm1,\\lambda}-{\\bf B}_0)\\in C([0,T],H^s(\\mathbb{R}^d))\\times C([0,T],H^s(\\mathbb{R}^d)),\n\\]\nwith the same initial bounds, and such that\n\\[\n\\lim_{\\lambda\\to\\infty}\\|u^{+1,\\lambda}(0)-u^{-1,\\lambda}(0)\\|_{H^s}=0,\n\\]\nwhile for every evolution time $t\\in[0,T]$,\n\\[\n\\liminf_{\\lambda\\to\\infty}\\|u^{+1,\\lambda}(t)-u^{-1,\\lambda}(t)\\|_{H^s}\\ge c(s,d)\\,\\gamma,\n\\]\nwhere $c(s,d)>0$ depends only on $s$ and $d$." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "B" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "diffusion cancellation", + "tampered_component": "velocity-only high-frequency instability promoted to joint $(u,b)$ separation", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped the explicit positive-time lower bound $\\ge c\\gamma|\\sin t|$ and retained only failure of uniform continuity", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "geometric_construction", + "tampered_component": "quantifier order in dependence on ${\\bf B}_0$", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "characteristic", + "tampered_component": "time-dependent lower bound $|\\sin t|$ replaced by uniform-in-time positive separation on $[0,T]$", + "template_used": "stronger_trap" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly reveal the correct option. It specifies the PDE setting and asks for the valid statement, but the decisive details—velocity-only separation, the exact quantifiers in ${\\bf B}_0$, and the $c(s,d)\\gamma|\\sin t|$ lower bound—are not given away." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recall question: it mainly asks for the precise statement of a known non-uniform continuity result. However, it is not a pure tautology, since the options differ in meaningful ways (quantifier order, weaker-vs-stronger conclusions, and which norms/sequences appear)." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish the exact theorem from plausible variants, especially between the correct sharp statement and nearby strengthenings/weakenings. Still, the task is primarily recognition/recall of a theorem statement rather than genuine generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target realistic failure modes: strengthening to joint $(u,b)$ separation, dropping the explicit lower bound to a weaker true statement, altering quantifier dependence on ${\\bf B}_0$, and replacing the $|\\sin t|$ dependence by an unrealistically uniform-in-time gap." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with strong distractors and little answer leakage, but it tests precise theorem recall more than generative reasoning." + } + }, + { + "id": "2602.06972v1", + "paper_link": "http://arxiv.org/abs/2602.06972v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "thm", + "content": "\\label{main}\nLet $S$ be a two-element ai-semiring, and let $n \\geq 2$ be an integer.\nThen the matrix semiring $\\mathbf{M}_n(S)$ is finitely based if and only if $S$ is not a distributive lattice.", + "start_pos": 12231, + "end_pos": 12447, + "label": "main" + }, + "ref_dict": { + "main": "\\begin{thm}\\label{main}\nLet $S$ be a two-element ai-semiring, and let $n \\geq 2$ be an integer.\nThen the matrix semiring $\\mathbf{M}_n(S)$ is finitely based if and only if $S$ is not a distributive lattice.\n\\end{thm}" + }, + "pre_theorem_intro_text_len": 5955, + "pre_theorem_intro_text": "An \\emph{additively idempotent semiring} (abbreviated as ai-semiring)\nis an algebraic structure $(S, +, \\cdot)$ equipped with two binary operations $+$ and $\\cdot$ such that:\n\\begin{itemize}\n\\item the additive reduct $(S, +)$ is a commutative idempotent semigroup;\n\\item the multiplicative reduct $(S, \\cdot)$ is a semigroup;\n\\item the distributive laws hold:\n\\[\n(x+y)z \\approx xz + yz, \\qquad x(y+z) \\approx xy + xz.\n\\]\n\\end{itemize}\nThis class of algebras includes well-known examples such as the Kleene semiring of regular languages~\\cite{con},\nthe max-plus algebra~\\cite{aei}, the power semiring of a semigroup~\\cite{dgv24},\nthe endomorphism semiring of a semilattice~\\cite{dgv25},\nthe semiring of all binary relations on a set \\cite{dolinka2009}, and distributive lattices~\\cite{burris1981}.\nThese and other similar algebras have found significant applications in several branches of mathematics,\nsuch as algebraic geometry~\\cite{cc}, tropical geometry~\\cite{ms}, information science~\\cite{gl},\ntheoretical computer science~\\cite{go}, and soft constraint solving and programming~\\cite{bis2004}.\n\nLet $S$ be an ai-semiring. Define a binary relation $\\leq$ on $S$ by\n\\[\na\\leq b\\Leftrightarrow a+b=b.\n\\]\nThen $\\leq$ is a partial order on $S$, and $(S, \\leq)$ becomes an upper semilattice in which\nthe supremum of any elements $a$ and $b$ is $a+b$.\nConsequently, the additive reduct $(S, +)$ is uniquely determined by this semilattice order.\nTherefore, it is often convenient to represent the addition operation using the Hasse diagram of $(S, \\leq)$.\nMoreover, one can easily verify that this semilattice order is compatible with multiplication.\nFor this reason, an ai-semiring is often called a \\emph{semilattice-ordered semigroup}.\n\nLet $n\\geq 1$ be an integer,\nlet $S$ be an ai-semiring, and let $\\mathbf{M}_n(S)$ denote the set of all $n \\times n$ matrices over $S$.\nThen $\\mathbf{M}_n(S)$ forms an ai-semiring under the usual matrix addition and multiplication, that is,\n\\[\nA+B = [a_{ij}+b_{ij}]_{n\\times n}, \\quad A \\cdot B = \\left[ \\sum_{k=1}^n a_{ik} b_{kj} \\right]_{n\\times n}\n\\]\nfor all $A=[a_{ij}]_{n\\times n}, B=[b_{ij}]_{n\\times n}\\in \\mathbf{M}_n(S)$.\nIt is obvious that $S$ is isomorphic to $\\mathbf{M}_1(S)$.\nFor $n\\geq 2$, $S$ can be embedded into $\\mathbf{M}_n(S)$,\nsince the mapping\n\\[\n\\varphi\\colon S \\rightarrow \\mathbf{M}_n(S),\\quad a \\mapsto [a]_{n\\times n},\n\\]\nwhere $[a]_{n\\times n}$ denotes the constant matrix with all entries equal to $a$,\nis an embedding mapping.\nMoreover, if $S$ has an additive identity that is also a multiplicative zero,\nthen we can easily obtain the following infinite strict ascending chain\n\\[\nS\\hookrightarrow \\mathbf{M}_2(S)\\hookrightarrow \\mathbf{M}_{3}(S)\\hookrightarrow \\mathbf{M}_{4}(S)\\hookrightarrow\\cdots.\n\\]\n\nLet $\\mathcal{V}$ be a class of algebras.\nThen $\\mathcal{V}$ is called a \\emph{variety} if it is closed under\ntaking subalgebras, homomorphic images, and arbitrary direct products.\nBy the celebrated Birkhoff's theorem \\cite{birkhoff1935}, $\\mathcal{V}$ is a variety if and only if\nit is an \\emph{equational class}, that is, the class of all algebras satisfying some set of identities.\n\nA variety is \\emph{finitely based} if it can be defined by a finite set of identities;\notherwise, it is \\emph{nonfinitely based}.\nAn algebra $A$ is finitely based or nonfinitely based if the variety $\\mathsf{V}(A)$\nit generates is finitely based or not.\nThe \\emph{finite basis problem} for a class of algebras,\none of the central problems in universal algebra,\nconcerns the classification of\nits members with respect to the property of being finitely based.\n\nA variety is \\emph{locally finite} if each of its finitely generated members is finite.\nA locally finite variety is \\emph{inherently nonfinitely based}\nif it is not contained in any finitely based locally finite variety.\nSince every finite algebra generates a locally finite variety,\nwe say that a finite algebra $A$ is inherently nonfinitely based if\nthe variety $\\mathsf{V}(A)$ is inherently nonfinitely based.\nIt follows immediately that every finite algebra whose variety contains\nan inherently nonfinitely based algebra is nonfinitely based.\nConsequently, every inherently nonfinitely based algebra must be nonfinitely based.\n\nOver the past two decades,\nthe finite basis problem for ai-semirings has attracted considerable attention,\nresulting in substantial\nprogress~\\cite{dol07, dolinka2009, dgv24, dgv25, gpz05, gv23, gv2501, gv2510, jrz, pas05, rz16, rzw, sr, volkov2024, yr25, yrzs, zrc}.\nIn particular, Dolinka~\\cite{dol07} constructed the first example of a nonfinitely based finite ai-semiring.\nPastijn et al.~\\cite{gpz05, pas05} proved that every ai-semiring satisfying the identity $x^2 \\approx x$ is finitely based;\nRen et al.~\\cite{rz16, rzw} later extended this to ai-semirings satisfying the identity $x^3 \\approx x$.\nShao and Ren~\\cite{sr} established that every algebra in the variety generated by all two-element ai-semirings is finitely based.\nSubsequently,\nJackson et al.~\\cite{jrz} and Zhao et al.~\\cite{zrc}\nprovided a complete classification of all three-element ai-semirings with respect to the finite basis property.\nMost recently,\nDolinka, Gusev and Volkov~\\cite{dgv25, gv2501} settled the finite basis problem for\nthe endomorphism semirings of finite semilattices.\n\nTo the best of our knowledge,\nthe study of the finite basis problem for matrix semirings was initiated by\nDolinka~\\cite{dolinka2009}, who considered the problem for $\\mathbf{M}_n(D_2)$ for $n\\geqslant 2$,\nwhere $D_2$ denotes the two-element distributive lattice.\nGusev and Volkov~\\cite{gv2510, volkov2024} are currently investigating\nthe same problem for upper triangular matrix semirings over $D_2$.\nMotivated by these works,\nwe undertake a systematic classification of matrix semirings $\\mathbf{M}_n(S)$,\nwhere $S$ ranges over all two-element ai-semirings.\nThe main result of this paper provides the following complete characterization.", + "context": "An \\emph{additively idempotent semiring} (abbreviated as ai-semiring)\nis an algebraic structure $(S, +, \\cdot)$ equipped with two binary operations $+$ and $\\cdot$ such that:\n\\begin{itemize}\n\\item the additive reduct $(S, +)$ is a commutative idempotent semigroup;\n\\item the multiplicative reduct $(S, \\cdot)$ is a semigroup;\n\\item the distributive laws hold:\n\\[\n(x+y)z \\approx xz + yz, \\qquad x(y+z) \\approx xy + xz.\n\\]\n\\end{itemize}\nThis class of algebras includes well-known examples such as the Kleene semiring of regular languages~\\cite{con},\nthe max-plus algebra~\\cite{aei}, the power semiring of a semigroup~\\cite{dgv24},\nthe endomorphism semiring of a semilattice~\\cite{dgv25},\nthe semiring of all binary relations on a set \\cite{dolinka2009}, and distributive lattices~\\cite{burris1981}.\nThese and other similar algebras have found significant applications in several branches of mathematics,\nsuch as algebraic geometry~\\cite{cc}, tropical geometry~\\cite{ms}, information science~\\cite{gl},\ntheoretical computer science~\\cite{go}, and soft constraint solving and programming~\\cite{bis2004}.\n\nLet $S$ be an ai-semiring. Define a binary relation $\\leq$ on $S$ by\n\\[\na\\leq b\\Leftrightarrow a+b=b.\n\\]\nThen $\\leq$ is a partial order on $S$, and $(S, \\leq)$ becomes an upper semilattice in which\nthe supremum of any elements $a$ and $b$ is $a+b$.\nConsequently, the additive reduct $(S, +)$ is uniquely determined by this semilattice order.\nTherefore, it is often convenient to represent the addition operation using the Hasse diagram of $(S, \\leq)$.\nMoreover, one can easily verify that this semilattice order is compatible with multiplication.\nFor this reason, an ai-semiring is often called a \\emph{semilattice-ordered semigroup}.\n\nLet $n\\geq 1$ be an integer,\nlet $S$ be an ai-semiring, and let $\\mathbf{M}_n(S)$ denote the set of all $n \\times n$ matrices over $S$.\nThen $\\mathbf{M}_n(S)$ forms an ai-semiring under the usual matrix addition and multiplication, that is,\n\\[\nA+B = [a_{ij}+b_{ij}]_{n\\times n}, \\quad A \\cdot B = \\left[ \\sum_{k=1}^n a_{ik} b_{kj} \\right]_{n\\times n}\n\\]\nfor all $A=[a_{ij}]_{n\\times n}, B=[b_{ij}]_{n\\times n}\\in \\mathbf{M}_n(S)$.\nIt is obvious that $S$ is isomorphic to $\\mathbf{M}_1(S)$.\nFor $n\\geq 2$, $S$ can be embedded into $\\mathbf{M}_n(S)$,\nsince the mapping\n\\[\n\\varphi\\colon S \\rightarrow \\mathbf{M}_n(S),\\quad a \\mapsto [a]_{n\\times n},\n\\]\nwhere $[a]_{n\\times n}$ denotes the constant matrix with all entries equal to $a$,\nis an embedding mapping.\nMoreover, if $S$ has an additive identity that is also a multiplicative zero,\nthen we can easily obtain the following infinite strict ascending chain\n\\[\nS\\hookrightarrow \\mathbf{M}_2(S)\\hookrightarrow \\mathbf{M}_{3}(S)\\hookrightarrow \\mathbf{M}_{4}(S)\\hookrightarrow\\cdots.\n\\]\n\nLet $\\mathcal{V}$ be a class of algebras.\nThen $\\mathcal{V}$ is called a \\emph{variety} if it is closed under\ntaking subalgebras, homomorphic images, and arbitrary direct products.\nBy the celebrated Birkhoff's theorem \\cite{birkhoff1935}, $\\mathcal{V}$ is a variety if and only if\nit is an \\emph{equational class}, that is, the class of all algebras satisfying some set of identities.\n\nOver the past two decades,\nthe finite basis problem for ai-semirings has attracted considerable attention,\nresulting in substantial\nprogress~\\cite{dol07, dolinka2009, dgv24, dgv25, gpz05, gv23, gv2501, gv2510, jrz, pas05, rz16, rzw, sr, volkov2024, yr25, yrzs, zrc}.\nIn particular, Dolinka~\\cite{dol07} constructed the first example of a nonfinitely based finite ai-semiring.\nPastijn et al.~\\cite{gpz05, pas05} proved that every ai-semiring satisfying the identity $x^2 \\approx x$ is finitely based;\nRen et al.~\\cite{rz16, rzw} later extended this to ai-semirings satisfying the identity $x^3 \\approx x$.\nShao and Ren~\\cite{sr} established that every algebra in the variety generated by all two-element ai-semirings is finitely based.\nSubsequently,\nJackson et al.~\\cite{jrz} and Zhao et al.~\\cite{zrc}\nprovided a complete classification of all three-element ai-semirings with respect to the finite basis property.\nMost recently,\nDolinka, Gusev and Volkov~\\cite{dgv25, gv2501} settled the finite basis problem for\nthe endomorphism semirings of finite semilattices.\n\nTo the best of our knowledge,\nthe study of the finite basis problem for matrix semirings was initiated by\nDolinka~\\cite{dolinka2009}, who considered the problem for $\\mathbf{M}_n(D_2)$ for $n\\geqslant 2$,\nwhere $D_2$ denotes the two-element distributive lattice.\nGusev and Volkov~\\cite{gv2510, volkov2024} are currently investigating\nthe same problem for upper triangular matrix semirings over $D_2$.\nMotivated by these works,\nwe undertake a systematic classification of matrix semirings $\\mathbf{M}_n(S)$,\nwhere $S$ ranges over all two-element ai-semirings.\nThe main result of this paper provides the following complete characterization.", + "full_context": "An \\emph{additively idempotent semiring} (abbreviated as ai-semiring)\nis an algebraic structure $(S, +, \\cdot)$ equipped with two binary operations $+$ and $\\cdot$ such that:\n\\begin{itemize}\n\\item the additive reduct $(S, +)$ is a commutative idempotent semigroup;\n\\item the multiplicative reduct $(S, \\cdot)$ is a semigroup;\n\\item the distributive laws hold:\n\\[\n(x+y)z \\approx xz + yz, \\qquad x(y+z) \\approx xy + xz.\n\\]\n\\end{itemize}\nThis class of algebras includes well-known examples such as the Kleene semiring of regular languages~\\cite{con},\nthe max-plus algebra~\\cite{aei}, the power semiring of a semigroup~\\cite{dgv24},\nthe endomorphism semiring of a semilattice~\\cite{dgv25},\nthe semiring of all binary relations on a set \\cite{dolinka2009}, and distributive lattices~\\cite{burris1981}.\nThese and other similar algebras have found significant applications in several branches of mathematics,\nsuch as algebraic geometry~\\cite{cc}, tropical geometry~\\cite{ms}, information science~\\cite{gl},\ntheoretical computer science~\\cite{go}, and soft constraint solving and programming~\\cite{bis2004}.\n\nLet $S$ be an ai-semiring. Define a binary relation $\\leq$ on $S$ by\n\\[\na\\leq b\\Leftrightarrow a+b=b.\n\\]\nThen $\\leq$ is a partial order on $S$, and $(S, \\leq)$ becomes an upper semilattice in which\nthe supremum of any elements $a$ and $b$ is $a+b$.\nConsequently, the additive reduct $(S, +)$ is uniquely determined by this semilattice order.\nTherefore, it is often convenient to represent the addition operation using the Hasse diagram of $(S, \\leq)$.\nMoreover, one can easily verify that this semilattice order is compatible with multiplication.\nFor this reason, an ai-semiring is often called a \\emph{semilattice-ordered semigroup}.\n\nLet $n\\geq 1$ be an integer,\nlet $S$ be an ai-semiring, and let $\\mathbf{M}_n(S)$ denote the set of all $n \\times n$ matrices over $S$.\nThen $\\mathbf{M}_n(S)$ forms an ai-semiring under the usual matrix addition and multiplication, that is,\n\\[\nA+B = [a_{ij}+b_{ij}]_{n\\times n}, \\quad A \\cdot B = \\left[ \\sum_{k=1}^n a_{ik} b_{kj} \\right]_{n\\times n}\n\\]\nfor all $A=[a_{ij}]_{n\\times n}, B=[b_{ij}]_{n\\times n}\\in \\mathbf{M}_n(S)$.\nIt is obvious that $S$ is isomorphic to $\\mathbf{M}_1(S)$.\nFor $n\\geq 2$, $S$ can be embedded into $\\mathbf{M}_n(S)$,\nsince the mapping\n\\[\n\\varphi\\colon S \\rightarrow \\mathbf{M}_n(S),\\quad a \\mapsto [a]_{n\\times n},\n\\]\nwhere $[a]_{n\\times n}$ denotes the constant matrix with all entries equal to $a$,\nis an embedding mapping.\nMoreover, if $S$ has an additive identity that is also a multiplicative zero,\nthen we can easily obtain the following infinite strict ascending chain\n\\[\nS\\hookrightarrow \\mathbf{M}_2(S)\\hookrightarrow \\mathbf{M}_{3}(S)\\hookrightarrow \\mathbf{M}_{4}(S)\\hookrightarrow\\cdots.\n\\]\n\nLet $\\mathcal{V}$ be a class of algebras.\nThen $\\mathcal{V}$ is called a \\emph{variety} if it is closed under\ntaking subalgebras, homomorphic images, and arbitrary direct products.\nBy the celebrated Birkhoff's theorem \\cite{birkhoff1935}, $\\mathcal{V}$ is a variety if and only if\nit is an \\emph{equational class}, that is, the class of all algebras satisfying some set of identities.\n\nOver the past two decades,\nthe finite basis problem for ai-semirings has attracted considerable attention,\nresulting in substantial\nprogress~\\cite{dol07, dolinka2009, dgv24, dgv25, gpz05, gv23, gv2501, gv2510, jrz, pas05, rz16, rzw, sr, volkov2024, yr25, yrzs, zrc}.\nIn particular, Dolinka~\\cite{dol07} constructed the first example of a nonfinitely based finite ai-semiring.\nPastijn et al.~\\cite{gpz05, pas05} proved that every ai-semiring satisfying the identity $x^2 \\approx x$ is finitely based;\nRen et al.~\\cite{rz16, rzw} later extended this to ai-semirings satisfying the identity $x^3 \\approx x$.\nShao and Ren~\\cite{sr} established that every algebra in the variety generated by all two-element ai-semirings is finitely based.\nSubsequently,\nJackson et al.~\\cite{jrz} and Zhao et al.~\\cite{zrc}\nprovided a complete classification of all three-element ai-semirings with respect to the finite basis property.\nMost recently,\nDolinka, Gusev and Volkov~\\cite{dgv25, gv2501} settled the finite basis problem for\nthe endomorphism semirings of finite semilattices.\n\nTo the best of our knowledge,\nthe study of the finite basis problem for matrix semirings was initiated by\nDolinka~\\cite{dolinka2009}, who considered the problem for $\\mathbf{M}_n(D_2)$ for $n\\geqslant 2$,\nwhere $D_2$ denotes the two-element distributive lattice.\nGusev and Volkov~\\cite{gv2510, volkov2024} are currently investigating\nthe same problem for upper triangular matrix semirings over $D_2$.\nMotivated by these works,\nwe undertake a systematic classification of matrix semirings $\\mathbf{M}_n(S)$,\nwhere $S$ ranges over all two-element ai-semirings.\nThe main result of this paper provides the following complete characterization.\n\n\\begin{abstract}\nWe provide a complete classification of matrix semirings $\\mathbf{M}_n(S)$ over two-element additively idempotent semirings $S$\nwith respect to the finite basis property.\nOur main theorem shows that for every integer $n \\geq 2$,\nthe semiring $\\mathbf{M}_n(S)$ is finitely based\nif and only if $S$ is distinct from a distributive lattice.\n\\end{abstract}\n\nTo the best of our knowledge,\nthe study of the finite basis problem for matrix semirings was initiated by\nDolinka~\\cite{dolinka2009}, who considered the problem for $\\mathbf{M}_n(D_2)$ for $n\\ge 2$,\nwhere $D_2$ denotes the two-element distributive lattice.\nGusev and Volkov~\\cite{gv2510, volkov2024} are currently investigating\nthe same problem for upper triangular matrix semirings over $D_2$.\nMotivated by these works,\nwe undertake a systematic classification of matrix semirings $\\mathbf{M}_n(S)$,\nwhere $S$ ranges over all two-element ai-semirings.\nThe main result of this paper provides the following complete characterization.\n\nFor the proof,\nwe first note that the case where $S$ is a distributive lattice follows from Dolinka's work.\nSpecifically,\n\\cite[Theorem B]{dolinka2009} states that the semiring $\\mathcal{R}_2$ of all binary relations on a two-element set is inherently nonfinitely based.\nSince $\\mathcal{R}_2 $ is isomorphic to $\\mathbf{M}_2(D_2)$,\nit follows immediately that $\\mathbf{M}_2(D_2)$ inherits this property.\nFurthermore, because $\\mathbf{M}_n(D_2)$ contains a copy of $\\mathbf{M}_2(D_2)$ for all $n \\geq 2$,\nthe same property extends to $\\mathbf{M}_n(D_2)$; this result is explicitly stated in \\cite[Corollary 6.2]{dolinka2009}.\n(We note that $D_2$ is also denoted $\\mathbb{B}_2$ and called the two-element Boolean semiring; our notation follows \\cite{sr}.)\n\nTherefore, to establish Theorem~\\ref{main}, it remains to prove the converse:\nthat the matrix semiring $\\mathbf{M}_n(S)$ is finitely based\nfor every two-element ai-semiring $S$ distinct from a distributive lattice and every $n \\geq 2$.\nThe necessary preliminaries are collected in Section 2, and the proof is carried out in Sections 3 and 4.\n\n\\begin{pro}\\label{pro21}\nLet $n\\geq 2$ be an integer. Then\n$\\mathsf{V}(\\mathbf{M}_n(L_2))$ is the ai-semiring variety determined by the identities \\eqref{eq:T22} and\n\\begin{align}\n&xy\\approx xz. \\label{f1}\n\\end{align}\n\\end{pro}\n\\begin{proof}\nWe first show that $\\mathbf{M}_n(L_2)$ satisfies identities \\eqref{eq:T22} and \\eqref{f1}.\nLet $A = [a_{ij}]$, $B = [b_{ij}]$, $C = [c_{ij}]$ be arbitrary matrices in $\\mathbf{M}_n(L_2)$, where all entries belong to $L_2$.\nFor any indices $1 \\leq i, j \\leq n$, we have\n\\[\n(AB)_{ij} = \\sum_{k=1}^{n} a_{ik}b_{kj} \\stackrel{\\eqrefbasis{eq:L2}}=\n\\sum_{k=1}^{n} a_{ik} \\stackrel{\\eqrefbasis{eq:L2}}= \\sum_{k=1}^{n} a_{ik}c_{kj} = (AC)_{ij},\n\\]\nand\n\\begin{align*}\n(A^2)_{ij} = \\sum_{k=1}^{n} a_{ik}a_{kj}\n&= \\left( \\sum_{k=1}^{n} a_{ik}a_{kj} \\right) + a_{ij}a_{jj} \\\\\n&\\stackrel{\\eqrefbasis{eq:L2}}= \\left( \\sum_{k=1}^{n} a_{ik}a_{kj} \\right) + a_{ij} \\\\\n&= (A^2)_{ij} + A_{ij} = (A^2 + A)_{ij}.\n\\end{align*}\nThis shows that $AB = AC$ and $A^2 = A^2 + A$.\nHence $\\mathbf{M}_n(L_2)$ satisfies the identities \\eqref{eq:T22} and \\eqref{f1}.\n\n\\begin{pro}\\label{pro22}\nLet $n\\geq 2$ be an integer. Then\n$\\mathsf{V}(\\mathbf{M}_n(R_2))$ coincides with $\\mathsf{V}(S_{56})$, properly contains $\\mathsf{V}(R_2)$,\nand is the ai-semiring variety determined by the identities \\eqref{eq:T22} and\n\\begin{align}\n&xy\\approx zy. \\label{f3}\n\\end{align}\n\\end{pro}\n\\begin{proof}\nObserve that $R_2$ and $L_2$ have dual multiplications.\nThe remaining argument parallels those of Proposition~\\ref{pro21} and Remark~\\ref{rem1}.\n\\end{proof}\n\n\\begin{pro}\\label{pro233}\nLet $n\\geq 1$ be an integer. Then\n$\\mathsf{V}(\\mathbf{M}_n(N_2))$ is the ai-semiring variety determined by the identities \\eqref{eq:N21} and \\eqref{eq:N22}.\n\\end{pro}\n\\begin{proof}\nLet $A = [a_{ij}]$, $B = [b_{ij}]$, $C = [c_{ij}]$, $D = [d_{ij}]$ be arbitrary matrices in $\\mathbf{M}_n(N_2)$.\nFor any $1 \\leq i, j \\leq n$, we have\n\\[\n(AB)_{ij} = \\sum_{k=1}^{n} a_{ik}b_{kj} \\stackrel{\\eqrefbasis{eq:N21}}= \\sum_{k=1}^{n} c_{ik}d_{kj} = (CD)_{ij},\n\\]\nand\n\\begin{align*}\n(A + A^2)_{ij}\n&= A_{ij} + (A^2)_{ij}=a_{ij} + \\left(\\sum_{k=1}^n a_{ik}a_{kj}\\right)\\\\\n&\\stackrel{\\eqrefbasis{eq:N21}}= a_{ij} + a_{ij}a_{ij}\n\\stackrel{\\eqrefbasis{eq:N22}}= a_{ij}= A_{ij}.\n\\end{align*}\nThis shows that $AB = CD$ and $A=A+A^2$.\nHence $\\mathbf{M}_n(N_2)$ satisfies both identities \\eqref{eq:N21} and \\eqref{eq:N22}.\n\n\\section{Conclusion}\nWe have provided a complete classification of\nmatrix semirings $\\mathbf{M}_n(S)$ over two-element ai-semirings $S$\nwith respect to the finite basis property.\nFrom Remark~\\ref{rem1} and \\cite[Figure 1]{yr25},\nthe variety $\\mathsf{V}(\\mathbf{M}_n(L_2))$ has precisely $5$ subvarieties;\nthey form a distributive lattice and are all finitely based.\nThe same conclusion holds for the variety $\\mathsf{V}(\\mathbf{M}_n(R_2))$.\nBy \\cite[Theorem 1]{polin} together with Propositions~\\ref{pro233} and \\ref{pro23},\nthe varieties $\\mathsf{V}(\\mathbf{M}_n(N_2))$ and $\\mathsf{V}(\\mathbf{M}_n(T_2))$\nare both minimal nontrivial varieties.\nIn contrast, the subvariety lattice of the variety $\\mathsf{V}(\\mathbf{M}_n(M_2))$\nremains to be explicitly described.\n\n\\begin{thm}\\label{main}\nLet $S$ be a two-element ai-semiring, and let $n \\geq 2$ be an integer.\nThen the matrix semiring $\\mathbf{M}_n(S)$ is finitely based if and only if $S$ is not a distributive lattice.\n\\end{thm}", + "post_theorem_intro_text_len": 1096, + "post_theorem_intro_text": "For the proof,\nwe first note that the case where $S$ is a distributive lattice follows from Dolinka's work.\nSpecifically,\n\\cite[Theorem B]{dolinka2009} states that the semiring $\\mathcal{R}_2$ of all binary relations on a two-element set is inherently nonfinitely based.\nSince $\\mathcal{R}_2 $ is isomorphic to $\\mathbf{M}_2(D_2)$,\nit follows immediately that $\\mathbf{M}_2(D_2)$ inherits this property.\nFurthermore, because $\\mathbf{M}_n(D_2)$ contains a copy of $\\mathbf{M}_2(D_2)$ for all $n \\geq 2$,\nthe same property extends to $\\mathbf{M}_n(D_2)$; this result is explicitly stated in \\cite[Corollary 6.2]{dolinka2009}.\n(We note that $D_2$ is also denoted $\\mathbb{B}_2$ and called the two-element Boolean semiring; our notation follows \\cite{sr}.)\n\nTherefore, to establish Theorem~\\ref{main}, it remains to prove the converse:\nthat the matrix semiring $\\mathbf{M}_n(S)$ is finitely based\nfor every two-element ai-semiring $S$ distinct from a distributive lattice and every $n \\geq 2$.\nThe necessary preliminaries are collected in Section 2, and the proof is carried out in Sections 3 and 4.", + "sketch": "For the proof of Theorem~\\ref{main}: (1) In the case where $S$ is a distributive lattice, the result follows from Dolinka's work: \\cite[Theorem B]{dolinka2009} shows that the semiring $\\mathcal{R}_2$ of all binary relations on a two-element set is inherently nonfinitely based. Since $\\mathcal{R}_2\\cong \\mathbf{M}_2(D_2)$, it follows that $\\mathbf{M}_2(D_2)$ is inherently nonfinitely based; and because $\\mathbf{M}_n(D_2)$ contains a copy of $\\mathbf{M}_2(D_2)$ for all $n\\ge2$, the same property extends to $\\mathbf{M}_n(D_2)$ (explicitly \\cite[Corollary 6.2]{dolinka2009}). (2) Thus, to establish Theorem~\\ref{main} it remains to prove the converse: $\\mathbf{M}_n(S)$ is finitely based for every two-element ai-semiring $S$ distinct from a distributive lattice and every $n\\ge2$. The preliminaries are in Section~2 and the proof is carried out in Sections~3 and~4.", + "expanded_sketch": "For the proof of the main theorem: (1) In the case where $S$ is a distributive lattice, the result follows from Dolinka's work: \\cite[Theorem B]{dolinka2009} shows that the semiring $\\mathcal{R}_2$ of all binary relations on a two-element set is inherently nonfinitely based. Since $\\mathcal{R}_2\\cong \\mathbf{M}_2(D_2)$, it follows that $\\mathbf{M}_2(D_2)$ is inherently nonfinitely based; and because $\\mathbf{M}_n(D_2)$ contains a copy of $\\mathbf{M}_2(D_2)$ for all $n\\ge2$, the same property extends to $\\mathbf{M}_n(D_2)$ (explicitly \\cite[Corollary 6.2]{dolinka2009}). (2) Thus, to establish the main theorem it remains to prove the converse: $\\mathbf{M}_n(S)$ is finitely based for every two-element ai-semiring $S$ distinct from a distributive lattice and every $n\\ge2$. The preliminaries are given next, and the proof is then carried out subsequently.", + "expanded_theorem": "\\label{main}\nLet $S$ be a two-element ai-semiring, and let $n \\geq 2$ be an integer.\nThen the matrix semiring $\\mathbf{M}_n(S)$ is finitely based if and only if $S$ is not a distributive lattice.", + "theorem_type": [ + "Biconditional or Equivalence", + "Universal" + ], + "mcq": { + "question": "An additively idempotent semiring (ai-semiring) is an algebra $(S,+,\\cdot)$ such that $(S,+)$ is a commutative idempotent semigroup, $(S,\\cdot)$ is a semigroup, and multiplication distributes over addition on both sides. Let $S$ be a two-element ai-semiring, and let $n\\ge 2$ be an integer. Write $\\mathbf{M}_n(S)$ for the ai-semiring of all $n\\times n$ matrices over $S$ with the usual matrix addition and multiplication. An algebra is called finitely based if the variety it generates is definable by finitely many identities. Which statement holds for every such $S$ and $n$?", + "correct_choice": { + "label": "A", + "text": "The matrix semiring $\\mathbf{M}_n(S)$ is finitely based if and only if $S$ is not a distributive lattice." + }, + "choices": [ + { + "label": "B", + "text": "The matrix semiring $\\mathbf{M}_n(S)$ is finitely based if and only if $S$ is a distributive lattice." + }, + { + "label": "C", + "text": "If $S$ is not a distributive lattice, then the matrix semiring $\\mathbf{M}_n(S)$ is finitely based." + }, + { + "label": "D", + "text": "For every two-element ai-semiring $S$, the matrix semiring $\\mathbf{M}_n(S)$ is finitely based for all sufficiently large integers $n\\ge 2$." + }, + { + "label": "E", + "text": "The matrix semiring $\\mathbf{M}_n(S)$ is finitely based if and only if $S$ itself is finitely based." + } + ], + "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": "distributive-lattice case direction", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped the converse implication excluding distributive lattices", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "finiteness", + "tampered_component": "uniformity in n from the embedding of $\\mathbf{M}_2(D_2)$ into all $\\mathbf{M}_n(D_2)$ for $n\\ge2$", + "template_used": "boundary_range" + }, + { + "label": "E", + "sketch_hook_type": "finiteness", + "tampered_component": "replacement of the matrix-semiring criterion by a criterion on the base semiring", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly reveal the correct criterion. It asks for an equivalent characterization, and the correct answer is not implied by the wording alone." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recall question: the correct option is essentially the theorem statement itself. However, the presence of nearby stronger, weaker, and boundary-case variants makes it slightly more than a pure restatement." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to reject the distractors by checking sufficiency vs. equivalence, the range restriction on n, and uniformity claims. Still, the question mainly tests precise recall of a characterization rather than substantial generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and well targeted: one weakens the biconditional, one overextends the range of n, one adds an unjustified uniform finite-basis claim, and one substitutes an unrelated finite-basis criterion on S itself." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it is primarily theorem recognition rather than a deeply generative reasoning task." + } + }, + { + "id": "2602.07167v1", + "paper_link": "http://arxiv.org/abs/2602.07167v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "proposition", + "content": "[Intermittency] \\label{prop:intermittency}For any $p\\geq 1$ integer, we have \n\\begin{align*}\n n \\exp\\left((p+\\frac{2p(p-1)}{n+2})\\tau \\right) \\leq \\mathbb{E} |F_\\tau|^{2p}\\leq n^{p} \\exp\\left((p+\\frac{2p(p-1)}{n+2})\\tau \\right),\n\\end{align*}\nwhere $|F|^2:={\\rm tr} F^*F$ denotes the square of the Frobenius norm.", + "start_pos": 6809, + "end_pos": 7148, + "label": "prop:intermittency" + }, + "ref_dict": { + "2pthmoment": "\\begin{align}\n \\E|F_\\tau|^{2p} \\sim_{n,p}\\footnotemark \\left(\\E^p |F_\\tau|^2\\right)^{1+\\frac{2(p-1)}{n+2}}. \\label{2pthmoment}\n\\end{align}", + "brownian1": "\\begin{enumerate}[label=(B\\arabic*)]\n \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n \\item \\label{brownian2} $\\bbE[B_{\\tau} B_\\tau] =0$, and\n \\item\\label{brownian3} $\\bbE[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n \\end{enumerate}", + "brownian3": "\\begin{enumerate}[label=(B\\arabic*)]\n \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n \\item \\label{brownian2} $\\bbE[B_{\\tau} B_\\tau] =0$, and\n \\item\\label{brownian3} $\\bbE[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n \\end{enumerate}", + "prop:intermittency": "\\begin{proposition}[Intermittency] \\label{prop:intermittency}For any $p\\geq 1$ integer, we have \n\\begin{align*}\n n \\exp\\left((p+\\frac{2p(p-1)}{n+2})\\tau \\right) \\leq \\E |F_\\tau|^{2p}\\leq n^{p} \\exp\\left((p+\\frac{2p(p-1)}{n+2})\\tau \\right),\n\\end{align*}\nwhere $|F|^2:=\\tr F^*F$ denotes the square of the Frobenius norm.\n\\end{proposition}", + "F": "\\begin{align}\\label{F}\n dF=F_\\tau\\circ dB, \\quad F_{\\tau=0}=id ,\n\\end{align}", + "prop:nontightness": "\\begin{proposition}[Non-tightness] \\label{prop:nontightness}\n For $\\tau \\gg_n 1$\\footnote{Here $\\tau \\gg_n 1$ means that there is a possibly large constant $C(n)$ depending only on $n$ such that the statement is true for any $\\tau$ with $\\tau \\geq C(n)$.}, we have \t\\begin{align}\\label{prop02}\n\t\t\\E | F_{ \\tau } |^2 I \\Big( { \\textstyle \\frac{ 1 }{ n } } | F_{ \\tau } |^2 \\leq { \\textstyle \\frac{ 1 }{ e } } \\big( { \\textstyle \\E \\frac{ 1 }{ n } } | F_{ \\tau } |^2 \\big)^{ \\frac{ n + 4 }{ n + 2 } } \\Big)\n\t\t\\leq \\Big( \\frac{ 1 }{ 2 } + \\frac{ C(n) }{ \\sqrt{\\tau} } \\Big) \\E| F_{ \\tau } |^2 .\n\t\\end{align}\n where $I(A)$ is the indicator function of the event $A$.\t\n\\end{proposition}", + "dd": "\\begin{align}\n dX=b(X_t)dt+\\sqrt{2}dW \\label{dd}\n\\end{align}" + }, + "pre_theorem_intro_text_len": 1023, + "pre_theorem_intro_text": "\\noindent\nWe are interested in the process $\\{F_\\tau\\}_{\\tau \\geq 0}$ that solves the Stratonovich stochastic differential equation (SDE)\n\\begin{align}\\label{F}\n dF=F_\\tau\\circ dB, \\quad F_{\\tau=0}=id ,\n\\end{align}\nwhere $\\{B_\\tau\\}_{\\tau \\geq 0}$ is a Brownian motion on ${\\frak{sl}}(n):=\\{{\\rm tr} B=0\\}$ that satisfies the following three assumptions:\n\\begin{enumerate}[label=(B\\arabic*)]\n \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n \\item \\label{brownian2} $\\mathbb E[B_{\\tau} B_\\tau] =0$, and\n \\item\\label{brownian3} $\\mathbb E[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n \\end{enumerate}\nSince the chain rule holds for Stratonovich SDEs, $F\\in \\textbf{SL}(n):=\\{\\text{det} F=1\\}$. In \\cite{MOW25}, it is proved that for $n\\geq 2$, there is a unique Brownian motion on $\\frak{sl}(n)$ that satisfies the Assumption \\ref{brownian1}-\\ref{brownian3}. We are interested in exploring the intermittent behavior of $F$ in \\eqref{F}. Our first result is on the moments:", + "context": "\\noindent\nWe are interested in the process $\\{F_\\tau\\}_{\\tau \\geq 0}$ that solves the Stratonovich stochastic differential equation (SDE)\n\\begin{align}\\label{F}\n dF=F_\\tau\\circ dB, \\quad F_{\\tau=0}=id ,\n\\end{align}\nwhere $\\{B_\\tau\\}_{\\tau \\geq 0}$ is a Brownian motion on ${\\frak{sl}}(n):=\\{{\\rm tr} B=0\\}$ that satisfies the following three assumptions:\n\\begin{enumerate}[label=(B\\arabic*)]\n \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n \\item \\label{brownian2} $\\mathbb E[B_{\\tau} B_\\tau] =0$, and\n \\item\\label{brownian3} $\\mathbb E[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n \\end{enumerate}\nSince the chain rule holds for Stratonovich SDEs, $F\\in \\textbf{SL}(n):=\\{\\text{det} F=1\\}$. In \\cite{MOW25}, it is proved that for $n\\geq 2$, there is a unique Brownian motion on $\\frak{sl}(n)$ that satisfies the Assumption \\ref{brownian1}-\\ref{brownian3}. We are interested in exploring the intermittent behavior of $F$ in \\eqref{F}. Our first result is on the moments:\n\n\\begin{align}\\label{F}\n dF=F_\\tau\\circ dB, \\quad F_{\\tau=0}=id ,\n\\end{align}\n\n\\begin{enumerate}[label=(B\\arabic*)]\n \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n \\item \\label{brownian2} $\\bbE[B_{\\tau} B_\\tau] =0$, and\n \\item\\label{brownian3} $\\bbE[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n \\end{enumerate}\n\n\\begin{enumerate}[label=(B\\arabic*)]\n \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n \\item \\label{brownian2} $\\bbE[B_{\\tau} B_\\tau] =0$, and\n \\item\\label{brownian3} $\\bbE[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n \\end{enumerate}", + "full_context": "\\noindent\nWe are interested in the process $\\{F_\\tau\\}_{\\tau \\geq 0}$ that solves the Stratonovich stochastic differential equation (SDE)\n\\begin{align}\\label{F}\n dF=F_\\tau\\circ dB, \\quad F_{\\tau=0}=id ,\n\\end{align}\nwhere $\\{B_\\tau\\}_{\\tau \\geq 0}$ is a Brownian motion on ${\\frak{sl}}(n):=\\{{\\rm tr} B=0\\}$ that satisfies the following three assumptions:\n\\begin{enumerate}[label=(B\\arabic*)]\n \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n \\item \\label{brownian2} $\\mathbb E[B_{\\tau} B_\\tau] =0$, and\n \\item\\label{brownian3} $\\mathbb E[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n \\end{enumerate}\nSince the chain rule holds for Stratonovich SDEs, $F\\in \\textbf{SL}(n):=\\{\\text{det} F=1\\}$. In \\cite{MOW25}, it is proved that for $n\\geq 2$, there is a unique Brownian motion on $\\frak{sl}(n)$ that satisfies the Assumption \\ref{brownian1}-\\ref{brownian3}. We are interested in exploring the intermittent behavior of $F$ in \\eqref{F}. Our first result is on the moments:\n\n\\begin{align}\\label{F}\n dF=F_\\tau\\circ dB, \\quad F_{\\tau=0}=id ,\n\\end{align}\n\n\\begin{enumerate}[label=(B\\arabic*)]\n \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n \\item \\label{brownian2} $\\bbE[B_{\\tau} B_\\tau] =0$, and\n \\item\\label{brownian3} $\\bbE[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n \\end{enumerate}\n\n\\begin{enumerate}[label=(B\\arabic*)]\n \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n \\item \\label{brownian2} $\\bbE[B_{\\tau} B_\\tau] =0$, and\n \\item\\label{brownian3} $\\bbE[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n \\end{enumerate}\n\n\\end{abstract}\n\\maketitle\n\n\\medskip \\noindent\nThe fact that $2p$-th moment scales with a rate much larger than the $p$-th power of the second moment amounts to strongly non-Gaussian and intermittent behavior. Proposition~\\ref{prop:intermittency} generalizes \\cite[Lemma 2]{MOW25} for $n=2$ to general $n\\ge 2$.\n\n\\medskip \\noindent\nWhile Proposition~\\ref{prop:intermittency} clearly expresses intermittency at the level of $F$, we are not able to transfer this higher-moment information to $\\nabla u$. This is because we can capture the proximity of $\\nabla u$ and $F$ only on the second moments. Hence we need to capture the shadow of the intermittency on the level of the second moment. In fact, it comes as a non-tightness result in the second moment in the sense that the extreme tails contain a substantial fraction of the second moment.\n \\begin{proposition}[Non-tightness] \\label{prop:nontightness}\n For $\\tau \\gg_n 1$\\footnote{Here $\\tau \\gg_n 1$ means that there is a possibly large constant $C(n)$ depending only on $n$ such that the statement is true for any $\\tau$ with $\\tau \\geq C(n)$.}, we have \\begin{align}\\label{prop02}\n \\E | F_{ \\tau } |^2 I \\Big( { \\textstyle \\frac{ 1 }{ n } } | F_{ \\tau } |^2 \\leq { \\textstyle \\frac{ 1 }{ e } } \\big( { \\textstyle \\E \\frac{ 1 }{ n } } | F_{ \\tau } |^2 \\big)^{ \\frac{ n + 4 }{ n + 2 } } \\Big)\n \\leq \\Big( \\frac{ 1 }{ 2 } + \\frac{ C(n) }{ \\sqrt{\\tau} } \\Big) \\E| F_{ \\tau } |^2 .\n \\end{align}\n where $I(A)$ is the indicator function of the event $A$. \n\\end{proposition}\n\\begin{remark}\\label{cor:nontightness} The information in Proposition~\\ref{prop:nontightness} can easily be upgraded to the lower bound as is done in \\cite[Theorem 3 \\& (118)]{MOW25}\n \\begin{align*}\n\\E |F_\\tau|^{2p} \\gtrsim_{n,p} \\bbE^{1+\\frac{n+4}{n+2}(p-1)} |F_\\tau|^{2},\n\\end{align*}\nwhich is weaker than \\eqref{2pthmoment} but not completely unrelated. This exponent is precisely the linearization of the quadratic exponent $p+\\frac{2p(p-1)}{n+2}$ around $p=1$. \n\\end{remark}\n\n\\subsection{Proof of Proposition~\\ref{prop:intermittency}}\nRecall from Lemma~\\ref{lem:odes}, we have for $p\\geq 1$ integer\n\\begin{align*}\n \\frac{ d\\bbE[\\tr G^p]}{d\\tau} =(p+\\frac{p(p-1)(n-2)}{\\alpha_n})\\bbE \\tr G_\\tau^p +\\frac{pn}{\\alpha_n}\\sum_{j=1}^{p-1}\\bbE \\tr G_\\tau^j \\tr G_\\tau^{p-j} \n\\end{align*}\n with initial condition $\\tr G_{\\tau=0}=\\tr id=n$. Using $\\tr G^j \\tr G^{p-j} \\geq \\tr G^p$ for $0\\leq j\\leq p$, we can bound the second term \n \\begin{align*}\n \\sum_{j=1}^{p-1}\\bbE \\tr G_\\tau^j \\tr G_\\tau^{p-j} \\geq (p-1) \\bbE \\tr G_\\tau^p\n\\end{align*}\nwhich leads to the differential inequality \n\\begin{align*}\n \\frac{ d\\bbE\\tr G^p}{d\\tau} \\geq (p+\\frac{p(p-1)(2n-2)}{\\alpha_n}) \\bbE\\tr G_\\tau^p\n\\end{align*}\nwhere $\\alpha_n$ is as defined \\eqref{alpha}. Hence we get \n\\begin{align*}\n \\bbE\\tr G_\\tau^p \\geq n e^{(p+\\frac{2p(p-1)}{n+2})\\tau} .\n\\end{align*}\nOn the other hand, recall again from Lemma~\\ref{lem:odes}, $\\bbE \\tr^p G_\\tau$ satisfies \n\\begin{align*}\n \\frac{ d\\bbE\\tr^p G}{d\\tau}& =(p-\\frac{2p(p-1)}{\\alpha_n}) \\bbE\\tr^p G_\\tau +\\frac{2p(p-1)n}{\\alpha_n} \\bbE\\tr^{p-2} G_\\tau\\tr G_\\tau^2 \n\\end{align*}\nwith initial condition $\\bbE\\tr^p G_{\\tau=0}=n^p$. This time using $\\tr G^2\\leq \\tr^2 G$, we get the differential inequality\n\\begin{align*}\n \\frac{d\\bbE \\tr^p G}{d\\tau}& \\leq (p+\\frac{p(p-1)(2n-2)}{\\alpha_n}) \\bbE\\tr^p G_\\tau \n\\end{align*}\nwhich leads to\n\\begin{align*}\n \\bbE\\tr^p G_\\tau \\leq n^pe^{(p+\\frac{2p(p-1)}{n+2})\\tau} .\n\\end{align*}\nHence we obtain \\eqref{lpestimates} which is a restatement of Proposition~\\ref{prop:intermittency}.\n\\qed\n\nFor parameters $ \\tau^* \\geq 0 $ and $ \\hat{\\sigma}^* $ that we will choose later we consider the terminal condition\n\\begin{align}\\label{tc}\n \\hat{\\zeta}( \\tau^*,\\hat{\\sigma})\\begin{cases}\n = 1, \\text{ if } \\hat{\\sigma} \\leq \\hat{\\sigma}^*,\\\\\n \\in [0,1] \\text{ for all } \\hat{\\sigma},\\\\\n =0, \\text{ if }\\hat{\\sigma}\\geq \\hat{\\sigma}^*+1,\n \\end{cases}\n \\text{ with } \\left| \\frac{\\partial^2 \\hat{\\zeta}( \\tau^* ,\\hat{\\sigma}) }{\\partial \\hat{\\sigma}^2}\\right| \\lesssim 1\n \\end{align}\n for the equation \\eqref{zetahat2} in Lemma~\\ref{lem:zeta}.\nIt is readily seen from \\eqref{tc} that we have\n \\begin{align}\n\\bbE \\hat R_{ \\tau^* } \\hat{\\zeta}(\\tau^*, \\hat R_{ \\tau^* }) \\stackrel{ (\\ref{tc}) }{ \\geq } \\bbE \\hat R_{ \\tau^* } I( \\hat R_{ \\tau^* } \\leq \\hat{R}^*) \n\\quad \\text{provided} \\quad \\hat\\sigma^* =: \\ln \\hat R^{ * }. \\label{ononehand}\n\\end{align}\nOur goal is to choose $ \\hat\\sigma^* $ large enough so that at initial time $ \\tau = 0 $ it holds\n\\begin{align}\\hat{\\zeta}( \\tau=0, \\hat\\sigma = 0 ) \\leq \\frac{1}{2} , \\label{initial}\n\\end{align}\nso that Lemma~\\ref{lem:zeta} implies\n\\begin{align}\n \\bbE \\hat R_{ \\tau^* } I( \\hat R_{ \\tau^* } \\leq \\hat{R}^*) - \\frac{ 1 }{ 2 }\n \\stackrel{ (\\ref{ononehand}) }{ \\leq } \\bbE \\hat{R}_{ \\tau^* } \\hat{\\zeta}( \\tau^*, \\hat R_{ \\tau^* }) - \\frac{ 1 }{ 2 }\n \\stackrel{ \\eqref{zetahatineq} \\& (\\ref{initial}) }{ \\lesssim_{ n } } \\int_0^{\\tau^*}d \\tau\\, e^{ - ( 1 - \\frac{ \\lambda_2 }{ 2 }) \\tau} \\sup_{\\hat \\sigma}\\left|\\frac{\\partial \\hat \\zeta}{\\partial\\hat \\sigma}+ \\frac{\\partial^2 \\hat \\zeta}{\\partial\\hat \\sigma^2} \\right| . \\label{zetahatineqb}\n\\end{align}\n\n\\noindent\nTo conclude the proof, it is left to show that the integral in \\eqref{zetahatineqb} is small. To this end, note that using the estimates for the terminal data \n\\begin{align*}\n \\sup_{\\hat\\sigma } \\left|\\frac{\\partial \\hat{\\zeta}( \\tau^*,\\hat{\\sigma})}{\\partial {\\hat\\sigma}}\\right|+\\left|\\frac{\\partial^2 \\hat{\\zeta}(\\tau^*,\\hat{\\sigma})}{\\partial^2 {\\hat \\sigma } }\\right| \n \\lesssim 1\n \\quad \\text{and} \\quad\n \\int d{\\hat{\\sigma}}\\left|\\frac{\\partial \\hat{\\zeta}( \\tau^*,\\hat{\\sigma})}{\\partial {\\hat\\sigma}}\\right| \n \\lesssim 1\n\\end{align*}\nwe learn (by convolution with the heat kernel) that \n\\begin{align*}\n \\sup_{ \\hat \\sigma } \\left|\\frac{\\partial \\hat{\\zeta} ( \\tau^* , \\hat\\sigma ) }{\\partial \\hat{\\sigma}}\\right| \\lesssim_{ n } \\frac{1}{ \\sqrt{1+ \\tau^*- \\tau}}, \\hskip 10pt \\sup_{ \\hat\\sigma } \\left|\\frac{\\partial^2 \\hat{\\zeta} ( \\tau^* , \\hat\\sigma ) }{\\partial \\hat{\\sigma}^2}\\right| \n \\lesssim_{ n } \\frac{1}{ 1+ \\tau^*- \\tau} .\n\\end{align*}\nHence, the integral term in \\eqref{zetahatineqb} is bounded by\n\\begin{align*} \\int_0^{\\tau*}d\\tau e^{ - ( 1 - \\frac{ \\lambda_2 }{ 2 } ) \\tau} \\sup_{\\hat \\sigma}\\left|\\frac{\\partial \\hat \\zeta}{\\partial\\hat \\sigma}+ \\frac{\\partial^2 \\hat \\zeta}{\\partial\\hat \\sigma^2} \\right| \\lesssim_{ n } \\int_0^{\\tau^*} d\\tau e^{-\\frac{4}{n-1} \\tau } \\frac{1}{\\sqrt{1+\\tau^*-\\tau}} \\lesssim_{ n } \\frac{ 1 }{\\sqrt{\\tau^*}},\n\\end{align*}\nand with some constant $ C = C ( n ) $ estimate \\eqref{zetahatineqb} turns into\n\\begin{align}\\label{ontheotherhand}\n \\bbE \\hat R_{ \\tau^* } I( \\hat R_{ \\tau^* } \\leq \\hat{R}^*) \\leq \\frac{ 1 }{ 2 } + \\frac{ C(n) }{ \\sqrt{ \\tau^* } }.\n\\end{align}\nFinally, undoing the change of variables we have\n\\begin{align*}\n \\hat R_{ \\tau^* }\n \\stackrel{ \\eqref{Rhat} }{ = } e^{ - \\tau^* } { \\textstyle \\frac{ 1 }{ n } } R_{ \\tau^* }\n \\stackrel{ \\ref{F3} , (\\ref{R}) \\& (\\ref{:G}) }{ = } \\frac{ | F_{ \\tau^* } |^2 }{ \\E | F_{ \\tau^* } |^2 } ,\n \\quad \n \\hat R^{ * }\n \\stackrel{ (\\ref{eqnHatRStar}) }{ \\leq } e^{ \\frac{ 2 }{ n + 2 } \\tau^* - 1 } \n \\stackrel{\\ref{F3} }{ = } \\frac{ 1 }{ e } \\Big( \\frac{ \\E | F_{ \\tau^* } |^2 }{ n } \\Big)^{ \\frac{ 2 }{ n + 2 } } ,\n\\end{align*}\nso that since $ \\frac{ 2 }{ n + 2 } + 1 = \\frac{ n + 4 }{ n + 2 } $ \n\\begin{align*}\n \\hat R_{ \\tau^* } I( \\hat R_{ \\tau^* } \\leq \\hat{R}^*)\n = \\frac{ | F_{ \\tau^* } |^2 }{ \\E | F_{ \\tau^* } |^2 } I \\Big( { \\textstyle \\frac{ 1 }{ n } } | F_{ \\tau^* } |^2 \\leq { \\textstyle \\frac{ 1 }{ e } } \\big( { \\textstyle \\frac{ \\E | F_{ \\tau^* } |^2 }{ n } } \\big)^{ \\frac{ n + 4 }{ n + 2 } } \\Big) .\n\\end{align*}\nThus \\eqref{ontheotherhand} turns into \\eqref{prop02}. \\qed", + "post_theorem_intro_text_len": 6008, + "post_theorem_intro_text": "\\noindent\nProposition~\\ref{prop:intermittency} implies that \n\\begin{align}\n \\mathbb{E}|F_\\tau|^{2p} \\sim_{n,p}\\footnotemark \\left(\\mathbb{E}^p |F_\\tau|^2\\right)^{1+\\frac{2(p-1)}{n+2}}. \\label{2pthmoment}\n\\end{align}\n\\footnotetext{Here $A \\sim_{n,p} B$ means that there exist constants $c=c(n,p)$ and $C=C(n,p)$ depending only on $n$ and $p$ such that $cB\\leq A\\leq CB$.} \n\n\\medskip \\noindent\nThe fact that $2p$-th moment scales with a rate much larger than the $p$-th power of the second moment amounts to strongly non-Gaussian and intermittent behavior. Proposition~\\ref{prop:intermittency} generalizes \\cite[Lemma 2]{MOW25} for $n=2$ to general $n\\ge 2$.\n\n\\medskip \\noindent\nThe interest in studying the geometric Brownian motion comes from its\nintriguing relation to the drift-diffusion process \n\\begin{align}\n dX=b(X_t)dt+\\sqrt{2}dW \\label{dd}\n\\end{align}\nwith $b$ being a divergence-free and time-independent vector field. One is interested in the case where $b$ is sampled from a stationary and isotropic Gaussian ensemble with the scaling \n \\begin{align}\n \\label{b} b(\\mu \\cdot)=\\frac{1}{\\mu}b \\text{ in law for all } \\mu>0,\n\\end{align}\nin which convection and diffusion balance at every scale. After implementing an ultraviolet cutoff, w.l.o.g.\\ at scale 1, $b$ is fixed up to a single constant which we describe in terms of $\\mathbb{E}|b|^2=\\epsilon^2 \\frac{ n }{ 4 } $, where $\\epsilon \\ll 1$ can be interpreted as the P\\'{e}clet number. It is now well-known that the mean-square displacement displays super-diffusive behavior $\\frac{1}{2t}\\mathbb E|X_t|^2 \\approx \\lambda(t)$ for $t \\gg 1$, where \n\\begin{align*}\n\\lambda(t):=\\sqrt{1+\\frac{\\epsilon^2}{2} \\ln(1+t)}.\n\\end{align*}\nThis behavior was rigorously established with increasing precision in \\cite{TV12, CHT22, CMOW22,ABK24, MOW25}. \n\n\\medskip \\noindent\nThe relationship between the geometric Brownian motion process $F$ in \\eqref{F} and the drift-diffusion process in \\eqref{dd} was discovered in \\cite{MOW25} on the level of the expected position, $u(x,t)$, (w.r.t. $W_t$) of the process $\\{X_t\\}_{t\\geq 0}$ starting from $X_{t=0}=x$. More precisely, one obtains $\\nabla u(0,t) \\approx F_{\\tau(T)}$ in law on average over $t\\in (0,T)$, where $$\\tau(T):=\\ln \\lambda(T), $$ see \\cite[Theorem 2 \\& (19)]{MOW25}.\n\n\\medskip \\noindent\nOur result shows that the intermittency of the drift-diffusion equation depends less on topology (stream lines of the divergence-free $b$ are closed iff $n=2$) and more on geometry ($\\textbf{SL}(n)$ has less curvature as $n$ increases). \n\n\\medskip \\noindent\nWhile Proposition~\\ref{prop:intermittency} clearly expresses intermittency at the level of $F$, we are not able to transfer this higher-moment information to $\\nabla u$. This is because we can capture the proximity of $\\nabla u$ and $F$ only on the second moments. Hence we need to capture the shadow of the intermittency on the level of the second moment. In fact, it comes as a non-tightness result in the second moment in the sense that the extreme tails contain a substantial fraction of the second moment.\n \\begin{proposition}[Non-tightness] \\label{prop:nontightness}\n For $\\tau \\gg_n 1$\\footnote{Here $\\tau \\gg_n 1$ means that there is a possibly large constant $C(n)$ depending only on $n$ such that the statement is true for any $\\tau$ with $\\tau \\geq C(n)$.}, we have \t\\begin{align}\\label{prop02}\n\t\t\\mathbb{E} | F_{ \\tau } |^2 I \\Big( { \\textstyle \\frac{ 1 }{ n } } | F_{ \\tau } |^2 \\leq { \\textstyle \\frac{ 1 }{ e } } \\big( { \\textstyle \\mathbb{E} \\frac{ 1 }{ n } } | F_{ \\tau } |^2 \\big)^{ \\frac{ n + 4 }{ n + 2 } } \\Big)\n\t\t\\leq \\Big( \\frac{ 1 }{ 2 } + \\frac{ C(n) }{ \\sqrt{\\tau} } \\Big) \\mathbb{E}| F_{ \\tau } |^2 .\n\t\\end{align}\n where $I(A)$ is the indicator function of the event $A$.\t\n\\end{proposition}\n\\begin{remark}\\label{cor:nontightness} The information in Proposition~\\ref{prop:nontightness} can easily be upgraded to the lower bound as is done in \\cite[Theorem 3 \\& (118)]{MOW25}\n \\begin{align*}\n\\mathbb{E} |F_\\tau|^{2p} \\gtrsim_{n,p} \\mathbb E^{1+\\frac{n+4}{n+2}(p-1)} |F_\\tau|^{2},\n\\end{align*}\nwhich is weaker than \\eqref{2pthmoment} but not completely unrelated. This exponent is precisely the linearization of the quadratic exponent $p+\\frac{2p(p-1)}{n+2}$ around $p=1$. \n\\end{remark}\n\n\\noindent\nThe results in Propositions~\\ref{prop:intermittency} and ~\\ref{prop:nontightness} were previously proven in the case $n=2$ (see \\cite[Lemma 2, Lemma 4]{MOW25}). The argument relied on the fact that the law of $R=\\frac{1}{2}{\\rm tr} F^*F$ can be identified as the Itô evolution \\begin{align*}\n dR= R_{\\tau} d\\tau +\\sqrt{R_\\tau^2-1} dw, \\quad R_{\\tau =0}=1,\n\\end{align*} \nwhere $\\{w_\\tau\\}_{\\tau\\geq 0}$ is a one-dimensional Brownian motion. In any dimension $n$, as a consequence of the $\\textbf{O}(n)$-invariance, it is not difficult to see that one obtains an evolution for the $(n-1)$ quantities ${\\rm tr} F^*F, {\\rm tr} (F^*F)^2,\\dots ,{\\rm tr} (F^*F)^{n-1}$. However, we are unable to leverage this evolution in order to characterize the evolution of $R$. In this paper, we take a different route: Relying on the observation that the quantities $\\mathbb{E} {\\rm tr} (F^*F)^2 $ and $\\mathbb{E}{\\rm tr}^2 F^*F $ satisfy a linear system of ODEs which is closed in any dimension $n$ and can be solved explicitly, we learn that $\\mathbb E{\\rm tr} (F^*F)^2$ can be well approximated by $\\mathbb E {\\rm tr}^2 F^* F$. This in turn allows us to approximately close the equation for $ R $ in dimensions $ n > 2 $.\n\n\\medskip\n \\noindent\nNote that $|F_\\tau|^2$ is close to a stochastic exponential, which is consistent with Proposition~\\ref{prop:intermittency} and \\ref{prop:nontightness}, namely $\\ln |F_\\tau|^2$ behaves like a Gaussian random variable with mean $(1-\\frac{2}{n+2})\\tau$ and variance $\\frac{4\\tau}{n+2}$. \n\n\\subsection*{Acknowledgment}\nWe thank Anna Wienhard and Corentin Le Bars for helpful discussions concerning random walks on groups. We thank Peter Morfe and Ofer Zeitouni for helpful discussions.", + "sketch": "For $n=2$, the previously used argument (\\cite[Lemma 2, Lemma 4]{MOW25}) identified the law of $R:=\\frac{1}{2}{\\rm tr}\\,F^*F$ via the one-dimensional It\\^o SDE\n\\[\n dR=R_{\\tau}\\,d\\tau+\\sqrt{R_{\\tau}^2-1}\\,dw,\\qquad R_{\\tau=0}=1,\n\\]\nwhich yields the needed moment information.\n\nIn general dimension $n$, by $\\mathbf O(n)$-invariance “one obtains an evolution for the $(n-1)$ quantities ${\\rm tr}F^*F,\\,{\\rm tr}(F^*F)^2,\\dots,{\\rm tr}(F^*F)^{n-1}$,” but the authors state they are “unable to leverage this evolution in order to characterize the evolution of $R$.”\n\nInstead, they “take a different route”: they use that “$\\mathbb E\\,{\\rm tr}(F^*F)^2$ and $\\mathbb E\\,{\\rm tr}^2 F^*F$ satisfy a linear system of ODEs which is closed in any dimension $n$ and can be solved explicitly.” From this explicit solution, they “learn that $\\mathbb E\\,{\\rm tr}(F^*F)^2$ can be well approximated by $\\mathbb E\\,{\\rm tr}^2 F^*F$,” which “in turn allows us to approximately close the equation for $R$ in dimensions $n>2$.”", + "expanded_sketch": "For $n=2$, the previously used argument (\\cite[Lemma 2, Lemma 4]{MOW25}) identified the law of $R:=\\frac{1}{2}{\\rm tr}\\,F^*F$ via the one-dimensional It\\^o SDE\n\\[\n dR=R_{\\tau}\\,d\\tau+\\sqrt{R_{\\tau}^2-1}\\,dw,\\qquad R_{\\tau=0}=1,\n\\]\nwhich yields the needed moment information.\n\nIn general dimension $n$, by $\\mathbf O(n)$-invariance “one obtains an evolution for the $(n-1)$ quantities ${\\rm tr}F^*F,\\,{\\rm tr}(F^*F)^2,\\dots,{\\rm tr}(F^*F)^{n-1}$,” but the authors state they are “unable to leverage this evolution in order to characterize the evolution of $R$.”\n\nInstead, they “take a different route”: they use that “$\\mathbb E\\,{\\rm tr}(F^*F)^2$ and $\\mathbb E\\,{\\rm tr}^2 F^*F$ satisfy a linear system of ODEs which is closed in any dimension $n$ and can be solved explicitly.” From this explicit solution, they “learn that $\\mathbb E\\,{\\rm tr}(F^*F)^2$ can be well approximated by $\\mathbb E\\,{\\rm tr}^2 F^*F$,” which “in turn allows us to approximately close the equation for $R$ in dimensions $n>2$.”,", + "expanded_theorem": "[Intermittency] \\label{prop:intermittency}For any $p\\geq 1$ integer, we have \n\\begin{align*}\n n \\exp\\left((p+\\frac{2p(p-1)}{n+2})\\tau \\right) \\leq \\mathbb{E} |F_\\tau|^{2p}\\leq n^{p} \\exp\\left((p+\\frac{2p(p-1)}{n+2})\\tau \\right),\n\\end{align*}\nwhere $|F|^2:={\\rm tr} F^*F$ denotes the square of the Frobenius norm.", + "theorem_type": [ + "Inequality or Bound", + "Universal" + ], + "mcq": { + "question": "Let $n\\ge 2$, let $\\mathfrak{sl}(n)=\\{A:\\operatorname{tr}A=0\\}$, and let $\\{B_\\tau\\}_{\\tau\\ge 0}$ be a Brownian motion on $\\mathfrak{sl}(n)$ such that for every $O\\in \\mathbf O(n)=\\{O^*O=\\mathrm{id}\\}$ one has $OBO^{-1}=_{\\mathrm{law}}B$, and moreover $\\mathbb E[B_\\tau B_\\tau]=0$ and $\\mathbb E[B_\\tau^*B_\\tau]=\\tau\\,\\mathrm{id}$. Let $\\{F_\\tau\\}_{\\tau\\ge 0}$ solve the Stratonovich SDE\n\\[\n dF = F_\\tau\\circ dB,\\qquad F_{\\tau=0}=\\mathrm{id}.\n\\]\nIf $|F|^2:=\\operatorname{tr}(F^*F)$ denotes the square of the Frobenius norm, which statement holds for every integer $p\\ge 1$ and every $\\tau\\ge 0$?", + "correct_choice": { + "label": "A", + "text": "\\[\n n\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n+2}\\right)\\tau\\right)\n \\le \\mathbb E|F_\\tau|^{2p}\n \\le n^p\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n+2}\\right)\\tau\\right).\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "\\[\n n\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n}\\right)\\tau\\right)\n \\le \\mathbb E|F_\\tau|^{2p}\n \\le n^p\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n}\\right)\\tau\\right).\n\\]" + }, + { + "label": "C", + "text": "\\[\n \\mathbb E|F_\\tau|^{2p}\n \\le n^p\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n+2}\\right)\\tau\\right).\n\\]" + }, + { + "label": "D", + "text": "\\[\n \\text{For each integer }p\\ge 1\\text{ there exists }C_p\\in[1,\\infty)\\text{ such that for all }\\tau\\ge 0,\n\\quad C_p^{-1}\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n+2}\\right)\\tau\\right)\n \\le \\mathbb E|F_\\tau|^{2p}\n \\le C_p\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n+2}\\right)\\tau\\right).\n\\]" + }, + { + "label": "E", + "text": "\\[\n n\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n+2}\\right)\\tau\\right)\n \\le \\mathbb E|F_\\tau|^{2p}\n \\le n^p\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n+2}\\right)\\tau\\right)\n\\]\nfor every real number \\(p\\ge 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": "explicit exponent denominator n+2", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped lower bound", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "explicit sharp prefactors n and n^p replaced by non-explicit constants depending on p", + "template_used": "uniformity_effectivity" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "range of p extended from integers to all reals", + "template_used": "boundary_range" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives only the stochastic setup and asks for the valid moment estimate; it does not explicitly state or strongly hint at the exact bound, the sharp denominator n+2, or the precise prefactors." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recall question: the correct choice appears to be the exact target estimate under the stated hypotheses. However, it is not a pure restatement because the options include nearby variants that force comparison of strength, sharpness, and parameter range." + }, + "GPS": { + "score": 2, + "justification": "Selecting the correct answer requires substantial mathematical judgment: one must track the exact exponent, distinguish sharp two-sided bounds from weaker true statements, and notice the restriction to integer p. The answer is not obvious from surface cues." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong and mathematically meaningful: one alters n+2 to n, one gives a weaker but still plausible one-sided bound, one replaces sharp constants by existential constants, and one improperly extends the range of p. These reflect realistic failure modes." + }, + "total_score": 7, + "overall_assessment": "A high-quality MCQ with no answer leakage and excellent distractors; its main limitation is that it leans somewhat toward exact theorem recall rather than fully independent generative reasoning." + } + }, + { + "id": "2602.07171v1", + "paper_link": "http://arxiv.org/abs/2602.07171v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "conjecture", + "content": "[Noferini--Williams \\cite{MR4418964}]\\label{conj}\nFor $n\\ge1$ and $0\\le m1$. By \\cite[Lemma~4.3]{MR4418964}, one of the following holds:\n\\[\nv\\mid m-1,\\quad v\\mid m-2,\\quad v\\mid 1.\n\\]\nSince $\\gcd(v,m-2)=1$ and $v>1$, the only possibility is $v\\mid m-1$. Write $m=1+\\alpha v$ for some $\\alpha\\in\\mathbb{N}$. Since $6\\mid m-2$, it follows that $6\\mid\\alpha v- 1$. Set $\\alpha_0 = n/v$. Since $\\alpha_0$ is even, an argument similar to that used in the proof of Lemma~\\ref{lucas} shows that $t^v + 1$ divides $t^n - 1$. Since $\\gcd(6, v) = 1$, we have\n\\[\n\\gcd(t^v+1,\\Phi_6(t))=1.\n\\]\nHence $t^v+1$ divides $G(t)$, and so for some $S(t)\\in\\mathbb{Z}[t]$, we may write\n\\[\nG(t)=(t^v+1)S(t).\n\\]\nBy \\eqref{eq:res-mult}, we have\n\\[\n\\operatorname{Res}(F,G)\n=\\operatorname{Res}(F,t^v+1)\\operatorname{Res}(F,S).\n\\]\nSince $\\operatorname{Res}(F,G)=1$, it follows that\n\\begin{equation}\\label{eq:resultant-F}\n\\operatorname{Res}(F,t^v+1)=1.\n\\end{equation}\n\n\\begin{corollary}\\label{special}\nLet $n=2\\cdot 3^s$ with $s\\geq 2$. Then $H(n,2\\pm n/3)^{ab}\\not\\cong \\mathbb{Z}^2.$\n\\end{corollary}\n\\begin{proof}\nThis is exactly the situation where $f(t)=t^m-t+1$ with $m=2\\pm n/3\\pmod n$. Set $K=\\pm n/3\\pmod n$ and apply Lemma~\\ref{lem:v1_reduction} and Lemma~\\ref{grow}.\n\\end{proof}\n\n\\begin{lemma}\\label{minimality}\nUnder Assumptions~\\ref{std}, let $u=\\tau v$ for some $\\tau\\ge 1$. If $a\\ge b+2$ or $s\\ge r+2$ or $(a=b+1$ and $s=r+1)$, then there exists an integer $n_1$ with $n_1\\mid n$ and $n_10$ such that for all $n\\in\\mathbb N$ and all $x_1,\\dots,x_n\\in \\M$,\n\\[\n\\W\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\le C_\\M\\left(\\frac1{\\sqrt n}\n+\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\\right)?\n\\]\n\\end{problem}", + "eq:GreenMeanZero": "\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{\\M} G(x,y)\\,dy=0.\n\\end{equation}", + "eq:Steinerberger2D": "\\begin{equation}\\label{eq:Steinerberger2D}\n\\W\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\lesssim_\\M\n\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\n+\n\\begin{cases}\n\\sqrt{\\frac{\\log n}{n}} & \\text{if } d=2,\\\\\nn^{-1/d} & \\text{if } d\\ge 3.\n\\end{cases}\n\\end{equation}", + "thm:main": "\\begin{theorem}[The $\\sqrt{\\log n}$ remainder]\\label{thm:main}\nLet $G$ be the symmetric mean-zero Green function of $-\\Delta$.\nThere does not exist a constant $C_{\\M}>0$ such that for all $n\\in\\mathbb{N}$ and all $x_1,\\dots,x_n\\in \\M$,\n\\begin{equation}\\label{eq:NoUniversalO1sqrtN}\n\\W\\!\\left(\\frac1n\\sum_{i=1}^n\\delta_{x_i},\\, dx\\right)\n\\le\nC_{\\M}\\left(\\frac{1}{\\sqrt{n}}+\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\\right).\n\\end{equation}\nIn particular, the $\\sqrt{\\log n}$ factor in the two-dimensional inequality \\eqref{eq:Steinerberger2D} cannot be removed if the unrenormalized Green-energy term were to be preserved.\n\\end{theorem}", + "def:green": "\\begin{definition}[Mean-zero Green function]\\label{def:green}\nA mean-zero Green function for $-\\Delta$ is a measurable function\n\\[\nG:\\M\\times \\M \\to \\mathbb{R}\\cup\\{+\\infty\\}\n\\]\nwith the following properties.\n\\begin{enumerate}\n\\item For each $x\\in \\M$, $y\\mapsto G(x,y)$ is locally integrable on $\\M$ and satisfies the normalization:\n\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{\\M} G(x,y)\\,dy=0.\n\\end{equation}\n\\item For each $f\\in C^\\infty(\\M)$ with $\\int_{\\M} f\\,dx=0$, the function\n\\[\nu(x)\\coloneqq \\int_{\\M} G(x,y)\\, f(y)\\,dy\n\\]\nis a weak solution of $-\\Delta u = f$ with $\\int_{\\M}u\\,dx=0$.\n\\end{enumerate}\n\\end{definition}" + }, + "pre_theorem_intro_text_len": 4896, + "pre_theorem_intro_text": "Let $(M,g)$ be a compact connected two-dimensional Riemannian manifold without boundary, $\\mathrm{d}_g$ is the induced distance, and let $G(x,y)$ denote the symmetric mean-zero Green function of the Laplacian (Definition \\ref{def:green}). We work with the normalized volume measure $dx=\\mathrm{vol}(M)^{-1}\\,\\mathrm{d}\\mathrm{vol}$. Whenever we integrate in the variable $y$, we write $dy$ for the same measure. For points $x_1,\\dots,x_n\\in M$, we define the empirical measure:\n\\[\n\\mu_n \\coloneqq \\frac1n\\sum_{i=1}^n \\delta_{x_i}.\n\\]\nSteinerberger \\cite{Ste21} showed that for $d\\ge 3$, the following Green--Wasserstein inequality holds:\n\\[\nW_2(\\mu_n,dx)\\ \\lesssim_{M}\\ n^{-1/d} + \\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}.\n\\]\nIn \\cite[Problem 53]{SteOpen}, he gives the following estimate:\n\\begin{equation}\\label{eq:Steinerberger2D}\nW_2\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\lesssim_M\n\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\n+\n\\begin{cases}\n\\sqrt{\\frac{\\log n}{n}} & \\text{if } d=2,\\\\\nn^{-1/d} & \\text{if } d\\ge 3.\n\\end{cases}\n\\end{equation}\nIn this note, we use\n\\[\n\\sum_{i\\neq j}(\\cdots)\\ \\coloneqq\\ \\sum_{\\substack{1\\le i,j\\le n\\\\ i\\neq j}}(\\cdots)\n\\]\nto denote the ordered sum over distinct indices. Notice that, by symmetry of $G$, it is twice the sum over $i0$ such that for all $n\\in\\mathbb N$ and all $x_1,\\dots,x_n\\in M$,\n\\[\nW_2\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\le C_M\\left(\\frac1{\\sqrt n}\n+\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\\right)?\n\\]\n\\end{problem}\n\\noindent Our main result, Theorem \\ref{thm:main}, gives a negative answer to Problem \\ref{prob:Steinerberger}.\n\n\\begin{remark}[The diagonal]\\label{rem:diagonal}\nWe note that the Green function $G$ is smooth on $(M\\times M)\\setminus \\{(x,x):x\\inM\\}$ with a logarithmic singularity along the diagonal.\nIn a deterministic context, if $x_i=x_j$ for some $i\\neq j$, then the term $G(x_i,x_j)$ is not finite. It is then natural to interpret the Green term in \\eqref{eq:Steinerberger2D} and Problem \\ref{prob:Steinerberger} as $+\\infty$, in which case the inequality holds trivially.\nIn the random settings that we use in this note, collisions occur with probability $0$ because $dx$ is non-atomic.\n\\end{remark}\n\nWe now recall some definitions that will be useful in the main argument.\nFor Borel probability measures $\\mu,\\nu$ on $M$, let us denote by $\\Gamma(\\mu,\\nu)$ the set of couplings of $\\mu$ and $\\nu$ on $M\\times M$.\nThe quadratic Wasserstein distance is\n\\[\nW_2(\\mu,\\nu)\\coloneqq \\left(\\inf_{\\gamma\\in \\Gamma(\\mu,\\nu)} \\int_{M\\times M} \\mathrm{d}_g(x,y)^2\\,\\,\\mathrm{d}\\gamma(x,y)\\right)^{1/2}.\n\\]\nGiven that $M$ is compact, it follows that $W_2(\\mu,\\nu)$ is always finite and satisfies $W_2(\\mu,\\nu)\\le \\mathrm{diam}(M)$. Let $\\Delta$ be the Laplace--Beltrami operator on $(M,g)$, which is realized here as a self-adjoint operator on $L^2(M,dx)$ with domain $H^2(M)$. Again, since $M$ is compact and connected, $\\ker(-\\Delta)$ consists of the constant functions.\n\n\\begin{definition}[Mean-zero Green function]\\label{def:green}\nA mean-zero Green function for $-\\Delta$ is a measurable function\n\\[\nG:M\\times M \\to \\mathbb{R}\\cup\\{+\\infty\\}\n\\]\nwith the following properties.\n\\begin{enumerate}\n\\item For each $x\\in M$, $y\\mapsto G(x,y)$ is locally integrable on $M$ and satisfies the normalization:\n\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{M} G(x,y)\\,dy=0.\n\\end{equation}\n\\item For each $f\\in C^\\infty(M)$ with $\\int_{M} f\\,dx=0$, the function\n\\[\nu(x)\\coloneqq \\int_{M} G(x,y)\\, f(y)\\,dy\n\\]\nis a weak solution of $-\\Delta u = f$ with $\\int_{M}u\\,dx=0$.\n\\end{enumerate}\n\\end{definition}\n\nRecall that the uniqueness and existence of the mean-zero Green function, as a distribution kernel, are guaranteed. We can also choose it to be symmetric (for example, see \\cite{Aubin98} or \\cite{Rosenberg97}). We now fix a symmetric mean-zero Green function $G$, so that $G(x,y)=G(y,x)$ holds.\nBy symmetry and \\eqref{eq:GreenMeanZero}, we obtain:\n\\[\n\\int_{M}G(y,x)\\,dy=0\\qquad\\text{for every }x\\in M.\n\\]\n\n\\begin{remark}[Local singularity]\\label{rem:logsing}\nIn dimension $2$, the following classical local expansion (in geodesic normal coordinates) is true:\n\\[\nG(x,y)= -\\frac{1}{2\\pi}\\log \\mathrm{d}_g(x,y) + H(x,y).\n\\]\nNote that $H$ extends continuously to the diagonal and is smooth off the diagonal.\nIn particular, we have $G(x,\\cdot)\\in L^1(M,dx)$ for each fixed $x$.\n\\end{remark}", + "context": "Let $(M,g)$ be a compact connected two-dimensional Riemannian manifold without boundary, $\\mathrm{d}_g$ is the induced distance, and let $G(x,y)$ denote the symmetric mean-zero Green function of the Laplacian (Definition \\ref{def:green}). We work with the normalized volume measure $dx=\\mathrm{vol}(M)^{-1}\\,\\mathrm{d}\\mathrm{vol}$. Whenever we integrate in the variable $y$, we write $dy$ for the same measure. For points $x_1,\\dots,x_n\\in M$, we define the empirical measure:\n\\[\n\\mu_n \\coloneqq \\frac1n\\sum_{i=1}^n \\delta_{x_i}.\n\\]\nSteinerberger \\cite{Ste21} showed that for $d\\ge 3$, the following Green--Wasserstein inequality holds:\n\\[\nW_2(\\mu_n,dx)\\ \\lesssim_{M}\\ n^{-1/d} + \\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}.\n\\]\nIn \\cite[Problem 53]{SteOpen}, he gives the following estimate:\n\\begin{equation}\\label{eq:Steinerberger2D}\nW_2\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\lesssim_M\n\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\n+\n\\begin{cases}\n\\sqrt{\\frac{\\log n}{n}} & \\text{if } d=2,\\\\\nn^{-1/d} & \\text{if } d\\ge 3.\n\\end{cases}\n\\end{equation}\nIn this note, we use\n\\[\n\\sum_{i\\neq j}(\\cdots)\\ \\coloneqq\\ \\sum_{\\substack{1\\le i,j\\le n\\\\ i\\neq j}}(\\cdots)\n\\]\nto denote the ordered sum over distinct indices. Notice that, by symmetry of $G$, it is twice the sum over $i0$ such that for all $n\\in\\mathbb N$ and all $x_1,\\dots,x_n\\in \\M$,\n\\[\n\\W\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\le C_\\M\\left(\\frac1{\\sqrt n}\n+\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\\right)?\n\\]\n\\end{problem}", + "full_context": "Let $(M,g)$ be a compact connected two-dimensional Riemannian manifold without boundary, $\\mathrm{d}_g$ is the induced distance, and let $G(x,y)$ denote the symmetric mean-zero Green function of the Laplacian (Definition \\ref{def:green}). We work with the normalized volume measure $dx=\\mathrm{vol}(M)^{-1}\\,\\mathrm{d}\\mathrm{vol}$. Whenever we integrate in the variable $y$, we write $dy$ for the same measure. For points $x_1,\\dots,x_n\\in M$, we define the empirical measure:\n\\[\n\\mu_n \\coloneqq \\frac1n\\sum_{i=1}^n \\delta_{x_i}.\n\\]\nSteinerberger \\cite{Ste21} showed that for $d\\ge 3$, the following Green--Wasserstein inequality holds:\n\\[\nW_2(\\mu_n,dx)\\ \\lesssim_{M}\\ n^{-1/d} + \\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}.\n\\]\nIn \\cite[Problem 53]{SteOpen}, he gives the following estimate:\n\\begin{equation}\\label{eq:Steinerberger2D}\nW_2\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\lesssim_M\n\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\n+\n\\begin{cases}\n\\sqrt{\\frac{\\log n}{n}} & \\text{if } d=2,\\\\\nn^{-1/d} & \\text{if } d\\ge 3.\n\\end{cases}\n\\end{equation}\nIn this note, we use\n\\[\n\\sum_{i\\neq j}(\\cdots)\\ \\coloneqq\\ \\sum_{\\substack{1\\le i,j\\le n\\\\ i\\neq j}}(\\cdots)\n\\]\nto denote the ordered sum over distinct indices. Notice that, by symmetry of $G$, it is twice the sum over $i0$ such that for all $n\\in\\mathbb N$ and all $x_1,\\dots,x_n\\in \\M$,\n\\[\n\\W\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\le C_\\M\\left(\\frac1{\\sqrt n}\n+\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\\right)?\n\\]\n\\end{problem}\n\nLet $(\\M,g)$ be a compact connected two-dimensional Riemannian manifold without boundary, $\\dist$ is the induced distance, and let $G(x,y)$ denote the symmetric mean-zero Green function of the Laplacian (Definition \\ref{def:green}). We work with the normalized volume measure $dx=\\mathrm{vol}(\\M)^{-1}\\dd\\mathrm{vol}$. Whenever we integrate in the variable $y$, we write $dy$ for the same measure. For points $x_1,\\dots,x_n\\in \\M$, we define the empirical measure:\n\\[\n\\mu_n \\coloneqq \\frac1n\\sum_{i=1}^n \\delta_{x_i}.\n\\]\nSteinerberger \\cite{Ste21} showed that for $d\\ge 3$, the following Green--Wasserstein inequality holds:\n\\[\n\\W(\\mu_n,dx)\\ \\lesssim_{\\M}\\ n^{-1/d} + \\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}.\n\\]\nIn \\cite[Problem 53]{SteOpen}, he gives the following estimate:\n\\begin{equation}\\label{eq:Steinerberger2D}\n\\W\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\lesssim_\\M\n\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\n+\n\\begin{cases}\n\\sqrt{\\frac{\\log n}{n}} & \\text{if } d=2,\\\\\nn^{-1/d} & \\text{if } d\\ge 3.\n\\end{cases}\n\\end{equation}\nIn this note, we use\n\\[\n\\sum_{i\\neq j}(\\cdots)\\ \\coloneqq\\ \\sum_{\\substack{1\\le i,j\\le n\\\\ i\\neq j}}(\\cdots)\n\\]\nto denote the ordered sum over distinct indices. Notice that, by symmetry of $G$, it is twice the sum over $i0$ be the injectivity radius of $(\\M,g)$ and set $r_0\\coloneqq \\mathrm{inj}(\\M)/4$. We now apply standard results on the singularity structure of the Green function in dimension $2$ (for instance, see \\cite{Aubin98}). In particular, the function\n\\[\nH(x,y)\\coloneqq G(x,y)+\\frac{1}{2\\pi}\\log \\dist(x,y),\n\\qquad x\\neq y,\n\\]\nextends continuously to the set $\\{(x,y)\\in \\M\\times\\M: \\dist(x,y)\\le 2r_0\\}$ (this also holds across the diagonal). Observe that this set is compact, so there exists $C_0>0$ such that\n\\[\n|H(x,y)|\\le C_0\\qquad\\text{whenever }\\dist(x,y)\\le 2r_0.\n\\]\nAs a result, for all $x\\neq y$ with $\\dist(x,y)<2r_0$, it follows that\n\\begin{equation}\\label{eq:GreenLogBound}\n|G(x,y)| \\le \\frac{1}{2\\pi}|\\log \\dist(x,y)|+C_0 \\le C\\bigl(1+|\\log \\dist(x,y)|\\bigr)\n\\end{equation}\nfor some constant $C>0$.\n\nWe now split $\\M\\times\\M$ into the near-diagonal region\n\\[\nD\\coloneqq \\{(x,y)\\in\\M\\times\\M: \\dist(x,y)0$ with:\n\\begin{equation}\\label{eq:JacobianBound}\nJ_x(\\rho,\\theta)\\le C_1\\rho\\qquad\\text{for all }x\\in\\M,\\ \\theta\\in\\mathbb{S}^1,\\ \\rho\\in(0,r_0)\n\\end{equation}\n(see, for instance, \\cite{Chavel84}).\nWe now apply the normalization of $dx$ and obtain:\n\\begin{align*}\n\\int_{B_{r_0}(x)} \\bigl(1+|\\log \\dist(x,y)|\\bigr)^2\\,dy\n&=\\frac{1}{\\mathrm{vol}(\\M)}\\int_{\\mathbb{S}^1}\\!\\!\\int_0^{r_0} \\bigl(1+|\\log \\rho|\\bigr)^2 J_x(\\rho,\\theta)\\,\\dd\\rho\\,\\dd\\theta\\\\\n&\\le \\frac{C_1}{\\mathrm{vol}(\\M)}\\int_{\\mathbb{S}^1}\\!\\!\\int_0^{r_0} \\bigl(1+|\\log \\rho|\\bigr)^2 \\rho\\,\\dd\\rho\\,\\dd\\theta.\n\\end{align*}\nThe right-hand side is finite since near $0$ the function $\\rho\\mapsto \\rho(\\log \\rho)^2$ is integrable. We apply the substitution $t=-\\log\\rho$ (so $\\rho=e^{-t}$ and $\\rho\\,\\dd\\rho=-e^{-2t}\\,\\dd t$) and get:\n\\[\n\\int_0^{r_0} \\bigl(1+|\\log \\rho|\\bigr)^2\\rho\\,\\dd\\rho\n=\\int_{\\infty}^{-\\log r_0} \\bigl(1+|t|\\bigr)^2(-e^{-2t})\\,\\dd t\n=\\int_{-\\log r_0}^{\\infty} \\bigl(1+|t|\\bigr)^2 e^{-2t}\\,\\dd t <\\infty.\n\\]\nIt follows that $\\int_{B_{r_0}(x)} (1+|\\log \\dist(x,y)|)^2\\,dy$ is bounded uniformly in $x$.\nIf we integrate over $x$, we arrive at $\\int_D G^2\\,dx\\,dy<\\infty$. Finally, we combine the estimates on $D$ and $D^c$. This finishes the proof of $\\sigma^2<\\infty$.\n\\end{proof}\n\n\\begin{theorem}[The $\\sqrt{\\log n}$ remainder]\\label{thm:main}\nLet $G$ be the symmetric mean-zero Green function of $-\\Delta$.\nThere does not exist a constant $C_{\\M}>0$ such that for all $n\\in\\mathbb{N}$ and all $x_1,\\dots,x_n\\in \\M$,\n\\begin{equation}\\label{eq:NoUniversalO1sqrtN}\n\\W\\!\\left(\\frac1n\\sum_{i=1}^n\\delta_{x_i},\\, dx\\right)\n\\le\nC_{\\M}\\left(\\frac{1}{\\sqrt{n}}+\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\\right).\n\\end{equation}\nIn particular, the $\\sqrt{\\log n}$ factor in the two-dimensional inequality \\eqref{eq:Steinerberger2D} cannot be removed if the unrenormalized Green-energy term were to be preserved.\n\\end{theorem}", + "post_theorem_intro_text_len": 3405, + "post_theorem_intro_text": "\\begin{proof}\nRecall that $\\mathrm{vol}$ denotes the Riemannian volume measure on $(M,g)$ and $dx = \\mathrm{vol}(M)^{-1}\\,\\mathrm{d}\\mathrm{vol}$. Let $\\mathrm{inj}(M)>0$ be the injectivity radius of $(M,g)$ and set $r_0\\coloneqq \\mathrm{inj}(M)/4$. We now apply standard results on the singularity structure of the Green function in dimension $2$ (for instance, see \\cite{Aubin98}). In particular, the function\n\\[\nH(x,y)\\coloneqq G(x,y)+\\frac{1}{2\\pi}\\log \\mathrm{d}_g(x,y),\n\\qquad x\\neq y,\n\\]\nextends continuously to the set $\\{(x,y)\\in M\\timesM: \\mathrm{d}_g(x,y)\\le 2r_0\\}$ (this also holds across the diagonal). Observe that this set is compact, so there exists $C_0>0$ such that\n\\[\n|H(x,y)|\\le C_0\\qquad\\text{whenever }\\mathrm{d}_g(x,y)\\le 2r_0.\n\\]\nAs a result, for all $x\\neq y$ with $\\mathrm{d}_g(x,y)<2r_0$, it follows that\n\\begin{equation}\\label{eq:GreenLogBound}\n|G(x,y)| \\le \\frac{1}{2\\pi}|\\log \\mathrm{d}_g(x,y)|+C_0 \\le C\\bigl(1+|\\log \\mathrm{d}_g(x,y)|\\bigr)\n\\end{equation}\nfor some constant $C>0$.\n\nWe now split $M\\timesM$ into the near-diagonal region\n\\[\nD\\coloneqq \\{(x,y)\\inM\\timesM: \\mathrm{d}_g(x,y)0$ with:\n\\begin{equation}\\label{eq:JacobianBound}\nJ_x(\\rho,\\theta)\\le C_1\\rho\\qquad\\text{for all }x\\inM,\\ \\theta\\in\\mathbb{S}^1,\\ \\rho\\in(0,r_0)\n\\end{equation}\n(see, for instance, \\cite{Chavel84}).\nWe now apply the normalization of $dx$ and obtain:\n\\begin{align*}\n\\int_{B_{r_0}(x)} \\bigl(1+|\\log \\mathrm{d}_g(x,y)|\\bigr)^2\\,dy\n&=\\frac{1}{\\mathrm{vol}(M)}\\int_{\\mathbb{S}^1}\\!\\!\\int_0^{r_0} \\bigl(1+|\\log \\rho|\\bigr)^2 J_x(\\rho,\\theta)\\,\\,\\mathrm{d}\\rho\\,\\,\\mathrm{d}\\theta\\\\\n&\\le \\frac{C_1}{\\mathrm{vol}(M)}\\int_{\\mathbb{S}^1}\\!\\!\\int_0^{r_0} \\bigl(1+|\\log \\rho|\\bigr)^2 \\rho\\,\\,\\mathrm{d}\\rho\\,\\,\\mathrm{d}\\theta.\n\\end{align*}\nThe right-hand side is finite since near $0$ the function $\\rho\\mapsto \\rho(\\log \\rho)^2$ is integrable. We apply the substitution $t=-\\log\\rho$ (so $\\rho=e^{-t}$ and $\\rho\\,\\,\\mathrm{d}\\rho=-e^{-2t}\\,\\,\\mathrm{d} t$) and get:\n\\[\n\\int_0^{r_0} \\bigl(1+|\\log \\rho|\\bigr)^2\\rho\\,\\,\\mathrm{d}\\rho\n=\\int_{\\infty}^{-\\log r_0} \\bigl(1+|t|\\bigr)^2(-e^{-2t})\\,\\,\\mathrm{d} t\n=\\int_{-\\log r_0}^{\\infty} \\bigl(1+|t|\\bigr)^2 e^{-2t}\\,\\,\\mathrm{d} t <\\infty.\n\\]\nIt follows that $\\int_{B_{r_0}(x)} (1+|\\log \\mathrm{d}_g(x,y)|)^2\\,dy$ is bounded uniformly in $x$.\nIf we integrate over $x$, we arrive at $\\int_D G^2\\,dx\\,dy<\\infty$. Finally, we combine the estimates on $D$ and $D^c$. This finishes the proof of $\\sigma^2<\\infty$.\n\\end{proof}", + "sketch": "Using the 2D Green function singularity, write\n\\[\nH(x,y)\\coloneqq G(x,y)+\\frac{1}{2\\pi}\\log \\mathrm{d}_g(x,y)\n\\]\nwhich “extends continuously” to a compact neighborhood of the diagonal, hence is bounded there. This gives the near-diagonal logarithmic bound\n\\[\n|G(x,y)|\\le C\\bigl(1+|\\log \\mathrm{d}_g(x,y)|\\bigr)\\qquad (\\mathrm{d}_g(x,y)<2r_0).\n\\]\nSplit \\(M\\times M\\) into the near-diagonal region \\(D=\\{\\mathrm{d}_g(x,y)0$ one has\n\\[\nG\\in L^2\\bigl(\\{(x,y)\\in M\\times M:\\ d_g(x,y)\\ge \\varepsilon\\},dx\\otimes dx\\bigr),\n\\]\nwhereas\n\\[\n\\int_M\\int_M G(x,y)^2\\,dx\\,dy=+\\infty.\n\\]" + }, + { + "label": "E", + "text": "For each fixed $x\\in M$, the map $y\\mapsto G(x,y)$ belongs to $L^p(M,dy)$ for every $p>2$, and consequently\n\\[\nG\\in L^p(M\\times M,dx\\otimes dx)\\qquad\\text{for every }p>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": "logarithmic singularity near the diagonal is bounded only after subtracting the log term, not in absolute size", + "template_used": "stronger_trap" + }, + { + "label": "C", + "sketch_hook_type": "regularity", + "tampered_component": "joint $L^2(M\\times M)$ conclusion weakened to fiberwise $L^2$ in the $y$-variable for each fixed $x$", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "case_split", + "tampered_component": "near-diagonal integrability of $\\rho(\\log\\rho)^2$", + "template_used": "boundary_range" + }, + { + "label": "E", + "sketch_hook_type": "counting_estimate", + "tampered_component": "critical exponent from the logarithmic singularity: $L^2$ holds but no gain to all $p>2$", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives only the setup and defining properties of the Green function; it does not reveal or strongly hint that square-integrability on M×M is the intended conclusion." + }, + "TAS": { + "score": 2, + "justification": "The item is not a bare restatement of a theorem in the stem. It asks the student to distinguish among several regularity/integrability claims (L1, L2, L∞, sectionwise bounds, divergence), so it involves competing conclusions." + }, + "GPS": { + "score": 1, + "justification": "A solver must reason about the 2D logarithmic singularity of the Green function and which integrability thresholds it satisfies, so some genuine mathematical reasoning is required. However, the pressure is weakened because choice C is also true, making the task less about uniquely generating the best conclusion." + }, + "DQS": { + "score": 1, + "justification": "B, D, and E are plausible and target common misconceptions about regularity and singularities. But C is a weaker true statement, so it is not a valid distractor under a single-answer format, which significantly harms the quality of the option set." + }, + "total_score": 6, + "overall_assessment": "Conceptually strong and free of answer leakage, but the MCQ is flawed as a single-choice item because at least one non-key option (C) is also true. This ambiguity weakens both the distractor set and the assessment of reasoning." + } + }, + { + "id": "2602.08158v1", + "paper_link": "http://arxiv.org/abs/2602.08158v1", + "theorems_cnt": 7, + "theorem": { + "env_name": "theorem", + "content": "\\label{DoldKan}\n Let $\\mathcal{A}$ be a weakly idempotent additive category. Let $M_\\bullet$ be a\n simplicial module. Every element of $M_n$ has a unique decomposition\n \\begin{equation*}\n x = \\sum_{k=0}^n \\sum_{0\\le i_k<\\cdots0} \\cup \\{ \\eta_{n+1}^n : [n{+}1] \\to\n [n]\\}_{n\\ge0} ,\n\\end{equation*}\nand $\\Lambda_\\infty$ is obtained from $\\Lambda_+$ by inverting\n$\\{ \\tau_n^{-1} : [n] \\to [n] \\}_{n\\ge0}$.\n\nThere is a contravariant isomorphism of categories $f\\mapsto f^\\circ$ from\n$\\Lambda_\\infty$ to its opposite $\\Lambda^\\op_\\infty$, which acts on generators by\ninterchanging $\\varepsilon^n_i$ and $\\eta^{n-1}_{n-i}$, and fixing\n$\\tau_n$. In particular, it exchanges the generators $\\varepsilon^n_0$ and\n$\\eta^{n-1}_n$ of $\\Lambda_\\infty$.\n\n\\begin{definition}\n Let $\\mathcal{A}$ be a pre-additive category (a category enriched in abelian\n groups). A \\textbf{paracyclic} (respectively \\textbf{duplicial} or\n \\textbf{simplicial}) module $M_\\bullet$ is a presheaf on\n $\\Lambda_\\infty$ (respectively $\\Lambda_+$ or $\\Delta$) taking values in $\\mathcal{A}$.\n\\end{definition}\n\nDenote the (co)action of the morphisms $\\varepsilon_i^n$ and\n$\\eta_i^n$ on a duplicial module $M_\\bullet$ by\n$\\partial_{n,i}:M_n\\to M_{n-1}$ and $s_{n,i}:M_n\\to M_n$, and the (co)action of\nthe morphisms $\\tau_n$ by $t_n:M_n\\to M_n$. The relations among the\ngenerators of $\\Lambda_+$ become the following relations on duplicial\nmodules $M_\\bullet$:\n\\begin{align*}\n \\partial_{n,k} \\partial_{n+1,j} &= \\partial_{n,j} \\partial_{n+1,k+1} , \\qquad \\,\\, 0\\le j\\le k\\le n , \\\\\n \\partial_{n+1,j} s_{n,k} &=\n \\begin{cases}\n t_n , & k-j=n+1 , \\\\\n s_{n-1,k-1} \\partial_{n,j} , & 1\\le k-j\\le n , \\\\\n 1 , & k-j=0,-1, \\\\\n s_{n-1,k} \\partial_{n,j-1} , & -n-1\\le k-j<-1 ,\n \\end{cases} \\\\\n s_{n,j} s_{n-1,k} &= s_{n,k+1} s_{n-1,j} , \\qquad \\!\\! 0\\le j\\le k\\le n .\n\\end{align*}\nIn particular,\n\\begin{align*}\n \\partial_{n,i} t_n &=\n \\begin{cases}\n t_{n-1} \\partial_{n,i+1} , & 0\\le i< n , \\\\\n \\partial_{n,0} , & i=n ,\n \\end{cases} \\intertext{and}\n t_{n+1} s_{n,i} &=\n \\begin{cases}\n s_{n,n+1} , & i=0 , \\\\\n s_{n,i-1} t_n , & 0< i\\le n+1 .\n \\end{cases}\n\\end{align*}\nDenote the action of $t_n^{n+1}$ on $M_n$ by $\\mathsf{T}$. We see that\n$\\T_{n-1}\\partial_{n,i}=\\partial_{n,i}\\T_n$ and $\\T_{n+1}s_{n,i}=s_{n,i}\\T_n$.\n\nAn \\textbf{idempotent} $(A,p)$ in a pre-additive category $\\mathcal{A}$ is a\npair consisting of an object $A$ of $\\mathcal{A}$ and $p:A\\to A$ an idempotent\nendomorphism of $A$, that is, $p^2=p$. If $p:A\\to A$ is an idempotent,\nthen so is $p^\\perp=1-p$, and the image of the idempotent $p$ is the\nkernel of $p^\\perp$ (and vice versa).\n\nLet $M_\\bullet$ be a simplicial module. Dold and Puppe \\cite{DP}*{Section 3}\ndefine idempotents\n\\begin{equation*}\n p_n = (1-s_{n-1,0}\\partial_{n,1})\\ldots(1-s_{n-1,n-1}\\partial_{n,n}) : M_n \\to M_n .\n\\end{equation*}\nThe kernel of the idempotent $p_n:M_n\\to M_n$, if it exists, is the\nmodule of degenerate chains $D_n(M)$. The image of $p_n$ is the module\n$N_n(M)$ of normalized chains. Provided these modules exist, we have\nthe direct sum decomposition\n\\begin{equation*}\n M_n = N_n(M) \\oplus D_n(M) .\n\\end{equation*}\n\nThe module $N_n(M)$ is the submodule on which the face maps\n$\\partial_{n,i}$, $1\\le i\\le n$, vanish:\n\\begin{equation*}\n N_n(M) = \\bigcap_{i=1}^n \\ker\\bigl( \\partial_{n,i}:M_n\\to M_{n-1} \\bigr) \\subset M_n .\n\\end{equation*}\nSimilarly, the module $D_n(M)$ is the sum of the images of the\ndegeneracy maps $s_{n-1,i}$, $0\\le i0 .\n \\end{cases}\n\\end{equation*}\n\nIt was proved by Eilenberg and Mac Lane \\cite{EM1} that the complexes\n$(M_\\bullet,b)$ and $(N_\\bullet(M),b)$ are homotopy equivalent. The following\nexpression for their homotopy is taken from Epstein\n\\cite{Epstein}*{Proposition 2.3}.\n\\begin{proposition}\n Let $M_\\bullet$ be a simplicial module. The operator\n \\begin{equation*}\n \\varphi_n = \\sum_{i=0}^n (-1)^i s_{n,i} p_{n,i} : M_n \\to M_{n+1}\n \\end{equation*}\n satisfies $b_{n+1}\\varphi_n + \\varphi_{n-1}b_n = p_n - 1$.\n\\end{proposition}\n\\begin{proof}\n For $0\\le i\\le n$,\n \\begin{align*}\n b_{n+1} s_{n,i} p_{n,i}\n &= \\sum_{k=0}^{i-1} (-1)^k \\partial_{n+1,k} s_{n,i} p_{n,i}\n + (-1)^i ( \\partial_{n+1,i} - \\partial_{n+1,i+1} ) s_{n,i} p_{n,i} \\\\\n & \\quad + \\sum_{k=i+2}^{n+1} (-1)^k \\partial_{n+1,k} s_{n,i} p_{n,i} \\\\\n &= \\sum_{k=0}^{i-1} (-1)^k s_{n-1,i-1} \\partial_{n,k} p_{n,i}\n + \\sum_{k=i+2}^{n+1} (-1)^k s_{n-1,i} \\partial_{n,k-1} p_{n,i} \\\\\n &= \\sum_{k=0}^n (-1)^k s_{n-1,i-1} \\partial_{n,k} p_{n,i}\n - (-1)^i s_{n-1,i-1} \\partial_{n,i} p_{n,i} \\\\\n &= s_{n-1,i-1} b_n p_{n,i} + (-1)^i ( p_{n,i-1} - p_{n,i} ) \\\\\n &= s_{n-1,i-1} p_{n-1,i-1} b_n + (-1)^i ( p_{n,i-1} - p_{n,i} ) .\n \\end{align*}\n Summing over $i$, the result follows\n\\end{proof}\n\nWe now come to the main result of this article.\n\\begin{theorem}\n $\\pi_n= p_n \\circ \\T_n = \\T_n \\circ p_n$\n\\end{theorem}\n\\begin{proof}\n \\begin{align*}\n \\pi_n &= (-1)^n \\partial_{n+1}(\\partial_{n+2}s_{n+1}-s_n\\partial_{n+1})^ns_n (-1)^{|j|-|i|} \\\\\n &= t_n^{n+1} + \\sum_{k=1}^{n-1} \\sum_{0\\le i_10}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n\t\\begin{align*}\n\t\t\t&d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{ \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\eps) d\\Haus \\ |\\ \\ell \\in \\curve^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n\t\\end{align*}\n\\end{definition}" + }, + "pre_theorem_intro_text_len": 5458, + "pre_theorem_intro_text": "This paper is devoted to the study of the functional introduced in \\cite{bonnivard2025phasefieldapproximationplateaus} to approximate a Plateau problem, which consists in finding a surface of minimal area spanning a collection of closed curves $\\gamma_0,..., \\gamma_n$ contained in the boundary of an open, bounded, convex set $\\Ccal_0 \\subset \\mathbb{R}^3$. \n\nMore precisely, the competitor surfaces are defined as the images of homotopies connecting the given curves. We define below the set of admissible homotopies connecting a curve $\\gamma_i$ to $\\gamma_j$:\n\\[\n\\mathrm{Hom}(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Lip}([0,1] \\times \\mathbb{S}^{1}, \\overline{\\Ccal_0}) \\text{ such that } \\ell(0) = \\gamma_i \\text{ and } \\ell(1) = \\gamma_{j} \\}.\n\\]\nNote that we require more than mere homotopies: we specifically consider Lipschitz homotopies. For any homotopy $\\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j)$, we define the associated surface $\\surf_\\ell$ as its image:\n\\begin{equation}\\label{Def:surface_image}\n\t\\surf_{\\ell} := \\ell([0,1] \\times \\mathbb{S}^{1}) \\subset \\overline{\\Ccal_0}.\n\\end{equation}\nWe emphasize that the surface $\\surf_{\\ell}$ is $\\mathcal{H}^{2}$-rectifiable and has finite $\\mathcal{H}^{2}$-measure. Moreover, for any $u\\in H^1(\\mathbb{R}^3)$, the trace of $u$ on $\\surf_{\\ell}$ is well defined, since the set of points that are not Lebesgue points of $u$ is $\\mathcal{H}^2$-negligible. \nHowever, to obtain the existence result, we need to impose additional properties on the competitors.\nFor $\\Lambda>0$, we consider\n\\[\n\\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j) \\text{ such that } \\;\\mathrm{Lip}(\\ell)\\leq \\Lambda \\text{ and } \\surf_\\ell \\text{ is } \\Lambda-\\text{upper Ahlfors regular} \\ \\}.\\]\n\nLet us recall the definition of upper Ahlfors regularity.\n\\begin{definition}\nLet $E\\subset \\mathbb{R}^3$ and $\\Lambda>0$. The set $E$ is said to be $\\Lambda$-upper Ahlfors regular if for all $x\\in E$ and all $r>0$ we have \n\\[\\mathcal{H}^{2}(E\\cap B(x,r)) \\leqslant \\Lambda \\pi r^2.\\]\n\\end{definition}\n\nThe definition of our functional is based on the notion of a generalized geodesic distance between curves, associated with a given weight function $u$, which is defined as follows.\n\\begin{definition}\\label{Def:geoDistance}\n\tLet $(\\delta_\\varepsilon)_{\\varepsilon>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n\t\\begin{align*}\n\t\t\t&d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{ \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2} \\ |\\ \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n\t\\end{align*}\n\\end{definition}\nNote that, in \\cite{bonnivard2025phasefieldapproximationplateaus}, the geodesic distance is defined for Lipschitz homotopies $\\ell$. The assumptions of uniform Lipschitz regularity and uniform upper Ahlfors regularity of the images are not required there. However, in the present work, we need to impose these assumptions in order to obtain the compactness result necessary for the existence theorem.\n\nLet $\\mathcal{C}$ an smooth open bounded convex set containing the closure $\\overline{\\Ccal_0}$. This set $\\mathcal{C}$ will serve as our domain of study. Indeed, since we rely on PDE techniques, it is convenient to work within a smooth bounded domain. We now recall the definition of the functional introduced in \\cite{bonnivard2025phasefieldapproximationplateaus}:\n\t\\begin{equation}\n\t\t\\label{functionalGeneral}\n\t\tF_{\\varepsilon}(u) := \\varepsilon \\int_{\\mathcal{C}}|\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2}dx + \\frac{1}{c_{\\varepsilon}}\\sum_{i=1}^{n}d_{u}(\\gamma_0, \\gamma_{i}).\n\t\\end{equation}\nWhen, the sequences of positive numbers $(\\delta_\\varepsilon)$ and $(c_\\varepsilon)$ are assumed to converge to zero, and to satisfy that $\\delta_\\varepsilon/c_\\varepsilon$ converges to zero as $\\varepsilon \\to 0$, \\cite{bonnivard2025phasefieldapproximationplateaus} establishes that this functional approximate some Plateau's problem through a $\\Gamma$-convergence result. \n\nFor simplicity, we will assume in the following that the prescribed boundary consists of only two curves, $\\gamma_0$ and $\\gamma_1$. However, the results established in the present article remain valid when the boundary contains more than two curves.\n\nIn this paper, we are specifically interested in the decoupled functional defined below.\n\\begin{definition}\n\tFor $u \\in H^{1}({\\mathcal C})$ such that $0\\leqslant u \\leqslant 1$, with $u = 1$ on $\\partial{{\\mathcal C}}$, and $\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)$,\n\t\\begin{align*}\n\t\t&E_{\\varepsilon}(u,\\ell) := \\varepsilon\\int_{\\mathcal{C}} |\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2} dx + \\frac{1}{c_{\\varepsilon}}\\int_{\\surf_{\\ell}} (u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2}.\n\t\\end{align*}\n\\end{definition}\n\n\\begin{remark}\n\tFrom Definition~\\ref{Def:geoDistance} of the geodesic distance between closed curves, we know that for all $u\\in H^{1}(\\mathcal{C})$ such that $0\\leqslant u \\leqslant 1$ and $u = 1$ on $\\partial{\\mathcal{C}}$, \n\t\t\\[F_{\\varepsilon}(u) = \\inf\\left \\{ E_{\\varepsilon}(u,\\ell)\\ ,\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1) \\right \\}.\\]\n\\end{remark}\n\nThe main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\\varepsilon$\n\\begin{equation*}\n \\inf_{(u,\\ell)} E_\\varepsilon(u,\\ell).\n\\end{equation*}", + "context": "More precisely, the competitor surfaces are defined as the images of homotopies connecting the given curves. We define below the set of admissible homotopies connecting a curve $\\gamma_i$ to $\\gamma_j$:\n\\[\n\\mathrm{Hom}(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Lip}([0,1] \\times \\mathbb{S}^{1}, \\overline{\\Ccal_0}) \\text{ such that } \\ell(0) = \\gamma_i \\text{ and } \\ell(1) = \\gamma_{j} \\}.\n\\]\nNote that we require more than mere homotopies: we specifically consider Lipschitz homotopies. For any homotopy $\\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j)$, we define the associated surface $\\surf_\\ell$ as its image:\n\\begin{equation}\\label{Def:surface_image}\n \\surf_{\\ell} := \\ell([0,1] \\times \\mathbb{S}^{1}) \\subset \\overline{\\Ccal_0}.\n\\end{equation}\nWe emphasize that the surface $\\surf_{\\ell}$ is $\\mathcal{H}^{2}$-rectifiable and has finite $\\mathcal{H}^{2}$-measure. Moreover, for any $u\\in H^1(\\mathbb{R}^3)$, the trace of $u$ on $\\surf_{\\ell}$ is well defined, since the set of points that are not Lebesgue points of $u$ is $\\mathcal{H}^2$-negligible. \nHowever, to obtain the existence result, we need to impose additional properties on the competitors.\nFor $\\Lambda>0$, we consider\n\\[\n\\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j) \\text{ such that } \\;\\mathrm{Lip}(\\ell)\\leq \\Lambda \\text{ and } \\surf_\\ell \\text{ is } \\Lambda-\\text{upper Ahlfors regular} \\ \\}.\\]\n\nThe definition of our functional is based on the notion of a generalized geodesic distance between curves, associated with a given weight function $u$, which is defined as follows.\n\\begin{definition}\\label{Def:geoDistance}\n Let $(\\delta_\\varepsilon)_{\\varepsilon>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n \\begin{align*}\n &d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{ \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2} \\ |\\ \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n \\end{align*}\n\\end{definition}\nNote that, in \\cite{bonnivard2025phasefieldapproximationplateaus}, the geodesic distance is defined for Lipschitz homotopies $\\ell$. The assumptions of uniform Lipschitz regularity and uniform upper Ahlfors regularity of the images are not required there. However, in the present work, we need to impose these assumptions in order to obtain the compactness result necessary for the existence theorem.\n\nLet $\\mathcal{C}$ an smooth open bounded convex set containing the closure $\\overline{\\Ccal_0}$. This set $\\mathcal{C}$ will serve as our domain of study. Indeed, since we rely on PDE techniques, it is convenient to work within a smooth bounded domain. We now recall the definition of the functional introduced in \\cite{bonnivard2025phasefieldapproximationplateaus}:\n \\begin{equation}\n \\label{functionalGeneral}\n F_{\\varepsilon}(u) := \\varepsilon \\int_{\\mathcal{C}}|\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2}dx + \\frac{1}{c_{\\varepsilon}}\\sum_{i=1}^{n}d_{u}(\\gamma_0, \\gamma_{i}).\n \\end{equation}\nWhen, the sequences of positive numbers $(\\delta_\\varepsilon)$ and $(c_\\varepsilon)$ are assumed to converge to zero, and to satisfy that $\\delta_\\varepsilon/c_\\varepsilon$ converges to zero as $\\varepsilon \\to 0$, \\cite{bonnivard2025phasefieldapproximationplateaus} establishes that this functional approximate some Plateau's problem through a $\\Gamma$-convergence result.\n\nIn this paper, we are specifically interested in the decoupled functional defined below.\n\\begin{definition}\n For $u \\in H^{1}({\\mathcal C})$ such that $0\\leqslant u \\leqslant 1$, with $u = 1$ on $\\partial{{\\mathcal C}}$, and $\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)$,\n \\begin{align*}\n &E_{\\varepsilon}(u,\\ell) := \\varepsilon\\int_{\\mathcal{C}} |\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2} dx + \\frac{1}{c_{\\varepsilon}}\\int_{\\surf_{\\ell}} (u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2}.\n \\end{align*}\n\\end{definition}\n\n\\begin{remark}\n From Definition~\\ref{Def:geoDistance} of the geodesic distance between closed curves, we know that for all $u\\in H^{1}(\\mathcal{C})$ such that $0\\leqslant u \\leqslant 1$ and $u = 1$ on $\\partial{\\mathcal{C}}$, \n \\[F_{\\varepsilon}(u) = \\inf\\left \\{ E_{\\varepsilon}(u,\\ell)\\ ,\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1) \\right \\}.\\]\n\\end{remark}\n\nThe main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\\varepsilon$\n\\begin{equation*}\n \\inf_{(u,\\ell)} E_\\varepsilon(u,\\ell).\n\\end{equation*}\n\n\\begin{definition}\\label{Def:geoDistance}\n\tLet $(\\delta_\\eps)_{\\eps>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n\t\\begin{align*}\n\t\t\t&d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{ \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\eps) d\\Haus \\ |\\ \\ell \\in \\curve^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n\t\\end{align*}\n\\end{definition}", + "full_context": "More precisely, the competitor surfaces are defined as the images of homotopies connecting the given curves. We define below the set of admissible homotopies connecting a curve $\\gamma_i$ to $\\gamma_j$:\n\\[\n\\mathrm{Hom}(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Lip}([0,1] \\times \\mathbb{S}^{1}, \\overline{\\Ccal_0}) \\text{ such that } \\ell(0) = \\gamma_i \\text{ and } \\ell(1) = \\gamma_{j} \\}.\n\\]\nNote that we require more than mere homotopies: we specifically consider Lipschitz homotopies. For any homotopy $\\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j)$, we define the associated surface $\\surf_\\ell$ as its image:\n\\begin{equation}\\label{Def:surface_image}\n \\surf_{\\ell} := \\ell([0,1] \\times \\mathbb{S}^{1}) \\subset \\overline{\\Ccal_0}.\n\\end{equation}\nWe emphasize that the surface $\\surf_{\\ell}$ is $\\mathcal{H}^{2}$-rectifiable and has finite $\\mathcal{H}^{2}$-measure. Moreover, for any $u\\in H^1(\\mathbb{R}^3)$, the trace of $u$ on $\\surf_{\\ell}$ is well defined, since the set of points that are not Lebesgue points of $u$ is $\\mathcal{H}^2$-negligible. \nHowever, to obtain the existence result, we need to impose additional properties on the competitors.\nFor $\\Lambda>0$, we consider\n\\[\n\\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j) \\text{ such that } \\;\\mathrm{Lip}(\\ell)\\leq \\Lambda \\text{ and } \\surf_\\ell \\text{ is } \\Lambda-\\text{upper Ahlfors regular} \\ \\}.\\]\n\nThe definition of our functional is based on the notion of a generalized geodesic distance between curves, associated with a given weight function $u$, which is defined as follows.\n\\begin{definition}\\label{Def:geoDistance}\n Let $(\\delta_\\varepsilon)_{\\varepsilon>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n \\begin{align*}\n &d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{ \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2} \\ |\\ \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n \\end{align*}\n\\end{definition}\nNote that, in \\cite{bonnivard2025phasefieldapproximationplateaus}, the geodesic distance is defined for Lipschitz homotopies $\\ell$. The assumptions of uniform Lipschitz regularity and uniform upper Ahlfors regularity of the images are not required there. However, in the present work, we need to impose these assumptions in order to obtain the compactness result necessary for the existence theorem.\n\nLet $\\mathcal{C}$ an smooth open bounded convex set containing the closure $\\overline{\\Ccal_0}$. This set $\\mathcal{C}$ will serve as our domain of study. Indeed, since we rely on PDE techniques, it is convenient to work within a smooth bounded domain. We now recall the definition of the functional introduced in \\cite{bonnivard2025phasefieldapproximationplateaus}:\n \\begin{equation}\n \\label{functionalGeneral}\n F_{\\varepsilon}(u) := \\varepsilon \\int_{\\mathcal{C}}|\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2}dx + \\frac{1}{c_{\\varepsilon}}\\sum_{i=1}^{n}d_{u}(\\gamma_0, \\gamma_{i}).\n \\end{equation}\nWhen, the sequences of positive numbers $(\\delta_\\varepsilon)$ and $(c_\\varepsilon)$ are assumed to converge to zero, and to satisfy that $\\delta_\\varepsilon/c_\\varepsilon$ converges to zero as $\\varepsilon \\to 0$, \\cite{bonnivard2025phasefieldapproximationplateaus} establishes that this functional approximate some Plateau's problem through a $\\Gamma$-convergence result.\n\nIn this paper, we are specifically interested in the decoupled functional defined below.\n\\begin{definition}\n For $u \\in H^{1}({\\mathcal C})$ such that $0\\leqslant u \\leqslant 1$, with $u = 1$ on $\\partial{{\\mathcal C}}$, and $\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)$,\n \\begin{align*}\n &E_{\\varepsilon}(u,\\ell) := \\varepsilon\\int_{\\mathcal{C}} |\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2} dx + \\frac{1}{c_{\\varepsilon}}\\int_{\\surf_{\\ell}} (u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2}.\n \\end{align*}\n\\end{definition}\n\n\\begin{remark}\n From Definition~\\ref{Def:geoDistance} of the geodesic distance between closed curves, we know that for all $u\\in H^{1}(\\mathcal{C})$ such that $0\\leqslant u \\leqslant 1$ and $u = 1$ on $\\partial{\\mathcal{C}}$, \n \\[F_{\\varepsilon}(u) = \\inf\\left \\{ E_{\\varepsilon}(u,\\ell)\\ ,\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1) \\right \\}.\\]\n\\end{remark}\n\nThe main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\\varepsilon$\n\\begin{equation*}\n \\inf_{(u,\\ell)} E_\\varepsilon(u,\\ell).\n\\end{equation*}\n\n\\begin{definition}\\label{Def:geoDistance}\n\tLet $(\\delta_\\eps)_{\\eps>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n\t\\begin{align*}\n\t\t\t&d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{ \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\eps) d\\Haus \\ |\\ \\ell \\in \\curve^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n\t\\end{align*}\n\\end{definition}\n\nThe main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\\varepsilon$\n\\begin{equation*}\n \\inf_{(u,\\ell)} E_\\varepsilon(u,\\ell).\n\\end{equation*}\n\nThere exists a minimizing couple $(u,\\ell)$ of the following problem in both variables\n\\begin{equation}\n \\label{pb min}\n \\min\\left \\{E_\\eps(u,\\ell), \\, u \\in H^{1}(\\C) \\text{ such that } 0\\leqslant u \\leqslant 1 , \\, u = 1 \\text{ on } \\partial{\\C} \\text{ and } \\ell \\in \\curve^\\Lambda(\\gamma_0,\\gamma_1) \\right\\}.\n\\end{equation}\n\nThe study of this energy is motivated by the $\\Gamma$-convergence result established in \\cite{bonnivard2025phasefieldapproximationplateaus}, where this decoupled functional was introduced both in the proof of the $\\Gamma$-convergence and for numerical applications. Recall that the phase-field approach they proposed is a generalization to Plateau’s problem of the method introduced in \\cite{bonnivard2015approximation} for Steiner’s problem. A similar study of the existence of solutions for the decoupled functional in the Steiner case was carried out in \\cite{bonnivard2018phase}. The present work is inspired by this paper, but there are some differences that we explain below.\n\nWe begin by studying the following minimization problem \n\\begin{equation}\n\\label{pb min l fixe}\n \\inf_{u} E_\\varepsilon(u,\\ell), \\text{ where } \\ell \\in \\curve^\\Lambda(\\gamma_0,\\gamma_1).\n\\end{equation}\nThe following Proposition will be proven in Section~\\ref{section l fixe}. \n\\begin{proposition}\nFor any given $\\ell \\in \\curve^\\Lambda(\\gamma_0, \\gamma_1)$, there exists a unique minimizer $u \\in H^1(\\Ccal)$ of $ E_\\varepsilon(\\cdot,l)$ which is globally $C^{0,\\alpha}$, for all $0< \\alpha <1$: there exists a constant $C_\\alpha>0$ such that \n\\[\\|u \\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]\n\\end{proposition}\nThe assumption that the surface $S_\\ell$ is $\\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity, and hence the existence of minimizers in this setting. We then apply classical results from elliptic regularity to derive Hölder estimates for the minimizer. Note that these estimates depend explicitly on $\\Lambda$ and $\\varepsilon$, but are independent of the specific surface $S_\\ell$. This part is strongly inspired by \\cite{bonnivard2018phase}.\n\n\\begin{proposition}\n\\label{prop holder}\nLet $\\ell \\in~\\curve^\\Lambda(\\gamma_0,\\gamma_1)$ and let $u$ be the minimizer of $E_\\varepsilon^\\ell$. Then, $u$ is $\\alpha$-Hölder continuous for all $0<\\alpha<1$ with the estimate:\n\\[\\|u\\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]\n\\end{proposition}\n\n\\begin{lemma}\n\\label{lem 2.10}\nRecall the definition of the parameter $\\eta_0 = \\dist(\\partial \\Ccal_0,\\partial \\Ccal)$ introduced in the previous Lemma.\nLet $\\ell \\in \\curve^\\Lambda(\\gamma_0, \\gamma_1)$, $0< \\rho < \\min(1,\\frac{\\eta_0}{4})$ and $x_0 \\in \\mathcal{C}$ such that $\\dist(x_0,\\partial \\mathcal{C}) \\geqslant \\frac{\\eta_0}{2} $. Then, $u \\in W^{1,p}(B(x_0,\\rho))$ for all $3\\leqslant p < \\infty$ and we have the following estimate on the rescaled function $u_\\rho (x) := u(x_0+\\rho x)$\n\\begin{equation}\n \\|\\nabla u_\\rho\\|_{L^p(B_1)} \\leqslant C_p\\left( \\frac{\\rho^2}{\\varepsilon^2}+1+\\frac{\\Lambda \\rho}{c_\\varepsilon \\varepsilon}\\right).\n\\end{equation}\n\\end{lemma}\n\n\\medskip\n\\emph{Step 3.} We define the rescaled function $u_\\rho$ on $B_2$ by $u_\\rho(x) := u(x_0+\\rho x)$. Then, for all $\\varphi \\in C^\\infty_c(B_2)$ \n\\begin{align*}\n \\int_{B_2}\\nabla u_\\rho \\cdot \\nabla\\varphi \\dm x &= \\frac{1}{\\rho}\\int_{B(x_0,2\\rho)}\\nabla u \\cdot \\nabla\\varphi_\\rho \\dm x,\\\\\n &= \\frac{1}{\\rho4\\varepsilon^2}\\int_{B(x_0,2\\rho)}(1- u)\\varphi_\\rho \\dm x - \\frac{1}{\\rho c_\\varepsilon \\varepsilon}B[\\ell](u,\\varphi_\\rho),\\\\\n &= \\frac{\\rho^2}{4\\varepsilon^2}\\int_{B_2}(1- u_\\rho)\\varphi \\dm x - \\frac{1}{\\rho c_\\varepsilon \\varepsilon}\\langle T_\\rho,\\varphi \\rangle.\n\\end{align*}\nHence, $u_\\rho$ satisfies the following PDE in $\\mathcal{D}'(B_2)$\n\\[-\\Delta u_\\rho = \\frac{\\rho^2}{4\\varepsilon^2}(1- u_\\rho) - \\frac{1}{\\rho c_\\varepsilon \\varepsilon} T_\\rho.\\]\nWe denote $w_\\rho := u_\\rho + \\frac{1}{\\rho c_\\varepsilon \\varepsilon}v_\\rho \\in H^1(B_2)\\cap L^\\infty(B_2)$. And we deduce that \n\\[-\\Delta w_\\rho = \\frac{\\rho^2}{4\\varepsilon^2}(1- u_\\rho) \\text{ in } \\mathcal{D}'(B_2). \\]\nThus, \\cite[Corollary 8.36]{gilbargelliptic} yields that $w_\\rho \\in C_{\\mathrm{loc}}^{1,\\alpha}(B_2)$ for all $\\alpha >0$ and \n\\begin{align*}\n \\|\\nabla w_\\rho\\|_{L^\\infty(B_1)} \\leqslant \\|w_\\rho\\|_{C^{1,\\alpha}(B_1)} &\\leqslant C\\left(\\| w_\\rho\\|_{L^\\infty(B_1)} +\\frac{\\rho^2}{4\\varepsilon^2}\\|1- u_\\rho\\|_{L^\\infty(B_1)}\\right),\\\\\n &= C\\left(\\| u_\\rho\\|_{L^\\infty(B_1)}+ \\frac{1}{\\rho c_\\varepsilon \\varepsilon}\\| v_\\rho\\|_{L^\\infty(B_1)} +\\frac{\\rho^2}{4\\varepsilon^2}\\right),\\\\\n &\\leqslant C\\left(1 + \\frac{\\Lambda \\rho}{c_\\varepsilon \\varepsilon} + \\frac{\\rho^2}{4\\varepsilon^2}\\right).\n\\end{align*}\nFinally, we can conclude that $u_\\rho = w_\\rho - \\frac{1}{\\rho c_\\varepsilon \\varepsilon}v_\\rho \\in W^{1,p}(B_1)$ with the desired estimate\n\\[ \\|\\nabla u_\\rho\\|_{L^p(B_1)} \\leqslant C_p\\left( \\frac{\\rho^2}{\\varepsilon^2}+1+\\frac{\\Lambda \\rho}{c_\\varepsilon \\varepsilon}\\right).\\]\nThis concludes the proof of this Lemma.\n\\end{proof}\n\nNow let's consider the case where $\\dist(x_0, \\partial \\mathcal{C}) > \\frac{\\eta_0}{2}> \\varepsilon$, in particular this implies that $B(x_0,\\varepsilon) \\subset \\C$. In this case, Lemma~\\ref{lem 2.10}, applied with $\\rho = \\varepsilon$ and $p = \\frac{3}{1-\\alpha}$ yields the estimate on the rescaled function $u_\\varepsilon$\n \\[\\|\\nabla u_\\varepsilon\\|_{L^p(B_1)} \\leqslant C_p\\left(1+ \\frac{\\Lambda}{c_\\varepsilon}\\right).\\]\n Since we have shown that the $L^\\infty$ norm of $u$ is smaller than $1$, we deduce that the $L^p$ norm of the scaled function $u_\\varepsilon$ is bounded by the volume of the domain. And we conclude by applying the Sobolev embedding Theorem (see for instance \\cite[Theorem 4.12]{adams2003sobolev}),\n \\begin{equation}\n \\|u_\\varepsilon\\|_{C^{0,\\alpha}(B_1)} \\leqslant C_\\alpha\\left(1+ \\frac{\\Lambda}{c_\\varepsilon}\\right).\n \\end{equation}\n Hence, scaling back leads to \n \\begin{equation}\n \\frac{|u(x)-u(y)|}{|x-y|^\\alpha} \\leqslant \\frac{1}{\\varepsilon^\\alpha} \\|u_\\varepsilon\\|_{C^{0,\\alpha}(B_1)} \\leqslant \\frac{C_\\alpha}{\\varepsilon^\\alpha}\\left(1+ \\frac{\\Lambda}{c_\\varepsilon}\\right).\n \\end{equation}\n\\end{itemize}\nFinally, since $\\|u\\|_{L^\\infty(\\Ccal)} \\leqslant 1$, we get the desired estimate on the Hölder norm of $u$ \n\\[ \\|u\\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\epsilon^{\\alpha}}.\\]\nThis achieves the proof of the Hölder regularity of the solution $u$ of $\\min E_\\eps(u,\\ell)$, when $\\ell$ is fixed in $\\curve^\\Lambda$.\n\\end{proof}", + "post_theorem_intro_text_len": 5460, + "post_theorem_intro_text": "The study of this energy is motivated by the $\\Gamma$-convergence result established in \\cite{bonnivard2025phasefieldapproximationplateaus}, where this decoupled functional was introduced both in the proof of the $\\Gamma$-convergence and for numerical applications. Recall that the phase-field approach they proposed is a generalization to Plateau’s problem of the method introduced in \\cite{bonnivard2015approximation} for Steiner’s problem. A similar study of the existence of solutions for the decoupled functional in the Steiner case was carried out in \\cite{bonnivard2018phase}. The present work is inspired by this paper, but there are some differences that we explain below.\n\nTo prove Theorem~\\ref{main th}, we first consider the optimization problems with either $u$ or $\\ell$ fixed. As usual, establishing the existence of minimizers requires both a compactness result and the lower semi-continuity of the functional. The compactness follows from the choice of the class of competitors, so the main issue is to establish lower semi-continuity.\n\nWe begin by studying the following minimization problem \n\\begin{equation}\n\\label{pb min l fixe}\n \\inf_{u} E_\\varepsilon(u,\\ell), \\text{ where } \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1).\n\\end{equation}\nThe following Proposition will be proven in Section~\\ref{section l fixe}. \n\\begin{proposition}\nFor any given $\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0, \\gamma_1)$, there exists a unique minimizer $u \\in H^1(\\mathcal{C})$ of $ E_\\varepsilon(\\cdot,l)$ which is globally $C^{0,\\alpha}$, for all $0< \\alpha <1$: there exists a constant $C_\\alpha>0$ such that \n\\[\\|u \\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]\n\\end{proposition}\nThe assumption that the surface $S_\\ell$ is $\\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity, and hence the existence of minimizers in this setting. We then apply classical results from elliptic regularity to derive Hölder estimates for the minimizer. Note that these estimates depend explicitly on $\\Lambda$ and $\\varepsilon$, but are independent of the specific surface $S_\\ell$. This part is strongly inspired by \\cite{bonnivard2018phase}.\n\nThen, in Section~\\ref{section u fixe} we consider the minimization problem\n\\begin{equation}\n\\label{pb min u fixe}\n \\inf_{\\ell \\in \\mathrm{Hom}^\\Lambda} E_\\varepsilon(u,\\ell), \\text{ where } u \\in H^1(\\mathcal{C})\\cap C(\\overline{\\mathcal{C}}),\n\\end{equation}\nand prove the following Proposition.\n\\begin{proposition}\n\\label{prop 1.2}\nFor all $u \\in H^1({\\mathcal C})\\cap C(\\overline{\\mathcal{C}})$ there exists a minimizer for the problem \\eqref{pb min u fixe}. \n\\end{proposition}\n\nAs previously mentionned, the main difficulty in the proof of Proposition~\\ref{prop 1.2} lies in establishing the lower semicontinuity of the functional, which relies on the following Lemma.\n\n \\begin{lemma}\n \\label{golab}\nLet $\\ell_n : [0,1] \\times \\mathbb{S}^1 \\to \\mathbb{R}^3$ be a sequence of uniformly Lipschitz functions, which converges uniformly to a Lipschitz function $\\ell : [0,1] \\times \\mathbb{S}^1 \\to \\mathbb{R}^3$.\nThen, for all open set $A$\n\n$$\\liminf \\mathcal{H}^{2} \\mathbin{\\vrule height 1.6ex depth 0pt width 0.13ex\\vrule height 0.13ex depth 0pt width 1.3ex} S_{\\ell_n}(A) \\geqslant \\mathcal{H}^{2} \\mathbin{\\vrule height 1.6ex depth 0pt width 0.13ex\\vrule height 0.13ex depth 0pt width 1.3ex} S_\\ell(A),$$\nwhere $S_\\ell$ (respectively $S_{\\ell_n}$) denotes the image of the Lipschitz function $\\ell$ (respectively $\\ell_n)$.\n\\end{lemma}\n\nThis Lemma, proven in Section~\\ref{section u fixe}, constitutes a central and original contribution of the present article and relies crucially on the specific definition of the class of surfaces under consideration. It can be viewed as a generalization of Go\\l{}\\k{a}b’s theorem to the two-dimensional Hausdorff measure, and its proof follows a similar strategy. In Go\\l{}\\k{a}b’s theorem, the first step consists in establishing the rectifiability of the limit set; in our setting, this property is immediate, since the surfaces are assumed to be images of Lipschitz functions. We then study the weak limit of the measures $(\\mathcal{H}^{2} \\mathbin{\\vrule height 1.6ex depth 0pt width 0.13ex\\vrule height 0.13ex depth 0pt width 1.3ex} S_{\\ell_n})$ and, using the tangent plane of the limit surface, derive a lower bound on the upper density of the limit measure to conclude. The key point of our proof relies on a topological argument, stating that a Lipschitz image, which is uniformly close to a disk, must have at least almost the density of a disk (see the proof of Lemma~\\ref{golab}).\n\nWe emphasize that the proof of this Lemma does not rely on the specific definition set $\\mathbb{S}^1 \\times [0,1]$; it only assumes it to be a two-dimensional subset of $\\mathbb{R}^3$. Moreover, the result is also expected to remain valid in dimensions higher than two, although the proof would then require more delicate topological arguments.\n\nFinally, in Section~\\ref{section rien fixe}, Theorem~\\ref{main th} is proved by combining these results, in particular the Hölder regularity and the preceding Lemma.\n\n\\textbf{Acknowledgments.}\n I would like to thank Antoine Lemenant for suggesting this problem to me as well as Matthieu Bonnivard for their help in this project. \nThis work was partially supported by the IUF grant of Antoine Lemenant and by the ANR project STOIQUES.", + "sketch": "To prove Theorem~\\ref{main th}, the authors “first consider the optimization problems with either $u$ or $\\ell$ fixed.” Existence requires “both a compactness result and the lower semi-continuity of the functional”; “the compactness follows from the choice of the class of competitors,” so “the main issue is to establish lower semi-continuity.”\n\n1) **Fix $\\ell$**: study \\eqref{pb min l fixe}. For any $\\ell\\in\\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)$ there is a “unique minimizer $u\\in H^1(\\mathcal C)$” which is “globally $C^{0,\\alpha}$,” with the stated estimate. Here, “the assumption that the surface $S_\\ell$ is $\\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity,” and then “classical results from elliptic regularity” give Hölder estimates (depending on $\\Lambda$ and $\\varepsilon$ but “independent of the specific surface $S_\\ell$”).\n\n2) **Fix $u$**: study \\eqref{pb min u fixe}. For $u\\in H^1(\\mathcal C)\\cap C(\\overline{\\mathcal C})$ there exists a minimizer in $\\ell$. The “main difficulty” is again lower semicontinuity, which “relies on” Lemma~\\ref{golab}, a Go\\l{}\\k{a}b-type lower semicontinuity statement for $\\mathcal H^2\\llcorner S_{\\ell_n}$ under uniform convergence of uniformly Lipschitz parametrizations. Its proof “follows a similar strategy” to Go\\l{}\\k{a}b: rectifiability is “immediate,” then one studies “the weak limit of the measures,” and “using the tangent plane of the limit surface, derive[s] a lower bound on the upper density of the limit measure to conclude”; the “key point” is “a topological argument” that a Lipschitz image uniformly close to a disk has “at least almost the density of a disk.”\n\n3) **Conclusion**: “Finally, in Section~\\ref{section rien fixe}, Theorem~\\ref{main th} is proved by combining these results, in particular the Hölder regularity and the preceding Lemma.”", + "expanded_sketch": "To prove the main theorem, the authors “first consider the optimization problems with either $u$ or $\\ell$ fixed.” Existence requires “both a compactness result and the lower semi-continuity of the functional”; “the compactness follows from the choice of the class of competitors,” so “the main issue is to establish lower semi-continuity.”\n\n1) **Fix $\\ell$**: study \\eqref{pb min l fixe}. For any $\\ell\\in\\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)$ there is a “unique minimizer $u\\in H^1(\\mathcal C)$” which is “globally $C^{0,\\alpha}$,” with the stated estimate. Here, “the assumption that the surface $S_\\ell$ is $\\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity,” and then “classical results from elliptic regularity” give Hölder estimates (depending on $\\Lambda$ and $\\varepsilon$ but “independent of the specific surface $S_\\ell$”).\n\n2) **Fix $u$**: study\n\\begin{equation}\n\\label{pb min u fixe}\n \\inf_{\\ell \\in \\curve^\\Lambda} E_\\varepsilon(u,\\ell), \\text{ where } u \\in H^1(\\Ccal)\\cap C(\\overline{\\Ccal}),\n\\end{equation}\nFor $u\\in H^1(\\mathcal C)\\cap C(\\overline{\\mathcal C})$ there exists a minimizer in $\\ell$. The “main difficulty” is again lower semicontinuity, which “relies on” the following lemma.\n\\begin{lemma}\n \\label{golab}\nLet $\\ell_n : [0,1] \\times \\mathbb{S}^1 \\to \\R^3$ be a sequence of uniformly Lipschitz functions, which converges uniformly to a Lipschitz function $\\ell : [0,1] \\times \\mathbb{S}^1 \\to \\R^3$.\nThen, for all open set $A$\n\n$$\\liminf \\Haus \\mres S_{\\ell_n}(A) \\geqslant \\Haus \\mres S_\\ell(A),$$\nwhere $S_\\ell$ (respectively $S_{\\ell_n}$) denotes the image of the Lipschitz function $\\ell$ (respectively $\\ell_n)$.\n\\end{lemma}\nIts proof “follows a similar strategy” to Go\\l{}\\k{a}b: rectifiability is “immediate,” then one studies “the weak limit of the measures,” and “using the tangent plane of the limit surface, derive[s] a lower bound on the upper density of the limit measure to conclude”; the “key point” is “a topological argument” that a Lipschitz image uniformly close to a disk has “at least almost the density of a disk.”\n\n3) **Conclusion**: “Finally, next we conclude the proof of the main theorem by combining these results, in particular the Hölder regularity and the preceding lemma.”", + "expanded_theorem": "\\label{main th}\n\nThere exists a minimizing couple $(u,\\ell)$ of the following problem in both variables\n\\begin{equation}\n \\label{pb min}\n \\min\\left \\{E_\\varepsilon(u,\\ell), \\, u \\in H^{1}({\\mathcal C}) \\text{ such that } 0\\leqslant u \\leqslant 1 , \\, u = 1 \\text{ on } \\partial{{\\mathcal C}} \\text{ and } \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1) \\right\\}.\n\\end{equation}\n\nMoreover, the minimizer $u$ is $\\alpha$-Hölder continuous for all $0<\\alpha<1$ with the estimate:\n\\[\\|u\\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]", + "theorem_type": [ + "Existence", + "Universal" + ], + "mcq": { + "question": "Fix \\(\\varepsilon>0\\), \\(\\Lambda>0\\), positive numbers \\(c_{\\varepsilon}\\) and \\(\\delta_{\\varepsilon}\\), a smooth bounded convex domain \\(\\mathcal C\\), a set \\(\\mathcal C_0\\) with \\(\\overline{\\mathcal C_0}\\subset \\mathcal C\\), and closed curves \\(\\gamma_0,\\gamma_1\\). For a Lipschitz homotopy \\(\\ell:[0,1]\\times \\mathbb S^1\\to \\overline{\\mathcal C_0}\\) with \\(\\ell(0)=\\gamma_0\\) and \\(\\ell(1)=\\gamma_1\\), let \\(\\Sigma_{\\ell}=\\ell([0,1]\\times \\mathbb S^1)\\). Define \\(\\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\) as the set of such \\(\\ell\\) satisfying \\(\\mathrm{Lip}(\\ell)\\le \\Lambda\\) and such that \\(\\Sigma_{\\ell}\\) is \\(\\Lambda\\)-upper Ahlfors regular (i.e. \\(\\mathcal H^2(\\Sigma_{\\ell}\\cap B(x,r))\\le \\Lambda r^2\\) for all balls). For admissible \\(u\\in H^1(\\mathcal C)\\) with \\(0\\le u\\le 1\\) and \\(u=1\\) on \\(\\partial\\mathcal C\\), define\n\\[\nE_{\\varepsilon}(u,\\ell)=\\varepsilon\\int_{\\mathcal C}|\\nabla u|^2\\,dx+\\frac{1}{4\\varepsilon}\\int_{\\mathcal C}(1-u)^2\\,dx+\\frac{1}{c_{\\varepsilon}}\\int_{\\Sigma_{\\ell}}(u^2+\\delta_{\\varepsilon})\\,d\\mathcal H^2.\n\\]\nWhich statement holds for the minimization problem over all admissible pairs \\((u,\\ell)\\)?", + "correct_choice": { + "label": "A", + "text": "There exists at least one admissible pair \\((u,\\ell)\\) that minimizes \\(E_{\\varepsilon}(u,\\ell)\\) jointly among all pairs with \\(u\\in H^1(\\mathcal C)\\), \\(0\\le u\\le 1\\), \\(u=1\\) on \\(\\partial\\mathcal C\\), and \\(\\ell\\in \\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\). Moreover, for such a minimizing pair, the function \\(u\\) belongs to \\(C^{0,\\alpha}(\\mathcal C)\\) for every \\(0<\\alpha<1\\), and it satisfies\n\\[\n\\|u\\|_{C^{0,\\alpha}(\\mathcal C)}\\le C_{\\alpha}\\,\\frac{1+\\Lambda c_{\\varepsilon}^{-1}}{\\varepsilon^{\\alpha}}.\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "There exists at least one admissible pair \\((u,\\ell)\\) that minimizes \\(E_{\\varepsilon}(u,\\ell)\\) jointly among all pairs with \\(u\\in H^1(\\mathcal C)\\), \\(0\\le u\\le 1\\), \\(u=1\\) on \\(\\partial\\mathcal C\\), and \\(\\ell\\in \\mathrm{Hom}(\\gamma_0,\\gamma_1)\\) (that is, without imposing the uniform Lipschitz bound or the \\(\\Lambda\\)-upper Ahlfors regularity on \\(\\Sigma_\\ell\\)). Moreover, for such a minimizing pair, the function \\(u\\) belongs to \\(C^{0,\\alpha}(\\mathcal C)\\) for every \\(0<\\alpha<1\\), and it satisfies\n\\[\n\\|u\\|_{C^{0,\\alpha}(\\mathcal C)}\\le C_{\\alpha}\\,\\frac{1+\\Lambda c_{\\varepsilon}^{-1}}{\\varepsilon^{\\alpha}}.\n\\]" + }, + { + "label": "C", + "text": "There exists at least one admissible pair \\((u,\\ell)\\) that minimizes \\(E_{\\varepsilon}(u,\\ell)\\) jointly among all pairs with \\(u\\in H^1(\\mathcal C)\\), \\(0\\le u\\le 1\\), \\(u=1\\) on \\(\\partial\\mathcal C\\), and \\(\\ell\\in \\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\)." + }, + { + "label": "D", + "text": "For every admissible \\(u\\in H^1(\\mathcal C)\\) with \\(0\\le u\\le 1\\) and \\(u=1\\) on \\(\\partial\\mathcal C\\), there exists \\(\\ell\\in \\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\) minimizing \\(E_{\\varepsilon}(u,\\ell)\\) over \\(\\ell\\); and for every admissible \\(\\ell\\in \\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\), there exists a minimizer \\(u\\) of \\(E_{\\varepsilon}(u,\\ell)\\) over \\(u\\). Consequently, there exists a pair \\((u,\\ell)\\) such that \\(u\\) minimizes \\(E_{\\varepsilon}(\\,b7\\,,\\ell)\\) and \\(\\ell\\) minimizes \\(E_{\\varepsilon}(u,\\,\\u00b7\\,)\\), and this pair minimizes \\(E_{\\varepsilon}\\) jointly over all admissible pairs. Moreover, for such a pair,\n\\[\n\\|u\\|_{C^{0,\\alpha}(\\mathcal C)}\\le C_{\\alpha}\\,\\frac{1+\\Lambda c_{\\varepsilon}^{-1}}{\\varepsilon^{\\alpha}}\n\\]\nfor every \\(0<\\alpha<1\\)." + }, + { + "label": "E", + "text": "There exists at least one admissible pair \\((u,\\ell)\\) that minimizes \\(E_{\\varepsilon}(u,\\ell)\\) jointly among all pairs with \\(u\\in H^1(\\mathcal C)\\), \\(0\\le u\\le 1\\), \\(u=1\\) on \\(\\partial\\mathcal C\\), and \\(\\ell\\in \\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\). Moreover, for such a minimizing pair, the function \\(u\\) belongs to \\(C^{0,\\alpha}(\\mathcal C)\\) for every \\(0<\\alpha<1\\), and it satisfies\n\\[\n\\|u\\|_{C^{0,\\alpha}(\\mathcal C)}\\le C_{\\alpha}\\,\\frac{1+\\Lambda c_{\\varepsilon}^{-1}}{\\varepsilon}.\n\\]" + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "D" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "regularity", + "tampered_component": "necessity_of_uniform_Lipschitz_and_Ahlfors_class_for_compactness_lsc", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "regularity", + "tampered_component": "dropped_Holder_regularity_conclusion_and_estimate", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "from_separate_minimizers_to_joint_minimizer", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "epsilon_scaling_in_Holder_estimate", + "template_used": "boundary_range" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not state or strongly hint at the exact correct conclusion; it only sets up the functional and asks which existence result is valid." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recall question: choice A essentially states the full theorem, while the others are perturbations of it. Still, the options do force comparison of different strengths of conclusion." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to reject the unrestricted class, unjustified uniqueness, and overstrong Lipschitz regularity, and to distinguish the strongest valid statement from a weaker true one. But the task remains largely recognition of the theorem statement." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically meaningful: one weakens the admissible class, one gives a weaker true statement, one overclaims uniqueness, and one overstates regularity. These reflect realistic failure modes." + }, + "total_score": 6, + "overall_assessment": "A reasonably strong MCQ with good distractors and little answer leakage, but it leans toward theorem recognition rather than deep generative reasoning." + } + }, + { + "id": "2602.08476v1", + "paper_link": "http://arxiv.org/abs/2602.08476v1", + "theorems_cnt": 7, + "theorem": { + "env_name": "theorem", + "content": "\\label{main th}\n\nThere exists a minimizing couple $(u,\\ell)$ of the following problem in both variables\n\\begin{equation}\n \\label{pb min}\n \\min\\left \\{E_\\varepsilon(u,\\ell), \\, u \\in H^{1}({\\mathcal C}) \\text{ such that } 0\\leqslant u \\leqslant 1 , \\, u = 1 \\text{ on } \\partial{{\\mathcal C}} \\text{ and } \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1) \\right\\}.\n\\end{equation}\n\nMoreover, the minimizer $u$ is $\\alpha$-Hölder continuous for all $0<\\alpha<1$ with the estimate:\n\\[\\|u\\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]", + "start_pos": 9246, + "end_pos": 9845, + "label": "main th" + }, + "ref_dict": { + "prop 1.2": "\\begin{proposition}\n\\label{prop 1.2}\nFor all $u \\in H^1(\\C)\\cap C(\\overline{\\Ccal})$ there exists a minimizer for the problem \\eqref{pb min u fixe}. \n\\end{proposition}", + "pb min u fixe": "\\begin{equation}\n\\label{pb min u fixe}\n \\inf_{\\ell \\in \\curve^\\Lambda} E_\\varepsilon(u,\\ell), \\text{ where } u \\in H^1(\\Ccal)\\cap C(\\overline{\\Ccal}),\n\\end{equation}", + "main th": "\\begin{theorem}\n\\label{main th}\n\nThere exists a minimizing couple $(u,\\ell)$ of the following problem in both variables\n\\begin{equation}\n \\label{pb min}\n \\min\\left \\{E_\\eps(u,\\ell), \\, u \\in H^{1}(\\C) \\text{ such that } 0\\leqslant u \\leqslant 1 , \\, u = 1 \\text{ on } \\partial{\\C} \\text{ and } \\ell \\in \\curve^\\Lambda(\\gamma_0,\\gamma_1) \\right\\}.\n\\end{equation}\n\nMoreover, the minimizer $u$ is $\\alpha$-Hölder continuous for all $0<\\alpha<1$ with the estimate:\n\\[\\|u\\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]\n\n\\end{theorem}", + "golab": "\\begin{lemma}\n \\label{golab}\nLet $\\ell_n : [0,1] \\times \\mathbb{S}^1 \\to \\R^3$ be a sequence of uniformly Lipschitz functions, which converges uniformly to a Lipschitz function $\\ell : [0,1] \\times \\mathbb{S}^1 \\to \\R^3$.\nThen, for all open set $A$\n\n$$\\liminf \\Haus \\mres S_{\\ell_n}(A) \\geqslant \\Haus \\mres S_\\ell(A),$$\nwhere $S_\\ell$ (respectively $S_{\\ell_n}$) denotes the image of the Lipschitz function $\\ell$ (respectively $\\ell_n)$.\n\\end{lemma}", + "Def:geoDistance": "\\begin{definition}\\label{Def:geoDistance}\n\tLet $(\\delta_\\eps)_{\\eps>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n\t\\begin{align*}\n\t\t\t&d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{ \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\eps) d\\Haus \\ |\\ \\ell \\in \\curve^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n\t\\end{align*}\n\\end{definition}" + }, + "pre_theorem_intro_text_len": 5458, + "pre_theorem_intro_text": "This paper is devoted to the study of the functional introduced in \\cite{bonnivard2025phasefieldapproximationplateaus} to approximate a Plateau problem, which consists in finding a surface of minimal area spanning a collection of closed curves $\\gamma_0,..., \\gamma_n$ contained in the boundary of an open, bounded, convex set $\\Ccal_0 \\subset \\mathbb{R}^3$. \n\nMore precisely, the competitor surfaces are defined as the images of homotopies connecting the given curves. We define below the set of admissible homotopies connecting a curve $\\gamma_i$ to $\\gamma_j$:\n\\[\n\\mathrm{Hom}(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Lip}([0,1] \\times \\mathbb{S}^{1}, \\overline{\\Ccal_0}) \\text{ such that } \\ell(0) = \\gamma_i \\text{ and } \\ell(1) = \\gamma_{j} \\}.\n\\]\nNote that we require more than mere homotopies: we specifically consider Lipschitz homotopies. For any homotopy $\\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j)$, we define the associated surface $\\surf_\\ell$ as its image:\n\\begin{equation}\\label{Def:surface_image}\n\t\\surf_{\\ell} := \\ell([0,1] \\times \\mathbb{S}^{1}) \\subset \\overline{\\Ccal_0}.\n\\end{equation}\nWe emphasize that the surface $\\surf_{\\ell}$ is $\\mathcal{H}^{2}$-rectifiable and has finite $\\mathcal{H}^{2}$-measure. Moreover, for any $u\\in H^1(\\mathbb{R}^3)$, the trace of $u$ on $\\surf_{\\ell}$ is well defined, since the set of points that are not Lebesgue points of $u$ is $\\mathcal{H}^2$-negligible. \nHowever, to obtain the existence result, we need to impose additional properties on the competitors.\nFor $\\Lambda>0$, we consider\n\\[\n\\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j) \\text{ such that } \\;\\mathrm{Lip}(\\ell)\\leq \\Lambda \\text{ and } \\surf_\\ell \\text{ is } \\Lambda-\\text{upper Ahlfors regular} \\ \\}.\\]\n\nLet us recall the definition of upper Ahlfors regularity.\n\\begin{definition}\nLet $E\\subset \\mathbb{R}^3$ and $\\Lambda>0$. The set $E$ is said to be $\\Lambda$-upper Ahlfors regular if for all $x\\in E$ and all $r>0$ we have \n\\[\\mathcal{H}^{2}(E\\cap B(x,r)) \\leqslant \\Lambda \\pi r^2.\\]\n\\end{definition}\n\nThe definition of our functional is based on the notion of a generalized geodesic distance between curves, associated with a given weight function $u$, which is defined as follows.\n\\begin{definition}\\label{Def:geoDistance}\n\tLet $(\\delta_\\varepsilon)_{\\varepsilon>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n\t\\begin{align*}\n\t\t\t&d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{ \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2} \\ |\\ \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n\t\\end{align*}\n\\end{definition}\nNote that, in \\cite{bonnivard2025phasefieldapproximationplateaus}, the geodesic distance is defined for Lipschitz homotopies $\\ell$. The assumptions of uniform Lipschitz regularity and uniform upper Ahlfors regularity of the images are not required there. However, in the present work, we need to impose these assumptions in order to obtain the compactness result necessary for the existence theorem.\n\nLet $\\mathcal{C}$ an smooth open bounded convex set containing the closure $\\overline{\\Ccal_0}$. This set $\\mathcal{C}$ will serve as our domain of study. Indeed, since we rely on PDE techniques, it is convenient to work within a smooth bounded domain. We now recall the definition of the functional introduced in \\cite{bonnivard2025phasefieldapproximationplateaus}:\n\t\\begin{equation}\n\t\t\\label{functionalGeneral}\n\t\tF_{\\varepsilon}(u) := \\varepsilon \\int_{\\mathcal{C}}|\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2}dx + \\frac{1}{c_{\\varepsilon}}\\sum_{i=1}^{n}d_{u}(\\gamma_0, \\gamma_{i}).\n\t\\end{equation}\nWhen, the sequences of positive numbers $(\\delta_\\varepsilon)$ and $(c_\\varepsilon)$ are assumed to converge to zero, and to satisfy that $\\delta_\\varepsilon/c_\\varepsilon$ converges to zero as $\\varepsilon \\to 0$, \\cite{bonnivard2025phasefieldapproximationplateaus} establishes that this functional approximate some Plateau's problem through a $\\Gamma$-convergence result. \n\nFor simplicity, we will assume in the following that the prescribed boundary consists of only two curves, $\\gamma_0$ and $\\gamma_1$. However, the results established in the present article remain valid when the boundary contains more than two curves.\n\nIn this paper, we are specifically interested in the decoupled functional defined below.\n\\begin{definition}\n\tFor $u \\in H^{1}({\\mathcal C})$ such that $0\\leqslant u \\leqslant 1$, with $u = 1$ on $\\partial{{\\mathcal C}}$, and $\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)$,\n\t\\begin{align*}\n\t\t&E_{\\varepsilon}(u,\\ell) := \\varepsilon\\int_{\\mathcal{C}} |\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2} dx + \\frac{1}{c_{\\varepsilon}}\\int_{\\surf_{\\ell}} (u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2}.\n\t\\end{align*}\n\\end{definition}\n\n\\begin{remark}\n\tFrom Definition~\\ref{Def:geoDistance} of the geodesic distance between closed curves, we know that for all $u\\in H^{1}(\\mathcal{C})$ such that $0\\leqslant u \\leqslant 1$ and $u = 1$ on $\\partial{\\mathcal{C}}$, \n\t\t\\[F_{\\varepsilon}(u) = \\inf\\left \\{ E_{\\varepsilon}(u,\\ell)\\ ,\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1) \\right \\}.\\]\n\\end{remark}\n\nThe main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\\varepsilon$\n\\begin{equation*}\n \\inf_{(u,\\ell)} E_\\varepsilon(u,\\ell).\n\\end{equation*}", + "context": "More precisely, the competitor surfaces are defined as the images of homotopies connecting the given curves. We define below the set of admissible homotopies connecting a curve $\\gamma_i$ to $\\gamma_j$:\n\\[\n\\mathrm{Hom}(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Lip}([0,1] \\times \\mathbb{S}^{1}, \\overline{\\Ccal_0}) \\text{ such that } \\ell(0) = \\gamma_i \\text{ and } \\ell(1) = \\gamma_{j} \\}.\n\\]\nNote that we require more than mere homotopies: we specifically consider Lipschitz homotopies. For any homotopy $\\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j)$, we define the associated surface $\\surf_\\ell$ as its image:\n\\begin{equation}\\label{Def:surface_image}\n \\surf_{\\ell} := \\ell([0,1] \\times \\mathbb{S}^{1}) \\subset \\overline{\\Ccal_0}.\n\\end{equation}\nWe emphasize that the surface $\\surf_{\\ell}$ is $\\mathcal{H}^{2}$-rectifiable and has finite $\\mathcal{H}^{2}$-measure. Moreover, for any $u\\in H^1(\\mathbb{R}^3)$, the trace of $u$ on $\\surf_{\\ell}$ is well defined, since the set of points that are not Lebesgue points of $u$ is $\\mathcal{H}^2$-negligible. \nHowever, to obtain the existence result, we need to impose additional properties on the competitors.\nFor $\\Lambda>0$, we consider\n\\[\n\\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j) \\text{ such that } \\;\\mathrm{Lip}(\\ell)\\leq \\Lambda \\text{ and } \\surf_\\ell \\text{ is } \\Lambda-\\text{upper Ahlfors regular} \\ \\}.\\]\n\nThe definition of our functional is based on the notion of a generalized geodesic distance between curves, associated with a given weight function $u$, which is defined as follows.\n\\begin{definition}\\label{Def:geoDistance}\n Let $(\\delta_\\varepsilon)_{\\varepsilon>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n \\begin{align*}\n &d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{ \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2} \\ |\\ \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n \\end{align*}\n\\end{definition}\nNote that, in \\cite{bonnivard2025phasefieldapproximationplateaus}, the geodesic distance is defined for Lipschitz homotopies $\\ell$. The assumptions of uniform Lipschitz regularity and uniform upper Ahlfors regularity of the images are not required there. However, in the present work, we need to impose these assumptions in order to obtain the compactness result necessary for the existence theorem.\n\nLet $\\mathcal{C}$ an smooth open bounded convex set containing the closure $\\overline{\\Ccal_0}$. This set $\\mathcal{C}$ will serve as our domain of study. Indeed, since we rely on PDE techniques, it is convenient to work within a smooth bounded domain. We now recall the definition of the functional introduced in \\cite{bonnivard2025phasefieldapproximationplateaus}:\n \\begin{equation}\n \\label{functionalGeneral}\n F_{\\varepsilon}(u) := \\varepsilon \\int_{\\mathcal{C}}|\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2}dx + \\frac{1}{c_{\\varepsilon}}\\sum_{i=1}^{n}d_{u}(\\gamma_0, \\gamma_{i}).\n \\end{equation}\nWhen, the sequences of positive numbers $(\\delta_\\varepsilon)$ and $(c_\\varepsilon)$ are assumed to converge to zero, and to satisfy that $\\delta_\\varepsilon/c_\\varepsilon$ converges to zero as $\\varepsilon \\to 0$, \\cite{bonnivard2025phasefieldapproximationplateaus} establishes that this functional approximate some Plateau's problem through a $\\Gamma$-convergence result.\n\nIn this paper, we are specifically interested in the decoupled functional defined below.\n\\begin{definition}\n For $u \\in H^{1}({\\mathcal C})$ such that $0\\leqslant u \\leqslant 1$, with $u = 1$ on $\\partial{{\\mathcal C}}$, and $\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)$,\n \\begin{align*}\n &E_{\\varepsilon}(u,\\ell) := \\varepsilon\\int_{\\mathcal{C}} |\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2} dx + \\frac{1}{c_{\\varepsilon}}\\int_{\\surf_{\\ell}} (u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2}.\n \\end{align*}\n\\end{definition}\n\n\\begin{remark}\n From Definition~\\ref{Def:geoDistance} of the geodesic distance between closed curves, we know that for all $u\\in H^{1}(\\mathcal{C})$ such that $0\\leqslant u \\leqslant 1$ and $u = 1$ on $\\partial{\\mathcal{C}}$, \n \\[F_{\\varepsilon}(u) = \\inf\\left \\{ E_{\\varepsilon}(u,\\ell)\\ ,\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1) \\right \\}.\\]\n\\end{remark}\n\nThe main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\\varepsilon$\n\\begin{equation*}\n \\inf_{(u,\\ell)} E_\\varepsilon(u,\\ell).\n\\end{equation*}\n\n\\begin{definition}\\label{Def:geoDistance}\n\tLet $(\\delta_\\eps)_{\\eps>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n\t\\begin{align*}\n\t\t\t&d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{ \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\eps) d\\Haus \\ |\\ \\ell \\in \\curve^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n\t\\end{align*}\n\\end{definition}", + "full_context": "More precisely, the competitor surfaces are defined as the images of homotopies connecting the given curves. We define below the set of admissible homotopies connecting a curve $\\gamma_i$ to $\\gamma_j$:\n\\[\n\\mathrm{Hom}(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Lip}([0,1] \\times \\mathbb{S}^{1}, \\overline{\\Ccal_0}) \\text{ such that } \\ell(0) = \\gamma_i \\text{ and } \\ell(1) = \\gamma_{j} \\}.\n\\]\nNote that we require more than mere homotopies: we specifically consider Lipschitz homotopies. For any homotopy $\\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j)$, we define the associated surface $\\surf_\\ell$ as its image:\n\\begin{equation}\\label{Def:surface_image}\n \\surf_{\\ell} := \\ell([0,1] \\times \\mathbb{S}^{1}) \\subset \\overline{\\Ccal_0}.\n\\end{equation}\nWe emphasize that the surface $\\surf_{\\ell}$ is $\\mathcal{H}^{2}$-rectifiable and has finite $\\mathcal{H}^{2}$-measure. Moreover, for any $u\\in H^1(\\mathbb{R}^3)$, the trace of $u$ on $\\surf_{\\ell}$ is well defined, since the set of points that are not Lebesgue points of $u$ is $\\mathcal{H}^2$-negligible. \nHowever, to obtain the existence result, we need to impose additional properties on the competitors.\nFor $\\Lambda>0$, we consider\n\\[\n\\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j) \\text{ such that } \\;\\mathrm{Lip}(\\ell)\\leq \\Lambda \\text{ and } \\surf_\\ell \\text{ is } \\Lambda-\\text{upper Ahlfors regular} \\ \\}.\\]\n\nThe definition of our functional is based on the notion of a generalized geodesic distance between curves, associated with a given weight function $u$, which is defined as follows.\n\\begin{definition}\\label{Def:geoDistance}\n Let $(\\delta_\\varepsilon)_{\\varepsilon>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n \\begin{align*}\n &d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{ \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2} \\ |\\ \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n \\end{align*}\n\\end{definition}\nNote that, in \\cite{bonnivard2025phasefieldapproximationplateaus}, the geodesic distance is defined for Lipschitz homotopies $\\ell$. The assumptions of uniform Lipschitz regularity and uniform upper Ahlfors regularity of the images are not required there. However, in the present work, we need to impose these assumptions in order to obtain the compactness result necessary for the existence theorem.\n\nLet $\\mathcal{C}$ an smooth open bounded convex set containing the closure $\\overline{\\Ccal_0}$. This set $\\mathcal{C}$ will serve as our domain of study. Indeed, since we rely on PDE techniques, it is convenient to work within a smooth bounded domain. We now recall the definition of the functional introduced in \\cite{bonnivard2025phasefieldapproximationplateaus}:\n \\begin{equation}\n \\label{functionalGeneral}\n F_{\\varepsilon}(u) := \\varepsilon \\int_{\\mathcal{C}}|\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2}dx + \\frac{1}{c_{\\varepsilon}}\\sum_{i=1}^{n}d_{u}(\\gamma_0, \\gamma_{i}).\n \\end{equation}\nWhen, the sequences of positive numbers $(\\delta_\\varepsilon)$ and $(c_\\varepsilon)$ are assumed to converge to zero, and to satisfy that $\\delta_\\varepsilon/c_\\varepsilon$ converges to zero as $\\varepsilon \\to 0$, \\cite{bonnivard2025phasefieldapproximationplateaus} establishes that this functional approximate some Plateau's problem through a $\\Gamma$-convergence result.\n\nIn this paper, we are specifically interested in the decoupled functional defined below.\n\\begin{definition}\n For $u \\in H^{1}({\\mathcal C})$ such that $0\\leqslant u \\leqslant 1$, with $u = 1$ on $\\partial{{\\mathcal C}}$, and $\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)$,\n \\begin{align*}\n &E_{\\varepsilon}(u,\\ell) := \\varepsilon\\int_{\\mathcal{C}} |\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2} dx + \\frac{1}{c_{\\varepsilon}}\\int_{\\surf_{\\ell}} (u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2}.\n \\end{align*}\n\\end{definition}\n\n\\begin{remark}\n From Definition~\\ref{Def:geoDistance} of the geodesic distance between closed curves, we know that for all $u\\in H^{1}(\\mathcal{C})$ such that $0\\leqslant u \\leqslant 1$ and $u = 1$ on $\\partial{\\mathcal{C}}$, \n \\[F_{\\varepsilon}(u) = \\inf\\left \\{ E_{\\varepsilon}(u,\\ell)\\ ,\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1) \\right \\}.\\]\n\\end{remark}\n\nThe main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\\varepsilon$\n\\begin{equation*}\n \\inf_{(u,\\ell)} E_\\varepsilon(u,\\ell).\n\\end{equation*}\n\n\\begin{definition}\\label{Def:geoDistance}\n\tLet $(\\delta_\\eps)_{\\eps>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n\t\\begin{align*}\n\t\t\t&d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{ \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\eps) d\\Haus \\ |\\ \\ell \\in \\curve^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n\t\\end{align*}\n\\end{definition}\n\nThe main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\\varepsilon$\n\\begin{equation*}\n \\inf_{(u,\\ell)} E_\\varepsilon(u,\\ell).\n\\end{equation*}\n\nThere exists a minimizing couple $(u,\\ell)$ of the following problem in both variables\n\\begin{equation}\n \\label{pb min}\n \\min\\left \\{E_\\eps(u,\\ell), \\, u \\in H^{1}(\\C) \\text{ such that } 0\\leqslant u \\leqslant 1 , \\, u = 1 \\text{ on } \\partial{\\C} \\text{ and } \\ell \\in \\curve^\\Lambda(\\gamma_0,\\gamma_1) \\right\\}.\n\\end{equation}\n\nThe study of this energy is motivated by the $\\Gamma$-convergence result established in \\cite{bonnivard2025phasefieldapproximationplateaus}, where this decoupled functional was introduced both in the proof of the $\\Gamma$-convergence and for numerical applications. Recall that the phase-field approach they proposed is a generalization to Plateau’s problem of the method introduced in \\cite{bonnivard2015approximation} for Steiner’s problem. A similar study of the existence of solutions for the decoupled functional in the Steiner case was carried out in \\cite{bonnivard2018phase}. The present work is inspired by this paper, but there are some differences that we explain below.\n\nWe begin by studying the following minimization problem \n\\begin{equation}\n\\label{pb min l fixe}\n \\inf_{u} E_\\varepsilon(u,\\ell), \\text{ where } \\ell \\in \\curve^\\Lambda(\\gamma_0,\\gamma_1).\n\\end{equation}\nThe following Proposition will be proven in Section~\\ref{section l fixe}. \n\\begin{proposition}\nFor any given $\\ell \\in \\curve^\\Lambda(\\gamma_0, \\gamma_1)$, there exists a unique minimizer $u \\in H^1(\\Ccal)$ of $ E_\\varepsilon(\\cdot,l)$ which is globally $C^{0,\\alpha}$, for all $0< \\alpha <1$: there exists a constant $C_\\alpha>0$ such that \n\\[\\|u \\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]\n\\end{proposition}\nThe assumption that the surface $S_\\ell$ is $\\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity, and hence the existence of minimizers in this setting. We then apply classical results from elliptic regularity to derive Hölder estimates for the minimizer. Note that these estimates depend explicitly on $\\Lambda$ and $\\varepsilon$, but are independent of the specific surface $S_\\ell$. This part is strongly inspired by \\cite{bonnivard2018phase}.\n\n\\begin{proposition}\n\\label{prop holder}\nLet $\\ell \\in~\\curve^\\Lambda(\\gamma_0,\\gamma_1)$ and let $u$ be the minimizer of $E_\\varepsilon^\\ell$. Then, $u$ is $\\alpha$-Hölder continuous for all $0<\\alpha<1$ with the estimate:\n\\[\\|u\\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]\n\\end{proposition}\n\n\\begin{lemma}\n\\label{lem 2.10}\nRecall the definition of the parameter $\\eta_0 = \\dist(\\partial \\Ccal_0,\\partial \\Ccal)$ introduced in the previous Lemma.\nLet $\\ell \\in \\curve^\\Lambda(\\gamma_0, \\gamma_1)$, $0< \\rho < \\min(1,\\frac{\\eta_0}{4})$ and $x_0 \\in \\mathcal{C}$ such that $\\dist(x_0,\\partial \\mathcal{C}) \\geqslant \\frac{\\eta_0}{2} $. Then, $u \\in W^{1,p}(B(x_0,\\rho))$ for all $3\\leqslant p < \\infty$ and we have the following estimate on the rescaled function $u_\\rho (x) := u(x_0+\\rho x)$\n\\begin{equation}\n \\|\\nabla u_\\rho\\|_{L^p(B_1)} \\leqslant C_p\\left( \\frac{\\rho^2}{\\varepsilon^2}+1+\\frac{\\Lambda \\rho}{c_\\varepsilon \\varepsilon}\\right).\n\\end{equation}\n\\end{lemma}\n\n\\medskip\n\\emph{Step 3.} We define the rescaled function $u_\\rho$ on $B_2$ by $u_\\rho(x) := u(x_0+\\rho x)$. Then, for all $\\varphi \\in C^\\infty_c(B_2)$ \n\\begin{align*}\n \\int_{B_2}\\nabla u_\\rho \\cdot \\nabla\\varphi \\dm x &= \\frac{1}{\\rho}\\int_{B(x_0,2\\rho)}\\nabla u \\cdot \\nabla\\varphi_\\rho \\dm x,\\\\\n &= \\frac{1}{\\rho4\\varepsilon^2}\\int_{B(x_0,2\\rho)}(1- u)\\varphi_\\rho \\dm x - \\frac{1}{\\rho c_\\varepsilon \\varepsilon}B[\\ell](u,\\varphi_\\rho),\\\\\n &= \\frac{\\rho^2}{4\\varepsilon^2}\\int_{B_2}(1- u_\\rho)\\varphi \\dm x - \\frac{1}{\\rho c_\\varepsilon \\varepsilon}\\langle T_\\rho,\\varphi \\rangle.\n\\end{align*}\nHence, $u_\\rho$ satisfies the following PDE in $\\mathcal{D}'(B_2)$\n\\[-\\Delta u_\\rho = \\frac{\\rho^2}{4\\varepsilon^2}(1- u_\\rho) - \\frac{1}{\\rho c_\\varepsilon \\varepsilon} T_\\rho.\\]\nWe denote $w_\\rho := u_\\rho + \\frac{1}{\\rho c_\\varepsilon \\varepsilon}v_\\rho \\in H^1(B_2)\\cap L^\\infty(B_2)$. And we deduce that \n\\[-\\Delta w_\\rho = \\frac{\\rho^2}{4\\varepsilon^2}(1- u_\\rho) \\text{ in } \\mathcal{D}'(B_2). \\]\nThus, \\cite[Corollary 8.36]{gilbargelliptic} yields that $w_\\rho \\in C_{\\mathrm{loc}}^{1,\\alpha}(B_2)$ for all $\\alpha >0$ and \n\\begin{align*}\n \\|\\nabla w_\\rho\\|_{L^\\infty(B_1)} \\leqslant \\|w_\\rho\\|_{C^{1,\\alpha}(B_1)} &\\leqslant C\\left(\\| w_\\rho\\|_{L^\\infty(B_1)} +\\frac{\\rho^2}{4\\varepsilon^2}\\|1- u_\\rho\\|_{L^\\infty(B_1)}\\right),\\\\\n &= C\\left(\\| u_\\rho\\|_{L^\\infty(B_1)}+ \\frac{1}{\\rho c_\\varepsilon \\varepsilon}\\| v_\\rho\\|_{L^\\infty(B_1)} +\\frac{\\rho^2}{4\\varepsilon^2}\\right),\\\\\n &\\leqslant C\\left(1 + \\frac{\\Lambda \\rho}{c_\\varepsilon \\varepsilon} + \\frac{\\rho^2}{4\\varepsilon^2}\\right).\n\\end{align*}\nFinally, we can conclude that $u_\\rho = w_\\rho - \\frac{1}{\\rho c_\\varepsilon \\varepsilon}v_\\rho \\in W^{1,p}(B_1)$ with the desired estimate\n\\[ \\|\\nabla u_\\rho\\|_{L^p(B_1)} \\leqslant C_p\\left( \\frac{\\rho^2}{\\varepsilon^2}+1+\\frac{\\Lambda \\rho}{c_\\varepsilon \\varepsilon}\\right).\\]\nThis concludes the proof of this Lemma.\n\\end{proof}\n\nNow let's consider the case where $\\dist(x_0, \\partial \\mathcal{C}) > \\frac{\\eta_0}{2}> \\varepsilon$, in particular this implies that $B(x_0,\\varepsilon) \\subset \\C$. In this case, Lemma~\\ref{lem 2.10}, applied with $\\rho = \\varepsilon$ and $p = \\frac{3}{1-\\alpha}$ yields the estimate on the rescaled function $u_\\varepsilon$\n \\[\\|\\nabla u_\\varepsilon\\|_{L^p(B_1)} \\leqslant C_p\\left(1+ \\frac{\\Lambda}{c_\\varepsilon}\\right).\\]\n Since we have shown that the $L^\\infty$ norm of $u$ is smaller than $1$, we deduce that the $L^p$ norm of the scaled function $u_\\varepsilon$ is bounded by the volume of the domain. And we conclude by applying the Sobolev embedding Theorem (see for instance \\cite[Theorem 4.12]{adams2003sobolev}),\n \\begin{equation}\n \\|u_\\varepsilon\\|_{C^{0,\\alpha}(B_1)} \\leqslant C_\\alpha\\left(1+ \\frac{\\Lambda}{c_\\varepsilon}\\right).\n \\end{equation}\n Hence, scaling back leads to \n \\begin{equation}\n \\frac{|u(x)-u(y)|}{|x-y|^\\alpha} \\leqslant \\frac{1}{\\varepsilon^\\alpha} \\|u_\\varepsilon\\|_{C^{0,\\alpha}(B_1)} \\leqslant \\frac{C_\\alpha}{\\varepsilon^\\alpha}\\left(1+ \\frac{\\Lambda}{c_\\varepsilon}\\right).\n \\end{equation}\n\\end{itemize}\nFinally, since $\\|u\\|_{L^\\infty(\\Ccal)} \\leqslant 1$, we get the desired estimate on the Hölder norm of $u$ \n\\[ \\|u\\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\epsilon^{\\alpha}}.\\]\nThis achieves the proof of the Hölder regularity of the solution $u$ of $\\min E_\\eps(u,\\ell)$, when $\\ell$ is fixed in $\\curve^\\Lambda$.\n\\end{proof}", + "post_theorem_intro_text_len": 5460, + "post_theorem_intro_text": "The study of this energy is motivated by the $\\Gamma$-convergence result established in \\cite{bonnivard2025phasefieldapproximationplateaus}, where this decoupled functional was introduced both in the proof of the $\\Gamma$-convergence and for numerical applications. Recall that the phase-field approach they proposed is a generalization to Plateau’s problem of the method introduced in \\cite{bonnivard2015approximation} for Steiner’s problem. A similar study of the existence of solutions for the decoupled functional in the Steiner case was carried out in \\cite{bonnivard2018phase}. The present work is inspired by this paper, but there are some differences that we explain below.\n\nTo prove Theorem~\\ref{main th}, we first consider the optimization problems with either $u$ or $\\ell$ fixed. As usual, establishing the existence of minimizers requires both a compactness result and the lower semi-continuity of the functional. The compactness follows from the choice of the class of competitors, so the main issue is to establish lower semi-continuity.\n\nWe begin by studying the following minimization problem \n\\begin{equation}\n\\label{pb min l fixe}\n \\inf_{u} E_\\varepsilon(u,\\ell), \\text{ where } \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1).\n\\end{equation}\nThe following Proposition will be proven in Section~\\ref{section l fixe}. \n\\begin{proposition}\nFor any given $\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0, \\gamma_1)$, there exists a unique minimizer $u \\in H^1(\\mathcal{C})$ of $ E_\\varepsilon(\\cdot,l)$ which is globally $C^{0,\\alpha}$, for all $0< \\alpha <1$: there exists a constant $C_\\alpha>0$ such that \n\\[\\|u \\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]\n\\end{proposition}\nThe assumption that the surface $S_\\ell$ is $\\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity, and hence the existence of minimizers in this setting. We then apply classical results from elliptic regularity to derive Hölder estimates for the minimizer. Note that these estimates depend explicitly on $\\Lambda$ and $\\varepsilon$, but are independent of the specific surface $S_\\ell$. This part is strongly inspired by \\cite{bonnivard2018phase}.\n\nThen, in Section~\\ref{section u fixe} we consider the minimization problem\n\\begin{equation}\n\\label{pb min u fixe}\n \\inf_{\\ell \\in \\mathrm{Hom}^\\Lambda} E_\\varepsilon(u,\\ell), \\text{ where } u \\in H^1(\\mathcal{C})\\cap C(\\overline{\\mathcal{C}}),\n\\end{equation}\nand prove the following Proposition.\n\\begin{proposition}\n\\label{prop 1.2}\nFor all $u \\in H^1({\\mathcal C})\\cap C(\\overline{\\mathcal{C}})$ there exists a minimizer for the problem \\eqref{pb min u fixe}. \n\\end{proposition}\n\nAs previously mentionned, the main difficulty in the proof of Proposition~\\ref{prop 1.2} lies in establishing the lower semicontinuity of the functional, which relies on the following Lemma.\n\n \\begin{lemma}\n \\label{golab}\nLet $\\ell_n : [0,1] \\times \\mathbb{S}^1 \\to \\mathbb{R}^3$ be a sequence of uniformly Lipschitz functions, which converges uniformly to a Lipschitz function $\\ell : [0,1] \\times \\mathbb{S}^1 \\to \\mathbb{R}^3$.\nThen, for all open set $A$\n\n$$\\liminf \\mathcal{H}^{2} \\mathbin{\\vrule height 1.6ex depth 0pt width 0.13ex\\vrule height 0.13ex depth 0pt width 1.3ex} S_{\\ell_n}(A) \\geqslant \\mathcal{H}^{2} \\mathbin{\\vrule height 1.6ex depth 0pt width 0.13ex\\vrule height 0.13ex depth 0pt width 1.3ex} S_\\ell(A),$$\nwhere $S_\\ell$ (respectively $S_{\\ell_n}$) denotes the image of the Lipschitz function $\\ell$ (respectively $\\ell_n)$.\n\\end{lemma}\n\nThis Lemma, proven in Section~\\ref{section u fixe}, constitutes a central and original contribution of the present article and relies crucially on the specific definition of the class of surfaces under consideration. It can be viewed as a generalization of Go\\l{}\\k{a}b’s theorem to the two-dimensional Hausdorff measure, and its proof follows a similar strategy. In Go\\l{}\\k{a}b’s theorem, the first step consists in establishing the rectifiability of the limit set; in our setting, this property is immediate, since the surfaces are assumed to be images of Lipschitz functions. We then study the weak limit of the measures $(\\mathcal{H}^{2} \\mathbin{\\vrule height 1.6ex depth 0pt width 0.13ex\\vrule height 0.13ex depth 0pt width 1.3ex} S_{\\ell_n})$ and, using the tangent plane of the limit surface, derive a lower bound on the upper density of the limit measure to conclude. The key point of our proof relies on a topological argument, stating that a Lipschitz image, which is uniformly close to a disk, must have at least almost the density of a disk (see the proof of Lemma~\\ref{golab}).\n\nWe emphasize that the proof of this Lemma does not rely on the specific definition set $\\mathbb{S}^1 \\times [0,1]$; it only assumes it to be a two-dimensional subset of $\\mathbb{R}^3$. Moreover, the result is also expected to remain valid in dimensions higher than two, although the proof would then require more delicate topological arguments.\n\nFinally, in Section~\\ref{section rien fixe}, Theorem~\\ref{main th} is proved by combining these results, in particular the Hölder regularity and the preceding Lemma.\n\n\\textbf{Acknowledgments.}\n I would like to thank Antoine Lemenant for suggesting this problem to me as well as Matthieu Bonnivard for their help in this project. \nThis work was partially supported by the IUF grant of Antoine Lemenant and by the ANR project STOIQUES.", + "sketch": "To prove Theorem~\\ref{main th}, the authors “first consider the optimization problems with either $u$ or $\\ell$ fixed.” Existence requires “both a compactness result and the lower semi-continuity of the functional”; “the compactness follows from the choice of the class of competitors,” so “the main issue is to establish lower semi-continuity.”\n\n1) **Fix $\\ell$**: study \\eqref{pb min l fixe}. For any $\\ell\\in\\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)$ there is a “unique minimizer $u\\in H^1(\\mathcal C)$” which is “globally $C^{0,\\alpha}$,” with the stated estimate. Here, “the assumption that the surface $S_\\ell$ is $\\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity,” and then “classical results from elliptic regularity” give Hölder estimates (depending on $\\Lambda$ and $\\varepsilon$ but “independent of the specific surface $S_\\ell$”).\n\n2) **Fix $u$**: study \\eqref{pb min u fixe}. For $u\\in H^1(\\mathcal C)\\cap C(\\overline{\\mathcal C})$ there exists a minimizer in $\\ell$. The “main difficulty” is again lower semicontinuity, which “relies on” Lemma~\\ref{golab}, a Go\\l{}\\k{a}b-type lower semicontinuity statement for $\\mathcal H^2\\llcorner S_{\\ell_n}$ under uniform convergence of uniformly Lipschitz parametrizations. Its proof “follows a similar strategy” to Go\\l{}\\k{a}b: rectifiability is “immediate,” then one studies “the weak limit of the measures,” and “using the tangent plane of the limit surface, derive[s] a lower bound on the upper density of the limit measure to conclude”; the “key point” is “a topological argument” that a Lipschitz image uniformly close to a disk has “at least almost the density of a disk.”\n\n3) **Conclusion**: “Finally, in Section~\\ref{section rien fixe}, Theorem~\\ref{main th} is proved by combining these results, in particular the Hölder regularity and the preceding Lemma.”", + "expanded_sketch": "To prove the main theorem, the authors “first consider the optimization problems with either $u$ or $\\ell$ fixed.” Existence requires “both a compactness result and the lower semi-continuity of the functional”; “the compactness follows from the choice of the class of competitors,” so “the main issue is to establish lower semi-continuity.”\n\n1) **Fix $\\ell$**: study \\eqref{pb min l fixe}. For any $\\ell\\in\\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)$ there is a “unique minimizer $u\\in H^1(\\mathcal C)$” which is “globally $C^{0,\\alpha}$,” with the stated estimate. Here, “the assumption that the surface $S_\\ell$ is $\\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity,” and then “classical results from elliptic regularity” give Hölder estimates (depending on $\\Lambda$ and $\\varepsilon$ but “independent of the specific surface $S_\\ell$”).\n\n2) **Fix $u$**: study\n\\begin{equation}\n\\label{pb min u fixe}\n \\inf_{\\ell \\in \\curve^\\Lambda} E_\\varepsilon(u,\\ell), \\text{ where } u \\in H^1(\\Ccal)\\cap C(\\overline{\\Ccal}),\n\\end{equation}\nFor $u\\in H^1(\\mathcal C)\\cap C(\\overline{\\mathcal C})$ there exists a minimizer in $\\ell$. The “main difficulty” is again lower semicontinuity, which “relies on” the following lemma.\n\\begin{lemma}\n \\label{golab}\nLet $\\ell_n : [0,1] \\times \\mathbb{S}^1 \\to \\R^3$ be a sequence of uniformly Lipschitz functions, which converges uniformly to a Lipschitz function $\\ell : [0,1] \\times \\mathbb{S}^1 \\to \\R^3$.\nThen, for all open set $A$\n\n$$\\liminf \\Haus \\mres S_{\\ell_n}(A) \\geqslant \\Haus \\mres S_\\ell(A),$$\nwhere $S_\\ell$ (respectively $S_{\\ell_n}$) denotes the image of the Lipschitz function $\\ell$ (respectively $\\ell_n)$.\n\\end{lemma}\nIts proof “follows a similar strategy” to Go\\l{}\\k{a}b: rectifiability is “immediate,” then one studies “the weak limit of the measures,” and “using the tangent plane of the limit surface, derive[s] a lower bound on the upper density of the limit measure to conclude”; the “key point” is “a topological argument” that a Lipschitz image uniformly close to a disk has “at least almost the density of a disk.”\n\n3) **Conclusion**: “Finally, next we conclude the proof of the main theorem by combining these results, in particular the Hölder regularity and the preceding lemma.”", + "expanded_theorem": "\\label{main th}\n\nThere exists a minimizing couple $(u,\\ell)$ of the following problem in both variables\n\\begin{equation}\n \\label{pb min}\n \\min\\left \\{E_\\varepsilon(u,\\ell), \\, u \\in H^{1}({\\mathcal C}) \\text{ such that } 0\\leqslant u \\leqslant 1 , \\, u = 1 \\text{ on } \\partial{{\\mathcal C}} \\text{ and } \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1) \\right\\}.\n\\end{equation}\n\nMoreover, the minimizer $u$ is $\\alpha$-Hölder continuous for all $0<\\alpha<1$ with the estimate:\n\\[\\|u\\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]", + "theorem_type": [ + "Existence", + "Universal" + ], + "mcq": { + "question": "Fix \\(\\varepsilon>0\\), \\(\\Lambda>0\\), positive numbers \\(c_{\\varepsilon}\\) and \\(\\delta_{\\varepsilon}\\), a smooth bounded convex domain \\(\\mathcal C\\), a set \\(\\mathcal C_0\\) with \\(\\overline{\\mathcal C_0}\\subset \\mathcal C\\), and closed curves \\(\\gamma_0,\\gamma_1\\). For a Lipschitz homotopy \\(\\ell:[0,1]\\times \\mathbb S^1\\to \\overline{\\mathcal C_0}\\) with \\(\\ell(0)=\\gamma_0\\) and \\(\\ell(1)=\\gamma_1\\), let \\(\\Sigma_{\\ell}=\\ell([0,1]\\times \\mathbb S^1)\\). Define \\(\\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\) as the set of such \\(\\ell\\) satisfying \\(\\mathrm{Lip}(\\ell)\\le \\Lambda\\) and such that \\(\\Sigma_{\\ell}\\) is \\(\\Lambda\\)-upper Ahlfors regular (i.e. \\(\\mathcal H^2(\\Sigma_{\\ell}\\cap B(x,r))\\le \\Lambda r^2\\) for all balls). For admissible \\(u\\in H^1(\\mathcal C)\\) with \\(0\\le u\\le 1\\) and \\(u=1\\) on \\(\\partial\\mathcal C\\), define\n\\[\nE_{\\varepsilon}(u,\\ell)=\\varepsilon\\int_{\\mathcal C}|\\nabla u|^2\\,dx+\\frac{1}{4\\varepsilon}\\int_{\\mathcal C}(1-u)^2\\,dx+\\frac{1}{c_{\\varepsilon}}\\int_{\\Sigma_{\\ell}}(u^2+\\delta_{\\varepsilon})\\,d\\mathcal H^2.\n\\]\nWhich statement holds for the minimization problem over all admissible pairs \\((u,\\ell)\\)?", + "correct_choice": { + "label": "A", + "text": "There exists at least one admissible pair \\((u,\\ell)\\) that minimizes \\(E_{\\varepsilon}(u,\\ell)\\) jointly among all pairs with \\(u\\in H^1(\\mathcal C)\\), \\(0\\le u\\le 1\\), \\(u=1\\) on \\(\\partial\\mathcal C\\), and \\(\\ell\\in \\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\). Moreover, for such a minimizing pair, the function \\(u\\) belongs to \\(C^{0,\\alpha}(\\mathcal C)\\) for every \\(0<\\alpha<1\\), and it satisfies\n\\[\n\\|u\\|_{C^{0,\\alpha}(\\mathcal C)}\\le C_{\\alpha}\\,\\frac{1+\\Lambda c_{\\varepsilon}^{-1}}{\\varepsilon^{\\alpha}}.\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "There exists at least one admissible pair \\((u,\\ell)\\) that minimizes \\(E_{\\varepsilon}(u,\\ell)\\) jointly among all pairs with \\(u\\in H^1(\\mathcal C)\\), \\(0\\le u\\le 1\\), \\(u=1\\) on \\(\\partial\\mathcal C\\), and \\(\\ell\\in \\mathrm{Hom}(\\gamma_0,\\gamma_1)\\) (that is, without imposing the uniform Lipschitz bound or the \\(\\Lambda\\)-upper Ahlfors regularity on \\(\\Sigma_\\ell\\)). Moreover, for such a minimizing pair, the function \\(u\\) belongs to \\(C^{0,\\alpha}(\\mathcal C)\\) for every \\(0<\\alpha<1\\), and it satisfies\n\\[\n\\|u\\|_{C^{0,\\alpha}(\\mathcal C)}\\le C_{\\alpha}\\,\\frac{1+\\Lambda c_{\\varepsilon}^{-1}}{\\varepsilon^{\\alpha}}.\n\\]" + }, + { + "label": "C", + "text": "There exists at least one admissible pair \\((u,\\ell)\\) that minimizes \\(E_{\\varepsilon}(u,\\ell)\\) jointly among all pairs with \\(u\\in H^1(\\mathcal C)\\), \\(0\\le u\\le 1\\), \\(u=1\\) on \\(\\partial\\mathcal C\\), and \\(\\ell\\in \\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\)." + }, + { + "label": "D", + "text": "For every admissible \\(u\\in H^1(\\mathcal C)\\) with \\(0\\le u\\le 1\\) and \\(u=1\\) on \\(\\partial\\mathcal C\\), there exists \\(\\ell\\in \\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\) minimizing \\(E_{\\varepsilon}(u,\\ell)\\) over \\(\\ell\\); and for every admissible \\(\\ell\\in \\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\), there exists a minimizer \\(u\\) of \\(E_{\\varepsilon}(u,\\ell)\\) over \\(u\\). Consequently, there exists a pair \\((u,\\ell)\\) such that \\(u\\) minimizes \\(E_{\\varepsilon}(\\,b7\\,,\\ell)\\) and \\(\\ell\\) minimizes \\(E_{\\varepsilon}(u,\\,\\u00b7\\,)\\), and this pair minimizes \\(E_{\\varepsilon}\\) jointly over all admissible pairs. Moreover, for such a pair,\n\\[\n\\|u\\|_{C^{0,\\alpha}(\\mathcal C)}\\le C_{\\alpha}\\,\\frac{1+\\Lambda c_{\\varepsilon}^{-1}}{\\varepsilon^{\\alpha}}\n\\]\nfor every \\(0<\\alpha<1\\)." + }, + { + "label": "E", + "text": "There exists at least one admissible pair \\((u,\\ell)\\) that minimizes \\(E_{\\varepsilon}(u,\\ell)\\) jointly among all pairs with \\(u\\in H^1(\\mathcal C)\\), \\(0\\le u\\le 1\\), \\(u=1\\) on \\(\\partial\\mathcal C\\), and \\(\\ell\\in \\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\). Moreover, for such a minimizing pair, the function \\(u\\) belongs to \\(C^{0,\\alpha}(\\mathcal C)\\) for every \\(0<\\alpha<1\\), and it satisfies\n\\[\n\\|u\\|_{C^{0,\\alpha}(\\mathcal C)}\\le C_{\\alpha}\\,\\frac{1+\\Lambda c_{\\varepsilon}^{-1}}{\\varepsilon}.\n\\]" + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "D" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "regularity", + "tampered_component": "necessity_of_uniform_Lipschitz_and_Ahlfors_class_for_compactness_lsc", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "regularity", + "tampered_component": "dropped_Holder_regularity_conclusion_and_estimate", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "from_separate_minimizers_to_joint_minimizer", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "epsilon_scaling_in_Holder_estimate", + "template_used": "boundary_range" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives only the setup and definitions of the admissible class and energy. It does not explicitly state the existence, Hölder regularity, or the precise estimate that appears in the correct choice." + }, + "TAS": { + "score": 1, + "justification": "The item is close to theorem recognition: the correct option essentially reproduces the theorem statement for the given setup. However, it is not a pure tautology because the alternatives modify key hypotheses and conclusions in nontrivial ways." + }, + "GPS": { + "score": 1, + "justification": "Selecting the correct answer requires some reasoning about compactness assumptions, strength of conclusions, and the scaling in the Hölder estimate. Still, the task is mainly identifying the exact theorem statement rather than deriving a conclusion from first principles." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong and mathematically meaningful: one drops needed compactness/regularity assumptions, one gives a weaker true statement, one confuses separate minimization with joint minimization, and one alters the epsilon scaling. These are plausible and distinct failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with no answer leakage and high-quality distractors, but it leans toward theorem-statement recognition rather than deep generative reasoning." + } + }, + { + "id": "2602.08644v1", + "paper_link": "http://arxiv.org/abs/2602.08644v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm:main}\n Let $(\\M_1,g_1)$ and $(\\M_2,g_2)$ be two n-dimensional smooth connected Riemannian manifolds with boundary. Let $\\omega_1\\subset \\M_1$ and $\\omega_2\\subset\\M_2$ be open and connected. For $j=1,2$, let $\\mathcal N_j$ be the local source-to-final value map\n \\eqref{eq:def_N} with $(M,g)=(\\M_j,g_j)$\n and $\\omega=\\omega_j$.\n In addition, suppose that there is a diffeomorphism $\\bm{\\Phi}:\\omega_1\\to \\omega_2$ such that \n \\begin{equation}\\label{NN}\n \\cN_1\\bm{\\Phi}^\\ast=\\bm{\\Phi}^\\ast \\cN_2.\n \\end{equation}\n Then $(\\M_1,g_1)$ and $(\\M_2,g_2)$ are isometric.", + "start_pos": 9075, + "end_pos": 9718, + "label": "thm:main" + }, + "ref_dict": { + "eq:problem_hypo": "\\begin{equation}\n \\label{eq:problem_hypo}\n \\begin{cases}\n (\\p_t^2-\\Delta_H)\\bu + d p=\\bf,\\ \\text{in } \\RR^+\\times \\M,\\\\\n d^* \\bu=0,\\ \\text{in } \\RR^+\\times \\M,\\\\\n \\bu|_{[0,T]\\times\\p\\M}=0,\\\\\n \\bu=0,\\ p=0,\\ \\text{in }\\RR^-\\times\\M.\n \\end{cases}\n \\end{equation}", + "thm:main": "\\begin{theorem}\n \\label{thm:main}\n Let $(\\M_1,g_1)$ and $(\\M_2,g_2)$ be two n-dimensional smooth connected Riemannian manifolds with boundary. Let $\\omega_1\\subset \\M_1$ and $\\omega_2\\subset\\M_2$ be open and connected. For $j=1,2$, let $\\mathcal N_j$ be the local source-to-final value map\n \\eqref{eq:def_N} with $(\\M,g)=(\\M_j,g_j)$\n and $\\omega=\\omega_j$.\n In addition, suppose that there is a diffeomorphism $\\bPhi:\\omega_1\\to \\omega_2$ such that \n \\begin{equation}\\label{NN}\n \\cN_1\\bPhi^\\ast=\\bPhi^\\ast \\cN_2.\n \\end{equation}\n Then $(\\M_1,g_1)$ and $(\\M_2,g_2)$ are isometric.\n \\end{theorem}", + "eq:NS1": "\\begin{equation}\n \\label{eq:NS1}\n \\begin{cases}\n \\p_t\\bu-\\Delta_H\\bu+R(\\bu) + d p=\\bf,\\ \\text{in } [0,T]\\times \\M,\\\\\n d^* \\bu=0,\\ \\text{in } [0,T]\\times \\M,\\\\\n \\bu|_{[0,T]\\times\\p\\M}=0,\\\\\n \\bu=0,\\ p=0,\\ \\text{in }\\{0\\}\\times\\M.\n \\end{cases}\n \\end{equation}", + "eq:def_N": "\\begin{align}\n\\label{eq:def_N}\n \\mathcal N:\\mathbb B_\\delta&\\to L^2\\Omega^1(\\omega),\\\\\n \\mathcal N(\\bf)&=\\bu_\\bf(T,\\cdot)|_\\omega,\n \\end{align}", + "eq:def_P": "\\begin{align}\n \\label{eq:def_P}\n \\P_i:C_0^\\infty((0,T);&C_0^\\infty\\Omega^1(\\omega_i))\\to L^2\\Omega^1(\\omega_i),\\\\\n \\P_i(\\bf)&=\\bu_i(T,\\cdot)|_{\\omega_i},\n \\end{align}", + "thm:NS": "\\begin{theorem}\n \\label{thm:NS}\nThere exists $\\delta>0$, depending on $(\\M,g)$ and $T$, such that for all $\\bf\\in \\mathbb B_\\delta$, problem \\eqref{eq:NS1} admits a unique solution $(\\bu,p)=(\\bu_\\bf,p_\\bf)$, $p$ being unique up to a map depending only on $t\\in[0,T]$, with $u_\\bf\\in \\mathcal C_\\alpha$, $dp_\\bf\\in C^{\\frac{\\alpha}{2}}([0,T];C^{\\alpha}\\Omega^1(\\M))$ and with $p_\\bf$ a solution of \\eqref{eq:p}. Moreover, the map $\\mathbb B_\\delta\\ni\\bf\\mapsto (u_\\bf,dp_\\bf)\\in \\cC_\\alpha(M)\\times L^2([0,T]; L^2\\Omega^1(\\M))$ is $C^1$.\n \\end{theorem}", + "linearization": "\\begin{proposition}\\label{linearization} \nFor $i=1,2$ and $\\bf\\in C^\\infty_0([0,T];C^\\infty_0\\Omega^1(\\omega_i))$, we fix $\\delta_{i,\\bf}=\\frac{\\delta}{1+\\|\\bf\\|_{\\mathcal C_\\alpha(M_i)}}$ with $\\delta$ introduced in Theorem \\ref{thm:NS}. Then the map\n$$(-\\delta_{i,\\bf},\\delta_{i,\\bf})\\ni s\\mapsto \\mathcal N_i(s\\bf)\\in L^2\\Omega^1(\\omega_i)$$\nis $C^1$ and we have\n\\begin{equation}\\label{lin1}\n \\partial_s\\mathcal N_i(s\\bf)|_{s=0}=\\mathcal P_i(\\bf).\n\\end{equation}\n\n\\end{proposition}" + }, + "pre_theorem_intro_text_len": 5628, + "pre_theorem_intro_text": "The Navier–Stokes equations are among the most fundamental mathematical models in fluid dynamics. \nIn this work, we consider the geometric Navier--Stokes equation on a Riemannian manifold $(M, g)$, see \\eqref{eq:NS1} below. When $M$ is a Euclidean domain and $g = c dx^2$, with $c > 0$ a constant, then \\eqref{eq:NS1} can be written as \nthe classical Navier-Stokes equations on $\\Omega$ with no-slip boundary condition \n\\begin{equation}\n\\begin{cases}\n \\p_t \\mathbf{u} + \\mathbf{u} \\nabla \\mathbf{u} - \\mu \\Delta \\mathbf{u} + \\nabla p = \\mathbf{f},\\text{ in }[0,T]\\timesM\\\\\n \\nabla\\cdot \\mathbf{u} = 0,\\text{ in }[0,T]\\timesM\\\\\n \\mathbf{u}=0,\\text{ on }[0,T]\\times \\partialM,\\\\\n \\mathbf{u}(0,\\cdot)=\\bu_0.\n\\end{cases}\n\\end{equation}\nHere $\\mathbf{u}: [0,T]\\times\\Omega\\to \\mathbb{R}^n$ is the velocity field of the fluid, $p: [0,T]\\times\\Omega\\to \\mathbb R$ is a scalar function representing the pressure, and \n$\\mu = c^{-1}$ is the coefficient of kinematic viscosity.\n\nThe generalization on a Riemannian manifold enables the study of fluid dynamics in complex geometric settings, thereby opening new perspectives for both theoretical analysis and practical applications \\cite{VlKh,EMa,MaRaSh,Sh}. The formulation of the Navier–Stokes equations on manifolds is motivated by several important areas, including: \n1) geophysical fluid dynamics \\cite{SaT,Val}, where the use of Riemannian geometry allows one to model the curvature of the Earth’s surface;\n2) General Relativity and cosmology \\cite{FMR}, modeling relativistic fluids, such as those present in neutron stars, black hole accretion disks, and the early universe;\n3)~fluid dynamics on biological membranes and microfluidic devices~\\cite{STONE}.\n\nIn all these contexts, the geometric structure of the manifold plays a central role in characterizing the physical properties of fluids and their surrounding media (e.g., density and viscosity). Motivated by these considerations, we investigate the inverse problem of identifying a Riemannian manifold from measurements of the fluid velocity field, associated with a source-to-final value map for the Navier–Stokes equations.\n\nInverse problems and, more broadly, identification problems related to fluid motion remain poorly understood. Only a limited number of studies have addressed this class of inverse problems, despite their strong physical motivation and mathematical significance. Among the few existing contributions devoted to inverse problems for the Stokes and Navier–Stokes equations, we may cite works concerning the determination of viscosity from boundary or internal measurements \\cite{FDJN,HLiWa,ImYa,LaUWa,LiWa,Liu}, the identification of objects immersed in a fluid \\cite{CaCo1,BCZ,CaCo2} as well as inverse source problems \\cite{BGKN,CIPY,ILY}. To the best of our knowledge, no prior studies have addressed inverse problems for the Navier–Stokes equations beyond the Euclidean setting.\n\n \\subsection{Notations and main results} \n Let $M$ be a n-dimensional, $n\\geq2$, smooth connected Riemannian manifold with boundary and $\\omega\\subset M$ be a nonempty open subset.\n We denote the space consisting of smooth sections of exterior $k$-form bundles on $M$ and $\\omega$ by $\\Omega^k(M)$ and $\\Omega^k(\\omega)$, respectively. Let\n \\begin{equation}\n \\ast:\\ \\Omega^k (M)\\to \\Omega^{n-k}(M)\n \\end{equation}\n be the Hodge star operator (see e.g. \\cite[section 14.1a]{frankel} for details). Then we define the codifferential operator $d^*: \\Omega^k(M)\\to \\Omega^{k-1}(M)$ as\n \\begin{equation}\n \\label{eq:def_codifferential}\n d^*=(-1)^{n(k+1)+1}\\ast d\\ast.\n \\end{equation}\n We denote the Hodge Laplacian $-dd^*-d^*d$ by $\\Delta_H$.\n Using the metric $g$, for every 1-form $u=u_idx_i$ on $M$ we associate the vector field $u_*=g^{ij}u_j\\partial_{x_i}\\in TM$. In the same way, for every vector field $v=v_i\\partial_{x_i}\\in TM$ we associate the 1-form $v^*=g_{ij}v_jdx_i$. With these notations, we write $R(w)=(\\nabla_{w_*}w_*)^*$,\nwhere $\\nabla$ denotes the Levi-Civita connection on $(M,g)$. Several works have been devoted to the study of the following geometric Navier-Stokes system \n \\begin{equation}\n \\label{eq:NS1}\n \\begin{cases}\n \\p_t\\mathbf{u}-\\Delta_H\\mathbf{u}+R(\\mathbf{u}) + d p=\\mathbf{f},\\ \\text{in } [0,T]\\times M,\\\\\n d^* \\mathbf{u}=0,\\ \\text{in } [0,T]\\times M,\\\\\n \\mathbf{u}|_{[0,T]\\times\\partialM}=0,\\\\\n \\mathbf{u}=0,\\ p=0,\\ \\text{in }\\{0\\}\\timesM.\n \\end{cases}\n \\end{equation}\n Without being exhaustive, one can refer to \\cite{EMa,AA91,CP92,TW93,CRT99} for more details.\n\n We denote by $\\mathbb B_r$ the set defined by\n $$\\mathbb B_r:=\\{\\mathbf{g} \\in \\cC_\\alpha(M):\\ \\supp(\\mathbf{g})\\subset (0,T]\\times\\omega,\\ \\|\\mathbf{g}\\|_{ \\cC_\\alpha(M)}\\leq r\\},$$\nwhere we refer to Section \\ref{sec:Nonlinear_NS} for the definition of the space $\\cC_\\alpha(M)$ and $\\norm{\\cdot}_{ \\cC_\\alpha(M)}$. We prove in Theorem \\ref{thm:NS} that \nthere exists $\\delta>0$, depending on $(M,g)$ and $T$, such that for all $\\mathbf{f}\\in \\mathbb B_\\delta$, problem \\eqref{eq:NS1} admits a solution $(\\mathbf{u},p)=(\\bu_\\mathbf{f},p_\\mathbf{f})$ with $\\mathbf{u}\\in \\mathcal C_\\alpha$. The solution is unique in the sense that $\\mathbf{u}$ is unique and $p$ is unique up to a map depending only on $t\\in[0,T]$. Then, we can define the local source-to-final value map\n\\begin{align}\n\\label{eq:def_N}\n \\mathcal N:\\mathbb B_\\delta&\\to L^2\\Omega^1(\\omega),\\\\\n \\mathcal N(\\mathbf{f})&=\\bu_\\mathbf{f}(T,\\cdot)|_\\omega,\n \\end{align}\n with $\\bu_\\mathbf{f}\\in \\cC_\\alpha(M)$ solving \\eqref{eq:NS1}.\n Our main result can be stated as follows.", + "context": "The Navier–Stokes equations are among the most fundamental mathematical models in fluid dynamics. \nIn this work, we consider the geometric Navier--Stokes equation on a Riemannian manifold $(M, g)$, see \\eqref{eq:NS1} below. When $M$ is a Euclidean domain and $g = c dx^2$, with $c > 0$ a constant, then \\eqref{eq:NS1} can be written as \nthe classical Navier-Stokes equations on $\\Omega$ with no-slip boundary condition \n\\begin{equation}\n\\begin{cases}\n \\p_t \\mathbf{u} + \\mathbf{u} \\nabla \\mathbf{u} - \\mu \\Delta \\mathbf{u} + \\nabla p = \\mathbf{f},\\text{ in }[0,T]\\timesM\\\\\n \\nabla\\cdot \\mathbf{u} = 0,\\text{ in }[0,T]\\timesM\\\\\n \\mathbf{u}=0,\\text{ on }[0,T]\\times \\partialM,\\\\\n \\mathbf{u}(0,\\cdot)=\\bu_0.\n\\end{cases}\n\\end{equation}\nHere $\\mathbf{u}: [0,T]\\times\\Omega\\to \\mathbb{R}^n$ is the velocity field of the fluid, $p: [0,T]\\times\\Omega\\to \\mathbb R$ is a scalar function representing the pressure, and \n$\\mu = c^{-1}$ is the coefficient of kinematic viscosity.\n\nThe generalization on a Riemannian manifold enables the study of fluid dynamics in complex geometric settings, thereby opening new perspectives for both theoretical analysis and practical applications \\cite{VlKh,EMa,MaRaSh,Sh}. The formulation of the Navier–Stokes equations on manifolds is motivated by several important areas, including: \n1) geophysical fluid dynamics \\cite{SaT,Val}, where the use of Riemannian geometry allows one to model the curvature of the Earth’s surface;\n2) General Relativity and cosmology \\cite{FMR}, modeling relativistic fluids, such as those present in neutron stars, black hole accretion disks, and the early universe;\n3)~fluid dynamics on biological membranes and microfluidic devices~\\cite{STONE}.\n\nIn all these contexts, the geometric structure of the manifold plays a central role in characterizing the physical properties of fluids and their surrounding media (e.g., density and viscosity). Motivated by these considerations, we investigate the inverse problem of identifying a Riemannian manifold from measurements of the fluid velocity field, associated with a source-to-final value map for the Navier–Stokes equations.\n\nInverse problems and, more broadly, identification problems related to fluid motion remain poorly understood. Only a limited number of studies have addressed this class of inverse problems, despite their strong physical motivation and mathematical significance. Among the few existing contributions devoted to inverse problems for the Stokes and Navier–Stokes equations, we may cite works concerning the determination of viscosity from boundary or internal measurements \\cite{FDJN,HLiWa,ImYa,LaUWa,LiWa,Liu}, the identification of objects immersed in a fluid \\cite{CaCo1,BCZ,CaCo2} as well as inverse source problems \\cite{BGKN,CIPY,ILY}. To the best of our knowledge, no prior studies have addressed inverse problems for the Navier–Stokes equations beyond the Euclidean setting.\n\n\\subsection{Notations and main results} \n Let $M$ be a n-dimensional, $n\\geq2$, smooth connected Riemannian manifold with boundary and $\\omega\\subset M$ be a nonempty open subset.\n We denote the space consisting of smooth sections of exterior $k$-form bundles on $M$ and $\\omega$ by $\\Omega^k(M)$ and $\\Omega^k(\\omega)$, respectively. Let\n \\begin{equation}\n \\ast:\\ \\Omega^k (M)\\to \\Omega^{n-k}(M)\n \\end{equation}\n be the Hodge star operator (see e.g. \\cite[section 14.1a]{frankel} for details). Then we define the codifferential operator $d^*: \\Omega^k(M)\\to \\Omega^{k-1}(M)$ as\n \\begin{equation}\n \\label{eq:def_codifferential}\n d^*=(-1)^{n(k+1)+1}\\ast d\\ast.\n \\end{equation}\n We denote the Hodge Laplacian $-dd^*-d^*d$ by $\\Delta_H$.\n Using the metric $g$, for every 1-form $u=u_idx_i$ on $M$ we associate the vector field $u_*=g^{ij}u_j\\partial_{x_i}\\in TM$. In the same way, for every vector field $v=v_i\\partial_{x_i}\\in TM$ we associate the 1-form $v^*=g_{ij}v_jdx_i$. With these notations, we write $R(w)=(\\nabla_{w_*}w_*)^*$,\nwhere $\\nabla$ denotes the Levi-Civita connection on $(M,g)$. Several works have been devoted to the study of the following geometric Navier-Stokes system \n \\begin{equation}\n \\label{eq:NS1}\n \\begin{cases}\n \\p_t\\mathbf{u}-\\Delta_H\\mathbf{u}+R(\\mathbf{u}) + d p=\\mathbf{f},\\ \\text{in } [0,T]\\times M,\\\\\n d^* \\mathbf{u}=0,\\ \\text{in } [0,T]\\times M,\\\\\n \\mathbf{u}|_{[0,T]\\times\\partialM}=0,\\\\\n \\mathbf{u}=0,\\ p=0,\\ \\text{in }\\{0\\}\\timesM.\n \\end{cases}\n \\end{equation}\n Without being exhaustive, one can refer to \\cite{EMa,AA91,CP92,TW93,CRT99} for more details.\n\nWe denote by $\\mathbb B_r$ the set defined by\n $$\\mathbb B_r:=\\{\\mathbf{g} \\in \\cC_\\alpha(M):\\ \\supp(\\mathbf{g})\\subset (0,T]\\times\\omega,\\ \\|\\mathbf{g}\\|_{ \\cC_\\alpha(M)}\\leq r\\},$$\nwhere we refer to Section \\ref{sec:Nonlinear_NS} for the definition of the space $\\cC_\\alpha(M)$ and $\\norm{\\cdot}_{ \\cC_\\alpha(M)}$. We prove in Theorem \\ref{thm:NS} that \nthere exists $\\delta>0$, depending on $(M,g)$ and $T$, such that for all $\\mathbf{f}\\in \\mathbb B_\\delta$, problem \\eqref{eq:NS1} admits a solution $(\\mathbf{u},p)=(\\bu_\\mathbf{f},p_\\mathbf{f})$ with $\\mathbf{u}\\in \\mathcal C_\\alpha$. The solution is unique in the sense that $\\mathbf{u}$ is unique and $p$ is unique up to a map depending only on $t\\in[0,T]$. Then, we can define the local source-to-final value map\n\\begin{align}\n\\label{eq:def_N}\n \\mathcal N:\\mathbb B_\\delta&\\to L^2\\Omega^1(\\omega),\\\\\n \\mathcal N(\\mathbf{f})&=\\bu_\\mathbf{f}(T,\\cdot)|_\\omega,\n \\end{align}\n with $\\bu_\\mathbf{f}\\in \\cC_\\alpha(M)$ solving \\eqref{eq:NS1}.\n Our main result can be stated as follows.", + "full_context": "The Navier–Stokes equations are among the most fundamental mathematical models in fluid dynamics. \nIn this work, we consider the geometric Navier--Stokes equation on a Riemannian manifold $(M, g)$, see \\eqref{eq:NS1} below. When $M$ is a Euclidean domain and $g = c dx^2$, with $c > 0$ a constant, then \\eqref{eq:NS1} can be written as \nthe classical Navier-Stokes equations on $\\Omega$ with no-slip boundary condition \n\\begin{equation}\n\\begin{cases}\n \\p_t \\mathbf{u} + \\mathbf{u} \\nabla \\mathbf{u} - \\mu \\Delta \\mathbf{u} + \\nabla p = \\mathbf{f},\\text{ in }[0,T]\\timesM\\\\\n \\nabla\\cdot \\mathbf{u} = 0,\\text{ in }[0,T]\\timesM\\\\\n \\mathbf{u}=0,\\text{ on }[0,T]\\times \\partialM,\\\\\n \\mathbf{u}(0,\\cdot)=\\bu_0.\n\\end{cases}\n\\end{equation}\nHere $\\mathbf{u}: [0,T]\\times\\Omega\\to \\mathbb{R}^n$ is the velocity field of the fluid, $p: [0,T]\\times\\Omega\\to \\mathbb R$ is a scalar function representing the pressure, and \n$\\mu = c^{-1}$ is the coefficient of kinematic viscosity.\n\nThe generalization on a Riemannian manifold enables the study of fluid dynamics in complex geometric settings, thereby opening new perspectives for both theoretical analysis and practical applications \\cite{VlKh,EMa,MaRaSh,Sh}. The formulation of the Navier–Stokes equations on manifolds is motivated by several important areas, including: \n1) geophysical fluid dynamics \\cite{SaT,Val}, where the use of Riemannian geometry allows one to model the curvature of the Earth’s surface;\n2) General Relativity and cosmology \\cite{FMR}, modeling relativistic fluids, such as those present in neutron stars, black hole accretion disks, and the early universe;\n3)~fluid dynamics on biological membranes and microfluidic devices~\\cite{STONE}.\n\nIn all these contexts, the geometric structure of the manifold plays a central role in characterizing the physical properties of fluids and their surrounding media (e.g., density and viscosity). Motivated by these considerations, we investigate the inverse problem of identifying a Riemannian manifold from measurements of the fluid velocity field, associated with a source-to-final value map for the Navier–Stokes equations.\n\nInverse problems and, more broadly, identification problems related to fluid motion remain poorly understood. Only a limited number of studies have addressed this class of inverse problems, despite their strong physical motivation and mathematical significance. Among the few existing contributions devoted to inverse problems for the Stokes and Navier–Stokes equations, we may cite works concerning the determination of viscosity from boundary or internal measurements \\cite{FDJN,HLiWa,ImYa,LaUWa,LiWa,Liu}, the identification of objects immersed in a fluid \\cite{CaCo1,BCZ,CaCo2} as well as inverse source problems \\cite{BGKN,CIPY,ILY}. To the best of our knowledge, no prior studies have addressed inverse problems for the Navier–Stokes equations beyond the Euclidean setting.\n\n\\subsection{Notations and main results} \n Let $M$ be a n-dimensional, $n\\geq2$, smooth connected Riemannian manifold with boundary and $\\omega\\subset M$ be a nonempty open subset.\n We denote the space consisting of smooth sections of exterior $k$-form bundles on $M$ and $\\omega$ by $\\Omega^k(M)$ and $\\Omega^k(\\omega)$, respectively. Let\n \\begin{equation}\n \\ast:\\ \\Omega^k (M)\\to \\Omega^{n-k}(M)\n \\end{equation}\n be the Hodge star operator (see e.g. \\cite[section 14.1a]{frankel} for details). Then we define the codifferential operator $d^*: \\Omega^k(M)\\to \\Omega^{k-1}(M)$ as\n \\begin{equation}\n \\label{eq:def_codifferential}\n d^*=(-1)^{n(k+1)+1}\\ast d\\ast.\n \\end{equation}\n We denote the Hodge Laplacian $-dd^*-d^*d$ by $\\Delta_H$.\n Using the metric $g$, for every 1-form $u=u_idx_i$ on $M$ we associate the vector field $u_*=g^{ij}u_j\\partial_{x_i}\\in TM$. In the same way, for every vector field $v=v_i\\partial_{x_i}\\in TM$ we associate the 1-form $v^*=g_{ij}v_jdx_i$. With these notations, we write $R(w)=(\\nabla_{w_*}w_*)^*$,\nwhere $\\nabla$ denotes the Levi-Civita connection on $(M,g)$. Several works have been devoted to the study of the following geometric Navier-Stokes system \n \\begin{equation}\n \\label{eq:NS1}\n \\begin{cases}\n \\p_t\\mathbf{u}-\\Delta_H\\mathbf{u}+R(\\mathbf{u}) + d p=\\mathbf{f},\\ \\text{in } [0,T]\\times M,\\\\\n d^* \\mathbf{u}=0,\\ \\text{in } [0,T]\\times M,\\\\\n \\mathbf{u}|_{[0,T]\\times\\partialM}=0,\\\\\n \\mathbf{u}=0,\\ p=0,\\ \\text{in }\\{0\\}\\timesM.\n \\end{cases}\n \\end{equation}\n Without being exhaustive, one can refer to \\cite{EMa,AA91,CP92,TW93,CRT99} for more details.\n\nWe denote by $\\mathbb B_r$ the set defined by\n $$\\mathbb B_r:=\\{\\mathbf{g} \\in \\cC_\\alpha(M):\\ \\supp(\\mathbf{g})\\subset (0,T]\\times\\omega,\\ \\|\\mathbf{g}\\|_{ \\cC_\\alpha(M)}\\leq r\\},$$\nwhere we refer to Section \\ref{sec:Nonlinear_NS} for the definition of the space $\\cC_\\alpha(M)$ and $\\norm{\\cdot}_{ \\cC_\\alpha(M)}$. We prove in Theorem \\ref{thm:NS} that \nthere exists $\\delta>0$, depending on $(M,g)$ and $T$, such that for all $\\mathbf{f}\\in \\mathbb B_\\delta$, problem \\eqref{eq:NS1} admits a solution $(\\mathbf{u},p)=(\\bu_\\mathbf{f},p_\\mathbf{f})$ with $\\mathbf{u}\\in \\mathcal C_\\alpha$. The solution is unique in the sense that $\\mathbf{u}$ is unique and $p$ is unique up to a map depending only on $t\\in[0,T]$. Then, we can define the local source-to-final value map\n\\begin{align}\n\\label{eq:def_N}\n \\mathcal N:\\mathbb B_\\delta&\\to L^2\\Omega^1(\\omega),\\\\\n \\mathcal N(\\mathbf{f})&=\\bu_\\mathbf{f}(T,\\cdot)|_\\omega,\n \\end{align}\n with $\\bu_\\mathbf{f}\\in \\cC_\\alpha(M)$ solving \\eqref{eq:NS1}.\n Our main result can be stated as follows.\n\nTo the best of our knowledge, Theorem \\ref{thm:main} is the first result establishing the recovery of the general geometric structure of a manifold from local velocity measurements, at the final time $t=T$, associated with solutions of the Navier-Stokes equations. By contrast, the existing literature is confined to the identification of the viscosity coefficient from measurements related to the stationary Navier-Stokes equations \\cite{HLiWa,LaUWa,LiWa}.\n\nAccording to the above discussion Theorem \\ref{thm:main} follows from the following theorem.\n \\begin{theorem}\n \\label{thm_main:hyperbolic}\n Suppose there is a diffeomorphism $\\bPhi:\\omega_1\\to\\omega_2$ such that \n \\begin{equation}\n \\bPhi^\\ast\\Lambda_{2,\\omega_2}=\\Lambda_{1,\\omega_1}\\bPhi^\\ast.\n \\end{equation}\n Then $(\\M_1,g_1)$ and $(\\M_2,g_2)$ are isometric.\n \\end{theorem}\n The remainder of this section is devoted to the proof of Theorem \\ref{thm_main:hyperbolic} using the BC method. We extend the BC method to the auxiliary hyperbolic system \\eqref{eq:problem_hypo}. This extension is not a direct application of existing results, due to the presence of the divergence-free constraint. Accordingly, each step of the BC method must be adapted. These adaptations are carried out below.\n\n\\subsection{Reconstruction of metric in the accessible domain}\n \\begin{proposition}\n \\label{prop:dist_omega}\n Suppose there is a diffeomorphism $\\bPhi:\\omega_1\\to\\omega_2$ such that $\\diam_1(\\omega_1)< T_{\\bPhi}(\\omega_1)$ and $\\bPhi^\\ast\\Lambda_{2,\\omega_2}=\\Lambda_{1,\\omega_1}\\bPhi^\\ast$, then for any points $x,y\\in \\omega_1$, there holds $d_{g_1}(x,y)=d_{g_2}(\\bPhi(x),\\bPhi(y))$. Moreover, $\\bPhi$ is an isometry.\n \\end{proposition}\n \\begin{proof}\n Let $\\varepsilon>0$ be small enough that $B_1(x,\\varepsilon), B_1(y,\\varepsilon)\\subset \\omega_1$ and $B^c_1(x,\\varepsilon)$, $B^c_1(y,\\varepsilon)$ are connected. Then we write \n \\begin{equation}\n \\mathcal{B}_k(p,\\varepsilon):=C_0^\\infty((0,\\infty);C_0^\\infty\\Omega^1(B_k(p,\\varepsilon))),\\ k=1,2,\n \\end{equation}\n and \n \\begin{equation}\n \\bPhi_\\ast (\\cB_1(x,\\varepsilon)):=C_0^\\infty((0,\\infty);C_0^\\infty\\Omega^1(\\bPhi(B_1(x,\\varepsilon)))).\n \\end{equation}\n We define\n \\begin{align}\n t_{1,\\varepsilon} := \\inf\\{t>0\\mid \\exists \\bf\\in \\mathcal{B}_1(x,\\varepsilon)\\text{ s.t. } \\supp(\\bu_1^\\bf(t,\\cdot))\\cap B_1(y,\\varepsilon)\\neq \\emptyset \\}\n \\end{align}\n \\begin{align}\n t_{2,\\varepsilon} := \\inf\\{t>0\\mid \\exists \\bf\\in \\mathcal{B}_2(\\bPhi(x),\\varepsilon)\\text{ s.t. } \\supp(\\bu_2^\\bf(t,\\cdot))\\cap B_2(\\bPhi(y),\\varepsilon)\\neq \\emptyset \\},\n \\end{align}\n and \n \\begin{align}\n \\widetilde{t_{2,\\varepsilon}} := \\inf\\{t>0\\mid \\exists \\bh\\in \\bPhi_\\ast (\\cB_1(x,\\varepsilon))\\text{ s.t. } \\supp(\\bu_2^\\bh(t,\\cdot))\\cap \\bPhi(B_1(y,\\varepsilon))\\neq \\emptyset \\}.\n \\end{align}\n As $\\bPhi:\\omega_1\\to \\omega_2$ is a diffeomorphism, we can find two positive functions $r(\\varepsilon),R(\\varepsilon)$ such that $\\lim_{\\varepsilon\\to 0}r(\\varepsilon)=\\lim_{\\varepsilon\\to 0}R(\\varepsilon)=0$, and\n \\begin{equation}\n B_2(\\bPhi(p),r(\\varepsilon))\\subset \\bPhi(B_1(p,\\varepsilon))\\subset B_2(\\bPhi(p),R(\\varepsilon))\n \\end{equation}\n for both points $p=x,y$. Hence, $t_{2,R(\\varepsilon)}\\leq \\widetilde{t_{2,\\varepsilon}}\\leq t_{2,r(\\varepsilon)}.$\n\nNext we show the local reconstruction can be glued together.\\par\n \\begin{lemma}\n \\label{lm:glue}\n Let $U_1,U_2\\subset \\M_1^\\inter$ be open sets with isometries \n \\begin{equation}\n \\bPhi:U_1\\to\\bPhi(U_1),\\ \\bPsi: U_2\\to \\bPsi(U_2).\n \\end{equation}\n Suppose $\\Lambda_{1,U_1}\\bPhi^\\ast=\\bPhi^\\ast \\Lambda_{2,\\bPhi(U_1)}$, $\\Lambda_{1,U_2}\\bPsi^\\ast=\\bPsi^\\ast \\Lambda_{2,\\bPsi(U_2)}$ and there exists an open set $\\cV\\subset U_1\\cap U_2$ such that $\\bPhi|_\\cV=\\bPsi|_\\cV$ and $U_1\\cap U_2$ is $\\cT$-exhaustive by $\\cV$. Then there holds\n \\begin{equation}\n \\label{eq:lm_6.14}\n \\bPhi|_{U_1\\cap U_2}=\\bPsi|_{U_1\\cap U_2}.\n \\end{equation}\n \\end{lemma}\n \\begin{proof}\n Let us show that \n \\begin{equation}\n \\label{eq:lm_6.14_1}\n \\bPhi|_{E^{(1)}_{\\cT}(\\cV,U_1\\cap U_2)}=\\bPsi|_{E^{(1)}_{\\cT}(\\cV,U_1\\cap U_2)}.\n \\end{equation}\n Let $y\\in U_1\\cap U_2\\cap M(\\cV,\\cT)$ be arbitrary. To get a contradiction, we assume that $\\bPhi(y)\\neq \\bPsi(y)$. Let $z\\in \\cV$ be such that $d_{g_1}(y,z)<\\cT$. For simplicity, we denote $d_{g_1}(y,z)$ by $s$. Since $\\bPhi$ is an isometry, we have $d_{g_2}(\\bPhi(y),\\bPhi(z))=s<\\cT$. Then we can choose a small enough $0<\\varepsilon<\\frac{1}{2}(\\cT-s)$ such that $B_2(\\bPhi(y),2\\varepsilon)\\subset \\bPhi(U_1)$, $B_2(\\bPsi(y),2\\varepsilon)\\subset \\bPsi(U_2)$, $B_2(\\bPhi(y),2\\varepsilon)\\cap B_2(\\bPsi(y),2\\varepsilon)=\\emptyset$, and $\\M_2\\setminus B_2(\\bPhi(y),\\varepsilon)$ and $\\M_2\\setminus B_2(\\bPsi(y),\\varepsilon)$ are connected.\n By approximate controllability, we may take $\\bf\\in C_0^\\infty((T-\\varepsilon,T);C_0^\\infty\\Omega^1(B_2(\\bPhi(y),\\varepsilon)))$ such that $\\bu_2^{\\bf}(T,\\cdot)\\neq 0$. For any \n \\begin{align}\n \\bh\\in C_0^\\infty((T-\\varepsilon,T);C_0^\\infty\\Omega^1(B_2(\\bPsi(y),\\varepsilon))),\n \\end{align}\n due to the finite speed of propagation, we have\n \\begin{equation}\n \\supp(\\bu_2^\\bf(T,\\cdot))\\cap \\supp(\\bu_2^\\bh(T,\\cdot))=\\emptyset.\n \\end{equation}\n Since\n \\begin{align}\n \\bPhi^\\ast: C_0^\\infty((T-\\varepsilon,T);C_0^\\infty\\Omega^1(B_2(\\bPhi(y),\\varepsilon)))\\to C_0^\\infty((T-\\varepsilon,T);C_0^\\infty\\Omega^1(B_1(y,\\varepsilon))),\\\\\n \\bPsi^\\ast: C_0^\\infty((T-\\varepsilon,T);C_0^\\infty\\Omega^1(B_2(\\bPsi(y),\\varepsilon)))\\to C_0^\\infty((T-\\varepsilon,T);C_0^\\infty\\Omega^1(B_1(y,\\varepsilon)))\n \\end{align}\n are vector space isomorphism, we can choose $\\bPsi^\\ast \\bh = \\bPhi^\\ast \\bf$. Let $\\delta>0$ be such that $B_1(z,\\delta)\\subset \\cV$. Then we have $B_2(\\bPhi(y),2\\varepsilon)\\subset M_2(B_2(\\bPhi(z),\\delta),s+2\\varepsilon-\\delta)$. Since \n $$T_2(B_2(\\bPhi(z),\\delta))\\geq T_{\\bPhi}(B_1(z,\\delta))\\geq \\cT-\\delta>s+2\\varepsilon-\\delta,$$ \n by the approximate controllability, there exists a sequence\n $$\\{\\bphi_n\\}_{n=1}^\\infty\\subset C_0^\\infty((T-(s+2\\varepsilon-\\delta),T);C_0^\\infty\\Omega^1(B_2(\\bPhi(z),\\delta))$$ \n such that\n $\\bu_2^{\\bphi_n}(T,\\cdot)\\to \\bu_2^{\\bf}(T,\\cdot)$ as $n\\to \\infty$. Since $$B_2(\\bPhi(y),2\\varepsilon)\\cap B_2(\\bPsi(y),2\\varepsilon)=\\emptyset,$$\n there holds\n \\begin{equation}\n \\lim_{n\\to \\infty}\\inner{\\bu_2^{\\bh}(T,\\cdot),\\bu_2^{\\bphi_n}(T,\\cdot)}=0,\n \\end{equation}\n and \n \\begin{equation}\n \\lim_{n\\to \\infty}\\inner{\\bu_2^{\\bf}(T,\\cdot),\\bu_2^{\\bphi_n}(T,\\cdot)}=\\norm{\\bu_2^{\\bf}(T,\\cdot)}_{L^2\\Omega^1(\\M_2)}>0.\n \\end{equation}\n Then for large enough $N$ we write $\\bpsi=\\bphi_N$, and we have\n \\begin{equation}\n \\label{eq:glue_contradiction}\n \\inner{\\bu_2^{\\bf-\\bh}(T,\\cdot),\\bu_2^{\\bpsi}(T,\\cdot)}>0.\n \\end{equation}\n Notice that $\\bPhi^\\ast \\bpsi=\\bPsi^\\ast \\bpsi$ as $\\bPhi|_\\cV=\\bPsi|_\\cV$. According to Lemma \\ref{lm:norm_equal} and the fact that $\\bPsi^\\ast \\bh = \\bPhi^\\ast \\bf$, we have\n \\begin{align}\n \\inner{\\bu_2^{\\bf-\\bh}(T,\\cdot),\\bu_2^{\\bpsi}(T,\\cdot)}&=\\inner{\\bu_2^{\\bf}(T,\\cdot),\\bu_2^{\\bpsi}(T,\\cdot)}-\\inner{\\bu_2^{\\bh}(T,\\cdot),\\bu_2^{\\bpsi}(T,\\cdot)}\\\\\n &=\\inner{\\bu_1^{\\bPhi^\\ast\\bf}(T,\\cdot),\\bu_1^{\\bPhi^\\ast\\bpsi}(T,\\cdot)}-\\inner{\\bu_1^{\\bPsi^\\ast\\bh}(T,\\cdot),\\bu_1^{\\bPsi^\\ast\\bpsi}(T,\\cdot)}\\\\\n &=\\inner{\\bu_1^{\\bPhi^\\ast\\bf-\\bPsi^\\ast\\bh}(T,\\cdot),\\bu_1^{\\bPhi^\\ast\\bpsi}(T,\\cdot)}=0.\n \\end{align}\n This is in contradiction with \\eqref{eq:glue_contradiction} and we have proved that \\eqref{eq:lm_6.14_1} holds as $y\\in E^{(1)}_{\\cT}(\\cV,U_1\\cap U_2)$ is arbitrary. Since $U_1\\cap U_2$ is $\\cT$-exhaustive by $\\cV$, there exists $N>0$ such that $U_1\\cap U_2=E^{(N)}_{\\cT}(\\cV,U_1\\cap U_2)$. Replacing $E^{(k)}_{\\cT}(\\cV,U_1\\cap U_2)$ by $E^{(k+1)}_{\\cT}(\\cV,U_1\\cap U_2)$ iteratively in above argument for $k=0,1,\\cdots,N-1$, we can obtain \\eqref{eq:lm_6.14}.\n \\end{proof}\n \\begin{proof}[Proof of Theorem \\ref{thm:main}]\n Provided $\\bPhi:\\omega_1\\to \\omega_2$ is a diffeomorphism and $\\Lambda_{1,\\omega_1}\\bPhi^\\ast=\\bPhi^\\ast \\Lambda_{2,\\omega_2}$. we have $\\bPhi$ is an isometry by Proposition \\ref{prop:dist_omega}.", + "post_theorem_intro_text_len": 7157, + "post_theorem_intro_text": "\\begin{remark}\n By making $\\omega_1$ and $\\omega_2$ smaller, connectedness can be assumed without loss of generality.\n \\end{remark}\n\\subsection{Comparision to previous literature}\n\nTo the best of our knowledge, Theorem \\ref{thm:main} is the first result establishing the recovery of the general geometric structure of a manifold from local velocity measurements, at the final time $t=T$, associated with solutions of the Navier-Stokes equations. By contrast, the existing literature is confined to the identification of the viscosity coefficient from measurements related to the stationary Navier-Stokes equations \\cite{HLiWa,LaUWa,LiWa}.\n\nOur proof of Theorem \\ref{thm:main} is based on application of the Boundary Control (BC) method, which remains one of the most powerful tools for solving inverse coefficient problems for partial differential equations (PDEs) in the time domain. The BC method was first introduced by Belishev in \\cite{Beli} and thereafter further developed for solving inverse coefficient determination problems related to the wave equation in Euclidean domains \\cite{ABI92,B87,B90,BK89,BK87,BKu89,BK91}. Later, this method was extended to manifold reconstruction \\cite{BeKu}. It has been shown that the BC method can be applied to other classes of scalar PDEs \\cite{FGKU,HLYZ,KaKuLaMa,KLLY,KOSY}.\n\nThe method appears to be less powerful in the context of systems of PDEs, however, inverse problems for the Maxwell \\cite{BeIs,KuLaSo} and Dirac \\cite{LK09} systems has been solved using it. Further, in \\cite{KOP18} the BC method was generalized for abstract wave equations associated to connection Laplacians on vector bundles (for example, a linearized equation satisfied by the Higgs field on the pre-quantum level). To the best of our knowledge, this work constitutes the first application of the BC method to the Stokes system. The approach consists in first establishing a rigorous correspondence between the data for the Stokes system and those for an auxiliary hyperbolic Stokes problem through spectral data, as developed in Section 5. The BC method is then implemented for this auxiliary hyperbolic Stokes problem in Section 7.\n\nIt is important to emphasize that the extension of the BC method to this setting cannot be derived from existing results in the literature. The principal obstruction arises from the divergence-free constraint inherent to the Stokes system, which introduces substantial technical and structural difficulties at every stage of the method. In Section 7, we provide a detailed exposition of the adaptation of the BC method to this framework, and we show how the difficulties encountered at each step can be resolved by means of carefully tailored geometric and analytic arguments.\n\nIn inverse problems concerning recovery of geometry on manifolds with boundary, measurements are typically taken on the boundary and modeled by the Dirichlet-to-Neumann map or the associated Cauchy data \\cite{AKKLT04,KK98,Oksanen2014,KaKuLaMa}. By contrast, inverse problems based on the local source-to-solution map have been predominantly studied on closed manifolds. In particular, it is shown in \\cite{Saksala2018,HLYZ,SS25} that the local source-to-solution map for various linear evolution equations determines the underlying manifold up to an isometry. A related result for the wave equation, where the source and observations are supported in disjoint sets, is established in \\cite{LNOY24}. \nIn this work, we consider the local source-to-solution map on manifolds with boundary.\n\nThe approach adopted in Sections 4 and 5 to address the nonlinearity relies on a linearization method that has proved effective in the analysis of numerous inverse problems for nonlinear PDEs. This method, originally introduced in \\cite{I93,IS94} in the context of parabolic and elliptic partial differential equations, consists in establishing that the first-order linearization of the data coincides with the data associated with an appropriate linearized equation. In contrast with the case of classical scalar partial differential equations (see, for instance, \\cite{L25} for a comprehensive review), the application of the linearization method to the Navier–Stokes system involves both the velocity field $\\mathbf{u}$ and the pressure $p$. This necessitates a suitable representation of the pressure $p$, which is introduced in Theorem \\ref{thm:NS}, together with the corresponding linearization result stated in Proposition \\ref{linearization}. To the best of our knowledge, the only existing works employing techniques closely related to this linearization approach for the Navier–Stokes system are \\cite{LaUWa,LiWa}, where the analysis is restricted to the Euclidean setting and to stationary Navier–Stokes equations in dimensions two and three. By contrast, our analysis is carried out in H\\\"older spaces, which allows for an extension to higher-dimensional settings.\n\nFollowing the linearization step, in Section 5 we make use of analytic and representation properties of solutions to the linear Stokes system in order to establish a connection between the local source-to-final-value map and a source-to-solution map associated with an auxiliary hyperbolic system. The inverse problem for this auxiliary hyperbolic system is subsequently analyzed by means of the BC method. This data transfer technique by mean of spectral data has previously been employed in the study of inverse coefficient problems for various scalar evolution partial differential equations \\cite{CK,HLYZ,KaKuLaMa,KLLY,KOSY}. As far as we know, the present work constitutes the first application of this methodology to Stokes systems.\n\n\\subsection{Outline of the paper}\n\nThe outline of the paper is as follows. In Section \\ref{sec:elliptic}, we establish regularity results for the stationary Stokes system on manifolds and analyze properties of the associated spectral data, which prepare the terrain for the study of the inverse problem. Section \\ref{sec:Nonlinear_NS} is devoted to the existence, uniqueness and appropriate regularity of the solution to the geometric Navier-Stokes equation \\eqref{eq:NS1}. In Section \\ref{sec:parabolic}, we consider the non-stationary Stokes equation, which arises as the linearization of \\eqref{eq:NS1}, and show that the data $\\mathcal N$ determines the corresponding restricted source-to-solution operator $\\mathcal P$, defined as \\eqref{eq:def_P}, for the non-stationary Stokes equation. Moreover, we show that $\\mathcal P$ is equivalent to the localized spectral projections of the stationary Stokes system, which in turn allows the data to be converted into the restricted source-to-solution operator for an auxiliary hyperbolic Stokes system \\eqref{eq:problem_hypo}. Section \\ref{sec:hyperbolic} analyzes \\eqref{eq:problem_hypo} and develops several technical tools, including conditional finite speed of propagation and approximate controllability. The new feature of these tools is that they hold only on a finite time interval determined by the underlying geometry. Combining these techniques with an adapted BC, our main result Theorem \\ref{thm:main} is proved in Section \\ref{sec:mian_proof}.", + "sketch": "The introduction states that the proof of Theorem~\\ref{thm:main} is based on the Boundary Control (BC) method. The approach is to (i) address the nonlinearity by a linearization method (Sections 4--5), establishing that the first-order linearization of the data coincides with the data for an appropriate linearized equation, which requires a suitable representation of the pressure $p$ (Theorem~\\ref{thm:NS}) and yields a linearization result (Proposition~\\ref{linearization}); then (ii) use analytic/representation properties of solutions to the linear Stokes system to connect the local source-to-final-value map $\\mathcal N$ to a (restricted) source-to-solution operator $\\mathcal P$ for the non-stationary Stokes equation, and show $\\mathcal P$ is equivalent to localized spectral projections of the stationary Stokes system; this (iii) allows converting the data into the restricted source-to-solution operator for an auxiliary hyperbolic Stokes system \\eqref{eq:problem_hypo} through spectral data (Section 5); finally (iv) analyze this auxiliary hyperbolic problem using tools developed in Section~\\ref{sec:hyperbolic} (including conditional finite speed of propagation and approximate controllability, valid on a finite time interval determined by the geometry) and then implement an adapted BC method (Section 7 / Section~\\ref{sec:mian_proof}) to conclude the manifold is determined up to an isometry.", + "expanded_sketch": "The introduction states that the proof of the main theorem is based on the Boundary Control (BC) method. The approach is to (i) address the nonlinearity by a linearization method (Sections 4--5), establishing that the first-order linearization of the data coincides with the data for an appropriate linearized equation, which requires a suitable representation of the pressure $p$. We first use the following theorem.\n\\begin{theorem}\n \\label{thm:NS}\nThere exists $\\delta>0$, depending on $(\\M,g)$ and $T$, such that for all $\\bf\\in \\mathbb B_\\delta$, problem \\eqref{eq:NS1} admits a unique solution $(\\bu,p)=(\\bu_\\bf,p_\\bf)$, $p$ being unique up to a map depending only on $t\\in[0,T]$, with $u_\\bf\\in \\mathcal C_\\alpha$, $dp_\\bf\\in C^{\\frac{\\alpha}{2}}([0,T];C^{\\alpha}\\Omega^1(\\M))$ and with $p_\\bf$ a solution of \\eqref{eq:p}. Moreover, the map $\\mathbb B_\\delta\\ni\\bf\\mapsto (u_\\bf,dp_\\bf)\\in \\cC_\\alpha(M)\\times L^2([0,T]; L^2\\Omega^1(\\M))$ is $C^1$.\n \\end{theorem}\nThis yields the following linearization result.\n\\begin{proposition}\\label{linearization} \nFor $i=1,2$ and $\\bf\\in C^\\infty_0([0,T];C^\\infty_0\\Omega^1(\\omega_i))$, we fix $\\delta_{i,\\bf}=\\frac{\\delta}{1+\\|\\bf\\|_{\\mathcal C_\\alpha(M_i)}}$ with $\\delta$ introduced in Theorem \\ref{thm:NS}. Then the map\n$$(-\\delta_{i,\\bf},\\delta_{i,\\bf})\\ni s\\mapsto \\mathcal N_i(s\\bf)\\in L^2\\Omega^1(\\omega_i)$$\nis $C^1$ and we have\n\\begin{equation}\\label{lin1}\n \\partial_s\\mathcal N_i(s\\bf)|_{s=0}=\\mathcal P_i(\\bf).\n\\end{equation}\n\n\\end{proposition}\nThen (ii) one uses analytic/representation properties of solutions to the linear Stokes system to connect the local source-to-final-value map $\\mathcal N$ to a (restricted) source-to-solution operator $\\mathcal P$ for the non-stationary Stokes equation, and show $\\mathcal P$ is equivalent to localized spectral projections of the stationary Stokes system; this (iii) allows converting the data into the restricted source-to-solution operator for an auxiliary hyperbolic Stokes system\n\\begin{equation}\n \\label{eq:problem_hypo}\n \\begin{cases}\n (\\p_t^2-\\Delta_H)\\bu + d p=\\bf,\\ \\text{in } \\RR^+\\times \\M,\\\\\n d^* \\bu=0,\\ \\text{in } \\RR^+\\times \\M,\\\\\n \\bu|_{[0,T]\\times\\p\\M}=0,\\\\\n \\bu=0,\\ p=0,\\ \\text{in }\\RR^-\\times\\M.\n \\end{cases}\n \\end{equation}\nthrough spectral data (Section 5); finally (iv) one analyzes this auxiliary hyperbolic problem using tools developed later (including conditional finite speed of propagation and approximate controllability, valid on a finite time interval determined by the geometry) and then implements an adapted BC method (Section 7 / later in the paper) to conclude the manifold is determined up to an isometry.", + "expanded_theorem": "\\label{thm:main}\n Let $(\\M_1,g_1)$ and $(\\M_2,g_2)$ be two n-dimensional smooth connected Riemannian manifolds with boundary. Let $\\omega_1\\subset \\M_1$ and $\\omega_2\\subset\\M_2$ be open and connected. For $j=1,2$, let $\\mathcal N_j$ be the local source-to-final value map\n \\begin{align}\n\\label{eq:def_N}\n \\mathcal N:\\mathbb B_\\delta&\\to L^2\\Omega^1(\\omega),\\\\\n \\mathcal N(\\bf)&=\\bu_\\bf(T,\\cdot)|_\\omega,\n \\end{align}\nwith $(M,g)=(\\M_j,g_j)$\n and $\\omega=\\omega_j$.\n In addition, suppose that there is a diffeomorphism $\\bm{\\Phi}:\\omega_1\\to \\omega_2$ such that \n \\begin{equation}\\label{NN}\n \\cN_1\\bm{\\Phi}^\\ast=\\bm{\\Phi}^\\ast \\cN_2.\n \\end{equation}\n Then $(\\M_1,g_1)$ and $(\\M_2,g_2)$ are isometric.,", + "theorem_type": [ + "Implication" + ], + "mcq": { + "question": "Let \\((\\mathcal M_1,g_1)\\) and \\((\\mathcal M_2,g_2)\\) be two smooth connected \\(n\\)-dimensional Riemannian manifolds with boundary, and let \\(\\omega_1\\subset \\mathcal M_1\\) and \\(\\omega_2\\subset \\mathcal M_2\\) be open connected subsets. For each \\(j=1,2\\), consider the geometric Navier--Stokes system on \\([0,T]\\times \\mathcal M_j\\) for a time-dependent 1-form \\(\\mathbf u\\) and pressure \\(p\\):\n\\[\n\\begin{cases}\n\\partial_t \\mathbf u-\\Delta_H\\mathbf u+R(\\mathbf u)+dp=\\mathbf f,\\\\\nd^*\\mathbf u=0,\\\\\n\\mathbf u|_{[0,T]\\times \\partial \\mathcal M_j}=0,\\\\\n\\mathbf u=0,\\ p=0 \\text{ at } t=0,\n\\end{cases}\n\\]\nwhere \\(\\Delta_H=-dd^*-d^*d\\) is the Hodge Laplacian and \\(R(\\mathbf u)=(\\nabla_{\\mathbf u_*}\\mathbf u_*)^*\\). For some \\(\\delta>0\\), let \\(\\mathbb B_\\delta\\) denote the admissible source 1-forms supported in \\((0,T]\\times \\omega_j\\) with \\(\\|\\mathbf f\\|_{C_\\alpha(\\mathcal M_j)}\\le \\delta\\), and for each such \\(\\mathbf f\\) let \\(\\mathbf u_{\\mathbf f}\\) be the corresponding solution. Define the local source-to-final value map\n\\[\n\\mathcal N_j(\\mathbf f)=\\mathbf u_{\\mathbf f}(T,\\cdot)|_{\\omega_j}\\in L^2\\Omega^1(\\omega_j).\n\\]\nAssume there exists a diffeomorphism \\(\\boldsymbol\\Phi:\\omega_1\\to \\omega_2\\) such that, with \\(\\boldsymbol\\Phi^*\\) denoting pullback of time-dependent 1-forms,\n\\[\n\\mathcal N_1\\,\\boldsymbol\\Phi^*=\\boldsymbol\\Phi^*\\mathcal N_2,\n\\]\ni.e. for every admissible source on \\(\\omega_2\\), the observed final 1-form on \\(\\omega_1\\) agrees after pullback. Which conclusion about \\((\\mathcal M_1,g_1)\\) and \\((\\mathcal M_2,g_2)\\) holds?", + "correct_choice": { + "label": "A", + "text": "The manifolds \\((\\mathcal M_1,g_1)\\) and \\((\\mathcal M_2,g_2)\\) are isometric; equivalently, there exists a diffeomorphism \\(F:\\mathcal M_1\\to \\mathcal M_2\\) such that \\(F^*g_2=g_1\\)." + }, + "choices": [ + { + "label": "B", + "text": "The manifolds \\((\\mathcal M_1,g_1)\\) and \\((\\mathcal M_2,g_2)\\) have isometric neighborhoods of \\(\\omega_1\\) and \\(\\omega_2\\); more precisely, there exist open sets \\(U_j\\subset \\mathcal M_j\\) with \\(\\omega_j\\subset U_j\\) and a diffeomorphism \\(F:U_1\\to U_2\\) such that \\(F|_{\\omega_1}=\\boldsymbol\\Phi\\) and \\(F^*g_2=g_1\\) on \\(U_1\\)." + }, + { + "label": "C", + "text": "The diffeomorphism \\(\\boldsymbol\\Phi:\\omega_1\\to \\omega_2\\) is an isometry between the observed regions; that is, \\(\\boldsymbol\\Phi^*g_2=g_1\\) on \\(\\omega_1\\)." + }, + { + "label": "D", + "text": "The manifolds \\((\\mathcal M_1,g_1)\\) and \\((\\mathcal M_2,g_2)\\) are isometric provided the intertwining identity \\(\\mathcal N_1\\,\\boldsymbol\\Phi^*=\\boldsymbol\\Phi^*\\mathcal N_2\\) holds for all smooth compactly supported sources on \\((0,T]\\times\\omega_2\\), without any smallness restriction \\(\\|\\mathbf f\\|_{C_\\alpha(\\mathcal M_2)}\\le \\delta\\)." + }, + { + "label": "E", + "text": "There exists a constant \\(c>0\\) and a diffeomorphism \\(F:\\mathcal M_1\\to \\mathcal M_2\\) extending \\(\\boldsymbol\\Phi\\) such that \\(F^*g_2=c\\,g_1\\); in particular, the manifolds are determined only up to a global conformal scaling by the local source-to-final value map." + } + ], + "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": "global_BC_gluing_to_full_manifold", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "geometric_construction", + "tampered_component": "dropped_global_extension_to_entire_manifolds", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "small_data_linearization_radius_delta", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "metric_identification_vs_conformal_gauge", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly reveal the correct choice. It states a technical intertwining hypothesis and asks for the resulting geometric conclusion, without giving away global isometry." + }, + "TAS": { + "score": 1, + "justification": "The item is very close to a theorem-recall question: the hypotheses are presented in theorem form and the student is asked for the conclusion. However, it is not a pure verbatim restatement, since the options force discrimination between global, local, weaker, and overstrong conclusions." + }, + "GPS": { + "score": 1, + "justification": "Moderate reasoning is required: the solver must identify the strongest valid consequence and reject tempting alternatives such as only local isometry, conformal ambiguity, or a stronger hypothesis change. Still, this mainly tests recognition of the theorem's precise conclusion rather than substantial derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong and mathematically meaningful. They reflect common failure modes: settling for a weaker true statement (local isometry on the observed region), inferring only neighborhood recovery, removing the small-data condition, or confusing metric recovery with conformal recovery." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-precision MCQ with strong distractors and little answer leakage, though it leans more toward recall of a stated result than genuinely generative mathematical reasoning." + } + }, + { + "id": "2602.08831v1", + "paper_link": "http://arxiv.org/abs/2602.08831v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "[{see Theorem \\ref{theorem l2 del delbar lemma}}]\\label{theorem intro l2 del delbar lemma}\n Let $(M,g)$ be a complete K\\\"ahler manifold such that the self-adjoint Laplacian $\\Delta$ has a spectral gap in $L^2\\Lambda^{k}_\\C$. Given a smooth $L^2$-form $\\alpha\\in L^2A^k_\\C$ which satisfies $\\partial\\alpha=\\delbar\\alpha=0$, then the following conditions are equivalent:\n \\begin{enumerate}\n \\item $\\alpha=\\partial\\delbar\\beta$, for $\\beta\\in L^2 A^{k-2}_\\C$;\n \\item $\\alpha=\\partial\\gamma$, for $\\gamma\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=\\partial\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=d\\lambda$, for $\\lambda\\in L^2 A^{k-1}_\\C$.\n \\end{enumerate}", + "start_pos": 8163, + "end_pos": 8958, + "label": "theorem intro l2 del delbar lemma" + }, + "ref_dict": { + "theorem l2 del delbar lemma": "\\begin{theorem}\\label{theorem l2 del delbar lemma}\n Let $(M,g)$ be a complete K\\\"ahler manifold such that $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{k}_\\C$. Given a smooth $L^2$-form $\\alpha\\in L^2A^k_\\C$ which satisfies $\\del\\alpha=\\delbar\\alpha=0$, then the following conditions are equivalent:\n \\begin{enumerate}\n \\item $\\alpha=\\del\\delbar\\beta$, for $\\beta\\in L^2 A^{k-2}_\\C$;\n \\item $\\alpha=\\del\\gamma$, for $\\gamma\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=\\del\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=d\\lambda$, for $\\lambda\\in L^2 A^{k-1}_\\C$.\n \\end{enumerate}\n\\end{theorem}", + "theorem spectral gap varouchas laplacians": "\\begin{theorem}\\label{theorem spectral gap varouchas laplacians}\n Let $(M,g)$ be a complete K\\\"ahler manifold. \n If $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$, $L^2\\Lambda^{p-1,q}$ and $L^2\\Lambda^{p,q-1}$, then $\\square_{A,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$. If $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$, $L^2\\Lambda^{p+1,q}$ and $L^2\\Lambda^{p,q+1}$, then $\\square_{BC,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$.\n\\end{theorem}", + "theorem intro l2 del delbar lemma": "\\begin{theorem}[{see Theorem \\ref{theorem l2 del delbar lemma}}]\\label{theorem intro l2 del delbar lemma}\n Let $(M,g)$ be a complete K\\\"ahler manifold such that the self-adjoint Laplacian $\\Delta$ has a spectral gap in $L^2\\Lambda^{k}_\\C$. Given a smooth $L^2$-form $\\alpha\\in L^2A^k_\\C$ which satisfies $\\del\\alpha=\\delbar\\alpha=0$, then the following conditions are equivalent:\n \\begin{enumerate}\n \\item $\\alpha=\\del\\delbar\\beta$, for $\\beta\\in L^2 A^{k-2}_\\C$;\n \\item $\\alpha=\\del\\gamma$, for $\\gamma\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=\\del\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=d\\lambda$, for $\\lambda\\in L^2 A^{k-1}_\\C$.\n \\end{enumerate}\n\\end{theorem}", + "theorem closed image del delbar": "\\begin{theorem}\\label{theorem closed image del delbar}\nLet $(M,g)$ be a complete K\\\"ahler manifold. If $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$, then for $b\\in\\{s,w\\}$ it follows that\n$\\im\\del\\delbar_b$ and $\\im\\delbar^*\\del^*_b$ are closed in $L^2\\Lambda^{p,q}$.\n\\end{theorem}", + "remark bounded geometry": "\\begin{remark}\\label{remark bounded geometry}\n Suppose that, in addition to the assumptions of Theorem \\ref{theorem l2 del delbar lemma}, the manifold $(M,g)$ is of bounded geometry and the elliptic Laplacians $\\Delta_{\\delbar}=\\Delta_{\\del}=\\frac12\\Delta_{d}$, $\\tilde\\Delta_{BC}$ and $\\tilde\\Delta_{A}$ are $\\cinf$-bounded and uniformly elliptic, with the bundles of differential forms which are of bounded geometry (cf. \\cite{Sh}). Examples of such manifolds are given by the universal coverings of K\\\"ahler hyperbolic manifolds \\cite{G}. Then, by \\cite[Proposition 1.1]{Sh} the domains of the above operators coincide with the Sobolev spaces\n \\[\n \\D(\\Delta_{\\delbar,sw})=\\D(\\Delta_{\\del,sw})=\\D(\\Delta_{d,sw})=W^2_2(M,\\Lambda^\\bullet_\\C)\n \\]\n and\n \\[\n \\D(\\tilde\\Delta_{A,sw})=\\D(\\tilde\\Delta_{BC,sw})=W^4_2(M,\\Lambda^\\bullet_\\C),\n \\]\n with the notation for Sobolev spaces as in \\cite{Sh}. Therefore, since the image of each Green operator is one of the above domains, and thus the image is a Sobolev space, in the statement of Theorem \\ref{theorem l2 del delbar lemma} we can choose $\\beta\\in W^2_2(M,\\Lambda^{k-2}_\\C)$ and $\\gamma,\\zeta,\\eta,\\theta,\\lambda\\in W^1_2(M,\\Lambda^{k-1}_\\C)$. In particular, $\\del\\beta,\\delbar\\beta\\in W^1_2(M,\\Lambda^{k-1}_\\C)\\subseteq L^2\\Lambda_\\C^{k-1}$.\n\\end{remark}", + "theorem spectral gap kodaira spencer laplacians": "\\begin{theorem}\\label{theorem spectral gap kodaira spencer laplacians}\nLet $(M,g)$ be a complete K\\\"ahler manifold. If $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$, then for $b\\in\\{s,w\\}$ the operators $\\tilde\\Delta_{A,sb}$ and $\\tilde\\Delta_{BC,bw}$ have a spectral gap in $L^2\\Lambda^{p,q}$.\n\\end{theorem}", + "theorem reduced l2 del delbar lemma": "\\begin{theorem}\\label{theorem reduced l2 del delbar lemma}\nGiven a complete K\\\"ahler manifold $(M,g)$, in both the spaces $L^2\\Lambda^k_\\C$ and $L^2\\Lambda^{p,q}$ the following equalities hold true\n\\begin{align*}\n\\c{\\im \\del\\delbar_s}\n&=\\c{\\im \\del_s}\\cap\\ker\\delbar_w=\\c{\\im \\delbar_s}\\cap\\ker\\del_w\\\\\n&=(\\c{\\im\\del_s}+\\c{\\im\\delbar_s})\\cap\\ker\\del_w\\cap\\ker\\delbar_w\\\\\n&=\\c{\\im d_s}\\cap\\ker\\del_w\\cap\\ker\\delbar_w.\n\\end{align*}\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 4719, + "pre_theorem_intro_text": "On compact K\\\"ahler manifolds, the $\\partial\\delbar$-Lemma \\cite[Lemmas 5.11, 5.15]{DGMS} states that for every smooth, complex-valued $k$-form $\\alpha\\in A^k_\\C$ which is $\\partial$- and $\\delbar$-closed, \\textit{i.e.}, $\\partial\\alpha=\\delbar\\alpha=0$, then the following conditions are equivalent:\n\\begin{enumerate}\n \\item $\\alpha=\\partial\\delbar\\beta$, for $\\beta\\in A^{k-2}_\\C$;\n \\item $\\alpha=\\partial\\gamma$, for $\\gamma\\in A^{k-1}_\\C$;\n \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in A^{k-1}_\\C$;\n \\item $\\alpha=\\partial\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in A^{k-1}_\\C$;\n \\item $\\alpha=d\\lambda$, for $\\lambda\\in A^{k-1}_\\C$,\n\\end{enumerate}\nwhere the exterior derivative decomposes as $d=\\partial+\\delbar$.\nThe validity of the above statement is invariant under holomorphic birational maps between compact complex manifolds \\cite[Theorem 5.22]{DGMS}, and has important cohomological and topological implications \\cite[Remark 5.15]{DGMS}: all the complex cohomology spaces (Aeppli, Bott-Chern, Dolbeault cohomology and its conjugate) are isomorphic, and the $k$-de Rham cohomology group is isomorphic to the direct sum of all the $(p,q)$-Dolbeault cohomology groups for $p+q=k$. A numerical characterisation of the validity of the $\\partial\\delbar$-Lemma involving the dimensions of de Rham, Aeppli and Bott-Chern cohomology was proven in \\cite{AT}.\n\nThe main purpose of this paper is to generalise the $\\partial\\delbar$-Lemma to an $L^2$-$\\partial\\delbar$-Lemma involving square integrable forms on a class of complete K\\\"ahler manifolds. We now introduce the result. Given a Hermitian manifold $(M,g)$, denote by $d^*,\\partial^*,\\delbar^*$ the $L^2$-formal adjoints of the differential operators $d,\\partial,\\delbar$, and by $\\Delta_D:=DD^*+D^*D$ for $D\\in\\{d,\\partial,\\delbar\\}$ the Hodge, $\\partial$- and Dolbeault Laplacians, which are second-order formally self-adjoint elliptic operators. It is well known that if the metric is complete, then all these second-order Laplacians are essentially self-adjoint, namely, their restrictions to smooth compactly supported forms have a unique self-adjoint extension as unbounded operators between the Hilbert spaces of $L^2$-forms $L^2\\Lambda^\\bullet_\\C:=\\oplus_{k\\in\\N}L^2\\Lambda^k_\\C$, \\textit{i.e.}, the space of possibly non-continuous, complex-valued forms with bounded $L^2$-norm. These self-adjoint operators are non-negative, therefore their spectrum is contained in $[0,+\\infty)$. We say that such an operator has a \\emph{spectral gap} if its spectrum is contained in $\\{0\\}\\cup[C,+\\infty)$ for some $C>0$. This is equivalent to the image of the operator being closed. \n\nOn the other hand, if the Hermitian metric $g$ is K\\\"ahler, \\textit{i.e.}, its fundamental $2$-form is closed, then by K\\\"ahler identities the above second-order Laplacians coincide up to a factor, namely, $\\Delta_d=2\\Delta_\\delbar=2\\Delta_\\partial$. As a consequence, if the metric is both K\\\"ahler and complete, then all the self-adjoint extensions of these Laplacians coincide up to a factor. In the introduction we denote the unique self-adjoint extension of the Hodge Laplacian by $\\Delta$. \n\nBesides compact K\\\"ahler manifolds, where the spectrum is discrete, examples of complete K\\\"ahler manifolds where $\\Delta$ has a spectral gap on the whole algebra of forms $L^2\\Lambda^\\bullet_\\C$ are given by complete K\\\"ahler $d$-bounded manifolds \\cite[Theorem 1.4.A]{G}, where $d$-bounded means that the fundamental form $\\omega$ is not only closed but also exact $\\omega=d\\eta$, and $\\eta$ is bounded in the pointwise norm. Explicit examples of complete K\\\"ahler $d$-bounded manifolds are: complete simply connected K\\\"ahler manifolds with sectional curvature bounded from above by a negative constant \\cite[0.1.B]{G} \\cite[Lemma 3.2]{CY}, Hermitian symmetric spaces of non compact type \\cite[0.1.C'']{G} \\cite[Proposition 8.6]{Ba}, hyperconvex domains in Stein manifolds \\cite[0.3.A(b)]{G} \\cite[Proposition 2.2]{Do}, strictly pseudoconvex domains in $\\C^n$ with the Bergman metric \\cite[Proposition 3.4]{Do}, bounded homogeneous domains in $\\C^n$ \\cite[Theorem 1]{KO}.\nWe expect this $L^2$-$\\partial\\delbar$-Lemma to be useful for further studies of the above classes of complete K\\\"ahler manifolds. For some recent results related to complete K\\\"ahler $d$-bounded manifolds we refer to \\cite{PT1,BDET,H}. The notion of an $L^2$-$\\partial\\delbar$-Lemma is also of interest on normal coverings of compact complex manifolds, in light of the inequalities between $L^2$ invariants proved in \\cite{HP,BP}.\n\nWe use the notation $L^2A^k_\\C$ to denote the space of smooth $L^2$-forms $L^2\\Lambda^k_\\C\\cap A^k_\\C$. The following theorem is the main result of the paper.", + "context": "On compact K\\\"ahler manifolds, the $\\partial\\delbar$-Lemma \\cite[Lemmas 5.11, 5.15]{DGMS} states that for every smooth, complex-valued $k$-form $\\alpha\\in A^k_\\C$ which is $\\partial$- and $\\delbar$-closed, \\textit{i.e.}, $\\partial\\alpha=\\delbar\\alpha=0$, then the following conditions are equivalent:\n\\begin{enumerate}\n \\item $\\alpha=\\partial\\delbar\\beta$, for $\\beta\\in A^{k-2}_\\C$;\n \\item $\\alpha=\\partial\\gamma$, for $\\gamma\\in A^{k-1}_\\C$;\n \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in A^{k-1}_\\C$;\n \\item $\\alpha=\\partial\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in A^{k-1}_\\C$;\n \\item $\\alpha=d\\lambda$, for $\\lambda\\in A^{k-1}_\\C$,\n\\end{enumerate}\nwhere the exterior derivative decomposes as $d=\\partial+\\delbar$.\nThe validity of the above statement is invariant under holomorphic birational maps between compact complex manifolds \\cite[Theorem 5.22]{DGMS}, and has important cohomological and topological implications \\cite[Remark 5.15]{DGMS}: all the complex cohomology spaces (Aeppli, Bott-Chern, Dolbeault cohomology and its conjugate) are isomorphic, and the $k$-de Rham cohomology group is isomorphic to the direct sum of all the $(p,q)$-Dolbeault cohomology groups for $p+q=k$. A numerical characterisation of the validity of the $\\partial\\delbar$-Lemma involving the dimensions of de Rham, Aeppli and Bott-Chern cohomology was proven in \\cite{AT}.\n\nThe main purpose of this paper is to generalise the $\\partial\\delbar$-Lemma to an $L^2$-$\\partial\\delbar$-Lemma involving square integrable forms on a class of complete K\\\"ahler manifolds. We now introduce the result. Given a Hermitian manifold $(M,g)$, denote by $d^*,\\partial^*,\\delbar^*$ the $L^2$-formal adjoints of the differential operators $d,\\partial,\\delbar$, and by $\\Delta_D:=DD^*+D^*D$ for $D\\in\\{d,\\partial,\\delbar\\}$ the Hodge, $\\partial$- and Dolbeault Laplacians, which are second-order formally self-adjoint elliptic operators. It is well known that if the metric is complete, then all these second-order Laplacians are essentially self-adjoint, namely, their restrictions to smooth compactly supported forms have a unique self-adjoint extension as unbounded operators between the Hilbert spaces of $L^2$-forms $L^2\\Lambda^\\bullet_\\C:=\\oplus_{k\\in\\N}L^2\\Lambda^k_\\C$, \\textit{i.e.}, the space of possibly non-continuous, complex-valued forms with bounded $L^2$-norm. These self-adjoint operators are non-negative, therefore their spectrum is contained in $[0,+\\infty)$. We say that such an operator has a \\emph{spectral gap} if its spectrum is contained in $\\{0\\}\\cup[C,+\\infty)$ for some $C>0$. This is equivalent to the image of the operator being closed.\n\nOn the other hand, if the Hermitian metric $g$ is K\\\"ahler, \\textit{i.e.}, its fundamental $2$-form is closed, then by K\\\"ahler identities the above second-order Laplacians coincide up to a factor, namely, $\\Delta_d=2\\Delta_\\delbar=2\\Delta_\\partial$. As a consequence, if the metric is both K\\\"ahler and complete, then all the self-adjoint extensions of these Laplacians coincide up to a factor. In the introduction we denote the unique self-adjoint extension of the Hodge Laplacian by $\\Delta$.\n\nBesides compact K\\\"ahler manifolds, where the spectrum is discrete, examples of complete K\\\"ahler manifolds where $\\Delta$ has a spectral gap on the whole algebra of forms $L^2\\Lambda^\\bullet_\\C$ are given by complete K\\\"ahler $d$-bounded manifolds \\cite[Theorem 1.4.A]{G}, where $d$-bounded means that the fundamental form $\\omega$ is not only closed but also exact $\\omega=d\\eta$, and $\\eta$ is bounded in the pointwise norm. Explicit examples of complete K\\\"ahler $d$-bounded manifolds are: complete simply connected K\\\"ahler manifolds with sectional curvature bounded from above by a negative constant \\cite[0.1.B]{G} \\cite[Lemma 3.2]{CY}, Hermitian symmetric spaces of non compact type \\cite[0.1.C'']{G} \\cite[Proposition 8.6]{Ba}, hyperconvex domains in Stein manifolds \\cite[0.3.A(b)]{G} \\cite[Proposition 2.2]{Do}, strictly pseudoconvex domains in $\\C^n$ with the Bergman metric \\cite[Proposition 3.4]{Do}, bounded homogeneous domains in $\\C^n$ \\cite[Theorem 1]{KO}.\nWe expect this $L^2$-$\\partial\\delbar$-Lemma to be useful for further studies of the above classes of complete K\\\"ahler manifolds. For some recent results related to complete K\\\"ahler $d$-bounded manifolds we refer to \\cite{PT1,BDET,H}. The notion of an $L^2$-$\\partial\\delbar$-Lemma is also of interest on normal coverings of compact complex manifolds, in light of the inequalities between $L^2$ invariants proved in \\cite{HP,BP}.\n\nWe use the notation $L^2A^k_\\C$ to denote the space of smooth $L^2$-forms $L^2\\Lambda^k_\\C\\cap A^k_\\C$. The following theorem is the main result of the paper.\n\n\\begin{theorem}\\label{theorem l2 del delbar lemma}\n Let $(M,g)$ be a complete K\\\"ahler manifold such that $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{k}_\\C$. Given a smooth $L^2$-form $\\alpha\\in L^2A^k_\\C$ which satisfies $\\del\\alpha=\\delbar\\alpha=0$, then the following conditions are equivalent:\n \\begin{enumerate}\n \\item $\\alpha=\\del\\delbar\\beta$, for $\\beta\\in L^2 A^{k-2}_\\C$;\n \\item $\\alpha=\\del\\gamma$, for $\\gamma\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=\\del\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=d\\lambda$, for $\\lambda\\in L^2 A^{k-1}_\\C$.\n \\end{enumerate}\n\\end{theorem}", + "full_context": "On compact K\\\"ahler manifolds, the $\\partial\\delbar$-Lemma \\cite[Lemmas 5.11, 5.15]{DGMS} states that for every smooth, complex-valued $k$-form $\\alpha\\in A^k_\\C$ which is $\\partial$- and $\\delbar$-closed, \\textit{i.e.}, $\\partial\\alpha=\\delbar\\alpha=0$, then the following conditions are equivalent:\n\\begin{enumerate}\n \\item $\\alpha=\\partial\\delbar\\beta$, for $\\beta\\in A^{k-2}_\\C$;\n \\item $\\alpha=\\partial\\gamma$, for $\\gamma\\in A^{k-1}_\\C$;\n \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in A^{k-1}_\\C$;\n \\item $\\alpha=\\partial\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in A^{k-1}_\\C$;\n \\item $\\alpha=d\\lambda$, for $\\lambda\\in A^{k-1}_\\C$,\n\\end{enumerate}\nwhere the exterior derivative decomposes as $d=\\partial+\\delbar$.\nThe validity of the above statement is invariant under holomorphic birational maps between compact complex manifolds \\cite[Theorem 5.22]{DGMS}, and has important cohomological and topological implications \\cite[Remark 5.15]{DGMS}: all the complex cohomology spaces (Aeppli, Bott-Chern, Dolbeault cohomology and its conjugate) are isomorphic, and the $k$-de Rham cohomology group is isomorphic to the direct sum of all the $(p,q)$-Dolbeault cohomology groups for $p+q=k$. A numerical characterisation of the validity of the $\\partial\\delbar$-Lemma involving the dimensions of de Rham, Aeppli and Bott-Chern cohomology was proven in \\cite{AT}.\n\nThe main purpose of this paper is to generalise the $\\partial\\delbar$-Lemma to an $L^2$-$\\partial\\delbar$-Lemma involving square integrable forms on a class of complete K\\\"ahler manifolds. We now introduce the result. Given a Hermitian manifold $(M,g)$, denote by $d^*,\\partial^*,\\delbar^*$ the $L^2$-formal adjoints of the differential operators $d,\\partial,\\delbar$, and by $\\Delta_D:=DD^*+D^*D$ for $D\\in\\{d,\\partial,\\delbar\\}$ the Hodge, $\\partial$- and Dolbeault Laplacians, which are second-order formally self-adjoint elliptic operators. It is well known that if the metric is complete, then all these second-order Laplacians are essentially self-adjoint, namely, their restrictions to smooth compactly supported forms have a unique self-adjoint extension as unbounded operators between the Hilbert spaces of $L^2$-forms $L^2\\Lambda^\\bullet_\\C:=\\oplus_{k\\in\\N}L^2\\Lambda^k_\\C$, \\textit{i.e.}, the space of possibly non-continuous, complex-valued forms with bounded $L^2$-norm. These self-adjoint operators are non-negative, therefore their spectrum is contained in $[0,+\\infty)$. We say that such an operator has a \\emph{spectral gap} if its spectrum is contained in $\\{0\\}\\cup[C,+\\infty)$ for some $C>0$. This is equivalent to the image of the operator being closed.\n\nOn the other hand, if the Hermitian metric $g$ is K\\\"ahler, \\textit{i.e.}, its fundamental $2$-form is closed, then by K\\\"ahler identities the above second-order Laplacians coincide up to a factor, namely, $\\Delta_d=2\\Delta_\\delbar=2\\Delta_\\partial$. As a consequence, if the metric is both K\\\"ahler and complete, then all the self-adjoint extensions of these Laplacians coincide up to a factor. In the introduction we denote the unique self-adjoint extension of the Hodge Laplacian by $\\Delta$.\n\nBesides compact K\\\"ahler manifolds, where the spectrum is discrete, examples of complete K\\\"ahler manifolds where $\\Delta$ has a spectral gap on the whole algebra of forms $L^2\\Lambda^\\bullet_\\C$ are given by complete K\\\"ahler $d$-bounded manifolds \\cite[Theorem 1.4.A]{G}, where $d$-bounded means that the fundamental form $\\omega$ is not only closed but also exact $\\omega=d\\eta$, and $\\eta$ is bounded in the pointwise norm. Explicit examples of complete K\\\"ahler $d$-bounded manifolds are: complete simply connected K\\\"ahler manifolds with sectional curvature bounded from above by a negative constant \\cite[0.1.B]{G} \\cite[Lemma 3.2]{CY}, Hermitian symmetric spaces of non compact type \\cite[0.1.C'']{G} \\cite[Proposition 8.6]{Ba}, hyperconvex domains in Stein manifolds \\cite[0.3.A(b)]{G} \\cite[Proposition 2.2]{Do}, strictly pseudoconvex domains in $\\C^n$ with the Bergman metric \\cite[Proposition 3.4]{Do}, bounded homogeneous domains in $\\C^n$ \\cite[Theorem 1]{KO}.\nWe expect this $L^2$-$\\partial\\delbar$-Lemma to be useful for further studies of the above classes of complete K\\\"ahler manifolds. For some recent results related to complete K\\\"ahler $d$-bounded manifolds we refer to \\cite{PT1,BDET,H}. The notion of an $L^2$-$\\partial\\delbar$-Lemma is also of interest on normal coverings of compact complex manifolds, in light of the inequalities between $L^2$ invariants proved in \\cite{HP,BP}.\n\nWe use the notation $L^2A^k_\\C$ to denote the space of smooth $L^2$-forms $L^2\\Lambda^k_\\C\\cap A^k_\\C$. The following theorem is the main result of the paper.\n\n\\begin{theorem}\\label{theorem l2 del delbar lemma}\n Let $(M,g)$ be a complete K\\\"ahler manifold such that $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{k}_\\C$. Given a smooth $L^2$-form $\\alpha\\in L^2A^k_\\C$ which satisfies $\\del\\alpha=\\delbar\\alpha=0$, then the following conditions are equivalent:\n \\begin{enumerate}\n \\item $\\alpha=\\del\\delbar\\beta$, for $\\beta\\in L^2 A^{k-2}_\\C$;\n \\item $\\alpha=\\del\\gamma$, for $\\gamma\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=\\del\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=d\\lambda$, for $\\lambda\\in L^2 A^{k-1}_\\C$.\n \\end{enumerate}\n\\end{theorem}\n\n\\begin{theorem}\\label{theorem closed image del delbar}\nLet $(M,g)$ be a complete K\\\"ahler manifold. If $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$, then for $b\\in\\{s,w\\}$ it follows that\n$\\im\\del\\delbar_b$ and $\\im\\delbar^*\\del^*_b$ are closed in $L^2\\Lambda^{p,q}$.\n\\end{theorem}\n\\begin{proof}\nBy Theorem \\ref{theorem spectral gap kodaira spencer laplacians}, $\\tilde\\Delta_{BC,bw}$ has a spectral gap in $L^2\\Lambda^{p,q}$. By Lemma \\ref{lemma spectral gap equiv}, this is equivalent to the closure of the image of $\\tilde\\Delta_{BC,bw}$. Therefore we can orthogonally decompose\n\\begin{align*}\nL^2\\Lambda^{p,q}&=\\ker\\tilde\\Delta_{BC,bw}\\overset{\\perp}{\\oplus}\\im\\tilde\\Delta_{BC,bw}\\\\\n&\\subseteq \\ker\\tilde\\Delta_{BC,bw}\\overset{\\perp}{\\oplus}\\left(\\im\\del\\delbar_b + \\im \\delbar^*\\del^*_s+\\im \\del^*\\delbar_s+\\im\\delbar^*\\del_s+\\im(\\del^*\\oplus\\delbar^*)_s\\right)\\\\\n&\\subseteq \\ker\\tilde\\Delta_{BC,bw}\\overset{\\perp}{\\oplus}\\im\\del\\delbar_b \\overset{\\perp}{\\oplus} (\\im\\del^*_s+\\im\\delbar^*_s)\\subseteq L^2\\Lambda^{p,q}.\n\\end{align*}\nThe first inclusion follows by definition of $\\tilde\\Delta_{BC,bw}$, while the second inclusion is justified by \\emph{(2)} of Proposition \\ref{proposition properties strong weak} and the fact that \n\\[\n\\im(\\del^*\\oplus\\delbar^*)_s\\subseteq\\c{\\im(\\del^*\\oplus\\delbar^*)_s}=\\c{\\im\\del^*_s}+\\c{\\im\\delbar^*_s}=\\im\\del^*_s+\\im\\delbar^*_s,\n\\]\nwhere the first equality is equation \\eqref{equation kernel image del* e delbar*} and the second is obtained via Lemmas \\ref{lemma spectral gap equiv 2} and \\ref{lemma im closed} using the spectral gap of $\\Delta_{\\delbar,sw}=\\Delta_{\\del,sw}$ in $L^2\\Lambda^{p,q}$.\nThe orthogonality of the above decompositions is easy to check.\nThe closure of $\\im\\del\\delbar_b$ now follows from the orthogonality of the decomposition of $L^2\\Lambda^{p,q}$ in the last line. Using the spectral gap of $\\tilde\\Delta_{A,sb}$ we similarly find the closure of $\\im\\delbar^*\\del^*_b$ in $L^2\\Lambda^{p,q}$.\n\\end{proof}\n\n\\begin{theorem}\\label{theorem l2 del delbar lemma}\n Let $(M,g)$ be a complete K\\\"ahler manifold such that $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{k}_\\C$. Given a smooth $L^2$-form $\\alpha\\in L^2A^k_\\C$ which satisfies $\\del\\alpha=\\delbar\\alpha=0$, then the following conditions are equivalent:\n \\begin{enumerate}\n \\item $\\alpha=\\del\\delbar\\beta$, for $\\beta\\in L^2 A^{k-2}_\\C$;\n \\item $\\alpha=\\del\\gamma$, for $\\gamma\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=\\del\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=d\\lambda$, for $\\lambda\\in L^2 A^{k-1}_\\C$.\n \\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n Since $\\Delta_{\\delbar,sw}=\\Delta_{\\del,sw}=\\frac12\\Delta_{d,sw}$ has a spectral gap in $L^2\\Lambda^{k}_\\C$, by Theorem \\ref{theorem spectral gap kodaira spencer laplacians} also $\\tilde\\Delta_{BC,sw}$ and $\\tilde\\Delta_{A,sw}$ have a spectral gap in $L^2\\Lambda^{k}_\\C$.\n We define the orthogonal projection\n \\[\n P_{BC}:L^2\\Lambda^{k}_\\C\\to\\ker \\tilde\\Delta_{BC,sw}\n \\]\n and the Green operator\n \\begin{align*}\n G_{BC}:&\\im \\tilde\\Delta_{BC,sw}\\to \\D(\\tilde\\Delta_{BC,sw})\\\\\n & \\tilde\\Delta_{BC,sw}\\beta\\mapsto\\beta.\n \\end{align*}\n Being $\\im \\tilde\\Delta_{BC,sw}$ closed, and recalling the orthogonal decomposition\n \\[\n L^2\\Lambda^{k}_\\C=\\ker \\tilde\\Delta_{BC,sw}\\oplus\\im \\tilde\\Delta_{BC,sw},\n \\]\n we can extend the Green operator on the whole space $L^2\\Lambda^{k}_\\C$ simply by setting $G_{BC}=0$ on $\\ker \\tilde\\Delta_{BC,sw}$. Given any form $\\alpha\\in L^2\\Lambda^{k}_\\C$, it decomposes as\n \\[\n \\alpha=P_{BC}\\alpha+(id-P_{BC})\\alpha,\n \\]\n where $(id-P_{BC})\\alpha=\\tilde\\Delta_{BC,sw}\\beta$. Now $G_{BC}\\alpha=\\beta$, therefore\n \\[\n \\alpha=P_{BC}\\alpha+\\tilde\\Delta_{BC,sw}G_{BC}\\alpha.\n \\]\n If, in particular, $\\alpha$ is smooth and so $\\alpha\\in L^2A^k_\\C$, then by elliptic regularity (see, \\textit{e.g.}, \\cite[Theorem 2.1]{HP}) also $G_{BC}\\alpha=\\beta\\in L^2A^k_\\C\\cap \\D(\\tilde\\Delta_{BC,sw})$ is smooth. In particular, by \\emph{(5)} of Proposition \\ref{proposition properties strong weak} we obtain \n \\[\n \\tilde\\Delta_{BC,sw}G_{BC}\\alpha=\\tilde\\Delta_{BC}G_{BC}\\alpha.\n \\]\n The same holds for analogue operators $P_\\delta$ and $G_\\delta$ for $\\delta\\in\\{A,\\del,\\delbar,d\\}$.\n Therefore, if $\\alpha\\in L^2A^k_\\C$, for $\\delta\\in\\{A,BC\\}$ and $\\epsilon\\in\\{\\del,\\delbar,d\\}$ we deduce decompositions\n \\[\n \\alpha=P_{\\delta}\\alpha+\\tilde\\Delta_{\\delta}G_{\\delta}\\alpha=P_\\epsilon\\alpha+\\Delta_{\\epsilon}G_{\\epsilon}\\alpha,\n \\]\n with smooth forms $G_{\\delta}\\alpha,G_{\\epsilon}\\alpha\\in L^2A^k_\\C$. Recall that by Theorem \\ref{theorem complete kahler equality harmonic} the kernels of $\\Delta_{\\delbar,sw}=\\Delta_{\\del,sw}=\\frac12\\Delta_{d,sw}$, $\\tilde\\Delta_{BC,sw}$ and $\\tilde\\Delta_{A,sw}$ coincide, thus $P_{BC}\\alpha=0$ iff $P_\\delta=0$ for all $\\delta\\in\\{A,\\del,\\delbar,d\\}$.\n\nNow assume that $\\alpha\\in L^2A^k_\\C$ satisfies $\\del\\alpha=\\delbar\\alpha=0$. We know $\\ker\\del_w\\cap\\ker\\delbar_w\\subseteq L^2A^k_\\C$ is $L^2$-orthogonal to\n \\begin{align*}\n &\\im d_s^*, &&\\im\\del_s^*, & &\\im\\delbar_s^*, &\\im(\\del^*\\oplus\\delbar^*)_s, &&\\im\\del\\delbar_s^*,\n \\end{align*}\n as in the proof of Theorem \\ref{theorem reduced l2 del delbar lemma}. Thanks to this and to \\emph{(5)} of Proposition \\ref{proposition properties strong weak}, we obtain\n \\begin{align*}\n &\\alpha=\\tilde\\Delta_{A}G_{A}\\alpha &&\\iff && \\alpha=(\\del\\del^*+\\delbar\\delbar^*)G_{A}\\alpha,\\\\\n &\\alpha=\\tilde\\Delta_{BC}G_{BC}\\alpha &&\\iff && \\alpha=\\del\\delbar\\delbar^*\\del^*G_{BC}\\alpha,\\\\\n &\\alpha=\\Delta_{\\del}G_{\\del}\\alpha &&\\iff && \\alpha=\\del\\del^*G_{\\del}\\alpha,\\\\\n &\\alpha=\\Delta_{\\delbar}G_{\\delbar}\\alpha &&\\iff && \\alpha=\\delbar\\delbar^*G_{\\delbar}\\alpha,\\\\\n &\\alpha=\\Delta_{d}G_{d}\\alpha &&\\iff && \\alpha=dd^*G_{d}\\alpha.\n \\end{align*}\n It is then enough to set $\\beta:=\\delbar^*\\del^*G_{BC}\\alpha$, $\\gamma:=\\del^*G_{\\del}\\alpha$, $\\zeta:=\\delbar^*G_{\\delbar}\\alpha$, $\\eta:=\\del^*G_{A}\\alpha$, $\\theta:=\\delbar^*G_{A}\\alpha$, $\\lambda:=d^*G_{d}\\alpha$, which lie in $L^2\\Lambda^\\bullet_\\C$ since the image of each Green operator is the domain of the corresponding Laplacian.\n This concludes the proof.\n\\end{proof}\n\n\\begin{remark}\\label{remark bounded geometry}\n Suppose that, in addition to the assumptions of Theorem \\ref{theorem l2 del delbar lemma}, the manifold $(M,g)$ is of bounded geometry and the elliptic Laplacians $\\Delta_{\\delbar}=\\Delta_{\\del}=\\frac12\\Delta_{d}$, $\\tilde\\Delta_{BC}$ and $\\tilde\\Delta_{A}$ are $\\cinf$-bounded and uniformly elliptic, with the bundles of differential forms which are of bounded geometry (cf. \\cite{Sh}). Examples of such manifolds are given by the universal coverings of K\\\"ahler hyperbolic manifolds \\cite{G}. Then, by \\cite[Proposition 1.1]{Sh} the domains of the above operators coincide with the Sobolev spaces\n \\[\n \\D(\\Delta_{\\delbar,sw})=\\D(\\Delta_{\\del,sw})=\\D(\\Delta_{d,sw})=W^2_2(M,\\Lambda^\\bullet_\\C)\n \\]\n and\n \\[\n \\D(\\tilde\\Delta_{A,sw})=\\D(\\tilde\\Delta_{BC,sw})=W^4_2(M,\\Lambda^\\bullet_\\C),\n \\]\n with the notation for Sobolev spaces as in \\cite{Sh}. Therefore, since the image of each Green operator is one of the above domains, and thus the image is a Sobolev space, in the statement of Theorem \\ref{theorem l2 del delbar lemma} we can choose $\\beta\\in W^2_2(M,\\Lambda^{k-2}_\\C)$ and $\\gamma,\\zeta,\\eta,\\theta,\\lambda\\in W^1_2(M,\\Lambda^{k-1}_\\C)$. In particular, $\\del\\beta,\\delbar\\beta\\in W^1_2(M,\\Lambda^{k-1}_\\C)\\subseteq L^2\\Lambda_\\C^{k-1}$.\n\\end{remark}", + "post_theorem_intro_text_len": 6129, + "post_theorem_intro_text": "The proof of Theorem \\ref{theorem intro l2 del delbar lemma} relies primarily on new spectral properties of some self-adjoint operators introduced in \\cite{HP} as part of the development of the Aeppli and Bott-Chern versions of the $L^2$-Hodge theory of general Hermitian manifolds and of complete K\\\"ahler manifolds. More precisely, we prove that if $\\Delta$ has a spectral gap, there are self-adjoint extensions of fourth-order elliptic Aeppli and Bott-Chern Laplacians having a spectral gap, thereby allowing one to use arguments based on elliptic regularity. Since this constitutes the core of the proof, we now discuss these Aeppli and Bott–Chern Laplacians in more detail.\n\nGiven a compact Hermitian manifold, there is a unique choice for the second-order Hodge, $\\partial$- and Dolbeault Laplacians $\\Delta_D=DD^*+D^*D$, for $D\\in\\{d,\\partial,\\delbar\\}$, such that the corresponding kernels are isomorphic to de Rham, $\\partial$- and Dolbeault cohomology, via classical Hodge theory. In the Aeppli and Bott-Chern cases the situation is slightly different: there are multiple possible choices for the associated Laplacians. We recall the definitions of Aeppli and Bott-Chern cohomology via the related differential complex. We denote the spaces of $(p,q)$-forms by $A^{p,q}$, so that $A^k_\\C=\\oplus_{p+q=k}A^{p,q}$, and the Hilbert space of $L^2$-$(p,q)$-forms by $L^2\\Lambda^{p,q}$. For any choice of integers $(p,q)$ we consider the complex\n\\[\n\\dots\\longrightarrow A^{p-1,q-2}\\oplus A^{p-2,q-1}\\overset{\\delbar\\oplus\\partial}{\\longrightarrow} A^{p-1,q-1}\\overset{\\partial\\delbar}{\\longrightarrow} A^{p,q}\\overset{\\partial+\\delbar}{\\longrightarrow} A^{p+1,q}\\oplus A^{p,q+1}{\\longrightarrow}\\dots\n\\]\nwhere $\\delbar\\oplus\\partial$ operates on $A^{p-1,q-2}\\oplus A^{p-2,q-1}$ as $\\delbar$ on $A^{p-1,q-2}$ plus $\\partial$ on $A^{p-2,q-1}$. The Aeppli and Bott-Chern cohomology spaces are defined as\n\\begin{align*}\nH^{p-1,q-1}_A:=\\frac{\\ker\\partial\\delbar\\cap A^{p-1,q-1}}{\\im\\delbar\\oplus\\partial},&&H^{p,q}_{BC}:=\\frac{\\ker(\\partial+\\delbar)\\cap A^{p,q}}{\\im\\partial\\delbar}.\n\\end{align*}\nThe \\lq\\lq natural\" Aeppli and Bott-Chern Laplacians are then defined as\n\\begin{align*}\n\\Delta_A&:=\\delbar^*\\partial^*\\partial\\delbar+(\\delbar\\oplus\\partial)(\\delbar\\oplus\\partial)^*=\\delbar^*\\partial^*\\partial\\delbar+\\partial\\partial^*+\\delbar\\delbar^*,\\\\\n\\Delta_{BC}&:=\\partial\\delbar\\delbar^*\\partial^*+(\\partial+\\delbar)^*(\\partial+\\delbar)=\\partial\\delbar\\delbar^*\\partial^*+\\partial^*\\partial+\\delbar^*\\delbar.\n\\end{align*}\nThe kernels of these operators are isomorphic to the Aeppli and Bott-Chern cohomology spaces, but they are not elliptic \\cite[Proposition 2.1]{S}. The first elliptic fourth-order operators whose kernels were shown to be isomorphic to the Aeppli and Bott-Chern cohomologies were defined in \\cite{KS} as\n\\begin{align*}\n\\tilde\\Delta_{A} &:=\n\\partial\\delbar\\delbar^*\\partial^*+\n\\delbar^*\\partial^*\\partial\\delbar+\n\\partial\\delbar^*\\delbar\\partial^*+\\delbar\\partial^*\\partial\\delbar^*+\n\\partial\\partial^*+\\delbar\\delbar^*,\\\\\n\\tilde\\Delta_{BC} &:=\n\\partial\\delbar\\delbar^*\\partial^*+\n\\delbar^*\\partial^*\\partial\\delbar+\\partial^*\\delbar\\delbar^*\\partial+\\delbar^*\\partial\\partial^*\\delbar\n+\\partial^*\\partial+\\delbar^*\\delbar.\n\\end{align*}\nWe will refer to these operators as the Kodaira-Spencer Laplacians. Another pair of elliptic fourth-order operators whose kernels are isomorphic to the Aeppli and Bott-Chern cohomologies were defined in \\cite{V} as\n\\begin{align*}\n\\square_{A}&:=\\delbar^*\\partial^*\\partial\\delbar+((\\delbar\\oplus\\partial)(\\delbar\\oplus\\partial)^*)^2=\\delbar^*\\partial^*\\partial\\delbar+(\\partial\\partial^*+\\delbar\\delbar^*)^2,\\\\\n\\square_{BC}&:=\\partial\\delbar\\delbar^*\\partial^*+((\\partial+\\delbar)^*(\\partial+\\delbar))^2=\\partial\\delbar\\delbar^*\\partial^*+(\\partial^*\\partial+\\delbar^*\\delbar)^2.\n\\end{align*}\nWe will refer to these operators as the Varouchas Laplacians.\n\nIn \\cite[Corollary 8.12]{HP} it was proven that on a complete K\\\"ahler manifold, if $\\Delta$ has a spectral gap in $L^2\\Lambda^\\bullet_\\C$, then there are self-adjoint extensions of $\\Delta_A$ and $\\Delta_{BC}$ having a spectral gap. Being $\\Delta_A$ and $\\Delta_{BC}$ non-elliptic, this was not sufficient to prove Theorem \\ref{theorem intro l2 del delbar lemma}. In fact, to prove our main result, we use the spectral gap of the elliptic Kodaira-Spencer Laplacians, which is shown in Theorem \\ref{theorem spectral gap kodaira spencer laplacians}. We also prove a spectral gap of the Varouchas Laplacians in Theorem \\ref{theorem spectral gap varouchas laplacians}. We remark that the spectral gap assumption of $\\Delta$ is essential for a statement which directly generalises the classical $\\partial\\delbar$-Lemma such as Theorem \\ref{theorem intro l2 del delbar lemma} does. Indeed, in the absence of a spectral gap, the images of the closed extensions of $d$, $\\partial$, $\\delbar$, and $\\partial\\delbar$ fail to be closed. For a weaker statement on complete K\\\"ahler manifolds without assuming a spectral gap, see Theorem \\ref{theorem reduced l2 del delbar lemma} (cf. \\cite[Corollary 8.6]{HP}).\n\nThe paper is organised as follows. In Section \\ref{section preliminaries}, we present preliminaries on Hilbert complexes, self-adjoint operators, spectral gaps, minimal and maximal closed extensions of differential operators, differential complexes on complex manifolds, self-adjoint extensions of second and fourth-order complex Laplacians, $L^2$ Hodge theory on complete K\\\"ahler manifolds. In Section \\ref{section spectral gap}, we prove the spectral gaps of the elliptic Aeppli and Bott-Chern Laplacians. In Theorem \\ref{theorem closed image del delbar} we also present a new proof of the closure of the image of the closed extensions of the operator $\\partial\\delbar$, originally proved in \\cite[Theorem 8.10]{HP}. Finally, in section \\ref{section l2 del delbar lemma}, we prove our main result Theorem \\ref{theorem intro l2 del delbar lemma}. In Remark \\ref{remark bounded geometry} we also point out an improvement of Theorem \\ref{theorem intro l2 del delbar lemma} when the manifold is of bounded geometry.", + "sketch": "The post-theorem introduction says that “The proof of Theorem \\ref{theorem intro l2 del delbar lemma} relies primarily on new spectral properties of some self-adjoint operators introduced in \\cite{HP}.” More precisely, they “prove that if $\\Delta$ has a spectral gap, there are self-adjoint extensions of fourth-order elliptic Aeppli and Bott-Chern Laplacians having a spectral gap, thereby allowing one to use arguments based on elliptic regularity,” and this is described as “the core of the proof.”\n\nBecause the “natural” Aeppli and Bott-Chern Laplacians $\\Delta_A,\\Delta_{BC}$ have kernels matching Aeppli/Bott--Chern cohomology but “are not elliptic,” the text explains that the spectral gap for self-adjoint extensions of $\\Delta_A,\\Delta_{BC}$ from \\cite[Corollary 8.12]{HP} “was not sufficient to prove Theorem \\ref{theorem intro l2 del delbar lemma}.” Instead, “to prove our main result, we use the spectral gap of the elliptic Kodaira-Spencer Laplacians,” establishing it in “Theorem \\ref{theorem spectral gap kodaira spencer laplacians},” and they “also prove a spectral gap of the Varouchas Laplacians in Theorem \\ref{theorem spectral gap varouchas laplacians}.”\n\nFinally, the introduction remarks that the “spectral gap assumption of $\\Delta$ is essential” for a direct generalisation like Theorem \\ref{theorem intro l2 del delbar lemma}, since “in the absence of a spectral gap, the images of the closed extensions of $d$, $\\partial$, $\\delbar$, and $\\partial\\delbar$ fail to be closed,” and it points to a weaker variant without a spectral gap (Theorem \\ref{theorem reduced l2 del delbar lemma}).", + "expanded_sketch": "The post-theorem introduction says that “The proof of the main theorem relies primarily on new spectral properties of some self-adjoint operators introduced in \\cite{HP}.” More precisely, they “prove that if $\\Delta$ has a spectral gap, there are self-adjoint extensions of fourth-order elliptic Aeppli and Bott-Chern Laplacians having a spectral gap, thereby allowing one to use arguments based on elliptic regularity,” and this is described as “the core of the proof.”\n\nBecause the “natural” Aeppli and Bott-Chern Laplacians $\\Delta_A,\\Delta_{BC}$ have kernels matching Aeppli/Bott--Chern cohomology but “are not elliptic,” the text explains that the spectral gap for self-adjoint extensions of $\\Delta_A,\\Delta_{BC}$ from \\cite[Corollary 8.12]{HP} “was not sufficient to prove the main theorem.” Instead, “to prove our main result, we use the spectral gap of the elliptic Kodaira-Spencer Laplacians,” establishing it in the following theorem.\n\n\\begin{theorem}\\label{theorem spectral gap kodaira spencer laplacians}\nLet $(M,g)$ be a complete K\\\"ahler manifold. If $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$, then for $b\\in\\{s,w\\}$ the operators $\\tilde\\Delta_{A,sb}$ and $\\tilde\\Delta_{BC,bw}$ have a spectral gap in $L^2\\Lambda^{p,q}$.\n\\end{theorem}\n\nThey “also prove a spectral gap of the Varouchas Laplacians” in the following theorem.\n\n\\begin{theorem}\\label{theorem spectral gap varouchas laplacians}\n Let $(M,g)$ be a complete K\\\"ahler manifold. \n If $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$, $L^2\\Lambda^{p-1,q}$ and $L^2\\Lambda^{p,q-1}$, then $\\square_{A,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$. If $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$, $L^2\\Lambda^{p+1,q}$ and $L^2\\Lambda^{p,q+1}$, then $\\square_{BC,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$.\n\\end{theorem}\n\nFinally, the introduction remarks that the “spectral gap assumption of $\\Delta$ is essential” for a direct generalisation like the main theorem, since “in the absence of a spectral gap, the images of the closed extensions of $d$, $\\partial$, $\\delbar$, and $\\partial\\delbar$ fail to be closed,” and it points to a weaker variant without a spectral gap given by the following theorem.\n\n\\begin{theorem}\\label{theorem reduced l2 del delbar lemma}\nGiven a complete K\\\"ahler manifold $(M,g)$, in both the spaces $L^2\\Lambda^k_\\C$ and $L^2\\Lambda^{p,q}$ the following equalities hold true\n\\begin{align*}\n\\c{\\im \\del\\delbar_s}\n&=\\c{\\im \\del_s}\\cap\\ker\\delbar_w=\\c{\\im \\delbar_s}\\cap\\ker\\del_w\\\\\n&=(\\c{\\im\\del_s}+\\c{\\im\\delbar_s})\\cap\\ker\\del_w\\cap\\ker\\delbar_w\\\\\n&=\\c{\\im d_s}\\cap\\ker\\del_w\\cap\\ker\\delbar_w.\n\\end{align*}\n\\end{theorem}", + "expanded_theorem": "\\begin{theorem}\\label{theorem l2 del delbar lemma}\n Let $(M,g)$ be a complete K\\\"ahler manifold such that $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{k}_\\C$. Given a smooth $L^2$-form $\\alpha\\in L^2A^k_\\C$ which satisfies $\\del\\alpha=\\delbar\\alpha=0$, then the following conditions are equivalent:\n \\begin{enumerate}\n \\item $\\alpha=\\del\\delbar\\beta$, for $\\beta\\in L^2 A^{k-2}_\\C$;\n \\item $\\alpha=\\del\\gamma$, for $\\gamma\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=\\del\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=d\\lambda$, for $\\lambda\\in L^2 A^{k-1}_\\C$.\n \\end{enumerate}\n\\end{theorem}\\label{theorem intro l2 del delbar lemma}\n Let $(M,g)$ be a complete K\\\"ahler manifold such that the self-adjoint Laplacian $\\Delta$ has a spectral gap in $L^2\\Lambda^{k}_\\C$. Given a smooth $L^2$-form $\\alpha\\in L^2A^k_\\C$ which satisfies $\\partial\\alpha=\\delbar\\alpha=0$, then the following conditions are equivalent:\n \\begin{enumerate}\n \\item $\\alpha=\\partial\\delbar\\beta$, for $\\beta\\in L^2 A^{k-2}_\\C$;\n \\item $\\alpha=\\partial\\gamma$, for $\\gamma\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=\\partial\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=d\\lambda$, for $\\lambda\\in L^2 A^{k-1}_\\C$.\n \\end{enumerate},", + "theorem_type": [ + "Biconditional or Equivalence", + "Existence" + ], + "mcq": { + "question": "Let $(M,g)$ be a complete K\\\"ahler manifold. Write $L^2A^k_{\\C}$ for the space of smooth complex-valued $k$-forms on $M$ with finite $L^2$ norm, and write $d=\\partial+\\bar\\partial$. Assume that the self-adjoint Dolbeault Laplacian $\\Delta_{\\bar\\partial,sw}$ on $L^2\\Lambda^k_{\\C}$ has a spectral gap, meaning that its spectrum is contained in $\\{0\\}\\cup[C,+\\infty)$ for some $C>0$ (equivalently, on a complete K\\\"ahler manifold, the self-adjoint Hodge Laplacian $\\Delta$ has a spectral gap on $L^2\\Lambda^k_{\\C}$). Let $\\alpha\\in L^2A^k_{\\C}$ satisfy $\\partial\\alpha=\\bar\\partial\\alpha=0$. Which statement about the existence of $L^2$-forms representing $\\alpha$ holds?", + "correct_choice": { + "label": "A", + "text": "The following five existence statements are equivalent: (1) there exists $\\beta\\in L^2A^{k-2}_{\\C}$ such that $\\alpha=\\partial\\bar\\partial\\beta$; (2) there exists $\\gamma\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\partial\\gamma$; (3) there exists $\\zeta\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\bar\\partial\\zeta$; (4) there exist $\\eta,\\theta\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\partial\\eta+\\bar\\partial\\theta$; and (5) there exists $\\lambda\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=d\\lambda$." + }, + "choices": [ + { + "label": "B", + "text": "The following five existence statements are equivalent: (1) there exists $\\beta\\in L^2A^{k-2}_{\\C}$ such that $\\alpha=\\partial\\bar\\partial\\beta$; (2) there exists $\\gamma\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\partial\\gamma$; (3) there exists $\\zeta\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\bar\\partial\\zeta$; (4) there exist $\\eta,\\theta\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\partial\\eta+\\bar\\partial\\theta$; and (5) there exists $\\lambda\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=d\\lambda$, and moreover one may always choose all of $\\beta,\\gamma,\\zeta,\\eta,\\theta,\\lambda$ to be smooth $L^2$-forms." + }, + { + "label": "C", + "text": "If there exists $\\beta\\in L^2A^{k-2}_{\\C}$ such that $\\alpha=\\partial\\bar\\partial\\beta$, then there exist $\\gamma,\\zeta,\\eta,\\theta,\\lambda\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\partial\\gamma=\\bar\\partial\\zeta=\\partial\\eta+\\bar\\partial\\theta=d\\lambda$." + }, + { + "label": "D", + "text": "The following five existence statements are equivalent: (1) there exists $\\beta\\in L^2A^{k-2}_{\\C}$ such that $\\alpha=\\partial\\bar\\partial\\beta$; (2) there exists $\\gamma\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\partial\\gamma$; (3) there exists $\\zeta\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\bar\\partial\\zeta$; (4) there exist $\\eta,\\theta\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\partial\\eta+\\bar\\partial\\theta$; and (5) there exists $\\lambda\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=d\\lambda$, even if one drops the spectral-gap assumption on $\\Delta_{\\bar\\partial,sw}$." + }, + { + "label": "E", + "text": "The following five existence statements are equivalent provided that $\\Delta_{\\bar\\partial,sw}$ has a spectral gap not only on $L^2\\Lambda^{k}_{\\C}$ but also on $L^2\\Lambda^{k-1}_{\\C}$: (1) there exists $\\beta\\in L^2A^{k-2}_{\\C}$ such that $\\alpha=\\partial\\bar\\partial\\beta$; (2) there exists $\\gamma\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\partial\\gamma$; (3) there exists $\\zeta\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\bar\\partial\\zeta$; (4) there exist $\\eta,\\theta\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\partial\\eta+\\bar\\partial\\theta$; and (5) there exists $\\lambda\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=d\\lambda$." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "B" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "regularity", + "tampered_component": "smoothness_of_L2_primitives", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped_reverse_implications_between_the_five_conditions", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "finiteness", + "tampered_component": "necessity_of_spectral_gap_for_closed_images_and_full_equivalence", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "case_split", + "tampered_component": "extra_degreewise_spectral_gap_hypothesis", + "template_used": "boundary_range" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not reveal the correct option. It states the hypotheses and asks which conclusion holds, but does not explicitly signal that the full five-way equivalence in choice A is the intended answer." + }, + "TAS": { + "score": 1, + "justification": "The item is still quite close to a theorem-statement recognition question: the correct choice is essentially the theorem's conclusion under the stated hypotheses. However, it is not a pure restatement because the alternatives vary the hypotheses, logical direction, and regularity claims." + }, + "GPS": { + "score": 2, + "justification": "To choose correctly, the solver must compare several nearby logical variants: full equivalence vs. one-way implication, necessity of the spectral-gap assumption, and whether extra smoothness or extra degreewise assumptions are justified. This creates real reasoning pressure rather than making the answer obvious." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong and mathematically meaningful. They reflect common failure modes: overclaiming regularity, accepting only a weaker true implication, dropping a necessary hypothesis, or adding an unnecessary extra assumption." + }, + "total_score": 7, + "overall_assessment": "A strong MCQ with little answer leakage and high-quality distractors. Its main weakness is that it remains close to theorem recall rather than fully independent generative problem solving." + } + }, + { + "id": "2602.09669v1", + "paper_link": "http://arxiv.org/abs/2602.09669v1", + "theorems_cnt": 5, + "theorem": { + "env_name": "thm", + "content": "\\label{Lions-theo}\nFor any $s$ in $\\mathbb R$, the space $\\mathcal H_s(\\Omega) $ is a RKHS and its two-point kernel $K_s$ is given, for any pair of $x$ and $y$ in $\\Omega$, by \n \\begin{equation} \\label{l-formu}\n K_s (x,y)= \\int_{\\partial\\Omega} \\Big((M^{-\\frac{s}{2}} \\, \\gamma_N \\big((G_1(x,\\cdot)\\big)\\Big)(z) \\, \\Big(M^{-\\frac{s}{2}} \\, \\gamma_N \\big((G_1(y,\\cdot)\\big) \\Big)(z)\\,d\\sigma(z) .\n \\end{equation}", + "start_pos": 14535, + "end_pos": 14965, + "label": "Lions-theo" + }, + "ref_dict": { + "Lions-theo": "\\begin{thm} \\label{Lions-theo}\nFor any $s$ in $\\R$, the space $\\mathcal H_s(\\Omega) $ is a RKHS and its two-point kernel $K_s$ is given, for any pair of $x$ and $y$ in $\\Omega$, by \n \\begin{equation} \\label{l-formu}\n K_s (x,y)= \\int_{\\partial\\Omega} \\Big((M^{-\\frac{s}{2}} \\, \\gamma_N \\big((G_1(x,\\cdot)\\big)\\Big)(z) \\, \\Big(M^{-\\frac{s}{2}} \\, \\gamma_N \\big((G_1(y,\\cdot)\\big) \\Big)(z)\\,d\\sigma(z) .\n \\end{equation}\n\\end{thm}", + "DIR": "\\begin{equation} \\label{DIR}\n -\\Delta G_1(x,\\cdot)=\\delta_x \\quad\\text{in}\\quad \\mathcal D'(\\Omega)\\qquad\\&\\qquad G_1(x,\\cdot)=0\\quad\\text{on}\\quad \\partial\\Omega, \n\\end{equation}", + "defRKHS": "\\begin{Definition} \\label{defRKHS}\nWe say that a Hilbert space $H$ of real-valued functions defined on $\\Omega$ is a reproducing kernel Hilbert space (RKHS for short) \n if for any $x$ in $\\Omega$, the evaluation mapping $E_x$ defined for any $u$ in $H$ by $E_x (u) = u(x)$ \nis linear and continuous from $H$ to $\\R$. Then, we denote by $K_x$ the unique element in $H$ such that for any $u$ in $H$, $E_x (u) = \\langle u,K_x\\rangle $, the inner product of $u$ and $K_x$ in $H$, given by Riesz' theorem. Finally, the two-point kernel is defined for any pair of $x$ and $y$ in $\\Omega$ by $K(x,y)= \\langle K_x,K_y\\rangle $. \n\\end{Definition}", + "Cani2": "\\begin{equation} \\label{Cani2}\nu(x) = \\langle \\gamma_D (u) ,\\, K_x \\rangle_s = \\langle M^{\\frac{s}{2}} \\, \\gamma_D (u) ,\\, M^{\\frac{s}{2}} \\, K_x \\rangle_{L^2 (\\partial \\Omega)}.\n\\end{equation}", + "Cani1": "\\begin{equation} \\label{Cani1}\n u(x)\n = \\Big\\langle \\gamma_D (u),\\, M^{-{s}} \\,\\gamma_N \\big( G_1 (x,\\cdot)\\big) \\Big\\rangle_s ,\n\\end{equation}", + "REg": "\\begin{equation} \\label{REg}\n \\forall x \\in \\Omega, \\quad \\forall s \\in \\R, \\quad M^{-\\frac{s}{2}} \\gamma_N \\big(G_1(x,\\cdot)\\big) \\in L^2 (\\partial\\Omega) ,\n\\end{equation}", + "l-formu": "\\begin{equation} \\label{l-formu}\n K_s (x,y)= \\int_{\\partial\\Omega} \\Big((M^{-\\frac{s}{2}} \\, \\gamma_N \\big((G_1(x,\\cdot)\\big)\\Big)(z) \\, \\Big(M^{-\\frac{s}{2}} \\, \\gamma_N \\big((G_1(y,\\cdot)\\big) \\Big)(z)\\,d\\sigma(z) .\n \\end{equation}", + "Hadamard": "\\begin{align} \\label{Hadamard}\n DG_\\Omega (\\alpha) (x,y) =\\int_{\\partial\\Omega} \\gamma_N \\big(G_1(x,\\cdot)\\big)(z) \\, \\gamma_N \\big(G_1(y,\\cdot)\\big)(z) \\, \\alpha(z)d\\sigma(z) .\n\\end{align}" + }, + "pre_theorem_intro_text_len": 7752, + "pre_theorem_intro_text": "\\subsection{Reproducing kernel Hilbert space}\n\nThe notion of reproducing kernel Hilbert space was first introduced in 1907 by Stanislaw Zaremba for boundary value problems for harmonic and biharmonic functions, and simultaneously by James Mercer in the theory of integral equations, before being more systematically tackled by Nachman Aronszajn and Stefan Bergman. These spaces have various applications in complex analysis, harmonic analysis, quantum mechanics, and statistical learning theory.\nThey are now defined as Hilbert spaces of functions in which pointwise evaluations are continuous linear functionals. It then follows from Riesz' theorem that each of these functionals can be represented as an inner product with an element of this Hilbert space. One defines a two-point kernel by considering the inner products of any pair of such elements. \\ \\par \\ \nMore precisely, we have the following definition, where $\\Omega$, as in all the paper, is a bounded connected open set of class $C^\\infty$ in $\\mathbb R^N$.\n\\begin{Definition} \\label{defRKHS}\nWe say that a Hilbert space $H$ of real-valued functions defined on $\\Omega$ is a reproducing kernel Hilbert space (RKHS for short) \n if for any $x$ in $\\Omega$, the evaluation mapping $E_x$ defined for any $u$ in $H$ by $E_x (u) = u(x)$ \nis linear and continuous from $H$ to $\\mathbb R$. Then, we denote by $K_x$ the unique element in $H$ such that for any $u$ in $H$, $E_x (u) = \\langle u,K_x\\rangle $, the inner product of $u$ and $K_x$ in $H$, given by Riesz' theorem. Finally, the two-point kernel is defined for any pair of $x$ and $y$ in $\\Omega$ by $K(x,y)= \\langle K_x,K_y\\rangle $. \n\\end{Definition}\nObserve that it follows from the properties of the inner product that the kernel $K$ is symmetric and positive definite. \nActually, a famous theorem by E. H. Moore and N. Aronszajn, see \\cite{Aronszajn}, states that every symmetric, positive definite kernel defines a unique reproducing kernel Hilbert space. \n\n\\subsection{Lions' formula for a reproducing kernel Hilbert space of harmonic functions}\nIn \\cite{Lions}, J.L. Lions has established an interesting formula for certain reproducing kernel Hilbert spaces of harmonic functions. \nTo recall his result, let us introduce the following differential operator on functions defined on the boundary $\\partial \\Omega$. \n\\begin{Definition} \\label{DefBel}\nLet $L$ be the negative of the Laplace-Beltrami operator on $\\partial \\Omega$, and let $M := L+ 1$. This operator $M$ is called the smoothed Laplace-Beltrami operator on $\\partial \\Omega$. \n\\end{Definition}\nLet $L^2 (\\partial \\Omega)$ the Hilbert space of square integrable functions on $\\partial \\Omega$\nwith respect to the $d-1$-dimensional measure $d\\sigma$ on $\\partial \\Omega$. \nThen the smoothed Laplace-Beltrami operator $M$ is a positive self-adjoint operator in $L^2 (\\partial \\Omega)$, so that, by Borel functional calculus, it makes sense to speak of its powers of any real order. This allows us to define some Sobolev spaces on $\\partial \\Omega$ for any $s$ in $\\mathbb R$ as follows.\n\\begin{Definition}\\label{def1.3}\nFor any $s$ in $\\mathbb R$, the Sobolev space $H^s(\\partial \\Omega)$ is the closure of the smooth functions on $\\partial \\Omega$ for the norm associated with the scalar product\n\\begin{equation} \\label{inpr}\n\\langle u,v\\rangle_s := \n\\big\\langle M^{\\frac{s}{2}} \\, u,M^{\\frac{s}{2}} \\,\\,v\\big\\rangle_{L^2 (\\partial \\Omega)} .\n\\end{equation}\n\\end{Definition}\nThe Dirichlet trace $\\gamma_D$ on the boundary $\\partial \\Omega$, defined by \n\\begin{equation} \\label{gammad}\n \\gamma_D: \\, u \\in C^\\infty (\\overline \\Omega) \\mapsto u\\vert_{\\partial \\Omega} \\in C^\\infty (\\partial \\Omega)\n\\end{equation} \nextends \nin a continuous linear map from the Sobolev space $H^s(\\Omega)$\nto the Sobolev space $H^{s-\\frac12} (\\partial \\Omega)$ for all $s > \\frac12$. It does not extend to the negative Sobolev spaces $H^{t} (\\partial \\Omega)$, with $t\\leqslant 0$; with such properties. \nHowever, it extends to the subspace of harmonic functions of the Sobolev space $H^s(\\Omega)$ for any $s \\in \\mathbb R$, again as a continuous linear map with values in $H^{s-\\frac12} (\\partial \\Omega)$. \nActually, this holds not only for harmonic functions but also for much more general \nsolutions of elliptic equations, as established by the works of Lions and Magenes, see \\cite{LM63,LM68}, by establishing some \n regularity properties in the case of Sobolev spaces with positive indexes first, then by using a duality argument and finally interpolation theory.\n Another approach, based on the theory of pseudodifferential operators for boundary value problems, \n emphasized that such a property can be seen as a variant of the partial hypoellipticity of the elliptic equations and originated from H\\\"ormander in \\cite[Theorem 2.5.6]{H63} and Boutet de Monvel \\cite{B71}, see also \\cite[Theorem 11.4]{Gr} for a more recent account of the topic. \nWith a slight abuse of notation, we keep the same notation $\\gamma_D$ for this trace map, as a continuous linear map from the subspace of harmonic functions of the Sobolev space $H^s(\\Omega)$ to $H^{s-\\frac12} (\\partial \\Omega)$ \nfor any $s \\in \\mathbb R$.\n\nNow, the space of harmonic distributions considered by Lions is the following.\n\\begin{Definition} \\label{def-Hs}\nFor any $s \\in \\mathbb R$, we define $\\mathcal H_s(\\Omega) $ as the space of harmonic functions $u$ in $ H^{s+\\frac12} (\\Omega)$ whose trace $\\gamma_D (u)$ on $\\partial \\Omega$ belongs to $H^s(\\partial \\Omega)$.\n\\end{Definition}\nIt follows from the references above that for any \n$s$ in $\\mathbb R$, any $u$ in $\\mathcal H_s (\\Omega)$ is actually a $C^\\infty$ function in $\\Omega$, and for any $x$ in $\\Omega$, \n we have the Poisson formula: \n\\begin{equation} \\label{Cani1}\n u(x)\n = \\Big\\langle \\gamma_D (u),\\, M^{-{s}} \\,\\gamma_N \\big( G_1 (x,\\cdot)\\big) \\Big\\rangle_s ,\n\\end{equation}\nwhere $G_1$ is the Green function, that is the function such that for any $x$ in $\\Omega$, \n$G_1(x,\\cdot)$ is the classical Green function with singularity at $x$, that is the solution to \n\\begin{equation} \\label{DIR}\n -\\Delta G_1(x,\\cdot)=\\delta_x \\quad\\text{in}\\quad \\mathcal D'(\\Omega)\\qquad\\&\\qquad G_1(x,\\cdot)=0\\quad\\text{on}\\quad \\partial\\Omega, \n\\end{equation}\nand where $\\gamma_N \\big(G_1(x,\\cdot)\\big)$ denotes its Neumann trace, that is \n$$\\gamma_N \\big(G_1(x,\\cdot)\\big)(z):= \\nabla_z G_1(x,z)\\cdot\\nu(z) \\quad \\text{for}\\quad z\\in\\partial\\Omega,$$ where $\\nu(z)$ is the outer unit normal at $z$.\nIn \\eqref{DIR}, $\\delta_x$ denotes the Dirac delta distribution at the position $x$. \nIn particular, it follows from elliptic regularity that \n\\begin{equation} \\label{REg}\n \\forall x \\in \\Omega, \\quad \\forall s \\in \\mathbb R, \\quad M^{-\\frac{s}{2}} \\gamma_N \\big(G_1(x,\\cdot)\\big) \\in L^2 (\\partial\\Omega) ,\n\\end{equation}\n so that the \nright-hand side of \\eqref{Cani1} \nmakes sense. Moreover, for any $s$ in $\\mathbb R$, the mapping that associates any $g$ in $H^s(\\partial \\Omega)$ to the function $\\mathcal P_s [g]$ defined for any $x$ in $\\Omega$, by \n\\begin{equation} \\label{poisson-iso}\n \\mathcal P_s[g] (x) := \\Big\\langle g, \\,M^{-{s}} \\,\\gamma_N \\big( G_1 (x,\\cdot)\\big) \\Big\\rangle_s ,\n\\end{equation}\nis a one-to-one isometry onto $\\mathcal H_s(\\Omega) $. This allows us to transport the Hilbert structure from $H^s(\\partial \\Omega)$ to $\\mathcal H_s(\\Omega) $, that is we \n equip $\\mathcal H_s(\\Omega) $ with the scalar product, defined for any $ u,v$ in $\\mathcal H_s(\\Omega) $, by \n $\\langle \\gamma_D (u),\\gamma_D (v) \\rangle_s $. \nActually, Lions' result, see \\cite[Formula (2.13)]{Lions}, is that it is a RKHS with a two-point kernel given as a boundary integral involving the \nGreen function $G_1$.", + "context": "\\subsection{Reproducing kernel Hilbert space}\n\n\\subsection{Lions' formula for a reproducing kernel Hilbert space of harmonic functions}\nIn \\cite{Lions}, J.L. Lions has established an interesting formula for certain reproducing kernel Hilbert spaces of harmonic functions. \nTo recall his result, let us introduce the following differential operator on functions defined on the boundary $\\partial \\Omega$. \n\\begin{Definition} \\label{DefBel}\nLet $L$ be the negative of the Laplace-Beltrami operator on $\\partial \\Omega$, and let $M := L+ 1$. This operator $M$ is called the smoothed Laplace-Beltrami operator on $\\partial \\Omega$. \n\\end{Definition}\nLet $L^2 (\\partial \\Omega)$ the Hilbert space of square integrable functions on $\\partial \\Omega$\nwith respect to the $d-1$-dimensional measure $d\\sigma$ on $\\partial \\Omega$. \nThen the smoothed Laplace-Beltrami operator $M$ is a positive self-adjoint operator in $L^2 (\\partial \\Omega)$, so that, by Borel functional calculus, it makes sense to speak of its powers of any real order. This allows us to define some Sobolev spaces on $\\partial \\Omega$ for any $s$ in $\\mathbb R$ as follows.\n\\begin{Definition}\\label{def1.3}\nFor any $s$ in $\\mathbb R$, the Sobolev space $H^s(\\partial \\Omega)$ is the closure of the smooth functions on $\\partial \\Omega$ for the norm associated with the scalar product\n\\begin{equation} \\label{inpr}\n\\langle u,v\\rangle_s := \n\\big\\langle M^{\\frac{s}{2}} \\, u,M^{\\frac{s}{2}} \\,\\,v\\big\\rangle_{L^2 (\\partial \\Omega)} .\n\\end{equation}\n\\end{Definition}\nThe Dirichlet trace $\\gamma_D$ on the boundary $\\partial \\Omega$, defined by \n\\begin{equation} \\label{gammad}\n \\gamma_D: \\, u \\in C^\\infty (\\overline \\Omega) \\mapsto u\\vert_{\\partial \\Omega} \\in C^\\infty (\\partial \\Omega)\n\\end{equation} \nextends \nin a continuous linear map from the Sobolev space $H^s(\\Omega)$\nto the Sobolev space $H^{s-\\frac12} (\\partial \\Omega)$ for all $s > \\frac12$. It does not extend to the negative Sobolev spaces $H^{t} (\\partial \\Omega)$, with $t\\leqslant 0$; with such properties. \nHowever, it extends to the subspace of harmonic functions of the Sobolev space $H^s(\\Omega)$ for any $s \\in \\mathbb R$, again as a continuous linear map with values in $H^{s-\\frac12} (\\partial \\Omega)$. \nActually, this holds not only for harmonic functions but also for much more general \nsolutions of elliptic equations, as established by the works of Lions and Magenes, see \\cite{LM63,LM68}, by establishing some \n regularity properties in the case of Sobolev spaces with positive indexes first, then by using a duality argument and finally interpolation theory.\n Another approach, based on the theory of pseudodifferential operators for boundary value problems, \n emphasized that such a property can be seen as a variant of the partial hypoellipticity of the elliptic equations and originated from H\\\"ormander in \\cite[Theorem 2.5.6]{H63} and Boutet de Monvel \\cite{B71}, see also \\cite[Theorem 11.4]{Gr} for a more recent account of the topic. \nWith a slight abuse of notation, we keep the same notation $\\gamma_D$ for this trace map, as a continuous linear map from the subspace of harmonic functions of the Sobolev space $H^s(\\Omega)$ to $H^{s-\\frac12} (\\partial \\Omega)$ \nfor any $s \\in \\mathbb R$.\n\nNow, the space of harmonic distributions considered by Lions is the following.\n\\begin{Definition} \\label{def-Hs}\nFor any $s \\in \\mathbb R$, we define $\\mathcal H_s(\\Omega) $ as the space of harmonic functions $u$ in $ H^{s+\\frac12} (\\Omega)$ whose trace $\\gamma_D (u)$ on $\\partial \\Omega$ belongs to $H^s(\\partial \\Omega)$.\n\\end{Definition}\nIt follows from the references above that for any \n$s$ in $\\mathbb R$, any $u$ in $\\mathcal H_s (\\Omega)$ is actually a $C^\\infty$ function in $\\Omega$, and for any $x$ in $\\Omega$, \n we have the Poisson formula: \n\\begin{equation} \\label{Cani1}\n u(x)\n = \\Big\\langle \\gamma_D (u),\\, M^{-{s}} \\,\\gamma_N \\big( G_1 (x,\\cdot)\\big) \\Big\\rangle_s ,\n\\end{equation}\nwhere $G_1$ is the Green function, that is the function such that for any $x$ in $\\Omega$, \n$G_1(x,\\cdot)$ is the classical Green function with singularity at $x$, that is the solution to \n\\begin{equation} \\label{DIR}\n -\\Delta G_1(x,\\cdot)=\\delta_x \\quad\\text{in}\\quad \\mathcal D'(\\Omega)\\qquad\\&\\qquad G_1(x,\\cdot)=0\\quad\\text{on}\\quad \\partial\\Omega, \n\\end{equation}\nand where $\\gamma_N \\big(G_1(x,\\cdot)\\big)$ denotes its Neumann trace, that is \n$$\\gamma_N \\big(G_1(x,\\cdot)\\big)(z):= \\nabla_z G_1(x,z)\\cdot\\nu(z) \\quad \\text{for}\\quad z\\in\\partial\\Omega,$$ where $\\nu(z)$ is the outer unit normal at $z$.\nIn \\eqref{DIR}, $\\delta_x$ denotes the Dirac delta distribution at the position $x$. \nIn particular, it follows from elliptic regularity that \n\\begin{equation} \\label{REg}\n \\forall x \\in \\Omega, \\quad \\forall s \\in \\mathbb R, \\quad M^{-\\frac{s}{2}} \\gamma_N \\big(G_1(x,\\cdot)\\big) \\in L^2 (\\partial\\Omega) ,\n\\end{equation}\n so that the \nright-hand side of \\eqref{Cani1} \nmakes sense. Moreover, for any $s$ in $\\mathbb R$, the mapping that associates any $g$ in $H^s(\\partial \\Omega)$ to the function $\\mathcal P_s [g]$ defined for any $x$ in $\\Omega$, by \n\\begin{equation} \\label{poisson-iso}\n \\mathcal P_s[g] (x) := \\Big\\langle g, \\,M^{-{s}} \\,\\gamma_N \\big( G_1 (x,\\cdot)\\big) \\Big\\rangle_s ,\n\\end{equation}\nis a one-to-one isometry onto $\\mathcal H_s(\\Omega) $. This allows us to transport the Hilbert structure from $H^s(\\partial \\Omega)$ to $\\mathcal H_s(\\Omega) $, that is we \n equip $\\mathcal H_s(\\Omega) $ with the scalar product, defined for any $ u,v$ in $\\mathcal H_s(\\Omega) $, by \n $\\langle \\gamma_D (u),\\gamma_D (v) \\rangle_s $. \nActually, Lions' result, see \\cite[Formula (2.13)]{Lions}, is that it is a RKHS with a two-point kernel given as a boundary integral involving the \nGreen function $G_1$.", + "full_context": "\\subsection{Reproducing kernel Hilbert space}\n\n\\subsection{Lions' formula for a reproducing kernel Hilbert space of harmonic functions}\nIn \\cite{Lions}, J.L. Lions has established an interesting formula for certain reproducing kernel Hilbert spaces of harmonic functions. \nTo recall his result, let us introduce the following differential operator on functions defined on the boundary $\\partial \\Omega$. \n\\begin{Definition} \\label{DefBel}\nLet $L$ be the negative of the Laplace-Beltrami operator on $\\partial \\Omega$, and let $M := L+ 1$. This operator $M$ is called the smoothed Laplace-Beltrami operator on $\\partial \\Omega$. \n\\end{Definition}\nLet $L^2 (\\partial \\Omega)$ the Hilbert space of square integrable functions on $\\partial \\Omega$\nwith respect to the $d-1$-dimensional measure $d\\sigma$ on $\\partial \\Omega$. \nThen the smoothed Laplace-Beltrami operator $M$ is a positive self-adjoint operator in $L^2 (\\partial \\Omega)$, so that, by Borel functional calculus, it makes sense to speak of its powers of any real order. This allows us to define some Sobolev spaces on $\\partial \\Omega$ for any $s$ in $\\mathbb R$ as follows.\n\\begin{Definition}\\label{def1.3}\nFor any $s$ in $\\mathbb R$, the Sobolev space $H^s(\\partial \\Omega)$ is the closure of the smooth functions on $\\partial \\Omega$ for the norm associated with the scalar product\n\\begin{equation} \\label{inpr}\n\\langle u,v\\rangle_s := \n\\big\\langle M^{\\frac{s}{2}} \\, u,M^{\\frac{s}{2}} \\,\\,v\\big\\rangle_{L^2 (\\partial \\Omega)} .\n\\end{equation}\n\\end{Definition}\nThe Dirichlet trace $\\gamma_D$ on the boundary $\\partial \\Omega$, defined by \n\\begin{equation} \\label{gammad}\n \\gamma_D: \\, u \\in C^\\infty (\\overline \\Omega) \\mapsto u\\vert_{\\partial \\Omega} \\in C^\\infty (\\partial \\Omega)\n\\end{equation} \nextends \nin a continuous linear map from the Sobolev space $H^s(\\Omega)$\nto the Sobolev space $H^{s-\\frac12} (\\partial \\Omega)$ for all $s > \\frac12$. It does not extend to the negative Sobolev spaces $H^{t} (\\partial \\Omega)$, with $t\\leqslant 0$; with such properties. \nHowever, it extends to the subspace of harmonic functions of the Sobolev space $H^s(\\Omega)$ for any $s \\in \\mathbb R$, again as a continuous linear map with values in $H^{s-\\frac12} (\\partial \\Omega)$. \nActually, this holds not only for harmonic functions but also for much more general \nsolutions of elliptic equations, as established by the works of Lions and Magenes, see \\cite{LM63,LM68}, by establishing some \n regularity properties in the case of Sobolev spaces with positive indexes first, then by using a duality argument and finally interpolation theory.\n Another approach, based on the theory of pseudodifferential operators for boundary value problems, \n emphasized that such a property can be seen as a variant of the partial hypoellipticity of the elliptic equations and originated from H\\\"ormander in \\cite[Theorem 2.5.6]{H63} and Boutet de Monvel \\cite{B71}, see also \\cite[Theorem 11.4]{Gr} for a more recent account of the topic. \nWith a slight abuse of notation, we keep the same notation $\\gamma_D$ for this trace map, as a continuous linear map from the subspace of harmonic functions of the Sobolev space $H^s(\\Omega)$ to $H^{s-\\frac12} (\\partial \\Omega)$ \nfor any $s \\in \\mathbb R$.\n\nNow, the space of harmonic distributions considered by Lions is the following.\n\\begin{Definition} \\label{def-Hs}\nFor any $s \\in \\mathbb R$, we define $\\mathcal H_s(\\Omega) $ as the space of harmonic functions $u$ in $ H^{s+\\frac12} (\\Omega)$ whose trace $\\gamma_D (u)$ on $\\partial \\Omega$ belongs to $H^s(\\partial \\Omega)$.\n\\end{Definition}\nIt follows from the references above that for any \n$s$ in $\\mathbb R$, any $u$ in $\\mathcal H_s (\\Omega)$ is actually a $C^\\infty$ function in $\\Omega$, and for any $x$ in $\\Omega$, \n we have the Poisson formula: \n\\begin{equation} \\label{Cani1}\n u(x)\n = \\Big\\langle \\gamma_D (u),\\, M^{-{s}} \\,\\gamma_N \\big( G_1 (x,\\cdot)\\big) \\Big\\rangle_s ,\n\\end{equation}\nwhere $G_1$ is the Green function, that is the function such that for any $x$ in $\\Omega$, \n$G_1(x,\\cdot)$ is the classical Green function with singularity at $x$, that is the solution to \n\\begin{equation} \\label{DIR}\n -\\Delta G_1(x,\\cdot)=\\delta_x \\quad\\text{in}\\quad \\mathcal D'(\\Omega)\\qquad\\&\\qquad G_1(x,\\cdot)=0\\quad\\text{on}\\quad \\partial\\Omega, \n\\end{equation}\nand where $\\gamma_N \\big(G_1(x,\\cdot)\\big)$ denotes its Neumann trace, that is \n$$\\gamma_N \\big(G_1(x,\\cdot)\\big)(z):= \\nabla_z G_1(x,z)\\cdot\\nu(z) \\quad \\text{for}\\quad z\\in\\partial\\Omega,$$ where $\\nu(z)$ is the outer unit normal at $z$.\nIn \\eqref{DIR}, $\\delta_x$ denotes the Dirac delta distribution at the position $x$. \nIn particular, it follows from elliptic regularity that \n\\begin{equation} \\label{REg}\n \\forall x \\in \\Omega, \\quad \\forall s \\in \\mathbb R, \\quad M^{-\\frac{s}{2}} \\gamma_N \\big(G_1(x,\\cdot)\\big) \\in L^2 (\\partial\\Omega) ,\n\\end{equation}\n so that the \nright-hand side of \\eqref{Cani1} \nmakes sense. Moreover, for any $s$ in $\\mathbb R$, the mapping that associates any $g$ in $H^s(\\partial \\Omega)$ to the function $\\mathcal P_s [g]$ defined for any $x$ in $\\Omega$, by \n\\begin{equation} \\label{poisson-iso}\n \\mathcal P_s[g] (x) := \\Big\\langle g, \\,M^{-{s}} \\,\\gamma_N \\big( G_1 (x,\\cdot)\\big) \\Big\\rangle_s ,\n\\end{equation}\nis a one-to-one isometry onto $\\mathcal H_s(\\Omega) $. This allows us to transport the Hilbert structure from $H^s(\\partial \\Omega)$ to $\\mathcal H_s(\\Omega) $, that is we \n equip $\\mathcal H_s(\\Omega) $ with the scalar product, defined for any $ u,v$ in $\\mathcal H_s(\\Omega) $, by \n $\\langle \\gamma_D (u),\\gamma_D (v) \\rangle_s $. \nActually, Lions' result, see \\cite[Formula (2.13)]{Lions}, is that it is a RKHS with a two-point kernel given as a boundary integral involving the \nGreen function $G_1$.\n\nOne quite explicit formula for the fractional Laplace operator of order $a$, with $a\\in (0,1)$, in $N\\in \\N^*$ dimensions, reads \n$$\n(-\\Delta)^a u (z) := c_{N,a}\\, \\pv \\int_{\\R^N}\\frac{u(z)-u(y)}{|z-y|^{N+2a}}dy,\n$$\nwith \n$$c_{N,a} :=\\pi^{N/2}a4^a\\frac{\\Gamma(\\frac{N+2a}{2})}{\\Gamma(1-a)},$$\nwhere $\\Gamma$ is the gamma function, and where $\\pv$ refers to the Cauchy principal value. \nAbove $u$ is a function from $\\R^N$ with real values. \nThe reason for the presence of the normalization constant $c_{N,a}$\n is to match with the Fourier definition which sets the fractional Laplace operator $(-\\Delta)^a$ as the Fourier multiplier of symbol \n$|\\xi|^{2a}$. \n The Green function $G_a(x,y)$ associated to the operator $(-\\Delta)^a$ and the homogeneous Dirichlet condition \nis the solution to \n\\begin{equation} \\label{DIRa}\n (-\\Delta)^a G_a(x,\\cdot)=\\delta_x \\quad\\text{in}\\quad \\mathcal D'(\\Omega)\\qquad\\&\\qquad G_a(x,\\cdot)=0\\quad\\text{in}\\quad \\R^N\\setminus\\Omega.\n\\end{equation}\nSuch a Green function plays the same role for the operator $(-\\Delta)^a$ as the Green function $G_1$ for the classical Laplace operator. \nIn the companion paper \\cite{sidy-franck_1}, we computed the following Hadamard-type variational formula for the fractional Green function $G_a(x,y)$: for any $x$ and $y$ in $\\Omega$, \n\\begin{align}\\label{var-green}\n DG_a (\\alpha) (x,y) = \\Gamma^2(1+a)\\int_{\\partial\\Omega} \\, \\big(\\gamma_0^a(G_a(x,\\cdot))\\big)(z) \\, \\big(\\gamma^a_0(G_a(y,\\cdot))\\big)(z)\n \\alpha(z) \\;d\\sigma(z).\n\\end{align}\nAbove, $ \\Gamma^2(1+a)$ denotes the square of $ \\Gamma(1+a)$ while $\\gamma_0^a$ denotes the fractional trace of order $a$, that is \n$$\\gamma_0^a(G_a(x,\\cdot):= \\gamma_D \\big(\\frac{G_a(x,\\cdot)}{d^a}\\big),$$ where $d$ is the distance function to the boundary $\\partial\\Omega$. \nThe right-hand side of \\eqref{var-green} makes sense because by boundary regularity, for any $x$ in $\\Omega$, $\\gamma_0^a(G_a(x,\\cdot))$ is $C^\\infty$ on ${\\partial\\Omega}$, see \\cite[Equation (2.14)]{Grubb-2014}. \nLet us point out that the analysis performed in \\cite{sidy-franck_1} allows for more general perturbations. \nWe also refer to the earlier works \\cite{DalibardVaret,DFW} on the subject.\n\n\\begin{proof}\nLet $u$ in $\\mathcal H_{a,s}(\\Omega)$. Recalling that $\\gamma^{a-1}_0(u)$ is in $H^{2\\theta}(\\partial\\Omega)$,\nby density, we have a sequence $(g_n)_{n\\in\\mathbb N}$ in $C^\\infty(\\partial\\Omega)$ such that\n\\begin{equation} \\label{ref-jeudi}\ng_n \\to \\gamma^{a-1}_0(u) \\quad \\text{in } H^{2\\theta}(\\partial\\Omega).\n\\end{equation}\nBy Theorem \\ref{f}, for any $n$ in $\\mathbb N$, we set \n $u_n := \\mathcal P_{a,s}[g_n]$. This \nis the unique $u_n$ in $H^{(a-1)(s+2a)}(\\Omega)$\nsolution of\n\\begin{equation*}\n(-\\Delta)^a u_n = 0 \\ \\text{in }\\Omega \\ \\ \\& \\ \\ \\gamma^{a-1}_0(u_n) = g_n \\ \\text{on }\\partial\\Omega.\n\\end{equation*}\nWe fix $x$ in $\\Omega$. Let $\\rho$ in $C^\\infty_0(\\mathbb R^N)$ with $\\rho \\ge 0$ and\n$\\int_{\\mathbb R^N}\\rho = 1$, and set $\\rho_\\varepsilon(z) := \\varepsilon^{-N}\\rho(z/\\varepsilon)$.\nFor $\\varepsilon>0$ small enough so that $\\mathrm{supp}\\,\\rho_\\varepsilon(\\cdot-x)\\subset \\Omega$,\nwe define\n\\begin{equation} \\label{*}\nv_{x,\\varepsilon}(z) := \\int_\\Omega G_a(z,y)\\rho_\\varepsilon(y-x)\\,dy.\n\\end{equation}\nThen $v_{x,\\varepsilon}=0$ in $\\mathbb R^N\\setminus\\Omega$, and in $\\mathcal D'(\\Omega)$ we have\n\\begin{equation} \\label{**}\n(-\\Delta)^a v_{x,\\varepsilon} = \\rho_\\varepsilon(\\cdot-x).\n\\end{equation}\nMoreover, since $\\rho_\\varepsilon(\\cdot-x)$ is smooth with compact support in $\\Omega$,\nthe function $v_{x,\\varepsilon}$ is regular enough for the hypotheses of \\cite[Theorem 5.1]{Grubb-2020}.\nWe apply Grubb's exact Green formula to the pair $(u_n,v_{x,\\varepsilon})$.\nSince $(-\\Delta)^a u_n = 0$ in $\\Omega$ and $(-\\Delta)^a v_{x,\\varepsilon}=\\rho_\\varepsilon(\\cdot-x)$ in $\\Omega$,\nwe obtain\n\\begin{equation} \\label{***}\n\\int_\\Omega u_n(y)\\rho_\\varepsilon(y-x)\\,dy\n= \\Gamma(a)\\Gamma(a+1)\\int_{\\partial\\Omega} g_n(z)\\,\\gamma_0^a(v_{x,\\varepsilon})(z)\\,d\\sigma(z).\n\\end{equation}\n\nOur main result establishes a counterpart of Theorem \\ref{Lions-theo} in the case of $a$-harmonic functions by showing that, for an appropriate range of indexes, the spaces $\\mathcal H_{a,s}(\\Omega)$ are RKHS with two-point kernels given by a boundary integral of some derivatives of the Green function $G_a(x,\\cdot)$ of the fractional Laplacian with a singularity at $x$.\nWe recall that $M$ is the smoothed Laplace-Beltrami operator on $\\partial \\Omega$, see Definition \\ref{DefBel}.\n\\begin{thm} \\label{s-Lions-theo}\nLet $a\\in (0,1)$, $s>-a-\\frac{1}{2}$, and\n $\\theta$ be given by \\eqref{theta}. \nThen the space $\\mathcal H_{a,s}(\\Omega)$ is a RKHS, and its two-point kernel, which we denote $K_{a,s}$, is given, for any $x$ and $y$ in $\\Omega$, by \n \\begin{equation} \\label{l-formu-s}\n K_{a,s}(x,y)= \\Gamma^2(a) \\Gamma^2(a+1) \\int_{\\partial \\Omega} \\Big( M^{-\\theta} \\, \n \\gamma_0^{a}(G_a(x,\\cdot))\n \\Big) \\Big(M^{-\\theta} \\, \\gamma_0^{a}(G_a(y,\\cdot)) \\Big) d\\sigma.\n \\end{equation}\n\\end{thm}\n\nWe also recall the following classical Poisson formula: for any solution $(u,p)$ of the steady Stokes problem \\eqref{eq_stoke} we have \n\\begin{equation} \\label{eq_Green-St}\nu(x)\n= \\int_{\\partial \\Omega} ( \\Sigma (\\mathfrak G(x,\\cdot),\\mathfrak P(x,\\cdot)) n) \\cdot u \\,d\\sigma.\n\\end{equation}\nwhere\n\\begin{equation} \\label{eq_Newt2}\n\\Sigma(u,p) = 2 D(u) - p \\mathbb{I}_3 \\quad \\text{ where } 2 D(u):= \\nabla u + (\\nabla u)^T , \n\\end{equation}\nand $\\mathbb{I}_3$ is the $3 \\times 3$ identity matrix \nand $(\\mathfrak G(x,\\cdot),\\mathfrak P(x,\\cdot))$ is the Green function associated with \n the Stokes system that is, the unique tensor such that for any $b \\in \\R^3$, $(u,p) := (\\mathfrak G(x,\\cdot)b,\\mathfrak P(x,\\cdot)b)$ is the unique solution to the problem: \n\\begin{equation*} \n\\left\\{\n\\begin{array}{rcl}\n- \\Delta u + \\nabla p &= b \\delta_x & \\, \\\\\n\\operatorname{div} u &= 0 &\\, \n\\end{array}\n\\right. \\quad \\text{ in $\\Omega$} , \\end{equation*}\nwith \\begin{equation*} \n u =0\\quad\\text{on}\\quad \\partial\\Omega.\n\\end{equation*}\n A reasoning similar to the ones above leads to the conclusion that the space $\\mathfrak H_s$ is a RKHS with the following\n Lions type formula for its $2$-points kernel $\\mathfrak K_s$: \n$$ \\mathfrak K_s (x,y) = \\int_{\\partial\\Omega} \\Big((M^{-\\frac{s}{2}} \\, \\gamma_D \\big(\\Sigma(\\mathfrak G(x,\\cdot),\\mathfrak P(x,\\cdot))n\\big)\\Big)(z) \\cdot \\Big(M^{-\\frac{s}{2}} \\, \\gamma_D \\big( \\Sigma(\\mathfrak G(y,\\cdot),\\mathfrak P(y,\\cdot))n \\big) \\Big)(z)\\,d\\sigma(z) .\n$$\n On the other hand, it is known since some works by Simon, see for instance \\cite{Simon} and the recent works by \\cite{KU,Oz}, \n that \n the Hadamard variation formula in the case of the Stokes equations reads: for any $b \\in \\R^3$,\n $$D(\\mathfrak G b) (\\alpha) (x,y) = \n \\int_{\\partial\\Omega} \\gamma_N \\big(\\mathfrak G(x,\\cdot) b\\big)(z) \\, \\gamma_N \\big(\\mathfrak G(y,\\cdot)b\\big)(z) \\, \\alpha(z)d\\sigma(z) , $$\nwhere $\\alpha\\in C^\\infty(\\partial\\Omega)$.", + "post_theorem_intro_text_len": 4751, + "post_theorem_intro_text": "Let us point out that the integral in \\eqref{l-formu} makes sense, and is finite, since for any $x$ in $\\Omega$, for any $s$ in $\\mathbb R$, \n$M^{-\\frac{s}{2}} \\gamma_N \\big((G_1(x,\\cdot)\\big) $\nis in $L^2 (\\partial\\Omega)$.\nActually, Lions dealt with some function spaces associated with second-order elliptic boundary value problems, see \\cite{Lions}. \nLions used a variational approach and obtained his formula through a penalization limit. \nAnother, more direct approach to Theorem \\ref{Lions-theo} was proposed later in \\cite{ELPL}, together with some extensions to more general elliptic operators of even orders. \nWe reproduce below their proof of \\eqref{l-formu} as a preparation for our extension to the case of fractional Laplace operators.\n\n \\begin{proof}\n It follows from the Poisson formula \\eqref{Cani1}, \\eqref{REg}\nand the Cauchy-Schwarz inequality that for any $x$ in $\\Omega$, the evaluation mapping $E_x$, see Definition \\ref{defRKHS}, is continuous from $\\mathcal H_s(\\Omega) $ to $\\mathbb R$. \nSince $\\mathcal H_s(\\Omega) $ is a Hilbert space, \n it follows from Riesz' theorem that for any $x$ in $\\Omega$, there exists $K_x$ in \n$\\mathcal H_s(\\Omega) $ \nsuch that for any $u$ in $\\mathcal H_s(\\Omega) $, \n\\begin{equation} \\label{Cani2}\nu(x) = \\langle \\gamma_D (u) ,\\, K_x \\rangle_s = \\langle M^{\\frac{s}{2}} \\, \\gamma_D (u) ,\\, M^{\\frac{s}{2}} \\, K_x \\rangle_{L^2 (\\partial \\Omega)}.\n\\end{equation}\nGathering \\eqref{Cani1} and \\eqref{Cani2} we deduce that for any $u$ in $\\mathcal H_s (\\Omega)$, \n\\begin{equation*} \n \\Big\\langle M^{\\frac{s}{2}} \\, \\gamma_D (u) , \\, M^{-\\frac{s}{2}} \\,\\gamma_N \\big( G_1 (x,\\cdot)\\big) - M^{\\frac{s}{2}} \\, K_x \\Big\\rangle_{L^2 (\\partial \\Omega)}=0.\n\\end{equation*}\n Then by a density argument on $u$, we conclude that \n $$ M^{\\frac{s}{2}} \\, K_x = M^{-\\frac{s}{2}} \\,\\gamma_N \\big( G_1 (x,\\cdot)\\big) ,$$\n almost everywhere on $\\partial\\Omega$.\n Therefore, for any $x$ and $y$ in $\\Omega$,\n \\begin{align*}\n K_s(x,y) &= \\Big\\langle M^{\\frac{s}{2}} \\, K_x, \\,M^{\\frac{s}{2}} \\, K_y \\Big\\rangle _{L^2 (\\partial \\Omega) } \n \\\\ &=\\Big\\langle M^{-\\frac{s}{2}} \\, \\gamma_N \\big( G_1 (x,\\cdot)\\big) ,\\, M^{-\\frac{s}{2}} \\, \\gamma_N \\big( G_1 (y,\\cdot)\\big) \\Big\\rangle _{L^2 (\\partial \\Omega) },\n \\end{align*}\n which is the desired conclusion.\n \\end{proof}\n\n\\subsection{Resemblance with the Hadamard variation formula}\nAn interesting observation in \\cite{ELPL} is the resemblance of \\eqref{l-formu} with the Hadamard variation formula \nfor the Green function of the classical Laplacian, which we now recall. \nThis formula expresses the derivative of the Green function $G_1$ with respect to the domain. \nMore precisely, one considers some perturbations $\\Omega_t$ of the domain $\\Omega$ in the normal direction by modifying the boundary $\\partial\\Omega$ of $\\Omega$ in \n$$\\partial\\Omega_t=\\Big\\{y=x+t \\alpha(x)\\nu(x), \\;x\\in\\partial\\Omega\\Big\\},$$\nwhere $\\alpha\\in C^\\infty(\\partial\\Omega)$, and $t$ runs in a open interval containing $0$. \nLet us rename $G_\\Omega$ the Green function\n $G_1$ associated with the domain $\\Omega$, as defined by \\eqref{DIR}, to emphasize here the dependence on the domain. Then we define the derivative of $G_\\Omega$ with respect to the domain in $\\alpha$ as \n\\begin{align}\n DG_\\Omega (\\alpha) (x,y):=\\lim_{t\\to 0, t \\neq 0} \\, \\frac{G_{\\Omega_t}(x,y)-G_\\Omega(x,y)}{t} .\n\\end{align}\nThen the pioneering discovery by Hadamard, see \\cite{Hadamard}, is that not only does this limit exist, but it is even given by the following explicit integral formula: for any $x$ and $y$ in $\\Omega$, \n\\begin{align} \\label{Hadamard}\n DG_\\Omega (\\alpha) (x,y) =\\int_{\\partial\\Omega} \\gamma_N \\big(G_1(x,\\cdot)\\big)(z) \\, \\gamma_N \\big(G_1(y,\\cdot)\\big)(z) \\, \\alpha(z)d\\sigma(z) .\n\\end{align}\nLet us mention that a rigorous proof of Hadamard's formula was given later on by Garabedian who also considered more general perturbations. After these pioneering works, shape derivative computations have received significant interest and several extensions of Hadamard's formula for more general elliptic boundary value problems were obtained, with various techniques. We refer here to \\cite{HenrotPierre,Sokolowski} for more. \n\\medskip\n\nFollowing \\cite{ELPL}, \n it is now clear that \\eqref{l-formu} and \\eqref{Hadamard} look very similar, in particular in the case where $s=0$ for which the only difference is the presence of the extra factor $\\alpha$ in \\eqref{Hadamard}.\n In \\cite{ELPL}, it is said furthermore that the authors \"are of the opinion that the connection between Hadamard variation formula and the reproducing kernel just indicated, in the classical harmonic case, seems to be an isolated phenomenon peculiar to the second order case.\"", + "sketch": "Using the Poisson formula \\eqref{Cani1}, \\eqref{REg} and the Cauchy--Schwarz inequality, one shows that for any $x\\in\\Omega$ the evaluation mapping $E_x$ is continuous from $\\mathcal H_s(\\Omega)$ to $\\mathbb R$. Since $\\mathcal H_s(\\Omega)$ is a Hilbert space, Riesz' theorem gives for each $x\\in\\Omega$ an element $K_x\\in\\mathcal H_s(\\Omega)$ such that for all $u\\in\\mathcal H_s(\\Omega)$,\n\\[\n u(x)=\\langle \\gamma_D(u),K_x\\rangle_s=\\langle M^{\\frac{s}{2}}\\gamma_D(u),\\,M^{\\frac{s}{2}}K_x\\rangle_{L^2(\\partial\\Omega)}. \\tag{\\eqref{Cani2}}\n\\]\nGathering \\eqref{Cani1} and \\eqref{Cani2} yields, for any $u\\in\\mathcal H_s(\\Omega)$,\n\\[\n\\Big\\langle M^{\\frac{s}{2}}\\gamma_D(u),\\, M^{-\\frac{s}{2}}\\gamma_N\\big(G_1(x,\\cdot)\\big)-M^{\\frac{s}{2}}K_x\\Big\\rangle_{L^2(\\partial\\Omega)}=0.\n\\]\nBy a density argument on $u$, one concludes that $M^{\\frac{s}{2}}K_x=M^{-\\frac{s}{2}}\\gamma_N\\big(G_1(x,\\cdot)\\big)$ a.e. on $\\partial\\Omega$. Therefore, for any $x,y\\in\\Omega$,\n\\[\nK_s(x,y)=\\langle M^{\\frac{s}{2}}K_x,\\,M^{\\frac{s}{2}}K_y\\rangle_{L^2(\\partial\\Omega)}\n=\\big\\langle M^{-\\frac{s}{2}}\\gamma_N\\big(G_1(x,\\cdot)\\big),\\,M^{-\\frac{s}{2}}\\gamma_N\\big(G_1(y,\\cdot)\\big)\\big\\rangle_{L^2(\\partial\\Omega)},\n\\]\nwhich is exactly \\eqref{l-formu}.", + "expanded_sketch": "Using the Poisson formula\n\\begin{equation} \\label{Cani1}\n u(x)\n = \\Big\\langle \\gamma_D (u),\\, M^{-{s}} \\,\\gamma_N \\big( G_1 (x,\\cdot)\\big) \\Big\\rangle_s ,\n\\end{equation}\n\\begin{equation} \\label{REg}\n \\forall x \\in \\Omega, \\quad \\forall s \\in \\R, \\quad M^{-\\frac{s}{2}} \\gamma_N \\big(G_1(x,\\cdot)\\big) \\in L^2 (\\partial\\Omega) ,\n\\end{equation}\nand the Cauchy--Schwarz inequality, one shows that for any $x\\in\\Omega$ the evaluation mapping $E_x$ is continuous from $\\mathcal H_s(\\Omega)$ to $\\mathbb R$. Since $\\mathcal H_s(\\Omega)$ is a Hilbert space, Riesz' theorem gives for each $x\\in\\Omega$ an element $K_x\\in\\mathcal H_s(\\Omega)$ such that for all $u\\in\\mathcal H_s(\\Omega)$,\n\\begin{equation} \\label{Cani2}\\nu(x) = \\langle \\gamma_D (u) ,\\, K_x \\rangle_s = \\langle M^{\\frac{s}{2}} \\, \\gamma_D (u) ,\\, M^{\\frac{s}{2}} \\, K_x \\rangle_{L^2 (\\partial \\Omega)}.\n\\end{equation}\nGathering the Poisson formula above and the preceding identity yields, for any $u\\in\\mathcal H_s(\\Omega)$,\n\\[\n\\Big\\langle M^{\\frac{s}{2}}\\gamma_D(u),\\, M^{-\\frac{s}{2}}\\gamma_N\\big(G_1(x,\\cdot)\\big)-M^{\\frac{s}{2}}K_x\\Big\\rangle_{L^2(\\partial\\Omega)}=0.\n\\]\nBy a density argument on $u$, one concludes that $M^{\\frac{s}{2}}K_x=M^{-\\frac{s}{2}}\\gamma_N\\big(G_1(x,\\cdot)\\big)$ a.e. on $\\partial\\Omega$. Therefore, for any $x,y\\in\\Omega$,\n\\[\nK_s(x,y)=\\langle M^{\\frac{s}{2}}K_x,\\,M^{\\frac{s}{2}}K_y\\rangle_{L^2(\\partial\\Omega)}\n=\\big\\langle M^{-\\frac{s}{2}}\\gamma_N\\big(G_1(x,\\cdot)\\big),\\,M^{-\\frac{s}{2}}\\gamma_N\\big(G_1(y,\\cdot)\\big)\\big\\rangle_{L^2(\\partial\\Omega)},\n\\]\nwhich is exactly\n\\begin{equation} \\label{l-formu}\n K_s (x,y)= \\int_{\\partial\\Omega} \\Big((M^{-\\frac{s}{2}} \\, \\gamma_N \\big((G_1(x,\\cdot)\\big)\\Big)(z) \\, \\Big(M^{-\\frac{s}{2}} \\, \\gamma_N \\big((G_1(y,\\cdot)\\big) \\Big)(z)\\,d\\sigma(z) .\n \\end{equation}\n.", + "expanded_theorem": "\\label{Lions-theo}\nFor any $s$ in $\\mathbb R$, the space $\\mathcal H_s(\\Omega) $ is a RKHS and its two-point kernel $K_s$ is given, for any pair of $x$ and $y$ in $\\Omega$, by \n \\begin{equation} \\label{l-formu}\n K_s (x,y)= \\int_{\\partial\\Omega} \\Big((M^{-\\frac{s}{2}} \\, \\gamma_N \\big((G_1(x,\\cdot)\\big)\\Big)(z) \\, \\Big(M^{-\\frac{s}{2}} \\, \\gamma_N \\big((G_1(y,\\cdot)\\big) \\Big)(z)\\,d\\sigma(z) .\n \\end{equation}", + "theorem_type": [ + "Universal", + "Classification or Bijection" + ], + "mcq": { + "question": "Let \\(\\Omega\\) be a domain with boundary \\(\\partial\\Omega\\). Let \\(L\\) denote the negative Laplace--Beltrami operator on \\(\\partial\\Omega\\), set \\(M:=L+1\\), and for \\(s\\in\\mathbb R\\) let \\(H^s(\\partial\\Omega)\\) be the Sobolev space on \\(\\partial\\Omega\\) defined from \\(M\\). Let \\(\\gamma_D\\) and \\(\\gamma_N\\) denote the Dirichlet and Neumann traces, respectively. For each \\(s\\in\\mathbb R\\), define\n\\[\n\\mathcal H_s(\\Omega):=\\{u\\in H^{s+\\frac12}(\\Omega): u \\text{ is harmonic in }\\Omega\\text{ and }\\gamma_D u\\in H^s(\\partial\\Omega)\\}.\n\\]\nFor \\(x\\in\\Omega\\), let \\(G_1(x,\\cdot)\\) be the Dirichlet Green function, i.e. the solution of\n\\[\n-\\Delta G_1(x,\\cdot)=\\delta_x\\quad\\text{in }\\mathcal D'(\\Omega),\n\\qquad\nG_1(x,\\cdot)=0\\quad\\text{on }\\partial\\Omega.\n\\]\nWhich statement holds for every real \\(s\\)?", + "correct_choice": { + "label": "A", + "text": "The space \\(\\mathcal H_s(\\Omega)\\) is a reproducing kernel Hilbert space, and its two-point kernel \\(K_s\\) is given for every \\(x,y\\in\\Omega\\) by\n\\[\nK_s(x,y)=\\int_{\\partial\\Omega}\\big(M^{-\\frac s2}\\gamma_N(G_1(x,\\cdot))\\big)(z)\\,\\big(M^{-\\frac s2}\\gamma_N(G_1(y,\\cdot))\\big)(z)\\,d\\sigma(z).\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "The space \\(\\mathcal H_s(\\Omega)\\) is a reproducing kernel Hilbert space, and its two-point kernel \\(K_s\\) is given for every \\(x,y\\in\\Omega\\) by\n\\[\nK_s(x,y)=\\int_{\\partial\\Omega}\\big(M^{-s}\\gamma_N(G_1(x,\\cdot))\\big)(z)\\,\\big(M^{-s}\\gamma_N(G_1(y,\\cdot))\\big)(z)\\,d\\sigma(z).\n\\]" + }, + { + "label": "C", + "text": "For every real \\(s\\), the space \\(\\mathcal H_s(\\Omega)\\) is a reproducing kernel Hilbert space." + }, + { + "label": "D", + "text": "For every real \\(s\\), the space \\(\\mathcal H_s(\\Omega)\\) is a reproducing kernel Hilbert space, and there exists a constant \\(c_s>0\\) such that for every \\(x,y\\in\\Omega\\),\n\\[\nK_s(x,y)=c_s\\int_{\\partial\\Omega}\\big(M^{-\\frac s2}\\gamma_N(G_1(x,\\cdot))\\big)(z)\\,\\big(M^{-\\frac s2}\\gamma_N(G_1(y,\\cdot))\\big)(z)\\,d\\sigma(z).\n\\]" + }, + { + "label": "E", + "text": "The space \\(\\mathcal H_s(\\Omega)\\) is a reproducing kernel Hilbert space, and its two-point kernel \\(K_s\\) is given for every \\(x,y\\in\\Omega\\) by\n\\[\nK_s(x,y)=\\int_{\\partial\\Omega}\\big(M^{-\\frac s2}\\gamma_D(G_1(x,\\cdot))\\big)(z)\\,\\big(M^{-\\frac s2}\\gamma_D(G_1(y,\\cdot))\\big)(z)\\,d\\sigma(z).\n\\]" + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "D" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "regularity", + "tampered_component": "half-power of M in the L^2 identification", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "explicit boundary integral formula for the reproducing kernel", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "exact normalization from the Riesz representer identity", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "Neumann trace in the Poisson formula replaced by Dirichlet trace", + "template_used": "property_confusion" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem introduces the objects and asks for the valid conclusion, but it does not explicitly reveal the reproducing-kernel formula or otherwise single out the correct option." + }, + "TAS": { + "score": 1, + "justification": "The item is close to theorem recall: one choice is essentially the full theorem statement verbatim. However, it is not a pure restatement because the student must distinguish it from nearby alternatives and a weaker true statement." + }, + "GPS": { + "score": 2, + "justification": "Selecting the correct answer requires meaningful discrimination among subtle variants: the power of M, Neumann vs. Dirichlet trace, exact normalization, and the stronger explicit formula versus a merely weaker true claim." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target realistic failure modes: exponent mismatch, trace confusion, loss of exact normalization, and choosing a weaker but true statement instead of the strongest correct one." + }, + "total_score": 7, + "overall_assessment": "A strong MCQ with high-quality distractors and genuine mathematical discrimination, though it remains somewhat close to direct theorem recognition rather than full derivational reasoning." + } + }, + { + "id": "2602.09749v1", + "paper_link": "http://arxiv.org/abs/2602.09749v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm:gen_upper_box_precise}\n Assume that $F\\subseteq \\mathbb{R}^p$ is a connected self-similar set with finitely many directions\n and $\\dim_H F = s > 1$.\n Then for the generic 1-Hölder-$\\alpha$ function $f:F\\to \\mathbb{R}$ and Lebesgue almost every $r\\in f(F)$,\n \\begin{displaymath}\n \\overline{\\dim}_B f^{-1}(r)=s-\\alpha.\n \\end{displaymath}", + "start_pos": 8348, + "end_pos": 8742, + "label": "thm:gen_upper_box_precise" + }, + "ref_dict": { + "thm:gen_upper_box_precise": "\\begin{theorem} \\label{thm:gen_upper_box_precise}\n Assume that $F\\subseteq \\mathbb{R}^p$ is a connected self-similar set with finitely many directions\n and $\\dim_H F = s > 1$.\n Then for the generic 1-Hölder-$\\alpha$ function $f:F\\to \\mathbb{R}$ and Lebesgue almost every $r\\in f(F)$,\n \\begin{displaymath}\n \\overline{\\dim}_B f^{-1}(r)=s-\\alpha.\n \\end{displaymath}\n\\end{theorem}", + "prop:wenxi": "\\begin{theorem}[{\\cite[Corollary~1]{wenxi}}] \\label{prop:wenxi}\n Assume that $A\\subseteq \\mathbb{R}^p$ is a self-similar set with finitely many directions and $\\dim_H A>1$.\n Then for a full measure set $\\mathcal{W}_A$ of the $(p-1)$-dimensional linear subspaces, if $W\\in \\mathcal{W}_A$, then \n $$\\mathcal{H}^1(\\mathrm{Pr}_{W^{\\perp}}(A)\\setminus \\{a\\in W^{\\perp}: \\dim (A\\cap (W+a))= \\dim_H A -1\\}) = 0,$$\n that is standard dimension drop occurs on a set of full measure. (Here $W^{\\perp}$ denotes the 1-dimensional orthocomplement of $W$, \n and $\\mathrm{Pr}_{W^{\\perp}}$ is the projection onto it.)\n\\end{theorem}", + "lemma:values_admitted_in_simplices": "\\begin{lemma} \\label{lemma:values_admitted_in_simplices}\n Let $F\\subseteq \\mathbb{R}^p$ be compact and $\\mathcal{S}$ be a finite family of simplices covering $F$ such that $\\inte (S) \\cap F\\neq \\emptyset$ for all $S\\in \\mathcal{S}$.\n Assume moreover that $f:\\bigcup_{S\\in\\mathcal{S}} S \\to \\mathbb{R}$ is a piecewise affine function, affine on members of $\\mathcal{S}$. \n Then for almost every $u\\in\\mathbb{R}^p$, we have\n \\begin{equation} \\label{eq:values_admitted_in_interior}\n \\bigcup_{S\\in\\mathcal{S}} f(S\\cap (F+u)) = \\bigcup_{S\\in\\mathcal{S}} f\\left(\\inte S\\cap (F+u)\\right) \\text{ a. e.}\n \\end{equation}\n i.e., for almost every $u$ almost every value admitted on $F+u$ by $f$ is admitted in the interior of a simplex.\n\\end{lemma}" + }, + "pre_theorem_intro_text_len": 2301, + "pre_theorem_intro_text": "Level sets of continuous functions defined on some domain $F\\subseteq \\mathbb{R}^p$ convey information about the dimensionality of $F$. Roughly speaking, a generic continuous function should have large level sets\nif $F$ itself is thick in an intuitive sense, and small otherwise. This heuristic prompted the investigation of the size of such level sets in terms of Hausdorff dimension\n(see \\cite{BBElevel, BBEtoph, BK}), with the quantity providing the answer (the topological Hausdorff dimension) attracting interest in its own right \\cite{MaZha}, even in physics\n\\cite{Balankintoph, Balankinfracspace, Balankintransport, Balankintoph2, Balankinfluid}.\n\nThe authors with Gáspár Vértesy initiated the study of the Hausdorff and box dimension of level sets of generic 1-Hölder-$\\alpha$ functions \\cite{ BuczolichMagaVertesy2025, sierc, sier, BUCZOLICH2024531}. \nUnder self-similarity conditions on the domain $F$, they proved that for every $\\alpha$ and for either of the Hausdorff/lower box/upper box dimension, there is a constant which is the dimension\nof Lebesgue almost every level set of the generic 1-Hölder-$\\alpha$ function. For any underlying dimension, these values form an interesting spectrum describing the geometry of $F$. \nHowever, dealing with this spectrum is far from trivial in any of the cases: the existing literature focuses on particular choices of $F$ and provides mostly estimates at the price of\na significant amount of technicalities, highly depending on the structure of $F$. In this direction, in \\cite{BuczolichMagaVertesy2025}, \nwe managed to provide lower and upper bounds on the Hausdorff spectrum for the Sierpiński triangle which asymptotically coincide as $\\alpha\\to 0+$.\nHowever, no explicit formulae or general machinery have been found before for describing any of the spectra for a larger class of fractals. In this respect this paper is the first one. \n\n\\subsection{The main result}\n\nWe give an explicit formula for the upper box dimension $\\overline{\\dim}_B$ of Lebesgue almost every level set of a generic 1-Hölder-$\\alpha$ function \nvalid for connected self-similar sets subject to the geometric condition of having {\\it finitely many directions}, i.e.,\nwhose defining similarities have orthogonal parts generating a finite subgroup of the orthogonal group.", + "context": "Level sets of continuous functions defined on some domain $F\\subseteq \\mathbb{R}^p$ convey information about the dimensionality of $F$. Roughly speaking, a generic continuous function should have large level sets\nif $F$ itself is thick in an intuitive sense, and small otherwise. This heuristic prompted the investigation of the size of such level sets in terms of Hausdorff dimension\n(see \\cite{BBElevel, BBEtoph, BK}), with the quantity providing the answer (the topological Hausdorff dimension) attracting interest in its own right \\cite{MaZha}, even in physics\n\\cite{Balankintoph, Balankinfracspace, Balankintransport, Balankintoph2, Balankinfluid}.\n\nThe authors with Gáspár Vértesy initiated the study of the Hausdorff and box dimension of level sets of generic 1-Hölder-$\\alpha$ functions \\cite{ BuczolichMagaVertesy2025, sierc, sier, BUCZOLICH2024531}. \nUnder self-similarity conditions on the domain $F$, they proved that for every $\\alpha$ and for either of the Hausdorff/lower box/upper box dimension, there is a constant which is the dimension\nof Lebesgue almost every level set of the generic 1-Hölder-$\\alpha$ function. For any underlying dimension, these values form an interesting spectrum describing the geometry of $F$. \nHowever, dealing with this spectrum is far from trivial in any of the cases: the existing literature focuses on particular choices of $F$ and provides mostly estimates at the price of\na significant amount of technicalities, highly depending on the structure of $F$. In this direction, in \\cite{BuczolichMagaVertesy2025}, \nwe managed to provide lower and upper bounds on the Hausdorff spectrum for the Sierpiński triangle which asymptotically coincide as $\\alpha\\to 0+$.\nHowever, no explicit formulae or general machinery have been found before for describing any of the spectra for a larger class of fractals. In this respect this paper is the first one.\n\n\\subsection{The main result}\n\nWe give an explicit formula for the upper box dimension $\\overline{\\dim}_B$ of Lebesgue almost every level set of a generic 1-Hölder-$\\alpha$ function \nvalid for connected self-similar sets subject to the geometric condition of having {\\it finitely many directions}, i.e.,\nwhose defining similarities have orthogonal parts generating a finite subgroup of the orthogonal group.", + "full_context": "Level sets of continuous functions defined on some domain $F\\subseteq \\mathbb{R}^p$ convey information about the dimensionality of $F$. Roughly speaking, a generic continuous function should have large level sets\nif $F$ itself is thick in an intuitive sense, and small otherwise. This heuristic prompted the investigation of the size of such level sets in terms of Hausdorff dimension\n(see \\cite{BBElevel, BBEtoph, BK}), with the quantity providing the answer (the topological Hausdorff dimension) attracting interest in its own right \\cite{MaZha}, even in physics\n\\cite{Balankintoph, Balankinfracspace, Balankintransport, Balankintoph2, Balankinfluid}.\n\nThe authors with Gáspár Vértesy initiated the study of the Hausdorff and box dimension of level sets of generic 1-Hölder-$\\alpha$ functions \\cite{ BuczolichMagaVertesy2025, sierc, sier, BUCZOLICH2024531}. \nUnder self-similarity conditions on the domain $F$, they proved that for every $\\alpha$ and for either of the Hausdorff/lower box/upper box dimension, there is a constant which is the dimension\nof Lebesgue almost every level set of the generic 1-Hölder-$\\alpha$ function. For any underlying dimension, these values form an interesting spectrum describing the geometry of $F$. \nHowever, dealing with this spectrum is far from trivial in any of the cases: the existing literature focuses on particular choices of $F$ and provides mostly estimates at the price of\na significant amount of technicalities, highly depending on the structure of $F$. In this direction, in \\cite{BuczolichMagaVertesy2025}, \nwe managed to provide lower and upper bounds on the Hausdorff spectrum for the Sierpiński triangle which asymptotically coincide as $\\alpha\\to 0+$.\nHowever, no explicit formulae or general machinery have been found before for describing any of the spectra for a larger class of fractals. In this respect this paper is the first one.\n\n\\subsection{The main result}\n\nWe give an explicit formula for the upper box dimension $\\overline{\\dim}_B$ of Lebesgue almost every level set of a generic 1-Hölder-$\\alpha$ function \nvalid for connected self-similar sets subject to the geometric condition of having {\\it finitely many directions}, i.e.,\nwhose defining similarities have orthogonal parts generating a finite subgroup of the orthogonal group.\n\n\\begin{abstract}\n In the previous decades, the size of level sets of functions have been extensively studied in various setups involving different regularity properties and\n size notions. In the case of Hölder functions, the authors have provided various bounds, but to date no explicit formulae have been found for any studied dimension \n and the results were valid only about very specific fractals.\n In this paper, for the first time, we have a result valid for a large class of self-similar sets, namely we prove that for these \n fractals Lebesgue almost every level set of the generic 1-Hölder-$\\alpha$ function defined on $F\\subseteq \\mathbb{R}^p$ has \n upper box dimension $\\dim_H F - \\alpha$. \n\\end{abstract}\n\nWe give an explicit formula for the upper box dimension $\\overline{\\dim}_B$ of Lebesgue almost every level set of a generic 1-Hölder-$\\alpha$ function \nvalid for connected self-similar sets subject to the geometric condition of having {\\it finitely many directions}, i.e.,\nwhose defining similarities have orthogonal parts generating a finite subgroup of the orthogonal group.\n\n\\subsection{Organization of the paper}\n\nGiven a function $f:F\\to \\mathbb{R}$, we put $D_{\\overline{B}}^f(r,F)=\\overline{\\dim}_B f^{-1}(r)$. For $f:F\\to \\mathbb{R}$, we denote by $D_{\\overline{B}*}^f(F)$ the essential infimum\nof the upper box dimension of non-empty level sets, i.e., \n\\begin{displaymath}\n D_{\\overline{B}*}^f(F)= \\begin{cases}\n \\inf\\{d: \\lambda\\{r : r\\in f(F) \\text{ and } D_{\\overline{B}}^f(r,F)\\leq{d}\\}>0\\}, & \\text{if } \\lambda(f(F))>0, \\\\\n 0, & \\text{if } \\lambda(f(F))=0.\n \\end{cases}\n\\end{displaymath}\nWe denote by $\\mg_{1,\\aaa}(F)$, or by simply $\\mg_{1,\\aaa}$ the set of dense $G_{\\ddd}$\nsets in $C_{1}^{\\aaa}(F)$. Whether $D_{\\overline{B}*}^f(F)$ has a generic value is a priori unclear, nevertheless the following quantity is always well-defined:\n\\begin{equation}\\label{def:generic_upper}\n D_{\\overline{B}*}(\\aaa,F)=\\inf_{\\cag\\in \\mg_{1,\\aaa}}\\sup\\{ D_{\\overline{B}*}^{f}(F):f\\in \\cag \\}.\n \\end{equation}\nAs established by \\cite[Theorem~3.8]{BUCZOLICH2024531}, subject to certain conditions on the connectivity properties of $F$, $D_{\\overline{B}*}(\\aaa,F)$ is indeed simply the generic value of $D_{\\overline{B}^*}^f(F)$.\nIn Section \\ref{sec:box_dimension_locally_connected} we cite these results and discuss how they extend to locally connected $F$. \nWe note that even if $D_{\\overline{B}*}(\\aaa,F)$ is the generic value of $D_{\\overline{B}^*}^f(F)$, it does not necessarily coincide with $D_{\\overline{B}}^f(r,F)$ for almost every $r\\in f(F)$.\nIn particular it is not clear a priori why in the setup of Theorem \\ref{thm:gen_upper_box_precise} almost every level set of the generic 1-Hölder-$\\alpha$ should have the same upper box dimension.\n\n\\begin{lemma}[{\\cite[Lemma~3.9]{BUCZOLICH2024531}}] \\label{lemma:from_covering_to_box}\n Assume that $0<\\alpha\\leq 1$, and the measurable set $F\\subseteq \\mathbb{R}^p$ has coverings $(\\mathcal{S}_n)_{n=1}^{\\infty}$, \n such that with some constants $C,l,\\rho>1$:\n \\begin{itemize}\n \\item the cardinality of $\\mathcal{S}_n$ is at most $Cl^n$ for some $C,l>1$,\n \\item if $S\\in \\mathcal{S}_n$, then $\\diam(S)\\leq C\\rho^{-n}$.\n \\end{itemize}\n Then for any $f\\in C_{\\alpha}^{1}(F)$ and almost every $r\\in \\mathbb{R}$, we have\n $\\overline{\\dim}_B(f^{-1}(r))\\leq \\frac{\\log l}{\\log \\rho} - \\alpha$. In particular,\n $D_{\\overline{B}*}(\\alpha, F)\\leq \\frac{\\log l}{\\log \\rho} - \\alpha$.\n\\end{lemma}\n\n\\begin{corollary} \\label{corollary:lower_box_dim_upper_bound}\n Assume that $0<\\alpha\\leq 1$, and let $F\\subseteq \\mathbb{R}^p$ be measurable. Then for every $f\\in C_{\\alpha}^{1}(F)$ and almost every $r\\in \\mathbb{R}$, we have\n $\\overline{\\dim}_B(f^{-1}(r))\\leq \\underline{\\dim}_B(F) - \\alpha$. In particular,\n $D_{\\overline{B}*}(\\alpha, F)\\leq \\underline{\\dim}_B(F) - \\alpha$.\n\\end{corollary}\n\n\\begin{theorem}[{\\cite[Corollary~1]{wenxi}}] \\label{prop:wenxi}\n Assume that $A\\subseteq \\mathbb{R}^p$ is a self-similar set with finitely many directions and $\\dim_H A>1$.\n Then for a full measure set $\\mathcal{W}_A$ of the $(p-1)$-dimensional linear subspaces, if $W\\in \\mathcal{W}_A$, then \n $$\\mathcal{H}^1(\\mathrm{Pr}_{W^{\\perp}}(A)\\setminus \\{a\\in W^{\\perp}: \\dim (A\\cap (W+a))= \\dim_H A -1\\}) = 0,$$\n that is standard dimension drop occurs on a set of full measure. (Here $W^{\\perp}$ denotes the 1-dimensional orthocomplement of $W$, \n and $\\mathrm{Pr}_{W^{\\perp}}$ is the projection onto it.)\n\\end{theorem}\n\n\\begin{lemma} \\label{lemma:values_admitted_in_simplices}\n Let $F\\subseteq \\mathbb{R}^p$ be compact and $\\mathcal{S}$ be a finite family of simplices covering $F$ such that $\\inte (S) \\cap F\\neq \\emptyset$ for all $S\\in \\mathcal{S}$.\n Assume moreover that $f:\\bigcup_{S\\in\\mathcal{S}} S \\to \\mathbb{R}$ is a piecewise affine function, affine on members of $\\mathcal{S}$. \n Then for almost every $u\\in\\mathbb{R}^p$, we have\n \\begin{equation} \\label{eq:values_admitted_in_interior}\n \\bigcup_{S\\in\\mathcal{S}} f(S\\cap (F+u)) = \\bigcup_{S\\in\\mathcal{S}} f\\left(\\inte S\\cap (F+u)\\right) \\text{ a. e.}\n \\end{equation}\n i.e., for almost every $u$ almost every value admitted on $F+u$ by $f$ is admitted in the interior of a simplex.\n\\end{lemma}\n\n\\begin{theorem} \\label{thm:gen_upper_box_precise}\n Assume that $F\\subseteq \\mathbb{R}^p$ is a connected self-similar set with finitely many directions\n and $\\dim_H F = s > 1$.\n Then for the generic 1-Hölder-$\\alpha$ function $f:F\\to \\mathbb{R}$ and Lebesgue almost every $r\\in f(F)$,\n \\begin{displaymath}\n \\overline{\\dim}_B f^{-1}(r)=s-\\alpha.\n \\end{displaymath}\n\\end{theorem}", + "post_theorem_intro_text_len": 1538, + "post_theorem_intro_text": "\\subsection{Organization of the paper}\n\nIn Section \\ref{sec:prelim} we introduce some notation and recall some relevant tools. \n\nIn Section \\ref{sec:box_dimension_locally_connected} we provide a lemma on how a dense set of functions over which the upper box dimension of level sets is controlled can be used\nto get bounds over a dense $G_\\delta$ set of functions. Such a lemma was previously provided by \\cite{BUCZOLICH2024531} assuming a more elaborate connectivity property,\nthis time it will be sufficient to assume local connectivity.\n\nIn Section \\ref{sec:lemmas} we prove two key lemmas, which do the heavy lifting for Section \\ref{sec:main}, in which we prove Theorem \\ref{thm:gen_upper_box_precise}. \nWe note that in \\cite[Theorem~3.6]{BUCZOLICH2024531}, we proved a lower bound for the upper box dimension of level sets under certain conditions using Theorem \\ref{prop:wenxi} (identical to\n\\cite[Proposition~2.6]{BUCZOLICH2024531}). A subtle point in applying this statement was not addressed explicitly there.\nLemma \\ref{lemma:values_admitted_in_simplices} fills this important geometric gap, essentially stating\nthat if we have a piecewise affine function, defined on simplices containing our fractal $F$ then with an arbitrarily small translation\nwe can make sure that the function takes Lebesgue almost all of its values in the interior of the simplices. This fact is quite easy if the Hausdorff dimension of $F$ is less than $2$,\nfor higher dimensions it needs some extra care.\n\nWe conclude the paper with some open problems.", + "sketch": "In Section \\ref{sec:lemmas} the authors “prove two key lemmas, which do the heavy lifting for Section \\ref{sec:main}, in which we prove Theorem \\ref{thm:gen_upper_box_precise}.” They also note that a “subtle point in applying” a previously used statement (Theorem \\ref{prop:wenxi}) “was not addressed explicitly” earlier, and that Lemma \\ref{lemma:values_admitted_in_simplices} “fills this important geometric gap,” namely: for a piecewise affine function on simplices containing the fractal $F$, “with an arbitrarily small translation we can make sure that the function takes Lebesgue almost all of its values in the interior of the simplices.” This geometric fact is said to be “quite easy if the Hausdorff dimension of $F$ is less than $2$,” but “for higher dimensions it needs some extra care.” Additionally, Section \\ref{sec:box_dimension_locally_connected} provides a lemma showing how “a dense set of functions over which the upper box dimension of level sets is controlled can be used to get bounds over a dense $G_\\delta$ set of functions,” now assuming local connectivity.", + "expanded_sketch": "Next the authors prove two key lemmas, which do the heavy lifting for the later section where they establish the main theorem. They also note that a subtle point in applying the following previously used statement was not addressed explicitly earlier.\n\n\\begin{theorem}[{\\cite[Corollary~1]{wenxi}}] \\label{prop:wenxi}\n Assume that $A\\subseteq \\mathbb{R}^p$ is a self-similar set with finitely many directions and $\\dim_H A>1$.\n Then for a full measure set $\\mathcal{W}_A$ of the $(p-1)$-dimensional linear subspaces, if $W\\in \\mathcal{W}_A$, then \n $$\\mathcal{H}^1(\\mathrm{Pr}_{W^{\\perp}}(A)\\setminus \\{a\\in W^{\\perp}: \\dim (A\\cap (W+a))= \\dim_H A -1\\}) = 0,$$\n that is standard dimension drop occurs on a set of full measure. (Here $W^{\\perp}$ denotes the 1-dimensional orthocomplement of $W$, \n and $\\mathrm{Pr}_{W^{\\perp}}$ is the projection onto it.)\n\\end{theorem}\n\nThey explain that the following lemma fills this important geometric gap, namely: for a piecewise affine function on simplices containing the fractal $F$, with an arbitrarily small translation we can make sure that the function takes Lebesgue almost all of its values in the interior of the simplices.\n\n\\begin{lemma} \\label{lemma:values_admitted_in_simplices}\n Let $F\\subseteq \\mathbb{R}^p$ be compact and $\\mathcal{S}$ be a finite family of simplices covering $F$ such that $\\inte (S) \\cap F\\neq \\emptyset$ for all $S\\in \\mathcal{S}$.\n Assume moreover that $f:\\bigcup_{S\\in\\mathcal{S}} S \\to \\mathbb{R}$ is a piecewise affine function, affine on members of $\\mathcal{S}$. \n Then for almost every $u\\in\\mathbb{R}^p$, we have\n \\begin{equation} \\label{eq:values_admitted_in_interior}\n \\bigcup_{S\\in\\mathcal{S}} f(S\\cap (F+u)) = \\bigcup_{S\\in\\mathcal{S}} f\\left(\\inte S\\cap (F+u)\\right) \\text{ a. e.}\n \\end{equation}\n i.e., for almost every $u$ almost every value admitted on $F+u$ by $f$ is admitted in the interior of a simplex.\n\\end{lemma}\n\nThis geometric fact is said to be quite easy if the Hausdorff dimension of $F$ is less than $2$, but for higher dimensions it needs some extra care. Additionally, later the authors provide a lemma showing how a dense set of functions over which the upper box dimension of level sets is controlled can be used to get bounds over a dense $G_\\delta$ set of functions, now assuming local connectivity.", + "expanded_theorem": "\\label{thm:gen_upper_box_precise}\n Assume that $F\\subseteq \\mathbb{R}^p$ is a connected self-similar set with finitely many directions\n and $\\dim_H F = s > 1$.\n Then for the generic 1-Hölder-$\\alpha$ function $f:F\\to \\mathbb{R}$ and Lebesgue almost every $r\\in f(F)$,\n \\begin{displaymath}\n \\overline{\\dim}_B f^{-1}(r)=s-\\alpha.\n \\end{displaymath}", + "theorem_type": [ + "Universal", + "Inequality or Bound" + ], + "mcq": { + "question": "Fix $0<\\alpha\\le 1$. Let $F\\subseteq \\mathbb{R}^p$ be a connected self-similar set with finitely many directions, meaning that in some defining iterated function system for $F$, the orthogonal parts of the similarities generate a finite subgroup of the orthogonal group, and assume that $\\dim_H F=s>1$. Let $C_\\alpha^1(F)$ denote the space of 1-H\\\"older-$\\alpha$ functions $f:F\\to\\mathbb{R}$, i.e. functions satisfying $|f(x)-f(y)|\\le |x-y|^\\alpha$ for all $x,y\\in F$. Which statement holds for generic $f\\in C_\\alpha^1(F)$ (that is, for all $f$ in a dense $G_\\delta$ subset of $C_\\alpha^1(F)$) concerning the level sets $f^{-1}(r)$ for Lebesgue almost every $r\\in f(F)$?", + "correct_choice": { + "label": "A", + "text": "For Lebesgue almost every $r\\in f(F)$, the upper box dimension of the level set satisfies\n$$\\overline{\\dim}_B f^{-1}(r)=s-\\alpha.$$" + }, + "choices": [ + { + "label": "B", + "text": "For Lebesgue almost every $r\\in f(F)$, the upper box dimension of the level set satisfies\n$$\\overline{\\dim}_B f^{-1}(r)=s-1.$$" + }, + { + "label": "C", + "text": "For Lebesgue almost every $r\\in f(F)$, the upper box dimension of the level set satisfies\n$$\\overline{\\dim}_B f^{-1}(r)\\le s-\\alpha.$$" + }, + { + "label": "D", + "text": "For every generic $f\\in C_\\alpha^1(F)$, there is a full measure set $R_f\\subseteq f(F)$ such that for every $r\\in R_f$,\n$$\\dim_H f^{-1}(r)=s-\\alpha.$$" + }, + { + "label": "E", + "text": "There exists a dense $G_\\delta$ set $\\mathcal G\\subseteq C_\\alpha^1(F)$ such that for every $f\\in \\mathcal G$ there exists a full measure set $R_f\\subseteq f(F)$ with\n$$\\overline{\\dim}_B f^{-1}(r)\\ge s-\\alpha$$\nfor all $r\\in R_f$." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "trace_identity", + "tampered_component": "dimension-drop amount replaced by fixed codimension one", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "counting_estimate", + "tampered_component": "dropped the lower-bound/equality conclusion, keeping only the upper bound", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "upper box dimension replaced by Hausdorff dimension", + "template_used": "property_confusion" + }, + { + "label": "E", + "sketch_hook_type": "geometric_construction", + "tampered_component": "retained only the lower-bound side coming from interior-value construction and omitted the matching upper-bound mechanism", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not reveal the exact conclusion. It specifies the hypotheses and asks which quantitative level-set statement is valid, but it does not explicitly give the equality, quantifiers, or generic/almost-every distinctions needed to identify the correct choice." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recall question: the stem lists the hypotheses and the options are variants of the theorem’s conclusion. However, it is not purely tautological, since the choices differ in dimension formula, equality vs. inequality, and quantifier strength." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is required to choose between nearby claims, especially the weaker true upper bound, the strengthened quantifiers, and the incorrect replacement of \u001falpha by 1. Still, the question mainly tests recognition of the precise theorem statement rather than deeper derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong and mathematically meaningful. They reflect common failure modes: confusing s-\u001falpha with s-1, accepting a weaker bound instead of the sharp equality, and overlooking the importance of 'generic' and 'almost every r' quantifiers." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with no answer leakage and high-quality distractors, though it leans more toward precise theorem recall than genuinely generative mathematical reasoning." + } + }, + { + "id": "2602.09797v1", + "paper_link": "http://arxiv.org/abs/2602.09797v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "thm", + "content": "\\label{thm:main}\nThe procyclic groups\n\\[\n \\prod_{p \\equiv 1 \\bmod 3} \\bbZ_p, \\quad \\prod_{p \\equiv 1 \\bmod 4} \\bbZ_p, \\quad \\prod_{p \\in \\{1,3\\} \\bmod 8} \\bbZ_p\n\\]\nhave Weil abscissa $2$.", + "start_pos": 4069, + "end_pos": 4280, + "label": "thm:main" + }, + "ref_dict": {}, + "pre_theorem_intro_text_len": 1282, + "pre_theorem_intro_text": "The \\emph{Weil abscissa} $\\alpha(G)$ of a profinite group $G$ is an intricate invariant that quantifies how the number of absolutely irreducible representations of $G$ over finite fields $\\mathbb{F}_{p^j}$ increases with the size of the fields. It is the abscissa of convergence of the Weil representation zeta function defined in \\cite{corob2024weil}. \nFor a finitely generated profinite \\emph{abelian} group $G$, the \\textit{Weil representation zeta function} admits the following simple definition\n\t\\begin{align*}\n\t\t\\zeta_G^W(s)=\\exp\\bigg(\\sum_{p\\in\\mathcal{P}}\\sum_{j=1}^{\\infty} |\\text{Hom}(G,\\mathbb{F}_{p^j}^\\times)|\\frac{p^{-sj}}{j}\\bigg),\n\t\\end{align*}\n for $s \\in \\mathbb{C}$; here $\\mathcal{P}$ denotes the set of all prime numbers and $|\\text{Hom}(G,\\mathbb{F}_{p^j}^\\times)|$ the number of continuous homomorphisms from $G$ to $\\mathbb{F}_{p^j}^\\times$. However, even for abelian groups the Weil abscissa is difficult to compute, since its exact value depends in a subtle way on the distribution of primes in arithmetic progressions; see \\cite{corob2024weil,kionke2024}. The purpose of this short note is to calculate the Weil abscissa of three procyclic groups building on ideas of the master thesis of the first author (written at FernUniversit\\\"at in Hagen in 2025).", + "context": "The \\emph{Weil abscissa} $\\alpha(G)$ of a profinite group $G$ is an intricate invariant that quantifies how the number of absolutely irreducible representations of $G$ over finite fields $\\mathbb{F}_{p^j}$ increases with the size of the fields. It is the abscissa of convergence of the Weil representation zeta function defined in \\cite{corob2024weil}. \nFor a finitely generated profinite \\emph{abelian} group $G$, the \\textit{Weil representation zeta function} admits the following simple definition\n \\begin{align*}\n \\zeta_G^W(s)=\\exp\\bigg(\\sum_{p\\in\\mathcal{P}}\\sum_{j=1}^{\\infty} |\\text{Hom}(G,\\mathbb{F}_{p^j}^\\times)|\\frac{p^{-sj}}{j}\\bigg),\n \\end{align*}\n for $s \\in \\mathbb{C}$; here $\\mathcal{P}$ denotes the set of all prime numbers and $|\\text{Hom}(G,\\mathbb{F}_{p^j}^\\times)|$ the number of continuous homomorphisms from $G$ to $\\mathbb{F}_{p^j}^\\times$. However, even for abelian groups the Weil abscissa is difficult to compute, since its exact value depends in a subtle way on the distribution of primes in arithmetic progressions; see \\cite{corob2024weil,kionke2024}. The purpose of this short note is to calculate the Weil abscissa of three procyclic groups building on ideas of the master thesis of the first author (written at FernUniversit\\\"at in Hagen in 2025).", + "full_context": "The \\emph{Weil abscissa} $\\alpha(G)$ of a profinite group $G$ is an intricate invariant that quantifies how the number of absolutely irreducible representations of $G$ over finite fields $\\mathbb{F}_{p^j}$ increases with the size of the fields. It is the abscissa of convergence of the Weil representation zeta function defined in \\cite{corob2024weil}. \nFor a finitely generated profinite \\emph{abelian} group $G$, the \\textit{Weil representation zeta function} admits the following simple definition\n \\begin{align*}\n \\zeta_G^W(s)=\\exp\\bigg(\\sum_{p\\in\\mathcal{P}}\\sum_{j=1}^{\\infty} |\\text{Hom}(G,\\mathbb{F}_{p^j}^\\times)|\\frac{p^{-sj}}{j}\\bigg),\n \\end{align*}\n for $s \\in \\mathbb{C}$; here $\\mathcal{P}$ denotes the set of all prime numbers and $|\\text{Hom}(G,\\mathbb{F}_{p^j}^\\times)|$ the number of continuous homomorphisms from $G$ to $\\mathbb{F}_{p^j}^\\times$. However, even for abelian groups the Weil abscissa is difficult to compute, since its exact value depends in a subtle way on the distribution of primes in arithmetic progressions; see \\cite{corob2024weil,kionke2024}. The purpose of this short note is to calculate the Weil abscissa of three procyclic groups building on ideas of the master thesis of the first author (written at FernUniversit\\\"at in Hagen in 2025).\n\n\\begin{abstract}\nHere we show that the Weil abscissa of \nthe procyclic groups $\\prod_{p \\in S} \\bbZ_p$ equals $2$ for three sets $S$: (i) the set of primes $p \\equiv 1 \\bmod 3$, (ii) the set of primes $p \\equiv 1 \\bmod 4$ and (iii) the set of primes $p \\equiv 1,3 \\bmod 8$. Our argument is based on the observation that integers all of whose prime factors lie in $S$ can be represented by a suitable binary quadratic form, which allows us to use a theorem of Iwaniec to exhibit a minorant for the Weil representation zeta function.\n\\end{abstract}\n\nThe \\emph{Weil abscissa} $\\alpha(G)$ of a profinite group $G$ is an intricate invariant that quantifies how the number of absolutely irreducible representations of $G$ over finite fields $\\mathbb{F}_{p^j}$ increases with the size of the fields. It is the abscissa of convergence of the Weil representation zeta function defined in \\cite{corob2024weil}. \nFor a finitely generated profinite \\emph{abelian} group $G$, the \\textit{Weil representation zeta function} admits the following simple definition\n \\begin{align*}\n \\zeta_G^W(s)=\\exp\\bigg(\\sum_{p\\in\\mathcal{P}}\\sum_{j=1}^{\\infty} |\\text{Hom}(G,\\mathbb{F}_{p^j}^\\times)|\\frac{p^{-sj}}{j}\\bigg),\n \\end{align*}\n for $s \\in \\mathbb{C}$; here $\\mathcal{P}$ denotes the set of all prime numbers and $|\\text{Hom}(G,\\mathbb{F}_{p^j}^\\times)|$ the number of continuous homomorphisms from $G$ to $\\mathbb{F}_{p^j}^\\times$. However, even for abelian groups the Weil abscissa is difficult to compute, since its exact value depends in a subtle way on the distribution of primes in arithmetic progressions; see \\cite{corob2024weil,kionke2024}. The purpose of this short note is to calculate the Weil abscissa of three procyclic groups building on ideas of the master thesis of the first author (written at FernUniversit\\\"at in Hagen in 2025).\n\n\\medskip\n\n\\begin{question}\nDo the procyclic groups $\\prod_{p \\equiv 2 \\bmod 3} \\bbZ_p$ and $\\prod_{p \\equiv 3 \\bmod 4} \\bbZ_p$ have Weil abscissa $2$?\n\\end{question}\n\nRepresentability of integers by quadratic forms is a classical topic in number theory. The sum of two squares problem is a famous example: which primes are sums of two squares? It was already known to Fermat that exactly $2$ and all primes in $S_1=\\{p\\in \\mathcal{P} \\mid p \\equiv 1 \\bmod 4\\}$ are the sum of two squares. We will need the following refined result on representations of integers as a sum of two \\emph{coprime} squares:\n\\begin{proposition}[{see \\cite[Corollary 3.2.2]{Fine2007}}]\\label{prop:1mod4}\n Let $n \\in \\mathbb{N}$. Then $n=x^2+y^2$ for coprime integers $x,y$ if and only if the prime decomposition of $n$ is $n=2^\\delta \\prod_{i=1}^k p_i^{\\alpha_i}$ where $\\delta \\in \\{0,1\\}$ and $p_1,\\dots, p_k \\in S_1$.\n\\end{proposition}\nThis result immediately implies\n\\begin{corollary}\\label{cor:1}\nDefine $P_{S_1}=\\{p\\in\\mathcal{P} \\mid p=x^2+y^2+1,\\,x,y\\in \\mathbb{Z}, \\; \\gcd(x,y)=1\\}$. All primes $p \\in P_{S_1}$ satisfy $(p-1)_{S_1}\\geq (p-1)/2$.\n\\end{corollary}\n\n\\begin{proposition}\n Let $n \\in \\mathbb{N}$ and $S_2=\\{p\\in \\mathcal{P} \\mid p \\equiv 1 \\bmod 8 \\text{ or } p \\equiv 3 \\bmod 8\\}$. If $n=x^2+2y^2$ for coprime $x,y \\in \\mathbb{Z}$, then $n=2^\\delta \\prod_{i=1}^k p_i^{\\alpha_i}$, where $\\delta \\in \\{0,1\\}$ and $p_1,\\dots,p_k\\in S_2$.\n\\end{proposition}\n\\begin{proof}\n First, we observe that for any even integer $n=x^2+2y^2$, where $\\gcd(x,y)=1$, $x$ must be an even integer. Therefore, $n$ has a residue of $2$ modulo $4$ and is only divisible by $2$, not $4$. For odd primes the result is given by an old theorem of Fermat \\cite[p. 7]{Cox1997}:\n Odd prime divisors $p$ of $n$ must satisfy the condition \\begin{align*}\n 1=\\begin{pmatrix}\n \\frac{-2}{p}\n\\end{pmatrix}=\\begin{pmatrix}\n \\frac{-1}{p}\n\\end{pmatrix}\\begin{pmatrix}\n \\frac{2}{p}\n\\end{pmatrix}=(-1)^{\\frac{p-1}{2}}(-1)^{\\frac{p^2-1}{8}}. \n \\end{align*} \n This implies, as one readily verifies, that all odd prime divisors of $n$ lie in~$S_2$.\n\\end{proof}\n\\begin{corollary}\\label{cor:2}\nDefine $P_{S_2}=\\{p\\in\\mathcal{P} \\mid p=x^2+2y^2+1,\\,x,y\\in \\mathbb{Z}, \\; \\gcd(x,y)=1\\}$.\nAll primes $p \\in P_{S_2}$ satisfy $(p-1)_{S_2}\\geq (p-1)/2$.\n\\end{corollary}\n\n\\begin{proposition}\n Let $n \\in \\mathbb{N}$ and $S_3=\\{p\\in \\mathcal{P} \\mid p \\equiv 1 \\bmod 3\\}$. If $n=x^2+3y^2$ for coprime $x,y \\in \\mathbb{Z}$, then $n=2^{\\delta_1} 3^{\\delta_2} \\prod_{i=1}^k p_i^{\\alpha_i}$, where $\\delta_1 \\in \\{0,1,2\\}$, $\\delta_2 \\in \\{0,1\\}$, $p_1,\\dots,p_k\\in S_3$.\n\\end{proposition}\n\\begin{proof}\n First, we observe that for any even integer $n=x^2+3y^2$, where $\\gcd(x,y)=1$, both $x$ and $y$ are odd integers.\n Since all odd squares have a residue of $1$ modulo $8$, $n$ cannot be divisible by $8$.\n Regarding the divisor $3^{\\delta_2}$, there are two cases.\n First, if $y$ is divisible by $3$, then $x$ is not and $\\delta_2=0$. Second, if $x$ is divisible by $3$, then $y$ is not, and thus, $\\delta_2=1$. For odd primes the result is again given by an old theorem of Fermat \\cite[p. 7]{Cox1997}:\n According to the law of quadratic reciprocity, odd prime divisors $p\\neq 3$ must satisfy the following condition: \\begin{align*}\n 1=\\begin{pmatrix}\n \\frac{-3}{p}\n\\end{pmatrix}=\\begin{pmatrix}\n \\frac{-1}{p}\n\\end{pmatrix}\\begin{pmatrix}\n \\frac{3}{p}\n\\end{pmatrix}=(-1)^{\\frac{p-1}{2}} \\begin{pmatrix}\n \\frac{3}{p}\n\\end{pmatrix}=\\begin{pmatrix}\n \\frac{p}{3}\n\\end{pmatrix}. \n \\end{align*} \n Since $1$ is the only quadratic residue of a prime $p\\neq3$ modulo $3$, the only possible odd prime divisors of $n$ are the primes in $S_3$. \n\\end{proof}\nAs before we deduce\n\\begin{corollary}\\label{cor:3}\nDefine $P_{S_3}=\\{p\\in\\mathcal{P} \\mid p=x^2+3y^2+1,\\,x,y\\in \\mathbb{Z}, \\; \\gcd(x,y)=1\\}$. \nAll primes $p \\in P_{S_3}$ satisfy $(p-1)_{S_3}\\geq (p-1)/12$. \n\\end{corollary}\nNext, we will need some results from the theory of quadratic forms.\n\\begin{definition}[see \\cite{Cox1997}]\n Two binary quadratic forms, $f(x,y)$ and $g(x,y$), are said to be \\emph{(properly) equivalent} if there are $a,b,c,d \\in \\mathbb{Z}$ such that $ad-bc=1$ and\n \\begin{align*}\n f(x,y)=g(ax+by,cx+dy) \\text{ for all } x,y \\in \\mathbb{Z}. \n \\end{align*}\n The \\emph{genus of a quadratic form} $f(x,y)$ is the set of all binary quadratic forms that represent the same numbers in $(\\mathbb{Z}/|\\Delta_f|\\mathbb{Z})^*$. It is denoted by $R_f$.\n\\end{definition}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main}]\nLet $S$ be any set of primes.\nThe free procyclic group $\\widehat{\\bbZ} = \\prod_{p \\in \\mathcal{P}} \\bbZ_p$ has Weil abscissa $2$; see \\cite[Theorem C (i)]{corob2024weil}. Hence the inequality \n\\[\\alpha(H_S) \\leq 2\\] follows immediately using that $H_S$ is a quotient of $\\widehat{\\bbZ}$.", + "post_theorem_intro_text_len": 2357, + "post_theorem_intro_text": "Let $S$ be a non-empty subset of $\\mathcal{P}$ and let $H_S=\\prod_{p\\in S} \\mathbb{Z}_p$; here $\\bbZ_p$ denotes the additive group of $p$-adic integers. Then $H_S$ is a procyclic group; this means, it is a profinite group that contains a dense cyclic subgroup.\nConjecture A of \\cite{kionke2024} predicts three equalities, one of which can be stated as follows\n\\begin{conjecture*}\n The shifted partial Riemann zeta function $\\zeta_S(s-1):= \\prod_{p \\in S} \\frac{1}{1-p^{-s+1}}$ has the same abscissa of convergence as $\\zeta^W_{H_S}(s)$.\n\\end{conjecture*}\nIf $S = \\{p \\in \\mathcal{P} \\mid p \\equiv a \\bmod d\\}$ is the set of primes in an arithmetic progression with $\\gcd(a,d)=1$, then $\\zeta_S(s)$ has abscissa $1$ - this can be seen for instance using the Siegel–Walfisz theorem \\cite{walfisz1936} - and hence our theorem confirms the conjecture for these three groups.\n\n\\medskip\n\nCurrently the conjecture is known only for very big or very small sets $S$.\nFirst, consider sets $S \\subset\\mathcal{P}$ for which $\\zeta_{\\overline{S}}$ converges for a real number $s<1$, where $\\overline{S}=\\mathcal{P}\\setminus S$. These sets are called \\textit{thick}. If $S$ is a thick set of primes, then the second author proved $\\alpha(H_S)=2$; see \\cite[Theorem 1.3]{kionke2024}. Second, the conjecture is known to hold if $\\zeta_S(s)$ has abscissa $0$ (\\cite[Lemma 2.1]{kionke2024}). For certain exponentially growing sequences $S$ of prime numbers this was already observed in \\cite[Remark 5.4]{corob2024weil}.\nTherefore, the question of the value of the Weil abscissa is of special interest in \"in-between\" cases.\n\n\\medskip\n\nIn this paper we analyze cases in which $S$ is the set of prime numbers that are part of one or more arithmetic progressions. To prove our main result, we will exploit the fact that primes in some arithmetic progressions are representable by certain binary quadratic forms and apply a theorem of Iwaniec \\cite{Iwaniec1972}. The methods can be extended to prove general results about certain unions of arithmetic progressions. However, the method is limited and we emphasize it does not apply to general arithmetic progressions. In particular, the following question remains open.\n\n\\begin{question}\nDo the procyclic groups $\\prod_{p \\equiv 2 \\bmod 3} \\bbZ_p$ and $\\prod_{p \\equiv 3 \\bmod 4} \\bbZ_p$ have Weil abscissa $2$?\n\\end{question}", + "sketch": "To prove Theorem~\\ref{thm:main}, the paper \"analyze[s] cases in which $S$ is the set of prime numbers that are part of one or more arithmetic progressions\" and \"exploit[s] the fact that primes in some arithmetic progressions are representable by certain binary quadratic forms\"; then the authors \"apply a theorem of Iwaniec \\cite{Iwaniec1972}.\" They note that these methods \"can be extended to prove general results about certain unions of arithmetic progressions,\" but that the approach is \"limited\" and \"does not apply to general arithmetic progressions.\"", + "expanded_sketch": "To prove the main theorem, the paper \"analyze[s] cases in which $S$ is the set of prime numbers that are part of one or more arithmetic progressions\" and \"exploit[s] the fact that primes in some arithmetic progressions are representable by certain binary quadratic forms\"; then the authors \"apply a theorem of Iwaniec \\cite{Iwaniec1972}.\" They note that these methods \"can be extended to prove general results about certain unions of arithmetic progressions,\" but that the approach is \"limited\" and \"does not apply to general arithmetic progressions.\"", + "expanded_theorem": "\\label{thm:main}\nThe procyclic groups\n\\[\n \\prod_{p \\equiv 1 \\bmod 3} \\bbZ_p, \\quad \\prod_{p \\equiv 1 \\bmod 4} \\bbZ_p, \\quad \\prod_{p \\in \\{1,3\\} \\bmod 8} \\bbZ_p\n\\]\nhave Weil abscissa $2$.", + "theorem_type": [ + "Universal" + ], + "mcq": { + "question": "For a finitely generated profinite abelian group \\(G\\), the Weil representation zeta function is\n\\[\n\\zeta_G^W(s)=\\exp\\!\\left(\\sum_{p\\in\\mathcal P}\\sum_{j=1}^{\\infty} |\\operatorname{Hom}(G,\\mathbb F_{p^j}^{\\times})|\\frac{p^{-sj}}{j}\\right),\n\\]\nand the Weil abscissa \\(\\alpha(G)\\) is the abscissa of convergence of this Dirichlet-type series. Here \\(\\mathcal P\\) is the set of primes and \\(\\mathbb Z_p\\) denotes the ring of \\(p\\)-adic integers. Consider the three procyclic groups\n\\[\n\\prod_{p\\equiv 1\\, (\\mathrm{mod}\\,3)} \\mathbb Z_p,\\qquad\n\\prod_{p\\equiv 1\\, (\\mathrm{mod}\\,4)} \\mathbb Z_p,\\qquad\n\\prod_{p\\equiv 1\\text{ or }3\\, (\\mathrm{mod}\\,8)} \\mathbb Z_p.\n\\]\nWhich statement holds for every group in this list?", + "correct_choice": { + "label": "A", + "text": "Each of these three procyclic groups has Weil abscissa \\(2\\); equivalently, if \\(G\\) is any one of them, then \\(\\alpha(G)=2\\)." + }, + "choices": [ + { + "label": "B", + "text": "Each of these three procyclic groups has Weil abscissa strictly less than \\(2\\); equivalently, if \\(G\\) is any one of them, then \\(\\alpha(G)<2\\)." + }, + { + "label": "C", + "text": "For each of these three procyclic groups \\(G\\), one has \\(\\alpha(G)\\le 2\\)." + }, + { + "label": "D", + "text": "Each of these three procyclic groups has Weil abscissa \\(2\\), and the same conclusion holds for every procyclic group of the form \\(\\prod_{p\\equiv a\\, (\\mathrm{mod}\\,m)} \\mathbb Z_p\\) arising from a single arithmetic progression of primes." + }, + { + "label": "E", + "text": "Exactly two of these three procyclic groups have Weil abscissa \\(2\\), while the remaining one has Weil abscissa different from \\(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": "lower-bound mechanism needed to upgrade from \\(\\alpha\\le 2\\) to equality", + "template_used": "stronger_trap" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped the matching lower bound \\(\\alpha(G)\\ge 2\\), retaining only the quotient-based upper bound", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "geometric_construction", + "tampered_component": "restricted applicability of quadratic-form/Iwaniec method to special progressions rather than arbitrary arithmetic progressions", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "case_split", + "tampered_component": "uniform conclusion across the three explicitly listed congruence classes", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not state or strongly hint that the Weil abscissa is 2. It gives definitions and the three groups, but the correct conclusion is not leaked." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recall question about the listed examples, so it is not fully non-tautological. However, the options do force discrimination between the exact claim, a weaker true statement, and stronger overgeneralizations." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to identify the strongest valid statement among nearby alternatives, especially versus the weaker true option and the overgeneralized option. Still, the item primarily tests recognition/application of a known result rather than substantial fresh derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically meaningful: one is a weaker true statement, one is an overreach to all arithmetic progressions, one asserts the wrong inequality, and one introduces an unjustified case split. These align with common failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it is somewhat close to theorem restatement and only moderately pressures genuine generative reasoning." + } + }, + { + "id": "2602.10519v1", + "paper_link": "http://arxiv.org/abs/2602.10519v1", + "theorems_cnt": 4, + "theorem": { + "env_name": "theorem", + "content": "\\label{main blocks}\nThe category $\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$ decomposes into blocks which are either trivial\n(that is equivalent to the category of vector spaces) or of type $\\Gamma$. In particular,\nthe category $\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$ is of tame representation type. Also the cohomology of\n$\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$ (that is the Ext algebra of the unit object ${\\bf 1}$) is isomorphic to the algebra\nof invariants $S^\\bullet(V)^\\Gamma$.", + "start_pos": 6618, + "end_pos": 7062, + "label": "main blocks" + }, + "ref_dict": { + "main blocks": "\\begin{theorem}\\label{main blocks}\nThe category $\\C(\\g,e_{sr},l,q)$ decomposes into blocks which are either trivial\n(that is equivalent to the category of vector spaces) or of type $\\Gamma$. In particular,\nthe category $\\C(\\g,e_{sr},l,q)$ is of tame representation type. Also the cohomology of\n$\\C(\\g,e_{sr},l,q)$ (that is the Ext algebra of the unit object $\\be$) is isomorphic to the algebra\nof invariants $S^\\bullet(V)^\\Gamma$.\n\\end{theorem}", + "main 7mod": "\\begin{theorem}\\label{main 7mod}\n(1) The category $\\bar \\C(G_2,G_2(a_1),7,q)$ has 12 trivial blocks and one block of type $\\tilde D_4$.\nIn particular, it has 17 simple objects.\n\n(2) We have $\\FP(\\bar \\C(G_2,G_2(a_1),7,q))=49(7+15[3]_7+12[5]_7)\\approx 3054.068811$.\n\n(3) The category $\\bar \\C(G_2,G_2(a_1),7,q)$ has stable Chevalley property.\n\n(4) The category $\\bar \\C(G_2,G_2(a_1),7,q)$ is completely anisotropic: it has no non-trivial\ncommutative exact algebras.\n\\end{theorem}", + "Witt": "\\label{Witt}\nIn this section $\\K$ is an algebraically closed field of arbitrary characteristic.\n\\subsection{Exact algebras} Let $\\C$ be a finite tensor category over $\\K$ and let $A=(A,m,u)\\in \\C$ be", + "main 7": "\\begin{theorem}\\label{main 7}\n(1) The category $\\C(G_2,G_2(a_1),7,q)$ has 15 trivial blocks and one block of type $\\tilde E_7$.\nIn particular it has 23 simple objects.\n\n(2) We have $\\FP(\\C(G_2,G_2(a_1),7,q))=294(7+15[3]_7+12[5]_7)\\approx 18324.416384$.\n\n(3) The category $\\C(G_2,G_2(a_1),7,q)$ has stable Chevalley property: tensor products of simple \nobjects are direct sums of simples and projectives.\n\n(4) The M\\\"uger center of the category $\\C(G_2,G_2(a_1),7,q)$ is equivalent to $\\Rep(S_3)$\n(where $S_3$ is the symmetric group on three letters).\n\\end{theorem}", + "genconj": "\\label{genconj}\n\\subsection{Categories $\\C(G_2,G_2(a_1),l,q)$}\nWe expect that Theorem \\ref{main 7} extends to the case of arbitrary undivisible (i.e. not\ndivisible by 3) $l\\ge 7$.\n\n\\begin{conjecture}\\", + "divisible": "\\begin{definition} \\label{divisible}\nWe say that $l\\in \\Z_{\\ge 1}$ is \\textit{divisible} if $l$ is divisible\nby $m$ and $l$ is \\textit{undivisible} otherwise. \\end{definition}", + "distconj": "\\begin{conjecture}\\label{g2conjm}\n Assume $l\\ge 12$ and $l$ is divisible by 3. Then\n\n $$\\FP(\\C(G_2, G_2(a_1),l,q))=\\frac29\\frac{l^4}{(2\\sin(\\pi/l))^4(2\\sin(2\\pi/l))^2(2\\sin(3\\pi/l))^4}.$$\nand for $l\\ge 15$\n$$\\FP(T(\\omega_1))=[3]_l+[5]_l-1,\\; \\FP(T(\\omega_2))=2[7]_l-[5]_l+[3]_l+2.$$\n\\end{conjecture}\n\nWe checked that the formulas work for $l=12,15,18,21$; this applies also for $l=9$ if we\ndefine category $\\C(G_2, G_2(a_1),9,q)$ to be a semisimplification of $\\T(\\g,q)$\n(for $q$ such that $q^2$ has order 9), see \\cite{RW}. The formula for $\\FP(T(\\omega_2))$ gives\nan incorrect result for $l=12$, namely the right hand side and the left hand side\ndiffer exactly by 1. This is likely to be explained by the fact that $T(\\omega_2)$\ndoes not coincide with Weyl module in this (and only in this) case.\n\n\\subsection{Categories $\\C(\\g,e,l,q)$ for distinguished nilpotent $e\\in \\g$}\\label{distconj}\nLet $Q=Q(e)$ be the centralizer of a $sl_2-$triple associated with $e$ in the simply\nconnected group $G$ with $\\mbox{Lie}(G)=\\g$. In the case of distinguished nilpotent $e$,\n$Q$ is a finite group. Let $2k_i+1, i\\in M=M(e)$ be the sizes of Jordan cells for\nthe adjoint action of $e$ on $\\g$ (it is well known that these sizes are odd for distinguished\nnilpotent elements; the number of these cells, that is the cardinality of set $M$, is\nthe dimension of the centralizer of $e$ in $\\g$). For any $k\\in \\Z_{>0}$ we set\n$$S_k(l):=\\frac{l}{(2^k\\sin(\\pi/l)\\sin(2\\pi/l)\\cdots \\sin(k\\pi/l))^2}.$$\n\n\\begin{conjecture}\\label{FPdim}\n Assume $m=1$ or $l$ is undivisible. Then\n $$\\FP(\\C(\\g,e,l,q))=|Q|\\prod_{i=1}^MS_{k_i}(l).$$\n\\end{conjecture}" + }, + "pre_theorem_intro_text_len": 4606, + "pre_theorem_intro_text": "\\subsection{}\nThis paper is a contribution to the theory of braided finite tensor categories. In the case\nof semisimple categories over $\\mathbb{C}$ many known examples arise from Wess-Zumino-Witten\nmodels in conformal field theory, see e.g. \\cite{BK}. One of the easiest algebraic constructions of these categories was given\nby H.~H.~Andersen \\cite{An} (see also \\cite{Saw}). In modern terms this construction can be summarized\nas follows: for a simple Lie algebra $\\mathfrak{g}$ and a root of unity $q$ such that $l$ is the order of $q^2$ one considers\nthe category $\\mathcal{T}(\\mathfrak{g},q)$ of {\\em tilting modules} over the quantum group at a root of unity $q$ associated \nwith Lie algebra $\\mathfrak{g}$. Then one defines a category $\\mathcal{C}(\\mathfrak{g}, l, q)$ as the {\\em semisimplification}\nof $\\mathcal{T}(\\mathfrak{g},q)$ (see e.g. \\cite{EOsemi}). The category $\\mathcal{C}(\\mathfrak{g}, l, q)$ is a semisimple braided tensor\ncategory; moreover this category is finite (i.e. it has only finitely many classes of simple objects) if\n$l$ is sufficiently large. The precise bounds for $l$ are given in \\cite[Figure 2]{Scho}; in this paper \nwe will always assume that $l$ is sufficiently large in this sense. Note that there is a combinatorial \ndifference between the case when $l$ is {\\em divisible} (see \\ref{divisible}) and when it is not. \n\nThe procedure of semisimplification above can be described as taking the quotient by a suitable\ntensor ideal (namely, by the ideal of negligible morphisms). However, the category $\\mathcal{T}(\\mathfrak{g},q)$ has \nmany other tensor ideals. In \\cite{CEO} for any {\\em distinguished} nilpotent element $e\\in \\mathfrak{g}$ (or, in the case\nwhen $l$ is divisible, $e\\in \\mathfrak{g}^L$ where $\\mathfrak{g}^L$ is the Langlands dual Lie algebra of $\\mathfrak{g}$),\na tensor ideal $\\cI_e$ was constructed such that the quotient category $\\mathcal{T}(\\mathfrak{g},q)/\\cI_e$ admits a {\\em monoidal\nabelian envelope} $\\mathcal{C}(\\mathfrak{g},e,l,q)$; moreover, the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ is a finite tensor category\nin the sense of \\cite{EOfinite}. For example when $e$ is a regular nilpotent element (which is always\ndistinguished), the ideal $\\cI_e$ is the ideal of negligible morphisms and $\\mathcal{C}(\\mathfrak{g},e,l,q)=\\mathcal{C}(\\mathfrak{g},l,q)$.\nUnfortunately, there is not much we can say about the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ for other nilpotent\nelements $e$ (however, see Section \\ref{distconj} for a conjectural formula for the Frobenius-Perron dimension\nof $\\mathcal{C}(\\mathfrak{g},e,l,q)$ and conjectural description of its cohomology). The goal of this paper is to give \nsome explicit information about the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ in the case when $e=e_{sr}$ is a subregular\nnilpotent element, see \\cite[4.2]{CMg}. Recall that $e_{sr}$ is distinguished if and only if $\\mathfrak{g}$ is not of\ntypes $A_n, n\\ge 1$ or $B_n, n\\ge 2$, see \\cite{CMg}.\n\nTo state our main results we need a bit more notation. Let $V$ be a two dimensional space and let\n$\\Gamma \\subset SL(V)$ be a finite subgroup of even order. By the classical McKay correspondence,\nthe finite subgroups of $SL(V)$ up to conjugacy are labelled by simply laced affine Dynkin diagrams (the only\nsubgroups of odd order correspond to affine Dynkin diagrams of type $\\tilde A_{2n}$). We associate\na subgroup $\\Gamma$ to $\\mathfrak{g}$ and $l$ as above as follows:\n\\renewcommand{\\arraystretch}{1.4}\n$$\n\\begin{array}{|c|c|c|c|c|c|c|}\n\\hline\n\\mathfrak{g}&B_n&C_n&D_n&E_n&F_4&G_2\\\\\n&n\\ge 3&n\\ge 3&n\\ge 4&n=6,7,8&&\\\\\n\\hline\nl&\\mbox{divisible}&\\mbox{not divisible}&\\mbox{any}&\\mbox{any}&\\mbox{any}&\\mbox{any}\\\\\n\\hline\n\\Gamma &\\tilde D_{2n} &\\tilde D_{2n}&\\tilde D_n&\\tilde E_n&\\tilde E_7&\\tilde E_7\\\\\n\\hline\n\\end{array}\n$$\n\nLet $\\wedge(V)$ be the exterior algebra of $V$. We can consider $\\wedge(V)$ as an algebra in the category\n$\\mbox{Rep}(\\Gamma)$ of finite dimensional representations of $\\Gamma$. We consider the abelian\ncategory of left $\\wedge(V)-$modules in the category $\\mbox{Rep}(\\Gamma)$ and we call it\n{\\em block of type} $\\Gamma$. Thus, a block of type $\\Gamma$ is the category equivalent to that of finite dimensional\nrepresentations of the cross product of $\\wedge(V)$ with the group algebra of $\\Gamma$.\nFinally let $S^\\bullet(V)$ be the symmetric algebra of $V$ which is graded by even integers (so $V \\subset\nS^\\bullet(V)$ is in degree 2). The group $\\Gamma$ acts on $S^\\bullet(V)$ preserving the grading.\nOur first main result describes the structure of abelian category $\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$.", + "context": "\\subsection{}\nThis paper is a contribution to the theory of braided finite tensor categories. In the case\nof semisimple categories over $\\mathbb{C}$ many known examples arise from Wess-Zumino-Witten\nmodels in conformal field theory, see e.g. \\cite{BK}. One of the easiest algebraic constructions of these categories was given\nby H.~H.~Andersen \\cite{An} (see also \\cite{Saw}). In modern terms this construction can be summarized\nas follows: for a simple Lie algebra $\\mathfrak{g}$ and a root of unity $q$ such that $l$ is the order of $q^2$ one considers\nthe category $\\mathcal{T}(\\mathfrak{g},q)$ of {\\em tilting modules} over the quantum group at a root of unity $q$ associated \nwith Lie algebra $\\mathfrak{g}$. Then one defines a category $\\mathcal{C}(\\mathfrak{g}, l, q)$ as the {\\em semisimplification}\nof $\\mathcal{T}(\\mathfrak{g},q)$ (see e.g. \\cite{EOsemi}). The category $\\mathcal{C}(\\mathfrak{g}, l, q)$ is a semisimple braided tensor\ncategory; moreover this category is finite (i.e. it has only finitely many classes of simple objects) if\n$l$ is sufficiently large. The precise bounds for $l$ are given in \\cite[Figure 2]{Scho}; in this paper \nwe will always assume that $l$ is sufficiently large in this sense. Note that there is a combinatorial \ndifference between the case when $l$ is {\\em divisible} (see \\ref{divisible}) and when it is not.\n\nThe procedure of semisimplification above can be described as taking the quotient by a suitable\ntensor ideal (namely, by the ideal of negligible morphisms). However, the category $\\mathcal{T}(\\mathfrak{g},q)$ has \nmany other tensor ideals. In \\cite{CEO} for any {\\em distinguished} nilpotent element $e\\in \\mathfrak{g}$ (or, in the case\nwhen $l$ is divisible, $e\\in \\mathfrak{g}^L$ where $\\mathfrak{g}^L$ is the Langlands dual Lie algebra of $\\mathfrak{g}$),\na tensor ideal $\\cI_e$ was constructed such that the quotient category $\\mathcal{T}(\\mathfrak{g},q)/\\cI_e$ admits a {\\em monoidal\nabelian envelope} $\\mathcal{C}(\\mathfrak{g},e,l,q)$; moreover, the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ is a finite tensor category\nin the sense of \\cite{EOfinite}. For example when $e$ is a regular nilpotent element (which is always\ndistinguished), the ideal $\\cI_e$ is the ideal of negligible morphisms and $\\mathcal{C}(\\mathfrak{g},e,l,q)=\\mathcal{C}(\\mathfrak{g},l,q)$.\nUnfortunately, there is not much we can say about the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ for other nilpotent\nelements $e$ (however, see Section \\ref{distconj} for a conjectural formula for the Frobenius-Perron dimension\nof $\\mathcal{C}(\\mathfrak{g},e,l,q)$ and conjectural description of its cohomology). The goal of this paper is to give \nsome explicit information about the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ in the case when $e=e_{sr}$ is a subregular\nnilpotent element, see \\cite[4.2]{CMg}. Recall that $e_{sr}$ is distinguished if and only if $\\mathfrak{g}$ is not of\ntypes $A_n, n\\ge 1$ or $B_n, n\\ge 2$, see \\cite{CMg}.\n\nTo state our main results we need a bit more notation. Let $V$ be a two dimensional space and let\n$\\Gamma \\subset SL(V)$ be a finite subgroup of even order. By the classical McKay correspondence,\nthe finite subgroups of $SL(V)$ up to conjugacy are labelled by simply laced affine Dynkin diagrams (the only\nsubgroups of odd order correspond to affine Dynkin diagrams of type $\\tilde A_{2n}$). We associate\na subgroup $\\Gamma$ to $\\mathfrak{g}$ and $l$ as above as follows:\n\\renewcommand{\\arraystretch}{1.4}\n$$\n\\begin{array}{|c|c|c|c|c|c|c|}\n\\hline\n\\mathfrak{g}&B_n&C_n&D_n&E_n&F_4&G_2\\\\\n&n\\ge 3&n\\ge 3&n\\ge 4&n=6,7,8&&\\\\\n\\hline\nl&\\mbox{divisible}&\\mbox{not divisible}&\\mbox{any}&\\mbox{any}&\\mbox{any}&\\mbox{any}\\\\\n\\hline\n\\Gamma &\\tilde D_{2n} &\\tilde D_{2n}&\\tilde D_n&\\tilde E_n&\\tilde E_7&\\tilde E_7\\\\\n\\hline\n\\end{array}\n$$\n\nLet $\\wedge(V)$ be the exterior algebra of $V$. We can consider $\\wedge(V)$ as an algebra in the category\n$\\mbox{Rep}(\\Gamma)$ of finite dimensional representations of $\\Gamma$. We consider the abelian\ncategory of left $\\wedge(V)-$modules in the category $\\mbox{Rep}(\\Gamma)$ and we call it\n{\\em block of type} $\\Gamma$. Thus, a block of type $\\Gamma$ is the category equivalent to that of finite dimensional\nrepresentations of the cross product of $\\wedge(V)$ with the group algebra of $\\Gamma$.\nFinally let $S^\\bullet(V)$ be the symmetric algebra of $V$ which is graded by even integers (so $V \\subset\nS^\\bullet(V)$ is in degree 2). The group $\\Gamma$ acts on $S^\\bullet(V)$ preserving the grading.\nOur first main result describes the structure of abelian category $\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$.", + "full_context": "\\subsection{}\nThis paper is a contribution to the theory of braided finite tensor categories. In the case\nof semisimple categories over $\\mathbb{C}$ many known examples arise from Wess-Zumino-Witten\nmodels in conformal field theory, see e.g. \\cite{BK}. One of the easiest algebraic constructions of these categories was given\nby H.~H.~Andersen \\cite{An} (see also \\cite{Saw}). In modern terms this construction can be summarized\nas follows: for a simple Lie algebra $\\mathfrak{g}$ and a root of unity $q$ such that $l$ is the order of $q^2$ one considers\nthe category $\\mathcal{T}(\\mathfrak{g},q)$ of {\\em tilting modules} over the quantum group at a root of unity $q$ associated \nwith Lie algebra $\\mathfrak{g}$. Then one defines a category $\\mathcal{C}(\\mathfrak{g}, l, q)$ as the {\\em semisimplification}\nof $\\mathcal{T}(\\mathfrak{g},q)$ (see e.g. \\cite{EOsemi}). The category $\\mathcal{C}(\\mathfrak{g}, l, q)$ is a semisimple braided tensor\ncategory; moreover this category is finite (i.e. it has only finitely many classes of simple objects) if\n$l$ is sufficiently large. The precise bounds for $l$ are given in \\cite[Figure 2]{Scho}; in this paper \nwe will always assume that $l$ is sufficiently large in this sense. Note that there is a combinatorial \ndifference between the case when $l$ is {\\em divisible} (see \\ref{divisible}) and when it is not.\n\nThe procedure of semisimplification above can be described as taking the quotient by a suitable\ntensor ideal (namely, by the ideal of negligible morphisms). However, the category $\\mathcal{T}(\\mathfrak{g},q)$ has \nmany other tensor ideals. In \\cite{CEO} for any {\\em distinguished} nilpotent element $e\\in \\mathfrak{g}$ (or, in the case\nwhen $l$ is divisible, $e\\in \\mathfrak{g}^L$ where $\\mathfrak{g}^L$ is the Langlands dual Lie algebra of $\\mathfrak{g}$),\na tensor ideal $\\cI_e$ was constructed such that the quotient category $\\mathcal{T}(\\mathfrak{g},q)/\\cI_e$ admits a {\\em monoidal\nabelian envelope} $\\mathcal{C}(\\mathfrak{g},e,l,q)$; moreover, the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ is a finite tensor category\nin the sense of \\cite{EOfinite}. For example when $e$ is a regular nilpotent element (which is always\ndistinguished), the ideal $\\cI_e$ is the ideal of negligible morphisms and $\\mathcal{C}(\\mathfrak{g},e,l,q)=\\mathcal{C}(\\mathfrak{g},l,q)$.\nUnfortunately, there is not much we can say about the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ for other nilpotent\nelements $e$ (however, see Section \\ref{distconj} for a conjectural formula for the Frobenius-Perron dimension\nof $\\mathcal{C}(\\mathfrak{g},e,l,q)$ and conjectural description of its cohomology). The goal of this paper is to give \nsome explicit information about the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ in the case when $e=e_{sr}$ is a subregular\nnilpotent element, see \\cite[4.2]{CMg}. Recall that $e_{sr}$ is distinguished if and only if $\\mathfrak{g}$ is not of\ntypes $A_n, n\\ge 1$ or $B_n, n\\ge 2$, see \\cite{CMg}.\n\nTo state our main results we need a bit more notation. Let $V$ be a two dimensional space and let\n$\\Gamma \\subset SL(V)$ be a finite subgroup of even order. By the classical McKay correspondence,\nthe finite subgroups of $SL(V)$ up to conjugacy are labelled by simply laced affine Dynkin diagrams (the only\nsubgroups of odd order correspond to affine Dynkin diagrams of type $\\tilde A_{2n}$). We associate\na subgroup $\\Gamma$ to $\\mathfrak{g}$ and $l$ as above as follows:\n\\renewcommand{\\arraystretch}{1.4}\n$$\n\\begin{array}{|c|c|c|c|c|c|c|}\n\\hline\n\\mathfrak{g}&B_n&C_n&D_n&E_n&F_4&G_2\\\\\n&n\\ge 3&n\\ge 3&n\\ge 4&n=6,7,8&&\\\\\n\\hline\nl&\\mbox{divisible}&\\mbox{not divisible}&\\mbox{any}&\\mbox{any}&\\mbox{any}&\\mbox{any}\\\\\n\\hline\n\\Gamma &\\tilde D_{2n} &\\tilde D_{2n}&\\tilde D_n&\\tilde E_n&\\tilde E_7&\\tilde E_7\\\\\n\\hline\n\\end{array}\n$$\n\nLet $\\wedge(V)$ be the exterior algebra of $V$. We can consider $\\wedge(V)$ as an algebra in the category\n$\\mbox{Rep}(\\Gamma)$ of finite dimensional representations of $\\Gamma$. We consider the abelian\ncategory of left $\\wedge(V)-$modules in the category $\\mbox{Rep}(\\Gamma)$ and we call it\n{\\em block of type} $\\Gamma$. Thus, a block of type $\\Gamma$ is the category equivalent to that of finite dimensional\nrepresentations of the cross product of $\\wedge(V)$ with the group algebra of $\\Gamma$.\nFinally let $S^\\bullet(V)$ be the symmetric algebra of $V$ which is graded by even integers (so $V \\subset\nS^\\bullet(V)$ is in degree 2). The group $\\Gamma$ acts on $S^\\bullet(V)$ preserving the grading.\nOur first main result describes the structure of abelian category $\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$.\n\nLet $\\wedge(V)$ be the exterior algebra of $V$. We can consider $\\wedge(V)$ as an algebra in the category\n$\\mbox{Rep}(\\Gamma)$ of finite dimensional representations of $\\Gamma$. We consider the abelian\ncategory of left $\\wedge(V)-$modules in the category $\\mbox{Rep}(\\Gamma)$ and we call it\n{\\em block of type} $\\Gamma$. Thus, a block of type $\\Gamma$ is the category equivalent to that of finite dimensional\nrepresentations of the cross product of $\\wedge(V)$ with the group algebra of $\\Gamma$.\nFinally let $S^\\bullet(V)$ be the symmetric algebra of $V$ which is graded by even integers (so $V \\subset\nS^\\bullet(V)$ is in degree 2). The group $\\Gamma$ acts on $S^\\bullet(V)$ preserving the grading.\nOur first main result describes the structure of abelian category $\\C(\\g,e_{sr},l,q)$.\n\n\\begin{remark} The proof of Theorem \\ref{main blocks} shows that the number of blocks of type $\\Gamma$\nis the same as the number of weights inside of the fundamental alcove (but not on its boundary). Thus, this\nnumber is the same as the number of simple objects in the category $\\C(\\g,l,q)$.\n\\end{remark}\n\n\\begin{theorem}\\label{main 7}\n(1) The category $\\C(G_2,G_2(a_1),7,q)$ has 15 trivial blocks and one block of type $\\tilde E_7$.\nIn particular it has 23 simple objects.\n\n\\begin{theorem}\\label{main 7mod}\n(1) The category $\\bar \\C(G_2,G_2(a_1),7,q)$ has 12 trivial blocks and one block of type $\\tilde D_4$.\nIn particular, it has 17 simple objects.\n\nWe consider the category $\\Rep(\\Gamma \\ltimes V, \\epsilon)$ of\nfinite dimensional representations of $\\Gamma \\ltimes V$ where $\\epsilon$ acts as the parity automorphism,\nsee e.g. \\cite[9.11]{EGNO}. Thus, $\\Rep(\\Gamma \\ltimes V, \\epsilon)$ is a symmetric finite tensor category.\nAs an abelian category it is equivalent to the category of representations of algebra\n$\\wedge(V)$ in the category $\\Rep(\\Gamma)$ (or, equivalently, representations of $\\wedge(V)$ with an action\nof $\\Gamma$ compatible in an obvious sense). The simple objects of $\\Rep(\\Gamma \\ltimes V, \\epsilon)$ are irreducible representations of $\\Gamma$ where $V\\subset \\wedge(V)$ acts by zero;\nlet $\\Irr(\\Gamma)$ be the set of isomorphism classes of such objects.\nThe projective cover of the unit object is $\\wedge(V)$; and the projective cover of $V_i\\in \\Irr(\\Gamma)$\nis $V_i\\otimes \\wedge(V)$. In particular, the composition factors of the projective cover of $V_i$ are\n$V_i$ appearing twice and the irreducible summands of $V_i\\otimes V$. Thus, the Cartan matrix of\nthe category $\\Rep(\\Gamma \\ltimes V, \\epsilon)$ is $2\\Id+A_\\Gamma$ where $A_\\Gamma$ is the adjacency\nmatrix of the McKay graph of $\\Gamma$ (recall that the vertices of the McKay graph are elements \nof $\\Irr(\\Gamma)$ and the number of edges between $V_i$ and $V_j$ is the multiplicity of $V_j$ as \na direct summand of $V_i\\otimes V$).\n\n\\begin{corollary} \\label{cor:self-dual}\n Let $\\mathfrak g$ be of type $B_n, C_n, F_4, G_2, D_{2n}, E_7$, or $E_8$.\n Then all simple objects of $\\C=\\C(\\mathfrak g, e_{sr}, l, q)$ are self-dual. \n\\end{corollary}\n\\begin{proof}\n It is well-known that in this case all simple representations of $\\mathfrak g$ are self-dual. It follows from properties (1) and (2) in Section \\ref{qgtitling} that the tilting modules $T(\\lambda)$ are also self-dual.\n\n\\subsection{Proof of Theorem \\ref{main blocks}} By Proposition \\ref{Cartansr} the Cartan\nmatrix of the principal block of $\\C(\\g,e_{sr},l,q)$ is $2\\Id+A(X_{sr})$ where $X_{sr}$ is\nsome affine Dynkin diagram not of type $A$, so it is a tree. Thus, by Proposition \\ref{tree}\n(applied to this block and to the category $\\Rep(\\Gamma \\ltimes V)$)\nwe see that the principal block is equivalent to $\\Rep(\\Gamma \\ltimes V)$ where \n$V$ is as in \\ref{toy} and $\\Gamma \\subset SL(V)$ is a finite subgroup with McKay graph $X_{sr}$.\nUsing the translation functors (which descend to the category $\\T(\\g,q)/\\cI_{e_{sr}}$) we see\nthat all the other blocks of $\\C(\\g,e_{sr},l,q)$ involving $T(w\\cdot \\lambda)$ where $\\lambda$\nis in the interior of the fundamental alcove are equivalent to the principal block, and hence\nto the category $\\Rep(\\Gamma \\ltimes V)$. The other blocks involving $T(\\lambda)$ with $\\lambda$\non the wall are trivial by Corollaries \\ref{nog222} and \\ref{cor:on the wall}. The remaining statements of \nTheorem \\ref{main blocks} follow from Proposition \\ref{3.1}.\n\nIt is a classical result of P.~Slodowy that the singularity of the nilpotent\ncone of $\\g$ at the point $e_{sr}$ is $\\BC^2/\\Gamma$ for a suitable subgroup\n$\\Gamma \\subset SL_2(\\BC)$, see \\cite{Sl}. Comparing this with Theorem \\ref{main blocks}\nwe expect that the cohomology of the category $\\C(\\g,e,l,q)$ is related\nwith the singularity of the nilpotent cone at point $e$. Thus let $\\Sl_e$\nbe the Slodowy slice at the point $e$, see e.g. \\cite{GG}. The variety $\\Sl_e$\nis equipped with an action of $Q\\times \\BC^\\times$. Thus, the algebra of functions\n$\\mO(Sl_e)$ is graded and is equipped with a grading-preserving action of $Q$;\nlet $\\mO(Sl_e)^Q\\subset \\mO(Sl_e)$ be the algebra of invariants.\n\n\\begin{theorem}\\label{main blocks}\nThe category $\\C(\\g,e_{sr},l,q)$ decomposes into blocks which are either trivial\n(that is equivalent to the category of vector spaces) or of type $\\Gamma$. In particular,\nthe category $\\C(\\g,e_{sr},l,q)$ is of tame representation type. Also the cohomology of\n$\\C(\\g,e_{sr},l,q)$ (that is the Ext algebra of the unit object $\\be$) is isomorphic to the algebra\nof invariants $S^\\bullet(V)^\\Gamma$.\n\\end{theorem}", + "post_theorem_intro_text_len": 4178, + "post_theorem_intro_text": "\\begin{remark} The proof of Theorem \\ref{main blocks} shows that the number of blocks of type $\\Gamma$\nis the same as the number of weights inside of the fundamental alcove (but not on its boundary). Thus, this\nnumber is the same as the number of simple objects in the category $\\mathcal{C}(\\mathfrak{g},l,q)$.\n\\end{remark}\n\n\\subsection{} Next, we study one specific example, the category $\\mathcal{C}(G_2,G_2(a_1),7,q)$ (thus, we\nconsider Lie algebra $\\mathfrak{g}$ of type $G_2$; also $G_2(a_1)$ is the standard notation for the subregular\nnilpotent orbit in type $G_2$). This is the simplest example of the categories considered above\n(at least for undivisible $l$). Recall the standard notation for the quantum numbers:\n$$[k]_l=\\frac{\\sin(k\\pi/l)}{\\sin(\\pi/l)}.$$\nIn particular, $[2]_7=[5]_7=2\\cos(\\pi/7)\\approx 1.801938$ and $[3]_7=[4]_7=\\frac{\\sin(3\\pi/7)}{\\sin(\\pi/7)}\\approx\n2.246980$.\n\n\\begin{theorem}\\label{main 7}\n(1) The category $\\mathcal{C}(G_2,G_2(a_1),7,q)$ has 15 trivial blocks and one block of type $\\tilde E_7$.\nIn particular it has 23 simple objects.\n\n(2) We have $\\text{FPdim}(\\mathcal{C}(G_2,G_2(a_1),7,q))=294(7+15[3]_7+12[5]_7)\\approx 18324.416384$.\n\n(3) The category $\\mathcal{C}(G_2,G_2(a_1),7,q)$ has stable Chevalley property: tensor products of simple \nobjects are direct sums of simples and projectives.\n\n(4) The M\\\"uger center of the category $\\mathcal{C}(G_2,G_2(a_1),7,q)$ is equivalent to $\\text{Rep}(S_3)$\n(where $S_3$ is the symmetric group on three letters).\n\\end{theorem}\n\nIn view of Theorem \\ref{main 7} (4), it makes sense to consider the de-equivariantization $\\bar \\mathcal{C}(G_2,G_2(a_1),7,q)$ \nof $\\mathcal{C}(G_2,G_2(a_1),7,q)$ with respect to its M\\\"uger center (so the category $\\bar \\mathcal{C}(G_2,G_2(a_1),7,q)$ \nis the Brugui\\`eres' modularisation of $\\mathcal{C}(G_2,G_2(a_1),7,q)$, see \\cite{Brug}). The category\n$\\bar \\mathcal{C}(G_2,G_2(a_1),7,q)$ is a non-semisimple modular tensor category in the sense of Shimizu,\nsee \\cite{Shim}.\n\n\\begin{theorem}\\label{main 7mod}\n(1) The category $\\bar \\mathcal{C}(G_2,G_2(a_1),7,q)$ has 12 trivial blocks and one block of type $\\tilde D_4$.\nIn particular, it has 17 simple objects.\n\n(2) We have $\\text{FPdim}(\\bar \\mathcal{C}(G_2,G_2(a_1),7,q))=49(7+15[3]_7+12[5]_7)\\approx 3054.068811$.\n\n(3) The category $\\bar \\mathcal{C}(G_2,G_2(a_1),7,q)$ has stable Chevalley property.\n\n(4) The category $\\bar \\mathcal{C}(G_2,G_2(a_1),7,q)$ is completely anisotropic: it has no non-trivial\ncommutative exact algebras.\n\\end{theorem}\n\nIn \\cite[Question 7.20]{ShYa} (see also \\cite[Question 6.25]{LaWa}) K.~Shimizu and H.~Yadav asked whether non-semisimple completely\nanisotropic categories exist; Theorem \\ref{main 7mod} (4) gives a positive answer to this question.\n\nIn \\cite[Definition 6.23]{LaWa} (see also \\cite[Definition 7.2]{ShYa}) R.~Laugwitz and C.~Walton defined an important Witt equivalence\nrelation on the set of non-degenerate braided finite tensor categories. In Section \\ref{Witt} we prove\nsome general properties of this relation which imply\n\n\\begin{theorem}\\label{main 7Witt}\nThe category $\\bar \\mathcal{C}(G_2,G_2(a_1),7,q)$ is not Witt equivalent to any semisimple category.\n\\end{theorem}\n\nThus, the non-semisimple Witt group is different from its semisimple version studied in \\cite{DMNO}.\n\n\\subsection{} In Section \\ref{genconj} we present some conjectures. Most importantly, we expect that the categories\n$\\mathcal{C}(\\mathfrak{g},e,l,q)$ make sense for all nilpotent elements $e\\in \\mathfrak{g}$ (or $e\\in \\mathfrak{g}^L$\nin the divisible case). If $e$ is not distinguished,\nthe category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ is not finite; however we expect that it is obtained from a finite tensor\ncategory by equivariantization. \n\n\\subsection{Acknowledgements} Some ideas that led to this paper were inspired by participation of one of us (V.O.) in American Institute of Mathematics SQuaRE ``Lie algebras in symmetric tensor categories''; we are very grateful to this institution and to the fellow participants Iv\\'an Angiono, Agustina Czenky, Pavel Etingof, Julia Plavnik, and Guillermo Sanmarco. We also thank Dmitri Nikshych and Kenichi Shimizu for useful discussions.", + "sketch": "The only proof-related information in the post-theorem text is the remark:\n\n\\begin{remark} The proof of Theorem \\ref{main blocks} shows that the number of blocks of type $\\Gamma$\nis the same as the number of weights inside of the fundamental alcove (but not on its boundary). Thus, this\nnumber is the same as the number of simple objects in the category $\\mathcal{C}(\\mathfrak{g},l,q)$.\\end{remark}\n\nNo further outline/steps of the proof of Theorem~\\ref{main blocks} are given.", + "expanded_sketch": "No expanded sketch found.", + "expanded_theorem": "\\label{main blocks}\nThe category $\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$ decomposes into blocks which are either trivial\n(that is equivalent to the category of vector spaces) or of type $\\Gamma$. In particular,\nthe category $\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$ is of tame representation type. Also the cohomology of\n$\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$ (that is the Ext algebra of the unit object ${\\bf 1}$) is isomorphic to the algebra\nof invariants $S^\\bullet(V)^\\Gamma$.", + "theorem_type": [ + "Classification or Bijection", + "Universal" + ], + "mcq": { + "question": "Let $\\mathfrak g$ be a simple Lie algebra and let $q$ be a root of unity such that $l$ is the order of $q^2$, with $l$ sufficiently large. Assume that $(\\mathfrak g,l)$ is one of the following cases: $\\mathfrak g=B_n$ ($n\\ge 3$) with $l$ divisible; $\\mathfrak g=C_n$ ($n\\ge 3$) with $l$ not divisible; $\\mathfrak g=D_n$ ($n\\ge 4$) with any $l$; $\\mathfrak g=E_n$ ($n=6,7,8$) with any $l$; $\\mathfrak g=F_4$ with any $l$; or $\\mathfrak g=G_2$ with any $l$. Let $e_{sr}$ be the subregular nilpotent element, and let $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ be the corresponding finite tensor category. Let $V$ be a $2$-dimensional vector space, and let $\\Gamma\\subset SL(V)$ be the finite subgroup attached to $(\\mathfrak g,l)$ by the McKay correspondence, namely of affine Dynkin type $\\tilde D_{2n}$ for $B_n$ and $C_n$, $\\tilde D_n$ for $D_n$, $\\tilde E_n$ for $E_n$, and $\\tilde E_7$ for $F_4$ and $G_2$. A block of type $\\Gamma$ means the abelian category of left $\\wedge(V)$-modules in $\\operatorname{Rep}(\\Gamma)$, equivalently the category of finite-dimensional representations of the crossed product algebra $\\wedge(V)\\rtimes \\Gamma$. If $S^\\bullet(V)$ denotes the symmetric algebra of $V$ (with the natural $\\Gamma$-action), which statement holds for every such category $\\mathcal C(\\mathfrak g,e_{sr},l,q)$?", + "correct_choice": { + "label": "A", + "text": "The category $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is either trivial (equivalent to the category of vector spaces) or of type $\\Gamma$. Consequently $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ has tame representation type, and its cohomology, i.e. the Ext algebra $\\operatorname{Ext}^\\bullet_{\\mathcal C}(\\mathbf 1,\\mathbf 1)$ of the unit object, is isomorphic to the invariant algebra $S^\\bullet(V)^\\Gamma$." + }, + "choices": [ + { + "label": "B", + "text": "The category $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is of type $\\Gamma$. Consequently $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ has tame representation type, and its cohomology, i.e. the Ext algebra $\\operatorname{Ext}^\\bullet_{\\mathcal C}(\\mathbf 1,\\mathbf 1)$ of the unit object, is isomorphic to the full symmetric algebra $S^\\bullet(V)$." + }, + { + "label": "C", + "text": "The category $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is either trivial (equivalent to the category of vector spaces) or of type $\\Gamma$. In particular, $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ has tame representation type." + }, + { + "label": "D", + "text": "The category $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is either trivial (equivalent to the category of vector spaces) or of type $\\Gamma$. Consequently $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ has finite representation type, and its cohomology, i.e. the Ext algebra $\\operatorname{Ext}^\\bullet_{\\mathcal C}(\\mathbf 1,\\mathbf 1)$ of the unit object, is isomorphic to the invariant algebra $S^\\bullet(V)^\\Gamma$." + }, + { + "label": "E", + "text": "The category $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is either trivial (equivalent to the category of vector spaces) or Morita equivalent to $\\operatorname{Rep}(\\Gamma)$. Consequently $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ has tame representation type, and its cohomology, i.e. the Ext algebra $\\operatorname{Ext}^\\bullet_{\\mathcal C}(\\mathbf 1,\\mathbf 1)$ of the unit object, is isomorphic to the invariant algebra $S^\\bullet(V)^\\Gamma$." + } + ], + "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": "invariants_vs_full_symmetric_and_presence_of_trivial_blocks", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped_cohomology_identification_with_S^\\bullet(V)^\\Gamma", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "tame_vs_finite_representation_type", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "block_type_Gamma_replaced_by_Rep(Gamma)", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly state the correct conclusion. It supplies technical setup and definitions, but it does not directly reveal the decomposition, tame type, and cohomology identification." + }, + "TAS": { + "score": 1, + "justification": "This is close to a theorem-recall item: the stem gives the hypotheses of a specific result and asks for the matching conclusion. It is not a pure restatement because the options differ in subtle but meaningful ways, yet it remains largely theorem recognition." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish between nearby statements (tame vs finite type, invariants vs full symmetric algebra, block type vs Morita equivalence, complete vs weaker true statement). However, it mainly tests precise recall/comparison rather than substantial generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically targeted. They reflect common failure modes: overstrengthening, weakening, confusing invariants with the full algebra, and replacing the intended block description with a nearby categorical notion." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with strong distractors and little answer leakage, but it is primarily a precise theorem-identification question rather than a genuinely generative reasoning task." + } + }, + { + "id": "2602.10676v1", + "paper_link": "http://arxiv.org/abs/2602.10676v1", + "theorems_cnt": 3, + "theorem": { + "env_name": "thmA", + "content": "\\label{thm:main}\nThere exists a constant $c>0$ such that the following holds. Let $M$ be a closed hyperbolic 3-manifold and let $\\gamma\\subset M$ be a simple closed geodesic. Suppose that $\\gamma$ has length $\\ell$ and has an embedded tubular neighborhood of radius at least $R\\in(0,1)$. Then the unique hyperbolic metric on $M-\\gamma$ has volume\n\\[\n{\\rm vol}(M-\\gamma)\\le{\\rm vol}(M)+c\\frac{\\ell}{R}.\n\\]", + "start_pos": 2288, + "end_pos": 2716, + "label": "thm:main" + }, + "ref_dict": { + "pro:metric on tube": "\\begin{pro}\n\\label{pro:metric on tube}\nThere exists a constant $c>0$ such that the following holds. Let $V$ be a solid torus. Suppose that the boundary $\\partial V$ is equipped with a flat metric of area $A$. Assume that the length of the flat geodesic representative on $\\partial V$ of the meridian $\\mu\\subset\\partial V$ is $\\ell>2\\pi$. Then there exists a smooth Riemannian metric $(V,g)$ with the following properties.\n\\begin{itemize}\n \\item{The metric $g$ is a hyperbolic cusp metric in a collar of $\\partial V$, the boundary $\\partial V$ is a horospherical section of the cusp, and the restriction of $g$ to the boundary agrees with the prescribed flat metric on $\\partial V$.}\n \\item{Denote by $S_g$ the scalar curvature of the metric $g$. We have\n \\[\n \\int_V{\\left(\\frac{|S_g|}{6}\\right)^{3/2}{\\rm dvol}_g}\\le\\frac{A}{2}\\left(1-\\frac{\\pi^2}{\\ell^2}\\right)\\left(1+c\\frac{\\pi^4}{\\ell^4}\\right).\n \\]\n }\n\\end{itemize} \n\\end{pro}", + "lem:functiondrill": "\\begin{lem}\n\\label{lem:functiondrill}\nThere exists a constant $c>0$ such that the following holds. For every $R>0$ there exist functions $a_{\\rm drill},b_{\\rm drill}:\\mb{R}\\to\\mb{R}$ such that \n\\begin{enumerate}\n \\item{$a_{\\rm drill},b_{\\rm drill}>0$.}\n \\item{$a_{\\rm drill}(r)=\\sinh(r),b_{\\rm drill}(r)=\\cosh(r)$ on $(2R/3,\\infty)$.}\n \\item{$a_{\\rm drill}(r)=\\sinh(R)e^r,\\cosh(R)b_{\\rm drill}(r)=e^r$ on $(-\\infty,R/3)$.}\n \\item{On $[R/3,2R/3]$ we have\n \\[\n \\left|\\frac{a_{\\rm drill}''}{a_{\\rm drill}}+\\frac{b_{\\rm drill}''}{b_{\\rm drill}}+\\frac{a_{\\rm drill}'b_{\\rm drill}'}{a_{\\rm drill}b_{\\rm drill}}\\right|\\le c\\frac{1}{R^2}.\n \\]\n }\n \\item{We have $a_{\\rm drill}b_{\\rm drill}\\le \\sinh(R)\\cosh(R)e^{2r}$.}\n\\end{enumerate}\n\\end{lem}", + "lem:functionfill": "\\begin{lem}\n\\label{lem:functionfill}\nThere exists a constant $c>0$ such that the following holds. For every $\\ell_1>2\\pi,\\ell_2>0$ there exist functions $a_{\\rm fill},b_{\\rm fill}:\\mb{R}\\to\\mb{R}$ that satisfy the following properties. Let $R=\\log(\\ell_1/\\pi)$ and $\\kappa=2\\pi\\ell_2/\\ell_1$.\n\\begin{enumerate}\n \\item{$a_{\\rm fill},b_{\\rm fill}>0$.}\n \\item{$a_{\\rm fill}(r)=2\\pi\\sinh(r+R),b(r+R)=\\kappa\\cosh(r+R)$ on $(-R,-2\\delta)$.}\n \\item{$a_{\\rm fill}(r)=\\ell_1e^r,b(r)=\\ell_2 e^r$ on $(-\\delta,0)$.}\n \\item{We have\n \\[\n \\left|\\frac{a_{\\rm fill}''}{a_{\\rm fill}}+\\frac{b_{\\rm fill}''}{b_{\\rm fill}}+\\frac{a_{\\rm fill}'b_{\\rm fill}'}{a_{\\rm fill}b_{\\rm fill}}-3\\right|\\le c\\frac{\\pi^4}{\\ell_1^4}.\n \\]\n }\n \\item{We have \n \\[\n a_{\\rm fill}b_{\\rm fill}\\le\\ell_1\\ell_2e^{2r}.\n \\]\n }\n\\end{enumerate}\n\\end{lem}", + "cor:main": "\\begin{corA}\n\\label{cor:main}\nThere exists a constant $c>0$ such that the following holds. Let $M$ be a closed hyperbolic 3-manifold and let $\\gamma\\subset M$ be a shortest non-trivial closed geodesic in $M$. Then the unique hyperbolic metric on $M-\\gamma$ has volume\n\\[\n{\\rm vol}(M-\\gamma)\\le{\\rm vol}(M)+c\\ell\n\\]\nwhere $\\ell=2\\cdot{\\rm inj}(M)$ is the length of $\\gamma$.\n\\end{corA}", + "thm:main": "\\begin{thmA}\n\\label{thm:main}\nThere exists a constant $c>0$ such that the following holds. Let $M$ be a closed hyperbolic 3-manifold and let $\\gamma\\subset M$ be a simple closed geodesic. Suppose that $\\gamma$ has length $\\ell$ and has an embedded tubular neighborhood of radius at least $R\\in(0,1)$. Then the unique hyperbolic metric on $M-\\gamma$ has volume\n\\[\n{\\rm vol}(M-\\gamma)\\le{\\rm vol}(M)+c\\frac{\\ell}{R}.\n\\]\n\\end{thmA}", + "thm:main'": "\\begin{thmA}\n\\label{thm:main'}\nThere exists a constant $c>0$ such that the following holds. Let $M$ be a finite volume hyperbolic 3-manifold with one cusp. Let $C$ be a cusp neighborhood of $M$ bounded by the horospherical torus $\\partial C$. Consider a simple closed geodesic $\\mu\\subset\\partial C$ for the intrinsic flat metric on $\\partial C$. If the length of $\\mu$ is $\\ell>\\sqrt{c}\\pi$ and the area of $\\partial C$ is $A$ then the Dehn filling of $M-C$ with slope $\\mu$, denoted by $M_\\mu$, has a hyperbolic metric of volume\n\\[\n{\\rm vol}(M_\\mu)+\\frac{A}{2}\\frac{\\pi^2}{\\ell^2}\\left(1-c\\frac{\\pi^2}{\\ell^2}\\right)\\le{\\rm vol}(M).\n\\]\n\\end{thmA}" + }, + "pre_theorem_intro_text_len": 560, + "pre_theorem_intro_text": "There are two important operations that one can perform on a finite volume hyperbolic 3-manifold. One can drill out a simple closed geodesic and one can Dehn fill a cusp. It is a well-known phenomenon that all drillings and most fillings give back a hyperbolizable 3-manifold whose volume is larger for the first operation and smaller for the second. The goal of this short note is to prove two results that give a {\\em linear} quantitative control on how much the volume can change.\n\n\\subsection*{Drilling}\nLet us start with drillings. We prove the following.", + "context": "There are two important operations that one can perform on a finite volume hyperbolic 3-manifold. One can drill out a simple closed geodesic and one can Dehn fill a cusp. It is a well-known phenomenon that all drillings and most fillings give back a hyperbolizable 3-manifold whose volume is larger for the first operation and smaller for the second. The goal of this short note is to prove two results that give a {\\em linear} quantitative control on how much the volume can change.\n\n\\subsection*{Drilling}\nLet us start with drillings. We prove the following.", + "full_context": "There are two important operations that one can perform on a finite volume hyperbolic 3-manifold. One can drill out a simple closed geodesic and one can Dehn fill a cusp. It is a well-known phenomenon that all drillings and most fillings give back a hyperbolizable 3-manifold whose volume is larger for the first operation and smaller for the second. The goal of this short note is to prove two results that give a {\\em linear} quantitative control on how much the volume can change.\n\n\\subsection*{Drilling}\nLet us start with drillings. We prove the following.\n\n\\subsection*{Drilling}\nLet us start with drillings. We prove the following.\n\n\\begin{corA}\n\\label{cor:main}\nThere exists a constant $c>0$ such that the following holds. Let $M$ be a closed hyperbolic 3-manifold and let $\\gamma\\subset M$ be a shortest non-trivial closed geodesic in $M$. Then the unique hyperbolic metric on $M-\\gamma$ has volume\n\\[\n{\\rm vol}(M-\\gamma)\\le{\\rm vol}(M)+c\\ell\n\\]\nwhere $\\ell=2\\cdot{\\rm inj}(M)$ is the length of $\\gamma$.\n\\end{corA}\n\nSimilar estimates on the volume increase under drilling operations on hyperbolic 3-manifolds were already considered by Bridgeman \\cite{B}, Agol \\cite{A}, Agol, Storm, Thurston, and Dunfield \\cite[Theorem 10.1]{AST} where they proved the existence of explicit functions $f,g>1$ such that ${\\rm vol}(M-\\gamma)\\le f(R){\\rm vol}(M)+g(R)\\ell$. So, the main achievement of this note is to get rid of the multiplicative constant in front of ${\\rm vol}(M)$. This shows that the increase in volume only depends on the {\\em local} geometry around $\\gamma$.\n\nTheorem \\ref{thm:main} goes in the direction of a conjecture of Bridgeman \\cite{B}, which asks whether there is a linear upper bound on the volume increase under drilling operations. While very good estimates are available when the drilled geodesics are short (see the work of Hodgson and Kerckhoff \\cite{HK}), it is still unclear what happens for long geodesics. Note that as long as we can uniformly bound from below $R$ the increase in volume is at most linear. This leads immediately to Corollary \\ref{cor:main}. In fact, it is well-known that a shortest geodesic in a closed hyperbolic 3-manifold has a tubular neighborhood of radius at least $\\ell/4$ where $\\ell$ is the length of the geodesic (see \\cite[Proposition 1.11]{GMT}). Combined with standard consequences of Margulis Lemma, this provides a universal constant $R_0>0$ such that a shortest geodesic in a closed hyperbolic 3-manifold $M$ always has a tubular neighborhood of radius $R\\ge R_0$ (for much more refined estimates see \\cite[Theorem 4.1]{GMT}).\n\n\\begin{thmA}\n\\label{thm:main'}\nThere exists a constant $c>0$ such that the following holds. Let $M$ be a finite volume hyperbolic 3-manifold with one cusp. Let $C$ be a cusp neighborhood of $M$ bounded by the horospherical torus $\\partial C$. Consider a simple closed geodesic $\\mu\\subset\\partial C$ for the intrinsic flat metric on $\\partial C$. If the length of $\\mu$ is $\\ell>\\sqrt{c}\\pi$ and the area of $\\partial C$ is $A$ then the Dehn filling of $M-C$ with slope $\\mu$, denoted by $M_\\mu$, has a hyperbolic metric of volume\n\\[\n{\\rm vol}(M_\\mu)+\\frac{A}{2}\\frac{\\pi^2}{\\ell^2}\\left(1-c\\frac{\\pi^2}{\\ell^2}\\right)\\le{\\rm vol}(M).\n\\]\n\\end{thmA}\n\nTo prove Theorem \\ref{thm:main}, we follow the approach of \\cite[Theorem 10.1]{AST} with a difference. We first change the metric of $M$ inside a tubular neighborhood $T$ of $\\gamma$ interpolating between the metric on the tube and the metric on a suitable hyperbolic cusp. We do so by keeping under control the scalar curvature (see Lemma \\ref{lem:functiondrill}). Then we compare the volume of the modified metric with that of the hyperbolic structure on $M-\\gamma$. Instead of relying on Perelman's monotonicity formula, we exploit some other consequences of Perelman's work \\cite{Pe1,Pe2}, namely, the exact computation of the sigma invariant of closed hyperbolic 3-manifolds as achieved by Anderson \\cite{And} and Kleiner and Lott \\cite{KL}. This allows us to keep track of the fact that we {\\em did not change} the metric outside the tubular neighborhood $T$ and obtain an additive error rather than a multiplicative one in the volume increase.\n\n\\begin{lem}\n\\label{lem:functionfill}\nThere exists a constant $c>0$ such that the following holds. For every $\\ell_1>2\\pi,\\ell_2>0$ there exist functions $a_{\\rm fill},b_{\\rm fill}:\\mb{R}\\to\\mb{R}$ that satisfy the following properties. Let $R=\\log(\\ell_1/\\pi)$ and $\\kappa=2\\pi\\ell_2/\\ell_1$.\n\\begin{enumerate}\n \\item{$a_{\\rm fill},b_{\\rm fill}>0$.}\n \\item{$a_{\\rm fill}(r)=2\\pi\\sinh(r+R),b(r+R)=\\kappa\\cosh(r+R)$ on $(-R,-2\\delta)$.}\n \\item{$a_{\\rm fill}(r)=\\ell_1e^r,b(r)=\\ell_2 e^r$ on $(-\\delta,0)$.}\n \\item{We have\n \\[\n \\left|\\frac{a_{\\rm fill}''}{a_{\\rm fill}}+\\frac{b_{\\rm fill}''}{b_{\\rm fill}}+\\frac{a_{\\rm fill}'b_{\\rm fill}'}{a_{\\rm fill}b_{\\rm fill}}-3\\right|\\le c\\frac{\\pi^4}{\\ell_1^4}.\n \\]\n }\n \\item{We have \n \\[\n a_{\\rm fill}b_{\\rm fill}\\le\\ell_1\\ell_2e^{2r}.\n \\]\n }\n\\end{enumerate}\n\\end{lem}\n\n\\begin{pro}\n\\label{pro:metric on tube}\nThere exists a constant $c>0$ such that the following holds. Let $V$ be a solid torus. Suppose that the boundary $\\partial V$ is equipped with a flat metric of area $A$. Assume that the length of the flat geodesic representative on $\\partial V$ of the meridian $\\mu\\subset\\partial V$ is $\\ell>2\\pi$. Then there exists a smooth Riemannian metric $(V,g)$ with the following properties.\n\\begin{itemize}\n \\item{The metric $g$ is a hyperbolic cusp metric in a collar of $\\partial V$, the boundary $\\partial V$ is a horospherical section of the cusp, and the restriction of $g$ to the boundary agrees with the prescribed flat metric on $\\partial V$.}\n \\item{Denote by $S_g$ the scalar curvature of the metric $g$. We have\n \\[\n \\int_V{\\left(\\frac{|S_g|}{6}\\right)^{3/2}{\\rm dvol}_g}\\le\\frac{A}{2}\\left(1-\\frac{\\pi^2}{\\ell^2}\\right)\\left(1+c\\frac{\\pi^4}{\\ell^4}\\right).\n \\]\n }\n\\end{itemize} \n\\end{pro}\n\n\\begin{lem}\n\\label{lem:functiondrill}\nThere exists a constant $c>0$ such that the following holds. For every $R>0$ there exist functions $a_{\\rm drill},b_{\\rm drill}:\\mb{R}\\to\\mb{R}$ such that \n\\begin{enumerate}\n \\item{$a_{\\rm drill},b_{\\rm drill}>0$.}\n \\item{$a_{\\rm drill}(r)=\\sinh(r),b_{\\rm drill}(r)=\\cosh(r)$ on $(2R/3,\\infty)$.}\n \\item{$a_{\\rm drill}(r)=\\sinh(R)e^r,\\cosh(R)b_{\\rm drill}(r)=e^r$ on $(-\\infty,R/3)$.}\n \\item{On $[R/3,2R/3]$ we have\n \\[\n \\left|\\frac{a_{\\rm drill}''}{a_{\\rm drill}}+\\frac{b_{\\rm drill}''}{b_{\\rm drill}}+\\frac{a_{\\rm drill}'b_{\\rm drill}'}{a_{\\rm drill}b_{\\rm drill}}\\right|\\le c\\frac{1}{R^2}.\n \\]\n }\n \\item{We have $a_{\\rm drill}b_{\\rm drill}\\le \\sinh(R)\\cosh(R)e^{2r}$.}\n\\end{enumerate}\n\\end{lem}\n\n\\begin{thmA}\n\\label{thm:main}\nThere exists a constant $c>0$ such that the following holds. Let $M$ be a closed hyperbolic 3-manifold and let $\\gamma\\subset M$ be a simple closed geodesic. Suppose that $\\gamma$ has length $\\ell$ and has an embedded tubular neighborhood of radius at least $R\\in(0,1)$. Then the unique hyperbolic metric on $M-\\gamma$ has volume\n\\[\n{\\rm vol}(M-\\gamma)\\le{\\rm vol}(M)+c\\frac{\\ell}{R}.\n\\]\n\\end{thmA}", + "post_theorem_intro_text_len": 6232, + "post_theorem_intro_text": "\\begin{corA}\n\\label{cor:main}\nThere exists a constant $c>0$ such that the following holds. Let $M$ be a closed hyperbolic 3-manifold and let $\\gamma\\subset M$ be a shortest non-trivial closed geodesic in $M$. Then the unique hyperbolic metric on $M-\\gamma$ has volume\n\\[\n{\\rm vol}(M-\\gamma)\\le{\\rm vol}(M)+c\\ell\n\\]\nwhere $\\ell=2\\cdot{\\rm inj}(M)$ is the length of $\\gamma$.\n\\end{corA}\n\nSimilar estimates on the volume increase under drilling operations on hyperbolic 3-manifolds were already considered by Bridgeman \\cite{B}, Agol \\cite{A}, Agol, Storm, Thurston, and Dunfield \\cite[Theorem 10.1]{AST} where they proved the existence of explicit functions $f,g>1$ such that ${\\rm vol}(M-\\gamma)\\le f(R){\\rm vol}(M)+g(R)\\ell$. So, the main achievement of this note is to get rid of the multiplicative constant in front of ${\\rm vol}(M)$. This shows that the increase in volume only depends on the {\\em local} geometry around $\\gamma$. \n\nTheorem \\ref{thm:main} goes in the direction of a conjecture of Bridgeman \\cite{B}, which asks whether there is a linear upper bound on the volume increase under drilling operations. While very good estimates are available when the drilled geodesics are short (see the work of Hodgson and Kerckhoff \\cite{HK}), it is still unclear what happens for long geodesics. Note that as long as we can uniformly bound from below $R$ the increase in volume is at most linear. This leads immediately to Corollary \\ref{cor:main}. In fact, it is well-known that a shortest geodesic in a closed hyperbolic 3-manifold has a tubular neighborhood of radius at least $\\ell/4$ where $\\ell$ is the length of the geodesic (see \\cite[Proposition 1.11]{GMT}). Combined with standard consequences of Margulis Lemma, this provides a universal constant $R_0>0$ such that a shortest geodesic in a closed hyperbolic 3-manifold $M$ always has a tubular neighborhood of radius $R\\ge R_0$ (for much more refined estimates see \\cite[Theorem 4.1]{GMT}). \n\nLet us mention the fact that the restriction to simple geodesics in closed hyperbolic 3-manifolds, rather than dealing with geodesic links in arbitrary finite volume ones, is not important. A straightforward generalization of Theorem \\ref{thm:main} holds in this larger setting with exactly the same arguments and more robust bookkeeping. We chose to restrict ourselves to the simple case to streamline the exposition. \n\n\\subsection*{Filling}\nNow we move to fillings. We give a new proof of the following lower bound due to Hodgson and Kerckhoff \\cite{HK} using completely different techniques.\n\n\\begin{thmA}\n\\label{thm:main'}\nThere exists a constant $c>0$ such that the following holds. Let $M$ be a finite volume hyperbolic 3-manifold with one cusp. Let $C$ be a cusp neighborhood of $M$ bounded by the horospherical torus $\\partial C$. Consider a simple closed geodesic $\\mu\\subset\\partial C$ for the intrinsic flat metric on $\\partial C$. If the length of $\\mu$ is $\\ell>\\sqrt{c}\\pi$ and the area of $\\partial C$ is $A$ then the Dehn filling of $M-C$ with slope $\\mu$, denoted by $M_\\mu$, has a hyperbolic metric of volume\n\\[\n{\\rm vol}(M_\\mu)+\\frac{A}{2}\\frac{\\pi^2}{\\ell^2}\\left(1-c\\frac{\\pi^2}{\\ell^2}\\right)\\le{\\rm vol}(M).\n\\]\n\\end{thmA}\n\nWe remark that our assumptions differ slightly from the ones of \\cite{HK} as we ask that $\\ell\\ge\\sqrt{c}\\pi$ whereas \\cite{HK} requires that $\\ell^2/A$ is bigger than a universal constant. \n\nThe problem of understanding how much the volume decreases under filling is better understood compared to the drilling case and has been studied with various methods. Thurston \\cite[Chapter 6]{ThuNotes} showed that ${\\rm vol}(M_\\mu)<{\\rm vol}(M)$ and that for every {\\em fixed} $M$ we have ${\\rm vol}(M_\\mu)\\to{\\rm vol}(M)$ as $\\ell\\to\\infty$. Neumann and Zagier \\cite{NZ} then clarified, for every {\\em fixed} $M$, the asymptotic behavior of ${\\rm vol}(M)-{\\rm vol}(M_\\mu)$ as a function of $A/\\ell^2$. Later, Hodgson and Kerckhoff \\cite{HK}, gave a {\\em universal} bound of the form ${\\rm vol}(M)-{\\rm vol}(M_\\mu)= A\\pi^2/2\\ell^2+O(A^2/\\ell^4)$.\n\nThe restriction to Dehn fillings of hyperbolic 3-manifolds of finite volume with one cusp rather than allowing multiple Dehn fillings of a selection of cusps of an arbitrary finite volume hyperbolic 3-manifold is not really important. A straightforward generalization of Theorem \\ref{thm:main'} holds in that larger setting with the same proof but more notation. In order to keep the exposition as linear and short as possible, we restricted ourselves to the simple setup.\n\n\\subsection*{On the proofs}\nLastly, a few words on the proofs. \n\nTo prove Theorem \\ref{thm:main}, we follow the approach of \\cite[Theorem 10.1]{AST} with a difference. We first change the metric of $M$ inside a tubular neighborhood $T$ of $\\gamma$ interpolating between the metric on the tube and the metric on a suitable hyperbolic cusp. We do so by keeping under control the scalar curvature (see Lemma \\ref{lem:functiondrill}). Then we compare the volume of the modified metric with that of the hyperbolic structure on $M-\\gamma$. Instead of relying on Perelman's monotonicity formula, we exploit some other consequences of Perelman's work \\cite{Pe1,Pe2}, namely, the exact computation of the sigma invariant of closed hyperbolic 3-manifolds as achieved by Anderson \\cite{And} and Kleiner and Lott \\cite{KL}. This allows us to keep track of the fact that we {\\em did not change} the metric outside the tubular neighborhood $T$ and obtain an additive error rather than a multiplicative one in the volume increase. \n\nThe proof of Theorem \\ref{thm:main'} follows exactly the same strategy. We change the metric of $M$ in a cusp neighborhood $C$ as in Gromov-Thurston's $2\\pi$-Theorem \\cite[Theorem 9]{BH} (of which we prove a quantitative version, see Proposition \\ref{pro:metric on tube}). This time the interpolation between the metric on the cusp and the metric on a tube is more delicate as we want to keep under control the scalar curvature up to higher order (see Lemma \\ref{lem:functionfill}). Then we use the sigma invariant comparison to conclude. \n\n\\subsection*{Acknowledgements}\nI warmly thank Ian Agol and Martin Bridgeman for useful discussions and generous feedback on a first draft of this article.", + "sketch": "To prove Theorem~\\ref{thm:main}, the authors “follow the approach of \\cite[Theorem 10.1]{AST} with a difference.” The steps described are:\n\\begin{itemize}\n\\item “We first change the metric of $M$ inside a tubular neighborhood $T$ of $\\gamma$ interpolating between the metric on the tube and the metric on a suitable hyperbolic cusp.”\n\\item This is done “by keeping under control the scalar curvature (see Lemma~\\ref{lem:functiondrill}).”\n\\item “Then we compare the volume of the modified metric with that of the hyperbolic structure on $M-\\gamma$.”\n\\item “Instead of relying on Perelman's monotonicity formula,” they “exploit some other consequences of Perelman's work \\cite{Pe1,Pe2}, namely, the exact computation of the sigma invariant of closed hyperbolic 3-manifolds as achieved by Anderson \\cite{And} and Kleiner and Lott \\cite{KL}.”\n\\item “This allows us to keep track of the fact that we \\emph{did not change} the metric outside the tubular neighborhood $T$ and obtain an additive error rather than a multiplicative one in the volume increase.”\n\\end{itemize}", + "expanded_sketch": "No expanded sketch found.", + "expanded_theorem": "\\label{thm:main}\nThere exists a constant $c>0$ such that the following holds. Let $M$ be a closed hyperbolic 3-manifold and let $\\gamma\\subset M$ be a simple closed geodesic. Suppose that $\\gamma$ has length $\\ell$ and has an embedded tubular neighborhood of radius at least $R\\in(0,1)$. Then the unique hyperbolic metric on $M-\\gamma$ has volume\n\\[\n{\\rm vol}(M-\\gamma)\\le{\\rm vol}(M)+c\\frac{\\ell}{R}.\n\\]", + "theorem_type": [ + "Existential–Universal", + "Inequality or Bound" + ], + "mcq": { + "question": "Let $M$ be a closed hyperbolic $3$-manifold, and let $\\gamma\\subset M$ be a simple closed geodesic of length $\\ell$. Assume that $\\gamma$ has an embedded tubular neighborhood of radius at least $R$, where $R\\in(0,1)$. For the drilled manifold $M-\\gamma$, equipped with its unique hyperbolic metric, which quantitative volume estimate holds?", + "correct_choice": { + "label": "A", + "text": "There exists a constant $c>0$, independent of $M$, $\\gamma$, $\\ell$, and $R$, such that\n\\[\n{\\rm vol}(M-\\gamma)\\le {\\rm vol}(M)+c\\frac{\\ell}{R}.\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "There exists a constant $c>0$, independent of $M$, $\\gamma$, $\\ell$, and $R$, such that\n\\[\n{\\rm vol}(M-\\gamma)\\le {\\rm vol}(M)+c\\frac{\\ell}{R^2}.\n\\]" + }, + { + "label": "C", + "text": "There exists a constant $c>0$, independent of $M$, $\\gamma$, $\\ell$, and $R$, such that\n\\[\n{\\rm vol}(M-\\gamma)\\le {\\rm vol}(M)+c\\ell.\n\\]" + }, + { + "label": "D", + "text": "For every $R\\in(0,1)$ there exists a constant $c_R>0$, depending only on $R$, such that for every closed hyperbolic $3$-manifold $M$ and every simple closed geodesic $\\gamma\\subset M$ of length $\\ell$ with an embedded tubular neighborhood of radius at least $R$,\n\\[\n{\\rm vol}(M-\\gamma)\\le {\\rm vol}(M)+c_R\\ell.\n\\]" + }, + { + "label": "E", + "text": "There exists a constant $c>0$, independent of $M$, $\\gamma$, $\\ell$, and $R$, such that\n\\[\n{\\rm vol}(M-\\gamma)\\le \\left(1+c\\frac{\\ell}{R}\\right){\\rm vol}(M).\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": "linear dependence on inverse tube radius", + "template_used": "stronger_trap" + }, + { + "label": "C", + "sketch_hook_type": "regularity", + "tampered_component": "dropped the factor $1/R$ using only that $R<1$", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "uniformity of the constant with respect to $R$", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "additive error versus multiplicative volume factor", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem only defines terms and notation; it does not reveal the specific bound, the dependence on R, or the quantifier structure that distinguishes the correct choice." + }, + "TAS": { + "score": 2, + "justification": "Although the correct option is theorem-like, the question is not a bare restatement: the choices differ by domain restrictions on R, quantifier order, and the error-term dependence, so the student must discriminate among competing formulations." + }, + "GPS": { + "score": 1, + "justification": "The item requires some reasoning about how changing R-ranges, swapping quantifiers, or replacing ell/R by ell or ell R affects validity. However, it still leans more toward precise theorem recognition than toward substantial generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically meaningful: they target common failure modes such as overextending parameter ranges, dropping a crucial factor, altering quantifier order, and using the wrong asymptotic dependence on R." + }, + "total_score": 7, + "overall_assessment": "A strong MCQ with no answer leakage and high-quality near-miss distractors. It avoids tautology reasonably well, though it tests careful theorem discrimination more than deep generative reasoning." + } + }, + { + "id": "2602.10700v1", + "paper_link": "http://arxiv.org/abs/2602.10700v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm:global_existence_cauchy}\nLet $N=2$ or $N=3$. Assume that $\\gamma$ satisfies\n\\begin{equation}\n \\begin{cases}\n \\gamma \\in [1, \\infty) & \\text{if } N=2, \\\\\n \\gamma \\in [1, 8/3) & \\text{if } N=3.\n \\end{cases}\n\\end{equation}\nLet $\\bar{\\rho} > 0$ be the far field behavior of density. Assume that the initial data $(\\rho_0, u_0)$ satisfies\n\\begin{equation}\n 0 < \\underline{\\varrho} \\le \\rho_0 \\le \\bar{\\varrho}, \\quad (\\rho_0 - \\bar{\\rho}) \\in H^3(\\mathbb{R}^N), \\quad u_0 \\in H^2(\\mathbb{R}^N),\n\\end{equation}\nwhere $\\underline{\\varrho}$ and $\\bar{\\varrho}$ are positive constants. Then the Cauchy problem \\eqref{NSK_simplified}-\\eqref{far field} admits a unique global strong solution $(\\rho, u)$ satisfying for any $0 < T < \\infty$ and $(x,t) \\in \\mathbb{R}^N \\times [0, T]$,\n\\begin{equation}\n (C(T))^{-1} \\le \\rho(x,t) \\le C(T),\n\\end{equation}\nand\n\\begin{equation}\n\\begin{aligned}\n &(\\rho - \\bar{\\rho}) \\in C([0, T]; H^3) \\cap L^2(0, T; H^4), \\quad \\rho_t \\in C([0, T]; H^1) \\cap L^2(0, T; H^2), \\\\\n &u \\in C([0, T]; H^2) \\cap L^2(0, T; H^3), \\quad u_t \\in L^\\infty(0, T; L^2) \\cap L^2(0, T; H^1),\n\\end{aligned}\n\\end{equation}\nwhere the constant $C(T) > 0$ depends on the initial data and $T$.", + "start_pos": 12442, + "end_pos": 13706, + "label": "thm:global_existence_cauchy" + }, + "ref_dict": { + "NSK": "\\begin{equation}\\label{NSK} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \n\\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) =\\nabla \\cdot \\mathbb{S} + \\nabla \\cdot \\mathbb{K}. \\end{cases} \\end{equation}", + "PRO3.4": "\\begin{proposition}\\label{PRO3.4}\nFor any $\\gamma \\in (1, \\frac{8}{3})$, there exists $q \\in (1, 4)$ such that\n\\begin{equation}\n \\gamma \\le \\frac{2q+6}{q+2}.\n\\end{equation}\n For this fixed $q$, there exists a positive constant $C$, depending on $T$, $q$, $\\gamma$, $E_0$, and the initial data $\\|\\rho_0^{1/(q+2)} v_0\\|_{L^{q+2}}$, such that\n\\begin{equation}\\label{L^{q+2}}\n\\sup_{0 \\le t \\le T} \\|\\rho^{\\frac{1}{q+2}} v\\|_{L^{q+2}} \\le C.\n\\end{equation}\n\\end{proposition}", + "NSK_simplified": "\\begin{equation}\\label{NSK_simplified} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) = \\nabla \\cdot \\left(2 \\rho \\mathcal{D}(u) \\right) + \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho). \\end{cases} \\end{equation}", + "far field": "\\begin{equation}\\label{far field}\n \\rho \\left( x,t \\right) \\rightarrow \\bar{\\rho}>0,\\quad u\\left( x,t \\right) \\rightarrow 0,\\quad \\mathrm{as}\\, |x|\\rightarrow \\infty .\n\\end{equation}" + }, + "pre_theorem_intro_text_len": 9174, + "pre_theorem_intro_text": "In this paper, we are concerned with the global well-posedness of the compressible Navier-Stokes-Korteweg equations in the whole space $\\mathbb{R}^N$. This system describes the motion of a viscous compressible fluid endowed with internal capillarity and reads as follows\n\\begin{equation}\\label{NSK} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \n\\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) =\\nabla \\cdot \\mathbb{S} + \\nabla \\cdot \\mathbb{K}. \\end{cases} \\end{equation}\nThe viscous stress tensor $\\mathbb{S}$ is given by Newton's rheological law\n\\begin{equation}\n \\mathbb{S} = 2\\mu(\\rho) \\mathcal{D}(u) + \\lambda(\\rho) \\nabla \\cdot u \\mathbb{I},\n\\end{equation}\nwhere $\\mathcal{D}(u) = \\frac{1}{2}(\\nabla u + \\nabla u^T)$ denotes the symmetric deformation tensor, and $\\mathbb{I}$ is the identity matrix in $\\mathbb{R}^N$. The scalar functions $\\mu(\\rho)$ and $\\lambda(\\rho)$ represent the shear and bulk viscosity coefficients, respectively, which satisfy the physical condition $\\mu(\\rho) > 0$ and $2\\mu(\\rho) + N\\lambda(\\rho) \\ge 0$.\nIn order to ensure the stability of the system and utilize the BD entropy structure, we assume the viscosity coefficients satisfy the relation $$\\lambda(\\rho) = 2(\\rho \\mu'(\\rho) - \\mu(\\rho)).$$\nIn the context of the Korteweg theory, the capillarity tensor $\\nabla \\cdot\\mathbb{K}$ is typically defined as\n\\begin{equation}\n \\nabla \\cdot \\mathbb{K} = \\nabla \\left( \\rho \\kappa(\\rho) \\Delta \\rho + \\frac{\\kappa(\\rho) + \\rho \\kappa'(\\rho)}{2} |\\nabla \\rho|^2 \\right) - \\nabla \\cdot \\left( \\kappa(\\rho) \\nabla \\rho \\otimes \\nabla \\rho \\right).\n\\end{equation}\nwhere $\\kappa(\\rho) > 0$ is the coefficient of capillarity.\\par\nThe study of capillary fluids originates from the pioneering work of Van der Waals and Korteweg \\cite{Korteweg,J.F. Van derWaals}. In their theory of capillarity, the fluid energy is assumed to depend not only on the standard thermodynamic variables but also on the gradient of the density. This concept was later formalized in the modern language of continuum mechanics by Dunn and Serrin in the 1980s, leading to the so-called Korteweg-type models. \\par\n When $\\kappa(\\rho) = 0$, $\\mu=\\rho$ and $\\lambda=0$, system \\eqref{NSK} reduces to the well-known viscous shallow water equations. \nSignificant progress has been made for the viscous shallow water equations in recent years. Regarding general multi-dimensional initial data, significant breakthroughs concerning the global well-posedness of weak solutions were achieved by Li-Xin \\cite{Li-Xin} and Vasseur-Yu \\cite{Vasseur-Yu}. Independently, these authors established the global existence of weak solutions for the compressible system with $\\mu(\\rho) = \\rho$ and $\\lambda(\\rho) = 0$, admitting arbitrarily large data and vacuum states. More specifically, they obtained global weak solutions for $\\gamma \\in (1, \\infty)$ when $N=2$, and for $\\gamma \\in (1, 3)$ when $N=3$. It is worth noting that Li-Xin \\cite{Li-Xin} extended their analysis to a wider class of viscosity coefficients satisfying the BD entropy relation. Concerning the global smooth solutions for multi-dimensional shallow water equations with arbitrarily large initial data, Huang-Meng-Zhang \\cite{Huang-Meng-Zhang-V} pioneered the proof of global classical solutions for the two-dimensional initial-boundary value problem under the assumption of radial symmetry with $\\gamma\\ge\\frac{3}{2}$. Subsequently, Gu-Huang \\cite{Gu-Huang} extended the range of the exponent to $\\gamma > 1$ and generalized the result to the three-dimensional case with $1 < \\gamma < 3$. Concurrently, the global existence of large solutions for the associated Cauchy problem was proved independently by Chen-Zhang-Zhu \\cite{Chen-Zhang-Zhu}. Moreover, Huang-Meng-Zhang \\cite{Huang-Meng-Zhang-V} successfully proved the global classical well-posedness for general isentropic compressible Navier-Stokes equations satisfying the BD entropy condition in 2D and 3D, treating the shallow water model as a specific instance.\\par\n When $\\kappa(\\rho) >0$, we review some related works regarding the well-posedness of the Navier-Stokes-Korteweg system with general viscosity coefficients. For the one-dimensional case, considering the system with specific density-dependent viscosity $\\mu(\\rho) = \\rho$ and capillarity $\\kappa(\\rho) = \\rho^{-1}$, Charve-Haspot \\cite{Charve-Haspot} established the existence of global strong solutions in the whole space, allowing for large non-vacuum initial data. Furthermore, they demonstrated that these solutions converge to the entropic weak solutions of the compressible Euler equations. Moreover, Germain-LeFloch \\cite{Germain-LeFloch} proved the global existence of finite energy weak solutions for the Cauchy problem with general density-dependent coefficients and demonstrated their convergence to the entropy solutions of the Euler system. For the case with power-law viscosity $\\mu(\\varrho) = \\varrho^\\alpha$ and capillarity $\\kappa(\\varrho) = \\varrho^\\beta$ satisfying specific conditions, Antonelli-Bresch-Spirito \\cite{Antonelli-Bresch-Spirito} established the existence of global weak solutions for the periodic problem with large data. Furthermore, investigating the Cauchy problem in Lagrangian coordinates, Chen et al. \\cite{Chen-Chai} obtained global classical solutions for large initial data away from vacuum, considering density-dependent viscosity and capillarity. For the multi-dimensional case, building upon their earlier local theory \\cite{Hattori-Li}, Hattori-Li \\cite{Hattori-Li-2} established the global existence of solutions for the constant-coefficient Cauchy problem with small, non-vacuum initial data. For the case where both viscosity and capillarity coefficients depend on the density, Danchin-Desjardins \\cite{Danchin_Desjardins} obtained global smooth solutions. Their result holds for small perturbations of a non-vacuum state in functional spaces that are critical with respect to the physical energy. Bresch and Desjardins \\cite{B-D} investigated the two-dimensional viscous shallow water equations extended by a capillary term. They established the global existence of weak solutions in the presence of vacuum and demonstrated their convergence towards the strong solution of the viscous quasi-geostrophic system with a free surface. \\par\nHowever, the global existence of strong solutions with arbitrarily large initial data for the multi-dimensional Navier-Stokes-Korteweg system has long remained an open problem. It was not until recently that the first author Huang \\cite{Huang-Meng-Zhang} in his newly preprint resolved this by establishing global strong solutions on the two- and three-dimensional periodic torus, provided that the initial density is bounded from above and below. However, extending their method to the whole space presents new difficulties. In this paper, we overcome these obstacles to establish the global existence of strong solutions for the multi-dimensional Navier-Stokes-Korteweg system with large initial data in the whole space. By employing the novel critical inequality established by the first author in his newly preprint \\cite{Huang-Meng-Zhang}, we successfully establish the lower bound of the density by employing a novel De Giorgi iteration method. It is worth noting that our work does not require any radial symmetry assumption. To the best of our knowledge, this can be regarded as the first result concerning global large solutions for the corresponding Cauchy problem.\n\\par\nThroughout the rest of this paper, we focus on the case of shallow-water viscosity coefficients\n$$\\mu(\\rho) = \\rho, \\quad \\lambda(\\rho) = 0,$$\nand we assume that $\\kappa(\\rho)$ satisfies \n$$\\kappa(\\rho)=\\frac{1}{\\rho}.$$\nWith the specific choice of the capillarity coefficient $\\kappa(\\rho)$, the divergence of the Korteweg tensor takes the following form\n\\begin{equation}\n\\nabla \\cdot \\mathbb{K} = \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho).\n\\end{equation}\nConsequently, under the assumption of shallow-water viscosity ($\\mu(\\rho)=\\rho, \\lambda(\\rho)=0$) and the specific capillarity coefficient $\\kappa(\\rho)=1/\\rho$, equation \\eqref{NSK} can be rewritten as\n\\begin{equation}\\label{NSK_simplified} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) = \\nabla \\cdot \\left(2 \\rho \\mathcal{D}(u) \\right) + \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho). \\end{cases} \\end{equation}\nNow we investigate the global existence of strong solutions to system \\eqref{NSK_simplified} in $\\mathbb{R}^N$, where $N=2, 3$. The system is supplemented with the prescribed initial data $(\\rho_0, u_0)$ satisfying\n\\begin{equation}\n\\rho(x, 0) = \\rho_0(x), \\quad u(x, 0) = u_0(x), \\quad \\text{for } x \\in \\mathbb{R}^N,\n\\end{equation}\nwith far field behavior \n\\begin{equation}\\label{far field}\n \\rho \\left( x,t \\right) \\rightarrow \\bar{\\rho}>0,\\quad u\\left( x,t \\right) \\rightarrow 0,\\quad \\mathrm{as}\\, |x|\\rightarrow \\infty .\n\\end{equation}\nWe now state the main result on the global existence of strong solutions to the Cauchy problem for system \\eqref{NSK_simplified}-\\eqref{far field} with arbitrarily large initial data.", + "context": "e assumption of radial symmetry with $\\gamma\\ge\\frac{3}{2}$. Subsequently, Gu-Huang \\cite{Gu-Huang} extended the range of the exponent to $\\gamma > 1$ and generalized the result to the three-dimensional case with $1 < \\gamma < 3$. Concurrently, the global existence of large solutions for the associated Cauchy problem was proved independently by Chen-Zhang-Zhu \\cite{Chen-Zhang-Zhu}. Moreover, Huang-Meng-Zhang \\cite{Huang-Meng-Zhang-V} successfully proved the global classical well-posedness for general isentropic compressible Navier-Stokes equations satisfying the BD entropy condition in 2D and 3D, treating the shallow water model as a specific instance.\\par\n When $\\kappa(\\rho) >0$, we review some related works regarding the well-posedness of the Navier-Stokes-Korteweg system with general viscosity coefficients. For the one-dimensional case, considering the system with specific density-dependent viscosity $\\mu(\\rho) = \\rho$ and capillarity $\\kappa(\\rho) = \\rho^{-1}$, Charve-Haspot \\cite{Charve-Haspot} established the existence of global strong solutions in the whole space, allowing for large non-vacuum initial data. Furthermore, they demonstrated that these solutions converge to the entropic weak solutions of the compressible Euler equations. Moreover, Germain-LeFloch \\cite{Germain-LeFloch} proved the global existence of finite energy weak solutions for the Cauchy problem with general density-dependent coefficients and demonstrated their convergence to the entropy solutions of the Euler system. For the case with power-law viscosity $\\mu(\\varrho) = \\varrho^\\alpha$ and capillarity $\\kappa(\\varrho) = \\varrho^\\beta$ satisfying specific conditions, Antonelli-Bresch-Spirito \\cite{Antonelli-Bresch-Spirito} established the existence of global weak solutions for the periodic problem with large data. Furthermore, investigating the Cauchy problem in Lagrangian coordinates, Chen et al. \\cite{Chen-Chai} obtained global classical solutions for large initial data away from vacuum, considering density-dependent viscosity and capillarity. For the multi-dimensional case, building upon their earlier local theory \\cite{Hattori-Li}, Hattori-Li \\cite{Hattori-Li-2} established the global existence of solutions for the constant-coefficient Cauchy problem with small, non-vacuum initial data. For the case where both viscosity and capillarity coefficients depend on the density, Danchin-Desjardins \\cite{Danchin_Desjardins} obtained global smooth solutions. Their result holds for small perturbations of a non-vacuum state in functional spaces that are critical with respect to the physical energy. Bresch and Desjardins \\cite{B-D} investigated the two-dimensional viscous shallow water equations extended by a capillary term. They established the global existence of weak solutions in the presence of vacuum and demonstrated their convergence towards the strong solution of the viscous quasi-geostrophic system with a free surface. \\par\nHowever, the global existence of strong solutions with arbitrarily large initial data for the multi-dimensional Navier-Stokes-Korteweg system has long remained an open problem. It was not until recently that the first author Huang \\cite{Huang-Meng-Zhang} in his newly preprint resolved this by establishing global strong solutions on the two- and three-dimensional periodic torus, provided that the initial density is bounded from above and below. However, extending their method to the whole space presents new difficulties. In this paper, we overcome these obstacles to establish the global existence of strong solutions for the multi-dimensional Navier-Stokes-Korteweg system with large initial data in the whole space. By employing the novel critical inequality established by the first author in his newly preprint \\cite{Huang-Meng-Zhang}, we successfully establish the lower bound of the density by employing a novel De Giorgi iteration method. It is worth noting that our work does not require any radial symmetry assumption. To the best of our knowledge, this can be regarded as the first result concerning global large solutions for the corresponding Cauchy problem.\n\\par\nThroughout the rest of this paper, we focus on the case of shallow-water viscosity coefficients\n$$\\mu(\\rho) = \\rho, \\quad \\lambda(\\rho) = 0,$$\nand we assume that $\\kappa(\\rho)$ satisfies \n$$\\kappa(\\rho)=\\frac{1}{\\rho}.$$\nWith the specific choice of the capillarity coefficient $\\kappa(\\rho)$, the divergence of the Korteweg tensor takes the following form\n\\begin{equation}\n\\nabla \\cdot \\mathbb{K} = \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho).\n\\end{equation}\nConsequently, under the assumption of shallow-water viscosity ($\\mu(\\rho)=\\rho, \\lambda(\\rho)=0$) and the specific capillarity coefficient $\\kappa(\\rho)=1/\\rho$, equation \\eqref{NSK} can be rewritten as\n\\begin{equation}\\label{NSK_simplified} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) = \\nabla \\cdot \\left(2 \\rho \\mathcal{D}(u) \\right) + \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho). \\end{cases} \\end{equation}\nNow we investigate the global existence of strong solutions to system \\eqref{NSK_simplified} in $\\mathbb{R}^N$, where $N=2, 3$. The system is supplemented with the prescribed initial data $(\\rho_0, u_0)$ satisfying\n\\begin{equation}\n\\rho(x, 0) = \\rho_0(x), \\quad u(x, 0) = u_0(x), \\quad \\text{for } x \\in \\mathbb{R}^N,\n\\end{equation}\nwith far field behavior \n\\begin{equation}\\label{far field}\n \\rho \\left( x,t \\right) \\rightarrow \\bar{\\rho}>0,\\quad u\\left( x,t \\right) \\rightarrow 0,\\quad \\mathrm{as}\\, |x|\\rightarrow \\infty .\n\\end{equation}\nWe now state the main result on the global existence of strong solutions to the Cauchy problem for system \\eqref{NSK_simplified}-\\eqref{far field} with arbitrarily large initial data.\n\n\\begin{equation}\\label{far field}\n \\rho \\left( x,t \\right) \\rightarrow \\bar{\\rho}>0,\\quad u\\left( x,t \\right) \\rightarrow 0,\\quad \\mathrm{as}\\, |x|\\rightarrow \\infty .\n\\end{equation}", + "full_context": "e assumption of radial symmetry with $\\gamma\\ge\\frac{3}{2}$. Subsequently, Gu-Huang \\cite{Gu-Huang} extended the range of the exponent to $\\gamma > 1$ and generalized the result to the three-dimensional case with $1 < \\gamma < 3$. Concurrently, the global existence of large solutions for the associated Cauchy problem was proved independently by Chen-Zhang-Zhu \\cite{Chen-Zhang-Zhu}. Moreover, Huang-Meng-Zhang \\cite{Huang-Meng-Zhang-V} successfully proved the global classical well-posedness for general isentropic compressible Navier-Stokes equations satisfying the BD entropy condition in 2D and 3D, treating the shallow water model as a specific instance.\\par\n When $\\kappa(\\rho) >0$, we review some related works regarding the well-posedness of the Navier-Stokes-Korteweg system with general viscosity coefficients. For the one-dimensional case, considering the system with specific density-dependent viscosity $\\mu(\\rho) = \\rho$ and capillarity $\\kappa(\\rho) = \\rho^{-1}$, Charve-Haspot \\cite{Charve-Haspot} established the existence of global strong solutions in the whole space, allowing for large non-vacuum initial data. Furthermore, they demonstrated that these solutions converge to the entropic weak solutions of the compressible Euler equations. Moreover, Germain-LeFloch \\cite{Germain-LeFloch} proved the global existence of finite energy weak solutions for the Cauchy problem with general density-dependent coefficients and demonstrated their convergence to the entropy solutions of the Euler system. For the case with power-law viscosity $\\mu(\\varrho) = \\varrho^\\alpha$ and capillarity $\\kappa(\\varrho) = \\varrho^\\beta$ satisfying specific conditions, Antonelli-Bresch-Spirito \\cite{Antonelli-Bresch-Spirito} established the existence of global weak solutions for the periodic problem with large data. Furthermore, investigating the Cauchy problem in Lagrangian coordinates, Chen et al. \\cite{Chen-Chai} obtained global classical solutions for large initial data away from vacuum, considering density-dependent viscosity and capillarity. For the multi-dimensional case, building upon their earlier local theory \\cite{Hattori-Li}, Hattori-Li \\cite{Hattori-Li-2} established the global existence of solutions for the constant-coefficient Cauchy problem with small, non-vacuum initial data. For the case where both viscosity and capillarity coefficients depend on the density, Danchin-Desjardins \\cite{Danchin_Desjardins} obtained global smooth solutions. Their result holds for small perturbations of a non-vacuum state in functional spaces that are critical with respect to the physical energy. Bresch and Desjardins \\cite{B-D} investigated the two-dimensional viscous shallow water equations extended by a capillary term. They established the global existence of weak solutions in the presence of vacuum and demonstrated their convergence towards the strong solution of the viscous quasi-geostrophic system with a free surface. \\par\nHowever, the global existence of strong solutions with arbitrarily large initial data for the multi-dimensional Navier-Stokes-Korteweg system has long remained an open problem. It was not until recently that the first author Huang \\cite{Huang-Meng-Zhang} in his newly preprint resolved this by establishing global strong solutions on the two- and three-dimensional periodic torus, provided that the initial density is bounded from above and below. However, extending their method to the whole space presents new difficulties. In this paper, we overcome these obstacles to establish the global existence of strong solutions for the multi-dimensional Navier-Stokes-Korteweg system with large initial data in the whole space. By employing the novel critical inequality established by the first author in his newly preprint \\cite{Huang-Meng-Zhang}, we successfully establish the lower bound of the density by employing a novel De Giorgi iteration method. It is worth noting that our work does not require any radial symmetry assumption. To the best of our knowledge, this can be regarded as the first result concerning global large solutions for the corresponding Cauchy problem.\n\\par\nThroughout the rest of this paper, we focus on the case of shallow-water viscosity coefficients\n$$\\mu(\\rho) = \\rho, \\quad \\lambda(\\rho) = 0,$$\nand we assume that $\\kappa(\\rho)$ satisfies \n$$\\kappa(\\rho)=\\frac{1}{\\rho}.$$\nWith the specific choice of the capillarity coefficient $\\kappa(\\rho)$, the divergence of the Korteweg tensor takes the following form\n\\begin{equation}\n\\nabla \\cdot \\mathbb{K} = \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho).\n\\end{equation}\nConsequently, under the assumption of shallow-water viscosity ($\\mu(\\rho)=\\rho, \\lambda(\\rho)=0$) and the specific capillarity coefficient $\\kappa(\\rho)=1/\\rho$, equation \\eqref{NSK} can be rewritten as\n\\begin{equation}\\label{NSK_simplified} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) = \\nabla \\cdot \\left(2 \\rho \\mathcal{D}(u) \\right) + \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho). \\end{cases} \\end{equation}\nNow we investigate the global existence of strong solutions to system \\eqref{NSK_simplified} in $\\mathbb{R}^N$, where $N=2, 3$. The system is supplemented with the prescribed initial data $(\\rho_0, u_0)$ satisfying\n\\begin{equation}\n\\rho(x, 0) = \\rho_0(x), \\quad u(x, 0) = u_0(x), \\quad \\text{for } x \\in \\mathbb{R}^N,\n\\end{equation}\nwith far field behavior \n\\begin{equation}\\label{far field}\n \\rho \\left( x,t \\right) \\rightarrow \\bar{\\rho}>0,\\quad u\\left( x,t \\right) \\rightarrow 0,\\quad \\mathrm{as}\\, |x|\\rightarrow \\infty .\n\\end{equation}\nWe now state the main result on the global existence of strong solutions to the Cauchy problem for system \\eqref{NSK_simplified}-\\eqref{far field} with arbitrarily large initial data.\n\n\\begin{equation}\\label{far field}\n \\rho \\left( x,t \\right) \\rightarrow \\bar{\\rho}>0,\\quad u\\left( x,t \\right) \\rightarrow 0,\\quad \\mathrm{as}\\, |x|\\rightarrow \\infty .\n\\end{equation}\n\n\\begin{document}\n\\maketitle\n\\begin{abstract}\nIn this paper, we establish global strong solutions for arbitrarily large initial data to the multi-dimensional compressible Navier-Stokes–Korteweg system, also referred to as the quantum Navier–Stokes equations, originally derived by Dunn and Serrin [Arch. Ration. Mech. Anal. 88(2):95–133, 1985]. Specifically, we prove the existence of global strong solutions for arbitrarily large initial data in the case $N=2$ when $\\gamma \\ge 1$, and $N=3$ with $1 \\le \\gamma < 8/3$ for the associated Cauchy problem. By employing techniques from Littlewood–Paley theory, range truncation analysis, refined Nash-Moser and De Giorgi iteration methods, we derive positive upper and lower bounds for the density. As a consequence, we are able to treat the whole-space case with strictly positive far-field density. To the best of our knowledge, this is the first result that establishes global strong solutions for physically relevant compressible Navier–Stokes equations in the whole space, without imposing any symmetry or special geometric assumptions on the initial data.\n\\end{abstract}\n\\bigskip\n\\noindent \\textbf{Keywords:} compressible Navier-Stokes-Korteweg system; quantum Navier-Stokes system; global large strong solutions; Nash-Moser iteration; De Giorgi iteration; Cauchy problem.\n\nIn the first phase, we work within the framework of Besov spaces, utilizing maximal regularity estimates for parabolic equations to secure the upper bound. The second and most crucial step is establishing the density lower bound. Although Haspot \\cite{Haspot 1} investigated the isothermal NSK system, his approach failed to yield a lower bound for the density. Our strategy draws inspiration from a key inequality introduced by the first author in \\cite{Huang-Meng-Zhang}. However, the technique in \\cite{Huang-Meng-Zhang} cannot be directly applied to the whole space problem with non-vacuum far-field conditions. To overcome this obstacle, we have developed a novel truncated De Giorgi iteration method.\n\n\\begin{proof}\nBy H\\\"{o}lder's inequality, Sobolev embedding and \\eqref{eq:19}, we have\n\\begin{equation}\\label{the initial estimate for Psi}\n\\begin{aligned}\n \\Psi(r(p+2)) &= \\int^{T}_{0}\\int_{\\mathbb{R}^{3}}\\rho|v|^{\\frac{5}{3}(p+2)}\\,dx\\,dt \\\\\n &\\leq \\int^{T}_{0}\\left( \\int \\rho^{\\frac{3}{2}}|v|^{p+2}\\,dx \\right)^{\\frac{2}{3}} \\left( \\int|v|^{3(p+2)}\\,dx \\right)^{\\frac{1}{3}}\\,dt \\\\\n &\\leq \\|\\rho\\|_{L_{T}^{\\infty}L^{\\infty}}^{\\frac{1}{3}} \\sup_{0 \\le t \\le T}\\left( \\int \\rho|v|^{p+2}\\,dx \\right)^{\\frac{2}{3}} \\int^{T}_{0} \\left\\| |v|^{\\frac{p+2}{2}} \\right\\|_{L^{6}}^{2}\\,dt \\\\\n &\\leq C \\sup_{0 \\le t \\le T}\\left( \\int \\rho|v|^{p+2}\\,dx\\right)^{\\frac{2}{3}} \\|\\rho^{-1}\\|_{L_{T}^{\\infty}L^{\\infty}} \\\\\n &\\quad \\times \\left( \\int^{T}_{0}\\int \\rho \\left| \\nabla|v|^{\\frac{p+2}{2}} \\right|^{2}\\,dx\\,dt + \\int^{T}_{0}\\int \\rho|v|^{p+2}\\,dx\\,dt \\right) \\\\\n &\\le CV_{T} \\left( \\sup_{0 \\le t \\le T}\\int \\rho|v|^{p+2}\\,dx + \\int^{T}_{0}\\int \\rho \\left| \\nabla|v|^{\\frac{p+2}{2}} \\right|^{2}\\,dx\\,dt \\right)^{r}.\n\\end{aligned}\n\\end{equation}\nTo estimate the right hand side, multiplying $\\eqref{parabolic_system}_2$ by $|v|^{p}v$ and integrating by parts yields\n\\begin{equation}\n \\frac{1}{p+2} \\frac{d}{dt}\\int \\rho|v|^{p+2}dx+p\\int \\rho|v|^{p}\\left| \\nabla \\left|v \\right| \\right| ^{2}dx+\\int \\rho|v|^{p}|\\nabla v|^{2}dx=\\int P(\\rho)\\nabla\\cdot\\left(\\left| v \\right| ^{p}v\\right)dx.\n\\end{equation}\nBy Young's inequality and \\eqref{eq:19}, we have\n\\begin{equation}\n \\begin{aligned}\n \\int P(\\rho)\\nabla \\cdot\\left(\\left| v \\right| ^{p}v\\right)dx\n=& \\int \\rho^{\\gamma}(\\left| v \\right| ^{p}\\nabla \\cdot v+ v\\cdot { \\nabla \\left( \\left| v \\right| ^{p} \\right) } )dx\\\\\n\\leq& \\int\\rho^{\\gamma}\\left| v \\right| ^{p} \\left| \\nabla v \\right|dx + p\\int \\rho^{\\gamma}|v|^{p}\\left| \\nabla \\left| v \\right| \\right|dx \\\\\n\\leq & \\frac{1}{2} \\int \\rho|v|^{p}|\\nabla v|^{2}dx+ \\frac{p}{2}\\int \\rho|v|^{p}|\\nabla \\left| v \\right| |^{2} dx+ C(p+1)\\int \\rho^{2\\gamma-1}|v|^{p}dx\\\\\n \\leq& \\frac{1}{2}\\int \\rho |v|^{p}\\left| \\nabla v \\right| ^{2}dx+ \\frac{p}{2}\\int \\rho|v|^{p}|\\nabla \\left| v \\right| |^{2}dx + C(p+1)\\left\\lVert \\rho \\right\\rVert _{L^{\\infty}}^{2(\\gamma-1)}\\int \\rho|v|^{p}dx\\\\ \n \\leq& \\frac{1}{2}\\int \\rho |v|^{p}\\left| \\nabla v \\right| ^{2}dx+ \\frac{p}{2}\\int \\rho|v|^{p}|\\nabla \\left| v \\right| |^{2}dx\\\\\n &+ C(p+1)\\sup_{0 \\le t \\le T}\\left( \\int \\rho|v|^{2}dx \\right)^{\\frac{2}{p}}\\left( \\int \\rho|v|^{p+2} dx\\right)^{1- \\frac{2}{p}}.\n\\end{aligned}\n\\end{equation}\nCombining \\eqref{v energy}, we have\n\\begin{equation}\n \\frac{1}{p+2}\\frac{d}{dt}\\int \\rho |v|^{p+2}dx + \\frac{1}{2}\\int \\rho |v|^p |\\nabla v|^2dx + \\frac{p}{2}\\int \\rho |v|^p |\\nabla |v||^2dx \\leq C(p+1)\\left( \\int \\rho |v|^{p+2} dx \\right)^{1-\\frac{2}{p}}.\n\\end{equation}\nIntegrating over $t\\in[0,T]$, we obtain\n\\begin{equation}\n \\begin{aligned}\n&\\sup_{0 \\le t \\le T}\\frac{1}{p+2}\\int{\\rho}|v|^{p+2}dx+ \\frac{1}{2}\\int_{0}^{T}\\int \\rho |v|^p |\\nabla v|^2dxdt + \\frac{p}{2}\\int_{0}^{T}\\int \\rho |v|^p |\\nabla |v||^2dxdt \\\\\n\\leq &C(p+1)\\int^{T}_{0}\\left( \\int \\rho |v|^{p+2} dx \\right)^{1-\\frac{2}{p}}dt+\\frac{2}{p+2}\\int \\rho_{0}|v_{0}|^{p+2}\\\\\n\\leq& C(p+1){T^{*}}^{\\frac{2}{p}}\\left( \\int^{T}_{0}\\int \\rho|v|^{p+2}dxdt \\right)^{1- \\frac{2}{p}}+ \\frac{2}{p+2}\\int \\rho_{0}|v_{0}|^{p+2}.\n\\end{aligned}\n\\end{equation}\nSubstituting this into the RHS of \\eqref{the initial estimate for Psi}, we get\n\\begin{equation}\n \\begin{aligned}\n\\Psi(r(p+2))&\\leq C{V}_{T} \\left( C(p+2)^2 \\left( \\int^{T}_{0}\\int \\rho |v|^{p+2} dxdt \\right)^{\\frac{p-2}{p}} + C \\int \\rho_0 |v_0|^{p+2} dx \\right) ^{r }\\\\\n&\\leq C{V}_{T} (p+2)^{2r } \\left( \\int^{T}_{0}\\int \\rho |v|^{p+2} dxdt \\right)^{r } + C{V}_{T} (p+2)^{2r } + C{V}_{T} \\left( \\int \\rho_0 |v_0|^{p+2} dx \\right)^{r }.\n\\end{aligned}\n\\end{equation}\nDenoting the constant in the RHS as $C_{3}\\geq1$, and noting that\n\\begin{equation}\n \\left\\| \\rho_0^{\\frac{1}{p+2}} v_0 \\right\\|_{L^{p+2}} \\leq \\left\\| \\rho_0^{\\frac{1}{2}} v_0 \\right\\|_{L^2}^{\\frac{p}{p+2}} \\|v_0\\|_{L^\\infty}^{\\frac{2}{p+2}} \\leq \\left\\| \\rho_0^{\\frac{1}{2}} v_0 \\right\\|_{L^2} + \\|v_0\\|_{L^\\infty},\n\\end{equation}\nthe proof is complete.\n\\end{proof}\nThe following proposition plays a pivotal role in deriving the positive lower bound for the density, as it establishes a crucial link between the density and the effective velocity.\n\\begin{proposition}\nThere exists a constant $c_{v}\\geq 1$ depending on $T^{*}$, $q$, $\\gamma$, $E_{0}$, $\\left\\lVert \\rho_{0}^{\\frac{1}{q+2}}v_{0} \\right\\rVert_{L^{q+2}}$, and $\\left\\lVert \\rho_{0} \\right\\rVert_{L^{\\infty}}$ such that\n\\begin{equation}\\label{eq:v_L_infty_est}\n \\left\\lVert v \\right\\rVert _{L_{T}^{\\infty}\\left(L^{\\infty}\\left(\\mathbb{R}^3\\right)\\right)}\\leq c_{v}(\\log V_{T} )^{\\frac{1}{2}}.\n\\end{equation}\n\\end{proposition}", + "post_theorem_intro_text_len": 6699, + "post_theorem_intro_text": "\\begin{remark}\n For the sake of brevity, we restrict our proof to the three-dimensional case where $\\gamma \\in (1, \\frac{8}{3})$. For the critical 3D case $(\\gamma=1)$ and the 2D case $(\\gamma \\ge 1)$, although the density upper bound estimates for these cases were addressed in Haspot \\cite{Haspot 1} and Yu-Wu \\cite{Yu-Wu}, neither work derived the density lower bound. Consequently, they were unable to establish the global existence of solutions. We emphasize that the approach developed in this paper can be successfully applied to establish the density lower bound for the critical 3D case $(\\gamma=1)$ and the 2D case $(\\gamma \\ge 1)$ as well. Therefore, we are able to obtain global solutions for both of these cases. It is worth noting that the technique used by Haspot \\cite{Haspot 1} relies on the assumption that the estimate for $\\|\\rho^{\\frac{1}{q+2}} v\\|_{L^\\infty_T L^{q+2}}$ is independent of $q$ to derive the density lower bound. Since such $q$-independence does not hold in the present context, Haspot's method is not applicable for securing a strictly positive lower bound.\n\\end{remark}\n\\begin{remark}\n Huang-Meng-Zhang \\cite{Huang-Meng-Zhang} established this result for the periodic domain. In this work, we extend their findings to the more challenging whole space setting by introducing a new De Giorgi iteration method to effectively handle the far field behavior. \n\\end{remark}\nProvided that the density remains strictly positive, we define the effective velocity $v$ as\n\\begin{equation}\nv = u + \\nabla \\log \\rho.\n\\end{equation}\nThis transformation converts the original system \\eqref{NSK} into the following parabolic system\n\\begin{equation}\\label{parabolic_system}\n\\begin{cases}\n\\partial_t \\rho + \\nabla \\cdot(\\rho v) - \\Delta \\rho = 0, \\\\\n\\rho \\partial_t v + \\rho u \\cdot \\nabla v + \\nabla P(\\rho) = \\nabla \\cdot(\\rho \\nabla v).\n\\end{cases}\n\\end{equation}\nConsequently, the initial effective velocity $v_0$ is defined by\n\\begin{equation}\\label{initial data}\nv_0 = u_0 + \\nabla \\log \\rho_0.\n\\end{equation}\nLet us outline the main strategy of the proof. Our proof is organized into three main parts: establishing the density upper bound, deriving the density lower bound, and finally, improving the regularity of the solution.\n\nIn the first phase, we work within the framework of Besov spaces, utilizing maximal regularity estimates for parabolic equations to secure the upper bound. The second and most crucial step is establishing the density lower bound. Although Haspot \\cite{Haspot 1} investigated the isothermal NSK system, his approach failed to yield a lower bound for the density. Our strategy draws inspiration from a key inequality introduced by the first author in \\cite{Huang-Meng-Zhang}. However, the technique in \\cite{Huang-Meng-Zhang} cannot be directly applied to the whole space problem with non-vacuum far-field conditions. To overcome this obstacle, we have developed a novel truncated De Giorgi iteration method.\n\nWe now proceed to outline the derivation of these density bounds.\\\\\n\\textbf{Upper bound of $\\rho$.} \\par\nTo derive the upper bound of the density, we fully exploit the parabolic structure of the system and employ maximal regularity estimates for the heat equation. Specifically, closing the density estimate requires bounding the norm $\\|\\rho^{1/(q+2)}v\\|_{L^{q+2}}$ for some $q > 1$. In Proposition \\ref{PRO3.4}, by means of a precise domain decomposition analysis, we established that for any $\\gamma \\in (1, \\frac{8}{3})$, there exists a $q$ such that the norm $\\|\\rho^{\\frac{1}{q+2}} v\\|_{L^{q+2}}$ remains bounded.\\\\\n\\textbf{Lower bound of $\\rho$.} \\par\n To derive the lower bound of the density following Haspot\\cite{Haspot 1}'s argument, it is essential to control the norm $\\|\\rho^{\\frac{1}{p+2}} v\\|_{L^\\infty_T L^{p+2}}$ by a constant independent of $p$. However, the bound for $\\|\\rho^{\\frac{1}{p+2}} v\\|_{L^\\infty_T L^{p+2}}$ tends to infinity as $p \\to \\infty$. To overcome this obstacle, Huang-Meng-Zhang \\cite{Huang-Meng-Zhang} employed the Moser iteration method to establish a control relationship between $\\|v\\|_{L^\\infty}$ and $\\sqrt{\\log (e^{\\frac{25}{9}}+\\|{\\rho}^{-1}\\|_{L^{\\infty}})}$. This critical inequality plays a pivotal role in our analysis. Although their approach can handle the density lower bound in a periodic domain, it fails in the whole space when the density exhibits non-zero far-field behavior at infinity. To address the difficulties arising in this context, we introduce the De Giorgi iteration technique to prove the lower bound of the density by employing a truncation level adapted to the far-field density. Specifically, we construct the following truncation and iteration sequence\n \\begin{equation}\n\\begin{split}\n \\rho^{-1}_{(k_n)}&:=\\max\\{\\rho^{-1}-{k_n},0\\},\n \\\\\n U_{n}^{T}&:=\\| \\rho _{(k_n)}^{-1}\\| _{L_{T}^{\\infty}L^2}^2+\\| \\nabla \\rho _{(k_n)}^{-1}\\| _{L_{T}^{2}L^2}^2.\n\\end{split}\n \\end{equation}\n where $k_n:=M\\left( 1-2^{-n} \\right)+2\\left\\| {\\rho}^{-1} _0 \\right\\| _{L^{\\infty}}.$ It is worth noting that the initial iteration value $k_0 = 2\\|1/\\rho_0\\|_{L^\\infty}$ is chosen specifically to handle the far-field behavior while ensuring that the initial energy $U_0^T$ satisfies the convergence condition\n \\begin{equation}\n U_0^T \\le K^{-\\frac{1}{\\nu}} A^{-\\frac{1}{\\nu^2}}.\n\\end{equation}\n We choose a suitable $T$ to bound the solution on $[0, T]$ and then employ a shifted iteration sequence for $[T, 2T]$. As long as the time step for each extension is uniform, this procedure proves the boundedness within the maximal lifespan $T^*$, which in turn establishes the time-dependent lower bound of the density.\\par\n In Section 2, we introduce some preliminaries, with a particular focus on Littlewood-Paley theory. In Section 3, we first demonstrate that for any $\\gamma \\in (1, \\frac{8}{3})$, there exists a $q>1$ satisfying the condition $\\gamma \\le \\frac{2q+6}{q+2}$ such that the quantity $\\sup_{0 \\le t \\le T} \\|\\rho^{\\frac{1}{q+2}} v\\|_{L^{q+2}}$ remains bounded. This allows us to establish the density upper bound by applying maximal regularity estimates for the heat equation. In Section 4, armed with the density upper bound, we extend the range of $q$ for which the boundedness of $\\sup_{0 \\le t \\le T} \\|\\rho^{\\frac{1}{q+2}} v\\|_{L^{q+2}}$ holds. Consequently, we employ the Moser iteration method to control $\\|v\\|_{L^{\\infty}}$ by a term involving $\\sqrt{\\log V_T}$, and subsequently utilize De Giorgi iteration to prove the existence of a density lower bound up to the maximal existence time $T^*$. Finally, in Section 5 and Section 6, we prove the main result by using the established density upper and lower bounds as blow-up criteria.", + "sketch": "To prove Theorem~\\ref{thm:global_existence_cauchy}, the authors assume the density stays strictly positive and introduce the \\emph{effective velocity} $v=u+\\nabla\\log\\rho$, which transforms \\eqref{NSK} into the parabolic system \\eqref{parabolic_system}. They then “outline the main strategy” in three parts: “establishing the density upper bound, deriving the density lower bound, and finally, improving the regularity of the solution.”\n\n(1) \\textbf{Upper bound of $\\rho$.} They “fully exploit the parabolic structure of the system and employ maximal regularity estimates for the heat equation.” Closing the estimate requires bounding $\\|\\rho^{1/(q+2)}v\\|_{L^{q+2}}$ for some $q>1$; in Proposition~\\ref{PRO3.4}, “by means of a precise domain decomposition analysis,” they show that for any $\\gamma\\in(1,\\frac{8}{3})$ there exists such a $q$ with $\\|\\rho^{\\frac{1}{q+2}}v\\|_{L^{q+2}}$ bounded.\n\n(2) \\textbf{Lower bound of $\\rho$.} They explain that Haspot’s route would need $\\|\\rho^{\\frac{1}{p+2}}v\\|_{L^\\infty_T L^{p+2}}$ bounded uniformly in $p$, but here it “tends to infinity as $p\\to\\infty$.” Instead they use the Huang–Meng–Zhang idea relating “$\\|v\\|_{L^\\infty}$ and $\\sqrt{\\log( e^{\\frac{25}{9}}+\\|\\rho^{-1}\\|_{L^\\infty})}$,” but since the periodic-domain method “fails in the whole space” with nonzero far-field density, they “developed a novel truncated De Giorgi iteration method.” They define truncations and energies\n\\[\n\\rho^{-1}_{(k_n)}:=\\max\\{\\rho^{-1}-k_n,0\\},\\qquad\nU_n^T:=\\|\\rho^{-1}_{(k_n)}\\|_{L_T^\\infty L^2}^2+\\|\\nabla\\rho^{-1}_{(k_n)}\\|_{L_T^2 L^2}^2,\n\\]\nwith truncation levels $k_n:=M(1-2^{-n})+2\\|\\rho_0^{-1}\\|_{L^\\infty}$, choosing $k_0=2\\|1/\\rho_0\\|_{L^\\infty}$ so that $U_0^T$ meets a convergence condition $U_0^T\\le K^{-1/\\nu}A^{-1/\\nu^2}$. They “choose a suitable $T$ to bound the solution on $[0,T]$ and then employ a shifted iteration sequence for $[T,2T]$,” and with a uniform time step this yields boundedness up to the maximal lifespan $T^*$, giving a “time-dependent lower bound of the density.”\n\n(3) \\textbf{Completion / global existence.} After establishing density upper and lower bounds, they “prove the main result by using the established density upper and lower bounds as blow-up criteria” (Sections 5–6), together with the earlier steps (Sections 3–4: maximal regularity for the upper bound; extending the $q$-range, then Moser to control $\\|v\\|_{L^\\infty}$ by $\\sqrt{\\log V_T}$, and De Giorgi for the lower bound up to $T^*$).", + "expanded_sketch": "To prove the main theorem, the authors assume the density stays strictly positive and introduce the \\emph{effective velocity} $v=u+\\nabla\\log\\rho$, which transforms\n\\begin{equation}\\label{NSK} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \n\\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) =\\nabla \\cdot \\mathbb{S} + \\nabla \\cdot \\mathbb{K}. \\end{cases} \\end{equation}\ninto the parabolic system \\eqref{parabolic_system}. They then “outline the main strategy” in three parts: “establishing the density upper bound, deriving the density lower bound, and finally, improving the regularity of the solution.”\n\n(1) \\textbf{Upper bound of $\\rho$.} They “fully exploit the parabolic structure of the system and employ maximal regularity estimates for the heat equation.” Closing the estimate requires bounding $\\|\\rho^{1/(q+2)}v\\|_{L^{q+2}}$ for some $q>1$. We first use the following proposition.\n\\begin{proposition}\\label{PRO3.4}\nFor any $\\gamma \\in (1, \\frac{8}{3})$, there exists $q \\in (1, 4)$ such that\n\\begin{equation}\n \\gamma \\le \\frac{2q+6}{q+2}.\n\\end{equation}\n For this fixed $q$, there exists a positive constant $C$, depending on $T$, $q$, $\\gamma$, $E_0$, and the initial data $\\|\\rho_0^{1/(q+2)} v_0\\|_{L^{q+2}}$, such that\n\\begin{equation}\\label{L^{q+2}}\n\\sup_{0 \\le t \\le T} \\|\\rho^{\\frac{1}{q+2}} v\\|_{L^{q+2}} \\le C.\n\\end{equation}\n\\end{proposition}\nIn particular, for any $\\gamma\\in(1,\\frac{8}{3})$ one can choose such a $q$ so that $\\|\\rho^{\\frac{1}{q+2}}v\\|_{L^{q+2}}$ is bounded.\n\n(2) \\textbf{Lower bound of $\\rho$.} They explain that Haspot’s route would need $\\|\\rho^{\\frac{1}{p+2}}v\\|_{L^\\infty_T L^{p+2}}$ bounded uniformly in $p$, but here it “tends to infinity as $p\\to\\infty$.” Instead they use the Huang–Meng–Zhang idea relating “$\\|v\\|_{L^\\infty}$ and $\\sqrt{\\log( e^{\\frac{25}{9}}+\\|\\rho^{-1}\\|_{L^\\infty})}$,” but since the periodic-domain method “fails in the whole space” with nonzero far-field density, they “developed a novel truncated De Giorgi iteration method.” They define truncations and energies\n\\[\n\\rho^{-1}_{(k_n)}:=\\max\\{\\rho^{-1}-k_n,0\\},\\qquad\nU_n^T:=\\|\\rho^{-1}_{(k_n)}\\|_{L_T^\\infty L^2}^2+\\|\\nabla\\rho^{-1}_{(k_n)}\\|_{L_T^2 L^2}^2,\n\\]\nwith truncation levels $k_n:=M(1-2^{-n})+2\\|\\rho_0^{-1}\\|_{L^\\infty}$, choosing $k_0=2\\|1/\\rho_0\\|_{L^\\infty}$ so that $U_0^T$ meets a convergence condition $U_0^T\\le K^{-1/\\nu}A^{-1/\\nu^2}$. They “choose a suitable $T$ to bound the solution on $[0,T]$ and then employ a shifted iteration sequence for $[T,2T]$,” and with a uniform time step this yields boundedness up to the maximal lifespan $T^*$, giving a “time-dependent lower bound of the density.”\n\n(3) \\textbf{Completion / global existence.} After establishing density upper and lower bounds, they complete the proof of the main theorem by using the established density upper and lower bounds as blow-up criteria (Sections 5–6), together with the earlier steps (Sections 3–4: maximal regularity for the upper bound; extending the $q$-range, then Moser to control $\\|v\\|_{L^\\infty}$ by $\\sqrt{\\log V_T}$, and De Giorgi for the lower bound up to $T^*$).", + "expanded_theorem": "\\label{thm:global_existence_cauchy}\nLet $N=2$ or $N=3$. Assume that $\\gamma$ satisfies\n\\begin{equation}\n \\begin{cases}\n \\gamma \\in [1, \\infty) & \\text{if } N=2, \\\\\n \\gamma \\in [1, 8/3) & \\text{if } N=3.\n \\end{cases}\n\\end{equation}\nLet $\\bar{\\rho} > 0$ be the far field behavior of density. Assume that the initial data $(\\rho_0, u_0)$ satisfies\n\\begin{equation}\n 0 < \\underline{\\varrho} \\le \\rho_0 \\le \\bar{\\varrho}, \\quad (\\rho_0 - \\bar{\\rho}) \\in H^3(\\mathbb{R}^N), \\quad u_0 \\in H^2(\\mathbb{R}^N),\n\\end{equation}\nwhere $\\underline{\\varrho}$ and $\\bar{\\varrho}$ are positive constants. Then the Cauchy problem\n\\begin{equation}\\label{NSK_simplified} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) = \\nabla \\cdot \\left(2 \\rho \\mathcal{D}(u) \\right) + \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho). \\end{cases} \\end{equation}\n\\begin{equation}\\label{far field}\n \\rho \\left( x,t \\right) \\rightarrow \\bar{\\rho}>0,\\quad u\\left( x,t \\right) \\rightarrow 0,\\quad \\mathrm{as}\\, |x|\\rightarrow \\infty .\n\\end{equation}\nadmits a unique global strong solution $(\\rho, u)$ satisfying for any $0 < T < \\infty$ and $(x,t) \\in \\mathbb{R}^N \\times [0, T]$,\n\\begin{equation}\n (C(T))^{-1} \\le \\rho(x,t) \\le C(T),\n\\end{equation}\nand\n\\begin{equation}\n\\begin{aligned}\n &(\\rho - \\bar{\\rho}) \\in C([0, T]; H^3) \\cap L^2(0, T; H^4), \\quad \\rho_t \\in C([0, T]; H^1) \\cap L^2(0, T; H^2), \\\\\n &u \\in C([0, T]; H^2) \\cap L^2(0, T; H^3), \\quad u_t \\in L^\\infty(0, T; L^2) \\cap L^2(0, T; H^1),\n\\end{aligned}\n\\end{equation}\nwhere the constant $C(T) > 0$ depends on the initial data and $T$.", + "theorem_type": [ + "Existence", + "Uniqueness" + ], + "mcq": { + "question": "Let $N=2$ or $N=3$, and assume\n\\[\n\\gamma\\in[1,\\infty)\\quad\\text{if }N=2,\n\\qquad\n\\gamma\\in[1,8/3)\\quad\\text{if }N=3.\n\\]\nLet $\\bar\\rho>0$ be the far-field density, and let the initial data $(\\rho_0,u_0)$ satisfy\n\\[\n0<\\underline\\varrho\\le \\rho_0\\le \\bar\\varrho,\n\\qquad\n\\rho_0-\\bar\\rho\\in H^3(\\mathbb R^N),\n\\qquad\nu_0\\in H^2(\\mathbb R^N),\n\\]\nfor some positive constants $\\underline\\varrho,\\bar\\varrho$. Consider the Cauchy problem for the shallow-water Navier-Stokes-Korteweg system on $\\mathbb R^N$:\n\\[\n\\begin{cases}\n\\partial_t\\rho+\\nabla\\cdot(\\rho u)=0,\\\\\n\\partial_t(\\rho u)+\\nabla\\cdot(\\rho u\\otimes u)+\\nabla P(\\rho)\n=\\nabla\\cdot\\bigl(2\\rho\\mathcal D(u)\\bigr)+\\nabla\\cdot\\bigl(\\rho\\nabla\\nabla\\log\\rho\\bigr),\n\\end{cases}\n\\]\nwith far-field condition\n\\[\n\\rho(x,t)\\to\\bar\\rho>0,\n\\qquad\nu(x,t)\\to0,\n\\qquad \\text{as }|x|\\to\\infty,\n\\]\nand initial data $(\\rho,u)|_{t=0}=(\\rho_0,u_0)$. Which statement holds about existence and uniqueness of solutions?", + "correct_choice": { + "label": "A", + "text": "There exists a unique global strong solution $(\\rho,u)$ to this Cauchy problem such that, for every $00$ depending on $T$ and the initial data with\n\\[\n(C(T))^{-1}\\le \\rho(x,t)\\le C(T)\n\\quad\\text{for all }(x,t)\\in\\mathbb R^N\\times[0,T],\n\\]\nand\n\\[\n\\rho-\\bar\\rho\\in C([0,T];H^3(\\mathbb R^N))\\cap L^2(0,T;H^4(\\mathbb R^N)),\n\\qquad\n\\rho_t\\in C([0,T];H^1(\\mathbb R^N))\\cap L^2(0,T;H^2(\\mathbb R^N)),\n\\]\n\\[\nu\\in C([0,T];H^2(\\mathbb R^N))\\cap L^2(0,T;H^3(\\mathbb R^N)),\n\\qquad\nu_t\\in L^\\infty(0,T;L^2(\\mathbb R^N))\\cap L^2(0,T;H^1(\\mathbb R^N)).\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "There exists a unique global strong solution $(\\rho,u)$ to this Cauchy problem such that there is a constant $C>0$, depending only on the initial data, with\n\\[\nC^{-1}\\le \\rho(x,t)\\le C\n\\quad\\text{for all }(x,t)\\in\\mathbb R^N\\times[0,\\infty),\n\\]\nand, for every $00$ depending on $T$ and the initial data with\n\\[\n(C(T))^{-1}\\le \\rho(x,t)\\le C(T)\n\\quad\\text{for all }(x,t)\\in\\mathbb R^N\\times[0,T],\n\\]\nand\n\\[\n\\rho-\\bar\\rho\\in C([0,T];H^3(\\mathbb R^N))\\cap L^2(0,T;H^4(\\mathbb R^N)),\n\\qquad\n\\rho_t\\in C([0,T];H^1(\\mathbb R^N))\\cap L^2(0,T;H^2(\\mathbb R^N)),\n\\]\n\\[\nu\\in C([0,T];H^2(\\mathbb R^N))\\cap L^2(0,T;H^3(\\mathbb R^N)),\n\\qquad\n\\nu_t\\in L^\\infty(0,T;L^2(\\mathbb R^N))\\cap L^2(0,T;H^1(\\mathbb R^N)).\n\\]" + }, + { + "label": "D", + "text": "There exists a unique global strong solution $(\\rho,u)$ to this Cauchy problem such that, for every $00$ depending on $T$ and the initial data with\n\\[\n(C(T))^{-1}\\le \\rho(x,t)\\le C(T)\n\\quad\\text{for all }(x,t)\\in\\mathbb R^N\\times[0,T],\n\\]\nand\n\\[\n\\rho-\\bar\\rho\\in C([0,T];H^3(\\mathbb R^N))\\cap L^2(0,T;H^4(\\mathbb R^N)),\n\\qquad\n\\rho_t\\in C([0,T];H^1(\\mathbb R^N))\\cap L^2(0,T;H^2(\\mathbb R^N)),\n\\]\n\\[\nu\\in C([0,T];H^2(\\mathbb R^N))\\cap L^2(0,T;H^3(\\mathbb R^N)),\n\\qquad\n\\nu_t\\in C([0,T];L^2(\\mathbb R^N))\\cap L^2(0,T;H^1(\\mathbb R^N)).\n\\]" + }, + { + "label": "E", + "text": "There exists a unique global strong solution $(\\rho,u)$ to this Cauchy problem for all\n\\[\n\ngamma\\in[1,\\infty)\\quad\\text{if }N=2\\text{ or }N=3,\n\\]\nsuch that, for every $00$ depending on $T$ and the initial data with\n\\[\n(C(T))^{-1}\\le \\rho(x,t)\\le C(T)\n\\quad\\text{for all }(x,t)\\in\\mathbb R^N\\times[0,T],\n\\]\nand\n\\[\n\\rho-\\bar\\rho\\in C([0,T];H^3(\\mathbb R^N))\\cap L^2(0,T;H^4(\\mathbb R^N)),\n\\qquad\n\\rho_t\\in C([0,T];H^1(\\mathbb R^N))\\cap L^2(0,T;H^2(\\mathbb R^N)),\n\\]\n\\[\nu\\in C([0,T];H^2(\\mathbb R^N))\\cap L^2(0,T;H^3(\\mathbb R^N)),\n\\qquad\n\\nu_t\\in L^\\infty(0,T;L^2(\\mathbb R^N))\\cap L^2(0,T;H^1(\\mathbb R^N)).\n\\]" + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "D" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "regularity", + "tampered_component": "time_dependent_density_bound_C(T)", + "template_used": "uniformity_effectivity" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "uniqueness", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "sharp_time_regularality_of_u_t", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "characteristic", + "tampered_component": "3D_gamma_upper_range_8_over_3", + "template_used": "boundary_range" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem states hypotheses only and does not explicitly reveal the conclusion. Although choice A matches the expected theorem statement closely, the stem itself does not leak that exact answer." + }, + "TAS": { + "score": 1, + "justification": "This is largely a theorem-recall question: the correct option is essentially the full existence theorem under the stated assumptions. However, it is not a pure restatement because the alternatives include stronger, weaker, and subtly altered conclusions." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is required to distinguish the exact theorem from nearby variants, especially the weaker true statement and the overstrong claims. Still, the task is mostly precision recall rather than substantial mathematical generation or derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically meaningful: one enlarges the 3D gamma range, one omits regularity details, one strengthens time-uniform density bounds, and one weakens global existence to finite-time statements. These reflect common failure modes in theorem interpretation." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no direct answer leakage, but it mainly tests accurate recall of a specific result rather than deeper generative reasoning." + } + }, + { + "id": "2602.10707v1", + "paper_link": "http://arxiv.org/abs/2602.10707v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm:intro}\nThe group of homeomorphisms of the real projective plane $\\mathbb{R}\\mathrm{P}^2$ and Möbius strip $M$ which are isotopic to the identity is stably unbounded with respect to commutator length in the sense of \\cite{BIP}.", + "start_pos": 2526, + "end_pos": 2773, + "label": "thm:intro" + }, + "ref_dict": { + "subsec:actions-on-cg": "\\begin{theorem}[Guessing geodesics]\\label{thm:bowditch-guessing-geodesics}\n Let $X$ be a graph and $D>0$ a number.\n Suppose that for each pair of distinct vertices $x,y$ of $X$ we have chosen a connected subgraph $P(x,y)$ containing $x$ and $y$.\n Suppose that if $d_X(x,y) = 1$, the diameter of $P(x,y)$ is at most $D$, and in addition we have for all pairwise distinct $x,y,z$:\n \\[ P(x,z) \\subseteq N_D(P(x,y)\\cup P(y,z)). \\]\n Then $X$ is $\\delta(D)$-hyperbolic, and each $P(x,y)$ is $B(D)$-Hausdorff close to a geodesic joining $x$ to $y$.\n\\end{theorem}\n\n\\subsection{Actions on curve graphs}\\label{subsec:actions-on-cg}\nOne can easily check that the groups $\\operatorname{Homeo}_0$ and $\\operatorname{Mcg}$ of a surface act on the fine curve graph respectively curve graph of that surface by isometries.\n\nIn \\cite{Bestvina}, Bestvina and Fujiwara introduced the notion of weak proper discontinuity, which will be the key to finding elements with positive stable commutator length.\n\nFirst, recall that for two hyperbolic isometries $f$ and $g$ of a $\\delta$-hyperbolic space $(X,d)$, we can find invariant quasi-axes $\\alpha$ and $\\beta$ of quality $L$ respectively $L'$ by taking the orbit of some point, e.g. $\\alpha = (a_k) \\coloneqq f^k(x_0)$ for $k\\in\\ZZ$ and $x_0\\in X$, and analogously for $\\beta = (b_k)$.\nWe then say that $f$ and $g$ (respectively their axes) are \\emph{quasi-equivalent}, if there exists a constant $B = B(L,L',\\delta)$ such that:\nfor any $D>0$, there is an isometry $h$ that sends a segment of length $D$ in $\\alpha$ into a $B$-neighbourhood of $\\beta$.\nIf $f$ is quasi-equivalent to its inverse, then we say that $f$ is \\emph{quasi-invertible}.\n\nThese quasi-geodesics define points in the Gromov boundary of $X$, and we can give an alternative definition of quasi-equivalence in terms of boundary points:\nlet $\\tau_\\pm \\coloneqq [a_{\\pm k}]$ and $\\upsilon_\\pm \\coloneqq [b_{\\pm k}]$ as equivalence classes of admissible sequences as in the previous section.\nThen $f$ and $g$ (respectively their axes) are quasi-equivalent, if there exists a sequence $h_k$ of isometries of $X$ such that $\\lim_{k\\rightarrow \\infty} h_k(\\tau_\\pm) = \\upsilon_\\pm$.\n\nFurther, we call two hyperbolic elements \\emph{dependent}, if they have quasi-axes which contain rays that are in finite distance of each other, and \\emph{independent} otherwise.\nIf $G$ contains two independent hyperbolic elements, we call the action \\emph{non-elementary}.\n\nThe central definition in this section is:\n\n\\begin{definition}\n Let $G$ be a group acting on the hyperbolic metric space $X$.\n This action is \\emph{weakly properly discontinuous (WPD)}, if the following holds: \\begin{itemize}\n \\item $G$ is not virtually cyclic\n \\item $\\exists g\\in G\\colon$ $g$ acts on $X$ as a hyperbolic isometry\n \\item $ \\forall x\\in X\\colon \\forall g \\in G \\textrm{ acting hyperbolically}\\colon \\forall C>0\\colon \\exists N>0\\colon \\{\\gamma \\in G \\mid d(x,\\gamma(x)) \\leq C, d(g^N(x),\\gamma(g^N(x)))\\leq C \\} \\textrm{ is finite.} $\n \\end{itemize}\n \\end{definition}\nNote that when we restrict an action $G\\curvearrowright X$ to a subgroup $HM$ that are oriented $B(K,L,\\delta)$-close, then $l_1$ and $l_2$ are oriented $B(K,L,\\delta)$-close and moreover the corresponding conjugates $g_1$ and $g_2$ of $g$ have positive powers which are equal,\n \\item the action of $G$ on $X$ is non-elementary,\n \\item two elements of $G$ are quasi-equivalent if and only if they have positive powers which are conjugate,\n \\item there exist two hyperbolic elements in $G$ which are not quasi-equivalent.\n \\end{enumerate}\n\\end{proposition}\n\nThe main result of \\cite{Bestvina} is:\n\\begin{theorem}[Theorem 1 of \\cite{Bestvina}]\\label{thm:bestvina-fujiwara-main-thm}\nSuppose a group $G$ acts on a $\\delta$-hyperbolic graph $X$ by isometries.\nSuppose also that the action is non-elementary and that there exist independent hyperbolic elements $g_1,g_2 \\in G$ which are not quasi-equivalent.\nThen, the space of non-trivial quasi-morphisms is infinite dimensional.\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 923, + "pre_theorem_intro_text": "In this article, we study the (un-)boundedness of norms on the group of homeomorphisms and diffeomorphisms of the Möbius strip and real projective plane that are isotopic to the identity.\nThis type of question first appeared in the work of Burago, Ivanov and Polterovich \\cite{BIP}, where the authors considered the identity components of groups of compactly supported $C^\\infty$-diffeomorphisms of various manifolds, and found that for all compact 3-manifolds and certain compact 4-manifolds these groups only admit conjugation-invariant norms that are bounded.\nTheir results have been extended to higher dimensional manifolds in \\cite{tsuboi}.\nFor surfaces, the situation is different:\nwhile the sphere again only admits conjugation-invariant norms that are bounded, almost all other surfaces have been found to have unbounded norms.\nThe main result of this article answers this question for the last two remaining cases:", + "context": "In this article, we study the (un-)boundedness of norms on the group of homeomorphisms and diffeomorphisms of the Möbius strip and real projective plane that are isotopic to the identity.\nThis type of question first appeared in the work of Burago, Ivanov and Polterovich \\cite{BIP}, where the authors considered the identity components of groups of compactly supported $C^\\infty$-diffeomorphisms of various manifolds, and found that for all compact 3-manifolds and certain compact 4-manifolds these groups only admit conjugation-invariant norms that are bounded.\nTheir results have been extended to higher dimensional manifolds in \\cite{tsuboi}.\nFor surfaces, the situation is different:\nwhile the sphere again only admits conjugation-invariant norms that are bounded, almost all other surfaces have been found to have unbounded norms.\nThe main result of this article answers this question for the last two remaining cases:", + "full_context": "In this article, we study the (un-)boundedness of norms on the group of homeomorphisms and diffeomorphisms of the Möbius strip and real projective plane that are isotopic to the identity.\nThis type of question first appeared in the work of Burago, Ivanov and Polterovich \\cite{BIP}, where the authors considered the identity components of groups of compactly supported $C^\\infty$-diffeomorphisms of various manifolds, and found that for all compact 3-manifolds and certain compact 4-manifolds these groups only admit conjugation-invariant norms that are bounded.\nTheir results have been extended to higher dimensional manifolds in \\cite{tsuboi}.\nFor surfaces, the situation is different:\nwhile the sphere again only admits conjugation-invariant norms that are bounded, almost all other surfaces have been found to have unbounded norms.\nThe main result of this article answers this question for the last two remaining cases:\n\n\\begin{abstract}\nUsing a recent result of Bowden, Hensel and Webb, we prove the existence of homeomorphisms with positive stable commutator length in the groups of homeomorphisms of the real projective plane and Möbius strip which are isotopic to the identity.\nThis completes the answer to a question posed by Burago, Ivanov and Polterovich on the boundedness of diffeomorphism groups of surfaces.\n\\end{abstract}\n\n\\section{Introduction}\nIn this article, we study the (un-)boundedness of norms on the group of homeomorphisms and diffeomorphisms of the Möbius strip and real projective plane that are isotopic to the identity.\nThis type of question first appeared in the work of Burago, Ivanov and Polterovich \\cite{BIP}, where the authors considered the identity components of groups of compactly supported $C^\\infty$-diffeomorphisms of various manifolds, and found that for all compact 3-manifolds and certain compact 4-manifolds these groups only admit conjugation-invariant norms that are bounded.\nTheir results have been extended to higher dimensional manifolds in \\cite{tsuboi}.\nFor surfaces, the situation is different:\nwhile the sphere again only admits conjugation-invariant norms that are bounded, almost all other surfaces have been found to have unbounded norms.\nThe main result of this article answers this question for the last two remaining cases:\n\nTherefore, all compact surfaces except the sphere admit unbounded norms.\nTo prove unboundedness, the central tool is Bavard duality \\cite{bavard}, which links a norm, namely commutator length, to the existence of non-trivial homogeneous quasi-morphisms.\nAn explicit construction of quasi-morphism was given by Brooks in \\cite{Brooks} for free groups.\nThis construction was generalised in \\cite{Bestvina} by Bestvina and Fujiwara to groups of isometries of a hyperbolic space, which they applied to (subgroups of) mapping class groups of hyperbolic surfaces.\n\n\\paragraph{Organisation of the paper.}\nIn the next section, we provide some background on the papers of Kuno \\cite{kuno-hyperbolicity} and Bowden, Hensel and Webb \\cite{BHW1} \\cite{BHW2}.\nIn the third section, we combine results of \\cite{kuno-hyperbolicity} and \\cite{BHW1} to show that the Möbius strip and real projective plane have hyperbolic and unbounded fine curve graphs.\nIn the fourth section, we use that result and the methods of \\cite{BHW2} to prove that both Möbius strip and real projective plane have stably unbounded $\\operatorname{Homeo}_0$.\n\n\\paragraph{Bavard duality.}\nIt is known that the groups $\\operatorname{Homeo}_0(M)$ of compact manifolds are perfect \\cite{anderson}, i.e. any element of the group can be written as a product of commutators, and the same is true for $\\operatorname{Diff}_0$, see e.g. \\cite{mann}.\nThe minimal number of commutators that appear in such a factorisation of a homeomorphism is called \\emph{commutator length}, usually denoted by cl, and it is an important example of a conjugation-invariant norm on the group $\\operatorname{Homeo}(M)$ as defined in \\cite{BIP}.\nSuch norms have a so-called \\emph{stabilisation}, which takes the following form for commutator length:\n\\[ \\operatorname{scl}(g) \\coloneqq \\lim_{n\\rightarrow \\infty} \\frac{\\operatorname{cl}(g^n)}{n} \\]\nfor $g\\in \\operatorname{Homeo}(M)$.\nWe call a group \\emph{stably unbounded} if the stabilisation of some norm takes positive values.\n\n\\subsection{Curve graphs and variations}\nCurve graphs and curve complexes were defined by Harvey in \\cite{Harvey}.\nThese are usually defined in terms of essential curves, but for our purposes it will be useful to only work with non-separating curves.\nThe main definition here is:\nthe \\emph{non-separating curve graph} $\\ncg(M)$ of a 2-manifold (possibly with boundary) $M$ is the graph with: \\begin{itemize}\n \\item vertices corresponding to the homotopy classes of non-separating closed curves in $M$\n \\item an edge between two vertices, if the corresponding homotopy classes have disjoint representatives\n\\end{itemize}\nTo study homeomorphisms that are isotopic to the identity, we have to get rid of the homotopy classes:\nthe \\emph{fine non-separating curve graph} accomplishes this.\nIt was introduced in \\cite{BHW1} and is defined by:\\begin{itemize}\n \\item vertices which each represent a non-separating closed curve in $M$\n \\item an edge between two vertices, if the corresponding curves are disjoint\n\\end{itemize}\n\nThe main result of \\cite{Bestvina} is:\n\\begin{theorem}[Theorem 1 of \\cite{Bestvina}]\\label{thm:bestvina-fujiwara-main-thm}\nSuppose a group $G$ acts on a $\\delta$-hyperbolic graph $X$ by isometries.\nSuppose also that the action is non-elementary and that there exist independent hyperbolic elements $g_1,g_2 \\in G$ which are not quasi-equivalent.\nThen, the space of non-trivial quasi-morphisms is infinite dimensional.\n\\end{theorem}\n\n\\begin{theorem}[cf. Theorem 4.1 in \\cite{BHW2}]\\label{thm:convergence-criterion}\nSuppose $(\\mathcal{F}_v,\\mathcal{F}_h,d)$ is a BSF structure with ending $\\mathcal{F}_v$.\nThen there is a point $\\tau_{\\mathcal{F}_v}$ on the Gromov boundary of $\\fncg(N)$ with the following properties: \\begin{enumerate}\n \\item The boundary point $\\tau_{\\mathcal{F}_v}$ depends only on the foliation $\\mathcal{F}_v$.\n \\item A homeomorphism of $N$ fixes $\\tau_{\\mathcal{F}_v}$ as a point on the Gromov boundary if and only if it preserves the foliation $\\mathcal{F}_v$.\n \\item A sequence of curves $(\\beta_i)$ in $\\fncg(N)$ converges to $\\tau_{\\mathcal{F}_v}$ if and only if \n the following conditions are satisfied: \\begin{itemize}\n \\item The sizes of the $\\beta_i$ diverge to infinity.\n \\item For any $B$, $\\epsilon$ there is a number $I$ so that if $b\\subseteq \\beta_i$ with $i>I$ is a segment of size at most $B$, then it $\\epsilon$-fellow travels a leaf segment of $\\mathcal{F}_v$ with respect to the metric $d$.\n \\end{itemize}", + "post_theorem_intro_text_len": 2390, + "post_theorem_intro_text": "Therefore, all compact surfaces except the sphere admit unbounded norms.\nTo prove unboundedness, the central tool is Bavard duality \\cite{bavard}, which links a norm, namely commutator length, to the existence of non-trivial homogeneous quasi-morphisms.\nAn explicit construction of quasi-morphism was given by Brooks in \\cite{Brooks} for free groups.\nThis construction was generalised in \\cite{Bestvina} by Bestvina and Fujiwara to groups of isometries of a hyperbolic space, which they applied to (subgroups of) mapping class groups of hyperbolic surfaces.\n\nIn their paper, they consider the action of the mapping class group on the curve graph (introduced by Harvey \\cite{Harvey}, shown to be hyperbolic by Masur and Minsky \\cite{masur-minsky}), which satisfies a newly identified property, namely weak proper discontinuity (see section \\ref{subsec:actions-on-cg}).\n\nAn analogous strategy was developed by Bowden, Hensel and Webb in \\cite{BHW1}, where the action of the identity component of the homeomorphism group of an orientable surface with positive genus on the newly introduced \\emph{fine} curve graph was studied.\n\nThis strategy was adapted to non-orientable surfaces of genus at least three in \\cite{kimura-kuno-qms}.\nIn \\cite{BHW2}, the action of these groups on the fine curve graph is studied further, and a tool is provided that characterises some of the elements whose powers have increasingly large commutator length.\nIn addition, \\cite{BHW2} provides a connection between certain foliations on surfaces and boundary points of the corresponding fine curve graph.\nIn this article, we aim to explain that the techniques and results of Bowden, Hensel and Webb generalise to non-orientable surfaces with genus 1.\n\n\\paragraph{Organisation of the paper.}\nIn the next section, we provide some background on the papers of Kuno \\cite{kuno-hyperbolicity} and Bowden, Hensel and Webb \\cite{BHW1} \\cite{BHW2}.\nIn the third section, we combine results of \\cite{kuno-hyperbolicity} and \\cite{BHW1} to show that the Möbius strip and real projective plane have hyperbolic and unbounded fine curve graphs.\nIn the fourth section, we use that result and the methods of \\cite{BHW2} to prove that both Möbius strip and real projective plane have stably unbounded $\\operatorname{Homeo}_0$.\n\n\\paragraph{Acknowledgements.}\nThe author wants to thank Sebastian Hensel and Javier de la Nuez Gonzalez.", + "sketch": "To prove Theorem~\\ref{thm:intro}, the introduction says the \"central tool is Bavard duality\" (linking commutator length to the existence of \"non-trivial homogeneous quasi-morphisms\"). The strategy follows the quasi-morphism constructions of Brooks and their generalisation by Bestvina--Fujiwara via actions on hyperbolic spaces, and the analogous approach of Bowden--Hensel--Webb using the action of $\\operatorname{Homeo}_0$ on a (fine) curve graph. Concretely, the paper is organised to (1) combine results of \\cite{kuno-hyperbolicity} and \\cite{BHW1} to show the Möbius strip and $\\mathbb{R}\\mathrm{P}^2$ have \"hyperbolic and unbounded fine curve graphs\", and then (2) \"use that result and the methods of \\cite{BHW2}\" to prove that both surfaces have \"stably unbounded $\\operatorname{Homeo}_0$\" (hence stably unbounded commutator length).", + "expanded_sketch": "To prove the main theorem, the introduction says the \"central tool is Bavard duality\" (linking commutator length to the existence of \"non-trivial homogeneous quasi-morphisms\"). The strategy follows the quasi-morphism constructions of Brooks and their generalisation by Bestvina--Fujiwara via actions on hyperbolic spaces, and the analogous approach of Bowden--Hensel--Webb using the action of $\\operatorname{Homeo}_0$ on a (fine) curve graph. Concretely, the paper is organised to (1) combine results of \\cite{kuno-hyperbolicity} and \\cite{BHW1} to show the Möbius strip and $\\mathbb{R}\\mathrm{P}^2$ have \"hyperbolic and unbounded fine curve graphs\", and then (2) use that result together with the methods of \\cite{BHW2} to prove that both surfaces have \"stably unbounded $\\operatorname{Homeo}_0$\" (hence stably unbounded commutator length).", + "expanded_theorem": "\\label{thm:intro}\nThe group of homeomorphisms of the real projective plane $\\mathbb{R}\\mathrm{P}^2$ and Möbius strip $M$ which are isotopic to the identity is stably unbounded with respect to commutator length in the sense of \\cite{BIP}.", + "theorem_type": [ + "Universal", + "Inequality or Bound" + ], + "mcq": { + "question": "Let \\(\\operatorname{Homeo}_0(S)\\) denote the group of homeomorphisms of a surface \\(S\\) that are isotopic to the identity. For \\(g\\in \\operatorname{Homeo}_0(S)\\), let \\(\\operatorname{cl}(g)\\) be its commutator length, i.e. the minimal number of commutators whose product is \\(g\\), and define the stable commutator length by\n\\[\n\\operatorname{scl}(g)=\\lim_{n\\to\\infty}\\frac{\\operatorname{cl}(g^n)}{n}.\n\\]\nSay that \\(\\operatorname{Homeo}_0(S)\\) is stably unbounded with respect to commutator length if there exists some \\(g\\in \\operatorname{Homeo}_0(S)\\) with \\(\\operatorname{scl}(g)>0\\). Which statement holds for every surface \\(S\\) in the class consisting of the real projective plane \\(\\mathbb{R}\\mathrm{P}^2\\) and the Möbius strip \\(M\\)?", + "correct_choice": { + "label": "A", + "text": "For each such surface \\(S\\in\\{\\mathbb{R}\\mathrm{P}^2,M\\}\\), the group \\(\\operatorname{Homeo}_0(S)\\) is stably unbounded with respect to commutator length; equivalently, there exists \\(g\\in \\operatorname{Homeo}_0(S)\\) such that \\(\\operatorname{scl}(g)>0\\)." + }, + "choices": [ + { + "label": "B", + "text": "For each such surface \\(S\\in\\{\\mathbb{R}\\mathrm{P}^2,M\\}\\), the group \\(\\operatorname{Homeo}_0(S)\\) is unbounded with respect to commutator length; equivalently, for every \\(N\\in\\mathbb{N}\\) there exists \\(g\\in \\operatorname{Homeo}_0(S)\\) with \\(\\operatorname{cl}(g)>N\\)." + }, + { + "label": "C", + "text": "For each such surface \\(S\\in\\{\\mathbb{R}\\mathrm{P}^2,M\\}\\), there exists \\(g\\in \\operatorname{Homeo}_0(S)\\) whose commutator lengths \\(\\operatorname{cl}(g^n)\\) are unbounded as \\(n\\to\\infty\\)." + }, + { + "label": "D", + "text": "There exists a single element \\(g\\) such that for every surface \\(S\\in\\{\\mathbb{R}\\mathrm{P}^2,M\\}\\), one has \\(g\\in \\operatorname{Homeo}_0(S)\\) and \\(\\operatorname{scl}(g)>0\\)." + }, + { + "label": "E", + "text": "For each such surface \\(S\\in\\{\\mathbb{R}\\mathrm{P}^2,M\\}\\), every nontrivial element \\(g\\in \\operatorname{Homeo}_0(S)\\) satisfies \\(\\operatorname{scl}(g)>0\\), so \\(\\operatorname{Homeo}_0(S)\\) is stably unbounded with respect to commutator length." + } + ], + "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": "positive_scl_vs_mere_unbounded_cl", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped_positive_linear_growth_rate_replaced_by_unboundedness_of_cl_gn", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "element_depends_on_surface", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "existence_of_some_element_strengthened_to_all_nontrivial_elements", + "template_used": "stronger_trap" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives definitions but does not explicitly or implicitly reveal that both groups are stably unbounded. The correct conclusion is not stated in advance." + }, + "TAS": { + "score": 1, + "justification": "The item is close to asking for the theorem’s conclusion directly: it asks which statement holds for these two specific groups under the given definitions. However, it is not a pure restatement because the options vary in strength and scope." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish positive stable commutator length from merely unbounded commutator length, and to reject overly strong uniform claims. Still, the item mainly tests recall of the result rather than substantial generative reasoning from first principles." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically meaningful: they target common confusions between unboundedness of cl and positivity of scl, existential versus uniform lower bounds, and whether the statement holds for one or both groups." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-recall MCQ with good distractors and no answer leakage, but it is only moderately non-tautological and does not strongly test generative mathematical reasoning." + } + }, + { + "id": "2602.10766v1", + "paper_link": "http://arxiv.org/abs/2602.10766v1", + "theorems_cnt": 3, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm:calderon_intro}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$.\nIf $\\psi \\in \\mathcal{B}_{\\pi}$ and\n$\n \\{\\pi(A^j P k, A^j) \\psi \\}_{ j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d } \n$\nis a Parseval frame for $L^2 (\\mathbb{R}^d)$, then $\\{ \\pi(x, A^j) \\psi \\}_{x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}}$ is a tight frame for $L^2 (\\mathbb{R}^d)$, and \n\\[\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det (P)| \\quad \\text{for a.e.} \\;\\; \\xi \\in \\mathbb{R}^d.\n\\]", + "start_pos": 8275, + "end_pos": 8773, + "label": "thm:calderon_intro" + }, + "ref_dict": { + "thm:calderon_intro": "\\begin{theorem} \\label{thm:calderon_intro}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$.\nIf $\\psi \\in \\mathcal{B}_{\\pi}$ and\n$\n \\{\\pi(A^j P k, A^j) \\psi \\}_{ j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d } \n$\nis a Parseval frame for $L^2 (\\mathbb{R}^d)$, then $\\{ \\pi(x, A^j) \\psi \\}_{x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}}$ is a tight frame for $L^2 (\\mathbb{R}^d)$, and \n\\[\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det (P)| \\quad \\text{for a.e.} \\;\\; \\xi \\in \\mathbb{R}^d.\n\\]\n\\end{theorem}", + "eq:discrete_wavelet": "\\begin{align} \\label{eq:discrete_wavelet}\n\\big\\{ |\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\big \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}.\n\\end{align}", + "eq:continuous_wavelet": "\\begin{align} \\label{eq:continuous_wavelet}\n\\pi(x, A^j) \\psi := |\\det(A)|^{-j/2} \\psi(A^{-j} ( \\cdot - x)), \\quad x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}.\n\\end{align}", + "conj:wavelet": "\\begin{conjecture}[\\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \\label{conj:wavelet}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$ and $\\psi \\in L^2 (\\mathbb{R}^d)$. Suppose that \\[ \\{|\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}\\]\nis an orthonormal basis, or more generally a Parseval frame, for $L^2 (\\mathbb{R}^d)$. Then the Calder\\'on sum formula holds:\n\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}\n\\end{conjecture}", + "eq:calderon": "\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}", + "rem:frame_bounds": "\\begin{remark} \\label{rem:frame_bounds}\n \\Cref{cor:quasi-lattice} implies, in particular, that if $\\pi(\\Lambda) \\psi$ is a Bessel sequence for some relatively dense set $\\Lambda \\subseteq G$, then $\\pi$ must be square-integrable, in the sense that $C_{\\psi} f \\in L^2 (G)$ for all $f \\in \\Hpi$. \nFor certain representations of semidirect product groups $G = \\mathbb{R}^d \\rtimes H$ of $\\mathbb{R}^d$ and a matrix group $H \\leq \\mathrm{GL}(d, \\mathbb{R})$, this was shown in \\cite{aniello2001discrete}. For that setting, the upper bound provided by \\Cref{cor:quasi-lattice} corresponds to \\cite[Proposition 1]{aniello2001discrete}. Although the corresponding lower bound is asserted as \\cite[Proposition 2]{aniello2001discrete} (even without any additional assumption on $\\psi$), the argument provided for the lower bound in \\cite{aniello2001discrete} (cf. \\cite[Theorem 1]{aniello2001discrete}) is incomplete.\n\\end{remark}", + "cor:quasi-lattice": "\\begin{corollary} \\label{cor:quasi-lattice}\nLet $G$ be amenable and let $U \\subseteq G$ and $K \\subseteq G$ be a nonempty open and compact set, respectively. Let $\\Lambda \\subseteq G$ be a $U$-separated, $K$-dense set and $\\psi \\in \\Hpi$.\n\nIf $\\pi(\\Lambda)\\psi$ is a Bessel sequence in $\\Hpi$ with Bessel bound $C_2 > 0$, then\n\\[ \\| C_{\\psi} f \\|^2 \\leq C_2 \\mu_G (K) \\| f \\|^2 \\quad \\text{for all} \\quad f \\in \\Hpi. \\]\n In addition, if $\\psi \\in \\mathcal{B}_{\\pi}$ and $\\pi(\\Lambda) \\psi$ also admits a lower frame bound $C_1 > 0$, then\n\\[ \\| C_{\\psi} f \\|^2 \\geq C_1 \\mu_G (U) \\| f \\|^2 \\quad \\text{for all} \\quad f \\in \\Hpi. \\]\nIn particular, if $\\pi(\\Lambda) \\psi$ is frame with $\\psi \\in \\mathcal{B}_{\\pi}$ for some quasi-lattice $\\Lambda \\subseteq G$ with a relatively compact Jordan measurable complement $C$, then \n\\[\nC_1 \\| f \\|^2 \\leq \\frac{\\|C_{\\psi} f\\|^2}{\\rho_G (C)} \\leq C_2 \\| f \\|^2\n\\]\nfor all $f \\in \\Hpi$.\n\\end{corollary}" + }, + "pre_theorem_intro_text_len": 4800, + "pre_theorem_intro_text": "For $\\psi \\in L^2 (\\mathbb{R}^d)$ and matrices $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$, the associated wavelet system is given by the collection of functions\n\\begin{align} \\label{eq:discrete_wavelet}\n\\big\\{ |\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\big \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}.\n\\end{align}\nIn higher dimensions, a classical condition in the study of wavelet systems is to assume that the matrix $A$ preserves the integer lattice $\\mathbb{Z}^d$, i.e., $A \\mathbb{Z}^d \\subseteq \\mathbb{Z}^d$, and that $A$ is expansive, i.e., all its eigenvalues are strictly greater than one in modulus. Under such conditions, a full characterization of Parseval wavelet frames, or more generally dual wavelet frames, was obtained in \\cite{chui2002characterization, bownik2000characterization, calogero2000characterization}, among others. In addition, the existence of wavelet bases for such dilations were shown in \\cite{dai1997wavelet}. Both types of results extend classical results from dimension one to arbitrary dimensions. \n\nBeyond the case of expansive dilation matrices, the theory of wavelet systems in higher dimensions is far less complete. Nevertheless, the existence of wavelet bases for nonexpansive dilation matrices has been studied in \\cite{speegle2003existence, bownik2017wavelets, ionascu2006simultaneous, wang2002wavelets} and culminated in the recent breakthrough \\cite{bownik2021simultaneous} that characterizes the dilation matrices admitting wavelet sets. On the other hand, the aforementioned characterization of Parseval wavelet frames is currently only known for special dilations, such as amplifying dilations \\cite{laugesen2002translational}, dilations expanding on a subspace \\cite{hernandez2002unified, guo2006some} or dilations satisfying the lattice counting estimate \\cite{bownik2017wavelets}. This has lead to the following conjecture \\cite[Conjecture 1]{bownik2017wavelets} and open problem \\cite[Problem 3.3]{bownik2020open}, which was already implicitly raised in \\cite[p. 177]{speegle2003existence}.\\footnote{The formulations in \\cite{bownik2017wavelets, bownik2020open} are under the implicit assumption that $|\\det(P)| = 1$.}\n\n\\begin{conjecture}[\\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \\label{conj:wavelet}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$ and $\\psi \\in L^2 (\\mathbb{R}^d)$. Suppose that \\[ \\{|\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}\\]\nis an orthonormal basis, or more generally a Parseval frame, for $L^2 (\\mathbb{R}^d)$. Then the Calder\\'on sum formula holds:\n\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}\n\\end{conjecture}\n\nThe Calder\\'on sum formula \\eqref{eq:calderon} is part of the aforementioned characterizion of Parseval wavelet frames known under additional assumptions on the dilation matrix. The upper bound for the Calder\\'on sum is known to hold for any Bessel sequence with bound $1$ (cf. \\cite[Proposition 4.1]{hernandez2002unified}).\n\nIn this paper, we present a new approach to \\Cref{conj:wavelet} which allows us to prove the Calder\\'on sum formula for arbitrary translation and dilation matrices under a mild condition on the wavelet function. Our approach is based on a relation between the frame properties of the discrete wavelet system \\eqref{eq:discrete_wavelet} and the semi-continuous wavelet system whose elements are given by\n\\begin{align} \\label{eq:continuous_wavelet}\n\\pi(x, A^j) \\psi := |\\det(A)|^{-j/2} \\psi(A^{-j} ( \\cdot - x)), \\quad x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}.\n\\end{align}\nThe action $\\pi$ forms a unitary group representation of the semi-direct product group $G = \\mathbb{R}^d \\rtimes \\langle A \\rangle$ of $\\mathbb{R}^d$ and the cyclic group $\\langle A \\rangle := \\{ A^j : j \\in \\mathbb{Z} \\}$ generated by $A \\in \\mathrm{GL}(d, \\mathbb{R})$. Observe that the wavelet system \\eqref{eq:discrete_wavelet} corresponds to $\\{\\pi(A^j P k, A^j) \\psi : j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d \\}$. We will assume a mild condition on the wavelet function. Namely, we assume that $\\psi \\in L^2 (\\mathbb{R}^d)$ is such that $\\pi(\\Lambda) \\psi$ is a Bessel sequence in $L^2 (\\mathbb{R}^d)$ for all relatively separated sets $\\Lambda $ in $ G = \\mathbb{R}^d \\rtimes \\langle A \\rangle$; in notation, $\\psi \\in \\mathcal{B}_{\\pi}$. This is a common assumption in the study of frames in the orbit of a group representation, see, e.g., \\cite{Gr08, FuGr07, enstad2025dynamical, fuehr2017density, caspers2023overcompleteness}.\nWe refer to Section \\ref{sec:coefficient} for an alternative description of the space $\\mathcal{B}_{\\pi}$ and further properties.\n\nUsing the notation from the previous paragraph, our main result is the following:", + "context": "For $\\psi \\in L^2 (\\mathbb{R}^d)$ and matrices $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$, the associated wavelet system is given by the collection of functions\n\\begin{align} \\label{eq:discrete_wavelet}\n\\big\\{ |\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\big \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}.\n\\end{align}\nIn higher dimensions, a classical condition in the study of wavelet systems is to assume that the matrix $A$ preserves the integer lattice $\\mathbb{Z}^d$, i.e., $A \\mathbb{Z}^d \\subseteq \\mathbb{Z}^d$, and that $A$ is expansive, i.e., all its eigenvalues are strictly greater than one in modulus. Under such conditions, a full characterization of Parseval wavelet frames, or more generally dual wavelet frames, was obtained in \\cite{chui2002characterization, bownik2000characterization, calogero2000characterization}, among others. In addition, the existence of wavelet bases for such dilations were shown in \\cite{dai1997wavelet}. Both types of results extend classical results from dimension one to arbitrary dimensions.\n\nBeyond the case of expansive dilation matrices, the theory of wavelet systems in higher dimensions is far less complete. Nevertheless, the existence of wavelet bases for nonexpansive dilation matrices has been studied in \\cite{speegle2003existence, bownik2017wavelets, ionascu2006simultaneous, wang2002wavelets} and culminated in the recent breakthrough \\cite{bownik2021simultaneous} that characterizes the dilation matrices admitting wavelet sets. On the other hand, the aforementioned characterization of Parseval wavelet frames is currently only known for special dilations, such as amplifying dilations \\cite{laugesen2002translational}, dilations expanding on a subspace \\cite{hernandez2002unified, guo2006some} or dilations satisfying the lattice counting estimate \\cite{bownik2017wavelets}. This has lead to the following conjecture \\cite[Conjecture 1]{bownik2017wavelets} and open problem \\cite[Problem 3.3]{bownik2020open}, which was already implicitly raised in \\cite[p. 177]{speegle2003existence}.\\footnote{The formulations in \\cite{bownik2017wavelets, bownik2020open} are under the implicit assumption that $|\\det(P)| = 1$.}\n\n\\begin{conjecture}[\\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \\label{conj:wavelet}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$ and $\\psi \\in L^2 (\\mathbb{R}^d)$. Suppose that \\[ \\{|\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}\\]\nis an orthonormal basis, or more generally a Parseval frame, for $L^2 (\\mathbb{R}^d)$. Then the Calder\\'on sum formula holds:\n\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}\n\\end{conjecture}\n\nThe Calder\\'on sum formula \\eqref{eq:calderon} is part of the aforementioned characterizion of Parseval wavelet frames known under additional assumptions on the dilation matrix. The upper bound for the Calder\\'on sum is known to hold for any Bessel sequence with bound $1$ (cf. \\cite[Proposition 4.1]{hernandez2002unified}).\n\nIn this paper, we present a new approach to \\Cref{conj:wavelet} which allows us to prove the Calder\\'on sum formula for arbitrary translation and dilation matrices under a mild condition on the wavelet function. Our approach is based on a relation between the frame properties of the discrete wavelet system \\eqref{eq:discrete_wavelet} and the semi-continuous wavelet system whose elements are given by\n\\begin{align} \\label{eq:continuous_wavelet}\n\\pi(x, A^j) \\psi := |\\det(A)|^{-j/2} \\psi(A^{-j} ( \\cdot - x)), \\quad x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}.\n\\end{align}\nThe action $\\pi$ forms a unitary group representation of the semi-direct product group $G = \\mathbb{R}^d \\rtimes \\langle A \\rangle$ of $\\mathbb{R}^d$ and the cyclic group $\\langle A \\rangle := \\{ A^j : j \\in \\mathbb{Z} \\}$ generated by $A \\in \\mathrm{GL}(d, \\mathbb{R})$. Observe that the wavelet system \\eqref{eq:discrete_wavelet} corresponds to $\\{\\pi(A^j P k, A^j) \\psi : j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d \\}$. We will assume a mild condition on the wavelet function. Namely, we assume that $\\psi \\in L^2 (\\mathbb{R}^d)$ is such that $\\pi(\\Lambda) \\psi$ is a Bessel sequence in $L^2 (\\mathbb{R}^d)$ for all relatively separated sets $\\Lambda $ in $ G = \\mathbb{R}^d \\rtimes \\langle A \\rangle$; in notation, $\\psi \\in \\mathcal{B}_{\\pi}$. This is a common assumption in the study of frames in the orbit of a group representation, see, e.g., \\cite{Gr08, FuGr07, enstad2025dynamical, fuehr2017density, caspers2023overcompleteness}.\nWe refer to Section \\ref{sec:coefficient} for an alternative description of the space $\\mathcal{B}_{\\pi}$ and further properties.\n\nUsing the notation from the previous paragraph, our main result is the following:\n\n\\begin{conjecture}[\\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \\label{conj:wavelet}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$ and $\\psi \\in L^2 (\\mathbb{R}^d)$. Suppose that \\[ \\{|\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}\\]\nis an orthonormal basis, or more generally a Parseval frame, for $L^2 (\\mathbb{R}^d)$. Then the Calder\\'on sum formula holds:\n\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}\n\\end{conjecture}\n\n\\begin{align} \\label{eq:discrete_wavelet}\n\\big\\{ |\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\big \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}.\n\\end{align}", + "full_context": "For $\\psi \\in L^2 (\\mathbb{R}^d)$ and matrices $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$, the associated wavelet system is given by the collection of functions\n\\begin{align} \\label{eq:discrete_wavelet}\n\\big\\{ |\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\big \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}.\n\\end{align}\nIn higher dimensions, a classical condition in the study of wavelet systems is to assume that the matrix $A$ preserves the integer lattice $\\mathbb{Z}^d$, i.e., $A \\mathbb{Z}^d \\subseteq \\mathbb{Z}^d$, and that $A$ is expansive, i.e., all its eigenvalues are strictly greater than one in modulus. Under such conditions, a full characterization of Parseval wavelet frames, or more generally dual wavelet frames, was obtained in \\cite{chui2002characterization, bownik2000characterization, calogero2000characterization}, among others. In addition, the existence of wavelet bases for such dilations were shown in \\cite{dai1997wavelet}. Both types of results extend classical results from dimension one to arbitrary dimensions.\n\nBeyond the case of expansive dilation matrices, the theory of wavelet systems in higher dimensions is far less complete. Nevertheless, the existence of wavelet bases for nonexpansive dilation matrices has been studied in \\cite{speegle2003existence, bownik2017wavelets, ionascu2006simultaneous, wang2002wavelets} and culminated in the recent breakthrough \\cite{bownik2021simultaneous} that characterizes the dilation matrices admitting wavelet sets. On the other hand, the aforementioned characterization of Parseval wavelet frames is currently only known for special dilations, such as amplifying dilations \\cite{laugesen2002translational}, dilations expanding on a subspace \\cite{hernandez2002unified, guo2006some} or dilations satisfying the lattice counting estimate \\cite{bownik2017wavelets}. This has lead to the following conjecture \\cite[Conjecture 1]{bownik2017wavelets} and open problem \\cite[Problem 3.3]{bownik2020open}, which was already implicitly raised in \\cite[p. 177]{speegle2003existence}.\\footnote{The formulations in \\cite{bownik2017wavelets, bownik2020open} are under the implicit assumption that $|\\det(P)| = 1$.}\n\n\\begin{conjecture}[\\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \\label{conj:wavelet}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$ and $\\psi \\in L^2 (\\mathbb{R}^d)$. Suppose that \\[ \\{|\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}\\]\nis an orthonormal basis, or more generally a Parseval frame, for $L^2 (\\mathbb{R}^d)$. Then the Calder\\'on sum formula holds:\n\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}\n\\end{conjecture}\n\nThe Calder\\'on sum formula \\eqref{eq:calderon} is part of the aforementioned characterizion of Parseval wavelet frames known under additional assumptions on the dilation matrix. The upper bound for the Calder\\'on sum is known to hold for any Bessel sequence with bound $1$ (cf. \\cite[Proposition 4.1]{hernandez2002unified}).\n\nIn this paper, we present a new approach to \\Cref{conj:wavelet} which allows us to prove the Calder\\'on sum formula for arbitrary translation and dilation matrices under a mild condition on the wavelet function. Our approach is based on a relation between the frame properties of the discrete wavelet system \\eqref{eq:discrete_wavelet} and the semi-continuous wavelet system whose elements are given by\n\\begin{align} \\label{eq:continuous_wavelet}\n\\pi(x, A^j) \\psi := |\\det(A)|^{-j/2} \\psi(A^{-j} ( \\cdot - x)), \\quad x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}.\n\\end{align}\nThe action $\\pi$ forms a unitary group representation of the semi-direct product group $G = \\mathbb{R}^d \\rtimes \\langle A \\rangle$ of $\\mathbb{R}^d$ and the cyclic group $\\langle A \\rangle := \\{ A^j : j \\in \\mathbb{Z} \\}$ generated by $A \\in \\mathrm{GL}(d, \\mathbb{R})$. Observe that the wavelet system \\eqref{eq:discrete_wavelet} corresponds to $\\{\\pi(A^j P k, A^j) \\psi : j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d \\}$. We will assume a mild condition on the wavelet function. Namely, we assume that $\\psi \\in L^2 (\\mathbb{R}^d)$ is such that $\\pi(\\Lambda) \\psi$ is a Bessel sequence in $L^2 (\\mathbb{R}^d)$ for all relatively separated sets $\\Lambda $ in $ G = \\mathbb{R}^d \\rtimes \\langle A \\rangle$; in notation, $\\psi \\in \\mathcal{B}_{\\pi}$. This is a common assumption in the study of frames in the orbit of a group representation, see, e.g., \\cite{Gr08, FuGr07, enstad2025dynamical, fuehr2017density, caspers2023overcompleteness}.\nWe refer to Section \\ref{sec:coefficient} for an alternative description of the space $\\mathcal{B}_{\\pi}$ and further properties.\n\nUsing the notation from the previous paragraph, our main result is the following:\n\n\\begin{conjecture}[\\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \\label{conj:wavelet}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$ and $\\psi \\in L^2 (\\mathbb{R}^d)$. Suppose that \\[ \\{|\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}\\]\nis an orthonormal basis, or more generally a Parseval frame, for $L^2 (\\mathbb{R}^d)$. Then the Calder\\'on sum formula holds:\n\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}\n\\end{conjecture}\n\n\\begin{align} \\label{eq:discrete_wavelet}\n\\big\\{ |\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\big \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}.\n\\end{align}\n\n\\begin{conjecture}[\\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \\label{conj:wavelet}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$ and $\\psi \\in L^2 (\\mathbb{R}^d)$. Suppose that \\[ \\{|\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}\\]\nis an orthonormal basis, or more generally a Parseval frame, for $L^2 (\\mathbb{R}^d)$. Then the Calder\\'on sum formula holds:\n\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}\n\\end{conjecture}\n\nUsing the notation from the previous paragraph, our main result is the following:\n\nA combination of \\Cref{thm:calderon_intro} with the known fact that tight frames of the form $\\{ \\pi(x, A^j) \\psi \\}_{x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}}$ exist only when $|\\det(A)| \\neq 1$ (cf. \\cite{larson2006explicit, laugesen2002characterization}) yields the following consequence.\n\n\\begin{corollary}\n Let $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$.\nIf there exists a function $\\psi \\in \\mathcal{B}_{\\pi}$ such that\n$\n\\big \\{\\pi(A^j P k, A^j) \\psi \\big\\}_{ j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d } \n$\nis a Parseval frame for $L^2 (\\mathbb{R}^d)$, then $|\\det(A)| \\neq 1$.\n\\end{corollary}\n\n\\subsection{Wavelet transform} \\label{sec:coefficient}\nLet $(\\pi, \\Hpi)$ be a projective unitary representation of $G$ on a Hilbert space $\\Hpi$, i.e., a strongly measurable map $\\pi : G \\to \\mathcal{U}(\\Hpi)$ satisfying $\\pi(e_G) = I_{\\Hpi}$ and\n\\[ \\pi(g_1) \\pi( g_2) = \\sigma(g_1, g_2) \\pi(g_1 g_2), \\quad g_1, g_2 \\in G ,\\]\nfor some (measurable) function $\\sigma : G \\times G \\to \\mathbb{T}$.\nFor $\\psi, f \\in \\Hpi$, we define the function\n\\[ C_\\psi f(g) = \\langle f, \\pi(g) \\psi \\rangle, \\quad g \\in G. \\]\n By \\cite[Lemma 7.1, Theorem 7.5]{varadarajan1985geometry}, the absolute value $|C_{\\psi} f| : G \\to [0, \\infty)$ is continuous for arbitrary $f, \\psi \\in \\Hpi$.\nWe say that a vector $\\psi \\in \\Hpi$ is \\emph{admissible} if the associated \\emph{wavelet transform} $C_{\\psi} : \\Hpi \\to L^{\\infty} (G)$ given by $f \\mapsto C_{\\psi} f$ is an isometry into $L^2 (G)$, that is, the orbit $\\pi(G) \\psi$ is a continuous Parseval frame for $\\Hpi$. Throughout, we always consider $L^2 (G)$ with respect to the left Haar measure $\\mu_G$.\n\n\\begin{proof}\nLet $\\varepsilon > 0$, $f \\in \\Hpi$ and let $U \\subseteq G$ be an open, relatively compact set such that $\\Lambda$ is $U$-separated.\nLet $\\Gamma \\in \\Omega(\\Lambda)$ and let $(\\Lambda_n)_{n \\in \\mathbb{N}}$ be a sequence of left translates $\\Lambda_n \\subseteq G$ of $\\Lambda$ such that $\\Gamma = \\lim_{n \\to \\infty} \\Lambda_n$. Clearly, each set $\\Lambda_n$, $n \\in \\mathbb{N}$, is $U$-separated and, by \\Cref{lem:relsep-reldense-passes-to-hull}, also the set $\\Gamma$ is $U$-separated. \n\\\\~\\\\\n(i) Let $\\pi(\\Lambda)\\psi$ be a Bessel sequence with Bessel bound $C_2$. Then $\\pi(\\Lambda_n)\\psi$ is a Bessel sequence with Bessel bound $C_2$ for all $n \\in \\N$. Let $K \\subseteq G$ be a compact set. Enlarging $K$, we may assume that $\\Gamma \\cap \\partial K = \\emptyset$. Write $\\Gamma \\cap K= \\{ \\gamma^{(1)}, ..., \\gamma^{(k)} \\} $. Then, by \\Cref{lem:convergence_sets}, there exists $n_0 \\in \\mathbb{N}$ such that $\\Lambda_n \\cap K = \\{\\lambda_n^{(1)}, ..., \\lambda_n^{(k)} \\}$ for all $n \\geq n_0$ and $\\lambda_n^{(j)} \\to \\gamma^{(j)}$ for all $1 \\leq j \\leq k$ as $n \\to \\infty$. This, combined with the continuity of the function $|C_{\\psi} f|$ on $G$, yields that there exists $n_1 \\geq n_0$ such that\n\\begin{align} \\label{eq:passes-to-hull_2}\n\\bigg| \\sum_{\\gamma \\in \\Gamma \\cap K} |(C_{\\psi} f)(\\gamma) |^2 - \\sum_{\\mu \\in \\Lambda_n \\cap K} |(C_{\\psi} f)(\\mu) |^2 \\bigg| \\leq \\frac{\\varepsilon}{2}, \\qquad n \\geq n_1 .\n\\end{align}\nThus, since $\\varepsilon > 0$ is arbitrary, we conclude that\n\\[ \\sum_{\\gamma \\in \\Gamma \\cap K} | (C_\\psi f)(\\gamma)|^2 \\leq \\sum_{\\mu \\in \\Lambda_n \\cap K} |(C_\\psi f)(\\mu)|^2 \\leq C_2 \\| f \\|^2 . \\]\nSince $f \\in \\Hpi$ and this holds for arbitrarily large compact sets $K \\subseteq G$, we deduce that $\\pi(\\Gamma)\\psi$ is a Bessel sequence with Bessel bound $C_2$.\n\\\\~\\\\\n(ii) Suppose $\\pi(\\Lambda)\\psi$ is a frame and $\\psi \\in \\mathcal{B}_{\\pi}$. By \\Cref{lem:sampling} there exists a compact set $K \\subseteq G$ such that\n\\[\n\\sum_{\\gamma \\in \\Gamma \\cap K^c} |(C_{\\psi} f)(\\gamma) |^2 < \\frac{\\varepsilon}{4} \\quad \\text{and} \\quad \\sum_{\\mu \\in \\Lambda_n \\cap K^c} |(C_{\\psi} f)(\\mu) |^2 < \\frac{\\varepsilon}{4}\n\\]\nfor all $n \\in \\mathbb{N}$. Therefore,\n\\begin{align} \\label{eq:passes-to-hull_1}\n\\bigg| \\sum_{\\gamma \\in \\Gamma \\cap K^c} |(C_{\\psi} f)(\\gamma) |^2 - \\sum_{\\mu \\in \\Lambda_n \\cap K^c} |(C_{\\psi} f)(\\mu) |^2 \\bigg| \\leq \\frac{\\varepsilon}{2}.\n\\end{align}\nBy enlarging $K$ if necessary, we may assume that \\eqref{eq:passes-to-hull_1} holds for some compact $K \\subseteq G$ with $\\Gamma \\cap \\partial K = \\emptyset$, see, e.g., the proof of \\cite[Theorem 3.9]{enstad2025dynamical}.\n\n\\begin{theorem} \nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$.\nIf $\\psi \\in \\mathcal{B}_{\\pi}$ and\n$\n \\{\\pi(A^j P k, A^j) \\psi \\}_{ j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d } \n$\nis a frame for $L^2 (\\mathbb{R}^d)$ with frame bounds $0 0$, then\n\\[ \\| C_{\\psi} f \\|^2 \\leq C_2 \\mu_G (K) \\| f \\|^2 \\quad \\text{for all} \\quad f \\in \\Hpi. \\]\n In addition, if $\\psi \\in \\mathcal{B}_{\\pi}$ and $\\pi(\\Lambda) \\psi$ also admits a lower frame bound $C_1 > 0$, then\n\\[ \\| C_{\\psi} f \\|^2 \\geq C_1 \\mu_G (U) \\| f \\|^2 \\quad \\text{for all} \\quad f \\in \\Hpi. \\]\nIn particular, if $\\pi(\\Lambda) \\psi$ is frame with $\\psi \\in \\mathcal{B}_{\\pi}$ for some quasi-lattice $\\Lambda \\subseteq G$ with a relatively compact Jordan measurable complement $C$, then \n\\[\nC_1 \\| f \\|^2 \\leq \\frac{\\|C_{\\psi} f\\|^2}{\\rho_G (C)} \\leq C_2 \\| f \\|^2\n\\]\nfor all $f \\in \\Hpi$.\n\\end{corollary}\n\nThe “actual proof for such inequalities is based on the simple idea of periodizing the norm of the wavelet transform and applying the corresponding frame inequalities to each integrand.” Finally, the main theorem can be derived from the general results proved later, with the derivation carried out later as well.", + "expanded_theorem": "\\label{thm:calderon_intro}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$.\nIf $\\psi \\in \\mathcal{B}_{\\pi}$ and\n$\n \\{\\pi(A^j P k, A^j) \\psi \\}_{ j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d } \n$\nis a Parseval frame for $L^2 (\\mathbb{R}^d)$, then $\\{ \\pi(x, A^j) \\psi \\}_{x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}}$ is a tight frame for $L^2 (\\mathbb{R}^d)$, and \n\\[\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det (P)| \\quad \\text{for a.e.} \\;\\; \\xi \\in \\mathbb{R}^d.\n\\]", + "theorem_type": [ + "Implication", + "Inequality or Bound" + ], + "mcq": { + "question": "Let $A,P\\in \\mathrm{GL}(d,\\mathbb{R})$, let $\\widehat\\psi$ denote the Fourier transform of $\\psi$, and define\n\\[\n\\pi(x,A^j)\\psi:=|\\det(A)|^{-j/2}\\,\\psi\\big(A^{-j}(\\cdot-x)\\big),\\qquad x\\in\\mathbb{R}^d,\\ j\\in\\mathbb{Z}.\n\\]\nAssume $\\psi\\in \\mathcal B_\\pi$, where $\\mathcal B_\\pi$ is the class of functions in $L^2(\\mathbb{R}^d)$ such that $\\pi(\\Lambda)\\psi$ is a Bessel sequence in $L^2(\\mathbb{R}^d)$ for every relatively separated set $\\Lambda$ in $G=\\mathbb{R}^d\\rtimes\\langle A\\rangle$. Suppose further that the discrete wavelet system\n\\[\n\\{\\pi(A^jPk,A^j)\\psi\\}_{j\\in\\mathbb{Z},\\,k\\in\\mathbb{Z}^d}\n\\]\nis a Parseval frame for $L^2(\\mathbb{R}^d)$. Which of the following statements holds?", + "correct_choice": { + "label": "A", + "text": "The semi-continuous system $\\{\\pi(x,A^j)\\psi\\}_{x\\in\\mathbb{R}^d,\\,j\\in\\mathbb{Z}}$ is a tight frame for $L^2(\\mathbb{R}^d)$, and\n\\[\n\\sum_{j\\in\\mathbb{Z}}\\big|\\widehat\\psi\\big((A^t)^j\\xi\\big)\\big|^2=|\\det(P)|\n\\qquad \\text{for a.e. }\\xi\\in\\mathbb{R}^d.\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "The semi-continuous system $\\{\\pi(x,A^j)\\psi\\}_{x\\in\\mathbb{R}^d,\\,j\\in\\mathbb{Z}}$ is a tight frame for $L^2(\\mathbb{R}^d)$, and\n\\[\n\\sum_{j\\in\\mathbb{Z}}\\big|\\widehat\\psi\\big((A^t)^j\\xi\\big)\\big|^2=1\n\\qquad \\text{for a.e. }\\xi\\in\\mathbb{R}^d.\n\\]" + }, + { + "label": "C", + "text": "The semi-continuous system $\\{\\pi(x,A^j)\\psi\\}_{x\\in\\mathbb{R}^d,\\,j\\in\\mathbb{Z}}$ is a tight frame for $L^2(\\mathbb{R}^d)$." + }, + { + "label": "D", + "text": "The semi-continuous system $\\{\\pi(x,A^j)\\psi\\}_{x\\in\\mathbb{R}^d,\\,j\\in\\mathbb{Z}}$ is a Parseval frame for $L^2(\\mathbb{R}^d)$, and\n\\[\n\\sum_{j\\in\\mathbb{Z}}\\big|\\widehat\\psi\\big((A^t)^j\\xi\\big)\\big|^2=1\n\\qquad \\text{for a.e. }\\xi\\in\\mathbb{R}^d.\n\\]" + }, + { + "label": "E", + "text": "For every $\\xi\\in\\mathbb{R}^d$ one has\n\\[\n\\sum_{j\\in\\mathbb{Z}}\\big|\\widehat\\psi\\big((A^t)^j\\xi\\big)\\big|^2=|\\det(P)|,\n\\]\nand consequently the semi-continuous system $\\{\\pi(x,A^j)\\psi\\}_{x\\in\\mathbb{R}^d,\\,j\\in\\mathbb{Z}}$ is a tight frame for $L^2(\\mathbb{R}^d)$." + } + ], + "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": "covolume_factor_|det(P)|", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "trace_identity", + "tampered_component": "dropped_Calderon_identity", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "trace_identity", + "tampered_component": "tight_bound_equals_|det(P)|_not_1", + "template_used": "property_confusion" + }, + { + "label": "E", + "sketch_hook_type": "trace_identity", + "tampered_component": "a.e._quantifier_strengthened_to_every_xi", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly reveal the correct option. It gives hypotheses and asks for the valid conclusion; the key distinctions among the options (tight vs. Parseval, frame bound |det(P)| vs. 1, a.e. vs. every ξ) are not leaked in the wording." + }, + "TAS": { + "score": 1, + "justification": "The item is very close to a theorem-recall question: under the stated hypotheses, the correct option appears to restate the standard conclusion. However, it is not a pure verbatim restatement, since the choices force discrimination among nearby variants of the conclusion." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is required to identify the strongest correct statement and reject subtle near-misses, especially around the normalization constant and quantifier strength. Still, the question mainly tests recognition of a known theorem rather than substantial derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target realistic errors: confusing tight with Parseval, dropping the Calderón identity, using 1 instead of |det(P)|, and strengthening 'for a.e. ξ' to 'for every ξ'. These are distinct and well aligned with common failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and little answer leakage, but only moderate generative depth because it closely tracks a standard result." + } + }, + { + "id": "2602.10970v1", + "paper_link": "http://arxiv.org/abs/2602.10970v1", + "theorems_cnt": 4, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm:main-strong-cover}\n Let $\\varepsilon > 0$. There exists a constant $C = C(\\varepsilon)>0$ such that if $G$ is an $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$, then $ST(G) \\leq (1+\\varepsilon)n\\log n$.", + "start_pos": 12724, + "end_pos": 12947, + "label": "thm:main-strong-cover" + }, + "ref_dict": { + "prop:cover-time": "\\begin{proposition}\\label{prop:cover-time}\n For every $\\varepsilon >0$, there exists $C = C(\\eps)$ such that if $G$ be an $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$, then $T(G) \\leq (1+\\eps) n\\log n$.\n\\end{proposition}", + "thm:main-strong-cover": "\\begin{theorem}\\label{thm:main-strong-cover}\n Let $\\eps > 0$. There exists a constant $C = C(\\eps)>0$ such that if $G$ is an $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$, then $ST(G) \\leq (1+\\eps)n\\log n$.\n\\end{theorem}", + "thm:main": "\\begin{theorem}\\label{thm:main}\n Let $\\varepsilon > 0$. There exists a constant $C = C(\\eps) >0$ such that the following holds: Let $G$ be an $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$ and $L = (1+\\eps) n\\log n$. \\textbf{whp} for every vertex $v\\in V(G)$, the trace of a simple random walk starting at $v$ of length $L$ is Hamiltonian.\n\\end{theorem}", + "lem:number-of-vists": "\\begin{lemma}\\label{lem:number-of-vists}\n Let $\\varepsilon > 0$. There exists constants $C = C(\\eps) >0$ and $\\rho = \\rho(\\eps) > 0$ such that the following holds: Let $G$ be a $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$ and $L = (1+\\eps) n\\log n$. Consider a random walk on $G$ starting from an arbitrary vertex and of length $L$. Then, \\textbf{whp} $\\gamma(v)\\geq \\rho \\log n$ for every vertex $v\\in V(G)$.\n\\end{lemma}" + }, + "pre_theorem_intro_text_len": 5711, + "pre_theorem_intro_text": "Let $G$ be a simple undirected connected graph. A \\emph{simple random walk} on $G$ is a stochastic process where, starting from a given vertex, each step consists of moving to a neighbor chosen uniformly at random. The set of edges traversed by this walk is called the \\emph{trace}. Random walks are fundamental objects in combinatorics, probability theory, and theoretical computer science. For a comprehensive treatment, we refer the reader to the textbook by Levin and Peres~\\cite{Levin-Peres-book} or the survey by Lov\\'{a}sz~\\cite{Lovasz-survey}.\n\nIn this paper, we study the graph formed by the trace of a simple random walk on a finite graph. This line of research was initiated by Frieze, Krivelevich, Michaeli, and Peled in~\\cite{Frieze-et-al}, who focused on the case where the base graph is random. Motivated by these results, in~\\cite{Frieze-et-al}, they asked to extend the inquiry to deterministic graphs that exhibit random-like properties, known as pseudorandom graphs. A prominent class of pseudorandom graphs is that of spectral expanders, or $(n, d, \\lambda)$-graphs. An $(n, d, \\lambda)$-graph is defined as a $d$-regular graph on $n$ vertices where the second largest eigenvalue in absolute value is at most $\\lambda$. The expander mixing lemma shows that $\\lambda$ governs the edge distribution: a smaller $\\lambda$ implies that the edge distribution of $G$ closely resembles that of the random graph $G(n, d/n)$. For a detailed introduction, we refer the reader to the survey by Krivelevich and Sudakov~\\cite{Krivelevich-Sudakov-survey}.\n\nWe focus our attention on the case where $G$ is an $(n,d,\\lambda)$-graph. Let $\\Gamma$ denote the subgraph on the vertex set of $G$ induced by the trace of the random walk. A fundamental question concerns the structural properties of $\\Gamma$, particularly its connectivity. This inquiry is intrinsically linked to the \\emph{cover time}, $T(G)$, defined as the expected number of steps required to visit every vertex, maximized over all possible starting vertices. Equivalently, $T(G)$ corresponds to the expectation of the hitting time for the property that $\\Gamma$ becomes connected. The cover time is a classical parameter in the study of random walks. A seminal result by Feige~\\cite{Feige-lower} establishes a universal lower bound of $(1-o(1))n\\log n$ for any connected graph on $n$ vertices. For Erd\\H{o}s-R\\'{e}nyi graph, random regular graph and more generally random graph with a fixed degree sequence, asymptotically sharp bounds have been proven~\\cite{Adullah-Copper-Frieze-degree-sequence,Cooper-Frieze-sparse-random, Cooper-Frieze-random-regular,Jonasson, Cooper-Frieze-Lubetzky}. In particular, Frieze, Krivelevich, Michaeli, and Peled~\\cite{Frieze-et-al} demonstrated that for random graphs $G(n,p)$ and $\\varepsilon>0$, there exists a constant $C_{\\varepsilon}$ such that if $p \\ge C_{\\varepsilon}\\log n/n$, the trace $\\Gamma$ generated by a random walk of length $(1+\\varepsilon)n\\log n$ is not merely connected, but Hamiltonian with high probability.\n\nGiven that $(n,d,\\lambda)$-graphs exhibit quasi-random behavior analogous to the random graph $G(n, d/n)$, it is natural to inquire whether the structural properties of the random walk trace keep in this setting. Specifically, we ask whether a random walk of length $(1+\\varepsilon)n\\log n$ yields a Hamiltonian trace on an $(n,d,\\lambda)$-graph. This question was raised by Frieze, Krivelevich, Michaeli, and Peled~\\cite{Frieze-et-al}. \nRecall that the connectivity threshold for $G(n,p)$ is $(1+ o(1))\\log n/n$ where as for the trace to be Hamiltonian, one needs $C_\\varepsilon\\log /n$ where $C_\\varepsilon$ is a big constant depending on $\\varepsilon$. For $(n,d,\\lambda)$-graph, the connectivity holds when $d/\\lambda > 1$. Thus, analogously, one would expect the answer holds when $d/\\lambda > C_\\varepsilon$ for some big constant $C_\\varepsilon$.\n\n\\subsection{Our results}\nBy the expander mixing lemma, $(n,d,\\lambda)$-graphs represent a class of graphs with expansion properties. The asymptotic behavior of random walks in this setting was investigated in the foundational work of Broder and Karlin~\\cite{Broder-Kralin}, who established that for any $d$-regular expander $G$, the cover time satisfies $T(G) = \\Theta(n\\log n)$ \\footnote{The notion of expander in~\\cite{Broder-Kralin} is different, but it applies to the $(n,d,\\lambda)$ setting.}. Our first contribution is a refinement of this bound for $(n,d,\\lambda)$-graphs. We demonstrate that, provided the spectral ratio is sufficiently large, the cover time is asymptotically optimal: $T(G) = (1+o(1))n\\log n$.\n\\begin{proposition}\\label{prop:cover-time}\n For every $\\varepsilon >0$, there exists $C = C(\\varepsilon)$ such that if $G$ be an $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$, then $T(G) \\leq (1+\\varepsilon) n\\log n$.\n\\end{proposition}\n\nWe now introduce a strengthrening of the cover time, termed the \\emph{strong cover time}. Let $ST(G)$ denote the minimum integer $t$ such that a simple random walk of length $t$ visits every vertex of $G$ with high probability (abbrviated as \\textbf{whp} throughout the paper), uniformly over all starting vertices. Note that by definition, $ST(G)$ provides a high-probability guarantee, whereas $T(G)$ captures only the expectation. \nDeterministically $ST(G)\\geq T(G)$.\nIn the context of random graphs $G(n,p)$, Jonasson~\\cite{Jonasson} established that $ST(G)= (1+o(1))n\\log n$ provided that $p = \\omega(\\log^3n/n)$. This result was subsequently refined by Frieze, Krivelevich, Michaeli, and Peled~\\cite{Frieze-et-al}, who extended the bound to the asymptotically optimal $p = \\omega(\\log n/n)$. We consider the problem analogous to the $(n,d,\\lambda)$-graphs.", + "context": "Let $G$ be a simple undirected connected graph. A \\emph{simple random walk} on $G$ is a stochastic process where, starting from a given vertex, each step consists of moving to a neighbor chosen uniformly at random. The set of edges traversed by this walk is called the \\emph{trace}. Random walks are fundamental objects in combinatorics, probability theory, and theoretical computer science. For a comprehensive treatment, we refer the reader to the textbook by Levin and Peres~\\cite{Levin-Peres-book} or the survey by Lov\\'{a}sz~\\cite{Lovasz-survey}.\n\nIn this paper, we study the graph formed by the trace of a simple random walk on a finite graph. This line of research was initiated by Frieze, Krivelevich, Michaeli, and Peled in~\\cite{Frieze-et-al}, who focused on the case where the base graph is random. Motivated by these results, in~\\cite{Frieze-et-al}, they asked to extend the inquiry to deterministic graphs that exhibit random-like properties, known as pseudorandom graphs. A prominent class of pseudorandom graphs is that of spectral expanders, or $(n, d, \\lambda)$-graphs. An $(n, d, \\lambda)$-graph is defined as a $d$-regular graph on $n$ vertices where the second largest eigenvalue in absolute value is at most $\\lambda$. The expander mixing lemma shows that $\\lambda$ governs the edge distribution: a smaller $\\lambda$ implies that the edge distribution of $G$ closely resembles that of the random graph $G(n, d/n)$. For a detailed introduction, we refer the reader to the survey by Krivelevich and Sudakov~\\cite{Krivelevich-Sudakov-survey}.\n\nWe focus our attention on the case where $G$ is an $(n,d,\\lambda)$-graph. Let $\\Gamma$ denote the subgraph on the vertex set of $G$ induced by the trace of the random walk. A fundamental question concerns the structural properties of $\\Gamma$, particularly its connectivity. This inquiry is intrinsically linked to the \\emph{cover time}, $T(G)$, defined as the expected number of steps required to visit every vertex, maximized over all possible starting vertices. Equivalently, $T(G)$ corresponds to the expectation of the hitting time for the property that $\\Gamma$ becomes connected. The cover time is a classical parameter in the study of random walks. A seminal result by Feige~\\cite{Feige-lower} establishes a universal lower bound of $(1-o(1))n\\log n$ for any connected graph on $n$ vertices. For Erd\\H{o}s-R\\'{e}nyi graph, random regular graph and more generally random graph with a fixed degree sequence, asymptotically sharp bounds have been proven~\\cite{Adullah-Copper-Frieze-degree-sequence,Cooper-Frieze-sparse-random, Cooper-Frieze-random-regular,Jonasson, Cooper-Frieze-Lubetzky}. In particular, Frieze, Krivelevich, Michaeli, and Peled~\\cite{Frieze-et-al} demonstrated that for random graphs $G(n,p)$ and $\\varepsilon>0$, there exists a constant $C_{\\varepsilon}$ such that if $p \\ge C_{\\varepsilon}\\log n/n$, the trace $\\Gamma$ generated by a random walk of length $(1+\\varepsilon)n\\log n$ is not merely connected, but Hamiltonian with high probability.\n\nGiven that $(n,d,\\lambda)$-graphs exhibit quasi-random behavior analogous to the random graph $G(n, d/n)$, it is natural to inquire whether the structural properties of the random walk trace keep in this setting. Specifically, we ask whether a random walk of length $(1+\\varepsilon)n\\log n$ yields a Hamiltonian trace on an $(n,d,\\lambda)$-graph. This question was raised by Frieze, Krivelevich, Michaeli, and Peled~\\cite{Frieze-et-al}. \nRecall that the connectivity threshold for $G(n,p)$ is $(1+ o(1))\\log n/n$ where as for the trace to be Hamiltonian, one needs $C_\\varepsilon\\log /n$ where $C_\\varepsilon$ is a big constant depending on $\\varepsilon$. For $(n,d,\\lambda)$-graph, the connectivity holds when $d/\\lambda > 1$. Thus, analogously, one would expect the answer holds when $d/\\lambda > C_\\varepsilon$ for some big constant $C_\\varepsilon$.\n\n\\subsection{Our results}\nBy the expander mixing lemma, $(n,d,\\lambda)$-graphs represent a class of graphs with expansion properties. The asymptotic behavior of random walks in this setting was investigated in the foundational work of Broder and Karlin~\\cite{Broder-Kralin}, who established that for any $d$-regular expander $G$, the cover time satisfies $T(G) = \\Theta(n\\log n)$ \\footnote{The notion of expander in~\\cite{Broder-Kralin} is different, but it applies to the $(n,d,\\lambda)$ setting.}. Our first contribution is a refinement of this bound for $(n,d,\\lambda)$-graphs. We demonstrate that, provided the spectral ratio is sufficiently large, the cover time is asymptotically optimal: $T(G) = (1+o(1))n\\log n$.\n\\begin{proposition}\\label{prop:cover-time}\n For every $\\varepsilon >0$, there exists $C = C(\\varepsilon)$ such that if $G$ be an $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$, then $T(G) \\leq (1+\\varepsilon) n\\log n$.\n\\end{proposition}\n\nWe now introduce a strengthrening of the cover time, termed the \\emph{strong cover time}. Let $ST(G)$ denote the minimum integer $t$ such that a simple random walk of length $t$ visits every vertex of $G$ with high probability (abbrviated as \\textbf{whp} throughout the paper), uniformly over all starting vertices. Note that by definition, $ST(G)$ provides a high-probability guarantee, whereas $T(G)$ captures only the expectation. \nDeterministically $ST(G)\\geq T(G)$.\nIn the context of random graphs $G(n,p)$, Jonasson~\\cite{Jonasson} established that $ST(G)= (1+o(1))n\\log n$ provided that $p = \\omega(\\log^3n/n)$. This result was subsequently refined by Frieze, Krivelevich, Michaeli, and Peled~\\cite{Frieze-et-al}, who extended the bound to the asymptotically optimal $p = \\omega(\\log n/n)$. We consider the problem analogous to the $(n,d,\\lambda)$-graphs.", + "full_context": "Let $G$ be a simple undirected connected graph. A \\emph{simple random walk} on $G$ is a stochastic process where, starting from a given vertex, each step consists of moving to a neighbor chosen uniformly at random. The set of edges traversed by this walk is called the \\emph{trace}. Random walks are fundamental objects in combinatorics, probability theory, and theoretical computer science. For a comprehensive treatment, we refer the reader to the textbook by Levin and Peres~\\cite{Levin-Peres-book} or the survey by Lov\\'{a}sz~\\cite{Lovasz-survey}.\n\nIn this paper, we study the graph formed by the trace of a simple random walk on a finite graph. This line of research was initiated by Frieze, Krivelevich, Michaeli, and Peled in~\\cite{Frieze-et-al}, who focused on the case where the base graph is random. Motivated by these results, in~\\cite{Frieze-et-al}, they asked to extend the inquiry to deterministic graphs that exhibit random-like properties, known as pseudorandom graphs. A prominent class of pseudorandom graphs is that of spectral expanders, or $(n, d, \\lambda)$-graphs. An $(n, d, \\lambda)$-graph is defined as a $d$-regular graph on $n$ vertices where the second largest eigenvalue in absolute value is at most $\\lambda$. The expander mixing lemma shows that $\\lambda$ governs the edge distribution: a smaller $\\lambda$ implies that the edge distribution of $G$ closely resembles that of the random graph $G(n, d/n)$. For a detailed introduction, we refer the reader to the survey by Krivelevich and Sudakov~\\cite{Krivelevich-Sudakov-survey}.\n\nWe focus our attention on the case where $G$ is an $(n,d,\\lambda)$-graph. Let $\\Gamma$ denote the subgraph on the vertex set of $G$ induced by the trace of the random walk. A fundamental question concerns the structural properties of $\\Gamma$, particularly its connectivity. This inquiry is intrinsically linked to the \\emph{cover time}, $T(G)$, defined as the expected number of steps required to visit every vertex, maximized over all possible starting vertices. Equivalently, $T(G)$ corresponds to the expectation of the hitting time for the property that $\\Gamma$ becomes connected. The cover time is a classical parameter in the study of random walks. A seminal result by Feige~\\cite{Feige-lower} establishes a universal lower bound of $(1-o(1))n\\log n$ for any connected graph on $n$ vertices. For Erd\\H{o}s-R\\'{e}nyi graph, random regular graph and more generally random graph with a fixed degree sequence, asymptotically sharp bounds have been proven~\\cite{Adullah-Copper-Frieze-degree-sequence,Cooper-Frieze-sparse-random, Cooper-Frieze-random-regular,Jonasson, Cooper-Frieze-Lubetzky}. In particular, Frieze, Krivelevich, Michaeli, and Peled~\\cite{Frieze-et-al} demonstrated that for random graphs $G(n,p)$ and $\\varepsilon>0$, there exists a constant $C_{\\varepsilon}$ such that if $p \\ge C_{\\varepsilon}\\log n/n$, the trace $\\Gamma$ generated by a random walk of length $(1+\\varepsilon)n\\log n$ is not merely connected, but Hamiltonian with high probability.\n\nGiven that $(n,d,\\lambda)$-graphs exhibit quasi-random behavior analogous to the random graph $G(n, d/n)$, it is natural to inquire whether the structural properties of the random walk trace keep in this setting. Specifically, we ask whether a random walk of length $(1+\\varepsilon)n\\log n$ yields a Hamiltonian trace on an $(n,d,\\lambda)$-graph. This question was raised by Frieze, Krivelevich, Michaeli, and Peled~\\cite{Frieze-et-al}. \nRecall that the connectivity threshold for $G(n,p)$ is $(1+ o(1))\\log n/n$ where as for the trace to be Hamiltonian, one needs $C_\\varepsilon\\log /n$ where $C_\\varepsilon$ is a big constant depending on $\\varepsilon$. For $(n,d,\\lambda)$-graph, the connectivity holds when $d/\\lambda > 1$. Thus, analogously, one would expect the answer holds when $d/\\lambda > C_\\varepsilon$ for some big constant $C_\\varepsilon$.\n\n\\subsection{Our results}\nBy the expander mixing lemma, $(n,d,\\lambda)$-graphs represent a class of graphs with expansion properties. The asymptotic behavior of random walks in this setting was investigated in the foundational work of Broder and Karlin~\\cite{Broder-Kralin}, who established that for any $d$-regular expander $G$, the cover time satisfies $T(G) = \\Theta(n\\log n)$ \\footnote{The notion of expander in~\\cite{Broder-Kralin} is different, but it applies to the $(n,d,\\lambda)$ setting.}. Our first contribution is a refinement of this bound for $(n,d,\\lambda)$-graphs. We demonstrate that, provided the spectral ratio is sufficiently large, the cover time is asymptotically optimal: $T(G) = (1+o(1))n\\log n$.\n\\begin{proposition}\\label{prop:cover-time}\n For every $\\varepsilon >0$, there exists $C = C(\\varepsilon)$ such that if $G$ be an $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$, then $T(G) \\leq (1+\\varepsilon) n\\log n$.\n\\end{proposition}\n\nWe now introduce a strengthrening of the cover time, termed the \\emph{strong cover time}. Let $ST(G)$ denote the minimum integer $t$ such that a simple random walk of length $t$ visits every vertex of $G$ with high probability (abbrviated as \\textbf{whp} throughout the paper), uniformly over all starting vertices. Note that by definition, $ST(G)$ provides a high-probability guarantee, whereas $T(G)$ captures only the expectation. \nDeterministically $ST(G)\\geq T(G)$.\nIn the context of random graphs $G(n,p)$, Jonasson~\\cite{Jonasson} established that $ST(G)= (1+o(1))n\\log n$ provided that $p = \\omega(\\log^3n/n)$. This result was subsequently refined by Frieze, Krivelevich, Michaeli, and Peled~\\cite{Frieze-et-al}, who extended the bound to the asymptotically optimal $p = \\omega(\\log n/n)$. We consider the problem analogous to the $(n,d,\\lambda)$-graphs.\n\n\\begin{theorem}\\label{thm:main}\n Let $\\varepsilon > 0$. There exists a constant $C = C(\\eps) >0$ such that the following holds: Let $G$ be an $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$ and $L = (1+\\eps) n\\log n$. \\textbf{whp} for every vertex $v\\in V(G)$, the trace of a simple random walk starting at $v$ of length $L$ is Hamiltonian.\n\\end{theorem}\n\nIt is well known that a random $d$-regular graph for any $d\\geq 3$ is \\textbf{whp} an $(n,d,\\lambda)$-graph for $\\lambda = O(\\sqrt{d})$~\\cite{Friedman}. We thus obtain the following corollary.\n\\begin{corollary}\\label{cor:random-d-regular}\n Let $\\varepsilon > 0$. There exists $d_0 = d_0(\\eps)$ that the following holds: Let $G$ be a random $d$-regular graph with $d\\geq d_0$ and let $L = (1+\\eps) n\\log n$. \\textbf{whp} $G$ satisfies that \\textbf{whp} for every vertex $v\\in V(G)$, the trace of a simple random walk starting at $v$ of length $L$ is Hamiltonian.\n\\end{corollary}\nWe remark that both results are asymptotically optimal due to the general lower bound on the cover time. Our proof strategy is as follows.\nFirst, since the second largest eigenvalue of the adjacency matrix of an $(n,d,\\lambda)$-graph is bounded away from $d$, we can control the transitional probability going from any vertex $u$ to another vertex $v$ at step $t$ by using the spectral decomposition of the transition matrix.\nWe show that after $(1+\\varepsilon)n\\log n$ steps, each vertex is visited $\\Theta(\\log n)$ times with high probability. Then, we use these visit counts to establish the expansion properties of the trace graph $\\Gamma$, which ensures the existence of a Hamiltonian cycle.\n\n\\subsection{Number of visits}\nFor a random walk on a graph $G$ of length $L$ and a vertex $v\\in V(G)$, let $\\gamma(v)$ denote the number of times the random walk visits $v$.\nThe goal of this section is to prove the following:\n\\begin{lemma}\\label{lem:number-of-vists}\n Let $\\varepsilon > 0$. There exists constants $C = C(\\eps) >0$ and $\\rho = \\rho(\\eps) > 0$ such that the following holds: Let $G$ be a $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$ and $L = (1+\\eps) n\\log n$. Consider a random walk on $G$ starting from an arbitrary vertex and of length $L$. Then, \\textbf{whp} $\\gamma(v)\\geq \\rho \\log n$ for every vertex $v\\in V(G)$.\n\\end{lemma}\nWe begin by establishing an auxiliary lemma, giving a lower bound on the probability a random walk starting from $u$ reaches $v$ in $n/\\sqrt{C}$ steps:\n\\begin{lemma}\\label{lem:lower-bound-return}\n Let $\\varepsilon > 0$. There exists a constant $C = C(\\eps) >0$ such that the following holds: Let $G$ be a $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$ and $T = n/\\sqrt{C}$. Then, for any pair of distinct vertices $u,v\\in V(G)$, the probability a simple random walk $X$ starting from $u$ reaches $v$ at least once within $T$ steps is at least $(1-0.1\\eps)/\\sqrt{C}$.\n\\end{lemma}\n\\begin{proof}\n Let $N$ denote the number of visits to $v$ in the first $T$ steps and $I_t$ the indicator random variable for $X_t = v$ so that $N = \\sum_{t = 1}^T I_i$. By linearity of expectation, we have \n \\[\n \\E[N] = \\sum_{t=1}^T \\Pr(X_t = v|X_0 = u) = \\sum_{t=1}^T (P^t)_{u,v}\\,.\n \\]\n Recall that the transition matrix $P = D^{-1}A$ and let $d = \\lambda_1\\geq \\dots\\geq \\lambda_n$ are eigenvalues of $A$. Define $S = D^{1/2}PD^{-1/2} = D^{-1/2}AD^{-1/2}$. Since $S$ is a symmetric real square matrix, its spectral decomposition can be written as $S = \\sum_{i=1}^n\\mu_i\\phi_i \\phi_i^T$, where $\\mu_i = \\lambda_i/d\\in [-1,1]$ are eigenvalues of $S$ and $\\{\\phi_i\\}_{i=1}^n$ forms an orthonormal basis. \n Observe that the stationary distribution of $X$ is $\\pi = (1/n,\\dots, 1/n)$, $\\mu_1 = 1$ and $\\phi_1(v) = (1/\\sqrt{n},\\dots, 1/\\sqrt{n})$.\n\nWe have the following immediate corollary for the stationary walk:\n\\begin{corollary}\\label{cor:stationary-walk}\n Let $\\varepsilon > 0$. There exists a constant $C = C(\\eps) >0$ such that the following holds: Let $G$ be a $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$ and $T = n/\\sqrt{C}$. Then, for any vertex $v\\in V(G)$, the probability a stationary walk visits $v$ at least once within $T$ steps is at least $(1-0.1\\eps)/\\sqrt{C}$.\n\\end{corollary}\n\\begin{proof}\n Recall that the stationary distribution for a $d$-regular graph is the uniform distribution over all vertices $(1/n,\\dots, 1/n)$. Also, the stationary walk $Y$ and the simple random walk $X$ have the same distribution conditionally on having the same starting vertex. \n Since~\\Cref{lem:lower-bound-return} holds for every starting vertex $u\\in V(G)$, the conclusion follows by integrating over the starting vertex. \n\\end{proof}\n\n\\subsection{C-expander}\nThe goal of this section is to prove the following:\n\\begin{lemma}\\label{lem:C-expander}\n Let $\\varepsilon > 0$. There exists a constant $C = C(\\eps) >0$ such that the following holds: Let $G$ be a $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$ and $L = (1+\\eps) n\\log n$. \n Let $\\Gamma$ be the trace graph obtained by a random walk of length $L$ starting from any vertex. Then,\n \\textbf{whp} $\\Gamma$ is a $C'$-expander for $C' = \\log \\log C$.\n\\end{lemma}\n\\begin{proof}\n To prove a graph is $C$-expander, we need to verify the expansion and joinedness properties. By~\\Cref{lem:number-of-vists}, there exists a constant $\\rho > 0$ such that $\\gamma(v) \\geq \\rho \\log n$ for all $v\\in V(G)$.\n\n\\section{Concluding remarks}\\label{sec:concluding-remarks}\nOne of the key tools we applied was~\\Cref{thm:Hamiltonicity}, which requires only $C$-expanders. One may wonder if~\\Cref{thm:main} is true with $(n,d,\\lambda)$-graphs replaced by $C$-expanders. This is unfortunately (very) incorrect:\n\\begin{proposition}\n For any $C \\leq 1.1n/\\log n$, there exists a $C$-expander for which the cover time is $a n \\log n$ for some $a > 1$.\n\\end{proposition}\n\\begin{proof}[Proof sketch]\n Consider a complete graph on $n-2$ vertices together with two special vertices $u,v$, each connecting to $C$ vertices of the complete graph and their neighborhoods are disjoint. Clearly, the graph is a $C$-expander. However, note that the expected number of steps needed to go from $u$ to $v$ is at least $1 + (n-1)/C)\\cdot (n-1)> n\\log n$. Indeed, the $1$ comes from going from $u$ to the clique. Then, in each step, it will have at most a $C/(n-1)$ chance of landing in $N(v)$ and thus in expectation $(n-1)/C$ steps are needed to land in $N(v)$. Starting from a vertex in $N(v)$, each time there is a $1/(n-1)$ chance of going to $v$ so in expectation it takes $n-1$ such trials. Clearly, the expected number of steps taken to go from $u$ to $v$ is a lower bound on the cover time, which finishes the proof.\n\\end{proof}\nOne natural strengthening of~\\Cref{thm:main} would be to consider following hitting time problem: Let $\\tau_{HC}(G)$ be the minimum $t\\in \\mathbb N$ for which the trace graph $\\Gamma$ becomes Hamiltonian and $\\tau_{1}$ the minimum $t\\in \\mathbb N$ such that $\\Gamma$ has minimum degree $1$.\nFor $(n,d,\\lambda)$-graphs with $d/\\lambda \\geq C$ or a random $d$-regular graph for $d$ sufficiently large, is it true that \\textbf{whp} $\\tau_{HC} = \\tau_1 + 1$? Note that $\\tau_{HC}\\geq \\tau_1 + 1$ is necessary since a Hamilton cycle uses two incident edges from each vertex.\nFor complete graph the corresponding hitting time result is known~\\cite{Frieze-et-al} .", + "post_theorem_intro_text_len": 5118, + "post_theorem_intro_text": "Again this is asymptotically optimal by the general lower bound of the cover time. In fact, we obtain~\\Cref{thm:main-strong-cover} as a direct consequence of a stronger result. \n\nThe connection between spectral expansion and the Hamiltonicity in pseudorandom graphs is well-established as in~\\cite{Kriveham,glockham,ferber2024hamiltonicity,chen2025robustness}. A recent breakthrough by Draganić, Montgomery, Munhá Correia, Pokrovskiy and Sudakov~\\cite{Draganic-et-al} demonstrates that any $(n,d,\\lambda)$-graph admits a Hamiltonian cycle, if the spectral ratio $d/\\lambda\\ge C$ for a sufficiently large constant $C$. Drawing a parallel to the trace of random walk landscape, Frieze, Krivelevich, Michaeli, and Peled~\\cite{Frieze-et-al} raised the question of whether the trace of a random walk on $(n,d,\\lambda)$-graphs and random regular graphs exhibits similar Hamiltonicity properties. \nWe answer these questions through the following two main results.\n\n\\begin{theorem}\\label{thm:main}\n Let $\\varepsilon > 0$. There exists a constant $C = C(\\varepsilon) >0$ such that the following holds: Let $G$ be an $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$ and $L = (1+\\varepsilon) n\\log n$. \\textbf{whp} for every vertex $v\\in V(G)$, the trace of a simple random walk starting at $v$ of length $L$ is Hamiltonian.\n\\end{theorem} \n\nWe remark that implicitly in~\\Cref{prop:cover-time},~\\Cref{thm:main-strong-cover} and~\\Cref{thm:main}, $d$ is sufficiently large. This follows from the fact that $\\lambda = \\Omega(d)$ (see~\\cite{Krivelevich-Sudakov-survey}), which means $1/C \\geq \\lambda/d\\geq \\Omega(1/\\sqrt{d})$ and therefore $d$ is sufficiently large as long as $C$ is.\n\nIt is well known that a random $d$-regular graph for any $d\\geq 3$ is \\textbf{whp} an $(n,d,\\lambda)$-graph for $\\lambda = O(\\sqrt{d})$~\\cite{Friedman}. We thus obtain the following corollary.\n\\begin{corollary}\\label{cor:random-d-regular}\n Let $\\varepsilon > 0$. There exists $d_0 = d_0(\\varepsilon)$ that the following holds: Let $G$ be a random $d$-regular graph with $d\\geq d_0$ and let $L = (1+\\varepsilon) n\\log n$. \\textbf{whp} $G$ satisfies that \\textbf{whp} for every vertex $v\\in V(G)$, the trace of a simple random walk starting at $v$ of length $L$ is Hamiltonian.\n\\end{corollary}\nWe remark that both results are asymptotically optimal due to the general lower bound on the cover time. Our proof strategy is as follows.\nFirst, since the second largest eigenvalue of the adjacency matrix of an $(n,d,\\lambda)$-graph is bounded away from $d$, we can control the transitional probability going from any vertex $u$ to another vertex $v$ at step $t$ by using the spectral decomposition of the transition matrix.\nWe show that after $(1+\\varepsilon)n\\log n$ steps, each vertex is visited $\\Theta(\\log n)$ times with high probability. Then, we use these visit counts to establish the expansion properties of the trace graph $\\Gamma$, which ensures the existence of a Hamiltonian cycle. \n\nCentral to our proof of \\Cref{thm:main} is the lemma that the trace of the random walk visits every vertex with logarithmic frequency. Specifically, \\Cref{lem:number-of-vists} establishes that there exists a constant $\\rho = \\rho(\\varepsilon) > 0$ such that with high probability, every vertex is visited at least $\\rho \\log n$ times. This phenomenon is intimately connected to the concept of \\emph{blanket time} in the probability theory, a parameter introduced by Winkler and Zuckerman~\\cite{Winkler-Zuckerman}. Let $\\pi$ denote the stationary distribution, where $\\pi_v = \\deg(v)/(2|E(G)|)$. For a parameter $\\delta \\in (0,1)$, the $\\delta$-blanket time, denoted $t_{bl}(G,\\delta)$, is defined as the minimum time $t$ required for the walk to visit every vertex $v$ at least $\\delta t \\pi_v$ times. While it is immediate that $t_{bl}(G,\\delta) \\ge T(G)$, Winkler and Zuckerman conjectured that the blanket time is, in fact, within a constant factor of the cover time; that is, $t_{bl}(G,\\delta) \\le C_\\delta T(G)$ for any $0<\\delta < 1$ and any graph $G$. \nKahn, Kim, Lov{\\'a}sz and Vu~\\cite{Kahn-Kim-Lovasz-Vu} showed that $t_{bl}(G,\\delta)= O(T(G)(\\log\\log n)^2)$ before the conjecture was affirmatively settled in a celebrated work of Ding, Lee, and Peres~\\cite{Ding-Lee-Peres}. \nOur results imply that there exists a $\\delta = \\delta(\\varepsilon)\\in (0,1)$ for which $t_{bl}(G,\\delta)\\leq (1+\\varepsilon)n\\log n$, which is the same as the cover time up to a $1+o(1)$ factor. In general, it is unclear for which graphs the same conclusion hold. In the concluding remarks we discuss and propose further questions along this direction.\n\n\\subsection{Paper organization} \n\\Cref{sec:preliminaries} contains the background of random walks and tools we use.~\\Cref{sec:proof} contains the proofs for all results and~\\Cref{sec:concluding-remarks} some remarks and open problems.\n\n\\subsection*{Acknowledgement}\nPart of this work was initiated during the visit of second author to Shanghai Center for Mathematical Sciences (SCMS) and Fudan University. The authors would like to thank Prof.\\ Hehui Wu and his group memebers for their support and hospitality.", + "sketch": "They state that \\Cref{thm:main-strong-cover} is obtained “as a direct consequence of a stronger result,” namely \\Cref{thm:main}. The proof strategy outlined is:\n\\begin{itemize}\n\\item Use that “the second largest eigenvalue of the adjacency matrix of an $(n,d,\\lambda)$-graph is bounded away from $d$” to “control the transitional probability going from any vertex $u$ to another vertex $v$ at step $t$ by using the spectral decomposition of the transition matrix.”\n\\item Show that “after $(1+\\varepsilon)n\\log n$ steps, each vertex is visited $\\Theta(\\log n)$ times with high probability.” (Central is the lemma that “with high probability, every vertex is visited at least $\\rho\\log n$ times,” i.e. \\Cref{lem:number-of-vists}.)\n\\item “Use these visit counts to establish the expansion properties of the trace graph $\\Gamma$, which ensures the existence of a Hamiltonian cycle.” Since \\Cref{thm:main-strong-cover} follows directly from this stronger Hamiltonicity statement, it follows as a corollary.\n\\end{itemize}\nThey further connect the logarithmic visit-frequency lemma to “blanket time,” noting their results imply the existence of $\\delta=\\delta(\\varepsilon)\\in(0,1)$ with $t_{bl}(G,\\delta)\\le (1+\\varepsilon)n\\log n$.", + "expanded_sketch": "They state that \\Cref{thm:main-strong-cover} is obtained “as a direct consequence of a stronger result,” namely the stronger Hamiltonicity statement.\n\nThe proof strategy outlined is:\n\\begin{itemize}\n\\item Use that “the second largest eigenvalue of the adjacency matrix of an $(n,d,\\lambda)$-graph is bounded away from $d$” to “control the transitional probability going from any vertex $u$ to another vertex $v$ at step $t$ by using the spectral decomposition of the transition matrix.”\n\\item Show that “after $(1+\\varepsilon)n\\log n$ steps, each vertex is visited $\\Theta(\\log n)$ times with high probability.” Central is the following lemma.\n\n\\begin{lemma}\\label{lem:number-of-vists}\n Let $\\varepsilon > 0$. There exists constants $C = C(\\eps) >0$ and $\\rho = \\rho(\\eps) > 0$ such that the following holds: Let $G$ be a $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$ and $L = (1+\\eps) n\\log n$. Consider a random walk on $G$ starting from an arbitrary vertex and of length $L$. Then, \\textbf{whp} $\\gamma(v)\\geq \\rho \\log n$ for every vertex $v\\in V(G)$.\n\\end{lemma}\n\n\\item “Use these visit counts to establish the expansion properties of the trace graph $\\Gamma$, which ensures the existence of a Hamiltonian cycle.” In establishing the main theorem, this yields Hamiltonicity of the trace, and hence the stated bound on $ST(G)$ follows as a corollary.\n\\end{itemize}\nThey further connect the logarithmic visit-frequency lemma to “blanket time,” noting their results imply the existence of $\\delta=\\delta(\\varepsilon)\\in(0,1)$ with $t_{bl}(G,\\delta)\\le (1+\\varepsilon)n\\log n$.", + "expanded_theorem": "\\label{thm:main-strong-cover}\n Let $\\varepsilon > 0$. There exists a constant $C = C(\\varepsilon)>0$ such that if $G$ is an $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$, then $ST(G) \\leq (1+\\varepsilon)n\\log n$.", + "theorem_type": [ + "Existential–Universal", + "Implication" + ], + "mcq": { + "question": "Let \\(\\varepsilon>0\\). An \\((n,d,\\lambda)\\)-graph is a \\(d\\)-regular graph on \\(n\\) vertices whose second-largest eigenvalue in absolute value is at most \\(\\lambda\\). Let \\(ST(G)\\) denote the strong cover time of \\(G\\), i.e. the least integer \\(t\\) such that a simple random walk of length \\(t\\), started from any vertex of \\(G\\), covers all vertices with high probability. Which conclusion is valid under the assumption that \\(G\\) is an \\((n,d,\\lambda)\\)-graph with \\(d/\\lambda\\ge C\\) for a suitable constant \\(C=C(\\varepsilon)>0\\)?", + "correct_choice": { + "label": "A", + "text": "For every \\(\\varepsilon>0\\), there exists a constant \\(C=C(\\varepsilon)>0\\) such that if \\(G\\) is an \\((n,d,\\lambda)\\)-graph with \\(d/\\lambda\\ge C\\), then \\(ST(G)\\le (1+\\varepsilon)n\\log n\\)." + }, + "choices": [ + { + "label": "B", + "text": "For every \\(\\varepsilon>0\\), there exists a constant \\(C=C(\\varepsilon)>0\\) such that if \\(G\\) is an \\((n,d,\\lambda)\\)-graph with \\(d/\\lambda\\ge C\\), then for every starting vertex, a simple random walk of length exactly \\((1+\\varepsilon)n\\log n\\) visits each vertex at least once with high probability." + }, + { + "label": "C", + "text": "For every \\(\\varepsilon>0\\), there exists a constant \\(C=C(\\varepsilon)>0\\) such that if \\(G\\) is an \\((n,d,\\lambda)\\)-graph with \\(d/\\lambda\\ge C\\), then the ordinary cover time of \\(G\\) is at most \\((1+\\varepsilon)n\\log n\\)." + }, + { + "label": "D", + "text": "There exists a universal constant \\(C>0\\) such that for every \\(\\varepsilon>0\\), if \\(G\\) is an \\((n,d,\\lambda)\\)-graph with \\(d/\\lambda\\ge C\\), then \\(ST(G)\\le (1+\\varepsilon)n\\log n\\)." + }, + { + "label": "E", + "text": "For every \\(\\varepsilon>0\\), there exists a constant \\(C=C(\\varepsilon)>0\\) such that if \\(G\\) is an \\((n,d,\\lambda)\\)-graph with \\(d/\\lambda\\ge C\\), then after \\((1+\\varepsilon)n\\log n\\) steps, a simple random walk started from any vertex visits every vertex at least \\(\\rho\\log n\\) times with high probability for some constant \\(\\rho=\\rho(\\varepsilon)>0\\)." + } + ], + "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": "least_time_vs_exact_time", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "strong_cover_replaced_by_cover_time", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "characteristic", + "tampered_component": "dependence_of_C_on_epsilon", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "counting_estimate", + "tampered_component": "lower_visit_bound_misread_as_each_vertex_visited_rho_log_n_times", + "template_used": "stronger_trap" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines the notation and asks which statement is true, but it does not reveal the theorem’s conclusion or single out choice A. There is no explicit answer leakage beyond necessary terminology." + }, + "TAS": { + "score": 1, + "justification": "The correct option is essentially the theorem statement itself, so the item is close to theorem recall. However, the alternatives vary quantifiers, exact-time vs least-time formulations, and strong vs ordinary cover time, so it is not a pure verbatim restatement." + }, + "GPS": { + "score": 1, + "justification": "Selecting the answer requires some reasoning about subtle logical differences between closely related statements, especially quantifier dependence and stronger/weaker variants. Still, it mainly tests recognition of the exact theorem rather than generating a conclusion from mathematical work." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically meaningful: they probe common confusions about stronger statements, weakened statements, dependence on epsilon, and fixed-time vs minimal-time formulations. They are distinct and well-targeted." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with no real answer leakage and strong distractors, but it leans toward precise theorem recognition rather than deeper generative reasoning." + } + }, + { + "id": "2602.11906v2", + "paper_link": "http://arxiv.org/abs/2602.11906v2", + "theorems_cnt": 1, + "theorem": { + "env_name": "maintheorem", + "content": "\\label[maintheorem]{thm:rt22-polynomially-simulated}\n$\\RCA_0 + \\BSig_2$ polynomially simulates $\\WKL_0 + \\RT_2^2$ with respect to $\\forall \\Sigma_3^0$ sentences.", + "start_pos": 6358, + "end_pos": 6556, + "label": null + }, + "ref_dict": {}, + "pre_theorem_intro_text_len": 4147, + "pre_theorem_intro_text": "If a computably axiomatized theory~$T_0$ is a conservative extension of another such theory~$T_1$ for a syntactical class of sentences $\\Gamma$, there is an algorithmic procedure which translates any $T_0$-proof~$p$ of a $\\Gamma$-sentence into its shortest $T_1$-proof~$\\hat p$. It is natural to wonder whether the translation $p \\mapsto \\hat p$ yields significantly longer proofs. If there exists a $T_0$-proof~$p$ of a $\\Gamma$-sentence such that the length $|\\hat p|$ of its translated $T_1$-proof is super-polynomial with respect to the length~$|p|$, we say that $T_0$ admits \\emph{non-trivial speedup} over $T_1$ for $\\Gamma$-formulas. In this case, the theory~$T_0$ is arguably useful for $\\Gamma$-sentences, in that it sometimes produces significantly shorter proofs than~$T_1$ for such sentences. If on the other hand, there exists a polynomial~$Q$ such that $|\\hat p| \\leq Q(|p|)$ for every $T_0$-proof~$p$ of any $\\Gamma$-sentence, then we consider that $T_0$ admits no significant speedup over~$T_1$ for $\\Gamma$-formulas, since the various proof systems are mutually polynomially simulated.\n\nThe study of proof size and proof speedups traces back to G\\\"odel~\\cite{davis1990kurt} (see Pudl\\'ak~\\cite{pudlak1998length} for an excellent survey). However, it was only in the 1990's that Avigad~\\cite{avigad1996formalizing} showed that model-theoretic conservation theorems based on forcing could be formalized into proof-theoretic arguments and yield proof size analysis. He proved in particular that $\\WKL_0$ is $\\Pi^1_1$-conservative over $\\RCA_0$ with no significant increase in the length of proofs. The technique was later used to formalize $\\Pi^1_1$-conservation theorems over~$\\RCA_0+\\ISig_n$~\\cite{Ikari-PhD,Ik-Yo} and $\\forall \\Sigma^0_2$-conservation theorems over~$\\RCA_0$~\\cite{kolodziejczyk2023ramsey} and over $\\RCA_0^*$~\\cite{katarzyna2025speedup}, where a $\\forall \\Sigma^0_n$-formula is of the form $\\forall X \\varphi(X)$ where $\\varphi$ is $\\Sigma^0_n$.\n\nWe are particularly interested in the first-order consequences of Ramsey's theorem for pairs and two colors ($\\RT^2_2$). Patey and Yokoyama~\\cite{patey2018proof} proved that $\\WKL_0 + \\RT^2_2$ is $\\forall \\Sigma^0_2$-conservative over~$\\RCA_0$ using the notion of $\\alpha$-largeness from Ketonen and Solovay~\\cite{ketonen1981rapidly}. The proof was later simplified by Ko{\\l}odziejczyk and Yokoyama~\\cite{kolo2020some} to give explicit polynomial bounds for this notion of largeness. Ko{\\l}odziejczyk, Wong and Yokoyama~\\cite{kolodziejczyk2023ramsey} then formalized the construction to prove that the conservation proof does not increase significantly the length of proofs. This constrasts with the fact that $\\WKL_0^* + \\RT^2_2$ yields a non-elementary speedup over~$\\RCA_0^*$ even for $\\Sigma_1$-sentences~\\cite{kolodziejczyk2023ramsey}. More recently, Le Houérou, Levy Patey and Yokoyama~\\cite{houerou2026pi} defined a parameterized version of Ketonen and Solovay's notion of largeness and proved that $\\WKL_0 + \\RT^2_2$ is $\\forall \\Sigma^0_3$-conservative over $\\RCA_0 + \\BSig_2$, with explicit bounds computation. However, these bounds are exponential, leaving open whether the conservation theorem yields a significant proof speedup.\n\nIn this article, we define a variant of the parameterized version of $\\alpha$-largeness \\cite{houerou2026pi} and prove that Ramsey's theorem for pairs admits polynomial bounds with respect to this notion of largeness. Then, using the now-called notion of forcing interpretation introduced by Avigad~\\cite{avigad1996formalizing} and developed in the context of $\\alpha$-largeness by Ko{\\l}odziejczyk, Wong and Yokoyama~\\cite{kolodziejczyk2023ramsey}, we prove that the conservation theorem for $\\WKL_0 + \\RT^2_2$ over $\\RCA_0 + \\BSig_2$ for $\\forall \\Sigma^0_3$-sentences does not yield any significant proof speedup. The polynomial bounds computation for this new notion of largeness is non-trivial, and require to prove a partition theorem for trees based on a finite version of Milliken's tree theorem~\\cite{milliken1979ramsey,dodos2016ramsey} with primitive recursive bounds. Our main theorem is therefore the following:", + "context": "If a computably axiomatized theory~$T_0$ is a conservative extension of another such theory~$T_1$ for a syntactical class of sentences $\\Gamma$, there is an algorithmic procedure which translates any $T_0$-proof~$p$ of a $\\Gamma$-sentence into its shortest $T_1$-proof~$\\hat p$. It is natural to wonder whether the translation $p \\mapsto \\hat p$ yields significantly longer proofs. If there exists a $T_0$-proof~$p$ of a $\\Gamma$-sentence such that the length $|\\hat p|$ of its translated $T_1$-proof is super-polynomial with respect to the length~$|p|$, we say that $T_0$ admits \\emph{non-trivial speedup} over $T_1$ for $\\Gamma$-formulas. In this case, the theory~$T_0$ is arguably useful for $\\Gamma$-sentences, in that it sometimes produces significantly shorter proofs than~$T_1$ for such sentences. If on the other hand, there exists a polynomial~$Q$ such that $|\\hat p| \\leq Q(|p|)$ for every $T_0$-proof~$p$ of any $\\Gamma$-sentence, then we consider that $T_0$ admits no significant speedup over~$T_1$ for $\\Gamma$-formulas, since the various proof systems are mutually polynomially simulated.\n\nThe study of proof size and proof speedups traces back to G\\\"odel~\\cite{davis1990kurt} (see Pudl\\'ak~\\cite{pudlak1998length} for an excellent survey). However, it was only in the 1990's that Avigad~\\cite{avigad1996formalizing} showed that model-theoretic conservation theorems based on forcing could be formalized into proof-theoretic arguments and yield proof size analysis. He proved in particular that $\\WKL_0$ is $\\Pi^1_1$-conservative over $\\RCA_0$ with no significant increase in the length of proofs. The technique was later used to formalize $\\Pi^1_1$-conservation theorems over~$\\RCA_0+\\ISig_n$~\\cite{Ikari-PhD,Ik-Yo} and $\\forall \\Sigma^0_2$-conservation theorems over~$\\RCA_0$~\\cite{kolodziejczyk2023ramsey} and over $\\RCA_0^*$~\\cite{katarzyna2025speedup}, where a $\\forall \\Sigma^0_n$-formula is of the form $\\forall X \\varphi(X)$ where $\\varphi$ is $\\Sigma^0_n$.\n\nWe are particularly interested in the first-order consequences of Ramsey's theorem for pairs and two colors ($\\RT^2_2$). Patey and Yokoyama~\\cite{patey2018proof} proved that $\\WKL_0 + \\RT^2_2$ is $\\forall \\Sigma^0_2$-conservative over~$\\RCA_0$ using the notion of $\\alpha$-largeness from Ketonen and Solovay~\\cite{ketonen1981rapidly}. The proof was later simplified by Ko{\\l}odziejczyk and Yokoyama~\\cite{kolo2020some} to give explicit polynomial bounds for this notion of largeness. Ko{\\l}odziejczyk, Wong and Yokoyama~\\cite{kolodziejczyk2023ramsey} then formalized the construction to prove that the conservation proof does not increase significantly the length of proofs. This constrasts with the fact that $\\WKL_0^* + \\RT^2_2$ yields a non-elementary speedup over~$\\RCA_0^*$ even for $\\Sigma_1$-sentences~\\cite{kolodziejczyk2023ramsey}. More recently, Le Houérou, Levy Patey and Yokoyama~\\cite{houerou2026pi} defined a parameterized version of Ketonen and Solovay's notion of largeness and proved that $\\WKL_0 + \\RT^2_2$ is $\\forall \\Sigma^0_3$-conservative over $\\RCA_0 + \\BSig_2$, with explicit bounds computation. However, these bounds are exponential, leaving open whether the conservation theorem yields a significant proof speedup.\n\nIn this article, we define a variant of the parameterized version of $\\alpha$-largeness \\cite{houerou2026pi} and prove that Ramsey's theorem for pairs admits polynomial bounds with respect to this notion of largeness. Then, using the now-called notion of forcing interpretation introduced by Avigad~\\cite{avigad1996formalizing} and developed in the context of $\\alpha$-largeness by Ko{\\l}odziejczyk, Wong and Yokoyama~\\cite{kolodziejczyk2023ramsey}, we prove that the conservation theorem for $\\WKL_0 + \\RT^2_2$ over $\\RCA_0 + \\BSig_2$ for $\\forall \\Sigma^0_3$-sentences does not yield any significant proof speedup. The polynomial bounds computation for this new notion of largeness is non-trivial, and require to prove a partition theorem for trees based on a finite version of Milliken's tree theorem~\\cite{milliken1979ramsey,dodos2016ramsey} with primitive recursive bounds. Our main theorem is therefore the following:", + "full_context": "If a computably axiomatized theory~$T_0$ is a conservative extension of another such theory~$T_1$ for a syntactical class of sentences $\\Gamma$, there is an algorithmic procedure which translates any $T_0$-proof~$p$ of a $\\Gamma$-sentence into its shortest $T_1$-proof~$\\hat p$. It is natural to wonder whether the translation $p \\mapsto \\hat p$ yields significantly longer proofs. If there exists a $T_0$-proof~$p$ of a $\\Gamma$-sentence such that the length $|\\hat p|$ of its translated $T_1$-proof is super-polynomial with respect to the length~$|p|$, we say that $T_0$ admits \\emph{non-trivial speedup} over $T_1$ for $\\Gamma$-formulas. In this case, the theory~$T_0$ is arguably useful for $\\Gamma$-sentences, in that it sometimes produces significantly shorter proofs than~$T_1$ for such sentences. If on the other hand, there exists a polynomial~$Q$ such that $|\\hat p| \\leq Q(|p|)$ for every $T_0$-proof~$p$ of any $\\Gamma$-sentence, then we consider that $T_0$ admits no significant speedup over~$T_1$ for $\\Gamma$-formulas, since the various proof systems are mutually polynomially simulated.\n\nThe study of proof size and proof speedups traces back to G\\\"odel~\\cite{davis1990kurt} (see Pudl\\'ak~\\cite{pudlak1998length} for an excellent survey). However, it was only in the 1990's that Avigad~\\cite{avigad1996formalizing} showed that model-theoretic conservation theorems based on forcing could be formalized into proof-theoretic arguments and yield proof size analysis. He proved in particular that $\\WKL_0$ is $\\Pi^1_1$-conservative over $\\RCA_0$ with no significant increase in the length of proofs. The technique was later used to formalize $\\Pi^1_1$-conservation theorems over~$\\RCA_0+\\ISig_n$~\\cite{Ikari-PhD,Ik-Yo} and $\\forall \\Sigma^0_2$-conservation theorems over~$\\RCA_0$~\\cite{kolodziejczyk2023ramsey} and over $\\RCA_0^*$~\\cite{katarzyna2025speedup}, where a $\\forall \\Sigma^0_n$-formula is of the form $\\forall X \\varphi(X)$ where $\\varphi$ is $\\Sigma^0_n$.\n\nWe are particularly interested in the first-order consequences of Ramsey's theorem for pairs and two colors ($\\RT^2_2$). Patey and Yokoyama~\\cite{patey2018proof} proved that $\\WKL_0 + \\RT^2_2$ is $\\forall \\Sigma^0_2$-conservative over~$\\RCA_0$ using the notion of $\\alpha$-largeness from Ketonen and Solovay~\\cite{ketonen1981rapidly}. The proof was later simplified by Ko{\\l}odziejczyk and Yokoyama~\\cite{kolo2020some} to give explicit polynomial bounds for this notion of largeness. Ko{\\l}odziejczyk, Wong and Yokoyama~\\cite{kolodziejczyk2023ramsey} then formalized the construction to prove that the conservation proof does not increase significantly the length of proofs. This constrasts with the fact that $\\WKL_0^* + \\RT^2_2$ yields a non-elementary speedup over~$\\RCA_0^*$ even for $\\Sigma_1$-sentences~\\cite{kolodziejczyk2023ramsey}. More recently, Le Houérou, Levy Patey and Yokoyama~\\cite{houerou2026pi} defined a parameterized version of Ketonen and Solovay's notion of largeness and proved that $\\WKL_0 + \\RT^2_2$ is $\\forall \\Sigma^0_3$-conservative over $\\RCA_0 + \\BSig_2$, with explicit bounds computation. However, these bounds are exponential, leaving open whether the conservation theorem yields a significant proof speedup.\n\nIn this article, we define a variant of the parameterized version of $\\alpha$-largeness \\cite{houerou2026pi} and prove that Ramsey's theorem for pairs admits polynomial bounds with respect to this notion of largeness. Then, using the now-called notion of forcing interpretation introduced by Avigad~\\cite{avigad1996formalizing} and developed in the context of $\\alpha$-largeness by Ko{\\l}odziejczyk, Wong and Yokoyama~\\cite{kolodziejczyk2023ramsey}, we prove that the conservation theorem for $\\WKL_0 + \\RT^2_2$ over $\\RCA_0 + \\BSig_2$ for $\\forall \\Sigma^0_3$-sentences does not yield any significant proof speedup. The polynomial bounds computation for this new notion of largeness is non-trivial, and require to prove a partition theorem for trees based on a finite version of Milliken's tree theorem~\\cite{milliken1979ramsey,dodos2016ramsey} with primitive recursive bounds. Our main theorem is therefore the following:\n\nWe are particularly interested in the first-order consequences of Ramsey's theorem for pairs and two colors ($\\RT^2_2$). Patey and Yokoyama~\\cite{patey2018proof} proved that $\\WKL_0 + \\RT^2_2$ is $\\forall \\Sigma^0_2$-conservative over~$\\RCA_0$ using the notion of $\\alpha$-largeness from Ketonen and Solovay~\\cite{ketonen1981rapidly}. The proof was later simplified by Ko{\\l}odziejczyk and Yokoyama~\\cite{kolo2020some} to give explicit polynomial bounds for this notion of largeness. Ko{\\l}odziejczyk, Wong and Yokoyama~\\cite{kolodziejczyk2023ramsey} then formalized the construction to prove that the conservation proof does not increase significantly the length of proofs. This constrasts with the fact that $\\WKL_0^* + \\RT^2_2$ yields a non-elementary speedup over~$\\RCA_0^*$ even for $\\Sigma_1$-sentences~\\cite{kolodziejczyk2023ramsey}. More recently, Le Houérou, Levy Patey and Yokoyama~\\cite{houerou2026pi} defined a parameterized version of Ketonen and Solovay's notion of largeness and proved that $\\WKL_0 + \\RT^2_2$ is $\\forall \\Sigma^0_3$-conservative over $\\RCA_0 + \\BSig_2$, with explicit bounds computation. However, these bounds are exponential, leaving open whether the conservation theorem yields a significant proof speedup.\n\nIn this article, we define a variant of the parameterized version of $\\alpha$-largeness \\cite{houerou2026pi} and prove that Ramsey's theorem for pairs admits polynomial bounds with respect to this notion of largeness. Then, using the now-called notion of forcing interpretation introduced by Avigad~\\cite{avigad1996formalizing} and developed in the context of $\\alpha$-largeness by Ko{\\l}odziejczyk, Wong and Yokoyama~\\cite{kolodziejczyk2023ramsey}, we prove that the conservation theorem for $\\WKL_0 + \\RT^2_2$ over $\\RCA_0 + \\BSig_2$ for $\\forall \\Sigma^0_3$-sentences does not yield any significant proof speedup. The polynomial bounds computation for this new notion of largeness is non-trivial, and require to prove a partition theorem for trees based on a finite version of Milliken's tree theorem~\\cite{milliken1979ramsey,dodos2016ramsey} with primitive recursive bounds. Our main theorem is therefore the following:\n\nThe article is divided into three parts : in \\Cref{sec:largeness}, we survey and compare three related notions of largeness, namely, Ketonen and Solvay's $\\alpha$-largeness, Le Houérou, Levy Patey and Yokoyama's $\\alpha$-largeness$(\\theta)$, and our new notion of $\\alpha$-largeness${}^*(\\theta)$, for $\\alpha$ of the form $\\bbomega^n \\cdot k$. In \\Cref{sec:combinatorics}, we develop the framework of $\\alpha$-largeness${}^*(\\theta)$ and prove polynomial upper bounds for $\\RT^2_2$-$\\bbomega^n$-largeness${}^*(\\theta)$. Last, in \\Cref{sec:translation}, we combine this upper bound with the framework of forcing interpretation to prove \\Cref{thm:rt22-polynomially-simulated}.\n\n\\Cref{fig:forcing-translations} summarizes the different intermediate theories we will consider between $\\RCA_0 + \\BSig_2$ and $\\RT_2^2 + \\WKL_0$, and the polynomial simulation between them. It is adapted from \\cite[Figure 1]{kolodziejczyk2023ramsey}, with $\\BSig_2$ added to every theory, and with a $\\forall \\Sigma_3^0$ polynomial simulation between \n$\\RCA_0 + \\BSig_2 + \\mathbb{I}'$ and $\\WKL_0 + \\RT_2^2$ instead of a $\\forall \\Sigma_2^0$ polynomial simulations.\n\n\\begin{lemma}\\label[lemma]{lem:poly-sim-I1}\n $\\RCA_0 + \\BSig_2 + (\\mathbb{I}1)$ is polynomially simulated by:\n \\begin{enumerate}[(a)]\n \\item $\\RCA_0 + \\ISig_2$ with respect to $\\forall \\Sigma_3^0$ sentences,\n \\item $\\RCA_0 + \\BSig_2 + \\neg \\ISig_2$ with respect to $\\mathcal{L}_2$ sentences.\n \\end{enumerate}\n\\end{lemma}\n\n\\begin{lemma}\\label[lemma]{lem:polynomially-reflecting}\n The forcing interpretation of $\\WKL_0 + \\RT_2^2$ in $\\RCA_0 + \\BSig_2 + \\mathbb{I}'$ given by $\\mathsf{Cond}, \\trianglelefteqslant$, $\\Vdash$ of \\Cref{def:forcing-interpretation-def} is polynomially $\\forall \\Sigma_3^0$-reflecting.\n\\end{lemma}\n\n\\begin{lemma}\\label[lemma]{lem:polynomial-simulation}\n $\\RCA_0 + \\mathbb{I}'$ polynomially simulates $\\WKL_0 + \\RT_2^2$ with respects to $\\forall \\Sigma_3^0$-sentences.\n\\end{lemma}\n\n\\begin{repmaintheorem}{thm:rt22-polynomially-simulated}\n$\\RCA_0 + \\BSig_2$ polynomially simulates $\\WKL_0 + \\RT_2^2$ with respect to $\\forall \\Sigma_3^0$ sentences.\n\\end{repmaintheorem}", + "post_theorem_intro_text_len": 1955, + "post_theorem_intro_text": "The article is divided into three parts : in \\Cref{sec:largeness}, we survey and compare three related notions of largeness, namely, Ketonen and Solvay's $\\alpha$-largeness, Le Houérou, Levy Patey and Yokoyama's $\\alpha$-largeness$(\\theta)$, and our new notion of $\\alpha$-largeness${}^*(\\theta)$, for $\\alpha$ of the form $\\bbomega^n \\cdot k$. In \\Cref{sec:combinatorics}, we develop the framework of $\\alpha$-largeness${}^*(\\theta)$ and prove polynomial upper bounds for $\\RT^2_2$-$\\bbomega^n$-largeness${}^*(\\theta)$. Last, in \\Cref{sec:translation}, we combine this upper bound with the framework of forcing interpretation to prove \\Cref{thm:rt22-polynomially-simulated}.\n\n\\subsection{Notation}\n\nMost of the proofs are over a weak subsystem of second-order arithmetic. We therefore distinguish the formal set of integers~$\\NN$ from the theory from the standard set of integers~$\\omega$ from the meta-theory. For instance, if $\\M = (M, S)$ is a non-standard model of second-order arithmetic, $\\NN^\\M = M$ and $\\omega^\\M = \\{0^\\M, 1^\\M, \\dots \\} \\subseteq M$. Similarly, we distinguish the formal ordinals of the theory and the ordinals from the meta-theory. We therefore write for instance $\\bbomega^\\bbomega$ for the formal counter-part of~$\\omega^\\omega$. Indeed, since $\\bbomega^{\\bbomega} = \\sup_{n \\in \\NN} \\bbomega^n$, if $\\omega \\subsetneq M$, then $\\bbomega$ and a fortiori $\\bbomega^\\bbomega$ are not well-founded from the viewpoint of the meta-theory.\n\nWe often identify an integer $k$ with its ordinal $\\{0, \\dots, k-1\\}$. \nGiven a set $X \\subseteq \\NN$ and an integer~$n \\in \\NN$, we write $[X]^n$ for the collection of all subsets of~$X$ of size~$n$.\nThe set $[X]^n$ is in one-to-one correspondence with the set of all increasing ordered $n$-tuples over~$X$. Therefore, for simplicity of notation, given a coloring $f : [X]^n \\to k$, we write $f(x_0, \\dots, x_{n-1})$ for $f(\\{x_0, \\dots, x_{n-1}\\})$ assuming that $x_0 < \\dots < x_{n-1}$.", + "sketch": "The post-theorem introduction gives a high-level proof plan: (1) in \\Cref{sec:largeness} it surveys and compares three notions of largeness—Ketonen and Solvay's $\\alpha$-largeness, Le Hou\\'erou–Levy Patey–Yokoyama's $\\alpha$-largeness$(\\theta)$, and the new notion $\\alpha$-largeness${}^*(\\theta)$—for $\\alpha$ of the form $\\bbomega^n \\cdot k$; (2) in \\Cref{sec:combinatorics} it develops the framework of $\\alpha$-largeness${}^*(\\theta)$ and proves polynomial upper bounds for $\\RT^2_2$-$\\bbomega^n$-largeness${}^*(\\theta)$; (3) in \\Cref{sec:translation} it \"combine[s] this upper bound with the framework of forcing interpretation\" to prove \\Cref{thm:rt22-polynomially-simulated}.", + "expanded_sketch": "No expanded sketch found.", + "expanded_theorem": "\\label[maintheorem]{thm:rt22-polynomially-simulated}\n$\\RCA_0 + \\BSig_2$ polynomially simulates $\\WKL_0 + \\RT_2^2$ with respect to $\\forall \\Sigma_3^0$ sentences.,", + "theorem_type": [ + "Implication" + ], + "mcq": { + "question": "Say that a theory \\(T_1\\) polynomially simulates a theory \\(T_0\\) with respect to a class \\(\\Gamma\\) of sentences if there exists a polynomial \\(Q\\) such that for every \\(T_0\\)-proof \\(p\\) of any sentence in \\(\\Gamma\\), the translated \\(T_1\\)-proof \\(\\hat p\\) of the same sentence satisfies \\(|\\hat p|\\le Q(|p|)\\). Here a \\(\\forall\\Sigma^0_3\\)-sentence means a sentence of the form \\(\\forall X\\,\\varphi(X)\\), where \\(\\varphi\\) is arithmetical \\(\\Sigma^0_3\\). Which statement holds for the second-order arithmetic theories \\(\\RCA_0+\\BSig_2\\) and \\(\\WKL_0+\\RT^2_2\\), where \\(\\BSig_2\\) is \\(\\Sigma^0_2\\)-bounding and \\(\\RT^2_2\\) is Ramsey's theorem for pairs and two colors?", + "correct_choice": { + "label": "A", + "text": "\\(\\RCA_0+\\BSig_2\\) polynomially simulates \\(\\WKL_0+\\RT^2_2\\) with respect to \\(\\forall\\Sigma^0_3\\) sentences; equivalently, there is a polynomial \\(Q\\) such that every \\((\\WKL_0+\\RT^2_2)\\)-proof \\(p\\) of a \\(\\forall\\Sigma^0_3\\)-sentence can be translated into an \\((\\RCA_0+\\BSig_2)\\)-proof \\(\\hat p\\) of the same sentence with \\(|\\hat p|\\le Q(|p|)\\)." + }, + "choices": [ + { + "label": "B", + "text": "\\(\\RCA_0+\\BSig_2\\) polynomially simulates \\(\\WKL_0+\\RT^2_2\\) with respect to all \\(\\mathcal L_2\\)-sentences; equivalently, there is a polynomial \\(Q\\) such that every \\((\\WKL_0+\\RT^2_2)\\)-proof \\(p\\) of an \\(\\mathcal L_2\\)-sentence can be translated into an \\((\\RCA_0+\\BSig_2)\\)-proof \\(\\hat p\\) of the same sentence with \\(|\\hat p|\\le Q(|p|)\\)." + }, + { + "label": "C", + "text": "\\(\\RCA_0+\\BSig_2\\) simulates \\(\\WKL_0+\\RT^2_2\\) with respect to \\(\\forall\\Sigma^0_3\\) sentences; equivalently, every \\((\\WKL_0+\\RT^2_2)\\)-proof of a \\(\\forall\\Sigma^0_3\\)-sentence can be translated into an \\((\\RCA_0+\\BSig_2)\\)-proof of the same sentence." + }, + { + "label": "D", + "text": "\\(\\WKL_0+\\RT^2_2\\) polynomially simulates \\(\\RCA_0+\\BSig_2\\) with respect to \\(\\forall\\Sigma^0_3\\) sentences; equivalently, there is a polynomial \\(Q\\) such that every \\((\\RCA_0+\\BSig_2)\\)-proof \\(p\\) of a \\(\\forall\\Sigma^0_3\\)-sentence can be translated into a \\((\\WKL_0+\\RT^2_2)\\)-proof \\(\\hat p\\) of the same sentence with \\(|\\hat p|\\le Q(|p|)\\)." + }, + { + "label": "E", + "text": "\\(\\RCA_0+\\BSig_2\\) simulates \\(\\WKL_0+\\RT^2_2\\) with respect to \\(\\forall\\Sigma^0_3\\) sentences, but no polynomial bound on the translation length is available; equivalently, for every translation of a \\((\\WKL_0+\\RT^2_2)\\)-proof \\(p\\) of a \\(\\forall\\Sigma^0_3\\)-sentence into an \\((\\RCA_0+\\BSig_2)\\)-proof \\(\\hat p\\) of the same sentence, one cannot in general ensure \\(|\\hat p|\\le Q(|p|)\\) for any polynomial \\(Q\\)." + } + ], + "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": "sentence-class restriction to \\forall\\Sigma^0_3", + "template_used": "stronger_trap" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "polynomial bound on translated proof length", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "direction of polynomial simulation", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "existence of a uniform polynomial bound", + "template_used": "uniformity_effectivity" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines polynomial simulation and the sentence class but does not reveal which direction of simulation is true, whether the result is polynomial or merely qualitative, or how broad the sentence class is." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recall question: the correct choice is essentially the exact theorem statement, with distractors formed by predictable modifications (stronger domain, weaker conclusion, reversed direction, denial of polynomial bound). It is not pure tautology, but it is only a mild reformulation." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to compare strength, direction, and uniformity across the options, but the question primarily tests recall/recognition of a known result rather than generating a conclusion from mathematical argument." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically meaningful: one overgeneralizes the sentence class, one weakens polynomial simulation to mere simulation, one reverses the simulation direction, and one denies the polynomial bound. These reflect common failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid recognition-style theorem question with good distractors and no answer leakage, but limited generative depth because it is very close to selecting the exact statement of a known result." + } + }, + { + "id": "2602.12122v1", + "paper_link": "http://arxiv.org/abs/2602.12122v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "\\sl \\label{th:Lq} Let $V_1,V_2\\in L^1(\\mathbb{R}^n)\\cap L^q(\\mathbb{R}^n)$ be time-independent potentials, where $q>1$ if $n=2$ and $q\\ge n/2$ if $n\\ge 3$. Let $\\mathcal U_T^1$ and $\\mathcal U_T^2$ denote the corresponding initial-to-final-state maps. Then\n\\[\n\\mathcal U_T^1=\\mathcal U_T^2 \\quad\\Longrightarrow\\quad V_1=V_2.\n\\]", + "start_pos": 8402, + "end_pos": 8742, + "label": "th:Lq" + }, + "ref_dict": { + "eq:orthogonality_intro": "\\begin{equation}\\label{eq:orthogonality_intro}\n\\int_{\\Sigma} (V_1-V_2)\\,u_1\\,\\overline{v_2}=0,\n\\end{equation}", + "th:Lq": "\\begin{theorem}\\sl \\label{th:Lq} Let $V_1,V_2\\in L^1(\\R^n)\\cap L^q(\\R^n)$ be time-independent potentials, where $q>1$ if $n=2$ and $q\\ge n/2$ if $n\\ge 3$. Let $\\mathcal U_T^1$ and $\\mathcal U_T^2$ denote the corresponding initial-to-final-state maps. Then\n\\[\n\\mathcal U_T^1=\\mathcal U_T^2 \\quad\\Longrightarrow\\quad V_1=V_2.\n\\]\n\\end{theorem}", + "eq:Schrodinger": "\\begin{equation}\\label{eq:Schrodinger}\n\\begin{cases}\ni \\partial_t u = -\\Delta u + V u & \\text{for } (t,x)\\in (0, T)\\times \\R^n \\eqqcolon \\Sigma,\n\\vspace{.6em}\n\\\\\nu(0,x) = f(x) & \\text{for } x\\in \\R^n.\n\\end{cases}\n\\end{equation}" + }, + "pre_theorem_intro_text_len": 3116, + "pre_theorem_intro_text": "In this paper, we study an inverse problem for the Schr\\\"odinger equation in which the available data consist of the map that sends any initial state $f$ at time $t=0$ to the solution at a fixed final time $t=T$.\n\nTo make this precise, let us consider the initial-value problem for the Schr\\\"odinger equation with a time-independent potential\n\\begin{equation}\\label{eq:Schrodinger}\n\\begin{cases}\ni \\partial_t u = -\\Delta u + V u & \\text{for } (t,x)\\in (0, T)\\times \\mathbb{R}^n \\eqqcolon \\Sigma,\n\\vspace{.6em}\n\\\\\nu(0,x) = f(x) & \\text{for } x\\in \\mathbb{R}^n.\n\\end{cases}\n\\end{equation}\nWe assume throughout the paper that $V=V(x)\\in L^q(\\mathbb{R}^n)$, where $q\\ge n/2$ if $n\\ge 3$ or $q>1$ if $n=2$. Then, by \\cites{zbMATH02204588,zbMATH00179225}, this direct problem is well posed, and for every $f\\in L^2(\\mathbb{R}^n)$ there exists a unique solution\n\\[\nu\\in C\\left([0,T];L^2(\\mathbb{R}^n)\\right).\n\\]\nThe evolution associated with \\eqref{eq:Schrodinger} therefore defines a bounded linear operator\n\\[\n\\mathcal U : f\\in L^2(\\mathbb{R}^n)\\mapsto u\\in C\\left([0,T];L^2(\\mathbb{R}^n)\\right).\n\\]\nConsequently, for any fixed time $t\\in[0,T]$, the operator\n\\[\n\\mathcal U_t : f \\in L^2(\\mathbb{R}^n) \\mapsto u(t,\\centerdot)\\in L^2(\\mathbb{R}^n)\n\\]\nis also bounded, uniformly in $t$. Solutions of the form $u=\\mathcal U f$, with $f\\in L^2(\\mathbb{R}^n)$, will be referred to as \\emph{physical solutions}, while we call $\\mathcal U_T$ the \\emph{initial-to-final-state map}.\n\nThe main question addressed in this paper is whether the initial-to-final-state map $\\mathcal U_T$ uniquely determines the Hamiltonian $-\\Delta+V$. This inverse problem was first studied for \\emph{time-dependent} potentials in \\cite{zbMATH07801151}. There, the authors show that if the potentials $V_1,V_2 \\in L^1((0,T);L^\\infty(\\mathbb{R}^n))$ satisfy a \\emph{super-exponential decay} condition at infinity, and if $\\mathcal U_T^j$ denotes the initial-to-final-state map associated with $-\\Delta+V_j$, then there holds:\n\\[\n\\mathcal U_T^1=\\mathcal U_T^2 \\quad\\Longrightarrow\\quad V_1=V_2.\n\\]\nMore recently, this uniqueness result was extended in \\cite{caro2025initialtofinalstateinverseproblemunbounded} to unbounded time-dependent potentials that are allowed to exhibit local $L^q$-type singularities, but still requiring the super-exponential decay assumption at infinity.\n\nThe case of time-independent potentials was previously considered in \\cite{zbMATH08122191}, where uniqueness was established under comparatively weaker decay assumptions than in the time-dependent setting, namely assuming only super-linear decay at infinity. The purpose of the present paper is to relax these assumptions even further. Specifically, we prove uniqueness for time-independent potentials that may exhibit singularities of $L^q$-type in sets of finite measure and satisfy only $L^1$-integrability in sets of infinite measure; in the time-independent setting, this represents a substantial improvement over the decay and integrability assumptions in \\cite{zbMATH08122191}, \\cite{zbMATH07801151}, and \\cite{caro2025initialtofinalstateinverseproblemunbounded}.", + "context": "In this paper, we study an inverse problem for the Schr\\\"odinger equation in which the available data consist of the map that sends any initial state $f$ at time $t=0$ to the solution at a fixed final time $t=T$.\n\nTo make this precise, let us consider the initial-value problem for the Schr\\\"odinger equation with a time-independent potential\n\\begin{equation}\\label{eq:Schrodinger}\n\\begin{cases}\ni \\partial_t u = -\\Delta u + V u & \\text{for } (t,x)\\in (0, T)\\times \\mathbb{R}^n \\eqqcolon \\Sigma,\n\\vspace{.6em}\n\\\\\nu(0,x) = f(x) & \\text{for } x\\in \\mathbb{R}^n.\n\\end{cases}\n\\end{equation}\nWe assume throughout the paper that $V=V(x)\\in L^q(\\mathbb{R}^n)$, where $q\\ge n/2$ if $n\\ge 3$ or $q>1$ if $n=2$. Then, by \\cites{zbMATH02204588,zbMATH00179225}, this direct problem is well posed, and for every $f\\in L^2(\\mathbb{R}^n)$ there exists a unique solution\n\\[\nu\\in C\\left([0,T];L^2(\\mathbb{R}^n)\\right).\n\\]\nThe evolution associated with \\eqref{eq:Schrodinger} therefore defines a bounded linear operator\n\\[\n\\mathcal U : f\\in L^2(\\mathbb{R}^n)\\mapsto u\\in C\\left([0,T];L^2(\\mathbb{R}^n)\\right).\n\\]\nConsequently, for any fixed time $t\\in[0,T]$, the operator\n\\[\n\\mathcal U_t : f \\in L^2(\\mathbb{R}^n) \\mapsto u(t,\\centerdot)\\in L^2(\\mathbb{R}^n)\n\\]\nis also bounded, uniformly in $t$. Solutions of the form $u=\\mathcal U f$, with $f\\in L^2(\\mathbb{R}^n)$, will be referred to as \\emph{physical solutions}, while we call $\\mathcal U_T$ the \\emph{initial-to-final-state map}.\n\nThe main question addressed in this paper is whether the initial-to-final-state map $\\mathcal U_T$ uniquely determines the Hamiltonian $-\\Delta+V$. This inverse problem was first studied for \\emph{time-dependent} potentials in \\cite{zbMATH07801151}. There, the authors show that if the potentials $V_1,V_2 \\in L^1((0,T);L^\\infty(\\mathbb{R}^n))$ satisfy a \\emph{super-exponential decay} condition at infinity, and if $\\mathcal U_T^j$ denotes the initial-to-final-state map associated with $-\\Delta+V_j$, then there holds:\n\\[\n\\mathcal U_T^1=\\mathcal U_T^2 \\quad\\Longrightarrow\\quad V_1=V_2.\n\\]\nMore recently, this uniqueness result was extended in \\cite{caro2025initialtofinalstateinverseproblemunbounded} to unbounded time-dependent potentials that are allowed to exhibit local $L^q$-type singularities, but still requiring the super-exponential decay assumption at infinity.\n\nThe case of time-independent potentials was previously considered in \\cite{zbMATH08122191}, where uniqueness was established under comparatively weaker decay assumptions than in the time-dependent setting, namely assuming only super-linear decay at infinity. The purpose of the present paper is to relax these assumptions even further. Specifically, we prove uniqueness for time-independent potentials that may exhibit singularities of $L^q$-type in sets of finite measure and satisfy only $L^1$-integrability in sets of infinite measure; in the time-independent setting, this represents a substantial improvement over the decay and integrability assumptions in \\cite{zbMATH08122191}, \\cite{zbMATH07801151}, and \\cite{caro2025initialtofinalstateinverseproblemunbounded}.\n\n\\begin{equation}\\label{eq:Schrodinger}\n\\begin{cases}\ni \\partial_t u = -\\Delta u + V u & \\text{for } (t,x)\\in (0, T)\\times \\R^n \\eqqcolon \\Sigma,\n\\vspace{.6em}\n\\\\\nu(0,x) = f(x) & \\text{for } x\\in \\R^n.\n\\end{cases}\n\\end{equation}", + "full_context": "In this paper, we study an inverse problem for the Schr\\\"odinger equation in which the available data consist of the map that sends any initial state $f$ at time $t=0$ to the solution at a fixed final time $t=T$.\n\nTo make this precise, let us consider the initial-value problem for the Schr\\\"odinger equation with a time-independent potential\n\\begin{equation}\\label{eq:Schrodinger}\n\\begin{cases}\ni \\partial_t u = -\\Delta u + V u & \\text{for } (t,x)\\in (0, T)\\times \\mathbb{R}^n \\eqqcolon \\Sigma,\n\\vspace{.6em}\n\\\\\nu(0,x) = f(x) & \\text{for } x\\in \\mathbb{R}^n.\n\\end{cases}\n\\end{equation}\nWe assume throughout the paper that $V=V(x)\\in L^q(\\mathbb{R}^n)$, where $q\\ge n/2$ if $n\\ge 3$ or $q>1$ if $n=2$. Then, by \\cites{zbMATH02204588,zbMATH00179225}, this direct problem is well posed, and for every $f\\in L^2(\\mathbb{R}^n)$ there exists a unique solution\n\\[\nu\\in C\\left([0,T];L^2(\\mathbb{R}^n)\\right).\n\\]\nThe evolution associated with \\eqref{eq:Schrodinger} therefore defines a bounded linear operator\n\\[\n\\mathcal U : f\\in L^2(\\mathbb{R}^n)\\mapsto u\\in C\\left([0,T];L^2(\\mathbb{R}^n)\\right).\n\\]\nConsequently, for any fixed time $t\\in[0,T]$, the operator\n\\[\n\\mathcal U_t : f \\in L^2(\\mathbb{R}^n) \\mapsto u(t,\\centerdot)\\in L^2(\\mathbb{R}^n)\n\\]\nis also bounded, uniformly in $t$. Solutions of the form $u=\\mathcal U f$, with $f\\in L^2(\\mathbb{R}^n)$, will be referred to as \\emph{physical solutions}, while we call $\\mathcal U_T$ the \\emph{initial-to-final-state map}.\n\nThe main question addressed in this paper is whether the initial-to-final-state map $\\mathcal U_T$ uniquely determines the Hamiltonian $-\\Delta+V$. This inverse problem was first studied for \\emph{time-dependent} potentials in \\cite{zbMATH07801151}. There, the authors show that if the potentials $V_1,V_2 \\in L^1((0,T);L^\\infty(\\mathbb{R}^n))$ satisfy a \\emph{super-exponential decay} condition at infinity, and if $\\mathcal U_T^j$ denotes the initial-to-final-state map associated with $-\\Delta+V_j$, then there holds:\n\\[\n\\mathcal U_T^1=\\mathcal U_T^2 \\quad\\Longrightarrow\\quad V_1=V_2.\n\\]\nMore recently, this uniqueness result was extended in \\cite{caro2025initialtofinalstateinverseproblemunbounded} to unbounded time-dependent potentials that are allowed to exhibit local $L^q$-type singularities, but still requiring the super-exponential decay assumption at infinity.\n\nThe case of time-independent potentials was previously considered in \\cite{zbMATH08122191}, where uniqueness was established under comparatively weaker decay assumptions than in the time-dependent setting, namely assuming only super-linear decay at infinity. The purpose of the present paper is to relax these assumptions even further. Specifically, we prove uniqueness for time-independent potentials that may exhibit singularities of $L^q$-type in sets of finite measure and satisfy only $L^1$-integrability in sets of infinite measure; in the time-independent setting, this represents a substantial improvement over the decay and integrability assumptions in \\cite{zbMATH08122191}, \\cite{zbMATH07801151}, and \\cite{caro2025initialtofinalstateinverseproblemunbounded}.\n\n\\begin{equation}\\label{eq:Schrodinger}\n\\begin{cases}\ni \\partial_t u = -\\Delta u + V u & \\text{for } (t,x)\\in (0, T)\\times \\R^n \\eqqcolon \\Sigma,\n\\vspace{.6em}\n\\\\\nu(0,x) = f(x) & \\text{for } x\\in \\R^n.\n\\end{cases}\n\\end{equation}\n\nThe case of time-independent potentials was previously considered in \\cite{zbMATH08122191}, where uniqueness was established under comparatively weaker decay assumptions than in the time-dependent setting, namely assuming only super-linear decay at infinity. The purpose of the present paper is to relax these assumptions even further. Specifically, we prove uniqueness for time-independent potentials that may exhibit singularities of $L^q$-type in sets of finite measure and satisfy only $L^1$-integrability in sets of infinite measure; in the time-independent setting, this represents a substantial improvement over the decay and integrability assumptions in \\cite{zbMATH08122191}, \\cite{zbMATH07801151}, and \\cite{caro2025initialtofinalstateinverseproblemunbounded}.\n\n\\medskip\n\\noindent\\textbf{Outline of the proof.}\nThe proof of Theorem~\\ref{th:Lq} proceeds by extracting information on the difference $V_1-V_2$ from $\\mathcal U_T^1 = \\mathcal U_T^2$, by testing it against suitable families of solutions.\n\nReturning to the solutions in \\eqref{eq:steadysol}, we can now use the invertibility of $\\mathrm{Id}-P_\\lambda\\circ V$ to complete the construction of the correction term $w^{\\mathrm{cor}}$. In the statement of the corollary below, the additional assumption $V\\in L^1(\\R^n)$ is used only to ensure that $Vw^{(0)}\\in L^{p'}(\\R^n)$, so that $P_\\lambda(Vw^{(0)})$ is well defined and belongs to $L^p(\\R^n)$; the invertibility of $\\mathrm{Id}-P_\\lambda\\circ V$ on $L^p(\\R^n)$ relies solely on the $L^q$ assumption.\n\\begin{corollary}\\label{cor:solutions}\\sl Let $n\\geq 2$ and consider a time-independent potential $V\\in L^1(\\R^n)\\cap L^q(\\R^n)$ for some $n/2 2\\lambda_V$ with $\\lambda_V$ as in the statement of Lemma~\\ref{lem:I-P inverse in Lp}, the functions\n\\[\nw(x)\\coloneqq e^{-i\\lambda\\omega\\cdot x}+w^{\\mathrm{cor}}(x)= w^{(0)}(x)+w^{\\mathrm{cor}}(x),\\qquad w^{\\mathrm{cor}}(x)\\coloneqq (\\mathrm{Id}-P_\\lambda\\circ V)^{-1}[P_\\lambda(Vw^{(0)})],\n\\]\nwith $ \\omega\\in\\mathbb S^{n-1}$ and $x\\in\\R^n$,\nare weak solutions of the time-independent Schr\\\"odinger equation \n\\[\n(\\Delta+\\lambda^2-V)w=0\\quad\\mathrm{in}\\quad\\R^n.\n\\]\nAdditionally, for $\\lambda>2\\lambda_V$ we have the estimate\n\\[\n\\left \\|w^{\\mathrm{cor}}\\right\\|_{L^p(\\R^n)}\\lesssim \\frac{1}{\\lambda^{2n\\left(\\frac1p-\\frac{1}{p_n}\\right)}} \\|V\\|_{L^{p'}(\\R^n)}\\leq \\frac{1}{\\lambda^{2n\\left(\\frac1p-\\frac{1}{p_n}\\right)}} \\|V\\|_{L^1(\\R^n)\\cap L^q(\\R^n)}.\n\\]\nIn particular, the function \n\\[\n\\psi_V ^{\\lambda,\\omega}(t,x)\\coloneqq e^{-i\\lambda^2 t}w(x)= e^{-i\\lambda^2 t}(w^{(0)}(x)+w^{\\mathrm{cor}}(x)),\\qquad (t,x)\\in\\R\\times \\R^n,\n\\]\nis a weak solution of \\eqref{sch_timeind}.\n\\end{corollary}\n\n\\begin{corollary}\\label{cor:solutionsVcrit}\\sl Let $n\\geq 3$ and consider a time-independent potential $V\\in L^1(\\R^n)\\cap L^{n/2}(\\R^n)$. Let $X_\\lambda,X_\\lambda ^*$ and $\\|V\\|_\\lambda$ as defined above. There exists a constant $\\lambda_V$ depending only on $V$ and the dimension such that for $\\lambda > \\lambda_V$, the functions\n\\[\nw(x)\\coloneqq e^{-i\\lambda\\omega\\cdot x} +w^{\\mathrm{cor}} (x)\\eqqcolon w^{(0)}(x)+w^{\\mathrm{cor}}(x),\\qquad w^{\\mathrm{cor}}(x)\\coloneqq (\\mathrm{Id}-P_\\lambda\\circ V)^{-1}[P_\\lambda(Vw^{(0)})],\n\\]\nwith $\\omega\\in\\mathbb S^{n-1} $ and $ x\\in\\R^n$,\nare weak solutions of the time-independent Schr\\\"odinger equation \n\\[\n(\\Delta+\\lambda^2-V)w=0\\quad\\mathrm{in}\\quad\\R^n.\n\\]\nAdditionally, the following estimates are satisfied\n\\[\n\\left\\|w^{\\mathrm{cor}} \\right\\|_{X_\\lambda ^*}\\lesssim \\|V\\|_{X_\\lambda}\\leq \\lambda^{-\\frac{1}{n+1}} \\|V\\|_{L^{q_n '}(\\R^n)}\\leq \\lambda^{-\\frac{1}{n+1}} \\|V\\|_{L^1(\\R^n)\\cap L^{n/2}(\\R^n)}\n\\]\nIn particular, the function \n\\[\n\\psi_V ^{\\lambda,\\omega}(t,x)\\coloneqq e^{-i\\lambda^2 t}w(x)= e^{-i\\lambda^2 t}(w^{\\mathrm{cor}}(x)+w^{(0)}(x)),\\qquad (t,x)\\in\\R\\times \\R^n,\n\\]\nis a weak solution of \\eqref{sch_timeind}.\n\\end{corollary}\n\n\\begin{proposition}\\label{pr:int_by_parts_time-harmonic_solutions}\\sl Let $V_1,V_2\\in L^1(\\R^n)\\cap L^q(\\R^n)$, where $q\\in(1,(n+1)/2]$ if $n=2$, or $q\\in[n/2,(n+1)/2]$ if $n\\geq 3$, and define $p$ by \n\\[\n \\frac{1}{q}+\\frac{2}{p}=1.\n\\]\nLet $\\mathcal U_T^1,\\mathcal U_T^2$ denote the initial-to-final-state maps corresponding to $V_1,V_2$, respectively.\n\n\\subsection{Proof of Theorem~\\ref{th:Lq}: the endpoint case}\\label{subsec: proof_uniqueness_endpoint} Let $V_1,V_2\\in L^1(\\R^n)\\cap L^{n/2}(\\R^n)$ and define $F\\coloneqq V_1 - V_2$. Given $\\xi \\in \\R^n$ we consider $\\nu \\in \\Sph^{n-1}$ such that $\\xi \\cdot \\nu = 0$ and $\\lambda \\geq \\max\\{\\lambda_{V_1},\\lambda_{V_2},|\\xi|/2\\}$, where $\\lambda_{V_1},\\lambda_{V_2}$ are the constants corresponding to $V_1,V_2$ from Corollary~\\ref{cor:solutionsVcrit}. As in \\S\\ref{subsec:proof_uniqueness_unbounded}, we consider the vectors\n\\[\n\\omega_1 \\coloneqq \\frac{1}{\\lambda} \\frac{\\xi}{2} + \\left( 1 - \\frac{|\\xi|^2}{4 \\lambda^2} \\right)^{1/2} \\nu,\\qquad \\omega_2 \\coloneqq -\\frac{1}{\\lambda} \\frac{\\xi}{2} + \\left( 1 - \\frac{|\\xi|^2}{4 \\lambda^2} \\right)^{1/2} \\nu,\n\\]\nand note that $\\omega_1,\\omega_2\\in\\mathbb S^{n-1}$ and $\\omega_1 - \\omega_2 = \\xi/\\lambda$. \nWe now construct the stationary state solutions as in Corollary~\\ref{cor:solutionsVcrit}\n\\[\n\\psi_j\\coloneqq \\psi_{V_j} ^{\\lambda,\\omega_j}=e^{-i\\lambda^2t}w_j(x)=e^{-i\\lambda^2t}\\left(w^{(0)}_j(x)+w^{\\mathrm{cor}}_j(x)\\right),\\qquad j\\in\\{1,2\\},\\qquad (t,x)\\in\\R\\times \\R^n,\n\\]\nand we recall the estimates\n\\begin{equation}\\label{eq:psiestimates}\n\\left\\|w^{\\mathrm{cor}}_j\\right\\|_{X_\\lambda ^*}\\lesssim \\|V_j\\|_{X_\\lambda}\\leq \\lambda^{-\\frac{1}{n+1}} \\|V_j\\|_{L^1(\\R^n)\\cap L^{n/2}(\\R^n)},\\qquad j\\in\\{1,2\\},\n\\end{equation}\nalso proved in Corollary~\\ref{cor:solutionsVcrit}. Since $ \\mathcal{U}^1_T=\\mathcal{U}^2_T$, plugging the solutions $\\psi_1,\\psi_2$ into the orthogonality relation of \nProposition~\\ref{pr:int_by_parts_time-harmonic_solutions} and repeating the calculations from \\S\\ref{subsec:proof_uniqueness_unbounded}, we have \n\\[\n\\begin{split}\n|\\wh{F}(\\xi)| &\\leq \\frac{1}{(2\\pi)^{n/2}} \\int_{\\R^n} \\left|F w^{(0)}_1 \\overline{w^{\\mathrm{cor}}_2}\\right| + \\int_{\\R^n} \\left|F \\overline{w^{(0)}_2} w^{\\mathrm{cor}}_1\\right| + \\int_{\\R^n} \\left|F w^{\\mathrm{cor}}_1 \\overline{w^{\\mathrm{cor}}_2}\\right| \n\\\\\n&\\lesssim \\left\\|Fw^{(0)}_1\\right\\|_{X_\\lambda} \\left\\|w^{\\mathrm{cor}}_2\\right\\|_{\\X_\\lambda ^*}+\\left\\|Fw^{(0)}_2\\right\\|_{X_\\lambda} \\left\\|w^{\\mathrm{cor}}_1\\right\\|_{\\X_\\lambda ^*}+\\|F\\|_\\lambda\\|w^{\\mathrm{cor}}_1\\|_{\\X_\\lambda ^*} \\left \\|w^{\\mathrm{cor}}_2\\right \\|_{\\X_\\lambda ^*},\n\\end{split}\n\\]\nwhere we have used the duality of $X_\\lambda$ and $X_\\lambda ^*$ and the H\\\"older-type inequality of Lemma~\\ref{lem:multiplication_V_endpoint_X_norm}. Using the estimates \\eqref{eq:psiestimates} and the fact that $|w^{(0)}_1|=|w^{(0)}_2|=1$ we get\n\\begin{align*}\n|\\wh{F}(\\xi)| &\\lesssim \n \\|F\\|_{\\X_\\lambda} \\lambda^{-\\frac{1}{n+1}}(\\|V_1\\|_{L^1(\\R^n)\\cap L^{n/2}(\\R^n)} + \\|V_2\\|_{L^1(\\R^n)\\cap L^{n/2}(\\R^n)}) \\\\\n&+ \\|F\\|_{\\lambda} \\lambda^{-\\frac{2}{n+1}} \\|V_1\\|_{L^1(\\R^n)\\cap L^{n/2}(\\R^n)} \\|V_2\\|_{L^1(\\R^n)\\cap L^{n/2}(\\R^n)} \\\\\n&\\lesssim\\lambda^{-\\frac{2}{n+1}} \\left(\\|V_1\\|_{L^1(\\R^n)\\cap L^{n/2}(\\R^n)} +\\|V_2\\|_{L^1(\\R^n)\\cap L^{n/2}(\\R^n)}\\right) ^2 \n(1 + \\|F\\|_{\\lambda}) \\to 0\n\\end{align*}\nas $\\lambda \\to \\infty$. Hence $\\wh{F}(\\xi) = 0$, and since $\\xi\\in\\R^n$ was arbitrary we conclude that $F=0$ and hence that $V_1=V_2$ almost everywhere in $\\R^n$, thus completing the proof of the endpoint case of Theorem~\\ref{th:Lq}.", + "post_theorem_intro_text_len": 6530, + "post_theorem_intro_text": "\\medskip\n\\noindent\\textbf{Outline of the proof.}\nThe proof of Theorem~\\ref{th:Lq} proceeds by extracting information on the difference $V_1-V_2$ from $\\mathcal U_T^1 = \\mathcal U_T^2$, by testing it against suitable families of solutions.\n\nThe first step is to show that the equality $\\mathcal U_T^1=\\mathcal U_T^2$ yields an \\emph{Alessandrini-type orthogonality relation}, in the sense of \\cite{Aless}, of the form\n\\begin{equation}\\label{eq:orthogonality_intro}\n\\int_{\\Sigma} (V_1-V_2)\\,u_1\\,\\overline{v_2}=0,\n\\end{equation}\nvalid for pairs of solutions associated with the potentials $V_1$ and $\\overline{V_2}$. This identity is initially available only for \\emph{physical solutions}, namely solutions belonging to $C([0,T];L^2(\\mathbb{R}^n))$. \n\nAt this stage, our first obstruction is that in order to recover pointwise information on $V_1-V_2$, we need to test \\eqref{eq:orthogonality_intro} against special \\emph{time-harmonic} solutions. Such solutions can be constructed as perturbations of time-harmonic solutions of the free Schr\\\"odinger equation; we refer to such solutions as \\emph{stationary states} associated with the given potential. However, these stationary states are not in $C([0, T]; L^2(\\mathbb{R}^n))$, and using them as test functions requires a substantial extension of the Alessandrini-type orthogonality relation beyond physical solutions.\n\nMore precisely, stationary states are obtained by inverting a resolvent-type operator and constructing correction terms via Neumann series. Concretely, they are of the form\n\\[\nu(t,x)=e^{-i|\\kappa|^2 t}\\left(e^{-i\\kappa\\cdot x}+w^{\\mathrm{cor}}(x)\\right),\n\\]\nwhere the leading term \\(e^{-i|\\kappa|^2 t}e^{-i\\kappa\\cdot x}\\) is a time-harmonic solution of the free Schr\\\"odinger equation, and the correction \\(w^{\\mathrm{cor}}\\) accounts for the presence of the potential \\(V\\).\n\nThe construction of \\(w^{\\mathrm{cor}}\\) reduces to inverting an operator of the form \\(\\mathrm{Id}-P_\\lambda\\circ V\\), where \\(P_\\lambda\\) denotes a solution operator for the Helmholtz equation \\((\\Delta+\\lambda^2)u=f\\). This inversion is carried out on suitable function spaces, chosen so that the operator \\(P_\\lambda\\circ V\\) is small, in a suitable sense.\n\nIn following the proof strategy outlined above, we encounter a second important obstruction. In the non-endpoint regime $V\\in L^q$ with $q>n/2$, decay in the energy parameter $\\lambda$ follows from the classical Kenig--Ruiz--Sogge resolvent estimate. The latter ensures that $P_\\lambda\\circ V$ is small on the relevant function spaces for large $\\lambda$. At the critical endpoint $q=n/2$, however, this estimate no longer yields decay in $\\lambda$, and the smallness of $P_\\lambda\\circ V$ cannot be obtained from the standard resolvent bound alone.\n\nTo address this issue, we introduce a new scale of Banach spaces that allows us to recover a decaying factor in the energy parameter, also at the endpoint. Within this framework, we establish an improved form of the Kenig--Ruiz--Sogge resolvent estimate, sufficient to construct stationary states for critical potentials despite the absence of quantitative decay at the endpoint. A similar idea has appeared in \\cite{CaroGarcia} for a related scattering problems with critically singular potentials; see also \\cite{zbMATH07867333}. When inserted into the extended Alessandrini-type orthogonality relation, these stationary states allow us to isolate the Fourier phase and hence recover the Fourier transform of $V_1-V_2$, proving uniqueness.\n\nThroughout the argument, the assumption of a stationary potential plays a crucial role. Indeed, in the time-dependent setting, the uniqueness results in \\cites{zbMATH07801151,caro2025initialtofinalstateinverseproblemunbounded} rely on constructing complex geometrical optics solutions whose leading terms are of the form\n\\[\n(t,x) \\longmapsto e^{it|\\kappa|^2} e^{\\kappa\\cdot x}, \\qquad \\kappa \\in \\mathbb{R}^n,\n\\]\nsupplemented by correction terms depending on the potential. These complex exponentials grow in certain directions. Using such solutions in an orthogonality relation of the form \\eqref{eq:orthogonality_intro} and controlling the corresponding correction terms naturally leads to assuming super-exponential decay of the potential at infinity.\n\nHowever, in the time-independent case considered here, we work with time-harmonic solutions whose main term is given in the form\n\\[\n(t,x) \\longmapsto e^{-i|\\kappa|^2 t} e^{-i\\kappa\\cdot x}, \\qquad \\kappa \\in \\mathbb{R}^n,\n\\]\nignoring again the perturbative terms correcting for the potential. Now the exponents are imaginary, so the leading term is purely oscillatory and unimodular. This allows us to restrict attention to a more regular class of solutions, which however is still sufficient to establish uniqueness in the case of time-independent potentials.\n\nInverse problems associated with the dynamical Schrödinger equation have been extensively studied; see, for example, \\cites{zbMATH01886353,zbMATH06733553,zbMATH05549395,zbMATH05655673,zbMATH05839237,zbMATH06864429}. Many of these works have also considered time-dependent Hamiltonians \\cites{zbMATH06769718,zbMATH06516179,zbMATH05379127,zbMATH07033617,zbMATH07242805}. A common feature of these investigations is the use of a dynamical Dirichlet-to-Neumann map, recorded on the boundary of a domain that contains the non-constant portions of the Hamiltonian. \n\nThis formulation stands in contrast to the one presented in our work. Here, the variable part of the Hamiltonian is not localized to a bounded region but is possibly present throughout the whole space. Moreover, our inverse problem is formulated with a distinct data requirement: knowledge only of the system's initial state and its corresponding state at a final time.\n\nThe rest of the paper is organized as follows. Section~\\ref{sec:aux} collects background material and auxiliary analytic tools, including well–posedness results and resolvent estimates. In Section~\\ref{sec:timeindependent_solutions} we construct stationary states for the Schr\\\"odinger equation with time-independent potentials, treating both the non-endpoint case $q>n/2$ and the endpoint case $q=n/2$ in dimensions $n\\ge3$. Section~\\ref{sec:orthorelation} is devoted to extending the Alessandrini-type orthogonality relation beyond physical solutions, and in particular to stationary-state solutions. Finally, in Section~\\ref{sec:unbounded_endpoint} we prove Theorem~\\ref{th:Lq} by applying the extended orthogonality relation to the stationary states constructed earlier.", + "sketch": "The proof of Theorem~\\ref{th:Lq} \\emph{extracts information on} $V_1-V_2$ from $\\mathcal U_T^1=\\mathcal U_T^2$ by testing against suitable families of solutions.\n\n1. From $\\mathcal U_T^1=\\mathcal U_T^2$, first derive an \\emph{Alessandrini-type orthogonality relation} (initially for \\emph{physical solutions} in $C([0,T];L^2(\\mathbb{R}^n))$):\n\\[\n\\int_{\\Sigma} (V_1-V_2)\\,u_1\\,\\overline{v_2}=0,\n\\]\nfor pairs of solutions associated with $V_1$ and $\\overline{V_2}$.\n\n2. To recover pointwise/Fourier information on $V_1-V_2$, test the orthogonality identity with special \\emph{time-harmonic} solutions (\\emph{stationary states}), constructed as perturbations of free time-harmonic Schr\\\"odinger solutions. These have the form\n\\[\n u(t,x)=e^{-i|\\kappa|^2 t}\\big(e^{-i\\kappa\\cdot x}+w^{\\mathrm{cor}}(x)\\big),\n\\]\nwhere $w^{\\mathrm{cor}}$ accounts for the potential. Since these stationary states are \\emph{not} in $C([0,T];L^2)$, one must \\emph{extend the Alessandrini-type orthogonality relation beyond physical solutions} to allow using them as test functions.\n\n3. Construct $w^{\\mathrm{cor}}$ by inverting a resolvent-type operator and building correction terms via Neumann series: reduce to inverting\n\\(\\mathrm{Id}-P_\\lambda\\circ V\\), where $P_\\lambda$ solves the Helmholtz equation $(\\Delta+\\lambda^2)u=f$, on function spaces where $P_\\lambda\\circ V$ is small.\n\n4. Handle the endpoint obstruction: for $q>n/2$, smallness for large $\\lambda$ comes from the classical Kenig--Ruiz--Sogge resolvent estimate, but at the critical endpoint $q=n/2$ it yields no decay in $\\lambda$. To fix this, \\emph{introduce a new scale of Banach spaces} that recovers a decaying factor in $\\lambda$ and establish an \\emph{improved} Kenig--Ruiz--Sogge resolvent estimate, enabling stationary-state construction even at the endpoint.\n\n5. Insert these stationary states into the \\emph{extended} orthogonality relation; this \\emph{isolates the Fourier phase} and thus \\emph{recovers the Fourier transform of} $V_1-V_2$, yielding $V_1=V_2$ (uniqueness).\n\n6. The argument uses crucially that the potential is time-independent: one can use time-harmonic solutions with leading term $e^{-i|\\kappa|^2 t}e^{-i\\kappa\\cdot x}$, which is \\emph{purely oscillatory and unimodular}, avoiding the growth issues of complex geometrical optics solutions used in the time-dependent setting.", + "expanded_sketch": "The proof of the main theorem extracts information on $V_1-V_2$ from $\\mathcal U_T^1=\\mathcal U_T^2$ by testing against suitable families of solutions.\n\n1. From $\\mathcal U_T^1=\\mathcal U_T^2$, first derive an \\emph{Alessandrini-type orthogonality relation} (initially for \\emph{physical solutions} in $C([0,T];L^2(\\mathbb{R}^n))$):\n\\[\n\\int_{\\Sigma} (V_1-V_2)\\,u_1\\,\\overline{v_2}=0,\n\\]\nfor pairs of solutions associated with $V_1$ and $\\overline{V_2}$.\n\n2. To recover pointwise/Fourier information on $V_1-V_2$, test the orthogonality identity with special \\emph{time-harmonic} solutions (\\emph{stationary states}), constructed as perturbations of free time-harmonic Schr\\\"odinger solutions. These have the form\n\\[\n u(t,x)=e^{-i|\\kappa|^2 t}\\big(e^{-i\\kappa\\cdot x}+w^{\\mathrm{cor}}(x)\\big),\n\\]\nwhere $w^{\\mathrm{cor}}$ accounts for the potential. Since these stationary states are \\emph{not} in $C([0,T];L^2)$, one must \\emph{extend the Alessandrini-type orthogonality relation beyond physical solutions} to allow using them as test functions.\n\n3. Construct $w^{\\mathrm{cor}}$ by inverting a resolvent-type operator and building correction terms via Neumann series: reduce to inverting\n\\(\\mathrm{Id}-P_\\lambda\\circ V\\), where $P_\\lambda$ solves the Helmholtz equation $(\\Delta+\\lambda^2)u=f$, on function spaces where $P_\\lambda\\circ V$ is small.\n\n4. Handle the endpoint obstruction: for $q>n/2$, smallness for large $\\lambda$ comes from the classical Kenig--Ruiz--Sogge resolvent estimate, but at the critical endpoint $q=n/2$ it yields no decay in $\\lambda$. To fix this, \\emph{introduce a new scale of Banach spaces} that recovers a decaying factor in $\\lambda$ and establish an \\emph{improved} Kenig--Ruiz--Sogge resolvent estimate, enabling stationary-state construction even at the endpoint.\n\n5. Insert these stationary states into the \\emph{extended} orthogonality relation; this \\emph{isolates the Fourier phase} and thus \\emph{recovers the Fourier transform of} $V_1-V_2$, yielding $V_1=V_2$ (uniqueness).\n\n6. The argument uses crucially that the potential is time-independent: one can use time-harmonic solutions with leading term $e^{-i|\\kappa|^2 t}e^{-i\\kappa\\cdot x}$, which is \\emph{purely oscillatory and unimodular}, avoiding the growth issues of complex geometrical optics solutions used in the time-dependent setting.", + "expanded_theorem": "\\sl \\label{th:Lq} Let $V_1,V_2\\in L^1(\\mathbb{R}^n)\\cap L^q(\\mathbb{R}^n)$ be time-independent potentials, where $q>1$ if $n=2$ and $q\\ge n/2$ if $n\\ge 3$. Let $\\mathcal U_T^1$ and $\\mathcal U_T^2$ denote the corresponding initial-to-final-state maps. Then\n\\[\n\\mathcal U_T^1=\\mathcal U_T^2 \\quad\\Longrightarrow\\quad V_1=V_2.\n\\]", + "theorem_type": [ + "Implication", + "Universal" + ], + "mcq": { + "question": "Fix $n\\ge 2$ and a final time $T>0$. For $j=1,2$, let $V_j=V_j(x)$ be a time-independent potential on $\\mathbb{R}^n$ such that $V_j\\in L^1(\\mathbb{R}^n)\\cap L^q(\\mathbb{R}^n)$, where $q>1$ if $n=2$ and $q\\ge n/2$ if $n\\ge 3$. For each initial datum $f\\in L^2(\\mathbb{R}^n)$, let $u_j$ be the solution of\n\\[\n\\begin{cases}\ni\\partial_t u = -\\Delta u + V_j u & \\text{for } (t,x)\\in(0,T)\\times \\mathbb{R}^n,\\\\\nu(0,x)=f(x) & \\text{for } x\\in \\mathbb{R}^n,\n\\end{cases}\n\\]\nand define the initial-to-final-state map $\\mathcal U_T^j:L^2(\\mathbb{R}^n)\\to L^2(\\mathbb{R}^n)$ by $\\mathcal U_T^j f=u_j(T,\\cdot)$. Which statement holds for every such pair of potentials?", + "correct_choice": { + "label": "A", + "text": "If $\\mathcal U_T^1=\\mathcal U_T^2$ as operators from $L^2(\\mathbb{R}^n)$ to $L^2(\\mathbb{R}^n)$, then $V_1=V_2$ almost everywhere on $\\mathbb{R}^n$." + }, + "choices": [ + { + "label": "B", + "text": "If $\\mathcal U_T^1=\\mathcal U_T^2$ as operators from $L^2(\\mathbb{R}^n)$ to $L^2(\\mathbb{R}^n)$, then $V_1=V_2$ almost everywhere on $\\mathbb{R}^n$, provided in addition that $q>n/2$ when $n\\ge 3$." + }, + { + "label": "C", + "text": "If $V_1=V_2$ almost everywhere on $\\mathbb{R}^n$, then $\\mathcal U_T^1=\\mathcal U_T^2$ as operators from $L^2(\\mathbb{R}^n)$ to $L^2(\\mathbb{R}^n)$." + }, + { + "label": "D", + "text": "If for every pair of physical solutions $u_1,u_2\\in C([0,T];L^2(\\mathbb{R}^n))$ associated with $V_1$ and $V_2$ one has\n\\[\n\\int_0^T\\!\\int_{\\mathbb{R}^n} (V_1-V_2)(x)\\,u_1(t,x)\\,\\overline{u_2(t,x)}\\,dx\\,dt=0,\n\\]\nthen $\\mathcal U_T^1=\\mathcal U_T^2$, and hence $V_1=V_2$ almost everywhere on $\\mathbb{R}^n$." + }, + { + "label": "E", + "text": "If $\\mathcal U_T^1=\\mathcal U_T^2$ as operators from $L^2(\\mathbb{R}^n)$ to $L^2(\\mathbb{R}^n)$, then the Fourier transform of $V_1-V_2$ vanishes almost everywhere on $\\mathbb{R}^n$, and therefore $V_1=V_2$ almost everywhere on $\\mathbb{R}^n$." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "characteristic", + "tampered_component": "endpoint case q=n/2", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "reverse implication only; drops uniqueness direction from map equality to potential equality", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "need to extend orthogonality beyond physical solutions and use conjugate-potential pairing", + "template_used": "property_confusion" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "Fourier recovery is pointwise in frequency via stationary-state tests, not merely almost everywhere", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly reveal the correct option. It states the setup and asks which claim is universally valid; the correct answer is not leaked directly, though option A closely matches the likely theorem-level conclusion." + }, + "TAS": { + "score": 1, + "justification": "This is close to a theorem-recall item: the correct choice is essentially the main uniqueness statement under the stated hypotheses. However, it is not a pure tautology because the alternatives modify endpoints, directions of implication, and related consequences." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish the exact valid statement from nearby variants, especially the endpoint case in B and the overclaims in D/E. Still, the item primarily tests recognition of the theorem rather than substantial generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically sophisticated and plausible: B tampers with the sharp range, C gives only the converse, D confuses an orthogonality condition with the full uniqueness mechanism, and E overstates Fourier-recovery consequences. They are distinct and target realistic failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with strong, nuanced distractors and little answer leakage, but it leans heavily on theorem recognition and is only moderately effective at testing genuine generative reasoning." + } + }, + { + "id": "2602.12190v1", + "paper_link": "http://arxiv.org/abs/2602.12190v1", + "theorems_cnt": 4, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm:highT}\n\t\tFix $\\beta<1$. Let $M=M(N)$ satisfy $M/N\\to0$, and let $k=k(N)\\to\\infty$\n\t\twith\n\t\t\\[\n\t\t\\frac{k(N)M(N)}{N}\\longrightarrow 0 \n\t\t\\]\n\t\t(which in particular allows $k=o(N)$ when $M$ is fixed).\n\n\t\tThen\n\t\t\\[\n\t\t\\TV\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\n\t\t\\longrightarrow 0\n\t\t\\qquad\\text{in }\\Pp_\\xi\\text{-probability}.\n\t\t\\]", + "start_pos": 11871, + "end_pos": 12237, + "label": "thm:highT" + }, + "ref_dict": { + "thm:stop_macro_Msmall": "\\begin{theorem}\t\t\\label{thm:stop_macro_Msmall}\n\t\tFix $\\beta\\in(0,1)$ and assume $M=M(N)\\to\\infty$ with $M=o(\\sqrt N)$.\n\t\tLet $k=k(N)$ satisfy $k/N\\to\\rho\\in(0,1)$.\n\t\tThen, in $\\P_\\xi$-probability,\n\t\t\\[\n\t\t\\TV\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 1.\n\t\t\\]\n\t\\end{theorem}", + "thm:crit_break_sqrtN": "\\begin{theorem}[Critical-window breakdown at $\\beta=1$ (fixed $M$)]\n\t\t\\label{thm:crit_break_sqrtN}\n\t\tFix $\\beta=1$ and let $M\\in\\mathbb N$ be fixed.\n\t\tLet $k=k(N)$ satisfy\n\t\t\\[\n\t\t\\frac{k}{\\sqrt N}\\longrightarrow c\\in(0,\\infty).\n\t\t\\]\n\t\tThen there exists a deterministic constant $b=b(c,M)>0$ such that\n\t\t\\[\n\t\t\\liminf_{N\\to\\infty} \\ d_{\\mathrm{TV}}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\ \\ge\\ b\n\t\t\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n\t\t\\]\n\t\tIn particular, propagation of chaos fails in the critical window $k\\asymp \\sqrt N$,\n\t\tshowing that the scaling in Theorem~\\ref{thm:crit_poc} is optimal.\n\n\t\\end{theorem}", + "thm:crit_poc": "\\begin{theorem}\t\t\\label{thm:crit_poc}\n\t\tFix $\\beta=1$ and let $M\\in\\mathbb N$ be fixed. Assume $k=k(N)\\to\\infty$ satisfies\n\t\t\\[\n\t\t\\frac{k(N)}{N^{1/2}}\\longrightarrow 0.\n\t\t\\]\n\t\tThen\n\t\t\\[\n\t\td_{\\mathrm{TV}}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\n\t\t\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n\t\t\\]\n\t\\end{theorem}", + "thm:highT": "\\begin{theorem}\t\t\\label{thm:highT}\n\t\tFix $\\beta<1$. Let $M=M(N)$ satisfy $M/N\\to0$, and let $k=k(N)\\to\\infty$\n\t\twith\n\t\t\\[\n\t\t\\frac{k(N)M(N)}{N}\\longrightarrow 0 \n\t\t\\]\n\t\t(which in particular allows $k=o(N)$ when $M$ is fixed).\n\n\t\tThen\n\t\t\\[\n\t\t\\TV\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\n\t\t\\longrightarrow 0\n\t\t\\qquad\\text{in }\\Pp_\\xi\\text{-probability}.\n\t\t\\]\n\t\\end{theorem}" + }, + "pre_theorem_intro_text_len": 9395, + "pre_theorem_intro_text": "Propagation of chaos is a central concept in the study of interacting\n\tparticle systems and mean-field models.\n\tRoughly speaking, it describes the phenomenon that,\n\tas the system size $N$ tends to infinity,\n\tfinite collections of particles behave asymptotically independently,\n\twith a common limiting law.\n\tPropagation of chaos originated in Kac's Markovian models for gas dynamics\n\t\\cite{Kac_foundations,Kac_probability}, an attempt to justify\n\tBoltzmann's ''Sto\\ss zahlenansatz''.\n\tPropagation of chaos has since become an important \n\tobject of study\n\tin probability, statistical mechanics, and mathematical physics.\n\tThe original approach by Kac was that,\n\tif at time 0 the finite marginal distributions of a system are product measures in the\n\tthermodynamic limit, then this should carry over to the time-evolved system. In equilibrium settings for mean-field\n\tGibbs measures (where the energy function is a \n\tfunction of the empirical measure) with a unique minimizer of the Helmholtz free energy, however, this was shown to\n\tfollow from the fact the extremal Gibbs measures locally look like product measures, i.e.\\\n\tthat any finite subset of spins forms a family of independent random variables in the thermodynamic limit \\cite[Theorem 3]{BAZ_chaos}.\n\tSuch results provide a rigorous justification of mean-field\n\tapproximations and explain why macroscopic behavior\n\tcan often be described by effective one-particle models. In the present note, we remain entirely within a static framework, and study propagation of chaos in the sense of asymptotic factorization of finite-dimensional marginals for families of (random) \n\tmean-field Gibbs measures.\n\n\tMoreover, for mean-field spin systems, propagation of chaos\n\tis closely tied to the high-temperature regime.\n\tIn classical models such as the Curie--Weiss Ising model,\n\tchaos holds when the inverse temperature is below the critical value,\n\twhile it breaks down in the low-temperature phase. There one has to replace\n\tthe product measure by a mixture of the (several) extremal limiting Gibbs measures.\n\tAn additional difficulty may arise when the Gibbs measures in question are random. To the best of our knowledge comparatively little is known in such situations (with the exception of of \\cite{BG98book} and \\cite{KL24}). The present paper contributes to the study of propagation of chaos for random mean-field Gibbs measures in the high-temperature and near-critical regimes.\n\n\tImportantly, in \\cite{BAZ_chaos}, the authors also introduced the concept of \\textit{increasing} propagation of chaos and showed that in the true high temperature regime of many mean-field models one can let the size of the marginals $k$ grow with the system size $N$ as long as $k=o(N)$. \n\tUnderstanding not only whether propagation of chaos holds,\n\tbut also the precise scales at which it breaks down,\n\thas become an important theme in recent work on mean-field Gibbs measures \\cite{BAZ_chaos,Lacker22, JKLM23,KL24,RS25,JKL25}.\n\n\tWhile increasing propagation of chaos has been established in several genuinely high-temperature regimes of ordered mean-field models, much less is known in the presence of random mean-field Gibbs measures, where already the analysis of critical fluctuations can become delicate (\\cite{gentzloewe, gentzloewe2, talagrand_critical_hopf}). \n\tIn the present setting of random mean-field Gibbs measures, we show that this high-temperature scaling persists for $\\beta <1$ (the critical value),\n\twhereas at criticality \n\t$\\beta=1$, the admissible growth drops to $k=o(\\sqrt N)$. Moreover, \n\tthese bounds on \n\t$k$ are optimal for the class of random mean-field Gibbs measures considered here.\n\n\tIn this article we study propagation of chaos\n\tin the Hopfield model, a paradigmatic example\n\tof a disordered mean-field spin system.\n\tThe Hopfield model has (at least) two distinct origins. It was first introduced by\n\tPastur and Figotin \\cite{PS84} as a solvable model of a disordered system. At about the same time it\n\twas independently invented by Hopfield as\n\ta model of associative memory \\cite{Hopfield1982}.\n\tBoth aspects have been intensively studied. For the disordered systems facet see e.g. \\cite{BGP94, BG97} or \\cite{talagrand}, while \n\tthe neural network aspect has found a renewed interest through two recent papers \\cite{KrotovHopfield2016,DHLUV17}.\n\n\tFor the purposes of the present paper, the \n\tprobabilistic viewpoint of seeing the Hopfield model\n\tas a mean-field Ising model with random, structured interactions is more appropriate.\n\tThe disorder, i.e.\\ the random interactions, is generated by a collection of random patterns,\n\twhich induces a random quadratic Hamiltonian\n\tand leads to a rich interplay between thermal fluctuations\n\tand quenched randomness.\n\tWhile the thermodynamic properties of the Hopfield model\n\tare well understood, much less is known about\n\tthe fine structure of its finite-dimensional marginals\n\tand their asymptotic independence properties. There is only one result concerning the propagation of chaos in the Hopfield model, see \\cite[Theorem 8.15]{BG98}. However, this result addresses fixed-dimensional marginals in the low-temperature regime, and relies on mechanisms different from those considered here.\n\n\tOur main goal is to analyze propagation of chaos\n\tfor the Hopfield Gibbs measure in the high-temperature\n\tand critical regimes, with particular emphasis\n\ton the size of the marginals.\n\tWe identify precise conditions under which propagation of chaos holds,\n\tas well as sharp thresholds for its breakdown.\n\tIn the high-temperature regime, we prove propagation of chaos\n\tfor growing marginals, provided their size grows sublinearly in the system size $N$.\n\tAt criticality, we show that chaos breaks down in a critical window,\n\twhose scale matches that of the dominant collective fluctuations.\n\treveal a clear transition between asymptotic independence and regimes of partial or complete breakdown of chaos,\n\tand highlight the role played by disorder-induced fluctuations\n\tin determining these regimes.\n\n\t\\subsection{The Model}\n\tLet us next describe the central model for the purpose of this note.\n\tLet $\\xi_i=(\\xi_i^1,\\dots,\\xi_i^M)\\in\\{-1,+1\\}^M$, $i=1,\\dots,N$, be i.i.d.\\ random vectors with independent coordinates,\n\t\\[\n\t\\mathbb E[\\xi_i^\\nu]=0, \\qquad \\mathbb E[(\\xi_i^\\nu)^2]=1 .\n\t\\]\n\t(In the interpretation of an associative memory, the vectors $(\\xi^\\mu)_{\\mu=1}^M=((\\xi_i^\\mu)_{i=1}^N)_{\\mu=1}^M$ are called images or patterns).\n\tIn what follows we will always assume that $M=M(N)$ may depend on $N$, but in such a way, that $M=o(N)$, which is the natural regime in which the Hopfield model exhibits mean-field behavior. \n\tFor fixed, i.e.\\ quenched, patterns $(\\xi^\\mu)_\\mu$\tdefine and for $\\sigma\\in\\{-1,+1\\}^N$, define the overlap vector\n\t\\[\n\tm_N(\\sigma) = \\big(m_N^1(\\sigma),\\dots,m_N^M(\\sigma)\\big),\n\t\\qquad\n\tm_N^\\nu(\\sigma)=\\frac1N\\sum_{i=1}^N \\sigma_i\\xi_i^\\nu .\n\t\\]\n\tThe Hopfield Gibbs measure at inverse temperature $\\beta>0$ is\n\t\\[\n\t\\mu_N(\\sigma)=\\frac{1}{Z_N}\n\t\\exp\\Big(\\frac{\\beta N}{2}\\|m_N(\\sigma)\\|^2\\Big).\n\t\\]\n\tThe Gibbs measure is fully determined by $m_N(\\sigma)$ (plus the inverse temperature), it thus is natural to consider \n\tthe behaviour of the overlap under the Gibbs measure.\n\tThis has been done in numerous papers: \n\tIn \\cite{BGP94} it was shown that -- similar to the Curie-Weiss model -- in the Hopfield model the critical (inverse) temperature is $\\beta=1$. While for $\\beta \\le 1$ the overlap vector gets concentrated in the $M$-dimensional 0-vector, for larger $\\beta$ the limit points of $m_N$ are associated with the 2$M$ vectors $\\pm z(\\beta) e_\\mu$ where \n\t$e_\\mu$ is the $\\mu$'th unit vector and $z(\\beta)$ is the largest solution of the equation\n\t$$\n\tz=\\tanh(\\beta z).\n\t$$\n\tCentral Limit theorems for $\\sqrt N m_N(\\cdot)$ were proven by Gentz \\cite{gentz_annals} or Bovier and Gayrard \\cite{BG_CLT}. All these results hold true for almost all realizations of the patterns and only need $M=o(N)$\n\t(for \\cite{gentz_annals} this is only true in the high temperature regime, which, however, is the most relevant for us in the present note.) Importantly, as in the Curie-Weiss model, at $\\beta=1$ the fluctuations are non-Gaussian. As was shown in \\cite{gentzloewe, gentzloewe2, talagrand_critical_hopf} at $\\beta=1$\n\tthe rescaled overlap vector $N^{1/4} m_N$ converges in distribution to a random limit (while in the other limit theorems the limit was deterministic). Also large and moderate deviations for the overlap vector are available (see \\cite{BG_LDP, EL_hopf}).\n\n\t\\subsection{Statement of the results}\n\tIn this subsection we state our main results on increasing propagation of chaos\n\tfor the Hopfield model in the high-temperature and critical regimes. In particular, we identify regimes in which propagation of chaos breaks down\n\tat explicit scales of the marginals. For $\\beta<1$, we allow the number of patterns $M=M(N)$ to diverge with $N$,\n\tand establish both increasing propagation of chaos and sharp breakdown results.\n\tAt criticality $\\beta=1$, we restrict attention to fixed $M$, and identify the\n\tcritical window in which propagation of chaos fails.\n\n\tIn order to formulate our results, let us agree on the following notation: \n\tFor $k\\le N$, let $\\mu_N^{(k)}$ denote the marginal of $\\mu_N$ on\n\t$(\\sigma_1,\\dots,\\sigma_k)$.\n\tLet $\\pi$ be the Rademacher law on $\\{-1,+1\\}$ with $\\pi(\\pm1)=\\frac12$.\n\n\tThen we will prove:", + "context": "Propagation of chaos is a central concept in the study of interacting\n particle systems and mean-field models.\n Roughly speaking, it describes the phenomenon that,\n as the system size $N$ tends to infinity,\n finite collections of particles behave asymptotically independently,\n with a common limiting law.\n Propagation of chaos originated in Kac's Markovian models for gas dynamics\n \\cite{Kac_foundations,Kac_probability}, an attempt to justify\n Boltzmann's ''Sto\\ss zahlenansatz''.\n Propagation of chaos has since become an important \n object of study\n in probability, statistical mechanics, and mathematical physics.\n The original approach by Kac was that,\n if at time 0 the finite marginal distributions of a system are product measures in the\n thermodynamic limit, then this should carry over to the time-evolved system. In equilibrium settings for mean-field\n Gibbs measures (where the energy function is a \n function of the empirical measure) with a unique minimizer of the Helmholtz free energy, however, this was shown to\n follow from the fact the extremal Gibbs measures locally look like product measures, i.e.\\\n that any finite subset of spins forms a family of independent random variables in the thermodynamic limit \\cite[Theorem 3]{BAZ_chaos}.\n Such results provide a rigorous justification of mean-field\n approximations and explain why macroscopic behavior\n can often be described by effective one-particle models. In the present note, we remain entirely within a static framework, and study propagation of chaos in the sense of asymptotic factorization of finite-dimensional marginals for families of (random) \n mean-field Gibbs measures.\n\nImportantly, in \\cite{BAZ_chaos}, the authors also introduced the concept of \\textit{increasing} propagation of chaos and showed that in the true high temperature regime of many mean-field models one can let the size of the marginals $k$ grow with the system size $N$ as long as $k=o(N)$. \n Understanding not only whether propagation of chaos holds,\n but also the precise scales at which it breaks down,\n has become an important theme in recent work on mean-field Gibbs measures \\cite{BAZ_chaos,Lacker22, JKLM23,KL24,RS25,JKL25}.\n\n\\subsection{The Model}\n Let us next describe the central model for the purpose of this note.\n Let $\\xi_i=(\\xi_i^1,\\dots,\\xi_i^M)\\in\\{-1,+1\\}^M$, $i=1,\\dots,N$, be i.i.d.\\ random vectors with independent coordinates,\n \\[\n \\mathbb E[\\xi_i^\\nu]=0, \\qquad \\mathbb E[(\\xi_i^\\nu)^2]=1 .\n \\]\n (In the interpretation of an associative memory, the vectors $(\\xi^\\mu)_{\\mu=1}^M=((\\xi_i^\\mu)_{i=1}^N)_{\\mu=1}^M$ are called images or patterns).\n In what follows we will always assume that $M=M(N)$ may depend on $N$, but in such a way, that $M=o(N)$, which is the natural regime in which the Hopfield model exhibits mean-field behavior. \n For fixed, i.e.\\ quenched, patterns $(\\xi^\\mu)_\\mu$ define and for $\\sigma\\in\\{-1,+1\\}^N$, define the overlap vector\n \\[\n m_N(\\sigma) = \\big(m_N^1(\\sigma),\\dots,m_N^M(\\sigma)\\big),\n \\qquad\n m_N^\\nu(\\sigma)=\\frac1N\\sum_{i=1}^N \\sigma_i\\xi_i^\\nu .\n \\]\n The Hopfield Gibbs measure at inverse temperature $\\beta>0$ is\n \\[\n \\mu_N(\\sigma)=\\frac{1}{Z_N}\n \\exp\\Big(\\frac{\\beta N}{2}\\|m_N(\\sigma)\\|^2\\Big).\n \\]\n The Gibbs measure is fully determined by $m_N(\\sigma)$ (plus the inverse temperature), it thus is natural to consider \n the behaviour of the overlap under the Gibbs measure.\n This has been done in numerous papers: \n In \\cite{BGP94} it was shown that -- similar to the Curie-Weiss model -- in the Hopfield model the critical (inverse) temperature is $\\beta=1$. While for $\\beta \\le 1$ the overlap vector gets concentrated in the $M$-dimensional 0-vector, for larger $\\beta$ the limit points of $m_N$ are associated with the 2$M$ vectors $\\pm z(\\beta) e_\\mu$ where \n $e_\\mu$ is the $\\mu$'th unit vector and $z(\\beta)$ is the largest solution of the equation\n $$\n z=\\tanh(\\beta z).\n $$\n Central Limit theorems for $\\sqrt N m_N(\\cdot)$ were proven by Gentz \\cite{gentz_annals} or Bovier and Gayrard \\cite{BG_CLT}. All these results hold true for almost all realizations of the patterns and only need $M=o(N)$\n (for \\cite{gentz_annals} this is only true in the high temperature regime, which, however, is the most relevant for us in the present note.) Importantly, as in the Curie-Weiss model, at $\\beta=1$ the fluctuations are non-Gaussian. As was shown in \\cite{gentzloewe, gentzloewe2, talagrand_critical_hopf} at $\\beta=1$\n the rescaled overlap vector $N^{1/4} m_N$ converges in distribution to a random limit (while in the other limit theorems the limit was deterministic). Also large and moderate deviations for the overlap vector are available (see \\cite{BG_LDP, EL_hopf}).\n\n\\subsection{Statement of the results}\n In this subsection we state our main results on increasing propagation of chaos\n for the Hopfield model in the high-temperature and critical regimes. In particular, we identify regimes in which propagation of chaos breaks down\n at explicit scales of the marginals. For $\\beta<1$, we allow the number of patterns $M=M(N)$ to diverge with $N$,\n and establish both increasing propagation of chaos and sharp breakdown results.\n At criticality $\\beta=1$, we restrict attention to fixed $M$, and identify the\n critical window in which propagation of chaos fails.\n\nIn order to formulate our results, let us agree on the following notation: \n For $k\\le N$, let $\\mu_N^{(k)}$ denote the marginal of $\\mu_N$ on\n $(\\sigma_1,\\dots,\\sigma_k)$.\n Let $\\pi$ be the Rademacher law on $\\{-1,+1\\}$ with $\\pi(\\pm1)=\\frac12$.\n\nThen we will prove:", + "full_context": "Propagation of chaos is a central concept in the study of interacting\n particle systems and mean-field models.\n Roughly speaking, it describes the phenomenon that,\n as the system size $N$ tends to infinity,\n finite collections of particles behave asymptotically independently,\n with a common limiting law.\n Propagation of chaos originated in Kac's Markovian models for gas dynamics\n \\cite{Kac_foundations,Kac_probability}, an attempt to justify\n Boltzmann's ''Sto\\ss zahlenansatz''.\n Propagation of chaos has since become an important \n object of study\n in probability, statistical mechanics, and mathematical physics.\n The original approach by Kac was that,\n if at time 0 the finite marginal distributions of a system are product measures in the\n thermodynamic limit, then this should carry over to the time-evolved system. In equilibrium settings for mean-field\n Gibbs measures (where the energy function is a \n function of the empirical measure) with a unique minimizer of the Helmholtz free energy, however, this was shown to\n follow from the fact the extremal Gibbs measures locally look like product measures, i.e.\\\n that any finite subset of spins forms a family of independent random variables in the thermodynamic limit \\cite[Theorem 3]{BAZ_chaos}.\n Such results provide a rigorous justification of mean-field\n approximations and explain why macroscopic behavior\n can often be described by effective one-particle models. In the present note, we remain entirely within a static framework, and study propagation of chaos in the sense of asymptotic factorization of finite-dimensional marginals for families of (random) \n mean-field Gibbs measures.\n\nImportantly, in \\cite{BAZ_chaos}, the authors also introduced the concept of \\textit{increasing} propagation of chaos and showed that in the true high temperature regime of many mean-field models one can let the size of the marginals $k$ grow with the system size $N$ as long as $k=o(N)$. \n Understanding not only whether propagation of chaos holds,\n but also the precise scales at which it breaks down,\n has become an important theme in recent work on mean-field Gibbs measures \\cite{BAZ_chaos,Lacker22, JKLM23,KL24,RS25,JKL25}.\n\n\\subsection{The Model}\n Let us next describe the central model for the purpose of this note.\n Let $\\xi_i=(\\xi_i^1,\\dots,\\xi_i^M)\\in\\{-1,+1\\}^M$, $i=1,\\dots,N$, be i.i.d.\\ random vectors with independent coordinates,\n \\[\n \\mathbb E[\\xi_i^\\nu]=0, \\qquad \\mathbb E[(\\xi_i^\\nu)^2]=1 .\n \\]\n (In the interpretation of an associative memory, the vectors $(\\xi^\\mu)_{\\mu=1}^M=((\\xi_i^\\mu)_{i=1}^N)_{\\mu=1}^M$ are called images or patterns).\n In what follows we will always assume that $M=M(N)$ may depend on $N$, but in such a way, that $M=o(N)$, which is the natural regime in which the Hopfield model exhibits mean-field behavior. \n For fixed, i.e.\\ quenched, patterns $(\\xi^\\mu)_\\mu$ define and for $\\sigma\\in\\{-1,+1\\}^N$, define the overlap vector\n \\[\n m_N(\\sigma) = \\big(m_N^1(\\sigma),\\dots,m_N^M(\\sigma)\\big),\n \\qquad\n m_N^\\nu(\\sigma)=\\frac1N\\sum_{i=1}^N \\sigma_i\\xi_i^\\nu .\n \\]\n The Hopfield Gibbs measure at inverse temperature $\\beta>0$ is\n \\[\n \\mu_N(\\sigma)=\\frac{1}{Z_N}\n \\exp\\Big(\\frac{\\beta N}{2}\\|m_N(\\sigma)\\|^2\\Big).\n \\]\n The Gibbs measure is fully determined by $m_N(\\sigma)$ (plus the inverse temperature), it thus is natural to consider \n the behaviour of the overlap under the Gibbs measure.\n This has been done in numerous papers: \n In \\cite{BGP94} it was shown that -- similar to the Curie-Weiss model -- in the Hopfield model the critical (inverse) temperature is $\\beta=1$. While for $\\beta \\le 1$ the overlap vector gets concentrated in the $M$-dimensional 0-vector, for larger $\\beta$ the limit points of $m_N$ are associated with the 2$M$ vectors $\\pm z(\\beta) e_\\mu$ where \n $e_\\mu$ is the $\\mu$'th unit vector and $z(\\beta)$ is the largest solution of the equation\n $$\n z=\\tanh(\\beta z).\n $$\n Central Limit theorems for $\\sqrt N m_N(\\cdot)$ were proven by Gentz \\cite{gentz_annals} or Bovier and Gayrard \\cite{BG_CLT}. All these results hold true for almost all realizations of the patterns and only need $M=o(N)$\n (for \\cite{gentz_annals} this is only true in the high temperature regime, which, however, is the most relevant for us in the present note.) Importantly, as in the Curie-Weiss model, at $\\beta=1$ the fluctuations are non-Gaussian. As was shown in \\cite{gentzloewe, gentzloewe2, talagrand_critical_hopf} at $\\beta=1$\n the rescaled overlap vector $N^{1/4} m_N$ converges in distribution to a random limit (while in the other limit theorems the limit was deterministic). Also large and moderate deviations for the overlap vector are available (see \\cite{BG_LDP, EL_hopf}).\n\n\\subsection{Statement of the results}\n In this subsection we state our main results on increasing propagation of chaos\n for the Hopfield model in the high-temperature and critical regimes. In particular, we identify regimes in which propagation of chaos breaks down\n at explicit scales of the marginals. For $\\beta<1$, we allow the number of patterns $M=M(N)$ to diverge with $N$,\n and establish both increasing propagation of chaos and sharp breakdown results.\n At criticality $\\beta=1$, we restrict attention to fixed $M$, and identify the\n critical window in which propagation of chaos fails.\n\nIn order to formulate our results, let us agree on the following notation: \n For $k\\le N$, let $\\mu_N^{(k)}$ denote the marginal of $\\mu_N$ on\n $(\\sigma_1,\\dots,\\sigma_k)$.\n Let $\\pi$ be the Rademacher law on $\\{-1,+1\\}$ with $\\pi(\\pm1)=\\frac12$.\n\nThen we will prove:\n\nIn order to formulate our results, let us agree on the following notation: \n For $k\\le N$, let $\\mu_N^{(k)}$ denote the marginal of $\\mu_N$ on\n $(\\sigma_1,\\dots,\\sigma_k)$.\n Let $\\pi$ be the Rademacher law on $\\{-1,+1\\}$ with $\\pi(\\pm1)=\\frac12$.\n\n\\begin{theorem} \\label{thm:stop_macro_Msmall}\n Fix $\\beta\\in(0,1)$ and assume $M=M(N)\\to\\infty$ with $M=o(\\sqrt N)$.\n Let $k=k(N)$ satisfy $k/N\\to\\rho\\in(0,1)$.\n Then, in $\\P_\\xi$-probability,\n \\[\n \\TV\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 1.\n \\]\n \\end{theorem}\n\n\\begin{theorem} \\label{thm:crit_poc}\n Fix $\\beta=1$ and let $M\\in\\mathbb N$ be fixed. Assume $k=k(N)\\to\\infty$ satisfies\n \\[\n \\frac{k(N)}{N^{1/2}}\\longrightarrow 0.\n \\]\n Then\n \\[\n d_{\\mathrm{TV}}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\n \\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n \\]\n \\end{theorem}\n\nAnd finally, we show\n \\begin{theorem}[Critical-window breakdown at $\\beta=1$ (fixed $M$)]\n \\label{thm:crit_break_sqrtN}\n Fix $\\beta=1$ and let $M\\in\\mathbb N$ be fixed.\n Let $k=k(N)$ satisfy\n \\[\n \\frac{k}{\\sqrt N}\\longrightarrow c\\in(0,\\infty).\n \\]\n Then there exists a deterministic constant $b=b(c,M)>0$ such that\n \\[\n \\liminf_{N\\to\\infty} \\ d_{\\mathrm{TV}}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\ \\ge\\ b\n \\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n \\]\n In particular, propagation of chaos fails in the critical window $k\\asymp \\sqrt N$,\n showing that the scaling in Theorem~\\ref{thm:crit_poc} is optimal.\n\n\\begin{lemma}\n \\label{lem:covconcentration}\n Let $\\xi_1,\\dots,\\xi_N\\in\\mathbb R^M$ be i.i.d.\\ centered subgaussian random vectors\n with covariance $\\Sigma:=\\E[\\xi_1\\xi_1^\\top]$.\n Let\n \\[\n \\widehat\\Sigma_N := \\frac1N\\sum_{i=1}^N \\xi_i\\xi_i^\\top .\n \\]\n Then there exists an absolute constant $C>0$ (depending only on the subgaussian\n norm of $\\xi_1$) such that for every $u\\ge 0$,\n \\begin{equation}\\label{eq:conc}\n \\Big\\|\\widehat\\Sigma_N-\\Sigma\\Big\\|_{\\mathrm{op}}\n \\le\n C\\Bigg(\\sqrt{\\frac{M+u}{N}}+\\frac{M+u}{N}\\Bigg)\\,\\|\\Sigma\\|_{\\mathrm{op}}\n \\qquad\\text{with probability at least }1-2e^{-u}.\n \\end{equation}\n In particular, if $\\Sigma=I_M$ (the $M$-dimensional identity matrix) and $M/N\\to 0$, then for every fixed $\\varepsilon>0$,\n \\[\n \\Pp\\Big(\\big\\|\\widehat\\Sigma_N-I_M\\big\\|_{\\mathrm{op}}\\le \\varepsilon\\Big)\\longrightarrow 1,\n \\qquad\\text{and hence}\\qquad\n \\Pp\\Big(\\|\\widehat\\Sigma_N\\|_{\\mathrm{op}}\\le 1+\\varepsilon\\Big)\\longrightarrow 1.\n \\]\n \\end{lemma}\n\n\\begin{lemma} \\label{lem:UI_LN_correct}\n Assume $k/N\\to\\rho\\in(0,1)$ and $M=o(k)$, and set\n $\\lambda_N:=\\tau^2 k/N$ with $\\tau^2=\\beta/(1-\\beta)$, so $\\lambda_N\\to\\lambda>0$.\n Let $L_N$ be given on $\\mathcal G_N$ by\n \\[\n L_N\n =\n (1+\\lambda_N)^{-M/2}\n \\exp\\!\\Big(\\alpha_N\\|S_{N,k}\\|^2\\Big),\n \\qquad \\text{with}\\quad\n \\alpha_N:=\\frac{\\lambda_N}{2(1+\\lambda_N)}.\n \\]\n Then there exists $\\eta>0$ such that\n \\[\n \\sup_N \\E_{\\pi^{\\otimes k}}\\big[L_N^{1+\\eta}\\mathbf 1_{\\mathcal G_N}\\big] <\\infty.\n \\]\n In particular, $(L_N)_N$ is uniformly integrable under $\\pi^{\\otimes k}$ on $\\mathcal G_N$.\n \\end{lemma}\n\nLet $r_\\ast^2:=\\frac{M(1+\\lambda)\\log(1+\\lambda)}{\\lambda}$be the unique radius where the two densities cross.\n Define the (optimal) set is $$A_\\ast:=\\{\\|x\\|\\ge r_\\ast\\}.$$ Then,\n \\[\n \\TV(\\mathcal N(0,I_M),\\mathcal N(0,(1+\\lambda)I_M))\n =\n \\P(Y\\in A_\\ast)-\\P(X\\in A_\\ast).\n \\]\n Since $\\chi^2_M/M\\to1$ in probability and concentrates at scale $\\sqrt M$,\n one has $\\|X\\|^2/M\\to1$ while $\\|Y\\|^2/M\\to 1+\\lambda$,\n and $r_\\ast^2/M$ lies strictly between $1$ and $1+\\lambda$.\n Therefore $\\P(X\\in A_\\ast)\\to0$ and $\\P(Y\\in A_\\ast)\\to1$, hence\n \\[\n \\TV(\\mathcal N(0,I_M),\\mathcal N(0,(1+\\lambda)I_M))\\longrightarrow 1\n \\qquad (M\\to\\infty).\n \\]\n Combining the previous steps yields $\\TV(\\mu_N^{(k)},\\pi^{\\otimes k})\\to1$ in $\\P_\\xi$-probability.\n \\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:crit_poc}]\n Let\n \\[\n L_N(\\sigma_{1:k})\n :=\\frac{d\\mu_N^{(k)}}{d\\pi^{\\otimes k}}(\\sigma_{1:k}).\n \\]\n Then, by the Cauchy--Schwarz inequality\n \\[\n d_{\\mathrm{TV}}(\\mu_N^{(k)},\\pi^{\\otimes k})\n =\\frac12\\,\\E_{\\pi^{\\otimes k}}\\big[|L_N-1|\\big]\n \\le \\frac12\\,\\Big(\\E_{\\pi^{\\otimes k}}\\big[(L_N-1)^2\\big]\\Big)^{1/2}\n =\\frac12\\,\\Big(\\E_{\\pi^{\\otimes k}}[L_N^2]-1\\Big)^{1/2}.\n \\]\n Thus it suffices to show that\n \\begin{equation}\\label{eq:chi2_to_0_beta1}\n \\E_{\\pi^{\\otimes k}}[L_N^2]\\longrightarrow 1\n \\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n \\end{equation}\n\n\\begin{theorem}\t\t\\label{thm:crit_poc}\n\t\tFix $\\beta=1$ and let $M\\in\\mathbb N$ be fixed. Assume $k=k(N)\\to\\infty$ satisfies\n\t\t\\[\n\t\t\\frac{k(N)}{N^{1/2}}\\longrightarrow 0.\n\t\t\\]\n\t\tThen\n\t\t\\[\n\t\td_{\\mathrm{TV}}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\n\t\t\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n\t\t\\]\n\t\\end{theorem}", + "post_theorem_intro_text_len": 4262, + "post_theorem_intro_text": "\\begin{theorem}\t\t\\label{thm:stop_macro_Msmall}\n\t\tFix $\\beta\\in(0,1)$ and assume $M=M(N)\\to\\infty$ with $M=o(\\sqrt N)$.\n\t\tLet $k=k(N)$ satisfy $k/N\\to\\rho\\in(0,1)$.\n\t\tThen, in $\\P_\\xi$-probability,\n\t\t\\[\n\t\t\\TV\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 1.\n\t\t\\]\n\t\\end{theorem}\n\n\t\\begin{remark}\n\t\tThe restriction $M=o(\\sqrt N)$ in Theorem~\\ref{thm:stop_macro_Msmall}\n\t\tis stronger than in Theorem~\\ref{thm:highT}.\n\t\tWhether this condition can be relaxed remains an open problem.\n\t\\end{remark}\t\n\n\t\\begin{theorem}\t\t\\label{thm:crit_poc}\n\t\tFix $\\beta=1$ and let $M\\in\\mathbb N$ be fixed. Assume $k=k(N)\\to\\infty$ satisfies\n\t\t\\[\n\t\t\\frac{k(N)}{N^{1/2}}\\longrightarrow 0.\n\t\t\\]\n\t\tThen\n\t\t\\[\n\t\td_{\\mathrm{TV}}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\n\t\t\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n\t\t\\]\n\t\\end{theorem}\n\n\tAnd finally, we show\n\t\\begin{theorem}[Critical-window breakdown at $\\beta=1$ (fixed $M$)]\n\t\t\\label{thm:crit_break_sqrtN}\n\t\tFix $\\beta=1$ and let $M\\in\\mathbb N$ be fixed.\n\t\tLet $k=k(N)$ satisfy\n\t\t\\[\n\t\t\\frac{k}{\\sqrt N}\\longrightarrow c\\in(0,\\infty).\n\t\t\\]\n\t\tThen there exists a deterministic constant $b=b(c,M)>0$ such that\n\t\t\\[\n\t\t\\liminf_{N\\to\\infty} \\ d_{\\mathrm{TV}}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\ \\ge\\ b\n\t\t\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n\t\t\\]\n\t\tIn particular, propagation of chaos fails in the critical window $k\\asymp \\sqrt N$,\n\t\tshowing that the scaling in Theorem~\\ref{thm:crit_poc} is optimal.\n\n\t\\end{theorem}\n\t\\subsection{Outline of the proofs}\n\n\tA central idea in our proofs is that the Hopfield Gibbs measure admits,\n\tvia a Hubbard--Stratonovich transformation, an explicit representation\n\tin terms of a mixture of product measures.\n\n\t\\medskip\n\t\\noindent\\textit{Step 1: Mixture-of-products representation.}\n\tFor fixed (quenched) patterns $(\\xi^\\mu)_\\mu$, the quadratic Hamiltonian\n\t$\\frac{\\beta N}{2}\\|m_N(\\sigma)\\|^2$ can be linearized by introducing an\n\tauxiliary Gaussian field. This standard technqie for quadratic Hamiltonians yields a representation of the form\n\t\\[\n\t\\mu_N(d\\sigma)\\;=\\;\\int \\nu_{N,\\beta}(d y)\\,\n\t\\bigotimes_{i=1}^N \\mu_{y,i}(d\\sigma_i),\n\t\\]\n\twhere $y\\in\\mathbb R^M$ is the Hubbard--Stratonovich field, $\\nu_{N,\\beta}$ is an\n\texplicit probability measure on $\\mathbb R^M$, and $\\mu_{y,i}$ is a Bernoulli law\n\twith bias depending on $y\\cdot \\xi_i$.\n\tConsequently, the $k$-spin marginal $\\mu_N^{(k)}$ is a mixture of product\n\tmeasures on $\\{-1,+1\\}^k$.\n\n\t\\medskip\n\t\\noindent\\textit{Step 2: Propagation of chaos reduces to stability of the mixture.}\n\tConditionally on $y$, the spins are independent.\n\tThus propagation of chaos for $\\mu_N$ is controlled by how strongly the\n\trandom field $y$ fluctuates under $\\nu_{N,\\beta}$ and by how sensitively\n\tthe single-spin biases $\\mu_{y,i}$ depend on $y$.\n\tQuantitatively, we compare the mixture to the unbiased product law\n\t$\\pi^{\\otimes k}$ in total variation distance.\n\n\t\\medskip\n\t\\noindent\\textit{Step 3: High temperature $\\beta<1$.}\n\tIn the regime $\\beta<1$, the Hubbard--Stratonovich field typically remains\n\tof order one and concentrates near the origin.\n\tA Taylor expansion of the single-spin biases and a control of\n\t$\\nu_{N,\\beta}$ show that correlations between $k$ spins are of order\n\t$kM/N$, yielding Theorem~\\ref{thm:highT} under the condition $kM/N\\to0$.\n\tFor macroscopic $k$ (with $k/N\\to\\rho$), the same representation allows us\n\tto exhibit order-one correlations induced by the random field, which leads\n\tto strong breakdown in total variation (Theorem~\\ref{thm:stop_macro_Msmall})\n\tunder the stated assumptions on $M$.\n\n\t\\medskip\n\t\\noindent\\textit{Step 4: Critical temperature $\\beta=1$ and the critical window.}\n\tAt criticality, the mixing measure $\\nu_{N,1}$ develops non-Gaussian\n\tfluctuations on the scale $N^{-1/4}$, reflecting the well-known\n\tcritical behavior of the overlap.\n\tThis amplification of the mixing fluctuations reduces the admissible growth\n\tof the marginals to $k=o(N^{1/4})$ (Theorem~\\ref{thm:crit_poc}).\n\tMoreover, when $k\\asymp \\sqrt N$, the mixture retains a nontrivial\n\tamount of randomness that produces correlations bounded away from zero,\n\tyielding a breakdown of propagation of chaos in the critical window\n\t(Theorem~\\ref{thm:crit_break_sqrtN}) and hence the optimality of the scaling.", + "sketch": "A central idea in our proofs is that the Hopfield Gibbs measure admits, via a Hubbard--Stratonovich transformation, an explicit representation in terms of a mixture of product measures.\n\n\\medskip\n\\noindent\\textit{Step 1: Mixture-of-products representation.}\nFor fixed (quenched) patterns $(\\xi^\\mu)_\\mu$, the quadratic Hamiltonian $\\frac{\\beta N}{2}\\|m_N(\\sigma)\\|^2$ is linearized by introducing an auxiliary Gaussian field (Hubbard--Stratonovich), yielding\n\\[\n\\mu_N(d\\sigma)=\\int \\nu_{N,\\beta}(d y)\\, \\bigotimes_{i=1}^N \\mu_{y,i}(d\\sigma_i),\n\\]\nwhere $y\\in\\R^M$, $\\nu_{N,\\beta}$ is an explicit probability measure on $\\R^M$, and $\\mu_{y,i}$ is a Bernoulli law with bias depending on $y\\cdot \\xi_i$. Consequently, the $k$-spin marginal $\\mu_N^{(k)}$ is a mixture of product measures on $\\{-1,+1\\}^k$.\n\n\\medskip\n\\noindent\\textit{Step 2: Propagation of chaos reduces to stability of the mixture.}\nConditionally on $y$, the spins are independent, so propagation of chaos is controlled by how strongly $y$ fluctuates under $\\nu_{N,\\beta}$ and by how sensitively the single-spin biases $\\mu_{y,i}$ depend on $y$. Quantitatively, one compares the mixture to the unbiased product law $\\pi^{\\otimes k}$ in total variation.\n\n\\medskip\n\\noindent\\textit{Step 3: High temperature $\\beta<1$ (Theorem~\\ref{thm:highT}).}\nIn the regime $\\beta<1$, the Hubbard--Stratonovich field “typically remains of order one and concentrates near the origin.” Using “a Taylor expansion of the single-spin biases and a control of $\\nu_{N,\\beta}$” shows that “correlations between $k$ spins are of order $kM/N$,” which yields Theorem~\\ref{thm:highT} under the condition $kM/N\\to0$.", + "expanded_sketch": "A central idea in our proofs is that the Hopfield Gibbs measure admits, via a Hubbard--Stratonovich transformation, an explicit representation in terms of a mixture of product measures.\n\n\\medskip\n\\noindent\\textit{Step 1: Mixture-of-products representation.}\nFor fixed (quenched) patterns $(\\xi^\\mu)_\\mu$, the quadratic Hamiltonian $\\frac{\\beta N}{2}\\|m_N(\\sigma)\\|^2$ is linearized by introducing an auxiliary Gaussian field (Hubbard--Stratonovich), yielding\n\\[\n\\mu_N(d\\sigma)=\\int \\nu_{N,\\beta}(d y)\\, \\bigotimes_{i=1}^N \\mu_{y,i}(d\\sigma_i),\n\\]\nwhere $y\\in\\R^M$, $\\nu_{N,\\beta}$ is an explicit probability measure on $\\R^M$, and $\\mu_{y,i}$ is a Bernoulli law with bias depending on $y\\cdot \\xi_i$. Consequently, the $k$-spin marginal $\\mu_N^{(k)}$ is a mixture of product measures on $\\{-1,+1\\}^k$.\n\n\\medskip\n\\noindent\\textit{Step 2: Propagation of chaos reduces to stability of the mixture.}\nConditionally on $y$, the spins are independent, so propagation of chaos is controlled by how strongly $y$ fluctuates under $\\nu_{N,\\beta}$ and by how sensitively the single-spin biases $\\mu_{y,i}$ depend on $y$. Quantitatively, one compares the mixture to the unbiased product law $\\pi^{\\otimes k}$ in total variation.\n\n\\medskip\n\\noindent\\textit{Step 3: High temperature $\\beta<1$ (main theorem).}\nIn the regime $\\beta<1$, the Hubbard--Stratonovich field “typically remains of order one and concentrates near the origin.” Using “a Taylor expansion of the single-spin biases and a control of $\\nu_{N,\\beta}$” shows that “correlations between $k$ spins are of order $kM/N$.” Under the condition $kM/N\\to0$, this yields the conclusion needed to establish the main theorem.", + "expanded_theorem": "\\label{thm:highT}\n\t\tFix $\\beta<1$. Let $M=M(N)$ satisfy $M/N\\to0$, and let $k=k(N)\\to\\infty$\n\t\twith\n\t\t\\[\n\t\t\\frac{k(N)M(N)}{N}\\longrightarrow 0 \n\t\t\\]\n\t\t(which in particular allows $k=o(N)$ when $M$ is fixed).\n\n\t\tThen\n\t\t\\[\n\t\t\\TV\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\n\t\t\\longrightarrow 0\n\t\t\\qquad\\text{in }\\Pp_\\xi\\text{-probability}.\n\t\t\\]", + "theorem_type": [ + "Asymptotic or Limit", + "Implication" + ], + "mcq": { + "question": "Let \\(\\xi_i=(\\xi_i^1,\\dots,\\xi_i^M)\\in\\{-1,+1\\}^M\\), \\(i=1,\\dots,N\\), be i.i.d. random vectors whose coordinates are independent and satisfy \\(\\mathbb E[\\xi_i^\\nu]=0\\) and \\(\\mathbb E[(\\xi_i^\\nu)^2]=1\\). For \\(\\sigma\\in\\{-1,+1\\}^N\\), define the overlap vector\n\\[\nm_N(\\sigma)=\\big(m_N^1(\\sigma),\\dots,m_N^M(\\sigma)\\big),\n\\qquad m_N^\\nu(\\sigma)=\\frac1N\\sum_{i=1}^N \\sigma_i\\xi_i^\\nu.\n\\]\nFor fixed inverse temperature \\(\\beta>0\\), the quenched Hopfield Gibbs measure on \\(\\{-1,+1\\}^N\\) is\n\\[\n\\mu_N(\\sigma)=\\frac1{Z_N}\\exp\\!\\Big(\\frac{\\beta N}{2}\\|m_N(\\sigma)\\|^2\\Big).\n\\]\nLet \\(\\mu_N^{(k)}\\) denote the marginal law of the first \\(k\\) spins under \\(\\mu_N\\), and let \\(\\pi\\) be the symmetric Bernoulli measure on \\(\\{-1,+1\\}\\), so that \\(\\pi^{\\otimes k}\\) is the product law of \\(k\\) i.i.d. fair \\(\\pm1\\) spins. Assume that \\(\\beta<1\\), that \\(M=M(N)\\) satisfies \\(M/N\\to0\\), and that \\(k=k(N)\\to\\infty\\) with\n\\[\n\\frac{k(N)M(N)}{N}\\longrightarrow 0.\n\\]\nHere total variation distance is \\(\\mathrm{TV}(\\nu,\\eta)=\\sup_A |\\nu(A)-\\eta(A)|\\), and \\(\\mathbb P_\\xi\\) denotes probability with respect to the random patterns \\((\\xi_i)\\). Under these assumptions, which statement about the \\(k\\)-spin marginal is valid?", + "correct_choice": { + "label": "A", + "text": "\\[\\mathrm{TV}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\\]" + }, + "choices": [ + { + "label": "B", + "text": "\\[\\mathrm{TV}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\\qquad\\text{almost surely in }\\mathbb P_\\xi.\\]" + }, + { + "label": "C", + "text": "\\[\\mu_N^{(k)}\\xrightarrow[]{\\ d\\ }\\pi^{\\otimes k}\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\\]" + }, + { + "label": "D", + "text": "If \\(M/N\\to0\\) and \\(k=o(N)\\), then for every fixed \\(\\beta<1\\),\n\\[\\mathrm{TV}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\\]" + }, + { + "label": "E", + "text": "Assume \\(\\beta<1\\), \\(M/N\\to0\\), and \\(k(N)M(N)/N\\to0\\). Then\n\\[N\\,\\mathrm{TV}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\\]" + } + ], + "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": "mode_of_convergence_in_patterns", + "template_used": "uniformity_effectivity" + }, + { + "label": "C", + "sketch_hook_type": "regularity", + "tampered_component": "total_variation_conclusion_replaced_by_weak_convergence", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "counting_estimate", + "tampered_component": "critical_scale_kM_over_N", + "template_used": "boundary_range" + }, + { + "label": "E", + "sketch_hook_type": "counting_estimate", + "tampered_component": "only_order_kM_over_N_control_not_rate_N_times_tv", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives the model, assumptions, and notation but does not state the conclusion or obviously reveal the correct option. The correct answer is not leaked explicitly or trivially." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-identification question: the correct choice is essentially the exact theorem conclusion under the stated assumptions. However, it is not a pure restatement because the alternatives vary in strength, mode of convergence, and scaling assumptions." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is required to distinguish total variation from weak convergence, probability from almost sure convergence, and the sharp condition kM/N -> 0 from weaker-looking but false variants. Still, the main task is recognizing the precise theorem statement rather than generating a new conclusion from scratch." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and well-targeted: one is stronger than the true result (almost sure), one is weaker but true, one relaxes the key scaling incorrectly, and one asserts an unjustified rate. These reflect realistic failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward precise recall/recognition of the result than deep generative reasoning." + } + }, + { + "id": "2602.12646v1", + "paper_link": "http://arxiv.org/abs/2602.12646v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theo", + "content": "\\label{theo.finite.top}\nFor fixed $n\\,,m \\in \\mathbb{Z}^+\\,,n \\geqslant 3\\,, m \\geq1\\,,$ and $ \\Gamma \\, ,\\Lambda \\in \\mathbb{R} \\,, \\Gamma\\,,\\Lambda \\geqslant 0\\,$, there exists $N = N(n,m,\\Gamma,\\Lambda) \\in \\mathbb{N}$ so that there are at most $N$ mutually non-diffeomorphic complete immersed minimal submanifolds $M^n$ in $\\mathbb{R}^{n+m}$ satisfying that $\\int_{M} \\lvertA\\rvert^{n}d\\mu_M \\leqslant \\Gamma$ and \n$\\vol_M ( B_R(0)) \\leqslant \\Lambda R^n$ for any $R>0\\,.$", + "start_pos": 11814, + "end_pos": 12264, + "label": "theo.finite.top" + }, + "ref_dict": { + "theo.finite.top": "\\begin{theo} \\label{theo.finite.top}\nFor fixed $n\\,,m \\in \\ZZ^+\\,,n \\geq 3\\,, m \\geq1\\,,$ and $ \\Gamma \\, ,\\Lambda \\in \\RR \\,, \\Gamma\\,,\\Lambda \\geq 0\\,$, there exists $N = N(n,m,\\Gamma,\\Lambda) \\in \\NN$ so that there are at most $N$ mutually non-diffeomorphic complete immersed minimal submanifolds $M^n$ in $\\RR^{n+m}$ satisfying that $\\int_{M} \\abs{A}^{n}d\\mu_M \\leq \\Gamma$ and \n$\\vol_M ( B_R(0)) \\leq \\Lambda R^n$ for any $R>0\\,.$\n\\end{theo}", + "lemm.Annular.composition": "\\begin{lemm}[Annular decomposition] \\label{lemm.Annular.composition}\nFor fixed $n\\,,m\\in \\ZZ^+\\,,n\\geq 2\\,,$ there is a $0 < \\sigma_0 <\\fh{1}{2} $ only depending on $n,m$ with the following property. Suppose that $M^n(\\iota : M^n \\to \\bar{B_2(0)} \\subset\\RR^{n+m})$ is a complete properly immersed submanifold with $\\iota(\\de M) \\subset \\de B_2(0)\\, . $ Assume that for some $\\sigma \\leq \\sigma_0$ and $p \\in B_{\\sigma_0}(0)\\,,$ we have:\n\\begin{enumerate}[itemsep=5pt, topsep=5pt]\n\\item For each component $M'$ of $M\\,,$ $M' \\cap B_{\\sigma}(p) \\neq \\emptyset \\,$.\n\\item The immersed submanifold $M$ intersects $\\de B_{\\sigma}(p)$ transversely, and $M \\cap \\de B_{\\sigma}(p)$ has $k$ components. Moreover, each component of $M \\cap \\de B_{\\sigma}(p)$ is diffeomorphic to $\\SS^{n-1}$ with the standard smooth structure.\n\\item The second fundamental form of $M$ satisfies $|A|(x)|\\iota(x)-p| \\leq \\fh{1}{4} $ for all $x \\in M \\cap \\left( \\overline{B_{1}(0)} \\setminus B_{\\sigma}(p) \\right)\\,.$\n\\end{enumerate}\nThen, $M$ intersects $\\de B_{1}(0)$ transversely. Both $M \\cap \\left(\\overline{B_{1}(0)}\\setminus {B_{\\sigma}(p)}\\right)$ and $M \\cap \\de B_{1}(0)$ have $k$ components. Moreover, each component of $M \\cap \\de B_{1}(0)$ is diffeomorphic to $\\SS^{n-1}$ with the standard smooth structure and\n each component of $M \\cap \\left(\\overline{B_{1}(0)}\\setminus {B_{\\sigma}(p)}\\right)$ is diffeomorphic to $\\SS^{n-1}\\times [0,1]$ with the standard smooth structure.\n\\end{lemm}", + "lemm:curv.est": "\\begin{lemm}\\label{lemm:curv.est}\nFor fixed $I \\in \\ZZ^+,0 0$ and a sequence of smooth blow-up sets $\\cB_{j} \\subset M_{j}$ so that \n\\begin{equation} \\label{eq:seq.cur.esti}\n |A_{M_{j}}|(x)d(\\iota_j(x),\\iota_j(\\cB_{j} \\cup \\de M_j) ) \\leq C\\, , \\ |\\cB_{j}|< I\\, , \\ \\iota_j(\\cB_j) \\subset B_{\\fh{3}{4}r_0}(0)\\, ,\n\\end{equation}\nfor all $x \\in M_{j}\\,.$ \n\\item $\\iota_j(\\cB_j)$ converges to $ \\widetilde \\cB_ \\infty \\subset \\RR^{n+m}$ in the Hausdorff distance sense and the Radon measure $\\nu_j$ converges to $ \\nu_\\infty$ in the Radon measure sense with $\\nu_\\infty(p_\\infty)\\geq 2K_0$ for any $p_\\infty \\in \\widetilde \\cB_\\infty\\,.$\n\\end{enumerate}\n\\end{lemm}", + "prop.diffeo": "\\begin{theo} \\label{prop.diffeo}\nGiven a sequence $M_{j}$ satisfying \\hyperlink{defi:beth}{$(\\beth)$} and each $M_j$ intersects $\\partial B_{1}(0)$ transversely. By passing to a subsequence, all of the $M_{j}\\cap B_{1}(0)$ are diffeomorphic. \n\\end{theo}" + }, + "pre_theorem_intro_text_len": 3642, + "pre_theorem_intro_text": "For minimal surfaces in $\\mathbb{R}^3$, finite total curvature means that the Gaussian curvature integral is finite. Chern and Osserman \\cite{Chern} proved that every minimal surface in $\\mathbb{R}^3$ is conformally equivalent to a compact Riemann surface $\\overline M$ punctured at a finite number of points, and the Gauss map on the surface can extend conformally to $\\overline M$.\n\nCollin \\cite{collin1} proved that any properly embedded minimal\nsurface in $\\mathbb{R}^3$ with finite topology and more than one end, has finite total curvature. \nColding and Minicozzi \\cite{ColdingMincozzi1} removed the proper condition, where they proved that a complete embedded minimal surface $\\Sigma$ with finite topology in $\\mathbb{R}^3$ must be proper.\nMeeks, Perez and Ros \\cite{Meeks2} showed that the number of ends of $\\Sigma$ is bounded by a constant depending on its genus.\n\nGiven an immersed minimal submanifold $M^{n}$ in $ \\mathbb{R}^{n+m}\\,,$ $M$ is said to have \\emph{finite total curvature} if\n $$\\int_{M} \\lvertA\\rvert^{n}d\\mu_M < \\infty\\,,$$\nwhere $A$ denotes the second fundamental form of $M$ in $ \\mathbb{R}^{n+m}$, and $\\mu_M$ denotes the volume element of $M$.\n\nAnderson \\cite{anderson1984compactification} gave a generalization of the\nChern-Osserman theorem \\cite{Chern} on minimal surfaces of finite total curvature: a complete minimal submanifold $M^n$ with finite total curvature is diffeomorphic to a compact $C^{\\infty} $ manifold $\\overline{M}^n$ punctured at a finite number of points $\\{p_i\\}_{1}^{\\ell} \\in \\overline{M}^n$ and the Gauss map $\\gamma:M^n \\to G_{n,m}$ extends to a $C^{n-2}$ map $\\overline \\gamma: \\overline M^n \\to G_{n,m} $ of the compactification(where $G_{n,m}$ denotes the Grassmann manifold of $n$-planes in Euclidean $(n+m)$-space). \nIn particular, $M$ has Euclidean volume growth with ratio bounded by a constant depending on $\\ell$.\nFor complete minimal hypersurfaces in $\\mathbb{R}^{n+1}$ with $3 \\leqslant n \\leqslant 6\\,,$ Tysk \\cite{Tysk} proved that finite index and Euclidean volume growth imply finite total curvature. \n\nChodosh, Ketover, and Maximo \\cite[Theorem 1.1]{Chodosh17} proved that\nfor a fixed closed Riemannian manifold $(M^n,g)(3 \\leqslant n \\leqslant 7)\\,,$ \n there can be at most $ N = N(M,g,\\Lambda,I)$ distinct diffeomorphism types in the set of\n embedded minimal hypersurfaces $\\Sigma \\subset (M,g)$ with $\\ind(\\Sigma) \\leqslant I$ and $\\vol_g(\\Sigma) \\leqslant \\Lambda\\,.$\n In particular, for $n=3\\,$, there is $r_0 = r_0(M,g,\\Lambda,I)$ so that\n any embedded minimal surface $\\Sigma$ in $(M^3,g)$ with $\\ind(\\Sigma) \\leqslant I$ and $\\mathrm{area}_g(\\Sigma) \\leqslant \\Lambda$ has\n $\\genus(\\Sigma) \\leqslant r_0$; for $4 \\leqslant n \\leqslant 7$, there is $N = N(n,I,\\Lambda) \\in \\mathbb{N}$ so that there are at most\n $N$ mutually non-diffeomorphic complete embedded minimal hypersurfaces $\\Sigma^{n-1} \\subset \\mathbb{R}^n$\n with $\\ind(\\Sigma) \\leqslant I$ and $\\vol(\\Sigma\\cap B_R(0)) \\leqslant \\Lambda R^{n-1}$ for all $R > 0$ \\cite[Theorem 1.2]{Chodosh17}.\n\nAntoine Song\\cite{songantoine1} showed that given a closed embedded minimal hypersurface $\\Sigma^n$ in a closed Riemannian manifold $(M^{n+1},g)$ with $2\\leqslant n \\leqslant 6\\,,$ the sum of dimensions over a fixed field of the cohomology groups of $\\Sigma$ can be bounded by the area and Morse index of $\\Sigma$ by a combinatorial argument.\n\nWe get a finiteness result for minimal submanifolds under the conditions of uniformly bound total curvature and Euclidean volume growth. This can be seen as a quantitative generalization of Anderson's Theorem \\cite{anderson1984compactification}.", + "context": "Collin \\cite{collin1} proved that any properly embedded minimal\nsurface in $\\mathbb{R}^3$ with finite topology and more than one end, has finite total curvature. \nColding and Minicozzi \\cite{ColdingMincozzi1} removed the proper condition, where they proved that a complete embedded minimal surface $\\Sigma$ with finite topology in $\\mathbb{R}^3$ must be proper.\nMeeks, Perez and Ros \\cite{Meeks2} showed that the number of ends of $\\Sigma$ is bounded by a constant depending on its genus.\n\nGiven an immersed minimal submanifold $M^{n}$ in $ \\mathbb{R}^{n+m}\\,,$ $M$ is said to have \\emph{finite total curvature} if\n $$\\int_{M} \\lvertA\\rvert^{n}d\\mu_M < \\infty\\,,$$\nwhere $A$ denotes the second fundamental form of $M$ in $ \\mathbb{R}^{n+m}$, and $\\mu_M$ denotes the volume element of $M$.\n\nAnderson \\cite{anderson1984compactification} gave a generalization of the\nChern-Osserman theorem \\cite{Chern} on minimal surfaces of finite total curvature: a complete minimal submanifold $M^n$ with finite total curvature is diffeomorphic to a compact $C^{\\infty} $ manifold $\\overline{M}^n$ punctured at a finite number of points $\\{p_i\\}_{1}^{\\ell} \\in \\overline{M}^n$ and the Gauss map $\\gamma:M^n \\to G_{n,m}$ extends to a $C^{n-2}$ map $\\overline \\gamma: \\overline M^n \\to G_{n,m} $ of the compactification(where $G_{n,m}$ denotes the Grassmann manifold of $n$-planes in Euclidean $(n+m)$-space). \nIn particular, $M$ has Euclidean volume growth with ratio bounded by a constant depending on $\\ell$.\nFor complete minimal hypersurfaces in $\\mathbb{R}^{n+1}$ with $3 \\leqslant n \\leqslant 6\\,,$ Tysk \\cite{Tysk} proved that finite index and Euclidean volume growth imply finite total curvature.\n\nChodosh, Ketover, and Maximo \\cite[Theorem 1.1]{Chodosh17} proved that\nfor a fixed closed Riemannian manifold $(M^n,g)(3 \\leqslant n \\leqslant 7)\\,,$ \n there can be at most $ N = N(M,g,\\Lambda,I)$ distinct diffeomorphism types in the set of\n embedded minimal hypersurfaces $\\Sigma \\subset (M,g)$ with $\\ind(\\Sigma) \\leqslant I$ and $\\vol_g(\\Sigma) \\leqslant \\Lambda\\,.$\n In particular, for $n=3\\,$, there is $r_0 = r_0(M,g,\\Lambda,I)$ so that\n any embedded minimal surface $\\Sigma$ in $(M^3,g)$ with $\\ind(\\Sigma) \\leqslant I$ and $\\mathrm{area}_g(\\Sigma) \\leqslant \\Lambda$ has\n $\\genus(\\Sigma) \\leqslant r_0$; for $4 \\leqslant n \\leqslant 7$, there is $N = N(n,I,\\Lambda) \\in \\mathbb{N}$ so that there are at most\n $N$ mutually non-diffeomorphic complete embedded minimal hypersurfaces $\\Sigma^{n-1} \\subset \\mathbb{R}^n$\n with $\\ind(\\Sigma) \\leqslant I$ and $\\vol(\\Sigma\\cap B_R(0)) \\leqslant \\Lambda R^{n-1}$ for all $R > 0$ \\cite[Theorem 1.2]{Chodosh17}.\n\nAntoine Song\\cite{songantoine1} showed that given a closed embedded minimal hypersurface $\\Sigma^n$ in a closed Riemannian manifold $(M^{n+1},g)$ with $2\\leqslant n \\leqslant 6\\,,$ the sum of dimensions over a fixed field of the cohomology groups of $\\Sigma$ can be bounded by the area and Morse index of $\\Sigma$ by a combinatorial argument.\n\nWe get a finiteness result for minimal submanifolds under the conditions of uniformly bound total curvature and Euclidean volume growth. This can be seen as a quantitative generalization of Anderson's Theorem \\cite{anderson1984compactification}.", + "full_context": "Collin \\cite{collin1} proved that any properly embedded minimal\nsurface in $\\mathbb{R}^3$ with finite topology and more than one end, has finite total curvature. \nColding and Minicozzi \\cite{ColdingMincozzi1} removed the proper condition, where they proved that a complete embedded minimal surface $\\Sigma$ with finite topology in $\\mathbb{R}^3$ must be proper.\nMeeks, Perez and Ros \\cite{Meeks2} showed that the number of ends of $\\Sigma$ is bounded by a constant depending on its genus.\n\nGiven an immersed minimal submanifold $M^{n}$ in $ \\mathbb{R}^{n+m}\\,,$ $M$ is said to have \\emph{finite total curvature} if\n $$\\int_{M} \\lvertA\\rvert^{n}d\\mu_M < \\infty\\,,$$\nwhere $A$ denotes the second fundamental form of $M$ in $ \\mathbb{R}^{n+m}$, and $\\mu_M$ denotes the volume element of $M$.\n\nAnderson \\cite{anderson1984compactification} gave a generalization of the\nChern-Osserman theorem \\cite{Chern} on minimal surfaces of finite total curvature: a complete minimal submanifold $M^n$ with finite total curvature is diffeomorphic to a compact $C^{\\infty} $ manifold $\\overline{M}^n$ punctured at a finite number of points $\\{p_i\\}_{1}^{\\ell} \\in \\overline{M}^n$ and the Gauss map $\\gamma:M^n \\to G_{n,m}$ extends to a $C^{n-2}$ map $\\overline \\gamma: \\overline M^n \\to G_{n,m} $ of the compactification(where $G_{n,m}$ denotes the Grassmann manifold of $n$-planes in Euclidean $(n+m)$-space). \nIn particular, $M$ has Euclidean volume growth with ratio bounded by a constant depending on $\\ell$.\nFor complete minimal hypersurfaces in $\\mathbb{R}^{n+1}$ with $3 \\leqslant n \\leqslant 6\\,,$ Tysk \\cite{Tysk} proved that finite index and Euclidean volume growth imply finite total curvature.\n\nChodosh, Ketover, and Maximo \\cite[Theorem 1.1]{Chodosh17} proved that\nfor a fixed closed Riemannian manifold $(M^n,g)(3 \\leqslant n \\leqslant 7)\\,,$ \n there can be at most $ N = N(M,g,\\Lambda,I)$ distinct diffeomorphism types in the set of\n embedded minimal hypersurfaces $\\Sigma \\subset (M,g)$ with $\\ind(\\Sigma) \\leqslant I$ and $\\vol_g(\\Sigma) \\leqslant \\Lambda\\,.$\n In particular, for $n=3\\,$, there is $r_0 = r_0(M,g,\\Lambda,I)$ so that\n any embedded minimal surface $\\Sigma$ in $(M^3,g)$ with $\\ind(\\Sigma) \\leqslant I$ and $\\mathrm{area}_g(\\Sigma) \\leqslant \\Lambda$ has\n $\\genus(\\Sigma) \\leqslant r_0$; for $4 \\leqslant n \\leqslant 7$, there is $N = N(n,I,\\Lambda) \\in \\mathbb{N}$ so that there are at most\n $N$ mutually non-diffeomorphic complete embedded minimal hypersurfaces $\\Sigma^{n-1} \\subset \\mathbb{R}^n$\n with $\\ind(\\Sigma) \\leqslant I$ and $\\vol(\\Sigma\\cap B_R(0)) \\leqslant \\Lambda R^{n-1}$ for all $R > 0$ \\cite[Theorem 1.2]{Chodosh17}.\n\nAntoine Song\\cite{songantoine1} showed that given a closed embedded minimal hypersurface $\\Sigma^n$ in a closed Riemannian manifold $(M^{n+1},g)$ with $2\\leqslant n \\leqslant 6\\,,$ the sum of dimensions over a fixed field of the cohomology groups of $\\Sigma$ can be bounded by the area and Morse index of $\\Sigma$ by a combinatorial argument.\n\nWe get a finiteness result for minimal submanifolds under the conditions of uniformly bound total curvature and Euclidean volume growth. This can be seen as a quantitative generalization of Anderson's Theorem \\cite{anderson1984compactification}.\n\nWe get a finiteness result for minimal submanifolds under the conditions of uniformly bound total curvature and Euclidean volume growth. This can be seen as a quantitative generalization of Anderson's Theorem \\cite{anderson1984compactification}.\n\nOur proof is inspired by the ideas in \\cite{Chodosh17}, but we need further research in some situations. For instance, one point of concentration is a plane in Proposition 7.1 of \\cite{Chodosh17}, while in our situation it may be a non-flat minimal submanifold with finite total curvature. In Theorem \\ref{prop.diffeo}, we can resolve it by an induction argument on the total curvature.\n\n\\begin{lemm}\\label{lemm:curv.est}\nFor fixed $I \\in \\ZZ^+,0 0$ and a sequence of smooth blow-up sets $\\cB_{j} \\subset M_{j}$ so that \n\\begin{equation} \\label{eq:seq.cur.esti}\n |A_{M_{j}}|(x)d(\\iota_j(x),\\iota_j(\\cB_{j} \\cup \\de M_j) ) \\leq C\\, , \\ |\\cB_{j}|< I\\, , \\ \\iota_j(\\cB_j) \\subset B_{\\fh{3}{4}r_0}(0)\\, ,\n\\end{equation}\nfor all $x \\in M_{j}\\,.$ \n\\item $\\iota_j(\\cB_j)$ converges to $ \\widetilde \\cB_ \\infty \\subset \\RR^{n+m}$ in the Hausdorff distance sense and the Radon measure $\\nu_j$ converges to $ \\nu_\\infty$ in the Radon measure sense with $\\nu_\\infty(p_\\infty)\\geq 2K_0$ for any $p_\\infty \\in \\widetilde \\cB_\\infty\\,.$\n\\end{enumerate}\n\\end{lemm}\n\\begin{proof}\nFirstly, we assume the conclusion in item(1) holds for the fixed $I\\,,$ and prove the conclusion in item(2) holds for the same fixed $I\\, $.\n Since $\\iota_j(\\cB_j) \\subset B_{\\fh{3}{4}r_0}(0) \\,,$ after passing to a subsequence, we can assume $\\iota_j(\\cB_j)$ converges to $\\widetilde \\cB_\\infty \\subset \\RR^{n+m}$ in the Hausdorff distance sense, i.e., $d_{\\cH}(\\iota_j(\\cB_j),\\widetilde \\cB_\\infty) \\to 0\\, $. Since $\\nu_j(\\RR^{n+m})=\\nu_j(B_{R_0}(0)) = \\int_{M_j} \\abs{A_{M_j}}^n d\\mu_{M_j}< IK_0\\,,$ after passing to a subsequence, we can assume $\\nu_j \\to \\nu_\\infty$ in the Radon measure sense.\n\nChodosh proved removal singularity theorem for embedded minimal hypersurfaces with finite total curvature in Euclidean space in his note \\cite{Chodosh1}. We adapt his method to study the immersed submanifolds of arbitrary codimension. \n\\begin{theo}[Removal singularity]\n\\label{theo.removal.singularity}\n Suppose that $\\iota: M^{n} \\to {B_2(0)}\\setminus\\{0\\} $ is a smooth minimal immersion, i.e.\\,,$$\\int_M \\div_MY d\\mu_{M} = 0\\, , \\text{ for any } Y\\in C^\\infty_c(B_2(0)\\setminus\\{0\\},\\RR^{n+m})\\,.$$ The minimal submanifold $M$ satisfies that $0 \\in \\bar {\\iota(M)}\\,,$ $\\vol(\\iota^{-1}(B_r(0)\\setminus\\{0\\})) \\leq \\Lambda r^n$ for any $00\\,.$ Moreover, $\\hat M_{\\infty}$ has bounded number of components due to the volume bound. By Theorem \\ref{eqival1}, every component of $\\hat M_{\\infty}$ is regular at infinity. Then we can choose $\\gamma \\geq 1$ large enough so that $\\partial B_{\\gamma}(0)$ intersects each component of $\\hat M_{\\infty}$ transversely, and each component of $\\hat M_{\\infty} \\cap \\partial B_{\\gamma}(0)$ is diffeomorphic to $\\SS^{n-1}$ with the standard smooth structure. Moreover the number of components of $\\hat M_{\\infty} \\cap \\partial B_{\\gamma}(0)$ is no more than $\\Lambda\\,.$ By the choice of $\\delta_{j}\\,$, the curvature estimates \\eqref{eq:prop.cur1} hold for all $x \\in M_{j} \\cap \\left( B_{2}(0) \\setminus B_{\\gamma \\delta_{j}}(\\iota_j(p_{j}))\\right)\\,$. Then by applying Lemma \\ref{lemm.Annular.composition}, we see that $M_{j} \\cap \\left( B_{2}(0) \\setminus B_{\\gamma \\delta_{j}}(\\iota_j(p_{j}))\\right)$ is diffeomorphic to the union of annular regions. In particular, $M_{j}\\cap B_{\\gamma\\delta_{j}}(\\iota_j(p_{j}))$ must be connected (because we have assumed that $M_{j}$ is connected in \\hyperlink{defi:beth}{$(\\beth)$}). Then we only need to prove $\\hat M_j \\cap B_\\gamma(0)$ are diffeomorphic to each other after passing to a subsequence.\n\n\\begin{proof}[Proof of Theorem \\ref{theo.finite.top}]\n We will prove Theorem \\ref{theo.finite.top} by contradiction. Since the volume bound and monotonicity formula imply that there exist at most finite number of components for any minimal submanifold satisfying the assumption of Theorem \\ref{theo.finite.top}, without loss of generality, we can assume the minimal submanifolds satisfying the assumption of Theorem \\ref{theo.finite.top} are connected. If $M_j^n$ is a sequence of pairwise non-diffeomorphic complete connected, immersed minimal submanifold in $\\RR^{n+m}$ with $\\vol(M_j\\cap B_{R}(0)) \\leq \\Lambda R^{n}$ for any $R>0$ and $$\\int_{M_j} \\abs{A_{M_j}}^n d \\mu_{M_j}\\leq \\Gamma < IK_0\\,\\,.$$ By rescaling $M_j,$ we can assume $$\\int_{M_j\\setminus B_{\\fh{1}{j}}(0)}\\abs{A_{M_j}}^n d \\mu_{M_j} <\\fh{1}{j}\\,,$$ and $M_j $ intersects $\\de B_1(0)$ transversely. By \n Theorem \\ref{eqival1},\n $M_j$ is properly immersed and regular at infinity. By rescaling $M_j\\,,$ we can assume $M_j \\setminus B_{\\fh1 2}(0) $ is the union of minimal graph and each minimal graph is defined over the exterior of a bounded region in an $n$-plane passing $0 \\in \\RR^{n+m}\\,.$ So after passing to a subsequence, we can assume the $M_j\\cap B_1(0) $ are pairwise non-diffeomorphic. By Lemma \\ref{lemm:curv.est}, the sequence $M_j\\cap B_2(0)$ satisfies \\hyperlink{defi:beth}{$(\\beth)$}. Then by Theorem \\ref{prop.diffeo}, after passing to a subsequence, all of the $M_j\\cap B_1(0)$ are diffeomorphic. This is a contradiction.\n \\end{proof}", + "post_theorem_intro_text_len": 1251, + "post_theorem_intro_text": "Our proof is inspired by the ideas in \\cite{Chodosh17}, but we need further research in some situations. For instance, one point of concentration is a plane in Proposition 7.1 of \\cite{Chodosh17}, while in our situation it may be a non-flat minimal submanifold with finite total curvature. In Theorem \\ref{prop.diffeo}, we can resolve it by an induction argument on the total curvature.\n\n\\subsection{Outline of the paper} \n\\begin{itemize}\n \\item In \\S \\ref{sec.pre}, we state several definitions\n and curvature estimates for minimal submanifolds which are needed in the following. \n \\item In \\S \\ref{sec.geo}, we describe the geometry of ends of complete immersed minimal submanifolds in $\\mathbb{R}^{n+m}$ with finite total curvature, enlightened by \\cite{schocen} and \\cite{anderson1984compactification}. \nThis helps us to derive curvature estimates away from finitely many points in Lemma \\ref{lemm:curv.est}.\n \\item In \\S \\ref{sec.top}, we prove a key topological result in Lemma \\ref{lemm.Annular.composition} allowing us to control the topology of the “intermediate regions”,\nthen combined the curvature estimates in \\S \\ref{sec.geo} we can prove Theorem \\ref{theo.finite.top} by an induction argument on the total curvature.\n\\end{itemize}", + "sketch": "The proof is “inspired by the ideas in \\cite{Chodosh17},” but differs because “in our situation [a concentration point] may be a non-flat minimal submanifold with finite total curvature” (rather than a plane as in Proposition 7.1 of \\cite{Chodosh17}). This issue is resolved “by an induction argument on the total curvature” (cf. Theorem \\ref{prop.diffeo}).\n\nThe stated structure to prove Theorem~\\ref{theo.finite.top} is:\n\\begin{itemize}\n\\item Develop “definitions and curvature estimates for minimal submanifolds” in \\S\\ref{sec.pre}.\n\\item In \\S\\ref{sec.geo}, “describe the geometry of ends of complete immersed minimal submanifolds in $\\mathbb{R}^{n+m}$ with finite total curvature,” which “helps us to derive curvature estimates away from finitely many points in Lemma~\\ref{lemm:curv.est}.”\n\\item In \\S\\ref{sec.top}, prove “a key topological result in Lemma~\\ref{lemm.Annular.composition} allowing us to control the topology of the \\lq intermediate regions\\rq,” and then, “combined [with] the curvature estimates in \\S\\ref{sec.geo} we can prove Theorem~\\ref{theo.finite.top} by an induction argument on the total curvature.”\n\\end{itemize}", + "expanded_sketch": "The proof is “inspired by the ideas in \\cite{Chodosh17},” but differs because “in our situation [a concentration point] may be a non-flat minimal submanifold with finite total curvature” (rather than a plane as in Proposition 7.1 of \\cite{Chodosh17}). This issue is resolved “by an induction argument on the total curvature,” using the following theorem:\n\n\\begin{theo} \\label{prop.diffeo}\nGiven a sequence $M_{j}$ satisfying \\hyperlink{defi:beth}{$(\\beth)$} and each $M_j$ intersects $\\partial B_{1}(0)$ transversely. By passing to a subsequence, all of the $M_{j}\\cap B_{1}(0)$ are diffeomorphic. \n\\end{theo}\n\nThe stated structure to prove the main theorem is:\n\\begin{itemize}\n\\item Develop “definitions and curvature estimates for minimal submanifolds” next.\n\\item After that, “describe the geometry of ends of complete immersed minimal submanifolds in $\\mathbb{R}^{n+m}$ with finite total curvature,” which “helps us to derive curvature estimates away from finitely many points” in the following lemma.\n\n\\begin{lemm}\\label{lemm:curv.est}\nFor fixed $I \\in \\ZZ^+,0 0$ and a sequence of smooth blow-up sets $\\cB_{j} \\subset M_{j}$ so that \n\\begin{equation} \\label{eq:seq.cur.esti}\n |A_{M_{j}}|(x)d(\\iota_j(x),\\iota_j(\\cB_{j} \\cup \\de M_j) ) \\leq C\\, , \\ |\\cB_{j}|< I\\, , \\ \\iota_j(\\cB_j) \\subset B_{\\fh{3}{4}r_0}(0)\\, ,\n\\end{equation}\nfor all $x \\in M_{j}\\,.$ \n\\item $\\iota_j(\\cB_j)$ converges to $ \\widetilde \\cB_ \\infty \\subset \\RR^{n+m}$ in the Hausdorff distance sense and the Radon measure $\\nu_j$ converges to $ \\nu_\\infty$ in the Radon measure sense with $\\nu_\\infty(p_\\infty)\\geq 2K_0$ for any $p_\\infty \\in \\widetilde \\cB_\\infty\\,.$\n\\end{enumerate}\n\\end{lemm}\n\n\\item Next, prove “a key topological result” in the following lemma, allowing control of the topology of the \\lq intermediate regions\\rq.\n\n\\begin{lemm}[Annular decomposition] \\label{lemm.Annular.composition}\nFor fixed $n\\,,m\\in \\ZZ^+\\,,n\\geq 2\\,,$ there is a $0 < \\sigma_0 <\\fh{1}{2} $ only depending on $n,m$ with the following property. Suppose that $M^n(\\iota : M^n \\to \\bar{B_2(0)} \\subset\\RR^{n+m})$ is a complete properly immersed submanifold with $\\iota(\\de M) \\subset \\de B_2(0)\\, . $ Assume that for some $\\sigma \\leq \\sigma_0$ and $p \\in B_{\\sigma_0}(0)\\,,$ we have:\n\\begin{enumerate}[itemsep=5pt, topsep=5pt]\n\\item For each component $M'$ of $M\\,,$ $M' \\cap B_{\\sigma}(p) \\neq \\emptyset \\,$.\n\\item The immersed submanifold $M$ intersects $\\de B_{\\sigma}(p)$ transversely, and $M \\cap \\de B_{\\sigma}(p)$ has $k$ components. Moreover, each component of $M \\cap \\de B_{\\sigma}(p)$ is diffeomorphic to $\\SS^{n-1}$ with the standard smooth structure.\n\\item The second fundamental form of $M$ satisfies $|A|(x)|\\iota(x)-p| \\leq \\fh{1}{4} $ for all $x \\in M \\cap \\left( \\overline{B_{1}(0)} \\setminus B_{\\sigma}(p) \\right)\\,.$\n\\end{enumerate}\nThen, $M$ intersects $\\de B_{1}(0)$ transversely. Both $M \\cap \\left(\\overline{B_{1}(0)}\\setminus {B_{\\sigma}(p)}\\right)$ and $M \\cap \\de B_{1}(0)$ have $k$ components. Moreover, each component of $M \\cap \\de B_{1}(0)$ is diffeomorphic to $\\SS^{n-1}$ with the standard smooth structure and\n each component of $M \\cap \\left(\\overline{B_{1}(0)}\\setminus {B_{\\sigma}(p)}\\right)$ is diffeomorphic to $\\SS^{n-1}\\times [0,1]$ with the standard smooth structure.\n\\end{lemm}\n\nCombined with the curvature estimates proved earlier, this yields an induction argument on the total curvature, completing the proof of the main theorem.\n\\end{itemize}", + "expanded_theorem": "\\label{theo.finite.top}\nFor fixed $n\\,,m \\in \\mathbb{Z}^+\\,,n \\geqslant 3\\,, m \\geq1\\,,$ and $ \\Gamma \\, ,\\Lambda \\in \\mathbb{R} \\, , \\Gamma\\,,\\Lambda \\geqslant 0\\,$, there exists $N = N(n,m,\\Gamma,\\Lambda) \\in \\mathbb{N}$ so that there are at most $N$ mutually non-diffeomorphic complete immersed minimal submanifolds $M^n$ in $\\mathbb{R}^{n+m}$ satisfying that $\\int_{M} \\lvertA\\rvert^{n}d\\mu_M \\leqslant \\Gamma$ and \n$\\vol_M ( B_R(0)) \\leqslant \\Lambda R^n$ for any $R>0\\,.$", + "theorem_type": [ + "Existential–Universal", + "Inequality or Bound" + ], + "mcq": { + "question": "Let n,m\\in\\mathbb{Z}^+ with n\\ge 3 and m\\ge 1, and let \\Gamma,\\Lambda\\ge 0. For an immersed minimal submanifold M^n\\subset \\mathbb{R}^{n+m}, let A denote its second fundamental form, let d\\mu_M be its induced volume element, and let \\operatorname{vol}_M(B_R(0)) denote the n-dimensional volume of M\\cap B_R(0), where B_R(0) is the Euclidean ball of radius R centered at the origin in \\mathbb{R}^{n+m}. Which quantitative finiteness statement holds for the class of complete immersed minimal submanifolds M^n in \\mathbb{R}^{n+m} satisfying\n\\[\n\\int_M |A|^n\\,d\\mu_M\\le \\Gamma\n\\quad\\text{and}\\quad\n\\operatorname{vol}_M(B_R(0))\\le \\Lambda R^n\\ \\text{for every }R>0?\n\\]", + "correct_choice": { + "label": "A", + "text": "There exists a number N=N(n,m,\\Gamma,\\Lambda)\\in\\mathbb{N} such that among all complete immersed minimal submanifolds M^n\\subset \\mathbb{R}^{n+m} satisfying\n\\(\\int_M |A|^n\\,d\\mu_M\\le \\Gamma\\) and \\(\\operatorname{vol}_M(B_R(0))\\le \\Lambda R^n\\) for every \\(R>0\\), there are at most N pairwise non-diffeomorphic examples." + }, + "choices": [ + { + "label": "B", + "text": "There exists a number $N=N(n,m,\\Gamma)\\in\\mathbb{N}$ such that among all complete immersed minimal submanifolds $M^n\\subset \\mathbb{R}^{n+m}$ satisfying\n\\(\\int_M |A|^n\\,d\\mu_M\\le \\Gamma\\) and \\(\\operatorname{vol}_M(B_R(0))\\le \\Lambda R^n\\) for every \\(R>0\\), there are at most $N$ pairwise non-diffeomorphic examples." + }, + { + "label": "C", + "text": "There are only finitely many pairwise non-diffeomorphic complete immersed minimal submanifolds $M^n\\subset \\mathbb{R}^{n+m}$ satisfying\n\\(\\int_M |A|^n\\,d\\mu_M\\le \\Gamma\\) and \\(\\operatorname{vol}_M(B_R(0))\\le \\Lambda R^n\\) for every \\(R>0\\)." + }, + { + "label": "D", + "text": "There exists a number $N=N(n,m,\\Gamma,\\Lambda)\\in\\mathbb{N}$ such that among all complete immersed minimal submanifolds $M^n\\subset \\mathbb{R}^{n+m}$ satisfying\n\\(\\int_M |A|^n\\,d\\mu_M\\le \\Gamma\\) and \\(\\operatorname{vol}_M(B_R(0))\\le \\Lambda R^n\\) for every \\(R>0\\), there are at most $N$ pairwise non-homeomorphic examples." + }, + { + "label": "E", + "text": "There exists a number $N=N(n,m,\\Gamma,\\Lambda)\\in\\mathbb{N}$ such that every complete immersed minimal submanifold $M^n\\subset \\mathbb{R}^{n+m}$ satisfying\n\\(\\int_M |A|^n\\,d\\mu_M\\le \\Gamma\\) and \\(\\operatorname{vol}_M(B_R(0))\\le \\Lambda R^n\\) for every \\(R>0\\) has at most $N$ ends." + } + ], + "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": "dependence_of_finiteness_constant_on_volume_growth", + "template_used": "quantifier_dependence" + }, + { + "label": "C", + "sketch_hook_type": "finiteness", + "tampered_component": "explicit_parameter_dependence_of_the_bound_N(n,m,\\Gamma,\\Lambda)", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "topological_equivalence_type_diffeomorphic_vs_homeomorphic", + "template_used": "property_confusion" + }, + { + "label": "E", + "sketch_hook_type": "case_split", + "tampered_component": "global_finiteness_of_diffeomorphism_types_replaced_by_uniform_end_count", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem states only the hypotheses and asks which finiteness conclusion is valid; it does not explicitly reveal the correct conclusion. Although the phrase 'quantitative finiteness statement' hints that an existence-of-N type answer is likely, it does not single out choice A." + }, + "TAS": { + "score": 1, + "justification": "This is close to theorem-recall: the correct option is essentially the theorem's conclusion under the stated hypotheses. However, it is not a pure restatement because the choices introduce meaningful variants in parameter dependence, topological category, and conclusion strength." + }, + "GPS": { + "score": 1, + "justification": "The item requires some reasoning or theorem-level discrimination: one must distinguish the exact dependence on parameters, the difference between diffeomorphic and homeomorphic, and stronger versus weaker finiteness claims. Still, it mainly tests recognition of the precise statement rather than substantial generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and well targeted. B tests incorrect parameter dependence, C is a weaker true statement, D swaps topological equivalence notions, and E replaces classification finiteness with a bound on ends. These reflect realistic mathematical confusions." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and little answer leakage, but it leans toward precise recall of a known result rather than deeper generative reasoning." + } + }, + { + "id": "2602.12762v1", + "paper_link": "http://arxiv.org/abs/2602.12762v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "\\label{thrm:beat-cactus-introversion}\n For all $n\\geq14$, minors of size $n(n-1)(n-2)+1$ of the tangency flattenings vanish on the $n$-th secant variety of the Segre variety in $\\P(\\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\mathbb{C}^n)$ but do not vanish on its $n$-th cactus variety.", + "start_pos": 10252, + "end_pos": 10549, + "label": "thrm:beat-cactus-introversion" + }, + "ref_dict": { + "thrm:beat-cactus-introversion": "\\begin{theorem}\n \\label{thrm:beat-cactus-introversion}\n For all $n\\geq14$, minors of size $n(n-1)(n-2)+1$ of the tangency flattenings vanish on the $n$-th secant variety of the Segre variety in $\\P(\\CC^n\\otimes\\CC^n\\otimes\\CC^n)$ but do not vanish on its $n$-th cactus variety.\n\\end{theorem}", + "prop:1de-koszul": "\\begin{proposition}\n \\label{prop:1de-koszul}\n Let $R$ be the apolar algebra to $e\\leq \\binom{d+1}2$ general quadrics in $d$ variables. If $e\\geq3$, then the structure tensor $T$ of $R$ has border cactus rank at least $\\floor{\\frac32d}+3$. In particular, if $3\\leq e\\leq \\floor{\\frac d2}+1$, then $R$ is nonsmoothable.\n\\end{proposition}", + "prop:quadratic-better-than-koszul": "\\begin{proposition}\n \\label{prop:quadratic-better-than-koszul}\n The space $\\im\\Phi_{\\mathrm Q}^C(T)\\subset A\\otimes B$ equals the sum of all $(T(\\alpha)\\otimes \\id_A)\\left(\\im\\koszul^{C,B}(T)\\right)$ over $\\alpha\\in A^*$, where we consider $T(\\alpha)$ as a map $C^*\\to B$. In particular, if $\\dim B=\\dim C$ and $T$ is $1_A$-generic, this implies $\\rank \\Phi_{\\mathrm Q}^C(T) \\geq \\rank\\koszul^{C,B}(T)$.\n\\end{proposition}", + "thrm:beat-cactus": "\\begin{theorem}\n \\label{thrm:beat-cactus}\n The $(n(n-1)(n-2)+1)$-minors of the tangency flattenings for border rank $n$ in $\\CC^n\\otimes \\CC^n\\otimes \\CC^n$ do not vanish on the $n$-th cactus variety for all $n\\geq 14$.\n\\end{theorem}", + "conj:tangency": "\\begin{conjecture}\n \\label{conj:tangency}\n Let $\\CC[x_1,\\dots,x_d]\\surjto R$ be a surjection to an $n$-dimensional algebra $R$, let $[R]$ be the corresponding point of $\\Hilb_n\\CC^d$ and let $T$ be the structure tensor of $R$. Then\n \\[\n \\rank\\tang^C(T) - n(n-1)(n-2) = dn - \\dim\\T_{[R]}\\Hilb_n\\CC^d.\n \\]\n\\end{conjecture}", + "subsec:mamu": "\\begin{proof}\n Let us prove only the second set of equalities of \\eqref{eq:koszul-quaddratic-algebra}, since the proof of the first one is very similar.\n\n Explicitly, the two flattenings in question map\n \\begin{align*}\n \\Phi_{\\mathrm Q}^A(T): \\bigwedge^2 R \\otimes R \\otimes R^* &\\to R \\otimes R^*,\\\\\n (a\\wedge a')\\otimes b\\otimes \\gamma &\\mapsto (ab)\\otimes \\gamma(a'\\cdot\\bullet) - (a'b)\\otimes \\gamma(a\\cdot\\bullet)\n \\end{align*}\n and\n \\begin{align*}\n \\koszul^{A,C}(T): \\bigwedge^2 R \\otimes R^* &\\to R \\otimes R^*,\\\\\n (a\\wedge a')\\otimes\\gamma &\\mapsto a\\otimes \\gamma(a'\\cdot\\bullet) - a'\\otimes \\gamma(a\\cdot\\bullet),\n \\end{align*}\n where we use $\\gamma(a\\cdot\\bullet)$ to denote the linear form $(x\\mapsto \\gamma(ax))\\in R^*$. We claim that in fact $\\Phi_{\\mathrm Q}^A(T)$ and $\\koszul^{A,C}(T)$ have the same images. The inclusion $\\im\\Phi^{A,C}(T)\\subseteq \\im\\Phi_{\\mathrm Q}^A$ is obvious by considering $b=1$ in the latter map.\n\n For the opposite inclusion, note that invertible elements form a dense set in $R$, so in particular, $R$ may be spanned by them, hence $\\im \\Phi_{\\mathrm Q}^A$ is spanned by images of $(a\\wedge a')\\otimes b\\otimes\\gamma$ with $b$ invertible. These lie in $\\im \\koszul^{A,C}$ by the following: we set\n \\[\n \\tilde a := ba, \\qquad \\tilde a' := ba', \\qquad \\tilde\\gamma := \\gamma(b^{-1}\\cdot\\bullet),\n \\]\n which then results in\n \\begin{multline*}\n \\koszul^{A,C}(T)\\left(\n (\\tilde a\\wedge \\tilde a') \\otimes \\tilde \\gamma\n \\right)\n =\n \\tilde a\\otimes\\tilde \\gamma(\\tilde a'\\cdot\\bullet) - \\tilde a'\\otimes\\tilde\\gamma(\\tilde a\\cdot\\bullet)\n =\n (ab)\\otimes\\gamma(b^{-1}ba'\\cdot\\bullet) - (a'b)\\otimes \\gamma(b^{-1}ba\\cdot\\bullet)\n =\\\\=\n (ab)\\otimes \\gamma(a'\\cdot\\bullet)-(a'b)\\otimes\\gamma(a\\cdot\\bullet)\n =\n \\Phi_{\\mathrm Q}^A(T)\\left(\n (a\\wedge a') \\otimes b\\otimes\\gamma\n \\right).\n \\end{multline*}\n This justifies $\\rank \\Phi_{\\mathrm Q}^A(T) = \\rank \\koszul^{A,C}(T)$. That this common rank equals $n(n-1)$ then follows from Lemma \\ref{lem:easy rank Koszul for algebras}.\n\\end{proof}\n\n\\section{Applications}\n\\label{sec:applications}\n\n\\subsection{Matrix multiplication}\n\\label{subsec:mamu}\n\nIt is well-known that the border rank of the $2\\times 2$ matrix multiplication tensor $M_2$ equals $7$ \\cite{Landsberg2006, Hauenstein-Ikenmeyer-Landsberg}. However, only recently the ``first hand-checkable algebraic proof'' appeared \\cite{Conner_Harper_Landsberg_2023}. Below we show that tangency flattenings provide an easy proof.\n\\begin{proposition}\\label{prop:M2=7}\n The border rank of $M_2$ equals seven.\n\\end{proposition}\n\\begin{proof}\nStrassen \\cite{strassen1969gaussian} provided an explicit rank seven decomposition of $M_2$. We focus on lower-bounding the border rank.\n\n By Corollary \\ref{cor:tan_flat} for $n=4$, $q=6$ we have to show that the tangency flattening has rank greater than $60$. We will prove that it is $64$. We fix a basis $(i,j)$ of $\\CC^4$, where $i,j\\in\\{0,1\\}$ and we use $\\neg i$ to denote the negation of $i$. Recall that $M_2=\\sum_{i,j,k\\in \\{0,1\\}} (i,j)\\otimes (j,k)\\otimes (k,i)$.\n\n We will prove that the tangency flattening is a block diagonal matrix with five blocks: a $48\\times 48$ identity matrix and four isomorphic, nondegenerate $4\\times 4$ matrices.\n\nThe tangency flattening of $M_2$ is the restriction to $\\bigwedge^3\\CC^4\\otimes \\CC^4\\otimes \\CC^4$ of\n\\begin{align}\n\\label{eq:mamu-tangency}\n(\\CC^4)^{\\otimes 3}\\otimes \\CC^4\\otimes \\CC^4 &\\to \\CC^4\\otimes \\CC^4\\otimes \\CC^4,\\\\\n\\nonumber\n(i_1,j_1)\\otimes (i_2,j_2)\\otimes (i_3,j_3)\\otimes (i_4,j_4)\\otimes (i_5,j_5) &\\mapsto\n\\begin{cases}\n(i_1,j_1)\\otimes (i_2,j_4)\\otimes (i_5,j_3), & \\text{if $j_2=i_4$ and $i_3 = j_5$,}\\\\\n0, & \\text{otherwise.}\n\\end{cases}\n\\end{align}\nWe divide the basis vectors in the domain into five groups.\n\nThe first group is formed by vectors of the form $\\bigl((\\neg j_5,\\neg i_4)\\wedge(i_2,i_4)\\wedge (j_5,j_3)\\bigr)\\otimes (i_4,j_4)\\otimes (i_5,j_5)$. By first choosing $i_4,j_4,i_5,j_5$ (16 options), and then choosing exterior products that contain $(\\neg j_5,\\neg i_4)$ (3 options), we see there are $48$ such (nonzero) vectors.\nNote that this already forces the exterior product to be of the form above, since the two remaining exterior factors are chosen from $(\\neg j_5, i_4)$, $(j_5, \\neg i_4)$ and $(j_5,i_4)$.\nAlthough the exterior product expands as a signed sum of six tensor products, only one of those -- the one ordered as in the presentation above -- maps to a nonzero vector in \\eqref{eq:mamu-tangency}. Thus, the image of each such basis vector is $(\\neg j_5,\\neg i_4)\\otimes (i_2,j_4)\\otimes (i_5,j_3)$. Note that such images must satisfy $(\\neg j_5,\\neg i_4)\\neq (\\neg i_2,\\neg j_3)$.\n\nThe next four groups are indexed by $j_4,i_5\\in \\{0,1\\}$. For fixed $j_4,i_5$ the group consists of the four basis vectors $\\bigl((j_5,i_4)\\wedge (\\neg j_5,i_4)\\wedge (j_5,\\neg i_4)\\bigr)\\otimes (i_4,j_4)\\otimes (i_5,j_5)$, where $i_4, j_5$ are arbitrary.\nThe image of each such vector is:\n\\[(j_5,i_4)\\otimes (\\neg j_5,j_4)\\otimes (i_5,\\neg i_4)-(\\neg j_5,i_4)\\otimes (j_5,j_4)\\otimes (i_5,\\neg i_4)-(j_5,\\neg i_4)\\otimes (\\neg j_5,j_4)\\otimes (i_5,i_4).\\]\nIn particular, the images are linear combinations of basis vectors that are distinct from images of the first group as well as other groups -- in each term, the first tensor factor consist exactly of negations of the first coordinate of the second factor and the second coordinate of the third factor. Meanwhile, the group is determined by reading off $j_4$ and $i_5$ from the second coordinate of the second factor and first coordinate of the third factor, respectively.\n\nThus indeed the tangency flattening is in block-diagonal form. It remains for us to prove that the isomorphic $4\\times 4$ blocks are of full rank.\nEach such matrix has the following form, where we label $(j_5,i_4)\\otimes (\\neg j_5,j_4)\\otimes (i_5,\\neg i_4)$ with the first factor $(j_5,i_4)$ and subsequently order rows and columns by $(j_5,i_4)=(0,0)$, $(0,1)$, $(1,0)$, $(1,1)$:\n\\[\\begin{pmatrix}\n 1 & -1&-1&0\\\\\n -1& 1&0&-1\\\\\n -1& 0&1&-1\\\\\n 0& -1&-1&1\\\\\n\\end{pmatrix}.\\]\nThe determinant of such a matrix is $-3$, thus indeed the matrix is of full rank, and so is the tangency flattening.\n\\end{proof}", + "cor:tan_flat": "\\begin{corollary}\\label{cor:tan_flat}\n If $\\dim V_1=\\dim V_2=\\dim V_3=n$, then $q(q-1)(n-2)+1$ minors of the tangency flattening provide equations for border rank $q$.\n\\end{corollary}" + }, + "pre_theorem_intro_text_len": 6726, + "pre_theorem_intro_text": "\\label{sec:intro}\nThe central problem concerning tensors is that of \\emph{tensor decomposition} -- presenting a tensor as a sum of simple tensors, or computing the smallest possible size of a decomposition, called the \\emph{tensor rank}. While the problem is NP-hard in general \\cite{RankNPcomplete, hillar2013most}, providing estimates for rank or related notions such as border rank or asymptotic rank has been a fruitful field. Perhaps the most famous application is the study of computational complexity of matrix multiplication (e.g. \\cite{strassen1969gaussian, coppersmith1982asymptotic, alman2024refined, williams2024new}). Further, a recent line of research relates low asymptotic rank of certain special tensors to faster than known algorithms for NP-hard problems \\cite{bjorklund2025fast, bjorklund2024asymptotic, pratt2024stronger, MaMu-SotA}.\nFor an introduction to tensors from an algebraic perspective, we refer the reader to \\cite[Chapter 9.2]{michalek2021invitation}, or to \\cite{landsberg2011tensors} for a more extensive exposition. For the connections to complexity theory, we refer to \\cite{burgisser2013algebraic, landsberg2017geometry}.\n\nThe main topic of this article is obtaining lower bounds on the border rank of tensors, a longstanding question in tensor geometry. The classical approach to the problem is by providing equations for secant varieties to the Segre variety. Flattenings, or maps which produce a matrix from a given tensor, have been a bountiful source of such lower bounds (e.g. \\cite{langsberg-ottaviani-equationsforsecant, landsberg2008generalizations, landsberg-ottaviani-newbounds, landsberg-michalek-mamu, landsberg-michalek-haystack}), with the equations thus obtained coming in the form of minors. In particular, \\emph{Strassen's equations}, further studied by Landsberg and Manivel and expanded by Landsberg and Ottaviani to \\emph{Koszul} and \\emph{Young flattenings}, became the gold standard in the field. Apart from flattenings, the techniques of \\emph{border apolarity} (see \\cite{Buczynscy-apolarity}) and \\emph{border substitution} (see \\cite{landsberg2017geometry}) have been introduced and used to prove several new lower bounds in recent years. However, all of these methods have quite restrictive limits of application.\n\nIndeed, current lower bound methods fall far short of being able to prove a tensor in $\\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\mathbb{C}^n$ to have the generic border rank, which is approximately $\\frac{n^2}3$. Instead, known flattening methods give maximal lower bounds of up to $(2-\\epsilon)n$ for small $\\epsilon>0$. Combining algebraic equations with other methods, it is possible to provide examples of tensors of border rank $2.02n$ \\cite{landsberg-michalek-haystack}. Still, there are well understood barriers which flattening methods cannot beat \\cite{jarek_cactus, Galazka-bundles, EGOW18, garg2019more}. In contrast, the applications to asymptotic rank in computer science would require superlinear lower bounds on border rank.\n\nThe disparity between the generic border rank and available lower bounds has been attributed to the inclusion of secant varieties in \\emph{cactus varieties} (see Section~\\ref{sec:preliminaries}). In particular, known flattening methods have been unable to distinguish secant from cactus varieties, even when the inclusion is known to be strict -- quoting from \\cite{Conner_Harper_Landsberg_2023}:\n\\begin{quote}\n``The geometric interpretation of the border rank lower bound barriers of \\cite{EGOW18, Galazka-bundles} is that all equations obtained by taking minors, called rank methods, are actually equations for a larger variety than $\\sigma_r(\\Seg(\\P A\\times\\P B\\times\\P C))$, called the $r$-th cactus variety \\cite{Buczynscy-apolarity}.''\n\\end{quote}\nIndeed, as proved in \\cite{jarek_cactus, Galazka-bundles, galazka-thesis} all of the known methods producing equations for the $r$-th secant variety are equations for the $r$-th cactus variety. Further, it is known that the $r$-th cactus variety of the Segre variety in $(\\mathbb{C}^n)^{\\ot 3}$ fills the ambient space for $r$ growing linearly in $n$ \\cite{ballico2019note, bernardi2018polynomials, galkazka2023multigraded}.\n\nFor these reasons, the existence of easily computable determinantal equations vanishing on secant varieties but not vanishing on cactus varieties, has long been considered unlikely or even impossible in the tensor-geometric community.\nIn this article, we illustrate that it is not the determinantal expression which creates the cactus barrier, but rather the linear embedding of the tensor product into a matrix space.\nThus the starting point of our approach in this article is focusing on \\emph{nonlinear} maps from tensor spaces to matrix spaces.\nTo obtain bounds on matrix rank with linear embeddings, it is enough to investigate the embedded image of the Segre variety and appeal to linearity. With nonlinear maps on the other hand, we need to work more generally, but this does yield stronger results. We obtain a new method, which we dub \\emph{Kronecker-Koszul flattenings}, or more generally \\emph{Kronecker-Young flattenings}, that allows one to produce\nexplicit and efficiently computable determinantal equations of secant varieties that do not generally vanish on cactus varieties. In particular, Kronecker-Koszul flattenings require less computation time and memory than methods based on a more direct identification of ideals defining the secant varieties (cf.~\\cite{Hauenstein-Ikenmeyer-Landsberg}).\n\n Broadly speaking, our method starts by tensoring together several copies of the same tensor. This, depending on the interpretation of the codomain, may be viewed as a Kronecker power or Veronese embedding \\cite{kaski2025universal}, and thus is the universal map for fixed degree polynomials. Next, we further tensor with several $2$-tensors corresponding to identity maps on various vector spaces. Finally, we group and contract some of the factors to exterior powers. This results in what we call a \\emph{Kronecker-Koszul tensor}. A Kronecker-Koszul flattening is then a classical flattening of the Kronecker-Koszul tensor. Note that due to initially taking a tensor power, Kronecker-Koszul tensors are in general nonlinear as functions of the starting tensor.\nSee Section~\\ref{sec:construction} for details of the construction. There we also comment on what we call {Kronecker-Young flattenings}, a further generalization which synthesizes the approach of Young flattenings into the method.\n\nOur main result uses the \\emph{tangency flattening}, a very particular Kronecker-Koszul flattening which depends quadratically on a tensor. See the paragraph preceding Corollary~\\ref{cor:tan_flat} for the definition.", + "context": "The main topic of this article is obtaining lower bounds on the border rank of tensors, a longstanding question in tensor geometry. The classical approach to the problem is by providing equations for secant varieties to the Segre variety. Flattenings, or maps which produce a matrix from a given tensor, have been a bountiful source of such lower bounds (e.g. \\cite{langsberg-ottaviani-equationsforsecant, landsberg2008generalizations, landsberg-ottaviani-newbounds, landsberg-michalek-mamu, landsberg-michalek-haystack}), with the equations thus obtained coming in the form of minors. In particular, \\emph{Strassen's equations}, further studied by Landsberg and Manivel and expanded by Landsberg and Ottaviani to \\emph{Koszul} and \\emph{Young flattenings}, became the gold standard in the field. Apart from flattenings, the techniques of \\emph{border apolarity} (see \\cite{Buczynscy-apolarity}) and \\emph{border substitution} (see \\cite{landsberg2017geometry}) have been introduced and used to prove several new lower bounds in recent years. However, all of these methods have quite restrictive limits of application.\n\nIndeed, current lower bound methods fall far short of being able to prove a tensor in $\\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\mathbb{C}^n$ to have the generic border rank, which is approximately $\\frac{n^2}3$. Instead, known flattening methods give maximal lower bounds of up to $(2-\\epsilon)n$ for small $\\epsilon>0$. Combining algebraic equations with other methods, it is possible to provide examples of tensors of border rank $2.02n$ \\cite{landsberg-michalek-haystack}. Still, there are well understood barriers which flattening methods cannot beat \\cite{jarek_cactus, Galazka-bundles, EGOW18, garg2019more}. In contrast, the applications to asymptotic rank in computer science would require superlinear lower bounds on border rank.\n\nThe disparity between the generic border rank and available lower bounds has been attributed to the inclusion of secant varieties in \\emph{cactus varieties} (see Section~\\ref{sec:preliminaries}). In particular, known flattening methods have been unable to distinguish secant from cactus varieties, even when the inclusion is known to be strict -- quoting from \\cite{Conner_Harper_Landsberg_2023}:\n\\begin{quote}\n``The geometric interpretation of the border rank lower bound barriers of \\cite{EGOW18, Galazka-bundles} is that all equations obtained by taking minors, called rank methods, are actually equations for a larger variety than $\\sigma_r(\\Seg(\\P A\\times\\P B\\times\\P C))$, called the $r$-th cactus variety \\cite{Buczynscy-apolarity}.''\n\\end{quote}\nIndeed, as proved in \\cite{jarek_cactus, Galazka-bundles, galazka-thesis} all of the known methods producing equations for the $r$-th secant variety are equations for the $r$-th cactus variety. Further, it is known that the $r$-th cactus variety of the Segre variety in $(\\mathbb{C}^n)^{\\ot 3}$ fills the ambient space for $r$ growing linearly in $n$ \\cite{ballico2019note, bernardi2018polynomials, galkazka2023multigraded}.\n\nFor these reasons, the existence of easily computable determinantal equations vanishing on secant varieties but not vanishing on cactus varieties, has long been considered unlikely or even impossible in the tensor-geometric community.\nIn this article, we illustrate that it is not the determinantal expression which creates the cactus barrier, but rather the linear embedding of the tensor product into a matrix space.\nThus the starting point of our approach in this article is focusing on \\emph{nonlinear} maps from tensor spaces to matrix spaces.\nTo obtain bounds on matrix rank with linear embeddings, it is enough to investigate the embedded image of the Segre variety and appeal to linearity. With nonlinear maps on the other hand, we need to work more generally, but this does yield stronger results. We obtain a new method, which we dub \\emph{Kronecker-Koszul flattenings}, or more generally \\emph{Kronecker-Young flattenings}, that allows one to produce\nexplicit and efficiently computable determinantal equations of secant varieties that do not generally vanish on cactus varieties. In particular, Kronecker-Koszul flattenings require less computation time and memory than methods based on a more direct identification of ideals defining the secant varieties (cf.~\\cite{Hauenstein-Ikenmeyer-Landsberg}).\n\nBroadly speaking, our method starts by tensoring together several copies of the same tensor. This, depending on the interpretation of the codomain, may be viewed as a Kronecker power or Veronese embedding \\cite{kaski2025universal}, and thus is the universal map for fixed degree polynomials. Next, we further tensor with several $2$-tensors corresponding to identity maps on various vector spaces. Finally, we group and contract some of the factors to exterior powers. This results in what we call a \\emph{Kronecker-Koszul tensor}. A Kronecker-Koszul flattening is then a classical flattening of the Kronecker-Koszul tensor. Note that due to initially taking a tensor power, Kronecker-Koszul tensors are in general nonlinear as functions of the starting tensor.\nSee Section~\\ref{sec:construction} for details of the construction. There we also comment on what we call {Kronecker-Young flattenings}, a further generalization which synthesizes the approach of Young flattenings into the method.\n\nOur main result uses the \\emph{tangency flattening}, a very particular Kronecker-Koszul flattening which depends quadratically on a tensor. See the paragraph preceding Corollary~\\ref{cor:tan_flat} for the definition.\n\n\\begin{corollary}\\label{cor:tan_flat}\n If $\\dim V_1=\\dim V_2=\\dim V_3=n$, then $q(q-1)(n-2)+1$ minors of the tangency flattening provide equations for border rank $q$.\n\\end{corollary}", + "full_context": "The main topic of this article is obtaining lower bounds on the border rank of tensors, a longstanding question in tensor geometry. The classical approach to the problem is by providing equations for secant varieties to the Segre variety. Flattenings, or maps which produce a matrix from a given tensor, have been a bountiful source of such lower bounds (e.g. \\cite{langsberg-ottaviani-equationsforsecant, landsberg2008generalizations, landsberg-ottaviani-newbounds, landsberg-michalek-mamu, landsberg-michalek-haystack}), with the equations thus obtained coming in the form of minors. In particular, \\emph{Strassen's equations}, further studied by Landsberg and Manivel and expanded by Landsberg and Ottaviani to \\emph{Koszul} and \\emph{Young flattenings}, became the gold standard in the field. Apart from flattenings, the techniques of \\emph{border apolarity} (see \\cite{Buczynscy-apolarity}) and \\emph{border substitution} (see \\cite{landsberg2017geometry}) have been introduced and used to prove several new lower bounds in recent years. However, all of these methods have quite restrictive limits of application.\n\nIndeed, current lower bound methods fall far short of being able to prove a tensor in $\\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\mathbb{C}^n$ to have the generic border rank, which is approximately $\\frac{n^2}3$. Instead, known flattening methods give maximal lower bounds of up to $(2-\\epsilon)n$ for small $\\epsilon>0$. Combining algebraic equations with other methods, it is possible to provide examples of tensors of border rank $2.02n$ \\cite{landsberg-michalek-haystack}. Still, there are well understood barriers which flattening methods cannot beat \\cite{jarek_cactus, Galazka-bundles, EGOW18, garg2019more}. In contrast, the applications to asymptotic rank in computer science would require superlinear lower bounds on border rank.\n\nThe disparity between the generic border rank and available lower bounds has been attributed to the inclusion of secant varieties in \\emph{cactus varieties} (see Section~\\ref{sec:preliminaries}). In particular, known flattening methods have been unable to distinguish secant from cactus varieties, even when the inclusion is known to be strict -- quoting from \\cite{Conner_Harper_Landsberg_2023}:\n\\begin{quote}\n``The geometric interpretation of the border rank lower bound barriers of \\cite{EGOW18, Galazka-bundles} is that all equations obtained by taking minors, called rank methods, are actually equations for a larger variety than $\\sigma_r(\\Seg(\\P A\\times\\P B\\times\\P C))$, called the $r$-th cactus variety \\cite{Buczynscy-apolarity}.''\n\\end{quote}\nIndeed, as proved in \\cite{jarek_cactus, Galazka-bundles, galazka-thesis} all of the known methods producing equations for the $r$-th secant variety are equations for the $r$-th cactus variety. Further, it is known that the $r$-th cactus variety of the Segre variety in $(\\mathbb{C}^n)^{\\ot 3}$ fills the ambient space for $r$ growing linearly in $n$ \\cite{ballico2019note, bernardi2018polynomials, galkazka2023multigraded}.\n\nFor these reasons, the existence of easily computable determinantal equations vanishing on secant varieties but not vanishing on cactus varieties, has long been considered unlikely or even impossible in the tensor-geometric community.\nIn this article, we illustrate that it is not the determinantal expression which creates the cactus barrier, but rather the linear embedding of the tensor product into a matrix space.\nThus the starting point of our approach in this article is focusing on \\emph{nonlinear} maps from tensor spaces to matrix spaces.\nTo obtain bounds on matrix rank with linear embeddings, it is enough to investigate the embedded image of the Segre variety and appeal to linearity. With nonlinear maps on the other hand, we need to work more generally, but this does yield stronger results. We obtain a new method, which we dub \\emph{Kronecker-Koszul flattenings}, or more generally \\emph{Kronecker-Young flattenings}, that allows one to produce\nexplicit and efficiently computable determinantal equations of secant varieties that do not generally vanish on cactus varieties. In particular, Kronecker-Koszul flattenings require less computation time and memory than methods based on a more direct identification of ideals defining the secant varieties (cf.~\\cite{Hauenstein-Ikenmeyer-Landsberg}).\n\nBroadly speaking, our method starts by tensoring together several copies of the same tensor. This, depending on the interpretation of the codomain, may be viewed as a Kronecker power or Veronese embedding \\cite{kaski2025universal}, and thus is the universal map for fixed degree polynomials. Next, we further tensor with several $2$-tensors corresponding to identity maps on various vector spaces. Finally, we group and contract some of the factors to exterior powers. This results in what we call a \\emph{Kronecker-Koszul tensor}. A Kronecker-Koszul flattening is then a classical flattening of the Kronecker-Koszul tensor. Note that due to initially taking a tensor power, Kronecker-Koszul tensors are in general nonlinear as functions of the starting tensor.\nSee Section~\\ref{sec:construction} for details of the construction. There we also comment on what we call {Kronecker-Young flattenings}, a further generalization which synthesizes the approach of Young flattenings into the method.\n\nOur main result uses the \\emph{tangency flattening}, a very particular Kronecker-Koszul flattening which depends quadratically on a tensor. See the paragraph preceding Corollary~\\ref{cor:tan_flat} for the definition.\n\n\\begin{corollary}\\label{cor:tan_flat}\n If $\\dim V_1=\\dim V_2=\\dim V_3=n$, then $q(q-1)(n-2)+1$ minors of the tangency flattening provide equations for border rank $q$.\n\\end{corollary}\n\nFor these reasons, the existence of easily computable determinantal equations vanishing on secant varieties but not vanishing on cactus varieties, has long been considered unlikely or even impossible in the tensor-geometric community.\nIn this article, we illustrate that it is not the determinantal expression which creates the cactus barrier, but rather the linear embedding of the tensor product into a matrix space.\nThus the starting point of our approach in this article is focusing on \\emph{nonlinear} maps from tensor spaces to matrix spaces.\nTo obtain bounds on matrix rank with linear embeddings, it is enough to investigate the embedded image of the Segre variety and appeal to linearity. With nonlinear maps on the other hand, we need to work more generally, but this does yield stronger results. We obtain a new method, which we dub \\emph{Kronecker-Koszul flattenings}, or more generally \\emph{Kronecker-Young flattenings}, that allows one to produce\nexplicit and efficiently computable determinantal equations of secant varieties that do not generally vanish on cactus varieties. In particular, Kronecker-Koszul flattenings require less computation time and memory than methods based on a more direct identification of ideals defining the secant varieties (cf.~\\cite{Hauenstein-Ikenmeyer-Landsberg}).\n\nOur main result uses the \\emph{tangency flattening}, a very particular Kronecker-Koszul flattening which depends quadratically on a tensor. See the paragraph preceding Corollary~\\ref{cor:tan_flat} for the definition.\n\nWhile Kronecker-Koszul flattenings generalize Koszul flattenings (cf. \\cite{langsberg-ottaviani-equationsforsecant, landsberg2008generalizations, landsberg-michalek-mamu, landsberg-michalek-haystack}), we also analyze in Section~\\ref{sec:koszul} bounds that may be extracted via Koszul flattenings for the border rank (and border cactus rank) of structure tensors of certain algebras. It is known \\cite{blaser2016degeneration} that a (finite) algebra is smoothable if and only if the associated structure tensor has minimal border rank. This is also one of the main challenges in application of the border apolarity method.\nFurther, two of the classical flattenings of structure tensors of algebras give rise to spaces of commuting matrices. Thus, many of the flattening methods cannot provide nontrivial bounds on border rank of such tensors. However, we show that it is possible to obtain lower bounds on border cactus rank of structure tensors of algebras strictly greater than the degree of the algebra, even using Koszul flattenings (see Proposition~\\ref{prop:1de-koszul}).\n\n\\begin{corollary}\\label{cor:flat bound}\n If $T$ has border rank at most $q$, then the rank of a Kronecker-Koszul flattening of $T$ is at most $F(q)\\cdot \\prod_{i=1}^r\\prod_{j=1}^{s_i} \\binom{d_i-|\\lambda_{i,j}|}{d_{i,j}'}$. In particular, minors of the Kronecker-Koszul flattening provide equations for secant varieties of the Segre variety.\n\\end{corollary}\n\\begin{proof}\n Follows from Theorem \\ref{thm:main}, as ranks of classical flattenings lower bound border rank of the tensor.\n\\end{proof}\nThe bounds in Theorem \\ref{thm:main} and Corollary \\ref{cor:flat bound} do not have to be tight. This can happen for various reasons, some more opaque than others:\n\nIn this section we present one explicit construction that we find particularly interesting. By \\cite{kaski2025universal} the linear span of image of the $k$-th Kronecker power of tensors in $A\\ot B \\ot C$ is $S^k(A\\ot B \\ot C)$. As a $\\GL(A)\\times \\GL(B)\\times \\GL(C)$ representation, the latter space has a decomposition into isotypic components\n \\[S^k(A\\ot B \\ot C)=\\bigoplus_{\\lambda,\\mu,\\rho\\vdash k}\\left(S^\\lambda(A)\\ot S^\\mu(B)\\ot S^\\rho(C)\\right)^{\\bigoplus K_{\\lambda,\\mu,\\rho}},\\]\n where $K_{\\lambda,\\mu,\\rho}$ is the Kronecker coefficient. In particular, when $\\lambda=\\mu=1^k$ and $\\rho=(k)$ we have $K_{\\lambda,\\mu,\\rho}=1$. This gives a canonical projection\n \\[\\pi:S^k(A\\ot B \\ot C)\\rightarrow \\bigwedge^k A\\ot \\bigwedge^k B\\ot S^k(C).\\]\n An example of Kronecker-Young flattening is the classical flattening of $\\pi(T^{\\otimes k})$ given by\n \\[\\Phi_{\\mathrm M,k}^C(T):\\bigwedge^kA^*\\ot\\bigwedge^kB^*\\rightarrow S^k(C).\\]\n The advantage of these choices is that the image of the map $\\Phi_{\\mathrm M,k}^C(T)$ has a direct interpretation in terms of the flattening $T^C:C^*\\rightarrow A\\ot B$. Namely, the image of $\\Phi_{\\mathrm M,k}^C(T)$ is the linear span of all $k\\times k$ minors of $T^C$ viewed as a matrix with entries from $C$. For this reason, we call $\\Phi_{\\mathrm M,k}$ the \\emph{$k$-minor flattening}.\n\\begin{lemma}\n If $\\dim A=\\dim B=\\dim C=n$ and a tensor $T\\in A\\ot B\\ot C$ has border rank at most $n$, then the rank of the $k$-minor flattening $\\Phi_{\\mathrm M,k}^C(T)$ is at most $\\binom{n}{k}$.\n\\end{lemma}\n\\begin{proof}\n The $n$-th secant variety of the Segre variety is the closure of the $\\GL(A)\\times \\GL(B)\\times \\GL(C)$ orbit of the unit tensor $I_n:=\\sum_{i=1}^n a_i\\ot b_i\\ot c_i$, where $a_i$, $b_i$, $c_i$ form respectively bases of $A$, $B$, $C$. As our constructions are continuous and equivariant, it is enough to prove that $\\rk \\Phi_{\\mathrm M,k}^C(I_n)=\\binom{n}{k} $. We note that $I_n^C(C^*)$ is the space of diagonal matrices. Clearly, the linear span of $k\\times k$ minors of the diagonal matrices is the linear span of squarefree monomials of degree $k$ in $n$ variables and thus has dimension $\\binom{n}{k}$.\n\\end{proof}\n\n\\begin{corollary}\n \\label{cor:alg-tangency-sum}\n If $T_i\\in R_i^*\\otimes R_i^*\\otimes R_i$, $i=1,2$ are structure tensors of $n_i$-dimensional commutative algebras $R_i$, then the structure tensor $T_1\\oplus T_2$ of the $n:=(n_1+n_2)$-dimensional algebra $R_1\\oplus R_2$ satisfies\n \\[\n \\rank\\tang^A(T_1\\oplus T_2) - n(n-1)(n-2) = \\Bigl(\\rank \\tang^A(T_1) - n_1(n_1-1)(n_1-2)\\Bigr) + \\Bigl(\\rank\\tang^A(T_2) - n_2(n_2-1)(n_2-2)\\Bigr).\n \\]\n\\end{corollary}\n\\begin{proof}\n Along with the conclusions of Proposition~\\ref{prop:koszul-equals-quad-on-algs}, observe that structure tensors of algebras are concise. Thus all but the first two terms of the right-hand side of \\eqref{eq:tangency-direct-sum} simplify, yielding\n \\[\n \\rank\\tang^A(T_1\\oplus T_2) = \\rank \\tang^A(T_1) + \\rank \\tang^A(T_2) + 3n_1n_2(n_1+n_2-2).\n \\]\n This rearranges to the Corollary.\n\\end{proof}\nThe following lemma is well-known to experts. It can be derived e.g.~from the results in \\cite{jelisiejew2024concise}. We include the proof for the sake of completeness.\n\\begin{lemma}\\label{lem:GorCactus}\n Let $R$ be a finite Gorenstein algebra and $T\\in (R^*)^{\\otimes k}\\ot R$ its structure tensor corresponding to the map $R^k\\rightarrow R$, $(r_1,\\dots,r_k)\\mapsto r_1\\cdots r_k$. The cactus rank of $T$ equals $d=\\dim_{\\CC} R$.\n\\end{lemma}\n\\begin{proof}\n As $T$ is concise it is enough to prove that it has cactus rank at most $d$. Thus we have to exhibit a scheme of length $d$, which will be $\\Spec R$, inside the Segre variety, so that $T$ is in its linear\nspan. As cactus rank is subadditive under direct sum we may assume $(R,\\mathfrak m)$ is local.\n\n\\begin{theorem}\n \\label{thrm:beat-cactus}\n The $(n(n-1)(n-2)+1)$-minors of the tangency flattenings for border rank $n$ in $\\CC^n\\otimes \\CC^n\\otimes \\CC^n$ do not vanish on the $n$-th cactus variety for all $n\\geq 14$.\n\\end{theorem}\n\\begin{proof}\nBy Lemma \\ref{lem:GorCactus}, structure tensors of Gorenstein algebras have minimal cactus rank.\n Thus, it suffices to produce examples of structure tensors of $n$-dimensional Gorenstein algebras whose tangency flattenings have ranks strictly higher than $n(n-1)(n-2)$.", + "post_theorem_intro_text_len": 2217, + "post_theorem_intro_text": "For example, for $n=14$ the minors in question are polynomials of degree $4370$ in $2744$ variables. Yet, they are explicit and we can determine their vanishing or nonvanishing very fast.\nWe will prove Theorem~\\ref{thrm:beat-cactus-introversion} by lower-bounding border rank via rank of the tangency flattening in Corollary~\\ref{cor:tan_flat} and exhibiting in Theorem~\\ref{thrm:beat-cactus} a sequence of tensors of cactus rank $n$ on which the above-named minors of the tangency flattening do not vanish. The name ``tangency'' flattening is motivated by a conjectural connection to tangent spaces of Hilbert schemes of points -- see Conjecture~\\ref{conj:tangency}. In Subsection~\\ref{subsec:mamu}, we additionally show that tangency flattenings provide a new and completely elementary proof of the fact that $2\\times2$ matrix multiplication tensor has border rank $7$.\n\nWhile Kronecker-Koszul flattenings generalize Koszul flattenings (cf. \\cite{langsberg-ottaviani-equationsforsecant, landsberg2008generalizations, landsberg-michalek-mamu, landsberg-michalek-haystack}), we also analyze in Section~\\ref{sec:koszul} bounds that may be extracted via Koszul flattenings for the border rank (and border cactus rank) of structure tensors of certain algebras. It is known \\cite{blaser2016degeneration} that a (finite) algebra is smoothable if and only if the associated structure tensor has minimal border rank. This is also one of the main challenges in application of the border apolarity method.\nFurther, two of the classical flattenings of structure tensors of algebras give rise to spaces of commuting matrices. Thus, many of the flattening methods cannot provide nontrivial bounds on border rank of such tensors. However, we show that it is possible to obtain lower bounds on border cactus rank of structure tensors of algebras strictly greater than the degree of the algebra, even using Koszul flattenings (see Proposition~\\ref{prop:1de-koszul}).\n\nFinally, we show that under mild assumptions, Koszul flattenings may be replaced and improved upon by certain quadratic Kronecker-Koszul flattenings for the purposes of lower-bounding border rank of tensor (see Proposition~\\ref{prop:quadratic-better-than-koszul}).", + "sketch": "We will prove Theorem~\\ref{thrm:beat-cactus-introversion} by (1) lower-bounding border rank via the rank of the tangency flattening (Corollary~\\ref{cor:tan_flat}), and (2) exhibiting (in Theorem~\\ref{thrm:beat-cactus}) a sequence of tensors of cactus rank $n$ on which the minors of size $n(n-1)(n-2)+1$ of the tangency flattening do not vanish.", + "expanded_sketch": "No expanded sketch found.", + "expanded_theorem": "\\label{thrm:beat-cactus-introversion}\n For all $n\\geq14$, minors of size $n(n-1)(n-2)+1$ of the tangency flattenings vanish on the $n$-th secant variety of the Segre variety in $\\P(\\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\mathbb{C}^n)$ but do not vanish on its $n$-th cactus variety.", + "theorem_type": [ + "Universal", + "Implication" + ], + "mcq": { + "question": "For each integer \\(n\\ge 14\\), consider the Segre variety \\(\\operatorname{Seg}(\\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n))\\subset \\mathbb P(\\mathbb C^n\\otimes \\mathbb C^n\\otimes \\mathbb C^n)\\). Let its \\(n\\)-th secant variety be the Zariski closure of the union of linear spans of \\(n\\) points of the Segre variety, and let its \\(n\\)-th cactus variety be the Zariski closure of the union of linear spans of length-\\(n\\) subschemes of the Segre variety. The paper associates to tensors in \\(\\mathbb C^n\\otimes \\mathbb C^n\\otimes \\mathbb C^n\\) certain matrix-valued maps called tangency flattenings; taking all minors of size \\(n(n-1)(n-2)+1\\) gives polynomial equations on the ambient projective space. Which statement holds for every such \\(n\\)?", + "correct_choice": { + "label": "A", + "text": "All minors of size \\(n(n-1)(n-2)+1\\) of the tangency flattenings vanish on the \\(n\\)-th secant variety of \\(\\operatorname{Seg}(\\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n))\\), but they do not vanish on its \\(n\\)-th cactus variety." + }, + "choices": [ + { + "label": "B", + "text": "All minors of size \\(n(n-1)(n-2)+1\\) of the tangency flattenings vanish on both the \\(n\\)-th secant variety and the \\(n\\)-th cactus variety of \\(\\operatorname{Seg}(\\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n))\\)." + }, + { + "label": "C", + "text": "All minors of size \\(n(n-1)(n-2)+1\\) of the tangency flattenings vanish on the \\(n\\)-th secant variety of \\(\\operatorname{Seg}(\\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n))\\)." + }, + { + "label": "D", + "text": "All minors of size \\(n(n-1)(n-2)\\) of the tangency flattenings vanish on the \\(n\\)-th secant variety of \\(\\operatorname{Seg}(\\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n))\\), but they do not vanish on its \\(n\\)-th cactus variety." + }, + { + "label": "E", + "text": "For every tensor in \\(\\mathbb C^n\\otimes \\mathbb C^n\\otimes \\mathbb C^n\\) outside the \\(n\\)-th cactus variety, some minor of size \\(n(n-1)(n-2)+1\\) of a tangency flattening is nonzero; equivalently, these minors cut out the \\(n\\)-th cactus variety set-theoretically." + } + ], + "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": "nonvanishing_on_cactus", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped_failure_on_cactus_variety", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "minor_size_threshold_plus_one", + "template_used": "boundary_range" + }, + { + "label": "E", + "sketch_hook_type": "finiteness", + "tampered_component": "mere_nonvanishing_vs_set_theoretic_definition", + "template_used": "stronger_trap" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives technical setup and asks for the resulting conclusion, but it does not explicitly state or strongly hint at the full correct claim. The key distinguishing feature—failure on the cactus variety—is not leaked." + }, + "TAS": { + "score": 1, + "justification": "The item is close to theorem recall: it asks which precise conclusion holds for a very specific construction. However, it is not a pure verbatim restatement, since the options vary meaningfully in scope, threshold, and quantifiers." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish the strongest correct statement from weaker, overgeneralized, or boundary-shifted alternatives. Still, the question mainly tests knowledge of a specific result rather than substantial generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and well-targeted: one is a weaker true-looking statement, others alter the cactus behavior, quantifiers, or the range n>=14. These reflect realistic mathematical failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with little answer leakage and strong distractors, but it leans more toward precise recall than genuine generative reasoning." + } + }, + { + "id": "2602.12762v1", + "paper_link": "http://arxiv.org/abs/2602.12762v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "\\label{thrm:beat-cactus-introversion}\n For all $n\\geq14$, minors of size $n(n-1)(n-2)+1$ of the tangency flattenings vanish on the $n$-th secant variety of the Segre variety in $\\P(\\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\mathbb{C}^n)$ but do not vanish on its $n$-th cactus variety.", + "start_pos": 10252, + "end_pos": 10549, + "label": "thrm:beat-cactus-introversion" + }, + "ref_dict": { + "thrm:beat-cactus-introversion": "\\begin{theorem}\n \\label{thrm:beat-cactus-introversion}\n For all $n\\geq14$, minors of size $n(n-1)(n-2)+1$ of the tangency flattenings vanish on the $n$-th secant variety of the Segre variety in $\\P(\\CC^n\\otimes\\CC^n\\otimes\\CC^n)$ but do not vanish on its $n$-th cactus variety.\n\\end{theorem}", + "prop:1de-koszul": "\\begin{proposition}\n \\label{prop:1de-koszul}\n Let $R$ be the apolar algebra to $e\\leq \\binom{d+1}2$ general quadrics in $d$ variables. If $e\\geq3$, then the structure tensor $T$ of $R$ has border cactus rank at least $\\floor{\\frac32d}+3$. In particular, if $3\\leq e\\leq \\floor{\\frac d2}+1$, then $R$ is nonsmoothable.\n\\end{proposition}", + "prop:quadratic-better-than-koszul": "\\begin{proposition}\n \\label{prop:quadratic-better-than-koszul}\n The space $\\im\\Phi_{\\mathrm Q}^C(T)\\subset A\\otimes B$ equals the sum of all $(T(\\alpha)\\otimes \\id_A)\\left(\\im\\koszul^{C,B}(T)\\right)$ over $\\alpha\\in A^*$, where we consider $T(\\alpha)$ as a map $C^*\\to B$. In particular, if $\\dim B=\\dim C$ and $T$ is $1_A$-generic, this implies $\\rank \\Phi_{\\mathrm Q}^C(T) \\geq \\rank\\koszul^{C,B}(T)$.\n\\end{proposition}", + "thrm:beat-cactus": "\\begin{theorem}\n \\label{thrm:beat-cactus}\n The $(n(n-1)(n-2)+1)$-minors of the tangency flattenings for border rank $n$ in $\\CC^n\\otimes \\CC^n\\otimes \\CC^n$ do not vanish on the $n$-th cactus variety for all $n\\geq 14$.\n\\end{theorem}", + "conj:tangency": "\\begin{conjecture}\n \\label{conj:tangency}\n Let $\\CC[x_1,\\dots,x_d]\\surjto R$ be a surjection to an $n$-dimensional algebra $R$, let $[R]$ be the corresponding point of $\\Hilb_n\\CC^d$ and let $T$ be the structure tensor of $R$. Then\n \\[\n \\rank\\tang^C(T) - n(n-1)(n-2) = dn - \\dim\\T_{[R]}\\Hilb_n\\CC^d.\n \\]\n\\end{conjecture}", + "subsec:mamu": "\\begin{proof}\n Let us prove only the second set of equalities of \\eqref{eq:koszul-quaddratic-algebra}, since the proof of the first one is very similar.\n\n Explicitly, the two flattenings in question map\n \\begin{align*}\n \\Phi_{\\mathrm Q}^A(T): \\bigwedge^2 R \\otimes R \\otimes R^* &\\to R \\otimes R^*,\\\\\n (a\\wedge a')\\otimes b\\otimes \\gamma &\\mapsto (ab)\\otimes \\gamma(a'\\cdot\\bullet) - (a'b)\\otimes \\gamma(a\\cdot\\bullet)\n \\end{align*}\n and\n \\begin{align*}\n \\koszul^{A,C}(T): \\bigwedge^2 R \\otimes R^* &\\to R \\otimes R^*,\\\\\n (a\\wedge a')\\otimes\\gamma &\\mapsto a\\otimes \\gamma(a'\\cdot\\bullet) - a'\\otimes \\gamma(a\\cdot\\bullet),\n \\end{align*}\n where we use $\\gamma(a\\cdot\\bullet)$ to denote the linear form $(x\\mapsto \\gamma(ax))\\in R^*$. We claim that in fact $\\Phi_{\\mathrm Q}^A(T)$ and $\\koszul^{A,C}(T)$ have the same images. The inclusion $\\im\\Phi^{A,C}(T)\\subseteq \\im\\Phi_{\\mathrm Q}^A$ is obvious by considering $b=1$ in the latter map.\n\n For the opposite inclusion, note that invertible elements form a dense set in $R$, so in particular, $R$ may be spanned by them, hence $\\im \\Phi_{\\mathrm Q}^A$ is spanned by images of $(a\\wedge a')\\otimes b\\otimes\\gamma$ with $b$ invertible. These lie in $\\im \\koszul^{A,C}$ by the following: we set\n \\[\n \\tilde a := ba, \\qquad \\tilde a' := ba', \\qquad \\tilde\\gamma := \\gamma(b^{-1}\\cdot\\bullet),\n \\]\n which then results in\n \\begin{multline*}\n \\koszul^{A,C}(T)\\left(\n (\\tilde a\\wedge \\tilde a') \\otimes \\tilde \\gamma\n \\right)\n =\n \\tilde a\\otimes\\tilde \\gamma(\\tilde a'\\cdot\\bullet) - \\tilde a'\\otimes\\tilde\\gamma(\\tilde a\\cdot\\bullet)\n =\n (ab)\\otimes\\gamma(b^{-1}ba'\\cdot\\bullet) - (a'b)\\otimes \\gamma(b^{-1}ba\\cdot\\bullet)\n =\\\\=\n (ab)\\otimes \\gamma(a'\\cdot\\bullet)-(a'b)\\otimes\\gamma(a\\cdot\\bullet)\n =\n \\Phi_{\\mathrm Q}^A(T)\\left(\n (a\\wedge a') \\otimes b\\otimes\\gamma\n \\right).\n \\end{multline*}\n This justifies $\\rank \\Phi_{\\mathrm Q}^A(T) = \\rank \\koszul^{A,C}(T)$. That this common rank equals $n(n-1)$ then follows from Lemma \\ref{lem:easy rank Koszul for algebras}.\n\\end{proof}\n\n\\section{Applications}\n\\label{sec:applications}\n\n\\subsection{Matrix multiplication}\n\\label{subsec:mamu}\n\nIt is well-known that the border rank of the $2\\times 2$ matrix multiplication tensor $M_2$ equals $7$ \\cite{Landsberg2006, Hauenstein-Ikenmeyer-Landsberg}. However, only recently the ``first hand-checkable algebraic proof'' appeared \\cite{Conner_Harper_Landsberg_2023}. Below we show that tangency flattenings provide an easy proof.\n\\begin{proposition}\\label{prop:M2=7}\n The border rank of $M_2$ equals seven.\n\\end{proposition}\n\\begin{proof}\nStrassen \\cite{strassen1969gaussian} provided an explicit rank seven decomposition of $M_2$. We focus on lower-bounding the border rank.\n\n By Corollary \\ref{cor:tan_flat} for $n=4$, $q=6$ we have to show that the tangency flattening has rank greater than $60$. We will prove that it is $64$. We fix a basis $(i,j)$ of $\\CC^4$, where $i,j\\in\\{0,1\\}$ and we use $\\neg i$ to denote the negation of $i$. Recall that $M_2=\\sum_{i,j,k\\in \\{0,1\\}} (i,j)\\otimes (j,k)\\otimes (k,i)$.\n\n We will prove that the tangency flattening is a block diagonal matrix with five blocks: a $48\\times 48$ identity matrix and four isomorphic, nondegenerate $4\\times 4$ matrices.\n\nThe tangency flattening of $M_2$ is the restriction to $\\bigwedge^3\\CC^4\\otimes \\CC^4\\otimes \\CC^4$ of\n\\begin{align}\n\\label{eq:mamu-tangency}\n(\\CC^4)^{\\otimes 3}\\otimes \\CC^4\\otimes \\CC^4 &\\to \\CC^4\\otimes \\CC^4\\otimes \\CC^4,\\\\\n\\nonumber\n(i_1,j_1)\\otimes (i_2,j_2)\\otimes (i_3,j_3)\\otimes (i_4,j_4)\\otimes (i_5,j_5) &\\mapsto\n\\begin{cases}\n(i_1,j_1)\\otimes (i_2,j_4)\\otimes (i_5,j_3), & \\text{if $j_2=i_4$ and $i_3 = j_5$,}\\\\\n0, & \\text{otherwise.}\n\\end{cases}\n\\end{align}\nWe divide the basis vectors in the domain into five groups.\n\nThe first group is formed by vectors of the form $\\bigl((\\neg j_5,\\neg i_4)\\wedge(i_2,i_4)\\wedge (j_5,j_3)\\bigr)\\otimes (i_4,j_4)\\otimes (i_5,j_5)$. By first choosing $i_4,j_4,i_5,j_5$ (16 options), and then choosing exterior products that contain $(\\neg j_5,\\neg i_4)$ (3 options), we see there are $48$ such (nonzero) vectors.\nNote that this already forces the exterior product to be of the form above, since the two remaining exterior factors are chosen from $(\\neg j_5, i_4)$, $(j_5, \\neg i_4)$ and $(j_5,i_4)$.\nAlthough the exterior product expands as a signed sum of six tensor products, only one of those -- the one ordered as in the presentation above -- maps to a nonzero vector in \\eqref{eq:mamu-tangency}. Thus, the image of each such basis vector is $(\\neg j_5,\\neg i_4)\\otimes (i_2,j_4)\\otimes (i_5,j_3)$. Note that such images must satisfy $(\\neg j_5,\\neg i_4)\\neq (\\neg i_2,\\neg j_3)$.\n\nThe next four groups are indexed by $j_4,i_5\\in \\{0,1\\}$. For fixed $j_4,i_5$ the group consists of the four basis vectors $\\bigl((j_5,i_4)\\wedge (\\neg j_5,i_4)\\wedge (j_5,\\neg i_4)\\bigr)\\otimes (i_4,j_4)\\otimes (i_5,j_5)$, where $i_4, j_5$ are arbitrary.\nThe image of each such vector is:\n\\[(j_5,i_4)\\otimes (\\neg j_5,j_4)\\otimes (i_5,\\neg i_4)-(\\neg j_5,i_4)\\otimes (j_5,j_4)\\otimes (i_5,\\neg i_4)-(j_5,\\neg i_4)\\otimes (\\neg j_5,j_4)\\otimes (i_5,i_4).\\]\nIn particular, the images are linear combinations of basis vectors that are distinct from images of the first group as well as other groups -- in each term, the first tensor factor consist exactly of negations of the first coordinate of the second factor and the second coordinate of the third factor. Meanwhile, the group is determined by reading off $j_4$ and $i_5$ from the second coordinate of the second factor and first coordinate of the third factor, respectively.\n\nThus indeed the tangency flattening is in block-diagonal form. It remains for us to prove that the isomorphic $4\\times 4$ blocks are of full rank.\nEach such matrix has the following form, where we label $(j_5,i_4)\\otimes (\\neg j_5,j_4)\\otimes (i_5,\\neg i_4)$ with the first factor $(j_5,i_4)$ and subsequently order rows and columns by $(j_5,i_4)=(0,0)$, $(0,1)$, $(1,0)$, $(1,1)$:\n\\[\\begin{pmatrix}\n 1 & -1&-1&0\\\\\n -1& 1&0&-1\\\\\n -1& 0&1&-1\\\\\n 0& -1&-1&1\\\\\n\\end{pmatrix}.\\]\nThe determinant of such a matrix is $-3$, thus indeed the matrix is of full rank, and so is the tangency flattening.\n\\end{proof}", + "cor:tan_flat": "\\begin{corollary}\\label{cor:tan_flat}\n If $\\dim V_1=\\dim V_2=\\dim V_3=n$, then $q(q-1)(n-2)+1$ minors of the tangency flattening provide equations for border rank $q$.\n\\end{corollary}" + }, + "pre_theorem_intro_text_len": 6726, + "pre_theorem_intro_text": "\\label{sec:intro}\nThe central problem concerning tensors is that of \\emph{tensor decomposition} -- presenting a tensor as a sum of simple tensors, or computing the smallest possible size of a decomposition, called the \\emph{tensor rank}. While the problem is NP-hard in general \\cite{RankNPcomplete, hillar2013most}, providing estimates for rank or related notions such as border rank or asymptotic rank has been a fruitful field. Perhaps the most famous application is the study of computational complexity of matrix multiplication (e.g. \\cite{strassen1969gaussian, coppersmith1982asymptotic, alman2024refined, williams2024new}). Further, a recent line of research relates low asymptotic rank of certain special tensors to faster than known algorithms for NP-hard problems \\cite{bjorklund2025fast, bjorklund2024asymptotic, pratt2024stronger, MaMu-SotA}.\nFor an introduction to tensors from an algebraic perspective, we refer the reader to \\cite[Chapter 9.2]{michalek2021invitation}, or to \\cite{landsberg2011tensors} for a more extensive exposition. For the connections to complexity theory, we refer to \\cite{burgisser2013algebraic, landsberg2017geometry}.\n\nThe main topic of this article is obtaining lower bounds on the border rank of tensors, a longstanding question in tensor geometry. The classical approach to the problem is by providing equations for secant varieties to the Segre variety. Flattenings, or maps which produce a matrix from a given tensor, have been a bountiful source of such lower bounds (e.g. \\cite{langsberg-ottaviani-equationsforsecant, landsberg2008generalizations, landsberg-ottaviani-newbounds, landsberg-michalek-mamu, landsberg-michalek-haystack}), with the equations thus obtained coming in the form of minors. In particular, \\emph{Strassen's equations}, further studied by Landsberg and Manivel and expanded by Landsberg and Ottaviani to \\emph{Koszul} and \\emph{Young flattenings}, became the gold standard in the field. Apart from flattenings, the techniques of \\emph{border apolarity} (see \\cite{Buczynscy-apolarity}) and \\emph{border substitution} (see \\cite{landsberg2017geometry}) have been introduced and used to prove several new lower bounds in recent years. However, all of these methods have quite restrictive limits of application.\n\nIndeed, current lower bound methods fall far short of being able to prove a tensor in $\\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\mathbb{C}^n$ to have the generic border rank, which is approximately $\\frac{n^2}3$. Instead, known flattening methods give maximal lower bounds of up to $(2-\\epsilon)n$ for small $\\epsilon>0$. Combining algebraic equations with other methods, it is possible to provide examples of tensors of border rank $2.02n$ \\cite{landsberg-michalek-haystack}. Still, there are well understood barriers which flattening methods cannot beat \\cite{jarek_cactus, Galazka-bundles, EGOW18, garg2019more}. In contrast, the applications to asymptotic rank in computer science would require superlinear lower bounds on border rank.\n\nThe disparity between the generic border rank and available lower bounds has been attributed to the inclusion of secant varieties in \\emph{cactus varieties} (see Section~\\ref{sec:preliminaries}). In particular, known flattening methods have been unable to distinguish secant from cactus varieties, even when the inclusion is known to be strict -- quoting from \\cite{Conner_Harper_Landsberg_2023}:\n\\begin{quote}\n``The geometric interpretation of the border rank lower bound barriers of \\cite{EGOW18, Galazka-bundles} is that all equations obtained by taking minors, called rank methods, are actually equations for a larger variety than $\\sigma_r(\\Seg(\\P A\\times\\P B\\times\\P C))$, called the $r$-th cactus variety \\cite{Buczynscy-apolarity}.''\n\\end{quote}\nIndeed, as proved in \\cite{jarek_cactus, Galazka-bundles, galazka-thesis} all of the known methods producing equations for the $r$-th secant variety are equations for the $r$-th cactus variety. Further, it is known that the $r$-th cactus variety of the Segre variety in $(\\mathbb{C}^n)^{\\ot 3}$ fills the ambient space for $r$ growing linearly in $n$ \\cite{ballico2019note, bernardi2018polynomials, galkazka2023multigraded}.\n\nFor these reasons, the existence of easily computable determinantal equations vanishing on secant varieties but not vanishing on cactus varieties, has long been considered unlikely or even impossible in the tensor-geometric community.\nIn this article, we illustrate that it is not the determinantal expression which creates the cactus barrier, but rather the linear embedding of the tensor product into a matrix space.\nThus the starting point of our approach in this article is focusing on \\emph{nonlinear} maps from tensor spaces to matrix spaces.\nTo obtain bounds on matrix rank with linear embeddings, it is enough to investigate the embedded image of the Segre variety and appeal to linearity. With nonlinear maps on the other hand, we need to work more generally, but this does yield stronger results. We obtain a new method, which we dub \\emph{Kronecker-Koszul flattenings}, or more generally \\emph{Kronecker-Young flattenings}, that allows one to produce\nexplicit and efficiently computable determinantal equations of secant varieties that do not generally vanish on cactus varieties. In particular, Kronecker-Koszul flattenings require less computation time and memory than methods based on a more direct identification of ideals defining the secant varieties (cf.~\\cite{Hauenstein-Ikenmeyer-Landsberg}).\n\n Broadly speaking, our method starts by tensoring together several copies of the same tensor. This, depending on the interpretation of the codomain, may be viewed as a Kronecker power or Veronese embedding \\cite{kaski2025universal}, and thus is the universal map for fixed degree polynomials. Next, we further tensor with several $2$-tensors corresponding to identity maps on various vector spaces. Finally, we group and contract some of the factors to exterior powers. This results in what we call a \\emph{Kronecker-Koszul tensor}. A Kronecker-Koszul flattening is then a classical flattening of the Kronecker-Koszul tensor. Note that due to initially taking a tensor power, Kronecker-Koszul tensors are in general nonlinear as functions of the starting tensor.\nSee Section~\\ref{sec:construction} for details of the construction. There we also comment on what we call {Kronecker-Young flattenings}, a further generalization which synthesizes the approach of Young flattenings into the method.\n\nOur main result uses the \\emph{tangency flattening}, a very particular Kronecker-Koszul flattening which depends quadratically on a tensor. See the paragraph preceding Corollary~\\ref{cor:tan_flat} for the definition.", + "context": "The main topic of this article is obtaining lower bounds on the border rank of tensors, a longstanding question in tensor geometry. The classical approach to the problem is by providing equations for secant varieties to the Segre variety. Flattenings, or maps which produce a matrix from a given tensor, have been a bountiful source of such lower bounds (e.g. \\cite{langsberg-ottaviani-equationsforsecant, landsberg2008generalizations, landsberg-ottaviani-newbounds, landsberg-michalek-mamu, landsberg-michalek-haystack}), with the equations thus obtained coming in the form of minors. In particular, \\emph{Strassen's equations}, further studied by Landsberg and Manivel and expanded by Landsberg and Ottaviani to \\emph{Koszul} and \\emph{Young flattenings}, became the gold standard in the field. Apart from flattenings, the techniques of \\emph{border apolarity} (see \\cite{Buczynscy-apolarity}) and \\emph{border substitution} (see \\cite{landsberg2017geometry}) have been introduced and used to prove several new lower bounds in recent years. However, all of these methods have quite restrictive limits of application.\n\nIndeed, current lower bound methods fall far short of being able to prove a tensor in $\\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\mathbb{C}^n$ to have the generic border rank, which is approximately $\\frac{n^2}3$. Instead, known flattening methods give maximal lower bounds of up to $(2-\\epsilon)n$ for small $\\epsilon>0$. Combining algebraic equations with other methods, it is possible to provide examples of tensors of border rank $2.02n$ \\cite{landsberg-michalek-haystack}. Still, there are well understood barriers which flattening methods cannot beat \\cite{jarek_cactus, Galazka-bundles, EGOW18, garg2019more}. In contrast, the applications to asymptotic rank in computer science would require superlinear lower bounds on border rank.\n\nThe disparity between the generic border rank and available lower bounds has been attributed to the inclusion of secant varieties in \\emph{cactus varieties} (see Section~\\ref{sec:preliminaries}). In particular, known flattening methods have been unable to distinguish secant from cactus varieties, even when the inclusion is known to be strict -- quoting from \\cite{Conner_Harper_Landsberg_2023}:\n\\begin{quote}\n``The geometric interpretation of the border rank lower bound barriers of \\cite{EGOW18, Galazka-bundles} is that all equations obtained by taking minors, called rank methods, are actually equations for a larger variety than $\\sigma_r(\\Seg(\\P A\\times\\P B\\times\\P C))$, called the $r$-th cactus variety \\cite{Buczynscy-apolarity}.''\n\\end{quote}\nIndeed, as proved in \\cite{jarek_cactus, Galazka-bundles, galazka-thesis} all of the known methods producing equations for the $r$-th secant variety are equations for the $r$-th cactus variety. Further, it is known that the $r$-th cactus variety of the Segre variety in $(\\mathbb{C}^n)^{\\ot 3}$ fills the ambient space for $r$ growing linearly in $n$ \\cite{ballico2019note, bernardi2018polynomials, galkazka2023multigraded}.\n\nFor these reasons, the existence of easily computable determinantal equations vanishing on secant varieties but not vanishing on cactus varieties, has long been considered unlikely or even impossible in the tensor-geometric community.\nIn this article, we illustrate that it is not the determinantal expression which creates the cactus barrier, but rather the linear embedding of the tensor product into a matrix space.\nThus the starting point of our approach in this article is focusing on \\emph{nonlinear} maps from tensor spaces to matrix spaces.\nTo obtain bounds on matrix rank with linear embeddings, it is enough to investigate the embedded image of the Segre variety and appeal to linearity. With nonlinear maps on the other hand, we need to work more generally, but this does yield stronger results. We obtain a new method, which we dub \\emph{Kronecker-Koszul flattenings}, or more generally \\emph{Kronecker-Young flattenings}, that allows one to produce\nexplicit and efficiently computable determinantal equations of secant varieties that do not generally vanish on cactus varieties. In particular, Kronecker-Koszul flattenings require less computation time and memory than methods based on a more direct identification of ideals defining the secant varieties (cf.~\\cite{Hauenstein-Ikenmeyer-Landsberg}).\n\nBroadly speaking, our method starts by tensoring together several copies of the same tensor. This, depending on the interpretation of the codomain, may be viewed as a Kronecker power or Veronese embedding \\cite{kaski2025universal}, and thus is the universal map for fixed degree polynomials. Next, we further tensor with several $2$-tensors corresponding to identity maps on various vector spaces. Finally, we group and contract some of the factors to exterior powers. This results in what we call a \\emph{Kronecker-Koszul tensor}. A Kronecker-Koszul flattening is then a classical flattening of the Kronecker-Koszul tensor. Note that due to initially taking a tensor power, Kronecker-Koszul tensors are in general nonlinear as functions of the starting tensor.\nSee Section~\\ref{sec:construction} for details of the construction. There we also comment on what we call {Kronecker-Young flattenings}, a further generalization which synthesizes the approach of Young flattenings into the method.\n\nOur main result uses the \\emph{tangency flattening}, a very particular Kronecker-Koszul flattening which depends quadratically on a tensor. See the paragraph preceding Corollary~\\ref{cor:tan_flat} for the definition.\n\n\\begin{corollary}\\label{cor:tan_flat}\n If $\\dim V_1=\\dim V_2=\\dim V_3=n$, then $q(q-1)(n-2)+1$ minors of the tangency flattening provide equations for border rank $q$.\n\\end{corollary}", + "full_context": "The main topic of this article is obtaining lower bounds on the border rank of tensors, a longstanding question in tensor geometry. The classical approach to the problem is by providing equations for secant varieties to the Segre variety. Flattenings, or maps which produce a matrix from a given tensor, have been a bountiful source of such lower bounds (e.g. \\cite{langsberg-ottaviani-equationsforsecant, landsberg2008generalizations, landsberg-ottaviani-newbounds, landsberg-michalek-mamu, landsberg-michalek-haystack}), with the equations thus obtained coming in the form of minors. In particular, \\emph{Strassen's equations}, further studied by Landsberg and Manivel and expanded by Landsberg and Ottaviani to \\emph{Koszul} and \\emph{Young flattenings}, became the gold standard in the field. Apart from flattenings, the techniques of \\emph{border apolarity} (see \\cite{Buczynscy-apolarity}) and \\emph{border substitution} (see \\cite{landsberg2017geometry}) have been introduced and used to prove several new lower bounds in recent years. However, all of these methods have quite restrictive limits of application.\n\nIndeed, current lower bound methods fall far short of being able to prove a tensor in $\\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\mathbb{C}^n$ to have the generic border rank, which is approximately $\\frac{n^2}3$. Instead, known flattening methods give maximal lower bounds of up to $(2-\\epsilon)n$ for small $\\epsilon>0$. Combining algebraic equations with other methods, it is possible to provide examples of tensors of border rank $2.02n$ \\cite{landsberg-michalek-haystack}. Still, there are well understood barriers which flattening methods cannot beat \\cite{jarek_cactus, Galazka-bundles, EGOW18, garg2019more}. In contrast, the applications to asymptotic rank in computer science would require superlinear lower bounds on border rank.\n\nThe disparity between the generic border rank and available lower bounds has been attributed to the inclusion of secant varieties in \\emph{cactus varieties} (see Section~\\ref{sec:preliminaries}). In particular, known flattening methods have been unable to distinguish secant from cactus varieties, even when the inclusion is known to be strict -- quoting from \\cite{Conner_Harper_Landsberg_2023}:\n\\begin{quote}\n``The geometric interpretation of the border rank lower bound barriers of \\cite{EGOW18, Galazka-bundles} is that all equations obtained by taking minors, called rank methods, are actually equations for a larger variety than $\\sigma_r(\\Seg(\\P A\\times\\P B\\times\\P C))$, called the $r$-th cactus variety \\cite{Buczynscy-apolarity}.''\n\\end{quote}\nIndeed, as proved in \\cite{jarek_cactus, Galazka-bundles, galazka-thesis} all of the known methods producing equations for the $r$-th secant variety are equations for the $r$-th cactus variety. Further, it is known that the $r$-th cactus variety of the Segre variety in $(\\mathbb{C}^n)^{\\ot 3}$ fills the ambient space for $r$ growing linearly in $n$ \\cite{ballico2019note, bernardi2018polynomials, galkazka2023multigraded}.\n\nFor these reasons, the existence of easily computable determinantal equations vanishing on secant varieties but not vanishing on cactus varieties, has long been considered unlikely or even impossible in the tensor-geometric community.\nIn this article, we illustrate that it is not the determinantal expression which creates the cactus barrier, but rather the linear embedding of the tensor product into a matrix space.\nThus the starting point of our approach in this article is focusing on \\emph{nonlinear} maps from tensor spaces to matrix spaces.\nTo obtain bounds on matrix rank with linear embeddings, it is enough to investigate the embedded image of the Segre variety and appeal to linearity. With nonlinear maps on the other hand, we need to work more generally, but this does yield stronger results. We obtain a new method, which we dub \\emph{Kronecker-Koszul flattenings}, or more generally \\emph{Kronecker-Young flattenings}, that allows one to produce\nexplicit and efficiently computable determinantal equations of secant varieties that do not generally vanish on cactus varieties. In particular, Kronecker-Koszul flattenings require less computation time and memory than methods based on a more direct identification of ideals defining the secant varieties (cf.~\\cite{Hauenstein-Ikenmeyer-Landsberg}).\n\nBroadly speaking, our method starts by tensoring together several copies of the same tensor. This, depending on the interpretation of the codomain, may be viewed as a Kronecker power or Veronese embedding \\cite{kaski2025universal}, and thus is the universal map for fixed degree polynomials. Next, we further tensor with several $2$-tensors corresponding to identity maps on various vector spaces. Finally, we group and contract some of the factors to exterior powers. This results in what we call a \\emph{Kronecker-Koszul tensor}. A Kronecker-Koszul flattening is then a classical flattening of the Kronecker-Koszul tensor. Note that due to initially taking a tensor power, Kronecker-Koszul tensors are in general nonlinear as functions of the starting tensor.\nSee Section~\\ref{sec:construction} for details of the construction. There we also comment on what we call {Kronecker-Young flattenings}, a further generalization which synthesizes the approach of Young flattenings into the method.\n\nOur main result uses the \\emph{tangency flattening}, a very particular Kronecker-Koszul flattening which depends quadratically on a tensor. See the paragraph preceding Corollary~\\ref{cor:tan_flat} for the definition.\n\n\\begin{corollary}\\label{cor:tan_flat}\n If $\\dim V_1=\\dim V_2=\\dim V_3=n$, then $q(q-1)(n-2)+1$ minors of the tangency flattening provide equations for border rank $q$.\n\\end{corollary}\n\nFor these reasons, the existence of easily computable determinantal equations vanishing on secant varieties but not vanishing on cactus varieties, has long been considered unlikely or even impossible in the tensor-geometric community.\nIn this article, we illustrate that it is not the determinantal expression which creates the cactus barrier, but rather the linear embedding of the tensor product into a matrix space.\nThus the starting point of our approach in this article is focusing on \\emph{nonlinear} maps from tensor spaces to matrix spaces.\nTo obtain bounds on matrix rank with linear embeddings, it is enough to investigate the embedded image of the Segre variety and appeal to linearity. With nonlinear maps on the other hand, we need to work more generally, but this does yield stronger results. We obtain a new method, which we dub \\emph{Kronecker-Koszul flattenings}, or more generally \\emph{Kronecker-Young flattenings}, that allows one to produce\nexplicit and efficiently computable determinantal equations of secant varieties that do not generally vanish on cactus varieties. In particular, Kronecker-Koszul flattenings require less computation time and memory than methods based on a more direct identification of ideals defining the secant varieties (cf.~\\cite{Hauenstein-Ikenmeyer-Landsberg}).\n\nOur main result uses the \\emph{tangency flattening}, a very particular Kronecker-Koszul flattening which depends quadratically on a tensor. See the paragraph preceding Corollary~\\ref{cor:tan_flat} for the definition.\n\nWhile Kronecker-Koszul flattenings generalize Koszul flattenings (cf. \\cite{langsberg-ottaviani-equationsforsecant, landsberg2008generalizations, landsberg-michalek-mamu, landsberg-michalek-haystack}), we also analyze in Section~\\ref{sec:koszul} bounds that may be extracted via Koszul flattenings for the border rank (and border cactus rank) of structure tensors of certain algebras. It is known \\cite{blaser2016degeneration} that a (finite) algebra is smoothable if and only if the associated structure tensor has minimal border rank. This is also one of the main challenges in application of the border apolarity method.\nFurther, two of the classical flattenings of structure tensors of algebras give rise to spaces of commuting matrices. Thus, many of the flattening methods cannot provide nontrivial bounds on border rank of such tensors. However, we show that it is possible to obtain lower bounds on border cactus rank of structure tensors of algebras strictly greater than the degree of the algebra, even using Koszul flattenings (see Proposition~\\ref{prop:1de-koszul}).\n\n\\begin{corollary}\\label{cor:flat bound}\n If $T$ has border rank at most $q$, then the rank of a Kronecker-Koszul flattening of $T$ is at most $F(q)\\cdot \\prod_{i=1}^r\\prod_{j=1}^{s_i} \\binom{d_i-|\\lambda_{i,j}|}{d_{i,j}'}$. In particular, minors of the Kronecker-Koszul flattening provide equations for secant varieties of the Segre variety.\n\\end{corollary}\n\\begin{proof}\n Follows from Theorem \\ref{thm:main}, as ranks of classical flattenings lower bound border rank of the tensor.\n\\end{proof}\nThe bounds in Theorem \\ref{thm:main} and Corollary \\ref{cor:flat bound} do not have to be tight. This can happen for various reasons, some more opaque than others:\n\nIn this section we present one explicit construction that we find particularly interesting. By \\cite{kaski2025universal} the linear span of image of the $k$-th Kronecker power of tensors in $A\\ot B \\ot C$ is $S^k(A\\ot B \\ot C)$. As a $\\GL(A)\\times \\GL(B)\\times \\GL(C)$ representation, the latter space has a decomposition into isotypic components\n \\[S^k(A\\ot B \\ot C)=\\bigoplus_{\\lambda,\\mu,\\rho\\vdash k}\\left(S^\\lambda(A)\\ot S^\\mu(B)\\ot S^\\rho(C)\\right)^{\\bigoplus K_{\\lambda,\\mu,\\rho}},\\]\n where $K_{\\lambda,\\mu,\\rho}$ is the Kronecker coefficient. In particular, when $\\lambda=\\mu=1^k$ and $\\rho=(k)$ we have $K_{\\lambda,\\mu,\\rho}=1$. This gives a canonical projection\n \\[\\pi:S^k(A\\ot B \\ot C)\\rightarrow \\bigwedge^k A\\ot \\bigwedge^k B\\ot S^k(C).\\]\n An example of Kronecker-Young flattening is the classical flattening of $\\pi(T^{\\otimes k})$ given by\n \\[\\Phi_{\\mathrm M,k}^C(T):\\bigwedge^kA^*\\ot\\bigwedge^kB^*\\rightarrow S^k(C).\\]\n The advantage of these choices is that the image of the map $\\Phi_{\\mathrm M,k}^C(T)$ has a direct interpretation in terms of the flattening $T^C:C^*\\rightarrow A\\ot B$. Namely, the image of $\\Phi_{\\mathrm M,k}^C(T)$ is the linear span of all $k\\times k$ minors of $T^C$ viewed as a matrix with entries from $C$. For this reason, we call $\\Phi_{\\mathrm M,k}$ the \\emph{$k$-minor flattening}.\n\\begin{lemma}\n If $\\dim A=\\dim B=\\dim C=n$ and a tensor $T\\in A\\ot B\\ot C$ has border rank at most $n$, then the rank of the $k$-minor flattening $\\Phi_{\\mathrm M,k}^C(T)$ is at most $\\binom{n}{k}$.\n\\end{lemma}\n\\begin{proof}\n The $n$-th secant variety of the Segre variety is the closure of the $\\GL(A)\\times \\GL(B)\\times \\GL(C)$ orbit of the unit tensor $I_n:=\\sum_{i=1}^n a_i\\ot b_i\\ot c_i$, where $a_i$, $b_i$, $c_i$ form respectively bases of $A$, $B$, $C$. As our constructions are continuous and equivariant, it is enough to prove that $\\rk \\Phi_{\\mathrm M,k}^C(I_n)=\\binom{n}{k} $. We note that $I_n^C(C^*)$ is the space of diagonal matrices. Clearly, the linear span of $k\\times k$ minors of the diagonal matrices is the linear span of squarefree monomials of degree $k$ in $n$ variables and thus has dimension $\\binom{n}{k}$.\n\\end{proof}\n\n\\begin{corollary}\n \\label{cor:alg-tangency-sum}\n If $T_i\\in R_i^*\\otimes R_i^*\\otimes R_i$, $i=1,2$ are structure tensors of $n_i$-dimensional commutative algebras $R_i$, then the structure tensor $T_1\\oplus T_2$ of the $n:=(n_1+n_2)$-dimensional algebra $R_1\\oplus R_2$ satisfies\n \\[\n \\rank\\tang^A(T_1\\oplus T_2) - n(n-1)(n-2) = \\Bigl(\\rank \\tang^A(T_1) - n_1(n_1-1)(n_1-2)\\Bigr) + \\Bigl(\\rank\\tang^A(T_2) - n_2(n_2-1)(n_2-2)\\Bigr).\n \\]\n\\end{corollary}\n\\begin{proof}\n Along with the conclusions of Proposition~\\ref{prop:koszul-equals-quad-on-algs}, observe that structure tensors of algebras are concise. Thus all but the first two terms of the right-hand side of \\eqref{eq:tangency-direct-sum} simplify, yielding\n \\[\n \\rank\\tang^A(T_1\\oplus T_2) = \\rank \\tang^A(T_1) + \\rank \\tang^A(T_2) + 3n_1n_2(n_1+n_2-2).\n \\]\n This rearranges to the Corollary.\n\\end{proof}\nThe following lemma is well-known to experts. It can be derived e.g.~from the results in \\cite{jelisiejew2024concise}. We include the proof for the sake of completeness.\n\\begin{lemma}\\label{lem:GorCactus}\n Let $R$ be a finite Gorenstein algebra and $T\\in (R^*)^{\\otimes k}\\ot R$ its structure tensor corresponding to the map $R^k\\rightarrow R$, $(r_1,\\dots,r_k)\\mapsto r_1\\cdots r_k$. The cactus rank of $T$ equals $d=\\dim_{\\CC} R$.\n\\end{lemma}\n\\begin{proof}\n As $T$ is concise it is enough to prove that it has cactus rank at most $d$. Thus we have to exhibit a scheme of length $d$, which will be $\\Spec R$, inside the Segre variety, so that $T$ is in its linear\nspan. As cactus rank is subadditive under direct sum we may assume $(R,\\mathfrak m)$ is local.\n\n\\begin{theorem}\n \\label{thrm:beat-cactus}\n The $(n(n-1)(n-2)+1)$-minors of the tangency flattenings for border rank $n$ in $\\CC^n\\otimes \\CC^n\\otimes \\CC^n$ do not vanish on the $n$-th cactus variety for all $n\\geq 14$.\n\\end{theorem}\n\\begin{proof}\nBy Lemma \\ref{lem:GorCactus}, structure tensors of Gorenstein algebras have minimal cactus rank.\n Thus, it suffices to produce examples of structure tensors of $n$-dimensional Gorenstein algebras whose tangency flattenings have ranks strictly higher than $n(n-1)(n-2)$.", + "post_theorem_intro_text_len": 2217, + "post_theorem_intro_text": "For example, for $n=14$ the minors in question are polynomials of degree $4370$ in $2744$ variables. Yet, they are explicit and we can determine their vanishing or nonvanishing very fast.\nWe will prove Theorem~\\ref{thrm:beat-cactus-introversion} by lower-bounding border rank via rank of the tangency flattening in Corollary~\\ref{cor:tan_flat} and exhibiting in Theorem~\\ref{thrm:beat-cactus} a sequence of tensors of cactus rank $n$ on which the above-named minors of the tangency flattening do not vanish. The name ``tangency'' flattening is motivated by a conjectural connection to tangent spaces of Hilbert schemes of points -- see Conjecture~\\ref{conj:tangency}. In Subsection~\\ref{subsec:mamu}, we additionally show that tangency flattenings provide a new and completely elementary proof of the fact that $2\\times2$ matrix multiplication tensor has border rank $7$.\n\nWhile Kronecker-Koszul flattenings generalize Koszul flattenings (cf. \\cite{langsberg-ottaviani-equationsforsecant, landsberg2008generalizations, landsberg-michalek-mamu, landsberg-michalek-haystack}), we also analyze in Section~\\ref{sec:koszul} bounds that may be extracted via Koszul flattenings for the border rank (and border cactus rank) of structure tensors of certain algebras. It is known \\cite{blaser2016degeneration} that a (finite) algebra is smoothable if and only if the associated structure tensor has minimal border rank. This is also one of the main challenges in application of the border apolarity method.\nFurther, two of the classical flattenings of structure tensors of algebras give rise to spaces of commuting matrices. Thus, many of the flattening methods cannot provide nontrivial bounds on border rank of such tensors. However, we show that it is possible to obtain lower bounds on border cactus rank of structure tensors of algebras strictly greater than the degree of the algebra, even using Koszul flattenings (see Proposition~\\ref{prop:1de-koszul}).\n\nFinally, we show that under mild assumptions, Koszul flattenings may be replaced and improved upon by certain quadratic Kronecker-Koszul flattenings for the purposes of lower-bounding border rank of tensor (see Proposition~\\ref{prop:quadratic-better-than-koszul}).", + "sketch": "We will prove Theorem~\\ref{thrm:beat-cactus-introversion} by (1) lower-bounding border rank via the rank of the tangency flattening (Corollary~\\ref{cor:tan_flat}), and (2) exhibiting (in Theorem~\\ref{thrm:beat-cactus}) a sequence of tensors of cactus rank $n$ on which the minors of size $n(n-1)(n-2)+1$ of the tangency flattening do not vanish.", + "expanded_sketch": "No expanded sketch found.", + "expanded_theorem": "\\label{thrm:beat-cactus-introversion}\n For all $n\\geq14$, minors of size $n(n-1)(n-2)+1$ of the tangency flattenings vanish on the $n$-th secant variety of the Segre variety in $\\P(\\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\mathbb{C}^n)$ but do not vanish on its $n$-th cactus variety.", + "theorem_type": [ + "Universal", + "Implication" + ], + "mcq": { + "question": "For each integer \\(n\\ge 14\\), consider the Segre variety \\(\\operatorname{Seg}(\\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n))\\subset \\mathbb P(\\mathbb C^n\\otimes \\mathbb C^n\\otimes \\mathbb C^n)\\). Let its \\(n\\)-th secant variety be the Zariski closure of the union of linear spans of \\(n\\) points of the Segre variety, and let its \\(n\\)-th cactus variety be the Zariski closure of the union of linear spans of length-\\(n\\) subschemes of the Segre variety. The paper associates to tensors in \\(\\mathbb C^n\\otimes \\mathbb C^n\\otimes \\mathbb C^n\\) certain matrix-valued maps called tangency flattenings; taking all minors of size \\(n(n-1)(n-2)+1\\) gives polynomial equations on the ambient projective space. Which statement holds for every such \\(n\\)?", + "correct_choice": { + "label": "A", + "text": "All minors of size \\(n(n-1)(n-2)+1\\) of the tangency flattenings vanish on the \\(n\\)-th secant variety of \\(\\operatorname{Seg}(\\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n))\\), but they do not vanish on its \\(n\\)-th cactus variety." + }, + "choices": [ + { + "label": "B", + "text": "All minors of size \\(n(n-1)(n-2)+1\\) of the tangency flattenings vanish on both the \\(n\\)-th secant variety and the \\(n\\)-th cactus variety of \\(\\operatorname{Seg}(\\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n))\\)." + }, + { + "label": "C", + "text": "All minors of size \\(n(n-1)(n-2)+1\\) of the tangency flattenings vanish on the \\(n\\)-th secant variety of \\(\\operatorname{Seg}(\\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n))\\)." + }, + { + "label": "D", + "text": "All minors of size \\(n(n-1)(n-2)\\) of the tangency flattenings vanish on the \\(n\\)-th secant variety of \\(\\operatorname{Seg}(\\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n))\\), but they do not vanish on its \\(n\\)-th cactus variety." + }, + { + "label": "E", + "text": "For every tensor in \\(\\mathbb C^n\\otimes \\mathbb C^n\\otimes \\mathbb C^n\\) outside the \\(n\\)-th cactus variety, some minor of size \\(n(n-1)(n-2)+1\\) of a tangency flattening is nonzero; equivalently, these minors cut out the \\(n\\)-th cactus variety set-theoretically." + } + ], + "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": "nonvanishing_on_cactus", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped_failure_on_cactus_variety", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "minor_size_threshold_plus_one", + "template_used": "boundary_range" + }, + { + "label": "E", + "sketch_hook_type": "finiteness", + "tampered_component": "mere_nonvanishing_vs_set_theoretic_definition", + "template_used": "stronger_trap" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not state the correct conclusion about secant versus cactus varieties. It gives the construction of the tangency flattenings and the minor size, but the key vanishing/nonvanishing behavior must still be identified from the choices." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a direct theorem-recall question: it asks which statement holds for every n after presenting the exact setup and the relevant minors. However, it is not completely tautological because the choices include weaker, stronger, and boundary-threshold variants rather than just a verbatim restatement." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish the precise theorem from nearby alternatives: a weaker true-looking statement, an off-by-one minor size, and an overstrong set-theoretic conclusion. Still, the task mainly tests recognition/recall of the exact result rather than substantial generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically meaningful. They reflect common failure modes: overgeneralizing vanishing to the cactus variety, dropping the sharper nonvanishing clause, using the wrong minor size threshold, and confusing existence of equations with a set-theoretic characterization." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with good distractors and no answer leakage, but it leans more toward precise recall of a stated result than toward deep generative reasoning." + } + }, + { + "id": "2602.13001v1", + "paper_link": "http://arxiv.org/abs/2602.13001v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "theorem", + "content": "\\label{mainth1}\nLet $c,d,s\\in\\mathbb Z^+$ with $d\\geq 2$, $1\\leq c,s\\leq d$ and $\\gcd(cs,d)=1$, and let $p\\geq5$ be a prime with $p\\equiv s\\pmod d$. Then, for any $r\\in\\mathbb Z^+$ with $(\\f12+\\alpha)^{\\ast_r}(\\f12+\\alpha^{\\ast_r})\\not\\eq0\\pmod{p}$, we have\n\\begin{equation}\\label{mainth1eq}\n\\sum_{k=0}^{p^r-1}(2k+\\alpha)\\frac{(\\alpha)_k^3(\\f12)_k}{(1)_k^3(\\f12+\\alpha)_k}\\equiv \\alpha^{\\ast_r}p^r -\\frac{(\\alpha^{\\ast_r})^3}{(\\f12+\\alpha)^{\\ast_r}}p^{r+2}H_{\\alpha^{\\ast_r}p-\\alpha^{\\ast_{r-1}}}^{(2)}\\pmod{p^{r+3}},\n\\end{equation}\nwhere $\\alpha=c/d$.", + "start_pos": 9805, + "end_pos": 10361, + "label": "mainth1" + }, + "ref_dict": { + "GuoZhaores1": "\\begin{equation}\\label{GuoZhaores1}\n\\sum_{k=0}^{(p^r-1)/2}(8k+1)\\f{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\eq p^r\\pmod{p^{r+3}},\n\\end{equation}", + "examples": "\\begin{table}[htbp]\n \\centering\n \\caption{Examples of the parameters in Theorem \\ref{mainth1}}\n \\label{examples}\n \\renewcommand{\\arraystretch}{1.5}\n \\begin{tabular}{>{\\centering\\arraybackslash}p{2cm}\n >{\\centering\\arraybackslash}p{2cm}\n >{\\centering\\arraybackslash}p{2cm}\n >{\\centering\\arraybackslash}p{3cm}\n >{\\centering\\arraybackslash}p{3cm}}\n \\toprule \n $d$ & $s$ & $\\alpha$ & $\\alpha^{\\ast_r}$ & $(1/2+\\alpha)^{\\ast_r}$\\\\\n \\midrule \n $2$ & $1$ & $1/2$ & $1/2$ & $1$\\\\\n $3$ & $1$ & $1/3$ & $1/3$ & $5/6$\\\\\n $3$ & $1$ & $2/3$ & $2/3$ & $1/6$\\\\\n $3$ & $1$ & $1/6$ & $1/6$ & $2/3$\\\\\n $3$ & $1$ & $5/6$ & $5/6$ & $1/3$\\\\\n $3$ & $2$ & $1/3$ & $(3-(-1)^r)/6$ & $(3+2(-1)^r)/6$\\\\\n $3$ & $2$ & $2/3$ & $(3+(-1)^r)/6$ & $(3-2(-1)^r)/6$\\\\\n $3$ & $2$ & $1/6$ & $(3-2(-1)^r)/6$ & $(3+(-1)^r)/6$\\\\\n $3$ & $2$ & $5/6$ & $(3+2(-1)^r)/6$ & $(3-(-1)^r)/6$\\\\\n $4$ & $1$ & $1/4$ & $1/4$ & $3/4$\\\\\n $4$ & $1$ & $3/4$ & $3/4$ & $1/4$\\\\\n $4$ & $3$ & $1/4$ & $(2-(-1)^r)/4$ & $(2+(-1)^r)/4$\\\\\n $4$ & $3$ & $3/4$ & $(2+(-1)^r)/4$ & $(2-(-1)^r)/4$\\\\\n \\bottomrule\n \\end{tabular}\n\\end{table}", + "mainth1eq": "\\begin{equation}\\label{mainth1eq}\n\\sum_{k=0}^{p^r-1}(2k+\\alpha)\\f{(\\alpha)_k^3(\\f12)_k}{(1)_k^3(\\f12+\\alpha)_k}\\eq \\alpha^{\\ast_r}p^r -\\f{(\\alpha^{\\ast_r})^3}{(\\f12+\\alpha)^{\\ast_r}}p^{r+2}H_{\\alpha^{\\ast_r}p-\\alpha^{\\ast_{r-1}}}^{(2)}\\pmod{p^{r+3}},\n\\end{equation}", + "RamaSer": "\\begin{equation}\\label{RamaSer}\n\\sum_{k=0}^{\\infty}(8k+1)\\f{(\\f14)_k^4}{(1)_k^4}=\\f{2\\sqrt2}{\\sqrt{\\pi}\\Gamma(\\f34)^2},\n\\end{equation}", + "cor1": "\\begin{equation}\\label{cor1}\n\\sum_{k=0}^{p^r-1}(4k+1)\\f{(\\f12)_k^4}{(1)_k^4}\\eq p^r \\pmod{p^{r+3}},\n\\end{equation}", + "GuoZhaoconj7.2": "\\begin{equation}\\label{GuoZhaoconj7.2}\n\\sum_{k=0}^{p^r-1}(8k+1)\\f{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\eq 3p^r+\\f{27}{4}p^{3r}\\sum_{j=1}^{(p^r-3)/4}\\f{1}{j^2}\\pmod{p^{r+3}}.\n\\end{equation}", + "2ordharmonic": "\\begin{equation}\\label{2ordharmonic}\nH_{p-1}^{(2)}\\eq H_{(p-1)/2}^{(2)}\\eq 0\\pmod{p},\n\\end{equation}", + "cor": "\\begin{corollary}\\label{cor}\nGuo and Zhao's conjectural supercongruence \\eqref{GuoZhaoconj7.2} \\cite[Conjecture 7.2]{Guo-Zhao2026} is true.\n\\end{corollary}", + "VanHammeconj": "\\begin{equation}\\label{VanHammeconj}\n\\sum_{k=0}^{(p-1)/4}(8k+1)\\f{(\\f14)_k^4}{(1)_k^4}\\eq p\\f{\\Gamma_p(\\f12)\\Gamma_p(\\f14)}{\\Gamma_p(\\f34)}\\pmod{p^3},\n\\end{equation}", + "Swisherextension": "\\begin{equation}\\label{Swisherextension}\n\\sum_{k=0}^{(3p-1)/4}(8k+1)\\f{(\\f14)_k^4}{(1)_k^4}\\eq -\\f{3}{2}p^2\\f{\\Gamma_p(\\f12)\\Gamma_p(\\f14)}{\\Gamma_p(\\f34)}\\pmod{p^4}.\n\\end{equation}", + "Whipple": "\\begin{align}\\label{Whipple}\n&\\sum_{k=0}^{\\infty}\\f{(a)_k(1+\\f{a}{2})_k(b)_k(c)_k(d)_k}{(1)_k(\\f a2)_k(1+a-b)_k(1+a-c)_k(1+a-d)_k}\\notag\\\\\n&\\qquad=\\f{\\Gamma(1+a-b)\\Gamma(1+a-c)\\Gamma(1+a-d)\\Gamma(1+a-b-c-d)}{\\Gamma(1+a)\\Gamma(1+a-b-c)\\Gamma(1+a-b-d)\\Gamma(1+a-c-d)}.\n\\end{align}", + "GuoZhaores2": "\\begin{equation}\\label{GuoZhaores2}\n\\sum_{k=0}^{p^r-1}(8k+1)\\f{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\eq 3p^r\\pmod{p^{r+2}}.\n\\end{equation}", + "Pan-Tauraso-Wangres": "\\begin{equation}\\label{Pan-Tauraso-Wangres}\n\\sum_{k=0}^{p-1}\\f{2k+\\alpha}{\\alpha}\\cdot\\f{(\\alpha)_k^4}{(1)_k^4}\\eq p^2\\alpha^{\\ast}(2\\alpha^{\\ast}-1)\\f{\\Gamma_p(1-2\\alpha)}{\\Gamma_p(1+\\alpha)\\Gamma_p(1-\\alpha)^3}\\pmod{p^4},\n\\end{equation}", + "mainth1": "\\begin{theorem}\\label{mainth1}\nLet $c,d,s\\in\\Z^+$ with $d\\geq 2$, $1\\leq c,s\\leq d$ and $\\gcd(cs,d)=1$, and let $p\\geq5$ be a prime with $p\\eq s\\pmod d$. Then, for any $r\\in\\Z^+$ with $(\\f12+\\alpha)^{\\ast_r}(\\f12+\\alpha^{\\ast_r})\\not\\eq0\\pmod{p}$, we have\n\\begin{equation}\\label{mainth1eq}\n\\sum_{k=0}^{p^r-1}(2k+\\alpha)\\f{(\\alpha)_k^3(\\f12)_k}{(1)_k^3(\\f12+\\alpha)_k}\\eq \\alpha^{\\ast_r}p^r -\\f{(\\alpha^{\\ast_r})^3}{(\\f12+\\alpha)^{\\ast_r}}p^{r+2}H_{\\alpha^{\\ast_r}p-\\alpha^{\\ast_{r-1}}}^{(2)}\\pmod{p^{r+3}},\n\\end{equation}\nwhere $\\alpha=c/d$.\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 5847, + "pre_theorem_intro_text": "In the 1910s, Ramanujan announced some convergent hypergeometric series related to $1/\\pi$ without proofs (cf. \\cite{Berndt1994}), such as\n\\begin{equation}\\label{RamaSer}\n\\sum_{k=0}^{\\infty}(8k+1)\\frac{(\\f14)_k^4}{(1)_k^4}=\\frac{2\\sqrt2}{\\sqrt{\\pi}\\Gamma(\\f34)^2},\n\\end{equation}\nwhere $(x)_n=x(x+1)\\cdots(x+n-1)$ is the Pochhammer symbol and $\\Gamma(x)$ is the classical Gamma function. \\eqref{RamaSer} was finally confirmed by Hardy \\cite{Hardy1923}.\n\nIn 1997, Van Hamme \\cite{VanHamme1997} observed that the truncated forms of original Ramanujan-type series possess good congruence properties. For example, corresponding to \\eqref{RamaSer}, Van Hamme conjectured the following supercongruence: for any prime $p\\eq1\\pmod4$,\n\\begin{equation}\\label{VanHammeconj}\n\\sum_{k=0}^{(p-1)/4}(8k+1)\\frac{(\\f14)_k^4}{(1)_k^4}\\equiv p\\frac{\\Gamma_p(\\f12)\\Gamma_p(\\f14)}{\\Gamma_p(\\f34)}\\pmod{p^3},\n\\end{equation}\nwhere $\\Gamma_p(x)$ denotes the $p$-adic Gamma function introduced by Morita \\cite{Morita1975}. Nowadays, we usually refer to \\eqref{VanHammeconj} as a $p$-adic analogue of \\eqref{RamaSer}. All of Van Hamme's observations have now been confirmed by different authors using various techniques (see, e.g., \\cite{Long2011,McCarthy-Osburn2008,Mortenson2008,Osburn-Zudilin2016,Swisher2015}). In particular, Swisher \\cite{Swisher2015} proved that \\eqref{VanHammeconj} holds modulo $p^4$ and established the following associated supercongruence: for any prime $p\\eq3\\pmod4$,\n\\begin{equation}\\label{Swisherextension}\n\\sum_{k=0}^{(3p-1)/4}(8k+1)\\frac{(\\f14)_k^4}{(1)_k^4}\\equiv -\\frac{3}{2}p^2\\frac{\\Gamma_p(\\f12)\\Gamma_p(\\f14)}{\\Gamma_p(\\f34)}\\pmod{p^4}.\n\\end{equation}\nNote that $(1/4)_k\\eq0\\pmod{p}$ for $(p+3)/4\\leq k\\leq p-1$ if $p\\eq1\\pmod4$ or $(3p+3)/4\\leq k\\leq p-1$ if $p\\eq3\\pmod4$. Therefore, both the upper limits of sums on the left-hand side of \\eqref{VanHammeconj} and \\eqref{Swisherextension} can be replaced with $p-1$. In 2022, Pan, Tauraso and Wang \\cite{Pan-Tauraso-Wang2022} established some parametric extensions of \\eqref{VanHammeconj} and \\eqref{Swisherextension}. For instance, they showed that for any odd prime $p$ and $p$-adic integer $\\alpha$ with $\\<-\\alpha\\>_p\\geq (p+1)/2$,\n\\begin{equation}\\label{Pan-Tauraso-Wangres}\n\\sum_{k=0}^{p-1}\\frac{2k+\\alpha}{\\alpha}\\cdot\\frac{(\\alpha)_k^4}{(1)_k^4}\\equiv p^2\\alpha^{\\ast}(2\\alpha^{\\ast}-1)\\frac{\\Gamma_p(1-2\\alpha)}{\\Gamma_p(1+\\alpha)\\Gamma_p(1-\\alpha)^3}\\pmod{p^4},\n\\end{equation}\nwhere for any $p$-adic integer $x$, $\\_{p^r}$ stands for the least nonnegative residue of $x$ modulo $p^r$ and $x^{\\ast}=(x+\\<-x\\>_p)/p$ denotes Dwork's dash operation (cf. \\cite{Dwork1969}). Clearly, \\eqref{Swisherextension} is the special case $\\alpha=1/4$ of \\eqref{Pan-Tauraso-Wangres}.\nFor other parametric extensions of \\eqref{VanHammeconj}, we refer the reader to \\cite{Barman-Saikia2020, Guo2025, Guo-Schlosser2020, Liu-Wang2021, Liu-Wang2022, Pan-Tauraso-Wang2022}.\n\nRecently, using the creative microscoping method (cf. \\cite{Guo-Zudilin2019}), Guo and Zhao \\cite{Guo-Zhao2026} studied some $q$-supercongruences from a very-well-poised ${}_6\\phi_5$ basic hypergeometric identity. As consequences, they obtained the following results: for any prime $p\\eq1\\pmod4$ and integer $r\\geq1$, \n\\begin{equation}\\label{GuoZhaores1}\n\\sum_{k=0}^{(p^r-1)/2}(8k+1)\\frac{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\equiv p^r\\pmod{p^{r+3}},\n\\end{equation}\nand for any prime $p\\eq3\\pmod4$ and odd integer $r\\geq1$,\n\\begin{equation}\\label{GuoZhaores2}\n\\sum_{k=0}^{p^r-1}(8k+1)\\frac{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\equiv 3p^r\\pmod{p^{r+2}}.\n\\end{equation}\nGuo and Zhao \\cite[Conjectures 7.1 and 7.2]{Guo-Zhao2026} also conjectured that \\eqref{GuoZhaores1} still holds modulo $p^{r+5}$ for $p>5$ and \\eqref{GuoZhaores2} can be extended to the modulus $p^{r+3}$ case as follows:\n\\begin{equation}\\label{GuoZhaoconj7.2}\n\\sum_{k=0}^{p^r-1}(8k+1)\\frac{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\equiv 3p^r+\\frac{27}{4}p^{3r}\\sum_{j=1}^{(p^r-3)/4}\\frac{1}{j^2}\\pmod{p^{r+3}}.\n\\end{equation}\nThis is our initial motivation. \n\nRecall Whipple's ${}_5F_4$ formula (cf. \\cite{Whipple1926})\n\\begin{align}\\label{Whipple}\n&\\sum_{k=0}^{\\infty}\\frac{(a)_k(1+\\frac{a}{2})_k(b)_k(c)_k(d)_k}{(1)_k(\\frac a2)_k(1+a-b)_k(1+a-c)_k(1+a-d)_k}\\notag\\\\\n&\\qquad=\\frac{\\Gamma(1+a-b)\\Gamma(1+a-c)\\Gamma(1+a-d)\\Gamma(1+a-b-c-d)}{\\Gamma(1+a)\\Gamma(1+a-b-c)\\Gamma(1+a-b-d)\\Gamma(1+a-c-d)}.\n\\end{align}\nClearly, \\eqref{RamaSer} is the special case $a=b=c=d=1/4$ of \\eqref{Whipple}. Meanwhile, \\eqref{Pan-Tauraso-Wangres} and \\eqref{GuoZhaores2} are $p$-adic analogues of \\eqref{Whipple} in the case $a=b=c=d=\\alpha$ and the case $a=b=c=1/4,\\ d=1/2$, respectively. Motivated by \\eqref{Pan-Tauraso-Wangres}, it is natural to ask whether \\eqref{GuoZhaores2} or \\eqref{GuoZhaoconj7.2} has a parametric extension. This is the second motivation.\n\nBefore stating our main result, we first introduce some notations. For $n\\in\\mathbb Z^+=\\{1,2,3,\\ldots\\}$ and $x\\in\\mathbb Z$, use $\\_n$ to denote the least nonnegative residue of $x$ modulo $n$. Let $p$ be an odd prime. Similarly as in $\\mathbb Z$, for any $p$-adic integer $x$ and $r\\in\\mathbb Z^+$, $\\_{p^r}$ stands for the least nonnegative residue of $x$ modulo $p^r$ and $x^{\\ast}=(x+\\<-x\\>_p)/p$ denotes Dwork's dash operation on $x$ (cf. \\cite{Dwork1969}). For convenience, for $n\\in\\mathbb Z^+$, use $x^{\\ast_n}$ to represent iterating the dash operation on $x$ $n$ times, that is,\n$$\nx^{\\ast_1}=x^{\\ast},\\quad x^{\\ast_n}=(x^{\\ast_{n-1}})^{\\ast}\\quad (n=2,3,4,\\ldots).\n$$\nIn particular, set $x^{\\ast_0}=x$. For $n\\in\\mathbb N=\\{0,1,2,\\ldots\\},\\ m\\in\\mathbb Z^+$, the $n$-th harmonic number of order $m$ is defined by\n$$\nH_n^{(m)}:=\\sum_{k=1}^n\\frac{1}{k^m}.\n$$\n\nOur main purpose is to establish a $p$-adic analogue of Whipple's formula \\eqref{Whipple} with $a=b=c=\\alpha$ and $d=\\f12$.", + "context": "In the 1910s, Ramanujan announced some convergent hypergeometric series related to $1/\\pi$ without proofs (cf. \\cite{Berndt1994}), such as\n\\begin{equation}\\label{RamaSer}\n\\sum_{k=0}^{\\infty}(8k+1)\\frac{(\\f14)_k^4}{(1)_k^4}=\\frac{2\\sqrt2}{\\sqrt{\\pi}\\Gamma(\\f34)^2},\n\\end{equation}\nwhere $(x)_n=x(x+1)\\cdots(x+n-1)$ is the Pochhammer symbol and $\\Gamma(x)$ is the classical Gamma function. \\eqref{RamaSer} was finally confirmed by Hardy \\cite{Hardy1923}.\n\nIn 1997, Van Hamme \\cite{VanHamme1997} observed that the truncated forms of original Ramanujan-type series possess good congruence properties. For example, corresponding to \\eqref{RamaSer}, Van Hamme conjectured the following supercongruence: for any prime $p\\eq1\\pmod4$,\n\\begin{equation}\\label{VanHammeconj}\n\\sum_{k=0}^{(p-1)/4}(8k+1)\\frac{(\\f14)_k^4}{(1)_k^4}\\equiv p\\frac{\\Gamma_p(\\f12)\\Gamma_p(\\f14)}{\\Gamma_p(\\f34)}\\pmod{p^3},\n\\end{equation}\nwhere $\\Gamma_p(x)$ denotes the $p$-adic Gamma function introduced by Morita \\cite{Morita1975}. Nowadays, we usually refer to \\eqref{VanHammeconj} as a $p$-adic analogue of \\eqref{RamaSer}. All of Van Hamme's observations have now been confirmed by different authors using various techniques (see, e.g., \\cite{Long2011,McCarthy-Osburn2008,Mortenson2008,Osburn-Zudilin2016,Swisher2015}). In particular, Swisher \\cite{Swisher2015} proved that \\eqref{VanHammeconj} holds modulo $p^4$ and established the following associated supercongruence: for any prime $p\\eq3\\pmod4$,\n\\begin{equation}\\label{Swisherextension}\n\\sum_{k=0}^{(3p-1)/4}(8k+1)\\frac{(\\f14)_k^4}{(1)_k^4}\\equiv -\\frac{3}{2}p^2\\frac{\\Gamma_p(\\f12)\\Gamma_p(\\f14)}{\\Gamma_p(\\f34)}\\pmod{p^4}.\n\\end{equation}\nNote that $(1/4)_k\\eq0\\pmod{p}$ for $(p+3)/4\\leq k\\leq p-1$ if $p\\eq1\\pmod4$ or $(3p+3)/4\\leq k\\leq p-1$ if $p\\eq3\\pmod4$. Therefore, both the upper limits of sums on the left-hand side of \\eqref{VanHammeconj} and \\eqref{Swisherextension} can be replaced with $p-1$. In 2022, Pan, Tauraso and Wang \\cite{Pan-Tauraso-Wang2022} established some parametric extensions of \\eqref{VanHammeconj} and \\eqref{Swisherextension}. For instance, they showed that for any odd prime $p$ and $p$-adic integer $\\alpha$ with $\\<-\\alpha\\>_p\\geq (p+1)/2$,\n\\begin{equation}\\label{Pan-Tauraso-Wangres}\n\\sum_{k=0}^{p-1}\\frac{2k+\\alpha}{\\alpha}\\cdot\\frac{(\\alpha)_k^4}{(1)_k^4}\\equiv p^2\\alpha^{\\ast}(2\\alpha^{\\ast}-1)\\frac{\\Gamma_p(1-2\\alpha)}{\\Gamma_p(1+\\alpha)\\Gamma_p(1-\\alpha)^3}\\pmod{p^4},\n\\end{equation}\nwhere for any $p$-adic integer $x$, $\\_{p^r}$ stands for the least nonnegative residue of $x$ modulo $p^r$ and $x^{\\ast}=(x+\\<-x\\>_p)/p$ denotes Dwork's dash operation (cf. \\cite{Dwork1969}). Clearly, \\eqref{Swisherextension} is the special case $\\alpha=1/4$ of \\eqref{Pan-Tauraso-Wangres}.\nFor other parametric extensions of \\eqref{VanHammeconj}, we refer the reader to \\cite{Barman-Saikia2020, Guo2025, Guo-Schlosser2020, Liu-Wang2021, Liu-Wang2022, Pan-Tauraso-Wang2022}.\n\nRecently, using the creative microscoping method (cf. \\cite{Guo-Zudilin2019}), Guo and Zhao \\cite{Guo-Zhao2026} studied some $q$-supercongruences from a very-well-poised ${}_6\\phi_5$ basic hypergeometric identity. As consequences, they obtained the following results: for any prime $p\\eq1\\pmod4$ and integer $r\\geq1$, \n\\begin{equation}\\label{GuoZhaores1}\n\\sum_{k=0}^{(p^r-1)/2}(8k+1)\\frac{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\equiv p^r\\pmod{p^{r+3}},\n\\end{equation}\nand for any prime $p\\eq3\\pmod4$ and odd integer $r\\geq1$,\n\\begin{equation}\\label{GuoZhaores2}\n\\sum_{k=0}^{p^r-1}(8k+1)\\frac{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\equiv 3p^r\\pmod{p^{r+2}}.\n\\end{equation}\nGuo and Zhao \\cite[Conjectures 7.1 and 7.2]{Guo-Zhao2026} also conjectured that \\eqref{GuoZhaores1} still holds modulo $p^{r+5}$ for $p>5$ and \\eqref{GuoZhaores2} can be extended to the modulus $p^{r+3}$ case as follows:\n\\begin{equation}\\label{GuoZhaoconj7.2}\n\\sum_{k=0}^{p^r-1}(8k+1)\\frac{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\equiv 3p^r+\\frac{27}{4}p^{3r}\\sum_{j=1}^{(p^r-3)/4}\\frac{1}{j^2}\\pmod{p^{r+3}}.\n\\end{equation}\nThis is our initial motivation.\n\nRecall Whipple's ${}_5F_4$ formula (cf. \\cite{Whipple1926})\n\\begin{align}\\label{Whipple}\n&\\sum_{k=0}^{\\infty}\\frac{(a)_k(1+\\frac{a}{2})_k(b)_k(c)_k(d)_k}{(1)_k(\\frac a2)_k(1+a-b)_k(1+a-c)_k(1+a-d)_k}\\notag\\\\\n&\\qquad=\\frac{\\Gamma(1+a-b)\\Gamma(1+a-c)\\Gamma(1+a-d)\\Gamma(1+a-b-c-d)}{\\Gamma(1+a)\\Gamma(1+a-b-c)\\Gamma(1+a-b-d)\\Gamma(1+a-c-d)}.\n\\end{align}\nClearly, \\eqref{RamaSer} is the special case $a=b=c=d=1/4$ of \\eqref{Whipple}. Meanwhile, \\eqref{Pan-Tauraso-Wangres} and \\eqref{GuoZhaores2} are $p$-adic analogues of \\eqref{Whipple} in the case $a=b=c=d=\\alpha$ and the case $a=b=c=1/4,\\ d=1/2$, respectively. Motivated by \\eqref{Pan-Tauraso-Wangres}, it is natural to ask whether \\eqref{GuoZhaores2} or \\eqref{GuoZhaoconj7.2} has a parametric extension. This is the second motivation.\n\nBefore stating our main result, we first introduce some notations. For $n\\in\\mathbb Z^+=\\{1,2,3,\\ldots\\}$ and $x\\in\\mathbb Z$, use $\\_n$ to denote the least nonnegative residue of $x$ modulo $n$. Let $p$ be an odd prime. Similarly as in $\\mathbb Z$, for any $p$-adic integer $x$ and $r\\in\\mathbb Z^+$, $\\_{p^r}$ stands for the least nonnegative residue of $x$ modulo $p^r$ and $x^{\\ast}=(x+\\<-x\\>_p)/p$ denotes Dwork's dash operation on $x$ (cf. \\cite{Dwork1969}). For convenience, for $n\\in\\mathbb Z^+$, use $x^{\\ast_n}$ to represent iterating the dash operation on $x$ $n$ times, that is,\n$$\nx^{\\ast_1}=x^{\\ast},\\quad x^{\\ast_n}=(x^{\\ast_{n-1}})^{\\ast}\\quad (n=2,3,4,\\ldots).\n$$\nIn particular, set $x^{\\ast_0}=x$. For $n\\in\\mathbb N=\\{0,1,2,\\ldots\\},\\ m\\in\\mathbb Z^+$, the $n$-th harmonic number of order $m$ is defined by\n$$\nH_n^{(m)}:=\\sum_{k=1}^n\\frac{1}{k^m}.\n$$\n\nOur main purpose is to establish a $p$-adic analogue of Whipple's formula \\eqref{Whipple} with $a=b=c=\\alpha$ and $d=\\f12$.", + "full_context": "In the 1910s, Ramanujan announced some convergent hypergeometric series related to $1/\\pi$ without proofs (cf. \\cite{Berndt1994}), such as\n\\begin{equation}\\label{RamaSer}\n\\sum_{k=0}^{\\infty}(8k+1)\\frac{(\\f14)_k^4}{(1)_k^4}=\\frac{2\\sqrt2}{\\sqrt{\\pi}\\Gamma(\\f34)^2},\n\\end{equation}\nwhere $(x)_n=x(x+1)\\cdots(x+n-1)$ is the Pochhammer symbol and $\\Gamma(x)$ is the classical Gamma function. \\eqref{RamaSer} was finally confirmed by Hardy \\cite{Hardy1923}.\n\nIn 1997, Van Hamme \\cite{VanHamme1997} observed that the truncated forms of original Ramanujan-type series possess good congruence properties. For example, corresponding to \\eqref{RamaSer}, Van Hamme conjectured the following supercongruence: for any prime $p\\eq1\\pmod4$,\n\\begin{equation}\\label{VanHammeconj}\n\\sum_{k=0}^{(p-1)/4}(8k+1)\\frac{(\\f14)_k^4}{(1)_k^4}\\equiv p\\frac{\\Gamma_p(\\f12)\\Gamma_p(\\f14)}{\\Gamma_p(\\f34)}\\pmod{p^3},\n\\end{equation}\nwhere $\\Gamma_p(x)$ denotes the $p$-adic Gamma function introduced by Morita \\cite{Morita1975}. Nowadays, we usually refer to \\eqref{VanHammeconj} as a $p$-adic analogue of \\eqref{RamaSer}. All of Van Hamme's observations have now been confirmed by different authors using various techniques (see, e.g., \\cite{Long2011,McCarthy-Osburn2008,Mortenson2008,Osburn-Zudilin2016,Swisher2015}). In particular, Swisher \\cite{Swisher2015} proved that \\eqref{VanHammeconj} holds modulo $p^4$ and established the following associated supercongruence: for any prime $p\\eq3\\pmod4$,\n\\begin{equation}\\label{Swisherextension}\n\\sum_{k=0}^{(3p-1)/4}(8k+1)\\frac{(\\f14)_k^4}{(1)_k^4}\\equiv -\\frac{3}{2}p^2\\frac{\\Gamma_p(\\f12)\\Gamma_p(\\f14)}{\\Gamma_p(\\f34)}\\pmod{p^4}.\n\\end{equation}\nNote that $(1/4)_k\\eq0\\pmod{p}$ for $(p+3)/4\\leq k\\leq p-1$ if $p\\eq1\\pmod4$ or $(3p+3)/4\\leq k\\leq p-1$ if $p\\eq3\\pmod4$. Therefore, both the upper limits of sums on the left-hand side of \\eqref{VanHammeconj} and \\eqref{Swisherextension} can be replaced with $p-1$. In 2022, Pan, Tauraso and Wang \\cite{Pan-Tauraso-Wang2022} established some parametric extensions of \\eqref{VanHammeconj} and \\eqref{Swisherextension}. For instance, they showed that for any odd prime $p$ and $p$-adic integer $\\alpha$ with $\\<-\\alpha\\>_p\\geq (p+1)/2$,\n\\begin{equation}\\label{Pan-Tauraso-Wangres}\n\\sum_{k=0}^{p-1}\\frac{2k+\\alpha}{\\alpha}\\cdot\\frac{(\\alpha)_k^4}{(1)_k^4}\\equiv p^2\\alpha^{\\ast}(2\\alpha^{\\ast}-1)\\frac{\\Gamma_p(1-2\\alpha)}{\\Gamma_p(1+\\alpha)\\Gamma_p(1-\\alpha)^3}\\pmod{p^4},\n\\end{equation}\nwhere for any $p$-adic integer $x$, $\\_{p^r}$ stands for the least nonnegative residue of $x$ modulo $p^r$ and $x^{\\ast}=(x+\\<-x\\>_p)/p$ denotes Dwork's dash operation (cf. \\cite{Dwork1969}). Clearly, \\eqref{Swisherextension} is the special case $\\alpha=1/4$ of \\eqref{Pan-Tauraso-Wangres}.\nFor other parametric extensions of \\eqref{VanHammeconj}, we refer the reader to \\cite{Barman-Saikia2020, Guo2025, Guo-Schlosser2020, Liu-Wang2021, Liu-Wang2022, Pan-Tauraso-Wang2022}.\n\nRecently, using the creative microscoping method (cf. \\cite{Guo-Zudilin2019}), Guo and Zhao \\cite{Guo-Zhao2026} studied some $q$-supercongruences from a very-well-poised ${}_6\\phi_5$ basic hypergeometric identity. As consequences, they obtained the following results: for any prime $p\\eq1\\pmod4$ and integer $r\\geq1$, \n\\begin{equation}\\label{GuoZhaores1}\n\\sum_{k=0}^{(p^r-1)/2}(8k+1)\\frac{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\equiv p^r\\pmod{p^{r+3}},\n\\end{equation}\nand for any prime $p\\eq3\\pmod4$ and odd integer $r\\geq1$,\n\\begin{equation}\\label{GuoZhaores2}\n\\sum_{k=0}^{p^r-1}(8k+1)\\frac{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\equiv 3p^r\\pmod{p^{r+2}}.\n\\end{equation}\nGuo and Zhao \\cite[Conjectures 7.1 and 7.2]{Guo-Zhao2026} also conjectured that \\eqref{GuoZhaores1} still holds modulo $p^{r+5}$ for $p>5$ and \\eqref{GuoZhaores2} can be extended to the modulus $p^{r+3}$ case as follows:\n\\begin{equation}\\label{GuoZhaoconj7.2}\n\\sum_{k=0}^{p^r-1}(8k+1)\\frac{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\equiv 3p^r+\\frac{27}{4}p^{3r}\\sum_{j=1}^{(p^r-3)/4}\\frac{1}{j^2}\\pmod{p^{r+3}}.\n\\end{equation}\nThis is our initial motivation.\n\nRecall Whipple's ${}_5F_4$ formula (cf. \\cite{Whipple1926})\n\\begin{align}\\label{Whipple}\n&\\sum_{k=0}^{\\infty}\\frac{(a)_k(1+\\frac{a}{2})_k(b)_k(c)_k(d)_k}{(1)_k(\\frac a2)_k(1+a-b)_k(1+a-c)_k(1+a-d)_k}\\notag\\\\\n&\\qquad=\\frac{\\Gamma(1+a-b)\\Gamma(1+a-c)\\Gamma(1+a-d)\\Gamma(1+a-b-c-d)}{\\Gamma(1+a)\\Gamma(1+a-b-c)\\Gamma(1+a-b-d)\\Gamma(1+a-c-d)}.\n\\end{align}\nClearly, \\eqref{RamaSer} is the special case $a=b=c=d=1/4$ of \\eqref{Whipple}. Meanwhile, \\eqref{Pan-Tauraso-Wangres} and \\eqref{GuoZhaores2} are $p$-adic analogues of \\eqref{Whipple} in the case $a=b=c=d=\\alpha$ and the case $a=b=c=1/4,\\ d=1/2$, respectively. Motivated by \\eqref{Pan-Tauraso-Wangres}, it is natural to ask whether \\eqref{GuoZhaores2} or \\eqref{GuoZhaoconj7.2} has a parametric extension. This is the second motivation.\n\nBefore stating our main result, we first introduce some notations. For $n\\in\\mathbb Z^+=\\{1,2,3,\\ldots\\}$ and $x\\in\\mathbb Z$, use $\\_n$ to denote the least nonnegative residue of $x$ modulo $n$. Let $p$ be an odd prime. Similarly as in $\\mathbb Z$, for any $p$-adic integer $x$ and $r\\in\\mathbb Z^+$, $\\_{p^r}$ stands for the least nonnegative residue of $x$ modulo $p^r$ and $x^{\\ast}=(x+\\<-x\\>_p)/p$ denotes Dwork's dash operation on $x$ (cf. \\cite{Dwork1969}). For convenience, for $n\\in\\mathbb Z^+$, use $x^{\\ast_n}$ to represent iterating the dash operation on $x$ $n$ times, that is,\n$$\nx^{\\ast_1}=x^{\\ast},\\quad x^{\\ast_n}=(x^{\\ast_{n-1}})^{\\ast}\\quad (n=2,3,4,\\ldots).\n$$\nIn particular, set $x^{\\ast_0}=x$. For $n\\in\\mathbb N=\\{0,1,2,\\ldots\\},\\ m\\in\\mathbb Z^+$, the $n$-th harmonic number of order $m$ is defined by\n$$\nH_n^{(m)}:=\\sum_{k=1}^n\\frac{1}{k^m}.\n$$\n\nOur main purpose is to establish a $p$-adic analogue of Whipple's formula \\eqref{Whipple} with $a=b=c=\\alpha$ and $d=\\f12$.\n\nOur main purpose is to establish a $p$-adic analogue of Whipple's formula \\eqref{Whipple} with $a=b=c=\\alpha$ and $d=\\f12$.\n\nTheorem \\ref{mainth1} seems quite strange. When $\\<-1/2-\\alpha\\>_p<\\min\\{\\<-\\alpha\\>_p,(p-1)/2\\}$, the summands on the left-hand side of \\eqref{mainth1eq} are not always $p$-adic integers. However, the sum of these summands is a $p$-adic integer. This phenomenon renders it difficult for us to prove Theorem \\ref{mainth1} directly using the formula \\eqref{Whipple} and conventional congruence techniques. To overcome this obstacle, we find a new parametric WZ pair (cf. \\cite{PWZ}) which allows us to transform the original sum to a computable form in the sense of congruence. This idea is crucial in our proof.\n\n\\begin{lemma}\\label{dashres}\nLet $c,d,s\\in\\Z^+$ with $d\\geq2$, $1\\leq c,s\\leq d$ and $\\gcd(cs,d)=1$. Then, for any prime $p\\eq s\\pmod{d}$, we have\n$$\n\\alpha^{\\ast}=\\f{\\_d}{d},\n$$\nwhere $s^{-1}$ stands for the inverse of $s$ modulo $d$.\n\\end{lemma}\n\n\\begin{proof}\nObviously,\n\\begin{align*}\n\\sum_{k=0}^{p^r-1}F(\\alpha^{\\ast_r}p^r,k)&=\\sum_{k=0}^{p^r-1}(2k+\\alpha^{\\ast_r}p^r)\\f{(\\alpha^{\\ast_r}p^r)_k^3(\\f12)_k}{(1)_k^3(\\f12+\\alpha^{\\ast_r}p^r)_k}\\\\\n&=\\alpha^{\\ast_r}p^r+p^{3r}{\\alpha^{\\ast_r}}^3\\sum_{k=1}^{p^r-1}\\f{2k+\\alpha^{\\ast_r}p^r}{k^3}\\f{(1+\\alpha^{\\ast_r}p^r)_{k-1}^3(\\f12)_k}{(1)_{k-1}^3(\\f12+\\alpha^{\\ast_r}p^r)_k}.\n\\end{align*}\nNote that for $k\\in\\{1,2,\\ldots,p^r-1\\}$, $\\ord_p(k)\\leq r-1$, where $\\ord_p$ stands for the $p$-adic order. Therefore,\n$$\n\\ord_p\\l(\\f{p^{3r}}{k^2}\\r)\\geq r+2\\quad \\t{and}\\quad \\ord_p\\l(\\f{p^{4r}}{k^3}\\r)\\geq r+3.\n$$\nMoreover, it is easy to see that\n$$\n\\f{(1+\\alpha^{\\ast_r}p^r)_{k-1}}{(1)_{k-1}^3}=\\prod_{j=1}^{k-1}\\f{j+\\alpha^{\\ast_r}p^r}{j}=\\prod_{j=1}^{k-1}\\l(1+\\f{\\alpha^{\\ast_r}p^r}{j}\\r)\\eq1\\pmod{p}\n$$\nand\n\\begin{align*}\n\\f{(\\f12)_k}{(\\f12+\\alpha^{\\ast_r}p^r)_k}&=\\prod_{j=0}^{k-1}\\f{\\f12+j}{\\f12+j+\\alpha^{\\ast_r}p^r}=\\prod_{j=0}^{k-1}\\f{1}{1+\\f{\\alpha^{\\ast_r}p^r}{\\f12+j}}\\\\\n&\\eq\\begin{cases}1/(2\\alpha^{\\ast_r}+1)\\pmod{p}\\quad&\\t{if}\\ k\\geq (p^r+1)/2,\\\\ 1\\pmod{p}\\quad&\\t{if}\\ k\\leq (p^r-1)/2.\\end{cases}\n\\end{align*}\nCombining the above and in view of Lemmas \\ref{u} and \\ref{2ordharmonicdown}, we arrive at\n$$\n\\sum_{k=0}^{p^r-1}F(\\alpha^{\\ast_r}p^r,k)\\eq \\alpha^{\\ast_r}p^r+2(\\alpha^{\\ast_r})^3p^{3r}\\sum_{k=1}^{(p^r-1)/2}\\f{1}{k^2}+\\f{2(\\alpha^{\\ast_r})^3p^{3r}}{2\\alpha^{\\ast_r}+1}\\sum_{k=(p^r+1)/2}^{p^r-1}\\f{1}{k^2}\\eq \\alpha^{\\ast_r}p^r \\pmod{p^{r+3}},\n$$\nas desired.\n\\end{proof}\n\n\\begin{lemma}\\label{evalG}\nUnder the conditions of Theorem \\ref{mainth1}, we have\n\\begin{equation}\\label{evalGeq}\n\\sum_{l=0}^{a-1}G(\\alpha+l,p^r)\\eq \\f{(\\alpha^{\\ast_r})^3}{(\\f12+\\alpha)^{\\ast_r}}p^{r+2}H_{\\alpha^{\\ast_r}p-\\alpha^{\\ast_{r-1}}}^{(2)}\\pmod{p^{r+3}}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nClearly,\n\\begin{align*}\n\\sum_{l=0}^{a-1}G(\\alpha+l,p^r)&=\\sum_{l=0}^{a-1}\\f{p^{3r}(p^r+2\\alpha+2l)}{(\\alpha+l)^3}\\f{(\\alpha+l)_{p^r}^3(\\f12)_{p^r}}{(1)_{p^r}^3(\\f12+\\alpha+l)_{p^r}}\\\\\n&=\\f{p^{3r}(\\alpha)_{p^r}^3(\\f12)_{p^r}}{(1)_{p^r}^3(\\f12+\\alpha)_{p^r}}\\sum_{l=0}^{a-1}\\f{(p^r+2\\alpha+2l)(\\alpha+p^r)_l^3(\\f12+\\alpha)_l}{(\\alpha+l)^3(\\alpha)_l^3(\\f12+\\alpha+p^r)_l}.\n\\end{align*}\nThen, by Lemma \\ref{pochreduce}, we have\n\\begin{align*}\n\\sum_{l=0}^{a-1}G(\\alpha+l,p^r)&=\\f{p^{3r}(\\alpha^{\\ast_r})^3}{2(\\f12+\\alpha)^{\\ast_r}}\\prod_{j=1}^r\\f{\\Gamma_p(\\alpha+p^j)^3\\Gamma_p(\\f12+p^j)\\Gamma_p(1)^3\\Gamma_p(\\f12+\\alpha)}{\\Gamma_p(\\alpha)^3\\Gamma_p(\\f12)\\Gamma_p(1+p^j)^3\\Gamma_p(\\f12+\\alpha+p^j)}\\\\\n&\\quad\\times \\sum_{l=0}^{a-1}\\f{(p^r+2\\alpha+2l)(\\alpha+p^r)_l^3(\\f12+\\alpha)_l}{(\\alpha+l)^3(\\alpha)_l^3(\\f12+\\alpha+p^r)_l},\n\\end{align*}\nwhere we have used the facts\n$$\n\\l(\\f12\\r)^{\\ast}=\\f12\\quad \\t{and}\\quad 1^{\\ast}=1.\n$$\nSince $(\\f12+\\alpha)^{\\ast_r}\\not\\eq0\\pmod{p}$, we have\n$$\n\\ord_p\\l(\\f{p^{3r}(\\alpha^{\\ast_r})^3}{(\\f12+\\alpha)^{\\ast_r}}\\r)\\geq 3r.\n$$\nIt is easy to see that $\\ord_p(\\alpha+l)\\leq r-1$ for $l\\in\\{0,1,2,\\ldots,a-1\\}$. It follows that\n$$\n\\ord_p\\l(\\f{p^{4r}}{(\\alpha+l)^3}\\r)\\geq r+3\n$$\nand\n$$\n\\ord_p\\l(\\f{p^{3r}}{(\\alpha+l)^2}\\r)\\geq r+2.\n$$\nFor $k\\in\\{0,1,2,\\ldots,a-1\\}$ we have\n$$\n\\f{(\\alpha+p^r)_l}{(\\alpha)_l}=\\prod_{j=0}^{l-1}\\f{\\alpha+j+p^r}{\\alpha+j}=\\prod_{j=0}^{l-1}\\l(1+\\f{p^r}{\\alpha+j}\\r)\\eq1\\pmod{p}.\n$$\nIn view of Lemma \\ref{1/2+alpha} and the above, we obtain\n\\begin{equation}\\label{evalGkey}\n\\sum_{l=0}^{a-1}G(\\alpha+l,p^r)\\eq\\f{p^{3r}(\\alpha^{\\ast_r})^3}{(\\f12+\\alpha)^{\\ast_r}}\\sum_{l=0}^{a-1}\\f{(\\f12+\\alpha)_l}{(\\alpha+l)^2(\\f12+\\alpha+p^r)_l}\\pmod{p^{r+3}}.\n\\end{equation}\n\nSuppose that $p=3$. By a similar argument as in the proof of Lemma \\ref{evalF}, we have\n\\begin{align*}\n\\sum_{k=0}^{p^r-1}F(\\alpha^{\\ast_r}p^r,k)&\\eq \\alpha^{\\ast_r}p^r+2(\\alpha^{\\ast_r})^3p^{3r}\\sum_{k=1}^{(p^r-1)/2}\\f{1}{k^2}+\\f{2(\\alpha^{\\ast_r})^3p^{3r}}{2\\alpha^{\\ast_r}+1}\\sum_{k=(p^r+1)/2}^{p^r-1}\\f{1}{k^2}\\pmod{p^{r+3}}.\n\\end{align*}\nNote that \n$$\n\\ord_p\\l(\\sum_{k=1}^{(p^r-1)/2}\\f{1}{k^2}\\r)\\geq -2(r-1)\\quad\\t{and}\\quad \\ord_p\\l(\\sum_{k=(p^r+1)/2}^{p^r-1}\\f{1}{k^2}\\r)\\geq -2(r-1).\n$$\nSince $\\alpha^{\\ast_r}=3/4\\eq0\\pmod{p}$ and $2\\alpha^{\\ast_r}+1=5/2\\not\\eq0\\pmod{p}$, we still have\n\\begin{equation}\\label{evalF'}\n\\sum_{k=0}^{p^r-1}F(\\alpha^{\\ast_r}p^r,k)\\eq \\alpha^{\\ast_r}p^r\\pmod{p^{r+3}}.\n\\end{equation}\nNow, $a=(3p^r-1)/4\\geq (p^r+1)/2$. From the proof of Lemma \\ref{evalG}, we know\n\\begin{align*}\n&\\sum_{l=0}^{a-1}G(\\alpha+l,p^r)\\\\\n&\\qquad\\eq \\f{p^{3r}(\\alpha^{\\ast_r})^3}{(\\f12+\\alpha)^{\\ast_r}}\\sum_{l=(p^r+1)/2}^{a}\\f{1}{(\\alpha+a-l)^2}+\\f{p^{3r}(\\alpha^{\\ast_r})^3(2\\alpha^{\\ast_r}-1)}{(\\f12+\\alpha)^{\\ast_r}(2\\alpha^{\\ast_r}+1)}\\sum_{l=1}^{(p^r-1)/2}\\f{1}{(\\alpha+a-l)^2}\\pmod{p^{r+3}}.\n\\end{align*}\nAlso, \n$$\n\\ord_p\\l(\\sum_{l=(p^r+1)/2}^{a}\\f{1}{(\\alpha+a-l)^2}\\r)\\geq -2(r-1)\\quad\\t{and}\\quad \\ord_p\\l(\\sum_{l=1}^{(p^r-1)/2}\\f{1}{(\\alpha+a-l)^2}\\r)\\geq -2(r-1).\n$$\nThen, in view of the facts $\\alpha^{\\ast_r}\\eq0\\pmod{p}$, $(1/2+\\alpha)^{\\ast_r}=1/4\\not\\eq0\\pmod{p}$ and $2\\alpha^{\\ast_r}+1\\not\\eq0\\pmod{p}$, we obtain\n$$\n\\sum_{l=0}^{a-1}G(\\alpha+l,p^r)\\eq0\\pmod{p^{r+3}}.\n$$\nThis, together with \\eqref{evalF'} gives\n$$\n\\sum_{k=0}^{p^r-1}F(\\alpha,k)=\\sum_{k=0}^{p^r-1}F(\\alpha^{\\ast_r}p^r,k)-\\sum_{l=0}^{a-1}G(\\alpha+l,p^r)\\eq\\alpha^{\\ast_r}p^r\\pmod{p^{r+3}},\n$$\nwhich implies\n\\begin{equation*}\n\\sum_{k=0}^{3^r-1}(8k+1)\\f{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\eq 3^{r+1}\\pmod{3^{r+3}}.\n\\end{equation*}\nMoreover, since \n$$\n\\ord_3\\l(\\sum_{j=1}^{(3^r-3)/4}\\f{1}{j^2}\\r)\\geq-2(r-1),\n$$\nwe have\n$$\n\\f{27}{4}3^{3r}\\sum_{j=1}^{(3^r-3)/4}\\f{1}{j^2}\\eq 0\\pmod{3^{r+3}}.\n$$\nTherefore, we arrive at\n\\begin{equation*}\n\\sum_{k=0}^{3^r-1}(8k+1)\\f{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\eq 3^{r+1}+\\f{27}{4}3^{3r}\\sum_{j=1}^{(3^r-3)/4}\\f{1}{j^2}\\pmod{3^{r+3}},\n\\end{equation*}\nas desired.", + "post_theorem_intro_text_len": 3020, + "post_theorem_intro_text": "Theorem \\ref{mainth1} seems quite strange. When $\\<-1/2-\\alpha\\>_p<\\min\\{\\<-\\alpha\\>_p,(p-1)/2\\}$, the summands on the left-hand side of \\eqref{mainth1eq} are not always $p$-adic integers. However, the sum of these summands is a $p$-adic integer. This phenomenon renders it difficult for us to prove Theorem \\ref{mainth1} directly using the formula \\eqref{Whipple} and conventional congruence techniques. To overcome this obstacle, we find a new parametric WZ pair (cf. \\cite{PWZ}) which allows us to transform the original sum to a computable form in the sense of congruence. This idea is crucial in our proof.\n\nIn particular, putting $d=4,s=3,c=1$ and requiring $r$ to be odd in Theorem \\ref{mainth1}, we have the following result.\n\\begin{corollary}\\label{cor}\nGuo and Zhao's conjectural supercongruence \\eqref{GuoZhaoconj7.2} \\cite[Conjecture 7.2]{Guo-Zhao2026} is true.\n\\end{corollary}\nMoreover, Theorem \\ref{mainth1} with $d=2,s=1,c=1$ gives that\n\\begin{equation}\\label{cor1}\n\\sum_{k=0}^{p^r-1}(4k+1)\\frac{(\\f12)_k^4}{(1)_k^4}\\equiv p^r \\pmod{p^{r+3}},\n\\end{equation}\nwhere we have used \\eqref{2ordharmonic}. Note that the $r=1$ case of \\eqref{cor1} is a stronger version of Van Hamme's (C.2) supercongruence \\cite{VanHamme1997} and was first proved by Long \\cite{Long2011}. \n\nFor the convenience of interested readers, in Table \\ref{examples}, we provide some concrete examples of the parameters in Theorem \\ref{mainth1} for future use.\n\n\\begin{table}[htbp]\n \\centering\n \\caption{Examples of the parameters in Theorem \\ref{mainth1}}\n \\label{examples}\n \\renewcommand{\\arraystretch}{1.5}\n \\begin{tabular}{>{\\centering\\arraybackslash}p{2cm}\n >{\\centering\\arraybackslash}p{2cm}\n >{\\centering\\arraybackslash}p{2cm}\n >{\\centering\\arraybackslash}p{3cm}\n >{\\centering\\arraybackslash}p{3cm}}\n \\toprule \n $d$ & $s$ & $\\alpha$ & $\\alpha^{\\ast_r}$ & $(1/2+\\alpha)^{\\ast_r}$\\\\\n \\midrule \n $2$ & $1$ & $1/2$ & $1/2$ & $1$\\\\\n $3$ & $1$ & $1/3$ & $1/3$ & $5/6$\\\\\n $3$ & $1$ & $2/3$ & $2/3$ & $1/6$\\\\\n $3$ & $1$ & $1/6$ & $1/6$ & $2/3$\\\\\n $3$ & $1$ & $5/6$ & $5/6$ & $1/3$\\\\\n $3$ & $2$ & $1/3$ & $(3-(-1)^r)/6$ & $(3+2(-1)^r)/6$\\\\\n $3$ & $2$ & $2/3$ & $(3+(-1)^r)/6$ & $(3-2(-1)^r)/6$\\\\\n $3$ & $2$ & $1/6$ & $(3-2(-1)^r)/6$ & $(3+(-1)^r)/6$\\\\\n $3$ & $2$ & $5/6$ & $(3+2(-1)^r)/6$ & $(3-(-1)^r)/6$\\\\\n $4$ & $1$ & $1/4$ & $1/4$ & $3/4$\\\\\n $4$ & $1$ & $3/4$ & $3/4$ & $1/4$\\\\\n $4$ & $3$ & $1/4$ & $(2-(-1)^r)/4$ & $(2+(-1)^r)/4$\\\\\n $4$ & $3$ & $3/4$ & $(2+(-1)^r)/4$ & $(2-(-1)^r)/4$\\\\\n \\bottomrule\n \\end{tabular}\n\\end{table}\n\nWe briefly outline this paper. In Section \\ref{sec2}, we prove some properties of Dwork's dash operation and give some immediate applications which play essential roles in the subsequent proof. We shall prove Theorem \\ref{mainth1} and Corollary \\ref{cor} in Section \\ref{sec3}.", + "sketch": "The introduction explains why a direct proof of Theorem~\\ref{mainth1} is hard and what method is used instead: when $\\langle-1/2-\\alpha\\rangle_p<\\min\\{\\langle-\\alpha\\rangle_p,(p-1)/2\\}$, “the summands on the left-hand side of \\eqref{mainth1eq} are not always $p$-adic integers,” although “the sum of these summands is a $p$-adic integer.” This “renders it difficult…to prove Theorem~\\ref{mainth1} directly using the formula \\eqref{Whipple} and conventional congruence techniques.”\n\nTo “overcome this obstacle,” the paper’s approach is to “find a new parametric WZ pair (cf. \\cite{PWZ}) which allows us to transform the original sum to a computable form in the sense of congruence,” and “this idea is crucial in our proof.” Additionally, Section~\\ref{sec2} establishes “properties of Dwork’s dash operation” with “immediate applications” that “play essential roles in the subsequent proof,” and then “We shall prove Theorem~\\ref{mainth1} and Corollary~\\ref{cor} in Section~\\ref{sec3}.”", + "expanded_sketch": "The introduction explains why a direct proof of the main theorem is hard and what method is used instead: when $\\langle-1/2-\\alpha\\rangle_p<\\min\\{\\langle-\\alpha\\rangle_p,(p-1)/2\\}$, “the summands on the left-hand side of\n\\begin{equation}\\label{mainth1eq}\n\\sum_{k=0}^{p^r-1}(2k+\\alpha)\\f{(\\alpha)_k^3(\\f12)_k}{(1)_k^3(\\f12+\\alpha)_k\\eq \\alpha^{\\ast_r}p^r -\\f{(\\alpha^{\\ast_r})^3}{(\\f12+\\alpha)^{\\ast_r}}p^{r+2}H_{\\alpha^{\\ast_r}p-\\alpha^{\\ast_{r-1}}}^{(2)}\\pmod{p^{r+3}},\n\\end{equation}\nare not always $p$-adic integers,” although “the sum of these summands is a $p$-adic integer.” This “renders it difficult…to prove the main theorem directly using the formula\n\\begin{align}\\label{Whipple}\n&\\sum_{k=0}^{\\infty}\\f{(a)_k(1+\\f{a}{2})_k(b)_k(c)_k(d)_k}{(1)_k(\\f a2)_k(1+a-b)_k(1+a-c)_k(1+a-d)_k}\\notag\\\\\n&\\qquad=\\f{\\Gamma(1+a-b)\\Gamma(1+a-c)\\Gamma(1+a-d)\\Gamma(1+a-b-c-d)}{\\Gamma(1+a)\\Gamma(1+a-b-c)\\Gamma(1+a-b-d)\\Gamma(1+a-c-d)}.\n\\end{align}\nand conventional congruence techniques.”\n\nTo “overcome this obstacle,” the paper’s approach is to “find a new parametric WZ pair (cf. \\cite{PWZ}) which allows us to transform the original sum to a computable form in the sense of congruence,” and “this idea is crucial in our proof.” Additionally, next the paper establishes “properties of Dwork’s dash operation” with “immediate applications” that “play essential roles in the subsequent proof,” and then later it proves the main theorem and also the following result.\n\n\\begin{corollary}\\label{cor}\nGuo and Zhao's conjectural supercongruence \\eqref{GuoZhaoconj7.2} \\cite[Conjecture 7.2]{Guo-Zhao2026} is true.\n\\end{corollary}", + "expanded_theorem": "\\label{mainth1}\nLet $c,d,s\\in\\mathbb Z^+$ with $d\\geq 2$, $1\\leq c,s\\leq d$ and $\\gcd(cs,d)=1$, and let $p\\geq5$ be a prime with $p\\equiv s\\pmod d$. Then, for any $r\\in\\mathbb Z^+$ with $(\\f12+\\alpha)^{\\ast_r}(\\f12+\\alpha^{\\ast_r})\\not\\eq0\\pmod{p}$, we have\n\\begin{equation}\\label{mainth1eq}\n\\sum_{k=0}^{p^r-1}(2k+\\alpha)\\frac{(\\alpha)_k^3(\\f12)_k}{(1)_k^3(\\f12+\\alpha)_k}\\equiv \\alpha^{\\ast_r}p^r -\\frac{(\\alpha^{\\ast_r})^3}{(\\f12+\\alpha)^{\\ast_r}}p^{r+2}H_{\\alpha^{\\ast_r}p-\\alpha^{\\ast_{r-1}}}^{(2)}\\pmod{p^{r+3}},\n\\end{equation}\nwhere $\\alpha=c/d$.", + "theorem_type": [ + "Implication", + "Inequality or Bound" + ], + "mcq": { + "question": "Let \\(\\alpha=c/d\\), where \\(c,d,s\\in\\mathbb Z^+\\) satisfy \\(d\\ge 2\\), \\(1\\le c,s\\le d\\), and \\(\\gcd(cs,d)=1\\). Let \\(p\\ge 5\\) be a prime with \\(p\\equiv s\\pmod d\\). For a \\(p\\)-adic integer \\(x\\), define Dwork’s dash operation by \\(x^\\ast=(x+\\langle -x\\rangle_p)/p\\), where \\(\\langle y\\rangle_p\\) is the least nonnegative residue of \\(y\\) modulo \\(p\\); define its iterates by \\(x^{\\ast_0}=x\\) and \\(x^{\\ast_m}=(x^{\\ast_{m-1}})^\\ast\\) for \\(m\\ge1\\). Also write \\((x)_k=x(x+1)\\cdots(x+k-1)\\) with \\((x)_0=1\\), and \\(H_n^{(2)}=\\sum_{j=1}^n 1/j^2\\). If \\(r\\in\\mathbb Z^+\\) satisfies \\((\\tfrac12+\\alpha)^{\\ast_r}(\\tfrac12+\\alpha^{\\ast_r})\\not\\equiv0\\pmod p\\), which quantitative congruence holds?", + "correct_choice": { + "label": "A", + "text": "For every such \\(r\\),\n\\[\n\\sum_{k=0}^{p^r-1}(2k+\\alpha)\\frac{(\\alpha)_k^3(\\tfrac12)_k}{(1)_k^3(\\tfrac12+\\alpha)_k}\\equiv \\alpha^{\\ast_r}p^r-\\frac{(\\alpha^{\\ast_r})^3}{(\\tfrac12+\\alpha)^{\\ast_r}}\\,p^{r+2}H_{\\alpha^{\\ast_r}p-\\alpha^{\\ast_{r-1}}}^{(2)}\\pmod{p^{r+3}}.\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "For every such \\(r\\),\n\\[\n\\sum_{k=0}^{p^r-1}(2k+\\alpha)\\frac{(\\alpha)_k^3(\\tfrac12)_k}{(1)_k^3(\\tfrac12+\\alpha)_k}\\equiv \\alpha^{\\ast_r}p^r-\\frac{(\\alpha^{\\ast_r})^3}{\\tfrac12+\\alpha^{\\ast_r}}\\,p^{r+2}H_{\\alpha^{\\ast_r}p-\\alpha^{\\ast_{r-1}}}^{(2)}\\pmod{p^{r+3}}.\n\\]" + }, + { + "label": "C", + "text": "For every such \\(r\\),\n\\[\n\\sum_{k=0}^{p^r-1}(2k+\\alpha)\\frac{(\\alpha)_k^3(\\tfrac12)_k}{(1)_k^3(\\tfrac12+\\alpha)_k}\\equiv \\alpha^{\\ast_r}p^r\\pmod{p^{r+2}}.\n\\]" + }, + { + "label": "D", + "text": "For every such \\(r\\),\n\\[\n\\sum_{k=0}^{p^r-1}(2k+\\alpha)\\frac{(\\alpha)_k^3(\\tfrac12)_k}{(1)_k^3(\\tfrac12+\\alpha)_k}\\equiv \\alpha^{\\ast_r}p^r-\\frac{(\\alpha^{\\ast_r})^3}{(\\tfrac12+\\alpha)^{\\ast_r}}\\,p^{r+2}H_{\\alpha^{\\ast_r}p-\\alpha^{\\ast_{r-1}}}^{(2)}\\pmod{p^{r+4}}.\n\\]" + }, + { + "label": "E", + "text": "If there exists \\(r\\in\\mathbb Z^+\\) such that \\((\\tfrac12+\\alpha)^{\\ast_r}(\\tfrac12+\\alpha^{\\ast_r})\\not\\equiv0\\pmod p\\), then\n\\[\n\\sum_{k=0}^{p^r-1}(2k+\\alpha)\\frac{(\\alpha)_k^3(\\tfrac12)_k}{(1)_k^3(\\tfrac12+\\alpha)_k}\\equiv \\alpha^{\\ast_r}p^r-\\frac{(\\alpha^{\\ast_r})^3}{(\\tfrac12+\\alpha)^{\\ast_r}}\\,p^{r+2}H_{\\alpha^{\\ast_r}p-\\alpha^{\\ast_r}}^{(2)}\\pmod{p^{r+3}}.\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": "dash_dependence_in_denominator", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "regularity", + "tampered_component": "dropped_second_order_correction_term_and_one_power_of_p_in_modulus", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "modulus_precision_p^{r+3}", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "harmonic_index_uses_alpha^{*_{r-1}}_not_alpha^{*_r}", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not reveal the correct congruence explicitly or through obvious wording cues; the answer must be selected from several close formulas." + }, + "TAS": { + "score": 1, + "justification": "The item is essentially asking for the exact stated conclusion of a specialized congruence theorem, with only slight perturbations in the alternatives. This is more theorem-recall than a genuinely new inferential task." + }, + "GPS": { + "score": 1, + "justification": "There is some reasoning pressure because the choices differ in subtle but meaningful ways (denominator, modulus strength, harmonic index, weaker/stronger form). However, success mainly depends on precise recall or recognition of the theorem rather than generating a conclusion from first principles." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target realistic failure modes: confusing dash placement, accepting a weaker true statement, overclaiming a stronger modulus, or altering the harmonic-sum index/quantifier." + }, + "total_score": 6, + "overall_assessment": "A technically strong MCQ with high-quality distractors and no direct answer leakage, but it leans heavily on theorem recognition/restatement rather than deeper generative mathematical reasoning." + } + }, + { + "id": "2602.13435v1", + "paper_link": "http://arxiv.org/abs/2602.13435v1", + "theorems_cnt": 5, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm:main_result} Let $E$ be a smooth complex elliptic curve with $\\End(E)=\\mathbb{Z}$. There exists a smooth projective complex surface $S$ with $\\Alb(S)\\cong E$ for which $\\Ch(S)$ is representable, while $S$ admits no universal $0$-cycle.", + "start_pos": 11714, + "end_pos": 11981, + "label": "thm:main_result" + }, + "ref_dict": { + "cor:integral_Hodge_S": "\\begin{corollary}\\label{cor:integral_Hodge_S} Let $E$ be a smooth elliptic curve over $\\bbC$ with $\\End(E)=\\Z$. Then there exists a bielliptic surface $S$ of type 2 whose Albanese variety $\\Alb(S)$ is isomorphic to $E,$ and for which the integral Hodge conjecture for $1$-cycles on $E\\times S$ fails. In particular, there exists a non-torsion integral Hodge class in $H^{4}(E\\times S,\\Z)$ that is not algebraic.\n\\end{corollary}", + "cor:obstruction": "\\begin{corollary}\\label{cor:obstruction}\nWe use the notation and assumptions of Theorem \\ref{thm:obstruction}, with condition \\eqref{it:correspondence} replaced by the following:\n\\begin{enumerate}\n \\item[\\((2')\\)]\\label{it:correspondence-prime} There exists an algebraically closed field extension $F/K$ and a correspondence\n$$\n[\\Gamma] \\in \\CH^{d}(C \\times_k X_{F})\n$$\ninducing a splitting of the pushforward map\n$$\n(\\phi_{X_F})_\\ast\\colon \\Ch(X_{F})_{\\hom}\\longrightarrow\\jac(C_{F}).\n$$\n\\end{enumerate}\nAssume moreover that $\\End(\\jac(C))=\\Z$. Then the sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(X_{0i})\\longrightarrow\\jac(C)$$\nadmits a section. \n\\end{corollary}", + "ques:2-dim_example": "\\begin{ques}\\label{ques:2-dim_example}\nDoes there exist a smooth projective complex surface $S$ whose Chow group of $0$-cycles $\\CH_0(S)$ is representable, but which admits no universal $0$-cycle?\n\\end{ques}", + "eq:fibration_X": "\\begin{equation}\\label{eq:fibration_X}\\phi_{X}\\colon X\\longrightarrow E\\end{equation}", + "eq:abel-jacobi-map": "\\begin{equation}\n \\label{eq:abel-jacobi-map} \\alpha_{X}\\colon\\Ch(X)_{\\hom}\\longrightarrow\\Alb(X).\n\\end{equation}", + "thm:main_result": "\\begin{theorem}\\label{thm:main_result} Let $E$ be a smooth complex elliptic curve with $\\End(E)=\\Z$. There exists a smooth projective complex surface $S$ with $\\Alb(S)\\cong E$ for which $\\Ch(S)$ is representable, while $S$ admits no universal $0$-cycle.\n\\end{theorem}", + "it:albanese_S_0": "\\begin{enumerate}\n \\item\\label{it:generic_fibre} The geometric generic fibre $S_{K}$ is a bielliptic surface of type $2$, and the induced morphism $\\phi_{S_{K}}\\colon S_{K}\\to E\\times_{k} K$ coincides with the Albanese fibration of $S_{K}$.\n \\item\\label{it:special_fibre} The special fibre $S_{0}=\\sum_{i=1}^{n}R_{i}$ is a reduced simple normal crossings divisor on $\\mathcal{S}$ whose dual graph is a chain; that is, each intersection $R_{i}\\cap R_{i+1}$ is a smooth irreducible curve, and $R_{i}\\cap R_{j}=\\varnothing$ whenever $j\\notin\\{i-1,i,i+1\\}$.\n \\item\\label{it:albanese_S_0} For each $i,$ let $\\al_{i}:\\Alb(R_{i})\\to E$ denote the homomorphism of Albanese varieties induced by the restriction of $\\phi_{\\mathcal{S}}$ to $R_{i}$. Then the sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(R_{i})\\longrightarrow E$$ does not admit a section. \n\\end{enumerate}", + "eq:psi": "\\begin{equation}\\label{eq:psi}\n \\psi_{(Z,[\\Gamma])}\\colon Z\\longrightarrow\\Alb(X),\\quad z\\longmapsto \\alpha_{X}\\circ[\\Gamma]_{\\ast}(z-z_{0})\n\\end{equation}", + "thm:obstruction": "\\begin{theorem}\n\\label{thm:obstruction}\nLet $k$ be an algebraically closed field and let $C$ be a smooth projective curve over $k$. Set $\\Delta := \\Spec k[[t]]$, and fix an algebraic closure $K$ of its function field. \n\nLet $p \\colon \\mathcal{X} \\to \\Delta$ be a flat, projective morphism of relative dimension $d$ with regular total space, and let $\\phi_{\\mathcal{X}} \\colon \\mathcal{X} \\to C \\times_k \\Delta$ be a surjective morphism.\nDenote by $X_0$ the special fibre of $p$, by\n$\\phi_{X_0} \\colon X_0 \\to C$ the induced morphism, and by $X_{K\n}$ the geometric generic fibre of $p$. Write $\\phi_{X_K} \\colon X_K \\to C_K$\nfor the base change of $\\phi_{\\mathcal{X}}$ to $K$.\\par\nWe further assume the following:\n\\begin{enumerate}\n \\item \\label{it:X_0} $X_{0}=\\sum_{i=1}^{n}X_{0i}$ is a reduced simple normal crossings divisor on $\\mathcal{X}$ whose dual graph is a chain. Precisely, each intersection $X_{0i}\\cap X_{0i+1}$ is smooth and irreducible of dimension $d-1,$ and $X_{0i}\\cap X_{0j}=\\varnothing$ whenever $j\\notin\\{i-1,i,i+1\\}$.\n \\item \\label{it:correspondence} \n There exists a correspondence\n$$\n[\\Gamma] \\in \\CH^{d}(C \\times_k X_{K})\n$$\ninducing a splitting of the pushforward map\n$$\n(\\phi_{X_K})_\\ast\\colon \\Ch(X_{K})_{\\hom}\\longrightarrow\\jac(C_{K}).\n$$\n\n\\end{enumerate}\n\\par \nFor each $i,$ let $\\al_{i}\\colon\\Alb(X_{0i})\\to \\jac(C)$ be the homomorphism of Albanese varieties induced by the restriction of $\\phi_{X_0}$ to $X_{0i}$. Then the induced sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(X_{0i})\\longrightarrow\\jac(C)$$\nadmits a section. \n\\end{theorem}", + "thm:intro_degeneration_S": "\\begin{theorem}\\label{thm:intro_degeneration_S} Let $k$ be an algebraically closed field of characteristic $\\neq2,$ and let $E$ be an elliptic curve over $k$ with $\\End(E)\\cong\\Z$. Set $\\Delta:=\\Spec k[[t]],$ and fix an algebraic closure $K$ of its function field. Then there exists a regular, flat, projective scheme $\\mathcal{S}\\to\\Delta,$ together with a morphism $$\\phi_{\\mathcal{S}}\\colon\\mathcal{S}\\longrightarrow E\\times\\Delta$$ such that the following properties hold:\n\\begin{enumerate}\n \\item\\label{it:generic_fibre} The geometric generic fibre $S_{K}$ is a bielliptic surface of type $2$, and the induced morphism $\\phi_{S_{K}}\\colon S_{K}\\to E\\times_{k} K$ coincides with the Albanese fibration of $S_{K}$.\n \\item\\label{it:special_fibre} The special fibre $S_{0}=\\sum_{i=1}^{n}R_{i}$ is a reduced simple normal crossings divisor on $\\mathcal{S}$ whose dual graph is a chain; that is, each intersection $R_{i}\\cap R_{i+1}$ is a smooth irreducible curve, and $R_{i}\\cap R_{j}=\\varnothing$ whenever $j\\notin\\{i-1,i,i+1\\}$.\n \\item\\label{it:albanese_S_0} For each $i,$ let $\\al_{i}:\\Alb(R_{i})\\to E$ denote the homomorphism of Albanese varieties induced by the restriction of $\\phi_{\\mathcal{S}}$ to $R_{i}$. Then the sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(R_{i})\\longrightarrow E$$ does not admit a section. \n\\end{enumerate} \n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 3797, + "pre_theorem_intro_text": "Let $X$ be a smooth projective complex variety of dimension $d=\\dim X$. Given a base-point $x_{0}\\in X(\\mathbf{C}),$ the universal morphism \\begin{equation*}\n \\al_{X}\\colon X\\longrightarrow\\Alb(X)\n\\end{equation*}\nto the Albanese variety of $X$ (see \\cite[Theorem 1.4.4]{Murre}) induces the so-called Abel--Jacobi map for $0$-cycles on $X$ \\begin{equation}\n \\label{eq:abel-jacobi-map} \\alpha_{X}\\colon\\Ch(X)_{\\hom}\\longrightarrow\\Alb(X).\n\\end{equation}\nThe group homomorphism $\\alpha_{X}$ no longer depends on the choice of the base-point and it is known to be surjective and regular. The latter asserts that, for any smooth projective complex variety $Z$ equipped with a base-point $z_{0}\\in Z(\\mathbf{C})$, and for any codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(Z\\times X)$, the assignment\n\\begin{equation}\\label{eq:psi}\n \\psi_{(Z,[\\Gamma])}\\colon Z\\longrightarrow\\Alb(X),\\quad z\\longmapsto \\alpha_{X}\\circ[\\Gamma]_{\\ast}(z-z_{0})\n\\end{equation}\ndefines a morphism of algebraic varieties; see \\cite[Definition 1.6.1]{Murre}. In fact, the Abel--Jacobi map $\\alpha_{X}$ also enjoys the universal property of being initial among all regular homomorphisms from $\\Ch(X)_{\\hom}$ to abelian varieties; see \\cite[Example 1.8(b)]{Murre}.\n\\par Since the homomorphism $\\alpha_{X}$ in \\eqref{eq:abel-jacobi-map} is regular and surjective, there exists a codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(\\Alb(X)\\times X)$ and a positive integer $n$ such that the morphism $\\psi_{(\\Alb(X),[\\Gamma])}$ in \\eqref{eq:psi}, with respect to the chosen base-point $0_{\\Alb(X)}\\in\\Alb(X)(\\mathbf{C})$, is equal to $n\\cdot\\Id_{\\Alb(X)}$; see \\cite[Corollary 1.6.3]{Murre}.\\par\n\nThe present work is concerned with a strengthening of this statement, in the sense of the following property of the variety $X$, first introduced and studied by Voisin; see \\cite{Voisin2024CycleCO,Voisin2025}.\n\n\\begin{definition}[\\cite{voisin_a}, Definition 1.1]\\label{def:universal_0-cycle} We say that $X$ admits a \\emph{universal $0$-cycle} if there exists a codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(\\Alb(X)\\times X)$ such that the morphism $\\psi_{(\\Alb(X),[\\Gamma])}$ in \\eqref{eq:psi}, with respect to the base-point $0_{\\Alb(X)}\\in\\Alb(X)(\\mathbf{C})$, is the identity map $\\Id_{\\Alb(X)}$ on the Albanese variety of $X$. \n\\end{definition}\n\nIf $X$ is a curve, then the Poincaré divisor on $\\Jac(X)\\times X$ induces a universal $0$-cycle on $X$. In contrast, Voisin showed in \\cite{Voisin2025} that for each $d\\geq2$, there exists a smooth projective complex $d$-fold $X$ admitting no universal $0$-cycle; see \\cite[Corollary 0.14]{Voisin2025}.\\par\nIn \\cite{colliotthélène2025notessurlapplicationdalbanese}, Colliot--Th\\'{e}l\\`ene considered whether the existence of a universal $0$-cycle persists under additional geometric assumptions on a smooth projective complex variety $X$. In particular, he asked whether this property holds for varieties with representable $\\Ch$-group, that is, for which the homomorphism $\\alpha_{X}$ in \\eqref{eq:abel-jacobi-map} is an isomorphism; see \\cite{Mumford1969,Roitman_1972}. Building on earlier ideas of Benoist--Ottem \\cite{benoist-ottem}, Voisin subsequently constructed in \\cite{voisin_a} a smooth projective complex threefold $X$ with representable $\\Ch$-group but admitting no universal $0$-cycle.\\par\n\nThis leads to the following question, raised by Totaro after Voisin’s talk at the workshop \\emph{Hodge Theory and Algebraic Cycles} (Clay Mathematics Institute, September 2025).\n\n\\begin{ques}\\label{ques:2-dim_example}\nDoes there exist a smooth projective complex surface $S$ whose Chow group of $0$-cycles $\\CH_0(S)$ is representable, but which admits no universal $0$-cycle?\n\\end{ques}\n\nThe main result of the present paper answers Question \\ref{ques:2-dim_example} in the affirmative.", + "context": "Let $X$ be a smooth projective complex variety of dimension $d=\\dim X$. Given a base-point $x_{0}\\in X(\\mathbf{C}),$ the universal morphism \\begin{equation*}\n \\al_{X}\\colon X\\longrightarrow\\Alb(X)\n\\end{equation*}\nto the Albanese variety of $X$ (see \\cite[Theorem 1.4.4]{Murre}) induces the so-called Abel--Jacobi map for $0$-cycles on $X$ \\begin{equation}\n \\label{eq:abel-jacobi-map} \\alpha_{X}\\colon\\Ch(X)_{\\hom}\\longrightarrow\\Alb(X).\n\\end{equation}\nThe group homomorphism $\\alpha_{X}$ no longer depends on the choice of the base-point and it is known to be surjective and regular. The latter asserts that, for any smooth projective complex variety $Z$ equipped with a base-point $z_{0}\\in Z(\\mathbf{C})$, and for any codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(Z\\times X)$, the assignment\n\\begin{equation}\\label{eq:psi}\n \\psi_{(Z,[\\Gamma])}\\colon Z\\longrightarrow\\Alb(X),\\quad z\\longmapsto \\alpha_{X}\\circ[\\Gamma]_{\\ast}(z-z_{0})\n\\end{equation}\ndefines a morphism of algebraic varieties; see \\cite[Definition 1.6.1]{Murre}. In fact, the Abel--Jacobi map $\\alpha_{X}$ also enjoys the universal property of being initial among all regular homomorphisms from $\\Ch(X)_{\\hom}$ to abelian varieties; see \\cite[Example 1.8(b)]{Murre}.\n\\par Since the homomorphism $\\alpha_{X}$ in \\eqref{eq:abel-jacobi-map} is regular and surjective, there exists a codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(\\Alb(X)\\times X)$ and a positive integer $n$ such that the morphism $\\psi_{(\\Alb(X),[\\Gamma])}$ in \\eqref{eq:psi}, with respect to the chosen base-point $0_{\\Alb(X)}\\in\\Alb(X)(\\mathbf{C})$, is equal to $n\\cdot\\Id_{\\Alb(X)}$; see \\cite[Corollary 1.6.3]{Murre}.\\par\n\nThe present work is concerned with a strengthening of this statement, in the sense of the following property of the variety $X$, first introduced and studied by Voisin; see \\cite{Voisin2024CycleCO,Voisin2025}.\n\n\\begin{definition}[\\cite{voisin_a}, Definition 1.1]\\label{def:universal_0-cycle} We say that $X$ admits a \\emph{universal $0$-cycle} if there exists a codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(\\Alb(X)\\times X)$ such that the morphism $\\psi_{(\\Alb(X),[\\Gamma])}$ in \\eqref{eq:psi}, with respect to the base-point $0_{\\Alb(X)}\\in\\Alb(X)(\\mathbf{C})$, is the identity map $\\Id_{\\Alb(X)}$ on the Albanese variety of $X$. \n\\end{definition}\n\nIf $X$ is a curve, then the Poincaré divisor on $\\Jac(X)\\times X$ induces a universal $0$-cycle on $X$. In contrast, Voisin showed in \\cite{Voisin2025} that for each $d\\geq2$, there exists a smooth projective complex $d$-fold $X$ admitting no universal $0$-cycle; see \\cite[Corollary 0.14]{Voisin2025}.\\par\nIn \\cite{colliotthélène2025notessurlapplicationdalbanese}, Colliot--Th\\'{e}l\\`ene considered whether the existence of a universal $0$-cycle persists under additional geometric assumptions on a smooth projective complex variety $X$. In particular, he asked whether this property holds for varieties with representable $\\Ch$-group, that is, for which the homomorphism $\\alpha_{X}$ in \\eqref{eq:abel-jacobi-map} is an isomorphism; see \\cite{Mumford1969,Roitman_1972}. Building on earlier ideas of Benoist--Ottem \\cite{benoist-ottem}, Voisin subsequently constructed in \\cite{voisin_a} a smooth projective complex threefold $X$ with representable $\\Ch$-group but admitting no universal $0$-cycle.\\par\n\n\\begin{ques}\\label{ques:2-dim_example}\nDoes there exist a smooth projective complex surface $S$ whose Chow group of $0$-cycles $\\CH_0(S)$ is representable, but which admits no universal $0$-cycle?\n\\end{ques}\n\nThe main result of the present paper answers Question \\ref{ques:2-dim_example} in the affirmative.\n\n\\begin{equation}\n \\label{eq:abel-jacobi-map} \\alpha_{X}\\colon\\Ch(X)_{\\hom}\\longrightarrow\\Alb(X).\n\\end{equation}\n\n\\begin{equation}\\label{eq:psi}\n \\psi_{(Z,[\\Gamma])}\\colon Z\\longrightarrow\\Alb(X),\\quad z\\longmapsto \\alpha_{X}\\circ[\\Gamma]_{\\ast}(z-z_{0})\n\\end{equation}", + "full_context": "Let $X$ be a smooth projective complex variety of dimension $d=\\dim X$. Given a base-point $x_{0}\\in X(\\mathbf{C}),$ the universal morphism \\begin{equation*}\n \\al_{X}\\colon X\\longrightarrow\\Alb(X)\n\\end{equation*}\nto the Albanese variety of $X$ (see \\cite[Theorem 1.4.4]{Murre}) induces the so-called Abel--Jacobi map for $0$-cycles on $X$ \\begin{equation}\n \\label{eq:abel-jacobi-map} \\alpha_{X}\\colon\\Ch(X)_{\\hom}\\longrightarrow\\Alb(X).\n\\end{equation}\nThe group homomorphism $\\alpha_{X}$ no longer depends on the choice of the base-point and it is known to be surjective and regular. The latter asserts that, for any smooth projective complex variety $Z$ equipped with a base-point $z_{0}\\in Z(\\mathbf{C})$, and for any codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(Z\\times X)$, the assignment\n\\begin{equation}\\label{eq:psi}\n \\psi_{(Z,[\\Gamma])}\\colon Z\\longrightarrow\\Alb(X),\\quad z\\longmapsto \\alpha_{X}\\circ[\\Gamma]_{\\ast}(z-z_{0})\n\\end{equation}\ndefines a morphism of algebraic varieties; see \\cite[Definition 1.6.1]{Murre}. In fact, the Abel--Jacobi map $\\alpha_{X}$ also enjoys the universal property of being initial among all regular homomorphisms from $\\Ch(X)_{\\hom}$ to abelian varieties; see \\cite[Example 1.8(b)]{Murre}.\n\\par Since the homomorphism $\\alpha_{X}$ in \\eqref{eq:abel-jacobi-map} is regular and surjective, there exists a codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(\\Alb(X)\\times X)$ and a positive integer $n$ such that the morphism $\\psi_{(\\Alb(X),[\\Gamma])}$ in \\eqref{eq:psi}, with respect to the chosen base-point $0_{\\Alb(X)}\\in\\Alb(X)(\\mathbf{C})$, is equal to $n\\cdot\\Id_{\\Alb(X)}$; see \\cite[Corollary 1.6.3]{Murre}.\\par\n\nThe present work is concerned with a strengthening of this statement, in the sense of the following property of the variety $X$, first introduced and studied by Voisin; see \\cite{Voisin2024CycleCO,Voisin2025}.\n\n\\begin{definition}[\\cite{voisin_a}, Definition 1.1]\\label{def:universal_0-cycle} We say that $X$ admits a \\emph{universal $0$-cycle} if there exists a codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(\\Alb(X)\\times X)$ such that the morphism $\\psi_{(\\Alb(X),[\\Gamma])}$ in \\eqref{eq:psi}, with respect to the base-point $0_{\\Alb(X)}\\in\\Alb(X)(\\mathbf{C})$, is the identity map $\\Id_{\\Alb(X)}$ on the Albanese variety of $X$. \n\\end{definition}\n\nIf $X$ is a curve, then the Poincaré divisor on $\\Jac(X)\\times X$ induces a universal $0$-cycle on $X$. In contrast, Voisin showed in \\cite{Voisin2025} that for each $d\\geq2$, there exists a smooth projective complex $d$-fold $X$ admitting no universal $0$-cycle; see \\cite[Corollary 0.14]{Voisin2025}.\\par\nIn \\cite{colliotthélène2025notessurlapplicationdalbanese}, Colliot--Th\\'{e}l\\`ene considered whether the existence of a universal $0$-cycle persists under additional geometric assumptions on a smooth projective complex variety $X$. In particular, he asked whether this property holds for varieties with representable $\\Ch$-group, that is, for which the homomorphism $\\alpha_{X}$ in \\eqref{eq:abel-jacobi-map} is an isomorphism; see \\cite{Mumford1969,Roitman_1972}. Building on earlier ideas of Benoist--Ottem \\cite{benoist-ottem}, Voisin subsequently constructed in \\cite{voisin_a} a smooth projective complex threefold $X$ with representable $\\Ch$-group but admitting no universal $0$-cycle.\\par\n\n\\begin{ques}\\label{ques:2-dim_example}\nDoes there exist a smooth projective complex surface $S$ whose Chow group of $0$-cycles $\\CH_0(S)$ is representable, but which admits no universal $0$-cycle?\n\\end{ques}\n\nThe main result of the present paper answers Question \\ref{ques:2-dim_example} in the affirmative.\n\n\\begin{equation}\n \\label{eq:abel-jacobi-map} \\alpha_{X}\\colon\\Ch(X)_{\\hom}\\longrightarrow\\Alb(X).\n\\end{equation}\n\n\\begin{equation}\\label{eq:psi}\n \\psi_{(Z,[\\Gamma])}\\colon Z\\longrightarrow\\Alb(X),\\quad z\\longmapsto \\alpha_{X}\\circ[\\Gamma]_{\\ast}(z-z_{0})\n\\end{equation}\n\nThe main result of the present paper answers Question \\ref{ques:2-dim_example} in the affirmative.\n\n\\subsection{Bielliptic surfaces}\\label{subsec:bielliptic} Throughout this subsection, the base field $k$ is assumed to be algebraically closed.\\par \nBielliptic surfaces constitute one of the classes of smooth projective surfaces of Kodaira dimension zero in the Enriques--Kodaira classification; see \\cite{beauville_surfaces,BombieriMumford1977}. Every bielliptic surface $S$ admits a presentation \\begin{equation*}\\label{eq:presentation_S}\n S\\cong\\bigl(E\\times_k F\\bigr)/\\Gi,\\end{equation*}\nwhere $E$ and $F$ are elliptic curves, and $\\Gi$ is a finite group acting freely on $E\\times_k F.$ The action is by translations on $E$ and by automorphisms on $F$. Moreover, one has an isomorphism $$F/\\Gi\\cong\\bbP^{1}_k.$$\\par\nBielliptic surfaces were classified by Bagnera--De Franchis into seven types; see \\cite{bielliptic,bielliptic_suwa}. The classification is summarized in Table \\ref{tab:bielliptic}.\n\\renewcommand{\\arraystretch}{1.5} \\begin{table}[h]\n\\centering\n\\caption{Classification of Bielliptic Surfaces (Bagnera--de Franchis)}\n\\label{tab:bielliptic}\n\\begin{tabular}{c c l c}\n\\hline\n\\textbf{Type} & \\textbf{Group $\\Gi$} & \\textbf{$\\NS(S)_{\\tor}$} & \\textbf{Order of $K_S$ in $\\Pic(S)$} \\\\\n\\hline\n1 & $\\mathbb{Z}/2$ & $\\Z/2\\times\\Z/2$ & 2 \\\\\n2 & $\\mathbb{Z}/2 \\times \\mathbb{Z}/2$ & $\\Z/2$ & 2 \\\\\n3 & $\\mathbb{Z}/4$ & $\\Z/2$ & 4 \\\\\n4 & $\\mathbb{Z}/4\\times \\mathbb{Z}/2$ & $0$ & 4 \\\\\n5 & $\\mathbb{Z}/3$ & $\\Z/3$ & 3 \\\\\n6 & $\\mathbb{Z}/3 \\times \\mathbb{Z}/3$ & $0$ & 3 \\\\\n7 & $\\mathbb{Z}/6$ & $0$ & 6 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\par The Albanese morphism of $S$ coincides with the projection $$\\phi_{S}\\colon S\\longrightarrow E/\\Gi,$$ whose fibres are elliptic curves isomorphic to $F$. The induced Abel--Jacobi map on $0$-cycles,\n$$(\\phi_{S})_{\\ast}\\colon\\Ch(S)_{\\hom}\\longrightarrow E/\\Gi$$\nis an isomorphism; see \\cite[Example (2)]{bloch-srinivas}. Equivalently, the Chow group of $0$-cycles on a bielliptic surface is representable.\\par\nThere is also a natural elliptic fibration $$\\psi_{S}\\colon S\\longrightarrow (F/\\Gi)\\cong\\bbP^{1}_{k}.$$ \nThis fibration admits multiple fibres corresponding to the branch points of the quotient morphism $F \\to F/\\Gi$, with multiplicities equal to the corresponding ramification indices.\\par\nLet $f$ and $e$ denote the classes in $\\NS(S)$ of the smooth fibres of $\\phi_S$ and $\\psi_S$, respectively. A direct computation shows \\begin{equation}\\label{eq:intersection_e_f}e^2=0,\\quad,f^2=0,\\quad e\\cdot f=|\\Gi|.\\end{equation}\n\n\\section{Main results}\\label{sec:final}\n\\subsection{A bielliptic surface of type 2 admitting no universal $0$-cycle} \nTheorem \\ref{thm:main_result} follows immediately from the result below.\n\\begin{theorem}\\label{thm:main_result_body}\nLet $k$ be an algebraically closed field of characteristic $\\neq 2$, and let $E$ be an elliptic curve over $k$ with $\\End(E)=\\mathbb{Z}$. Set $\\Delta := \\Spec k[[t]]$, and fix an algebraic closure $K$ of its function field. Let $\\mathcal S \\to \\Delta$ be the degeneration constructed in Theorem \\ref{thm:intro_degeneration_S}, together with the induced morphism\n$$\n\\phi_{\\mathcal S} \\colon \\mathcal S \\longrightarrow E \\times_k \\Delta .\n$$\nThen for any algebraically closed field extension $F/K$, there exists no correspondence\n$$\n[\\Gamma] \\in \\CH^{2}(E \\times_k S_F)\n$$\nthat induces a splitting of the pushforward map\n\\begin{equation*}\n(\\phi_{S_F})_{\\ast} \\colon \\CH_0(S_F)_{\\hom} \\longrightarrow E_{F} .\n\\end{equation*}\n\\end{theorem}\n\\begin{proof} Suppose, for contradiction, that there exists an algebraically closed field extension $F/K$ and a correspondence\n$$\n[\\Gamma] \\in \\CH^{2}(E \\times_k S_F)\n$$\nsuch that the induced endomorphism $$(\\phi_{S_F})_{\\ast}\\circ[\\Gamma]_{\\ast}\\in\\End\\bigl(E_{F}\\bigr)$$ is the identity.\\par\nLet $$S_0=\\sum_{i=1}^nR_i$$ denote the special fibre of the family $\\ca S\\to\\Delta,$ as in \\eqref{it:special_fibre} of Theorem \\ref{thm:intro_degeneration_S}. For each $i,$ let $\\al_i\\colon\\Alb(R_i)\\to E$ be the morphism of Albanese varieties induced by the restriction of $\\phi_{\\mathcal S}$ to $R_i$. By Corollary \\ref{cor:obstruction}, the assumption above implies that the sum morphism \n$$\\sum_{i=1}^n\\al_i\\colon\\bigoplus_{i=1}^n\\Alb(R_i)\\longrightarrow E$$\nadmits a section. \\par\nThis contradicts condition \\eqref{it:albanese_S_0} of Theorem \\ref{thm:intro_degeneration_S}. The resulting contradiction completes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main_result}] Let $E$ be an elliptic curve over $\\bbC$ with $\\End(E)=\\Z$. Choose a countable algebraically closed subfield $k_0\\subset\\bbC$ over which $E$ is defined; thus $$E\\cong E_0\\times_{k_{0}}\\bbC$$ for some elliptic curve $E_{0}$ over $k_{0}$.\\par\nLet $$\\ca S\\longrightarrow\\Delta:=\\Spec k_{0}[[t]]$$ be the strictly semistable degeneration constructed in Theorem \\ref{thm:intro_degeneration_S}, together with the induced morphism $$\\phi_{\\mathcal S} \\colon \\mathcal S \\longrightarrow E_0 \\times_{k_{0}} \\Delta .$$\\par\nSince $k_0$ is countable, the inclusion $k_0\\subset\\bbC$ factors through the Laurent series field $k_0((t))$. Set $$S:=S_{k_{0}((t))}\\times_{k_{0}((t))}\\bbC.$$\nBy \\eqref{it:generic_fibre} of Theorem \\ref{thm:intro_degeneration_S}, the surface $S$ is a bielliptic surface of type 2, and the morphism $$\\phi_{S}\\colon S\\longrightarrow E$$ induced by $\\phi_{\\ca S}$ coincides with the Albanese fibration of $S$.\\par\nIn particular, the pushforward homomorphism $$(\\phi_{S})_{\\ast} \\colon \\CH_0(S)_{\\hom} \\longrightarrow E $$ agrees with the Abel--Jacobi map for $0$-cycles on $S$. It is well known that bielliptic surfaces have representable $\\Ch$-group; see \\cite[Example (2)]{bloch-srinivas}. Applying Theorem \\ref{thm:main_result_body}, we conclude that there exists no correspondence $$\n[\\Gamma] \\in \\CH^{2}(E \\times S)\n$$\nthat induces a splitting of the Abel--Jacobi map \n\\begin{equation*}\n(\\phi_{S})_{\\ast} \\colon \\CH_0(S)_{\\hom} \\longrightarrow E.\n\\end{equation*}\nEquivalently, the surface $S$ admits no universal $0$-cycle; cf. Definition \\ref{def:universal_0-cycle}. This completes the proof of Theorem \\ref{thm:main_result}. \n\\end{proof}\n\\subsection{A counterexample to the integral Hodge conjecture}\nIn this final section, we apply Theorem \\ref{thm:main_result} to prove Corollary \\ref{cor:integral_Hodge_S}. In particular, we construct a threefold $X$ of Kodaira dimension zero for which the integral Hodge conjecture fails for $1$-cycles. In contrast to the counterexamples of Benoist--Ottem \\cite{benoist-ottem}, the non-algebraic Hodge class in degree $4$ arising from our construction is non-torsion.\\par\nLet $X$ be a smooth projective variety over $\\bbC$ of dimension $d= \\dim X$. We begin by recalling how the non-existence of a universal $0$-cycle on $X$ is related to the failure of the integral Hodge conjecture. Throughout, all (co)homology groups are taken to be Betti (co)homology, and the subscript \"tf\" denotes the quotient by torsion.\\par\n\n\\begin{proof}[Proof of Corollary \\ref{cor:integral_Hodge_S}]Let $E$ be an elliptic curve over $\\bbC$ with $\\End(E)=\\Z$. \nBy Theorem \\ref{thm:main_result}, there exists a bielliptic surface $S$ of type 2 with $\\Alb(S)\\cong E$ such that $S$ admits no universal $0$-cycle (cf. Definition \\ref{def:universal_0-cycle}).\nBy Lemma \\ref{lem:int_Hodge_conj_vs_uni_0_cycl}, it follows that the cohomology class $$\\delta_{S}\\in H^{4}\\bigl(E\\times S,\\Z\\bigr)_{\\mathrm{tf}}$$ constructed in \\eqref{eq:Hodge_class} is a Hodge class which is not algebraic.\\par\nThis provides a counterexample to the integral Hodge conjecture in degree $4$\nfor $E\\times S$.\\end{proof}", + "post_theorem_intro_text_len": 5482, + "post_theorem_intro_text": "Recall that ruled surfaces have representable Chow groups of $0$-cycles and do admit a universal $0$-cycle; this follows directly from the corresponding result for curves.\\par \nMore generally, let $X$ be a non-uniruled smooth projective surface with geometric genus $p_{g}(X)=0$ and irregularity $q(X)\\neq0$. By work of Beauville, the structure of such surfaces is well understood; see \\cite[Chapter VI]{beauville_surfaces}. Specifically, $X$ is birational to a quotient \\begin{equation*}\\label{eq:structure_X}\\bigl(C\\times D\\bigr)/\\Gi,\\end{equation*}\nwhere $C$ and $D$ are smooth projective curves and $\\Gi$ is a finite group acting faithfully on both curves, without fixed points on $C \\times D$. Moreover, at least one of the curves is elliptic. Up to exchanging $C$ and $D$, one has $$C/\\Gi\\cong E,\\qquad D/\\Gi\\cong\\mathbf{P}^{1}_{\\mathbf{C}},$$ where $E$ is an elliptic curve. In this situation, the induced fibration \\begin{equation}\\label{eq:fibration_X}\\phi_{X}\\colon X\\longrightarrow E\\end{equation} coincides with the Albanese morphism of $X$.\\par\nIt was shown in \\cite{voisin_a} that if the index of the fibration $\\phi_X$ in \\eqref{eq:fibration_X}, defined as the greatest common divisor of the degrees $\\deg(C/E)$, where $C \\subset X$ ranges over irreducible curves dominating $E$, is equal to one, then $X$ admits a universal $0$-cycle. In particular, this condition is satisfied when the group $\\Gi$ is cyclic; see \\cite[Proposition 4.4]{voisin_a}.\\par \nOur example in Theorem \\ref{thm:main_result} arises from a very general bielliptic surface $S$ of type 2 (see \\cite{bielliptic, bielliptic_suwa}) whose Albanese variety $\\Alb(S)$ is isomorphic to the prescribed elliptic curve $E$. In this situation, the Albanese fibration $\\phi_{S}\\colon S\\to \\Alb(S)\\cong E$ has index 2. As observed in \\cite[$\\S 2.2$]{voisin_a}, this condition alone does not suffice to imply that $S$ admits no universal $0$-cycle. The proof of Theorem \\ref{thm:main_result} therefore rests on the following geometric result.\n\n\\begin{theorem}\\label{thm:intro_degeneration_S} Let $k$ be an algebraically closed field of characteristic $\\neq2,$ and let $E$ be an elliptic curve over $k$ with $\\End(E)\\cong\\mathbb{Z}$. Set $\\Delta:=\\Spec k[[t]],$ and fix an algebraic closure $K$ of its function field. Then there exists a regular, flat, projective scheme $\\mathcal{S}\\to\\Delta,$ together with a morphism $$\\phi_{\\mathcal{S}}\\colon\\mathcal{S}\\longrightarrow E\\times\\Delta$$ such that the following properties hold:\n\\begin{enumerate}\n \\item\\label{it:generic_fibre} The geometric generic fibre $S_{K}$ is a bielliptic surface of type $2$, and the induced morphism $\\phi_{S_{K}}\\colon S_{K}\\to E\\times_{k} K$ coincides with the Albanese fibration of $S_{K}$.\n \\item\\label{it:special_fibre} The special fibre $S_{0}=\\sum_{i=1}^{n}R_{i}$ is a reduced simple normal crossings divisor on $\\mathcal{S}$ whose dual graph is a chain; that is, each intersection $R_{i}\\cap R_{i+1}$ is a smooth irreducible curve, and $R_{i}\\cap R_{j}=\\varnothing$ whenever $j\\notin\\{i-1,i,i+1\\}$.\n \\item\\label{it:albanese_S_0} For each $i,$ let $\\al_{i}:\\Alb(R_{i})\\to E$ denote the homomorphism of Albanese varieties induced by the restriction of $\\phi_{\\mathcal{S}}$ to $R_{i}$. Then the sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(R_{i})\\longrightarrow E$$ does not admit a section. \n\\end{enumerate} \n\\end{theorem}\nAs we shall explain, the non-existence of a section in \\eqref{it:albanese_S_0} provides the key obstruction to the existence of a universal $0$-cycle on the geometric generic fibre of the family $\\mathcal{S}\\to\\Delta$; see Theorem \\ref{thm:obstruction} and Corollary \\ref{cor:obstruction}.\\par\nWe conclude the introduction with the following application of Theorem \\ref{thm:main_result} to the integral Hodge conjecture.\n\\begin{corollary}\\label{cor:integral_Hodge_S} Let $E$ be a smooth elliptic curve over $\\mathbf{C}$ with $\\End(E)=\\mathbb{Z}$. Then there exists a bielliptic surface $S$ of type 2 whose Albanese variety $\\Alb(S)$ is isomorphic to $E,$ and for which the integral Hodge conjecture for $1$-cycles on $E\\times S$ fails. In particular, there exists a non-torsion integral Hodge class in $H^{4}(E\\times S,\\mathbb{Z})$ that is not algebraic.\n\\end{corollary}\nCorollary \\ref{cor:integral_Hodge_S} adds a further example to the list of threefolds with torsion canonical bundle for which the integral Hodge conjecture fails for $1$-cycles. This should be compared with the counterexamples of Benoist--Ottem \\cite{benoist-ottem}, where the obstruction arises from a non-algebraic torsion cohomology class in degree 4. In contrast, our construction relies on a different degeneration strategy: rather than degenerating the elliptic curve $E,$ we degenerate the surface $S$. This idea is inspired by recent work of the author \\cite{Alexandrou,alexandrou2025torsionhigherchowcycles} and by work of Schreieder \\cite{Schreieder2025}. \\par\nThe paper is organized as follows. In $\\S \\ref{sec:preliminaries}$ we review the necessary background material. Section \\ref{sec:obstruction} introduces an obstruction to the existence of a universal $0$-cycle on a smooth projective variety $X$ with representable Chow group of $0$-cycles. In $\\S \\ref{sec:degen_bielliptic}$ we prove Theorem \\ref{thm:intro_degeneration_S}. Finally, $\\S \\ref{sec:final}$ is devoted to the proofs of our main results, including Theorem \\ref{thm:main_result} and Corollary \\ref{cor:integral_Hodge_S}.", + "sketch": "For surfaces of the form \\eqref{eq:structure_X} with Albanese fibration \\eqref{eq:fibration_X}, it is recalled that Voisin shows: if the index of \\(\\phi_X\\) is \\(1\\), then \\(X\\) admits a universal \\(0\\)-cycle (and this holds in particular when \\(\\Gi\\) is cyclic).\n\nFor Theorem~\\ref{thm:main_result}, the surface \\(S\\) is taken to be “a very general bielliptic surface of type 2” with \\(\\Alb(S)\\cong E\\), for which the Albanese fibration \\(\\phi_S\\) has index \\(2\\); and it is noted (following \\cite[\\S 2.2]{voisin_a}) that “this condition alone does not suffice to imply that \\(S\\) admits no universal \\(0\\)-cycle.”\n\nThe stated strategy is instead to use a degeneration: “The proof of Theorem~\\ref{thm:main_result} therefore rests on” Theorem~\\ref{thm:intro_degeneration_S}, constructing a regular flat projective family \\(\\mathcal S\\to\\Delta\\) whose geometric generic fibre \\(S_K\\) is such a bielliptic surface (with its Albanese fibration), and whose special fibre \\(S_0=\\sum R_i\\) is an SNC divisor with chain dual graph. The key geometric input is that for the induced maps \\(\\al_i:\\Alb(R_i)\\to E\\), “the sum morphism \\(\\sum_i \\al_i:\\bigoplus_i \\Alb(R_i)\\to E\\) does not admit a section” \\eqref{it:albanese_S_0}.\n\nAs announced, “the non-existence of a section in \\eqref{it:albanese_S_0} provides the key obstruction to the existence of a universal \\(0\\)-cycle on the geometric generic fibre of the family \\(\\mathcal S\\to\\Delta\\)” (via Theorem~\\ref{thm:obstruction} and Corollary~\\ref{cor:obstruction}), yielding the nonexistence of a universal \\(0\\)-cycle for \\(S\\) in Theorem~\\ref{thm:main_result}.", + "expanded_sketch": "For surfaces of the form \\eqref{eq:structure_X} with Albanese fibration \\begin{equation}\\label{eq:fibration_X}\\phi_{X}\\colon X\\longrightarrow E\\end{equation}, it is recalled that Voisin shows: if the index of \\(\\phi_X\\) is \\(1\\), then \\(X\\) admits a universal \\(0\\)-cycle (and this holds in particular when \\(\\Gi\\) is cyclic).\n\nTo prove the main theorem, the surface \\(S\\) is taken to be “a very general bielliptic surface of type 2” with \\(\\Alb(S)\\cong E\\), for which the Albanese fibration \\(\\phi_S\\) has index \\(2\\); and it is noted (following Voisin, \\S 2.2) that “this condition alone does not suffice to imply that \\(S\\) admits no universal \\(0\\)-cycle.”\n\nThe stated strategy is instead to use a degeneration: “The proof of the main theorem therefore rests on” the following theorem.\n\n\\begin{theorem}\\label{thm:intro_degeneration_S} Let $k$ be an algebraically closed field of characteristic $\\neq2,$ and let $E$ be an elliptic curve over $k$ with $\\End(E)\\cong\\Z$. Set $\\Delta:=\\Spec k[[t]],$ and fix an algebraic closure $K$ of its function field. Then there exists a regular, flat, projective scheme $\\mathcal{S}\\to\\Delta,$ together with a morphism $$\\phi_{\\mathcal{S}}\\colon\\mathcal{S}\\longrightarrow E\\times\\Delta$$ such that the following properties hold:\n\\begin{enumerate}\n \\item\\label{it:generic_fibre} The geometric generic fibre $S_{K}$ is a bielliptic surface of type $2$, and the induced morphism $\\phi_{S_{K}}\\colon S_{K}\\to E\\times_{k} K$ coincides with the Albanese fibration of $S_{K}$.\n \\item\\label{it:special_fibre} The special fibre $S_{0}=\\sum_{i=1}^{n}R_{i}$ is a reduced simple normal crossings divisor on $\\mathcal{S}$ whose dual graph is a chain; that is, each intersection $R_{i}\\cap R_{i+1}$ is a smooth irreducible curve, and $R_{i}\\cap R_{j}=\\varnothing$ whenever $j\\notin\\{i-1,i,i+1\\}$.\n \\item\\label{it:albanese_S_0} For each $i,$ let $\\al_{i}:\\Alb(R_{i})\\to E$ denote the homomorphism of Albanese varieties induced by the restriction of $\\phi_{\\mathcal{S}}$ to $R_{i}$. Then the sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(R_{i})\\longrightarrow E$$ does not admit a section. \n\\end{enumerate} \n\\end{theorem}\n\nThus one constructs a regular flat projective family \\(\\mathcal S\\to\\Delta\\) whose geometric generic fibre \\(S_K\\) is such a bielliptic surface (with its Albanese fibration), and whose special fibre \\(S_0=\\sum R_i\\) is an SNC divisor with chain dual graph. The key geometric input is the preceding theorem’s assertion that for the induced maps \\(\\al_i:\\Alb(R_i)\\to E\\), the sum morphism \\(\\sum_i \\al_i:\\bigoplus_i \\Alb(R_i)\\to E\\) does not admit a section.\n\nAs announced, “the non-existence of a section in” this property provides the key obstruction to the existence of a universal \\(0\\)-cycle on the geometric generic fibre of the family \\(\\mathcal S\\to\\Delta\\) (via the following theorem and corollary), yielding the nonexistence of a universal \\(0\\)-cycle for \\(S\\) in the main theorem.\n\n\\begin{theorem}\n\\label{thm:obstruction}\nLet $k$ be an algebraically closed field and let $C$ be a smooth projective curve over $k$. Set $\\Delta := \\Spec k[[t]]$, and fix an algebraic closure $K$ of its function field. \n\nLet $p \\colon \\mathcal{X} \\to \\Delta$ be a flat, projective morphism of relative dimension $d$ with regular total space, and let $\\phi_{\\mathcal{X}} \\colon \\mathcal{X} \\to C \\times_k \\Delta$ be a surjective morphism.\nDenote by $X_0$ the special fibre of $p$, by\n$\\phi_{X_0} \\colon X_0 \\to C$ the induced morphism, and by $X_{K\n}$ the geometric generic fibre of $p$. Write $\\phi_{X_K} \\colon X_K \\to C_K$\nfor the base change of $\\phi_{\\mathcal{X}}$ to $K$.\\par\nWe further assume the following:\n\\begin{enumerate}\n \\item \\label{it:X_0} $X_{0}=\\sum_{i=1}^{n}X_{0i}$ is a reduced simple normal crossings divisor on $\\mathcal{X}$ whose dual graph is a chain. Precisely, each intersection $X_{0i}\\cap X_{0i+1}$ is smooth and irreducible of dimension $d-1,$ and $X_{0i}\\cap X_{0j}=\\varnothing$ whenever $j\\notin\\{i-1,i,i+1\\}$.\n \\item \\label{it:correspondence} \n There exists a correspondence\n$$\n[\\Gamma] \\in \\CH^{d}(C \\times_k X_{K})\n$$\ninducing a splitting of the pushforward map\n$$\n(\\phi_{X_K})_\\ast\\colon \\Ch(X_{K})_{\\hom}\\longrightarrow\\jac(C_{K}).\n$$\n\n\\end{enumerate}\n\\par \nFor each $i,$ let $\\al_{i}\\colon\\Alb(X_{0i})\\to \\jac(C)$ be the homomorphism of Albanese varieties induced by the restriction of $\\phi_{X_0}$ to $X_{0i}$. Then the induced sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(X_{0i})\\longrightarrow\\jac(C)$$\nadmits a section. \n\\end{theorem}\n\n\\begin{corollary}\\label{cor:obstruction}\nWe use the notation and assumptions of Theorem \\ref{thm:obstruction}, with condition \\eqref{it:correspondence} replaced by the following:\n\\begin{enumerate}\n \\item[\\((2')\\)]\\label{it:correspondence-prime} There exists an algebraically closed field extension $F/K$ and a correspondence\n$$\n[\\Gamma] \\in \\CH^{d}(C \\times_k X_{F})\n$$\ninducing a splitting of the pushforward map\n$$\n(\\phi_{X_F})_\\ast\\colon \\Ch(X_{F})_{\\hom}\\longrightarrow\\jac(C_{F}).\n$$\n\\end{enumerate}\nAssume moreover that $\\End(\\jac(C))=\\Z$. Then the sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(X_{0i})\\longrightarrow\\jac(C)$$\nadmits a section. \n\\end{corollary}", + "expanded_theorem": "\\label{thm:main_result} Let $E$ be a smooth complex elliptic curve with $\\End(E)=\\mathbb{Z}$. There exists a smooth projective complex surface $S$ with $\\Alb(S)\\cong E$ for which $\\Ch(S)$ is representable, while $S$ admits no universal $0$-cycle.,", + "theorem_type": [ + "Existence", + "Implication" + ], + "mcq": { + "question": "Let $E$ be a smooth complex elliptic curve with $\\operatorname{End}(E)=\\mathbb{Z}$. For a smooth projective complex surface $S$, let\n\\[\n\\alpha_S\\colon \\CH_0(S)_{\\mathrm{hom}}\\longrightarrow \\operatorname{Alb}(S)\n\\]\nbe the Abel--Jacobi map for $0$-cycles. Say that $\\CH_0(S)$ is representable if $\\alpha_S$ is an isomorphism. Say that $S$ admits a universal $0$-cycle if there exists a codimension-$2$ cycle $[\\Gamma]\\in \\CH^2(\\operatorname{Alb}(S)\\times S)$ such that the induced morphism\n\\[\n\\psi_{(\\operatorname{Alb}(S),[\\Gamma])}\\colon \\operatorname{Alb}(S)\\to \\operatorname{Alb}(S),\\qquad a\\mapsto \\alpha_S\\bigl([\\Gamma]_*(a-0_{\\operatorname{Alb}(S)})\\bigr),\n\\]\nis the identity map on $\\operatorname{Alb}(S)$. Under these assumptions on $E$, which conclusion holds?", + "correct_choice": { + "label": "A", + "text": "There exists a smooth projective complex surface $S$ with $\\operatorname{Alb}(S)\\cong E$ such that $\\CH_0(S)$ is representable, i.e. $\\alpha_S$ is an isomorphism, but $S$ does not admit a universal $0$-cycle; equivalently, there is no codimension-$2$ cycle $[\\Gamma]\\in \\CH^2(\\operatorname{Alb}(S)\\times S)$ for which $\\psi_{(\\operatorname{Alb}(S),[\\Gamma])}=\\operatorname{Id}_{\\operatorname{Alb}(S)}$." + }, + "choices": [ + { + "label": "B", + "text": "For every smooth projective complex surface $S$ with $\\operatorname{Alb}(S)\\cong E$, if $\\CH_0(S)$ is representable, then $S$ admits a universal $0$-cycle; equivalently, whenever $\\alpha_S$ is an isomorphism there exists a codimension-$2$ cycle $[\\Gamma]\\in \\CH^2(\\operatorname{Alb}(S)\\times S)$ such that $\\psi_{(\\operatorname{Alb}(S),[\\Gamma])}=\\operatorname{Id}_{\\operatorname{Alb}(S)}$." + }, + { + "label": "C", + "text": "There exists a smooth projective complex surface $S$ with $\\operatorname{Alb}(S)\\cong E$ such that $\\CH_0(S)$ is representable, i.e. $\\alpha_S$ is an isomorphism." + }, + { + "label": "D", + "text": "There exists a smooth projective complex surface $S$ with $\\operatorname{Alb}(S)\\cong E$ such that $\\CH_0(S)$ is representable and $S$ admits a universal $0$-cycle after base change to some algebraically closed field extension; equivalently, there is an algebraically closed extension $F/\\mathbb{C}$ and a codimension-$2$ cycle $[\\Gamma]\\in \\CH^2(\\operatorname{Alb}(S)_F\\times S_F)$ for which $\\psi_{(\\operatorname{Alb}(S)_F,[\\Gamma])}=\\operatorname{Id}_{\\operatorname{Alb}(S)_F}$." + }, + { + "label": "E", + "text": "There exists a smooth projective complex surface $S$ with $\\operatorname{Alb}(S)\\cong E$ such that $\\CH_0(S)$ is representable, but $S$ does not admit a universal $0$-cycle only in the weaker sense that no codimension-$2$ cycle $[\\Gamma]\\in \\CH^2(\\operatorname{Alb}(S)\\times S)$ makes $\\psi_{(\\operatorname{Alb}(S),[\\Gamma])}$ equal to $n\\,\\operatorname{Id}_{\\operatorname{Alb}(S)}$ for any positive integer $n$." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "geometric_construction", + "tampered_component": "index-1/cyclic sufficient criterion promoted to universal statement for all representable surfaces", + "template_used": "quantifier_dependence" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped the failure of a universal $0$-cycle", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "finiteness", + "tampered_component": "existence over an algebraically closed extension $F/K$ treated as if it could occur over a base change from $\\mathbb{C}$ despite the obstruction using $\\operatorname{End}(E)=\\mathbb{Z}$", + "template_used": "uniformity_effectivity" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "confusion between absence of identity splitting and absence of any Murre multiple $n\\operatorname{Id}$", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives definitions and hypotheses but does not explicitly or implicitly reveal which of the logically nearby conclusions is correct. No choice is singled out by wording." + }, + "TAS": { + "score": 1, + "justification": "This is close to theorem-recall: the stem sets up the exact notions and asks which conclusion holds under the stated assumptions. However, it is not a pure restatement because the options vary meaningfully by quantifier strength, base change, and weakening/strengthening." + }, + "GPS": { + "score": 2, + "justification": "Selecting the right answer requires distinguishing subtle logical variants: existence vs universality, exact failure of identity splitting vs stronger failures, and whether base change could remove the obstruction. That creates real reasoning pressure." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically targeted: B overgeneralizes, C is a weaker true statement, D tests misunderstanding about base change, and E confuses failure of identity with failure of all multiples. They are distinct and aligned with likely failure modes." + }, + "total_score": 7, + "overall_assessment": "A strong MCQ with subtle, high-quality distractors and little answer leakage. Its main weakness is that it remains somewhat theorem-recall in flavor rather than a fully generative problem." + } + }, + { + "id": "2602.13582v1", + "paper_link": "http://arxiv.org/abs/2602.13582v1", + "theorems_cnt": 3, + "theorem": { + "env_name": "theorem", + "content": "\\label{thMain}\nThere exist constants $\\delta, \\eps > 0$ and a sequence of finite groups $(G_n)_{n \\geq 1}$ with $|G_n| \\to \\infty$\nwith bounded-size generating sets $X_n, Y_n \\subseteq G_n$ such that\n\\begin{enumerate}[\\normalfont(a)]\n \\item $\\op{Cay}(G_n,X_n)$ are $\\eps$-expander graphs, and\n \\item $\\op{diam}( \\op{Cay}(G_n,Y_n) ) \\geq \\exp(\\delta \\sqrt{\\log|G_n|})$.\n\\end{enumerate}", + "start_pos": 5700, + "end_pos": 6108, + "label": "thMain" + }, + "ref_dict": { + "quesPS": "\\begin{question}[Pyber \\& Szab\\'o~\\cite{PS13}*{Question~23}]\n \\label{quesPS}\n Let $(G_n)_{n \\geq 1}$ be a sequence of finite groups and suppose that\n $\\Cay(G_n,X_n)$ are expander graphs for some sets $(X_n)_{n \\geq 1}$ of bounded cardinality.\n Is it true that for every sequence of generating sets $(Y_n)_{n \\geq 1}$ we have\n $$ \\diam( \\Cay(G_n,Y_n) ) \\> \\le \\> C (\\log |G_n|)^C $$\n for some constant $C$?\n\\end{question}", + "thMain": "\\begin{theorem} \\label{thMain}\nThere exist constants $\\delta, \\eps > 0$ and a sequence of finite groups $(G_n)_{n \\geq 1}$ with $|G_n| \\to \\infty$\nwith bounded-size generating sets $X_n, Y_n \\subseteq G_n$ such that\n\\begin{enumerate}[\\normalfont(a)]\n \\item $\\Cay(G_n,X_n)$ are $\\eps$-expander graphs, and\n \\item $\\diam( \\Cay(G_n,Y_n) ) \\geq \\exp(\\delta \\sqrt{\\log|G_n|})$.\n\\end{enumerate}\n\\end{theorem}", + "thExpSum": "\\begin{theorem} \\label{thExpSum}\n There exist constants $\\delta, \\eps >0$ such that the following holds.\n For every integer $n \\ge 1$ and prime $p \\le e^{\\delta n}$,\n there exists $v \\in V_0$ such that, for all nonconstant $w \\in V$,\n \\[\n \\left| \\frac1{n!} \\sum_{\\sigma \\in S_n} e_p\\br{\\gen{v, w^\\sigma}} \\right|\n \\> \\le \\> 1 - \\eps.\n \\]\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 2441, + "pre_theorem_intro_text": "Let $G$ be an infinite group generated by a finite set $X$.\nSeveral important properties of the Cayley graph $\\op{Cay}(G,X)$ do not depend on the choice of $X$,\nand are actually group properties:\nthe growth type, the number of ends, hyperbolicity, amenability (via the F{\\o}lner condition), and property (T).\nThe key observation behind these facts is that, for any other finite subset $Y$ of $G$,\nthere exists a constant $c$ such that every element in $Y$ can be written as the product of at most $c$ elements in $X$.\n\nThe situation is different for a sequence of finite groups.\nAnswering a question of Lubotzky--Weiss~\\cite{LW93}, Alon, Lubotzky, and Wigderson~\\cite{ALW01}\nconstructed finite groups whose Cayley graphs are expanders with respect to one bounded-size generating set but not with respect to another.\nSpecifically, they showed that the groups $C_2^{p+1} \\rtimes \\PSL_2(p)$ have this property.\nIn the seminal paper~\\cite{Kas07}, Kassabov showed that the symmetric groups also have this property,\nand subsequently Kassabov, Lubotzky, and Nikolov~\\cite{KLN06} proved that almost all nonabelian finite simple groups are expanders with respect to suitable bounded-size generating sets.\n\nHowever, all of the groups listed above likely have worst-case polylogarithmic diameter.\nIn the case of the finite simple groups, this is the content of the famous conjecture of Babai~\\cite{BS92}*{Conjecture~1.7}.\nThis observation prompted Pyber and Szab\\'o to ask the following question:\n\n\\begin{question}[Pyber \\& Szab\\'o~\\cite{PS13}*{Question~23}]\n \\label{quesPS}\n Let $(G_n)_{n \\geq 1}$ be a sequence of finite groups and suppose that\n $\\op{Cay}(G_n,X_n)$ are expander graphs for some sets $(X_n)_{n \\geq 1}$ of bounded cardinality.\n Is it true that for every sequence of generating sets $(Y_n)_{n \\geq 1}$ we have\n $$ \\op{diam}( \\op{Cay}(G_n,Y_n) ) \\> \\le \\> C (\\log |G_n|)^C $$\n for some constant $C$?\n\\end{question}\n\nA similar question was asked by the second author at the 2024 Oberwolfach meeting \\emph{Growth and Expansion in Groups} \\cite{BD24}*{p.~1028, Question 5}:\nthere it was asked whether it is enough to assume that $\\op{diam}(\\op{Cay}(G_n, X_n))$ is polylogarithmic, i.e., whether polylogarithmic diameter is a group property.\n\nIn this note we answer both questions negatively. In fact we show that the diameter of $G$ with respect to the second generating set can be as large as $\\exp(c \\sqrt{\\log |G|})$.", + "context": "Let $G$ be an infinite group generated by a finite set $X$.\nSeveral important properties of the Cayley graph $\\op{Cay}(G,X)$ do not depend on the choice of $X$,\nand are actually group properties:\nthe growth type, the number of ends, hyperbolicity, amenability (via the F{\\o}lner condition), and property (T).\nThe key observation behind these facts is that, for any other finite subset $Y$ of $G$,\nthere exists a constant $c$ such that every element in $Y$ can be written as the product of at most $c$ elements in $X$.\n\nThe situation is different for a sequence of finite groups.\nAnswering a question of Lubotzky--Weiss~\\cite{LW93}, Alon, Lubotzky, and Wigderson~\\cite{ALW01}\nconstructed finite groups whose Cayley graphs are expanders with respect to one bounded-size generating set but not with respect to another.\nSpecifically, they showed that the groups $C_2^{p+1} \\rtimes \\PSL_2(p)$ have this property.\nIn the seminal paper~\\cite{Kas07}, Kassabov showed that the symmetric groups also have this property,\nand subsequently Kassabov, Lubotzky, and Nikolov~\\cite{KLN06} proved that almost all nonabelian finite simple groups are expanders with respect to suitable bounded-size generating sets.\n\nHowever, all of the groups listed above likely have worst-case polylogarithmic diameter.\nIn the case of the finite simple groups, this is the content of the famous conjecture of Babai~\\cite{BS92}*{Conjecture~1.7}.\nThis observation prompted Pyber and Szab\\'o to ask the following question:\n\n\\begin{question}[Pyber \\& Szab\\'o~\\cite{PS13}*{Question~23}]\n \\label{quesPS}\n Let $(G_n)_{n \\geq 1}$ be a sequence of finite groups and suppose that\n $\\op{Cay}(G_n,X_n)$ are expander graphs for some sets $(X_n)_{n \\geq 1}$ of bounded cardinality.\n Is it true that for every sequence of generating sets $(Y_n)_{n \\geq 1}$ we have\n $$ \\op{diam}( \\op{Cay}(G_n,Y_n) ) \\> \\le \\> C (\\log |G_n|)^C $$\n for some constant $C$?\n\\end{question}\n\nA similar question was asked by the second author at the 2024 Oberwolfach meeting \\emph{Growth and Expansion in Groups} \\cite{BD24}*{p.~1028, Question 5}:\nthere it was asked whether it is enough to assume that $\\op{diam}(\\op{Cay}(G_n, X_n))$ is polylogarithmic, i.e., whether polylogarithmic diameter is a group property.\n\nIn this note we answer both questions negatively. In fact we show that the diameter of $G$ with respect to the second generating set can be as large as $\\exp(c \\sqrt{\\log |G|})$.", + "full_context": "Let $G$ be an infinite group generated by a finite set $X$.\nSeveral important properties of the Cayley graph $\\op{Cay}(G,X)$ do not depend on the choice of $X$,\nand are actually group properties:\nthe growth type, the number of ends, hyperbolicity, amenability (via the F{\\o}lner condition), and property (T).\nThe key observation behind these facts is that, for any other finite subset $Y$ of $G$,\nthere exists a constant $c$ such that every element in $Y$ can be written as the product of at most $c$ elements in $X$.\n\nThe situation is different for a sequence of finite groups.\nAnswering a question of Lubotzky--Weiss~\\cite{LW93}, Alon, Lubotzky, and Wigderson~\\cite{ALW01}\nconstructed finite groups whose Cayley graphs are expanders with respect to one bounded-size generating set but not with respect to another.\nSpecifically, they showed that the groups $C_2^{p+1} \\rtimes \\PSL_2(p)$ have this property.\nIn the seminal paper~\\cite{Kas07}, Kassabov showed that the symmetric groups also have this property,\nand subsequently Kassabov, Lubotzky, and Nikolov~\\cite{KLN06} proved that almost all nonabelian finite simple groups are expanders with respect to suitable bounded-size generating sets.\n\nHowever, all of the groups listed above likely have worst-case polylogarithmic diameter.\nIn the case of the finite simple groups, this is the content of the famous conjecture of Babai~\\cite{BS92}*{Conjecture~1.7}.\nThis observation prompted Pyber and Szab\\'o to ask the following question:\n\n\\begin{question}[Pyber \\& Szab\\'o~\\cite{PS13}*{Question~23}]\n \\label{quesPS}\n Let $(G_n)_{n \\geq 1}$ be a sequence of finite groups and suppose that\n $\\op{Cay}(G_n,X_n)$ are expander graphs for some sets $(X_n)_{n \\geq 1}$ of bounded cardinality.\n Is it true that for every sequence of generating sets $(Y_n)_{n \\geq 1}$ we have\n $$ \\op{diam}( \\op{Cay}(G_n,Y_n) ) \\> \\le \\> C (\\log |G_n|)^C $$\n for some constant $C$?\n\\end{question}\n\nA similar question was asked by the second author at the 2024 Oberwolfach meeting \\emph{Growth and Expansion in Groups} \\cite{BD24}*{p.~1028, Question 5}:\nthere it was asked whether it is enough to assume that $\\op{diam}(\\op{Cay}(G_n, X_n))$ is polylogarithmic, i.e., whether polylogarithmic diameter is a group property.\n\nIn this note we answer both questions negatively. In fact we show that the diameter of $G$ with respect to the second generating set can be as large as $\\exp(c \\sqrt{\\log |G|})$.\n\n\\begin{abstract}\n We study how the spectral gap and diameter of Cayley graphs depend strongly on the choice of generating set.\n We answer a question of Pyber and Szab\\'o (2013)\n by exhibiting a sequence of finite groups $G_n$ with $|G_n| \\to \\infty$ admitting bounded generating sets $X_n,Y_n$ such that $\\operatorname{Cay}(G_n,X_n)$ is an expander\n while $\\operatorname{Cay}(G_n,Y_n)$ has super-polylogarithmic diameter.\n The construction uses the semidirect product $G_n = C_p^{n-1} \\rtimes S_n$ with $p$ exponentially large in $n$,\n and the analysis reduces to bounding some exponential sums of permutational type.\n\\end{abstract}\n\n\\begin{question}[Pyber \\& Szab\\'o~\\cite{PS13}*{Question~23}]\n \\label{quesPS}\n Let $(G_n)_{n \\geq 1}$ be a sequence of finite groups and suppose that\n $\\Cay(G_n,X_n)$ are expander graphs for some sets $(X_n)_{n \\geq 1}$ of bounded cardinality.\n Is it true that for every sequence of generating sets $(Y_n)_{n \\geq 1}$ we have\n $$ \\diam( \\Cay(G_n,Y_n) ) \\> \\le \\> C (\\log |G_n|)^C $$\n for some constant $C$?\n\\end{question}\n\nIn this note we answer both questions negatively. In fact we show that the diameter of $G$ with respect to the second generating set can be as large as $\\exp(c \\sqrt{\\log |G|})$.\n\nBy contrast having polynomially large diameter \\emph{is} a group property,\nand such groups are called \\emph{almost flat}: see \\cite{BT16}*{Theorem~4.1, Corollary~4.16}.\nIn particular, if (a) holds then $\\diam(\\Cay(G_n, Y_n)) \\le |G_n|^{o(1)}$,\nand it is an interesting open question to find the optimal bound in (b).\n\n\\begin{theorem} \\label{thExpSum}\n There exist constants $\\delta, \\eps >0$ such that the following holds.\n For every integer $n \\ge 1$ and prime $p \\le e^{\\delta n}$,\n there exists $v \\in V_0$ such that, for all nonconstant $w \\in V$,\n \\[\n \\left| \\frac1{n!} \\sum_{\\sigma \\in S_n} e_p\\br{\\gen{v, w^\\sigma}} \\right|\n \\> \\le \\> 1 - \\eps.\n \\]\n\\end{theorem}\n\n\\begin{lemma}\n \\label{lem:kaz-basics}\n Let $G$ be a finite group and let $S, T \\subseteq G$.\n \\begin{enumerate}\n \\item $\\kaz(G, S) \\le \\kaz(G, T)$ if $S \\subseteq T$;\n \\item $\\kaz(G, S) \\le 2$;\n \\item $\\kaz(G, G) \\ge \\sqrt 2$;\n \\item $\\kaz(G, S) \\ge \\frac1n \\kaz(G, S^n)$.\n \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n (1)--(2) are clear. For (3)--(4) see \\cite{Kas07}*{Propositions~1.3--1.4}.\n\\end{proof}\n\n\\begin{lemma} \\label{lemSean1}\n Let $\\pi \\colon G \\to U(V)$ and let $\\xi \\in V$ be $\\eps$-almost $S$-invariant.\n Assume $\\kaz(G, S) > 0$ and let $\\eps' = \\eps / \\kaz(G, S)$.\n Then\n \\begin{enumerate}[(i)]\n \\item there is a $G$-invariant vector $\\xi_1 \\in V$ such that $\\|\\xi - \\xi_1\\| < \\eps'\\|\\xi\\|$;\n \\item $\\xi$ is $2\\eps'$-almost $G$-invariant.\n \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n \\emph{(i)}\n Orthogonally decompose $\\xi = \\xi_1 + \\xi_2$ where $\\xi_1 \\in V^G$ and $\\xi_2 \\in (V^G)^\\perp$.\n Since $((V^G)^\\perp)^G = 0$, there is some $s \\in S$ such that\n \\[\n \\kaz(G, S) \\|\\xi_2\\| \\le \\|\\pi(s) \\xi_2 - \\xi_2\\| = \\|\\pi(s) \\xi - \\xi\\| < \\eps \\|\\xi\\|.\n \\]\n Hence $\\|\\xi_2\\| < \\eps' \\|\\xi\\|$.\n\nWe now prove Theorem~\\ref{thMain}(a) assuming Theorem~\\ref{thExpSum} (which will be proved in the next section).\nLet $V= \\F_p^n$ and let $V_0 < V$ be the deleted permutation module of dimension $n-1$.\nLet $G = V_0 \\rtimes S_n$ and $v \\in V_0$.\nBy Kassabov~\\cite{Kas07} there is a bounded-size generating set $T \\subseteq S_n$ such that $\\kaz(S_n, T)$ is bounded away from zero.\nIf $X = \\{v\\} \\cup T$,\nthen by Proposition~\\ref{prop:semidirect} and Lemma~\\ref{lemKazSG} we have\n\\begin{align*}\n \\kaz(G, X)\n &\\ge \\frac{\\sqrt 2}{48} \\kaz(V_0, v^{S_n}) \\kaz(S_n, T)\n \\\\\n &\\ge \\frac{1}{24} \\gap(V_0, v^{S_n})^{1/2} \\kaz(S_n, T).\n \\end{align*}\nWe now focus on $\\gap(V_0, v^{S_n})$.\nObserve that $\\Cay(V,v^{S_n})$ is the disjoint union of $p$ graphs all isomorphic to $\\Cay(V_0,v^{S_n})$.\nIt follows that the eigenvalues of the adjacency operator of $\\Cay(V_0, v^{S_n})$ are the same as for\n$\\Cay(V, v^{S_n})$, the only difference being their multiplicities.\nSince $V$ is abelian, its regular representation is isomorphic to a direct sum of one-dimensional representations\nindexed by the irreducible complex characters.\nWe can identify these characters with vectors $w \\in V$, using the formula\n\\[\n \\chi_w(u) = e_p(\\gen{u, w}) \\qquad (u,w \\in V).\n\\]\n(Recall that $e_p(x) := \\exp(2 \\pi i x / p)$.)\nIt follows that the eigenvalues of $\\Cay(V,v^{S_n})$ and so of $\\Cay(V_0,v^{S_n})$ are given by\n\\[\n \\frac1{2|S_n|} \\sum_{\\sigma \\in S_n} \\br{\\chi_w(v^\\sigma) + \\chi_w(-v^{\\sigma})} \\hspace{1cm} (w \\in V) .\n\\]\nThese are precisely the real values $\\{ \\Re[\\lambda_{v,w}] \\}_{w \\in V}$, where\n\\[\n \\lambda_{v,w} \\> := \\>\n \\frac1{n!} \\sum_{\\sigma \\in S_n} e_p (\\gen{v, w^\\sigma}) .\n\\]\nThe trivial eigenvalues are associated to the constant vectors, and so\n$\\gap(V_0,v^{S_n}) = 1 - \\max_{w \\in V \\setminus \\gen \\one} \\Re[\\lambda_{v,w}]$.\n Since $\\Re[\\lambda_{v,w}] \\le |\\lambda_{v,w}|$, the reduction follows.\n\n\\begin{theorem} \\label{thExpSum}\n There exist constants $\\delta, \\eps >0$ such that the following holds.\n For every integer $n \\ge 1$ and prime $p \\le e^{\\delta n}$,\n there exists $v \\in V_0$ such that, for all nonconstant $w \\in V$,\n \\[\n \\left| \\frac1{n!} \\sum_{\\sigma \\in S_n} e_p\\br{\\gen{v, w^\\sigma}} \\right|\n \\> \\le \\> 1 - \\eps.\n \\]\n\\end{theorem}\n\n\\begin{theorem} \\label{thMain}\nThere exist constants $\\delta, \\eps > 0$ and a sequence of finite groups $(G_n)_{n \\geq 1}$ with $|G_n| \\to \\infty$\nwith bounded-size generating sets $X_n, Y_n \\subseteq G_n$ such that\n\\begin{enumerate}[\\normalfont(a)]\n \\item $\\Cay(G_n,X_n)$ are $\\eps$-expander graphs, and\n \\item $\\diam( \\Cay(G_n,Y_n) ) \\geq \\exp(\\delta \\sqrt{\\log|G_n|})$.\n\\end{enumerate}\n\\end{theorem}", + "post_theorem_intro_text_len": 3242, + "post_theorem_intro_text": "By contrast having polynomially large diameter \\emph{is} a group property,\nand such groups are called \\emph{almost flat}: see \\cite{BT16}*{Theorem~4.1, Corollary~4.16}.\nIn particular, if (a) holds then $\\op{diam}(\\op{Cay}(G_n, Y_n)) \\le |G_n|^{o(1)}$,\nand it is an interesting open question to find the optimal bound in (b).\n\nWe now describe the construction.\nNotably, there are elements in common with both Alon--Lubotzky--Wigderson and Kassabov.\nFrom now on we suppress the subscript $n$, which will be clear from context.\nThe group is just\n\\[G \\> = \\> V_0 \\rtimes S_n,\\]\nwhere $V_0 < V = \\F_p^n$ is the deleted permutation module of dimension $n-1$,\nand $p \\sim e^{cn}$ is a large prime.\nNote that\n\\[\n \\> |G| \\> = \\> p^{n-1} n! \\> = \\> \\exp\\br{cn^2 + O(n \\log n)} .\n\\]\nThe two generating sets $X$ and $Y$ both have the form $\\{v\\} \\cup T$ where $v \\in V_0$ and $T \\subset S_n$.\nFor $Y$ we make the unimaginative choice\n\\[\n v = (1, -1, 0, \\dots, 0), \\qquad T = \\{(1, 2), (1, 2, \\dots, n) \\}.\n\\]\nClearly $\\op{diam}(\\op{Cay}(G, Y)) \\ge \\floor{pn/2} \\ge p \\sim e^{cn}$, so (b) holds.\n\nThe nontrivial part of the proof is the demonstration that there is a choice of $v$ and $T$ such that $\\op{Cay}(G, X)$ has a uniform spectral gap.\nFor $T$ we take a bounded-size expanding generating set for $S_n$ (which exists by the result of Kassabov).\nBy an analysis similar to that in \\cite{ALW01},\nthe problem of finding $v \\in V_0$ reduces to the following ``permutational exponential sum'' estimate.\n\nFor $v, w \\in V = \\F_p^n$, let $\\gen {v,w} = \\sum_{i=1}^n v_i w_i$.\nLet $\\mathbf{1} = (1, \\dots, 1)$ denote the all-one vector, so that $V_0 = \\mathbf{1}^\\perp$,\nand let $e_p(x) := \\exp(2 \\pi i x / p)$.\n\n\\begin{theorem} \\label{thExpSum}\n There exist constants $\\delta, \\eps >0$ such that the following holds.\n For every integer $n \\ge 1$ and prime $p \\le e^{\\delta n}$,\n there exists $v \\in V_0$ such that, for all nonconstant $w \\in V$,\n \\[\n \\left| \\frac1{n!} \\sum_{\\sigma \\in S_n} e_p\\br{\\langle{v, w^\\sigma}\\rangle} \\right|\n \\> \\le \\> 1 - \\eps.\n \\]\n\\end{theorem}\n\nIn fact we show that a random $v \\in V_0$ works.\nWhen $p$ is bounded, this can be proved by a straightforward modification of the analysis in \\cite{ALW01}.\nThe main novelty in our result is that $p$ is allowed to be enormous:\nin technical terms, the reason we can take $p$ exponential in $n$ is that we do not need a union bound over all vectors,\nbut only over support-one vectors, and we can then use a deterministic switching argument based on the Cauchy--Schwarz inequality to deal with all other vectors.\n\n\\begin{remark}\n We have chosen to present the simplest argument that answers Question~\\ref{quesPS}, but\n a more elaborate analysis shows that Theorems~\\ref{thMain} and \\ref{thExpSum} are true for any constant $\\delta > 0$ and some suitable $\\eps = \\eps(\\delta) > 0$.\n Moreover, $\\eps(\\delta)$ can be taken to be a continuous monotonic function such that $\\eps(\\delta) \\to 1$ as $\\delta \\to 0$.\n On the other hand, a standard ``rectification'' argument in additive combinatorics (see \\cite{BLR98}*{Section~3} for example) shows that necessarily $\\eps(\\delta) \\to 0$ as $\\delta \\to \\infty$ in Theorem~\\ref{thExpSum}.\n\\end{remark}", + "sketch": "We “describe the construction” of the groups in Theorem~\\ref{thMain} as follows. “The group is just”\n\\[G = V_0 \\rtimes S_n,\\]\nwhere $V_0<\\F_p^n$ is the deleted permutation module and “$p\\sim e^{cn}$ is a large prime,” so $|G|=p^{n-1}n!=\\exp(cn^2+O(n\\log n))$. The generating sets “both have the form $\\{v\\}\\cup T$ where $v\\in V_0$ and $T\\subset S_n$.”\n\nFor the large-diameter generating set $Y$, one takes\n\\[v=(1,-1,0,\\dots,0),\\qquad T=\\{(1,2),(1,2,\\dots,n)\\},\\]\nand then “clearly $\\operatorname{diam}(\\operatorname{Cay}(G,Y))\\ge \\lfloor pn/2\\rfloor\\ge p\\sim e^{cn}$, so (b) holds.”\n\n“The nontrivial part of the proof” is to choose $v$ and $T$ so that $\\operatorname{Cay}(G,X)$ has “a uniform spectral gap.” For $T$ one uses “a bounded-size expanding generating set for $S_n$ (which exists by the result of Kassabov).” Then, “by an analysis similar to that in \\cite{ALW01}, the problem of finding $v\\in V_0$ reduces to” the exponential-sum estimate stated as Theorem~\\ref{thExpSum}. The text adds that “in fact we show that a random $v\\in V_0$ works.” The key point enabling $p$ to be “enormous” (exponential in $n$) is that “we do not need a union bound over all vectors, but only over support-one vectors, and we can then use a deterministic switching argument based on the Cauchy--Schwarz inequality to deal with all other vectors.”", + "expanded_sketch": "We “describe the construction” of the groups in establishing the main theorem as follows. “The group is just”\n\\[G = V_0 \\rtimes S_n,\\]\nwhere $V_0<\\F_p^n$ is the deleted permutation module and “$p\\sim e^{cn}$ is a large prime,” so $|G|=p^{n-1}n!=\\exp(cn^2+O(n\\log n))$. The generating sets “both have the form $\\{v\\}\\cup T$ where $v\\in V_0$ and $T\\subset S_n$.”\n\nFor the large-diameter generating set $Y$, one takes\n\\[v=(1,-1,0,\\dots,0),\\qquad T=\\{(1,2),(1,2,\\dots,n)\\},\\]\nand then “clearly $\\operatorname{diam}(\\operatorname{Cay}(G,Y))\\ge \\lfloor pn/2\\rfloor\\ge p\\sim e^{cn}$, so (b) holds.”\n\n“The nontrivial part of the proof” is to choose $v$ and $T$ so that $\\operatorname{Cay}(G,X)$ has “a uniform spectral gap.” For $T$ one uses “a bounded-size expanding generating set for $S_n$ (which exists by the result of Kassabov).” Then, “by an analysis similar to that in A. Alon, Y. Roichman, and N. Linial, \\emph{The spectral gap of random Cayley graphs} (2001), the problem of finding $v\\in V_0$ reduces to” the following theorem.\n\n\\begin{theorem} \\label{thExpSum}\n There exist constants $\\delta, \\eps >0$ such that the following holds.\n For every integer $n \\ge 1$ and prime $p \\le e^{\\delta n}$,\n there exists $v \\in V_0$ such that, for all nonconstant $w \\in V$,\n \\[\n \\left| \\frac1{n!} \\sum_{\\sigma \\in S_n} e_p\\br{\\gen{v, w^\\sigma}} \\right|\n \\> \\le \\> 1 - \\eps.\n \\]\n\\end{theorem}\n\nThe text adds that “in fact we show that a random $v\\in V_0$ works.” The key point enabling $p$ to be “enormous” (exponential in $n$) is that “we do not need a union bound over all vectors, but only over support-one vectors, and we can then use a deterministic switching argument based on the Cauchy--Schwarz inequality to deal with all other vectors.”", + "expanded_theorem": "\\label{thMain}\nThere exist constants $\\delta, \\eps > 0$ and a sequence of finite groups $(G_n)_{n \\geq 1}$ with $|G_n| \\to \\infty$\nwith bounded-size generating sets $X_n, Y_n \\subseteq G_n$ such that\n\\begin{enumerate}[\\normalfont(a)]\n \\item $\\op{Cay}(G_n,X_n)$ are $\\eps$-expander graphs, and\n \\item $\\op{diam}( \\op{Cay}(G_n,Y_n) ) \\geq \\exp(\\delta \\sqrt{\\log|G_n|})$.\n\\end{enumerate}", + "theorem_type": [ + "Existence", + "Existential–Universal" + ], + "mcq": { + "question": "Which statement holds? Here, for a finite group $G$ and a generating set $S \\subseteq G$, $\\operatorname{Cay}(G,S)$ denotes the Cayley graph of $G$ with respect to $S$, $\\operatorname{diam}(\\operatorname{Cay}(G,S))$ is its graph diameter, and saying that $\\operatorname{Cay}(G,S)$ is an $\\varepsilon$-expander means that the family has expansion uniformly bounded below by $\\varepsilon>0$. Also, “bounded-size generating sets” means that the cardinalities of the generating sets are uniformly bounded independently of $n$.", + "correct_choice": { + "label": "A", + "text": "There exist constants $\\delta,\\varepsilon>0$ and a sequence of finite groups $(G_n)_{n\\ge 1}$ with $|G_n|\\to\\infty$, together with generating sets $X_n,Y_n\\subseteq G_n$ of uniformly bounded size, such that for every $n\\ge 1$, the Cayley graph $\\operatorname{Cay}(G_n,X_n)$ is an $\\varepsilon$-expander graph and $$\\operatorname{diam}(\\operatorname{Cay}(G_n,Y_n))\\ge \\exp\\!\\bigl(\\delta\\sqrt{\\log|G_n|}\\bigr).$$" + }, + "choices": [ + { + "label": "B", + "text": "There exist constants $\\delta,\\varepsilon>0$ and a sequence of finite groups $(G_n)_{n\\ge 1}$ with $|G_n|\\to\\infty$, together with generating sets $X_n,Y_n\\subseteq G_n$ of uniformly bounded size, such that for every $n\\ge 1$, both Cayley graphs $\\operatorname{Cay}(G_n,X_n)$ and $\\operatorname{Cay}(G_n,Y_n)$ are $\\varepsilon$-expander graphs and $$\\operatorname{diam}(\\operatorname{Cay}(G_n,Y_n))\\ge \\exp\\!\\bigl(\\delta\\sqrt{\\log|G_n|}\\bigr).$$" + }, + { + "label": "C", + "text": "There exist constants $\\delta,\\varepsilon>0$ and a sequence of finite groups $(G_n)_{n\\ge 1}$ with $|G_n|\\to\\infty$, together with generating sets $X_n,Y_n\\subseteq G_n$ of uniformly bounded size, such that for every $n\\ge 1$, the Cayley graph $\\operatorname{Cay}(G_n,X_n)$ is an $\\varepsilon$-expander graph and $$\\operatorname{diam}(\\operatorname{Cay}(G_n,Y_n))\\ge (\\log|G_n|)^{\\delta}.$$" + }, + { + "label": "D", + "text": "There exist constants $\\delta,\\varepsilon>0$ such that for every sequence of finite groups $(G_n)_{n\\ge 1}$ with $|G_n|\\to\\infty$ and every choice of uniformly bounded-size generating sets $X_n,Y_n\\subseteq G_n$, if the Cayley graphs $\\operatorname{Cay}(G_n,X_n)$ are $\\varepsilon$-expander graphs, then one has $$\\operatorname{diam}(\\operatorname{Cay}(G_n,Y_n))\\ge \\exp\\!\\bigl(\\delta\\sqrt{\\log|G_n|}\\bigr)$$ for every $n\\ge 1$." + }, + { + "label": "E", + "text": "There exist constants $\\delta,\\varepsilon>0$ and a sequence of finite groups $(G_n)_{n\\ge 1}$ with $|G_n|\\to\\infty$, together with generating sets $X_n,Y_n\\subseteq G_n$ of uniformly bounded size, such that for every $n\\ge 1$, the Cayley graph $\\operatorname{Cay}(G_n,X_n)$ is an $\\varepsilon$-expander graph and $$\\operatorname{diam}(\\operatorname{Cay}(G_n,Y_n))\\ge \\exp\\!\\bigl(\\delta\\log|G_n|\\bigr).$$" + } + ], + "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": "separation between expanding set and large-diameter set", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "lower-bound strength on the diameter", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "geometric_construction", + "tampered_component": "existential choice of groups and generating sets", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "counting_estimate", + "tampered_component": "size of the diameter lower bound relative to $|G_n|$", + "template_used": "stronger_trap" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem only supplies definitions and notation. It does not explicitly or implicitly signal which of the closely related existence statements is correct." + }, + "TAS": { + "score": 2, + "justification": "The item is not a bare restatement of a theorem in the stem. The choices differ in quantifiers, strength of the diameter bound, and whether both generating sets must expand, so the student must discriminate among genuinely competing conclusions." + }, + "GPS": { + "score": 1, + "justification": "The question requires some reasoning about logical strength and asymptotic bounds, but it mainly tests precise recognition/recall of a theorem statement rather than generating a conclusion from given premises." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target common errors: strengthening the conclusion too far, weakening it, changing existential to universal quantifiers, or incorrectly imposing expansion on both generating sets." + }, + "total_score": 7, + "overall_assessment": "A strong MCQ with no answer leakage and high-quality distractors. Its main limitation is that it leans more toward theorem recognition than fully generative mathematical reasoning." + } + }, + { + "id": "2602.13750v1", + "paper_link": "http://arxiv.org/abs/2602.13750v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "[\\cite{FCW}]\\label{Kn}\nLet $n$ be a positive number. Then\n$$\\tau_{o}(K_{n})=\\frac{1}{2^{n}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}.$$", + "start_pos": 4186, + "end_pos": 4347, + "label": "Kn" + }, + "ref_dict": {}, + "pre_theorem_intro_text_len": 1521, + "pre_theorem_intro_text": "Let $G=(V(G), E(G))$ be a graph without self-loops and $\\mathcal{T}(G)$ be the collection of spanning trees of $G$.\nWe usually call $\\tau(G)=|\\mathcal{T}(G)|$ the number of spanning trees of $G$. Denote by $K_n$, $K_{m,n}$, and $K_{n_1,n_2,\\ldots, n_s}$ \nbe the complete graph of $n$ vertices, the complete bipartite graph that one partite has $m$ vertices and the other partite has $n$ vertices, \nand the complete $s$-partite graph that each partite has $n_i$ vertices.\n\nCounting spanning trees is a classic problem originating in the 19th century that continues to be a hot topic today. \nSome recent developments can be found in \\cite{DG,DKM,GJ,LCY,LY,Yan,YT} for example. \n\nIn 2025, motivated by Gallai's classical theorem (\\cite{Lovasz}, Section 5, Problem 17) that the vertices of \nany graph can be partitioned into two sets \nwhere one induces an even-degree subgraph and the other induces an odd-degree subgraph, and also the concept of \nhomeomorphically irreducible spanning trees (HISTs, spanning trees containing no vertices of degree two), \nZheng and Wu \\cite{ZW} introduced the concept of odd spanning trees. An odd spanning tree of a graph $G$ \nis a spanning tree $T$ of $G$ where each vertex has odd degree in $T$.\n\nThe famous Cayley's formula \\cite{Cayley} states that $\\tau(K_n)=n^{n-2}$. \nLet $\\tau_o(G)$ be the number of odd spanning trees of $G$. \nRecently, Feng, Chen and Wu counted the number of odd spanning trees in complete graphs via Prüfer code and the exponential generating function as follows.", + "context": "Let $G=(V(G), E(G))$ be a graph without self-loops and $\\mathcal{T}(G)$ be the collection of spanning trees of $G$.\nWe usually call $\\tau(G)=|\\mathcal{T}(G)|$ the number of spanning trees of $G$. Denote by $K_n$, $K_{m,n}$, and $K_{n_1,n_2,\\ldots, n_s}$ \nbe the complete graph of $n$ vertices, the complete bipartite graph that one partite has $m$ vertices and the other partite has $n$ vertices, \nand the complete $s$-partite graph that each partite has $n_i$ vertices.\n\nCounting spanning trees is a classic problem originating in the 19th century that continues to be a hot topic today. \nSome recent developments can be found in \\cite{DG,DKM,GJ,LCY,LY,Yan,YT} for example.\n\nIn 2025, motivated by Gallai's classical theorem (\\cite{Lovasz}, Section 5, Problem 17) that the vertices of \nany graph can be partitioned into two sets \nwhere one induces an even-degree subgraph and the other induces an odd-degree subgraph, and also the concept of \nhomeomorphically irreducible spanning trees (HISTs, spanning trees containing no vertices of degree two), \nZheng and Wu \\cite{ZW} introduced the concept of odd spanning trees. An odd spanning tree of a graph $G$ \nis a spanning tree $T$ of $G$ where each vertex has odd degree in $T$.\n\nThe famous Cayley's formula \\cite{Cayley} states that $\\tau(K_n)=n^{n-2}$. \nLet $\\tau_o(G)$ be the number of odd spanning trees of $G$. \nRecently, Feng, Chen and Wu counted the number of odd spanning trees in complete graphs via Prüfer code and the exponential generating function as follows.", + "full_context": "Let $G=(V(G), E(G))$ be a graph without self-loops and $\\mathcal{T}(G)$ be the collection of spanning trees of $G$.\nWe usually call $\\tau(G)=|\\mathcal{T}(G)|$ the number of spanning trees of $G$. Denote by $K_n$, $K_{m,n}$, and $K_{n_1,n_2,\\ldots, n_s}$ \nbe the complete graph of $n$ vertices, the complete bipartite graph that one partite has $m$ vertices and the other partite has $n$ vertices, \nand the complete $s$-partite graph that each partite has $n_i$ vertices.\n\nCounting spanning trees is a classic problem originating in the 19th century that continues to be a hot topic today. \nSome recent developments can be found in \\cite{DG,DKM,GJ,LCY,LY,Yan,YT} for example.\n\nIn 2025, motivated by Gallai's classical theorem (\\cite{Lovasz}, Section 5, Problem 17) that the vertices of \nany graph can be partitioned into two sets \nwhere one induces an even-degree subgraph and the other induces an odd-degree subgraph, and also the concept of \nhomeomorphically irreducible spanning trees (HISTs, spanning trees containing no vertices of degree two), \nZheng and Wu \\cite{ZW} introduced the concept of odd spanning trees. An odd spanning tree of a graph $G$ \nis a spanning tree $T$ of $G$ where each vertex has odd degree in $T$.\n\nThe famous Cayley's formula \\cite{Cayley} states that $\\tau(K_n)=n^{n-2}$. \nLet $\\tau_o(G)$ be the number of odd spanning trees of $G$. \nRecently, Feng, Chen and Wu counted the number of odd spanning trees in complete graphs via Prüfer code and the exponential generating function as follows.\n\nThe famous Cayley's formula \\cite{Cayley} states that $\\tau(K_n)=n^{n-2}$. \nLet $\\tau_o(G)$ be the number of odd spanning trees of $G$. \nRecently, Feng, Chen and Wu counted the number of odd spanning trees in complete graphs via Prüfer code and the exponential generating function as follows.\n\nNote: in \\cite{FCW}, the original theorem is of the form\n\\[\n\\tau_{o}(K_{n})=\n\\begin{cases}\n0, & \\text{if } n \\text{ is odd;} \\\\\n\\displaystyle\\frac{1}{2^{n}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}, & \\text{if } n \\text{ is even.}\n\\end{cases}\n\\]\nWe slightly revised the theorem since when $n$ is odd, it is easy to check that $\\frac{1}{2^{n}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}=0$.\n\n\\begin{lemma}\\label{bo}\n$$\n \\sum_{\\substack{y \\in \\{\\pm1\\}^n}} \\left(\\sum_{i=1}^n a_i y_i \\right)^m=\n 2^n\\cdot\\sum\\limits_{\\substack{k_1+k_2+\\cdots+k_n=m\\\\k_i\\in 2\\mathbb{N} }}\\frac{m!}{k_1!k_2!\\cdots k_n!}a_1^{k_1}a_2^{k_2}\\cdots a_n^{k_n}.\n$$ \n\\end{lemma}\n\n\\begin{theorem}[\\cite{Berge,Lovasz}]\\label{Kn-degree}\nLet $d_1,d_2,\\dots,d_n$ be positive integers summing up to $2n-2$. Then the number of spanning trees of $K_n$ in which the vertex $i$ has degree exactly $d_i$ for all $i=1,2,\\dots,n$ equals$$\n \\dfrac{(n-2)!}{(d_1-1)!(d_2-1)!\\cdots(d_n-1)!}.$$\n\\end{theorem}\n\n\\begin{theorem}\\label{Kmn-degree}\nLet $K_{m,n}$ be the complete bipartite graph with bipartition $A=\\{u_1,u_2,\\ldots,u_m\\}$ and $B=\\{v_1,v_2,\\ldots,v_n\\}$. Let $a_i$ and $b_j$ be positive integers which $\\sum_{i=1}^{m}a_i=\\sum_{j=1}^{n}b_j=m+n-1$. Then the number of spanning trees of $K_{m,n}$ in which the vertex $u_i$ and $v_j$ have degree exactly $a_i$ and $b_j$ for $i\\in\\{1,2,\\ldots,m\\}$ and $j\\in\\{1,2,\\ldots,n\\}$ equals \n $$\n \\frac{(m-1)!(n-1)!}{\\prod_{i=1}^{m}(a_i-1)!\\prod_{j=1}^{n}(b_j-1)!},\n $$\n\\end{theorem}\n\\begin{proof}\n We use induction on $m+n$. For $m=1$ or $n=1$, $K_{m,n}$ is a star and the statement is trivial. \nSince $\\sum_{i=1}^{m}a_i=\\sum_{j=1}^{n}b_j=m+n-1$, $\\sum_{i=1}^{m}a_i+\\sum_{j=1}^{n}b_j=2(m+n)-2<2(m+n)$, \nthere must exist a leaf, and we may assume that $a_m=1$ where $m>1$. \nRemove $u_m$, in any tree under consideration, $u_m$ is adjacent to some $v_j$, $1\\leq j\\leq n$ and the removal of $u_m$ results in another tree on $\\{u_1,\\dots,u_{m-1},v_1,\\dots,v_n\\}$ with degrees $a_1,\\dots,a_{m-1},b_1,\\cdots,b_j-1,\\dots,b_n$. \nBy the induction hypothesis, the number of trees in $K_{m-1,n}$ that $\\{u_1,\\dots,u_{m-1},v_1,\\dots,v_n\\}$ has degrees $a_1,\\dots,a_{m-1},b_1,\\cdots,b_j-1,\\dots,b_n$ is\n $$\\frac{(m-2)!(n-1)!}{(a_1-1)!\\cdots(a_{m-1}-1)!(b_1-1)!\\cdots(b_j-2)!\\cdots(b_n-1)!}\n =\\frac{(m-2)!(n-1)!(b_j-1)}{\\prod_{i=1}^{m}(a_i-1)!\\prod_{j=1}^{n}(b_j-1)!}.\n $$ \n Thus, the number of trees on $\\{u_1,\\dots,u_{m},v_1,\\dots,v_n\\}$ with degrees $a_1,\\dots,a_{m},b_1,\\dots,b_n$ is\n\n\\noindent\n{\\bf Proof of Theorem \\ref{Kn}.}\nBy Theorem \\ref{Kn-degree}, the number of odd spanning trees of $K_n$ in which \nthe vertex $i$ has degree exactly $d_i=2d_i'+1$ for all $i=1,2,\\dots,n$ equals$$\n \\dfrac{(n-2)!}{(2d_1')!(2d_2')!\\cdots(2d_n')!},$$\n where $d_i\\geq0$, and $\\sum_{i=1}^{n}(2d_i')=n-2$.\nTherefore, \n$$\\tau_o(K_n)=\\sum_{\\substack{k_1+k_2+\\cdots+k_n=n-2\\\\k_i\\in 2\\mathbb{N} }}\\frac{(n-2)!}{k_1!k_2!\\cdots k_n!}.$$\nIt follows from Lemma \\ref{bo} that\n$$\n \\sum_{\\substack{\\varepsilon_i \\in \\{\\pm1\\}}}(\\varepsilon_1x_1+\\varepsilon_2x_2+\\cdots\\varepsilon_nx_n)^{n-2}= 2^n\\cdot\\sum_{\\substack{k_1+k_2+\\cdots+k_n=n-2\\\\k_i\\in 2\\mathbb{N} }}\\frac{(n-2)!}{k_1!k_2!\\cdots k_n!}x_1^{k_1}x_2^{k_2}\\cdots x_n^{k_n}.\n $$\nLet $x_1=x_2=\\cdots=x_n=1$, we obtain that\n $$\n \\tau_o(K_n)=\\frac{1}{2^n} \\sum_{\\substack{\\varepsilon_i \\in \\{\\pm1\\}}}(\\varepsilon_1+\\varepsilon_2+\\cdots\\varepsilon_n)^{n-2}\\\\\n =\\frac{1}{2^n}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}.\n $$\n{\\hfill$\\Box$}\n\n\\begin{theorem}\\label{Kmn}\nLet $m$ and $n$ be positive numbers. Then\n$$\\tau_o(K_{m,n})=\\frac{1}{2^{m+n}}\\left[\\sum_{i=0}^{m}\\binom{m}{i}(2i-m)^{n-1}\\right]\\left[\\sum_{j=0}^{n}\\binom{n}{j}(2j-n)^{m-1}\\right].$$ \n\\end{theorem}\n\n\\begin{proof}\n The notation and symbols used below are consistent with those in Theorem \\ref{Kmn-degree}. \nBy Theorem \\ref{Kmn-degree}, the number of odd spanning trees of $K_{m,n}$ \nin which the vertex $u_i\\in A$ and $v_j\\in B$ have degree exactly $d(u_i)=2d'(u_i)+1$ and $d(v_j)=2d'(v_j)+1$ \nfor all $i\\in\\{1,2,\\dots,m\\}$ and $j\\in\\{1,2,\\dots,n\\}$ equals\n$$\n \\frac{(m-1)!(n-1)!}{\\prod_{i=1}^{m}2d'(u_i)!\\prod_{j=1}^{n}2d'(v_j)!},\n $$\n where all $d'\\geq0$, $\\sum_{i=1}^{m}2d'(u_i)=n-1$ and $\\sum_{j=1}^{n}2d'(v_j)=m-1$. \nTherefore, \n$$\\tau_o(K_{m,n})=\\sum_{\\substack{k_1+k_2+\\cdots+k_m=n-1\\\\k_i\\in 2\\mathbb{N} }}\n \\sum_{\\substack{l_1+l_2+\\cdots+l_n=m-1\\\\l_i\\in 2\\mathbb{N} }}\\frac{(m-1)!(n-1)!}{\\prod_{i=1}^{m}k_i!\\prod_{j=1}^{n}l_j!}.$$", + "post_theorem_intro_text_len": 551, + "post_theorem_intro_text": "Note: in \\cite{FCW}, the original theorem is of the form\n\\[\n\\tau_{o}(K_{n})=\n\\begin{cases}\n0, & \\text{if } n \\text{ is odd;} \\\\\n\\displaystyle\\frac{1}{2^{n}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}, & \\text{if } n \\text{ is even.}\n\\end{cases}\n\\]\nWe slightly revised the theorem since when $n$ is odd, it is easy to check that $\\frac{1}{2^{n}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}=0$.\n\nIn this note, we we give a simple proof via a classical spanning tree enumeration formula and the Boolean function. \nWe also generalize it to complete bipartite graphs.", + "sketch": "To prove Theorem~\\ref{Kn}, the note says it will “give a simple proof via a classical spanning tree enumeration formula and the Boolean function.” It also notes that the statement is revised from \\cite{FCW} because “when $n$ is odd, it is easy to check that $\\frac{1}{2^{n}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}=0$.”", + "expanded_sketch": "No expanded sketch found.", + "expanded_theorem": "[F.C.W. \\cite{FCW}]\\label{Kn}\nLet $n$ be a positive number. Then\n$$\\tau_{o}(K_{n})=\\frac{1}{2^{n}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}.$$,", + "theorem_type": [ + "Classification or Bijection", + "Universal" + ], + "mcq": { + "question": "Let \\(K_n\\) denote the complete graph on \\(n\\) vertices, and let \\(\\tau_o(G)\\) denote the number of odd spanning trees of a graph \\(G\\), where an odd spanning tree is a spanning tree in which every vertex has odd degree. For every positive integer \\(n\\), which statement gives \\(\\tau_o(K_n)\\)?", + "correct_choice": { + "label": "A", + "text": "\\[\\tau_o(K_n)=\\frac{1}{2^n}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}.\\]" + }, + "choices": [ + { + "label": "B", + "text": "\\[\\tau_o(K_n)=\\frac{1}{2^n}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-1}.\\]" + }, + { + "label": "C", + "text": "\\[\\tau_o(K_n)=0\\quad\\text{for every odd }n.\\]" + }, + { + "label": "D", + "text": "\\[\\tau_o(K_n)=\\frac{1}{2^{n-1}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}.\\]" + }, + { + "label": "E", + "text": "\\[\\tau_o(K_n)=\\frac{1}{2^n}\\sum_{k=0}^{n}\\binom{n}{k}\\lvert 2k-n\\rvert^{\\,n-2}.\\]" + } + ], + "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": "exponent_from_Prufer_degree_sum", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "counting_estimate", + "tampered_component": "closed_formula_replaced_by_odd_n_vanishing_consequence", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "counting_estimate", + "tampered_component": "normalizing_factor_2^n", + "template_used": "uniformity_effectivity" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "signed_cancellation_in_binomial_sum", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines the notation and asks for the value of the quantity, but it does not reveal the formula or give direct hints pointing uniquely to choice A." + }, + "TAS": { + "score": 1, + "justification": "The item is essentially a theorem-identification question asking for the exact closed form of \u0007u03c4_o(K_n). It is not a verbatim restatement of the answer, but it is close to recall of a known formula rather than a substantially transformed application." + }, + "GPS": { + "score": 1, + "justification": "There is some reasoning pressure because several options are structurally similar and one distractor is a weaker true statement, so the student must distinguish the strongest correct conclusion. Still, success depends largely on recalling or recognizing the exact formula rather than generating a new argument." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically targeted: wrong exponent, wrong normalization factor, loss of sign cancellation via absolute value, and a weaker true parity consequence. These reflect realistic failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it mainly tests recognition of a known formula rather than deep generative reasoning." + } + }, + { + "id": "2602.13968v1", + "paper_link": "http://arxiv.org/abs/2602.13968v1", + "theorems_cnt": 4, + "theorem": { + "env_name": "thm", + "content": "\\label{thm:intro-cap-comp} \nLet $\\Omega \\subset \\subset \\mathbb{C}^n$ be a strictly pseudoconvex domain. \n\\begin{itemize}\n\\item[(a)] There exists a constant $A>0$ such that for every Borel set $E\\subset \\Omega$,\n$$\n\t{\\tt c}(E) \\leq A [cap(E,\\Omega)]^\\frac{1}{n}.\n$$\n\\item[(b)]\nAssume $D \\subset \\subset \\Omega$ be a subdomain. There exists a constant $A'$ such that for every Borel set $E\\subset D$,\n$$\n\t\\frac{1}{A'} cap(E,\\Omega) \\leq {\\tt c} (E).\n$$\n\\end{itemize}\nMoreover, the above inequalities are sharp as far as the exponents are concerned.", + "start_pos": 11254, + "end_pos": 11793, + "label": "thm:intro-cap-comp" + }, + "ref_dict": { + "thm:intro-cap-comp": "\\begin{thm} \\label{thm:intro-cap-comp} \nLet $\\Om \\subset \\subset \\bC^n$ be a strictly pseudoconvex domain. \n\\begin{itemize}\n\\item[(a)] There exists a constant $A>0$ such that for every Borel set $E\\subset \\Om$,\n$$\n\t\\tc(E) \\leq A [cap(E,\\Om)]^\\frac{1}{n}.\n$$\n\\item[(b)]\nAssume $D \\subset \\subset \\Om$ be a subdomain. There exists a constant $A'$ such that for every Borel set $E\\subset D$,\n$$\n\t\\frac{1}{A'} cap(E,\\Om) \\leq \\tc (E).\n$$\n\\end{itemize}\nMoreover, the above inequalities are sharp as far as the exponents are concerned.\n\\end{thm}", + "rmk:sharp-exp": "\\begin{remark}[Proof of the last statement in Theorem~\\ref{thm:intro-cap-comp}]\n\\label{rmk:sharp-exp} The Alexander-Taylor inequality \\cite{AT84} reads\n$$\n\t\\exp (-A_r/cap(K,\\Om)) \\leq T_R(K) \\leq \\exp(-2\\pi/[cap(K,\\Om)]^\\frac{1}{n}),\n$$\nwhere the exponents in both inequalities are sharp (\\cite[Remark~2]{AT84}).\nTogether with $cap(K) \\leq A_r \\tc(K)$ in Lemma~\\ref{lem:cap-c}\nand the first inequality in Lemma~\\ref{lem:AT-type} we derive \n$$\n\t\\exp (-A/cap(K,\\Om)) \\leq \\exp (-A/\\tc(K,\\Om)) \\leq \\exp(-2\\pi/[cap(K,\\Om)]^\\frac{1}{n})\n$$\nwith the sharp exponents. So are the ones in Lemma~\\ref{lem:c-cap} and Lemma~\\ref{lem:cap-c}.\n\\end{remark}", + "thm:subextension": "\\begin{thm} \\label{thm:subextension} \nLet $f \\in W^*(B(0,1))$ be such that $f\\leq -1$. Assume $\\|f\\|_*\\leq 1$. Let $\\veps>0$ and $0 s\\}$ for $s > 0$. Then,\n\\[\\label{eq:c-sublevel-set-est}\n\t\\tc (E_s) \\leq \\frac{1}{s^2} \\|f\\|_*^2.\n\\]\n\\end{lem}", + "rmk:sharp-DMV": "\\begin{remark}\\label{rmk:sharp-DMV} The inequalities in Proposition~\\ref{prop:equiv-est} are sharp as far as the exponents are concerned. In fact, it follows from the first inequality (a) and Lemma~\\ref{lem:c-cap} that for a Borel set $E\\subset \\Om$, \n$$\n\tV_{2n} (E) \\leq A_1 e^{-\\al/ \\tc (E)} \\leq A_1 e^{-\\al/ [cap(E, \\Om)]^\\frac{1}{n}}.\n$$\nWe know from \\cite[Theorem~A]{ACKPZ09} that the exponent $1/n$ in the inequality between the volume and Bedford-Taylor capacity is sharp. So are the exponents in the inequalities of the proposition. Consequently, the exponent 2 in Proposition~\\ref{prop:DMV} and \\cite[Theorem~1.2]{DMV} is the optimal one for all $n\\geq 1$.\n\\end{remark}", + "thm:VV": "\\begin{thm} \\label{thm:VV} Let $\\Om \\subset \\subset \\bC^n$ be an open subset and let $f\\in W^*(\\Om)$. There exists a Borel subset $E\\subset \\Om$ such that $\\tc (E) =0$ and it satisfies \n\\begin{itemize}\n\\item\n[(i)] for each $x\\in \\Om\\setminus E$,\n$$\n\t\\lim_{r\\to 0} \\intavg_{B(x,r)} f dy = f^\\star(x).\n$$\n\\item\n[(ii)] Moreover, for each $x\\in \\Om\\setminus E$,\n$$\n\t\\lim_{r\\to 0} \\intavg_{B(x,r)} |f-f^\\star(x)|^2 dy =0.\n$$\n\\end{itemize}\nConsequently, the precise representative $f^\\star$ is quasi-continuous.\n\\end{thm}", + "lem:c-cap": "\\begin{lem}[dominated by Bedford-Taylor capacity] \\label{lem:c-cap} Assume $\\Om$ is strictly pseudoconvex. There exists a constant $A>0$ such that for every Borel set $E\\subset\\subset \\Om$,\n$$\n\t\\tc(E) \\leq A \\left[cap(E,\\Om)\\right]^\\frac{1}{n}.\n$$\n\\end{lem}", + "eq:DMV-intro": "\\begin{thm}\\label{thm:intro-choquet-c} Let $\\Om$ be a bounded open set in $\\bC^n$. The set function on Borel subsets $E\\mapsto \\tc (E)= \\tc (E,\\Om)$ is a Choquet capacity.\n\\end{thm}\n\nThis result is the local analogue of \\cite{Vigny}. On compact K\\\"ahler manifolds the proof of \\cite[Theorem~30]{Vigny} employed the capacity notion in the Dirichlet spaces \\cite[Lemma~23]{Vigny} and the characterization of pluripolar sets via a family of capacity associated with closed positive current \\cite{FO84}. Here we use a similar strategy. Notably, we are able to give a simpler proof of the characterization in Corollary~\\ref{cor:FO-polar} by using only pluripotential theory. \n\nAnother remarkable property of functions in the complex Sobolev space is the uniformly exponential integrability due to Dinh, Marinescu and Vu \\cite{DMV}, which is very close to the one of psh functions. Namely, \nthere exist positive constants $\\al$ and $A$ such that for every $f \\in W^*(B(0,1))$ with $\\|f\\|_*\\leq 1$, \n\\[\\label{eq:DMV-intro}\n\t\\int_{\\bar B(0,\\frac{1}{8})} e^{\\al |f|^2} d x \\leq A.\n\\]\nThis result should be compared with the well-known exponential integrability of psh functions due to H\\\"ormander \\cite[Proposition~4.2.9]{Ho07} and Skoda \\cite{Sk72}. In the proof of \\cite[Theorem~1.2]{DMV} the authors used the induction argument in dimension and slicing theory for positive currents. We provide a new and simpler one in Proposition~\\ref{prop:DMV} by reducing the inequality to one dimensional case. In this way, it is enough to work with smooth functions after taking the standard convolution with smooth kernels. The constants can be explicitly computed though they are suboptimal. Additionally, we point out in Remark~\\ref{rmk:sharp-DMV} that the exponent 2 of $|f|$ in \\eqref{eq:DMV-intro} is the optimal for all $n\\geq 1$, which was only known for $n=1$ in \\cite{DMV}. \n\nThe third main result fully characterizes the inequality \\eqref{eq:DMV-intro} by the volume-capacity inequality of sublevel sets.\n\n\\begin{thm} \\label{prop:equiv-est-intro} Let $K\\subset\\subset \\Om$ be a compact subset. The following statements hold and they are equivalent to each other.\n\\begin{itemize}\n\\item[(a)] \nThere exist constants $A_1>0$ and $\\al>0$ depending only on $K, \\Om$ such that for every Borel set $E\\subset K$ $$V_{2n}(E) \\leq A_1 e^{-\\al/\\tc (E)}.$$\n\n\\item[(b)] There exist uniform constants $A_1>0$ and $\\al>0$ depending only on $K, \\Om$ such that for every $f\\in W^*(\\Om)$ and $f\\leq 0$,\n $$V_{2n}(\\{f< -1\\} \\cap K) \\leq A_1 e^{\\frac{-\\al}{\\|f\\|_*^{2}}}.$$\n\\item\n[(c)] There exist uniform constants $A_1>0$ and $\\al>0$ depending only on $K, \\Om$ such that for every $f\\in W^*(\\Om)$ and $f\\leq 0$ whose norm $\\|f\\|_* \\leq \n\\ka$,\n$$V_{2n} (\\{f < -s\\} \\cap K) \\leq A_1 e^{-\\al s^2/\\ka^2}, \\quad \\forall s>0.$$\n\n\\item \n[(d)] There exist uniform constants $A_1>0$ and $\\al>0$ depending only on $K, \\Om$ such that for every $f\\in W^*(\\Om)$ and $f\\leq 0$ with $\\|f\\|_* \\leq 1$,\n$$V_{2n} (\\{f < -s\\} \\cap K) \\leq A_1 e^{-\\al s^2}, \\quad \\forall s>0.$$\n\\end{itemize}\n\\end{thm}", + "prop:ae-qe": "\\begin{prop} \\label{prop:ae-qe} Let $D\\subset \\Om$ be an open set and $f$ is quasi-continuous on $D$. If $f\\geq 0$ a.e. on $D$, then $f\\geq 0$ quasi-everywhere on $D$.\n\\end{prop}", + "prop:DMV": "\\begin{prop} \\label{prop:DMV} Let $B(0,1) \\subset \\bC^n$ be the unit ball. There exist positive constants $\\al$ and $A$ such that for all $f \\in W^*(B(0,1))$ with $\\|f\\|_*\\leq 1$, \n$$\n\t\\int_{\\bar B(0,\\frac{1}{8})} e^{\\al |f|^2} dx \\leq A.\n$$\n\\end{prop}", + "prop:density": "\\begin{prop} \\label{prop:density} Let $\\Om\\subset \\bC^n$ be a bounded domain. \n\\begin{itemize}\n\\item[(a)] If $n=1$, then $W^*(\\Om) = W^{1,2}(\\Om)$ together with its Sobolev norm. \n\\item[(b)] If $n\\geq 2$, then the continuous functions in $C^0(\\Om) \\cap W^*(\\Om)$ are not dense in $W^*(\\Om)$ with respect to $W^*$-norm. \n\\end{itemize}\n\\end{prop}", + "lem:outer-reg": "\\begin{lem}[outer regularity] \\label{lem:outer-reg} For a Borel subset $E\\subset \\Om$,\n\\[\\notag\t\\tc(E) = \\inf\\{ \\tc(G) : E \\subset G,\\; G \\text{ is open}\\}.\n\\]\n\\end{lem}", + "thm:star-norm": "\\begin{thm}\\label{thm:star-norm} Let $\\Om$ be a bounded domain in $\\bC^n$.\n\\begin{itemize}\n\\item[(a)] $\\| \\cdot\\|_*$ is a norm in $W^*(\\Om)$ and $\\| \\cdot \\|_{W^{1,2}(\\Om)} \\leq \\|\\cdot \\|_*$.\n\\item[(b)] $W^*(\\Om)$ and $W^*_0(\\Om)$ are Banach spaces with this norm.\n\\end{itemize}\n\\end{thm}", + "cor:reflexive": "\\begin{cor} \\label{cor:reflexive} Let $\\Om$ be an open set in $\\bC^n$, $n\\geq 2$. Then, $W^*(\\Om)$ is not reflexive.\n\\end{cor}", + "rmk:inner-reg": "\\begin{remark}\\label{rmk:inner-reg} The relative compactness condition of $E$ will be removed later after proving that $\\tc(\\cdot,\\Om)$ is a capacity in the sense of Choquet (Theorem~\\ref{thm:choquet-c}).\n\\end{remark}", + "rmk:DMV-holder": "\\begin{remark}\\label{rmk:DMV-holder} Let $\\Om$ be a bounded strictly psedocovex domain and $K\\subset \\subset \\Om$ be a compact subset. Let $u_1,...,u_n$ be H\\\"older continuous psh functions in $\\Om$. It follows from \\cite[Proposition~2.4]{DKN} that the Radon measure \n$$\\mu = {\\bf 1}_K \\, dd^c u_1 \\wed \\cdots \\wed dd^c u_n$$\nis $W^*(\\Om)$-H\\\"older continuous. That means there exist positive constants $c>0$ and $\\al>0$ such that for $f\\in W^*(\\Om)$ and $\\|f\\|_* \\leq 1$,\n$$\n\t\\left| \\int_\\Om f d\\mu \\right| \\leq c \\|f\\|_{*}^\\al.\n$$\nCombining this inequality and the arguments in the proof of \\cite[Proposition~2.9]{Ng18} we get that there exist $\\al_1>0$ and $c_1>0$ such that for $f \\in W^*(\\Om)$, $f\\leq 0$ and $\\|f\\|_* \\leq 1$, \n$$\n\t\\mu (f < -s) \\leq c_1 e^{-\\al_1 s^2}, \\quad \\forall s>0.\n$$\nIn other words, the equivalent inequalities in Proposition~\\ref{prop:equiv-est} holds for a very large family of Monge-Amp\\`ere measures associated with H\\\"older continuous psh functions. This result is also equivalent to the statement of \\cite[Theorem~1.2]{DMV}.\n\\end{remark}", + "defn:L2-ma": "\\begin{defn}\n\\label{defn:L2-ma} Let $\\phi \\in PSH\\cap L^\\infty(\\Om)$.\nDefine $\\mu: = (dd^c\\phi)^n$. Let $g \\in W^*(\\Om)$ and $K \\subset \\subset \\Om$ be a compact subset. If $g\\geq 0$, then\n$$\n\t\\int_K g d\\mu := \\int_K \\wt g d\\mu,\n$$\nwhere $\\wt g$ is a quasi-continuous modification. \nGenerally, \n$$\n\t\\int_K g d\\mu := \\int_K g^+ d\\mu - \\int_K g^- d\\mu.\n$$\n\\end{defn}", + "lem:AT-type": "\\begin{lem} \\label{lem:AT-type} There exist positive constants $A_r$ and $A_R$ depending on $r,R$ respectively such that for every compact set $K\\subset B_r$,\n$$ \\exp(- A_R \\; \\tc (K,B_R)^{-1}) \\leq T_R(K) \\leq \t\\exp(- A_r\\;\\tc (K, B_R)^{-1/n}).\n$$ \nEquivalently,\n\\[\\label{eq:AT}\n\t\\frac{1}{A_r} \\frac{1}{M_K^n} \\leq \\tc (K,B_R) \\leq \\frac{A_R }{M_K}.\n\\]\n\\end{lem}", + "cor:FO-polar": "\\begin{cor} \\label{cor:FO-polar}\nLet $E\\subset \\Om$ be a Borel subset. Then, $cap (E) =0$ if and only if $\\tc_\\te(E) =0$ for all $\\te $ of the form \\eqref{eq:current-te} with $0<\\de \\leq 1$.\n\\end{cor}" + }, + "pre_theorem_intro_text_len": 5859, + "pre_theorem_intro_text": "Let $(X,\\omega)$ be a compact K\\\"ahler manifold. A natural subspace of the Sobolev space $W^{1,2}(X)$, called the complex Sobolev space $W^*$, was introduced by Dinh and Sibony \\cite{DS} in their work on complex dynamics. \nThis new space turned out to be a good complex version of the former one as it takes into account the complex structure of manifolds. Many important families of functions which carry useful properties of underlying manifolds belong to the space. Among them are Lipschitz functions, bounded quasi plurisubharmonic (psh) functions or more general bounded delta quasi-psh function studied in \\cite{CW05}. Basic functional aspects of this space were studied by Vigny \\cite{Vigny} where he showed that it is a Banach space with a corresponding norm $\\| \\cdot\\|_*$, this norm is stronger than the Sobolev norm. However, it is not reflexive and smooth functions are not dense in $W^*$ with respect to the strong topology of that norm. He also defined a functional capacity for this space and showed that it is a capacity in sense of Choquet. Moreover, the functional capacity is qualitative comparable to the relative capacity of Bedford and Taylor in global pluripotential theory defined in \\cite{ko03}. \n\nThanks to flexible properties this space has found many of applications in complex dynamic in high dimensions, complex Monge-Amp\\`ere equations and other areas, see e.g., \\cite{BiD23}, \\cite{DKN}, \\cite{DKW}, \\cite{DNV25}, \\cite{Vig15}, \\cite{Vu20, Vu24, Vu26} and \\cite{WZ}. Also a higher complex Sobolev space is proposed in \\cite{DoN25}. \n\nAn element in a Sobolev space is defined almost everywhere up to a set of Lebesgue measure zero which is unlike psh or quasi-psh functions. Its fine properties with respect to the Sobolev capacity are classical ones \\cite{EG92}, \\cite{KLV-book}. However, this capacity is not good enough to characterize pluripolar sets, therefore it is not suitable for studying functions in $W^*$. One needs to work with the corresponding capacity of $W^*$ which dominates the Sobolev one. However, the new capacity in complex Sobolev spaces is harder to analyze. A crucial difference is that we can no longer rely on maximal function techniques, which are very powerful, in these spaces (see, e.g \\cite{KLV-book}). \n\nRecently, it has been showed in \\cite{DMV}, \\cite{Vigny} and \\cite{VV24} that pluripotential theory is a suitable and powerful tool for studying $W^*$. As a result they made major progresses on study local properties of functions in such spaces. \n\nWe continue to study local complex Sobolev spaces which has been considered partially in \\cite{DMV} and \\cite{Vigny}. Our goal is to develop further this approach by focusing on new and optimal inequalities between the functional capacity and classical capacities of Bedford and Taylor \\cite{BT82} and of Alexander and Taylor \\cite{AT84} in pluripotential theory.\n\nLet $\\Omega \\subset \\mathbb{C}^n$ be a bounded domain and $\\omega = dd^c |z|^2$ the standard K\\\"ahler form in $\\mathbb{C}^n$. Denote by $W^{1,2}(\\Omega, \\mathbb{R})$ the usual Sobolev space. For $f\\in W^{1,2}(\\Omega, \\mathbb{R})$ we define $\\Ga_f$ to be the set of all positive closed $(1,1)$-current $T$ satisfying\n\\[\\label{eq:test-currents}\\notag\n\tdf \\wedge d^c f \\leq T \\quad\\text{weakly in } \\Omega.\n\\]\nHere we use the normalization \n$$d^c = \\frac{\\sqrt{-1}}{2\\pi} (\\overline{\\d} -\\partial), \\quad dd^c = \\frac{\\sqrt{-1}}{\\pi}\\partial\\overline{\\d}.\n$$\nConsider the subspace\n\\[ \\label{eq:sobolev-space} \\notag\nW^*(\\Omega)= \\left\\{f \\in W^{1,2}(\\Omega,\\mathbb{R}): \\text{there exists $T\\in \\Ga_f$ with } \\|T\\|_\\Omega <+\\infty\n\\right\\},\n\\]\nwhere \n$\n\t \\|T \\|_\\Omega := \\int_\\Omega T \\wedge \\omega^{n-1}.\n$\nFor $f \\in W^*(\\Omega)$ one defines the $W^*$-norm\n\\[\\label{eq:norm} \\notag\n\t\\|f\\|_*^2 = \\| f\\|_{L^2(\\Omega)}^2 + \\inf_{T\\in \\Ga_f} \\|T\\|_{\\Omega}.\n\\]\nWe then reprove basic functional results obtained in \\cite{Vigny}. Namely, $(W^*(\\Omega), \\|\\cdot\\|_*)$ is a Banach space (Theorem~\\ref{thm:star-norm}) and continuous functions are not dense in $W^*(\\Omega)$ with respect to strong topology of the norm $\\|\\cdot\\|_*$ (Proposition~\\ref{prop:density}). If $n=1$, then $W^*(\\Omega) = W^{1,2}(\\Omega)$, otherwise $W^*(\\Omega)$ is not reflexive (Corollary~\\ref{cor:reflexive}). It should be pointed out that there are some similarity of these properties with the ones of the space of delta psh functions studied by Cegrell and Wiklund \\cite{CW05}.\n\nInspired by the functional capacity in \\cite{Vigny} we define for a Borel subset $E\\subset \\Omega$ the local capacity ${\\tt c}(E):= {\\tt c}(E,\\Omega)$ as follows:\n\\[\\label{eq:intro-cap-defn}\\notag\n\t{\\tt c}(E) = \\inf \n\t\\left\\{ \\| v \\|_*^2 \\;\\rvert\\; v\\in \\cali{K}(E)\n\t\\right\\},\n\\] \nwhere\n\\[\\label{eq:intro-ke-cap}\\notag\n \t\\cali{K}(E) = \\left\\{ v \\in W^*(\\Omega) \\;\\rvert\\; \\{v \\leq -1\\}^o \\supset E \\text{ and } v\\leq 0 \\right\\}.\n\\]\nHere $ \\{v \\leq -1\\}^o \\supset E$ means $v\\leq -1$ a.e in a neighborhood of $E$. \n\nThe functional capacity in the Sobolev spaces is very well-understood which is also an effective tool to study fine properties of its element, see e.g. \\cite[Chapters~4.7-4.8]{EG92}. In contrast, the use of ${\\tt c}(\\cdot)$ for the complex Sobolev space is not effective so far. \nOur goal is to investigate systematically local properties of functions in $W^*(\\Omega)$ by using this capacity. This is a different perspective compared to \\cite{Vigny} and \\cite{DMV} who used the relative capacity for psh functions due to Bedford and Taylor \\cite{BT82}. Namely, for a Borel set $E\\subset \\Omega$, \n\\[\\label{eq:intro-BT-cap}\\notag\n cap(E,\\Omega) = \\sup\\left\\{ \\int_E (dd^c u)^n : u\\in PSH(\\Omega), \\;-1 \\leq u \\leq 0\\right\\}.\n\\]\nOur first main result explains the reason why $cap(\\cdot, \\Omega)$ could be effectively used in previous works. Roughly speaking they are equivalent to each other.", + "context": "Let $(X,\\omega)$ be a compact K\\\"ahler manifold. A natural subspace of the Sobolev space $W^{1,2}(X)$, called the complex Sobolev space $W^*$, was introduced by Dinh and Sibony \\cite{DS} in their work on complex dynamics. \nThis new space turned out to be a good complex version of the former one as it takes into account the complex structure of manifolds. Many important families of functions which carry useful properties of underlying manifolds belong to the space. Among them are Lipschitz functions, bounded quasi plurisubharmonic (psh) functions or more general bounded delta quasi-psh function studied in \\cite{CW05}. Basic functional aspects of this space were studied by Vigny \\cite{Vigny} where he showed that it is a Banach space with a corresponding norm $\\| \\cdot\\|_*$, this norm is stronger than the Sobolev norm. However, it is not reflexive and smooth functions are not dense in $W^*$ with respect to the strong topology of that norm. He also defined a functional capacity for this space and showed that it is a capacity in sense of Choquet. Moreover, the functional capacity is qualitative comparable to the relative capacity of Bedford and Taylor in global pluripotential theory defined in \\cite{ko03}.\n\nAn element in a Sobolev space is defined almost everywhere up to a set of Lebesgue measure zero which is unlike psh or quasi-psh functions. Its fine properties with respect to the Sobolev capacity are classical ones \\cite{EG92}, \\cite{KLV-book}. However, this capacity is not good enough to characterize pluripolar sets, therefore it is not suitable for studying functions in $W^*$. One needs to work with the corresponding capacity of $W^*$ which dominates the Sobolev one. However, the new capacity in complex Sobolev spaces is harder to analyze. A crucial difference is that we can no longer rely on maximal function techniques, which are very powerful, in these spaces (see, e.g \\cite{KLV-book}).\n\nRecently, it has been showed in \\cite{DMV}, \\cite{Vigny} and \\cite{VV24} that pluripotential theory is a suitable and powerful tool for studying $W^*$. As a result they made major progresses on study local properties of functions in such spaces.\n\nLet $\\Omega \\subset \\mathbb{C}^n$ be a bounded domain and $\\omega = dd^c |z|^2$ the standard K\\\"ahler form in $\\mathbb{C}^n$. Denote by $W^{1,2}(\\Omega, \\mathbb{R})$ the usual Sobolev space. For $f\\in W^{1,2}(\\Omega, \\mathbb{R})$ we define $\\Ga_f$ to be the set of all positive closed $(1,1)$-current $T$ satisfying\n\\[\\label{eq:test-currents}\\notag\n df \\wedge d^c f \\leq T \\quad\\text{weakly in } \\Omega.\n\\]\nHere we use the normalization \n$$d^c = \\frac{\\sqrt{-1}}{2\\pi} (\\overline{\\d} -\\partial), \\quad dd^c = \\frac{\\sqrt{-1}}{\\pi}\\partial\\overline{\\d}.\n$$\nConsider the subspace\n\\[ \\label{eq:sobolev-space} \\notag\nW^*(\\Omega)= \\left\\{f \\in W^{1,2}(\\Omega,\\mathbb{R}): \\text{there exists $T\\in \\Ga_f$ with } \\|T\\|_\\Omega <+\\infty\n\\right\\},\n\\]\nwhere \n$\n \\|T \\|_\\Omega := \\int_\\Omega T \\wedge \\omega^{n-1}.\n$\nFor $f \\in W^*(\\Omega)$ one defines the $W^*$-norm\n\\[\\label{eq:norm} \\notag\n \\|f\\|_*^2 = \\| f\\|_{L^2(\\Omega)}^2 + \\inf_{T\\in \\Ga_f} \\|T\\|_{\\Omega}.\n\\]\nWe then reprove basic functional results obtained in \\cite{Vigny}. Namely, $(W^*(\\Omega), \\|\\cdot\\|_*)$ is a Banach space (Theorem~\\ref{thm:star-norm}) and continuous functions are not dense in $W^*(\\Omega)$ with respect to strong topology of the norm $\\|\\cdot\\|_*$ (Proposition~\\ref{prop:density}). If $n=1$, then $W^*(\\Omega) = W^{1,2}(\\Omega)$, otherwise $W^*(\\Omega)$ is not reflexive (Corollary~\\ref{cor:reflexive}). It should be pointed out that there are some similarity of these properties with the ones of the space of delta psh functions studied by Cegrell and Wiklund \\cite{CW05}.\n\nInspired by the functional capacity in \\cite{Vigny} we define for a Borel subset $E\\subset \\Omega$ the local capacity ${\\tt c}(E):= {\\tt c}(E,\\Omega)$ as follows:\n\\[\\label{eq:intro-cap-defn}\\notag\n {\\tt c}(E) = \\inf \n \\left\\{ \\| v \\|_*^2 \\;\\rvert\\; v\\in \\cali{K}(E)\n \\right\\},\n\\] \nwhere\n\\[\\label{eq:intro-ke-cap}\\notag\n \\cali{K}(E) = \\left\\{ v \\in W^*(\\Omega) \\;\\rvert\\; \\{v \\leq -1\\}^o \\supset E \\text{ and } v\\leq 0 \\right\\}.\n\\]\nHere $ \\{v \\leq -1\\}^o \\supset E$ means $v\\leq -1$ a.e in a neighborhood of $E$.\n\nThe functional capacity in the Sobolev spaces is very well-understood which is also an effective tool to study fine properties of its element, see e.g. \\cite[Chapters~4.7-4.8]{EG92}. In contrast, the use of ${\\tt c}(\\cdot)$ for the complex Sobolev space is not effective so far. \nOur goal is to investigate systematically local properties of functions in $W^*(\\Omega)$ by using this capacity. This is a different perspective compared to \\cite{Vigny} and \\cite{DMV} who used the relative capacity for psh functions due to Bedford and Taylor \\cite{BT82}. Namely, for a Borel set $E\\subset \\Omega$, \n\\[\\label{eq:intro-BT-cap}\\notag\n cap(E,\\Omega) = \\sup\\left\\{ \\int_E (dd^c u)^n : u\\in PSH(\\Omega), \\;-1 \\leq u \\leq 0\\right\\}.\n\\]\nOur first main result explains the reason why $cap(\\cdot, \\Omega)$ could be effectively used in previous works. Roughly speaking they are equivalent to each other.\n\n\\begin{cor} \\label{cor:reflexive} Let $\\Om$ be an open set in $\\bC^n$, $n\\geq 2$. Then, $W^*(\\Om)$ is not reflexive.\n\\end{cor}\n\n\\begin{prop} \\label{prop:density} Let $\\Om\\subset \\bC^n$ be a bounded domain. \n\\begin{itemize}\n\\item[(a)] If $n=1$, then $W^*(\\Om) = W^{1,2}(\\Om)$ together with its Sobolev norm. \n\\item[(b)] If $n\\geq 2$, then the continuous functions in $C^0(\\Om) \\cap W^*(\\Om)$ are not dense in $W^*(\\Om)$ with respect to $W^*$-norm. \n\\end{itemize}\n\\end{prop}\n\n\\begin{thm}\\label{thm:star-norm} Let $\\Om$ be a bounded domain in $\\bC^n$.\n\\begin{itemize}\n\\item[(a)] $\\| \\cdot\\|_*$ is a norm in $W^*(\\Om)$ and $\\| \\cdot \\|_{W^{1,2}(\\Om)} \\leq \\|\\cdot \\|_*$.\n\\item[(b)] $W^*(\\Om)$ and $W^*_0(\\Om)$ are Banach spaces with this norm.\n\\end{itemize}\n\\end{thm}", + "full_context": "Let $(X,\\omega)$ be a compact K\\\"ahler manifold. A natural subspace of the Sobolev space $W^{1,2}(X)$, called the complex Sobolev space $W^*$, was introduced by Dinh and Sibony \\cite{DS} in their work on complex dynamics. \nThis new space turned out to be a good complex version of the former one as it takes into account the complex structure of manifolds. Many important families of functions which carry useful properties of underlying manifolds belong to the space. Among them are Lipschitz functions, bounded quasi plurisubharmonic (psh) functions or more general bounded delta quasi-psh function studied in \\cite{CW05}. Basic functional aspects of this space were studied by Vigny \\cite{Vigny} where he showed that it is a Banach space with a corresponding norm $\\| \\cdot\\|_*$, this norm is stronger than the Sobolev norm. However, it is not reflexive and smooth functions are not dense in $W^*$ with respect to the strong topology of that norm. He also defined a functional capacity for this space and showed that it is a capacity in sense of Choquet. Moreover, the functional capacity is qualitative comparable to the relative capacity of Bedford and Taylor in global pluripotential theory defined in \\cite{ko03}.\n\nAn element in a Sobolev space is defined almost everywhere up to a set of Lebesgue measure zero which is unlike psh or quasi-psh functions. Its fine properties with respect to the Sobolev capacity are classical ones \\cite{EG92}, \\cite{KLV-book}. However, this capacity is not good enough to characterize pluripolar sets, therefore it is not suitable for studying functions in $W^*$. One needs to work with the corresponding capacity of $W^*$ which dominates the Sobolev one. However, the new capacity in complex Sobolev spaces is harder to analyze. A crucial difference is that we can no longer rely on maximal function techniques, which are very powerful, in these spaces (see, e.g \\cite{KLV-book}).\n\nRecently, it has been showed in \\cite{DMV}, \\cite{Vigny} and \\cite{VV24} that pluripotential theory is a suitable and powerful tool for studying $W^*$. As a result they made major progresses on study local properties of functions in such spaces.\n\nLet $\\Omega \\subset \\mathbb{C}^n$ be a bounded domain and $\\omega = dd^c |z|^2$ the standard K\\\"ahler form in $\\mathbb{C}^n$. Denote by $W^{1,2}(\\Omega, \\mathbb{R})$ the usual Sobolev space. For $f\\in W^{1,2}(\\Omega, \\mathbb{R})$ we define $\\Ga_f$ to be the set of all positive closed $(1,1)$-current $T$ satisfying\n\\[\\label{eq:test-currents}\\notag\n df \\wedge d^c f \\leq T \\quad\\text{weakly in } \\Omega.\n\\]\nHere we use the normalization \n$$d^c = \\frac{\\sqrt{-1}}{2\\pi} (\\overline{\\d} -\\partial), \\quad dd^c = \\frac{\\sqrt{-1}}{\\pi}\\partial\\overline{\\d}.\n$$\nConsider the subspace\n\\[ \\label{eq:sobolev-space} \\notag\nW^*(\\Omega)= \\left\\{f \\in W^{1,2}(\\Omega,\\mathbb{R}): \\text{there exists $T\\in \\Ga_f$ with } \\|T\\|_\\Omega <+\\infty\n\\right\\},\n\\]\nwhere \n$\n \\|T \\|_\\Omega := \\int_\\Omega T \\wedge \\omega^{n-1}.\n$\nFor $f \\in W^*(\\Omega)$ one defines the $W^*$-norm\n\\[\\label{eq:norm} \\notag\n \\|f\\|_*^2 = \\| f\\|_{L^2(\\Omega)}^2 + \\inf_{T\\in \\Ga_f} \\|T\\|_{\\Omega}.\n\\]\nWe then reprove basic functional results obtained in \\cite{Vigny}. Namely, $(W^*(\\Omega), \\|\\cdot\\|_*)$ is a Banach space (Theorem~\\ref{thm:star-norm}) and continuous functions are not dense in $W^*(\\Omega)$ with respect to strong topology of the norm $\\|\\cdot\\|_*$ (Proposition~\\ref{prop:density}). If $n=1$, then $W^*(\\Omega) = W^{1,2}(\\Omega)$, otherwise $W^*(\\Omega)$ is not reflexive (Corollary~\\ref{cor:reflexive}). It should be pointed out that there are some similarity of these properties with the ones of the space of delta psh functions studied by Cegrell and Wiklund \\cite{CW05}.\n\nInspired by the functional capacity in \\cite{Vigny} we define for a Borel subset $E\\subset \\Omega$ the local capacity ${\\tt c}(E):= {\\tt c}(E,\\Omega)$ as follows:\n\\[\\label{eq:intro-cap-defn}\\notag\n {\\tt c}(E) = \\inf \n \\left\\{ \\| v \\|_*^2 \\;\\rvert\\; v\\in \\cali{K}(E)\n \\right\\},\n\\] \nwhere\n\\[\\label{eq:intro-ke-cap}\\notag\n \\cali{K}(E) = \\left\\{ v \\in W^*(\\Omega) \\;\\rvert\\; \\{v \\leq -1\\}^o \\supset E \\text{ and } v\\leq 0 \\right\\}.\n\\]\nHere $ \\{v \\leq -1\\}^o \\supset E$ means $v\\leq -1$ a.e in a neighborhood of $E$.\n\nThe functional capacity in the Sobolev spaces is very well-understood which is also an effective tool to study fine properties of its element, see e.g. \\cite[Chapters~4.7-4.8]{EG92}. In contrast, the use of ${\\tt c}(\\cdot)$ for the complex Sobolev space is not effective so far. \nOur goal is to investigate systematically local properties of functions in $W^*(\\Omega)$ by using this capacity. This is a different perspective compared to \\cite{Vigny} and \\cite{DMV} who used the relative capacity for psh functions due to Bedford and Taylor \\cite{BT82}. Namely, for a Borel set $E\\subset \\Omega$, \n\\[\\label{eq:intro-BT-cap}\\notag\n cap(E,\\Omega) = \\sup\\left\\{ \\int_E (dd^c u)^n : u\\in PSH(\\Omega), \\;-1 \\leq u \\leq 0\\right\\}.\n\\]\nOur first main result explains the reason why $cap(\\cdot, \\Omega)$ could be effectively used in previous works. Roughly speaking they are equivalent to each other.\n\n\\begin{cor} \\label{cor:reflexive} Let $\\Om$ be an open set in $\\bC^n$, $n\\geq 2$. Then, $W^*(\\Om)$ is not reflexive.\n\\end{cor}\n\n\\begin{prop} \\label{prop:density} Let $\\Om\\subset \\bC^n$ be a bounded domain. \n\\begin{itemize}\n\\item[(a)] If $n=1$, then $W^*(\\Om) = W^{1,2}(\\Om)$ together with its Sobolev norm. \n\\item[(b)] If $n\\geq 2$, then the continuous functions in $C^0(\\Om) \\cap W^*(\\Om)$ are not dense in $W^*(\\Om)$ with respect to $W^*$-norm. \n\\end{itemize}\n\\end{prop}\n\n\\begin{thm}\\label{thm:star-norm} Let $\\Om$ be a bounded domain in $\\bC^n$.\n\\begin{itemize}\n\\item[(a)] $\\| \\cdot\\|_*$ is a norm in $W^*(\\Om)$ and $\\| \\cdot \\|_{W^{1,2}(\\Om)} \\leq \\|\\cdot \\|_*$.\n\\item[(b)] $W^*(\\Om)$ and $W^*_0(\\Om)$ are Banach spaces with this norm.\n\\end{itemize}\n\\end{thm}\n\nThe functional capacity in the Sobolev spaces is very well-understood which is also an effective tool to study fine properties of its element, see e.g. \\cite[Chapters~4.7-4.8]{EG92}. In contrast, the use of $\\tc(\\cdot)$ for the complex Sobolev space is not effective so far. \nOur goal is to investigate systematically local properties of functions in $W^*(\\Om)$ by using this capacity. This is a different perspective compared to \\cite{Vigny} and \\cite{DMV} who used the relative capacity for psh functions due to Bedford and Taylor \\cite{BT82}. Namely, for a Borel set $E\\subset \\Om$, \n\\[\\label{eq:intro-BT-cap}\\notag\n cap(E,\\Om) = \\sup\\left\\{ \\int_E (dd^c u)^n : u\\in PSH(\\Om), \\;-1 \\leq u \\leq 0\\right\\}.\n\\]\nOur first main result explains the reason why $cap(\\cdot, \\Om)$ could be effectively used in previous works. Roughly speaking they are equivalent to each other.\n\nIn the local setting the proofs are different. The key idea in the proof of the first inequality is making use of Cegrell's inequality \\cite{Ce04} and the one for the second inequality comes from the comparison principle. On the other hand, the optimality of the exponents is extracted from the Alexander-Taylor type inequality which is proved in Lemma~\\ref{lem:AT-type}. The latter inequality is of independent interest. One of its applications is the existence of an entire psh subextension for a given $f \\in W^*(B(0,1))$ in Theorem~\\ref{thm:subextension}. This psh subextension result is an important ingredient to derive the property that the set Lebesgue points of such functions are of capacity zero. This result is due to Vigny and Vu \\cite{VV24} where the bounded case was proved earlier in \\cite{Vigny}. We improve the statement of that result and its proof is also simplified in Theorem~\\ref{thm:VV}.\n\n\\begin{thm} \\label{prop:equiv-est-intro} Let $K\\subset\\subset \\Om$ be a compact subset. The following statements hold and they are equivalent to each other.\n\\begin{itemize}\n\\item[(a)] \nThere exist constants $A_1>0$ and $\\al>0$ depending only on $K, \\Om$ such that for every Borel set $E\\subset K$ $$V_{2n}(E) \\leq A_1 e^{-\\al/\\tc (E)}.$$\n\n\\begin{remark} \\label{rmk:c-equiv} \\mbox{}\n\\begin{enumerate}\n\\item[(a)] Let $E \\subset E'$ be Borel sets in $\\Om$. If $V_{2n}(E'\\setminus E)=0$, then $\\tc(E) = \\tc(E')$. It is non-decreasing under holomorphic maps, i.e., if $h: G \\to \\Om$ is a holomorphic map, then\n$$\n \\tc (E,G) \\leq \\tc (h(E), \\Om). \n$$\n\\item[(b)] Let $D\\subset\\subset \\Om' \\subset \\Om$. \nWe do not know if there exists a constant $A>0$ depending only on $\\Om$ and $\\Om'$ such that for every Borel set $ E\\subset D$, \n\\[\\label{eq:c-comparablity}\n \\tc(E,\\Om) \\leq A\\; \\tc(E, \\Om').\n\\]\nHowever, we will have a weaker version with some exponent on the right hand side later \nin Corollary~\\ref{cor:dom-equiv-c} for the new capacity $\\tc (\\cdot, \\Om)$.\n\\end{enumerate}\n\\end{remark}\n\n\\begin{lem}[dominated by Bedford-Taylor capacity] \\label{lem:c-cap} Assume $\\Om$ is strictly pseudoconvex. There exists a constant $A>0$ such that for every Borel set $E\\subset\\subset \\Om$,\n$$\n \\tc(E) \\leq A \\left[cap(E,\\Om)\\right]^\\frac{1}{n}.\n$$\n\\end{lem}\n\n\\begin{proof} Let $u_E^*$ denote the relative extremal for $E$ in \\eqref{eq:rel-ext-fct}.\nSince both capacities are outer regular (Lemma~\\ref{lem:outer-reg} and \\cite{BT82}), we may assume that $E=G$ is an open set and hence $u_G = u_G^*$. Then, $-1\\leq u_G \\leq 0$ and $u_G =-1$ on $G$. It follows that\n\\[\\label{eq:BT-dom-a}\n \\tc(G) \\leq \\|u_G\\|_*^2.\n\\]\nLet $\\rho$ be a strictly psh defining function for $\\Om$. We may assume that $dd^c \\rho \\geq \\om$. Since $G\\subset \\subset \\Om$, it implies that $u_G =0$ on $\\d\\Om$ (see, e.g., \\cite[Properties (13.11)]{De89}). Using $-1 \\leq u_G \\leq 0$ and integration by parts,\n$$\n \\|u_G\\|_{L^2}^2 = \\int (u_G)^2 (dd^c \\rho)^n \\leq \\int - u_G (dd^c\\rho)^n = \\int -\\rho dd^c u_G \\wed (dd^c\\rho)^{n-1}.\n$$\nHere and in the proof below we skip writing $\\Om$ in the integral symbols \nfor simplicity. The last integral is less than \n$$\n \\|\\rho\\|_{L^\\infty(\\Om)} \\int dd^c u_G \\wed (dd^c\\rho)^{n-1}.\n$$\nSince $0\\leq u_G+1 \\in PSH(\\Om)$, we have\n$d u_G \\wed d^c u_G \\leq \\frac{1}{2} dd^c (1+u_G)^2.$\nHence, \n\\[\\label{eq:BT-dom-b}\\begin{aligned}\n \\|u_G\\|_*^2 \n&\\leq \\int (u_G)^2 (dd^c \\rho)^n + \\frac{1}{2}\\int dd^c (1+u_G)^2 \\wed (dd^c \\rho)^{n-1} \\\\\n&\\leq A_1 \\int dd^c u_G \\wed (dd^c\\rho)^{n-1} + \\int dd^c u_G \\wed (dd^c \\rho)^{n-1},\n\\end{aligned}\\]\nwhere $A_1 = \\|\\rho\\|_{L^\\infty(\\Om)}$ and we used the fact that \n$$\\begin{aligned}\n& \\frac{1}{2}\\int dd^c (1+u_G)^2 \\wed (dd^c \\rho)^{n-1} \\\\\n&=\\int du_G \\wed d^c u_G \\wed (dd^c\\rho)^{n-1} + \\int (1+u_G) dd^c u_G \\wed (dd^c \\rho)^{n-1} \\\\\n &= \\int dd^c u_G \\wed (dd^c \\rho)^{n-1}.\n\\end{aligned}\n$$\nInvoking an inequality of Cegrell \\cite{Ce04} we have\n\\[\\label{eq:BT-dom-c}\\begin{aligned}\n \\int dd^c u_G \\wed (dd^c \\rho)^{n-1}& \\leq \\left[\\int (dd^c\\rho)^n\\right]^\\frac{n-1}{n} \\left[\\int (dd^c u_G)^n\\right]^\\frac{1}{n} \\\\ &= A_0 [cap(G,\\Om)]^\\frac{1}{n},\n\\end{aligned}\\]\nwhere $A_0$ is a uniform constant as $\\rho$ is smooth on $\\bar\\Om$ and we used the formula \\eqref{eq:BT-cap-id} for the last identity.\n\nBy \\cite[Th\\'eor\\`em~1]{Okada82} we have a Poincar\\'e type inequality, for $f\\in C^\\infty_c(\\Om)$,\n$$\n \\int f^2 d\\mu = \\int f^2 \\te \\wed dd^c |z|^2 \\leq 8 \\sup_\\Om |z| \\int df \\wed d^c f \\wed \\te.\n$$\nIt follows that $\\cE_1^\\te(f) = \\| f \\|^2_{L^2(d\\mu)} + \\cE^\\te(f)$ is equivalent to $\\cE^\\te(f)$. As noted in \\cite[page 211]{FO84} that potential theory of \\cite[Chapter~2]{FOT94} can be formulated in terms of $\\cE^\\te$ instead of $\\cE^\\te_1$. For an open subset $E\\subset \\Om$ its capacity is given by\n\\[\\label{eq:cap-te}\n \\tc_\\te (E) = \\inf\\{\\cE(v) : v\\in \\cF, \\; v\\geq 1 \\quad \\mu-\\text{\\rm a.e.} \\text{ on } E\\}.\n\\]\nFor a general Borel set we let\n$$\n \\tc_\\te(E) = \\inf \\{\\tc_\\te(G): E \\subset G, \\quad G \\text{ is open subset in }\\Om\\}.\n$$\nIn particular, for a compact subset $K\\subset \\Om$,\n$$\n \\tc_\\te(K) = \\inf\\{ \\cE^\\te(v): v\\in \\cF\\cap C^0(\\Om), \\; v\\geq 1 \\text{ on }K\\}.\n$$\nThank to special feature of the functional capacity in a Hilbert space it is a Choquet capcity \\cite[Theorem~2.1.1]{FOT94}. \nMoreover, we have a comparison between this capacity and the Bedford-Taylor capacity as follows.\n\\begin{lem}\\label{lem:FO-polar} Let $\\Om \\subset \\subset \\bC^n$ be a strictly pseudoconvex domain and $0<\\de \\leq 1$.\n\\begin{itemize}\n\\item[(a)] For every compact subset $K\\subset \\Om$, \n$$\n cap (K) \\leq 8 \\tc_\\te (K),\n$$\nwhere $\\te= (dd^c u_K^* +\\de \\om)^{n-1}$.\n\\item[(b)] Let $D\\subset \\subset \\Om$ be a subdomain. There exists a constant $A_0= A(D,\\Om)$ such that for every open subset $G\\subset D$,\n$$\n \\tc_\\te{(G)} \\leq A_0 \\left[cap (G)\\right]^\\frac{1}{n}.\n$$ \n\\end{itemize}\n\\end{lem}\n\n\\begin{remark}\\label{rmk:sharp-DMV} The inequalities in Proposition~\\ref{prop:equiv-est} are sharp as far as the exponents are concerned. In fact, it follows from the first inequality (a) and Lemma~\\ref{lem:c-cap} that for a Borel set $E\\subset \\Om$, \n$$\n V_{2n} (E) \\leq A_1 e^{-\\al/ \\tc (E)} \\leq A_1 e^{-\\al/ [cap(E, \\Om)]^\\frac{1}{n}}.\n$$\nWe know from \\cite[Theorem~A]{ACKPZ09} that the exponent $1/n$ in the inequality between the volume and Bedford-Taylor capacity is sharp. So are the exponents in the inequalities of the proposition. Consequently, the exponent 2 in Proposition~\\ref{prop:DMV} and \\cite[Theorem~1.2]{DMV} is the optimal one for all $n\\geq 1$.\n\\end{remark}", + "post_theorem_intro_text_len": 7916, + "post_theorem_intro_text": "The statements of the theorem consist of Lemmas~\\ref{lem:c-cap}, ~\\ref{lem:cap-c} and Remarks~\\ref{rmk:inner-reg},~\\ref{rmk:sharp-exp}. \nOn compact K\\\"ahler manifolds both inequalities was obtained in \\cite[Proposition~5.1]{DKN} where ${\\tt c}(E)$ is defined by another equivalent norm with $W^*$-norm, but the optimality was unknown. A qualitative version of the first inequality on such manifolds was obtained early in \\cite{Vigny} and the second inequality was also proved there without showing that the exponents are sharp.\n\nIn the local setting the proofs are different. The key idea in the proof of the first inequality is making use of Cegrell's inequality \\cite{Ce04} and the one for the second inequality comes from the comparison principle. On the other hand, the optimality of the exponents is extracted from the Alexander-Taylor type inequality which is proved in Lemma~\\ref{lem:AT-type}. The latter inequality is of independent interest. One of its applications is the existence of an entire psh subextension for a given $f \\in W^*(B(0,1))$ in Theorem~\\ref{thm:subextension}. This psh subextension result is an important ingredient to derive the property that the set Lebesgue points of such functions are of capacity zero. This result is due to Vigny and Vu \\cite{VV24} where the bounded case was proved earlier in \\cite{Vigny}. We improve the statement of that result and its proof is also simplified in Theorem~\\ref{thm:VV}.\n\nAn immediate consequence of Theorem~\\ref{thm:intro-cap-comp} and \\cite{BT82} is that a Borel set $P\\subset \\Omega$ is pluripolar if and only if ${\\tt c}(P)=0$. Provided the equivalence of capacities the results in \\cite{DMV} can be restated in terms of ${\\tt c} (\\cdot)$ where they are obtained previously for $cap(\\cdot, \\Omega)$. However, as emphasized above we would take another route by using advantages of the functional capacity ${\\tt c}(\\cdot)$ (see e.g., Lemma~\\ref{lem:outer-reg}, Proposition~\\ref{prop:ae-qe}, Lemma~\\ref{lem:cap-sublevel-set}) and hence providing alternative proofs for the ones obtained in \\cite{DMV}. In particular, using the ideas in \\cite{DMV} and \\cite{Vigny}, we simplify the proof of quasi-continuous representative result proved earlier in \\cite[Theorem~2.10]{DMV} (see also \\cite[Theorem~22]{Vigny}).\n\n\\begin{cor} \\label{cor:intro-quasi-mod} Let $f\\in W^*(\\Omega)$. There exists a quasi-continuous modification $\\wt f$ such that $\\wt f = f$ a.e. Moreover, if $g$ is another quasi-continuous modification of $f$, then $g =\\wt f$ outside a pluripolar set.\n\\end{cor}\n\nThe most technical part of the proof lies in delicate integral estimates. Along the way we refine the ones obtained in \\cite{DMV}. It turns out in Theorem~\\ref{thm:VV} that the precise representative, in the sense of Sobolev spaces \\cite[Chapter~4]{EG92},\n\\[\\label{eq:rep} f^*(x) \\equiv\n\\begin{cases}\n\\lim_{r\\to 0} \\intavg_{B(x,r)} f \\; d y &\\quad \\text{if this limit exits,} \\\\\n=0 &\\quad \\text{otherwise},\n\\end{cases}\t\n\\] \nis a quasi-continuous modification, where $d y$ is the Lebesgue measure on $\\mathbb{C}^n = \\mathbb{R}^{2n}$.\n\nThanks to the quasi-continuity result \n we can define the integration of a function $g\\in W^*(\\Omega)$ against Monge-Amp\\`ere measures associated with a bounded psh function (Definition~\\ref{defn:L2-ma}). This result is pushed much further in Section~9 where we show that $W^*(\\Omega)$ is naturally embedded into a Dirichlet space associated with a closed positive $(n-1,n-1)$ current $\n\\theta = [dd^c (\\phi + \\delta |z|^2)]^{n-1}$ where $\\phi \\in PSH \\cap L^\\infty(\\Omega)$ and $\\delta>0$ are given.\nTherefore, the integral $\\int_K df \\wedge d^c g\\wedge \\theta$ for $f,g\\in W^*(\\Omega)$ is well-defined on every compact set $K\\subset \\Omega$. This aspect has been considered recently by Do, Nguyen and Vu \\cite{DNV25}.\n\nThe second main result is as follows. \n\\begin{thm}\\label{thm:intro-choquet-c} Let $\\Omega$ be a bounded open set in $\\mathbb{C}^n$. The set function on Borel subsets $E\\mapsto {\\tt c} (E)= {\\tt c} (E,\\Omega)$ is a Choquet capacity.\n\\end{thm}\n\nThis result is the local analogue of \\cite{Vigny}. On compact K\\\"ahler manifolds the proof of \\cite[Theorem~30]{Vigny} employed the capacity notion in the Dirichlet spaces \\cite[Lemma~23]{Vigny} and the characterization of pluripolar sets via a family of capacity associated with closed positive current \\cite{FO84}. Here we use a similar strategy. Notably, we are able to give a simpler proof of the characterization in Corollary~\\ref{cor:FO-polar} by using only pluripotential theory. \n\nAnother remarkable property of functions in the complex Sobolev space is the uniformly exponential integrability due to Dinh, Marinescu and Vu \\cite{DMV}, which is very close to the one of psh functions. Namely, \nthere exist positive constants $\\alpha$ and $A$ such that for every $f \\in W^*(B(0,1))$ with $\\|f\\|_*\\leq 1$, \n\\[\\label{eq:DMV-intro}\n\t\\int_{\\bar B(0,\\frac{1}{8})} e^{\\alpha |f|^2} d x \\leq A.\n\\]\nThis result should be compared with the well-known exponential integrability of psh functions due to H\\\"ormander \\cite[Proposition~4.2.9]{Ho07} and Skoda \\cite{Sk72}. In the proof of \\cite[Theorem~1.2]{DMV} the authors used the induction argument in dimension and slicing theory for positive currents. We provide a new and simpler one in Proposition~\\ref{prop:DMV} by reducing the inequality to one dimensional case. In this way, it is enough to work with smooth functions after taking the standard convolution with smooth kernels. The constants can be explicitly computed though they are suboptimal. Additionally, we point out in Remark~\\ref{rmk:sharp-DMV} that the exponent 2 of $|f|$ in \\eqref{eq:DMV-intro} is the optimal for all $n\\geq 1$, which was only known for $n=1$ in \\cite{DMV}. \n\nThe third main result fully characterizes the inequality \\eqref{eq:DMV-intro} by the volume-capacity inequality of sublevel sets.\n\n\\begin{thm} \\label{prop:equiv-est-intro} Let $K\\subset\\subset \\Omega$ be a compact subset. The following statements hold and they are equivalent to each other.\n\\begin{itemize}\n\\item[(a)] \nThere exist constants $A_1>0$ and $\\alpha>0$ depending only on $K, \\Omega$ such that for every Borel set $E\\subset K$ $$V_{2n}(E) \\leq A_1 e^{-\\alpha/{\\tt c} (E)}.$$\n\n\\item[(b)] There exist uniform constants $A_1>0$ and $\\alpha>0$ depending only on $K, \\Omega$ such that for every $f\\in W^*(\\Omega)$ and $f\\leq 0$,\n $$V_{2n}(\\{f< -1\\} \\cap K) \\leq A_1 e^{\\frac{-\\alpha}{\\|f\\|_*^{2}}}.$$\n\\item\n[(c)] There exist uniform constants $A_1>0$ and $\\alpha>0$ depending only on $K, \\Omega$ such that for every $f\\in W^*(\\Omega)$ and $f\\leq 0$ whose norm $\\|f\\|_* \\leq \n\\kappa$,\n$$V_{2n} (\\{f < -s\\} \\cap K) \\leq A_1 e^{-\\alpha s^2/\\kappa^2}, \\quad \\forall s>0.$$\n\n\\item \n[(d)] There exist uniform constants $A_1>0$ and $\\alpha>0$ depending only on $K, \\Omega$ such that for every $f\\in W^*(\\Omega)$ and $f\\leq 0$ with $\\|f\\|_* \\leq 1$,\n$$V_{2n} (\\{f < -s\\} \\cap K) \\leq A_1 e^{-\\alpha s^2}, \\quad \\forall s>0.$$\n\\end{itemize}\n\\end{thm}\n\nThe characterization is partially inspired by the work of \\AA hag, Cegrell, Ko\\l odziej, Pham and Zeriahi \\cite{ACKPZ09}. Lastly, we point out in Remark~\\ref{rmk:DMV-holder} that the inequalities holds for a very large family of Monge-Amp\\`ere measures associated with H\\\"older continuous psh functions.\n\n\\bigskip\n\n{\\em Acknowledgement.} I would like to thank G. Marinescu and D.-V. Vu for a kind invitation to visit University of Cologne in September 2025 and to whom I had many helpful discussions on the results of \\cite{DMV}. I am grateful to S. Ko\\l odziej and D.-V. Vu for useful comments on the draft of the paper. It was written while the author visited the Center for Complex Geometry (Daejeon). He would like to thank Jun-Muk Hwang and Yongnam Lee for their kind support and exceptional hospitality. He is also grateful to the institution for providing perfect working conditions.", + "sketch": "For Theorem~\\ref{thm:intro-cap-comp}, the text says that its statements are provided by Lemmas~\\ref{lem:c-cap}, \\ref{lem:cap-c} and Remarks~\\ref{rmk:inner-reg}, \\ref{rmk:sharp-exp}. In the local setting, “the key idea in the proof of the first inequality is making use of Cegrell's inequality \\cite{Ce04}” while “the one for the second inequality comes from the comparison principle.” The “optimality of the exponents is extracted from the Alexander-Taylor type inequality which is proved in Lemma~\\ref{lem:AT-type}.”", + "expanded_sketch": "To prove the main theorem, the text says that its statements are provided by the following lemmas and remarks.\n\n\\begin{lem}[dominated by Bedford-Taylor capacity] \\label{lem:c-cap} Assume $\\Om$ is strictly pseudoconvex. There exists a constant $A>0$ such that for every Borel set $E\\subset\\subset \\Om$,\n$$\n\t\\tc(E) \\leq A \\left[cap(E,\\Om)\\right]^\\frac{1}{n}.\n$$\n\\end{lem}\n\n\\begin{lem} \\label{lem:cap-c} Let $D \\subset \\subset \\Om$. There exists a constant $A'$ depending only on $D$ and $\\Om$ such that for every Borel set $E \\subset D$,\n$$\tcap(E,\\Om) \\leq A'\\, \\tc(E)\n$$\n\\end{lem}\n\n\\begin{remark}\\label{rmk:inner-reg} The relative compactness condition of $E$ will be removed later after proving that $\\tc(\\cdot,\\Om)$ is a capacity in the sense of Choquet (Theorem~\\ref{thm:choquet-c}).\n\\end{remark}\n\n\\begin{remark}[Proof of the last statement in Theorem~\\ref{thm:intro-cap-comp}]\n\\label{rmk:sharp-exp} The Alexander-Taylor inequality \\cite{AT84} reads\n$$\n\t\\exp (-A_r/cap(K,\\Om)) \\leq T_R(K) \\leq \\exp(-2\\pi/[cap(K,\\Om)]^\\frac{1}{n}),\n$$\nwhere the exponents in both inequalities are sharp (\\cite[Remark~2]{AT84}).\nTogether with $cap(K) \\leq A_r \\tc(K)$ in Lemma~\\ref{lem:cap-c}\nand the first inequality in Lemma~\\ref{lem:AT-type} we derive \n$$\n\t\\exp (-A/cap(K,\\Om)) \\leq \\exp (-A/\\tc(K,\\Om)) \\leq \\exp(-2\\pi/[cap(K,\\Om)]^\\frac{1}{n})\n$$\nwith the sharp exponents. So are the ones in Lemma~\\ref{lem:c-cap} and Lemma~\\ref{lem:cap-c}.\n\\end{remark}\n\nIn the local setting, “the key idea in the proof of the first inequality is making use of Cegrell's inequality \\cite{Ce04}” while “the one for the second inequality comes from the comparison principle.” The “optimality of the exponents is extracted from the Alexander-Taylor type inequality which is proved in the following lemma.”\n\n\\begin{lem} \\label{lem:AT-type} There exist positive constants $A_r$ and $A_R$ depending on $r,R$ respectively such that for every compact set $K\\subset B_r$,\n$$ \\exp(- A_R \\; \\tc (K,B_R)^{-1}) \\leq T_R(K) \\leq \t\\exp(- A_r\\;\\tc (K, B_R)^{-1/n}).\n$$ \nEquivalently,\n\\[\\label{eq:AT}\n\t\\frac{1}{A_r} \\frac{1}{M_K^n} \\leq \\tc (K,B_R) \\leq \\frac{A_R }{M_K}.\n\\]\n\\end{lem}", + "expanded_theorem": "\\label{thm:intro-cap-comp} \nLet $\\Omega \\subset \\subset \\mathbb{C}^n$ be a strictly pseudoconvex domain. \n\\begin{itemize}\n\\item[(a)] There exists a constant $A>0$ such that for every Borel set $E\\subset \\Omega$,\n$$\n\t{\\tt c}(E) \\leq A [cap(E,\\Omega)]^\\frac{1}{n}.\n$$\n\\item[(b)]\nAssume $D \\subset \\subset \\Omega$ be a subdomain. There exists a constant $A'$ such that for every Borel set $E\\subset D$,\n$$\n\t\\frac{1}{A'} cap(E,\\Omega) \\leq {\\tt c} (E).\n$$\n\\end{itemize}\nMoreover, the above inequalities are sharp as far as the exponents are concerned.,", + "theorem_type": [ + "Existential–Universal", + "Inequality or Bound" + ], + "mcq": { + "question": "Let \\(\\Omega\\Subset \\mathbb C^n\\) be a strictly pseudoconvex domain. For a Borel set \\(E\\subset \\Omega\\), define the Bedford--Taylor capacity by\n\\[\n\\operatorname{cap}(E,\\Omega)=\\sup\\left\\{\\int_E (dd^c u)^n:\\ u\\in PSH(\\Omega),\\ -1\\le u\\le 0\\right\\}.\n\\]\nAlso define the local functional capacity\n\\[\n\\mathbf c(E)=\\inf\\{\\|v\\|_*^2:\\ v\\in \\mathcal K(E)\\},\n\\]\nwhere\n\\[\n\\mathcal K(E)=\\left\\{v\\in W^*(\\Omega): v\\le 0\\text{ on }\\Omega,\\ \\{v\\le -1\\}^o\\supset E\\right\\},\n\\]\nwith \\(\\{v\\le -1\\}^o\\supset E\\) meaning that \\(v\\le -1\\) almost everywhere in a neighborhood of \\(E\\), and where\n\\[\nW^*(\\Omega)=\\left\\{f\\in W^{1,2}(\\Omega,\\mathbb R): \\exists\\text{ a positive closed }(1,1)\\text{-current }T\\text{ with }df\\wedge d^cf\\le T\\text{ weakly and }\\|T\\|_\\Omega<\\infty\\right\\},\n\\]\n\\[\n\\|f\\|_*^2=\\|f\\|_{L^2(\\Omega)}^2+\\inf_{T\\in \\Gamma_f}\\|T\\|_\\Omega,\n\\qquad \\|T\\|_\\Omega=\\int_\\Omega T\\wedge \\omega^{n-1}.\n\\]\nUnder these assumptions, which quantitative comparison between \\(\\mathbf c(E)\\) and \\(\\operatorname{cap}(E,\\Omega)\\) holds?", + "correct_choice": { + "label": "A", + "text": "There exists a constant \\(A>0\\) such that for every Borel set \\(E\\subset \\Omega\\),\n\\[\n\\mathbf c(E)\\le A\\,[\\operatorname{cap}(E,\\Omega)]^{1/n}.\n\\]\nMoreover, if \\(D\\Subset \\Omega\\) is any subdomain, then there exists a constant \\(A'>0\\) such that for every Borel set \\(E\\subset D\\),\n\\[\n\\frac{1}{A'}\\operatorname{cap}(E,\\Omega)\\le \\mathbf c(E).\n\\]\nIn addition, these inequalities are sharp with respect to the exponents." + }, + "choices": [ + { + "label": "B", + "text": "There exists a constant \\(A>0\\) such that for every Borel set \\(E\\subset \\Omega\\),\n\\[\n\\mathbf c(E)\\le A\\,\\operatorname{cap}(E,\\Omega).\n\\]\nMoreover, if \\(D\\Subset \\Omega\\) is any subdomain, then there exists a constant \\(A'>0\\) such that for every Borel set \\(E\\subset D\\),\n\\[\n\\frac{1}{A'}\\,[\\operatorname{cap}(E,\\Omega)]^{1/n}\\le \\mathbf c(E).\n\\]\nIn addition, these inequalities are sharp with respect to the exponents." + }, + { + "label": "C", + "text": "There exists a constant \\(A>0\\) such that for every Borel set \\(E\\subset \\Omega\\),\n\\[\n\\mathbf c(E)\\le A\\,[\\operatorname{cap}(E,\\Omega)]^{1/n}.\n\\]\nMoreover, if \\(D\\Subset \\Omega\\) is any subdomain, then there exists a constant \\(A'>0\\) such that for every Borel set \\(E\\subset D\\),\n\\[\n\\frac{1}{A'}\\operatorname{cap}(E,\\Omega)\\le \\mathbf c(E).\n\\]" + }, + { + "label": "D", + "text": "There exists a constant \\(A>0\\), depending only on \\(\\Omega\\), such that for every subdomain \\(D\\Subset \\Omega\\) and every Borel set \\(E\\subset D\\),\n\\[\n\\mathbf c(E)\\le A\\,[\\operatorname{cap}(E,\\Omega)]^{1/n}\n\\quad\\text{and}\\quad\n\\frac{1}{A}\\operatorname{cap}(E,\\Omega)\\le \\mathbf c(E).\n\\]\nIn particular, the same constant works uniformly for all choices of \\(D\\Subset \\Omega\\), and the exponents are sharp." + }, + { + "label": "E", + "text": "There exists a constant \\(A>0\\) such that for every Borel set \\(E\\subset \\Omega\\),\n\\[\n\\mathbf c(E)\\le A\\,[\\operatorname{cap}(E,\\Omega)]^{1/n}.\n\\]\nMoreover, there exists a constant \\(A'>0\\) such that for every Borel set \\(E\\subset \\Omega\\),\n\\[\n\\frac{1}{A'}\\operatorname{cap}(E,\\Omega)\\le \\mathbf c(E).\n\\]\nIn addition, these inequalities are sharp with respect to the exponents." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "sharp exponents 1/n versus 1", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped the sharpness conclusion", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "dependence of lower-bound constant on the subdomain D", + "template_used": "uniformity_effectivity" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "restriction of the reverse inequality to E\\subset D\\Subset\\Omega", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives only the definitions of the two capacities and asks for the valid comparison statement; it does not explicitly state the correct inequalities, exponents, or quantifier structure." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-identification question: it asks for the exact comparison result rather than an application or derivation. However, it is not a pure restatement because the options differ in exponents, constant dependence, and sharpness claims." + }, + "GPS": { + "score": 1, + "justification": "Solving it requires moderate reasoning or precise theorem recall to distinguish subtle alternatives, especially the exponent swap, the need to restrict the lower bound to subdomains, and the sharpness clause. It does not strongly force original generative reasoning beyond discriminating among variants." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically meaningful: they target common failure modes such as swapped exponents, omission of one inequality, incorrect uniformity in constants, and replacing comparison by exact equality." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward precise recall/recognition than deep generative reasoning." + } + }, + { + "id": "2602.14213v1", + "paper_link": "http://arxiv.org/abs/2602.14213v1", + "theorems_cnt": 5, + "theorem": { + "env_name": "theorem", + "content": "\\label{main}\n\tLet $G$ be a controllable graph and $p$ be an odd prime such that $\\rank_p W(G)=n-1$. Then $v_p(L(G))\\le \\frac{1}{2} v_p(\\det W(G))$.", + "start_pos": 7243, + "end_pos": 7420, + "label": "main" + }, + "ref_dict": { + "pf": "\\label{pf}\nWe begin with the fundamental matrix characterization of generalized cospectrality for graphs.\n\n\\begin{lemma}[\\cite{johnson,wang2006}]\n\tLet $G$ and $H$ be two graphs of the same order. Then", + "evencase": "\\begin{proposition}[\\cite{wang2017JCTB}]\\label{evencase}\n\tIf $2^{-\\lfloor\\frac{n}{2}\\rfloor}$ is odd then $L(G)$ is odd.\n\\end{proposition}", + "pre": "\\begin{theorem}\\label{main}\n\tLet $G$ be a controllable graph and $p$ be an odd prime such that $\\rank_p W(G)=n-1$. Then $v_p(L(G))\\le \\frac{1}{2} v_p(\\det W(G))$.\n\\end{theorem}\n\tThe main innovation of this paper lies in the shift of the underlying algebraic structure from the vector space over the field $\\mathbb{Z}/p\\mathbb{Z}$ (as used in \\cite{wang2013EJC,qiu2023}) to the module over the local ring $\\mathbb{Z}/p^k\\mathbb{Z}$. While working over a ring introduces the complexity of zero divisors, many basic results from linear algebra over fields still hold for the ring $\\mathbb{Z}/p^k\\mathbb{Z}$. In particular, it possesses the ``Basis Extension Property'' (Steinitz Property), which allows us to generalize arguments from linear algebra. \n\n\tThe rest of this paper is organized as follows. In Sec.~\\ref{pre}, we recall some preliminaries on Smith normal forms and modules over $\\mathbb{Z}/p^k \\mathbb{Z}$. The proof of Theorem \\ref{main} is given in Sec.~\\ref{pf}, which can be seen as a module-theoretical version of the original argument of Wang \\cite{wang2006}, combined with some improvements and simplifications developed in \\cite{wang2020}. Some direct applications of Theorem \\ref{main} are presented in Sec.~\\ref{dis}. The paper concludes with a conjecture proposing a further possible improvement to Theorem \\ref{main}.\n\\section{Preliminaries}\n\\label{pre}\nLet $R$ be a commutative principal ideal ring with identity. It is well known \\cite{stanley2016JCTA} that any matrix $M$ over $R$ has a Smith normal form (SNF); that is, there exist invertible matrices $U$ and $V$ over $R$ such that \n\\begin{equation*}\n\tUMV=\\begin{bmatrix}\n\t\t\\diag(d_1,\\ldots,d_r)&0\\\\\n\t\t0&0\n\t\\end{bmatrix},\n\\end{equation*}\nwhere the elements $d_i$ ($i=1,\\ldots,r$) are nonzero and satisfy $d_1\\mid d_2\\mid \\cdots\\mid d_r$. The nonzero elements $d_1,\\ldots,d_r$ are called the \\emph{invariant factors} of $M$.\n\nThroughout this paper, we fix an odd prime $p$ and consider three kinds of rings: $\\mathbb{Z}$, $\\mathbb{Z}/p\\mathbb{Z}$ and $\\mathbb{Z}/p^k\\mathbb{Z}$, where $k\\ge 2$. Let $\\mathbb{Z} \\to \\mathbb{Z}/p^k \\mathbb{Z} \\to \\mathbb{Z}/p\\mathbb{Z}$ be the natural projections. For a matrix $M$ over $\\mathbb{Z}$, the SNFs of the projections of $M$ are determined naturally by the SNF of $M$ over $\\mathbb{Z}$ via $p$-adic valuations. For example, suppose $M$ has the Smith normal form $\\diag(2,10,30,270)$ over $\\mathbb{Z}$. Then, over the field $\\mathbb{Z}/3\\mathbb{Z}$, the SNF of $M$ is $\\diag(1,1,0,0)$, whereas over the local ring $\\mathbb{Z}/3^2\\mathbb{Z}$, its SNF is $\\diag(1,1,3,0)$. \n\nIn this paper, we are mainly concerned with the SNFs of matrices over the local ring $\\mathbb{Z}/p^k \\mathbb{Z}$. Note that the invariant factors are unique only up to multiplication by units. To ensure uniqueness, we conventionally assume that each nonzero invariant factor takes the form $p^{c}$ for some integer $0 \\le c < k$. We formalize this in the following lemma.\n\n\\begin{lemma}\n\tFor any matrix $M$ over $\\mathbb{Z}/p^k\\mathbb{Z}$, there exist invertible matrices $U$ and $V$ over $\\mathbb{Z}/p^k\\mathbb{Z}$ such that \n\t\\begin{equation*}\n\t\tUMV = \\begin{bmatrix}\n\t\t\t\\diag(p^{c_1}, \\ldots, p^{c_r})&0\\\\\n\t\t\t0&0\n\t\t\\end{bmatrix},\n\t\\end{equation*}\n\twhere $0 \\le c_1 \\le \\cdots \\le c_r < k$.\n\\end{lemma}\nA classic result in linear algebra states that the linear system $Mx=b$ over a field has a solution if and only if $\\rank M = \\rank (M,b)$. The corresponding question over an arbitrary ring is generally more complicated, but for our purposes, the existence of the SNF gives a straightforward extension.\n\n\\begin{proposition}\\label{ssnf}\n\tThe linear system $Mx=b$ over $\\mathbb{Z}/p^k\\mathbb{Z}$ has a solution if and only if $M$ and $(M,b)$ have the same invariant factors.\n\\end{proposition} \n\n\\begin{proof}\n\tThe ``only if'' part is clear. Let $\\mathcal{C}(M)$ and $\\mathcal{C}(M,b)$ be the modules generated by the columns of $M$ and $(M,b)$, respectively. Since $M$ and $(M,b)$ have the same invariant factors, the two modules $\\mathcal{C}(M)$ and $\\mathcal{C}(M,b)$ are isomorphic. As $\\mathcal{C}(M,b)$ is finite, it cannot be isomorphic to any proper submodule. Since $\\mathcal{C}(M)$ is a submodule of $\\mathcal{C}(M,b)$, the isomorphism $\\mathcal{C}(M)\\cong \\mathcal{C}(M,b)$ implies that they must be equal. Thus, $b\\in\\mathcal{C}(M)$, i.e., $Mx=b$ has a solution. \n\\end{proof}\nThe SNF is particularly useful for determining the structure of the solution space (kernel) of a linear system. It is important to note that the kernel is not always a free module. However, under specific conditions on the invariant factors, it is.\n\n\\begin{proposition}\\label{kf}\n\tLet $M$ be an $m \\times n$ matrix over the ring $\\mathbb{Z}/p^k\\mathbb{Z}$. Let the invariant factors of $M$ be $p^{c_1}, \\ldots, p^{c_r}$ satisfying $0 \\le c_1 \\le \\dots \\le c_r < k$. Then the kernel of $M$ is given, up to isomorphism, by:\n\t\\begin{equation*}\n\t\t\\ker(M) \\cong \\bigoplus_{i=1}^r \\left( \\mathbb{Z}/p^{c_i}\\mathbb{Z} \\right) \\oplus \\left( \\mathbb{Z}/p^k \\mathbb{Z} \\right)^{n-r}.\n\t\\end{equation*}\n\tIn particular, if $c_1=\\cdots=c_r=0$, then $\\ker(M)\\cong (\\mathbb{Z}/p^k\\mathbb{Z})^{n-r}$, which is a free module of rank $n-r$.\n\\end{proposition}\n\nThe following proposition is a direct generalization of \\cite[Lemma 7]{wang2013EJC}, where the case $m=n,k=2$ was considered. The original proof in \\cite{wang2013EJC} remains valid for the general setting and is therefore omitted here.\n\n\\begin{proposition}[\\cite{wang2013EJC}]\\label{dnk}\n\tLet $M$ be an $m\\times n~(m\\ge n)$ integral matrix whose SNF is \n\t\\begin{equation*}\n\t\t\\begin{bmatrix}\n\t\t\\diag(d_1,\\ldots,d_n)\\\\\n\t\t0_{(m-n)\\times n}\n\t\t\\end{bmatrix}.\n\t\\end{equation*} Then the equation $Mz\\equiv 0\\pmod{p^{k}}$ has a solution $z\\not\\equiv 0\\pmod{p}$ if and only if $p^{k}\\mid d_n.$\n\\end{proposition}\n\nA set of vectors $v_1, \\ldots, v_m \\in (\\mathbb{Z}/p^k\\mathbb{Z})^n$ is called \\emph{linearly independent} if $\\sum_{i=1}^m c_i v_i = 0$ implies $c_i = 0$ for all $i$. It is known that $v_1,\\ldots, v_m$ are linearly independent if and only if their projections in the vector space $(\\mathbb{Z}/p\\mathbb{Z})^n$ are linearly independent (over the field $\\mathbb{Z}/p\\mathbb{Z}$). A key property of the ring $\\mathbb{Z}/p^k\\mathbb{Z}$ is the so-called Basis Extension Property (or Steinitz Property \\cite{mcdonald}), which states that any linearly independent vectors in a free $\\mathbb{Z}/p^k\\mathbb{Z}$-module $M$ can always be extended to a basis of $M$. Since we only require the finite-rank case, we formalize it in the following proposition.\n\n\\begin{proposition}\\label{bas}\n\tLet $M$ be a free submodule of rank $m$ in $(\\mathbb{Z}/p^k\\mathbb{Z})^n$. Let $v_1, \\ldots, v_k \\in M$ ($k\\le m$) be linearly independent. Then:\n\n\t\\textup{(i)} If $k=m$, the set $\\{v_1, \\ldots, v_k\\}$ constitutes a basis of $M$.\n\n\t\\textup{(ii)} If $k< m$, there exist $m-k$ vectors $v_{k+1}, \\ldots, v_{m} \\in M$ such that $\\{v_1,\\ldots,v_m\\}$ constitutes a basis of $M$.\n\\end{proposition}\n\\section{Proof of Theorem \\ref{main}}\\label{pf}\nWe begin with the fundamental matrix characterization of generalized cospectrality for graphs.\n\n\\begin{lemma}[\\cite{johnson,wang2006}]\n\tLet $G$ and $H$ be two graphs of the same order. Then $G$ and $H$ are generalized cospectral if and only if there exists a regular orthogonal matrix $Q$ such that $Q^\\mathsf{T} A(G) Q=A(H)$. Moreover, if $G$ is controllable, then $Q^\\mathsf{T}=W(H)(W(G))^{-1}$, and hence $Q$ is unique and rational.\n\\end{lemma}\n\nIn the following, let $G$ be an $n$-vertex controllable graph and $p$ be an odd prime factor of $\\det W(G)$ such that $\\rank_p W(G)=n-1$. Denote $\\tau=v_p(L(G))$, i.e., $\\tau=\\max\\{v_p (\\ell(Q))\\colon\\, Q\\in \\mathcal{Q}(G)\\}$. Note that if $\\tau=0$, then Theorem \\ref{main} clearly holds. Thus, we may assume that $\\tau\\ge 1$. For simplicity, the adjacency matrix $A(G)$ and the walk matrix $W(G)$ will be denoted by $A$ and $W$, respectively, when there is no confusion.\n\n\\begin{lemma}\\label{fourcong}\n\tThere exist an integral vector $z_0$ with $z_0\\not\\equiv 0\\pmod{p}$ and an integer $\\lambda_0$ such that $z_0^\\mathsf{T} z_0\\equiv 0\\pmod{p^{2\\tau}}$, $z_0^\\mathsf{T} A z_0\\equiv 0\\pmod{p^{2\\tau}}$, $W^\\mathsf{T} z_0\\equiv 0\\pmod{p^\\tau}$, and $Az_0\\equiv \\lambda_0z_0\\pmod{p^\\tau}$.\n\\end{lemma}\n\n\\begin{proof}\n\tLet $Q\\in \\mathcal{Q}(G)$ be such that $v_p(\\ell(Q))=\\tau$, and let $H$ be the corresponding graph. Let $\\ell=\\ell(Q)$ and $\\hat{Q}=\\ell Q$. Then $\\hat{Q}$ is an integral matrix and $\\hat{Q}\\not\\equiv 0\\pmod{p}$. Let $z_0$ be a column of $\\hat{Q}$ such that $z_0\\not\\equiv 0\\pmod{p}$. Since $Q$ is an orthogonal matrix and $p^\\tau\\mid \\ell$, we have $\\hat{Q}^\\mathsf{T} \\hat{Q}=\\ell^2 I$, and hence $z_0^\\mathsf{T} z_0\\equiv 0\\pmod{p^{2\\tau}}$. Similarly, as $Q^\\mathsf{T} A Q$ is a $(0,1)$-matrix, we have $\\hat{Q}^\\mathsf{T} A\\hat{Q}\\equiv 0\\pmod{p^{2\\tau}}$, and hence $z_0^\\mathsf{T} A z_0\\equiv 0\\pmod{p^{2\\tau}}$. Moreover, noting that $W^\\mathsf{T} Q$ equals $(W(H))^\\mathsf{T}$, which is an integral matrix, we find that $W^\\mathsf{T} \\hat{Q}\\equiv 0\\pmod{\\ell}$, and hence $W^\\mathsf{T} z_0\\equiv 0\\pmod{p^\\tau}$.\n\n\tLet $M=\\{z\\in \\mathbb{Z}^n\\colon\\, W^\\mathsf{T} z\\equiv 0\\pmod {p^\\tau}\\}$. Since $\\rank_p W=n-1$ and $p^\\tau \\mid \\det W$, we see that, over $\\mathbb{Z}/p^\\tau \\mathbb{Z}$, the SNF of $W^\\mathsf{T}$ is $\\diag(1,1,\\ldots,1,0)$. It follows from Proposition \\ref{kf} that $M$ is a free $\\mathbb{Z}/p^\\tau \\mathbb{Z}$-module of rank one. Since $z_0\\in M$ and $z_0\\not\\equiv 0\\pmod{p}$, we see that $\\{z_0\\}$ forms a basis for $M$ over $\\mathbb{Z}/p^\\tau \\mathbb{Z}$ by Proposition \\ref{bas} (i). On the other hand, by the Cayley-Hamilton Theorem, it is easy to see that $M$ is an $A$-invariant $\\mathbb{Z}/p^\\tau \\mathbb{Z}$-submodule. This implies that there exists an integer $\\lambda_0$ such that $Az_0\\equiv \\lambda_0 z_0\\pmod{p^\\tau}$. This completes the proof.\t\n\\end{proof}\n\\begin{lemma}\\label{wy}\n\tIf $(A-\\lambda_0 I)y\\equiv s p^j z_0\\pmod{p^{j+\\tau}}$ for some integer $s$ and integer $j\\ge 0$, then\n\t\\begin{equation*}\n\t\tW^\\mathsf{T} y\\equiv (e^\\mathsf{T} y)(1,\\lambda_0,\\ldots,\\lambda_0^{n-1})^\\mathsf{T} \\pmod{p^{j+\\tau}}.\n\t\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\n\tWe proceed by induction on $k$ to show that $e^\\mathsf{T} A^k y\\equiv \\lambda_0^k e^\\mathsf{T} y\\pmod{p^{j+\\tau}}$ for $k=0,1,\\ldots,n-1$. The base case $k=0$ is trivial. Assume that the congruence holds for some integer $k$ with $0 \\le k < n-1$; we proceed to verify it for $k+1$. \n\n\tBy Lemma \\ref{fourcong}, we have $W^\\mathsf{T} z_0\\equiv 0\\pmod{p^\\tau}$, which implies that $e^\\mathsf{T} A^k z_0\\equiv 0\\pmod{p^\\tau}$. Consequently, we obtain $e^\\mathsf{T} A^k (sp^j z_0)\\equiv 0\\pmod{p^{j+\\tau}}$. By the hypothesis of this lemma, $Ay \\equiv sp^j z_0+\\lambda_0 y \\pmod{p^{j+\\tau}}$, which implies that\n\t\\[\n\te^\\mathsf{T} A^{k+1}y \\equiv e^\\mathsf{T} A^k (sp^j z_0+\\lambda_0 y) \\equiv \\lambda_0 e^\\mathsf{T} A^k y \\equiv \\lambda_0^{k+1}e^\\mathsf{T} y \\pmod{p^{j+\\tau}}.\n\t\\]\n\tThis completes the proof.\t\n\\end{proof}\n\\begin{lemma}\\label{rA}\n\tLet $S=\\diag(f_1,\\ldots,f_n)$ be the SNF of $A-\\lambda_0 I$. Then $f_{n-2}\\not \\equiv 0\\pmod{p}$ and $f_n\\equiv 0\\pmod{p^\\tau}$.\n\\end{lemma}\n\n\\begin{proof}\n\tBy Lemma \\ref{fourcong}, we have $(A-\\lambda_0 I)z_0\\equiv 0\\pmod{p^\\tau}$ and $z_0\\not\\equiv 0\\pmod{p}$. It follows from Proposition \\ref{dnk} that $f_n\\equiv 0\\pmod{p^\\tau}$. \tIt remains to show that $f_{n-2}\\not \\equiv 0\\pmod{p}$. Suppose to the contrary that $f_{n-2}\\equiv 0\\pmod{p}$. Then $\\rank_p (A-\\lambda_0 I)\\le n-3$. \n\tLet \\begin{equation*}\n\tB=\\begin{bmatrix} A-\\lambda_0 I\\\\e^\\mathsf{T}\\end{bmatrix}.\\end{equation*}\n\tThen, we have $\\rank_p B\\le n-2$. Recall from Lemma \\ref{fourcong} that $W^\\mathsf{T} z_0 \\equiv 0 \\pmod{p}$, which implies $e^\\mathsf{T} z_0 \\equiv 0 \\pmod{p}$. Together with $(A-\\lambda_0 I)z_0 \\equiv 0 \\pmod{p}$, this yields $Bz_0 \\equiv 0 \\pmod{p}$. Since the nullity of $B$ over $\\mathbb{Z}/p\\mathbb{Z}$ is at least $n - (n-2) = 2$, the equation $Bz\\equiv 0\\pmod {p}$ must have a solution $z_1$ such that $z_0$ and $z_1$ are linearly independent over $\\mathbb{Z}/p\\mathbb{Z}$. \n\n\tNoting that $Az_1\\equiv \\lambda_0 z_1 \\pmod{p}$ and $e^\\mathsf{T} z_1\\equiv 0\\pmod{p}$, we clearly have \n\t\\[\n\te^\\mathsf{T} A^{k}z_1\\equiv \\lambda_0^k e^\\mathsf{T} z_1\\equiv 0\\pmod{p}\n\t\\]\n\tfor $k=0,1,\\ldots, n-1$. This means that $W^\\mathsf{T} z_1\\equiv 0\\pmod{p}$. By Lemma \\ref{fourcong}, we also have $W^\\mathsf{T} z_0\\equiv 0\\pmod{p}$. Therefore, $\\rank_p W^\\mathsf{T} \\le n-2$. This contradicts the assumption that $\\rank_p W = n-1$, which completes the proof.\n\\end{proof}\n\\begin{lemma} \\label{lc}\n\t\tLet $M = [A-\\lambda_0 I, z_0]$. Then $z_0^\\mathsf{T} M \\equiv 0 \\pmod{p^\\tau}$ and the SNF of $M$ is $[\\diag(I_{n-1},0),0]$. Moreover, any integral vector $z$ satisfying $z_0^\\mathsf{T} z \\equiv 0 \\pmod{p^\\tau}$ can be expressed as a linear combination of the columns of $M$ over $\\mathbb{Z}/p^\\tau \\mathbb{Z}$.\n\t\\end{lemma}\n\t\\begin{proof} By Lemma \\ref{fourcong}, we have $(A-\\lambda_0 I)z_0\\equiv 0\\pmod{p^\\tau}$ and $z_0^\\mathsf{T} z_0\\equiv 0\\pmod{p^\\tau}$. As $A-\\lambda_0 I$ is symmetric, we find that\n\t\t\\begin{equation}\\label{mz}\n\t\t\tM^\\mathsf{T} z_0=\\begin{bmatrix}\n\t\t\t\tA-\\lambda_0 I\\\\\n\t\t\t\tz_0^\\mathsf{T}\n\t\t\t\\end{bmatrix}z_0\\equiv 0\\pmod{p^\\tau}.\n\t\t\\end{equation} Let the SNF of $M$ be $S= [\\diag(p^{c_1},p^{c_2}, \\dots, p^{c_n}), 0]$. Clearly, the SNF of $M^\\mathsf{T}$ is $S^\\mathsf{T}$. Note that $z_0\\not\\equiv 0\\pmod{p}$. It follows from Eq.~\\eqref{mz} and Proposition \\ref{dnk} that $p^{c_n}$ is zero over $R$. Thus, the SNF of $M$ can be simplified as \n\t\t\\begin{equation}\\label{ss}\n\t\t\tS=[\\diag(p^{c_1},\\ldots,p^{c_{n-1}},0),0].\n\t\t\\end{equation}\n\n\\noindent\\textbf{Claim}: $p^{c_{n-1}}$ is a unit in $R$, i.e., $c_{n-1}= 0$.\n\n\t\tSuppose to the contrary that $c_{n-1}\\ge 1$. Then, we have $\\rank_p S\\le n-2$, or equivalently, $\\rank_p M\\le n-2$. On the other hand, by Lemma \\ref{rA}, we see that $\\rank_p (A-\\lambda_0 I)\\ge n-2$ and hence $\\rank_p M \\ge n-2$. Thus, we must have $\\rank_p M=\\rank_p (A-\\lambda_0 I)=n-2$. It follows that there exists an integral vector $z_1$ such that $(A-\\lambda_0 I)z_1\\equiv z_0\\pmod{p}$.\t\n\n\t\tAs $\\rank_p (A-\\lambda_0 I)=n-2$, $(A-\\lambda_0I)z\\equiv 0\\pmod{p}$ has two solutions $z_2$ and $z_3$ that are linearly independent over $\\mathbb{Z}/p\\mathbb{Z}$. Since $(A-\\lambda_0 I)z_1\\equiv z_0\\not \\equiv 0\\pmod{p}$, $z_1$ cannot be written as a linear combination of $z_2$ and $z_3$. This implies that $z_1,z_2,z_3$ are linearly independent. Consider the equation $e^\\mathsf{T}(k_1z_1+k_2z_2+k_3z_3)\\equiv 0 \\pmod{p}$ with three unknowns $k_1,k_2,k_3$. Clearly, it has at least two independent solutions over $\\mathbb{Z}/p\\mathbb{Z}$. Let $(a_1,a_2,a_3)^\\mathsf{T}$ and $(b_1,b_2,b_3)^\\mathsf{T}$ be two such solutions and write $\\alpha=a_1z_1+a_2z_2+a_3z_3$ and $\\beta=b_1z_1+b_2z_2+b_3z_3$. It is easy to see that $\\alpha$ and $\\beta$ are linearly independent over $\\mathbb{Z}/p\\mathbb{Z}$. Note that $(A-\\lambda_0I)\\alpha\\equiv a_1z_0$ and $e^\\mathsf{T}\\alpha\\equiv 0\\pmod{p}$. Using a similar argument as in the proof of Lemma~\\ref{wy}, we find that $W^\\mathsf{T}\\alpha \\equiv 0\\pmod{p}$. Also, $W^\\mathsf{T} \\beta\\equiv 0\\pmod{p}$. Thus, we have found two linearly independent solutions of $W^\\mathsf{T} z\\equiv 0\\pmod{p}$. This contradicts the fact that $\\rank_p W^\\mathsf{T}=n-1$ and hence completes the proof of the Claim.\n\n\t\tBy the Claim, we see that Eq.~\\eqref{ss} can be further reduced to \n\t\t\\begin{equation}\\label{s3}\n\t\t\tS=[\\diag(I_{n-1},0),0].\n\t\t\\end{equation}\n\t\tLet $\\mathcal{C}(M)$ be the $R$-module generated by the columns of $M$ and $N$ be the module $\\{z\\in R^n\\colon\\, z_0^\\mathsf{T} z=0\\}$. By Eq.~\\eqref{s3}, we know that $\\mathcal{C}(M)$ is isomorphic to the free module $R^{n-1}$. Since $z_0\\not\\equiv 0\\pmod{p}$, we find that $N$ is also isomorphic to $R^{n-1}$. Thus, the two $R$-modules $\\mathcal{C}(M)$ and $N$ are isomorphic. Since $z_0^\\mathsf{T} M=0$ over $R$, we know that $\\mathcal{C}(M)$ is a submodule of $N$. As $R$ is a finite ring, we must have $\\mathcal{C}(M)=N$, which completes the proof of Lemma \\ref{lc}.\n\t\\end{proof}\n\n\\begin{lemma}\\label{ez}\n\tThere exists an integral vector $z_1$ such that $e^\\mathsf{T} z_1\\not\\equiv 0\\pmod {p}$ and $(A-\\lambda_0 I)z_1\\equiv p^cz_0\\pmod{p^\\tau}$ for some $c\\in \\{0,1,\\ldots,\\tau\\}$.\n\\end{lemma}\n\\begin{proof}\n\tLet $R=\\mathbb{Z}/p^\\tau \\mathbb{Z}$. By Lemma \\ref{rA}, we know that the SNF of $A-\\lambda_0 I$ is\n\t\\begin{equation}\\label{smA}\n\t\tS=\\diag(1,1,\\ldots,1,p^c,0) \\text{~for some~} c\\in\\{0,1,\\ldots,\\tau\\}.\n\t\\end{equation}\nLet $M_i=[A-\\lambda_0 I, p^i z_0]$ for $i\\in\\{0,1,\\ldots,c\\}$. \n\n\\noindent\\textbf{Claim 1}: The SNF of $M_i$ is $[\\diag(1,1,\\ldots,1,p^i,0),0]$ for $i\\in\\{0,1,\\ldots,c\\}$.\n\nLet $S_i$ be the SNF of $M_i$. Noting that $\\rank_p M_i\\ge \\rank_p (A-\\lambda_0 I)\\ge n-2$, we conclude that $S_i$ must have the form\n\\begin{equation}\\label{si}\n\tS_i=[\\diag(1,\\ldots,1,p^{t_{i}},p^{t'_{i}}),0].\n\\end{equation}\nSince $ M_i^\\mathsf{T} z_0\\equiv 0\\pmod{p^\\tau}$ and $z_0\\not\\equiv 0\\pmod{p}$, Proposition \\ref{dnk} indicates that $p^{t'_i}$ is the zero element in $R$. Thus, Eq.~\\eqref{si} can be simplified as\n\\begin{equation}\\label{st}\n\tS_i=[\\diag(1,\\ldots,1,p^{t_{i}},0),0].\n\\end{equation}\nWe need to show that $t_i=i$ for each $i\\in\\{0,1,\\ldots,c\\}$. Clearly, $t_0=0$ by Lemma \\ref{lc}. Thus, it suffices to show that the sequence $\\{t_i\\}_{0\\le i\\le c}$ satisfies \n\\begin{equation}\\label{tc}\n\tt_c=c\n\\end{equation}\nand \n\\begin{equation}\\label{icr}\n\tt_i\\le t_{i+1}\\le t_{i}+1\\text{~for~} 0\\le i0$ since otherwise we have nothing to show. Let $D^{(i)}$ and $D^{(i+1)}$ be any two corresponding $(n-1)$-th minors of $M_i$ and $M_{i+1}$, respectively. By the constructions of $M_i$ and $M_{i+1}$, we have either $D^{(i+1)}=D^{(i)}$ or $D^{(i+1)}=pD^{(i)}$. It follows that $v_p(\\Delta^{(i)})\\le v_p(\\Delta^{(i+1)}) \\le v_p(\\Delta^{(i)}) +1$, i.e., $t_i\\le t_{i+1}\\le t_i+1$. Thus, Claim 1 follows.\n\nTo complete the proof of Lemma \\ref{ez}, we consider the following three cases:\n\n\\noindent\\emph{Case 1}: $c=0$. Since $A-\\lambda_0 I$ and $M_0=[A-\\lambda_0 I,z_0]$ have the same invariant factors over $R$, Proposition \\ref{ssnf} implies that there exists an integral vector $z_1$ such that \\begin{equation}\\label{as}\n(A-\\lambda_0 I)z_1\\equiv z_0\\pmod{p^\\tau}.\n\\end{equation} By Lemma \\ref{wy}, we have \n\t\\begin{equation*}\n\tW^\\mathsf{T} z_1\\equiv e^\\mathsf{T} z_1(1,\\lambda_0,\\ldots,\\lambda_0^{n-1})^\\mathsf{T} \\pmod{p^{\\tau}}.\n\\end{equation*}\nSuppose to the contrary that $e^\\mathsf{T} z_1\\equiv 0\\pmod {p}$. Then we have $W^\\mathsf{T} z_1\\equiv 0\\pmod{p}$. Since $\\rank_p W=n-1$ and $z_0$ is a nonzero (over $\\mathbb{Z}/p\\mathbb{Z}$) solution of $W^\\mathsf{T} z\\equiv 0\\pmod{p}$, we conclude that $z_1\\equiv kz_0\\pmod{p}$ for some integer $k$. But this would imply \n\\begin{equation*}\n(A-\\lambda_0 I) z_1\\equiv k(A-\\lambda_0 I) z_0\\equiv 0\\pmod{p},\n\\end{equation*}\nwhich, combining with Eq.~\\eqref{as}, leads to $z_0\\equiv 0\\pmod{p}$. This is a contradiction and hence we must have $e^\\mathsf{T} z_1\\not\\equiv 0\\pmod {p}$.\n\n\\noindent\\emph{Case 2}: $c=\\tau$. As the SNF of $A-\\lambda_0 I$ is $S=\\diag(I_{n-1},0,0)$, we know that the kernel of $(A-\\lambda_0 I)$ is a free $R$-module of rank $2$ by Proposition \\ref{kf}. Let $K=\\ker (A-\\lambda_0 I)$ be the kernel. Since $z_0\\in K$ and $z_0\\not\\equiv 0\\pmod{p}$, Proposition \\ref{bas} (ii) implies that there exists an integer vector $z_1$ such that $\\{z_0,z_1\\}$ (over $R$) constitutes a basis of the free module $K$. In particular, $(A-\\lambda_0 I)z_1\\equiv 0\\pmod{p^\\tau}$. Using Lemma \\ref{wy}, we conclude that \n\t\\begin{equation*}\n\tW^\\mathsf{T} z_1\\equiv e^\\mathsf{T} z_1(1,\\lambda_0,\\ldots,\\lambda_0^{n-1})^\\mathsf{T} \\pmod{p^{\\tau}}.\n\\end{equation*}\nIf $e^\\mathsf{T} z_1\\equiv 0\\pmod{p}$ then we would have \t$W^\\mathsf{T} z_1\\equiv 0\\pmod{p}$. But since $z_0$ and $z_1$ are independent over the field $\\mathbb{Z}/p\\mathbb{Z}$, we must have $\\rank_p W^\\mathsf{T}\\le n-2$. This is a contradiction and hence $e^\\mathsf{T} z_1\\not\\equiv 0\\pmod{p}$.\n\n\\noindent{\\emph{Case 3}}: $0 0$\nsuch that\n\\[\n\\|\\nabla u\\|_{L^{{p},\\lambda}(\\Omega')} \\leq C.\n\\]", + "start_pos": 8951, + "end_pos": 9651, + "label": "MainThm" + }, + "ref_dict": { + "FJNL": "\\begin{Prop}\\label{FJNL}\nLet $u$ be a local minimizer of the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined\nin \\eqref{MainF} under the assumption of Theorem~\\ref{MainThm}.\nThen there exists a constant $0 < \\tau_0 < 1$ such that the following holds:\nfor every $\\tau \\in (0,\\tau_0)$ there exists $\\varepsilon_0 = \\varepsilon_0(\\tau) > 0$\nwith the property that, if $B_r(x_0) \\Subset \\Omega$ and one of the following\nconditions is satisfied,\n\\begin{enumerate}[label=(\\roman*)]\n\\item $|E \\cap B_r(x_0)| < \\varepsilon_0 |B_r|$,\n\\item $|B_r(x_0) \\setminus E| < \\varepsilon_0 |B_r|$,\n\\item there exists a half-space $H$ such that\n\\[\n\\frac{|(E \\Delta H) \\cap B_r(x_0)|}{|B_r|} < \\varepsilon_0,\n\\]\n\\end{enumerate}\nthen, for every $0 < \\delta < n$, the estimate\n\\[\n\\int_{B_{\\tau r}(x_0)} |\\nabla u|^p \\, dx\n\\leq\nC_0 \\tau^{n-\\delta}\n\\left(\n\\int_{B_r(x_0)} |\\nabla u|^p \\, dx + r^n\n\\right)\n\\]\nholds, where the constant $C_0$ depends only on\n$n, \\nu, L, \\alpha,\\beta, \\delta$ and $\\|\\nabla u\\|_{L^2(\\Omega)}$.\n\\end{Prop}", + "MainF": "\\begin{equation}\\label{MainF}\n\\mathcal{F}(u;E) := \\int_{\\Omega} \\sigma_E(x)\\, F(\\nabla u)\\, dx,\n\\end{equation}", + "MainThm": "\\begin{Thm}[Gradient regularity in Morrey spaces]\\label{MainThm}\nLet $u \\in W^{1,p}(\\Omega)$ be a local minimizer of the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined\nin\\eqref{MainF}, and assume that\n$\\partial E$ is a hypersurface of class $C^1$.\nSuppose that assumptions \\emph{(H1)}--\\emph{(H2)} are satisfied where $2 0$\nsuch that\n\\[\n\\|\\nabla u\\|_{L^{{p},\\lambda}(\\Omega')} \\leq C.\n\\]\n\\end{Thm}" + }, + "pre_theorem_intro_text_len": 5502, + "pre_theorem_intro_text": "Transmission problems arise naturally in the study of diffusive processes in\nheterogeneous media, where the governing laws change abruptly across fixed interfaces.\nTypical applications include elasticity theory, composite materials, and conductivity\nphenomena. This kind of problems appear also in the study of thermal insulation of bodies under sources and prescribed boundary conditions (see for example \\cite{AC,DPNST})\nFrom a mathematical perspective, such problems are characterised by the presence of\npiecewise-defined operators or energy densities, together with suitable transmission\nconditions prescribed along the separating interface.\nThe first systematic investigation of transmission problems in elasticity theory dates\nback to the seminal work of M.~Picone in the 1950s (see \\cite{Pico}).\n\nIn this paper, we investigate a class of \\emph{variational transmission problems}\nassociated with nonlinear integral functionals exhibiting discontinuous coefficients\nacross a prescribed interface.\nOur focus is on the regularity properties of local minimizers, with particular attention\nto the behaviour of the gradient near the interface.\n\nLet $\\Omega \\subset \\mathbb{R}^n$ be a bounded domain and let $E \\subset \\Omega$ be\na measurable subset.\nWe assume that the interface $\\partial E$ is a hypersurface of class $C^1$.\nWe fix two constants $0 < \\alpha < \\beta < +\\infty$ and define\n\\[\n\\sigma_E := \\beta \\mathbbm{1}_E + \\alpha \\mathbbm{1}_{E^c},\n\\]\nwhere $\\mathbbm{1}_E$ denotes the characteristic function of the set $E$ and $E^c$ is the complementary of $E$.\nWe consider the integral functional\n\\begin{equation}\\label{MainF}\n\\mathcal{F}(u;E) := \\int_{\\Omega} \\sigma_E(x)\\, F(\\nabla u)\\, dx,\n\\end{equation}\ndefined for functions $u \\in W^{1,p}(\\Omega)$.\n\nThe integrand $F \\colon \\mathbb{R}^n \\to \\mathbb{R}$ is assumed to belong to\n$C^2(\\mathbb{R}^n)$ and to satisfy standard $p$-growth and monotonicity conditions:\nfor all $\\xi, \\eta \\in \\mathbb{R}^n$,\n\\begin{equation}\n\\label{Monotonicity}\\tag{H1}\n\\langle \\nabla F(\\xi) - \\nabla F(\\eta), \\xi - \\eta \\rangle\n\\geq \\nu \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|^2,\n\\end{equation}\n\\begin{equation}\n\\label{Growth}\\tag{H2}\n|\\nabla F(\\xi) - \\nabla F(\\eta)|\n\\leq L \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|,\n\\end{equation}\nfor some constants $\\nu, L > 0$, $\\mu \\in (0,1]$, and $p > 2$.\n\nLocal minimizers of the functional $\\mathcal{F}$ are understood in the usual\nvariational sense.\n\n\\begin{Def}\nLet $u \\in W^{1,p}(\\Omega)$.\nWe say that $u$ is a \\emph{local minimizer} of $\\mathcal{F}(\\cdot;\\Omega)$ in $\\Omega$\nif, for every ball $B_r(x_0) \\subset \\Omega$ and every\n$\\phi \\in W^{1,p}_0(B_r(x_0))$, one has\n\\[\n\\mathcal{F}(u;B_r(x_0)) \\leq \\mathcal{F}(u + \\phi;B_r(x_0)).\n\\]\n\\end{Def}\n\nBefore stating our main result, let us place it in the context of the existing literature.\nIn the linear case $p = 2$, transmission problems with piecewise constant coefficients\nhave been extensively studied, and gradient regularity across sufficiently smooth\ninterfaces is by now classical (see, e.g.,~\\cite{GT,LV,Grisvard}).\nSuch results follow from the elliptic regularity theory for divergence-form operators\nwith discontinuous coefficients (see for instance\n\\cite{Evans, GT, LM, Grisvard} and \\cite[Theorem~7.53]{AFP}.\nIn this setting, local $W^{1,2}$- and H\\\"older-type regularity for the gradient of minimizers is well\nunderstood under mild geometric assumptions on the interface.\n\nMore recently, fine regularity properties for linear transmission problems across\n$C^{1,\\alpha}$ interfaces have been obtained by Caffarelli, Soria-Carro, and Stinga\n\\cite{CSS}.\n\nBy contrast, the nonlinear case $p \\neq 2$ remains far less developed.\nTo the best of our knowledge, the only available regularity results for nonlinear\ntransmission problems concern equations subject to \\emph{additive} transmission\nconditions on the interface.\nIn particular, in the recent work \\cite{BPU}, BMO regularity of the gradient is obtained\nfor degenerate quasilinear equations under transmission conditions of the form\n\\[\ng(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n-\ng(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n= f \\quad \\text{on } \\partial E.\n\\]\n\nIn contrast, the transmission condition naturally associated with the variational\nfunctional \\eqref{MainF} is of \\emph{multiplicative} type, namely\n\\[\ng_1(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n=\ng_2(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n\\quad \\text{on } \\partial E,\n\\]\nwhich arises intrinsically from the Euler--Lagrange equations and expresses the\ncontinuity of the nonlinear flux across the interface. We stress that this multiplicative transmission condition is intrinsic to the\nvariational structure of the functional \\eqref{MainF}, as it arises directly from\nthe Euler--Lagrange equations and cannot be prescribed independently.\nThe presence of a discontinuous coefficient combined with nonlinear growth prevents\na direct application of standard techniques such as freezing arguments or difference\nquotient estimates across the interface.\nTo the best of our knowledge, no Morrey-space regularity results for gradients of local\nminimizers are available in this genuinely nonlinear variational transmission setting.\n\nDenoting by $L^{{p},\\lambda}$ the classical Morrey spaces, we can now state the main\nresult of this paper, which establishes local Morrey regularity for the gradient of\nlocal minimizers under minimal geometric assumptions on the interface.", + "context": "Let $\\Omega \\subset \\mathbb{R}^n$ be a bounded domain and let $E \\subset \\Omega$ be\na measurable subset.\nWe assume that the interface $\\partial E$ is a hypersurface of class $C^1$.\nWe fix two constants $0 < \\alpha < \\beta < +\\infty$ and define\n\\[\n\\sigma_E := \\beta \\mathbbm{1}_E + \\alpha \\mathbbm{1}_{E^c},\n\\]\nwhere $\\mathbbm{1}_E$ denotes the characteristic function of the set $E$ and $E^c$ is the complementary of $E$.\nWe consider the integral functional\n\\begin{equation}\\label{MainF}\n\\mathcal{F}(u;E) := \\int_{\\Omega} \\sigma_E(x)\\, F(\\nabla u)\\, dx,\n\\end{equation}\ndefined for functions $u \\in W^{1,p}(\\Omega)$.\n\nThe integrand $F \\colon \\mathbb{R}^n \\to \\mathbb{R}$ is assumed to belong to\n$C^2(\\mathbb{R}^n)$ and to satisfy standard $p$-growth and monotonicity conditions:\nfor all $\\xi, \\eta \\in \\mathbb{R}^n$,\n\\begin{equation}\n\\label{Monotonicity}\\tag{H1}\n\\langle \\nabla F(\\xi) - \\nabla F(\\eta), \\xi - \\eta \\rangle\n\\geq \\nu \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|^2,\n\\end{equation}\n\\begin{equation}\n\\label{Growth}\\tag{H2}\n|\\nabla F(\\xi) - \\nabla F(\\eta)|\n\\leq L \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|,\n\\end{equation}\nfor some constants $\\nu, L > 0$, $\\mu \\in (0,1]$, and $p > 2$.\n\n\\begin{Def}\nLet $u \\in W^{1,p}(\\Omega)$.\nWe say that $u$ is a \\emph{local minimizer} of $\\mathcal{F}(\\cdot;\\Omega)$ in $\\Omega$\nif, for every ball $B_r(x_0) \\subset \\Omega$ and every\n$\\phi \\in W^{1,p}_0(B_r(x_0))$, one has\n\\[\n\\mathcal{F}(u;B_r(x_0)) \\leq \\mathcal{F}(u + \\phi;B_r(x_0)).\n\\]\n\\end{Def}\n\nBy contrast, the nonlinear case $p \\neq 2$ remains far less developed.\nTo the best of our knowledge, the only available regularity results for nonlinear\ntransmission problems concern equations subject to \\emph{additive} transmission\nconditions on the interface.\nIn particular, in the recent work \\cite{BPU}, BMO regularity of the gradient is obtained\nfor degenerate quasilinear equations under transmission conditions of the form\n\\[\ng(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n-\ng(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n= f \\quad \\text{on } \\partial E.\n\\]\n\nIn contrast, the transmission condition naturally associated with the variational\nfunctional \\eqref{MainF} is of \\emph{multiplicative} type, namely\n\\[\ng_1(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n=\ng_2(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n\\quad \\text{on } \\partial E,\n\\]\nwhich arises intrinsically from the Euler--Lagrange equations and expresses the\ncontinuity of the nonlinear flux across the interface. We stress that this multiplicative transmission condition is intrinsic to the\nvariational structure of the functional \\eqref{MainF}, as it arises directly from\nthe Euler--Lagrange equations and cannot be prescribed independently.\nThe presence of a discontinuous coefficient combined with nonlinear growth prevents\na direct application of standard techniques such as freezing arguments or difference\nquotient estimates across the interface.\nTo the best of our knowledge, no Morrey-space regularity results for gradients of local\nminimizers are available in this genuinely nonlinear variational transmission setting.\n\nDenoting by $L^{{p},\\lambda}$ the classical Morrey spaces, we can now state the main\nresult of this paper, which establishes local Morrey regularity for the gradient of\nlocal minimizers under minimal geometric assumptions on the interface.", + "full_context": "Let $\\Omega \\subset \\mathbb{R}^n$ be a bounded domain and let $E \\subset \\Omega$ be\na measurable subset.\nWe assume that the interface $\\partial E$ is a hypersurface of class $C^1$.\nWe fix two constants $0 < \\alpha < \\beta < +\\infty$ and define\n\\[\n\\sigma_E := \\beta \\mathbbm{1}_E + \\alpha \\mathbbm{1}_{E^c},\n\\]\nwhere $\\mathbbm{1}_E$ denotes the characteristic function of the set $E$ and $E^c$ is the complementary of $E$.\nWe consider the integral functional\n\\begin{equation}\\label{MainF}\n\\mathcal{F}(u;E) := \\int_{\\Omega} \\sigma_E(x)\\, F(\\nabla u)\\, dx,\n\\end{equation}\ndefined for functions $u \\in W^{1,p}(\\Omega)$.\n\nThe integrand $F \\colon \\mathbb{R}^n \\to \\mathbb{R}$ is assumed to belong to\n$C^2(\\mathbb{R}^n)$ and to satisfy standard $p$-growth and monotonicity conditions:\nfor all $\\xi, \\eta \\in \\mathbb{R}^n$,\n\\begin{equation}\n\\label{Monotonicity}\\tag{H1}\n\\langle \\nabla F(\\xi) - \\nabla F(\\eta), \\xi - \\eta \\rangle\n\\geq \\nu \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|^2,\n\\end{equation}\n\\begin{equation}\n\\label{Growth}\\tag{H2}\n|\\nabla F(\\xi) - \\nabla F(\\eta)|\n\\leq L \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|,\n\\end{equation}\nfor some constants $\\nu, L > 0$, $\\mu \\in (0,1]$, and $p > 2$.\n\n\\begin{Def}\nLet $u \\in W^{1,p}(\\Omega)$.\nWe say that $u$ is a \\emph{local minimizer} of $\\mathcal{F}(\\cdot;\\Omega)$ in $\\Omega$\nif, for every ball $B_r(x_0) \\subset \\Omega$ and every\n$\\phi \\in W^{1,p}_0(B_r(x_0))$, one has\n\\[\n\\mathcal{F}(u;B_r(x_0)) \\leq \\mathcal{F}(u + \\phi;B_r(x_0)).\n\\]\n\\end{Def}\n\nBy contrast, the nonlinear case $p \\neq 2$ remains far less developed.\nTo the best of our knowledge, the only available regularity results for nonlinear\ntransmission problems concern equations subject to \\emph{additive} transmission\nconditions on the interface.\nIn particular, in the recent work \\cite{BPU}, BMO regularity of the gradient is obtained\nfor degenerate quasilinear equations under transmission conditions of the form\n\\[\ng(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n-\ng(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n= f \\quad \\text{on } \\partial E.\n\\]\n\nIn contrast, the transmission condition naturally associated with the variational\nfunctional \\eqref{MainF} is of \\emph{multiplicative} type, namely\n\\[\ng_1(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n=\ng_2(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n\\quad \\text{on } \\partial E,\n\\]\nwhich arises intrinsically from the Euler--Lagrange equations and expresses the\ncontinuity of the nonlinear flux across the interface. We stress that this multiplicative transmission condition is intrinsic to the\nvariational structure of the functional \\eqref{MainF}, as it arises directly from\nthe Euler--Lagrange equations and cannot be prescribed independently.\nThe presence of a discontinuous coefficient combined with nonlinear growth prevents\na direct application of standard techniques such as freezing arguments or difference\nquotient estimates across the interface.\nTo the best of our knowledge, no Morrey-space regularity results for gradients of local\nminimizers are available in this genuinely nonlinear variational transmission setting.\n\nDenoting by $L^{{p},\\lambda}$ the classical Morrey spaces, we can now state the main\nresult of this paper, which establishes local Morrey regularity for the gradient of\nlocal minimizers under minimal geometric assumptions on the interface.\n\n\\begin{Prop}\\label{FJNL}\nLet $u$ be a local minimizer of the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined\nin \\eqref{MainF} under the assumption of Theorem~\\ref{MainThm}.\nThen there exists a constant $0 < \\tau_0 < 1$ such that the following holds:\nfor every $\\tau \\in (0,\\tau_0)$ there exists $\\varepsilon_0 = \\varepsilon_0(\\tau) > 0$\nwith the property that, if $B_r(x_0) \\Subset \\Omega$ and one of the following\nconditions is satisfied,\n\\begin{enumerate}[label=(\\roman*)]\n\\item $|E \\cap B_r(x_0)| < \\varepsilon_0 |B_r|$,\n\\item $|B_r(x_0) \\setminus E| < \\varepsilon_0 |B_r|$,\n\\item there exists a half-space $H$ such that\n\\[\n\\frac{|(E \\Delta H) \\cap B_r(x_0)|}{|B_r|} < \\varepsilon_0,\n\\]\n\\end{enumerate}\nthen, for every $0 < \\delta < n$, the estimate\n\\[\n\\int_{B_{\\tau r}(x_0)} |\\nabla u|^p \\, dx\n\\leq\nC_0 \\tau^{n-\\delta}\n\\left(\n\\int_{B_r(x_0)} |\\nabla u|^p \\, dx + r^n\n\\right)\n\\]\nholds, where the constant $C_0$ depends only on\n$n, \\nu, L, \\alpha,\\beta, \\delta$ and $\\|\\nabla u\\|_{L^2(\\Omega)}$.\n\\end{Prop}\n\\begin{Rem}\n\\label{Reg}\n For the proof of Theorem \\ref{MainThm}, it would be sufficient to state the proposition only in case (i). We nevertheless include the other two cases for the sake of completeness.\n\\end{Rem}\n\n\\begin{Lem}[Standard $p$-growth bounds]\\label{lem:p-growth}\nAssume \\((H1)\\)--\\((H2)\\). Then there exists a constant $C=C(n,p,\\nu,L)>0$ such that for all $\\xi\\in\\R^n$\n\\[\nC^{-1}(\\mu^2+|\\xi|^2)^{\\frac p2}-C\\mu^p \\le F(\\xi)\\le C(\\mu^2+|\\xi|^2)^{\\frac p2}+C\\mu^p .\n\\]\n\\end{Lem}\nThe following results on higher integrability and local Hölder continuity\nof minimizers are classical. Since their proofs are standard, we omit them and refer the reader to\n\\cite{Giu,GiaMar}.\n{\n\\begin{Thm}[Higher integrability]\n\\label{HigherInt}\nLet $u \\in W^{1,p}(\\Omega)$ be a local minimizer of the functional \\eqref{MainF}.\nThere exist $s > 1$ and $C = C(n,p,\\nu,L,\\beta)$ such that for any ball\n$B_{2r}(x_{0})\\subset\\subset\\Omega$\n\\begin{equation}\n \\int_{B_r(x_0)}|\\nabla u|^{sp}\\,dx\\leq C\\bigg(\\int_{B_{2r}(x_0)}\\big(|\\nabla u|^p+\\mu^p\\big)\\,dx\n \\bigg)^s.\n\\end{equation}\n\\end{Thm}\n\\begin{Thm}[H\\\"older continuity]Let $u \\in W^{1,p}(\\Omega)$ be a local minimizer of the functional \\eqref{MainF}. Then\n\\begin{enumerate}[label=(\\roman*)]\n\\item For any open set $\\Omega'\\Subset \\Omega$ the quantity $\\norm{u}_{L^{\\infty}(\\Omega')}$ is bounded by a constant $C = C(n,p,\\nu,L,\\beta)\\norm{u}_{L^2(\\Omega)}$.\n\\item u is locally H\\\"older continuous in $\\Omega$.\n\\end{enumerate}\n\\end{Thm}\n}\n\n\\begin{Thm}\\label{DS}\nLet $u$ be a minimizer of $\\mathcal{F}$ in $B_r(x_0)$. Then $\\partial_i u\\in W^{1,2}_{loc}(B_{r})$ for every $i=1\\dots n-1$. Moreover the following properties hold:\n\\begin{enumerate}[label=\\roman*)]\n\\item $u \\in W^{2,2}_{\\mathrm{loc}}(B_r^+)$ e $u \\in W^{2,2}_{\\mathrm{loc}}(B_r^-)$;\n\\item for every $x_0 \\in \\Gamma_r$ and $R>0$ such that $B_{4R}(x_0)\\subset B_r$, it holds that\n\\begin{align}\n\\int_{B_{R/2}} V^{p-2}\\,|\\nabla\\nabla' u|^2 \\,dx\n&\\leq \\frac{C}{R^2}\\int_{B_{2R}} V^p \\,dx, \\label{eq:tangential-estimate}\n\\end{align}\nwhere $C=C\\!\\left(\\tfrac{L}{\\nu},\\,n,\\,p,\\,\\alpha,\\,\\beta,\\mu\\right)$ is a positive constant.\n\\end{enumerate}\n\\end{Thm}\n\n\\section{Proof of Proposition 1.3}\n\\begin{proof}\nFix $\\tau\\in(0,\\tau_0)$ (to be chosen small only depending on $n,p,\\nu,L,\\beta/\\alpha$) and assume without loss of generality that $x_0=0$.\nWe prove that there exists $\\e_0=\\e_0(\\tau)>0$ such that if $B_r(x_0)\\Subset\\Om$ and one of {\\rm(i)}--{\\rm(iii)}\nholds, then for every $0<\\delta0$.\\\\\n\\indent \n\\noindent\\textbf{Step 1: the flat comparison in case (iii).}\nAssume that (iii) holds true. Up to a rigid motion we may assume that $H=\\{x_n>0\\}$ and set\n\\[\nE_H:=H\\cap B_{r},\\quad \\sigma_H:=\\beta\\mathbbm 1_{E_H}+\\alpha\\mathbbm 1_{B_r\\setminus E_H}.\n\\]\nLet $u_H\\in u+W^{1,p}_0(B_\\frac{r}{2})$ be the unique minimizer of the functional\n\\[\n\\mathcal F_H(w;B_\\frac{r}{2}):=\\int_{B_\\frac{r}{2}}\\sigma_H(x)\\,F(\\nabla w)\\,dx .\n\\]\nSubtracting the Euler-Lagrange equation satisfied by $u$ from the one satisfied by $u_H$ and testing them with $u-u_H$, we get\n\\begin{align}\n \\int_{B_{\\frac{r}{2}}} \\sigma_E\n\\big(DF(\\nabla u)-DF(\\nabla u_H)\\big)\n\\cdot(\\nabla u-\\nabla u_H)\\,dx\n& =\n\\int_{B_{\\frac{r}{2}}} (\\sigma_H-\\sigma_E)\\,\nDF(\\nabla u_H)\\cdot(\\nabla u-\\nabla u_H)\\,dx\n\\end{align}\nUsing assumptions \\eqref{Monotonicity} and \\eqref{Growth} and H\\\"older's inequality, for every $\\varepsilon>0$ it holds that\n\\begin{align}\n \\int_{B_{\\frac{r}{2}}}|\\D u_H-\\D u|^p\\,dx\n & \\leq C\\int_{(E\\Delta E_H)\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D u_H|^2\\big)^{\\frac{p-2}{2}}|\\D u-\\D u_H|\\,dx\\\\\n& \\leq C\\bigg(\\frac{1}{\\varepsilon}\\int_{(E\\Delta E_H)\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D u_H|^2\\big)^{\\frac{p}{2}}\\,dx+\\varepsilon\\int_{B_\\frac{r}{2}}|\\D u-\\D u_H|^p\\,dx \\bigg).\n\\end{align}\nChoosing $\\varepsilon$ sufficiently small, we get \n\\begin{align}\n \\int_{B_{\\frac{r}{2}}}|\\D u_H-\\D u|^p\\,dx\n& \\leq C\\int_{(E\\Delta E_H)\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D u_H|^2\\big)^{\\frac{p}{2}}\\,dx.\n\\end{align}\nThus, the minimality of $u_H$ with respect to $u$, H\\\"older's inequality and Theorem \\ref{HigherInt} yield\n\\begin{align}\n\\label{eqq1}\n\\int_{B_{\\frac{r}{2}}}|\\D u_H-\\D u|^p\\,dx& \\leq \\int_{(E\\Delta E_H)\\cap B_\\frac{r}{2}} (1+|\\nabla u|^2)^{\\frac p2}\\,dx\\\\\n& \\le \\bigg(\\frac{|(E\\Delta E_H)\\cap B_r|}{|B_r|}\\bigg)^{1-\\frac1s}|B_r|\\bigg(\\dashint_{B_r}(1+|\\nabla u|^2)^{\\frac{sp}{2}}\\,dx\\bigg)^{\\frac1s}\\\\\n& \\le C\\,\\e_0^{1-\\frac1s}\\int_{B_r}(1+|\\nabla u|^2)^{\\frac p2}\\,dx,\n\\end{align}\nwhere we have used assumption \\emph{iii)} and $C=C(n,p,\\beta/\\alpha,\\nu/L)$ is a positive constant.\n\n\\medskip\n\\noindent\\textbf{Step 3: the one-phase cases (i) and (ii).}\nAssume (i): $|E\\cap B_r|<\\e_0|B_r|$ (the case (ii) is analogous). Let us denote by $v\\in u+W^{1,p}_0(B_\\frac{r}{2})$ be the unique minimizer of the functional\n\\[\n\\mathcal G(w):=\\int_{B_\\frac{r}{2}}\\,F(\\nabla w)\\,dx .\n\\]\nWe can proceed as in step $(iii)$ subtracting the Euler-Lagrange equation satisfied by $u$ from the one satisfied by $v$ and testing them with $u-v$, getting\n\\begin{align}\n \\int_{B_{\\frac{r}{2}}} \\sigma_E\n\\big(DF(\\nabla u)-DF(\\nabla v)\\big)\n\\cdot(\\nabla u-\\nabla v)\\,dx\n& =\n\\int_{B_{\\frac{r}{2}}} (\\alpha-\\sigma_E)\\,\nDF(\\nabla v)\\cdot(\\nabla u-\\nabla v)\\,dx\n\\end{align}\nUsing assumptions \\eqref{Monotonicity} and \\eqref{Growth} and H\\\"older's inequality, for every $\\varepsilon>0$ it holds that\n\\begin{align}\n \\int_{B_{\\frac{r}{2}}}|\\D v -\\D u|^p\\,dx\n & \\leq C\\int_{E\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D v|^2\\big)^{\\frac{p-2}{2}}|\\D u-\\D v|\\,dx\\\\\n& \\leq C\\bigg(\\frac{1}{\\varepsilon}\\int_{E\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D v|^2\\big)^{\\frac{p}{2}}\\,dx+\\varepsilon\\int_{B_\\frac{r}{2}}|\\D u-\\D v|^p\\,dx \\bigg).\n\\end{align}\nChoosing $\\varepsilon$ sufficiently small, we get \n\\begin{align}\n \\int_{B_{\\frac{r}{2}}}|\\D v-\\D u|^p\\,dx\n& \\leq C\\int_{E\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D v|^2\\big)^{\\frac{p}{2}}\\,dx.\n\\end{align}\nUsing the minimality of $v$ compared with $u$ and Theorem\n\\ref{HigherInt} we deduce,\n\\begin{align}\n\\label{eqq11}\n\\int_{B_{\\frac{r}{2}}}|\\D v-\\D u|^p\\,dx& \\leq \\int_{E\\cap B_{\\frac{r}{2}}} (1+|\\nabla u|^2)^{\\frac p2}\\,dx\\\\\n& \\le \\bigg(\\frac{E\\cap B_{r}}{|B_r|}\\bigg)^{1-\\frac1s}|B_r|\\bigg(\\dashint_{B_r}(1+|\\nabla u|^2)^{\\frac{sp}{2}}\\,dx\\bigg)^{\\frac1s}\\\\\n& \\le C\\,\\e_0^{1-\\frac1s}\\int_{B_r}(1+|\\nabla u|^2)^{\\frac p2}\\,dx,\n\\end{align}\nwhere we have used assumption \\emph{i)} and $C=C(n,p,\\beta/\\alpha,\\nu/L)$ is a positive constant. Thereafter we can argue as in the previous step using the classical decay estimate for minimizers of regular integrals in the Calculus of Variations\n\\[\n\\int_{B_{\\tau \\rho}}|\\nabla v|^p\\,dx\n\\le C\\tau^{n}\\int_{B_{ \\rho}}|\\nabla u|^p\\,dx,\n\\]\nfor every $\\rho\\leq \\frac{r}{2}$.\n\\end{proof}", + "post_theorem_intro_text_len": 3397, + "post_theorem_intro_text": "\\begin{Rem}\nWithin the present variational transmission framework, Morrey regularity represents\nthe natural level of regularity that can be expected for the gradient of local\nminimizers, in view of the discontinuous coefficients and the intrinsic transmission\ncondition.\n\\end{Rem}\nThe proof of Theorem~\\ref{MainThm} is obtained as a consequence of the following\ndecay estimate for the energy.\n\n\\begin{Prop}\\label{FJNL}\nLet $u$ be a local minimizer of the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined\nin \\eqref{MainF} under the assumption of Theorem~\\ref{MainThm}.\nThen there exists a constant $0 < \\tau_0 < 1$ such that the following holds:\nfor every $\\tau \\in (0,\\tau_0)$ there exists $\\varepsilon_0 = \\varepsilon_0(\\tau) > 0$\nwith the property that, if $B_r(x_0) \\Subset \\Omega$ and one of the following\nconditions is satisfied,\n\\begin{enumerate}[label=(\\roman*)]\n\\item $|E \\cap B_r(x_0)| < \\varepsilon_0 |B_r|$,\n\\item $|B_r(x_0) \\setminus E| < \\varepsilon_0 |B_r|$,\n\\item there exists a half-space $H$ such that\n\\[\n\\frac{|(E \\Delta H) \\cap B_r(x_0)|}{|B_r|} < \\varepsilon_0,\n\\]\n\\end{enumerate}\nthen, for every $0 < \\delta < n$, the estimate\n\\[\n\\int_{B_{\\tau r}(x_0)} |\\nabla u|^p \\, dx\n\\leq\nC_0 \\tau^{n-\\delta}\n\\left(\n\\int_{B_r(x_0)} |\\nabla u|^p \\, dx + r^n\n\\right)\n\\]\nholds, where the constant $C_0$ depends only on\n$n, \\nu, L, \\alpha,\\beta, \\delta$ and $\\|\\nabla u\\|_{L^2(\\Omega)}$.\n\\end{Prop}\n\\begin{Rem}\n\\label{Reg}\n For the proof of Theorem \\ref{MainThm}, it would be sufficient to state the proposition only in case (i). We nevertheless include the other two cases for the sake of completeness.\n\\end{Rem}\n\nWe give an outline of the proof of Proposition~\\ref{FJNL}. It relies on decay estimates for the gradient\nnear the interface, first established in a flat configuration, i.e.\\ when the interface coincides\nwith a hyperplane.\nSection~3 is devoted to this ``flat case''.\n\nThe starting point is a local boundedness estimate for the tangential gradient\n$\\nabla' u$, where $\\nabla'=(\\partial_{x_1},\\dots,\\partial_{x_{n-1}})$ denotes the vector of derivatives parallel to\nthe interface.\nThis is proved in Proposition~3.2 via a difference-quotient argument with test functions involving\nonly tangential increments.\nA key ingredient is a Moser iteration applied to\n$Z:=|\\Delta_{i,h}u|^{m/p}$.\nDuring the estimates, the full gradient $\\nabla u$ enters through the structure conditions\n\\emph{(H1)}--\\emph{(H2)} and is treated as a weight; this is the origin of the restriction\n$2 0$\nwith the property that, if $B_r(x_0) \\Subset \\Omega$ and one of the following\nconditions is satisfied,\n\\begin{enumerate}[label=(\\roman*)]\n\\item $|E \\cap B_r(x_0)| < \\varepsilon_0 |B_r|$,\n\\item $|B_r(x_0) \\setminus E| < \\varepsilon_0 |B_r|$,\n\\item there exists a half-space $H$ such that\n\\[\n\\frac{|(E \\Delta H) \\cap B_r(x_0)|}{|B_r|} < \\varepsilon_0,\n\\]\n\\end{enumerate}\nthen, for every $0 < \\delta < n$, the estimate\n\\[\n\\int_{B_{\\tau r}(x_0)} |\\nabla u|^p \\, dx\n\\leq\nC_0 \\tau^{n-\\delta}\n\\left(\n\\int_{B_r(x_0)} |\\nabla u|^p \\, dx + r^n\n\\right)\n\\]\nholds, where the constant $C_0$ depends only on\n$n, \\nu, L, \\alpha,\\beta, \\delta$ and $\\|\\nabla u\\|_{L^2(\\Omega)}$.\n\\end{Prop}\n\nAn outline is given for the proof of Proposition~\\ref{FJNL}:\n\\begin{itemize}\n\\item The argument \\emph{relies on decay estimates for the gradient near the interface}, first established in a \\emph{flat configuration} (interface coincides with a hyperplane), treated next.\n\\item Starting point: a \\emph{local boundedness estimate for the tangential gradient} $\\nabla' u$ (derivatives parallel to the interface). This is proved in Proposition~3.2 via a \\emph{difference-quotient argument} with test functions involving only tangential increments.\n\\item Key ingredient: a \\emph{Moser iteration} applied to $Z:=|\\Delta_{i,h}u|^{m/p}$. In the estimates, the full gradient $\\nabla u$ appears through (H1)--(H2) and is \\emph{treated as a weight}; this is cited as the origin of the restriction $2 0$\nsuch that\n\\[\n\\|\\nabla u\\|_{L^{{p},\\lambda}(\\Omega')} \\leq C.\n\\]", + "theorem_type": [ + "Implication", + "Existential–Universal" + ], + "mcq": { + "question": "Let \\(\\Omega\\subset\\mathbb{R}^n\\) be a bounded domain, let \\(E\\subset\\Omega\\) be measurable with \\(\\partial E\\) a \\(C^1\\) hypersurface, and fix constants \\(0<\\alpha<\\beta<\\infty\\). Define \\(\\sigma_E:=\\beta\\mathbbm{1}_E+\\alpha\\mathbbm{1}_{E^c}\\) and, for \\(v\\in W^{1,p}(\\Omega)\\),\n\\[\\mathcal F(v;E):=\\int_{\\Omega}\\sigma_E(x)F(\\nabla v)\\,dx.\\]\nAssume \\(F\\in C^2(\\mathbb{R}^n)\\) and that for some \\(\\nu,L>0\\), \\(\\mu\\in(0,1]\\), and every \\(\\xi,\\eta\\in\\mathbb{R}^n\\),\n\\[\\langle \\nabla F(\\xi)-\\nabla F(\\eta),\\xi-\\eta\\rangle\\ge \\nu\\big(\\mu^2+|\\xi|^2+|\\eta|^2\\big)^{\\frac{p-2}{2}}|\\xi-\\eta|^2,\\]\n\\[|\\nabla F(\\xi)-\\nabla F(\\eta)|\\le L\\big(\\mu^2+|\\xi|^2+|\\eta|^2\\big)^{\\frac{p-2}{2}}|\\xi-\\eta|,\\]\nwith \\(20\\) such that\n\\[\\|\\nabla u\\|_{L^{p,\\lambda}(\\Omega')}\\le C.\\]" + }, + "choices": [ + { + "label": "B", + "text": "For every \\(0\\le \\lambda0\\), independent of \\(\\lambda\\), such that\n\\[\\|\\nabla u\\|_{L^{p,\\lambda}(\\Omega')}\\le C.\\]" + }, + { + "label": "C", + "text": "One has \\(\\nabla u\\in L^{p,\\lambda}_{\\mathrm{loc}}(\\Omega)\\) for at least one exponent \\(\\lambda\\in[0,n)\\)." + }, + { + "label": "D", + "text": "For every \\(0\\le \\lambda0\\) such that\n\\[\\|\\nabla u\\|_{L^{p,\\lambda}(\\Omega')}\\le C\\,\\|\\nabla u\\|_{L^{p}(\\Omega)}.\\]" + }, + { + "label": "E", + "text": "There exists \\(\\lambda_0\\in(0,n)\\), depending only on \\(n,p,\\nu,L,\\alpha,\\beta\\), such that \\(\\nabla u\\in L^{p,\\lambda_0}_{\\mathrm{loc}}(\\Omega)\\). Moreover, for every \\(\\Omega'\\Subset\\Omega\\), there exists a constant \\(C=C\\big(n,p,\\nu,L,\\beta,E,\\operatorname{diam}(\\Omega),\\operatorname{dist}(\\Omega',\\partial\\Omega)\\big)>0\\) with\n\\[\\|\\nabla u\\|_{L^{p,\\lambda_0}(\\Omega')}\\le C.\\]" + } + ], + "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": "dependence on Morrey exponent in the local bound", + "template_used": "uniformity_effectivity" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "universal quantifier over all \\(\\lambda 2). Still, the item mainly tests recognition of the precise theorem statement rather than deep derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target realistic failure modes: weakening the quantifier, adding the forbidden endpoint λ = n, claiming uniformity in E, and overextending the p-range. They are distinct and nontrivial." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-identification MCQ with strong distractors and no answer leakage, but it leans more toward precise recall of a known result than toward genuinely generative mathematical reasoning." + } + }, + { + "id": "2602.14658v1", + "paper_link": "http://arxiv.org/abs/2602.14658v1", + "theorems_cnt": 5, + "theorem": { + "env_name": "Thm", + "content": "[Gradient regularity in Morrey spaces]\\label{MainThm}\nLet $u \\in W^{1,p}(\\Omega)$ be a local minimizer of the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined\nin\\eqref{MainF}, and assume that\n$\\partial E$ is a hypersurface of class $C^1$.\nSuppose that assumptions \\emph{(H1)}--\\emph{(H2)} are satisfied where $2 0$\nsuch that\n\\[\n\\|\\nabla u\\|_{L^{{p},\\lambda}(\\Omega')} \\leq C.\n\\]", + "start_pos": 8951, + "end_pos": 9651, + "label": "MainThm" + }, + "ref_dict": { + "FJNL": "\\begin{Prop}\\label{FJNL}\nLet $u$ be a local minimizer of the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined\nin \\eqref{MainF} under the assumption of Theorem~\\ref{MainThm}.\nThen there exists a constant $0 < \\tau_0 < 1$ such that the following holds:\nfor every $\\tau \\in (0,\\tau_0)$ there exists $\\varepsilon_0 = \\varepsilon_0(\\tau) > 0$\nwith the property that, if $B_r(x_0) \\Subset \\Omega$ and one of the following\nconditions is satisfied,\n\\begin{enumerate}[label=(\\roman*)]\n\\item $|E \\cap B_r(x_0)| < \\varepsilon_0 |B_r|$,\n\\item $|B_r(x_0) \\setminus E| < \\varepsilon_0 |B_r|$,\n\\item there exists a half-space $H$ such that\n\\[\n\\frac{|(E \\Delta H) \\cap B_r(x_0)|}{|B_r|} < \\varepsilon_0,\n\\]\n\\end{enumerate}\nthen, for every $0 < \\delta < n$, the estimate\n\\[\n\\int_{B_{\\tau r}(x_0)} |\\nabla u|^p \\, dx\n\\leq\nC_0 \\tau^{n-\\delta}\n\\left(\n\\int_{B_r(x_0)} |\\nabla u|^p \\, dx + r^n\n\\right)\n\\]\nholds, where the constant $C_0$ depends only on\n$n, \\nu, L, \\alpha,\\beta, \\delta$ and $\\|\\nabla u\\|_{L^2(\\Omega)}$.\n\\end{Prop}", + "MainF": "\\begin{equation}\\label{MainF}\n\\mathcal{F}(u;E) := \\int_{\\Omega} \\sigma_E(x)\\, F(\\nabla u)\\, dx,\n\\end{equation}", + "MainThm": "\\begin{Thm}[Gradient regularity in Morrey spaces]\\label{MainThm}\nLet $u \\in W^{1,p}(\\Omega)$ be a local minimizer of the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined\nin\\eqref{MainF}, and assume that\n$\\partial E$ is a hypersurface of class $C^1$.\nSuppose that assumptions \\emph{(H1)}--\\emph{(H2)} are satisfied where $2 0$\nsuch that\n\\[\n\\|\\nabla u\\|_{L^{{p},\\lambda}(\\Omega')} \\leq C.\n\\]\n\\end{Thm}" + }, + "pre_theorem_intro_text_len": 5502, + "pre_theorem_intro_text": "Transmission problems arise naturally in the study of diffusive processes in\nheterogeneous media, where the governing laws change abruptly across fixed interfaces.\nTypical applications include elasticity theory, composite materials, and conductivity\nphenomena. This kind of problems appear also in the study of thermal insulation of bodies under sources and prescribed boundary conditions (see for example \\cite{AC,DPNST})\nFrom a mathematical perspective, such problems are characterised by the presence of\npiecewise-defined operators or energy densities, together with suitable transmission\nconditions prescribed along the separating interface.\nThe first systematic investigation of transmission problems in elasticity theory dates\nback to the seminal work of M.~Picone in the 1950s (see \\cite{Pico}).\n\nIn this paper, we investigate a class of \\emph{variational transmission problems}\nassociated with nonlinear integral functionals exhibiting discontinuous coefficients\nacross a prescribed interface.\nOur focus is on the regularity properties of local minimizers, with particular attention\nto the behaviour of the gradient near the interface.\n\nLet $\\Omega \\subset \\mathbb{R}^n$ be a bounded domain and let $E \\subset \\Omega$ be\na measurable subset.\nWe assume that the interface $\\partial E$ is a hypersurface of class $C^1$.\nWe fix two constants $0 < \\alpha < \\beta < +\\infty$ and define\n\\[\n\\sigma_E := \\beta \\mathbbm{1}_E + \\alpha \\mathbbm{1}_{E^c},\n\\]\nwhere $\\mathbbm{1}_E$ denotes the characteristic function of the set $E$ and $E^c$ is the complementary of $E$.\nWe consider the integral functional\n\\begin{equation}\\label{MainF}\n\\mathcal{F}(u;E) := \\int_{\\Omega} \\sigma_E(x)\\, F(\\nabla u)\\, dx,\n\\end{equation}\ndefined for functions $u \\in W^{1,p}(\\Omega)$.\n\nThe integrand $F \\colon \\mathbb{R}^n \\to \\mathbb{R}$ is assumed to belong to\n$C^2(\\mathbb{R}^n)$ and to satisfy standard $p$-growth and monotonicity conditions:\nfor all $\\xi, \\eta \\in \\mathbb{R}^n$,\n\\begin{equation}\n\\label{Monotonicity}\\tag{H1}\n\\langle \\nabla F(\\xi) - \\nabla F(\\eta), \\xi - \\eta \\rangle\n\\geq \\nu \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|^2,\n\\end{equation}\n\\begin{equation}\n\\label{Growth}\\tag{H2}\n|\\nabla F(\\xi) - \\nabla F(\\eta)|\n\\leq L \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|,\n\\end{equation}\nfor some constants $\\nu, L > 0$, $\\mu \\in (0,1]$, and $p > 2$.\n\nLocal minimizers of the functional $\\mathcal{F}$ are understood in the usual\nvariational sense.\n\n\\begin{Def}\nLet $u \\in W^{1,p}(\\Omega)$.\nWe say that $u$ is a \\emph{local minimizer} of $\\mathcal{F}(\\cdot;\\Omega)$ in $\\Omega$\nif, for every ball $B_r(x_0) \\subset \\Omega$ and every\n$\\phi \\in W^{1,p}_0(B_r(x_0))$, one has\n\\[\n\\mathcal{F}(u;B_r(x_0)) \\leq \\mathcal{F}(u + \\phi;B_r(x_0)).\n\\]\n\\end{Def}\n\nBefore stating our main result, let us place it in the context of the existing literature.\nIn the linear case $p = 2$, transmission problems with piecewise constant coefficients\nhave been extensively studied, and gradient regularity across sufficiently smooth\ninterfaces is by now classical (see, e.g.,~\\cite{GT,LV,Grisvard}).\nSuch results follow from the elliptic regularity theory for divergence-form operators\nwith discontinuous coefficients (see for instance\n\\cite{Evans, GT, LM, Grisvard} and \\cite[Theorem~7.53]{AFP}.\nIn this setting, local $W^{1,2}$- and H\\\"older-type regularity for the gradient of minimizers is well\nunderstood under mild geometric assumptions on the interface.\n\nMore recently, fine regularity properties for linear transmission problems across\n$C^{1,\\alpha}$ interfaces have been obtained by Caffarelli, Soria-Carro, and Stinga\n\\cite{CSS}.\n\nBy contrast, the nonlinear case $p \\neq 2$ remains far less developed.\nTo the best of our knowledge, the only available regularity results for nonlinear\ntransmission problems concern equations subject to \\emph{additive} transmission\nconditions on the interface.\nIn particular, in the recent work \\cite{BPU}, BMO regularity of the gradient is obtained\nfor degenerate quasilinear equations under transmission conditions of the form\n\\[\ng(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n-\ng(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n= f \\quad \\text{on } \\partial E.\n\\]\n\nIn contrast, the transmission condition naturally associated with the variational\nfunctional \\eqref{MainF} is of \\emph{multiplicative} type, namely\n\\[\ng_1(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n=\ng_2(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n\\quad \\text{on } \\partial E,\n\\]\nwhich arises intrinsically from the Euler--Lagrange equations and expresses the\ncontinuity of the nonlinear flux across the interface. We stress that this multiplicative transmission condition is intrinsic to the\nvariational structure of the functional \\eqref{MainF}, as it arises directly from\nthe Euler--Lagrange equations and cannot be prescribed independently.\nThe presence of a discontinuous coefficient combined with nonlinear growth prevents\na direct application of standard techniques such as freezing arguments or difference\nquotient estimates across the interface.\nTo the best of our knowledge, no Morrey-space regularity results for gradients of local\nminimizers are available in this genuinely nonlinear variational transmission setting.\n\nDenoting by $L^{{p},\\lambda}$ the classical Morrey spaces, we can now state the main\nresult of this paper, which establishes local Morrey regularity for the gradient of\nlocal minimizers under minimal geometric assumptions on the interface.", + "context": "Let $\\Omega \\subset \\mathbb{R}^n$ be a bounded domain and let $E \\subset \\Omega$ be\na measurable subset.\nWe assume that the interface $\\partial E$ is a hypersurface of class $C^1$.\nWe fix two constants $0 < \\alpha < \\beta < +\\infty$ and define\n\\[\n\\sigma_E := \\beta \\mathbbm{1}_E + \\alpha \\mathbbm{1}_{E^c},\n\\]\nwhere $\\mathbbm{1}_E$ denotes the characteristic function of the set $E$ and $E^c$ is the complementary of $E$.\nWe consider the integral functional\n\\begin{equation}\\label{MainF}\n\\mathcal{F}(u;E) := \\int_{\\Omega} \\sigma_E(x)\\, F(\\nabla u)\\, dx,\n\\end{equation}\ndefined for functions $u \\in W^{1,p}(\\Omega)$.\n\nThe integrand $F \\colon \\mathbb{R}^n \\to \\mathbb{R}$ is assumed to belong to\n$C^2(\\mathbb{R}^n)$ and to satisfy standard $p$-growth and monotonicity conditions:\nfor all $\\xi, \\eta \\in \\mathbb{R}^n$,\n\\begin{equation}\n\\label{Monotonicity}\\tag{H1}\n\\langle \\nabla F(\\xi) - \\nabla F(\\eta), \\xi - \\eta \\rangle\n\\geq \\nu \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|^2,\n\\end{equation}\n\\begin{equation}\n\\label{Growth}\\tag{H2}\n|\\nabla F(\\xi) - \\nabla F(\\eta)|\n\\leq L \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|,\n\\end{equation}\nfor some constants $\\nu, L > 0$, $\\mu \\in (0,1]$, and $p > 2$.\n\n\\begin{Def}\nLet $u \\in W^{1,p}(\\Omega)$.\nWe say that $u$ is a \\emph{local minimizer} of $\\mathcal{F}(\\cdot;\\Omega)$ in $\\Omega$\nif, for every ball $B_r(x_0) \\subset \\Omega$ and every\n$\\phi \\in W^{1,p}_0(B_r(x_0))$, one has\n\\[\n\\mathcal{F}(u;B_r(x_0)) \\leq \\mathcal{F}(u + \\phi;B_r(x_0)).\n\\]\n\\end{Def}\n\nBy contrast, the nonlinear case $p \\neq 2$ remains far less developed.\nTo the best of our knowledge, the only available regularity results for nonlinear\ntransmission problems concern equations subject to \\emph{additive} transmission\nconditions on the interface.\nIn particular, in the recent work \\cite{BPU}, BMO regularity of the gradient is obtained\nfor degenerate quasilinear equations under transmission conditions of the form\n\\[\ng(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n-\ng(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n= f \\quad \\text{on } \\partial E.\n\\]\n\nIn contrast, the transmission condition naturally associated with the variational\nfunctional \\eqref{MainF} is of \\emph{multiplicative} type, namely\n\\[\ng_1(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n=\ng_2(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n\\quad \\text{on } \\partial E,\n\\]\nwhich arises intrinsically from the Euler--Lagrange equations and expresses the\ncontinuity of the nonlinear flux across the interface. We stress that this multiplicative transmission condition is intrinsic to the\nvariational structure of the functional \\eqref{MainF}, as it arises directly from\nthe Euler--Lagrange equations and cannot be prescribed independently.\nThe presence of a discontinuous coefficient combined with nonlinear growth prevents\na direct application of standard techniques such as freezing arguments or difference\nquotient estimates across the interface.\nTo the best of our knowledge, no Morrey-space regularity results for gradients of local\nminimizers are available in this genuinely nonlinear variational transmission setting.\n\nDenoting by $L^{{p},\\lambda}$ the classical Morrey spaces, we can now state the main\nresult of this paper, which establishes local Morrey regularity for the gradient of\nlocal minimizers under minimal geometric assumptions on the interface.", + "full_context": "Let $\\Omega \\subset \\mathbb{R}^n$ be a bounded domain and let $E \\subset \\Omega$ be\na measurable subset.\nWe assume that the interface $\\partial E$ is a hypersurface of class $C^1$.\nWe fix two constants $0 < \\alpha < \\beta < +\\infty$ and define\n\\[\n\\sigma_E := \\beta \\mathbbm{1}_E + \\alpha \\mathbbm{1}_{E^c},\n\\]\nwhere $\\mathbbm{1}_E$ denotes the characteristic function of the set $E$ and $E^c$ is the complementary of $E$.\nWe consider the integral functional\n\\begin{equation}\\label{MainF}\n\\mathcal{F}(u;E) := \\int_{\\Omega} \\sigma_E(x)\\, F(\\nabla u)\\, dx,\n\\end{equation}\ndefined for functions $u \\in W^{1,p}(\\Omega)$.\n\nThe integrand $F \\colon \\mathbb{R}^n \\to \\mathbb{R}$ is assumed to belong to\n$C^2(\\mathbb{R}^n)$ and to satisfy standard $p$-growth and monotonicity conditions:\nfor all $\\xi, \\eta \\in \\mathbb{R}^n$,\n\\begin{equation}\n\\label{Monotonicity}\\tag{H1}\n\\langle \\nabla F(\\xi) - \\nabla F(\\eta), \\xi - \\eta \\rangle\n\\geq \\nu \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|^2,\n\\end{equation}\n\\begin{equation}\n\\label{Growth}\\tag{H2}\n|\\nabla F(\\xi) - \\nabla F(\\eta)|\n\\leq L \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|,\n\\end{equation}\nfor some constants $\\nu, L > 0$, $\\mu \\in (0,1]$, and $p > 2$.\n\n\\begin{Def}\nLet $u \\in W^{1,p}(\\Omega)$.\nWe say that $u$ is a \\emph{local minimizer} of $\\mathcal{F}(\\cdot;\\Omega)$ in $\\Omega$\nif, for every ball $B_r(x_0) \\subset \\Omega$ and every\n$\\phi \\in W^{1,p}_0(B_r(x_0))$, one has\n\\[\n\\mathcal{F}(u;B_r(x_0)) \\leq \\mathcal{F}(u + \\phi;B_r(x_0)).\n\\]\n\\end{Def}\n\nBy contrast, the nonlinear case $p \\neq 2$ remains far less developed.\nTo the best of our knowledge, the only available regularity results for nonlinear\ntransmission problems concern equations subject to \\emph{additive} transmission\nconditions on the interface.\nIn particular, in the recent work \\cite{BPU}, BMO regularity of the gradient is obtained\nfor degenerate quasilinear equations under transmission conditions of the form\n\\[\ng(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n-\ng(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n= f \\quad \\text{on } \\partial E.\n\\]\n\nIn contrast, the transmission condition naturally associated with the variational\nfunctional \\eqref{MainF} is of \\emph{multiplicative} type, namely\n\\[\ng_1(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n=\ng_2(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n\\quad \\text{on } \\partial E,\n\\]\nwhich arises intrinsically from the Euler--Lagrange equations and expresses the\ncontinuity of the nonlinear flux across the interface. We stress that this multiplicative transmission condition is intrinsic to the\nvariational structure of the functional \\eqref{MainF}, as it arises directly from\nthe Euler--Lagrange equations and cannot be prescribed independently.\nThe presence of a discontinuous coefficient combined with nonlinear growth prevents\na direct application of standard techniques such as freezing arguments or difference\nquotient estimates across the interface.\nTo the best of our knowledge, no Morrey-space regularity results for gradients of local\nminimizers are available in this genuinely nonlinear variational transmission setting.\n\nDenoting by $L^{{p},\\lambda}$ the classical Morrey spaces, we can now state the main\nresult of this paper, which establishes local Morrey regularity for the gradient of\nlocal minimizers under minimal geometric assumptions on the interface.\n\n\\begin{Prop}\\label{FJNL}\nLet $u$ be a local minimizer of the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined\nin \\eqref{MainF} under the assumption of Theorem~\\ref{MainThm}.\nThen there exists a constant $0 < \\tau_0 < 1$ such that the following holds:\nfor every $\\tau \\in (0,\\tau_0)$ there exists $\\varepsilon_0 = \\varepsilon_0(\\tau) > 0$\nwith the property that, if $B_r(x_0) \\Subset \\Omega$ and one of the following\nconditions is satisfied,\n\\begin{enumerate}[label=(\\roman*)]\n\\item $|E \\cap B_r(x_0)| < \\varepsilon_0 |B_r|$,\n\\item $|B_r(x_0) \\setminus E| < \\varepsilon_0 |B_r|$,\n\\item there exists a half-space $H$ such that\n\\[\n\\frac{|(E \\Delta H) \\cap B_r(x_0)|}{|B_r|} < \\varepsilon_0,\n\\]\n\\end{enumerate}\nthen, for every $0 < \\delta < n$, the estimate\n\\[\n\\int_{B_{\\tau r}(x_0)} |\\nabla u|^p \\, dx\n\\leq\nC_0 \\tau^{n-\\delta}\n\\left(\n\\int_{B_r(x_0)} |\\nabla u|^p \\, dx + r^n\n\\right)\n\\]\nholds, where the constant $C_0$ depends only on\n$n, \\nu, L, \\alpha,\\beta, \\delta$ and $\\|\\nabla u\\|_{L^2(\\Omega)}$.\n\\end{Prop}\n\\begin{Rem}\n\\label{Reg}\n For the proof of Theorem \\ref{MainThm}, it would be sufficient to state the proposition only in case (i). We nevertheless include the other two cases for the sake of completeness.\n\\end{Rem}\n\n\\begin{Lem}[Standard $p$-growth bounds]\\label{lem:p-growth}\nAssume \\((H1)\\)--\\((H2)\\). Then there exists a constant $C=C(n,p,\\nu,L)>0$ such that for all $\\xi\\in\\R^n$\n\\[\nC^{-1}(\\mu^2+|\\xi|^2)^{\\frac p2}-C\\mu^p \\le F(\\xi)\\le C(\\mu^2+|\\xi|^2)^{\\frac p2}+C\\mu^p .\n\\]\n\\end{Lem}\nThe following results on higher integrability and local Hölder continuity\nof minimizers are classical. Since their proofs are standard, we omit them and refer the reader to\n\\cite{Giu,GiaMar}.\n{\n\\begin{Thm}[Higher integrability]\n\\label{HigherInt}\nLet $u \\in W^{1,p}(\\Omega)$ be a local minimizer of the functional \\eqref{MainF}.\nThere exist $s > 1$ and $C = C(n,p,\\nu,L,\\beta)$ such that for any ball\n$B_{2r}(x_{0})\\subset\\subset\\Omega$\n\\begin{equation}\n \\int_{B_r(x_0)}|\\nabla u|^{sp}\\,dx\\leq C\\bigg(\\int_{B_{2r}(x_0)}\\big(|\\nabla u|^p+\\mu^p\\big)\\,dx\n \\bigg)^s.\n\\end{equation}\n\\end{Thm}\n\\begin{Thm}[H\\\"older continuity]Let $u \\in W^{1,p}(\\Omega)$ be a local minimizer of the functional \\eqref{MainF}. Then\n\\begin{enumerate}[label=(\\roman*)]\n\\item For any open set $\\Omega'\\Subset \\Omega$ the quantity $\\norm{u}_{L^{\\infty}(\\Omega')}$ is bounded by a constant $C = C(n,p,\\nu,L,\\beta)\\norm{u}_{L^2(\\Omega)}$.\n\\item u is locally H\\\"older continuous in $\\Omega$.\n\\end{enumerate}\n\\end{Thm}\n}\n\n\\begin{Thm}\\label{DS}\nLet $u$ be a minimizer of $\\mathcal{F}$ in $B_r(x_0)$. Then $\\partial_i u\\in W^{1,2}_{loc}(B_{r})$ for every $i=1\\dots n-1$. Moreover the following properties hold:\n\\begin{enumerate}[label=\\roman*)]\n\\item $u \\in W^{2,2}_{\\mathrm{loc}}(B_r^+)$ e $u \\in W^{2,2}_{\\mathrm{loc}}(B_r^-)$;\n\\item for every $x_0 \\in \\Gamma_r$ and $R>0$ such that $B_{4R}(x_0)\\subset B_r$, it holds that\n\\begin{align}\n\\int_{B_{R/2}} V^{p-2}\\,|\\nabla\\nabla' u|^2 \\,dx\n&\\leq \\frac{C}{R^2}\\int_{B_{2R}} V^p \\,dx, \\label{eq:tangential-estimate}\n\\end{align}\nwhere $C=C\\!\\left(\\tfrac{L}{\\nu},\\,n,\\,p,\\,\\alpha,\\,\\beta,\\mu\\right)$ is a positive constant.\n\\end{enumerate}\n\\end{Thm}\n\n\\section{Proof of Proposition 1.3}\n\\begin{proof}\nFix $\\tau\\in(0,\\tau_0)$ (to be chosen small only depending on $n,p,\\nu,L,\\beta/\\alpha$) and assume without loss of generality that $x_0=0$.\nWe prove that there exists $\\e_0=\\e_0(\\tau)>0$ such that if $B_r(x_0)\\Subset\\Om$ and one of {\\rm(i)}--{\\rm(iii)}\nholds, then for every $0<\\delta0$.\\\\\n\\indent \n\\noindent\\textbf{Step 1: the flat comparison in case (iii).}\nAssume that (iii) holds true. Up to a rigid motion we may assume that $H=\\{x_n>0\\}$ and set\n\\[\nE_H:=H\\cap B_{r},\\quad \\sigma_H:=\\beta\\mathbbm 1_{E_H}+\\alpha\\mathbbm 1_{B_r\\setminus E_H}.\n\\]\nLet $u_H\\in u+W^{1,p}_0(B_\\frac{r}{2})$ be the unique minimizer of the functional\n\\[\n\\mathcal F_H(w;B_\\frac{r}{2}):=\\int_{B_\\frac{r}{2}}\\sigma_H(x)\\,F(\\nabla w)\\,dx .\n\\]\nSubtracting the Euler-Lagrange equation satisfied by $u$ from the one satisfied by $u_H$ and testing them with $u-u_H$, we get\n\\begin{align}\n \\int_{B_{\\frac{r}{2}}} \\sigma_E\n\\big(DF(\\nabla u)-DF(\\nabla u_H)\\big)\n\\cdot(\\nabla u-\\nabla u_H)\\,dx\n& =\n\\int_{B_{\\frac{r}{2}}} (\\sigma_H-\\sigma_E)\\,\nDF(\\nabla u_H)\\cdot(\\nabla u-\\nabla u_H)\\,dx\n\\end{align}\nUsing assumptions \\eqref{Monotonicity} and \\eqref{Growth} and H\\\"older's inequality, for every $\\varepsilon>0$ it holds that\n\\begin{align}\n \\int_{B_{\\frac{r}{2}}}|\\D u_H-\\D u|^p\\,dx\n & \\leq C\\int_{(E\\Delta E_H)\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D u_H|^2\\big)^{\\frac{p-2}{2}}|\\D u-\\D u_H|\\,dx\\\\\n& \\leq C\\bigg(\\frac{1}{\\varepsilon}\\int_{(E\\Delta E_H)\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D u_H|^2\\big)^{\\frac{p}{2}}\\,dx+\\varepsilon\\int_{B_\\frac{r}{2}}|\\D u-\\D u_H|^p\\,dx \\bigg).\n\\end{align}\nChoosing $\\varepsilon$ sufficiently small, we get \n\\begin{align}\n \\int_{B_{\\frac{r}{2}}}|\\D u_H-\\D u|^p\\,dx\n& \\leq C\\int_{(E\\Delta E_H)\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D u_H|^2\\big)^{\\frac{p}{2}}\\,dx.\n\\end{align}\nThus, the minimality of $u_H$ with respect to $u$, H\\\"older's inequality and Theorem \\ref{HigherInt} yield\n\\begin{align}\n\\label{eqq1}\n\\int_{B_{\\frac{r}{2}}}|\\D u_H-\\D u|^p\\,dx& \\leq \\int_{(E\\Delta E_H)\\cap B_\\frac{r}{2}} (1+|\\nabla u|^2)^{\\frac p2}\\,dx\\\\\n& \\le \\bigg(\\frac{|(E\\Delta E_H)\\cap B_r|}{|B_r|}\\bigg)^{1-\\frac1s}|B_r|\\bigg(\\dashint_{B_r}(1+|\\nabla u|^2)^{\\frac{sp}{2}}\\,dx\\bigg)^{\\frac1s}\\\\\n& \\le C\\,\\e_0^{1-\\frac1s}\\int_{B_r}(1+|\\nabla u|^2)^{\\frac p2}\\,dx,\n\\end{align}\nwhere we have used assumption \\emph{iii)} and $C=C(n,p,\\beta/\\alpha,\\nu/L)$ is a positive constant.\n\n\\medskip\n\\noindent\\textbf{Step 3: the one-phase cases (i) and (ii).}\nAssume (i): $|E\\cap B_r|<\\e_0|B_r|$ (the case (ii) is analogous). Let us denote by $v\\in u+W^{1,p}_0(B_\\frac{r}{2})$ be the unique minimizer of the functional\n\\[\n\\mathcal G(w):=\\int_{B_\\frac{r}{2}}\\,F(\\nabla w)\\,dx .\n\\]\nWe can proceed as in step $(iii)$ subtracting the Euler-Lagrange equation satisfied by $u$ from the one satisfied by $v$ and testing them with $u-v$, getting\n\\begin{align}\n \\int_{B_{\\frac{r}{2}}} \\sigma_E\n\\big(DF(\\nabla u)-DF(\\nabla v)\\big)\n\\cdot(\\nabla u-\\nabla v)\\,dx\n& =\n\\int_{B_{\\frac{r}{2}}} (\\alpha-\\sigma_E)\\,\nDF(\\nabla v)\\cdot(\\nabla u-\\nabla v)\\,dx\n\\end{align}\nUsing assumptions \\eqref{Monotonicity} and \\eqref{Growth} and H\\\"older's inequality, for every $\\varepsilon>0$ it holds that\n\\begin{align}\n \\int_{B_{\\frac{r}{2}}}|\\D v -\\D u|^p\\,dx\n & \\leq C\\int_{E\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D v|^2\\big)^{\\frac{p-2}{2}}|\\D u-\\D v|\\,dx\\\\\n& \\leq C\\bigg(\\frac{1}{\\varepsilon}\\int_{E\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D v|^2\\big)^{\\frac{p}{2}}\\,dx+\\varepsilon\\int_{B_\\frac{r}{2}}|\\D u-\\D v|^p\\,dx \\bigg).\n\\end{align}\nChoosing $\\varepsilon$ sufficiently small, we get \n\\begin{align}\n \\int_{B_{\\frac{r}{2}}}|\\D v-\\D u|^p\\,dx\n& \\leq C\\int_{E\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D v|^2\\big)^{\\frac{p}{2}}\\,dx.\n\\end{align}\nUsing the minimality of $v$ compared with $u$ and Theorem\n\\ref{HigherInt} we deduce,\n\\begin{align}\n\\label{eqq11}\n\\int_{B_{\\frac{r}{2}}}|\\D v-\\D u|^p\\,dx& \\leq \\int_{E\\cap B_{\\frac{r}{2}}} (1+|\\nabla u|^2)^{\\frac p2}\\,dx\\\\\n& \\le \\bigg(\\frac{E\\cap B_{r}}{|B_r|}\\bigg)^{1-\\frac1s}|B_r|\\bigg(\\dashint_{B_r}(1+|\\nabla u|^2)^{\\frac{sp}{2}}\\,dx\\bigg)^{\\frac1s}\\\\\n& \\le C\\,\\e_0^{1-\\frac1s}\\int_{B_r}(1+|\\nabla u|^2)^{\\frac p2}\\,dx,\n\\end{align}\nwhere we have used assumption \\emph{i)} and $C=C(n,p,\\beta/\\alpha,\\nu/L)$ is a positive constant. Thereafter we can argue as in the previous step using the classical decay estimate for minimizers of regular integrals in the Calculus of Variations\n\\[\n\\int_{B_{\\tau \\rho}}|\\nabla v|^p\\,dx\n\\le C\\tau^{n}\\int_{B_{ \\rho}}|\\nabla u|^p\\,dx,\n\\]\nfor every $\\rho\\leq \\frac{r}{2}$.\n\\end{proof}", + "post_theorem_intro_text_len": 3397, + "post_theorem_intro_text": "\\begin{Rem}\nWithin the present variational transmission framework, Morrey regularity represents\nthe natural level of regularity that can be expected for the gradient of local\nminimizers, in view of the discontinuous coefficients and the intrinsic transmission\ncondition.\n\\end{Rem}\nThe proof of Theorem~\\ref{MainThm} is obtained as a consequence of the following\ndecay estimate for the energy.\n\n\\begin{Prop}\\label{FJNL}\nLet $u$ be a local minimizer of the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined\nin \\eqref{MainF} under the assumption of Theorem~\\ref{MainThm}.\nThen there exists a constant $0 < \\tau_0 < 1$ such that the following holds:\nfor every $\\tau \\in (0,\\tau_0)$ there exists $\\varepsilon_0 = \\varepsilon_0(\\tau) > 0$\nwith the property that, if $B_r(x_0) \\Subset \\Omega$ and one of the following\nconditions is satisfied,\n\\begin{enumerate}[label=(\\roman*)]\n\\item $|E \\cap B_r(x_0)| < \\varepsilon_0 |B_r|$,\n\\item $|B_r(x_0) \\setminus E| < \\varepsilon_0 |B_r|$,\n\\item there exists a half-space $H$ such that\n\\[\n\\frac{|(E \\Delta H) \\cap B_r(x_0)|}{|B_r|} < \\varepsilon_0,\n\\]\n\\end{enumerate}\nthen, for every $0 < \\delta < n$, the estimate\n\\[\n\\int_{B_{\\tau r}(x_0)} |\\nabla u|^p \\, dx\n\\leq\nC_0 \\tau^{n-\\delta}\n\\left(\n\\int_{B_r(x_0)} |\\nabla u|^p \\, dx + r^n\n\\right)\n\\]\nholds, where the constant $C_0$ depends only on\n$n, \\nu, L, \\alpha,\\beta, \\delta$ and $\\|\\nabla u\\|_{L^2(\\Omega)}$.\n\\end{Prop}\n\\begin{Rem}\n\\label{Reg}\n For the proof of Theorem \\ref{MainThm}, it would be sufficient to state the proposition only in case (i). We nevertheless include the other two cases for the sake of completeness.\n\\end{Rem}\n\nWe give an outline of the proof of Proposition~\\ref{FJNL}. It relies on decay estimates for the gradient\nnear the interface, first established in a flat configuration, i.e.\\ when the interface coincides\nwith a hyperplane.\nSection~3 is devoted to this ``flat case''.\n\nThe starting point is a local boundedness estimate for the tangential gradient\n$\\nabla' u$, where $\\nabla'=(\\partial_{x_1},\\dots,\\partial_{x_{n-1}})$ denotes the vector of derivatives parallel to\nthe interface.\nThis is proved in Proposition~3.2 via a difference-quotient argument with test functions involving\nonly tangential increments.\nA key ingredient is a Moser iteration applied to\n$Z:=|\\Delta_{i,h}u|^{m/p}$.\nDuring the estimates, the full gradient $\\nabla u$ enters through the structure conditions\n\\emph{(H1)}--\\emph{(H2)} and is treated as a weight; this is the origin of the restriction\n$2 0$\nwith the property that, if $B_r(x_0) \\Subset \\Omega$ and one of the following\nconditions is satisfied,\n\\begin{enumerate}[label=(\\roman*)]\n\\item $|E \\cap B_r(x_0)| < \\varepsilon_0 |B_r|$,\n\\item $|B_r(x_0) \\setminus E| < \\varepsilon_0 |B_r|$,\n\\item there exists a half-space $H$ such that\n\\[\n\\frac{|(E \\Delta H) \\cap B_r(x_0)|}{|B_r|} < \\varepsilon_0,\n\\]\n\\end{enumerate}\nthen, for every $0 < \\delta < n$, the estimate\n\\[\n\\int_{B_{\\tau r}(x_0)} |\\nabla u|^p \\, dx\n\\leq\nC_0 \\tau^{n-\\delta}\n\\left(\n\\int_{B_r(x_0)} |\\nabla u|^p \\, dx + r^n\n\\right)\n\\]\nholds, where the constant $C_0$ depends only on\n$n, \\nu, L, \\alpha,\\beta, \\delta$ and $\\|\\nabla u\\|_{L^2(\\Omega)}$.\n\\end{Prop}\n\nAn outline is given for the proof of Proposition~\\ref{FJNL}:\n\\begin{itemize}\n\\item The argument \\emph{relies on decay estimates for the gradient near the interface}, first established in a \\emph{flat configuration} (interface coincides with a hyperplane), treated next.\n\\item Starting point: a \\emph{local boundedness estimate for the tangential gradient} $\\nabla' u$ (derivatives parallel to the interface). This is proved in Proposition~3.2 via a \\emph{difference-quotient argument} with test functions involving only tangential increments.\n\\item Key ingredient: a \\emph{Moser iteration} applied to $Z:=|\\Delta_{i,h}u|^{m/p}$. In the estimates, the full gradient $\\nabla u$ appears through (H1)--(H2) and is \\emph{treated as a weight}; this is cited as the origin of the restriction $2 0$\nsuch that\n\\[\n\\|\\nabla u\\|_{L^{{p},\\lambda}(\\Omega')} \\leq C.\n\\]", + "theorem_type": [ + "Implication", + "Existential–Universal" + ], + "mcq": { + "question": "Let \\(\\Omega\\subset\\mathbb{R}^n\\) be a bounded domain, let \\(E\\subset\\Omega\\) be measurable with \\(\\partial E\\) a \\(C^1\\) hypersurface, and fix constants \\(0<\\alpha<\\beta<\\infty\\). Define \\(\\sigma_E:=\\beta\\mathbbm{1}_E+\\alpha\\mathbbm{1}_{E^c}\\) and, for \\(v\\in W^{1,p}(\\Omega)\\),\n\\[\\mathcal F(v;E):=\\int_{\\Omega}\\sigma_E(x)F(\\nabla v)\\,dx.\\]\nAssume \\(F\\in C^2(\\mathbb{R}^n)\\) and that for some \\(\\nu,L>0\\), \\(\\mu\\in(0,1]\\), and every \\(\\xi,\\eta\\in\\mathbb{R}^n\\),\n\\[\\langle \\nabla F(\\xi)-\\nabla F(\\eta),\\xi-\\eta\\rangle\\ge \\nu\\big(\\mu^2+|\\xi|^2+|\\eta|^2\\big)^{\\frac{p-2}{2}}|\\xi-\\eta|^2,\\]\n\\[|\\nabla F(\\xi)-\\nabla F(\\eta)|\\le L\\big(\\mu^2+|\\xi|^2+|\\eta|^2\\big)^{\\frac{p-2}{2}}|\\xi-\\eta|,\\]\nwith \\(20\\) such that\n\\[\\|\\nabla u\\|_{L^{p,\\lambda}(\\Omega')}\\le C.\\]" + }, + "choices": [ + { + "label": "B", + "text": "For every \\(0\\le \\lambda0\\), independent of \\(\\lambda\\), such that\n\\[\\|\\nabla u\\|_{L^{p,\\lambda}(\\Omega')}\\le C.\\]" + }, + { + "label": "C", + "text": "One has \\(\\nabla u\\in L^{p,\\lambda}_{\\mathrm{loc}}(\\Omega)\\) for at least one exponent \\(\\lambda\\in[0,n)\\)." + }, + { + "label": "D", + "text": "For every \\(0\\le \\lambda0\\) such that\n\\[\\|\\nabla u\\|_{L^{p,\\lambda}(\\Omega')}\\le C\\,\\|\\nabla u\\|_{L^{p}(\\Omega)}.\\]" + }, + { + "label": "E", + "text": "There exists \\(\\lambda_0\\in(0,n)\\), depending only on \\(n,p,\\nu,L,\\alpha,\\beta\\), such that \\(\\nabla u\\in L^{p,\\lambda_0}_{\\mathrm{loc}}(\\Omega)\\). Moreover, for every \\(\\Omega'\\Subset\\Omega\\), there exists a constant \\(C=C\\big(n,p,\\nu,L,\\beta,E,\\operatorname{diam}(\\Omega),\\operatorname{dist}(\\Omega',\\partial\\Omega)\\big)>0\\) with\n\\[\\|\\nabla u\\|_{L^{p,\\lambda_0}(\\Omega')}\\le C.\\]" + } + ], + "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": "dependence on Morrey exponent in the local bound", + "template_used": "uniformity_effectivity" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "universal quantifier over all \\(\\lambda4$;\nirreducible \nrepresentations\nof $SO(n)$\nwill be denoted\nby \n$\\pi_{n}^{\\mu}$\nas in \\cite[Theorem 19.22]{FulHar}.\n $\\tau=\\pi_{n}^{0}\\otimes \\pi_{2}^\\nu$ \n if $n$ is odd, i.e., $\\tau$ is a character; for $n$ even $\\tau=\n \\pi_{n}^{\\mu}$ \n or its \n tensor product\n with a character\n$\\tau=\\pi_{n}^{\\mu}\n\\otimes \\pi_{2}^\\nu$, \n where $\\mu=(a,\\dots,a,\\pm a)$ for some $a\\ge 0$.\nIf $(G,K) = (Spin_0(n,2), \nSpin(n) \\times SO(2)/\\Z_2)$, where $Spin_0(n,2)$ is the double cover of $SO_0(n,2)$\nthen up to a character $\\tau$ is the spin representation $(\\frac{1}{2}, \\dots, \\frac{1}{2})$ when $n$ is odd, and\n$\\tau= \\pi_{n}^{\\mu}$, $\\mu=(a,\\dots,a,\\pm a)$ for some integer $a\\ge 0$ a half-integer\nwhen $n$ is even.\n\n\\item Type $E_6$, $G=E_{6(-14)}, K = Spin(10) \\times U(1) / \\Gamma$, where $\\Gamma$ is a finite subgroup.\n$\\tau=\\pi_{\\omega_1} \\otimes \\chi_n$ for $n \\equiv 1 \\mod 4$, or $\\pi_{\\omega_2} \\otimes \\chi_n$ for $n \\equiv 3 \\mod 4$. Here $\\omega_1$ and $\\omega_2$ are the positive and negative spin representations of $Spin(10)$ and $U(1) = \\{e^{it \\frac{Z}{2}} \\ | \\ t \\in \\R \\}$.\n\n\\item Type $E_7$, $G=E_{7(-25)}, K = E_6 U(1)$.\n$\\tau$ is a character.\n\n\\end{enumerate} \n\\end{theorem}", + "finiterepprop": "\\begin{proposition}\n\\label{finiterepprop}\n(1) Let $(W_{\\lambda}, G^{\\C})$ be a highest weight representation of $G^{\\C}$ and its Lie algebra $\\mathfrak{g}^{\\C}$. For the multiplicities we have $[V_{\\tau} : W_{\\lambda}|_K] \\leq 1$, and if $V_{\\tau} \\subseteq W_{\\lambda}|_K$ we have $[W_{\\lambda} : I_{\\sigma,-\\lambda|_{\\af} - \\rho_{\\af}}^0 ] \\leq 1$, viewed as Lie algebra representations. \n\n(2)\nLet $(W_\\lambda, G ^{\\C})$ be\na finite-dimensional\nirreducible representation\nsuch that \n$V_{\\tau} \\subseteq W_{\\lambda}|_K$. Up to constants there is a unique $G$-equivariant map\n$$F : W_{\\lambda} \\rightarrow C^{\\infty}(G/K,V_{\\tau})$$\ndefined by\n$$F(w)(g) = P_{\\tau} (g^{-1} \\cdot w), g\\in G, w\\in W,$$\nwhere $P_{\\tau}$ is the projection onto $V_{\\tau}$. Moreover, if we normalize $J_{\\lambda}$ so that for all $v \\in V_{\\tau}$ we have $J_{\\lambda}(v) = f_v$, where $f_v(kan) = a^{\\lambda|_{\\mathfrak{a}}} P_{\\sigma} \\tau(k)^{-1} v$,\nthen there is a factorization\nof $F$ \nas a product $G$-equivariant maps,\n$$\nF=\nS_{\\sigma, -\\lambda|_{\\mathfrak{a}} - \\rho_{\\mathfrak{a}}}\nJ_{\\lambda}: (W_\\lambda, G^{\\C})\\to I_{\\sigma, -\\lambda|_{\\mathfrak{a}} - \\rho_{\\mathfrak{a}}}^0\n\\to C^\\infty(G/K,V_{\\tau}).\n$$\n\\end{proposition}", + "egvaluethm": "\\begin{theorem}\n\\label{egvaluethm}\nAssume the irreducible finite-dimensional representation $V_\\tau$ \nof $K$ is multiplicity-free when restricted to $M$.\n\n(1) For any $M$-irreducible subrepresentation $(\\sigma,U_\\sigma)$ in $\\tau|_M$, $f \\in I_{\\sigma,\\nu}^0$ and $D \\in \\mathcal{D}^G(G/K, V_{\\tau})$, we have $DS_{\\sigma, \\nu}(f) = \\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}})S_{\\sigma, \\nu}(f)$ for a constant $\\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}})$.\nFurthermore, we have the invariance property:\n$$\\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}}) = \\omega(D)(s \\cdot \\sigma,s \\cdot \\nu + \\rho_{\\mathfrak{a}}), \\ \\forall \\ s\\in W.$$\n\n(2) Let $(W_\\lambda, G^{\\C})$\nbe a finite-dimensional\nrepresentation of $\nG^{\\C}$ containing $V_\\tau$. Let $D \\in \\mathcal{D}^G(G/K,V_{\\tau})$, then $D$ acts on \n$F(W_\\lambda)$\nas the constant\n$$\nDF(w) = \\omega(D)(\\sigma_{\\lambda},-\\lambda|_{\\mathfrak{a}})F(w), \n\\quad \\ \\forall \\ w \\in W_\\lambda.\n$$\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 2244, + "pre_theorem_intro_text": "In the present paper we shall study invariant differential operators on homogeneous vector bundles over Hermitian symmetric domains, their eigenfunctions and eigenvalues. \nLet $G$ be a\nnon-compact semisimple \nHermitian Lie group and $K$ its maximal compact subgroup. Let\n$V_\\tau$\nbe a finite-dimensional irreducible \nrepresentation of $K$ and $C^\\infty(G/K,V_{\\tau})$ \nbe the space of smooth sections \nof the homogeneous vector \nbundle \n$G\\times_{(K, \\tau)} V_\\tau$\ndefined by $V_\\tau$.\nThe algebraic properties\nof \nthe ring\n$\\mathcal D^G(G/K,V_{\\tau})$ \nof invariant\ndifferential operators on\n$C^\\infty(G/K,V_{\\tau})$ is of fundamental interest, both in analysis on symmetric spaces \n and in the study of universal\n enveloping algebras \\cite{deitmar, helg, ricc, Sh}.\nWhen $(K, \\tau)$\nis a character, i.e., a one-dimensional\nrepresentation, the ring is commutative, and \nShimura has constructed\na linear basis \n$\\{\\mathcal L_{\\mu}\\}$\nconsisting of formally positive \noperators\nby using the Schmid decomposition\nof the symmetric tensor\nalgebra\n$S(\\mathfrak p^+)$\nunder $K$, where \n$\\mathfrak p^+$\nis the holomorphic tangent space of $G/K$\nat $o=K\\in G/K$.\nFor general \n$V_\\tau$ it is proved by Deitmar\n\\cite{deitmar} \nthat \n$\\mathcal D^G(G/K, V_\\tau)$\nis commutative\nif and only if \nthe restriction $\\tau|_M$\nto $M$ is multiplicity-free, where $M$\nis the centralizer in $K$ of the real Cartan group of $G$. \nIn the present paper we shall\nclassify all such representations\n$(K, \\tau)$. \nWe construct \n eigenfunctions of invariant\ndifferential operators\nusing the Szegö transform and\nwe study the invariance\nproperties of the eigenvalues\nof invariant differential\noperators. This question can be posed for any symmetric space, we study the Hermitian case because there are canonical constructions of invariant differential operators related to the Hua-Kostant-Schmid decomposition \\cite{Sh} of invariant differential operator. It is an interesting problem to further study the eigenvalue problem for those operators;\nsee \\cite{SZ}\nfor the case of one-dimensional representations $V_\\tau$ of $K$. \n\n\\subsection{Main\nresults and methods}\n\nThe main result of this paper\nis the following theorem; see\nbelow for the exact notation\nand definitions.", + "context": "In the present paper we shall study invariant differential operators on homogeneous vector bundles over Hermitian symmetric domains, their eigenfunctions and eigenvalues. \nLet $G$ be a\nnon-compact semisimple \nHermitian Lie group and $K$ its maximal compact subgroup. Let\n$V_\\tau$\nbe a finite-dimensional irreducible \nrepresentation of $K$ and $C^\\infty(G/K,V_{\\tau})$ \nbe the space of smooth sections \nof the homogeneous vector \nbundle \n$G\\times_{(K, \\tau)} V_\\tau$\ndefined by $V_\\tau$.\nThe algebraic properties\nof \nthe ring\n$\\mathcal D^G(G/K,V_{\\tau})$ \nof invariant\ndifferential operators on\n$C^\\infty(G/K,V_{\\tau})$ is of fundamental interest, both in analysis on symmetric spaces \n and in the study of universal\n enveloping algebras \\cite{deitmar, helg, ricc, Sh}.\nWhen $(K, \\tau)$\nis a character, i.e., a one-dimensional\nrepresentation, the ring is commutative, and \nShimura has constructed\na linear basis \n$\\{\\mathcal L_{\\mu}\\}$\nconsisting of formally positive \noperators\nby using the Schmid decomposition\nof the symmetric tensor\nalgebra\n$S(\\mathfrak p^+)$\nunder $K$, where \n$\\mathfrak p^+$\nis the holomorphic tangent space of $G/K$\nat $o=K\\in G/K$.\nFor general \n$V_\\tau$ it is proved by Deitmar\n\\cite{deitmar} \nthat \n$\\mathcal D^G(G/K, V_\\tau)$\nis commutative\nif and only if \nthe restriction $\\tau|_M$\nto $M$ is multiplicity-free, where $M$\nis the centralizer in $K$ of the real Cartan group of $G$. \nIn the present paper we shall\nclassify all such representations\n$(K, \\tau)$. \nWe construct \n eigenfunctions of invariant\ndifferential operators\nusing the Szegö transform and\nwe study the invariance\nproperties of the eigenvalues\nof invariant differential\noperators. This question can be posed for any symmetric space, we study the Hermitian case because there are canonical constructions of invariant differential operators related to the Hua-Kostant-Schmid decomposition \\cite{Sh} of invariant differential operator. It is an interesting problem to further study the eigenvalue problem for those operators;\nsee \\cite{SZ}\nfor the case of one-dimensional representations $V_\\tau$ of $K$.\n\n\\subsection{Main\nresults and methods}\n\nThe main result of this paper\nis the following theorem; see\nbelow for the exact notation\nand definitions.", + "full_context": "In the present paper we shall study invariant differential operators on homogeneous vector bundles over Hermitian symmetric domains, their eigenfunctions and eigenvalues. \nLet $G$ be a\nnon-compact semisimple \nHermitian Lie group and $K$ its maximal compact subgroup. Let\n$V_\\tau$\nbe a finite-dimensional irreducible \nrepresentation of $K$ and $C^\\infty(G/K,V_{\\tau})$ \nbe the space of smooth sections \nof the homogeneous vector \nbundle \n$G\\times_{(K, \\tau)} V_\\tau$\ndefined by $V_\\tau$.\nThe algebraic properties\nof \nthe ring\n$\\mathcal D^G(G/K,V_{\\tau})$ \nof invariant\ndifferential operators on\n$C^\\infty(G/K,V_{\\tau})$ is of fundamental interest, both in analysis on symmetric spaces \n and in the study of universal\n enveloping algebras \\cite{deitmar, helg, ricc, Sh}.\nWhen $(K, \\tau)$\nis a character, i.e., a one-dimensional\nrepresentation, the ring is commutative, and \nShimura has constructed\na linear basis \n$\\{\\mathcal L_{\\mu}\\}$\nconsisting of formally positive \noperators\nby using the Schmid decomposition\nof the symmetric tensor\nalgebra\n$S(\\mathfrak p^+)$\nunder $K$, where \n$\\mathfrak p^+$\nis the holomorphic tangent space of $G/K$\nat $o=K\\in G/K$.\nFor general \n$V_\\tau$ it is proved by Deitmar\n\\cite{deitmar} \nthat \n$\\mathcal D^G(G/K, V_\\tau)$\nis commutative\nif and only if \nthe restriction $\\tau|_M$\nto $M$ is multiplicity-free, where $M$\nis the centralizer in $K$ of the real Cartan group of $G$. \nIn the present paper we shall\nclassify all such representations\n$(K, \\tau)$. \nWe construct \n eigenfunctions of invariant\ndifferential operators\nusing the Szegö transform and\nwe study the invariance\nproperties of the eigenvalues\nof invariant differential\noperators. This question can be posed for any symmetric space, we study the Hermitian case because there are canonical constructions of invariant differential operators related to the Hua-Kostant-Schmid decomposition \\cite{Sh} of invariant differential operator. It is an interesting problem to further study the eigenvalue problem for those operators;\nsee \\cite{SZ}\nfor the case of one-dimensional representations $V_\\tau$ of $K$.\n\n\\subsection{Main\nresults and methods}\n\nThe main result of this paper\nis the following theorem; see\nbelow for the exact notation\nand definitions.\n\nIn the present paper we shall study invariant differential operators on homogeneous vector bundles over Hermitian symmetric domains, their eigenfunctions and eigenvalues. \nLet $G$ be a\nnon-compact semisimple \nHermitian Lie group and $K$ its maximal compact subgroup. Let\n$V_\\tau$\nbe a finite-dimensional irreducible \nrepresentation of $K$ and $C^\\infty(G/K,V_{\\tau})$ \nbe the space of smooth sections \nof the homogeneous vector \nbundle \n$G\\times_{(K, \\tau)} V_\\tau$\ndefined by $V_\\tau$.\nThe algebraic properties\nof \nthe ring\n$\\mathcal D^G(G/K,V_{\\tau})$ \nof invariant\ndifferential operators on\n$C^\\infty(G/K,V_{\\tau})$ is of fundamental interest, both in analysis on symmetric spaces \n and in the study of universal\n enveloping algebras \\cite{deitmar, helg, ricc, Sh}.\nWhen $(K, \\tau)$\nis a character, i.e., a one-dimensional\nrepresentation, the ring is commutative, and \nShimura has constructed\na linear basis \n$\\{\\mathcal L_{\\mu}\\}$\nconsisting of formally positive \noperators\nby using the Schmid decomposition\nof the symmetric tensor\nalgebra\n$S(\\mathfrak p^+)$\nunder $K$, where \n$\\mathfrak p^+$\nis the holomorphic tangent space of $G/K$\nat $o=K\\in G/K$.\nFor general \n$V_\\tau$ it is proved by Deitmar\n\\cite{deitmar} \nthat \n$\\mathcal D^G(G/K, V_\\tau)$\nis commutative\nif and only if \nthe restriction $\\tau|_M$\nto $M$ is multiplicity-free, where $M$\nis the centralizer in $K$ of the real Cartan group of $G$. \nIn the present paper we shall\nclassify all such representations\n$(K, \\tau)$. \nWe construct \n eigenfunctions of invariant\ndifferential operators\nusing the Szegö transform and\nwe study the invariance\nproperties of the eigenvalues\nof invariant differential\noperators. This question can be posed for any symmetric space, we study the Hermitian case because there are canonical constructions of invariant differential operators related to the Hua-Kostant-Schmid decomposition \\cite{Sh} of invariant differential operator. It is an interesting problem to further study the eigenvalue problem for those operators;\nsee \\cite{SZ}\nfor the case of one-dimensional representations $V_\\tau$ of $K$.\n\nThe main result of this paper\nis the following theorem; see\nbelow for the exact notation\nand definitions.\n\nThe proof of the \nclassification\nis through a case by case\ncomputation\nand is technically involved.\nWe shall repeatedly use the classical branching rules in\n\\cite{good},\nthe classifications\nof weight-free\nrepresentations\nin \\cite{Howe}\nand multiplicity-free results in\n\\cite{St}.\nThe proof \nfor the uniqueness\nof $F: W_\\lambda \\to \nC^\\infty(G/K, V_\\tau)\n$ is somewhat\nnatural,\nit is done through\nthe realization\n$J: W_\\lambda \n\\to Ind_{MAN}^G(\\sigma\\otimes \\alpha\\otimes 1)$\nof finite-dimensional\nrepresentations $W_\\lambda$\nin the induced representation\nin $Ind_{MAN}^G(\\sigma\\otimes \\alpha\\otimes 1)$,\na factorization\nof $F=SJ$\nas a product of $J$\nand the Poisson-Szeg\\\"o{}\ntransform, and the multiplicity-free property of $V_\\tau$\nunder $M$. The result\non the invariance\nof eigenvalues\nof differential\noperators is \na consequence of a\ngeneral result of Lepowsky \\cite{L}\nand the factorization.\n\n(2)\nLet $(W_\\lambda, G ^{\\C})$ be\na finite-dimensional\nirreducible representation\nsuch that \n$V_{\\tau} \\subseteq W_{\\lambda}|_K$. Up to constants there is a unique $G$-equivariant map\n$$F : W_{\\lambda} \\rightarrow C^{\\infty}(G/K,V_{\\tau})$$\ndefined by\n$$F(w)(g) = P_{\\tau} (g^{-1} \\cdot w), g\\in G, w\\in W,$$\nwhere $P_{\\tau}$ is the projection onto $V_{\\tau}$. Moreover, if we normalize $J_{\\lambda}$ so that for all $v \\in V_{\\tau}$ we have $J_{\\lambda}(v) = f_v$, where $f_v(kan) = a^{\\lambda|_{\\mathfrak{a}}} P_{\\sigma} \\tau(k)^{-1} v$,\nthen there is a factorization\nof $F$ \nas a product $G$-equivariant maps,\n$$\nF=\nS_{\\sigma, -\\lambda|_{\\mathfrak{a}} - \\rho_{\\mathfrak{a}}}\nJ_{\\lambda}: (W_\\lambda, G^{\\C})\\to I_{\\sigma, -\\lambda|_{\\mathfrak{a}} - \\rho_{\\mathfrak{a}}}^0\n\\to C^\\infty(G/K,V_{\\tau}).\n$$\n\\end{proposition}\n\nWe want to prove invariance properties on the eigenvalues of differential operators. These invariance properties have been used in \\cite{SZ} to compute the eigenvalues of Shimura differential operators for one-dimensional representations $\\tau$. We first state some facts derived from \\cite[Section 3]{L}. Note that from the Iwasawa decomposition for the Lie algebra we have the decomposition \n$$U(\\mathfrak g^{\\mathbb C}\n)\n=U(\\mathfrak a^{\\mathbb C}) U(\\mathfrak k^{\\mathbb C})\\oplus\n\\mathfrak n^{\\mathbb C}\nU(\\mathfrak g^{\\mathbb C}).$$\nLet \n$$\n\\Omega:U( \\mathfrak g^{\\mathbb C})\\rightarrow U( \\mathfrak a^{\\mathbb C}) U(\\mathfrak k^{\\mathbb C})\n$$\nbe the correponding\nprojection. We give $U(\\mathfrak a^{\\mathbb C}) U(\\mathfrak k^{\\mathbb C})$ an algebra structure by identifying it with the algebra $U(\\mathfrak a^{\\mathbb C}) \\otimes U(\\mathfrak k^{\\mathbb C})$ and regard $\\Omega$ as a map to $U( \\mathfrak a^{\\mathbb C}) \\otimes U(\\mathfrak k^{\\mathbb C})$. Then $\\Omega(uv)=\\Omega(v)\\Omega(u), $ for any $u \\in U(\\mathfrak g^{\\mathbb C}), v \\in (U(\\mathfrak g^{\\mathbb C})^{K}$ and $$\\Omega(U(\\mathfrak g^{\\mathbb C})^{M})\\subseteq U( \\mathfrak a^{\\mathbb C}) \\otimes U(\\mathfrak k^{\\mathbb C})^{M}.$$\nLet $T:U(\\mathfrak k^{\\mathbb C})\\rightarrow U(\n\\mathfrak k^{\\mathbb C})$ be the canonical anti-automorphism of $U(\\mathfrak k^{\\mathbb C})$ defined by :\n$$T(1)=1, T(x)=-x, T(xy)=T(y)T(x).$$\nPut $$\\Omega_{\\tau}=(1\\otimes \\tau) \\circ (1 \\otimes T)\\circ \\Omega$$\nThen $ \\Omega_{\\tau}:U(\\mathfrak g^{\\mathbb C})^{K}\\rightarrow \nU(\n\\mathfrak a^{\\mathbb C}\n)\\otimes \\mbox{End}_{M}V_{\\tau}$ is an algebra homomorphism.\nAny $\\lambda \\in \\mathfrak (a^{\\mathbb C})'$\ncan be extended to an evaluation on \n$\nU(\\mathfrak a^{\\mathbb C})$, which we also denote by $\\lambda$.\nLet\n$$\\Omega_{\\tau, \\lambda}=(\\lambda\\otimes 1)\\circ \\Omega_{\\tau}:U(\\mathfrak g^{\\mathbb C})^{K}\\rightarrow \\mbox{End}_{M}V_{\\tau}.$$\n\n\\begin{lemma}\n\\label{egvalue}\nFor $v \\in V_{\\tau}$ let $f_{v, \\lambda}\\in C^\\infty(G/K, V_\\tau)$\nbe defined by $f(x) = \\Psi_{\\tau, \\lambda}(x) v, x\\in G$,\nwhere\n$\\Psi_{\\tau, \\lambda}\n$ is defined in (\\ref{Psi}). For any $D \\in U(\\mathfrak{g}^{\\C})^{K}\n\\subset\n(\nU(\\mathfrak{g}^{\\C}\\otimes End(V_\\tau))^{K}\n$\nrealized \nas differential\noperator on $C^\\infty(G/K, V_\\tau)$\nvia $\\nabla(D)$ in\n(\\ref{diffopUnAlg})\nwe have $$(\\nabla(D)f_{v, \\lambda})(x)= \\Psi_{\\tau, \\lambda}(x) \\Omega_{\n\\tau, -\\lambda+\\rho_{\\mathfrak{a}}\n}(D)v, \\ \\forall \\ x \\in G.$$\n \\end{lemma}\n\n(1) For any $M$-irreducible subrepresentation $(\\sigma,U_\\sigma)$ in $\\tau|_M$, $f \\in I_{\\sigma,\\nu}^0$ and $D \\in \\mathcal{D}^G(G/K, V_{\\tau})$, we have $DS_{\\sigma, \\nu}(f) = \\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}})S_{\\sigma, \\nu}(f)$ for a constant $\\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}})$.\nFurthermore, we have the invariance property:\n$$\\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}}) = \\omega(D)(s \\cdot \\sigma,s \\cdot \\nu + \\rho_{\\mathfrak{a}}), \\ \\forall \\ s\\in W.$$\n\n\\begin{proof}(1) Recall that any invariant differential operator $D \\in \\mathcal{D}^G(G/K, V_{\\tau})$ can be represented by an element of $U(\\mathfrak{g}^{\\mathbb C})^K$ via\nthe natural\nmap (\\ref{diffopUnAlg});\nsee also \\cite[Proposition 2.1]{Sh},\nand \n$\\Omega_{\\nu,\\tau}(D)$ \nis well-defined using\nthe representative in $U(\\mathfrak{g}^{\\C})$.\nAs $\\tau|_M$ is multiplicity-free and $\\Omega_{-\\nu + \\rho_{\\mathfrak{a}}}(D)|_{U_{\\sigma}} \\in \\mbox{End}_{M}(U_\\sigma) = \\C I_{U_{\\sigma}}$, by Lemma \\ref{egvalue},\nwe have $D.S_{\\sigma, \\nu}(f) =\\Omega_{-\\nu + \\rho_{\\mathfrak{a}},U_{\\sigma}}(D)S_{\\sigma, \\nu}(f) = \\omega(D)(\\sigma, \\nu + \\rho_{\\mathfrak{a}}) S_{\\sigma,\\nu}(f)$.\nThe invariance property follows directly from\n\\cite[Theorem 9.8(2)]{L}.", + "post_theorem_intro_text_len": 5239, + "post_theorem_intro_text": "The proof of the \nclassification\nis through a case by case\ncomputation\nand is technically involved.\nWe shall repeatedly use the classical branching rules in\n\\cite{good},\nthe classifications\nof weight-free\nrepresentations\nin \\cite{Howe}\nand multiplicity-free results in\n\\cite{St}.\nThe proof \nfor the uniqueness\nof $F: W_\\lambda \\to \nC^\\infty(G/K, V_\\tau)\n$ is somewhat\nnatural,\nit is done through\nthe realization\n$J: W_\\lambda \n\\to Ind_{MAN}^G(\\sigma\\otimes \\alpha\\otimes 1)$\nof finite-dimensional\nrepresentations $W_\\lambda$\nin the induced representation\nin $Ind_{MAN}^G(\\sigma\\otimes \\alpha\\otimes 1)$,\na factorization\nof $F=SJ$\nas a product of $J$\nand the Poisson-Szeg\\\"o{}\ntransform, and the multiplicity-free property of $V_\\tau$\nunder $M$. The result\non the invariance\nof eigenvalues\nof differential\noperators is \na consequence of a\ngeneral result of Lepowsky \\cite{L}\nand the factorization.\n\n\\subsection{Related\nresults and questions}\n\nWhen $ \\tau$ is a character Shimura \\cite{Sh} \nhas constructed\na system of $r$ generators\nfor the ring $\\mathcal D^G(G/K, V_\\tau)$ \nand proposed several\nquestions on determining\nthe domains of positivity\nfor the eigenvalues\nof the generators\nand of the \nwhole linear basis\n$\\{\\mathcal L_{\\mu}\\}$\nof formally positive\noperators. A partial\nanswer to these\nquestions has been obtained\nin \\cite{SZ, Zhang}. Here invariance of eigenvalues under the Weyl group was used. \nWe may also pose similar\nquestions as above for $(K, \\tau)$\nin the list of our classification. \nThis is related to the antihomomorphism $\\Omega:U(\\mathfrak{g})^{K}\\rightarrow U( \\mathfrak a^{\\mathbb C}) \\otimes U(\\mathfrak k^{\\mathbb C})^{M}$ \nin Section \\ref{sect-4}; see \\cite{ACT1, ACT2}\nfor the case\nof real rank \none groups.\n\nThe computation\nof eigenvalues\nof invariant \ndifferential\noperators is closely\nrelated to the \nproblem of characterizing\nfinite-dimensional representations\nof $G$ containing\na fixed $K$-type $\\tau$.\nWhen $\\tau$\nis a character there\nis the Cartan-Helgason\ntheorem and its generalization\nby Schlichtkrull \\cite{schlichtkrull}. On\nthe analytic side there is the explicit Plancherel formula for\nthe $L^2(G/K, V_\\tau)$\nby Shimeno \\cite{Shimeno}.\nIt might be possible\nto find a characterization\nof the finite-dimensional\n$G$-representations\ncontaining $(\\tau, K)$\nand to find\nthe discrete\nseries $(\\pi, G)$\ncontained in \n$L^2(G/K, V_\\tau)$,\nin this case \nthey appear with multiplicity\nat most one (see e.g. \\cite[Prop. 2.2, (40)]{Campo},\n\\cite{ricc} or \\cite[Vol. II, Proposition 6.1.1.6]{war}).\n\n\\subsection{List of notation and\nsymbols}\n\n\\begin{enumerate}\n\\item $G/K$: Hermitian\nsymmetric space.\n\n\\item \n$\n\\mathfrak g=\\mathfrak\nk +\\mathfrak p$,\n$\\mathfrak g^{\\mathbb C}=\n\\mathfrak p^{-}\n+\\mathfrak\nk^{\\mathbb C}\n+\\mathfrak p^+\n$: Cartan and\nHarish-Chandra decompositions of the Lie algebra $\\mathfrak g$ of $G$ and its\ncomplexification $\n\\mathfrak g^{\\mathbb C}$.\n\n\\item $\\mathfrak t^{\\mathbb C}\n=\\mathfrak\n(t^-)^{\\mathbb C} \n\\oplus \n\\mathfrak (t^+)^{\\mathbb C} \n\\subset \n\\mathfrak\nk^{\\mathbb C}$:\nCartan\nsubalgebra of $\\mathfrak{k}^{\\mathbb C}$, and\n$\\gamma_r>\\cdots >\\gamma_1$ are the\nHarish-Chandra strongly\nortogonal roots\nwith $\\gamma_j|_{\n\\mathfrak t_{\\mathbb C}^+}=0 $ and \n $\\gamma_r$ being\n the highest non-compact\n root.\n\n\\item \n $\\mathfrak a\n \\subset\\mathfrak p$:\n maximal abelian \nsubspace (Cartan subspace) of $\\mathfrak p$;\n$c: \\mathfrak a^{\\mathbb C}\n\\to \\mathfrak (t^{-})^{\\mathbb C}\n$ is then the Cayley transform and\n$\\alpha_j = \\gamma_j\\circ c$ is the\nCayley transform\nof the Harish-Chandra\northogonal roots $\\gamma_j$\nto $\\mathfrak a^{\\mathbb C}$. Furthermore,\n$M\\subset K, M'\\subset K$, $W=W(\\mathfrak g, \\mathfrak a)$ are respectively the centralizer\nand normalizer of $\\mathfrak a$\nin $K$, and the Weyl group. Also, $\\mathfrak h^{\\mathbb C}=\n\\mathfrak a^{\\mathbb C}\n+\\mathfrak (t^{+})^{\\mathbb C}$.\n\n\\item \n$(W_{\\lambda}, G^{\\mathbb{C}}),\n(V_\\tau, K),\n(U_\\sigma, M)$:\n Finite-dimensional\nirreducible\nrepresentations\nof the respective\ngroups, the highest\nweight $\\lambda$\nbeing defined\non the Cartan subalgebra $\\mathfrak{h}^{\\mathbb{C}}$ of $\\mathfrak{g}^{\\mathbb{C}}$, and\nhighest weight of\n$\\tau$ being defined\non the Cartan subalgebra $\\mathfrak{t}^{\\mathbb{C}}$.\n\n\\item $G=NAK$, \n$g=n(g)e^{H(g)}k(g)$: the Iwasawa decomposition \nof $G$ with $n(g)\\in N, H(g)\\in \\mathfrak{a}, k(g)\\in K$ and $g \\in G$.\n\n\\item $P=MAN$:\nminimal \nparabolic subgroup of $G$.\n\n\\item $I_{\\sigma, \\nu}=Ind_{P}^G(\\sigma\\otimes e^{\\nu}\\otimes 1)$\n: normalized\ninduced representation\nof $G$ from $P$,\nthe corresponding\nrepresentation \nof $\\mathfrak g^{\\mathbb C}$ on the subspace of $K$-finite vectors will\nalso be written as\n$I_{\\sigma, \\nu}^0$.\n\n\\item $C^\\infty(G/K,V_{\\tau})$: \nspace of\nsmooth sections\nof the vector bundle\n$G\\times_K V_\\tau$\nover $G/K$\ndefined by\nthe representation $(K, \\tau)$. We let $\\mathcal D^G(G/K,V_{\\tau})$ then be the ring \nof $G$-invariant\ndifferential operators on \n$C^\\infty(G/K,V_{\\tau})$.\n\n\\end{enumerate}\n\n\\subsection*{Acknowledgments}\nWe would like to thank Pavle Pandzic for informing \nus that some of our results\non the classifications\nof $(V_\\tau, \\tau)$\nhave also been obtained\nearlier by his joint work with \nSoo-Teck Lee and for some further discussions.", + "sketch": "The proof breaks into three parts corresponding to (1)--(3).\n\n(1) For the classification, “the proof of the classification is through a case by case computation and is technically involved,” repeatedly using “the classical branching rules in \\cite{good}, the classifications of weight-free representations in \\cite{Howe} and multiplicity-free results in \\cite{St}.”\n\n(2) For uniqueness of the $G$-equivariant map $F:W_\\lambda\\to C^\\infty(G/K,V_\\tau)$, the argument “is done through the realization $J:W_\\lambda\\to \\Ind_{MAN}^G(\\sigma\\otimes\\alpha\\otimes 1)$ of finite-dimensional representations $W_\\lambda$ in the induced representation,” together with “a factorization of $F=SJ$ as a product of $J$ and the Poisson-Szeg\\\"o{} transform,” and using “the multiplicity-free property of $V_\\tau$ under $M$.”\n\n(3) The Weyl-group invariance of eigenvalues of invariant differential operators “is a consequence of a general result of Lepowsky \\cite{L} and the factorization.”", + "expanded_sketch": "No expanded sketch found.", + "expanded_theorem": "(\\begin{theorem}\n\\label{KresThm}\nLet $(G, K)$ be an irreducible Hermitian symmetric pair\nand\n $M = C_K(\\mathfrak{a})$\n be the centralizer of the\n Cartan subalgebra \n $\\mathfrak{a}\\subset \\mathfrak p$ in $K$. \nThe complete list of $(K, \\tau)$ with $\\tau|_M$ being multiplicity-free is given as follows:\n\n\\begin{enumerate}\n \\item Type A, \n $G = SU(r+b,r), \n K = S(U(r+b) \\times U(r))$; irreducible representations of $U(n)$ with highest\n weights $\\mu$\n will be denoted by \n $\\pi^{\\mu}_n$ as in \\cite[Theorem 5.5.22]{good}. \n\\begin{enumerate}\n \\item $r=1$. $\\tau$ is arbitrary.\n\\item $r=2$. $\\tau= \\pi^{\\mu}_{2+b} \\otimes \\pi^{\\nu}_2$ where $\\pi_{2+b}^{\\mu}$ is\na character and $\\pi_2^{\\nu}$ is any representations, or $\\pi_2^{\\nu}$ is \na character and\n$$\n\\mu = (\\overbrace{l_1+l_2,\\cdots, l_1+l_2}^{{j }},\\overbrace{l_2,\\cdots,l_2}^{{b+2-j }}).\n$$\n\n\\item $r \\geq 3$. $\\tau= \\pi^{\\mu}_{2+b} \\otimes \\pi^{\\nu}_2$ where $\\pi_{r+b}^{\\mu}$\nor $\\pi_r^{\\nu}$ is a character and the other must be the symmetric\ntensor powers\n$S^m(\\mathbb C^{p})$ \nor an exterior power\n$\n\\wedge^j \\mathbb C^p$ when restricted to $SL(p)$\n($p=r+b$ or $p=r$\naccordingly).\n\\end{enumerate} \n\n\\item Type C, $G=Sp(n,\\R)$, $K = U(n)$. $\\tau$ is a character \nor its tensor product\nwith the exterior power representations\n$\n\\wedge^j \\mathbb C^n$. The same holds for the double cover $(G,K) = (Mp(n,\\R),\\widehat{U(n)})$.\n\n\\item Type D, $G = SO^*(2n), K = U(n)$. $\\tau$ \nis a character\nor its tensor product\nwith the symmetric tensor\npowers\n$S^m(\\mathbb C^{p})$ or their duals.\n\n\\item Type BD, $G=SO_0(n,2)$, \n $K = SO(n) \\times SO(2)$, $n>4$;\nirreducible \nrepresentations\nof $SO(n)$\nwill be denoted\nby \n$\\pi_{n}^{\\mu}$\nas in \\cite[Theorem 19.22]{FulHar}.\n $\\tau=\\pi_{n}^{0}\\otimes \\pi_{2}^\\nu$ \n if $n$ is odd, i.e., $\\tau$ is a character; for $n$ even $\\tau=\n \\pi_{n}^{\\mu}$ \n or its \n tensor product\n with a character\n$\\tau=\\pi_{n}^{\\mu}\n\\otimes \\pi_{2}^\\nu$, \n where $\\mu=(a,\\dots,a,\\pm a)$ for some $a\\ge 0$.\nIf $(G,K) = (Spin_0(n,2), \nSpin(n) \\times SO(2)/\\Z_2)$, where $Spin_0(n,2)$ is the double cover of $SO_0(n,2)$\nthen up to a character $\\tau$ is the spin representation $(\\frac{1}{2}, \\dots, \\frac{1}{2})$ when $n$ is odd, and\n$\\tau= \\pi_{n}^{\\mu}$, $\\mu=(a,\\dots,a,\\pm a)$ for some integer $a\\ge 0$ a half-integer\nwhen $n$ is even.\n\n\\item Type $E_6$, $G=E_{6(-14)}, K = Spin(10) \\times U(1) / \\Gamma$, where $\\Gamma$ is a finite subgroup.\n$\\tau=\\pi_{\\omega_1} \\otimes \\chi_n$ for $n \\equiv 1 \\mod 4$, or $\\pi_{\\omega_2} \\otimes \\chi_n$ for $n \\equiv 3 \\mod 4$. Here $\\omega_1$ and $\\omega_2$ are the positive and negative spin representations of $Spin(10)$ and $U(1) = \\{e^{it \\frac{Z}{2}} \\ | \\ t \\in \\R \\}$.\n\n\\item Type $E_7$, $G=E_{7(-25)}, K = E_6 U(1)$.\n$\\tau$ is a character.\n\n\\end{enumerate} \n\\end{theorem},\n\\begin{proposition}\n\\label{finiterepprop}\n(1) Let $(W_{\\lambda}, G^{\\C})$ be a highest weight representation of $G^{\\C}$ and its Lie algebra $\\mathfrak{g}^{\\C}$. For the multiplicities we have $[V_{\\tau} : W_{\\lambda}|_K] \\leq 1$, and if $V_{\\tau} \\subseteq W_{\\lambda}|_K$ we have $[W_{\\lambda} : I_{\\sigma,-\\lambda|_{\\af} - \\rho_{\\af}}^0 ] \\leq 1$, viewed as Lie algebra representations. \n\n(2)\nLet $(W_\\lambda, G ^{\\C})$ be\na finite-dimensional\nirreducible representation\nsuch that \n$V_{\\tau} \\subseteq W_{\\lambda}|_K$. Up to constants there is a unique $G$-equivariant map\n$$F : W_{\\lambda} \\rightarrow C^{\\infty}(G/K,V_{\\tau})$$\ndefined by\n$$F(w)(g) = P_{\\tau} (g^{-1} \\cdot w), g\\in G, w\\in W,$$\nwhere $P_{\\tau}$ is the projection onto $V_{\\tau}$. Moreover, if we normalize $J_{\\lambda}$ so that for all $v \\in V_{\\tau}$ we have $J_{\\lambda}(v) = f_v$, where $f_v(kan) = a^{\\lambda|_{\\mathfrak{a}}} P_{\\sigma} \\tau(k)^{-1} v$,\nthen there is a factorization\nof $F$ \nas a product $G$-equivariant maps,\n$$\nF=\nS_{\\sigma, -\\lambda|_{\\mathfrak{a}} - \\rho_{\\mathfrak{a}}}\nJ_{\\lambda}: (W_\\lambda, G^{\\C})\\to I_{\\sigma, -\\lambda|_{\\mathfrak{a}} - \\rho_{\\mathfrak{a}}}^0\n\\to C^\\infty(G/K,V_{\\tau}).\n$$\n\\end{proposition},\n\\begin{theorem}\n\\label{egvaluethm}\nAssume the irreducible finite-dimensional representation $V_\\tau$ \nof $K$ is multiplicity-free when restricted to $M$.\n\n(1) For any $M$-irreducible subrepresentation $(\\sigma,U_\\sigma)$ in $\\tau|_M$, $f \\in I_{\\sigma,\\nu}^0$ and $D \\in \\mathcal{D}^G(G/K, V_{\\tau})$, we have $DS_{\\sigma, \\nu}(f) = \\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}})S_{\\sigma, \\nu}(f)$ for a constant $\\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}})$.\nFurthermore, we have the invariance property:\n$$\\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}}) = \\omega(D)(s \\cdot \\sigma,s \\cdot \\nu + \\rho_{\\mathfrak{a}}), \\ \\forall \\ s\\in W.$$\n\n(2) Let $(W_\\lambda, G^{\\C})$\nbe a finite-dimensional\nrepresentation of $\nG^{\\C}$ containing $V_\\tau$. Let $D \\in \\mathcal{D}^G(G/K,V_{\\tau})$, then $D$ acts on \n$F(W_\\lambda)$\nas the constant\n$$\nDF(w) = \\omega(D)(\\sigma_{\\lambda},-\\lambda|_{\\mathfrak{a}})F(w), \n\\quad \\ \\forall \\ w \\in W_\\lambda.\n$$\n\\end{theorem})\n(1) The representation \n$\\tau|_M$ is\nmultiplicity-free if and only if \n$(K, \\tau)$ is in the list above.\n\n(2) Let $(K, \\tau)$ be as\nin (1) acting on $V_\\tau$. For any finite-dimensional irreducible representation\n$(W_\\lambda, G^{\\mathbb{C}})$ there is up to scalar at most one $G$-equivariant map into $C^\\infty(G/K, V_\\tau)$.\n\n(3)\nLet $(K, \\tau)\n$ be as in (1), $\\sigma$\nan irreducible representation in $\\tau|_M$\nand \n$(W_\\lambda, G^{\\mathbb C})$ \nas\nin (2). Let $D$ \nbe an invariant\ndifferential operator\non $C^\\infty(G/K, V_\\tau)$.\nWrite the eigenvalues \nof $D$ on $W_\\lambda$\nas $\\omega(D)(\\sigma_{\\lambda},-\\lambda|_{\\mathfrak{a}}\n)$\nwhich can be extended uniquely to a polynomial of $\n\\lambda|_{\\mathfrak{a}}\n\\in \n(\\mathfrak a^{\\mathbb C})'\n$.\nThen the eigenvalue\n$\\omega(D)$ has following\ninvariance \nproperty,\n$$\\omega(D)(\\sigma,\\kappa + \\rho_{\\mathfrak{a}}) = \\omega(D)(s \\cdot \\sigma,s \\cdot \\kappa + \\rho_{\\mathfrak{a}}), \\ \\forall \\ \\kappa \\in \n(\\mathfrak a^{\\mathbb C})', s\\in W.$$\n", + "theorem_type": [ + "Biconditional or Equivalence", + "Universal" + ], + "mcq": { + "question": "Let (G,K) be an irreducible Hermitian symmetric pair with Cartan decomposition \\(\\mathfrak g=\\mathfrak k\\oplus\\mathfrak p\\), let \\(\\mathfrak a\\subset \\mathfrak p\\) be a Cartan subalgebra, and let \\(M=C_K(\\mathfrak a)\\). Let \\(\\tau\\) be an irreducible finite-dimensional representation of \\(K\\) on \\(V_\\tau\\), let \\(C^\\infty(G/K,V_\\tau)\\) be the smooth sections of the homogeneous bundle \\(G\\times_{(K,\\tau)}V_\\tau\\), and let \\(\\mathcal D^G(G/K,V_\\tau)\\) be the algebra of \\(G\\)-invariant differential operators on that space. Write \\(W\\) for the Weyl group of \\((\\mathfrak g,\\mathfrak a)\\) and \\(\\rho_{\\mathfrak a}\\) for the half-sum of positive restricted roots. In type A, \\(\\pi_n^\\mu\\) denotes the irreducible \\(U(n)\\)-representation of highest weight \\(\\mu\\); in type BD, \\(\\pi_n^\\mu\\) denotes the irreducible \\(SO(n)\\)-representation of highest weight \\(\\mu\\); in type \\(E_6\\), \\(\\chi_n\\) denotes the weight-\\(n\\) character of \\(U(1)\\), and \\(\\omega_1,\\omega_2\\) are the positive and negative half-spin representations of \\(\\mathrm{Spin}(10)\\). Which statement holds for every such \\((G,K,\\tau)\\)?", + "correct_choice": { + "label": "A", + "text": "The restriction \\(\\tau|_M\\) is multiplicity-free if and only if \\((K,\\tau)\\) is one of the following: (1) Type A: \\(G=SU(r+b,r)\\), \\(K=S(U(r+b)\\times U(r))\\): (a) if \\(r=1\\), then \\(\\tau\\) is arbitrary; (b) if \\(r=2\\), then \\(\\tau=\\pi_{b+2}^{\\mu}\\otimes \\pi_2^{\\nu}\\), where either \\(\\pi_{b+2}^{\\mu}\\) is a character and \\(\\pi_2^{\\nu}\\) is arbitrary, or \\(\\pi_2^{\\nu}\\) is a character and \\(\\mu=(\\overbrace{l_1+l_2,\\dots,l_1+l_2}^{j},\\overbrace{l_2,\\dots,l_2}^{b+2-j})\\); (c) if \\(r\\ge 3\\), then \\(\\tau=\\pi_{r+b}^{\\mu}\\otimes \\pi_r^{\\nu}\\), where one of \\(\\pi_{r+b}^{\\mu}\\) or \\(\\pi_r^{\\nu}\\) is a character and the other, upon restriction to \\(SL(p)\\) (with \\(p=r+b\\) or \\(p=r\\), respectively), is either a symmetric power \\(S^m(\\mathbb C^p)\\) or an exterior power \\(\\wedge^j\\mathbb C^p\\); (2) Type C: \\(G=Sp(n,\\mathbb R)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with an exterior power \\(\\wedge^j\\mathbb C^n\\); the same holds for the double cover \\((G,K)=(Mp(n,\\mathbb R),\\widehat{U(n)})\\); (3) Type D: \\(G=SO^*(2n)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with a symmetric tensor power \\(S^m(\\mathbb C^n)\\) or its dual; (4) Type BD: \\(G=SO_0(n,2)\\), \\(K=SO(n)\\times SO(2)\\), \\(n>4\\); if \\(n\\) is odd, then \\(\\tau=\\pi_n^0\\otimes \\pi_2^\\nu\\), i.e. \\(\\tau\\) is a character; if \\(n\\) is even, then \\(\\tau=\\pi_n^\\mu\\) or \\(\\tau=\\pi_n^\\mu\\otimes \\pi_2^\\nu\\), where \\(\\mu=(a,\\dots,a,\\pm a)\\) for some \\(a\\ge 0\\); for the double cover \\((G,K)=(\\mathrm{Spin}_0(n,2),\\,\\mathrm{Spin}(n)\\times SO(2)/\\mathbb Z_2)\\), up to a character \\(\\tau\\) is the spin representation \\((\\tfrac12,\\dots,\\tfrac12)\\) when \\(n\\) is odd, and when \\(n\\) is even one has \\(\\tau=\\pi_n^\\mu\\) with \\(\\mu=(a,\\dots,a,\\pm a)\\) for some nonnegative integral or half-integral \\(a\\); (5) Type \\(E_6\\): \\(G=E_{6(-14)}\\), \\(K=\\mathrm{Spin}(10)\\times U(1)/\\Gamma\\), and \\(\\tau=\\pi_{\\omega_1}\\otimes \\chi_n\\) for \\(n\\equiv 1\\pmod 4\\), or \\(\\tau=\\pi_{\\omega_2}\\otimes \\chi_n\\) for \\(n\\equiv 3\\pmod 4\\); (6) Type \\(E_7\\): \\(G=E_{7(-25)}\\), \\(K=E_6U(1)\\), and \\(\\tau\\) is a character. Moreover, for every \\((K,\\tau)\\) in this list and every finite-dimensional irreducible representation \\((W_\\lambda,G^{\\mathbb C})\\), there is, up to scalar, at most one \\(G\\)-equivariant map \\(W_\\lambda\\to C^\\infty(G/K,V_\\tau)\\). Finally, if \\(\\sigma\\) is an irreducible summand of \\(\\tau|_M\\), \\(D\\in \\mathcal D^G(G/K,V_\\tau)\\), and the eigenvalue of \\(D\\) on \\(W_\\lambda\\) is written as \\(\\omega(D)(\\sigma_\\lambda,-\\lambda|_{\\mathfrak a})\\) and extended uniquely to a polynomial in \\(\\lambda|_{\\mathfrak a}\\in (\\mathfrak a^{\\mathbb C})'\\), then \\(\\omega(D)\\) satisfies the Weyl-group invariance property \\(\\omega(D)(\\sigma,\\kappa+\\rho_{\\mathfrak a})=\\omega(D)(s\\cdot \\sigma,s\\cdot \\kappa+\\rho_{\\mathfrak a})\\) for all \\(\\kappa\\in (\\mathfrak a^{\\mathbb C})'\\) and all \\(s\\in W\\)." + }, + "choices": [ + { + "label": "B", + "text": "The restriction \\(\\tau|_M\\) is multiplicity-free if and only if \\((K,\\tau)\\) is one of the following: (1) Type A: \\(G=SU(r+b,r)\\), \\(K=S(U(r+b)\\times U(r))\\): (a) if \\(r=1\\), then \\(\\tau\\) is arbitrary; (b) if \\(r=2\\), then \\(\\tau=\\pi_{b+2}^{\\mu}\\otimes \\pi_2^{\\nu}\\), where either \\(\\pi_{b+2}^{\\mu}\\) is a character and \\(\\pi_2^{\\nu}\\) is arbitrary, or \\(\\pi_2^{\\nu}\\) is a character and \\(\\mu=(\\overbrace{l_1+l_2,\\dots,l_1+l_2}^{j},\\overbrace{l_2,\\dots,l_2}^{b+2-j})\\); (c) if \\(r\\ge 3\\), then \\(\\tau=\\pi_{r+b}^{\\mu}\\otimes \\pi_r^{\\nu}\\), where one of \\(\\pi_{r+b}^{\\mu}\\) or \\(\\pi_r^{\\nu}\\) is a character and the other, upon restriction to \\(SL(p)\\) (with \\(p=r+b\\) or \\(p=r\\), respectively), is either a symmetric power \\(S^m(\\mathbb C^p)\\) or an exterior power \\(\\wedge^j\\mathbb C^p\\); (2) Type C: \\(G=Sp(n,\\mathbb R)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with an exterior power \\(\\wedge^j\\mathbb C^n\\); the same holds for the double cover \\((G,K)=(Mp(n,\\mathbb R),\\widehat{U(n)})\\); (3) Type D: \\(G=SO^*(2n)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with a symmetric tensor power \\(S^m(\\mathbb C^n)\\), an exterior power \\(\\wedge^j\\mathbb C^n\\), or their duals; (4) Type BD: \\(G=SO_0(n,2)\\), \\(K=SO(n)\\times SO(2)\\), \\(n>4\\); if \\(n\\) is odd, then \\(\\tau=\\pi_n^0\\otimes \\pi_2^\\nu\\), i.e. \\(\\tau\\) is a character; if \\(n\\) is even, then \\(\\tau=\\pi_n^\\mu\\) or \\(\\tau=\\pi_n^\\mu\\otimes \\pi_2^\\nu\\), where \\(\\mu=(a,\\dots,a,\\pm a)\\) for some \\(a\\ge 0\\); for the double cover \\((G,K)=(\\mathrm{Spin}_0(n,2),\\,\\mathrm{Spin}(n)\\times SO(2)/\\mathbb Z_2)\\), up to a character \\(\\tau\\) is the spin representation \\((\\tfrac12,\\dots,\\tfrac12)\\) when \\(n\\) is odd, and when \\(n\\) is even one has \\(\\tau=\\pi_n^\\mu\\) with \\(\\mu=(a,\\dots,a,\\pm a)\\) for some nonnegative integral or half-integral \\(a\\); (5) Type \\(E_6\\): \\(G=E_{6(-14)}\\), \\(K=\\mathrm{Spin}(10)\\times U(1)/\\Gamma\\), and \\(\\tau=\\pi_{\\omega_1}\\otimes \\chi_n\\) for \\(n\\equiv 1\\pmod 4\\), or \\(\\tau=\\pi_{\\omega_2}\\otimes \\chi_n\\) for \\(n\\equiv 3\\pmod 4\\); (6) Type \\(E_7\\): \\(G=E_{7(-25)}\\), \\(K=E_6U(1)\\), and \\(\\tau\\) is a character. Moreover, for every \\((K,\\tau)\\) in this list and every finite-dimensional irreducible representation \\((W_\\lambda,G^{\\mathbb C})\\), there exists a unique \\(G\\)-equivariant map \\(W_\\lambda\\to C^\\infty(G/K,V_\\tau)\\). Finally, if \\(\\sigma\\) is an irreducible summand of \\(\\tau|_M\\), \\(D\\in \\mathcal D^G(G/K,V_\\tau)\\), and the eigenvalue of \\(D\\) on \\(W_\\lambda\\) is written as \\(\\omega(D)(\\sigma_\\lambda,-\\lambda|_{\\mathfrak a})\\) and extended uniquely to a polynomial in \\(\\lambda|_{\\mathfrak a}\\in (\\mathfrak a^{\\mathbb C})'\\), then \\(\\omega(D)\\) satisfies the Weyl-group invariance property \\(\\omega(D)(\\sigma,\\kappa+\\rho_{\\mathfrak a})=\\omega(D)(s\\cdot \\sigma,s\\cdot \\kappa+\\rho_{\\mathfrak a})\\) for all \\(\\kappa\\in (\\mathfrak a^{\\mathbb C})'\\) and all \\(s\\in W\\)." + }, + { + "label": "C", + "text": "If \\(\\tau|_M\\) is multiplicity-free, then \\((K,\\tau)\\) is one of the representations in the classification listed above. Moreover, for every such \\((K,\\tau)\\) and every finite-dimensional irreducible representation \\((W_\\lambda,G^{\\mathbb C})\\), there is, up to scalar, at most one \\(G\\)-equivariant map \\(W_\\lambda\\to C^\\infty(G/K,V_\\tau)\\). Finally, if \\(\\sigma\\) is an irreducible summand of \\(\\tau|_M\\), \\(D\\in \\mathcal D^G(G/K,V_\\tau)\\), and the eigenvalue of \\(D\\) on \\(W_\\lambda\\) is written as \\(\\omega(D)(\\sigma_\\lambda,-\\lambda|_{\\mathfrak a})\\) and extended uniquely to a polynomial in \\(\\lambda|_{\\mathfrak a}\\in (\\mathfrak a^{\\mathbb C})'\\), then \\(\\omega(D)(\\sigma,\\kappa+\\rho_{\\mathfrak a})=\\omega(D)(s\\cdot \\sigma,s\\cdot \\kappa+\\rho_{\\mathfrak a})\\) for all \\(\\kappa\\in (\\mathfrak a^{\\mathbb C})'\\) and all \\(s\\in W\\)." + }, + { + "label": "D", + "text": "The restriction \\(\\tau|_M\\) is multiplicity-free if and only if \\((K,\\tau)\\) is one of the following: (1) Type A: \\(G=SU(r+b,r)\\), \\(K=S(U(r+b)\\times U(r))\\): (a) if \\(r=1\\), then \\(\\tau\\) is arbitrary; (b) if \\(r=2\\), then \\(\\tau=\\pi_{b+2}^{\\mu}\\otimes \\pi_2^{\\nu}\\), where either \\(\\pi_{b+2}^{\\mu}\\) is a character and \\(\\pi_2^{\\nu}\\) is arbitrary, or \\(\\pi_2^{\\nu}\\) is a character and \\(\\mu=(\\overbrace{l_1+l_2,\\dots,l_1+l_2}^{j},\\overbrace{l_2,\\dots,l_2}^{b+2-j})\\); (c) if \\(r\\ge 3\\), then \\(\\tau=\\pi_{r+b}^{\\mu}\\otimes \\pi_r^{\\nu}\\), where one of \\(\\pi_{r+b}^{\\mu}\\) or \\(\\pi_r^{\\nu}\\) is a character and the other, upon restriction to \\(SL(p)\\) (with \\(p=r+b\\) or \\(p=r\\), respectively), is either a symmetric power \\(S^m(\\mathbb C^p)\\) or an exterior power \\(\\wedge^j\\mathbb C^p\\); (2) Type C: \\(G=Sp(n,\\mathbb R)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with an exterior power \\(\\wedge^j\\mathbb C^n\\); the same holds for the double cover \\((G,K)=(Mp(n,\\mathbb R),\\widehat{U(n)})\\); (3) Type D: \\(G=SO^*(2n)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with a symmetric tensor power \\(S^m(\\mathbb C^n)\\) or its dual; (4) Type BD: \\(G=SO_0(n,2)\\), \\(K=SO(n)\\times SO(2)\\), \\(n\\ge 4\\); if \\(n\\) is odd, then \\(\\tau=\\pi_n^0\\otimes \\pi_2^\\nu\\), i.e. \\(\\tau\\) is a character; if \\(n\\) is even, then \\(\\tau=\\pi_n^\\mu\\) or \\(\\tau=\\pi_n^\\mu\\otimes \\pi_2^\\nu\\), where \\(\\mu=(a,\\dots,a,\\pm a)\\) for some \\(a\\ge 0\\); for the double cover \\((G,K)=(\\mathrm{Spin}_0(n,2),\\,\\mathrm{Spin}(n)\\times SO(2)/\\mathbb Z_2)\\), up to a character \\(\\tau\\) is the spin representation \\((\\tfrac12,\\dots,\\tfrac12)\\) when \\(n\\) is odd, and when \\(n\\) is even one has \\(\\tau=\\pi_n^\\mu\\) with \\(\\mu=(a,\\dots,a,\\pm a)\\) for some nonnegative integral or half-integral \\(a\\); (5) Type \\(E_6\\): \\(G=E_{6(-14)}\\), \\(K=\\mathrm{Spin}(10)\\times U(1)/\\Gamma\\), and \\(\\tau=\\pi_{\\omega_1}\\otimes \\chi_n\\) for \\(n\\equiv 1\\pmod 4\\), or \\(\\tau=\\pi_{\\omega_2}\\otimes \\chi_n\\) for \\(n\\equiv 3\\pmod 4\\); (6) Type \\(E_7\\): \\(G=E_{7(-25)}\\), \\(K=E_6U(1)\\), and \\(\\tau\\) is a character. Moreover, for every \\((K,\\tau)\\) in this list and every finite-dimensional irreducible representation \\((W_\\lambda,G^{\\mathbb C})\\), there is, up to scalar, at most one \\(G\\)-equivariant map \\(W_\\lambda\\to C^\\infty(G/K,V_\\tau)\\). Finally, if \\(\\sigma\\) is an irreducible summand of \\(\\tau|_M\\), \\(D\\in \\mathcal D^G(G/K,V_\\tau)\\), and the eigenvalue of \\(D\\) on \\(W_\\lambda\\) is written as \\(\\omega(D)(\\sigma_\\lambda,-\\lambda|_{\\mathfrak a})\\) and extended uniquely to a polynomial in \\(\\lambda|_{\\mathfrak a}\\in (\\mathfrak a^{\\mathbb C})'\\), then \\(\\omega(D)\\) satisfies the Weyl-group invariance property \\(\\omega(D)(\\sigma,\\kappa+\\rho_{\\mathfrak a})=\\omega(D)(s\\cdot \\sigma,s\\cdot \\kappa+\\rho_{\\mathfrak a})\\) for all \\(\\kappa\\in (\\mathfrak a^{\\mathbb C})'\\) and all \\(s\\in W\\)." + }, + { + "label": "E", + "text": "The restriction \\(\\tau|_M\\) is multiplicity-free if and only if \\((K,\\tau)\\) is one of the following: (1) Type A: \\(G=SU(r+b,r)\\), \\(K=S(U(r+b)\\times U(r))\\): (a) if \\(r=1\\), then \\(\\tau\\) is arbitrary; (b) if \\(r=2\\), then \\(\\tau=\\pi_{b+2}^{\\mu}\\otimes \\pi_2^{\\nu}\\), where either \\(\\pi_{b+2}^{\\mu}\\) is a character and \\(\\pi_2^{\\nu}\\) is arbitrary, or \\(\\pi_2^{\\nu}\\) is a character and \\(\\mu=(\\overbrace{l_1+l_2,\\dots,l_1+l_2}^{j},\\overbrace{l_2,\\dots,l_2}^{b+2-j})\\); (c) if \\(r\\ge 3\\), then \\(\\tau=\\pi_{r+b}^{\\mu}\\otimes \\pi_r^{\\nu}\\), where one of \\(\\pi_{r+b}^{\\mu}\\) or \\(\\pi_r^{\\nu}\\) is a character and the other, upon restriction to \\(SL(p)\\) (with \\(p=r+b\\) or \\(p=r\\), respectively), is either a symmetric power \\(S^m(\\mathbb C^p)\\) or an exterior power \\(\\wedge^j\\mathbb C^p\\); (2) Type C: \\(G=Sp(n,\\mathbb R)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with an exterior power \\(\\wedge^j\\mathbb C^n\\); the same holds for the double cover \\((G,K)=(Mp(n,\\mathbb R),\\widehat{U(n)})\\); (3) Type D: \\(G=SO^*(2n)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with a symmetric tensor power \\(S^m(\\mathbb C^n)\\) or its dual; (4) Type BD: \\(G=SO_0(n,2)\\), \\(K=SO(n)\\times SO(2)\\), \\(n>4\\); if \\(n\\) is odd, then \\(\\tau=\\pi_n^0\\otimes \\pi_2^\\nu\\), i.e. \\(\\tau\\) is a character; if \\(n\\) is even, then \\(\\tau=\\pi_n^\\mu\\) or \\(\\tau=\\pi_n^\\mu\\otimes \\pi_2^\\nu\\), where \\(\\mu=(a,\\dots,a,\\pm a)\\) for some \\(a\\ge 0\\); for the double cover \\((G,K)=(\\mathrm{Spin}_0(n,2),\\,\\mathrm{Spin}(n)\\times SO(2)/\\mathbb Z_2)\\), up to a character \\(\\tau\\) is the spin representation \\((\\tfrac12,\\dots,\\tfrac12)\\) when \\(n\\) is odd, and when \\(n\\) is even one has \\(\\tau=\\pi_n^\\mu\\) with \\(\\mu=(a,\\dots,a,\\pm a)\\) for some nonnegative integral or half-integral \\(a\\); (5) Type \\(E_6\\): \\(G=E_{6(-14)}\\), \\(K=\\mathrm{Spin}(10)\\times U(1)/\\Gamma\\), and \\(\\tau=\\pi_{\\omega_1}\\otimes \\chi_n\\) for \\(n\\equiv 1\\pmod 4\\), or \\(\\tau=\\pi_{\\omega_2}\\otimes \\chi_n\\) for \\(n\\equiv 3\\pmod 4\\); (6) Type \\(E_7\\): \\(G=E_{7(-25)}\\), \\(K=E_6U(1)\\), and \\(\\tau\\) is a character. Moreover, for every \\((K,\\tau)\\) in this list and every finite-dimensional irreducible representation \\((W_\\lambda,G^{\\mathbb C})\\), there is, up to scalar, at most one \\(G\\)-equivariant map \\(W_\\lambda\\to C^\\infty(G/K,V_\\tau)\\). Finally, if \\(\\sigma\\) is an irreducible summand of \\(\\tau|_M\\), \\(D\\in \\mathcal D^G(G/K,V_\\tau)\\), and the eigenvalue of \\(D\\) on \\(W_\\lambda\\) is written as \\(\\omega(D)(\\sigma_\\lambda,-\\lambda|_{\\mathfrak a})\\) and extended uniquely to a polynomial in \\(\\lambda|_{\\mathfrak a}\\in (\\mathfrak a^{\\mathbb C})'\\), then \\(\\omega(D)\\) satisfies the stronger invariance property \\(\\omega(D)(\\sigma,\\kappa)=\\omega(D)(s\\cdot \\sigma,s\\cdot \\kappa)\\) for all \\(\\kappa\\in (\\mathfrak a^{\\mathbb C})'\\) and all \\(s\\in W\\)." + } + ], + "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": "uniqueness up to scalar replaced by absolute existence and uniqueness", + "template_used": "stronger_trap" + }, + { + "label": "C", + "sketch_hook_type": "case_split", + "tampered_component": "dropped the converse direction in the classification equivalence", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "case_split", + "tampered_component": "type BD boundary changed from n>4 to n\\ge 4", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "Weyl-invariance shift by \\rho_{\\mathfrak a} removed", + "template_used": "property_confusion" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem introduces notation and context only; it does not state the classification, uniqueness, or Weyl-invariance conclusion. There is no explicit or trivial cue pointing to choice A." + }, + "TAS": { + "score": 1, + "justification": "The item is not a literal restatement of the stem, but it is essentially a theorem-recognition question: the student is asked to identify the exact global statement of a classification theorem from near-verbatim variants." + }, + "GPS": { + "score": 1, + "justification": "The question requires careful comparison of subtle logical and technical differences across options, but it mainly tests precise recall/discrimination rather than generative mathematical reasoning from the setup." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically meaningful: they alter converse direction, uniqueness strength, boundary cases, and the rho-shift in Weyl invariance. These are distinct and aligned with realistic failure modes." + }, + "total_score": 6, + "overall_assessment": "A strong theorem-discrimination MCQ with excellent distractors and little answer leakage, but it leans more toward recognition of an exact classification statement than genuine generative reasoning." + } + }, + { + "id": "2602.16329v1", + "paper_link": "http://arxiv.org/abs/2602.16329v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "thm", + "content": "\\label{MainThm}\n\t\t$(T_t^{(2)})_{t\\geq 0}$ is hypercontractive. Moreover, the optimal time $t_p$ satisfies\n\t\t\\[\n\t\t\\widetilde{c}(\\beta) (p - 1) \\leq e^{2\\tau t_p} \\leq \\widetilde{C}(\\beta) (p - 1), \\quad \\forall\\, 2 < p < \\infty,\n\t\t\\]\n\t\twhere $\\widetilde{c}(\\beta)$ and $\\widetilde{C}(\\beta)$ are positive constants depending only on $\\beta$.", + "start_pos": 13047, + "end_pos": 13381, + "label": "MainThm" + }, + "ref_dict": { + "MainThm": "\\begin{thm}\\label{MainThm}\n\t\t$(T_t^{(2)})_{t\\geq 0}$ is hypercontractive. Moreover, the optimal time $t_p$ satisfies\n\t\t\\[\n\t\t\\wt{c}(\\beta) (p - 1) \\leq e^{2\\tau t_p} \\leq \\wt{C}(\\beta) (p - 1), \\quad \\forall\\, 2 < p < \\infty,\n\t\t\\]\n\t\twhere $\\wt{c}(\\beta)$ and $\\wt{C}(\\beta)$ are positive constants depending only on $\\beta$.\n\t\\end{thm}", + "Main-Lem": "\\begin{thm}\\label{Main-Lem}\n\t\tLet $f_{k, n, m}$ be defined as before. For all $k \\geq 1$, $-k \\leq m \\leq k$, there exists a constant $C(\\beta)$ such that\n\t\t\\begin{equation}\n\t\t\t\\left( \\sum_{n=0}^\\8 e^{-n\\beta}| f_{k, n, m} |^p \\right)^{1/p} \\leq e^{m\\beta(1/2p-1/4)} \\mk{ C(\\beta)p }^{ k/2 } \\left( \\sum_{n=0}^\\8 e^{-n\\beta} | f_{k, n, m} |^2 \\right)^{1/2}.\n\t\t\\end{equation}\n\t\\end{thm}", + "gap": "\\begin{equation}\\label{gap}\n\t\t\\tau=\\min\\{\\tau_1, \\tau_2\\}>0. \n\t\\end{equation}", + "Prop2.1": "\\begin{proposition}\\label{Prop2.1}\n\t\t$T_t^{(2)}$ is hypercontractive if for any $20$ such that for any $x \\in L_k$, $k\\in \\nat$,\n\t\t\\begin{align}\\label{KL}\n\t\t\t\\left\\| \\rho^{1/(2p) - 1/4} x \\rho^{1/(2p) - 1/4}\\right\\|_{p} \\leq c_p^k \\left\\| x \\right\\|_{2}. \n\t\t\\end{align}\n\t\tMoreover, the optimal time $t_p$ satisfies\n\t\t\\begin{equation}\\label{optimal-1}\n\t\t\te^{2\\tau t_p} \\leq 2c_p^2, \\quad \\forall \\, 20$ is a fixed inverse temperature. Note that $e^{-\\beta N}$ is a trace class operator in $B(H)$. Let \n\t$$ \\rho=\\dfrac{e^{-\\beta N}}{\\text{Tr}(e^{-\\beta N})}=(1-e^{-\\beta})e^{-\\beta N}. $$\n\tIndeed, $\\omega$ is a Gibbs state, and for any $x\\in B(H)$,\n\t$$ \\omega(x)=\\text{Tr}(\\rho x). $$\n\tWe refer the interested reader to \\cite{BR} for more details. Now we introduce the quantum Ornstein-Uhlenbeck semigroups constructed in \\cite{KP}. Let the parameters $\\alpha_1, \\alpha_2, \\alpha_3 \\in \\mathbb{R}$ satisfy the following relations\n\t\\begin{align*}\n\t\t& \\frac{1}{2}\\left(1+\\alpha_2^2\\right) \\sinh (\\beta / 2)=-\\alpha_1 \\cosh (\\beta / 2), \\\\\n\t\t& \\frac{1}{2}\\left(\\alpha_1^2+\\alpha_3^2\\right) \\sinh (\\beta / 2)=\\alpha_2 \\alpha_3 \\cosh (\\beta / 2).\n\t\\end{align*}\n\tLet $G$ be the elliptic operator on $B(H)$ given by\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\tG(A)=\\frac{\\gamma}{2}\\left(1+\\alpha_2^2\\right)[P,[P, A]]&+\\frac{\\gamma}{2}\\left(\\alpha_1^2+\\alpha_3^2\\right)[Q,[Q, A]]-i \\gamma \\alpha_1(Q[P, A]+[P, A] Q)\\\\\n\t\t\t&-i \\gamma \\alpha_2 \\alpha_3(P[Q, A]+[Q, A] P), \\quad \\quad \\forall \\, A\\in B(H),\n\t\t\\end{aligned}\n\t\\end{equation*}\n\twhere $\\gamma=\\left(1+\\alpha_2^2\\right)^{-1}$ is the normalized constant. Since $P$ and $Q$ are unbounded operators affiliated to $\\mathcal{M}$, the above is a formal expression.\n\n\tFor $00$ such that for any $t \\geq t_p$, and for any $ x\\in B(H) $ with $\\omega(x)=0$,\n\t\\begin{equation}\\label{hypercontractivity}\n\t\t\\| T_t(\\rho^{1/2}x\\rho^{1/2}) \\|_{L_p(\\rho)} \\leq \\|\\rho^{1/2}x\\rho^{1/2}\\|_{L_2(\\rho)}.\n\t\\end{equation}\n\tMoreover, the least $t_p$ is called the optimal time of $T_t$ with respect to the parameter $p$.\n\n\tOur result is the following theorem.", + "context": "\\label{Introduction}\n In the celebrated papers \\cite{EN1966,EN1973}, Nelson showed that the classical Ornstein-Uhlenbeck semigroup is hypercontractive and utilized such property to establish the existence and uniqueness of the ground state of a special semigroup which arises from the constructive quantum field theory. Since then, hypercontractivity becomes a useful and powerful tool in quantum field theory, quantum statistical mechanics and quantum information.\n\nLet $H=\\ell_2(\\mathbb{N})$, where $\\mathbb{N}=\\{0,1,2, \\cdots\\}$ is the set of all natural numbers. Let $\\{e_n\\}_{n=0}^\\infty$ be the canonical orthonormal basis of $H$. Then we define the creation and annihilation operators $a^*$ and $a$ as follows\n \\[\n a^* e_n = \\sqrt{n+1} \\, e_{n+1}, \\quad a e_n = \\sqrt{n} \\, e_{n-1}, \\quad (e_{-1} := 0).\n \\]\n It is clear that they satisfy the canonical commutation relation (CCR)\n \\[\n [a, a^*] = 1,\n \\]\n where $1$ denotes the identity operator by a slight abuse of notation. The number operator $N := a^* a$ acts as\n \\[\n N e_n = n \\cdot e_n.\n \\]\n The position and momentum operators are defined as\n \\[\n Q = \\frac{1}{\\sqrt{2}}(a + a^*), \\quad P = \\frac{1}{\\sqrt{2}i}(a - a^*),\n \\]\n and so\n \\[\n [Q, P] = i \\cdot 1.\n \\]\n For any $z \\in \\mathbb{C}$, define the Weyl operators\n \\[\n W(z) = e^{\\frac {i}{\\sqrt{2}}(z a^* + \\bar{z} a)} = e^{i ( \\Re z \\cdot Q + \\Im z \\cdot P ) }.\n \\]\n These operators satisfy the Weyl relation\n \\[\n W(z)^*=W(-z), \\quad \\text{and} \\quad \\forall z, w\\in\\mathbb{C}, \\ W(z) W(w) = e^{-\\frac{i}{2} \\Im (\\bar{z} w)} W(z + w).\n \\]\n Let $\\mathcal{A}$ be the $C^*$-algebra generated by all Weyl operators. It is well-known that $\\mathcal{A}\\subsetneqq B(H)$ and $\\mathcal{A}{''}=B(H)$. Let $\\omega$ be the normal faithful state on $B(H)$, which is defined by\n $$ \\omega(W(z))=e^{-\\frac{|z|^2}{4} (1+e^{-\\beta})(1-e^{-\\beta})^{-1}}, \\quad \\forall \\, z\\in \\mathbb{C} $$\n where $\\beta>0$ is a fixed inverse temperature. Note that $e^{-\\beta N}$ is a trace class operator in $B(H)$. Let \n $$ \\rho=\\dfrac{e^{-\\beta N}}{\\text{Tr}(e^{-\\beta N})}=(1-e^{-\\beta})e^{-\\beta N}. $$\n Indeed, $\\omega$ is a Gibbs state, and for any $x\\in B(H)$,\n $$ \\omega(x)=\\text{Tr}(\\rho x). $$\n We refer the interested reader to \\cite{BR} for more details. Now we introduce the quantum Ornstein-Uhlenbeck semigroups constructed in \\cite{KP}. Let the parameters $\\alpha_1, \\alpha_2, \\alpha_3 \\in \\mathbb{R}$ satisfy the following relations\n \\begin{align*}\n & \\frac{1}{2}\\left(1+\\alpha_2^2\\right) \\sinh (\\beta / 2)=-\\alpha_1 \\cosh (\\beta / 2), \\\\\n & \\frac{1}{2}\\left(\\alpha_1^2+\\alpha_3^2\\right) \\sinh (\\beta / 2)=\\alpha_2 \\alpha_3 \\cosh (\\beta / 2).\n \\end{align*}\n Let $G$ be the elliptic operator on $B(H)$ given by\n \\begin{equation*}\n \\begin{aligned}\n G(A)=\\frac{\\gamma}{2}\\left(1+\\alpha_2^2\\right)[P,[P, A]]&+\\frac{\\gamma}{2}\\left(\\alpha_1^2+\\alpha_3^2\\right)[Q,[Q, A]]-i \\gamma \\alpha_1(Q[P, A]+[P, A] Q)\\\\\n &-i \\gamma \\alpha_2 \\alpha_3(P[Q, A]+[Q, A] P), \\quad \\quad \\forall \\, A\\in B(H),\n \\end{aligned}\n \\end{equation*}\n where $\\gamma=\\left(1+\\alpha_2^2\\right)^{-1}$ is the normalized constant. Since $P$ and $Q$ are unbounded operators affiliated to $\\mathcal{M}$, the above is a formal expression.\n\nFor $00$ such that for any $t \\geq t_p$, and for any $ x\\in B(H) $ with $\\omega(x)=0$,\n \\begin{equation}\\label{hypercontractivity}\n \\| T_t(\\rho^{1/2}x\\rho^{1/2}) \\|_{L_p(\\rho)} \\leq \\|\\rho^{1/2}x\\rho^{1/2}\\|_{L_2(\\rho)}.\n \\end{equation}\n Moreover, the least $t_p$ is called the optimal time of $T_t$ with respect to the parameter $p$.\n\nOur result is the following theorem.\n\n\\begin{equation}\\label{gap}\n\t\t\\tau=\\min\\{\\tau_1, \\tau_2\\}>0. \n\t\\end{equation}", + "full_context": "\\label{Introduction}\n In the celebrated papers \\cite{EN1966,EN1973}, Nelson showed that the classical Ornstein-Uhlenbeck semigroup is hypercontractive and utilized such property to establish the existence and uniqueness of the ground state of a special semigroup which arises from the constructive quantum field theory. Since then, hypercontractivity becomes a useful and powerful tool in quantum field theory, quantum statistical mechanics and quantum information.\n\nLet $H=\\ell_2(\\mathbb{N})$, where $\\mathbb{N}=\\{0,1,2, \\cdots\\}$ is the set of all natural numbers. Let $\\{e_n\\}_{n=0}^\\infty$ be the canonical orthonormal basis of $H$. Then we define the creation and annihilation operators $a^*$ and $a$ as follows\n \\[\n a^* e_n = \\sqrt{n+1} \\, e_{n+1}, \\quad a e_n = \\sqrt{n} \\, e_{n-1}, \\quad (e_{-1} := 0).\n \\]\n It is clear that they satisfy the canonical commutation relation (CCR)\n \\[\n [a, a^*] = 1,\n \\]\n where $1$ denotes the identity operator by a slight abuse of notation. The number operator $N := a^* a$ acts as\n \\[\n N e_n = n \\cdot e_n.\n \\]\n The position and momentum operators are defined as\n \\[\n Q = \\frac{1}{\\sqrt{2}}(a + a^*), \\quad P = \\frac{1}{\\sqrt{2}i}(a - a^*),\n \\]\n and so\n \\[\n [Q, P] = i \\cdot 1.\n \\]\n For any $z \\in \\mathbb{C}$, define the Weyl operators\n \\[\n W(z) = e^{\\frac {i}{\\sqrt{2}}(z a^* + \\bar{z} a)} = e^{i ( \\Re z \\cdot Q + \\Im z \\cdot P ) }.\n \\]\n These operators satisfy the Weyl relation\n \\[\n W(z)^*=W(-z), \\quad \\text{and} \\quad \\forall z, w\\in\\mathbb{C}, \\ W(z) W(w) = e^{-\\frac{i}{2} \\Im (\\bar{z} w)} W(z + w).\n \\]\n Let $\\mathcal{A}$ be the $C^*$-algebra generated by all Weyl operators. It is well-known that $\\mathcal{A}\\subsetneqq B(H)$ and $\\mathcal{A}{''}=B(H)$. Let $\\omega$ be the normal faithful state on $B(H)$, which is defined by\n $$ \\omega(W(z))=e^{-\\frac{|z|^2}{4} (1+e^{-\\beta})(1-e^{-\\beta})^{-1}}, \\quad \\forall \\, z\\in \\mathbb{C} $$\n where $\\beta>0$ is a fixed inverse temperature. Note that $e^{-\\beta N}$ is a trace class operator in $B(H)$. Let \n $$ \\rho=\\dfrac{e^{-\\beta N}}{\\text{Tr}(e^{-\\beta N})}=(1-e^{-\\beta})e^{-\\beta N}. $$\n Indeed, $\\omega$ is a Gibbs state, and for any $x\\in B(H)$,\n $$ \\omega(x)=\\text{Tr}(\\rho x). $$\n We refer the interested reader to \\cite{BR} for more details. Now we introduce the quantum Ornstein-Uhlenbeck semigroups constructed in \\cite{KP}. Let the parameters $\\alpha_1, \\alpha_2, \\alpha_3 \\in \\mathbb{R}$ satisfy the following relations\n \\begin{align*}\n & \\frac{1}{2}\\left(1+\\alpha_2^2\\right) \\sinh (\\beta / 2)=-\\alpha_1 \\cosh (\\beta / 2), \\\\\n & \\frac{1}{2}\\left(\\alpha_1^2+\\alpha_3^2\\right) \\sinh (\\beta / 2)=\\alpha_2 \\alpha_3 \\cosh (\\beta / 2).\n \\end{align*}\n Let $G$ be the elliptic operator on $B(H)$ given by\n \\begin{equation*}\n \\begin{aligned}\n G(A)=\\frac{\\gamma}{2}\\left(1+\\alpha_2^2\\right)[P,[P, A]]&+\\frac{\\gamma}{2}\\left(\\alpha_1^2+\\alpha_3^2\\right)[Q,[Q, A]]-i \\gamma \\alpha_1(Q[P, A]+[P, A] Q)\\\\\n &-i \\gamma \\alpha_2 \\alpha_3(P[Q, A]+[Q, A] P), \\quad \\quad \\forall \\, A\\in B(H),\n \\end{aligned}\n \\end{equation*}\n where $\\gamma=\\left(1+\\alpha_2^2\\right)^{-1}$ is the normalized constant. Since $P$ and $Q$ are unbounded operators affiliated to $\\mathcal{M}$, the above is a formal expression.\n\nFor $00$ such that for any $t \\geq t_p$, and for any $ x\\in B(H) $ with $\\omega(x)=0$,\n \\begin{equation}\\label{hypercontractivity}\n \\| T_t(\\rho^{1/2}x\\rho^{1/2}) \\|_{L_p(\\rho)} \\leq \\|\\rho^{1/2}x\\rho^{1/2}\\|_{L_2(\\rho)}.\n \\end{equation}\n Moreover, the least $t_p$ is called the optimal time of $T_t$ with respect to the parameter $p$.\n\nOur result is the following theorem.\n\n\\begin{equation}\\label{gap}\n\t\t\\tau=\\min\\{\\tau_1, \\tau_2\\}>0. \n\t\\end{equation}\n\nOur result is the following theorem.\n\nWe can also obtain some estimate of the optimal time of $(T_t^{(2)})_{t\\geq 0}$ for all $ x\\in B(H) $ without the restriction $\\omega(x)=0$ with the help of Theorem \\ref{MainThm}. For any $20$ be the least constant such that for any $t \\geq t_p'$, and for any $ x\\in B(H) $,\n \\begin{equation*}\n \\| T_t^{(2)}(\\rho^{1/2}x\\rho^{1/2}) \\|_{L_p(\\rho)} \\leq \\|\\rho^{1/2}x\\rho^{1/2}\\|_{L_2(\\rho)}.\n \\end{equation*}\n\n\\begin{corollary}\\label{MainThm2}\n For $20$ such that for any $x \\in L_k$, $k\\in \\nat$,\n \\begin{align}\\label{KL}\n \\left\\| \\rho^{1/(2p) - 1/4} x \\rho^{1/(2p) - 1/4}\\right\\|_{p} \\leq c_p^k \\left\\| x \\right\\|_{2}. \n \\end{align}\n Moreover, the optimal time $t_p$ satisfies\n \\begin{equation}\\label{optimal-1}\n e^{2\\tau t_p} \\leq 2c_p^2, \\quad \\forall \\, 20$ such that for any $x \\in L_k$, $k\\in \\nat$,\n\t\t\\begin{align}\\label{KL}\n\t\t\t\\left\\| \\rho^{1/(2p) - 1/4} x \\rho^{1/(2p) - 1/4}\\right\\|_{p} \\leq c_p^k \\left\\| x \\right\\|_{2}. \n\t\t\\end{align}\n\t\tMoreover, the optimal time $t_p$ satisfies\n\t\t\\begin{equation}\\label{optimal-1}\n\t\t\te^{2\\tau t_p} \\leq 2c_p^2, \\quad \\forall \\, 20$ be the least constant such that for any $t \\geq t_p'$, and for any $ x\\in B(H) $,\n\t\\begin{equation*}\n\t\t\\| T_t^{(2)}(\\rho^{1/2}x\\rho^{1/2}) \\|_{L_p(\\rho)} \\leq \\|\\rho^{1/2}x\\rho^{1/2}\\|_{L_2(\\rho)}.\n\t\\end{equation*}\n\n\t\\begin{corollary}\\label{MainThm2}\n\t\tFor $20$. Let $H=\\ell_2(\\mathbb N)$, let $N=a^*a$ be the number operator, and let\n\\[\n\\rho=(1-e^{-\\beta})e^{-\\beta N},\\qquad \\omega(x)=\\operatorname{Tr}(\\rho x).\n\\]\nLet $\\alpha_1,\\alpha_2,\\alpha_3\\in\\mathbb R$ satisfy\n\\[\n\\tfrac12(1+\\alpha_2^2)\\sinh(\\beta/2)=-\\alpha_1\\cosh(\\beta/2),\n\\]\n\\[\n\\tfrac12(\\alpha_1^2+\\alpha_3^2)\\sinh(\\beta/2)=\\alpha_2\\alpha_3\\cosh(\\beta/2).\n\\]\nUsing these parameters, form the corresponding quantum Ornstein-Uhlenbeck semigroup $\\bigl(T_t^{(2)}\\bigr)_{t\\ge 0}$ on $L_2(\\rho)$, where $T_t^{(2)}$ is induced from $P_t=e^{-t\\mathcal G^{(2)}}$ via\n\\[\n\\mathcal G^{(2)}(\\rho^{1/4}x\\rho^{1/4})=\\rho^{1/4}G(x)\\rho^{1/4}.\n\\]\nLet $\\tau$ denote the spectral gap of $T_t^{(2)}$. For $20$ such that for all $t\\ge t_p$ and all $x\\in B(H)$ with $\\omega(x)=0$,\n\\[\n\\|T_t^{(2)}(\\rho^{1/2}x\\rho^{1/2})\\|_{L_p(\\rho)}\\le \\|\\rho^{1/2}x\\rho^{1/2}\\|_{L_2(\\rho)}.\n\\]\nWhen this holds, let $t_p$ be the optimal such time. Which statement holds for this semigroup for every $2 1/{(m-1)} $, where $m$ is the mean of $p$. Thus, if $\\mu$ is fixed, then the critical value for $\\alpha$ is $\\mu/(m-1)$.", + "start_pos": 13144, + "end_pos": 13549, + "label": "thm:single" + }, + "ref_dict": { + "rates table": "\\begin{table}[h]\n \\label{rates table}\n\\begin{center}\n\\begin{tabular}{c|c|c|c}\n\\hline\n& original infection rate & increased infection rate & recovery rate\\\\\n\\hline \ndisease $A$ & $\\alpha_1$ & $\\beta_1\\in [\\alpha_1,\\infty]$ & $\\mu_1$\\\\\n\\hline \ndisease $B$ & $\\alpha_2$ & $\\beta_2\\in [\\alpha_2, \\infty]$ & $\\mu_2$\\\\\n\\hline \n\\end{tabular}\n\\end{center}\n\\caption{Parameters for the two-type SIR process.}\n \\end{table}", + "thm:single": "\\begin{theorem}\\label{thm:single}\n Let the infection rate be $\\alpha$ and the recovery rate be $\\mu$ for a single-type SIR process on $GW (p)$. At time 0, only the root is infected. The probability of survival is greater than 0 if and only if $ \\alpha/\\mu > 1/{(m-1)} $, where $m$ is the mean of $p$. Thus, if $\\mu$ is fixed, then the critical value for $\\alpha$ is $\\mu/(m-1)$. \n \\end{theorem}", + "eq:ode": "\\begin{equation} \\label{eq:ode}\n \\begin{cases}\n s'(t)=-2\\alpha s(t)x(t), \\\\ \n q'(t)=(\\alpha s(t)-C \\alpha q(t))x(t), \\\\\n x'(t)=(\\alpha s(t)+C \\alpha q(t))x(t)-x(t). \n \\end{cases}\n\\end{equation}", + "rmk:c=infty": "\\begin{remark}\\label{rmk:c=infty}\nIn the special case of equal infection/recovery rates for both diseases (denoted by $\\alpha,\\beta=C\\alpha$ and $\\mu$) with $C=\\infty$, the conclusion of Theorem \\ref{general} is consistent with the prediction made in \\cite{grassberger2016phase} regarding infinite tree graphs, where the authors claimed that (for $C=\\infty$), \n$$\n\\mathbb{P}(\\mbox{infinitely many vertices are infected with both diseases})>0\\, \\mbox{ if and only if }\\, \\frac{\\alpha}{\\mu}>\\frac{1}{m-1}. \n$$ See also the discussions above Table \\ref{rates table}.\n\\end{remark}", + "general": "\\begin{theorem}\n \\label{general}\nConsider the two-type SIR process on $GW (p)$ with parameters $\\alpha_i$, $\\beta_i$, $\\mu_i$, $i=1,2$. At time $0$, the root is infected with both disease $A$ and disease $B$. All other vertices are susceptible to both $A$ and $B$. Let $m$ be the mean of $p$. If\n \\begin{equation}\\label{eq:assump}\n \\max\\left\\{\\frac{\\alpha_1}{\\mu_1},\\frac{\\alpha_2}{\\mu_2}\\right\\}\\leq \\frac{1}{m-1}\n \\textnormal {,}\n \\end{equation}\nthen the probability of survival is $0$.\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 9852, + "pre_theorem_intro_text": "SIR models and their variants have been widely used to predict and control the disease spreading process. In the standard single-type SIR process based on a graph, each node can be in one of three states: S (representing `susceptible'), I (representing `infected') and R (representing `recovered'). Vertices in state I try to infect their neighbors at rate $\\alpha$, independently across each edge, and recover (turn into state R) at rate $\\mu$. Alternatively, in terms of species evolution, one might imagine that each individual gives birth after a random amount of time distributed as Exp($\\alpha$) and dies after time distributed as Exp($\\mu$), where all the exponential random variables are independent. A closely related model is the SIS model, also called the contact process, where a vertex turns into state S instead of R when it recovers.\n\nResearchers have also been fascinated by the behavior of epidemic models that incorporate interactions (competition or cooperation) among multiple species. Two-type SIS models with competitions\nfor space on the $d$-dimensional integer lattice $\\mathbb{Z}^d$\nwere investigated in \\cite{MounTford2019,Neuhauser1992,Stover2020}.\nFor the case of equal death rates, Neuhauser \\cite{Neuhauser1992} proved that the species with the smaller birth rate dies out locally, while \nMountford, Pantoja, and Valesin \\cite{MounTford2019} proved that the winner takes over a ball whose radius grows linearly over time.\nNeuhauser conjectured that for general death rates, the species with the higher birth/death ratio wins the competition, which has been verified by Stover \\cite{Stover2020} for certain cases but still remains largely open.\nFor cooperative interactions, Durrett and Yao \\cite{DurrettYao2020} considered a symbiotic contact process on $\\mathbb{Z}^d$\nwhere the presence of one species can reduce the death rate of the other type at the same site. It was proved that strong symbiosis can lower the critical value, but the general case (especially weak symbiosis) remains unclear.\n\nIn another direction, Lanchier and Neuhauser\n\\cite{lanchier2010stochastic,lanchier2006stochastic} studied stochastic models with hosts and symbionts. In these models, each host can be infected with a symbiont, while a few species of hosts compete against each other. Durrett and Lanchier \\cite{durrett2008coexistence} studied another case where only one of two species of hosts can be infected with one type of symbionts. In these papers, conditions for survival and coexistence have been studied, and the main results describe the long-term behavior of the models under certain conditions.\nA few years later, Lanchier and Zhang \\cite{lanchier2016some} studied the ``stacked contact process'', and Ma \\cite{ma2022complete} studied the ``two-level contact process''. In these two models, there are uninfected hosts and infected hosts. The results focused on the phase transition and the limiting distribution of the models. \n\n\tMotivated by a series of works by Ghanbarnejad and coauthors\n \\cite{cai2015avalanche, chen2013outbreaks,grassberger2016phase,zarei2019exact},\n we consider a \\emph{two-type cooperative SIR process} on a network. Suppose that two diseases, $A$ and $B$, spread simultaneously on the network, each type acting as an SIR process. Vertices in the graph can be infected by both types. Thus, there are nine possible states for each vertex,\n \\begin{equation}\\label{eq:statespace}\n S\\text{, }A \\text{, } B \\text{, } a \\text{, } b \\text{, } Ab \\text{, } aB \\text{, } AB\\text{ and }ab.\n \\end{equation}\n Here, capital $A$ and $B$ indicate active infection, and lower case $a$ and $b$ indicate recovery from $A$ and $B$. We assume that the two diseases interact in a cooperative way. \n When a vertex $x$ is susceptible to both diseases, then it can be infected with $A$ at rate $\\alpha_1$. On the other hand, \n if $x$ has been infected with $B$ before (regardless of whether it has recovered from $B$ or not), then $x$ acquires $A$ at a higher rate $\\beta_1 \\in [\\alpha_1, \\infty]$.\n Note that an infinite rate means that an infection transmits immediately. The constant $C _ 1:= \\beta _1 / \\alpha _1 \\in [1,\\infty]$ is called the \\emph{cooperativity coefficient} for $A$. Likewise, a vertex $x$ can be infected by $B$ at a higher rate $\\beta _2 \\geq \\alpha _2$ if it has been infected with $A$ before. The cooperativity coefficient of $B$ is $C _ 2 = \\beta _2 / \\alpha _2$. Our definition of the two-type SIR process simulates the situation in which the immune system weakens because of a prior infection.\n\nChen et al.\\@ \\cite{chen2013outbreaks} analyzed a deterministic version of this model with mean-field methods, involving a homogeneously mixed population of infinitely many agents. \nIn total, there are nine ordinary differential equations for the evolution of all states. Let $[z](t)$ be the fraction of agents in state $z$ at time $t$.\nUnder the assumption of equal infection rates for both diseases, and letting the recovery rate be 1, \\cite{chen2013outbreaks} reduced the system of nine equations to three:\n\\begin{equation} \\label{eq:ode}\n \\begin{cases}\n s'(t)=-2\\alpha s(t)x(t), \\\\ \n q'(t)=(\\alpha s(t)-C \\alpha q(t))x(t), \\\\\n x'(t)=(\\alpha s(t)+C \\alpha q(t))x(t)-x(t). \n \\end{cases}\n\\end{equation}\nIn these equations, $s(t)=[S](t)$ is the fraction of susceptible agents, \n$$x(t)=[A](t)+[AB](t)+[Ab](t)=[B](t)+[AB](t)+[aB](t)$$\nis the fraction of agents that are actively infected by disease $A$ (or $B$) at time $t$, and $$q(t)=[A](t)+[a](t)=[B](t)+[b](t)$$ accounts for the agents which have an infection history of only one disease. The initial condition is set to $[A](0)=[B](0)=\\epsilon/2$, $[S] (0) = 1 - \\epsilon$ and the fractions of all other states are $0$. Hence, $s(0)=1-\\epsilon$ and $q(0)=x(0)=\\epsilon/2$. \n\nDenote the final epidemic size of the system of equations \\eqref{eq:ode} by $$\\mathcal{R}(\\alpha, C, \\epsilon) = 1-\\lim_{t\\to\\infty} s(t).$$\nZarei et al.\\@ \\cite{zarei2019exact} found\n$\\mathcal R(\\alpha, C, \\epsilon)=1-s(0)\\exp(-2\\alpha T_0)$, where $T_0$ is given by\n$$\n\\inf\\left\\{t>0: t+s(0)\\exp(-2\\alpha t)+q(0)\\exp(-C\\beta t)-\\frac{ s(0)}{C-2}(\\exp(-C\\alpha t)-\\exp(-2\\alpha t) )=1 \\right\\}\n$$\nif $C\\neq 2$.\nNote that this expression already implies the criticality at $C=2$.\nLet\n$$\n\\mathcal{R}_*(\\alpha, C) = \\lim_{\\epsilon\\to 0} \\mathcal R(\\alpha, C, \\epsilon)\n$$\nwhen the limit exists (otherwise, one may replace the limit with upper limit). Based on numerical experiments and some non-rigorous arguments, \\cite{chen2013outbreaks, zarei2019exact} claimed the following:\n\\begin{enumerate}[label = (\\roman*)]\n\t\\item If $C \\leq 2$, then $\\mathcal R(\\alpha, C, \\epsilon)$ is continuous in $\\alpha$. Moreover, $\\mathcal R_*(\\alpha, C)=0$ for $\\alpha \\leq 1$, and behaves like $\\alpha-1$ ($C<2$) or $\\sqrt{\\alpha-1}$ ($C=2$) for $\\alpha$ close to 1.\n\t\\item If $C>2$, then $\\mathcal R(\\alpha, C, \\epsilon)$ is discontinuous in $\\alpha$ at some $\\alpha_0=\\alpha_0(C,\\epsilon)$, and\n\\begin{equation}\\label{eq:discont}\n \\lim_{\\epsilon \\to 0} \\mathcal R(\\alpha_0-, C, \\epsilon)=0 \\quad \\textnormal{ and } \\quad \n\\lim_{\\epsilon \\to 0} \\mathcal R(\\alpha_0+, C, \\epsilon)>0. \n\\end{equation}\n\\item For $C>2$, the critical infection rate $\\alpha_0(C, \\epsilon)=1-\\sqrt{(C-2)\\epsilon}+O(\\epsilon)$ as $\\epsilon\\to 0$.\n\\item The quantity $\\mathcal R_*(\\alpha, C)\\equiv 0$ for $\\alpha \\leq 1/2$, and $$\\lim_{\\epsilon \\to 0}\\lim_{C\\to\\infty}R(\\alpha, C,\\epsilon) \\sim \\alpha-\\frac{1}{2}\\quad \\textnormal{ as } \\alpha \\to 1/2 \\textnormal{.}$$\n\\end{enumerate}\n\nIt is generally expected that\nmean-field ODEs may give good approximations for the stochastic particle systems on the complete graph.\n In real-world networks, nontrivial spatial structures are often present. Grassberger et al.\\@ \\cite{grassberger2016phase} conducted simulations for the two-type SIR model on Erd\\H{o}s-R\\'enyi graphs and the integer lattice $\\mathbb{Z}^d$. They argued that \n Erd\\H{o}s-R\\'enyi graphs have a discontinuous \n phase transition for the fraction of eventually infected vertices when the cooperativity is sufficiently strong, as predicted by the mean-field ODEs. For the case of $\\mathbb{Z}^d$ with $d\\geq 2$, simulations in \\cite{grassberger2016phase} showed that the two-type model shares similar critical and near-critical features with the single-type SIR process if and only if $d\\leq 3$. \n \\cite{grassberger2016phase} also considered the near-critical behavior of the two-type SIR model on trees. In particular, \\cite{grassberger2016phase} suggested that the critical value (for infinitely many vertices to be infected with both diseases) remains the same as the single-type model for $C=\\infty$. See also Remark \\ref{rmk:c=infty} below. \n\n In this paper, we consider the two-type SIR process on Galton-Watson trees with cooperative interactions defined before. The rates are not necessarily the same for $A$ and $B$, and there are six parameters in total, shown in Table \\ref{rates table}.\n\n \\begin{table}[h]\n \\label{rates table}\n\\begin{center}\n\\begin{tabular}{c|c|c|c}\n\\hline\n& original infection rate & increased infection rate & recovery rate\\\\\n\\hline \ndisease $A$ & $\\alpha_1$ & $\\beta_1\\in [\\alpha_1,\\infty]$ & $\\mu_1$\\\\\n\\hline \ndisease $B$ & $\\alpha_2$ & $\\beta_2\\in [\\alpha_2, \\infty]$ & $\\mu_2$\\\\\n\\hline \n\\end{tabular}\n\\end{center}\n\\caption{Parameters for the two-type SIR process.}\n \\end{table}\n\nLet $GW (p)$ be the Galton-Watson tree with the offspring distribution $p$. Let $m \\in (1,\\infty)$ be the finite mean of $p$. (We let $m>1$ so that $GW(p)$ itself is supercritical.) Consider the single-type SIR process on $GW(p)$. We say that it \\textit{survives} if for all $t\\geq 0$, there is at least one infected vertex at time $t$. The following result is classic.", + "context": "Motivated by a series of works by Ghanbarnejad and coauthors\n \\cite{cai2015avalanche, chen2013outbreaks,grassberger2016phase,zarei2019exact},\n we consider a \\emph{two-type cooperative SIR process} on a network. Suppose that two diseases, $A$ and $B$, spread simultaneously on the network, each type acting as an SIR process. Vertices in the graph can be infected by both types. Thus, there are nine possible states for each vertex,\n \\begin{equation}\\label{eq:statespace}\n S\\text{, }A \\text{, } B \\text{, } a \\text{, } b \\text{, } Ab \\text{, } aB \\text{, } AB\\text{ and }ab.\n \\end{equation}\n Here, capital $A$ and $B$ indicate active infection, and lower case $a$ and $b$ indicate recovery from $A$ and $B$. We assume that the two diseases interact in a cooperative way. \n When a vertex $x$ is susceptible to both diseases, then it can be infected with $A$ at rate $\\alpha_1$. On the other hand, \n if $x$ has been infected with $B$ before (regardless of whether it has recovered from $B$ or not), then $x$ acquires $A$ at a higher rate $\\beta_1 \\in [\\alpha_1, \\infty]$.\n Note that an infinite rate means that an infection transmits immediately. The constant $C _ 1:= \\beta _1 / \\alpha _1 \\in [1,\\infty]$ is called the \\emph{cooperativity coefficient} for $A$. Likewise, a vertex $x$ can be infected by $B$ at a higher rate $\\beta _2 \\geq \\alpha _2$ if it has been infected with $A$ before. The cooperativity coefficient of $B$ is $C _ 2 = \\beta _2 / \\alpha _2$. Our definition of the two-type SIR process simulates the situation in which the immune system weakens because of a prior infection.\n\nChen et al.\\@ \\cite{chen2013outbreaks} analyzed a deterministic version of this model with mean-field methods, involving a homogeneously mixed population of infinitely many agents. \nIn total, there are nine ordinary differential equations for the evolution of all states. Let $[z](t)$ be the fraction of agents in state $z$ at time $t$.\nUnder the assumption of equal infection rates for both diseases, and letting the recovery rate be 1, \\cite{chen2013outbreaks} reduced the system of nine equations to three:\n\\begin{equation} \\label{eq:ode}\n \\begin{cases}\n s'(t)=-2\\alpha s(t)x(t), \\\\ \n q'(t)=(\\alpha s(t)-C \\alpha q(t))x(t), \\\\\n x'(t)=(\\alpha s(t)+C \\alpha q(t))x(t)-x(t). \n \\end{cases}\n\\end{equation}\nIn these equations, $s(t)=[S](t)$ is the fraction of susceptible agents, \n$$x(t)=[A](t)+[AB](t)+[Ab](t)=[B](t)+[AB](t)+[aB](t)$$\nis the fraction of agents that are actively infected by disease $A$ (or $B$) at time $t$, and $$q(t)=[A](t)+[a](t)=[B](t)+[b](t)$$ accounts for the agents which have an infection history of only one disease. The initial condition is set to $[A](0)=[B](0)=\\epsilon/2$, $[S] (0) = 1 - \\epsilon$ and the fractions of all other states are $0$. Hence, $s(0)=1-\\epsilon$ and $q(0)=x(0)=\\epsilon/2$.\n\nDenote the final epidemic size of the system of equations \\eqref{eq:ode} by $$\\mathcal{R}(\\alpha, C, \\epsilon) = 1-\\lim_{t\\to\\infty} s(t).$$\nZarei et al.\\@ \\cite{zarei2019exact} found\n$\\mathcal R(\\alpha, C, \\epsilon)=1-s(0)\\exp(-2\\alpha T_0)$, where $T_0$ is given by\n$$\n\\inf\\left\\{t>0: t+s(0)\\exp(-2\\alpha t)+q(0)\\exp(-C\\beta t)-\\frac{ s(0)}{C-2}(\\exp(-C\\alpha t)-\\exp(-2\\alpha t) )=1 \\right\\}\n$$\nif $C\\neq 2$.\nNote that this expression already implies the criticality at $C=2$.\nLet\n$$\n\\mathcal{R}_*(\\alpha, C) = \\lim_{\\epsilon\\to 0} \\mathcal R(\\alpha, C, \\epsilon)\n$$\nwhen the limit exists (otherwise, one may replace the limit with upper limit). Based on numerical experiments and some non-rigorous arguments, \\cite{chen2013outbreaks, zarei2019exact} claimed the following:\n\\begin{enumerate}[label = (\\roman*)]\n \\item If $C \\leq 2$, then $\\mathcal R(\\alpha, C, \\epsilon)$ is continuous in $\\alpha$. Moreover, $\\mathcal R_*(\\alpha, C)=0$ for $\\alpha \\leq 1$, and behaves like $\\alpha-1$ ($C<2$) or $\\sqrt{\\alpha-1}$ ($C=2$) for $\\alpha$ close to 1.\n \\item If $C>2$, then $\\mathcal R(\\alpha, C, \\epsilon)$ is discontinuous in $\\alpha$ at some $\\alpha_0=\\alpha_0(C,\\epsilon)$, and\n\\begin{equation}\\label{eq:discont}\n \\lim_{\\epsilon \\to 0} \\mathcal R(\\alpha_0-, C, \\epsilon)=0 \\quad \\textnormal{ and } \\quad \n\\lim_{\\epsilon \\to 0} \\mathcal R(\\alpha_0+, C, \\epsilon)>0. \n\\end{equation}\n\\item For $C>2$, the critical infection rate $\\alpha_0(C, \\epsilon)=1-\\sqrt{(C-2)\\epsilon}+O(\\epsilon)$ as $\\epsilon\\to 0$.\n\\item The quantity $\\mathcal R_*(\\alpha, C)\\equiv 0$ for $\\alpha \\leq 1/2$, and $$\\lim_{\\epsilon \\to 0}\\lim_{C\\to\\infty}R(\\alpha, C,\\epsilon) \\sim \\alpha-\\frac{1}{2}\\quad \\textnormal{ as } \\alpha \\to 1/2 \\textnormal{.}$$\n\\end{enumerate}\n\n\\begin{table}[h]\n \\label{rates table}\n\\begin{center}\n\\begin{tabular}{c|c|c|c}\n\\hline\n& original infection rate & increased infection rate & recovery rate\\\\\n\\hline \ndisease $A$ & $\\alpha_1$ & $\\beta_1\\in [\\alpha_1,\\infty]$ & $\\mu_1$\\\\\n\\hline \ndisease $B$ & $\\alpha_2$ & $\\beta_2\\in [\\alpha_2, \\infty]$ & $\\mu_2$\\\\\n\\hline \n\\end{tabular}\n\\end{center}\n\\caption{Parameters for the two-type SIR process.}\n \\end{table}\n\nLet $GW (p)$ be the Galton-Watson tree with the offspring distribution $p$. Let $m \\in (1,\\infty)$ be the finite mean of $p$. (We let $m>1$ so that $GW(p)$ itself is supercritical.) Consider the single-type SIR process on $GW(p)$. We say that it \\textit{survives} if for all $t\\geq 0$, there is at least one infected vertex at time $t$. The following result is classic.", + "full_context": "Motivated by a series of works by Ghanbarnejad and coauthors\n \\cite{cai2015avalanche, chen2013outbreaks,grassberger2016phase,zarei2019exact},\n we consider a \\emph{two-type cooperative SIR process} on a network. Suppose that two diseases, $A$ and $B$, spread simultaneously on the network, each type acting as an SIR process. Vertices in the graph can be infected by both types. Thus, there are nine possible states for each vertex,\n \\begin{equation}\\label{eq:statespace}\n S\\text{, }A \\text{, } B \\text{, } a \\text{, } b \\text{, } Ab \\text{, } aB \\text{, } AB\\text{ and }ab.\n \\end{equation}\n Here, capital $A$ and $B$ indicate active infection, and lower case $a$ and $b$ indicate recovery from $A$ and $B$. We assume that the two diseases interact in a cooperative way. \n When a vertex $x$ is susceptible to both diseases, then it can be infected with $A$ at rate $\\alpha_1$. On the other hand, \n if $x$ has been infected with $B$ before (regardless of whether it has recovered from $B$ or not), then $x$ acquires $A$ at a higher rate $\\beta_1 \\in [\\alpha_1, \\infty]$.\n Note that an infinite rate means that an infection transmits immediately. The constant $C _ 1:= \\beta _1 / \\alpha _1 \\in [1,\\infty]$ is called the \\emph{cooperativity coefficient} for $A$. Likewise, a vertex $x$ can be infected by $B$ at a higher rate $\\beta _2 \\geq \\alpha _2$ if it has been infected with $A$ before. The cooperativity coefficient of $B$ is $C _ 2 = \\beta _2 / \\alpha _2$. Our definition of the two-type SIR process simulates the situation in which the immune system weakens because of a prior infection.\n\nChen et al.\\@ \\cite{chen2013outbreaks} analyzed a deterministic version of this model with mean-field methods, involving a homogeneously mixed population of infinitely many agents. \nIn total, there are nine ordinary differential equations for the evolution of all states. Let $[z](t)$ be the fraction of agents in state $z$ at time $t$.\nUnder the assumption of equal infection rates for both diseases, and letting the recovery rate be 1, \\cite{chen2013outbreaks} reduced the system of nine equations to three:\n\\begin{equation} \\label{eq:ode}\n \\begin{cases}\n s'(t)=-2\\alpha s(t)x(t), \\\\ \n q'(t)=(\\alpha s(t)-C \\alpha q(t))x(t), \\\\\n x'(t)=(\\alpha s(t)+C \\alpha q(t))x(t)-x(t). \n \\end{cases}\n\\end{equation}\nIn these equations, $s(t)=[S](t)$ is the fraction of susceptible agents, \n$$x(t)=[A](t)+[AB](t)+[Ab](t)=[B](t)+[AB](t)+[aB](t)$$\nis the fraction of agents that are actively infected by disease $A$ (or $B$) at time $t$, and $$q(t)=[A](t)+[a](t)=[B](t)+[b](t)$$ accounts for the agents which have an infection history of only one disease. The initial condition is set to $[A](0)=[B](0)=\\epsilon/2$, $[S] (0) = 1 - \\epsilon$ and the fractions of all other states are $0$. Hence, $s(0)=1-\\epsilon$ and $q(0)=x(0)=\\epsilon/2$.\n\nDenote the final epidemic size of the system of equations \\eqref{eq:ode} by $$\\mathcal{R}(\\alpha, C, \\epsilon) = 1-\\lim_{t\\to\\infty} s(t).$$\nZarei et al.\\@ \\cite{zarei2019exact} found\n$\\mathcal R(\\alpha, C, \\epsilon)=1-s(0)\\exp(-2\\alpha T_0)$, where $T_0$ is given by\n$$\n\\inf\\left\\{t>0: t+s(0)\\exp(-2\\alpha t)+q(0)\\exp(-C\\beta t)-\\frac{ s(0)}{C-2}(\\exp(-C\\alpha t)-\\exp(-2\\alpha t) )=1 \\right\\}\n$$\nif $C\\neq 2$.\nNote that this expression already implies the criticality at $C=2$.\nLet\n$$\n\\mathcal{R}_*(\\alpha, C) = \\lim_{\\epsilon\\to 0} \\mathcal R(\\alpha, C, \\epsilon)\n$$\nwhen the limit exists (otherwise, one may replace the limit with upper limit). Based on numerical experiments and some non-rigorous arguments, \\cite{chen2013outbreaks, zarei2019exact} claimed the following:\n\\begin{enumerate}[label = (\\roman*)]\n \\item If $C \\leq 2$, then $\\mathcal R(\\alpha, C, \\epsilon)$ is continuous in $\\alpha$. Moreover, $\\mathcal R_*(\\alpha, C)=0$ for $\\alpha \\leq 1$, and behaves like $\\alpha-1$ ($C<2$) or $\\sqrt{\\alpha-1}$ ($C=2$) for $\\alpha$ close to 1.\n \\item If $C>2$, then $\\mathcal R(\\alpha, C, \\epsilon)$ is discontinuous in $\\alpha$ at some $\\alpha_0=\\alpha_0(C,\\epsilon)$, and\n\\begin{equation}\\label{eq:discont}\n \\lim_{\\epsilon \\to 0} \\mathcal R(\\alpha_0-, C, \\epsilon)=0 \\quad \\textnormal{ and } \\quad \n\\lim_{\\epsilon \\to 0} \\mathcal R(\\alpha_0+, C, \\epsilon)>0. \n\\end{equation}\n\\item For $C>2$, the critical infection rate $\\alpha_0(C, \\epsilon)=1-\\sqrt{(C-2)\\epsilon}+O(\\epsilon)$ as $\\epsilon\\to 0$.\n\\item The quantity $\\mathcal R_*(\\alpha, C)\\equiv 0$ for $\\alpha \\leq 1/2$, and $$\\lim_{\\epsilon \\to 0}\\lim_{C\\to\\infty}R(\\alpha, C,\\epsilon) \\sim \\alpha-\\frac{1}{2}\\quad \\textnormal{ as } \\alpha \\to 1/2 \\textnormal{.}$$\n\\end{enumerate}\n\n\\begin{table}[h]\n \\label{rates table}\n\\begin{center}\n\\begin{tabular}{c|c|c|c}\n\\hline\n& original infection rate & increased infection rate & recovery rate\\\\\n\\hline \ndisease $A$ & $\\alpha_1$ & $\\beta_1\\in [\\alpha_1,\\infty]$ & $\\mu_1$\\\\\n\\hline \ndisease $B$ & $\\alpha_2$ & $\\beta_2\\in [\\alpha_2, \\infty]$ & $\\mu_2$\\\\\n\\hline \n\\end{tabular}\n\\end{center}\n\\caption{Parameters for the two-type SIR process.}\n \\end{table}\n\nLet $GW (p)$ be the Galton-Watson tree with the offspring distribution $p$. Let $m \\in (1,\\infty)$ be the finite mean of $p$. (We let $m>1$ so that $GW(p)$ itself is supercritical.) Consider the single-type SIR process on $GW(p)$. We say that it \\textit{survives} if for all $t\\geq 0$, there is at least one infected vertex at time $t$. The following result is classic.\n\n\\begin{table}[h]\n \\label{rates table}\n\\begin{center}\n\\begin{tabular}{c|c|c|c}\n\\hline\n& original infection rate & increased infection rate & recovery rate\\\\\n\\hline \ndisease $A$ & $\\alpha_1$ & $\\beta_1\\in [\\alpha_1,\\infty]$ & $\\mu_1$\\\\\n\\hline \ndisease $B$ & $\\alpha_2$ & $\\beta_2\\in [\\alpha_2, \\infty]$ & $\\mu_2$\\\\\n\\hline \n\\end{tabular}\n\\end{center}\n\\caption{Parameters for the two-type SIR process.}\n \\end{table}\n\n\\begin{theorem}\n \\label{general}\nConsider the two-type SIR process on $GW (p)$ with parameters $\\alpha_i$, $\\beta_i$, $\\mu_i$, $i=1,2$. At time $0$, the root is infected with both disease $A$ and disease $B$. All other vertices are susceptible to both $A$ and $B$. Let $m$ be the mean of $p$. If\n \\begin{equation}\\label{eq:assump}\n \\max\\left\\{\\frac{\\alpha_1}{\\mu_1},\\frac{\\alpha_2}{\\mu_2}\\right\\}\\leq \\frac{1}{m-1}\n \\textnormal {,}\n \\end{equation}\nthen the probability of survival is $0$.\n\\end{theorem} \n\\begin{remark}\\label{rmk:c=infty}\nIn the special case of equal infection/recovery rates for both diseases (denoted by $\\alpha,\\beta=C\\alpha$ and $\\mu$) with $C=\\infty$, the conclusion of Theorem \\ref{general} is consistent with the prediction made in \\cite{grassberger2016phase} regarding infinite tree graphs, where the authors claimed that (for $C=\\infty$), \n$$\n\\mathbb{P}(\\mbox{infinitely many vertices are infected with both diseases})>0\\, \\mbox{ if and only if }\\, \\frac{\\alpha}{\\mu}>\\frac{1}{m-1}. \n$$ See also the discussions above Table \\ref{rates table}.\n\\end{remark}\nCombining Theorems \\ref{thm:single} and \\ref{general}, the two-type SIR process survives with a positive probability if and only if\n $$\n \\max\\left\\{\\frac{\\alpha_1}{\\mu_1},\\frac{\\alpha_2}{\\mu_2}\\right\\}> \\frac{1}{m-1}.\n $$\n Since the Galton-Watson tree and its variants arise as local limits of some random graph models such as the configuration model and the {\\ER} graph, our result may be useful for studying the two-type SIR processes on these random graphs. We leave further investigations as future work.\n\nThough Theorem \\ref{thm:single} is more or less well-known, here we give a proof for the sake of completeness and also to illustrate the basic idea that will also be used in the proof of Theorem \\ref{general}. We couple the Galton-Watson tree with an epidemic process by \n revealing the number of children of a vertex after it becomes infected. At time 0 only the root $o$ is infected and \n the only available information about the tree is the degree of $o$. Since $o$ infects each of its children at rate $\\alpha$ and recovers at rate $\\mu$, the probability that any given child is infected before $o$ recovers is equal to $\\alpha/(\\alpha+\\mu)$ by standard properties of the exponential distribution. \n For any vertex $x$,\n we let $N_x$ be the number of children of $x$ that are eventually infected by $x$. Since the degree $D_o$ of $o$ is distributed according to $p$, we see that the mean of $N_o$ is \n \\begin{equation}\\label{eq:edo}\n \\E\\left( \\frac{\\alpha}{\\alpha + \\mu} D_o\\right)=\\frac{m\\alpha}{\\alpha+ \\mu}.\n \\end{equation}\n Denote the distribution of $N_o$ by $q$. \n Since the number of children of any given vertex has the same distribution $p$, we see that the number of infected children of any infected vertex must have the distribution $q$, and must be independent of its ancestors.\n\nLet $Y_n$ be the number of infected vertices in the $n$-th generation of $GW (p)$. Then\n$Y_n$ forms a branching process with the initial value $Y_0=1$ and branching distribution $q$, whose \nmean is equal to $m\\alpha/(\\alpha+ \\mu)$.\n Standard results on the branching process imply that \n \\begin{equation}\\label{eq:thm}\n \\P(Y_n>0, \\forall n)>0 \\Leftrightarrow \n \\frac{\\alpha}{\\alpha+ \\mu}m>1 \\Leftrightarrow \\alpha>\\frac{ \\mu}{m-1}\n \\textnormal {,}\n \\end{equation}\nwhich proves Theorem \\ref{thm:single}.\n\nLet $\\mathcal{X}_n$ be the set of vertices in the $n$-th generation of $GW (p)$ that eventually receive both infections, and let $X_n=\\abs{\\mathcal{X}_n}$, for all $n \\geq 0$. Note that the root is the single vertex in the zeroth generation. Let $\\mathcal{F}_n$ be the $\\sigma$-algebra generated by the number of children of all $x$ up to the $ (n-1)$-th generation and all $\\tau_A(x)$, $\\tau_B(x)$, for $x$ up to the $n$-th generation. By Lemma \\ref{t}, \n\\begin{equation*}\n \\E(X_{n+1}|\\mathcal{F}_n)\\leq X_n\\textnormal{ a.s., for }n\\geq 0.\n\\end{equation*}\nThus $\\{X_n,n\\geq 0\\}$ is a non-negative integer-valued supermartingale. Hence $X_n$ must converge almost surely to some limit $X$. Moreover, there exists a constant $c>0$ such that for all $k,n\\geq 1$,\n\\begin{equation}\\label{xn=0}\n\\P(X_{n+1}=0|X_n=k)\\geq c^k,\n\\end{equation}\nby considering the event where all vertices in $\\mathcal{X}_n$ recover before infecting any child. Let $\\hat{\\Omega}$ be the event\n $$\n \\left\\{ \\lim_{n\\to\\infty} X_n=0 \\right\\}=\\{X_n=0\\textnormal { for some }n\\}.\n $$\nBy Levy's 0-1 Law \\cite[Theorem 4.6.9] {durrett2019probability} and equation \\eqref{xn=0},\n \\begin{equation*}\n \\P ( \\mathbf{1}_{\\hat{\\Omega}} \\mid \\mathcal{F}_n) \\to 1=1_{\\hat{\\Omega}} \\text{ a.s.}\n \\end{equation*}\nThus, $\\P(\\hat{\\Omega})=1$ and $X=0$ a.s. Now define\n\\begin{equation*}\n N=\\inf\\{n: X_n=0\\}.\n\\end{equation*}\nAll vertices $z$ in the $N$-th generation of $GW (p)$ are either infected by one type of disease, or never infected. Since the subtree of $GW (p)$ re-rooted at $z$ has the same distribution as $GW (p)$, we can apply Theorem \\ref{thm:single} to deduce that the number of infected vertices in that subtree is almost surely finite. Thus, we conclude that the two-type SIR process \nsurvives with probability 0.\n\\end{proof}\n\n\\begin{itemize}\n \\item \\textbf{Rigorous analysis for the ODE system \\eqref{eq:ode}.} It would be interesting to inspect the claims made by physicists \\cite{chen2013outbreaks} regarding properties of final epidemic size $\\mathcal R(\\alpha, C, \\epsilon)$, particularly the discontinuous transition \\eqref{eq:discont}. One can also study whether the mean-field equations approximate the true dynamics on the complete graph. \n \\item \\textbf{Survival of the weaker species in the asymmetric case.} If one species is (sub)critical while the other is supercritical, can the weaker species survive with positive probability when the cooperation is strong? See simulations (Figures \\ref{sim growth} and \\ref{sim_surv}) and discussions below.\n \\item \\textbf{Effects of the graph structure on the two-type SIR process.} As mentioned in the introduction, the simulations in \\cite{grassberger2016phase} found that the structure of the underlying graph (particularly the existence of loops in the graph) has a major impact on the two-type SIR process thereon. We do expect a qualitatively different situation for the survival probability if the Galton-Watson tree is replaced with the integer lattice.\n \\item \\textbf{Different cooperativity mechanism.}\n One can also consider the cooperativity mechanism as in \\cite{DurrettYao2020}, where the recovery rates decrease from $\\mu_i$ to $\\hat{\\mu}_i$ if a node has two infections. By comparison with two independent single-type SIR processes, it can be shown that the critical value for survival matches the single-type model if $\\hat{\\mu}_i$ is sufficiently close to $\\mu_i$. However, the case where $\\hat{\\mu}_i$ is small remains unknown.\n \\item \\textbf{Cooperative SIS models.} If we consider the SIS dynamics on a Galton-Watson tree, by an oriented percolation argument, the critical value becomes smaller if the cooperation is sufficiently strong. As indicated by \\cite{DurrettYao2020}, the general case may be rather challenging. \n\\end{itemize}", + "post_theorem_intro_text_len": 2338, + "post_theorem_intro_text": "Given Theorem \\ref{thm:single}, it is natural to probe whether the critical value gets smaller in the cooperative SIR process. We say that the two-type SIR process survives if for all $t\\geq 0$, there is at least one infected vertex (with $A$ or $B$) at time $t$. According to the following Theorem \\ref{general}, the answer is no. \n\n\\begin{theorem}\n \\label{general}\nConsider the two-type SIR process on $GW (p)$ with parameters $\\alpha_i$, $\\beta_i$, $\\mu_i$, $i=1,2$. At time $0$, the root is infected with both disease $A$ and disease $B$. All other vertices are susceptible to both $A$ and $B$. Let $m$ be the mean of $p$. If\n \\begin{equation}\\label{eq:assump}\n \\max\\left\\{\\frac{\\alpha_1}{\\mu_1},\\frac{\\alpha_2}{\\mu_2}\\right\\}\\leq \\frac{1}{m-1}\n \\textnormal {,}\n \\end{equation}\nthen the probability of survival is $0$.\n\\end{theorem} \n\\begin{remark}\\label{rmk:c=infty}\nIn the special case of equal infection/recovery rates for both diseases (denoted by $\\alpha,\\beta=C\\alpha$ and $\\mu$) with $C=\\infty$, the conclusion of Theorem \\ref{general} is consistent with the prediction made in \\cite{grassberger2016phase} regarding infinite tree graphs, where the authors claimed that (for $C=\\infty$), \n$$\n\\mathbb{P}(\\mbox{infinitely many vertices are infected with both diseases})>0\\, \\mbox{ if and only if }\\, \\frac{\\alpha}{\\mu}>\\frac{1}{m-1}. \n$$ See also the discussions above Table \\ref{rates table}.\n\\end{remark}\nCombining Theorems \\ref{thm:single} and \\ref{general}, the two-type SIR process survives with a positive probability if and only if\n $$\n \\max\\left\\{\\frac{\\alpha_1}{\\mu_1},\\frac{\\alpha_2}{\\mu_2}\\right\\}> \\frac{1}{m-1}.\n $$\n Since the Galton-Watson tree and its variants arise as local limits of some random graph models such as the configuration model and the {Erd\\H{o}s-R\\'enyi} graph, our result may be useful for studying the two-type SIR processes on these random graphs. We leave further investigations as future work. \n\nIn the remainder of this paper, we prove\nTheorems \\ref{thm:single} and \\ref{general} in Section \\ref{sec:single} and Section \\ref{sec:two}, respectively, by coupling the Galton-Watson tree with the epidemic process. Section \\ref{sec:conclusion} summarizes the main findings of this paper and states many open problems as well as some simulation results.", + "sketch": "In the remainder of this paper, we prove Theorems \\ref{thm:single} and \\ref{general} in Section \\ref{sec:single} and Section \\ref{sec:two}, respectively, \\emph{by coupling the Galton-Watson tree with the epidemic process}.", + "expanded_sketch": "In the remainder of this paper, we prove the main theorem and we first prove the following theorem.\n\n\\begin{theorem}\n \\label{general}\nConsider the two-type SIR process on $GW (p)$ with parameters $\\alpha_i$, $\\beta_i$, $\\mu_i$, $i=1,2$. At time $0$, the root is infected with both disease $A$ and disease $B$. All other vertices are susceptible to both $A$ and $B$. Let $m$ be the mean of $p$. If\n \\begin{equation}\\label{eq:assump}\n \\max\\left\\{\\frac{\\alpha_1}{\\mu_1},\\frac{\\alpha_2}{\\mu_2}\\right\\}\\leq \\frac{1}{m-1}\n \\textnormal {,}\n \\end{equation}\nthen the probability of survival is $0$.\n\\end{theorem}\n\nWe do so \\emph{by coupling the Galton-Watson tree with the epidemic process}.", + "expanded_theorem": "\\label{thm:single}\n Let the infection rate be $\\alpha$ and the recovery rate be $\\mu$ for a single-type SIR process on $GW (p)$. At time 0, only the root is infected. The probability of survival is greater than 0 if and only if $ \\alpha/\\mu > 1/{(m-1)} $, where $m$ is the mean of $p$. Thus, if $\\mu$ is fixed, then the critical value for $\\alpha$ is $\\mu/(m-1)$.", + "theorem_type": [ + "Biconditional or Equivalence", + "Existence" + ], + "mcq": { + "question": "Let \\(GW(p)\\) be a Galton--Watson tree with offspring distribution \\(p\\) and mean \\(m\\). Consider a single-type SIR process on \\(GW(p)\\) with infection rate \\(\\alpha\\) and recovery rate \\(\\mu\\), started at time \\(0\\) with only the root infected. Here, “survival” means that the infection does not die out. Which statement holds about when survival has positive probability?", + "correct_choice": { + "label": "A", + "text": "The survival probability is greater than \\(0\\) if and only if \\(\\alpha/\\mu > 1/(m-1)\\). Equivalently, if \\(\\mu\\) is fixed, then the critical value of the infection rate is \\(\\mu/(m-1)\\)." + }, + "choices": [ + { + "label": "B", + "text": "The survival probability is greater than \\(0\\) if and only if \\(\\alpha/\\mu \\ge 1/(m-1)\\). Equivalently, if \\(\\mu\\) is fixed, then the critical value of the infection rate is attained already at \\(\\alpha=\\mu/(m-1)\\)." + }, + { + "label": "C", + "text": "If \\(\\alpha/\\mu > 1/(m-1)\\), then the survival probability is greater than \\(0\\). Equivalently, if \\(\\mu\\) is fixed and \\(\\alpha>\\mu/(m-1)\\), then survival has positive probability." + }, + { + "label": "D", + "text": "The survival probability is greater than \\(0\\) if and only if \\(\\alpha/\\mu > 1/m\\). Equivalently, if \\(\\mu\\) is fixed, then the critical value of the infection rate is \\(\\mu/m\\)." + }, + { + "label": "E", + "text": "The survival probability is greater than \\(0\\) whenever the mean offspring satisfies \\(m\\,\\alpha/\\mu > 1\\), and in particular the critical value for \\(\\alpha\\) is \\(\\mu/m\\)." + } + ], + "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": "strictness_of_threshold", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "counting_estimate", + "tampered_component": "only_if_direction", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "counting_estimate", + "tampered_component": "effective_branching_factor_m_minus_1", + "template_used": "property_confusion" + }, + { + "label": "E", + "sketch_hook_type": "geometric_construction", + "tampered_component": "root_vs_nonroot_branching_count", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not state or strongly hint at the threshold formula; it only asks for the correct survival criterion. The correct answer is not leaked directly." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recall question, since the correct option essentially states the exact threshold result. However, it is not a pure tautology because the alternatives vary in strictness, logical strength, and the branching-factor term." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish the exact 'if and only if' statement from the weaker true implication and from the common confusion between m and m-1. Still, the item mainly tests recognition/recall of the threshold rather than deep derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically meaningful: B tests boundary strictness, C is a weaker-but-true statement, and D/E reflect a natural m versus m-1 branching-factor confusion. These align well with common failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with strong distractors and no answer leakage, but it leans more toward theorem recognition than genuinely generative reasoning." + } + }, + { + "id": "2602.18969v1", + "paper_link": "http://arxiv.org/abs/2602.18969v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "thm", + "content": "The Prym maps \n$$\n\\mathcal{P}^{iso}_3 : \\cRH_3^{iso} \\rightarrow \\cA_6^{(1,1,1,2,2,4)} \\qquad \\mathcal{P}^{ni}_3 : \\cRH_3^{ni} \\rightarrow \\cA_6^{(1,1,1,1,4,4)}\n$$\nare injective on each irreducible component of the source moduli space.", + "start_pos": 7298, + "end_pos": 7529, + "label": null + }, + "ref_dict": { + "4components": "\\begin{lem}\\label{4components}\nThe moduli space $\\cRH_3^{V_4}$ consists of four irreducible components, corresponding to the configuration of the Weierstrass points in the generators $ \\eta, \\ \\xi$ as shown \nin I.1, I.2, II.1 and II.2. Moreover, the degree of the forgetful \nmap $\\cRH_3^{V_4} \\ra \\cH_3$ on each component is 56, 280, 105, 210, respectively. \n\\end{lem}", + "autoofP": "\\begin{prop}\\label{autoofP}\nLet $(P, \\Xi)$ be an element in the image of the Prym map \n$\\cP^{iso}_3$, respectively in $\\cP^{ni}_3$ Case I.2. Then it holds \n$$\nB:=\\{\\psi \\in \\Aut (P,\\Xi) \\ : \\ \\psi^2=id, \\ \n\\psi_{|K(\\Xi)}= \\pm id \\} = \\langle \\sigma, \\tau, j, -1 \\rangle \\simeq \\ZZ_2^4 \n$$\nwhere $j$ is any lift of the hyperelliptic involution from $H$.\n\\end{prop}", + "FullKlein": "\\begin{thm}\\label{FullKlein}\n The Klein Prym maps $$\n\\cP^{iso}_3 : \\cR_3^{iso} \\ra \\cA_6^{(1,1,1,2,2,4)} \\qquad \\cP^{ni}_3 : \\cR_3^{ni} \\ra \\cA_6^{(1,1,1,1,4,4)}\n$$ are generically finite. Moreover they are of degree $1$ on images of coverings of hyperelliptic curves of types $I.2, II.1, II.2$.\n\\end{thm}" + }, + "pre_theorem_intro_text_len": 2700, + "pre_theorem_intro_text": "Let $H$ be a hyperelliptic smooth projective curve over $\\mathbb C$ of genus 3. We consider Klein coverings $f:\\widetilde{C} \\rightarrow H$, that is, \n$\\widetilde{C}$ admits an action by a group of fixed point free automorphisms $V_4$ isomorphic to $\\ZZ_2 \\times \\ZZ_2$ such that $H = \\widetilde{C} / V_4$. \nOne can associate to such a covering a polarised abelian sixfold in the following way. Let $\\Nm_f : J\\widetilde{C} \\rightarrow JH$ be the norm map, sending the divisor class $[\\sum_i n_ip_i]$ to $[\\sum_i n_if(p_i)]$,\nwith $p_i \\in \\widetilde{C}$ and $n_i\\in \\mathbb Z$. The Prym variety $P(f)$ of the covering $f$ is defined as the connected component of 0 of the kernel of $\\Nm_f$. It is an abelian subvariety of $J\\widetilde{C}$ of dimension the difference of the genera of the curves, which in this case is 6 and the restriction of the principal polarisation on $J\\widetilde{C}$ induces a non-principal one on $P(f)$. \n\nPrym varieties of étale double covers have been extensively studied in the past and more recently it has been proven that \nthe Prym map is injective for double covers ramified in at least six points (\\cite{NO22}). For étale cyclic coverings over a hyperelliptic curve, the (generic) injectivity \nof the Prym has been shown for infinitely many degrees (\\cite{NOS24}, \n\\cite{NOPS25}). There are also new results on the fiber of the Prym map for non-cyclic coverings in low genera (\\cite{S25}). \n\nKlein coverings can be defined from the curve $H$ by choosing a subgroup \n$\\langle \\eta, \\xi\\rangle $ of order four of the group $JH[2]$ of 2-torsion points in $JH$. \nWe distinguish two types of Klein coverings depending whether the subgroup is isotropic or not with respect to the Weil pairing on $JH[2]$. \nLet $\\cRH_3^{iso}$, respectively $\\cRH_3^{ni}$, denote the moduli space parametrising isotropic, respectively non-isotropic, Klein \ncoverings on hyperelliptic curves of genus 3. \nEach of these moduli spaces consists of two irreducible components (see Lemma \\ref{4components}). \n\nConsider now the Prym map that sends $[f:\\widetilde{C} \\rightarrow H] \\in \\cRH_3^{V_4}=\\cRH_3^{iso} \\sqcup \\cRH_3^{ni} $ to the \npolarised Prym variety $(P,\\Xi)$. In this article we continue the study of the injectivity of the Prym map for Klein \ncoverings of genus $3$ curves, which was started in \\cite{BO24}, where we prove the injectivity on the locus of hyperelliptic coverings $f:\\widetilde{C} \\rightarrow H $ (i.e. $\\widetilde{C}$ is also hyperelliptic) \ncorresponding to the case I.1 below. Let us denote by\n$\\cA_g^{\\delta}$ the moduli space of polarised abelian varieties of dimension $g$ with polarisation type $\\delta$.\nThe main theorem of the paper is the following.", + "context": "Let $H$ be a hyperelliptic smooth projective curve over $\\mathbb C$ of genus 3. We consider Klein coverings $f:\\widetilde{C} \\rightarrow H$, that is, \n$\\widetilde{C}$ admits an action by a group of fixed point free automorphisms $V_4$ isomorphic to $\\ZZ_2 \\times \\ZZ_2$ such that $H = \\widetilde{C} / V_4$. \nOne can associate to such a covering a polarised abelian sixfold in the following way. Let $\\Nm_f : J\\widetilde{C} \\rightarrow JH$ be the norm map, sending the divisor class $[\\sum_i n_ip_i]$ to $[\\sum_i n_if(p_i)]$,\nwith $p_i \\in \\widetilde{C}$ and $n_i\\in \\mathbb Z$. The Prym variety $P(f)$ of the covering $f$ is defined as the connected component of 0 of the kernel of $\\Nm_f$. It is an abelian subvariety of $J\\widetilde{C}$ of dimension the difference of the genera of the curves, which in this case is 6 and the restriction of the principal polarisation on $J\\widetilde{C}$ induces a non-principal one on $P(f)$.\n\nPrym varieties of étale double covers have been extensively studied in the past and more recently it has been proven that \nthe Prym map is injective for double covers ramified in at least six points (\\cite{NO22}). For étale cyclic coverings over a hyperelliptic curve, the (generic) injectivity \nof the Prym has been shown for infinitely many degrees (\\cite{NOS24}, \n\\cite{NOPS25}). There are also new results on the fiber of the Prym map for non-cyclic coverings in low genera (\\cite{S25}).\n\nKlein coverings can be defined from the curve $H$ by choosing a subgroup \n$\\langle \\eta, \\xi\\rangle $ of order four of the group $JH[2]$ of 2-torsion points in $JH$. \nWe distinguish two types of Klein coverings depending whether the subgroup is isotropic or not with respect to the Weil pairing on $JH[2]$. \nLet $\\cRH_3^{iso}$, respectively $\\cRH_3^{ni}$, denote the moduli space parametrising isotropic, respectively non-isotropic, Klein \ncoverings on hyperelliptic curves of genus 3. \nEach of these moduli spaces consists of two irreducible components (see Lemma \\ref{4components}).\n\nConsider now the Prym map that sends $[f:\\widetilde{C} \\rightarrow H] \\in \\cRH_3^{V_4}=\\cRH_3^{iso} \\sqcup \\cRH_3^{ni} $ to the \npolarised Prym variety $(P,\\Xi)$. In this article we continue the study of the injectivity of the Prym map for Klein \ncoverings of genus $3$ curves, which was started in \\cite{BO24}, where we prove the injectivity on the locus of hyperelliptic coverings $f:\\widetilde{C} \\rightarrow H $ (i.e. $\\widetilde{C}$ is also hyperelliptic) \ncorresponding to the case I.1 below. Let us denote by\n$\\cA_g^{\\delta}$ the moduli space of polarised abelian varieties of dimension $g$ with polarisation type $\\delta$.\nThe main theorem of the paper is the following.\n\n\\begin{lem}\\label{4components}\nThe moduli space $\\cRH_3^{V_4}$ consists of four irreducible components, corresponding to the configuration of the Weierstrass points in the generators $ \\eta, \\ \\xi$ as shown \nin I.1, I.2, II.1 and II.2. Moreover, the degree of the forgetful \nmap $\\cRH_3^{V_4} \\ra \\cH_3$ on each component is 56, 280, 105, 210, respectively. \n\\end{lem}", + "full_context": "Let $H$ be a hyperelliptic smooth projective curve over $\\mathbb C$ of genus 3. We consider Klein coverings $f:\\widetilde{C} \\rightarrow H$, that is, \n$\\widetilde{C}$ admits an action by a group of fixed point free automorphisms $V_4$ isomorphic to $\\ZZ_2 \\times \\ZZ_2$ such that $H = \\widetilde{C} / V_4$. \nOne can associate to such a covering a polarised abelian sixfold in the following way. Let $\\Nm_f : J\\widetilde{C} \\rightarrow JH$ be the norm map, sending the divisor class $[\\sum_i n_ip_i]$ to $[\\sum_i n_if(p_i)]$,\nwith $p_i \\in \\widetilde{C}$ and $n_i\\in \\mathbb Z$. The Prym variety $P(f)$ of the covering $f$ is defined as the connected component of 0 of the kernel of $\\Nm_f$. It is an abelian subvariety of $J\\widetilde{C}$ of dimension the difference of the genera of the curves, which in this case is 6 and the restriction of the principal polarisation on $J\\widetilde{C}$ induces a non-principal one on $P(f)$.\n\nPrym varieties of étale double covers have been extensively studied in the past and more recently it has been proven that \nthe Prym map is injective for double covers ramified in at least six points (\\cite{NO22}). For étale cyclic coverings over a hyperelliptic curve, the (generic) injectivity \nof the Prym has been shown for infinitely many degrees (\\cite{NOS24}, \n\\cite{NOPS25}). There are also new results on the fiber of the Prym map for non-cyclic coverings in low genera (\\cite{S25}).\n\nKlein coverings can be defined from the curve $H$ by choosing a subgroup \n$\\langle \\eta, \\xi\\rangle $ of order four of the group $JH[2]$ of 2-torsion points in $JH$. \nWe distinguish two types of Klein coverings depending whether the subgroup is isotropic or not with respect to the Weil pairing on $JH[2]$. \nLet $\\cRH_3^{iso}$, respectively $\\cRH_3^{ni}$, denote the moduli space parametrising isotropic, respectively non-isotropic, Klein \ncoverings on hyperelliptic curves of genus 3. \nEach of these moduli spaces consists of two irreducible components (see Lemma \\ref{4components}).\n\nConsider now the Prym map that sends $[f:\\widetilde{C} \\rightarrow H] \\in \\cRH_3^{V_4}=\\cRH_3^{iso} \\sqcup \\cRH_3^{ni} $ to the \npolarised Prym variety $(P,\\Xi)$. In this article we continue the study of the injectivity of the Prym map for Klein \ncoverings of genus $3$ curves, which was started in \\cite{BO24}, where we prove the injectivity on the locus of hyperelliptic coverings $f:\\widetilde{C} \\rightarrow H $ (i.e. $\\widetilde{C}$ is also hyperelliptic) \ncorresponding to the case I.1 below. Let us denote by\n$\\cA_g^{\\delta}$ the moduli space of polarised abelian varieties of dimension $g$ with polarisation type $\\delta$.\nThe main theorem of the paper is the following.\n\n\\begin{lem}\\label{4components}\nThe moduli space $\\cRH_3^{V_4}$ consists of four irreducible components, corresponding to the configuration of the Weierstrass points in the generators $ \\eta, \\ \\xi$ as shown \nin I.1, I.2, II.1 and II.2. Moreover, the degree of the forgetful \nmap $\\cRH_3^{V_4} \\ra \\cH_3$ on each component is 56, 280, 105, 210, respectively. \n\\end{lem}\n\nKlein coverings can be defined from the curve $H$ by choosing a subgroup \n$\\langle \\eta, \\xi\\rangle $ of order four of the group $JH[2]$ of 2-torsion points in $JH$. \nWe distinguish two types of Klein coverings depending whether the subgroup is isotropic or not with respect to the Weil pairing on $JH[2]$. \nLet $\\cRH_3^{iso}$, respectively $\\cRH_3^{ni}$, denote the moduli space parametrising isotropic, respectively non-isotropic, Klein \ncoverings on hyperelliptic curves of genus 3. \nEach of these moduli spaces consists of two irreducible components (see Lemma \\ref{4components}).\n\nConsider now the Prym map that sends $[f:\\tC \\ra H] \\in \\cRH_3^{V_4}=\\cRH_3^{iso} \\sqcup \\cRH_3^{ni} $ to the \npolarised Prym variety $(P,\\Xi)$. In this article we continue the study of the injectivity of the Prym map for Klein \ncoverings of genus $3$ curves, which was started in \\cite{BO24}, where we prove the injectivity on the locus of hyperelliptic coverings $f:\\tC \\ra H $ (i.e. $\\tC$ is also hyperelliptic) \ncorresponding to the case I.1 below. Let us denote by\n$\\cA_g^{\\delta}$ the moduli space of polarised abelian varieties of dimension $g$ with polarisation type $\\delta$.\nThe main theorem of the paper is the following.\n\nActually, the injectivity also holds on the entire moduli space $\\cRH_3^{V_4}$ since the dimensions of the factors appearing in the isotypical decomposition of the associated Prym variety, depends on the type of the Klein covering and it is different for each irreducible component.\n\n\\begin{thm}\\label{thmI.2}\nThe Prym map \n$\\cP^{ni}_3 : \\cRH_3^{ni} \\ra \\cA_6^{(1,1,1,1,4,4)}$ is injective on the irreducible component consisting of Klein coverings of type I.2.\n\n\\begin{thm}\nThe Prym map \n$\\cP^{iso}_3 : \\cRH_3^{iso} \\ra \\cA_6^{(1,1,1,2,2,4)}$ is injective on the irreducible component consisting of Klein coverings of type II.1.\n\\end{thm}\n\\begin{proof}\nLet $(P, \\Xi)$ be the image of a Klein covering of type $II.1$ under $\\cP^{iso}_3$. First note that, by Proposition \\ref{autoofP}, $(P, \\Xi)$ determines the group of involutions $B\\simeq \\ZZ^4_2$ on $P$.\nOne then look at the fixed loci of the involutions together with the restricted polarisation types to distinguish $j,j\\s,j\\tau,j\\s\\tau$ and $JC_j,JC_{j\\s},JC_{j\\tau},JC_{j\\s\\tau}$ respectively. Now, one chooses any two of these, say $j$ and $j\\s$ and by Torelli Theorem one recovers $C_j$ and $C_{j\\s}$.\nBy Lemma \\ref{intersection_Jacobians_II.1} and Lemma \\ref{key_lemma} we obtain the maps $C_j\\to E$ and $C_{j\\s}\\to E$, so we have the following fibered product.\n\n\\begin{thm}\nThe Prym map \n$\\cP^{iso}_3 : \\cRH_3^{iso} \\ra \\cA_6^{(1,1,1,2,2,4)}$ is injective on the irreducible component consisting of Klein coverings of type II.2.\n\\end{thm}\n\\begin{proof}\nLet $(P, \\Xi)$ be an element on the image of $\\cP^{ni}_3$. Then the group $\\langle \\sigma, \\tau,j, -1 \\rangle \\simeq \\ZZ_2^4$ acts on $P$ and according to Proposition \\ref{isotyp_II.2}, this action induces the isotypical decomposition \n$$\nP = P^*(C_{\\s}/H) \\boxplus P^*(C_{\\tau}/H) \\boxplus E_j \\boxplus F\n$$\nwith a distinguished elliptic curve $F$ of exponent $(4)$. \nMoreover, by construction \n$$\nP^*(C_{\\s}/H)+F \\sim (JC_{j\\s}, 2\\Theta_{C_{j\\s}}), \\quad P^*(C_{\\tau}/H)+F \\sim (JC_{j\\tau}, 2\\Theta_{C_{j\\tau}}).\n$$\nSo the isotypical components determine uniquely the \npolarised Jacobians called $JC_{j\\s}$ and $JD_{j\\tau}$ and\nby Torelli Theorem the genus 3 curves $C_{j\\s}$ and $ C_{j\\tau}$ are also determined. Since $F$ is embedded in the Jacobians by Lemma \\ref{key_lemma}, we consider the fibered product \n\\begin{equation}\\label{fibred_product_II.2}\n\\xymatrix@R=.9cm@C=1cm{\n \\widetilde{C_{j\\s} \\times_{F} C_{j\\tau}}\\ar[d]_{2:1} \\ar[r]^{ \\ 2:1} & C_{j\\s} \\ar[d]^{h_\\s} \\\\\nC_{j\\tau} \\ar[r]_{h_\\tau} & F \n}\n\\end{equation}\n\n\\begin{thm}\\label{FullKlein}\n The Klein Prym maps $$\n\\cP^{iso}_3 : \\cR_3^{iso} \\ra \\cA_6^{(1,1,1,2,2,4)} \\qquad \\cP^{ni}_3 : \\cR_3^{ni} \\ra \\cA_6^{(1,1,1,1,4,4)}\n$$ are generically finite. Moreover they are of degree $1$ on images of coverings of hyperelliptic curves of types $I.2, II.1, II.2$.\n\\end{thm}\n\\begin{proof}\n Let us consider the non-isotropic case. By Theorems \\ref{thmI.1} and \\ref{thmI.2} the Prym map ${\\cP^{ni}_3}_{|\\cRH_3^{ni}}$ is injective on each component. Moreover, images of respective components are disjoint as the isotypical decomposition yields components of different dimensions. Hence, the image of the Prym map, being an irreducible variety and containing $2$ disjoint subvarieties of dimension $5$ has to be of dimension at least $6=\\dim(\\cR_3^{ni})$. This shows that the map is generically finite.\n\nThe above theorem is an evidence to a positive answer of the following question.\n\\begin{question}\nAre the following Prym maps\n$$\n\\cP^{iso}_3 : \\cR_3^{iso} \\ra \\cA_6^{(1,1,1,2,2,4)} \\qquad \\cP^{ni}_3 : \\cR_3^{ni} \\ra \\cA_6^{(1,1,1,1,4,4)}\n$$\n injective?\n\\end{question}\n\n\\begin{lem}\\label{4components}\nThe moduli space $\\cRH_3^{V_4}$ consists of four irreducible components, corresponding to the configuration of the Weierstrass points in the generators $ \\eta, \\ \\xi$ as shown \nin I.1, I.2, II.1 and II.2. Moreover, the degree of the forgetful \nmap $\\cRH_3^{V_4} \\ra \\cH_3$ on each component is 56, 280, 105, 210, respectively. \n\\end{lem}\n\n\\begin{prop}\\label{autoofP}\nLet $(P, \\Xi)$ be an element in the image of the Prym map \n$\\cP^{iso}_3$, respectively in $\\cP^{ni}_3$ Case I.2. Then it holds \n$$\nB:=\\{\\psi \\in \\Aut (P,\\Xi) \\ : \\ \\psi^2=id, \\ \n\\psi_{|K(\\Xi)}= \\pm id \\} = \\langle \\sigma, \\tau, j, -1 \\rangle \\simeq \\ZZ_2^4 \n$$\nwhere $j$ is any lift of the hyperelliptic involution from $H$.\n\\end{prop}", + "post_theorem_intro_text_len": 2121, + "post_theorem_intro_text": "Actually, the injectivity also holds on the entire moduli space $\\cRH_3^{V_4}$ since the dimensions of the factors appearing in the isotypical decomposition of the associated Prym variety, depends on the type of the Klein covering and it is different for each irreducible component. \n\nThe general idea of the proof, with some variations in each case, is the following. A Prym variety $(P,\\Xi)$ in the image of the Prym map determines a group of automorphisms of $P$ isomorphic to \n$\\ZZ_2^4$ (see Proposition \\ref{autoofP}). This gives us the isotypical decomposition of $P$ and one can identify the Jacobians by means of the type of the restricted polarisation and using the action of the involutions on the kernel of the polarisation. With this information we can reconstruct the quotients curves and finally recover $\\widetilde{C}$ as a fibered product. \n\nAs an application, we can show the following theorem for Klein coverings over {\\it any} curve of genus 3, see Theorem \\ref{FullKlein}.\n\\begin{thm}\n The Klein Prym maps $$\n\\mathcal{P}^{iso}_3 : \\cR_3^{iso} \\rightarrow \\cA_6^{(1,1,1,2,2,4)} \\qquad \\mathcal{P}^{ni}_3 : \\cR_3^{ni} \\rightarrow \\cA_6^{(1,1,1,1,4,4)}\n$$ are generically finite. Moreover they are of degree $1$ on images of coverings of hyperelliptic curves of types $I.2, II.1, II.2$.\n\\end{thm}\n\nThe structure of the paper is as follows. In Section 1, we state basic definitions and key lemmas and show that the moduli space have 4 components, called I.1, I.2, II.1, II.2. Each one of the Sections 2,\\ 3,\\ 4 and 5 is devoted to one of these cases and to the proof of the injectivity of the corresponding Prym map. Finally, Section 6 is devoted to Klein coverings of any genus 3 curve.\n\n\\subsection*{Acknowledgments}\nThe authors would like to thank Anatoli Shatsila for finding a reference that shortened considerably one of the proofs. The first author has been supported by the Polish National Science Centre project number 2024/54/E/ST1/00330. Some results of the paper were obtained during his visit to Humboldt University in Berlin. He would like to thank the university for hospitality.", + "sketch": "Actually, the injectivity also holds on the entire moduli space $\\cRH_3^{V_4}$ since the dimensions of the factors appearing in the isotypical decomposition of the associated Prym variety depends on the type of the Klein covering and it is different for each irreducible component.\n\nThe general idea of the proof (``with some variations in each case'') is:\n\\begin{itemize}\n\\item A Prym variety $(P,\\Xi)$ in the image of the Prym map determines a group of automorphisms of $P$ isomorphic to $\\ZZ_2^4$ (Proposition \\ref{autoofP}).\n\\item This yields the isotypical decomposition of $P$.\n\\item ``One can identify the Jacobians\" using the type of the restricted polarisation and the action of the involutions on the kernel of the polarisation.\n\\item From this data, ``we can reconstruct the quotients curves\" and ``finally recover $\\widetilde{C}$ as a fibered product.\"\n\\end{itemize}", + "expanded_sketch": "Actually, the injectivity also holds on the entire moduli space $\\cRH_3^{V_4}$ since the dimensions of the factors appearing in the isotypical decomposition of the associated Prym variety depends on the type of the Klein covering and it is different for each irreducible component.\n\nThe general idea of the proof (``with some variations in each case'') is:\n\\begin{itemize}\n\\item A Prym variety $(P,\\Xi)$ in the image of the Prym map determines a group of automorphisms of $P$ isomorphic to $\\ZZ_2^4$. More precisely:\n\\begin{prop}\\label{autoofP}\nLet $(P, \\Xi)$ be an element in the image of the Prym map \n$\\cP^{iso}_3$, respectively in $\\cP^{ni}_3$ Case I.2. Then it holds \n$$\nB:=\\{\\psi \\in \\Aut (P,\\Xi) \\ : \\ \\psi^2=id, \\ \n\\psi_{|K(\\Xi)}= \\pm id \\} = \\langle \\sigma, \\tau, j, -1 \\rangle \\simeq \\ZZ_2^4 \n$$\nwhere $j$ is any lift of the hyperelliptic involution from $H$.\n\\end{prop}\n\\item This yields the isotypical decomposition of $P$.\n\\item ``One can identify the Jacobians\" using the type of the restricted polarisation and the action of the involutions on the kernel of the polarisation.\n\\item From this data, ``we can reconstruct the quotients curves\" and ``finally recover $\\widetilde{C}$ as a fibered product.\"\n\\end{itemize}", + "expanded_theorem": "The Prym maps \n$$\n\\mathcal{P}^{iso}_3 : \\cRH_3^{iso} \\rightarrow \\cA_6^{(1,1,1,2,2,4)} \\qquad \\mathcal{P}^{ni}_3 : \\cRH_3^{ni} \\rightarrow \\cA_6^{(1,1,1,1,4,4)}\n$$\nare injective on each irreducible component of the source moduli space.", + "theorem_type": [ + "Universal", + "Implication" + ], + "mcq": { + "question": "Let \\(\\cRH_3^{iso}\\) (respectively \\(\\cRH_3^{ni}\\)) be the moduli space of Klein coverings \\(f:\\widetilde C\\to H\\) of hyperelliptic smooth projective curves \\(H\\) of genus \\(3\\) over \\(\\mathbb C\\), where \\(\\widetilde C\\) admits a fixed-point-free action of \\(V_4\\simeq \\mathbb Z_2\\times \\mathbb Z_2\\) with quotient \\(H=\\widetilde C/V_4\\), and where the associated order-four subgroup \\(\\langle \\eta,\\xi\\rangle\\subset JH[2]\\) is isotropic (respectively non-isotropic) with respect to the Weil pairing. For such a covering, let \\(P(f)\\) be the Prym variety, namely the connected component of \\(0\\) in \\(\\ker(\\operatorname{Nm}_f:J\\widetilde C\\to JH)\\), endowed with the induced polarization. Consider the Prym maps\n\\[\n\\mathcal P^{iso}_3: \\cRH_3^{iso}\\to \\cA_6^{(1,1,1,2,2,4)},\\qquad\n\\mathcal P^{ni}_3: \\cRH_3^{ni}\\to \\cA_6^{(1,1,1,1,4,4)},\n\\]\nwhere \\(\\cA_6^{\\delta}\\) denotes the moduli space of polarized abelian sixfolds of polarization type \\(\\delta\\). Which statement holds for every irreducible component of the source moduli spaces?", + "correct_choice": { + "label": "A", + "text": "For both maps, the restriction to any irreducible component of the source is injective; that is, on each irreducible component of \\(\\cRH_3^{iso}\\) and on each irreducible component of \\(\\cRH_3^{ni}\\), two Klein coverings with the same polarized Prym variety represent the same point of that component." + }, + "choices": [ + { + "label": "B", + "text": "For both maps, the Prym variety determines the Klein covering globally on the whole source moduli space; that is, \\(\\mathcal P^{iso}_3\\) is injective on all of \\(\\cRH_3^{iso}\\) and \\(\\mathcal P^{ni}_3\\) is injective on all of \\(\\cRH_3^{ni}\\), so two Klein coverings with the same polarized Prym variety necessarily coincide even if they lie in different irreducible components." + }, + { + "label": "C", + "text": "For both maps, the restriction to each irreducible component has finite fibers; equivalently, on every irreducible component of \\(\\cRH_3^{iso}\\) and of \\(\\cRH_3^{ni}\\), only finitely many Klein coverings can have the same polarized Prym variety." + }, + { + "label": "D", + "text": "For both maps, there is a single reconstruction procedure from the polarized Prym variety that is uniform across all irreducible components, in the sense that the same isotypical decomposition and the same pair of quotient Jacobians recover the Klein covering in every isotropic and non-isotropic case." + }, + { + "label": "E", + "text": "For both maps, the polarized Prym variety determines the underlying hyperelliptic base curve \\(H\\) on each irreducible component, but not necessarily the full Klein covering; in other words, equality of polarized Prym varieties forces the same quotient curve \\(H\\), although the corresponding order-four subgroup \\(\\langle\\eta,\\xi\\rangle\\subset JH[2]\\) may still vary." + } + ], + "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": "componentwise injectivity versus global injectivity across distinct irreducible components", + "template_used": "stronger_trap" + }, + { + "label": "C", + "sketch_hook_type": "finiteness", + "tampered_component": "dropped injective and kept finite fibers on each irreducible component", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "case_split", + "tampered_component": "case-dependent isotypical decomposition and reconstruction data", + "template_used": "uniformity_effectivity" + }, + { + "label": "E", + "sketch_hook_type": "geometric_construction", + "tampered_component": "reconstruction of the full covering from quotient curves and fibered product, not merely the base curve", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines the Prym maps and asks which conclusion holds, but it does not explicitly or implicitly reveal that the correct property is injectivity on each irreducible component." + }, + "TAS": { + "score": 1, + "justification": "The item is largely a theorem-identification question: it asks for the precise valid statement about the Prym maps among nearby variants. This is not a pure restatement, since the options include stronger and weaker alternatives, but it remains close to recalling the theorem's formulation." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish full injectivity, componentwise injectivity, generic finiteness, and generically injective behavior. However, the task mainly tests precise theorem recall rather than substantial generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically meaningful: they vary the quantifiers and strength of the conclusion in ways that reflect common confusion between injective, generically injective, and generically finite, as well as componentwise versus global statements." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward precise recall of a known result than toward deep generative reasoning." + } + }, + { + "id": "2602.19175v1", + "paper_link": "http://arxiv.org/abs/2602.19175v1", + "theorems_cnt": 4, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm0.1}\n\t\tLet $(X,{\\mathrm d},\\mathfrak m)$ be an ${\\rm RCD}(K,N)$ metric measure space. Let $S\\subseteq X$ be a John domain and $Y\\subseteq X$ be compact with $\\mathfrak m(Y)>0$. Let $\\rho\\in \\mathcal{P}(S)$ be with $a_1\\mathfrak m\\restr{S} \\leq \\rho\\leq a_2\\mathfrak m\\restr{S}$ for some positive constants $a_1,a_2$. Then there exists a constant $C>0$, depending on $K,N,a_1,a_2, S, \\mathop{\\rm diam}\\nolimits(S\\cup Y)$, such that for any $\\mu,\\nu\\in\\mathcal{P}(Y)$, \n\t\t\\begin{equation}\n\t\t\t\\|\\phi_\\mu-\\phi_\\nu\\|_{L^1(\\rho)}\\leq C W_1^\\frac{1}{2}(\\mu,\\nu),\n\t\t\\end{equation}\n\t\twhere $\\phi_\\mu$ and $\\phi_\\nu$ are the Kantorovich potentials from $\\rho$ to $\\mu$ and $\\rho$ to $\\nu$ respectively.", + "start_pos": 11594, + "end_pos": 12265, + "label": "thm0.1" + }, + "ref_dict": { + "A": "\\begin{appendices}\n\t\\section{Poincar\\'e inequality: local to global}\\label{A}\n\t\t\\begin{lemma}\\label{p3.10}\n\tLet $x_0\\in S$, $01\\) if there is a covering \\(\\mathcal{F}\\) of $S$ by open balls $B \\in \\mathcal{F}$ such that:\n\n\t\t\\begin{enumerate}\n\t\t\t\t\\item For any \\(x \\in S\\),\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\sum_{B \\in \\mathcal{F}} \\chi_{2B}(x) \\leq E \\chi_{S}(x).\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\\item For some fixed ball \\(B_0\\) in \\(\\mathcal{F}\\), called the \\emph{central ball}, and for every \\(B \\in \\mathcal{F}\\), there exists a chain \\(B_0, B_1, \\ldots, B_N = B\\) of distinct balls from \\(\\mathcal{F}\\) such that\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\tB \\subset F B_j,~~~\\forall j \\in \\{0, \\ldots, N-1\\}.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\\item Consecutive balls of the above chain overlap quantitatively:\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\rho(B_j \\cap B_{j+1}) \\geq G^{-1} \\max(\\rho(B_j), \\rho(B_{j+1})),~~~\\forall j \\in \\{0, \\ldots, N-1\\}.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\\end{enumerate}\n\t\\end{definition}\n\n\t\\begin{lemma}\\label{3.1}\n\tFor any $f\\in L^1(\\rho)$, it holds that\n\t\\begin{equation}\\label{A.5}\n\t\t\\int_S |f(x)-\\E_{\\rho}(f)|\\,\\d\\rho(x)\\leq C_5 \\sum_{B\\in\\mathcal{F}}\\rho(B)\\int_B |f(x)-\\E_{\\rho_B}(f)|\\,\\d\\rho_B(x),\n\t\\end{equation}\n\twhere $C_5$ depends on $K,N,\\diam(S),S$.\n\t\\end{lemma}\n\n \\begin{proof} \n \tSince $S$ is a John domain, and $\\rho$ is a doubling measure on $S$, by \\cite[Proposition 3.7]{arXiv:2504.05412}, $\\rho$ satisfies the Boman chain condition. Hence there exists a covering $\\mathcal{F}$ of $S$ satisfying Definition \\ref{Boman chain condition}.\n\n \tFor the central ball $B_0$, note that\n \t\\begin{equation}\\label{A.2}\n \t\t\\begin{aligned}\n \t\t\t\\int_S |f(x)-\\E_{\\rho}(f)|\\,\\d\\rho(x)\n \t\t\t\\leq & \\int_S |f(x)-\\E_{\\rho_{B_0}}(f)|\\,\\d\\rho(x)+\\int_S |\\E_{\\rho_{B_0}}(f)-\\E_{\\rho}(f)|\\,\\d\\rho(x)\\\\\n \t\t\t\\leq & 2\\int_S |f(x)-\\E_{\\rho_{B_0}}(f)|\\,\\d\\rho(x).\n \t\t\\end{aligned}\n \t\\end{equation}\n\n For $B\\in\\mathcal{F}$, \tdenote $a_B=\\int_B |f-\\E_{\\rho_{B}}(f)|\\,\\d\\rho=\\rho(B)\\int_B |f-\\E_{\\rho_{B}}(f)|\\,\\d\\rho_B$. Since $\\mathcal{F}$ is a covering of $S$, we have\n \t\\begin{equation}\\label{A.3}\n \t\t\\begin{aligned}\n \t\t\t&\\int_S |f(x)-\\E_{\\rho_{B_0}}(f)|\\,\\d\\rho(x)\n \t\t\t\\leq \\sum_{B \\in \\mathcal{F}} \\int_B |f(x)-\\E_{\\rho_{B_0}}(f)|\\,\\d\\rho(x)\\\\\n \t\t\t\\leq & \\sum_{B \\in \\mathcal{F}} \\left(\\int_B |f(x)-\\E_{\\rho_{B}}(f)|\\,\\d\\rho(x)+\\int_B |\\E_{\\rho_{B}}(f)-\\E_{\\rho_{B_0}}(f)|\\,\\d\\rho(x)\\right)\\\\\n \t\t\t\\leq & \\sum_{B \\in \\mathcal{F}} \\left(a_B+\\rho(B) |\\E_{\\rho_{B}}(f)-\\E_{\\rho_{B_0}}(f)|\\right).\n \t\t\\end{aligned}\n \t\\end{equation}\n\n \tFor any $B\\in\\mathcal{F}$, by Boman chain condition, there exists a chain $B_0, B_1,\\dots, B_N=B$ of distinct balls from \\(\\mathcal{F}\\), such that for any \\(j \\in \\{0, \\ldots, N-1\\}\\),\n \t\\begin{equation}\n \t\t\\begin{aligned}\n \t\t|\\E_{\\rho_{B_j}}(f)-\\E_{\\rho_{B_{j+1}}}(f)|= &\\left|\\frac{1}{\\rho(B_j\\cap B_{j+1})}\\int_{B_j\\cap B_{j+1}}\\left(\\E_{\\rho_{B_j}}(f)-\\E_{\\rho_{B_{j+1}}}(f)\\right)\\,\\d\\rho\\right|\\\\\n \t\t\\leq &\\frac{1}{\\rho(B_j\\cap B_{j+1})}\\int_{B_j\\cap B_{j+1}}\\left|\\E_{\\rho_{B_j}}(f)-\\E_{\\rho_{B_{j+1}}}(f)\\right|\\,\\d\\rho\\\\\n \t\t\\leq &\\frac{1}{\\rho(B_j\\cap B_{j+1})}\\left(\\int_{B_j\\cap B_{j+1}}\\left|f-\\E_{\\rho_{B_j}}(f)\\right|+\\left|f-\\E_{\\rho_{B_{j+1}}}(f)\\right|\\,\\d\\rho\\right)\\\\\n \t\t\\leq & \\frac{a_{B_j}+a_{B_{j+1}}}{\\rho(B_j\\cap B_{j+1})}\n \t\t\\overset{*}{\\leq} G \\left(\\frac{a_{B_j}}{\\rho(B_j)}+\\frac{a_{B_{j+1}}}{\\rho(B_{j+1})}\\right),\n \t\t\\end{aligned}\n \t\\end{equation}\n \twhere $(*)$ follows from the quantitative chain overlap of Boman chain condition $3$.\n\n \t Thus, we have\n \t \\begin{equation}\n \t |\\E_{\\rho_{B}}(f)-\\E_{\\rho_{B_0}}(f)|\\leq \\sum_{j=0}^{N-1}|\\E_{\\rho_{B_j}}(f)-\\E_{\\rho_{B_{j+1}}}(f)|\\leq 2G \\sum_{j=0}^{N}\\frac{a_{B_j}}{\\rho(B_j)}\\overset{*}{\\leq} 2G \\sum_{B\\subset F \\bar{B}}\\frac{a_{\\bar{B}}}{\\rho(\\bar{B})},\n \t \\end{equation}\n \t where $ \\sum_{B\\subset F \\bar{B}}$ means that the sum runs over all $\\bar{B}\\in\\mathcal{F}$ satisfying $B\\subset F \\bar{B}$, and $(*)$ follows from the Boman chain condition $2$.\n \t Then by Fubini--Tonelli theorem, \n \t \\begin{equation}\\label{A.6}\n \t \t\\begin{aligned}\n \t \t \t \\sum_{B \\in \\mathcal{F}} \\rho(B) |\\E_{\\rho_{B}}(f)-\\E_{\\rho_{B_0}}(f)|\n \t \t \t \\leq & 2G\\sum_{B \\in \\mathcal{F}} \\rho(B) \\sum_{B\\subset F \\overline{B}}\\frac{a_{\\overline{B}}}{\\rho(\\overline{B})}\n \t \t \t \\leq 2G\\sum_{\\bar{B} \\in \\mathcal{F}} \\frac{a_{\\bar{B}}}{\\rho(\\bar{B})}\\sum_{B\\subset F \\bar{B}}\\rho(B).\n \t \t\\end{aligned}\n \t \\end{equation}\n\nBy Boman chain condition $1$ and the doubling property of $\\rho$, we have \n \t \\begin{equation}\\label{aa}\n \\sum_{B\\subset F \\bar{B}}\\rho(B)\\leq E\\rho(F\\bar{B})\\leq E\\beta^2F^\\frac{\\log \\beta}{\\log 2}\\rho(\\bar{B}),\n \t \\end{equation}\nwhere $\\beta=\\beta(K,N,\\diam(S))$ is the doubling constant.\n\n \t Combining \\eqref{A.2}, \\eqref{A.3}, \\eqref{A.6} and \\eqref{aa}, we obtain\n \t \\begin{equation}\\label{A_A}\n \t \t\\begin{aligned}\n \t \t\t \\int_S |f(x)-\\E_{\\rho}(f)|\\,\\d\\rho(x)\\leq & 2 \\sum_{B \\in \\mathcal{F}} \\left(a_B+\\rho(B) |\\E_{\\rho_{B}}(f)-\\E_{\\rho_{B_0}}(f)|\\right)\\\\\n \t \t\t \\leq &2(1+2\\beta^2 EF^\\frac{\\log \\beta}{\\log 2}G) \\sum_{B \\in \\mathcal{F}} \\rho(B)\\int_B |f-\\E_{\\rho_{B}}(f)|\\,\\d\\rho_B,\n \t \t\\end{aligned}\n \t \\end{equation}\n which is the thesis.\n \\end{proof}\n\n\t\\end{appendices}", + "thm0.1": "\\begin{theorem}\\label{thm0.1}\n\t\tLet $(X,\\d,\\mm)$ be an ${\\rm RCD}(K,N)$ metric measure space. Let $S\\subseteq X$ be a John domain and $Y\\subseteq X$ be compact with $\\mm(Y)>0$. Let $\\rho\\in \\mathcal{P}(S)$ be with $a_1\\mm\\restr{S} \\leq \\rho\\leq a_2\\mm\\restr{S}$ for some positive constants $a_1,a_2$. Then there exists a constant $C>0$, depending on $K,N,a_1,a_2, S, \\diam(S\\cup Y)$, such that for any $\\mu,\\nu\\in\\mathcal{P}(Y)$, \n\t\t\\begin{equation}\n\t\t\t\\|\\phi_\\mu-\\phi_\\nu\\|_{L^1(\\rho)}\\leq C W_1^\\frac{1}{2}(\\mu,\\nu),\n\t\t\\end{equation}\n\t\twhere $\\phi_\\mu$ and $\\phi_\\nu$ are the Kantorovich potentials from $\\rho$ to $\\mu$ and $\\rho$ to $\\nu$ respectively.\n\t\\end{theorem}" + }, + "pre_theorem_intro_text_len": 6021, + "pre_theorem_intro_text": "\\subsection{Motivation and setting}\n\t\\paragraph{Optimal transport.}Optimal transport, initiated by Monge \\cite{monge1781} and reframed by Kantorovich \\cite{zbMATH03099866}, seeks the most efficient way to redistribute mass between two probability distributions. Precisely, given two probability measures $\\rho$ and $\\mu$ defined on Polish spaces $X$ and $Y$ respectively, and a cost function $c:X\\times Y\\rightarrow \\mathbb{R}$, the Monge optimal transport problem aims to find a minimizer $T$, called an optimal transport map, of the following optimization problem among all measurable maps $H: X \\to Y$ pushing $\\rho$ forward to $\\mu$:\n\t\\begin{equation}\\label{2.1}\n\t\t\\int_X c(x, T(x)) \\,{\\mathrm d}\\rho(x)=\\inf_{H_\\sharp \\rho = \\mu}\\int_X c(x, H(x)) \\,{\\mathrm d}\\rho(x), \\tag{MP}\n\t\\end{equation}\n The Monge problem prevents mass splitting, sometimes making the problem ill-posed. Its relaxation, called Kantorovich problem, optimizes over joint couplings instead of deterministic maps. A fundamental result known as Kantorovich duality, asserts that the Kantorovich problem is equivalent to the following dual problem:\n\t\\begin{equation}\n\t\t\\sup_{ \\phi(x) + \\psi(y) \\leq c(x, y)} \\left\\{ \\int_X \\phi(x) \\, {\\mathrm d}\\rho(x) + \\int_Y \\psi(y) \\, {\\mathrm d}\\mu(y) \\right\\}.\\tag{KD}\n\t\\end{equation}\n\tThe optimal functions $\\phi,\\psi$, which always exist under standard assumptions, are called Kantorovich potentials \\cite{AG-U, V-O}. \n\n\tIn the Euclidean space, for $c(x,y)=\\frac{1}{2}|x-y|^2$, Brenier's landmark result \\cite{brenier1991polar} established that, for absolutely continuous source measures, the optimal transport map is unique and takes the form $T(x)=\\nabla u(x)$ for a convex function $u$. This was later extended to Riemannian manifolds by McCann \\cite{mccann2001polar}, who showed that for $c(x,y)=\\frac{1}{2}{\\mathrm d}^2(x,y)$, the optimal transport map is given almost everywhere by $T(x)=\\exp_x(-\\nabla\\varphi(x))$ for a Kantorovich potential $\\varphi$.\n\n\t A fundamental question in both theoretical and applied contexts of optimal transport is the quantitative stability of optimal transport maps and Kantorovich potentials under perturbations of the target measure (see Letrouit's lecture note \\cite{LetrouitLecture} and the references therein). Based on recent breakthroughs in the quantitative stability of optimal transport on Euclidean spaces \\cite{zbMATH07794624, arXiv:2411.04908}, on boundaries of convex bodies \\cite{Kitagawa12, Kitagawa25} and on Riemannian manifolds \\cite{arXiv:2504.05412}, \nKitagawa--Letrouit--M\\'erigot \\cite[\\S 1.2]{arXiv:2504.05412} conjecture that:\n\t\\begin{center}\n\t\t\\emph{the quantitative stability results are also true in more general metric measure spaces with synthetic curvature bounds.}\n\t\\end{center}\n\n\t\\paragraph{General setting.}\n\tIn this paper, we confirm the conjecture of Kitagawa-- Letrouit--M\\'erigot for the following spaces:\n\t\\begin{itemize}\n\t\t\\item RCD$(K,N)$ spaces (metric measure spaces with synthetic Ricci curvature lower bound): we prove quantitative $L^1$-stability of Kantorovich potentials with respect to Wasserstein perturbations of the target measure;\n\t\t\\item Alexandrov spaces with (sectional) curvature lower bound: we establish stability estimates for optimal transport maps in terms of the $L^2$-norm of the potential difference.\n\t\\end{itemize}\n\n\tAn Alexandrov space is a geodesic space of finite Hausdorff dimension and of curvature bounded from below (cf. \\cite{zbMATH01626771}). An ${\\rm RCD}(K,N)$ space is a metric measure spaces verifying the synthetic Riemannian curvature-dimension condition \\cite{zbMATH05049052,zbMATH05578758, zbMATH06303881, G-O}. An $n$-dimensional Alexandrov space with curvature bounded from below by $k$, equipped with its $n$-Hausdorff measure, is an ${\\rm RCD}(k(n-1), n)$ space \\cite{zbMATH06032507,zbMATH05962558}.\n\t\t After \\cite{brenier1991polar, mccann2001polar}, Gigli--Rajala--Sturm \\cite[Theorem 1.1]{ gigli2016optimal} proved the existence and uniqueness of the optimal transport map on an $\\rm{RCD}(K,N)$ space $(X,{\\mathrm d},\\mathfrak m)$, for the quadratic cost $c(x,y)=\\frac{1}{2}{\\mathrm d}^2(x,y)$ and the source measure $\\rho\\ll \\mathfrak m$. \n\n\t John domains, named after F. John \\cite{zbMATH03166766}, encompass many cases of interest, such as bounded Lipschitz domains, bounded domains satisfying a cone condition and certain fractal domains (see \\cite{zbMATH00858510} for more discussions). \n\t\\begin{definition}[John domain]\n\t\tA bounded open subset $S$ of a metric space is called a \\emph{John domain} if there is a distinguished point $x_0 \\in S$ and a constant $\\eta > 0$ such that, for every $x \\in S$, there is a rectifiable curve $\\gamma \\colon [0, \\ell(\\gamma)] \\to S$ parametrized by arc length, such that $\\gamma(0) = x$, $\\gamma(\\ell(\\gamma)) = x_0$, and\n\t\t\\begin{equation*}\n\t\t\t{\\mathrm d}(\\gamma(t), S^c) \\geq \\eta t,~~~\\forall t \\in [0, \\ell(\\gamma)],\n\t\t\\end{equation*}\n\t\twhere $S^c$ denotes the complement of $S$.\n\t\\end{definition}\n\tIn \\cite[Theorem 1.9]{arXiv:2411.04908}, it has been shown that when the source measure $\\rho$ is the uniform density on some non-John domain $S\\subset \\mathbb{R}^n$, then no quantitative stability estimates of the form $\\|\\phi_\\mu-\\phi_\\nu\\|_{L^2(\\rho)}\\leq C W_p^\\alpha(\\mu,\\nu)$ \n\tcan hold for any $C, \\alpha, p>0$. \n\tMoreover, examples in \\cite{letrouit2025unstableoptimaltransportmaps} indicates that both unboundedness of the density and openness of $S$ may cause instability of optimal transport maps.\n\n\t In this paper, we always assume that the Kantorovich potential $\\phi_\\mu$ from $\\rho$ to $\\mu$ satisfies $\\E_\\rho(\\phi_\\mu)=\\int_S\\phi_\\mu\\,{\\mathrm d}\\rho=0$. Together with the uniqueness of the optimal transport map, such $\\phi_\\mu$ is unique and thus makes sense to study its stability.\n\n\t\\subsection{Main results}\n\tOur main theorem concerns the quantitative $L^1$ stability of Kantorovich potentials on ${\\rm RCD}(K,N)$ spaces.", + "context": "\\subsection{Motivation and setting}\n \\paragraph{Optimal transport.}Optimal transport, initiated by Monge \\cite{monge1781} and reframed by Kantorovich \\cite{zbMATH03099866}, seeks the most efficient way to redistribute mass between two probability distributions. Precisely, given two probability measures $\\rho$ and $\\mu$ defined on Polish spaces $X$ and $Y$ respectively, and a cost function $c:X\\times Y\\rightarrow \\mathbb{R}$, the Monge optimal transport problem aims to find a minimizer $T$, called an optimal transport map, of the following optimization problem among all measurable maps $H: X \\to Y$ pushing $\\rho$ forward to $\\mu$:\n \\begin{equation}\\label{2.1}\n \\int_X c(x, T(x)) \\,{\\mathrm d}\\rho(x)=\\inf_{H_\\sharp \\rho = \\mu}\\int_X c(x, H(x)) \\,{\\mathrm d}\\rho(x), \\tag{MP}\n \\end{equation}\n The Monge problem prevents mass splitting, sometimes making the problem ill-posed. Its relaxation, called Kantorovich problem, optimizes over joint couplings instead of deterministic maps. A fundamental result known as Kantorovich duality, asserts that the Kantorovich problem is equivalent to the following dual problem:\n \\begin{equation}\n \\sup_{ \\phi(x) + \\psi(y) \\leq c(x, y)} \\left\\{ \\int_X \\phi(x) \\, {\\mathrm d}\\rho(x) + \\int_Y \\psi(y) \\, {\\mathrm d}\\mu(y) \\right\\}.\\tag{KD}\n \\end{equation}\n The optimal functions $\\phi,\\psi$, which always exist under standard assumptions, are called Kantorovich potentials \\cite{AG-U, V-O}.\n\nA fundamental question in both theoretical and applied contexts of optimal transport is the quantitative stability of optimal transport maps and Kantorovich potentials under perturbations of the target measure (see Letrouit's lecture note \\cite{LetrouitLecture} and the references therein). Based on recent breakthroughs in the quantitative stability of optimal transport on Euclidean spaces \\cite{zbMATH07794624, arXiv:2411.04908}, on boundaries of convex bodies \\cite{Kitagawa12, Kitagawa25} and on Riemannian manifolds \\cite{arXiv:2504.05412}, \nKitagawa--Letrouit--M\\'erigot \\cite[\\S 1.2]{arXiv:2504.05412} conjecture that:\n \\begin{center}\n \\emph{the quantitative stability results are also true in more general metric measure spaces with synthetic curvature bounds.}\n \\end{center}\n\nAn Alexandrov space is a geodesic space of finite Hausdorff dimension and of curvature bounded from below (cf. \\cite{zbMATH01626771}). An ${\\rm RCD}(K,N)$ space is a metric measure spaces verifying the synthetic Riemannian curvature-dimension condition \\cite{zbMATH05049052,zbMATH05578758, zbMATH06303881, G-O}. An $n$-dimensional Alexandrov space with curvature bounded from below by $k$, equipped with its $n$-Hausdorff measure, is an ${\\rm RCD}(k(n-1), n)$ space \\cite{zbMATH06032507,zbMATH05962558}.\n After \\cite{brenier1991polar, mccann2001polar}, Gigli--Rajala--Sturm \\cite[Theorem 1.1]{ gigli2016optimal} proved the existence and uniqueness of the optimal transport map on an $\\rm{RCD}(K,N)$ space $(X,{\\mathrm d},\\mathfrak m)$, for the quadratic cost $c(x,y)=\\frac{1}{2}{\\mathrm d}^2(x,y)$ and the source measure $\\rho\\ll \\mathfrak m$.\n\nJohn domains, named after F. John \\cite{zbMATH03166766}, encompass many cases of interest, such as bounded Lipschitz domains, bounded domains satisfying a cone condition and certain fractal domains (see \\cite{zbMATH00858510} for more discussions). \n \\begin{definition}[John domain]\n A bounded open subset $S$ of a metric space is called a \\emph{John domain} if there is a distinguished point $x_0 \\in S$ and a constant $\\eta > 0$ such that, for every $x \\in S$, there is a rectifiable curve $\\gamma \\colon [0, \\ell(\\gamma)] \\to S$ parametrized by arc length, such that $\\gamma(0) = x$, $\\gamma(\\ell(\\gamma)) = x_0$, and\n \\begin{equation*}\n {\\mathrm d}(\\gamma(t), S^c) \\geq \\eta t,~~~\\forall t \\in [0, \\ell(\\gamma)],\n \\end{equation*}\n where $S^c$ denotes the complement of $S$.\n \\end{definition}\n In \\cite[Theorem 1.9]{arXiv:2411.04908}, it has been shown that when the source measure $\\rho$ is the uniform density on some non-John domain $S\\subset \\mathbb{R}^n$, then no quantitative stability estimates of the form $\\|\\phi_\\mu-\\phi_\\nu\\|_{L^2(\\rho)}\\leq C W_p^\\alpha(\\mu,\\nu)$ \n can hold for any $C, \\alpha, p>0$. \n Moreover, examples in \\cite{letrouit2025unstableoptimaltransportmaps} indicates that both unboundedness of the density and openness of $S$ may cause instability of optimal transport maps.\n\nIn this paper, we always assume that the Kantorovich potential $\\phi_\\mu$ from $\\rho$ to $\\mu$ satisfies $\\E_\\rho(\\phi_\\mu)=\\int_S\\phi_\\mu\\,{\\mathrm d}\\rho=0$. Together with the uniqueness of the optimal transport map, such $\\phi_\\mu$ is unique and thus makes sense to study its stability.\n\n\\subsection{Main results}\n Our main theorem concerns the quantitative $L^1$ stability of Kantorovich potentials on ${\\rm RCD}(K,N)$ spaces.", + "full_context": "\\subsection{Motivation and setting}\n \\paragraph{Optimal transport.}Optimal transport, initiated by Monge \\cite{monge1781} and reframed by Kantorovich \\cite{zbMATH03099866}, seeks the most efficient way to redistribute mass between two probability distributions. Precisely, given two probability measures $\\rho$ and $\\mu$ defined on Polish spaces $X$ and $Y$ respectively, and a cost function $c:X\\times Y\\rightarrow \\mathbb{R}$, the Monge optimal transport problem aims to find a minimizer $T$, called an optimal transport map, of the following optimization problem among all measurable maps $H: X \\to Y$ pushing $\\rho$ forward to $\\mu$:\n \\begin{equation}\\label{2.1}\n \\int_X c(x, T(x)) \\,{\\mathrm d}\\rho(x)=\\inf_{H_\\sharp \\rho = \\mu}\\int_X c(x, H(x)) \\,{\\mathrm d}\\rho(x), \\tag{MP}\n \\end{equation}\n The Monge problem prevents mass splitting, sometimes making the problem ill-posed. Its relaxation, called Kantorovich problem, optimizes over joint couplings instead of deterministic maps. A fundamental result known as Kantorovich duality, asserts that the Kantorovich problem is equivalent to the following dual problem:\n \\begin{equation}\n \\sup_{ \\phi(x) + \\psi(y) \\leq c(x, y)} \\left\\{ \\int_X \\phi(x) \\, {\\mathrm d}\\rho(x) + \\int_Y \\psi(y) \\, {\\mathrm d}\\mu(y) \\right\\}.\\tag{KD}\n \\end{equation}\n The optimal functions $\\phi,\\psi$, which always exist under standard assumptions, are called Kantorovich potentials \\cite{AG-U, V-O}.\n\nA fundamental question in both theoretical and applied contexts of optimal transport is the quantitative stability of optimal transport maps and Kantorovich potentials under perturbations of the target measure (see Letrouit's lecture note \\cite{LetrouitLecture} and the references therein). Based on recent breakthroughs in the quantitative stability of optimal transport on Euclidean spaces \\cite{zbMATH07794624, arXiv:2411.04908}, on boundaries of convex bodies \\cite{Kitagawa12, Kitagawa25} and on Riemannian manifolds \\cite{arXiv:2504.05412}, \nKitagawa--Letrouit--M\\'erigot \\cite[\\S 1.2]{arXiv:2504.05412} conjecture that:\n \\begin{center}\n \\emph{the quantitative stability results are also true in more general metric measure spaces with synthetic curvature bounds.}\n \\end{center}\n\nAn Alexandrov space is a geodesic space of finite Hausdorff dimension and of curvature bounded from below (cf. \\cite{zbMATH01626771}). An ${\\rm RCD}(K,N)$ space is a metric measure spaces verifying the synthetic Riemannian curvature-dimension condition \\cite{zbMATH05049052,zbMATH05578758, zbMATH06303881, G-O}. An $n$-dimensional Alexandrov space with curvature bounded from below by $k$, equipped with its $n$-Hausdorff measure, is an ${\\rm RCD}(k(n-1), n)$ space \\cite{zbMATH06032507,zbMATH05962558}.\n After \\cite{brenier1991polar, mccann2001polar}, Gigli--Rajala--Sturm \\cite[Theorem 1.1]{ gigli2016optimal} proved the existence and uniqueness of the optimal transport map on an $\\rm{RCD}(K,N)$ space $(X,{\\mathrm d},\\mathfrak m)$, for the quadratic cost $c(x,y)=\\frac{1}{2}{\\mathrm d}^2(x,y)$ and the source measure $\\rho\\ll \\mathfrak m$.\n\nJohn domains, named after F. John \\cite{zbMATH03166766}, encompass many cases of interest, such as bounded Lipschitz domains, bounded domains satisfying a cone condition and certain fractal domains (see \\cite{zbMATH00858510} for more discussions). \n \\begin{definition}[John domain]\n A bounded open subset $S$ of a metric space is called a \\emph{John domain} if there is a distinguished point $x_0 \\in S$ and a constant $\\eta > 0$ such that, for every $x \\in S$, there is a rectifiable curve $\\gamma \\colon [0, \\ell(\\gamma)] \\to S$ parametrized by arc length, such that $\\gamma(0) = x$, $\\gamma(\\ell(\\gamma)) = x_0$, and\n \\begin{equation*}\n {\\mathrm d}(\\gamma(t), S^c) \\geq \\eta t,~~~\\forall t \\in [0, \\ell(\\gamma)],\n \\end{equation*}\n where $S^c$ denotes the complement of $S$.\n \\end{definition}\n In \\cite[Theorem 1.9]{arXiv:2411.04908}, it has been shown that when the source measure $\\rho$ is the uniform density on some non-John domain $S\\subset \\mathbb{R}^n$, then no quantitative stability estimates of the form $\\|\\phi_\\mu-\\phi_\\nu\\|_{L^2(\\rho)}\\leq C W_p^\\alpha(\\mu,\\nu)$ \n can hold for any $C, \\alpha, p>0$. \n Moreover, examples in \\cite{letrouit2025unstableoptimaltransportmaps} indicates that both unboundedness of the density and openness of $S$ may cause instability of optimal transport maps.\n\nIn this paper, we always assume that the Kantorovich potential $\\phi_\\mu$ from $\\rho$ to $\\mu$ satisfies $\\E_\\rho(\\phi_\\mu)=\\int_S\\phi_\\mu\\,{\\mathrm d}\\rho=0$. Together with the uniqueness of the optimal transport map, such $\\phi_\\mu$ is unique and thus makes sense to study its stability.\n\n\\subsection{Main results}\n Our main theorem concerns the quantitative $L^1$ stability of Kantorovich potentials on ${\\rm RCD}(K,N)$ spaces.\n\nJohn domains, named after F. John \\cite{zbMATH03166766}, encompass many cases of interest, such as bounded Lipschitz domains, bounded domains satisfying a cone condition and certain fractal domains (see \\cite{zbMATH00858510} for more discussions). \n \\begin{definition}[John domain]\n A bounded open subset $S$ of a metric space is called a \\emph{John domain} if there is a distinguished point $x_0 \\in S$ and a constant $\\eta > 0$ such that, for every $x \\in S$, there is a rectifiable curve $\\gamma \\colon [0, \\ell(\\gamma)] \\to S$ parametrized by arc length, such that $\\gamma(0) = x$, $\\gamma(\\ell(\\gamma)) = x_0$, and\n \\begin{equation*}\n \\d(\\gamma(t), S^c) \\geq \\eta t,~~~\\forall t \\in [0, \\ell(\\gamma)],\n \\end{equation*}\n where $S^c$ denotes the complement of $S$.\n \\end{definition}\n In \\cite[Theorem 1.9]{arXiv:2411.04908}, it has been shown that when the source measure $\\rho$ is the uniform density on some non-John domain $S\\subset \\R^n$, then no quantitative stability estimates of the form $\\|\\phi_\\mu-\\phi_\\nu\\|_{L^2(\\rho)}\\leq C W_p^\\alpha(\\mu,\\nu)$ \n can hold for any $C, \\alpha, p>0$. \n Moreover, examples in \\cite{letrouit2025unstableoptimaltransportmaps} indicates that both unboundedness of the density and openness of $S$ may cause instability of optimal transport maps.\n\n\\subsection{Main results}\n Our main theorem concerns the quantitative $L^1$ stability of Kantorovich potentials on ${\\rm RCD}(K,N)$ spaces.\n\nIf $(S, \\d, \\mm)$ is a compact $\\rcdkn$ space, it supports a global $(1,1)$-Poincar\\'e inequality (cf.\\cite{zbMATH01474795, zbMATH06043352}) and we can remove the John domain assumption:\n \\begin{corollary}\\label{cor}\n Let $(X,\\d,\\mm)$ be a compact ${\\rm RCD}(K,N)$ metric measure space. Let $\\rho\\in \\mathcal{P}(X)$ be with $a_1\\mm \\leq \\rho\\leq a_2\\mm$ for some positive constants $a_1,a_2$. Then the conclusion of Theorem \\ref{thm0.1} holds.\n \\end{corollary}\n\nAdapting the strategy of \\cite{arXiv:2504.05412}, we can also prove the stability of optimal transport maps on Alexandrov spaces.\n \\begin{theorem}\\label{thm0.2}\n Let $(X, \\d)$ be an $n$-dimensional Alexandrov space, $\\mm$ be the $n$-Hausdorff measure. Under the same assumption for $S, Y$ and $\\rho$ as in Theorem \\ref{thm0.1}, and if $S$ additionally has finite perimeter, then there exists a constant $C>0$, depending on $k,n,a_1, a_2, \\diam(S\\cup Y), {\\rm{Per}}(S), S$, such that for any $\\mu,\\nu\\in\\mathcal{P}_2(Y)$, we have\n \\begin{equation}\n \\int_S \\d^2(T_\\mu(x), T_\\nu(x))\\,\\d \\rho(x) \\leq CW_1^{1/6}(\\mu, \\nu),\n \\end{equation}\n where $T_\\mu$ and $T_\\nu$ are the optimal transport maps from $\\rho$ to $\\mu$ and $\\rho$ to $\\nu$ respectively.\n \\end{theorem}\n\n\\begin{proposition}\\label{2.11}\n It holds that\n \\begin{equation}\n \\int_{S}\\big| \\E_{\\mu_x^t[\\psi_*]}(v)-\\E_{\\mu^t[\\psi_*]}(v))\\big|\\,\\d\\rho(x)\\leq \\frac{\\kappa}{\\sqrt{t}}\\left(\\int_S {\\rm{Var}}_{ \\mu_x^t[\\psi_*]}(v)\\,\\d\\rho(x)\\right)^\\frac{1}{2},\n \\end{equation}\n where $\\mu^t[\\psi_*]$ is defined as in \\eqref{mu} and $\\kappa$ depends on $K,N,\\Lambda_\\psi,a_1,a_2, S$.\n \\end{proposition}\n \\begin{proof}\n Since $a_1\\mm\\leq \\rho\\leq a_2\\mm$, by \\cite[Corollary 2.4]{zbMATH05049052}, $\\rho$ is a doubling measure. Moreover, \n since $S$ is a John domain, by \\cite[Proposition 3.7]{arXiv:2504.05412}, $\\rho$ satisfies the Boman chain condition (see Definition \\ref{Boman chain condition}) and we can choose a covering $\\mathcal{F}$, such that for any $B\\in \\mathcal{F}$, $r_B\\leq 1$. Then \n \\begin{equation}\n \\begin{aligned}\n &\\int_{S}\\big| \\E_{\\mu_x^t[\\psi_*]}(v)-\\E_{\\mu^t[\\psi_*]}(v))\\big|\\,\\d\\rho(x)\\\\\n \\mathop{\\leq}^{\\text {Lemma \\ref{3.1}}} &C_5 \\sum_{B\\in\\mathcal{F}}\\rho(B)\\int_B \\big| \\E_{\\mu_x^t[\\psi_*]}(v)-{\\rho(B)}^{-1}\\E_{\\mu^t[\\psi_*]}(v))\\big|\\,\\d\\rho_B(x)\\\\\n \\mathop{\\leq }^{\\text{Lemma \\ref{p3.10}}}&C_5 \\sum_{B\\in\\mathcal{F}}\\rho(B)C_4 r_B\\int_B \\big|\\nabla_x \\E_{\\mu_x^t[\\psi_*]}(v)\\big|\\,\\d\\rho_B(x)\\\\\n \\mathop{\\leq}^{\\text{Proposition \\ref{3.9}} }& \\frac{C_3C_4C_5}{\\sqrt{t}}\\sum_{B\\in\\mathcal{F}}\\sqrt{\\mm(B_{2r_B})}\\sqrt{r_B}\\left(\\int_B {\\rm{Var}}_{ \\mu_x^t[\\psi_*]}(v)\\,\\d\\rho(x)\\right)^\\frac{1}{2}\\\\\n \\mathop{\\leq}^{\\text{H\\\"{o}lder}~~}&\\frac{C_3C_4C_5}{\\sqrt{t}}\\left(\\sum_{B\\in\\mathcal{F}}\\mm(B_{2r_B})\\right)^\\frac{1}{2}\\left(\\sum_{B\\in\\mathcal{F}}\\int_B {\\rm{Var}}_{ \\mu_x^t[\\psi_*]}(v)\\,\\d\\rho(x)\\right)^\\frac{1}{2}\\\\\n \\overset{\\text{Boman chain~}}{\\leq}&\\frac{\\kappa}{\\sqrt{t}}\\left(\\int_S {\\rm{Var}}_{ \\mu_x^t[\\psi_*]}(v)\\,\\d\\rho(x)\\right)^\\frac{1}{2}\n \\end{aligned}\n \\end{equation}\n which is the thesis.\n \\end{proof}\n\n\\begin{theorem}\\label{thm2}\n Let $(X, \\d)$ be an Alexandrov space with no boundary. Then under the same assumptions of Theorem \\ref{thm0.2}, there exists a constant $\\bar{C}>0$, depending on $k,n,a_1, a_2, \\diam(S\\cup Y), {\\rm{Per}}(S), S$, such that for any $\\mu,\\nu\\in\\mathcal{P}_2(Y)$, we have \n \\begin{equation}\n \\int_S |\\nabla\\phi_\\mu(x)-\\nabla\\phi_\\nu(x)|^2\\,\\d\\rho(x)\\leq \\bar{C} \\left(\\int_S|\\phi_\\mu(x)-\\phi_\\nu(x)|^2\\,\\d\\rho(x)\\right)^\\frac{1}{3},\n \\end{equation}\n where $\\phi_\\mu$ and $\\phi_\\nu$ are the Kantorovich potentials from $\\rho$ to $\\mu$ and $\\rho$ to $\\nu$ respectively.\n \\end{theorem}\n\n\\begin{lemma}[\\cite{zbMATH07794624}, Lemma 5.1]\\label{l0.18}\n Let $I\\subseteq \\R$ be a compact segment and let $u,v: I\\rightarrow\\R$ be two convex functions such that $|u'|$ and $|v'|$ (defined a.e. on $I$) are uniformly bounded on $I$. Then\n \\begin{equation*}\n \\|u'-v'\\|_{L^2(I)}^2\\leq 8 (\\|u'\\|_{L^\\infty(I)}+\\|v'\\|_{L^\\infty(I)})^\\frac{4}{3}\\|u-v\\|_{L^2(I)}^\\frac{2}{3}.\n \\end{equation*}\n \\end{lemma}\n Therefore, applying Lemma \\ref{l0.18} to the functions $\\zeta|s|^2-u_\\mu^{(x,v)}$ and $\\zeta|s|^2-u_\\nu^{(x,v)}$ on each compact segment $[\\alpha_i(x,v),\\beta_i(x,v)]$ and using H\\\"{o}lder inequality, we obtain\n \\begin{equation}\\label{0.55}\n \\begin{aligned}\n &\\int_S |\\nabla\\phi_\\mu(x)-\\nabla\\phi_\\nu(x)|^2\\,\\d\\rho(x)\\\\\n \\mathop{\\leq}^{\\eqref{0.54}}& \\frac{a_2}{c_nd_n}\\int_{{\\rm S}X}\\sum_{i\\in I_S(x,v)}\\int_{\\alpha_i(x,v)}^{\\beta_i(x,v)}\\langle\\nabla\\phi_\\mu(b_s(x,v))-\\nabla\\phi_\\nu(b_s(x,v)),t_s(x,v)\\rangle^2\\,\\d s \\,\\d\\mm_S(x,v)\\\\\n \\leq&C_1 \\int_{{\\rm S}X}\\sum_{i\\in I_S(x,v)}\\left(\\int_{\\alpha_i(x,v)}^{\\beta_i(x,v)}|\\phi_\\mu(b_s(x,v))-\\phi_\\nu(b_s(x,v))|^2\\,\\d s\\right)^\\frac{1}{3}\\,\\d\\mm_S(x,v)\\\\\n \\leq&C_1 \\int_{{\\rm S}X}\\left(\\# I_S\\right)^\\frac{2}{3}\\left(\\int_0^1|\\phi_\\mu(b_s)-\\phi_\\nu(b_s)|^2\\chi_S(b_s)\\,\\d s\\right)^\\frac{1}{3}\\d\\mm_S\\\\\n \\leq&C_1 \\left(\\int_{{\\rm S}X}\\# I_S\\,\\d\\mm_S\\right)^\\frac{2}{3}\\left(\\int_{{\\rm S}X}\\int_0^1|\\phi_\\mu(b_s)-\\phi_\\nu(b_s)|^2\\chi_S(b_s)\\,\\d s \\,\\d\\mm_S\\right)^\\frac{1}{3}\\\\\n \\leq&C_1 \\left(\\frac{c_n}{a_1}\\right)^\\frac{1}{3}\\left(\\int_{{\\rm S}X}\\# I_S\\,\\d\\mm_S\\right)^\\frac{2}{3}\\left(\\int_S|\\phi_\\mu-\\phi_\\nu|^2\\,\\d\\rho\\right)^\\frac{1}{3},\n \\end{aligned}\n \\end{equation}\n where $C_1$ depends on $k,n,a_2, \\diam(S\\cup Y)$, and the last inequality follows from the invariance of $\\mm_S$ under the geodesic flow.\n\nLet $\\phi_\\mu$ and $\\phi_\\nu$ be the Kantorovich potentials from $\\rho$ to $\\mu$ and $\\rho$ to $\\nu$ respectively. From \\cite{zbMATH05349267,zbMATH06032507} we know that $\\nabla \\phi_\\mu(x), \\nabla \\phi_\\nu(x) \\in {\\rm{T}}_xX$ and\n $T_\\mu(x)= \\exp_x(-\\nabla \\phi_\\mu(x))$, $T_\\nu(x)= \\exp_x(-\\nabla \\phi_\\nu(x))$ are well-defined for almost every $x\\in S$. By triangle comparison condition (cf. \\cite{zbMATH05342782}), it holds\n \\begin{equation}\\label{exp}\n \\d \\big(T_\\mu(x), T_\\nu(x) \\big) \\leq C_2|\\nabla \\phi_\\mu(x)-\\nabla \\phi_\\nu(x)|,\n \\end{equation}\n for some constant $C_2>0$ which depends on $k, \\diam(S\\cup Y)$. \n \\medskip", + "post_theorem_intro_text_len": 3816, + "post_theorem_intro_text": "If $(S, {\\mathrm d}, \\mathfrak m)$ is a compact ${\\rm RCD}(K, N)$ space, it supports a global $(1,1)$-Poincar\\'e inequality (cf.\\cite{zbMATH01474795, zbMATH06043352}) and we can remove the John domain assumption:\n\t\\begin{corollary}\\label{cor}\n\t\tLet $(X,{\\mathrm d},\\mathfrak m)$ be a compact ${\\rm RCD}(K,N)$ metric measure space. Let $\\rho\\in \\mathcal{P}(X)$ be with $a_1\\mathfrak m \\leq \\rho\\leq a_2\\mathfrak m$ for some positive constants $a_1,a_2$. Then the conclusion of Theorem \\ref{thm0.1} holds.\n\t\\end{corollary}\n\n\tAdapting the strategy of \\cite{arXiv:2504.05412}, we can also prove the stability of optimal transport maps on Alexandrov spaces.\n\t\t\\begin{theorem}\\label{thm0.2}\n\t\tLet $(X, {\\mathrm d})$ be an $n$-dimensional Alexandrov space, $\\mathfrak m$ be the $n$-Hausdorff measure. Under the same assumption for $S, Y$ and $\\rho$ as in Theorem \\ref{thm0.1}, and if $S$ additionally has finite perimeter, then there exists a constant $C>0$, depending on $k,n,a_1, a_2, \\mathop{\\rm diam}\\nolimits(S\\cup Y), {\\rm{Per}}(S), S$, such that for any $\\mu,\\nu\\in\\mathcal{P}_2(Y)$, we have\n\t\t\\begin{equation}\n\t\t\t\\int_S {\\mathrm d}^2(T_\\mu(x), T_\\nu(x))\\,{\\mathrm d} \\rho(x) \\leq CW_1^{1/6}(\\mu, \\nu),\n\t\t\\end{equation}\n\t\twhere $T_\\mu$ and $T_\\nu$ are the optimal transport maps from $\\rho$ to $\\mu$ and $\\rho$ to $\\nu$ respectively.\n\t\\end{theorem}\n\n\t\\subsection{Strategy: heat kernel-regularized \\texorpdfstring{$c$}{c}-transform}\n\nMotivated by regularized $c$-transforms using Gibbs kernels $e^{-c(x, y)/\\varepsilon}$ \\cite{arXiv:2504.05412}, entropic optimal transport \\cite{GTJEMS, GTT25} and Varadhan's formula\n\\[\n\\lim_{t \\to 0} -t \\log p_{t/2}(x, y) = \\frac{1}{2} {\\mathrm d}^2(x, y)=c(x,y),\n\\]\n we make use of {heat kernel-regularized $c$-transform} (Hopf--Cole transform):\n\\[\n\\mathop{\\rm Lip}\\nolimits(X, {\\mathrm d}) \\ni \\psi \\mapsto \\Phi_t[\\psi](x) = -t \\log \\int_X e^{\\frac{\\psi(y)}{t}} p_{t/2}(x, y) \\, {\\mathrm d}\\mathfrak m(y).\n\\]\nThis approach allows us to bypass the low regularity of Kantorovich potentials in the non-smooth setting. \n\nWe define the heat kernel regularized Kantorovich functional as\n\t\\begin{equation*}\n\t\t\\K_t[\\varphi]:=\\int_S \\Phi_t[\\varphi]\\,{\\mathrm d}\\rho.\n\t\\end{equation*}\nSimilar to \\cite{arXiv:2504.05412}, the key in our proof is the strong concavity of the functional $\\K_t$.\nTo achieve this, we first derive a local strong concavity estimate using heat kernel estimate, then globalize the estimate on the support of the source measure using a Boman chain argument for John domains.\n\nUnlike the regularized $c$-transform used in \\cite{arXiv:2504.05412}, the existence of the boundary of $Y$ may lead to the failure in our heat kernel regularization argument. This possibility is ruled out by using the measure concentration property of the heat kernel and by making a careful choice of Lipschitz extension.\n\n\\medskip\n\n\t\\noindent {\\bf Organization.} This paper is structured as follows. In Section~\\ref{sec3}, we prove quantitative stability of Kantorovich potentials on $\\mathrm{RCD}(K,N)$ spaces. In Section \\ref{sec4}, we establish stability of optimal transport maps on Alexandrov spaces. The Appendix \\ref{A} contains technical lemmas about Poincar\\'e inequalities.\n\n\t\\bigskip\n\n\\noindent {\\bf Declaration.} The authors declare no conflict of interest and that the manuscript has no associated data.\n\n\\medskip\n\n\\noindent {\\bf Acknowledgement.} This work is supported by the Ministry of Science \\& Technology of China (2021YFA1000900, 2021YFA1002200), National Natural Science Foundation of China (12201596) and Shandong Provincial Natural Science Foundation (ZR2025QB05). The authors thank Nicola Gigli, Jun Kitagawa, Nan Li, Cyril Letrouit, Quentin M\\'erigot, Luca Tamanini for helpful discussions and suggestions on the bibliography.", + "sketch": "Motivated by “regularized $c$-transforms … entropic optimal transport … and Varadhan's formula”, the proof of Theorem~\\ref{thm0.1} uses a **heat kernel-regularized $c$-transform** (Hopf--Cole transform)\n\\[\n\\psi\\mapsto \\Phi_t[\\psi](x)=-t\\log\\int_X e^{\\frac{\\psi(y)}{t}}\\,p_{t/2}(x,y)\\,\\mathrm d\\mathfrak m(y),\n\\]\nwhich “allows us to bypass the low regularity of Kantorovich potentials in the non-smooth setting.” One then defines the “heat kernel regularized Kantorovich functional”\n\\[\n\\mathcal K_t[\\varphi]:=\\int_S \\Phi_t[\\varphi]\\,\\mathrm d\\rho.\n\\]\n“Similar to \\cite{arXiv:2504.05412}, the key in our proof is the strong concavity of the functional $\\mathcal K_t$.” To obtain this strong concavity, they “first derive a local strong concavity estimate using heat kernel estimate, then globalize the estimate on the support of the source measure using a Boman chain argument for John domains.”\n\nThey also note a technical issue specific to this regularization: “the existence of the boundary of $Y$ may lead to the failure in our heat kernel regularization argument,” and this is “ruled out by using the measure concentration property of the heat kernel and by making a careful choice of Lipschitz extension.”", + "expanded_sketch": "Motivated by “regularized $c$-transforms … entropic optimal transport … and Varadhan's formula”, in establishing the main theorem, the proof uses a **heat kernel-regularized $c$-transform** (Hopf--Cole transform)\n\\[\n\\psi\\mapsto \\Phi_t[\\psi](x)=-t\\log\\int_X e^{\\frac{\\psi(y)}{t}}\\,p_{t/2}(x,y)\\,\\mathrm d\\mathfrak m(y),\n\\]\nwhich “allows us to bypass the low regularity of Kantorovich potentials in the non-smooth setting.” One then defines the “heat kernel regularized Kantorovich functional”\n\\[\n\\mathcal K_t[\\varphi]:=\\int_S \\Phi_t[\\varphi]\\,\\mathrm d\\rho.\n\\]\n“Similar to arXiv:2504.05412, the key in our proof is the strong concavity of the functional $\\mathcal K_t$.” To obtain this strong concavity, they “first derive a local strong concavity estimate using heat kernel estimate, then globalize the estimate on the support of the source measure using a Boman chain argument for John domains.”\n\nThey also note a technical issue specific to this regularization: “the existence of the boundary of $Y$ may lead to the failure in our heat kernel regularization argument,” and this is “ruled out by using the measure concentration property of the heat kernel and by making a careful choice of Lipschitz extension.”,", + "expanded_theorem": "\\label{thm0.1}\n\t\tLet $(X,{\\mathrm d},\\mathfrak m)$ be an ${\\rm RCD}(K,N)$ metric measure space. Let $S\\subseteq X$ be a John domain and $Y\\subseteq X$ be compact with $\\mathfrak m(Y)>0$. Let $\\rho\\in \\mathcal{P}(S)$ be with $a_1\\mathfrak m\\restr{S} \\leq \\rho\\leq a_2\\mathfrak m\\restr{S}$ for some positive constants $a_1,a_2$. Then there exists a constant $C>0$, depending on $K,N,a_1,a_2, S, \\mathop{\\rm diam}\\nolimits(S\\cup Y)$, such that for any $\\mu,\\nu\\in\\mathcal{P}(Y)$, \n\t\t\\begin{equation}\n\t\t\t\\|\\phi_\\mu-\\phi_\\nu\\|_{L^1(\\rho)}\\leq C W_1^\\frac{1}{2}(\\mu,\\nu),\n\t\t\\end{equation}\n\t\twhere $\\phi_\\mu$ and $\\phi_\\nu$ are the Kantorovich potentials from $\\rho$ to $\\mu$ and $\\rho$ to $\\nu$ respectively.,", + "theorem_type": [ + "Existential–Universal", + "Inequality or Bound" + ], + "mcq": { + "question": "Let \\((X,{\\mathrm d},\\mathfrak m)\\) be an \\({\\rm RCD}(K,N)\\) metric measure space. Let \\(S\\subseteq X\\) be a John domain, meaning that \\(S\\) is a bounded open set for which there exist \\(x_0\\in S\\) and \\(\\eta>0\\) such that every \\(x\\in S\\) can be joined to \\(x_0\\) by a rectifiable curve \\(\\gamma:[0,\\ell(\\gamma)]\\to S\\), parametrized by arc length, with \\(\\gamma(0)=x\\), \\(\\gamma(\\ell(\\gamma))=x_0\\), and \\({\\mathrm d}(\\gamma(t),X\\setminus S)\\ge \\eta t\\) for all \\(t\\in[0,\\ell(\\gamma)]\\). Let \\(Y\\subseteq X\\) be compact with \\(\\mathfrak m(Y)>0\\). Assume \\(\\rho\\in \\mathcal P(S)\\) satisfies \\(a_1\\,\\mathfrak m\\!\\restriction_S \\le \\rho \\le a_2\\,\\mathfrak m\\!\\restriction_S\\) for some constants \\(a_1,a_2>0\\). For each \\(\\mu\\in\\mathcal P(Y)\\), let \\(\\phi_\\mu\\) denote a Kantorovich potential for the quadratic cost \\(c(x,y)=\\tfrac12{\\mathrm d}(x,y)^2\\) from \\(\\rho\\) to \\(\\mu\\), normalized by \\(\\int_S \\phi_\\mu\\,{\\mathrm d}\\rho=0\\). Also let\n\\[\nW_1(\\mu,\\nu):=\\inf_{\\pi\\in\\Pi(\\mu,\\nu)}\\int_{Y\\times Y}{\\mathrm d}(y,z)\\,{\\mathrm d}\\pi(y,z)\n\\]\nbe the 1-Wasserstein distance. Which quantitative estimate holds for all \\(\\mu,\\nu\\in\\mathcal P(Y)\\)?", + "correct_choice": { + "label": "A", + "text": "There exists a constant \\(C>0\\), depending only on \\(K,N,a_1,a_2,S\\), and \\(\\operatorname{diam}(S\\cup Y)\\), such that for every \\(\\mu,\\nu\\in\\mathcal P(Y)\\),\n\\[\n\\|\\phi_\\mu-\\phi_\\nu\\|_{L^1(\\rho)}=\\int_S |\\phi_\\mu-\\phi_\\nu|\\,{\\mathrm d}\\rho \\le C\\,W_1(\\mu,\\nu)^{1/2}.\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "There exists a constant \\(C>0\\), depending only on \\(K,N,a_1,a_2,S\\), and \\(\\operatorname{diam}(S\\cup Y)\\), such that for every \\(\\mu,\\nu\\in\\mathcal P(Y)\\),\n\\[\n\\|\\phi_\\mu-\\phi_\\nu\\|_{L^1(\\rho)}=\\int_S |\\phi_\\mu-\\phi_\\nu|\\,{\\mathrm d}\\rho \\le C\\,W_1(\\mu,\\nu).\n\\]" + }, + { + "label": "C", + "text": "There exists a constant \\(C>0\\), depending only on \\(K,N,a_1,a_2,S\\), and \\(\\operatorname{diam}(S\\cup Y)\\), such that for every \\(\\mu,\\nu\\in\\mathcal P(Y)\\),\n\\[\n\\|\\phi_\\mu-\\phi_\\nu\\|_{L^1(\\rho)}=\\int_S |\\phi_\\mu-\\phi_\\nu|\\,{\\mathrm d}\\rho \\le C.\n\\]" + }, + { + "label": "D", + "text": "There exists a constant \\(C>0\\), depending only on \\(K,N,a_1,a_2\\), such that for every John domain \\(S\\subseteq X\\), every compact \\(Y\\subseteq X\\) with \\(\\mathfrak m(Y)>0\\), and every \\(\\mu,\\nu\\in\\mathcal P(Y)\\),\n\\[\n\\|\\phi_\\mu-\\phi_\\nu\\|_{L^1(\\rho)}=\\int_S |\\phi_\\mu-\\phi_\\nu|\\,{\\mathrm d}\\rho \\le C\\,W_1(\\mu,\\nu)^{1/2}.\n\\]" + }, + { + "label": "E", + "text": "There exists a constant \\(C>0\\), depending only on \\(K,N,a_1,a_2,S\\), and \\(\\operatorname{diam}(S\\cup Y)\\), such that for every \\(\\mu,\\nu\\in\\mathcal P(Y)\\),\n\\[\n\\|\\phi_\\mu-\\phi_\\nu\\|_{L^\\infty(S)}\\le C\\,W_1(\\mu,\\nu)^{1/2}.\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": "sharp_holder_exponent_one_half", + "template_used": "stronger_trap" + }, + { + "label": "C", + "sketch_hook_type": "regularity", + "tampered_component": "wasserstein_rate_factor", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "geometric_construction", + "tampered_component": "dependence_on_domain_geometry_and_diameter", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "L1_control_replaced_by_Linfty_control", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem states hypotheses and asks for the resulting uniform stability estimate, but it does not explicitly reveal the correct exponent, norm, or quantifier structure. There is no direct answer leakage beyond the standard setup." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recall question: it essentially asks for the exact conclusion under a detailed set of assumptions. The answer is not literally restated in the stem, but the task is mainly to recognize the theorem's precise formulation." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is required to distinguish the sharp statement from nearby variants involving a stronger Lipschitz rate, weaker boundedness-only claim, dependence of constants on the measures, or an incorrect L^infty norm. However, this is mostly discriminating among theorem variants rather than generating a conclusion from first principles." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target common errors: replacing the sharp 1/2 exponent by 1, dropping the Wasserstein dependence, weakening uniformity via target-dependent constants, and strengthening L^1 to L^infty. They are distinct and well aligned with likely failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no obvious leakage, but it mainly tests precise recall/recognition of a known result rather than deeper generative reasoning." + } + }, + { + "id": "2602.19195v1", + "paper_link": "http://arxiv.org/abs/2602.19195v1", + "theorems_cnt": 5, + "theorem": { + "env_name": "theorem", + "content": "\\label{theorem1.01}\n(Zagier \\cite{[Z-AM]}) For non-negative integers $a$ and $b$,\n\\begin{align}\nH(a,b)=2\\sum_{k=1}^{a+b+1}(-1)^kc_{a,b}\\zeta(2k+1)\n\\zeta(\\{2\\}^{a+b+1-k}),\n\\end{align}\nwhere\n$$c_{a,b}:=\\binom{2k}{2a+2}-\\Big(1-\\frac{1}{2^{2k}}\\Big)\\binom{2k}{2b+1}.$$", + "start_pos": 5505, + "end_pos": 5798, + "label": "theorem1.01" + }, + "ref_dict": { + "eq1.1": "\\begin{align}\\label{eq1.1}\nK(a,b)=\\sum_{k=1}^{a+b+1}(-1)^{k-1}d_{a,b}K(a+b-k+1)\\zeta(2k+1),\n\\end{align}", + "theorem1.01": "\\begin{theorem}\\label{theorem1.01}\n(Zagier \\cite{[Z-AM]}) For non-negative integers $a$ and $b$,\n\\begin{align}\nH(a,b)=2\\sum_{k=1}^{a+b+1}(-1)^kc_{a,b}\\zeta(2k+1)\n\\zeta(\\{2\\}^{a+b+1-k}),\n\\end{align}\nwhere\n$$c_{a,b}:=\\binom{2k}{2a+2}-\\Big(1-\\frac{1}{2^{2k}}\\Big)\\binom{2k}{2b+1}.$$\n\\end{theorem}", + "theorem1.2": "\\begin{theorem}\\label{theorem1.2}\nFor nonnegative integers $a$ and $b$, we have $T(a,b)=\\hat L(a,b)$, where\n\\begin{align*}\n\\hat L(a,b):=\\frac{-2}{(2a+1)!}\\left(\\frac{\\pi}{2}\\right)^{2a+2b+2}\n\\sum_{n=0}^{\\infty}\\frac{\\zeta(2n)}{(2n+2a+1)(2n+2a+2)\\cdots(2n+2a+2b+2)2^{2n}}.\n\\end{align*}\n\\end{theorem}", + "con1": "\\begin{conjecture}\\label{con1}\nFor nonnegative integers $a$ and $b$, we have\n\\begin{align}\nH(a,b)=\\frac{-4\\pi^{2a+2b+2}}{(2a+2)!}\\sum_{n=0}^{\\infty}\n\\frac{\\zeta(2n)}{(2n+2a+2)(2n+2a+3)\\cdots(2n+2a+2b+3)2^{2n}}.\n\\end{align}\n\\end{conjecture}", + "con2": "\\begin{conjecture}\\label{con2}\nFor nonnegative integers $a$ and $b$, we have\n\\begin{align}\nT(a,b)=\\frac{-2}{(2a+1)!}\\left(\\frac{\\pi}{2}\\right)^{2a+2b+2}\n\\sum_{n=0}^{\\infty}\\frac{\\zeta(2n)}{(2n+2a+1)(2n+2a+2)\\cdots(2n+2a+2b+2)2^{2n}}.\n\\end{align}\n\\end{conjecture}", + "theorem1.1": "\\begin{theorem}\\label{theorem1.1}\nFor nonnegative integers $a$ and $b$, we have $H(a,b)=L(a,b)$, where\n\\begin{align*}\nL(a,b):=\\frac{-4\\pi^{2a+2b+2}}{(2a+2)!}\\sum_{n=0}^{\\infty}\n\\frac{\\zeta(2n)}{(2n+2a+2)(2n+2a+3)\\cdots(2n+2a+2b+3)2^{2n}}.\n\\end{align*}\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 1592, + "pre_theorem_intro_text": "The systematic study of multiple zeta values (MZVs) started\nin the early 1990s with the work of Hoffman \\cite{[H-PJM]}\nand Zagier \\cite{[Z-FECM]}. Let $r\\ge 1$ be an integer. For any multiple index\n$\\mathbf{s}=(s_1,s_2,\\cdots,s_r)\\in \\mathbb{Z}_{\\geq 1}^r$ with\n$s_r\\textgreater 1$, the {\\it multiple zeta value (MZV for short)}\nis defined by\n\\begin{align*}\n\\zeta(s_1,s_2,\\cdots,s_r):=\\sum_{1\\leq k_10$, these two conjectures\nare still kept open so far.\n\n\\begin{theorem}\\label{theorem1.1}\nFor nonnegative integers $a$ and $b$, we have $H(a,b)=L(a,b)$, where\n\\begin{align*}\nL(a,b):=\\frac{-4\\pi^{2a+2b+2}}{(2a+2)!}\\sum_{n=0}^{\\infty}\n\\frac{\\zeta(2n)}{(2n+2a+2)(2n+2a+3)\\cdots(2n+2a+2b+3)2^{2n}}.\n\\end{align*}\n\\end{theorem}\n\n\\begin{theorem}\\label{theorem1.2}\nFor nonnegative integers $a$ and $b$, we have $T(a,b)=\\hat L(a,b)$, where\n\\begin{align*}\n\\hat L(a,b):=\\frac{-2}{(2a+1)!}\\left(\\frac{\\pi}{2}\\right)^{2a+2b+2}\n\\sum_{n=0}^{\\infty}\\frac{\\zeta(2n)}{(2n+2a+1)(2n+2a+2)\\cdots(2n+2a+2b+2)2^{2n}}.\n\\end{align*}\n\\end{theorem}\n\nFirst, we consider the integral\n$$\\int_{0}^{\\frac{\\pi}{2}} x^{2a + 2}\\left(1 - \\frac{2}{\\pi} x\\right)^{2b + 1} \\cot(x)\\mathrm{d}x.$$\nBy using the well-known identity (see, for example, Theorem 1.2.4 of \\cite{[AAR]})\n\\begin{align}\\label{eq3.1'}\nx\\cot(x)=-2\\sum_{n=0}^{\\infty}\\frac{\\zeta(2n)}{\\pi^{2n}}x^{2n},\n\\end{align}\nwe derive that\n\\begin{align}\\label{eq3.2}\n&\\int_{0}^{\\frac{\\pi}{2}}x^{2a+2}\\Big(1-\\frac{2}{\\pi} x\\Big)^{2b+1}\n\\cot(x)\\mathrm{d}x\\nonumber\\\\\n=&-2\\int_{0}^{\\frac{\\pi}{2}}x^{2a+1}\\Big(1-\\frac{2}{\\pi} x\\Big)^{2b+1}\n\\sum_{n=0}^{\\infty}\\frac{\\zeta(2n)}{\\pi^{2n}}x^{2n} \\mathrm{d}x \\nonumber\\\\\n=&-2\\sum_{n=0}^{\\infty}\\frac{\\zeta(2n)}{\\pi^{2n}}\n\\int_{0}^{\\frac{\\pi}{2}}x^{2n+2a+1}\\Big(1-\\frac{2}{\\pi} x\\Big)^{2b+1}\\mathrm{d}x.\n\\end{align}\n\n\\begin{theorem}\\label{theorem1.01}\n(Zagier \\cite{[Z-AM]}) For non-negative integers $a$ and $b$,\n\\begin{align}\nH(a,b)=2\\sum_{k=1}^{a+b+1}(-1)^kc_{a,b}\\zeta(2k+1)\n\\zeta(\\{2\\}^{a+b+1-k}),\n\\end{align}\nwhere\n$$c_{a,b}:=\\binom{2k}{2a+2}-\\Big(1-\\frac{1}{2^{2k}}\\Big)\\binom{2k}{2b+1}.$$\n\\end{theorem}", + "post_theorem_intro_text_len": 4374, + "post_theorem_intro_text": "\\noindent The proof of Theorem \\ref{theorem1.01} is given by computing the associated\ngenerating functions of both sides in a closed form, and then showing\nthey are entire functions of exponential growth that agree at\nsufficiently many points to force their equality. Later on, his proof\nwas simplified by Li \\cite{[L-MRL]}. In 2017, Hessami Pilehrood\nand Hessami Pilehrood \\cite{[HHT-JMAA]} gave an alternative proof of Theorem\n\\ref{theorem1.01}.\n\nIn a similar way, Hoffman \\cite{[H-CNTP]} defined the multiple\n$t$-values (``odd variant'' of MZVs). For any multiple index\n$\\mathbf{s}=(s_1,s_2,...,s_r)\\in \\mathbb{Z}_{\\geq 1}^r$, the\n{\\it multiple $t$-value } is defined by\n\\begin{align*}\nt(s_1,s_2,...,s_r):=\\sum_{1\\leq k_10$, Lupu\n\\cite{[L-MZ]} proposed the following two conjectures.\n\n\\begin{conjecture}\\label{con1}\nFor nonnegative integers $a$ and $b$, we have\n\\begin{align}\nH(a,b)=\\frac{-4\\pi^{2a+2b+2}}{(2a+2)!}\\sum_{n=0}^{\\infty}\n\\frac{\\zeta(2n)}{(2n+2a+2)(2n+2a+3)\\cdots(2n+2a+2b+3)2^{2n}}.\n\\end{align}\n\\end{conjecture}\n\n\\begin{conjecture}\\label{con2}\nFor nonnegative integers $a$ and $b$, we have\n\\begin{align}\nT(a,b)=\\frac{-2}{(2a+1)!}\\left(\\frac{\\pi}{2}\\right)^{2a+2b+2}\n\\sum_{n=0}^{\\infty}\\frac{\\zeta(2n)}{(2n+2a+1)(2n+2a+2)\\cdots(2n+2a+2b+2)2^{2n}}.\n\\end{align}\n\\end{conjecture}\n\\noindent Notice that Conjectures 1.2 and 1.3 are shown in \\cite{[L-MZ]}\nto be true when $b=0$. But for the general case $b>0$, these two conjectures\nare still kept open so far.\n\nIn 2026, Lai, Lupu, and Orr \\cite{[LLO-PAMS]} gave elementary and direct proofs\nfor both $H(a, b)$ and $T(a, b)$. In fact, they \\cite{[LLO-PAMS]} first provided\nan integral expression for $H(a, b)$, and then showed that this integral expression\nis equal to Zagier's result. The same proof method was applied to $T(a, b)$.\n\nIn this paper, our main goal is to investigate Conjectures \\ref{con1} and \\ref{con2}.\nWe will prove that both conjectures of Lupu are true. Actually, by employing the integral\nexpressions presented in \\cite{[LLO-PAMS]} for $H(a,b)$ and $T(a,b)$ and using\nthe properties of Beta and Gamma functions, we show the following main results\nof this paper.\n\n\\begin{theorem}\\label{theorem1.1}\nFor nonnegative integers $a$ and $b$, we have $H(a,b)=L(a,b)$, where\n\\begin{align*}\nL(a,b):=\\frac{-4\\pi^{2a+2b+2}}{(2a+2)!}\\sum_{n=0}^{\\infty}\n\\frac{\\zeta(2n)}{(2n+2a+2)(2n+2a+3)\\cdots(2n+2a+2b+3)2^{2n}}.\n\\end{align*}\n\\end{theorem}\n\n\\begin{theorem}\\label{theorem1.2}\nFor nonnegative integers $a$ and $b$, we have $T(a,b)=\\hat L(a,b)$, where\n\\begin{align*}\n\\hat L(a,b):=\\frac{-2}{(2a+1)!}\\left(\\frac{\\pi}{2}\\right)^{2a+2b+2}\n\\sum_{n=0}^{\\infty}\\frac{\\zeta(2n)}{(2n+2a+1)(2n+2a+2)\\cdots(2n+2a+2b+2)2^{2n}}.\n\\end{align*}\n\\end{theorem}\n\nThis paper is organized as follows. In Section 2, we supply several preliminary lemmas\nthat are needed in the proofs of our main results. Sections 3 and 4 are devoted to\nthe proofs of Theorems \\ref{theorem1.1} and \\ref{theorem1.2}, respectively.", + "sketch": "The post-theorem introduction sketches Zagier's strategy for proving Theorem~\\ref{theorem1.01}: compute the associated generating functions of both sides \"in a closed form,\" then show these generating functions are \"entire functions of exponential growth\" and that they \"agree at sufficiently many points to force their equality.\" It is also noted that \"his proof was simplified by Li\" and that Hessami Pilehrood and Hessami Pilehrood gave \"an alternative proof\" of Theorem~\\ref{theorem1.01}.", + "expanded_sketch": "The post-theorem introduction sketches Zagier's strategy for proving the main theorem: compute the associated generating functions of both sides \"in a closed form,\" then show these generating functions are \"entire functions of exponential growth\" and that they \"agree at sufficiently many points to force their equality.\" It is also noted that \"his proof was simplified by Li\" and that Hessami Pilehrood and Hessami Pilehrood gave \"an alternative proof\" of the main theorem.", + "expanded_theorem": "\\label{theorem1.01}\n(Zagier \\cite{[Z-AM]}) For non-negative integers $a$ and $b$,\n\\begin{align}\nH(a,b)=2\\sum_{k=1}^{a+b+1}(-1)^kc_{a,b}\\zeta(2k+1)\n\\zeta(\\{2\\}^{a+b+1-k}),\n\\end{align}\nwhere\n$$c_{a,b}:=\\binom{2k}{2a+2}-\\Big(1-\\frac{1}{2^{2k}}\\Big)\\binom{2k}{2b+1}.$$", + "theorem_type": [ + "Universal", + "Classification or Bijection" + ], + "mcq": { + "question": "Let \\(a,b\\in \\mathbb Z_{\\ge 0}\\). For a multiple index \\((s_1,\\dots,s_r)\\) with \\(s_r>1\\), define the multiple zeta value\n\\[\n\\zeta(s_1,\\dots,s_r)=\\sum_{1\\le k_1<\\cdots 1$, $\\mu+\\lambda>0$, $\\mu>0$, $\\kappa>0$, and $R>0$.\nHere, $ \\rho, v, w $, and $ p $ represent the density, the horizontal velocity, vertical velocity, and pressure, respectively. $ \\mathrm{div}_{h}\\,, \\nabla_{h} $, and $\\Delta_{h} $ are, and will be, the divergence, gradient, and Laplace operators in the horizontal variables $ (x,y) $, respectively.\nWe investigate system \\eqref{eq:CPE} in the horizontally periodic channel\n$$ \\mathcal O:= \\mathbb T^2 \\times (0,1) = \\lbrace (x,y,z) | (x,y) \\in \\mathbb T^2, z \\in (0,1) \\rbrace,\n$$\nand consider the following\nboundary conditions:\n\\begin{equation}\\label{bc:CPE}\n\t\\partial_z \\theta\\big|_{z=0,1} = 0, \\quad \\partial_z v\\big|_{z=0,1} = 0, \\quad w\\big|_{z=0,1} = 0.\n\\end{equation}\n\nThe isentropic compressible primitive equations have been investigated by the last two authors in \\cite{LT2018a} for local strong solutions and \\cite{LT2018b} for global weak solutions. The existence of global weak solutions is also studied by \\cite{Wang2020}, independently. See also, \\cite{Jiu2018,Ersoy2011a,Ersoy2012,Gatapov2005}.\n\nOn the other hand, the incompressible primitive equations have been the subject of intensive mathematical research since the introduction\nby Lions, Temam, and Wang in \\cite{Lions1992,JLLions1992}. For instance, Guill\\'en-Gonz\\'alez, Masmoudi, and Rodr\\'iguez-Bellido in\n\\cite{GuillenGonzalez2001} study the local existence of strong solutions and global existence of strong solutions with small initial data. In\n\\cite{HuTemamZiane2003}, the authors address the global existence of strong solutions in a domain with small depth. The first breakthrough\nconcerning the global well-posedness of incompressible primitive equations is obtained by Cao and Titi in \\cite{Cao2007}. See also,\n\\cite{Cao2014a,Cao2014b,Cao2016,Cao2016a,CaoLiTiti2020,CaoLiTitiWang2024,Boling,MHATK,MHTKa,GMK,IKMZ,Li2017a,LiYuan2022} and the references therein for\nrelated literatures; in particular, global well-posedness of strong solutions was proved in \\cite{Cao2016,Cao2016a,CaoLiTiti2020,CaoLiTitiWang2024} to the primitive equations with only horizontal viscosity.\n\nWe have introduced the PE diagram in \\cite{LT2018LowMach1}, which concerns the low Mach number limit and the small aspect ratio (between the vertical and horizontal scales) limit. In \\cite{LT2018LowMach1,LT2018LowMach2}, we also establish the low Mach number limit for the isentropic compressible primitive equations. However, the counterpart study of the compressible Navier-Stokes-Fourier equations (see, e.g., \\cite{Alazard2006}) indicates that the PE diagram might be completely different for non-isentropic flows, due to the additional heat conductivity. We leave such a subject to future study.\n\nFor interested readers, we refer to \\cite{Temam1977,Lions1996,Lions1998,Feireisl2004,Feireisl2009a} for the study of hydrodynamic equations.\n\nOne can easily see that, compared to the Navier-Stokes equations, the evolutionary equation for vertical velocity is missing in system\n\\eqref{eq:CPE}. In fact, this is one of the main challenges in the study of the compressible primitive equations. In order to have a better\nunderstanding of the system, we derive a representation of vertical velocity in terms of horizontal velocity, density, and pressure.\nConsequently, we will reformulate system \\eqref{eq:CPE} to an equivalent one.\n\nUse \\subeqref{eq:CPE}{1} to rewrite \\subeqref{eq:CPE}{2} as\n$$\n\\rho(\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv)+\\nabla_hp=\\mu\\Delta v+(\\mu+\\lambda)\\nabla_h\\text{div}_hv\n$$\nand define\n\\begin{equation}\\label{def:rho-1}\n\t\\sigma := \\frac1\\rho.\n\\end{equation}\nThen, it is clear that\n\\begin{equation}\n\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma \\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma \\nabla_h\\text{div}_hv. \\label{EQv00}\n\\end{equation}\n\nNote that \\subeqref{eq:CPE}{3} implies that $p$ is independent of the vertical variable $z$ and one can use \\subeqref{eq:CPE}{3} to rewrite \\subeqref{eq:CPE}{4} as\n\\begin{equation*}\n \\partial_tp+v\\cdot\\nabla_hp+\\gamma p(\\text{div}_hv+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)Q(\\nabla v).\n\\end{equation*}\nSeparating the $z$-average part and the fluctuation part of the above equation, and recalling the boundary conditions $\\partial_z\\theta|_{z=0,1}=w|_{z=0,1}=0$, one obtains that\n\\begin{eqnarray}\n \\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)},\\label{EQp00}\\\\\n\\tilde v\\cdot\\nabla_hp+\\gamma p(\\text{div}_h\\tilde v+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)\\widetilde{Q(\\nabla v)}. \\label{EQw000}\n\\end{eqnarray}\nHere, for a function $f$, we use\n\\begin{equation}\\label{TBf}\n\\bar f:=\\int_0^1f dz\\quad\\mbox{and}\\quad \\tilde f:=f-\\bar f,\n\\end{equation}\nto represent the vertical integral (equivalently, average) and the vertical fluctuation of the quantity considered, respectively.\n\nSince $\\theta = R^{-1} \\rho^{-1} p=R^{-1}\\sigma p,$\nit follows from \\eqref{EQw000} and \\subeqref{eq:CPE}{3} that\n\\begin{eqnarray}\\label{eq:dz-w}\n\t\t \\partial_z w &=&-\\text{div}_h\\tilde v+\\frac{\\gamma-1}{\\gamma p}\\left(\\kappa\\partial_z^2\\theta+\\widetilde{Q(\\nabla v)}\\right)-\\frac{1}{\\gamma p}\\tilde v\\cdot\\nabla_hp\\nonumber\\\\\n&=&\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}\\partial_{zz} \\sigma -\\text{div}_h\\tilde v-\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right)\\nonumber\\\\\n&=&\\nu\\partial_z^2\\sigma-\\phi(v,p),\\label{EQw00}\n\\end{eqnarray}\nwhere $\\nu=\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}$ and\n\\begin{eqnarray}\n \\phi(v,p):=\\text{div}_h\\tilde v+\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right).\\label{phi}\n\\end{eqnarray}\nRecalling that $p$ is independent of $z$, it holds that $\\partial_z\\sigma=R\\partial_z(p^{-1}\\theta)= R p^{-1}\\partial_z\\theta$. Then, the boundary condition \\eqref{bc:CPE} implies\n\\begin{equation}\\label{bc:rf-CPE}\n\\partial_z \\sigma\\big|_{z=0,1} = 0.\n\\end{equation}\nThanks to this and recalling that $w|_{z=0}$, it follows from \\eqref{EQw00} that\n\\begin{equation}\n w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz'.\\label{EQw00'}\n\\end{equation}\n\nIt follows from \\subeqref{eq:CPE}{1} that\n\\begin{equation*}\\label{eq:rho-1}\n\t\\partial_t \\rho^{-1} + v \\cdot \\nabla_{h} \\rho^{-1} - \\rho^{-1} \\mathrm{div}_{h}\\, v = \\rho^{-2} \\partial_z (\\rho w) = \\rho^{-1} \\partial_z w - w \\partial_z \\rho^{-1},\n\\end{equation*}\nfrom which, using \\eqref{eq:dz-w} and recalling $\\sigma=\\frac1\\rho$, one gets\n\\begin{equation}\\label{EQs00}\n\t\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma.\n\\end{equation}\n\nNow, collecting \\eqref{EQv00}, \\eqref{EQp00}, \\eqref{EQw00'}, \\eqref{EQs00}, we have the following reformulated system which is equivalent to \\eqref{eq:CPE}\n\\begin{eqnarray}\n &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n \\label{EQs0}\\\\\n &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}\nwhere $\\phi(v,p)$ and $Q(\\nabla v)$ are expressed as in (\\ref{phi}) and (\\ref{Q}), respectively. The boundary conditions read as\n\\begin{eqnarray}\n &v, \\sigma, \\mbox{ and } p \\mbox{ are periodic in }x, y, \\label{BC1}\\\\\n &\\partial_zv|_{z=0,1}=0, \\quad\\partial_z\\sigma|_{z=0,1}=0. \\label{BC2}\n\\end{eqnarray}\nThe initial condition is\n\\begin{equation}\n \\label{IC}\n (v, \\sigma, p)|_{t=0}=(v_{0}, \\sigma_{0}, p_{0}).\n\\end{equation}\n\nBefore stating the main result, we recall some standard notations. For positive integer $k$ and $q\\in[1,\\infty]$, $L^q(\\mathcal O)$ and\n$L^q(\\mathbb T^2)$ are the Lebesgue spaces,\n$W^{k,q}(\\mathcal O)$ and $W^{k,q}(\\mathbb T^2)$ are the Sobolev spaces. If $q=2$, we use $H^k$ instead of $W^{k,2}$. For simplicity,\nwe will use $\\|\\cdot\\|_q$ to denote $L^q(\\mathcal O)$ or $L^q(\\mathbb T^2)$ norms of the corresponding function,\nif the meaning is clear from the context.\n\nWe are now in the position to state the main result of this paper.", + "context": "In this work, our goal is to investigate the fundamental problem of the well-posedness of solutions to the compressible primitive equations with non-trivial entropy. In particular, we consider the non-isentropic compressible primitive equations with only vertical diffusion for the temperature:\n\\begin{equation}\\label{eq:CPE}\n \\begin{cases}\n \\partial_t \\rho + \\mathrm{div}_{h}\\, (\\rho v) + \\partial_z (\\rho w) = 0, \\\\\n \\partial_t (\\rho v) + \\mathrm{div}_{h}\\,(\\rho v\\otimes v) + \\partial_z(\\rho w v) + \\nabla_{h} p = \\mathrm{div}_{h}\\, \\mathbb S_h + \\mu \\partial_{zz} v, \\\\\n \\partial_z p = 0, \\\\\n \\dfrac{1}{\\gamma-1}\\bigl(\\partial_t p + \\mathrm{div}_{h}\\,(p v) + \\partial_z (pw) \\bigr) + p(\\mathrm{div}_{h}\\, v + \\partial_z w) = \\partial_z (\\kappa \\partial_z \\theta )+Q(\\nabla v),\n \\end{cases}\n\\end{equation}\nwith\n\\begin{eqnarray}\n &\\mathbb S_h = \\mu (\\nabla_{h} v + \\nabla_{h}^\\top v) + \\lambda \\mathrm{div}_{h}\\, v \\mathbb I_2, \\quad\n p = R \\rho \\theta, \\nonumber\\\\\n &Q(\\nabla v)=\\mathbb S_h:\\nabla_hv+\\mu|\\partial_zv|^2,\\label{Q}\n\\end{eqnarray}\nwith constant coefficients $\\gamma$, $\\mu$, $\\lambda$, $\\kappa,$ and $R$ satisfying $\\gamma > 1$, $\\mu+\\lambda>0$, $\\mu>0$, $\\kappa>0$, and $R>0$.\nHere, $ \\rho, v, w $, and $ p $ represent the density, the horizontal velocity, vertical velocity, and pressure, respectively. $ \\mathrm{div}_{h}\\,, \\nabla_{h} $, and $\\Delta_{h} $ are, and will be, the divergence, gradient, and Laplace operators in the horizontal variables $ (x,y) $, respectively.\nWe investigate system \\eqref{eq:CPE} in the horizontally periodic channel\n$$ \\mathcal O:= \\mathbb T^2 \\times (0,1) = \\lbrace (x,y,z) | (x,y) \\in \\mathbb T^2, z \\in (0,1) \\rbrace,\n$$\nand consider the following\nboundary conditions:\n\\begin{equation}\\label{bc:CPE}\n \\partial_z \\theta\\big|_{z=0,1} = 0, \\quad \\partial_z v\\big|_{z=0,1} = 0, \\quad w\\big|_{z=0,1} = 0.\n\\end{equation}\n\nNote that \\subeqref{eq:CPE}{3} implies that $p$ is independent of the vertical variable $z$ and one can use \\subeqref{eq:CPE}{3} to rewrite \\subeqref{eq:CPE}{4} as\n\\begin{equation*}\n \\partial_tp+v\\cdot\\nabla_hp+\\gamma p(\\text{div}_hv+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)Q(\\nabla v).\n\\end{equation*}\nSeparating the $z$-average part and the fluctuation part of the above equation, and recalling the boundary conditions $\\partial_z\\theta|_{z=0,1}=w|_{z=0,1}=0$, one obtains that\n\\begin{eqnarray}\n \\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)},\\label{EQp00}\\\\\n\\tilde v\\cdot\\nabla_hp+\\gamma p(\\text{div}_h\\tilde v+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)\\widetilde{Q(\\nabla v)}. \\label{EQw000}\n\\end{eqnarray}\nHere, for a function $f$, we use\n\\begin{equation}\\label{TBf}\n\\bar f:=\\int_0^1f dz\\quad\\mbox{and}\\quad \\tilde f:=f-\\bar f,\n\\end{equation}\nto represent the vertical integral (equivalently, average) and the vertical fluctuation of the quantity considered, respectively.\n\nSince $\\theta = R^{-1} \\rho^{-1} p=R^{-1}\\sigma p,$\nit follows from \\eqref{EQw000} and \\subeqref{eq:CPE}{3} that\n\\begin{eqnarray}\\label{eq:dz-w}\n \\partial_z w &=&-\\text{div}_h\\tilde v+\\frac{\\gamma-1}{\\gamma p}\\left(\\kappa\\partial_z^2\\theta+\\widetilde{Q(\\nabla v)}\\right)-\\frac{1}{\\gamma p}\\tilde v\\cdot\\nabla_hp\\nonumber\\\\\n&=&\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}\\partial_{zz} \\sigma -\\text{div}_h\\tilde v-\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right)\\nonumber\\\\\n&=&\\nu\\partial_z^2\\sigma-\\phi(v,p),\\label{EQw00}\n\\end{eqnarray}\nwhere $\\nu=\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}$ and\n\\begin{eqnarray}\n \\phi(v,p):=\\text{div}_h\\tilde v+\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right).\\label{phi}\n\\end{eqnarray}\nRecalling that $p$ is independent of $z$, it holds that $\\partial_z\\sigma=R\\partial_z(p^{-1}\\theta)= R p^{-1}\\partial_z\\theta$. Then, the boundary condition \\eqref{bc:CPE} implies\n\\begin{equation}\\label{bc:rf-CPE}\n\\partial_z \\sigma\\big|_{z=0,1} = 0.\n\\end{equation}\nThanks to this and recalling that $w|_{z=0}$, it follows from \\eqref{EQw00} that\n\\begin{equation}\n w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz'.\\label{EQw00'}\n\\end{equation}\n\nNow, collecting \\eqref{EQv00}, \\eqref{EQp00}, \\eqref{EQw00'}, \\eqref{EQs00}, we have the following reformulated system which is equivalent to \\eqref{eq:CPE}\n\\begin{eqnarray}\n &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n \\label{EQs0}\\\\\n &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}\nwhere $\\phi(v,p)$ and $Q(\\nabla v)$ are expressed as in (\\ref{phi}) and (\\ref{Q}), respectively. The boundary conditions read as\n\\begin{eqnarray}\n &v, \\sigma, \\mbox{ and } p \\mbox{ are periodic in }x, y, \\label{BC1}\\\\\n &\\partial_zv|_{z=0,1}=0, \\quad\\partial_z\\sigma|_{z=0,1}=0. \\label{BC2}\n\\end{eqnarray}\nThe initial condition is\n\\begin{equation}\n \\label{IC}\n (v, \\sigma, p)|_{t=0}=(v_{0}, \\sigma_{0}, p_{0}).\n\\end{equation}\n\nBefore stating the main result, we recall some standard notations. For positive integer $k$ and $q\\in[1,\\infty]$, $L^q(\\mathcal O)$ and\n$L^q(\\mathbb T^2)$ are the Lebesgue spaces,\n$W^{k,q}(\\mathcal O)$ and $W^{k,q}(\\mathbb T^2)$ are the Sobolev spaces. If $q=2$, we use $H^k$ instead of $W^{k,2}$. For simplicity,\nwe will use $\\|\\cdot\\|_q$ to denote $L^q(\\mathcal O)$ or $L^q(\\mathbb T^2)$ norms of the corresponding function,\nif the meaning is clear from the context.\n\nWe are now in the position to state the main result of this paper.", + "full_context": "In this work, our goal is to investigate the fundamental problem of the well-posedness of solutions to the compressible primitive equations with non-trivial entropy. In particular, we consider the non-isentropic compressible primitive equations with only vertical diffusion for the temperature:\n\\begin{equation}\\label{eq:CPE}\n \\begin{cases}\n \\partial_t \\rho + \\mathrm{div}_{h}\\, (\\rho v) + \\partial_z (\\rho w) = 0, \\\\\n \\partial_t (\\rho v) + \\mathrm{div}_{h}\\,(\\rho v\\otimes v) + \\partial_z(\\rho w v) + \\nabla_{h} p = \\mathrm{div}_{h}\\, \\mathbb S_h + \\mu \\partial_{zz} v, \\\\\n \\partial_z p = 0, \\\\\n \\dfrac{1}{\\gamma-1}\\bigl(\\partial_t p + \\mathrm{div}_{h}\\,(p v) + \\partial_z (pw) \\bigr) + p(\\mathrm{div}_{h}\\, v + \\partial_z w) = \\partial_z (\\kappa \\partial_z \\theta )+Q(\\nabla v),\n \\end{cases}\n\\end{equation}\nwith\n\\begin{eqnarray}\n &\\mathbb S_h = \\mu (\\nabla_{h} v + \\nabla_{h}^\\top v) + \\lambda \\mathrm{div}_{h}\\, v \\mathbb I_2, \\quad\n p = R \\rho \\theta, \\nonumber\\\\\n &Q(\\nabla v)=\\mathbb S_h:\\nabla_hv+\\mu|\\partial_zv|^2,\\label{Q}\n\\end{eqnarray}\nwith constant coefficients $\\gamma$, $\\mu$, $\\lambda$, $\\kappa,$ and $R$ satisfying $\\gamma > 1$, $\\mu+\\lambda>0$, $\\mu>0$, $\\kappa>0$, and $R>0$.\nHere, $ \\rho, v, w $, and $ p $ represent the density, the horizontal velocity, vertical velocity, and pressure, respectively. $ \\mathrm{div}_{h}\\,, \\nabla_{h} $, and $\\Delta_{h} $ are, and will be, the divergence, gradient, and Laplace operators in the horizontal variables $ (x,y) $, respectively.\nWe investigate system \\eqref{eq:CPE} in the horizontally periodic channel\n$$ \\mathcal O:= \\mathbb T^2 \\times (0,1) = \\lbrace (x,y,z) | (x,y) \\in \\mathbb T^2, z \\in (0,1) \\rbrace,\n$$\nand consider the following\nboundary conditions:\n\\begin{equation}\\label{bc:CPE}\n \\partial_z \\theta\\big|_{z=0,1} = 0, \\quad \\partial_z v\\big|_{z=0,1} = 0, \\quad w\\big|_{z=0,1} = 0.\n\\end{equation}\n\nNote that \\subeqref{eq:CPE}{3} implies that $p$ is independent of the vertical variable $z$ and one can use \\subeqref{eq:CPE}{3} to rewrite \\subeqref{eq:CPE}{4} as\n\\begin{equation*}\n \\partial_tp+v\\cdot\\nabla_hp+\\gamma p(\\text{div}_hv+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)Q(\\nabla v).\n\\end{equation*}\nSeparating the $z$-average part and the fluctuation part of the above equation, and recalling the boundary conditions $\\partial_z\\theta|_{z=0,1}=w|_{z=0,1}=0$, one obtains that\n\\begin{eqnarray}\n \\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)},\\label{EQp00}\\\\\n\\tilde v\\cdot\\nabla_hp+\\gamma p(\\text{div}_h\\tilde v+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)\\widetilde{Q(\\nabla v)}. \\label{EQw000}\n\\end{eqnarray}\nHere, for a function $f$, we use\n\\begin{equation}\\label{TBf}\n\\bar f:=\\int_0^1f dz\\quad\\mbox{and}\\quad \\tilde f:=f-\\bar f,\n\\end{equation}\nto represent the vertical integral (equivalently, average) and the vertical fluctuation of the quantity considered, respectively.\n\nSince $\\theta = R^{-1} \\rho^{-1} p=R^{-1}\\sigma p,$\nit follows from \\eqref{EQw000} and \\subeqref{eq:CPE}{3} that\n\\begin{eqnarray}\\label{eq:dz-w}\n \\partial_z w &=&-\\text{div}_h\\tilde v+\\frac{\\gamma-1}{\\gamma p}\\left(\\kappa\\partial_z^2\\theta+\\widetilde{Q(\\nabla v)}\\right)-\\frac{1}{\\gamma p}\\tilde v\\cdot\\nabla_hp\\nonumber\\\\\n&=&\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}\\partial_{zz} \\sigma -\\text{div}_h\\tilde v-\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right)\\nonumber\\\\\n&=&\\nu\\partial_z^2\\sigma-\\phi(v,p),\\label{EQw00}\n\\end{eqnarray}\nwhere $\\nu=\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}$ and\n\\begin{eqnarray}\n \\phi(v,p):=\\text{div}_h\\tilde v+\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right).\\label{phi}\n\\end{eqnarray}\nRecalling that $p$ is independent of $z$, it holds that $\\partial_z\\sigma=R\\partial_z(p^{-1}\\theta)= R p^{-1}\\partial_z\\theta$. Then, the boundary condition \\eqref{bc:CPE} implies\n\\begin{equation}\\label{bc:rf-CPE}\n\\partial_z \\sigma\\big|_{z=0,1} = 0.\n\\end{equation}\nThanks to this and recalling that $w|_{z=0}$, it follows from \\eqref{EQw00} that\n\\begin{equation}\n w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz'.\\label{EQw00'}\n\\end{equation}\n\nNow, collecting \\eqref{EQv00}, \\eqref{EQp00}, \\eqref{EQw00'}, \\eqref{EQs00}, we have the following reformulated system which is equivalent to \\eqref{eq:CPE}\n\\begin{eqnarray}\n &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n \\label{EQs0}\\\\\n &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}\nwhere $\\phi(v,p)$ and $Q(\\nabla v)$ are expressed as in (\\ref{phi}) and (\\ref{Q}), respectively. The boundary conditions read as\n\\begin{eqnarray}\n &v, \\sigma, \\mbox{ and } p \\mbox{ are periodic in }x, y, \\label{BC1}\\\\\n &\\partial_zv|_{z=0,1}=0, \\quad\\partial_z\\sigma|_{z=0,1}=0. \\label{BC2}\n\\end{eqnarray}\nThe initial condition is\n\\begin{equation}\n \\label{IC}\n (v, \\sigma, p)|_{t=0}=(v_{0}, \\sigma_{0}, p_{0}).\n\\end{equation}\n\nBefore stating the main result, we recall some standard notations. For positive integer $k$ and $q\\in[1,\\infty]$, $L^q(\\mathcal O)$ and\n$L^q(\\mathbb T^2)$ are the Lebesgue spaces,\n$W^{k,q}(\\mathcal O)$ and $W^{k,q}(\\mathbb T^2)$ are the Sobolev spaces. If $q=2$, we use $H^k$ instead of $W^{k,2}$. For simplicity,\nwe will use $\\|\\cdot\\|_q$ to denote $L^q(\\mathcal O)$ or $L^q(\\mathbb T^2)$ norms of the corresponding function,\nif the meaning is clear from the context.\n\nWe are now in the position to state the main result of this paper.\n\nWe are now in the position to state the main result of this paper.\n\n\\begin{proposition}\n \\label{PROP-APR}\nGiven $(v_{0}, \\sigma_{0}, p_{0})$ satisfying (\\ref{ASSUMIC1})--(\\ref{ASSUMIC2}). Let $X_0$ be an arbitrary constant such that\n$\\|v_{0}\\|_{H^3}^2+\\|\\sigma_{0}\\|_{H^2}^2+\\|p_0\\|_{H^3}^2\\leq X_0$. Then, there is a positive time $\\mathcal T_0$,\ndepending only on $\\gamma$, $\\nu$, $\\mu$, $\\lambda$, $\\underline\\sigma$, $\\underline p$, and\n $X_0$, such that system (\\ref{EQv})--(\\ref{EQp}),\n subject to (\\ref{BC1})--(\\ref{IC}), has a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$,\n satisfying\n \\begin{eqnarray*}\n &&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\geq0.5\\underline\\sigma,\\quad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\geq0.5\\underline p,\\\\\n &&\\sup_{0\\leq t\\leq\\mathcal T_0}(\\|v\\|_{H^3}^2+\\|\\sigma\\|_{H^2}^2+\\|p\\|_{H^3}^2)(t)+\\epsilon\\int_0^{\\mathcal T_0}(\\|\\nabla_h\\sigma\\|_{H^2}^2+\\|\\nabla_hp\\|_{H^3}^2)dt\\nonumber\\\\\n &&\\, \\, ~~~+\\int_0^{\\mathcal T_0}(\\|v\\|_{H^4}^2+\\|\\partial_z\\sigma\\|_{H^2}^2+\\|\\partial_tv\\|_{H^2}^2+\\|\\partial_t\\sigma\\|_{H^1}^2+\\|\\partial_tp\\|_{H^2}^2)dt \\leq K_0,\n \\end{eqnarray*}\n for a positive constant $K_0$ depending only on $\\gamma$, $\\nu$, $\\mu$, $\\lambda$, $\\underline\\sigma$, $\\underline p$, and\n $X_0$.\n\\end{proposition}\n\nBy Proposition \\ref{PROP-APR}, there is a positive time $\\mathcal T_0$ depending only on $\\gamma,$ $\\nu,$ $\\mu$, $\\lambda$,\n$\\underline\\sigma$, $\\underline p$, and $X_0$, such that for any $\\epsilon\\in(0,\\epsilon_0)$, system \\eqref{EQv}--\\eqref{EQp}, subject to \\eqref{BC1}--\\eqref{IC}, has a unique solution $(v_\\epsilon,\n\\sigma_\\epsilon, p_\\epsilon)$ on $\\mathcal O\\times(0,\\mathcal T_0)$,\nwith initial data $(v_{0,\\epsilon}, \\sigma_{0,\\epsilon}, p_{0,\\epsilon})$, satisfying\n \\begin{eqnarray}\n &&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma_\\epsilon\\geq0.5\\underline\\sigma,\\quad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p_\\epsilon\\geq0.5\\underline p,\\label{APRLBsp}\\\\\n &&\\sup_{0\\leq t\\leq\\mathcal T_0}(\\|v_\\epsilon\\|_{H^3}^2+\\|\\sigma_\\epsilon\\|_{H^2}^2+\\|p_\\epsilon\\|_{H^3}^2)(t)\n +\\epsilon\\int_0^{\\mathcal T_0}(\\|\\nabla_h\\sigma_\\epsilon\\|_{H^2}^2+\\|\\nabla_hp_\\epsilon\\|_{H^3}^2)dt\\nonumber\\\\\n &&\\, \\, ~~~+\\int_0^{\\mathcal T_0}(\\|v_\\epsilon\\|_{H^4}^2+\\|\\partial_z\\sigma_\\epsilon\\|_{H^2}^2+\\|\\partial_tv_\\epsilon\\|_{H^2}^2+\\|\\partial_t\\sigma_\\epsilon\n \\|_{H^1}^2+\\|\\partial_tp_\\epsilon\\|_{H^2}^2)dt \\leq K_0,\\label{APRHk}\n \\end{eqnarray}\n for a positive constant $K_0$ depending only on $\\gamma$, $\\nu$, $\\mu$, $\\lambda$, $\\underline\\sigma$, $\\underline p$, and\n $X_0$.\n\nThanks to \\eqref{APRHk}, by the Banach-Alaoglu theorem, and using Cantor's diagonal argument,\n there is a subsequence $(v_{\\epsilon_n}, \\sigma_{\\epsilon_n}, p_{\\epsilon_n})$ and a triple $(v,\\sigma,p)$,\n such that\n \\begin{eqnarray}\n &&v_{\\epsilon_n}\\stackrel{\\ast}{\\rightharpoonup}v,\\quad\\mbox{in }L^\\infty(0,\\mathcal T_0; H^3(\\mathcal O)),\\label{WeaC1}\\\\\n &&\\sigma_{\\epsilon_n}\\stackrel{\\ast}{\\rightharpoonup}\\sigma,\\quad\\mbox{in }L^\\infty(0,\\mathcal T_0; H^2(\\mathcal O)),\\label{WeaC2}\\\\\n &&p_{\\epsilon_n}\\stackrel{\\ast}{\\rightharpoonup}p,\\quad\\mbox{in }L^\\infty(0,\\mathcal T_0; H^3(\\mathbb T^2)),\\label{WeaC3}\\\\\n &&v_{\\epsilon_n}{\\rightharpoonup}v,\\quad\\mbox{in }L^2(0,\\mathcal T_0; H^4(\\mathcal O)),\\label{WeaC4}\\\\\n &&\\partial_z\\sigma_{\\epsilon_n} {\\rightharpoonup}\\partial_z\\sigma,\\quad\\mbox{in }L^2(0,\\mathcal T_0; H^2(\\mathcal O)),\\label{WeaC5}\\\\\n &&\\partial_tv_{\\epsilon_n} {\\rightharpoonup}\\partial_tv,\\quad\\mbox{in }L^2(0,\\mathcal T_0; H^2(\\mathcal O)),\\label{WeaC6}\\\\\n &&\\partial_t\\sigma_{\\epsilon_n} {\\rightharpoonup}\\partial_t\\sigma,\\quad\\mbox{in }L^2(0,\\mathcal T_0; H^1(\\mathcal O)),\\label{WeaC7}\\\\\n &&\\partial_tp_{\\epsilon_n} {\\rightharpoonup}\\partial_tp,\\quad\\mbox{in }L^2(0,\\mathcal T_0; H^2(\\mathbb T^2)),\\label{WeaC8}\n \\end{eqnarray}\n where $\\stackrel{\\ast}{\\rightharpoonup}$ and $\\rightharpoonup$ represent the weak-* and weak convergence, respectively, in the corresponding spaces. With the aid of the above convergence, noticing that $H^3(\\mathcal O)\\hookrightarrow\\hookrightarrow H^2(\\mathcal O)$, $H^2(\\mathcal O)\\hookrightarrow\\hookrightarrow H^1(\\mathcal O)\\cap C(\\overline{\\mathcal O})$, and $H^3(\\mathbb T^2)\\hookrightarrow\\hookrightarrow H^2(\\mathbb T^2)\\cap C(\\mathbb T^2)$,\n it follows from the Aubin-Lions lemma that\n \\begin{eqnarray}\n &&v_{\\epsilon_n} \\rightarrow v,\\quad\\mbox{in }C([0,\\mathcal T_0]; H^2(\\mathcal O)),\\label{StrC1}\\\\\n &&\\sigma_{\\epsilon_n} \\rightarrow\\sigma,\\quad\\mbox{in }C([0,\\mathcal T_0]; H^1(\\mathcal O)\\cap C(\\overline{\\mathcal O})),\\label{StrC2}\\\\\n &&\\partial_z\\sigma_{\\epsilon_n} \\rightarrow\\partial_z\\sigma,\\quad\\mbox{in }L^2(0,\\mathcal T_0; H^1(\\mathcal O)),\\\\\n &&p_{\\epsilon_n} \\rightarrow p,\\quad\\mbox{in }C([0,\\mathcal T_0]; H^2(\\mathbb T^2)\\cap C(\\mathbb T^2)).\\label{StrC3}\n \\end{eqnarray}\n These imply that $(v,\\sigma,p)$ has initial data $(v_0,\\sigma_0,p_0)$, enjoys all regularities stated in Theorem \\ref{thmmain}, and fulfills the boundary conditions \\eqref{BC1} and \\eqref{BC2}.\n Besides, by \\eqref{StrC2} and \\eqref{StrC3}, it follows from \\eqref{APRLBsp} that\n \\begin{equation*}\n \\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma \\geq0.5\\underline\\sigma,\\quad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p \\geq0.5\\underline p.\n \\end{equation*}\n By \\eqref{WeaC1}--\\eqref{StrC3}, one can take the limit as $n\\rightarrow\\infty$ to \\eqref{EQv}--\\eqref{EQp} to show that $(v,\\sigma,p)$ satisfies system \\eqref{EQv0}--\\eqref{EQp0} in the sense of distribution and further pointwisely, by the regularities of $(v,\\sigma,p)$. Therefore, $(v,\\sigma,p)$ is a strong solution\n to system \\eqref{EQv}--\\eqref{EQp}, subject to \\eqref{BC1}--\\eqref{IC}, on $\\mathcal O\\times(0,\\mathcal T_0)$.\n\n\\section{Appendix B: local existence of the regularized system}\n\\label{APPENDIXB}\nDenote\n\\begin{eqnarray*}\n N_1(v,\\sigma,p):=-[(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp],\\label{N1}\\\\\n N_2(v,\\sigma,p):=-[v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)],\\label{N2}\\\\\n N_3(v,p):=-(\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v)+(\\gamma-1)\\overline{Q(\\nabla v)},\\label{N3}\n\\end{eqnarray*}\nwhere $w, \\phi(v,p), \\bar f,$ and $Q(\\nabla v)$ are expressed, respectively, by\\eqref{EQw00'}, \\eqref{phi}, \\eqref{TBf}, and \\eqref{Q}.\nRecall $\\mathcal O=\\mathbb T^2\\times(0,1)$. Define a solution mapping\n\\begin{equation}\\label{DEFF}\n(v,\\sigma,p)\\mapsto(V,\\Sigma,P)=:\\mathfrak{F}(v,\\sigma,p),\n\\end{equation}\nwhere $(V,\\Sigma,P)$ is the unique solution to the following linear system\n\\begin{eqnarray}\n \\partial_tV-\\mu\\sigma\\Delta V-(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hV&=&N_1(v,\\sigma,p),\\label{AEQV}\\\\\n \\partial_t\\Sigma-\\nu\\sigma\\partial_z^2\\Sigma-\\epsilon\\Delta_h\\Sigma&=&N_2(v,\\sigma,p),\\label{AEQSIGMA}\\\\\n \\partial_tP-\\epsilon\\Delta_hP&=&N_3(v,p), \\label{AEQP}\n\\end{eqnarray}\nin $\\mathcal O\\times(0,T)$, subject to\n\\begin{eqnarray}\n &V,\\Sigma, P\\text{ are periodic in }x,y, \\quad\\partial_zV|_{z=0,1}=0,\\quad \\partial_z\\Sigma|_{z=0,1}=0,\\label{ABC}\\\\\n &(V,\\Sigma,P)|_{t=0}=(v_0,\\sigma_0,p_0),\\label{AIC}\n\\end{eqnarray}\nwhere $(v_0,\\sigma_0)\\in H^3(\\mathcal O)$, $p_0\\in H^3(\\mathbb T^2),$ and\n\\begin{equation*}\n \\partial_zv_0|_{z=0,1}=0,\\quad \\partial_z\\sigma_0|_{z=0,1}=0,\\quad\\sigma_0\\geq\\underline\\sigma, \\quad p_0\\geq\\underline p,\\label{A1}\n\\end{equation*}\nfor two positive numbers $\\underline\\sigma$ and $\\underline p$.", + "post_theorem_intro_text_len": 2475, + "post_theorem_intro_text": "Some comments on the proof of Theorem \\ref{thmmain} are presented as follows.\nRegularity assumption $\\sigma_0\\in H^2(\\mathcal O)$ comes from the following observation: for the toy model\n$$\n\\partial_t\\sigma=\\nu\\sigma\\partial_z^2\\sigma,\n$$\nin order that $H^k(\\mathcal O)$ regularity propagates, one needs $k\\geq2$.\nFortunately, such $H^2(\\mathcal O)$ regularity on $\\sigma_0$ is still sufficient for the more complicated equation \\eqref{EQs0}.\nWhile for the aim of getting $H^2(\\mathcal O)$ estimate on $\\sigma$, one may apply $\\Delta$ to \\eqref{EQs0} and, as a result, the\nquantity $\\Delta\\phi(v,p)$ and further, recalling the expression of $\\phi(v,p)$, $\\nabla_h\\Delta_hp$ will be encountered.\nIn order to get information on $\\nabla_h\\Delta_hp$, by equation \\eqref{EQp0}, one requires $p_0\\in H^3(\\mathbb T^2)$ and moreover\nthe quantity $\\nabla_h^4v$ will be encountered. Due to this and by \\eqref{EQv0}, one has to assume $v_0\\in H^3(\\mathcal O)$.\nThe existence of solutions to system \\eqref{EQv0}--\\eqref{EQp0}, subject to \\eqref{BC1}--\\eqref{IC}, is established by parabolic\nregularization argument. A regularized system is introduced by adding the regularizing dissipative terms $\\epsilon\\Delta_h\\sigma$ and $\\epsilon\\Delta_hp$\nto \\eqref{EQs0} and \\eqref{EQp0}, respectively, see \\eqref{EQv}--\\eqref{EQp}, below. Such a regularized system is a semi-linear\nparabolic system. The main effort of the proof is then to carry out\nsuitable $\\epsilon$-independent {\\it a priori} estimates and the local existence\nfollows passing to the limit $\\epsilon\\rightarrow0$.\nContinuous dependence on the initial data, which in particular implies\nthe uniqueness, is proved by performing energy estimates to the difference between two solutions. We employ $L^2$ energy estimate to the\nsubtracted $\\sigma$ equation which contains a term involving $\\nabla_hp$. This, due to the absence of dissipation\nin the pressure equation, forces us to perform $H^1(\\mathbb T^2)$ type energy estimate to the subtracted pressure equation and\nfurther $H^1(\\mathcal O)$ energy to the subtracted velocity equations.\n\nThe rest of this paper is arranged as follows: in the next section, Section \\ref{sec3}, we consider the $\\epsilon$-regularized system\nand perform suitable a priori estimates; the main result, Theorem \\ref{thmmain}, is proved in Section \\ref{sec4}; some commutator estimates are proved in Appendix A; local existence and uniqueness of the $\\epsilon$-regularized system is proved in Appendix B.", + "sketch": "For Theorem~\\ref{thmmain}, the authors explain the proof strategy and why the initial regularities are chosen.\n\n- **Why $\\sigma_0\\in H^2(\\mathcal O)$:** From the toy model $\\partial_t\\sigma=\\nu\\sigma\\partial_z^2\\sigma$, “in order that $H^k(\\mathcal O)$ regularity propagates, one needs $k\\ge 2$.” This $H^2$ assumption is said to be “still sufficient for the more complicated equation \\eqref{EQs0}.”\n\n- **Why $p_0\\in H^3(\\mathbb T^2)$ and $v_0\\in H^3(\\mathcal O)$:** To get an “$H^2(\\mathcal O)$ estimate on $\\sigma$,” one applies “$\\Delta$ to \\eqref{EQs0},” which leads to “$\\Delta\\phi(v,p)$” and hence to “$\\nabla_h\\Delta_h p$.” To control $\\nabla_h\\Delta_h p$ using \\eqref{EQp0}, one needs “$p_0\\in H^3(\\mathbb T^2)$,” and in the process “the quantity $\\nabla_h^4 v$ will be encountered,” which (via \\eqref{EQv0}) “has to assume $v_0\\in H^3(\\mathcal O)$.”\n\n- **Existence via parabolic regularization and $\\epsilon$-independent estimates:** Existence is established by a “parabolic regularization argument.” They introduce a regularized system by adding “the regularizing dissipative terms $\\epsilon\\Delta_h\\sigma$ and $\\epsilon\\Delta_h p$ to \\eqref{EQs0} and \\eqref{EQp0}, respectively.” The resulting system is “a semi-linear parabolic system.” The “main effort” is to derive “suitable $\\epsilon$-independent {\\it a priori} estimates,” and then “the local existence follows passing to the limit $\\epsilon\\to 0$.”\n\n- **Uniqueness/continuous dependence by energy estimates on differences:** “Continuous dependence on the initial data, which in particular implies the uniqueness,” is proved by “performing energy estimates to the difference between two solutions.” Specifically, they use an “$L^2$ energy estimate to the subtracted $\\sigma$ equation” but this equation has “a term involving $\\nabla_h p$.” Because of “the absence of dissipation in the pressure equation,” they are “force[d]” to do an “$H^1(\\mathbb T^2)$ type energy estimate to the subtracted pressure equation” and “further $H^1(\\mathcal O)$ energy to the subtracted velocity equations.”", + "expanded_sketch": "For the main theorem, the authors explain the proof strategy and why the initial regularities are chosen.\n\n- **Why $\\sigma_0\\in H^2(\\mathcal O)$:** From the toy model $\\partial_t\\sigma=\\nu\\sigma\\partial_z^2\\sigma$, “in order that $H^k(\\mathcal O)$ regularity propagates, one needs $k\\ge 2$.” This $H^2$ assumption is said to be “still sufficient for the more complicated equation\n\\begin{eqnarray}\n &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n \\label{EQs0}\\\\\n &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}\n.”\n\n- **Why $p_0\\in H^3(\\mathbb T^2)$ and $v_0\\in H^3(\\mathcal O)$:** To get an “$H^2(\\mathcal O)$ estimate on $\\sigma$,” one applies “$\\Delta$ to” the $\\sigma$-equation in\n\\begin{eqnarray}\n &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n \\label{EQs0}\\\\\n &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}\nwhich leads to “$\\Delta\\phi(v,p)$” and hence to “$\\nabla_h\\Delta_h p$.” To control $\\nabla_h\\Delta_h p$ using the pressure equation in the same system (i.e. the $p$-equation displayed above), one needs “$p_0\\in H^3(\\mathbb T^2)$,” and in the process “the quantity $\\nabla_h^4 v$ will be encountered,” which (via the velocity equation in the displayed system above) “has to assume $v_0\\in H^3(\\mathcal O)$.”\n\n- **Existence via parabolic regularization and $\\epsilon$-independent estimates:** Existence is established by a “parabolic regularization argument.” They introduce a regularized system by adding “the regularizing dissipative terms $\\epsilon\\Delta_h\\sigma$ and $\\epsilon\\Delta_h p$” to the $\\sigma$- and $p$-equations in\n\\begin{eqnarray}\n &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n \\label{EQs0}\\\\\n &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}\nrespectively. The resulting system is “a semi-linear parabolic system.” The “main effort” is to derive “suitable $\\epsilon$-independent {\\it a priori} estimates,” and then “the local existence follows passing to the limit $\\epsilon\\to 0$.”\n\n- **Uniqueness/continuous dependence by energy estimates on differences:** “Continuous dependence on the initial data, which in particular implies the uniqueness,” is proved by “performing energy estimates to the difference between two solutions.” Specifically, they use an “$L^2$ energy estimate to the subtracted $\\sigma$ equation” but this equation has “a term involving $\\nabla_h p$.” Because of “the absence of dissipation in the pressure equation” (namely, in the $p$-equation in the displayed system above), they are “force[d]” to do an “$H^1(\\mathbb T^2)$ type energy estimate to the subtracted pressure equation” and “further $H^1(\\mathcal O)$ energy to the subtracted velocity equations.”", + "expanded_theorem": "\\label{thmmain}\nGiven $v_{0}\\in H^3(\\mathcal O)$, $\\sigma_{0} \\in H^2(\\mathcal O)$, and $p_{0}\\in H^3(\\mathbb T^2),$ such that\n$$\n\\sigma_{0}\\geq\\underline\\sigma, \\quad p_{0}\\geq\\underline p,\\quad \\partial_zv_{0}|_{z=0,1}=0,\\quad \\partial_z\\sigma_{0}|_{z=0,1}=0,\n$$\nfor two positive numbers $\\underline\\sigma$ and $\\underline p$.\nThen, there is a positive time $\\mathcal T_0$,\ndepending only on $\\gamma$, $\\nu$, $\\mu$, $\\lambda$, $\\underline\\sigma$, $\\underline p$, and\n $\\|v_{0}\\|_{H^3}^2+\\|\\sigma_{0}\\|_{H^2}^2+\\|p_0\\|_{H^3}^2$, such that the system\n\\begin{eqnarray}\n &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n \\label{EQs0}\\\\\n &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}\n subject to\n\\begin{eqnarray}\n &v, \\sigma, \\mbox{ and } p \\mbox{ are periodic in }x, y, \\label{BC1}\\\\\n &\\partial_zv|_{z=0,1}=0, \\quad\\partial_z\\sigma|_{z=0,1}=0. \\label{BC2}\n\\end{eqnarray}\nand\n\\begin{equation}\n \\label{IC}\n (v, \\sigma, p)|_{t=0}=(v_{0}, \\sigma_{0}, p_{0}).\n\\end{equation}\nhas a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$,\n satisfying\n \\begin{eqnarray*}\n &&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\geq0.5\\underline\\sigma,\\quad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\geq0.5\\underline p,\\\\\n &&v\\in C([0,\\mathcal T_0]; H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0; H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0; H^4(\\mathcal O)),\\\\\n &&\\sigma\\in C([0,\\mathcal T_0]; H^1(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0; H^2(\\mathcal O)),\\quad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0; H^2(\\mathcal O)),\\\\\n &&\\partial_tv\\in L^2(0,\\mathcal T_0; H^2(\\mathcal O)),\\quad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0; H^1(\\mathcal O)),\\\\\n &&p\\in C([0,\\mathcal T_0]; H^2(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0; H^3(\\mathbb T^2)),\\quad\\partial_tp\\in L^2(0,\\mathcal T_0; H^2(\\mathbb T^2)).\n \\end{eqnarray*},", + "theorem_type": [ + "Uniqueness", + "Existence" + ], + "mcq": { + "question": "Let $\\mathcal O=\\mathbb T^2\\times(0,1)$. For a function $f(x,y,z)$, define\n$$\\bar f(x,y):=\\int_0^1 f(x,y,z)\\,dz,\\qquad \\tilde f:=f-\\bar f.$$ \nLet\n$$Q(\\nabla v):=\\bigl[\\mu(\\nabla_h v+\\nabla_h^{\\top}v)+\\lambda\\,\\mathrm{div}_h v\\,\\mathbb I_2\\bigr]:\\nabla_h v+\\mu|\\partial_z v|^2,$$\n$$\\phi(v,p):=\\mathrm{div}_h\\tilde v+\\frac{1}{\\gamma p}\\Bigl(\\tilde v\\cdot\\nabla_h p-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\Bigr),$$\n$$w=\\nu\\partial_z\\sigma-\\int_0^z \\phi(v,p)(x,y,z',t)\\,dz'.$$\nAssume\n$$v_0\\in H^3(\\mathcal O),\\qquad \\sigma_0\\in H^2(\\mathcal O),\\qquad p_0\\in H^3(\\mathbb T^2),$$\nand for some positive constants $\\underline\\sigma,\\underline p$,\n$$\\sigma_0\\ge \\underline\\sigma,\\qquad p_0\\ge \\underline p,\\qquad \\partial_z v_0|_{z=0,1}=0,\\qquad \\partial_z\\sigma_0|_{z=0,1}=0.$$ \nWhich statement holds for the problem of finding $v,\\sigma$ on $\\mathcal O\\times(0,T)$ and $p$ on $\\mathbb T^2\\times(0,T)$ such that\n\\begin{eqnarray*}\n&&\\partial_t v+(v\\cdot\\nabla_h)v+w\\partial_z v+\\sigma\\nabla_h p=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\mathrm{div}_h v,\\\\\n&&\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma\\bigl(\\phi(v,p)-\\mathrm{div}_h v\\bigr)=\\nu\\sigma\\partial_z^2\\sigma,\\\\\n&&\\partial_t p+\\bar v\\cdot\\nabla_h p+\\gamma p\\,\\mathrm{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)},\n\\end{eqnarray*}\nwith $v,\\sigma,p$ periodic in $(x,y)$, boundary conditions $\\partial_z v|_{z=0,1}=0$ and $\\partial_z\\sigma|_{z=0,1}=0$, and initial data $(v,\\sigma,p)|_{t=0}=(v_0,\\sigma_0,p_0)$?", + "correct_choice": { + "label": "A", + "text": "There exists a time $\\mathcal T_0>0$, depending only on $\\gamma,\\nu,\\mu,\\lambda,\\underline\\sigma,\\underline p$, and $\\|v_0\\|_{H^3}^2+\\|\\sigma_0\\|_{H^2}^2+\\|p_0\\|_{H^3}^2$, such that the above system has a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$ satisfying\n\\begin{eqnarray*}\n&&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\ge 0.5\\,\\underline\\sigma,\\qquad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\ge 0.5\\,\\underline p,\\\\\n&&v\\in C([0,\\mathcal T_0];H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0;H^4(\\mathcal O)),\\\\\n&&\\sigma\\in C([0,\\mathcal T_0];H^1(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\\\\n&&\\partial_t v\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0;H^1(\\mathcal O)),\\\\\n&&p\\in C([0,\\mathcal T_0];H^2(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathbb T^2)),\\qquad \\partial_t p\\in L^2(0,\\mathcal T_0;H^2(\\mathbb T^2)).\n\\end{eqnarray*}" + }, + "choices": [ + { + "label": "B", + "text": "There exists a time $\\mathcal T_0>0$, depending only on $\\gamma,\\nu,\\mu,\\lambda,\\underline\\sigma,\\underline p$, and $\\|v_0\\|_{H^3}^2+\\|\\sigma_0\\|_{H^1}^2+\\|p_0\\|_{H^2}^2$, such that the above system has a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$ satisfying\n\\begin{eqnarray*}\n&&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\ge 0.5\\,\\underline\\sigma,\\qquad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\ge 0.5\\,\\underline p,\\\\\n&&v\\in C([0,\\mathcal T_0];H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0;H^4(\\mathcal O)),\\\\\n&&\\sigma\\in C([0,\\mathcal T_0];L^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^1(\\mathcal O)),\\qquad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0;H^1(\\mathcal O)),\\\\\n&&\\partial_t v\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0;L^2(\\mathcal O)),\\\\\n&&p\\in C([0,\\mathcal T_0];H^1(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0;H^2(\\mathbb T^2)),\\qquad \\partial_t p\\in L^2(0,\\mathcal T_0;H^1(\\mathbb T^2)).\n\\end{eqnarray*}" + }, + { + "label": "C", + "text": "There exists a time $\\mathcal T_0>0$, depending only on $\\gamma,\\nu,\\mu,\\lambda,\\underline\\sigma,\\underline p$, and $\\|v_0\\|_{H^3}^2+\\|\\sigma_0\\|_{H^2}^2+\\|p_0\\|_{H^3}^2$, such that the above system has at least one local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$ satisfying\n\\begin{eqnarray*}\n&&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\ge 0.5\\,\\underline\\sigma,\\qquad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\ge 0.5\\,\\underline p,\\\\\n&&v\\in C([0,\\mathcal T_0];H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0;H^4(\\mathcal O)),\\\\\n&&\\sigma\\in C([0,\\mathcal T_0];H^1(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\\\\n&&\\partial_t v\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0;H^1(\\mathcal O)),\\\\\n&&p\\in C([0,\\mathcal T_0];H^2(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathbb T^2)),\\qquad \\partial_t p\\in L^2(0,\\mathcal T_0;H^2(\\mathbb T^2)).\n\\end{eqnarray*}" + }, + { + "label": "D", + "text": "There exists a universal time $\\mathcal T_0>0$, depending only on $\\gamma,\\nu,\\mu,\\lambda,\\underline\\sigma,$ and $\\underline p$ but independent of $\\|v_0\\|_{H^3},\\|\\sigma_0\\|_{H^2},\\|p_0\\|_{H^3}$, such that the above system has a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$ satisfying\n\\begin{eqnarray*}\n&&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\ge 0.5\\,\\underline\\sigma,\\qquad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\ge 0.5\\,\\underline p,\\\\\n&&v\\in C([0,\\mathcal T_0];H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0;H^4(\\mathcal O)),\\\\\n&&\\sigma\\in C([0,\\mathcal T_0];H^1(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\\\\n&&\\partial_t v\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0;H^1(\\mathcal O)),\\\\\n&&p\\in C([0,\\mathcal T_0];H^2(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathbb T^2)),\\qquad \\partial_t p\\in L^2(0,\\mathcal T_0;H^2(\\mathbb T^2)).\n\\end{eqnarray*}" + }, + { + "label": "E", + "text": "There exists a time $\\mathcal T_0>0$, depending only on $\\gamma,\\nu,\\mu,\\lambda,\\underline\\sigma,\\underline p$, and $\\|v_0\\|_{H^3}^2+\\|\\sigma_0\\|_{H^2}^2+\\|p_0\\|_{H^3}^2$, such that the above system has a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$ satisfying\n\\begin{eqnarray*}\n&&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\ge \\underline\\sigma,\\qquad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\ge \\underline p,\\\\\n&&v\\in C([0,\\mathcal T_0];H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0;H^4(\\mathcal O)),\\\\\n&&\\sigma\\in C([0,\\mathcal T_0];H^1(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\\\\n&&\\partial_t v\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0;H^1(\\mathcal O)),\\\\\n&&p\\in C([0,\\mathcal T_0];H^2(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathbb T^2)),\\qquad \\partial_t p\\in L^2(0,\\mathcal T_0;H^2(\\mathbb T^2)).\n\\end{eqnarray*}" + } + ], + "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": "minimal initial regularity for $\\sigma_0$ and $p_0$", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped uniqueness while keeping the same existence/regularity conclusion", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "dependence of lifespan on Sobolev norms of initial data", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "strict preservation of the full lower bounds instead of the halved bounds", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly state the correct existence theorem. It presents the PDE system and assumptions, but the reader must still distinguish among uniqueness, regularity, and lifespan-dependence claims." + }, + "TAS": { + "score": 1, + "justification": "This is close to a theorem-identification item: the task is essentially to pick the correct local well-posedness statement for the displayed system. The alternatives introduce some variation, but the question still largely tests recognition of the theorem rather than a genuinely independent conclusion." + }, + "GPS": { + "score": 1, + "justification": "Moderate reasoning is needed to assess which statement is most plausible, especially regarding uniqueness, top-order time continuity, and dependence on positivity lower bounds. However, it does not strongly force derivation or synthesis beyond theorem-level matching." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target realistic failure modes: weakened initial regularity, loss of uniqueness, incorrect parameter dependence, and an overly strong regularity upgrade. They are distinct and well aligned with common PDE theorem confusions." + }, + "total_score": 6, + "overall_assessment": "A solid but theorem-recognition-heavy MCQ: little answer leakage and strong distractors, but only moderate generative reasoning and limited avoidance of theorem-restatement structure." + } + }, + { + "id": "2602.20035v1", + "paper_link": "http://arxiv.org/abs/2602.20035v1", + "theorems_cnt": 3, + "theorem": { + "env_name": "thm", + "content": "[Communication-efficient similarity search for signals]\\label{thm:jl_signals_receiver}\nFix $d\\ge 1$ and $n \\ge 8,$ and let $f_1,\\dots,f_n\\colon [0,1]^d\\to[-1,1]$ be continuous.\nA \\emph{sketching protocol} consists of an encoding map\n\\[\n\\Enc\\colon C([0,1]^d)\\to\\mathbb R^k\n\\]\nand a decoding rule\n\\[\n\\Dec\\colon \\mathbb R^k\\times \\mathbb R^k\\to\\mathbb R,\n\\]\nwhere $\\Dec(\\Enc(f_i),\\Enc(f_j))$ is interpreted as an estimate of the squared $L_2$-distance\n\\[\nD(i,j):=\\int_{[0,1]^d}\\abs{f_i(x)-f_j(x)}^2\\,\\,\\mathrm{d} x.\n\\]\n\nThen for every $\\varepsilon\\in(0,1)$ there exist $k\\le C\\,\\frac{\\ln n}{\\varepsilon^2}$ and maps $\\Enc,\\Dec$\nsuch that for all $i,j\\in[n]$,\n\\[\n\\abs{\\Dec(\\Enc(f_i),\\Enc(f_j)) - D(i,j)} \\le \\varepsilon.\n\\]\nMoreover, one may take $\\Dec(u,v)=\\left|u-v\\right|^2$ (i.e.\\ decoding is just squared Euclidean distance $\\left|\\cdot\\right|^2$) and $\\Enc$ to be linear.", + "start_pos": 7434, + "end_pos": 8281, + "label": "thm:jl_signals_receiver" + }, + "ref_dict": {}, + "pre_theorem_intro_text_len": 3469, + "pre_theorem_intro_text": "The theorems of Helly~\\cite{helly1923mengen} and Carath\\'eodory~\\cite{caratheodory1911variabilitatsbereich} are two celebrated results that form part of the foundation of modern convexity and its adjacent areas.\nAt a basic level, these theorems reveal fundamental combinatorial and geometric properties of convex sets in finite-dimensional linear spaces and, in particular, they can be used to characterize the dimension of the ambient space.\nBoth results have given rise to extensive and largely independent lines of research; we refer to~\\cite{barany2022helly, barany2021combinatorial} for recent surveys.\n\nIn a breakthrough paper~\\cite{adiprasito2020theorems}, Adiprasito, B\\'ar\\'any, Mustafa, and Terpai obtained \\emph{no-dimensional} (or \\emph{approximate}) versions of Helly's and Carath\\'eodory's theorems in Euclidean spaces.\nThese results establish a striking connection between combinatorial convexity and approximation theory: instead of exact intersection/containment statements whose quantitative form depends on the dimension,\none gets approximate conclusions controlled by a single natural parameter.\nMotivated by potential applications to algorithmic questions (such as linear sketching and local-to-global guarantees in optimization), it is natural to ask for analogous results beyond Euclidean spaces.\n\nThe goal of this note is to bring attention to this circle of problems.\nWe formulate several open questions and record some straightforward corollaries of the no-dimensional results in Banach spaces.\nAt the same time, we make one additional tweak and ``return the dimension to the picture'': we combine no-dimensional results with a standard substitution trick from Banach space theory,\nwhich allows one to pass from $\\ell_1$-type or $\\ell_\\infty$-type geometry to an appropriately chosen $\\ell_p$-space with desired geometric properties.\n\nWe will treat the needed Banach space facts as a black box.\nFor convenience, we recall all relevant definitions and inequalities in \\Href{Section}{sec:banach_space_likbez}, in order to keep the paper self-contained.\nThen we explain in detail what we mean by no-dimensional versions of Helly's and Carath\\'eodory's theorems, survey recent results on the topic, and formulate several open problems.\nFinally, we describe a standard substitution trick and show that a number of ``local-to-global'' statements become straightforward corollaries of the no-dimensional theory.\n\n\\subsection{Applications of no-dimensional results}\n\nIn many applications \\cite{Garofalakis2013,Wang2016} one stores or transmits large objects\n(signals, images, fields), while the downstream task uses only pairwise similarity queries,\noften measured by the squared $L_2$-distance (equivalently, the $L_2$-energy).\nThe following theorem shows that, for a fixed collection of $n$ signals, one can replace each\nsignal by a short ``fingerprint'' in $\\mathbb R^k$ for sufficiently small $k$, while preserving all\npairwise squared $L_2$-distances up to additive error~$\\varepsilon$.\n\nThe result can be viewed as a weak additive variant of the celebrated Johnson--Lindenstrauss\nflattening lemma \\cite{johnson1984extensions}; see also \\cite{vershynin2018high} for modern\nexpositions. While the classical Johnson--Lindenstrauss lemma is typically proved by a random\nprojection argument, our proof is based on a deterministic construction arising from a\nno-dimensional Carath\\'eodory-type lemma. This provides a simple illustration of the method.", + "context": "The theorems of Helly~\\cite{helly1923mengen} and Carath\\'eodory~\\cite{caratheodory1911variabilitatsbereich} are two celebrated results that form part of the foundation of modern convexity and its adjacent areas.\nAt a basic level, these theorems reveal fundamental combinatorial and geometric properties of convex sets in finite-dimensional linear spaces and, in particular, they can be used to characterize the dimension of the ambient space.\nBoth results have given rise to extensive and largely independent lines of research; we refer to~\\cite{barany2022helly, barany2021combinatorial} for recent surveys.\n\nIn a breakthrough paper~\\cite{adiprasito2020theorems}, Adiprasito, B\\'ar\\'any, Mustafa, and Terpai obtained \\emph{no-dimensional} (or \\emph{approximate}) versions of Helly's and Carath\\'eodory's theorems in Euclidean spaces.\nThese results establish a striking connection between combinatorial convexity and approximation theory: instead of exact intersection/containment statements whose quantitative form depends on the dimension,\none gets approximate conclusions controlled by a single natural parameter.\nMotivated by potential applications to algorithmic questions (such as linear sketching and local-to-global guarantees in optimization), it is natural to ask for analogous results beyond Euclidean spaces.\n\nThe goal of this note is to bring attention to this circle of problems.\nWe formulate several open questions and record some straightforward corollaries of the no-dimensional results in Banach spaces.\nAt the same time, we make one additional tweak and ``return the dimension to the picture'': we combine no-dimensional results with a standard substitution trick from Banach space theory,\nwhich allows one to pass from $\\ell_1$-type or $\\ell_\\infty$-type geometry to an appropriately chosen $\\ell_p$-space with desired geometric properties.\n\nWe will treat the needed Banach space facts as a black box.\nFor convenience, we recall all relevant definitions and inequalities in \\Href{Section}{sec:banach_space_likbez}, in order to keep the paper self-contained.\nThen we explain in detail what we mean by no-dimensional versions of Helly's and Carath\\'eodory's theorems, survey recent results on the topic, and formulate several open problems.\nFinally, we describe a standard substitution trick and show that a number of ``local-to-global'' statements become straightforward corollaries of the no-dimensional theory.\n\nIn many applications \\cite{Garofalakis2013,Wang2016} one stores or transmits large objects\n(signals, images, fields), while the downstream task uses only pairwise similarity queries,\noften measured by the squared $L_2$-distance (equivalently, the $L_2$-energy).\nThe following theorem shows that, for a fixed collection of $n$ signals, one can replace each\nsignal by a short ``fingerprint'' in $\\mathbb R^k$ for sufficiently small $k$, while preserving all\npairwise squared $L_2$-distances up to additive error~$\\varepsilon$.\n\nThe result can be viewed as a weak additive variant of the celebrated Johnson--Lindenstrauss\nflattening lemma \\cite{johnson1984extensions}; see also \\cite{vershynin2018high} for modern\nexpositions. While the classical Johnson--Lindenstrauss lemma is typically proved by a random\nprojection argument, our proof is based on a deterministic construction arising from a\nno-dimensional Carath\\'eodory-type lemma. This provides a simple illustration of the method.", + "full_context": "The theorems of Helly~\\cite{helly1923mengen} and Carath\\'eodory~\\cite{caratheodory1911variabilitatsbereich} are two celebrated results that form part of the foundation of modern convexity and its adjacent areas.\nAt a basic level, these theorems reveal fundamental combinatorial and geometric properties of convex sets in finite-dimensional linear spaces and, in particular, they can be used to characterize the dimension of the ambient space.\nBoth results have given rise to extensive and largely independent lines of research; we refer to~\\cite{barany2022helly, barany2021combinatorial} for recent surveys.\n\nIn a breakthrough paper~\\cite{adiprasito2020theorems}, Adiprasito, B\\'ar\\'any, Mustafa, and Terpai obtained \\emph{no-dimensional} (or \\emph{approximate}) versions of Helly's and Carath\\'eodory's theorems in Euclidean spaces.\nThese results establish a striking connection between combinatorial convexity and approximation theory: instead of exact intersection/containment statements whose quantitative form depends on the dimension,\none gets approximate conclusions controlled by a single natural parameter.\nMotivated by potential applications to algorithmic questions (such as linear sketching and local-to-global guarantees in optimization), it is natural to ask for analogous results beyond Euclidean spaces.\n\nThe goal of this note is to bring attention to this circle of problems.\nWe formulate several open questions and record some straightforward corollaries of the no-dimensional results in Banach spaces.\nAt the same time, we make one additional tweak and ``return the dimension to the picture'': we combine no-dimensional results with a standard substitution trick from Banach space theory,\nwhich allows one to pass from $\\ell_1$-type or $\\ell_\\infty$-type geometry to an appropriately chosen $\\ell_p$-space with desired geometric properties.\n\nWe will treat the needed Banach space facts as a black box.\nFor convenience, we recall all relevant definitions and inequalities in \\Href{Section}{sec:banach_space_likbez}, in order to keep the paper self-contained.\nThen we explain in detail what we mean by no-dimensional versions of Helly's and Carath\\'eodory's theorems, survey recent results on the topic, and formulate several open problems.\nFinally, we describe a standard substitution trick and show that a number of ``local-to-global'' statements become straightforward corollaries of the no-dimensional theory.\n\nIn many applications \\cite{Garofalakis2013,Wang2016} one stores or transmits large objects\n(signals, images, fields), while the downstream task uses only pairwise similarity queries,\noften measured by the squared $L_2$-distance (equivalently, the $L_2$-energy).\nThe following theorem shows that, for a fixed collection of $n$ signals, one can replace each\nsignal by a short ``fingerprint'' in $\\mathbb R^k$ for sufficiently small $k$, while preserving all\npairwise squared $L_2$-distances up to additive error~$\\varepsilon$.\n\nThe result can be viewed as a weak additive variant of the celebrated Johnson--Lindenstrauss\nflattening lemma \\cite{johnson1984extensions}; see also \\cite{vershynin2018high} for modern\nexpositions. While the classical Johnson--Lindenstrauss lemma is typically proved by a random\nprojection argument, our proof is based on a deterministic construction arising from a\nno-dimensional Carath\\'eodory-type lemma. This provides a simple illustration of the method.\n\nThe result can be viewed as a weak additive variant of the celebrated Johnson--Lindenstrauss\nflattening lemma \\cite{johnson1984extensions}; see also \\cite{vershynin2018high} for modern\nexpositions. While the classical Johnson--Lindenstrauss lemma is typically proved by a random\nprojection argument, our proof is based on a deterministic construction arising from a\nno-dimensional Carath\\'eodory-type lemma. This provides a simple illustration of the method.\n\nWe emphasize that this result is much weaker than the Johnson--Lindenstrauss flattening lemma.\nFirst, we only obtain an additive approximation to the squared $L_2$-distance. Second, the range\nof the functions is a fixed interval, so the supremum norm is uniformly bounded (and we exploit\nthis in the proof). Third, the functions are defined on a domain of measure one, which is also\nimportant for our argument since we work on a probability space.\n\n\\begin{thm}[Local-to-global Chebyshev regression]\\label{thm:chebyshev_ball}\nLet $a_1,\\dots,a_m\\in [-1,1]^d$ and $b_1,\\dots,b_m\\in \\R$.\nFix $k\\in[m]$ and parameters $R\\ge 1$ and $r\\ge 0$.\nSuppose that for every subset $J\\subset[m]$ with $\\card{J}=k$ there exists a point $x_J\\in R\\ball{}_{\\ell_1^d}$ such that\n\\[\n\\max_{j\\in J}\\abs{\\iprod{a_j}{x_J}-b_j}\\ \\le\\ r.\n\\]\nThen there exists a point $x\\in eR\\ball{}_{\\ell_1^d}$ such that\n\\[\n\\max_{i\\in[m]}\\abs{\\iprod{a_i}{x}-b_i}\n\\ \\le\\\nr + 21\\,R\\,\\sqrt{\\frac{\\ln d}{k}}.\n\\]\n\\end{thm}\n\n\\begin{thm}[Quantum feasibility from local consistency]\\label{thm:quantum_psd_local_to_global}\nLet $d\\ge 3$ and $k\\in[m]$. Let $A_1,\\dots,A_m\\in\\mathcal{H}_d$ be Hermitian matrices satisfying\n\\[\n\\normsch[\\infty]{A_i}\\le 1 \\qquad\\text{for all } i\\in[m],\n\\]\nand let $b_1,\\dots,b_m\\in\\R$. Fix $t\\ge 0$.\nAssume that for every subset $J\\subset[m]$ with $\\abs{J}=k$, there exists a density matrix\n$\\rho_J\\succeq 0$ with $\\mathrm{Tr}(\\rho_J)=1$ such that\n\\[\n\\abs{\\iprod{A_j}{\\rho_J} - b_j}\n\\le t \\qquad\\text{for all } j\\in J.\n\\]\nThen there exists a density matrix $\\rho\\succeq 0$ with $\\mathrm{Tr}(\\rho)=1$ such that\n\\[\n\\abs{\\iprod{A_i}{\\rho} - b_i}\n\\ \\le\\\nt + 21\\,\\sqrt{\\frac{\\ln d}{k}},\n\\]\nfor all $i\\in[m]$.\n\\end{thm}\n\n\\begin{lem}\\label{lem:Caratheodory_sequence_l_p}\nLet $2\\le p<+\\infty$.\nThen the sequence $\\braces{21\\sqrt{\\frac{p-1}{k}}}_{k\\ge 1}$ is a Carath\\'eodory sequence for $\\ell_p$ and for $S_p$.\nMoreover, whenever the convex hull of a subset $Q$ of the unit ball $\\ball{}_X$ contains the origin,\nthere exists a greedy algorithm that selects points $x_1,\\dots,x_k\\in Q$ successively so that\n\\[\n\\norm{\\frac{x_1+\\cdots+x_k}{k}}\n\\le R_k(X)\n\\]\nat each step.\n\\end{lem}\n\n\\section{Dimension strikes back. Carath\\'eodory-type results and problems}\n\\subsection{Johnson--Lindenstrauss-type sketching}\n\\Href{Theorem}{thm:jl_signals_receiver} follows from the following result \nsince the measure of $[0,1]^d$ is one. We note that similar results were obtained in \\cite{Eskenazis2023}.\n\\begin{thm}[Johnson--Lindenstrauss sketching for bounded functions]\\label{thm:JL_for_functions}\nLet $(\\Omega,\\mu)$ be a probability space and let $f_1,\\dots,f_n\\colon \\Omega\\to[-1,1]$ be functions in $L_2(\\Omega,\\mu)$.\nFix $\\varepsilon\\in(0,1)$. Then there exists an integer\n\\[\nk \\ \\le\\ C\\,\\frac{\\ln n}{\\varepsilon^2}\n\\]\nand vectors $x_1,\\dots,x_n\\in\\R^k$ such that for all $i,j\\in[n]$,\n\\[\n\\enorm{x_i-x_j}^2- \\varepsilon \\ \\le\\ \n\\norm{f_i-f_j}_2^2 \\ \\le\\ \n\\enorm{x_i-x_j}^2 + \\varepsilon .\n\\]\nHere $C>0$ is an absolute constant.\n\\end{thm}\n\nApply \\Href{Lemma}{lem:Caratheodory_sequence_l_p} in the space $\\ell_\\infty^{ {n\\choose 2}}$ (via the substitution from \\Href{Proposition}{prp:l_p_trick_bounds})\nto obtain points $\\omega_1,\\dots,\\omega_k\\in\\Omega$ such that the empirical average\n\\[\n\\frac{1}{k}\\sum_{s=1}^k v(\\omega_s)\n\\]\napproximates $\\int_\\Omega v(\\omega)\\,d\\mu(\\omega)$ in $\\ell_\\infty^{ {n\\choose 2}}$ up to error $\\varepsilon$.\nEquivalently, for all $i0$ an absolute constant independent of $d$, $n$, $\\varepsilon$, and of the range bound on the functions." + }, + { + "label": "C", + "text": "There exist an integer $k\\le C\\,\\frac{\\ln n}{\\varepsilon^2}$ and maps $\\mathrm{Enc}$ and $\\mathrm{Dec}$ such that for every $i,j\\in[n]$,\n\\[\n\\big|\\mathrm{Dec}(\\mathrm{Enc}(f_i),\\mathrm{Enc}(f_j)) - D(i,j)\\big|\\le \\varepsilon.\n\\]" + }, + { + "label": "D", + "text": "There exist an integer $k\\le C\\,\\frac{\\ln n}{\\varepsilon^2}$ and maps $\\mathrm{Enc}$ and $\\mathrm{Dec}$ such that for every $i,j\\in[n]$,\n\\[\n\\big|\\mathrm{Dec}(\\mathrm{Enc}(f_i),\\mathrm{Enc}(f_j)) - D(i,j)\\big|\\le \\varepsilon.\n\\]\nMoreover, one may choose $\\mathrm{Dec}(u,v)=|u-v|^2$ and choose $\\mathrm{Enc}$ to be linear even when the functions $f_1,\\dots,f_n\\colon [0,1]^d\\to\\mathbb R$ are arbitrary continuous functions, with no assumption that their values lie in $[-1,1]$." + }, + { + "label": "E", + "text": "There exist maps $\\mathrm{Enc}$ and $\\mathrm{Dec}$ such that for every $\\varepsilon\\in(0,1)$ and every $i,j\\in[n]$,\n\\[\n\\big|\\mathrm{Dec}(\\mathrm{Enc}(f_i),\\mathrm{Enc}(f_j)) - D(i,j)\\big|\\le \\varepsilon,\n\\]\nwith a sketch dimension satisfying\n\\[\nk\\le C\\,\\frac{\\ln n}{\\varepsilon},\n\\]\nand one may take $\\mathrm{Dec}(u,v)=|u-v|^2$ and $\\mathrm{Enc}$ linear." + } + ], + "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": "hidden dependence on bounded range normalization", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped the additional conclusion specifying Euclidean decoding and linear encoding", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "uniform bounded-range hypothesis $f_i\\in[-1,1]$", + "template_used": "property_confusion" + }, + { + "label": "E", + "sketch_hook_type": "counting_estimate", + "tampered_component": "the $\\varepsilon^{-2}$ dependence in the sketch dimension bound", + "template_used": "boundary_range" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines the setup and asks which conclusion is valid, but it does not explicitly reveal additive vs. relative error, linearity, decoder form, or quantifier dependence. There is no direct answer leakage." + }, + "TAS": { + "score": 2, + "justification": "This is not a bare restatement of a theorem in the stem. The choices differ in meaningful ways: additive vs. relative error, linear vs. arbitrary encoding, instance-dependent vs. uniform construction, and dimension scaling." + }, + "GPS": { + "score": 1, + "justification": "A solver must reason about which theorem-strengthening is justified, so there is some real mathematical pressure. However, because choice C is a weaker statement that is also true if A is true, the item does not cleanly force selection of a unique strongest conclusion unless the stem explicitly asked for the strongest statement." + }, + "DQS": { + "score": 1, + "justification": "B, D, and E are plausible theorem-variant distractors tied to common overclaim errors. But C is not a genuine distractor: it is a weaker true statement implied by A, making the item ambiguous under the wording 'Which statement holds'." + }, + "total_score": 6, + "overall_assessment": "Good on leakage and non-tautology, but flawed as a single-answer MCQ because C is also true. It tests theorem-comparison reasoning, yet the ambiguity weakens its validity." + } + }, + { + "id": "2602.20143v1", + "paper_link": "http://arxiv.org/abs/2602.20143v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm:isopery}\n We have $\\gamma(\\alpha, n) \\leqslant \\frac{(1-\\alpha)(1-(1-\\alpha)^n)}{\\alpha n}$ for all $n\\geqslant 1$ and $\\alpha \\in (0, 1)$.", + "start_pos": 5335, + "end_pos": 5505, + "label": "thm:isopery" + }, + "ref_dict": { + "thm:isopery": "\\begin{theorem}\\label{thm:isopery}\n We have $\\gamma(\\alpha, n) \\le \\frac{(1-\\alpha)(1-(1-\\alpha)^n)}{\\alpha n}$ for all $n\\ge 1$ and $\\alpha \\in (0, 1)$.\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 2390, + "pre_theorem_intro_text": "Let $\\Omega$ be a finite set and let $n\\geqslant 2$. Let $\\mu=\\mu_n$ be the uniform probability measure on $\\Omega^n$. We say that an ordered pair of words $(w, u) \\in \\Omega^n \\times \\Omega^n$ {\\em overlaps} if a final segment of $w$ coincides with an initial segment of $u$. That is, if we denote $w=(w_1, \\ldots, w_n)$, $u=(u_1, \\ldots, u_n)$ then for some $j\\in \\{1, \\ldots, n\\}$ we have $(w_{n-j+1}, \\ldots, w_n) = (u_1, \\ldots, u_j)$. Note that we in particular allow $u = w$. \nWe are interested in the following extremal question: suppose that $A, B\\subset \\Omega^n$ are sets of words such that no two words $w \\in A$ and $u \\in B$ overlap. For what pairs of densities $\\alpha, \\beta \\in (0,1)$ is it possible to have $\\mu(A) \\geqslant \\alpha$ and $\\mu(B) \\geqslant \\beta$?\n\nThere is a related question about non-overlapping codes (also known as `cross-bifix-free' codes) that has been extensively studied in the computer science literature \\cite{bernini2017gray, blackburn2015non, chee2013cross, levenshteindecoding, stanovnik2024search}. In our notation, the question is to determine the size of a largest code $A \\subset \\Omega^n$ such that no two distinct words in $A$ overlap (see \\cite{chee2013cross} for an asymptotically sharp construction). So the question we consider can be thought of as a bipartite variant of this and, to the best of our knowledge, it has not been studied before.\n\nDefine the shift map $s=s_n: \\Omega^n \\rightarrow \\Omega^{n-1}$ by\n\\[\ns(w_1, \\ldots, w_n) = (w_2, \\ldots, w_n).\n\\]\nFor a subset $A \\subset \\Omega^n$ we can define the set of words which do not overlap with $A$ as follows:\n\\[\nU= U(A) = \\Omega^n \\setminus \\bigcup_{j=0}^{n-1} s^j(A) \\times \\Omega^{j}.\n\\]\nIt is easy to see that $U(A)$ is precisely the set of all $u$ such that the pair $(w,u)$ does not overlap for all $w\\in A$. \nLet $\\gamma(\\alpha, n)$ be the largest possible measure of the set $U(A)$ over all $A\\subset \\Omega^n $ of measure $\\alpha$ and all finite sets $\\Omega$. Here we consider the uniform measure on the space $\\Omega^n$. \nNote that if $A \\subset A'$ then we have the inclusion $U(A') \\subset U(A)$. This means that \n$\\gamma(\\alpha, n)$ is a monotone decreasing function in $\\alpha$. For example, it is an easy exercise to show that $\\gamma(\\alpha,2) = \\max\\{ (1-\\alpha)^2, ~1-\\alpha^{1/2} \\}$ holds for any $\\alpha \\in (0,1)$.\n\nOur result is the following estimate.", + "context": "Let $\\Omega$ be a finite set and let $n\\geqslant 2$. Let $\\mu=\\mu_n$ be the uniform probability measure on $\\Omega^n$. We say that an ordered pair of words $(w, u) \\in \\Omega^n \\times \\Omega^n$ {\\em overlaps} if a final segment of $w$ coincides with an initial segment of $u$. That is, if we denote $w=(w_1, \\ldots, w_n)$, $u=(u_1, \\ldots, u_n)$ then for some $j\\in \\{1, \\ldots, n\\}$ we have $(w_{n-j+1}, \\ldots, w_n) = (u_1, \\ldots, u_j)$. Note that we in particular allow $u = w$. \nWe are interested in the following extremal question: suppose that $A, B\\subset \\Omega^n$ are sets of words such that no two words $w \\in A$ and $u \\in B$ overlap. For what pairs of densities $\\alpha, \\beta \\in (0,1)$ is it possible to have $\\mu(A) \\geqslant \\alpha$ and $\\mu(B) \\geqslant \\beta$?\n\nThere is a related question about non-overlapping codes (also known as `cross-bifix-free' codes) that has been extensively studied in the computer science literature \\cite{bernini2017gray, blackburn2015non, chee2013cross, levenshteindecoding, stanovnik2024search}. In our notation, the question is to determine the size of a largest code $A \\subset \\Omega^n$ such that no two distinct words in $A$ overlap (see \\cite{chee2013cross} for an asymptotically sharp construction). So the question we consider can be thought of as a bipartite variant of this and, to the best of our knowledge, it has not been studied before.\n\nDefine the shift map $s=s_n: \\Omega^n \\rightarrow \\Omega^{n-1}$ by\n\\[\ns(w_1, \\ldots, w_n) = (w_2, \\ldots, w_n).\n\\]\nFor a subset $A \\subset \\Omega^n$ we can define the set of words which do not overlap with $A$ as follows:\n\\[\nU= U(A) = \\Omega^n \\setminus \\bigcup_{j=0}^{n-1} s^j(A) \\times \\Omega^{j}.\n\\]\nIt is easy to see that $U(A)$ is precisely the set of all $u$ such that the pair $(w,u)$ does not overlap for all $w\\in A$. \nLet $\\gamma(\\alpha, n)$ be the largest possible measure of the set $U(A)$ over all $A\\subset \\Omega^n $ of measure $\\alpha$ and all finite sets $\\Omega$. Here we consider the uniform measure on the space $\\Omega^n$. \nNote that if $A \\subset A'$ then we have the inclusion $U(A') \\subset U(A)$. This means that \n$\\gamma(\\alpha, n)$ is a monotone decreasing function in $\\alpha$. For example, it is an easy exercise to show that $\\gamma(\\alpha,2) = \\max\\{ (1-\\alpha)^2, ~1-\\alpha^{1/2} \\}$ holds for any $\\alpha \\in (0,1)$.\n\nOur result is the following estimate.", + "full_context": "Let $\\Omega$ be a finite set and let $n\\geqslant 2$. Let $\\mu=\\mu_n$ be the uniform probability measure on $\\Omega^n$. We say that an ordered pair of words $(w, u) \\in \\Omega^n \\times \\Omega^n$ {\\em overlaps} if a final segment of $w$ coincides with an initial segment of $u$. That is, if we denote $w=(w_1, \\ldots, w_n)$, $u=(u_1, \\ldots, u_n)$ then for some $j\\in \\{1, \\ldots, n\\}$ we have $(w_{n-j+1}, \\ldots, w_n) = (u_1, \\ldots, u_j)$. Note that we in particular allow $u = w$. \nWe are interested in the following extremal question: suppose that $A, B\\subset \\Omega^n$ are sets of words such that no two words $w \\in A$ and $u \\in B$ overlap. For what pairs of densities $\\alpha, \\beta \\in (0,1)$ is it possible to have $\\mu(A) \\geqslant \\alpha$ and $\\mu(B) \\geqslant \\beta$?\n\nThere is a related question about non-overlapping codes (also known as `cross-bifix-free' codes) that has been extensively studied in the computer science literature \\cite{bernini2017gray, blackburn2015non, chee2013cross, levenshteindecoding, stanovnik2024search}. In our notation, the question is to determine the size of a largest code $A \\subset \\Omega^n$ such that no two distinct words in $A$ overlap (see \\cite{chee2013cross} for an asymptotically sharp construction). So the question we consider can be thought of as a bipartite variant of this and, to the best of our knowledge, it has not been studied before.\n\nDefine the shift map $s=s_n: \\Omega^n \\rightarrow \\Omega^{n-1}$ by\n\\[\ns(w_1, \\ldots, w_n) = (w_2, \\ldots, w_n).\n\\]\nFor a subset $A \\subset \\Omega^n$ we can define the set of words which do not overlap with $A$ as follows:\n\\[\nU= U(A) = \\Omega^n \\setminus \\bigcup_{j=0}^{n-1} s^j(A) \\times \\Omega^{j}.\n\\]\nIt is easy to see that $U(A)$ is precisely the set of all $u$ such that the pair $(w,u)$ does not overlap for all $w\\in A$. \nLet $\\gamma(\\alpha, n)$ be the largest possible measure of the set $U(A)$ over all $A\\subset \\Omega^n $ of measure $\\alpha$ and all finite sets $\\Omega$. Here we consider the uniform measure on the space $\\Omega^n$. \nNote that if $A \\subset A'$ then we have the inclusion $U(A') \\subset U(A)$. This means that \n$\\gamma(\\alpha, n)$ is a monotone decreasing function in $\\alpha$. For example, it is an easy exercise to show that $\\gamma(\\alpha,2) = \\max\\{ (1-\\alpha)^2, ~1-\\alpha^{1/2} \\}$ holds for any $\\alpha \\in (0,1)$.\n\nOur result is the following estimate.\n\nOur result is the following estimate.\n\nSo if we have a pair of non-overlapping sets $A, B\\subset \\Omega^n$ with densities $\\alpha$ and $\\beta$ then we have $\\alpha\\beta n\\le (1-\\alpha)(1-(1-\\alpha)^n) \\le 1 $.\n\nWe can interpolate between these two examples by taking $A = \\Omega^{n-k} \\times S^k$ for some $1 \\le k \\le n$.\nThen the set $U = U(A)$ is given by\n\\begin{align*}\nU = \\{ (w_1, \\ldots, w_n):~ w_1 \\not\\in S, \\quad \\{w_{j+1}, \\ldots, w_{j+k}\\}\\not\\subset S,~j=0, \\ldots, n-k \\}. \n\\end{align*}\nThe exact formula for $\\mu(U)$ is a bit complicated (it involves generalized Fibonacci numbers, see \\cite{chee2013cross}) but we can use a simple Poisson approximation inequality due to \\cite{arratia1989two} (see also \\cite{godbole1991poisson, godbole1993improved, guibas1980long}) to get a good estimate of the measure of $U$.\nDenote $p = |S|/|\\Omega|$ so that $\\alpha = p^k$. \nLet $Z_1, \\ldots, Z_{n-1} \\sim \\operatorname{Ber}(p)$ be iid Bernoulli random variables. Let $R_{n-1}$ be the length of the longest run of 1-s in the sequence $(Z_1, \\ldots, Z_{n-1})$. The measure of $U$ can then be computed in terms of $R_{n-1}$:\n\\[\n\\mu(U) = (1-p)\\Pr[ R_{n-1} < k].\n\\]\nIndeed, we can view $w_2, \\ldots, w_n \\in \\Omega$ as iid variables uniformly distributed on $\\Omega$ and select $Z_i = 1_{w_{i+1} \\in S}$. \nBy \\cite[Example 3]{arratia1989two} we have the following estimate on this probability:\n\\[\n\\left|\\Pr[ R_{n-1} < k] - e^{-\\lambda}\\right| \\le \\frac{\\lambda (2k+1)}{n-1} + 2 p^k, \\quad \\lambda = p^k ((n-2)(1-p)+1)\n\\]\nLet $k = [n\\alpha \\log(1/\\alpha) ]$. Then for $1/n \\ll \\alpha \\ll 1$ we have $p = \\alpha^{1/k} = e^{-\\frac{\\log(1/\\alpha)}{k}} = 1 - \\frac{1+o(1)}{n\\alpha}$. This gives $\\lambda = 1+o(1)$ and $\\Pr[R_{n-1} < k] = e^{-1} + o(1)$ and so we have\n\\[\n\\mu(U) = \\frac{e^{-1}+o(1)}{\\alpha n},\n\\] \nwhere $o(1)$ tends to zero as $\\min(\\alpha^{-1}, n\\alpha) \\to \\infty$. This matches the bound in Theorem \\ref{thm:isopery} for all $\\alpha \\in (1/n, 1/2)$ up to a constant factor.\n\nWe trivially have $\\beta_{1}=\\alpha_{1}$. Since $A_{j} = s(A_{j+1})$, we have the inclusion $A_{j+1} \\subset \\Omega\\times A_{j} $. We have $B_{j+1} = (B_j \\times \\Omega) \\cup A_{j+1}$ and, in particular $B_j \\times \\Omega \\subset B_{j+1}$. Together these observations imply the following chain of inequalities:\n\\[\n\\beta_n \\ge \\ldots \\ge \\beta_1 = \\alpha_1 \\ge \\ldots \\ge \\alpha_n.\n\\]\nNow let us define sets $D_j$ as follows:\n\\[\nD_j = A_j \\setminus (B_{j-1} \\times \\Omega) = B_j \\setminus (B_{j-1} \\times \\Omega),\n\\]\nwhere for $j=1$ we put $D_1 = A_1=B_1$. \nIn particular, since $B_{j-1} \\times \\Omega \\subset B_{j}$, we can write $B_j$ as a disjoint union $(B_{j-1}\\times \\Omega) \\sqcup D_j$ and\n$\\mu(D_j) = \\beta_j-\\beta_{j-1}$ for all $j=1, \\ldots, n$ (where we set $\\beta_0=0$).\nNote that we can write \n\\begin{align*}\nB_j &= D_j \\sqcup (B_{j-1}\\times \\Omega) = D_j \\sqcup (D_{j-1}\\times \\Omega) \\sqcup (B_{j-2}\\times \\Omega^2) = \\ldots \\\\\n&= D_j \\sqcup (D_{j-1}\\times \\Omega) \\sqcup (D_{j-2}\\times \\Omega^2) \\sqcup \\ldots \\sqcup (D_{1}\\times \\Omega^{j-1}) \\\\\n& = A_j \\cup \\left( (D_{j-1}\\times \\Omega) \\sqcup (D_{j-2}\\times \\Omega^2) \\sqcup \\ldots \\sqcup (D_{1}\\times \\Omega^{j-1}) \\right).\n\\end{align*}\nUsing Observation \\ref{obs1}, we have the following bounds for $i=1, \\ldots, j-1$:\n\\[\n\\mu(A_j \\cap (D_i \\times \\Omega^{j-i})) \\le \\lambda_{j, i}\\cdot \\mu(D_i) \\le \\alpha_{j-i} \\cdot \\mu(D_i)\n\\]\nSo since sets $D_i \\times \\Omega^{j-i}$ are pairwise disjoint, we obtain\n\\[\n\\mu(B_j) = \\mu(A_j) + \\sum_{i=1}^{j-1} \\mu(D_i \\times \\Omega^{j-i} \\setminus A_j) \\ge \\mu(A_j) + \\sum_{i=1}^{j-1} (1- \\alpha_{j- i}) \\mu(D_{i})\n\\]\ngiving the following relation between $\\alpha$-s and $\\beta$-s:\n\\begin{equation}\\label{eq:beta}\n \\beta_j \\ge \\alpha_j + \\sum_{i=1}^{j-1} (1-\\alpha_{j-i}) (\\beta_{i}-\\beta_{i-1}).\n\\end{equation}\nDenote $\\gamma_i = 1-\\beta_i = \\mu(U(A_i))$ and let $\\delta_i = \\alpha_{i-1}-\\alpha_i$ for $i=1, \\ldots, n$ where we put $\\alpha_0=1$ and $\\gamma_0=1$. Then (\\ref{eq:beta}) can be rewritten as follows:\n\\begin{equation}\\label{eq:gamma}\n \\gamma_j \\le \\sum_{i=0}^{j-1} \\gamma_i\\delta_{j-i}.\n\\end{equation}\nWe also have the following information about $\\gamma_i, \\delta_i$: \n\\[\n\\gamma_n \\le \\gamma_{n-1} \\le \\ldots \\le \\gamma_1 \\le \\gamma_0 = 1,\n\\]\n\\[\n\\delta_1+\\ldots+\\delta_n = 1-\\alpha, \\quad \\delta_i \\ge 0, \\quad i=1, \\ldots, n.\n\\]\nWe will use these properties to upper bound $\\gamma_n$. The idea is to use (\\ref{eq:gamma}) to compare $\\gamma$ with a random walk on $\\Z$.\n\nLet $p_i = \\frac{\\delta_i}{1-\\alpha}$ and let $Z$ be the random variable supported on $\\{1, \\ldots, n\\}$ given by the following distribution:\n\\[\n\\Pr[Z = i] = p_i.\n\\]\nLet $Z_1, \\ldots, Z_n$ be i.i.d. copies of $Z$.\n\\begin{obs}\n For every $j=0, \\ldots, n$ we have the following:\n \\begin{equation}\\label{eq:probability}\n \\gamma_j \\le \\sum_{s= 0}^n (1-\\alpha)^s \\Pr[Z_1+\\ldots+Z_s = j]. \n \\end{equation}\n\\end{obs}\n\\begin{proof}\n We prove this by induction on $j$. The base case $j=0$ is clear since only $s=0$ term contributes. Now for $j\\ge 1$ we have by (\\ref{eq:gamma}):\n \\begin{align*}\n \\gamma_j &\\le \\sum_{i=0}^{j-1} \\gamma_i \\delta_{j-i} = \n (1-\\alpha) \\sum_{i=0}^{j-1} \\gamma_i p_{j-i} \\\\\n &\\le (1-\\alpha) \\sum_{i=0}^{j-1} \\sum_{s \\ge 0} (1-\\alpha)^s \\Pr[Z_1+\\ldots+Z_s = i] \\Pr[Z_{s+1} = j-i] \\\\\n &=\\sum_{s\\ge 0} (1-\\alpha)^s\\Pr[Z_1+\\ldots+Z_s = j].\n \\end{align*}\n\\end{proof}\n\n\\begin{cor}\\label{cor:isoperimetry}\n Let $A \\subset \\Omega^n$ be a set of measure $\\alpha \\in (0,1)$. For $w \\in \\Omega^n$ define $f(w)$ to be the number of indices $j\\in \\{0, \\ldots, n-1\\}$ such that $w \\in s^j(A) \\times \\Omega^{j}$.\n Then for every integer $t \\in [1, n/4]$ we have the following level set estimate on $f$:\n \\begin{equation}\n \\mu(\\{w\\in \\Omega^n:~ f(w) \\le t\\}) \\le \\frac{8t}{\\alpha n}.\n \\end{equation}\n\\end{cor}\n\nLet $\\tilde n = [n/2t]$ and $r = n-2t \\tilde n$. By the assumption on $t$ we have $\\tilde n\\ge 1$. Consider a new alphabet $\\tilde \\Omega = \\Omega^{2t}$ and\n let $\\tilde s = \\tilde s_j: \\tilde \\Omega^{j} \\rightarrow \\tilde \\Omega^{j-1}$ denote the shift map defined on words over the alphabet $\\tilde \\Omega$. By identifying $\\tilde \\Omega^j = \\Omega^{2t j}$, we get that $\\tilde s = s^{2t}$. For $i=0, \\ldots, 2t-1$ let \n \\[\n \\tilde A_i = s^{r+i}(A) \\times \\Omega^{i} \\subset \\Omega^{2t \\tilde n} = \\tilde \\Omega^{\\tilde n}.\n \\]\n Note that $\\mu(\\tilde A_i) \\ge \\mu(A) =\\alpha$ for $i=0, \\ldots, 2t-1$. For an arbitrary subset $\\tilde A \\subset \\tilde \\Omega^{\\tilde n}$ we denote $\\tilde U(\\tilde A) = \\tilde \\Omega^{\\tilde n} \\setminus \\bigcup_{j=0}^{\\tilde n-1} \\tilde s^j(\\tilde A) \\times \\tilde \\Omega^{j}$, that is the analogue of $U(A)$ over the new alphabet.\n Let $w \\in \\Omega^n$ and denote $\\tilde w = s^r(w) \\in \\tilde\\Omega^{\\tilde n}$.\n Note that we have $\\tilde w \\not \\in \\tilde U(\\tilde A_i)$ precisely when there exists $j \\in \\{0, \\ldots, \\tilde n-1\\}$ such that $\\tilde w \\in \\tilde s^j(\\tilde A_i) \\times \\tilde \\Omega^j$. The latter is in turn equivalent to $w \\in s^{2t j + i + r}(A)$. It follows that we have\n \\[\n \\#\\{i \\in \\{0, \\ldots, 2t-1\\}: \\tilde w \\not \\in \\tilde U(\\tilde A_i)\\} \\le \\#\\{ i\\in \\{0, \\ldots, n-1\\}: w \\in s^{i}(A)\\times \\Omega^i \\} = f(w)\n \\]\n Thus, if $f(w) \\le t$ then there are at least $2t-t=t$ indices $i \\in \\{0, \\ldots, 2t-1\\}$ such that $\\tilde w \\in \\tilde U(\\tilde A_i)$. So by the union bound and Theorem \\ref{thm:isopery} applied to each $\\tilde U(\\tilde A_i)$ we have \n \\[\n t\\mu(\\{w:~f(w) \\le t\\}) \\le \\sum_{i=0}^{2t-1} \\mu(\\tilde U(\\tilde A_i)) \\le 2t \\frac{(1-\\alpha) (1 - (1-\\alpha)^{\\tilde n})}{\\alpha \\tilde n} \\le \\frac{2t}{\\alpha \\tilde n}.\n \\]\n So recalling that $\\tilde n = [n / 2t]$ we get\n \\[\n \\mu(\\{w:~f(w) \\le t\\}) \\le \\frac{2}{\\alpha [n/2t]} \\le \\frac{8t}{\\alpha n}\n \\]\n provided that $n \\ge 4t$, concluding the proof.\n\\end{proof}", + "post_theorem_intro_text_len": 3064, + "post_theorem_intro_text": "So if we have a pair of non-overlapping sets $A, B\\subset \\Omega^n$ with densities $\\alpha$ and $\\beta$ then we have $\\alpha\\beta n\\leqslant (1-\\alpha)(1-(1-\\alpha)^n) \\leqslant 1 $. \n\nThe proof of Theorem \\ref{thm:isopery} is presented in the next section. The rough idea is to split the set $\\Omega^n \\setminus U(A)$ into several disjoint pieces and use inclusion-exclusion to lower bound the size of each piece. This then gives a certain recursive relationship between various densities associated with $A$ and $U(A)$ and their shifts. This relationship can be interpreted in terms of a certain random walk leading to the desired estimate.\n\nWe close this section by considering some examples essentially matching the upper bound in Theorem \\ref{thm:isopery}. Let $S \\subset \\Omega$ be an arbitrary subset.\nThen for $A = S^n$ one can check that\n\\[\nU(A) = (\\Omega\\setminus S)\\times \\Omega^{n-1}\n\\]\nand so we get\n\\[\n\\mu(U(A)) = 1-\\alpha^{1/n} = \\frac{\\log(1/\\alpha)}{n} + O\\left(\\frac{\\log^2(1/\\alpha)}{n^2}\\right).\n\\]\nThis is a good bound for $\\alpha \\in (1/2, 1)$. \n\nSimilarly, for $A = \\Omega^{n-1} \\times S$ one can check that\n\\[\nU(A) = (\\Omega\\setminus S)^n\n\\]\nso that we get\n\\[\n\\mu(U(A)) = (1-\\alpha)^n.\n\\]\nThis is a good bound for $\\alpha \\in (0, 1/n)$. \n\nWe can interpolate between these two examples by taking $A = \\Omega^{n-k} \\times S^k$ for some $1 \\leqslant k \\leqslant n$.\nThen the set $U = U(A)$ is given by\n\\begin{align*}\nU = \\{ (w_1, \\ldots, w_n):~ w_1 \\not\\in S, \\quad \\{w_{j+1}, \\ldots, w_{j+k}\\}\\not\\subset S,~j=0, \\ldots, n-k \\}. \n\\end{align*}\nThe exact formula for $\\mu(U)$ is a bit complicated (it involves generalized Fibonacci numbers, see \\cite{chee2013cross}) but we can use a simple Poisson approximation inequality due to \\cite{arratia1989two} (see also \\cite{godbole1991poisson, godbole1993improved, guibas1980long}) to get a good estimate of the measure of $U$.\nDenote $p = |S|/|\\Omega|$ so that $\\alpha = p^k$. \nLet $Z_1, \\ldots, Z_{n-1} \\sim \\operatorname{Ber}(p)$ be iid Bernoulli random variables. Let $R_{n-1}$ be the length of the longest run of 1-s in the sequence $(Z_1, \\ldots, Z_{n-1})$. The measure of $U$ can then be computed in terms of $R_{n-1}$:\n\\[\n\\mu(U) = (1-p)\\Pr[ R_{n-1} < k].\n\\]\nIndeed, we can view $w_2, \\ldots, w_n \\in \\Omega$ as iid variables uniformly distributed on $\\Omega$ and select $Z_i = 1_{w_{i+1} \\in S}$. \nBy \\cite[Example 3]{arratia1989two} we have the following estimate on this probability:\n\\[\n\\left|\\Pr[ R_{n-1} < k] - e^{-\\lambda}\\right| \\leqslant \\frac{\\lambda (2k+1)}{n-1} + 2 p^k, \\quad \\lambda = p^k ((n-2)(1-p)+1)\n\\]\nLet $k = [n\\alpha \\log(1/\\alpha) ]$. Then for $1/n \\ll \\alpha \\ll 1$ we have $p = \\alpha^{1/k} = e^{-\\frac{\\log(1/\\alpha)}{k}} = 1 - \\frac{1+o(1)}{n\\alpha}$. This gives $\\lambda = 1+o(1)$ and $\\Pr[R_{n-1} < k] = e^{-1} + o(1)$ and so we have\n\\[\n\\mu(U) = \\frac{e^{-1}+o(1)}{\\alpha n},\n\\] \nwhere $o(1)$ tends to zero as $\\min(\\alpha^{-1}, n\\alpha) \\to \\infty$. This matches the bound in Theorem \\ref{thm:isopery} for all $\\alpha \\in (1/n, 1/2)$ up to a constant factor.", + "sketch": "The post-theorem introduction says: \"The proof of Theorem \\ref{thm:isopery} is presented in the next section. The rough idea is to split the set $\\Omega^n \\setminus U(A)$ into several disjoint pieces and use inclusion-exclusion to lower bound the size of each piece.\" This \"then gives a certain recursive relationship between various densities associated with $A$ and $U(A)$ and their shifts.\" Finally, \"[t]his relationship can be interpreted in terms of a certain random walk leading to the desired estimate.\"", + "expanded_sketch": "The post-theorem introduction says: \"The proof of the main theorem is presented in the next section. The rough idea is to split the set $\\Omega^n \\setminus U(A)$ into several disjoint pieces and use inclusion-exclusion to lower bound the size of each piece.\" This \"then gives a certain recursive relationship between various densities associated with $A$ and $U(A)$ and their shifts.\" Finally, \"[t]his relationship can be interpreted in terms of a certain random walk leading to the desired estimate.\"", + "expanded_theorem": "\\label{thm:isopery}\n We have $\\gamma(\\alpha, n) \\leqslant \\frac{(1-\\alpha)(1-(1-\\alpha)^n)}{\\alpha n}$ for all $n\\geqslant 1$ and $\\alpha \\in (0, 1)$.", + "theorem_type": [ + "Inequality or Bound", + "Universal" + ], + "mcq": { + "question": "For a finite set $\\Omega$ and an integer $n\\ge 1$, let $\\mu$ be the uniform probability measure on $\\Omega^n$. Let $s: \\Omega^n\\to\\Omega^{n-1}$ be the shift map $s(w_1,\\dots,w_n)=(w_2,\\dots,w_n)$, and let $s^j$ denote its $j$-fold iterate (with $s^0$ the identity). For a set $A\\subset \\Omega^n$, define\n\\[\nU(A)=\\Omega^n\\setminus \\bigcup_{j=0}^{n-1} s^j(A)\\times \\Omega^j,\n\\]\nso $U(A)$ is the set of words in $\\Omega^n$ that do not overlap any word of $A$. For $\\alpha\\in(0,1)$, let $\\gamma(\\alpha,n)$ be the largest possible value of $\\mu(U(A))$ over all finite alphabets $\\Omega$ and all sets $A\\subset \\Omega^n$ with $\\mu(A)=\\alpha$. Which statement holds for every $n\\ge 1$ and every $\\alpha\\in(0,1)$?", + "correct_choice": { + "label": "A", + "text": "$\\gamma(\\alpha,n)\\le \\dfrac{(1-\\alpha)\\bigl(1-(1-\\alpha)^n\\bigr)}{\\alpha n}$." + }, + "choices": [ + { + "label": "B", + "text": "$\\gamma(\\alpha,n)\\le \\dfrac{(1-\\alpha)\\bigl(1-(1-\\alpha)^{n-1}\\bigr)}{\\alpha n}$." + }, + { + "label": "C", + "text": "$\\gamma(\\alpha,n)\\le \\dfrac{1-\\alpha}{\\alpha n}$." + }, + { + "label": "D", + "text": "$\\gamma(\\alpha,n)\\le \\dfrac{(1-\\alpha)\\bigl(1-(1-\\alpha)^n\\bigr)}{\\alpha}$." + }, + { + "label": "E", + "text": "$\\gamma(\\alpha,n)= \\dfrac{(1-\\alpha)\\bigl(1-(1-\\alpha)^n\\bigr)}{\\alpha n}$." + } + ], + "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": "number_of_shift_levels", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "counting_estimate", + "tampered_component": "dropped_factor_1-(1-\\alpha)^n", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "random_walk", + "tampered_component": "averaging_by_n", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "upper_bound_not_sharp_for_all_sets", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines the objects carefully but does not reveal the bound itself or give strong clues uniquely pointing to choice A." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recall format: after introducing the definitions, it asks which universal bound holds. This is not a literal restatement inside the stem, but it is largely selecting the exact theorem conclusion from nearby variants." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to compare the candidate inequalities (e.g., weaker true bound, missing factor of n, false equality), but the question mainly rewards recognition or recall of the precise bound rather than generating a conclusion from first principles." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically structured: B is an off-by-one variant, C is a weaker true statement, D omits the averaging-by-n factor, and E incorrectly strengthens an upper bound to equality." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it is somewhat theorem-recall driven and only moderately tests generative mathematical reasoning." + } + }, + { + "id": "2602.20462v1", + "paper_link": "http://arxiv.org/abs/2602.20462v1", + "theorems_cnt": 3, + "theorem": { + "env_name": "thm", + "content": "\\label{thm:mainisoperim}\nFor all $A\\subset \\{0,1\\}^n$ with $|A|\\le \\frac12$,\n\\begin{equation}\\label{eqn:isoperimhalf}\n\\mathbf{E} \\sqrt{h_A} \\ge |A| \\sqrt{\\log_2(1/|A|)}.\n\\end{equation}", + "start_pos": 2578, + "end_pos": 2783, + "label": "thm:mainisoperim" + }, + "ref_dict": { + "eqn:isoperimhalf": "\\begin{equation}\\label{eqn:isoperimhalf}\n\\mathbf{E} \\sqrt{h_A} \\ge |A| \\sqrt{\\log_2(1/|A|)}.\n\\end{equation}", + "thm:mainisoperim": "\\begin{thm}\\label{thm:mainisoperim}\nFor all $A\\subset \\{0,1\\}^n$ with $|A|\\le \\frac12$,\n\\begin{equation}\\label{eqn:isoperimhalf}\n\\mathbf{E} \\sqrt{h_A} \\ge |A| \\sqrt{\\log_2(1/|A|)}.\n\\end{equation}\n\\end{thm}", + "thm:poincare": "\\begin{thm}\\label{thm:poincare} For all $f:\\{0,1\\}^n\\to \\{0,1\\}$,\n\\begin{equation}\\label{eqn:poincareL1}\n\\|\\nabla f\\|_1\\ge \\|f-\\mathbf{E}f\\|_1.\n\\end{equation}\nEquality is achieved when $f=\\mathbf{1}_A$ for a half-cube $A$.\n\\end{thm}", + "eqn:sensitivity": "\\begin{equation}\\label{eqn:sensitivity}\n\\mathbf{E}\\sqrt{s_f} \\ge 1\n\\end{equation}", + "eqn:poincareL1": "\\begin{equation}\\label{eqn:poincareL1}\n\\|\\nabla f\\|_1\\ge \\|f-\\mathbf{E}f\\|_1.\n\\end{equation}", + "cor:kahnpark": "\\begin{thm}\\label{cor:kahnpark}\nLet $(A,B,W)$ be a partition of $\\{0,1\\}^n$ and assume $|A|=\\frac12$. Then\n\\[ |\\nabla(A,B)| + \\sqrt{n}\\,|W|\\ge \\tfrac12. \\]\n\\end{thm}", + "eqn:isoperim2": "\\begin{equation}\\label{eqn:isoperim2}\n\\mathbf{E} \\sqrt{h_A} \\ge B_{w_1}(|A|),\n\\end{equation}" + }, + "pre_theorem_intro_text_len": 357, + "pre_theorem_intro_text": "Let $n\\ge 1$ be an integer.\nFor $A\\subset \\{0,1\\}^n$ and $x\\in A$ let\n$h_A(x)$ denote the number of edges connecting $x$ with $\\{0,1\\}^n\\setminus A$. In other words, $h_A(x)$ is the number of single-bit flips of $x$ that leave $A$.\nIf $x\\not\\in A$, then let $h_A(x)=0$.\nHere $\\mathbf{E}f=2^{-n} \\sum_{x\\in \\{0,1\\}^n} f(x)$ and $|A|=\\mathbf{E} \\mathbf{1}_A$.", + "context": "Let $n\\ge 1$ be an integer.\nFor $A\\subset \\{0,1\\}^n$ and $x\\in A$ let\n$h_A(x)$ denote the number of edges connecting $x$ with $\\{0,1\\}^n\\setminus A$. In other words, $h_A(x)$ is the number of single-bit flips of $x$ that leave $A$.\nIf $x\\not\\in A$, then let $h_A(x)=0$.\nHere $\\mathbf{E}f=2^{-n} \\sum_{x\\in \\{0,1\\}^n} f(x)$ and $|A|=\\mathbf{E} \\mathbf{1}_A$.", + "full_context": "Let $n\\ge 1$ be an integer.\nFor $A\\subset \\{0,1\\}^n$ and $x\\in A$ let\n$h_A(x)$ denote the number of edges connecting $x$ with $\\{0,1\\}^n\\setminus A$. In other words, $h_A(x)$ is the number of single-bit flips of $x$ that leave $A$.\nIf $x\\not\\in A$, then let $h_A(x)=0$.\nHere $\\mathbf{E}f=2^{-n} \\sum_{x\\in \\{0,1\\}^n} f(x)$ and $|A|=\\mathbf{E} \\mathbf{1}_A$.\n\n\\section{Introduction}\nLet $n\\ge 1$ be an integer.\nFor $A\\subset \\{0,1\\}^n$ and $x\\in A$ let\n$h_A(x)$ denote the number of edges connecting $x$ with $\\{0,1\\}^n\\setminus A$. In other words, $h_A(x)$ is the number of single-bit flips of $x$ that leave $A$.\nIf $x\\not\\in A$, then let $h_A(x)=0$.\nHere $\\mathbf{E}f=2^{-n} \\sum_{x\\in \\{0,1\\}^n} f(x)$ and $|A|=\\mathbf{E} \\mathbf{1}_A$.\n\nThis inequality is sharp in two different ways: first, it is an equality when $A$ is a subcube.\nSecond, there is no corresponding dimension-free lower bound when $\\sqrt{h_A}$ is replaced by $h_A^\\beta$ for any $\\beta<\\frac12$, which can be seen by Hamming ball examples.\nThe study of $\\mathbf{E} \\sqrt{h_A}$ was first initiated by Talagrand \\cite{Tal93} who proved lower bounds such as $\\mathbf{E}\\sqrt{h_A}\\ge \\sqrt{2}|A|(1-|A|)$. This was later improved by Bobkov--G\\\"otze \\cite{BG99}, as a consequence of more general results.\n\nSharp lower bounds for $\\mathbf{E} h_A^\\beta$ for all $\\beta\\ge \\frac12$ follow from \\eqref{eqn:isoperimhalf} and H\\\"older's inequality; see e.g. \\cite[Lemma 5.1]{DIR24}.\nSuch bounds were previously proved for $\\beta\\ge 0.50057$ in \\cite{DIR24} and prior to that for $\\beta\\ge \\log_2(3/2)\\approx 0.585$ in \\cite{KP20,BIM23}.\nFor $\\beta=1$ one has Harper's classical isoperimetric inequality \\cite{Har66,Ber67,Hart76}.\nBy the Cauchy-Schwarz inequality, \\eqref{eqn:isoperimhalf} implies the following sharp strengthening of the classical isoperimetric inequality for $|A|\\le \\frac12$,\n\\begin{equation}\\label{eqn:classicalsharpening}\n\\mathbf{E} h_A \\ge \\frac{|A|}{|\\partial A|} |A|\\log_2(1/|A|),\n\\end{equation}\nwhere $\\partial A$ is the support of $h_A$, i.e. the vertex boundary of $A$.\nThis is an equality if $A$ is a subcube. It improves on the classical isoperimetric inequality because of the factor $\\frac{|A|}{|\\partial A|}\\ge 1$. Moreover, it fails when the factor is replaced by $(\\frac{|A|}{|\\partial A|})^{\\gamma}$ for any $\\gamma>1$, again by Hamming ball examples.\nSuch an inequality was observed by Beltran--Ivanisvili--Madrid \\cite[Cor. 1.2]{BIM23} for $\\gamma=\\frac1{\\log_2 (3/2)}-1\\approx 0.709$.\n\n\\subsection*{Applications}\nAs an immediate consequence of Theorem \\ref{thm:mainisoperim} we settle a conjecture of Kahn and Park \\cite[Conjecture 1.3]{KP20} on cube partitions:\n\\begin{thm}\\label{cor:kahnpark}\nLet $(A,B,W)$ be a partition of $\\{0,1\\}^n$ and assume $|A|=\\frac12$. Then\n\\[ |\\nabla(A,B)| + \\sqrt{n}\\,|W|\\ge \\tfrac12. \\]\n\\end{thm}\nHere $|W|=\\mathbf{E} \\mathbf{1}_W$ and the edge boundary measure $|\\nabla(A,B)|$ is defined by $2^{-n} \\# \\{(x,y)\\in \\mathcal{E}\\,:\\,x\\in A, y\\in B\\}$, where $\\mathcal{E}$ is the set of edges in the Hamming cube $\\{0,1\\}^n$.\nTheorem \\ref{cor:kahnpark} follows from Theorem \\ref{thm:mainisoperim} because for $|B\\cup W|=|A|=\\frac12$, we have\n\\[\\tfrac12\\le \\mathbf{E} \\sqrt{h_{B\\cup W}} \\le \\mathbf{E} (h_{B\\cup W}\\mathbf{1}_B) + \\mathbf{E} (\\sqrt{h_{B\\cup W}}\\mathbf{1}_W),\\]\nwhich is no greater than $|\\nabla(A,B)| + \\sqrt{n} |W|$ as required.\n\nOur next application is a sharp $L^1$ Poincar\\'e inequality for Boolean-valued functions.\n\\begin{thm}\\label{thm:poincare} For all $f:\\{0,1\\}^n\\to \\{0,1\\}$,\n\\begin{equation}\\label{eqn:poincareL1}\n\\|\\nabla f\\|_1\\ge \\|f-\\mathbf{E}f\\|_1.\n\\end{equation}\nEquality is achieved when $f=\\mathbf{1}_A$ for a half-cube $A$.\n\\end{thm}\nHere $|\\nabla f(x)|=\\sqrt{\\sum_{i=1}^n (\\tfrac12(f(x)-f(x\\oplus e_i)))^2}$ and $\\|f\\|_1=\\mathbf{E} |f|$. Note that it is still open to determine the sharp constant in this inequality for {\\it real-valued} $f$, but it is known that this constant lies in the interval $(\\frac2{\\pi},\\sqrt{\\frac{2}{\\pi}}]$; see \\cite{BELP08}, \\cite{ILvHV}, also \\cite{haonan1}, \\cite{Esken1} for quantum and vector-valued inequalities. In particular, \\eqref{eqn:poincareL1} shows that the lower bound improves for Boolean-valued functions, which was first observed in \\cite{BIM23}.\nTheorem \\ref{thm:poincare} does not follow directly from Theorem \\ref{thm:mainisoperim}, but from the more technical isoperimetric inequality \\eqref{eqn:isoperim2} proved below.\n\nFinally, isoperimetric inequalities for the Hamming cube are also related to a family of conjectures in information theory concerning noisy channels (Chen--Nair \\cite{CN24}). One of these is the most informative Boolean function conjecture, also known as Courtade--Kumar conjecture \\cite{KC13}.\nThe latter would be implied by a stronger conjecture, known as Hellinger conjecture \\cite{ABCJN17, CGN25}. Roughly speaking, these conjectures concern optimality questions for Boolean functions under standard noisy-channel models on $\\{0,1\\}^n$, where each bit is independently perturbed.\nIn the low-noise limit, the Hellinger conjecture would in particular imply that\n\\begin{equation}\\label{eqn:sensitivity}\n\\mathbf{E}\\sqrt{s_f} \\ge 1\n\\end{equation}\nholds for balanced Boolean functions $f:\\{0,1\\}^n\\to \\{0,1\\}$ (i.e. $\\mathbf{E}f=\\tfrac12$), where $s_f(x)$ is the sensitivity of $f$ at $x$, which is defined as the number of single-bit flips of $x$ that change the value of $f(x)$, i.e. $s_f(x)=h_A(x)+h_{A^c}(x)$.\nThis implication is well-known, but for convenience of the reader we provide details in \\S \\ref{sec:it}.\nSince $h_A$ and $h_{A^c}$ have disjoint support, $\\sqrt{s_f}=\\sqrt{h_A}+\\sqrt{h_{A^c}}$, so Theorem \\ref{thm:mainisoperim} with $|A|=\\frac12$ immediately implies \\eqref{eqn:sensitivity}.\n\nFor $0\\le x\\le y\\le 1$ and $B:[0,1]\\to [0,\\infty)$ let\n\\begin{equation}\\label{eqn:Gdef}\nG_1[B](x,y) = \\sqrt{(y-x)^{2}+B(y)^{2}} + B(x) - 2 B(\\tfrac{x+y}2),\n\\end{equation}\n\\[ G_2[B](x,y) = y-x + (\\sqrt{2}-1) B(y) + B(x) - 2 B(\\tfrac{x+y}2),\\]\nand $G = \\max(G_1, G_2).$\nKahn and Park \\cite{KP20} proved that if $B$ satisfies $B(0)=B(1)=0$ and the two-point inequality\n\\[G[B](x,y)\\ge 0\\]\nholds for all $0\\le x\\le y\\le 1$, then $\\mathbf{E} \\sqrt{h_A}\\ge B(|A|)$\nfor all $A\\subset \\{0,1\\}^n$ and $n\\ge 1$.\n\nThis reduces the proof of Theorem \\ref{thm:mainisoperim} to finding a Bellman function $B$ and verifying the two-point inequality.\nWe use the following refinement of the Bellman function used in \\cite{DIR24}.\nFor $x\\in (0,1]$ define\n\\[ L(x) = x \\sqrt{\\log_2(1/x)} \\]\nand $L(0)=0$.\nLet $Q(x)$ be the unique cubic interpolating polynomial such that $Q(0)=Q(1)=0$, $Q(\\frac12)=\\frac12$ and $Q(\\frac14)=2^{-3/2}$, i.e.\n\\[ Q(x) = \\tfrac23 x (1-x)(2^{5/2}-3+4(3-2^{3/2}) x). \\]\nThe idea of using the polynomial $Q$ goes back to Beltran--Ivanisvili--Madrid \\cite{BIM23}.\nLet $I(x)$ denote the Gaussian isoperimetric profile, i.e. the unique function on $[0,1]$ such that\n$I(0)=I(1)=0, I\\cdot I''=-1$.\nFor a parameter $w\\in (\\frac12,1]$ let\n\\[ {\\mathrm{J}}_w(x) = \\tfrac12 I(\\tfrac1{2w})^{-1} I(\\tfrac{1-x}{w}). \\]\nNote that ${\\mathrm{J}}_w(\\frac12)=\\frac12$ and ${\\mathrm{J}}_w(1)=0$.\nThen define\n\\begin{equation}\\label{eqn:bbdef}\nB_w(x) = \\left\\{ \\begin{array}{ll}\nL(x) & \\text{for } x\\in [0, \\frac14],\\\\\nQ(x)\n& \\text{for }x\\in [\\frac14, \\frac12],\\\\\n{\\mathrm{J}}_w(x) & \\text{for }x\\in [\\frac12, 1].\n\\end{array}\\right.\n\\end{equation}\nFor some motivation on the choice of such functions, see \\cite[\\S 2]{DIR24}.\nIn the following we shall fix\n\\begin{equation}\\label{eqn:w1def}\nw=w_1=\\tfrac{29}{32}\n\\end{equation}\nand write ${\\mathrm{J}}={\\mathrm{J}}_{w_1}$.\nNote that since $B_w(x)\\ge L(x)$ for $x\\in [0,\\frac12]$ (see \\cite[Lemma 5.9 (2)]{DIR24}), in order to prove Theorem \\ref{thm:mainisoperim} it suffices to prove that there exists $w$ such that\n\\begin{equation}\\label{eqn:maintwopt}\nG[B_w](x,y)\\ge 0\n\\end{equation}\nfor all $0\\le x\\le y\\le 1$.\nAs a consequence of this argument we obtain the isoperimetric inequality\n\\begin{equation}\\label{eqn:isoperim2}\n\\mathbf{E} \\sqrt{h_A} \\ge B_{w_1}(|A|),\n\\end{equation}\nvalid for all $A\\subset \\{0,1\\}^n$.", + "post_theorem_intro_text_len": 5932, + "post_theorem_intro_text": "This inequality is sharp in two different ways: first, it is an equality when $A$ is a subcube.\nSecond, there is no corresponding dimension-free lower bound when $\\sqrt{h_A}$ is replaced by $h_A^\\beta$ for any $\\beta<\\frac12$, which can be seen by Hamming ball examples.\nThe study of $\\mathbf{E} \\sqrt{h_A}$ was first initiated by Talagrand \\cite{Tal93} who proved lower bounds such as $\\mathbf{E}\\sqrt{h_A}\\ge \\sqrt{2}|A|(1-|A|)$. This was later improved by Bobkov--G\\\"otze \\cite{BG99}, as a consequence of more general results.\n\nSharp lower bounds for $\\mathbf{E} h_A^\\beta$ for all $\\beta\\ge \\frac12$ follow from \\eqref{eqn:isoperimhalf} and H\\\"older's inequality; see e.g. \\cite[Lemma 5.1]{DIR24}.\nSuch bounds were previously proved for $\\beta\\ge 0.50057$ in \\cite{DIR24} and prior to that for $\\beta\\ge \\log_2(3/2)\\approx 0.585$ in \\cite{KP20,BIM23}.\nFor $\\beta=1$ one has Harper's classical isoperimetric inequality \\cite{Har66,Ber67,Hart76}.\nBy the Cauchy-Schwarz inequality, \\eqref{eqn:isoperimhalf} implies the following sharp strengthening of the classical isoperimetric inequality for $|A|\\le \\frac12$,\n\\begin{equation}\\label{eqn:classicalsharpening}\n\\mathbf{E} h_A \\ge \\frac{|A|}{|\\partial A|} |A|\\log_2(1/|A|),\n\\end{equation}\nwhere $\\partial A$ is the support of $h_A$, i.e. the vertex boundary of $A$.\nThis is an equality if $A$ is a subcube. It improves on the classical isoperimetric inequality because of the factor $\\frac{|A|}{|\\partial A|}\\ge 1$. Moreover, it fails when the factor is replaced by $(\\frac{|A|}{|\\partial A|})^{\\gamma}$ for any $\\gamma>1$, again by Hamming ball examples.\nSuch an inequality was observed by Beltran--Ivanisvili--Madrid \\cite[Cor. 1.2]{BIM23} for $\\gamma=\\frac1{\\log_2 (3/2)}-1\\approx 0.709$.\n\nThe proof of Theorem \\ref{thm:mainisoperim} relies on a crucial induction argument of Kahn and Park \\cite{KP20}, which reduced proving lower bounds for $\\mathbf{E} h_A^\\beta$ to finding a certain Bellman function and verifying a two-point inequality.\nOur Bellman function is a further refinement of the function used in \\cite{DIR24}, which in turn builds on important work by Beltran--Ivanisvili--Madrid \\cite{BIM23}.\nIn the proof of the two-point inequality we draw from estimates and tools introduced in \\cite{DIR24}.\n\n\\subsection*{Applications}\nAs an immediate consequence of Theorem \\ref{thm:mainisoperim} we settle a conjecture of Kahn and Park \\cite[Conjecture 1.3]{KP20} on cube partitions:\n\\begin{thm}\\label{cor:kahnpark}\nLet $(A,B,W)$ be a partition of $\\{0,1\\}^n$ and assume $|A|=\\frac12$. Then\n\\[ |\\nabla(A,B)| + \\sqrt{n}\\,|W|\\ge \\tfrac12. \\]\n\\end{thm}\nHere $|W|=\\mathbf{E} \\mathbf{1}_W$ and the edge boundary measure $|\\nabla(A,B)|$ is defined by $2^{-n} \\# \\{(x,y)\\in \\mathcal{E}\\,:\\,x\\in A, y\\in B\\}$, where $\\mathcal{E}$ is the set of edges in the Hamming cube $\\{0,1\\}^n$.\nTheorem \\ref{cor:kahnpark} follows from Theorem \\ref{thm:mainisoperim} because for $|B\\cup W|=|A|=\\frac12$, we have\n\\[\\tfrac12\\le \\mathbf{E} \\sqrt{h_{B\\cup W}} \\le \\mathbf{E} (h_{B\\cup W}\\mathbf{1}_B) + \\mathbf{E} (\\sqrt{h_{B\\cup W}}\\mathbf{1}_W),\\]\nwhich is no greater than $|\\nabla(A,B)| + \\sqrt{n} |W|$ as required.\n\nOur next application is a sharp $L^1$ Poincar\\'e inequality for Boolean-valued functions.\n\\begin{thm}\\label{thm:poincare} For all $f:\\{0,1\\}^n\\to \\{0,1\\}$,\n\\begin{equation}\\label{eqn:poincareL1}\n\\|\\nabla f\\|_1\\ge \\|f-\\mathbf{E}f\\|_1.\n\\end{equation}\nEquality is achieved when $f=\\mathbf{1}_A$ for a half-cube $A$.\n\\end{thm}\nHere $|\\nabla f(x)|=\\sqrt{\\sum_{i=1}^n (\\tfrac12(f(x)-f(x\\oplus e_i)))^2}$ and $\\|f\\|_1=\\mathbf{E} |f|$. Note that it is still open to determine the sharp constant in this inequality for {\\it real-valued} $f$, but it is known that this constant lies in the interval $(\\frac2{\\pi},\\sqrt{\\frac{2}{\\pi}}]$; see \\cite{BELP08}, \\cite{ILvHV}, also \\cite{haonan1}, \\cite{Esken1} for quantum and vector-valued inequalities. In particular, \\eqref{eqn:poincareL1} shows that the lower bound improves for Boolean-valued functions, which was first observed in \\cite{BIM23}.\nTheorem \\ref{thm:poincare} does not follow directly from Theorem \\ref{thm:mainisoperim}, but from the more technical isoperimetric inequality \\eqref{eqn:isoperim2} proved below.\n\nFinally, isoperimetric inequalities for the Hamming cube are also related to a family of conjectures in information theory concerning noisy channels (Chen--Nair \\cite{CN24}). One of these is the most informative Boolean function conjecture, also known as Courtade--Kumar conjecture \\cite{KC13}.\nThe latter would be implied by a stronger conjecture, known as Hellinger conjecture \\cite{ABCJN17, CGN25}. Roughly speaking, these conjectures concern optimality questions for Boolean functions under standard noisy-channel models on $\\{0,1\\}^n$, where each bit is independently perturbed.\nIn the low-noise limit, the Hellinger conjecture would in particular imply that\n\\begin{equation}\\label{eqn:sensitivity}\n\\mathbf{E}\\sqrt{s_f} \\ge 1\n\\end{equation}\nholds for balanced Boolean functions $f:\\{0,1\\}^n\\to \\{0,1\\}$ (i.e. $\\mathbf{E}f=\\tfrac12$), where $s_f(x)$ is the sensitivity of $f$ at $x$, which is defined as the number of single-bit flips of $x$ that change the value of $f(x)$, i.e. $s_f(x)=h_A(x)+h_{A^c}(x)$.\nThis implication is well-known, but for convenience of the reader we provide details in \\S \\ref{sec:it}.\nSince $h_A$ and $h_{A^c}$ have disjoint support, $\\sqrt{s_f}=\\sqrt{h_A}+\\sqrt{h_{A^c}}$, so Theorem \\ref{thm:mainisoperim} with $|A|=\\frac12$ immediately implies \\eqref{eqn:sensitivity}.\n\n\\subsection*{Acknowledgments} P.D., P.I. and J.R. thank the American Institute of Mathematics (AIM) for funding our SQuaRE project. The authors were also supported in part by grants from the National Science Foundation CAREER-DMS-2152401 (P.I.), DMS-2154835 (J.R.) and by the Simons Foundation\nSFI-MPS-TSM-00013943 (P.D.), SFI-MPS-TSM-00014075 (J.R.), Simons Fellowship, and Humboldt Research Fellowship (P.I.).", + "sketch": "The post-theorem introduction states that the proof of Theorem~\\ref{thm:mainisoperim} \"relies on a crucial induction argument of Kahn and Park \\cite{KP20}, which reduced proving lower bounds for $\\mathbf{E} h_A^\\beta$ to finding a certain Bellman function and verifying a two-point inequality.\" It further says: \"Our Bellman function is a further refinement of the function used in \\cite{DIR24}, which in turn builds on important work by Beltran--Ivanisvili--Madrid \\cite{BIM23}.\" Finally, it notes that \"[i]n the proof of the two-point inequality we draw from estimates and tools introduced in \\cite{DIR24}.\"", + "expanded_sketch": "The post-theorem introduction states that the proof of Theorem~\\ref{thm:mainisoperim} “relies on a crucial induction argument of Kahn and Park \\cite{KP20}, which reduced proving lower bounds for $\\mathbf{E} h_A^\\beta$ to finding a certain Bellman function and verifying a two-point inequality.” It further says: “Our Bellman function is a further refinement of the function used in \\cite{DIR24}, which in turn builds on important work by Beltran--Ivanisvili--Madrid \\cite{BIM23}.” Finally, it notes that “[i]n the proof of the two-point inequality we draw from estimates and tools introduced in \\cite{DIR24}.”", + "expanded_theorem": "\\label{thm:mainisoperim}\nFor all $A\\subset \\{0,1\\}^n$ with $|A|\\le \\frac12$,\n\\begin{equation}\\label{eqn:isoperimhalf}\n\\mathbf{E} \\sqrt{h_A} \\ge |A| \\sqrt{\\log_2(1/|A|)}.\n\\end{equation}", + "theorem_type": [ + "Inequality or Bound", + "Universal" + ], + "mcq": { + "question": "Let n\\ge 1 be an integer, and let A\\subset \\{0,1\\}^n satisfy |A|\\le \\tfrac12, where\n\\[\\mathbf{E}f=2^{-n}\\sum_{x\\in\\{0,1\\}^n} f(x),\\qquad |A|=\\mathbf{E}\\mathbf{1}_A=2^{-n}\\#A.\\]\nFor x\\in A, let h_A(x) be the number of single-bit flips of x that leave A, equivalently the number of edges from x to \\{0,1\\}^n\\setminus A; and for x\\notin A set h_A(x)=0. Which statement holds for every such set A?", + "correct_choice": { + "label": "A", + "text": "\\[\\mathbf{E}\\sqrt{h_A}\\ge |A|\\sqrt{\\log_2(1/|A|)}.\\]" + }, + "choices": [ + { + "label": "B", + "text": "\\[\\mathbf{E}\\sqrt{h_A}\\ge |A|\\sqrt{\\log_2\\!\\bigl(1/(2|A|)\\bigr)}.\\]" + }, + { + "label": "C", + "text": "\\[\\mathbf{E}\\sqrt{h_A}\\ge |A|(1-|A|).\\]" + }, + { + "label": "D", + "text": "\\[\\mathbf{E}\\sqrt{h_A}\\ge |A|\\sqrt{\\log_2(1/|A|)}\\qquad\\text{for all }A\\subset\\{0,1\\}^n.\\]" + }, + { + "label": "E", + "text": "\\[\\mathbf{E}h_A\\ge |A|\\sqrt{\\log_2(1/|A|)}.\\]" + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "sharp Bellman profile normalization inside the logarithm", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "replaced the sharp logarithmic lower bound by a weaker elementary bound", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "hypothesis restriction |A|\\le 1/2", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "moment/order of the boundary term, replacing \\sqrt{h_A} by h_A", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines the objects carefully but does not reveal the sharp inequality or otherwise point directly to choice A. There is no explicit answer cue beyond the general setup." + }, + "TAS": { + "score": 1, + "justification": "The correct option is essentially the exact target inequality, so the item does have a theorem-recognition flavor. However, it is not a pure restatement because the alternatives vary the hypothesis, normalization, and strength of conclusion in meaningful ways." + }, + "GPS": { + "score": 2, + "justification": "To select A, a student must distinguish a sharp universal bound from a weaker true statement (C), a close but incorrect logarithmic normalization (B), an overgeneralization beyond |A|<=1/2 (D), and a moment mismatch (E). This creates genuine pressure to reason about sharpness and scope." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target realistic failure modes: weakening the bound, changing the domain of validity, altering the logarithmic term, or confusing E[sqrt(h_A)] with E[h_A]. They are distinct and nontrivial." + }, + "total_score": 7, + "overall_assessment": "A strong MCQ with no answer leakage and high-quality distractors; its main limitation is that the correct choice is still close to a direct statement of the underlying theorem, so it tests theorem discrimination more than full derivation." + } + }, + { + "id": "2602.20938v1", + "paper_link": "http://arxiv.org/abs/2602.20938v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "\\label{t1} Let $\\Omega \\subset \\mathbb{R}^2$ be a smooth bounded convex domain such that $\\mathrm{inr}(\\Omega)=r$ and $D(\\Omega)=D$. Then if $D$ is large enough, there exists $\\lambda>0$ for which problem \\eqref{con} admits a pattern.", + "start_pos": 3111, + "end_pos": 3375, + "label": "t1" + }, + "ref_dict": { + "t1": "\\begin{theorem}\\label{t1} Let $\\Omega \\subset \\mathbb{R}^2$ be a smooth bounded convex domain such that $\\mathrm{inr}(\\Omega)=r$ and $D(\\Omega)=D$. Then if $D$ is large enough, there exists $\\lambda>0$ for which problem \\eqref{con} admits a pattern. \\end{theorem}", + "fig1": "\\begin{figure}[ht]\n\\centering\n\\def\\svgwidth{1\\linewidth}\n\\input{fig3.pdf_tex}\n\\caption{Convex domain\n$\\Omega \\subset \\mathbb{R}^2$ with inradius $r$ and diameter $D$.}\n\\label{fig1}\n\\end{figure}", + "tmi": "\\begin{theorem}\\label{tmi}\nLet $\\Omega \\subset \\mathbb{R}^2$ be a bounded convex domain and again let\n$\nD := D(\\Omega).\n$\nThen there exists a constant $c>0$ depending only on the dimension such that\n\\[\n\\mu_1(\\Omega) \\ge \\frac{c}{D}.\n\\]\n\\end{theorem}", + "con": "\\begin{equation}\\label{con}\n\\begin{cases}\nu_t - \\Delta u = 0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+, \\\\\n\\partial_{\\nu} u = \\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+, \\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\end{equation}" + }, + "pre_theorem_intro_text_len": 1183, + "pre_theorem_intro_text": "Consider the following problem with nonlinear Neumann boundary condition\n\\begin{equation}\\label{con}\n\\begin{cases}\nu_t - \\Delta u = 0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+, \\\\\n\\partial_{\\nu} u = \\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+, \\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\end{equation}\nwhere $\\Omega \\subset \\mathbb{R}^2$ is a smooth, bounded, and convex domain, \n$\\nu$ denotes the outward unit normal vector to $\\partial\\Omega$, \n$\\lambda$ is a positive parameter, and $g$ is a bistable nonlinearity, \nhere taken in its prototypical form $g(u)=u-u^3$.\n\nA non-constant stationary solution of \\eqref{con} that is linearly stable will be called a {\\it pattern}.\n\nIn order to state our main result, we recall that the \\emph{inradius} of a set \n$\\Omega \\subset \\mathbb{R}^2$ is defined by\n\\[\n\\mathrm{inr}(\\Omega) := \\sup \\left\\{ r > 0 \\; ; \\; \\exists\\, x \\in \\Omega \n\\ \\text{such that} \\ B_r(x) \\subset \\Omega \\right\\},\n\\]\nwhere $B_r(x)$ denotes the open ball of radius $r$ centered at $x$.\n\nThe \\emph{diameter} of $\\Omega$ is defined by\n\\[\nD(\\Omega) := \\sup \\left\\{ |x-y| \\; ; \\; x,y \\in \\Omega \\right\\}.\n\\]\n\nThe main result of this work is stated as follows.", + "context": "Consider the following problem with nonlinear Neumann boundary condition\n\\begin{equation}\\label{con}\n\\begin{cases}\nu_t - \\Delta u = 0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+, \\\\\n\\partial_{\\nu} u = \\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+, \\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\end{equation}\nwhere $\\Omega \\subset \\mathbb{R}^2$ is a smooth, bounded, and convex domain, \n$\\nu$ denotes the outward unit normal vector to $\\partial\\Omega$, \n$\\lambda$ is a positive parameter, and $g$ is a bistable nonlinearity, \nhere taken in its prototypical form $g(u)=u-u^3$.\n\nA non-constant stationary solution of \\eqref{con} that is linearly stable will be called a {\\it pattern}.\n\nIn order to state our main result, we recall that the \\emph{inradius} of a set \n$\\Omega \\subset \\mathbb{R}^2$ is defined by\n\\[\n\\mathrm{inr}(\\Omega) := \\sup \\left\\{ r > 0 \\; ; \\; \\exists\\, x \\in \\Omega \n\\ \\text{such that} \\ B_r(x) \\subset \\Omega \\right\\},\n\\]\nwhere $B_r(x)$ denotes the open ball of radius $r$ centered at $x$.\n\nThe \\emph{diameter} of $\\Omega$ is defined by\n\\[\nD(\\Omega) := \\sup \\left\\{ |x-y| \\; ; \\; x,y \\in \\Omega \\right\\}.\n\\]\n\nThe main result of this work is stated as follows.", + "full_context": "Consider the following problem with nonlinear Neumann boundary condition\n\\begin{equation}\\label{con}\n\\begin{cases}\nu_t - \\Delta u = 0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+, \\\\\n\\partial_{\\nu} u = \\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+, \\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\end{equation}\nwhere $\\Omega \\subset \\mathbb{R}^2$ is a smooth, bounded, and convex domain, \n$\\nu$ denotes the outward unit normal vector to $\\partial\\Omega$, \n$\\lambda$ is a positive parameter, and $g$ is a bistable nonlinearity, \nhere taken in its prototypical form $g(u)=u-u^3$.\n\nA non-constant stationary solution of \\eqref{con} that is linearly stable will be called a {\\it pattern}.\n\nIn order to state our main result, we recall that the \\emph{inradius} of a set \n$\\Omega \\subset \\mathbb{R}^2$ is defined by\n\\[\n\\mathrm{inr}(\\Omega) := \\sup \\left\\{ r > 0 \\; ; \\; \\exists\\, x \\in \\Omega \n\\ \\text{such that} \\ B_r(x) \\subset \\Omega \\right\\},\n\\]\nwhere $B_r(x)$ denotes the open ball of radius $r$ centered at $x$.\n\nThe \\emph{diameter} of $\\Omega$ is defined by\n\\[\nD(\\Omega) := \\sup \\left\\{ |x-y| \\; ; \\; x,y \\in \\Omega \\right\\}.\n\\]\n\nThe main result of this work is stated as follows.\n\n\\abstract{We consider the heat equation in a smooth bounded convex domain \n$\\Omega \\subset \\mathbb{R}^2$ with nonlinear Neumann boundary condition \n$\\partial_\\nu u = \\lambda (u - u^3)$. \nStable non-constant stationary solutions do not exist when $\\Omega$ is a ball. \nWe show that this behavior is not a consequence of convexity alone. \nMore precisely, if the inradius of $\\Omega$ is fixed and its diameter is sufficiently large, then there exists $\\lambda>0$ for which the problem admits such a solution. \nThe result reveals a geometric mechanism for the emergence of stable non-constant stationary solutions in elongated convex domains.}\n\nConsider the following problem with nonlinear Neumann boundary condition\n\\begin{equation}\\label{con}\n\\begin{cases}\nu_t - \\Delta u = 0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+, \\\\\n\\partial_{\\nu} u = \\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+, \\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\end{equation}\nwhere $\\Omega \\subset \\mathbb{R}^2$ is a smooth, bounded, and convex domain, \n$\\nu$ denotes the outward unit normal vector to $\\partial\\Omega$, \n$\\lambda$ is a positive parameter, and $g$ is a bistable nonlinearity, \nhere taken in its prototypical form $g(u)=u-u^3$.\n\nThe main result of this work is stated as follows.\n\nTheorem \\ref{t1} reveals a purely geometric mechanism for pattern\nformation. While the inradius remains fixed, increasing the diameter\nforces the emergence of patterns for appropriate values\nof $\\lambda$. In other words, sufficiently elongated convex domains (see Figure \\ref{fig1})\nnecessarily admit patterns. This phenomenon is independent of symmetry assumptions and depends\nonly on the convexity and aspect ratio of the domain.\n\n\\begin{lemma}\\label{LFU}\nLet $\\Omega \\subset \\mathbb{R}^n$ be a bounded convex domain,\nand denote $D=D(\\Omega)$.\nThen there exists a constant $C=C(n)>0$ such that,\nfor every $u\\in H^1(\\Omega)$,\n\\[\n\\|u-u_{\\partial\\Omega}\\|_{L^2(\\partial\\Omega)}\n\\le\nC D^{1/2}\\|\\nabla u\\|_{L^2(\\Omega)},\n\\]\nwhere\n\\[\nu_{\\partial\\Omega}\n=\n\\frac{1}{|\\partial\\Omega|}\n\\int_{\\partial\\Omega} u\\ d\\sigma.\n\\]\n\\end{lemma}\n\n\\begin{theorem}\\label{tmi}\nLet $\\Omega \\subset \\mathbb{R}^2$ be a bounded convex domain and again let\n$\nD := D(\\Omega).\n$\nThen there exists a constant $c>0$ depending only on the dimension such that\n\\[\n\\mu_1(\\Omega) \\ge \\frac{c}{D}.\n\\]\n\\end{theorem}\n\n\\begin{lemma}\\label{le1}\nLet $\\Omega \\subset \\mathbb{R}^n$ ($n \\ge 2$) be a smooth bounded domain,\n$\\Omega_l$ and $\\Omega_r$ two disjoint subdomains of $\\Omega$\nwith smooth boundaries, and\n$S_j = \\partial\\Omega \\cap \\partial\\Omega_j$,\n$\\mathcal{H}^{n-1}(S_j) > 0$ $(j=l,r)$.\nFor $p > n$, we define the set\n\\begin{equation}\\label{L}\n\\Lambda(\\Omega)\n=\n\\left\\{\\begin{aligned}\nv \\in W^{1,p}(\\Omega) :\\quad\n&-1 \\le v(x) \\le 1,\\; x \\in \\overline{\\Omega},\\\\\n&\\int_{S_l} v \\, d\\sigma < 0,\\qquad\n\\int_{S_r} v \\, d\\sigma > 0,\\\\\n&E(v) < \\varepsilon_0 - G(1)\\mathcal{H}^{n-1}(\\partial\\Omega)\n\\end{aligned}\\right\\},\n\\end{equation}\nwhere\n\\[\n\\varepsilon_0\n=\nG(1)\n\\min\n\\left\\{\\begin{aligned}\n&\\mathcal{H}^{n-1}(S_l)\\min\\{\\lambda,\\,\\mu_1(\\Omega_l)\\},\\\\\n&\\mathcal{H}^{n-1}(S_r)\\min\\{\\lambda,\\,\\mu_1(\\Omega_r)\\}\n\\end{aligned}\\right\\},\n\\]\nand $\\mu_1(\\Omega_j)$ is the first positive eigenvalue of the Steklov\nproblem \\ref{ST} defined in $\\Omega_j$ $(j=l,r)$.\n\nFor convenience, in this section we denote a point in $\\mathbb{R}^2$ by $(x,y)$. Our objective is to prove that if $\\Omega \\subset \\mathbb{R}^2$ is smooth bounded convex domain\n with inradius $r$ and diameter $D$ (see Figure \\ref{fig1}) large enough, then one can find $\\lambda > 0$ such that\n$\n\\Lambda(\\Omega) \\neq \\emptyset.\n$, where $\\Lambda(\\Omega)$ is the set defined in \\eqref{L}.\n\n\\begin{figure}[ht]\n\\centering\n\\def\\svgwidth{1\\linewidth}\n\\input{fig3.pdf_tex}\n\\caption{Convex domain\n$\\Omega \\subset \\mathbb{R}^2$ with inradius $r$ and diameter $D$.}\n\\label{fig1}\n\\end{figure}\n\n\\begin{theorem}\\label{t1} Let $\\Omega \\subset \\mathbb{R}^2$ be a smooth bounded convex domain such that $\\mathrm{inr}(\\Omega)=r$ and $D(\\Omega)=D$. Then if $D$ is large enough, there exists $\\lambda>0$ for which problem \\eqref{con} admits a pattern. \\end{theorem}", + "post_theorem_intro_text_len": 4620, + "post_theorem_intro_text": "Theorem \\ref{t1} reveals a purely geometric mechanism for pattern\nformation. While the inradius remains fixed, increasing the diameter\nforces the emergence of patterns for appropriate values\nof $\\lambda$. In other words, sufficiently elongated convex domains (see Figure \\ref{fig1})\nnecessarily admit patterns. This phenomenon is independent of symmetry assumptions and depends\nonly on the convexity and aspect ratio of the domain.\n\nFrom a biological perspective, Theorem~\\ref{t1} suggests that\ngeometric elongation of the domain can counterbalance\nthe homogenizing action of diffusion in the interior.\nWhen the diameter becomes sufficiently large compared to the inradius,\nthe global geometric structure weakens diffusive stabilization\nand allows patterns to arise.\n\nThe relation between convexity and stability of non-constant stationary solutions \nhas been a central theme in the study of semilinear parabolic equations. \nIn the classical works of Casten--Holland \\cite{CH} and Matano \\cite{matano}, the nonlinearity\nacts in the interior of the domain and homogeneous Neumann boundary\nconditions are imposed. In that setting, it was shown that in convex\ndomains patterns cannot exist under very general assumptions on the\nnonlinearity. These results reveal a strong rigidity\nphenomenon induced by convexity.\n\nIn contrast, in the present work the nonlinearity acts through the\nboundary condition. Theorem~\\ref{t1} shows that, in this setting,\nconvexity by itself does not constitute a structural obstruction\nto pattern formation.\nAlthough it is known that no pattern can exist when $\\Omega$ is a ball\n(see \\cite{NLM}), this reflects the special symmetry of the ball rather\nthan convexity per se.\n\nTo the best of our knowledge, no previous result establishes a direct\ncriterion for the existence or absence of patterns in terms of global\ngeometric quantities such as the inradius and the diameter. Conversely,\nmuch of the existing literature connects the existence of patterns to\ncurvature effects or other local geometric properties of the boundary, for instance, see \\cite{CH,SAE,FA,SN} and references therein.\n\nIt is worth mentioning that in \\cite{CJ}, through a computer-assisted analysis\ncombined with bifurcation techniques, strong evidence was provided for\nthe existence of patterns for problem \\eqref{con} in the case\n$g(u)=u-u^3$, $\\lambda>2,84083164$, and $\\Omega$ equal to the unit square,\nwhich is a convex planar domain. The authors further conjectured that the same phenomenon should persist\nfor smooth convex domains obtained by suitably rounding the corners of the square.\n\nIn addition to \\cite{CJ}, a proof in the convex case was mentioned in \\cite{CS,DP}\nas part of work in preparation. To the best of our knowledge, a\ncomplete proof has not yet appeared in the literature.\n\nIn the same framework of boundary reactions in a bidimensional domain, Del Pino {\\it et al.} \\cite{DP}\nconstructed solutions with multiple boundary transitions, addressing existence and qualitative properties, but not stability. The existence of patterns has been obtained under stronger geometric or structural assumptions. \nIn \\cite{AG,C2,C3}, existence results were proved in special classes of non-convex bounded domains, such as dumbbell-type and perfored domains. \nMoreover, when the boundary nonlinearity depends explicitly on the spatial variable, namely $g=g(x,u)$, existence of patterns has been established in \\cite{ARRIETA} and for problems with variable diffusivity, see \\cite{ACM}.\n\nThe proof of Theorem \\ref{t1} is based on the construction of a specific set, denoted by $\\Lambda(\\Omega)$, associated with the geometry of $\\Omega$ and the positive flow generated by \\eqref{con}. The underlying strategy ensures that, whenever $\\Lambda(\\Omega)$ is nonempty, it contains the desired pattern. Thus, the core of the argument consists in proving that $\\Lambda(\\Omega)\\neq\\emptyset$. This approach appears to have been introduced in \\cite{matano} and has since been employed in different contexts, for instance in \\cite{ar1, C2}, as well as in some subsequent works \\cite{ACM}.\n\nThe present work is organized as follows. Section 2 is devoted to the preliminary results needed for the proof of Theorem \\ref{t1}. In particular, in Theorem \\ref{tmi} we establish a lower bound for the first positive Steklov eigenvalue, expressed in terms of the diameter of the domain and a constant depending only on the dimension. This estimate, which may be of independent interest in its own right, plays a key role in the proof of Theorem \\ref{t1}, carried out in Section 3. Section 4 concludes the paper with some final remarks.", + "sketch": "The proof of Theorem \\ref{t1} is based on “the construction of a specific set, denoted by $\\Lambda(\\Omega)$, associated with the geometry of $\\Omega$ and the positive flow generated by \\eqref{con}.” The strategy is that “whenever $\\Lambda(\\Omega)$ is nonempty, it contains the desired pattern,” so “the core of the argument consists in proving that $\\Lambda(\\Omega)\\neq\\emptyset$.”\n\nA key ingredient used in this program is that “in Theorem \\ref{tmi} we establish a lower bound for the first positive Steklov eigenvalue, expressed in terms of the diameter of the domain and a constant depending only on the dimension,” and “this estimate … plays a key role in the proof of Theorem \\ref{t1}.”", + "expanded_sketch": "In establishing the main theorem, the proof is based on “the construction of a specific set, denoted by $\\Lambda(\\Omega)$, associated with the geometry of $\\Omega$ and the positive flow generated by\n\\begin{equation}\\label{con}\n\\begin{cases}\\nu_t - \\Delta u = 0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+, \\\\\n\\partial_{\\nu} u = \\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+, \\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\end{equation}\n.” The strategy is that “whenever $\\Lambda(\\Omega)$ is nonempty, it contains the desired pattern,” so “the core of the argument consists in proving that $\\Lambda(\\Omega)\\neq\\emptyset$.”\n\nWe first prove the following theorem.\n\\begin{theorem}\\label{tmi}\nLet $\\Omega \\subset \\mathbb{R}^2$ be a bounded convex domain and again let\n$\nD := D(\\Omega).\n$\nThen there exists a constant $c>0$ depending only on the dimension such that\n\\[\n\\mu_1(\\Omega) \\ge \\frac{c}{D}.\n\\]\n\\end{theorem}\nThis estimate plays a key role in the proof of the main theorem.", + "expanded_theorem": "\\label{t1} Let $\\Omega \\subset \\mathbb{R}^2$ be a smooth bounded convex domain such that $\\mathrm{inr}(\\Omega)=r$ and $D(\\Omega)=D$. Then if $D$ is large enough, there exists $\\lambda>0$ for which the problem\n\\begin{equation}\\label{con}\n\\begin{cases}\\nu_t - \\Delta u = 0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+, \\\\\n\\partial_{\\nu} u = \\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+, \\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\end{equation}\nadmits a pattern.", + "theorem_type": [ + "Implication", + "Existence" + ], + "mcq": { + "question": "Let \\(\\Omega \\subset \\mathbb{R}^2\\) be a smooth bounded convex domain with inradius\n\\[\n\\mathrm{inr}(\\Omega):=\\sup\\{r'>0:\\exists x\\in\\Omega\\text{ such that }B_{r'}(x)\\subset \\Omega\\}=r\n\\]\nand diameter\n\\[\nD(\\Omega):=\\sup\\{|x-y|:x,y\\in\\Omega\\}=D.\n\\]\nConsider the heat equation with nonlinear Neumann boundary condition\n\\[\n\\begin{cases}\nu_t-\\Delta u=0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+,\\\\\n\\partial_\\nu u=\\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+,\\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\]\nwhere \\(\\nu\\) is the outward unit normal and \\(g(u)=u-u^3\\). A pattern means a non-constant stationary solution that is linearly stable. If \\(D\\) is sufficiently large, which conclusion about this problem is valid?", + "correct_choice": { + "label": "A", + "text": "There exists \\(\\lambda>0\\) such that the problem admits a pattern; equivalently, for some positive \\(\\lambda\\), it has a linearly stable non-constant stationary solution." + }, + "choices": [ + { + "label": "B", + "text": "There exists a constant \\(\\lambda_0>0\\), depending only on the dimension, such that for every \\(\\lambda\\ge \\lambda_0\\) the problem admits a pattern whenever \\(D\\) is sufficiently large." + }, + { + "label": "C", + "text": "There exists \\(\\lambda>0\\) such that the problem admits a stationary solution that is non-constant." + }, + { + "label": "D", + "text": "For every sufficiently large diameter \\(D\\), and for every smooth bounded convex domain \\(\\Omega\\subset\\mathbb{R}^2\\) with \\(\\mathrm{inr}(\\Omega)=r\\) and \\(D(\\Omega)=D\\), the problem admits a pattern for every \\(\\lambda>0\\)." + }, + { + "label": "E", + "text": "If \\(D\\) is sufficiently large, then for every \\(\\lambda>0\\) there exists a linearly stable non-constant stationary solution of the problem." + } + ], + "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": "existential dependence of the admissible parameter \\(\\lambda\\)", + "template_used": "uniformity_effectivity" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "linear stability requirement in the definition of pattern", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "finiteness", + "tampered_component": "quantifier order on \\(\\lambda\\): existence replaced by universality", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "parameter-selection mechanism via nonemptiness of \\(\\Lambda(\\Omega)\\) for some suitable \\(\\lambda\\)", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not state or strongly hint at the correct quantifier structure. It gives the PDE setup and definition of pattern, but the correct choice must be identified from the options." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-identification question: the correct option is essentially the precise theorem conclusion. However, it is not a pure tautology because the alternatives vary meaningfully in stability and quantifiers over λ." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish existence for some λ from stronger universal claims and from the weaker non-constant stationary conclusion. Still, the question mainly tests recognition of the exact valid statement rather than substantial generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target common errors: dropping linear stability, strengthening existence to universality, and asserting uniform parameter bounds. They are distinct and well aligned with likely failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it leans more toward precise theorem recall/quantifier discrimination than deep generative reasoning." + } + }, + { + "id": "2602.21115v1", + "paper_link": "http://arxiv.org/abs/2602.21115v1", + "theorems_cnt": 3, + "theorem": { + "env_name": "thm", + "content": "\\label{thm}\nLet $0\\leq f\\in C^1([0,1))$ satisfying $F(1)=+\\infty$.\nAssume that $0\\le u\\le 1$ is a stable radial solution to \\eqref{d-p}.\nIf $2\\le n\\le 6$, then $u$ is regular and satisfies\n$$\\frac 12 u_r(1)^2\\le F(\\|u\\|_{L^\\infty(B_1)})\\le Cu_r(1)^2.$$\nAdditionally, if $f$ is nondecreasing, then\n $$F(\\|u\\|_{L^\\infty(B_1)})\\le C,$$\nwhere $C>0$ is a universal constant.", + "start_pos": 8818, + "end_pos": 9202, + "label": "thm" + }, + "ref_dict": { + "un-bd": "\\begin{lem}\\label{un-bd}\nLet $0\\le f\\in C^1([0,1])$ be nondecreasing. Assume\nthat $0\\le u\\le 1$ is a stable radial solution to\n\\eqref{d-p}. Then\n$$|u_r(1)|\\le 2.$$\n\n\\end{lem}", + "gz": "\\begin{align}\\label{gz}\n\\gz:=\\liminf_{t\\to 1^{-}}\\frac{f(t)f''(t)}{f'(t)^2}>1\n\\end{align}", + "thm": "\\begin{thm}\\label{thm}\nLet $0\\leq f\\in C^1([0,1))$ satisfying $F(1)=+\\fz$.\nAssume that $0\\le u\\le 1$ is a stable radial solution to \\eqref{d-p}.\nIf $2\\le n\\le 6$, then $u$ is regular and satisfies\n$$\\frac 12 u_r(1)^2\\le F(\\|u\\|_{L^\\fz(B_1)})\\le Cu_r(1)^2.$$\nAdditionally, if $f$ is nondecreasing, then\n $$F(\\|u\\|_{L^\\fz(B_1)})\\le C,$$\nwhere $C>0$ is a universal constant.\n\\end{thm}", + "g-dp": "\\begin{align}\\label{g-dp}\n\\left\\{\n\\begin{aligned}\n-\\bdz u & = \\lz f(u)&\\quad{\\rm in}&\\quad B_1 \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}", + "thm-2": "\\begin{thm}\\label{thm-2}\nLet $f\\in C^1([0,1))$ be nondecreasing, convex, and\n$F(1)=+\\fz$. Suppose that $0\\le u^{\\star}\\le 1$ is the extremal solution to\n\\eqref{g-dp}. Then\n$$\\|u^{\\star}\\|_{L^\\fz(B_1)}<1\\quad \\forall \\, 2\\le n\\le 6.$$\n\n\\end{thm}", + "d-p": "\\begin{align}\\label{d-p}\n\\left\\{\n\\begin{aligned}\n-\\bdz u & = f(u)&\\quad{\\rm in}&\\quad B_1 \\setminus\\{ 0\\} \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}" + }, + "pre_theorem_intro_text_len": 1602, + "pre_theorem_intro_text": "This paper focuses on the issues associated with MEMS problem:\n\\begin{align}\\label{d-p}\n\\left\\{\n\\begin{aligned}\n-\\Delta u & = f(u)&\\quad{\\rm in}&\\quad B_1 \\setminus\\{ 0\\} \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}\nwhere $B_1$ is the unit ball centered at the origin in ${\\mathbb R}^n$($n\\ge 2$), $0\\le u\\le 1$ is a stable radial solution, and $f:[0,1)\\to [0,+\\infty)$ is\n$C^1$-function which blow-up at\n$u=1$. Throughout the paper, for each $0\\le t\\le 1$ we define $F(t):=\\int^t_0 f(s)\\,ds$ and require that $F(1)=\\int^{1}_0f(s)\\,ds=+\\infty$.\n\nRecall that a radial solution $u$ of \\eqref{d-p} is called stable if\n$$\\int_{B_1}f'(u)\\xi^2\\,dx\\le \\int_{B_1}|\\nabla \\xi|^2\\,dx$$\nfor every $\\xi \\in C^\\infty(B_1)$ with compact support in $B_1\\backslash\\{0\\}$, where\n$f'(s)$ denotes the derivative of $f$ at the point $s$. Furthermore, if\n$\\|u\\|_{L^\\infty(B_1)}<1$, we refer to $u$ as regular in the sense of Bruera and Cabr\\'e \\cite{bc}.\nNotably, the blow-up of $f(u)$ at $u=1$ implies that the stable radial solution $u$ does not need to be regular. For instance, when $n\\ge 7$ and $f(t)=\\frac{n-1}{1-t}$, the radial function $1-|x|$ is always stable yet satisfies\n$\\|u\\|_{L^\\infty(B_1)}=1$. In contrast, for dimensions $2\\le n\\le 6$, this function is not a stable solution. For further details, see Bruera and Cabr\\'e \\cite{bc} and Meadows \\cite{m04}.\n\nThis example naturally leads to the following question: in dimensions $2\\le n\\le 6$, is every stable radial solution of \\eqref{d-p} regular?\n\nIn the radial case, we provide a complete answer to this question.", + "context": "This paper focuses on the issues associated with MEMS problem:\n\\begin{align}\\label{d-p}\n\\left\\{\n\\begin{aligned}\n-\\Delta u & = f(u)&\\quad{\\rm in}&\\quad B_1 \\setminus\\{ 0\\} \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}\nwhere $B_1$ is the unit ball centered at the origin in ${\\mathbb R}^n$($n\\ge 2$), $0\\le u\\le 1$ is a stable radial solution, and $f:[0,1)\\to [0,+\\infty)$ is\n$C^1$-function which blow-up at\n$u=1$. Throughout the paper, for each $0\\le t\\le 1$ we define $F(t):=\\int^t_0 f(s)\\,ds$ and require that $F(1)=\\int^{1}_0f(s)\\,ds=+\\infty$.\n\nRecall that a radial solution $u$ of \\eqref{d-p} is called stable if\n$$\\int_{B_1}f'(u)\\xi^2\\,dx\\le \\int_{B_1}|\\nabla \\xi|^2\\,dx$$\nfor every $\\xi \\in C^\\infty(B_1)$ with compact support in $B_1\\backslash\\{0\\}$, where\n$f'(s)$ denotes the derivative of $f$ at the point $s$. Furthermore, if\n$\\|u\\|_{L^\\infty(B_1)}<1$, we refer to $u$ as regular in the sense of Bruera and Cabr\\'e \\cite{bc}.\nNotably, the blow-up of $f(u)$ at $u=1$ implies that the stable radial solution $u$ does not need to be regular. For instance, when $n\\ge 7$ and $f(t)=\\frac{n-1}{1-t}$, the radial function $1-|x|$ is always stable yet satisfies\n$\\|u\\|_{L^\\infty(B_1)}=1$. In contrast, for dimensions $2\\le n\\le 6$, this function is not a stable solution. For further details, see Bruera and Cabr\\'e \\cite{bc} and Meadows \\cite{m04}.\n\nThis example naturally leads to the following question: in dimensions $2\\le n\\le 6$, is every stable radial solution of \\eqref{d-p} regular?\n\nIn the radial case, we provide a complete answer to this question.\n\n\\begin{align}\\label{d-p}\n\\left\\{\n\\begin{aligned}\n-\\bdz u & = f(u)&\\quad{\\rm in}&\\quad B_1 \\setminus\\{ 0\\} \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}", + "full_context": "This paper focuses on the issues associated with MEMS problem:\n\\begin{align}\\label{d-p}\n\\left\\{\n\\begin{aligned}\n-\\Delta u & = f(u)&\\quad{\\rm in}&\\quad B_1 \\setminus\\{ 0\\} \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}\nwhere $B_1$ is the unit ball centered at the origin in ${\\mathbb R}^n$($n\\ge 2$), $0\\le u\\le 1$ is a stable radial solution, and $f:[0,1)\\to [0,+\\infty)$ is\n$C^1$-function which blow-up at\n$u=1$. Throughout the paper, for each $0\\le t\\le 1$ we define $F(t):=\\int^t_0 f(s)\\,ds$ and require that $F(1)=\\int^{1}_0f(s)\\,ds=+\\infty$.\n\nRecall that a radial solution $u$ of \\eqref{d-p} is called stable if\n$$\\int_{B_1}f'(u)\\xi^2\\,dx\\le \\int_{B_1}|\\nabla \\xi|^2\\,dx$$\nfor every $\\xi \\in C^\\infty(B_1)$ with compact support in $B_1\\backslash\\{0\\}$, where\n$f'(s)$ denotes the derivative of $f$ at the point $s$. Furthermore, if\n$\\|u\\|_{L^\\infty(B_1)}<1$, we refer to $u$ as regular in the sense of Bruera and Cabr\\'e \\cite{bc}.\nNotably, the blow-up of $f(u)$ at $u=1$ implies that the stable radial solution $u$ does not need to be regular. For instance, when $n\\ge 7$ and $f(t)=\\frac{n-1}{1-t}$, the radial function $1-|x|$ is always stable yet satisfies\n$\\|u\\|_{L^\\infty(B_1)}=1$. In contrast, for dimensions $2\\le n\\le 6$, this function is not a stable solution. For further details, see Bruera and Cabr\\'e \\cite{bc} and Meadows \\cite{m04}.\n\nThis example naturally leads to the following question: in dimensions $2\\le n\\le 6$, is every stable radial solution of \\eqref{d-p} regular?\n\nIn the radial case, we provide a complete answer to this question.\n\n\\begin{align}\\label{d-p}\n\\left\\{\n\\begin{aligned}\n-\\bdz u & = f(u)&\\quad{\\rm in}&\\quad B_1 \\setminus\\{ 0\\} \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}\n\nRecall that a radial solution $u$ of \\eqref{d-p} is called stable if\n$$\\int_{B_1}f'(u)\\xi^2\\,dx\\le \\int_{B_1}|\\nabla \\xi|^2\\,dx$$\nfor every $\\xi \\in C^\\fz(B_1)$ with compact support in $B_1\\backslash\\{0\\}$, where\n$f'(s)$ denotes the derivative of $f$ at the point $s$. Furthermore, if\n$\\|u\\|_{L^\\fz(B_1)}<1$, we refer to $u$ as regular in the sense of Bruera and Cabr\\'e \\cite{bc}.\nNotably, the blow-up of $f(u)$ at $u=1$ implies that the stable radial solution $u$ does not need to be regular. For instance, when $n\\ge 7$ and $f(t)=\\frac{n-1}{1-t}$, the radial function $1-|x|$ is always stable yet satisfies\n$\\|u\\|_{L^\\fz(B_1)}=1$. In contrast, for dimensions $2\\le n\\le 6$, this function is not a stable solution. For further details, see Bruera and Cabr\\'e \\cite{bc} and Meadows \\cite{m04}.\n\nThis example naturally leads to the following question: in dimensions $2\\le n\\le 6$, is every stable radial solution of \\eqref{d-p} regular?\n\n(ii) When dimension\n$3\\le n\\le 6$, $F(1)=+\\infty$ is necessary for regular stable solutions to exist for certain nonlinearities $f$; counterexamples can be found in Bruera and Cabr\\'e \\cite{bc}. Indeed, Bruera and Cabr\\'e \\cite{bc}\nconstructed a singular stable solution $u(x)=1-|x|^{2/(1+p)}$ for $n\\ge 3$ and for some\n $p\\in (0,1)$ satisfying\n$$-\\bdz u=\\frac{2}{1+p}\\left(\\frac{2}{1+p}+n-2\\right)(1-u)^{-p}$$\nwith $F(1)<+\\fz$. For dimension $n=2$ (without assuming $F(1)=+\\infty$), Luo, Ye and Zhou \\cite{lyz} showed that the stable radial solution $u$ of $-\\bdz u+ c(x)\\cdot \\nabla u=f(u)$ is regular if $f$ is nonnegative, nondecreasing, and convex, where $c(x)$ is smooth vector function. Observe that when $c(x)$ is a zero vector, this reduces to the case $-\\Delta u = f(u)$.\n\nRecently, inspired by a key stability inequality established \\cite{cfrs},\nBruera and Cabr\\'e \\cite{bc} investigated the case of a nonlinearity $f(u)$\nthat blows up as $u\\to 1^{-}$. Specifically, under the following Crandall-Rabinowitz type condition:\n\\begin{align}\\label{gz}\n\\gz:=\\liminf_{t\\to 1^{-}}\\frac{f(t)f''(t)}{f'(t)^2}>1\n\\end{align}\nfor all $f\\in C^2([0,1))$ satisfying $f\\ge 0$,\n$f'\\ge 0$ and $F(1)=+\\fz$, they established the following local estimate for all stable solutions\n(not necessarily radial) up to optimal dimension:\n$$\\|u\\|_{L^{\\fz}(B_{1/2})}\\le F^{-1}(C\\|u\\|^2_{L^1(B_1)})<1\\quad\n\\forall \\, 2\\le n\\le 6,$$\nwhere constant $C$ depending only on $n$ and $\\gz$. Additionally, a forthcoming study by Figalli and Franceschini \\cite{ff} provides an explicit upper bound for the Hausdorff dimension of the singular set for stable solutions expressed in terms of $f$, $f'$, and $f''$. The result covers both globally defined nonlinearities and nonlinearities with finite blow-up.\n\nIn order to show Theorem \\ref{thm}, we recall the following important result concerning stable radial solutions, which was established by Villegas \\cite[Theorem 1.7]{v12}.\n\\begin{lem}\\label{vi-lem}\nLet $n\\geq 2$ and $f:[0,1)\\to [0,+\\fz)$ be a $C^1$ function satisfying $f\\ge 0$. Let $0\\le u\\le 1$ be a stable radial solution to\n\\eqref{d-p} in $W^{1,2}(B_1)$. Then there exists a constant $K_n$ depending only on $n$ such that\n\\begin{align}\\label{vi-le1}\n|u_r(t)|\\le K_n \\|\\nabla u\\|_{L^2(B_1\\backslash B_{1/2})}t^{-n/2+\\sqrt{n-1}+1},\n\\quad \\forall \\, 00$.\nHence we finish this proof.\n\\end{proof}\nWe apply now Theorem \\ref{thm} to prove Theorem\n\\ref{thm-2}.\n\\begin{proof}[Proof of Theorem \\ref{thm-2}]\nLet $0<\\lz<\\lz^{\\star}$ and let $u_{\\lz}\\in C^2(\\overline B_1)$ be a stable radial\nsolution to \\eqref{g-dp}. It follows from Theorem \\ref{thm} that\n$$F(\\|u_{\\lz}\\|_{L^\\fz(B_1)})\\le C(n)\n\\left(\\frac{d u_{\\lz}(1)}{dr}\\right)^2.$$\nUsing Lemma 2.3 in \\cite{v12} by Villegas, one has that\n$$F(\\|u_{\\lz}\\|_{L^\\fz(B_1)})\\le C(n)\n\\left(\\frac{d u_{\\lz}(1)}{dr}\\right)^2\n\\le C(n)\\left(\\frac{d u_{\\lz}(1/2)}{dr}\\right)^2\n\\le C(n)\\int^{3/4}_{1/2}\\left(\\frac{d u_{\\lz}(t)}{dr}\\right)^2\\,dt,$$\nwhere we used the fact that $t^{2n-2}\\left(\\frac{d u_{\\lz}(t)}{dr}\\right)^2$\nis nondecreasing in the last inequality. Since\n$\\lim_{\\lz\\to \\lz^{\\star}}u_{\\lz}=u^{\\star}$ and\n$\\nabla u_{\\lz}\\to \\nabla u^{\\star}$ in $L^2(B_{3/4})$ as\n$\\lz \\to \\lz^{\\star}$, passing to the limit $\\lz\\to \\lz^{\\star}$ we conclude that\n$$F(\\|u^{\\star}\\|_{L^\\fz(B_1)})\n\\le C(n)\\int^{3/4}_{1/2}\\left(\\frac{d u^{\\star}(t)}{dr}\\right)^2\\,dt<+\\fz.$$\n\n\\begin{align}\\label{d-p}\n\\left\\{\n\\begin{aligned}\n-\\bdz u & = f(u)&\\quad{\\rm in}&\\quad B_1 \\setminus\\{ 0\\} \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}\n\n\\begin{align}\\label{g-dp}\n\\left\\{\n\\begin{aligned}\n-\\bdz u & = \\lz f(u)&\\quad{\\rm in}&\\quad B_1 \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}\n\n\\begin{thm}\\label{thm}\nLet $0\\leq f\\in C^1([0,1))$ satisfying $F(1)=+\\fz$.\nAssume that $0\\le u\\le 1$ is a stable radial solution to \\eqref{d-p}.\nIf $2\\le n\\le 6$, then $u$ is regular and satisfies\n$$\\frac 12 u_r(1)^2\\le F(\\|u\\|_{L^\\fz(B_1)})\\le Cu_r(1)^2.$$\nAdditionally, if $f$ is nondecreasing, then\n $$F(\\|u\\|_{L^\\fz(B_1)})\\le C,$$\nwhere $C>0$ is a universal constant.\n\\end{thm}\n\n\\begin{thm}\\label{thm-2}\nLet $f\\in C^1([0,1))$ be nondecreasing, convex, and\n$F(1)=+\\fz$. Suppose that $0\\le u^{\\star}\\le 1$ is the extremal solution to\n\\eqref{g-dp}. Then\n$$\\|u^{\\star}\\|_{L^\\fz(B_1)}<1\\quad \\forall \\, 2\\le n\\le 6.$$\n\n\\end{thm}\n\n\\begin{lem}\\label{un-bd}\nLet $0\\le f\\in C^1([0,1])$ be nondecreasing. Assume\nthat $0\\le u\\le 1$ is a stable radial solution to\n\\eqref{d-p}. Then\n$$|u_r(1)|\\le 2.$$\n\n\\end{lem}", + "post_theorem_intro_text_len": 8000, + "post_theorem_intro_text": "We now provide some remarks on the main results.\n\\begin{rem}\n\\rm\n(i) The dimension $n \\leq 6$ in Theorem \\ref{thm} is optimal: for $n \\geq 7$ and $f(t) = \\frac{n-1}{1-t}$, the function $1 - |x|$ is always stable but not regular. Additional examples can be found in Bruera and Cabr\\'e \\cite{bc}.\n\n(ii) When dimension\n$3\\le n\\le 6$, $F(1)=+\\infty$ is necessary for regular stable solutions to exist for certain nonlinearities $f$; counterexamples can be found in Bruera and Cabr\\'e \\cite{bc}. Indeed, Bruera and Cabr\\'e \\cite{bc}\nconstructed a singular stable solution $u(x)=1-|x|^{2/(1+p)}$ for $n\\ge 3$ and for some\n $p\\in (0,1)$ satisfying\n$$-\\Delta u=\\frac{2}{1+p}\\left(\\frac{2}{1+p}+n-2\\right)(1-u)^{-p}$$\nwith $F(1)<+\\infty$. For dimension $n=2$ (without assuming $F(1)=+\\infty$), Luo, Ye and Zhou \\cite{lyz} showed that the stable radial solution $u$ of $-\\Delta u+ c(x)\\cdot \\nabla u=f(u)$ is regular if $f$ is nonnegative, nondecreasing, and convex, where $c(x)$ is smooth vector function. Observe that when $c(x)$ is a zero vector, this reduces to the case $-\\Delta u = f(u)$.\n\n(iii) In Theorem \\ref{thm} above, with slight modifications, it can be shown that the stable radial solution is regular under the conditions $0\\le u\\le 1$, $0\\le f$ and $F(1)=+\\infty$,\nwithout requiring the boundary condition: $u=0$ on $\\partial B_1$.\n\\end{rem}\n\nLet us now review some relevant work in this direction. In the non-radial setting with a nonlinearity $f(u)$ that is unbounded as\n$u\\to +\\infty$, a pioneering study was first conducted by Crandall and Rabinowitz \\cite{cr75}\n with\n$f(t)=e^t$ and $f(t)=(1+t)^p$, for $p>1$. Subsequently, the boundedness of stable solutions\n$u$ to $-\\Delta u=f(u)$ in general domains $\\Omega$ has been extensively investigated over the past three decades; see \\cite{cc06,c10,n20,v13}. Notably, Cabr\\'e, Figalli, Ros-Oton and Serra \\cite{cfrs} made a breakthrough:\nSuppose $f$ is locally Lipschitz, nonnegative, and nondecreasing, and $u\\in W^{1,2}_0(\\Omega)$\nis stable solution of $-\\Delta u=f(u)$ in $C^3$ domain $\\Omega$, they showed that\n$$\\|u\\|_{C^{0,\\alpha}(\\overline \\Omega)}\\le C(n)\\|u\\|_{L^\\infty(\\Omega)}\\quad\n2\\le n\\le 9$$\nfor some $\\alpha=\\alpha(n)\\in (0,1)$. They also established the local H\\\"older continuity of\n$u$ for $n\\le 9$. Note that the dimension $n\\le 9$ is optimal for nonlinearities\n$f(u)$ that blow-up as $u\\to \\infty$. In addition, Cabr\\'e provided a quantitative proof of these results in \\cite{c23,c24}.\nWe also mention that, as established in\nthe work of Erneta \\cite{e23},\nthe optimal boundary regularity of $u$ in $C^3$ domains attainable already in the class of\n$C^{1,1}$-domains.\n\nRecently, inspired by a key stability inequality established \\cite{cfrs},\nBruera and Cabr\\'e \\cite{bc} investigated the case of a nonlinearity $f(u)$\nthat blows up as $u\\to 1^{-}$. Specifically, under the following Crandall-Rabinowitz type condition:\n\\begin{align}\\label{gz}\n{\\gamma}:=\\liminf_{t\\to 1^{-}}\\frac{f(t)f''(t)}{f'(t)^2}>1\n\\end{align}\nfor all $f\\in C^2([0,1))$ satisfying $f\\ge 0$,\n$f'\\ge 0$ and $F(1)=+\\infty$, they established the following local estimate for all stable solutions\n(not necessarily radial) up to optimal dimension:\n$$\\|u\\|_{L^{\\infty}(B_{1/2})}\\le F^{-1}(C\\|u\\|^2_{L^1(B_1)})<1\\quad\n\\forall \\, 2\\le n\\le 6,$$\nwhere constant $C$ depending only on $n$ and ${\\gamma}$. Additionally, a forthcoming study by Figalli and Franceschini \\cite{ff} provides an explicit upper bound for the Hausdorff dimension of the singular set for stable solutions expressed in terms of $f$, $f'$, and $f''$. The result covers both globally defined nonlinearities and nonlinearities with finite blow-up.\n\nOn the other hand, for smooth domains $\\Omega$, Bruera and Cabr\\'e \\cite{bc} also derived global estimates for stable solutions\n$u\\in C^2( \\Omega)\\cap C^0(\\overline \\Omega)$ of the Dirichlet boundary value problem:\n\\begin{align*}\n\\left\\{\n\\begin{aligned}\n-\\Delta u & = f(u)&\\quad{\\rm in}&\\quad \\Omega \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial \\Omega,\\\\\n\\end{aligned}\n\\right.\n\\end{align*}\nand showed the following:\n\\begin{itemize}\n\\item[$\\bullet$] If $n\\le 2$, then $\\|u\\|_{L^\\infty(\\Omega)}<1$ without condition \\eqref{gz}.\n\n\\item[$\\bullet$] If $3\\le n\\le 6$ and $f$ satisfies the condition \\eqref{gz}, then $\\|u\\|_{L^\\infty(\\Omega)}<1$.\n\n\\end{itemize}\nIt is worth noting that additional partial results related to this direction can be found in\n\\cite{ces,cgg,lyz,m04}. In particular, for dimension $n=2$ (without requiring $F(1)=+\\infty$), Luo, Ye and Zhou \\cite{lyz} proved that the stable radial solution $u$ of $-\\Delta u+c(x)\\cdot \\nabla u=f(u)$\n is regular provided that $f$ is nonnegative, nondecreasing, and convex. Motivated by these findings, Bruera and Cabr\\'e \\cite{bc} proposed the following open problem:\n\\begin{center}\n{\\bf Open problem}. Does regularity of stable solutions hold up to the optimal\ndimension for singular nonlinearities under no Crandall-Rabinowitz type condition?\n\\end{center}\nIn the radial setting,\nwe provide a positive answer to this open problem in\nTheorem \\ref{thm}. However, the non-radial case remains open.\n\nTheorem \\ref{thm} has an important application to the Gelfand-type problem; see for example\nin \\cite{bv97,b03}. Precisely, let $f\\in C^1([0,1))$ satisfy $f(0)>0$ and be nondecreasing, convex, and $F(1)=+\\infty$.\n\nGiven a constant $\\lambda>0$ consider the nonlinear elliptic problem:\n\\begin{align}\\label{g-dp}\n\\left\\{\n\\begin{aligned}\n-\\Delta u & = \\lambda f(u)&\\quad{\\rm in}&\\quad B_1 \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}\nA fundamental question is to determine a constant $\\lambda^{\\star}>0$ such that the problem\n\\eqref{g-dp} has a unique $L^1$-solution $u^{\\star}\\in L^1(B_1)$ in the sense that\n$$-\\int_{B_1}u^{\\star}\\Delta \\xi \\,dx=\n\\lambda^{\\star}\\int_{B_1}f(u^{\\star})\\xi\\,dx\\quad\\forall \\, \\xi\\in C^{0,1}_c(B_1)$$\nSuch $L^1$-solution solution $u^{\\star}$ is called extremal solution.\nThe uniqueness and existence of extremal solution were established by\nCastorina, Esposito and Sciunzi \\cite{ces}. Moreover, they\nshowed that for every $0<\\lambda<\\lambda^{\\star}$, there exist a stable solution\n$u_{\\lambda}\\in C^2(\\overline B_1)$ such that\n$$\\lim_{\\lambda\\to \\lambda^{\\star}}u_{\\lambda}\n=u^{\\star}.$$\nCombining this result with Theorem \\ref{thm}, we conclude that the extremal solution is always regular in dimensions $2\\le n\\le 6$.\n\n\\begin{thm}\\label{thm-2}\nLet $f\\in C^1([0,1))$ be nondecreasing, convex, and\n$F(1)=+\\infty$. Suppose that $0\\le u^{\\star}\\le 1$ is the extremal solution to\n\\eqref{g-dp}. Then\n$$\\|u^{\\star}\\|_{L^\\infty(B_1)}<1\\quad \\forall \\, 2\\le n\\le 6.$$\n\n\\end{thm}\n\nFinally, we outline the main ideas for proving the theorems.\nSince Theorem \\ref{thm-2} follows directly from Theorem \\ref{thm},\nit suffices to explain the proof strategy for Theorem \\ref{thm}.\n\nA key starting point is the following observation for general radial solutions:\n$$\nF(\\|u\\|_{L^\\infty(B_1)})= (n-1)\\int^1_0\\frac{u_r(t)^2}{t}\\,dt+\\frac{u_r(1)^2}2.\n$$\nSince $u\\in C^1(\\overline B_1\\backslash\\{0\\})$,\nthe main task is reduced to bounding the term $\\int^1_0\\frac{u_r(t)^2}{t}\\,dt$. By a key\nestimate due to Villegas \\cite[Theorem 1.7]{v12} (see also \\cite{cc06}), we deduce\n\\begin{align*}\n\\vert u_r(t)\\vert \\le C_n \\vert u_r(1)\\vert t^{-n/2+\\sqrt{n-1}+1},\n\\quad \\forall \\, 00\\quad{\\rm if \\ and\\ only\\ if}\\quad 2\\le n\\le 6.$$\nThis implies that $u_r(t)$ is H\\\"older continuous near $t=0$ and hence\nthe behavior of\n $t^{-1}u_r(t)^2$ is like $t^{-\\alpha}$ for some\n$\\alpha\\in (0,1)$ whenever $2\\le n\\le 6$. Therefore, the integral\n$\\int^1_0\\frac{u_r(t)^2}{t}\\,dt$ is finite, and we conclude\n\n$$F(\\|u\\|_{L^\\infty(B_1)})\\leq C_nu_r(1)^2<+\\infty.$$\nCombining this with the condition $F(1)=\\infty$, we obtain $\\|u\\|_{L^\\infty(B_1)}<1$.\nMoreover, if $f$ is nondecreasing, the universal bound on $F(\\|u\\|_{L^\\infty(B_1)})$ follows\nfrom the upper bound on $|u_r(1)|$; see Lemma \\ref{un-bd} for details.", + "sketch": "To prove Theorem~\\ref{thm}, since Theorem~\\ref{thm-2} follows directly from it, the strategy starts from the identity for general radial solutions\n\\[\nF(\\|u\\|_{L^\\infty(B_1)})= (n-1)\\int^1_0\\frac{u_r(t)^2}{t}\\,dt+\\frac{u_r(1)^2}2.\n\\]\nThus (with $u\\in C^1(\\overline B_1\\backslash\\{0\\})$) the main task is to bound $\\int^1_0\\frac{u_r(t)^2}{t}\\,dt$. Using Villegas' estimate \\cite[Theorem 1.7]{v12} (see also \\cite{cc06}),\n\\[\n|u_r(t)|\\le C_n|u_r(1)|\\, t^{-n/2+\\sqrt{n-1}+1},\\qquad 00$ iff $2\\le n\\le 6$, this gives H\\\"older continuity of $u_r(t)$ near $t=0$; hence $t^{-1}u_r(t)^2$ behaves like $t^{-\\alpha}$ for some $\\alpha\\in(0,1)$ when $2\\le n\\le 6$. Therefore $\\int^1_0\\frac{u_r(t)^2}{t}\\,dt<\\infty$, yielding\n\\[\nF(\\|u\\|_{L^\\infty(B_1)})\\le C_n u_r(1)^2<+\\infty.\n\\]\nCombining this with $F(1)=\\infty$ gives $\\|u\\|_{L^\\infty(B_1)}<1$ (regularity). Finally, if $f$ is nondecreasing, the claimed universal bound for $F(\\|u\\|_{L^\\infty(B_1)})$ follows from an upper bound on $|u_r(1)|$ (see Lemma~\\ref{un-bd}).", + "expanded_sketch": "To prove the main theorem, since\n\\begin{thm}\\label{thm-2}\nLet $f\\in C^1([0,1))$ be nondecreasing, convex, and\n$F(1)=+\\fz$. Suppose that $0\\le u^{\\star}\\le 1$ is the extremal solution to\n\\eqref{g-dp}. Then\n$$\\|u^{\\star}\\|_{L^\\fz(B_1)}<1\\quad \\forall \\, 2\\le n\\le 6.$$\n\n\\end{thm}\nfollows directly from it, the strategy starts from the identity for general radial solutions\n\\[\nF(\\|u\\|_{L^\\infty(B_1)})= (n-1)\\int^1_0\\frac{u_r(t)^2}{t}\\,dt+\\frac{u_r(1)^2}2.\n\\]\nThus (with $u\\in C^1(\\overline B_1\\backslash\\{0\\})$) the main task is to bound $\\int^1_0\\frac{u_r(t)^2}{t}\\,dt$. Using Villegas' estimate \\cite[Theorem 1.7]{v12} (see also \\cite{cc06}),\n\\[\n|u_r(t)|\\le C_n|u_r(1)|\\, t^{-n/2+\\sqrt{n-1}+1},\\qquad 00$ iff $2\\le n\\le 6$, this gives H\\\"older continuity of $u_r(t)$ near $t=0$; hence $t^{-1}u_r(t)^2$ behaves like $t^{-\\alpha}$ for some $\\alpha\\in(0,1)$ when $2\\le n\\le 6$. Therefore $\\int^1_0\\frac{u_r(t)^2}{t}\\,dt<\\infty$, yielding\n\\[\nF(\\|u\\|_{L^\\infty(B_1)})\\le C_n u_r(1)^2<+\\infty.\n\\]\nCombining this with $F(1)=\\infty$ gives $\\|u\\|_{L^\\infty(B_1)}<1$ (regularity). Finally, if $f$ is nondecreasing, the claimed universal bound for $F(\\|u\\|_{L^\\infty(B_1)})$ follows from an upper bound on $|u_r(1)|$. We use the following lemma.\n\\begin{lem}\\label{un-bd}\nLet $0\\le f\\in C^1([0,1])$ be nondecreasing. Assume\nthat $0\\le u\\le 1$ is a stable radial solution to\n\\eqref{d-p}. Then\n$$|u_r(1)|\\le 2.$$\n\n\\end{lem}\n", + "expanded_theorem": "\\label{thm}\nLet $0\\leq f\\in C^1([0,1))$ satisfying $F(1)=+\\infty$.\nAssume that $0\\le u\\le 1$ is a stable radial solution to\n\\begin{align}\\label{d-p}\n\\left\\{\n\\begin{aligned}\n-\\bdz u & = f(u)&\\quad{\\rm in}&\\quad B_1 \\setminus\\{ 0\\} \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}\nIf $2\\le n\\le 6$, then $u$ is regular and satisfies\n$$\\frac 12 u_r(1)^2\\le F(\\|u\\|_{L^\\infty(B_1)})\\le Cu_r(1)^2.$$\nAdditionally, if $f$ is nondecreasing, then\n $$F(\\|u\\|_{L^\\infty(B_1)})\\le C,$$\nwhere $C>0$ is a universal constant.,", + "theorem_type": [ + "Implication", + "Inequality or Bound" + ], + "mcq": { + "question": "Let $B_1\\subset \\mathbb{R}^n$ be the unit ball, let $f:[0,1)\\to[0,\\infty)$ be a $C^1$ function, and define\n$$F(t):=\\int_0^t f(s)\\,ds.$$ \nAssume $F(1)=+\\infty$, $2\\le n\\le 6$, and that $u$ is a radial function with $0\\le u\\le 1$ solving\n$$\\begin{cases}\n-\\Delta u=f(u) & \\text{in } B_1\\setminus\\{0\\},\\\\\nu=0 & \\text{on } \\partial B_1,\n\\end{cases}$$\nwhere $u$ is stable in the sense that\n$$\\int_{B_1} f'(u)\\,\\xi^2\\,dx\\le \\int_{B_1} |\\nabla \\xi|^2\\,dx$$\nfor every $\\xi\\in C^\\infty(B_1)$ compactly supported in $B_1\\setminus\\{0\\}$. Here $u_r(1)$ denotes the radial derivative at $r=1$, and “$u$ is regular” means $\\|u\\|_{L^\\infty(B_1)}<1$. Which of the following statements is valid?", + "correct_choice": { + "label": "A", + "text": "The solution $u$ is regular, and it satisfies\n$$\\frac12\\,u_r(1)^2\\le F\\bigl(\\|u\\|_{L^\\infty(B_1)}\\bigr)\\le C\\,u_r(1)^2.$$ \nMoreover, if $f$ is nondecreasing, then\n$$F\\bigl(\\|u\\|_{L^\\infty(B_1)}\\bigr)\\le C,$$\nfor a universal constant $C>0$." + }, + "choices": [ + { + "label": "B", + "text": "The solution $u$ is regular, and it satisfies\n$$\\frac12\\,u_r(1)^2\\le F\\bigl(\\|u\\|_{L^\\infty(B_1)}\\bigr)\\le C\\,u_r(1)^2.$$ \nMoreover, if $f$ is nondecreasing, then the same conclusion holds with a universal constant $C>0$ for every dimension $n\\ge 2$." + }, + { + "label": "C", + "text": "The solution $u$ is regular, and it satisfies\n$$F\\bigl(\\|u\\|_{L^\\infty(B_1)}\\bigr)\\le C\\,u_r(1)^2.$$" + }, + { + "label": "D", + "text": "The solution $u$ is regular, and it satisfies\n$$\\frac12\\,u_r(1)^2\\le F\\bigl(\\|u\\|_{L^\\infty(B_1)}\\bigr)\\le C\\,u_r(1)^2.$$ \nMoreover, if $f$ is nondecreasing, then\n$$F\\bigl(\\|u\\|_{L^\\infty(B_1)}\\bigr)\\le C,$$\nwhere $C>0$ is a constant depending only on $n$ and not on $f$ or $u$." + }, + { + "label": "E", + "text": "If $f$ is nondecreasing, then every stable radial solution $u$ satisfies\n$$\\|u\\|_{L^\\infty(B_1)}<1$$\nfor all $n\\ge 2$, and moreover\n$$\\frac12\\,u_r(1)^2\\le F\\bigl(\\|u\\|_{L^\\infty(B_1)}\\bigr)\\le C\\,u_r(1)^2.$$" + } + ], + "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 from integrability exponent", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped lower bound and dropped universal nondecreasing conclusion", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "dependence of the universal constant in the monotone case", + "template_used": "uniformity_effectivity" + }, + { + "label": "E", + "sketch_hook_type": "characteristic", + "tampered_component": "uses monotonicity to bypass the sharp restriction $2\\le n\\le 6$", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly state the conclusion in choice A, nor does it contain obvious wording cues that uniquely identify the correct option. The hypotheses are substantial, but they do not directly leak the exact form of the final estimates." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recall question: it presents the full hypothesis set and asks for the valid conclusion. However, it is not a pure tautological restatement, since the choices differ in meaningful ways (dimension range, sharpness of inequalities, and dependence/universality of constants)." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish the strongest correct conclusion from nearby variants, especially regarding the role of the dimension bound, the lower bound involving u_r(1), and universality of constants. Still, the task mainly tests recognition of the precise theorem statement rather than deeper derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target common failure modes: overextending the dimension range, weakening the conclusion, incorrectly changing constant dependence, or swapping the structure of the estimates. They are distinct and well aligned with how a learner might misremember the result." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-identification MCQ with strong distractors and little answer leakage, but it leans more toward precise recall than genuine generative mathematical reasoning." + } + }, + { + "id": "2602.21659v1", + "paper_link": "http://arxiv.org/abs/2602.21659v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "[Fundamental inequality]\\label{thm:main_intro}\nLet $F\\subset S^3$ be a closed surface. Then every knot $K\\subset S^3$ satisfies:\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1,\n\\]\nwhere $t(K)$ is the tunnel number of $K$, and $\\delta(F)=g(M_1)+g(M_2)-g(F)$ is the Heegaard deficiency of $F$.", + "start_pos": 2789, + "end_pos": 3106, + "label": "thm:main_intro" + }, + "ref_dict": { + "thm:main_intro": "\\begin{theorem}[Fundamental inequality]\\label{thm:main_intro}\nLet $F\\subset S^3$ be a closed surface. Then every knot $K\\subset S^3$ satisfies:\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1,\n\\]\nwhere $t(K)$ is the tunnel number of $K$, and $\\delta(F)=g(M_1)+g(M_2)-g(F)$ is the Heegaard deficiency of $F$.\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 1144, + "pre_theorem_intro_text": "\\subsection{Background and Motivation}\nLow-dimensional topology quantifies knot complexity through numerical invariants. While classical invariants, such as the standard crossing number, rely on planar diagrams, these two-dimensional projections compress spatial information which does not fully reflect the global topology of the knot exterior within the ambient space $S^3$.\n\nTo capture this three-dimensional structure, we extend the notion of diagrammatic complexity to arbitrary closed surfaces. For a knot $K$ and a closed surface $F \\subset S^3$, we define the \\emph{surface crossing number} $c(K;F)$ as the minimal number of crossings among all regular diagrams of $K$ obtained by isotoping the knot into a regular neighborhood of $F$. \n\nIn this paper, we establish an inequality demonstrating that $c(K;F)$ is bounded below by the tunnel number of $K$ and the Heegaard deficiency of $F$.\n\n\\subsection{Main Results}\nThe objective of this paper is to establish a quantitative relationship between a knot's three-dimensional complexity, measured via its tunnel number, and its diagrammatic complexity on an arbitrary fixed closed surface.", + "context": "\\subsection{Background and Motivation}\nLow-dimensional topology quantifies knot complexity through numerical invariants. While classical invariants, such as the standard crossing number, rely on planar diagrams, these two-dimensional projections compress spatial information which does not fully reflect the global topology of the knot exterior within the ambient space $S^3$.\n\nTo capture this three-dimensional structure, we extend the notion of diagrammatic complexity to arbitrary closed surfaces. For a knot $K$ and a closed surface $F \\subset S^3$, we define the \\emph{surface crossing number} $c(K;F)$ as the minimal number of crossings among all regular diagrams of $K$ obtained by isotoping the knot into a regular neighborhood of $F$.\n\nIn this paper, we establish an inequality demonstrating that $c(K;F)$ is bounded below by the tunnel number of $K$ and the Heegaard deficiency of $F$.\n\n\\subsection{Main Results}\nThe objective of this paper is to establish a quantitative relationship between a knot's three-dimensional complexity, measured via its tunnel number, and its diagrammatic complexity on an arbitrary fixed closed surface.", + "full_context": "\\subsection{Background and Motivation}\nLow-dimensional topology quantifies knot complexity through numerical invariants. While classical invariants, such as the standard crossing number, rely on planar diagrams, these two-dimensional projections compress spatial information which does not fully reflect the global topology of the knot exterior within the ambient space $S^3$.\n\nTo capture this three-dimensional structure, we extend the notion of diagrammatic complexity to arbitrary closed surfaces. For a knot $K$ and a closed surface $F \\subset S^3$, we define the \\emph{surface crossing number} $c(K;F)$ as the minimal number of crossings among all regular diagrams of $K$ obtained by isotoping the knot into a regular neighborhood of $F$.\n\nIn this paper, we establish an inequality demonstrating that $c(K;F)$ is bounded below by the tunnel number of $K$ and the Heegaard deficiency of $F$.\n\n\\subsection{Main Results}\nThe objective of this paper is to establish a quantitative relationship between a knot's three-dimensional complexity, measured via its tunnel number, and its diagrammatic complexity on an arbitrary fixed closed surface.\n\n\\documentclass[11pt]{amsart}\n\\usepackage{amsmath,amssymb,amsthm}\n\\usepackage{hyperref}\n\\usepackage{tikz}\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{remark}[theorem]{Remark}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{question}[theorem]{Question}\n\\title[Crossing Numbers of Knots on Closed Surfaces]\n{Crossing Numbers of Knots on Closed Surfaces}\n\\author{Makoto Ozawa}\n\\address{Department of Natural Sciences, Faculty of Arts and Sciences, Komazawa University, 1-23-1 Komazawa, Setagaya-ku, Tokyo, 154-8525, Japan}\n\\email{w3c@komazawa-u.ac.jp}\n\\date{\\today}\n\\dedicatory{Dedicated to Professor Kanji Morimoto on the occasion of his retirement.}\n\\keywords{knot, surface crossing number, tunnel number, Heegaard deficiency, Heegaard splitting, surface bridge number, surface ascending number}\n\\begin{document}\n\\begin{abstract}\nLet $c(K;F)$ denote the surface crossing number of a knot $K$ with respect\nto a closed surface $F \\subset S^{3}$. We establish the lower bound\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1,\n\\]\nwhere $t(K)$ is the tunnel number of $K$ and $\\delta(F)$ is the Heegaard\ndeficiency of $F$. In particular, for any fixed closed surface $F$, the\nsurface crossing number $c(K;F)$ is unbounded over all knots $K$.\nFurthermore, we construct a family of knots $K_m$ demonstrating that $c(K_m;F) = \\Theta(t(K_m))$, which shows that this lower bound is asymptotically sharp.\n\\end{abstract}\n\\maketitle\n\n\\subsection{Main Results}\nThe objective of this paper is to establish a quantitative relationship between a knot's three-dimensional complexity, measured via its tunnel number, and its diagrammatic complexity on an arbitrary fixed closed surface.\n\nThe proof relies on a chain of inequalities relating the surface crossing number $c(K;F)$, the surface ascending number $a(K;F)$, and the surface bridge number $b(K;F)$:\n$$\\frac{c(K;F)-1}{2} \\ge a(K;F) \\ge b(K;F)-1 \\ge t(K)-\\delta(F).$$\nThis sequence demonstrates that the geometric obstruction to simplifying a knot diagram is bounded below by a combination of the knot exterior's complexity, $t(K)$, and the ambient surface's Heegaard deficiency, $\\delta(F)$. Unlike the classical planar crossing number, $c(K;F)$ directly reflects the Heegaard-theoretic complexity of the knot.\n\nFurthermore, we prove that the linear lower bound in Theorem \\ref{thm:main_intro} is asymptotically optimal. For iterated connected sums $K_m$ of a prime knot (such as the trefoil), we obtain the linear growth estimate:\n$$c(K_m;F) \\ge 2m - 2\\delta(F) + 1.$$\nPairing this with a general upper bound derived from the subadditivity of the planar crossing number yields $c(K_m;F) = \\Theta(t(K_m))$. This confirms that no general lower bound of higher order exists, and that the surface crossing number grows linearly with respect to the tunnel number.\n\n\\begin{theorem}[Fundamental inequality]\\label{thm:fundamental}\nLet $F\\subset S^{3}$ be a closed separating surface, and let $K\\subset S^{3}$ be a knot. Then\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1.\n\\]\n\\end{theorem}\n\nApplying the fundamental inequality to $K_m$ yields:\n\\begin{proposition}\nFor every closed surface $F\\subset S^{3}$,\n\\[\nc(K_m;F) \\ge 2\\bigl(m-\\delta(F)\\bigr)+1 \\to \\infty \\quad \\text{as } m \\to \\infty.\n\\]\n\\end{proposition}\n\n\\section{Proof of the Main Inequality}\nIn this section, we prove Theorem \\ref{thm:fundamental} by establishing the chain\n$$\\frac{c(K;F)-1}{2} \\ge a(K;F) \\ge b(K;F)-1 \\ge t(K)-\\delta(F).$$\nThroughout the proof, let $F\\subset S^{3}$ be a closed separating surface with decomposition\n$S^{3}=M_{1}\\cup_{F}M_{2}$.\n\n\\subsection{Generalization to spatial graphs}\nThe framework extends to spatial graphs. A spatial graph $G$ is a 1-dimensional CW complex embedded in $S^{3}$. The key arguments rely on the 1-dimensionality of the object: in the proof of Lemma \\ref{lem:amalgamation}, a general position argument shows that the 2-dimensional attaching disks of the 1-handles can be made disjoint from $G$. The tunnel number $t(G)$, bridge position, and ascending number extend to this setting, and one expects a lower bound of the form\n\\[\nc(G;F) \\ge 2\\bigl(t(G)-\\delta(F)\\bigr) + C_G,\n\\]\nwhere $C_G$ is a constant depending on the combinatorial structure of $G$ (such as its Euler characteristic or vertex degrees).\n\n\\begin{theorem}[Fundamental inequality]\\label{thm:main_intro}\nLet $F\\subset S^3$ be a closed surface. Then every knot $K\\subset S^3$ satisfies:\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1,\n\\]\nwhere $t(K)$ is the tunnel number of $K$, and $\\delta(F)=g(M_1)+g(M_2)-g(F)$ is the Heegaard deficiency of $F$.\n\\end{theorem}", + "post_theorem_intro_text_len": 2183, + "post_theorem_intro_text": "The proof relies on a chain of inequalities relating the surface crossing number $c(K;F)$, the surface ascending number $a(K;F)$, and the surface bridge number $b(K;F)$:\n$$\\frac{c(K;F)-1}{2} \\ge a(K;F) \\ge b(K;F)-1 \\ge t(K)-\\delta(F).$$\nThis sequence demonstrates that the geometric obstruction to simplifying a knot diagram is bounded below by a combination of the knot exterior's complexity, $t(K)$, and the ambient surface's Heegaard deficiency, $\\delta(F)$. Unlike the classical planar crossing number, $c(K;F)$ directly reflects the Heegaard-theoretic complexity of the knot.\n\nFurthermore, we prove that the linear lower bound in Theorem \\ref{thm:main_intro} is asymptotically optimal. For iterated connected sums $K_m$ of a prime knot (such as the trefoil), we obtain the linear growth estimate:\n$$c(K_m;F) \\ge 2m - 2\\delta(F) + 1.$$\nPairing this with a general upper bound derived from the subadditivity of the planar crossing number yields $c(K_m;F) = \\Theta(t(K_m))$. This confirms that no general lower bound of higher order exists, and that the surface crossing number grows linearly with respect to the tunnel number.\n\n\\subsection{Organization of the Paper}\nThe paper is organized as follows. Section 2 recalls basic definitions and establishes the monotonicity of $c(K;F)$ under surface compression. Section 3 states the main results. Section 4 proves the fundamental inequality using the ascending number, bridge position, and the $(g,b)$-decomposition framework of \\cite{MSY}. Section 5 provides the explicit construction of the knot family $K_m$ to demonstrate the asymptotic sharpness of our lower bound. Finally, Section 6 discusses further implications and open questions. These include the hierarchy of surface invariants, their behavior under connected sums, and a comparison with the surface trunk. We outline generalizations to surfaces with boundary and spatial graphs, and remark on the necessity of the genuine manifold condition compared to singular spaces \\cite{D,G}. The section concludes with questions regarding the asymptotic behavior for prime knots, lower bounds by the Haken number, and the minimality of alternating projections on closed surfaces.", + "sketch": "To prove Theorem~\\ref{thm:main_intro}, the argument “relies on a chain of inequalities relating the surface crossing number $c(K;F)$, the surface ascending number $a(K;F)$, and the surface bridge number $b(K;F)$”:\n\\[\n\\frac{c(K;F)-1}{2} \\ge a(K;F) \\ge b(K;F)-1 \\ge t(K)-\\delta(F).\n\\]\nThis sequence yields the stated lower bound for $c(K;F)$ in terms of “a combination of the knot exterior's complexity, $t(K)$, and the ambient surface's Heegaard deficiency, $\\delta(F)$.”\n\nFor optimality, the introduction says they “prove that the linear lower bound in Theorem~\\ref{thm:main_intro} is asymptotically optimal” by taking iterated connected sums $K_m$ of a prime knot and obtaining\n\\[\nc(K_m;F) \\ge 2m - 2\\delta(F) + 1.\n\\]\n“Pairing this with a general upper bound derived from the subadditivity of the planar crossing number yields $c(K_m;F)=\\Theta(t(K_m))$,” so “no general lower bound of higher order exists,” and $c(K;F)$ grows linearly with tunnel number.", + "expanded_sketch": "To prove the main theorem, the argument “relies on a chain of inequalities relating the surface crossing number $c(K;F)$, the surface ascending number $a(K;F)$, and the surface bridge number $b(K;F)$”:\n\\[\n\\frac{c(K;F)-1}{2} \\ge a(K;F) \\ge b(K;F)-1 \\ge t(K)-\\delta(F).\n\\]\nThis sequence yields the stated lower bound for $c(K;F)$ in terms of “a combination of the knot exterior's complexity, $t(K)$, and the ambient surface's Heegaard deficiency, $\\delta(F)$.”\n\nFor optimality, the introduction says they “prove that the linear lower bound in the main theorem is asymptotically optimal” by taking iterated connected sums $K_m$ of a prime knot and obtaining\n\\[\nc(K_m;F) \\ge 2m - 2\\delta(F) + 1.\n\\]\n“Pairing this with a general upper bound derived from the subadditivity of the planar crossing number yields $c(K_m;F)=\\Theta(t(K_m))$,” so “no general lower bound of higher order exists,” and $c(K;F)$ grows linearly with tunnel number.", + "expanded_theorem": "[Fundamental inequality]\\label{thm:main_intro}\nLet $F\\subset S^3$ be a closed surface. Then every knot $K\\subset S^3$ satisfies:\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1,\n\\]\nwhere $t(K)$ is the tunnel number of $K$, and $\\delta(F)=g(M_1)+g(M_2)-g(F)$ is the Heegaard deficiency of $F$.", + "theorem_type": [ + "Inequality or Bound", + "Universal" + ], + "mcq": { + "question": "Let $F\\subset S^3$ be any closed surface, and write $S^3=M_1\\cup_F M_2$. Define the Heegaard deficiency of $F$ by $\\delta(F)=g(M_1)+g(M_2)-g(F)$, where $g(\\cdot)$ denotes genus. For a knot $K\\subset S^3$, let $t(K)$ be its tunnel number, and let $c(K;F)$ denote the surface crossing number of $K$ with respect to $F$, meaning the minimum number of crossings among all regular diagrams obtained by isotoping $K$ into a regular neighborhood of $F$. Which statement holds for every such closed surface $F$ and every knot $K\\subset S^3$?", + "correct_choice": { + "label": "A", + "text": "$c(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1$." + }, + "choices": [ + { + "label": "B", + "text": "$c(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)$." + }, + { + "label": "C", + "text": "$c(K;F) \\ge t(K)-\\delta(F)+1$." + }, + { + "label": "D", + "text": "$c(K;F) \\ge 2\\bigl(t(K)+\\delta(F)\\bigr)+1$." + }, + { + "label": "E", + "text": "$c(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+3$." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "constant-term-from-half-integer-chain", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped leading factor 2 while keeping the same dependence on $t(K)$ and $\\delta(F)$", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "sign of Heegaard deficiency term", + "template_used": "property_confusion" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "sharp additive constant in the final bound", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives definitions and asks for the correct universal estimate, but it does not explicitly or implicitly reveal the exact bound. The correct additive constant and the use of \u0003(F) must be inferred from the options." + }, + "TAS": { + "score": 1, + "justification": "This is largely a theorem-recall item: the task is to identify the exact published inequality among close variants. It is not a pure tautology because the choices compete on sharpness, constants, and the correct invariant, but it still mainly tests recognition of the theorem statement." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is required to compare a sharp bound, a weaker true bound, and false strengthenings or parameter substitutions. However, the item does not demand substantial derivation; it primarily rewards remembering or recognizing the exact estimate." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong: one is a stronger-but-false variant, one is a weaker true statement, one swaps in the wrong invariant, and one suggests an unjustified uniform constant. These are plausible and mathematically meaningful failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with no answer leakage and high-quality distractors, but it mainly tests recognition of a theorem statement rather than deeper generative reasoning." + } + }, + { + "id": "2602.21659v1", + "paper_link": "http://arxiv.org/abs/2602.21659v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "[Fundamental inequality]\\label{thm:main_intro}\nLet $F\\subset S^3$ be a closed surface. Then every knot $K\\subset S^3$ satisfies:\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1,\n\\]\nwhere $t(K)$ is the tunnel number of $K$, and $\\delta(F)=g(M_1)+g(M_2)-g(F)$ is the Heegaard deficiency of $F$.", + "start_pos": 2789, + "end_pos": 3106, + "label": "thm:main_intro" + }, + "ref_dict": { + "thm:main_intro": "\\begin{theorem}[Fundamental inequality]\\label{thm:main_intro}\nLet $F\\subset S^3$ be a closed surface. Then every knot $K\\subset S^3$ satisfies:\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1,\n\\]\nwhere $t(K)$ is the tunnel number of $K$, and $\\delta(F)=g(M_1)+g(M_2)-g(F)$ is the Heegaard deficiency of $F$.\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 1144, + "pre_theorem_intro_text": "\\subsection{Background and Motivation}\nLow-dimensional topology quantifies knot complexity through numerical invariants. While classical invariants, such as the standard crossing number, rely on planar diagrams, these two-dimensional projections compress spatial information which does not fully reflect the global topology of the knot exterior within the ambient space $S^3$.\n\nTo capture this three-dimensional structure, we extend the notion of diagrammatic complexity to arbitrary closed surfaces. For a knot $K$ and a closed surface $F \\subset S^3$, we define the \\emph{surface crossing number} $c(K;F)$ as the minimal number of crossings among all regular diagrams of $K$ obtained by isotoping the knot into a regular neighborhood of $F$. \n\nIn this paper, we establish an inequality demonstrating that $c(K;F)$ is bounded below by the tunnel number of $K$ and the Heegaard deficiency of $F$.\n\n\\subsection{Main Results}\nThe objective of this paper is to establish a quantitative relationship between a knot's three-dimensional complexity, measured via its tunnel number, and its diagrammatic complexity on an arbitrary fixed closed surface.", + "context": "\\subsection{Background and Motivation}\nLow-dimensional topology quantifies knot complexity through numerical invariants. While classical invariants, such as the standard crossing number, rely on planar diagrams, these two-dimensional projections compress spatial information which does not fully reflect the global topology of the knot exterior within the ambient space $S^3$.\n\nTo capture this three-dimensional structure, we extend the notion of diagrammatic complexity to arbitrary closed surfaces. For a knot $K$ and a closed surface $F \\subset S^3$, we define the \\emph{surface crossing number} $c(K;F)$ as the minimal number of crossings among all regular diagrams of $K$ obtained by isotoping the knot into a regular neighborhood of $F$.\n\nIn this paper, we establish an inequality demonstrating that $c(K;F)$ is bounded below by the tunnel number of $K$ and the Heegaard deficiency of $F$.\n\n\\subsection{Main Results}\nThe objective of this paper is to establish a quantitative relationship between a knot's three-dimensional complexity, measured via its tunnel number, and its diagrammatic complexity on an arbitrary fixed closed surface.", + "full_context": "\\subsection{Background and Motivation}\nLow-dimensional topology quantifies knot complexity through numerical invariants. While classical invariants, such as the standard crossing number, rely on planar diagrams, these two-dimensional projections compress spatial information which does not fully reflect the global topology of the knot exterior within the ambient space $S^3$.\n\nTo capture this three-dimensional structure, we extend the notion of diagrammatic complexity to arbitrary closed surfaces. For a knot $K$ and a closed surface $F \\subset S^3$, we define the \\emph{surface crossing number} $c(K;F)$ as the minimal number of crossings among all regular diagrams of $K$ obtained by isotoping the knot into a regular neighborhood of $F$.\n\nIn this paper, we establish an inequality demonstrating that $c(K;F)$ is bounded below by the tunnel number of $K$ and the Heegaard deficiency of $F$.\n\n\\subsection{Main Results}\nThe objective of this paper is to establish a quantitative relationship between a knot's three-dimensional complexity, measured via its tunnel number, and its diagrammatic complexity on an arbitrary fixed closed surface.\n\n\\documentclass[11pt]{amsart}\n\\usepackage{amsmath,amssymb,amsthm}\n\\usepackage{hyperref}\n\\usepackage{tikz}\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{remark}[theorem]{Remark}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{question}[theorem]{Question}\n\\title[Crossing Numbers of Knots on Closed Surfaces]\n{Crossing Numbers of Knots on Closed Surfaces}\n\\author{Makoto Ozawa}\n\\address{Department of Natural Sciences, Faculty of Arts and Sciences, Komazawa University, 1-23-1 Komazawa, Setagaya-ku, Tokyo, 154-8525, Japan}\n\\email{w3c@komazawa-u.ac.jp}\n\\date{\\today}\n\\dedicatory{Dedicated to Professor Kanji Morimoto on the occasion of his retirement.}\n\\keywords{knot, surface crossing number, tunnel number, Heegaard deficiency, Heegaard splitting, surface bridge number, surface ascending number}\n\\begin{document}\n\\begin{abstract}\nLet $c(K;F)$ denote the surface crossing number of a knot $K$ with respect\nto a closed surface $F \\subset S^{3}$. We establish the lower bound\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1,\n\\]\nwhere $t(K)$ is the tunnel number of $K$ and $\\delta(F)$ is the Heegaard\ndeficiency of $F$. In particular, for any fixed closed surface $F$, the\nsurface crossing number $c(K;F)$ is unbounded over all knots $K$.\nFurthermore, we construct a family of knots $K_m$ demonstrating that $c(K_m;F) = \\Theta(t(K_m))$, which shows that this lower bound is asymptotically sharp.\n\\end{abstract}\n\\maketitle\n\n\\subsection{Main Results}\nThe objective of this paper is to establish a quantitative relationship between a knot's three-dimensional complexity, measured via its tunnel number, and its diagrammatic complexity on an arbitrary fixed closed surface.\n\nThe proof relies on a chain of inequalities relating the surface crossing number $c(K;F)$, the surface ascending number $a(K;F)$, and the surface bridge number $b(K;F)$:\n$$\\frac{c(K;F)-1}{2} \\ge a(K;F) \\ge b(K;F)-1 \\ge t(K)-\\delta(F).$$\nThis sequence demonstrates that the geometric obstruction to simplifying a knot diagram is bounded below by a combination of the knot exterior's complexity, $t(K)$, and the ambient surface's Heegaard deficiency, $\\delta(F)$. Unlike the classical planar crossing number, $c(K;F)$ directly reflects the Heegaard-theoretic complexity of the knot.\n\nFurthermore, we prove that the linear lower bound in Theorem \\ref{thm:main_intro} is asymptotically optimal. For iterated connected sums $K_m$ of a prime knot (such as the trefoil), we obtain the linear growth estimate:\n$$c(K_m;F) \\ge 2m - 2\\delta(F) + 1.$$\nPairing this with a general upper bound derived from the subadditivity of the planar crossing number yields $c(K_m;F) = \\Theta(t(K_m))$. This confirms that no general lower bound of higher order exists, and that the surface crossing number grows linearly with respect to the tunnel number.\n\n\\begin{theorem}[Fundamental inequality]\\label{thm:fundamental}\nLet $F\\subset S^{3}$ be a closed separating surface, and let $K\\subset S^{3}$ be a knot. Then\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1.\n\\]\n\\end{theorem}\n\nApplying the fundamental inequality to $K_m$ yields:\n\\begin{proposition}\nFor every closed surface $F\\subset S^{3}$,\n\\[\nc(K_m;F) \\ge 2\\bigl(m-\\delta(F)\\bigr)+1 \\to \\infty \\quad \\text{as } m \\to \\infty.\n\\]\n\\end{proposition}\n\n\\section{Proof of the Main Inequality}\nIn this section, we prove Theorem \\ref{thm:fundamental} by establishing the chain\n$$\\frac{c(K;F)-1}{2} \\ge a(K;F) \\ge b(K;F)-1 \\ge t(K)-\\delta(F).$$\nThroughout the proof, let $F\\subset S^{3}$ be a closed separating surface with decomposition\n$S^{3}=M_{1}\\cup_{F}M_{2}$.\n\n\\subsection{Generalization to spatial graphs}\nThe framework extends to spatial graphs. A spatial graph $G$ is a 1-dimensional CW complex embedded in $S^{3}$. The key arguments rely on the 1-dimensionality of the object: in the proof of Lemma \\ref{lem:amalgamation}, a general position argument shows that the 2-dimensional attaching disks of the 1-handles can be made disjoint from $G$. The tunnel number $t(G)$, bridge position, and ascending number extend to this setting, and one expects a lower bound of the form\n\\[\nc(G;F) \\ge 2\\bigl(t(G)-\\delta(F)\\bigr) + C_G,\n\\]\nwhere $C_G$ is a constant depending on the combinatorial structure of $G$ (such as its Euler characteristic or vertex degrees).\n\n\\begin{theorem}[Fundamental inequality]\\label{thm:main_intro}\nLet $F\\subset S^3$ be a closed surface. Then every knot $K\\subset S^3$ satisfies:\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1,\n\\]\nwhere $t(K)$ is the tunnel number of $K$, and $\\delta(F)=g(M_1)+g(M_2)-g(F)$ is the Heegaard deficiency of $F$.\n\\end{theorem}", + "post_theorem_intro_text_len": 2183, + "post_theorem_intro_text": "The proof relies on a chain of inequalities relating the surface crossing number $c(K;F)$, the surface ascending number $a(K;F)$, and the surface bridge number $b(K;F)$:\n$$\\frac{c(K;F)-1}{2} \\ge a(K;F) \\ge b(K;F)-1 \\ge t(K)-\\delta(F).$$\nThis sequence demonstrates that the geometric obstruction to simplifying a knot diagram is bounded below by a combination of the knot exterior's complexity, $t(K)$, and the ambient surface's Heegaard deficiency, $\\delta(F)$. Unlike the classical planar crossing number, $c(K;F)$ directly reflects the Heegaard-theoretic complexity of the knot.\n\nFurthermore, we prove that the linear lower bound in Theorem \\ref{thm:main_intro} is asymptotically optimal. For iterated connected sums $K_m$ of a prime knot (such as the trefoil), we obtain the linear growth estimate:\n$$c(K_m;F) \\ge 2m - 2\\delta(F) + 1.$$\nPairing this with a general upper bound derived from the subadditivity of the planar crossing number yields $c(K_m;F) = \\Theta(t(K_m))$. This confirms that no general lower bound of higher order exists, and that the surface crossing number grows linearly with respect to the tunnel number.\n\n\\subsection{Organization of the Paper}\nThe paper is organized as follows. Section 2 recalls basic definitions and establishes the monotonicity of $c(K;F)$ under surface compression. Section 3 states the main results. Section 4 proves the fundamental inequality using the ascending number, bridge position, and the $(g,b)$-decomposition framework of \\cite{MSY}. Section 5 provides the explicit construction of the knot family $K_m$ to demonstrate the asymptotic sharpness of our lower bound. Finally, Section 6 discusses further implications and open questions. These include the hierarchy of surface invariants, their behavior under connected sums, and a comparison with the surface trunk. We outline generalizations to surfaces with boundary and spatial graphs, and remark on the necessity of the genuine manifold condition compared to singular spaces \\cite{D,G}. The section concludes with questions regarding the asymptotic behavior for prime knots, lower bounds by the Haken number, and the minimality of alternating projections on closed surfaces.", + "sketch": "To prove Theorem~\\ref{thm:main_intro}, the argument “relies on a chain of inequalities relating the surface crossing number $c(K;F)$, the surface ascending number $a(K;F)$, and the surface bridge number $b(K;F)$”:\n\\[\n\\frac{c(K;F)-1}{2} \\ge a(K;F) \\ge b(K;F)-1 \\ge t(K)-\\delta(F).\n\\]\nThis sequence yields the stated lower bound for $c(K;F)$ in terms of “a combination of the knot exterior's complexity, $t(K)$, and the ambient surface's Heegaard deficiency, $\\delta(F)$.”\n\nFor optimality, the introduction says they “prove that the linear lower bound in Theorem~\\ref{thm:main_intro} is asymptotically optimal” by taking iterated connected sums $K_m$ of a prime knot and obtaining\n\\[\nc(K_m;F) \\ge 2m - 2\\delta(F) + 1.\n\\]\n“Pairing this with a general upper bound derived from the subadditivity of the planar crossing number yields $c(K_m;F)=\\Theta(t(K_m))$,” so “no general lower bound of higher order exists,” and $c(K;F)$ grows linearly with tunnel number.", + "expanded_sketch": "To prove the main theorem, the argument “relies on a chain of inequalities relating the surface crossing number $c(K;F)$, the surface ascending number $a(K;F)$, and the surface bridge number $b(K;F)$”:\n\\[\n\\frac{c(K;F)-1}{2} \\ge a(K;F) \\ge b(K;F)-1 \\ge t(K)-\\delta(F).\n\\]\nThis sequence yields the stated lower bound for $c(K;F)$ in terms of “a combination of the knot exterior's complexity, $t(K)$, and the ambient surface's Heegaard deficiency, $\\delta(F)$.”\n\nFor optimality, the introduction says they “prove that the linear lower bound in the main theorem is asymptotically optimal” by taking iterated connected sums $K_m$ of a prime knot and obtaining\n\\[\nc(K_m;F) \\ge 2m - 2\\delta(F) + 1.\n\\]\n“Pairing this with a general upper bound derived from the subadditivity of the planar crossing number yields $c(K_m;F)=\\Theta(t(K_m))$,” so “no general lower bound of higher order exists,” and $c(K;F)$ grows linearly with tunnel number.", + "expanded_theorem": "[Fundamental inequality]\\label{thm:main_intro}\nLet $F\\subset S^3$ be a closed surface. Then every knot $K\\subset S^3$ satisfies:\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1,\n\\]\nwhere $t(K)$ is the tunnel number of $K$, and $\\delta(F)=g(M_1)+g(M_2)-g(F)$ is the Heegaard deficiency of $F$.", + "theorem_type": [ + "Inequality or Bound", + "Universal" + ], + "mcq": { + "question": "Let $F\\subset S^3$ be any closed surface, and write $S^3=M_1\\cup_F M_2$. Define the Heegaard deficiency of $F$ by $\\delta(F)=g(M_1)+g(M_2)-g(F)$, where $g(\\cdot)$ denotes genus. For a knot $K\\subset S^3$, let $t(K)$ be its tunnel number, and let $c(K;F)$ denote the surface crossing number of $K$ with respect to $F$, meaning the minimum number of crossings among all regular diagrams obtained by isotoping $K$ into a regular neighborhood of $F$. Which statement holds for every such closed surface $F$ and every knot $K\\subset S^3$?", + "correct_choice": { + "label": "A", + "text": "$c(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1$." + }, + "choices": [ + { + "label": "B", + "text": "$c(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)$." + }, + { + "label": "C", + "text": "$c(K;F) \\ge t(K)-\\delta(F)+1$." + }, + { + "label": "D", + "text": "$c(K;F) \\ge 2\\bigl(t(K)+\\delta(F)\\bigr)+1$." + }, + { + "label": "E", + "text": "$c(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+3$." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "constant-term-from-half-integer-chain", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped leading factor 2 while keeping the same dependence on $t(K)$ and $\\delta(F)$", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "sign of Heegaard deficiency term", + "template_used": "property_confusion" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "sharp additive constant in the final bound", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem only defines the quantities and asks for a universal inequality; it does not explicitly reveal the correct bound or give strong hints about the exact coefficients or constant term." + }, + "TAS": { + "score": 1, + "justification": "This is close to a theorem-statement recognition item: the task is essentially to identify the correct universal inequality. However, the nearby alternative bounds mean it is not a completely verbatim restatement." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish the sharp bound from weaker true or false variants, especially because the options differ by subtle coefficient/sign/constant changes. Still, the item mainly tests precise recall or recognition rather than substantial derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target realistic errors: dropping the factor of 2, changing the sign of the deficiency term, or altering the additive constant. Including a weaker true statement also improves quality." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it remains primarily a theorem-recognition question rather than one that strongly tests generative mathematical reasoning." + } + }, + { + "id": "2602.22167v1", + "paper_link": "http://arxiv.org/abs/2602.22167v1", + "theorems_cnt": 5, + "theorem": { + "env_name": "theorem", + "content": "[Konyagin \\cite{Kon}]\nLet $\\varepsilon>0$ and suppose $H_i>p^{1/4+\\varepsilon}$ for all $1\\leq i\\leq n$. Then\n\\[\n\\left|\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\\right|\n\\ll_{n,\\varepsilon} p^{-\\varepsilon^2/2}|B|.\n\\]", + "start_pos": 5390, + "end_pos": 5631, + "label": null + }, + "ref_dict": { + "l2": "\\begin{Lemma}\\label{l2}\nWe have\n\\[\nL_{2} \\;\\ll\\; |B|^{2}(\\log p)^{n}.\n\\]\n\\end{Lemma}", + "l1": "\\begin{Lemma}\\label{l1}\nWe have\n\\[\nL_{1} \\;\\ll\\; |B|^{2}\\log p.\n\\]\n\\end{Lemma}", + "KT": "\\begin{Proposition}[Katz]\\label{KT}\nLet $\\chi$ be a non-trivial multiplicative character of $\\mathbb{F}_{p^n}$ and let\n$g\\in\\mathbb{F}_{p^n}$ generate the extension\n$\\mathbb{F}_{p^n}=\\mathbb{F}_p(g)$. Then, for any interval\n$I\\subseteq [1,p]\\cap\\mathbb{Z}$,\n\\[\n\\left|\\sum_{t\\in I}\\chi(g+t)\\right|\n\\le c(n)\\sqrt{p}\\log p .\n\\]\n\\end{Proposition}", + "MT": "\\begin{theorem}[Main Theorem]\\label{MT}\nLet $n\\geq2$, let $\\chi$ be a nontrivial multiplicative character of $\\mathbb{F}_{p^n}$, and suppose\n\\[\n|B|\\geq p^{n(1/4+\\varepsilon)},\\qquad\nH_1\\leq H_2\\leq\\cdots\\leq H_n\\leq \\sqrt{p/2}.\n\\]\nThen\n\\[\n\\left|\\sum_{x\\in B}\\chi(x)\\right|\n\\ll |B|\\, p^{-\\varepsilon^2\n\\frac{1-\\frac{1}{2n}}{\\left(1+\\frac{1}{4n}\\right)\\left(2-\\frac{1}{2n}\\right)}}.\n\\]\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 2571, + "pre_theorem_intro_text": "Let $p$ be a prime, and let $\\mathbb{F}_{p^{n}}$ denote the finite field with $p^{n}$ elements. Let us define intervals of length $H_i$ in $\\mathbb{Z}$ as \n\\[ I_i=[N_i+1,N_i+H_i]\\cap \\mathbb{Z}.\\] Fix a basis $\\{\\omega_{1},\\ldots,\\omega_{n}\\}$ of $\\mathbb{F}_{p^{n}}$ over $\\mathbb{F}_{p}$. For integers $N_i$ and $H_i$ satisfying\n\\[\n1 \\leq H_i \\leq p \\qquad (1 \\leq i \\leq n),\n\\]\nwe define the box\n\\[\nB=\\left\\{ \\sum_{i=1}^{n} x_i\\omega_i :\nN_i+1 \\leq x_i \\leq N_i+H_i,\\; 1 \\leq i \\leq n \\right\\}\n\\subseteq \\mathbb{F}_{p^{n}}.\n\\]\nand the volume of the box as \n\\[|B|=H_1\\cdot H_2 \\cdots H_n. \\]\nLet $\\chi$ be a multiplicative character on $\\mathbb{F}_{p^n}$.\n\n\\medskip\n\nWe begin with some historical background. In the case $n=1$, Burgess's bound \\cite{B1} remains the strongest result to date. It asserts that for every $\\varepsilon>0$ there exists $\\delta>0$ such that\n\\[\n\\left|\\sum_{x=N+1}^{N+H}\\chi(x)\\right|\\ll_{\\varepsilon} p^{-\\delta}H\n\\]\nwhenever $H\\geq p^{1/4+\\varepsilon}$. In subsequent work \\cite{B2}, Burgess also obtained an analogue of this bound for $n=2$ in the case of certain special bases. Karatsuba \\cite{K} later extended these ideas to general finite fields $\\mathbb{F}_{p^n}$, under the assumption that the chosen basis arises from a root of an irreducible polynomial of degree $n$ over $\\mathbb{F}_p$.\n\nFor arbitrary bases, the first nontrivial bounds were obtained by Davenport and Lewis \\cite{DL}. They studied character sums over $\\mathbb{F}_{p^n}$ of the form\n\\[\n\\sum_{x_1\\in I_1,\\ldots,x_d\\in I_n} \\chi\\!\\left(x_1\\omega_1+\\cdots+x_n\\omega_n\\right),\n\\]\nwhere $\\{\\omega_1,\\ldots,\\omega_n\\}$ is a fixed basis of $\\mathbb{F}_{p^d}$ over $\\mathbb{F}_p$ and $I_i$ are intervals in $\\mathbb{Z}$. They proved that \n\\[\n\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\n=O\\!\\left(H_1\\cdots H_n\\,p^{-\\delta(\\varepsilon)}\\right),\n\\]\nprovided that $H_i=p^{\\rho_n+\\varepsilon}$ for all $i$, where\n\\[\n\\rho_d=\\frac12-\\frac{1}{2(d+1)}.\n\\]\n\nThis result was later improved by M. C. Chang \\cite{C1}, who showed that\n\\[\n\\left|\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\\right|\n\\ll_{d,\\varepsilon} p^{-\\varepsilon^2/4}|B|,\n\\]\nwhenever $|H_i|>p^{2/5+\\varepsilon}$ for all $1\\leq i\\leq d$. In particular, this improves upon the Davenport–Lewis bound for $d\\geq5$. Chang \\cite{Cha} also obtained Burgess-strength bounds in each coordinate for $d=2$, assuming $H_1,H_2>p^{1/4+\\varepsilon}$.\n\nSubsequently, S. Konyagin generalized Chang's two-dimensional result to arbitrary finite fields of degree $n$ over $\\mathbb{F}_p$, establishing Burgess-strength estimates in every coordinate.", + "context": "Let $p$ be a prime, and let $\\mathbb{F}_{p^{n}}$ denote the finite field with $p^{n}$ elements. Let us define intervals of length $H_i$ in $\\mathbb{Z}$ as \n\\[ I_i=[N_i+1,N_i+H_i]\\cap \\mathbb{Z}.\\] Fix a basis $\\{\\omega_{1},\\ldots,\\omega_{n}\\}$ of $\\mathbb{F}_{p^{n}}$ over $\\mathbb{F}_{p}$. For integers $N_i$ and $H_i$ satisfying\n\\[\n1 \\leq H_i \\leq p \\qquad (1 \\leq i \\leq n),\n\\]\nwe define the box\n\\[\nB=\\left\\{ \\sum_{i=1}^{n} x_i\\omega_i :\nN_i+1 \\leq x_i \\leq N_i+H_i,\\; 1 \\leq i \\leq n \\right\\}\n\\subseteq \\mathbb{F}_{p^{n}}.\n\\]\nand the volume of the box as \n\\[|B|=H_1\\cdot H_2 \\cdots H_n. \\]\nLet $\\chi$ be a multiplicative character on $\\mathbb{F}_{p^n}$.\n\nWe begin with some historical background. In the case $n=1$, Burgess's bound \\cite{B1} remains the strongest result to date. It asserts that for every $\\varepsilon>0$ there exists $\\delta>0$ such that\n\\[\n\\left|\\sum_{x=N+1}^{N+H}\\chi(x)\\right|\\ll_{\\varepsilon} p^{-\\delta}H\n\\]\nwhenever $H\\geq p^{1/4+\\varepsilon}$. In subsequent work \\cite{B2}, Burgess also obtained an analogue of this bound for $n=2$ in the case of certain special bases. Karatsuba \\cite{K} later extended these ideas to general finite fields $\\mathbb{F}_{p^n}$, under the assumption that the chosen basis arises from a root of an irreducible polynomial of degree $n$ over $\\mathbb{F}_p$.\n\nFor arbitrary bases, the first nontrivial bounds were obtained by Davenport and Lewis \\cite{DL}. They studied character sums over $\\mathbb{F}_{p^n}$ of the form\n\\[\n\\sum_{x_1\\in I_1,\\ldots,x_d\\in I_n} \\chi\\!\\left(x_1\\omega_1+\\cdots+x_n\\omega_n\\right),\n\\]\nwhere $\\{\\omega_1,\\ldots,\\omega_n\\}$ is a fixed basis of $\\mathbb{F}_{p^d}$ over $\\mathbb{F}_p$ and $I_i$ are intervals in $\\mathbb{Z}$. They proved that \n\\[\n\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\n=O\\!\\left(H_1\\cdots H_n\\,p^{-\\delta(\\varepsilon)}\\right),\n\\]\nprovided that $H_i=p^{\\rho_n+\\varepsilon}$ for all $i$, where\n\\[\n\\rho_d=\\frac12-\\frac{1}{2(d+1)}.\n\\]\n\nThis result was later improved by M. C. Chang \\cite{C1}, who showed that\n\\[\n\\left|\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\\right|\n\\ll_{d,\\varepsilon} p^{-\\varepsilon^2/4}|B|,\n\\]\nwhenever $|H_i|>p^{2/5+\\varepsilon}$ for all $1\\leq i\\leq d$. In particular, this improves upon the Davenport–Lewis bound for $d\\geq5$. Chang \\cite{Cha} also obtained Burgess-strength bounds in each coordinate for $d=2$, assuming $H_1,H_2>p^{1/4+\\varepsilon}$.\n\nSubsequently, S. Konyagin generalized Chang's two-dimensional result to arbitrary finite fields of degree $n$ over $\\mathbb{F}_p$, establishing Burgess-strength estimates in every coordinate.", + "full_context": "Let $p$ be a prime, and let $\\mathbb{F}_{p^{n}}$ denote the finite field with $p^{n}$ elements. Let us define intervals of length $H_i$ in $\\mathbb{Z}$ as \n\\[ I_i=[N_i+1,N_i+H_i]\\cap \\mathbb{Z}.\\] Fix a basis $\\{\\omega_{1},\\ldots,\\omega_{n}\\}$ of $\\mathbb{F}_{p^{n}}$ over $\\mathbb{F}_{p}$. For integers $N_i$ and $H_i$ satisfying\n\\[\n1 \\leq H_i \\leq p \\qquad (1 \\leq i \\leq n),\n\\]\nwe define the box\n\\[\nB=\\left\\{ \\sum_{i=1}^{n} x_i\\omega_i :\nN_i+1 \\leq x_i \\leq N_i+H_i,\\; 1 \\leq i \\leq n \\right\\}\n\\subseteq \\mathbb{F}_{p^{n}}.\n\\]\nand the volume of the box as \n\\[|B|=H_1\\cdot H_2 \\cdots H_n. \\]\nLet $\\chi$ be a multiplicative character on $\\mathbb{F}_{p^n}$.\n\nWe begin with some historical background. In the case $n=1$, Burgess's bound \\cite{B1} remains the strongest result to date. It asserts that for every $\\varepsilon>0$ there exists $\\delta>0$ such that\n\\[\n\\left|\\sum_{x=N+1}^{N+H}\\chi(x)\\right|\\ll_{\\varepsilon} p^{-\\delta}H\n\\]\nwhenever $H\\geq p^{1/4+\\varepsilon}$. In subsequent work \\cite{B2}, Burgess also obtained an analogue of this bound for $n=2$ in the case of certain special bases. Karatsuba \\cite{K} later extended these ideas to general finite fields $\\mathbb{F}_{p^n}$, under the assumption that the chosen basis arises from a root of an irreducible polynomial of degree $n$ over $\\mathbb{F}_p$.\n\nFor arbitrary bases, the first nontrivial bounds were obtained by Davenport and Lewis \\cite{DL}. They studied character sums over $\\mathbb{F}_{p^n}$ of the form\n\\[\n\\sum_{x_1\\in I_1,\\ldots,x_d\\in I_n} \\chi\\!\\left(x_1\\omega_1+\\cdots+x_n\\omega_n\\right),\n\\]\nwhere $\\{\\omega_1,\\ldots,\\omega_n\\}$ is a fixed basis of $\\mathbb{F}_{p^d}$ over $\\mathbb{F}_p$ and $I_i$ are intervals in $\\mathbb{Z}$. They proved that \n\\[\n\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\n=O\\!\\left(H_1\\cdots H_n\\,p^{-\\delta(\\varepsilon)}\\right),\n\\]\nprovided that $H_i=p^{\\rho_n+\\varepsilon}$ for all $i$, where\n\\[\n\\rho_d=\\frac12-\\frac{1}{2(d+1)}.\n\\]\n\nThis result was later improved by M. C. Chang \\cite{C1}, who showed that\n\\[\n\\left|\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\\right|\n\\ll_{d,\\varepsilon} p^{-\\varepsilon^2/4}|B|,\n\\]\nwhenever $|H_i|>p^{2/5+\\varepsilon}$ for all $1\\leq i\\leq d$. In particular, this improves upon the Davenport–Lewis bound for $d\\geq5$. Chang \\cite{Cha} also obtained Burgess-strength bounds in each coordinate for $d=2$, assuming $H_1,H_2>p^{1/4+\\varepsilon}$.\n\nSubsequently, S. Konyagin generalized Chang's two-dimensional result to arbitrary finite fields of degree $n$ over $\\mathbb{F}_p$, establishing Burgess-strength estimates in every coordinate.\n\nWe begin with some historical background. In the case $n=1$, Burgess's bound \\cite{B1} remains the strongest result to date. It asserts that for every $\\varepsilon>0$ there exists $\\delta>0$ such that\n\\[\n\\left|\\sum_{x=N+1}^{N+H}\\chi(x)\\right|\\ll_{\\varepsilon} p^{-\\delta}H\n\\]\nwhenever $H\\geq p^{1/4+\\varepsilon}$. In subsequent work \\cite{B2}, Burgess also obtained an analogue of this bound for $n=2$ in the case of certain special bases. Karatsuba \\cite{K} later extended these ideas to general finite fields $\\mathbb{F}_{p^n}$, under the assumption that the chosen basis arises from a root of an irreducible polynomial of degree $n$ over $\\mathbb{F}_p$.\n\nFor arbitrary bases, the first nontrivial bounds were obtained by Davenport and Lewis \\cite{DL}. They studied character sums over $\\mathbb{F}_{p^n}$ of the form\n\\[\n\\sum_{x_1\\in I_1,\\ldots,x_d\\in I_n} \\chi\\!\\left(x_1\\omega_1+\\cdots+x_n\\omega_n\\right),\n\\]\nwhere $\\{\\omega_1,\\ldots,\\omega_n\\}$ is a fixed basis of $\\mathbb{F}_{p^d}$ over $\\mathbb{F}_p$ and $I_i$ are intervals in $\\mathbb{Z}$. They proved that \n\\[\n\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\n=O\\!\\left(H_1\\cdots H_n\\,p^{-\\delta(\\varepsilon)}\\right),\n\\]\nprovided that $H_i=p^{\\rho_n+\\varepsilon}$ for all $i$, where\n\\[\n\\rho_d=\\frac12-\\frac{1}{2(d+1)}.\n\\]\n\nThis result was later improved by M. C. Chang \\cite{C1}, who showed that\n\\[\n\\left|\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\\right|\n\\ll_{d,\\varepsilon} p^{-\\varepsilon^2/4}|B|,\n\\]\nwhenever $|H_i|>p^{2/5+\\varepsilon}$ for all $1\\leq i\\leq d$. In particular, this improves upon the Davenport–Lewis bound for $d\\geq5$. Chang \\cite{Cha} also obtained Burgess-strength bounds in each coordinate for $d=2$, assuming $H_1,H_2>p^{1/4+\\varepsilon}$.\n\nSubsequently, S. Konyagin generalized Chang's two-dimensional result to arbitrary finite fields of degree $n$ over $\\mathbb{F}_p$, establishing Burgess-strength estimates in every coordinate.\n\nWhile this theorem provides strong cancellation, it requires each interval to be of Burgess length, precluding situations in which some intervals are short while others are long. This limitation was partially overcome by M. R. Gabdullin \\cite{GB}, who generalized Konyagin's result for $n=2,3$ under a weaker product condition.\n\n\\begin{theorem}[Gabdullin \\cite{GB}]\nLet $n\\in\\{2,3\\}$, let $\\chi$ be a nontrivial multiplicative character of $\\mathbb{F}_{p^n}$, and suppose\n\\[\n|B|\\geq p^{n(1/4+\\varepsilon)}, \\qquad H_1\\leq H_2\\leq\\cdots\\leq H_n.\n\\]\nThen\n\\[\n\\left|\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\\right|\n\\ll_{n,\\varepsilon} |B| p^{-\\varepsilon^2/12}.\n\\]\n\\end{theorem}\n\n\\begin{theorem}[Main Theorem]\\label{MT}\nLet $n\\geq2$, let $\\chi$ be a nontrivial multiplicative character of $\\mathbb{F}_{p^n}$, and suppose\n\\[\n|B|\\geq p^{n(1/4+\\varepsilon)},\\qquad\nH_1\\leq H_2\\leq\\cdots\\leq H_n\\leq \\sqrt{p/2}.\n\\]\nThen\n\\[\n\\left|\\sum_{x\\in B}\\chi(x)\\right|\n\\ll |B|\\, p^{-\\varepsilon^2\n\\frac{1-\\frac{1}{2n}}{\\left(1+\\frac{1}{4n}\\right)\\left(2-\\frac{1}{2n}\\right)}}.\n\\]\n\\end{theorem}\n\nAs a consequence of our Main Theorem, we can now treat sublattices of co-dimension at least $o(n)$, while allowing each interval to have length as small as $p^{1/4+\\varepsilon}$. Moreover, we are no longer restricted to a special choice of basis; the result holds for an arbitrary basis of $\\mathbb{F}_{p^n}$ over $\\mathbb{F}_p$, in contrast with our previous work.\n\\begin{Corollary}\nLet $\\chi$ be a nontrivial multiplicative character of $\\mathbb{F}_{p^n}$, let $I_i \\subset \\mathbb{Z}$ be intervals of lengths $H_i$, and let $\\{\\omega_1,\\ldots,\\omega_n\\}$ be a basis of $\\mathbb{F}_{p^n}$ over $\\mathbb{F}_p$. \nFor every $\\varepsilon>0$, there exists $\\delta(\\varepsilon)>0$ such that\n\\[\n\\left|\\sum_{x_1\\in I_1,\\ldots,x_k\\in I_k}\n\\chi(x_1\\omega_1+\\cdots+x_k\\omega_k)\\right|\n\\ll_{n,\\varepsilon}\nH_1\\cdots H_k\\,p^{-\\delta(\\varepsilon)},\n\\]\nprovided that\n\\[\nH_i \\ge p^{\\frac{n}{k}\\left(\\frac14+\\varepsilon\\right)}\n\\quad (1\\le i\\le k),\n\\]\nwhere\n\\[\n\\delta(\\varepsilon)\n=\n\\varepsilon^2\n\\frac{1-\\frac{1}{2n}}{\\left(1+\\frac{1}{4n}\\right)\\left(2-\\frac{1}{2n}\\right)}.\n\\]\nIn particular, if $k=n-o(n)$, then the condition becomes\n$H_i\\geq p^{\\frac{n}{n-o(n)}(1/4+\\varepsilon)}$, and this exponent tends to $1/4+\\varepsilon$ as $n\\to\\infty$. Thus, in the limit of large dimension, we obtain Burgess-strength bounds in each coordinate.\n\\end{Corollary}", + "post_theorem_intro_text_len": 6012, + "post_theorem_intro_text": "While this theorem provides strong cancellation, it requires each interval to be of Burgess length, precluding situations in which some intervals are short while others are long. This limitation was partially overcome by M. R. Gabdullin \\cite{GB}, who generalized Konyagin's result for $n=2,3$ under a weaker product condition.\n\n\\begin{theorem}[Gabdullin \\cite{GB}]\nLet $n\\in\\{2,3\\}$, let $\\chi$ be a nontrivial multiplicative character of $\\mathbb{F}_{p^n}$, and suppose\n\\[\n|B|\\geq p^{n(1/4+\\varepsilon)}, \\qquad H_1\\leq H_2\\leq\\cdots\\leq H_n.\n\\]\nThen\n\\[\n\\left|\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\\right|\n\\ll_{n,\\varepsilon} |B| p^{-\\varepsilon^2/12}.\n\\]\n\\end{theorem}\n\nIn this paper, we generalize Gabdullin's result to arbitrary dimension $n$. Our main theorem establishes nontrivial cancellation for character sums over general boxes in $\\mathbb{F}_{p^n}$, assuming only a lower bound on the volume of the box.\n\n\\begin{theorem}[Main Theorem]\\label{MT}\nLet $n\\geq2$, let $\\chi$ be a nontrivial multiplicative character of $\\mathbb{F}_{p^n}$, and suppose\n\\[\n|B|\\geq p^{n(1/4+\\varepsilon)},\\qquad\nH_1\\leq H_2\\leq\\cdots\\leq H_n\\leq \\sqrt{p/2}.\n\\]\nThen\n\\[\n\\left|\\sum_{x\\in B}\\chi(x)\\right|\n\\ll |B|\\, p^{-\\varepsilon^2\n\\frac{1-\\frac{1}{2n}}{\\left(1+\\frac{1}{4n}\\right)\\left(2-\\frac{1}{2n}\\right)}}.\n\\]\n\\end{theorem}\n\n\\begin{remark}\nIf the extension degree $n$ is prime, the monotonicity assumption $H_1\\leq\\cdots\\leq H_n\\leq \\sqrt{p/2}$ may be removed. A sketch of the argument is given in Case 3 of the proof of Theorem \\ref{MT}.\n\\end{remark}\n\nIn a previous work \\cite{Ch}, the author studied character sums over sublattices of codimension one in $\\mathbb{F}_{p^n}$, obtaining nontrivial bounds under significantly weaker length conditions.\n\n\\begin{theorem}[Chattopadhyay \\cite{Ch}]\nLet $\\chi$ be a nontrivial multiplicative character of $\\mathbb{F}_{p^d}$, and let $\\omega$ be a generator of $\\mathbb{F}_{p^n}^\\ast$. Given $\\varepsilon>0$, there exists $\\delta(\\varepsilon)>0$ such that if $I_0,\\ldots,I_{n-2}$ are intervals of length $p^{\\rho''_n+\\varepsilon}$, where\n\\[\n\\rho''_n=\\frac{1460-1000n+\\sqrt{1000000n^2-1960000n+490000}}{960},\n\\]\nthen\n\\[\n\\sum_{x_0\\in I_0,\\ldots,x_{n-2}\\in I_{n-2}}\n\\chi(x_0+x_1\\omega+\\cdots+x_{n-2}\\omega^{n-2})\n\\ll p^{-\\delta(\\varepsilon)} |I_0|\\cdots|I_{n-2}|.\n\\]\n\\end{theorem}\n\nThe novelty of this result lied in treating character sums over sublattices of codimension one.\n\n\\medskip\n\nAs a consequence of our Main Theorem, we can now treat sublattices of co-dimension at least $o(n)$, while allowing each interval to have length as small as $p^{1/4+\\varepsilon}$. Moreover, we are no longer restricted to a special choice of basis; the result holds for an arbitrary basis of $\\mathbb{F}_{p^n}$ over $\\mathbb{F}_p$, in contrast with our previous work.\n\\begin{Corollary}\nLet $\\chi$ be a nontrivial multiplicative character of $\\mathbb{F}_{p^n}$, let $I_i \\subset \\mathbb{Z}$ be intervals of lengths $H_i$, and let $\\{\\omega_1,\\ldots,\\omega_n\\}$ be a basis of $\\mathbb{F}_{p^n}$ over $\\mathbb{F}_p$. \nFor every $\\varepsilon>0$, there exists $\\delta(\\varepsilon)>0$ such that\n\\[\n\\left|\\sum_{x_1\\in I_1,\\ldots,x_k\\in I_k}\n\\chi(x_1\\omega_1+\\cdots+x_k\\omega_k)\\right|\n\\ll_{n,\\varepsilon}\nH_1\\cdots H_k\\,p^{-\\delta(\\varepsilon)},\n\\]\nprovided that\n\\[\nH_i \\ge p^{\\frac{n}{k}\\left(\\frac14+\\varepsilon\\right)}\n\\quad (1\\le i\\le k),\n\\]\nwhere\n\\[\n\\delta(\\varepsilon)\n=\n\\varepsilon^2\n\\frac{1-\\frac{1}{2n}}{\\left(1+\\frac{1}{4n}\\right)\\left(2-\\frac{1}{2n}\\right)}.\n\\]\nIn particular, if $k=n-o(n)$, then the condition becomes\n$H_i\\geq p^{\\frac{n}{n-o(n)}(1/4+\\varepsilon)}$, and this exponent tends to $1/4+\\varepsilon$ as $n\\to\\infty$. Thus, in the limit of large dimension, we obtain Burgess-strength bounds in each coordinate.\n\\end{Corollary}\n\n\\medskip\n\nThe proof of the Main Theorem closely follows the approach of \\cite{GB}, employing techniques from \\cite{C1}, \\cite{K} and \\cite{Kon} to estimate character sums over boxes in finite fields whose side lengths are at most $\\sqrt{p/2}$. Our argument begins with the classical additive shifting technique $x\\mapsto x+yz$ introduced by Burgess \\cite{B2}. Applying H\\\"older's inequality twice to the resulting sum of shifted characters produces three terms: one is related to the multiplicative energy of the box, while the remaining terms can be bounded using Weil's bound and trivial estimates. Consequently, achieving a nontrivial saving for the character sum reduces to obtaining a bound on the multiplicative energy of a set in $\\mathbb{F}_{p^n}$ which are described in Lemma \\ref{l1} and Lemma \\ref{l2}. More specifically, we employ tools from the geometry of numbers, in particular the concepts of successive minima and Minkowski's second theorem, together with an auxiliary lemma that provides upper and lower bounds for the product of successive minimas which turns out to be an essential ingredient in our energy estimate.\\\\\n\nIn fact, we have also removed the boundedness assumption on the side lengths of the boxes in Theorem~\\ref{MT}, as described in \\textbf{Remark~1}, when the degree of the extension is prime. The key ingredient in the proof of \\textbf{Remark~1} is Theorem~\\ref{KT}, which provides square-root cancellation for the corresponding complete character sum over $\\mathbb{F}_{p}$. Under the remaining assumption that the larger side length of the box exceeds $p^{1/2+\\varepsilon}$, we obtain a non-trivial saving for the character sum along that direction. Estimating the remaining summations trivially then yields an overall non-trivial saving over the volume of the box. For further details, see \\textbf{Case~3} in the proof of the main theorem. \n\n\\medskip\n\n{\\bf Acknowledgments.} The author expresses his sincere gratitude to Professor Stephan Baier for his valuable conversations. The author thanks the Ramakrishna Mission Vivekananda Educational and Research Institute for an excellent work environment. The research of the author was supported by a CSIR Ph.D. fellowship under file number 09/0934(13170)/2022-EMR-I.", + "sketch": "The proof of Theorem~\\ref{MT} “closely follows the approach of \\cite{GB},” using “techniques from \\cite{C1}, \\cite{K} and \\cite{Kon}.” It begins with “the classical additive shifting technique $x\\mapsto x+yz$ introduced by Burgess \\cite{B2}$,” applied to the character sum over a box (with side lengths “at most $\\sqrt{p/2}$”). Then, “applying H\\\"older's inequality twice to the resulting sum of shifted characters produces three terms”: one term “is related to the multiplicative energy of the box,” and “the remaining terms can be bounded using Weil's bound and trivial estimates.” Hence, getting cancellation “reduces to obtaining a bound on the multiplicative energy of a set in $\\mathbb{F}_{p^n}$,” as given in “Lemma~\\ref{l1} and Lemma~\\ref{l2}.” For the energy bound, the argument uses “tools from the geometry of numbers,” specifically “successive minima and Minkowski's second theorem,” together with “an auxiliary lemma that provides upper and lower bounds for the product of successive minimas,” described as “an essential ingredient” in the energy estimate.\n\nFor the prime-degree refinement mentioned after Theorem~\\ref{MT} (Remark~1), the boundedness on side lengths can be removed: “The key ingredient” is “Theorem~\\ref{KT}, which provides square-root cancellation for the corresponding complete character sum over $\\mathbb{F}_{p}$.” If “the larger side length of the box exceeds $p^{1/2+\\varepsilon}$,” one gets “a non-trivial saving for the character sum along that direction,” and then “estimating the remaining summations trivially” yields “an overall non-trivial saving over the volume of the box” (cf. “Case~3 in the proof of the main theorem”).", + "expanded_sketch": "The proof of\n\\begin{theorem}[Main Theorem]\\label{MT}\nLet $n\\geq2$, let $\\chi$ be a nontrivial multiplicative character of $\\mathbb{F}_{p^n}$, and suppose\n\\[\n|B|\\geq p^{n(1/4+\\varepsilon)},\\qquad\nH_1\\leq H_2\\leq\\cdots\\leq H_n\\leq \\sqrt{p/2}.\n\\]\nThen\n\\[\n\\left|\\sum_{x\\in B}\\chi(x)\\right|\n\\ll |B|\\, p^{-\\varepsilon^2\n\\frac{1-\\frac{1}{2n}}{\\left(1+\\frac{1}{4n}\\right)\\left(2-\\frac{1}{2n}\\right)}}.\n\\]\n\\end{theorem}\n“closely follows the approach of \\cite{GB},” using “techniques from \\cite{C1}, \\cite{K} and \\cite{Kon}.” It begins with “the classical additive shifting technique $x\\mapsto x+yz$ introduced by Burgess \\cite{B2}$,” applied to the character sum over a box (with side lengths “at most $\\sqrt{p/2}$”). Then, “applying H\\\"older's inequality twice to the resulting sum of shifted characters produces three terms”: one term “is related to the multiplicative energy of the box,” and “the remaining terms can be bounded using Weil's bound and trivial estimates.” Hence, getting cancellation “reduces to obtaining a bound on the multiplicative energy of a set in $\\mathbb{F}_{p^n}$,” as given in the following lemmas.\n\n\\begin{Lemma}\\label{l1}\nWe have\n\\[\nL_{1} \\;\\ll\\; |B|^{2}\\log p.\n\\]\n\\end{Lemma}\n\n\\begin{Lemma}\\label{l2}\nWe have\n\\[\nL_{2} \\;\\ll\\; |B|^{2}(\\log p)^{n}.\n\\]\n\\end{Lemma}\n\nFor the energy bound, the argument uses “tools from the geometry of numbers,” specifically “successive minima and Minkowski's second theorem,” together with “an auxiliary lemma that provides upper and lower bounds for the product of successive minimas,” described as “an essential ingredient” in the energy estimate.\n\nFor the prime-degree refinement mentioned after the main theorem (Remark~1), the boundedness on side lengths can be removed: “The key ingredient” is the following proposition, which provides square-root cancellation for the corresponding complete character sum over $\\mathbb{F}_{p}$.\n\n\\begin{Proposition}[Katz]\\label{KT}\nLet $\\chi$ be a non-trivial multiplicative character of $\\mathbb{F}_{p^n}$ and let\n$g\\in\\mathbb{F}_{p^n}$ generate the extension\n$\\mathbb{F}_{p^n}=\\mathbb{F}_p(g)$. Then, for any interval\n$I\\subseteq [1,p]\\cap\\mathbb{Z}$,\n\\[\n\\left|\\sum_{t\\in I}\\chi(g+t)\\right|\n\\le c(n)\\sqrt{p}\\log p .\n\\]\n\\end{Proposition}\n\nIf “the larger side length of the box exceeds $p^{1/2+\\varepsilon}$,” one gets “a non-trivial saving for the character sum along that direction,” and then “estimating the remaining summations trivially” yields “an overall non-trivial saving over the volume of the box” (cf. “Case~3 in the proof of the main theorem”).", + "expanded_theorem": "[Konyagin \\cite{Kon}]\nLet $\\varepsilon>0$ and suppose $H_i>p^{1/4+\\varepsilon}$ for all $1\\leq i\\leq n$. Then\n\\[\n\\left|\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\\right|\n\\ll_{n,\\varepsilon} p^{-\\varepsilon^2/2}|B|.\n\\],", + "theorem_type": [ + "Inequality or Bound", + "Implication" + ], + "mcq": { + "question": "Let p be a prime, let \\(\\mathbb{F}_{p^n}\\) be the finite field with \\(p^n\\) elements, and fix a basis \\(\\{\\omega_1,\\ldots,\\omega_n\\}\\) of \\(\\mathbb{F}_{p^n}\\) over \\(\\mathbb{F}_p\\). For integers \\(N_i\\) and \\(H_i\\), define intervals \\(I_i=[N_i+1,N_i+H_i]\\cap\\mathbb{Z}\\) and the box\n\\[\nB=\\left\\{\\sum_{i=1}^n x_i\\omega_i: N_i+1\\le x_i\\le N_i+H_i\\text{ for }1\\le i\\le n\\right\\}\\subseteq \\mathbb{F}_{p^n},\n\\]\nwith volume \\(|B|=H_1\\cdots H_n\\). Let \\(\\chi\\) be a nontrivial multiplicative character on \\(\\mathbb{F}_{p^n}\\). If \\(\\varepsilon>0\\) and \\(H_i>p^{1/4+\\varepsilon}\\) for every \\(1\\le i\\le n\\), which quantitative estimate holds for the character sum over \\(B\\)?", + "correct_choice": { + "label": "A", + "text": "\\[\\left|\\sum_{x\\in B}\\chi(x)\\right|\\ll_{n,\\varepsilon} p^{-\\varepsilon^2/2}\\,|B|.\\]" + }, + "choices": [ + { + "label": "B", + "text": "\\[\\left|\\sum_{x\\in B}\\chi(x)\\right|\\ll_{n,\\varepsilon} p^{-\\varepsilon^2/2}\\,|B|,\\qquad \\text{provided }H_i\\ge p^{1/4+\\varepsilon}\\text{ for every }1\\le i\\le n.\\]" + }, + { + "label": "C", + "text": "\\[\\left|\\sum_{x\\in B}\\chi(x)\\right|=o_{n,\\varepsilon}(|B|)\\quad\\text{as }p\\to\\infty.\\]" + }, + { + "label": "D", + "text": "\\[\\left|\\sum_{x\\in B}\\chi(x)\\right|\\ll_{\\varepsilon} p^{-\\varepsilon^2/2}\\,|B|.\\]" + }, + { + "label": "E", + "text": "\\[\\left|\\sum_{x\\in B}\\chi(x)\\right|\\ll_{n,\\varepsilon} p^{-\\varepsilon^2}\\,|B|.\\]" + } + ], + "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 on each side length", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "regularity", + "tampered_component": "explicit power saving factor p^{-\\varepsilon^2/2}", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "dependence of the implied constant on n", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "exact exponent of the power saving", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives the setup and hypotheses but does not disclose the exact conclusion. The correct option is not signaled by wording in the question itself." + }, + "TAS": { + "score": 1, + "justification": "This is close to theorem recall: the task is essentially to identify the precise stated conclusion under the given hypotheses. However, it is not a verbatim restatement because the student must distinguish among nearby alternatives." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to compare exact, weaker, and stronger bounds and note subtle quantifier issues, but the item mainly tests recognition of the known result rather than genuinely generating an argument." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and well targeted: one is weaker but true, one adds unnecessary side conditions, one alters dependence of the saving exponent, and one overstates the exponent. These reflect realistic failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-identification MCQ with strong distractors and little answer leakage, but it leans more toward precise recall than toward deep generative reasoning." + } + }, + { + "id": "2602.22191v1", + "paper_link": "http://arxiv.org/abs/2602.22191v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "maintheorem", + "content": "\\label{thm:maintheorem}(cf. Theorem \\ref{thm:cdbound})\nLet $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\\operatorname{cd}(A,I)\\leq d-\\floor*{\\frac{d-1}{c}}.$$", + "start_pos": 8062, + "end_pos": 8357, + "label": "thm:maintheorem" + }, + "ref_dict": { + "thm:cdbound": "\\begin{theorem}\\label{thm:cdbound}\n Let $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\\cd(A,I)\\leq d-\\floor*{\\frac{d-1}{c}}.$$\n\\end{theorem}", + "lemma:p_zd_on_H_I^n": "\\begin{lemma}\\label{lemma:p_zd_on_H_I^n}\n Let $I$, $A$, $x$, $I_1$ and $I_2$ be as in Lemma \\ref{lemma:radical(I+p)}, $M$ be an $A$-module and $n>\\max(\\cd(M,I_1),\\cd(M,I_2))$. Then $H_I^n(M)$ is $x$-power torsion, that is, every element of $H^n_I(M)$ is annihilated by some power of $x$.\n\\end{lemma}", + "thm:induction": "\\begin{theorem}\\label{thm:induction}\n Let $(A,\\mathfrak{m},k)$ be an unramified regular local ring of mixed characteristic $(0,p)$, and let $I$ be an ideal of $A$. Suppose $\\height(I)<\\height(I:(I:p))$ (for example, this holds if $p$ is a nonzerodivisor on $A/I)$.\\footnote{We use the convention that $\\height(A)=\\infty.$}\n\n Set $c=\\bight(I)$. Let $n>c$ be an integer. Assume that for all integers $s$, with $1\\leq s\\leq c-1,$ and for all $q\\geq n-s$, $H^q_{IA_P}(A_P)=0$ for all $P\\in \\Spec(A)$ such that $I\\subseteq P$ and $\\dim(A/P)\\geq s+1.$ Then $H^q_I(A)=0$ for all $q\\geq n.$\n\\end{theorem}", + "example:sharpness": "\\begin{example}\\label{example:sharpness}\n Given an unramified regular local ring $(A,\\mathfrak{m})$ of dimension $d>0$ and a positive integer $c\\leq d$, there exists an ideal $I$ of $A$ such that $\\cd(A,I)=d-\\floor*{\\frac{d-1}{c}}.$ We construct this ideal as in \\cite{Lyu2}. \n\n Let $N=\\floor*{\\frac{d-1}{c}}.$ Let $I_0,\\dots,I_N$ be ideals of pure height $c$ in $A$ such that $I_0+\\dots+I_N$ is $\\mathfrak{m}$-primary and let $I=I_0\\cap\\dots\\cap I_N$. Then $\\cd(A,I)=d-N.$ The proof of this statement is identical to \\cite{Lyu2}.\n\n To resolve any conflict with the previous remark, we show that in this example, if $p\\in I$, then $\\floor*{\\frac{d-2}{c-1}}= \\floor*{\\frac{d-1}{c}}.$ Indeed, if $p\\in I,$ we have $p\\in I_j$ for all $j\\in\\{0,\\dots,N\\}$ and by Theorem \\ref{thm:Serre}, $$d=\\height(I_0+\\dots+I_N)\\leq 1+(N+1)(c-1)=Nc+c-N.$$ Let $d-1=Nc+k$, where $0\\leq kc$ be an integer. Assume that for all integers $s$, with $1\\leq s\\leq c-1,$ and for all $q\\geq n-s$, $H^q_{IA_P}(A_P)=0$ for all $P\\in \\Spec(A)$ such that $I\\subseteq P$ and $\\dim(A/P)\\geq s+1.$ Then $H^q_I(A)=0$ for all $q\\geq n.$\n\\end{maintheorem}" + }, + "pre_theorem_intro_text_len": 3311, + "pre_theorem_intro_text": "Local cohomology, introduced by Grothendieck in the early 1960s, plays a pivotal role in commutative algebra and algebraic geometry. Its vanishing behavior reflects subtle geometric and topological properties of the underlying scheme and has been studied extensively in the work of Grothendieck, Hartshorne, Faltings, and others. Understanding when local cohomology modules vanish remains a fundamental and delicate problem. \n\nLet $A$ be a commutative Noetherian ring of Krull dimension $d$ and $I$ be an ideal of $A$. The \\textit{cohomological dimension} of $I$ in $A$ is $$\\operatorname{cd}(A,I):=\\max\\{n\\:|\\:H^n_I(A)\\neq 0\\}.$$ Alternatively, $\\operatorname{cd}(A,I)$ is the least integer $n$ such that $H^q_I(M)=0$ for all $A$-modules $M$ and $q>n$ \\cite[Theorem 9.6]{24h}. As $H^q_I(M)=0$ for all $A$-modules $M$ and $q>d$ \\cite{Gr}, $\\operatorname{cd}(A,I)$ always exists and is bounded above by $d$.\n\nThe computation of better upper bounds on cohomological dimension has been of great interest over the past few decades. It is well-known that $$\\operatorname{cd}(A,I)\\leq\\operatorname{ara}(I),$$ where $\\operatorname{ara}(I)$, the $\\textit{arithmetic rank}$ of $I$, is the least number of equations required to generate $I$ up to radical. However, the arithmetic rank of an ideal is difficult to compute and can be far from the cohomological dimension \\cite{BS,Bar,JPSW,BMMP}. \n\nA complementary approach is to bound the cohomological dimension using topological invariants of the spectrum of $A/I$. The most famous result in this direction is the Hartshorne--Lichtenbaum Vanishing Theorem \\cite{Ha1}, which states that for a complete local domain $(A,\\mathfrak{m})$, we have $\\operatorname{cd}(A,I)\\leq d-1$ if and only if $I$ is not $\\mathfrak{m}$-primary. Furthermore, the Second Vanishing Theorem \\cite{Ogus,PS,HunLyu,Zhang} asserts that in equal characteristic or unramified mixed characteristic, for a complete regular local ring $(A,\\mathfrak{m})$ with separably closed residue field, we have $\\operatorname{cd}(A,I)\\leq d-2$ if and only if $\\dim(A/I)\\geq 2$ and the punctured spectrum of $A/I$ is connected. Peskine and Szpiro \\cite{PS} proved that if $(A,\\mathfrak{m)}$ is a regular local ring of equal characteristic $p$ and $A/I$ is Cohen-Macaulay, then $\\operatorname{cd}(A,I)=\\operatorname{ht}(I).$ Beyond the results discussed here, further vanishing statements have been proved in various settings \\cite{MR3078644,DaoTakagi, Burke}, though the conditions required become progressively harder to check. Ogus \\cite{Ogus}, Hartshorne and Speiser \\cite{HarSpe}, Lyubeznik \\cite{Lyu3}, Mustață and Popa \\cite{MuPo}, and Reichelt, Saito and Walther \\cite{RTW} obtained concrete formulae for cohomological dimension in equal characteristic in terms of singularity invariants, see also \\cite{BBLSZ}. However, these methods require the understanding of de Rham cohomologies, Hodge filtrations, or Frobenius structures, and in practice they can be difficult to compute.\n\nIn equal characteristic, Faltings \\cite{Faltings} established a general bound on cohomological dimension in terms of the \\textit{big height} of an ideal. The big height of an ideal is the maximum of the heights of all its minimal primes. In this article, we extend Faltings' result to the unramified mixed characteristic setting.", + "context": "Local cohomology, introduced by Grothendieck in the early 1960s, plays a pivotal role in commutative algebra and algebraic geometry. Its vanishing behavior reflects subtle geometric and topological properties of the underlying scheme and has been studied extensively in the work of Grothendieck, Hartshorne, Faltings, and others. Understanding when local cohomology modules vanish remains a fundamental and delicate problem.\n\nLet $A$ be a commutative Noetherian ring of Krull dimension $d$ and $I$ be an ideal of $A$. The \\textit{cohomological dimension} of $I$ in $A$ is $$\\operatorname{cd}(A,I):=\\max\\{n\\:|\\:H^n_I(A)\\neq 0\\}.$$ Alternatively, $\\operatorname{cd}(A,I)$ is the least integer $n$ such that $H^q_I(M)=0$ for all $A$-modules $M$ and $q>n$ \\cite[Theorem 9.6]{24h}. As $H^q_I(M)=0$ for all $A$-modules $M$ and $q>d$ \\cite{Gr}, $\\operatorname{cd}(A,I)$ always exists and is bounded above by $d$.\n\nThe computation of better upper bounds on cohomological dimension has been of great interest over the past few decades. It is well-known that $$\\operatorname{cd}(A,I)\\leq\\operatorname{ara}(I),$$ where $\\operatorname{ara}(I)$, the $\\textit{arithmetic rank}$ of $I$, is the least number of equations required to generate $I$ up to radical. However, the arithmetic rank of an ideal is difficult to compute and can be far from the cohomological dimension \\cite{BS,Bar,JPSW,BMMP}.\n\nA complementary approach is to bound the cohomological dimension using topological invariants of the spectrum of $A/I$. The most famous result in this direction is the Hartshorne--Lichtenbaum Vanishing Theorem \\cite{Ha1}, which states that for a complete local domain $(A,\\mathfrak{m})$, we have $\\operatorname{cd}(A,I)\\leq d-1$ if and only if $I$ is not $\\mathfrak{m}$-primary. Furthermore, the Second Vanishing Theorem \\cite{Ogus,PS,HunLyu,Zhang} asserts that in equal characteristic or unramified mixed characteristic, for a complete regular local ring $(A,\\mathfrak{m})$ with separably closed residue field, we have $\\operatorname{cd}(A,I)\\leq d-2$ if and only if $\\dim(A/I)\\geq 2$ and the punctured spectrum of $A/I$ is connected. Peskine and Szpiro \\cite{PS} proved that if $(A,\\mathfrak{m)}$ is a regular local ring of equal characteristic $p$ and $A/I$ is Cohen-Macaulay, then $\\operatorname{cd}(A,I)=\\operatorname{ht}(I).$ Beyond the results discussed here, further vanishing statements have been proved in various settings \\cite{MR3078644,DaoTakagi, Burke}, though the conditions required become progressively harder to check. Ogus \\cite{Ogus}, Hartshorne and Speiser \\cite{HarSpe}, Lyubeznik \\cite{Lyu3}, Mustață and Popa \\cite{MuPo}, and Reichelt, Saito and Walther \\cite{RTW} obtained concrete formulae for cohomological dimension in equal characteristic in terms of singularity invariants, see also \\cite{BBLSZ}. However, these methods require the understanding of de Rham cohomologies, Hodge filtrations, or Frobenius structures, and in practice they can be difficult to compute.\n\nIn equal characteristic, Faltings \\cite{Faltings} established a general bound on cohomological dimension in terms of the \\textit{big height} of an ideal. The big height of an ideal is the maximum of the heights of all its minimal primes. In this article, we extend Faltings' result to the unramified mixed characteristic setting.\n\n\\begin{theorem}\\label{thm:cdbound}\n Let $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\\cd(A,I)\\leq d-\\floor*{\\frac{d-1}{c}}.$$\n\\end{theorem}", + "full_context": "Local cohomology, introduced by Grothendieck in the early 1960s, plays a pivotal role in commutative algebra and algebraic geometry. Its vanishing behavior reflects subtle geometric and topological properties of the underlying scheme and has been studied extensively in the work of Grothendieck, Hartshorne, Faltings, and others. Understanding when local cohomology modules vanish remains a fundamental and delicate problem.\n\nLet $A$ be a commutative Noetherian ring of Krull dimension $d$ and $I$ be an ideal of $A$. The \\textit{cohomological dimension} of $I$ in $A$ is $$\\operatorname{cd}(A,I):=\\max\\{n\\:|\\:H^n_I(A)\\neq 0\\}.$$ Alternatively, $\\operatorname{cd}(A,I)$ is the least integer $n$ such that $H^q_I(M)=0$ for all $A$-modules $M$ and $q>n$ \\cite[Theorem 9.6]{24h}. As $H^q_I(M)=0$ for all $A$-modules $M$ and $q>d$ \\cite{Gr}, $\\operatorname{cd}(A,I)$ always exists and is bounded above by $d$.\n\nThe computation of better upper bounds on cohomological dimension has been of great interest over the past few decades. It is well-known that $$\\operatorname{cd}(A,I)\\leq\\operatorname{ara}(I),$$ where $\\operatorname{ara}(I)$, the $\\textit{arithmetic rank}$ of $I$, is the least number of equations required to generate $I$ up to radical. However, the arithmetic rank of an ideal is difficult to compute and can be far from the cohomological dimension \\cite{BS,Bar,JPSW,BMMP}.\n\nA complementary approach is to bound the cohomological dimension using topological invariants of the spectrum of $A/I$. The most famous result in this direction is the Hartshorne--Lichtenbaum Vanishing Theorem \\cite{Ha1}, which states that for a complete local domain $(A,\\mathfrak{m})$, we have $\\operatorname{cd}(A,I)\\leq d-1$ if and only if $I$ is not $\\mathfrak{m}$-primary. Furthermore, the Second Vanishing Theorem \\cite{Ogus,PS,HunLyu,Zhang} asserts that in equal characteristic or unramified mixed characteristic, for a complete regular local ring $(A,\\mathfrak{m})$ with separably closed residue field, we have $\\operatorname{cd}(A,I)\\leq d-2$ if and only if $\\dim(A/I)\\geq 2$ and the punctured spectrum of $A/I$ is connected. Peskine and Szpiro \\cite{PS} proved that if $(A,\\mathfrak{m)}$ is a regular local ring of equal characteristic $p$ and $A/I$ is Cohen-Macaulay, then $\\operatorname{cd}(A,I)=\\operatorname{ht}(I).$ Beyond the results discussed here, further vanishing statements have been proved in various settings \\cite{MR3078644,DaoTakagi, Burke}, though the conditions required become progressively harder to check. Ogus \\cite{Ogus}, Hartshorne and Speiser \\cite{HarSpe}, Lyubeznik \\cite{Lyu3}, Mustață and Popa \\cite{MuPo}, and Reichelt, Saito and Walther \\cite{RTW} obtained concrete formulae for cohomological dimension in equal characteristic in terms of singularity invariants, see also \\cite{BBLSZ}. However, these methods require the understanding of de Rham cohomologies, Hodge filtrations, or Frobenius structures, and in practice they can be difficult to compute.\n\nIn equal characteristic, Faltings \\cite{Faltings} established a general bound on cohomological dimension in terms of the \\textit{big height} of an ideal. The big height of an ideal is the maximum of the heights of all its minimal primes. In this article, we extend Faltings' result to the unramified mixed characteristic setting.\n\n\\begin{theorem}\\label{thm:cdbound}\n Let $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\\cd(A,I)\\leq d-\\floor*{\\frac{d-1}{c}}.$$\n\\end{theorem}\n\nIn equal characteristic, Faltings \\cite{Faltings} established a general bound on cohomological dimension in terms of the \\textit{big height} of an ideal. The big height of an ideal is the maximum of the heights of all its minimal primes. In this article, we extend Faltings' result to the unramified mixed characteristic setting.\n\nWe also establish an inductive criterion for the vanishing of local cohomology modules. See Theorem \\ref{thm:induction} for a more general statement than the one stated below.\n\n\\begin{theorem}\\label{thm:cdbound}\n Let $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\\cd(A,I)\\leq d-\\floor*{\\frac{d-1}{c}}.$$\n\\end{theorem}\n\\begin{proof} We may assume that $A$ is complete and $I$ is radical as local cohomology and big height remain unchanged. If $c=1$, $I$ is principal and $\\cd(A,I)\\leq 1$. Thus we assume $c>1.$ We also assume $c1,$ let $I'=I_1+\\dots+I_{N-1}$, $I=I'+I_N$ and $J=I_1\\cap I_t+\\dots+I_{N-1}\\cap I_N$. Note that each $I_j\\cap I_t$ has big height at most $c$ by Remark \\ref{rmk:bight}. As $\\sqrt{J}=\\sqrt{I'\\cap I_N}$, the Mayer-Vietoris sequence for local cohomology yields $$\\cdots\\to H_J^{i-1}(A)\\to H_I^{i}(A)\\to H_{I'}^{i}(A)\\oplus H_{I_N}^{i}(A)\\to \\cdots$$ \n For $i>d-\\floor*{\\frac{d-1}{c}}+N-1$, $H_J^{i-1}(A)=H_{I'}^{i}(A)=H_{I_N}^{i}(A)=0$ by the induction hypothesis. Hence, $H_I^i(A)=0$ for $i>d-\\floor*{\\frac{d-1}{c}}+N-1$.\n\\end{proof}\n\n\\begin{theorem}[cf. {\\cite[Theorem 5.2]{HunLyu}}]\\label{thm:example}\n Let $(A,\\mathfrak{m})$ be a $d$-dimensional regular local ring of equal characteristic or unramified mixed characteristic. Let $c0$ and a positive integer $c\\leq d$, there exists an ideal $I$ of $A$ such that $\\cd(A,I)=d-\\floor*{\\frac{d-1}{c}}.$ We construct this ideal as in \\cite{Lyu2}. \n\n Let $N=\\floor*{\\frac{d-1}{c}}.$ Let $I_0,\\dots,I_N$ be ideals of pure height $c$ in $A$ such that $I_0+\\dots+I_N$ is $\\mathfrak{m}$-primary and let $I=I_0\\cap\\dots\\cap I_N$. Then $\\cd(A,I)=d-N.$ The proof of this statement is identical to \\cite{Lyu2}.\n\n To resolve any conflict with the previous remark, we show that in this example, if $p\\in I$, then $\\floor*{\\frac{d-2}{c-1}}= \\floor*{\\frac{d-1}{c}}.$ Indeed, if $p\\in I,$ we have $p\\in I_j$ for all $j\\in\\{0,\\dots,N\\}$ and by Theorem \\ref{thm:Serre}, $$d=\\height(I_0+\\dots+I_N)\\leq 1+(N+1)(c-1)=Nc+c-N.$$ Let $d-1=Nc+k$, where $0\\leq kc$ be an integer. Assume that for all integers $s$, with $1\\leq s\\leq c-1,$ and for all $q\\geq n-s$, $H^q_{IA_P}(A_P)=0$ for all $P\\in \\Spec(A)$ such that $I\\subseteq P$ and $\\dim(A/P)\\geq s+1.$ Then $H^q_I(A)=0$ for all $q\\geq n.$\n\\end{theorem}", + "post_theorem_intro_text_len": 3988, + "post_theorem_intro_text": "We also establish an inductive criterion for the vanishing of local cohomology modules. See Theorem \\ref{thm:induction} for a more general statement than the one stated below.\n\n\\begin{maintheorem}\\label{thm:indcrit}(cf. Theorem \\ref{thm:induction})\n Let $(A,\\mathfrak{m})$ be an unramified regular local ring of mixed characteristic $(0,p)$ and $I$ be an ideal of $A$ of big height $c$. Suppose $p$ is a nonzerodivisor on $A/I.$\n\n Let $n>c$ be an integer. Assume that for all integers $s$, with $1\\leq s\\leq c-1,$ and for all $q\\geq n-s$, $H^q_{IA_P}(A_P)=0$ for all $P\\in \\operatorname{Spec}(A)$ such that $I\\subseteq P$ and $\\dim(A/P)\\geq s+1.$ Then $H^q_I(A)=0$ for all $q\\geq n.$\n\\end{maintheorem}\n\nWe now outline the strategy for proving our results.\n\n\\begin{enumerate}\n \\item In equal characteristic, Faltings proved a criterion \\cite[Satz 1]{Faltings} of the nature of Theorem \\ref{thm:indcrit}, and used it as an inductive tool to establish an upper bound on $\\operatorname{cd}(A,I).$ We generalize Faltings' criteria to the unramified mixed characteristic setting under the assumption that $p$ is a nonzerodivisor on $A/I$.\\footnote{This assumption can be further relaxed: see Theorem \\ref{thm:induction} for details.}\n\n In the proof, we reduce the problem to the case of a local cohomology module supported only at the maximal ideal. We handle this remaining case by introducing an auxiliary tensor construction over the base discrete valuation ring and analyzing local cohomology with respect to the diagonal ideal. This allows us to exploit precise dimension bounds and numerical inequalities to force the collapse of a natural spectral sequence, yielding a vanishing statement in the auxiliary setting.\n We then leverage a result of Zhou on the injective dimension of local cohomology modules in unramified mixed characteristic \\cite[Theorem 5.1]{Zhou}, together with the finiteness of Bass numbers for such modules established in \\cite{Lyu4, NB}.\n \\item We split our proof of Theorem \\ref{thm:maintheorem} into three cases. If $p$ is a nonzerodivisor on $A/I$, we can use the inductive criterion above to prove that the local cohomology modules $H_I^n(A)$ vanish for $n$ greater than the desired bound. If $p\\in I,$ we reduce our analysis to $A/p$, a regular local ring of equal characteristic $p$, and employ the already known bound in equal characteristic.\n \\item If $p$ is a zerodivisor on $I$, we build on the analysis of the preceding two cases to once again reduce to equal characteristic $p$. The guiding observation is that $H^n_I(A)$ is $p$-power torsion for $n$ exceeding the desired bound (Lemma \\ref{lemma:p_zd_on_H_I^n}). \n\n At this stage, the argument must be refined to control the behavior of the big height of the ideal $I(A/p).$ A systemic analysis reveals that all cases can be resolved except when $\\operatorname{bight}(I)$ divides $\\dim(A)-1$. In this remaining case, a key input is the structural description of the critical local cohomology module $H^q_J(A/p)$ \\cite[Theorem 4.1]{Lyu1}, which allows us to reduce to the previously treated situation in which $p$ is a nonzerodivisor on $A/I$.\n\n\\end{enumerate}\n\nThe paper is organized as follows. In Section \\ref{section: InductionTheorem}, we collect the principal properties and a number of subtle features of the big height of an ideal. We then prove the inductive criteria for the vanishing of local cohomology in the unramified mixed characteristic setting (Theorem \\ref{thm:induction}). In Section \\ref{section:cdbound}, we use Theorem \\ref{thm:induction} to prove the upper bound on cohomological dimension (Theorem \\ref{thm:cdbound}). We also address the sharpness of our bound (Example \\ref{example:sharpness}) and illustrate how the bound can be used to compute the exact cohomological dimension of suitably chosen ideals. In Section \\ref{section:questions}, we collect natural questions suggested by our results, and comment on current progress and related directions.", + "sketch": "To prove Theorem~\\ref{thm:maintheorem}, the authors first “generalize Faltings' criteria to the unramified mixed characteristic setting under the assumption that $p$ is a nonzerodivisor on $A/I$,” i.e. they prove an inductive vanishing criterion of the form Theorem~\\ref{thm:indcrit}. In that proof, they “reduce the problem to the case of a local cohomology module supported only at the maximal ideal,” then handle it by “introducing an auxiliary tensor construction over the base discrete valuation ring and analyzing local cohomology with respect to the diagonal ideal,” using “dimension bounds and numerical inequalities to force the collapse of a natural spectral sequence, yielding a vanishing statement in the auxiliary setting,” and finally “leverage a result of Zhou on the injective dimension of local cohomology modules… together with the finiteness of Bass numbers.”\n\nThey then “split [the] proof of Theorem~\\ref{thm:maintheorem} into three cases:”\n(1) If $p$ is a nonzerodivisor on $A/I$, they “use the inductive criterion above to prove that the local cohomology modules $H_I^n(A)$ vanish for $n$ greater than the desired bound.”\n(2) If “$p\\in I$,” they “reduce [the] analysis to $A/p$… of equal characteristic $p$, and employ the already known bound in equal characteristic.”\n(3) If $p$ is a zerodivisor on $A/I$, they “once again reduce to equal characteristic $p$,” using that “$H^n_I(A)$ is $p$-power torsion for $n$ exceeding the desired bound,” and then refine the argument to control “the behavior of the big height of the ideal $I(A/p)$”; after a “systemic analysis” they reduce remaining cases (notably when $\\operatorname{bight}(I)$ divides $\\dim(A)-1$) using “the structural description of the critical local cohomology module $H^q_J(A/p)$… which allows us to reduce to the previously treated situation in which $p$ is a nonzerodivisor on $A/I$.”", + "expanded_sketch": "No expanded sketch found.", + "expanded_theorem": "\\label{thm:maintheorem}(To prove the main theorem, we use the following result. \\begin{theorem}\\label{thm:cdbound}\n Let $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\\cd(A,I)\\leq d-\\floor*{\\frac{d-1}{c}}.$$\n\\end{theorem})\nLet $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\\operatorname{cd}(A,I)\\leq d-\\floor*{\\frac{d-1}{c}}.$$, ", + "theorem_type": [ + "Inequality or Bound", + "Universal" + ], + "mcq": { + "question": "Let \\(A\\) be a \\(d\\)-dimensional unramified regular local ring of mixed characteristic \\((0,p)\\), and let \\(I\\subset A\\) be a nonzero proper ideal. Write \\(c=\\operatorname{bight}(I)\\), where the big height is the maximum of the heights of the minimal primes of \\(I\\). Also, let \\(\\operatorname{cd}(A,I)=\\max\\{n\\mid H_I^n(A)\\neq 0\\}\\) denote the cohomological dimension of \\(I\\) in \\(A\\). Which statement holds for every such pair \\((A,I)\\)?", + "correct_choice": { + "label": "A", + "text": "\\(\\operatorname{cd}(A,I)\\le d-\\left\\lfloor \\frac{d-1}{c}\\right\\rfloor.\\)" + }, + "choices": [ + { + "label": "B", + "text": "\\(\\operatorname{cd}(A,I)\\le d-\\left\\lfloor \\frac{d}{c}\\right\\rfloor.\\)" + }, + { + "label": "C", + "text": "\\(\\operatorname{cd}(A,I)\\le d-c+1.\\)" + }, + { + "label": "D", + "text": "If \\(c 6n+3$ and $\\phi$ is arbitrary; \n\\item[(C)]\n$H = \\Sp_{2n}$, $p > 6n$ and $\\phi$ satisfies $(\\star)$; \n\\item[(D)]\n$H = \\SO_{2n}$, $p > 6n$ and $\\phi$ satisfies $(\\star)$. \n\\end{description}\nSet $\\psi = \\widehat\\phi$. \n\n\\begin{enumerate}\n\\item\nThere exists $\\pi \\in \\Pi_\\psi$ such that \n\\[\nd_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi) \\in \\bar\\mathcal{N}(\\pi)^{\\max}.\n\\]\n\\item\nFor $\\pi \\in \\Pi_\\psi$ and for $\\oo_H^\\mathrm{st} \\in \\bar\\mathcal{N}(\\pi)$, \nif $\\oo_H^\\mathrm{st} \\not= d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)$, then \n\\[\n\\dim(\\oo_H^\\mathrm{st}) < \\dim(d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)).\n\\]\nHere, $\\dim(\\oo_H^\\mathrm{st})$ is the Zariski dimension of the orbit as an algebraic variety.\n\\item\nIf we further assume Hypothesis \\ref{hypo} below, then Conjecture \\ref{upper} is true for $\\phi$, i.e., \n\\[\n\\bar\\mathcal{N}(\\pi)^{\\max} \\leq d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\n\\]\nfor any $\\pi \\in \\Pi_\\psi$.\n\\end{enumerate}", + "start_pos": 10589, + "end_pos": 11637, + "label": "main" + }, + "ref_dict": { + "upper": "\\begin{conj}\\label{upper}\nLet $\\phi \\in \\Phi(H(F))$. \nThen for any $\\pi \\in \\Pi_\\phi$, we have\n\\[\n\\bar\\NN(\\hat\\pi)^{\\max} \\leq d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi).\n\\]\nMoreover, if $\\phi$ is tempered, \nthen the equality holds for some $\\pi \\in \\Pi_\\phi$. \n\\end{conj}", + "twist": "\\begin{thm}\\label{twist}\nLet $\\phi \\colon W_F \\times \\SL_2(\\C) \\rightarrow G^\\vee$ \nbe a self-dual tempered $L$-parameter for $G(F) = \\GL_m(F)$, \nand let $\\pi_\\psi$ be the Zelevinsky dual of the irreducible tempered representation \n$\\pi_\\phi$ corresponding to $\\phi$. \nThen \n\\[\n\\bar\\NN_\\theta(\\tl\\pi_\\psi)^{\\max} = \\{d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi)\\}.\n\\]\nMoreover, $c_{\\oo_G}(\\tl\\pi_\\psi) = c_{\\oo_G'}(\\tl\\pi_\\psi)$ \nfor any rational orbits $\\oo_G$ and $\\oo_G'$ contained in $d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi)$.\n\\end{thm}", + "transfer": "\\begin{thm}\\label{transfer}\nLet $\\oo_H^\\st \\in \\NN(\\h(\\overline{F}))$ be a special nilpotent orbit.\nDefine $\\oo_G^\\st \\in \\NN(\\g^\\theta(\\overline{F}))$ by \n\\[\n\\oo_G^\\st = d_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st)). \n\\]\nThen for each rational orbit $\\oo_H \\subset \\oo_H^\\st$, \nthere exists a constant $\\gamma_{\\oo_H} \\in \\C$ such that \n\\[\n\\sum_{\\oo_G \\subset \\oo_G^\\st} \\hat\\mu_{\\oo_G}\n\\quad\\text{is equal to the transfer of }\\quad\n\\sum_{\\oo_H \\subset \\oo_H^\\st} \\gamma_{\\oo_H} \\hat\\mu_{\\oo_H}\n\\]\non a small enough neighborhood of $\\theta \\in G(F) \\rtimes \\theta$.\n\\end{thm}", + "main": "\\begin{thm}\\label{main}\nLet $H$ be a split classical group over $F$, \nand let $\\phi \\in \\Phi_\\temp(H(F))$. \nSuppose that one of the following holds: \n\\begin{description}\n\\item[(B)]\n$H = \\SO_{2n+1}$, $p > 6n+3$ and $\\phi$ is arbitrary; \n\\item[(C)]\n$H = \\Sp_{2n}$, $p > 6n$ and $\\phi$ satisfies $(\\star)$; \n\\item[(D)]\n$H = \\SO_{2n}$, $p > 6n$ and $\\phi$ satisfies $(\\star)$. \n\\end{description}\nSet $\\psi = \\widehat\\phi$. \n\n\\begin{enumerate}\n\\item\nThere exists $\\pi \\in \\Pi_\\psi$ such that \n\\[\nd_{H^\\vee}(\\Ad(H^\\vee)N_\\phi) \\in \\bar\\NN(\\pi)^{\\max}.\n\\]\n\\item\nFor $\\pi \\in \\Pi_\\psi$ and for $\\oo_H^\\st \\in \\bar\\NN(\\pi)$, \nif $\\oo_H^\\st \\not= d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi)$, then \n\\[\n\\dim(\\oo_H^\\st) < \\dim(d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi)).\n\\]\nHere, $\\dim(\\oo_H^\\st)$ is the Zariski dimension of the orbit as an algebraic variety.\n\\item\nIf we further assume Hypothesis \\ref{hypo} below, then Conjecture \\ref{upper} is true for $\\phi$, i.e., \n\\[\n\\bar\\NN(\\pi)^{\\max} \\leq d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi)\n\\]\nfor any $\\pi \\in \\Pi_\\psi$.\n\\end{enumerate}\n\\end{thm}", + "WS": "\\begin{prop}\\label{WS}\nFor $i \\in \\{1,2\\}$, let $\\oo_{H_i}^\\st \\in \\NN(\\h_i(\\overline{F}))$ be a special nilpotent orbit.\nThen \n\\[\nW(\\oo_{H_1}^\\st, \\oo_{H_2}^\\st) \n\\leq d_{H^\\vee}(\\Ad(H^\\vee)\\xi(d_{H_1}(\\oo_{H_1}^\\st), d_{H_2}(\\oo_{H_2}^\\st))). \n\\]\n\\end{prop}", + "hypo": "\\begin{hyp}\\label{hypo}\nLet $\\oo_H^\\st \\in \\NN(\\h(\\overline{F}))$ be a special nilpotent orbit.\nDefine $\\oo_G^\\st \\in \\NN(\\g^\\theta(\\overline{F}))$ by \n\\[\n\\oo_G^\\st = d_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st)). \n\\]\nThen for any constants $\\{c_{\\oo_H} \\;|\\; \\oo_H \\subset \\oo_H^\\st\\}$ such that \n$\\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H} \\hat\\mu_{\\oo_H}$ is stable, \nthere exist constants $\\{a_{\\oo_G} \\;|\\; \\oo_G \\subset \\oo_G^\\st\\}$ such that \n\\[\n\\sum_{\\oo_G \\subset \\oo_G^\\st} a_{\\oo_G} \\hat\\mu_{\\oo_G}\n\\quad\\text{is equal to the transfer of}\\quad\n\\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H} \\hat\\mu_{\\oo_H} \n\\]\non a small enough neighborhood of $\\theta \\in G(F) \\rtimes \\theta$.\n\\end{hyp}" + }, + "pre_theorem_intro_text_len": 8707, + "pre_theorem_intro_text": "This work is motivated by the elusive relation \nbetween the irreducible admissible Harish-Chandra characters of a reductive $p$-adic group \nand the Langlands-Arthur parameters. \nArguably the best-known instance of this relation is the Hiraga--Ichino--Ikeda formal degree formula \nfor square-integrable characters in terms of adjoint $\\gamma$-factors \\cite{HII}. \nThe formal degree concerns the coefficient of the distribution attached to the zero nilpotent orbit \nin the Harish-Chandra--Howe local character expansion. \nAt the opposite end, \nthe largest orbits that contribute to the character expansion determine the wavefront set of the character distribution. \nThey are the focus of the present paper. \nUnder a technical assumption, which is unnecessary for $\\SO_{2n+1}$, and for $p$ large (see Theorem \\ref{main} for the precise conditions), we prove that \nthe largest-dimensional orbits in the union of the geometric wavefront sets of the representations \nin the Aubert--Zelevinsky dual of a tempered $L$-packet for a split classical $p$-adic group \n{\\it equal} the Spaltenstein dual of the corresponding tempered nilpotent Langlands parameter. \n\\par\n\nA relation of this sort, Conjecture \\ref{upper}, was proposed in \\cite{CK,HLLS}, \nmotivated by the results of \\cite{CMBO1,CMBO2} for unipotent representations of split (and `inner-to-split') $p$-adic groups. \nThe fact that such a link should exist had already been apparent from \nthe foundational work of Adams--Barbasch--Vogan on the definition and construction of microlocal Arthur packets \\cite{ABV}. \nIn Theorem \\ref{main}, we also show that the full Conjecture \\ref{upper} follows by further assuming Hypothesis \\ref{hypo}.\n\\vskip 10pt\n\nWe fix a non-archimedean local field $F$ of characteristic zero and of residue characteristic $p \\gg 0$.\nLet $\\overline{F}$ be a fixed algebraic closure of $F$. \nIn this section, we state our main theorem (Theorem \\ref{main})\nand explain our idea for the proof. \n\n\\subsection{Wavefront set}\nLet $H$ be a connected reductive group over $F$ with the Lie algebra $\\mathfrak{h}$.\nWe denote by $\\mathcal{N}(\\mathfrak{h}(F))$ (resp.~ $\\mathcal{N}(\\mathfrak{h}(\\overline{F}))$) \nthe set of nilpotent $\\mathrm{Ad}(H(F))$-orbits (resp.~ $\\mathrm{Ad}(H(\\overline{F}))$-orbits) \nin $\\mathfrak{h}(F)$ (resp.~ $\\mathfrak{h}(\\overline{F})$).\nFor an irreducible admissible representation $\\pi$ of $H(F)$, \nwe denote its distribution character by $\\Theta_\\pi$. \nThe Harish-Chandra--Howe local character expansion (\\cite[Theorem 16.2]{HC}) states that \n\\[\n\\Theta_\\pi(\\exp(X)) = \\sum_{\\oo_H \\in \\mathcal{N}(\\mathfrak{h}(F))} c_{\\oo_H}(\\pi) \\hat\\mu_{\\oo_H}(\\exp(X))\n\\]\nfor $X \\in \\mathfrak{h}(F)$ sufficiently near to $0$, \nwhere $\\hat\\mu_{\\oo_H}$ is the Fourier transform of the orbital integral, \nand $c_{\\oo_H}(\\pi)$ is a constant. \nSet $\\mathcal{N}(\\pi) = \\{ \\oo_H \\in \\mathcal{N}(\\mathfrak{h}(F)) \\;|\\; c_{\\oo_H}(\\pi)\\not= 0\\}$ \nand let $\\bar\\mathcal{N}(\\pi)$ be the set of $\\oo_H^\\mathrm{st} \\in \\mathcal{N}(\\mathfrak{h}(\\overline{F}))$\nsuch that $\\oo_H^\\mathrm{st}$ contains a rational orbit $\\oo_H$ belonging to $\\mathcal{N}(\\pi)$.\nWe denote by $\\bar\\mathcal{N}(\\pi)^{\\max}$ the subset of $\\bar\\mathcal{N}(\\pi)$ consisting of maximal orbits\nwith respect to the closure ordering\n\\[\n\\oo_H^\\mathrm{st} \\leq \\oo_H'^\\mathrm{st} \\overset{\\text{def}}{\\iff} \\oo_H^\\mathrm{st} \\subset \\overline{\\oo_H'^\\mathrm{st}}.\n\\]\nWe call $\\bar\\mathcal{N}(\\pi)^{\\max}$ the \\emph{(geometric) wavefront set} of $\\pi$. \n\n\\subsection{$L$-parameters}\nDenote by $H^\\vee$ the complex Langlands dual group, \nand by $\\mathfrak{h}^\\vee$ the Lie algebra of $H^\\vee$. \nLet $W_F$ be the Weil group of $F$. \nAn \\emph{$L$-parameter} for $H(F)$ is a homomorphism \n\\[\n\\phi \\colon W_F \\times \\SL_2(\\mathbb{C}) \\rightarrow H^\\vee\n\\]\nsuch that $\\phi(W_F)$ consists of semisimple elements, $\\phi|_{W_F}$ is smooth and $\\phi|_{\\SL_2(\\mathbb{C})}$ is algebraic. \nIt is called \\emph{tempered} if $\\phi(W_F)$ is bounded. \nLet $\\Phi(H(F))$ be the set of $H^\\vee$-conjugacy classes of $L$-parameters for $H(F)$, \nand let $\\Phi_\\mathrm{temp}(H(F))$ be its subset consisting of tempered $L$-parameters. \nFor $\\phi \\in \\Phi(H(F))$, by the derivative of $\\phi|_{\\SL_2(\\mathbb{C})}$, \nwe have a linear map\n\\[\nd\\phi|_{\\SL_2(\\mathbb{C})} \\colon \\sl_2(\\mathbb{C}) \\rightarrow \\mathfrak{h}^\\vee.\n\\]\nWe denote by $N_{\\phi} \\in \\mathfrak{h}^\\vee$ the image of \n$\\begin{pmatrix}\n0 & 1 \\\\ 0 & 0\n\\end{pmatrix}$ under this map.\nThe $\\mathrm{Ad}(H^\\vee)$-orbit of $N_\\phi$ is denoted by $\\mathrm{Ad}(H^\\vee) N_\\phi$.\n\\par \n\nLet $\\mathcal{N}(\\mathfrak{h}^\\vee)$ be the set of nilpotent $\\mathrm{Ad}(H^\\vee)$-orbits in $\\mathfrak{h}^\\vee$.\nWe denote by \n\\[\nd_{H} \\colon \\mathcal{N}(\\mathfrak{h}(\\overline{F})) \\rightarrow \\mathcal{N}(\\mathfrak{h}^\\vee), \\quad\nd_{H^\\vee} \\colon \\mathcal{N}(\\mathfrak{h}^\\vee) \\rightarrow \\mathcal{N}(\\mathfrak{h}(\\overline{F}))\n\\]\nthe Spaltenstein duality maps (\\cite[Section 10]{Sp}). \nThe images of these maps are the sets of \\emph{special nilpotent conjugacy classes}.\nMoreover, $\\oo_H^\\mathrm{st} \\leq d_{H^\\vee}(d_H(\\oo_H^\\mathrm{st}))$, \nand the equality holds if and only if $\\oo_H^\\mathrm{st}$ is special. \nSee \\cite[Appendix A]{BV} for more details.\nAn $L$-parameter $\\phi \\in \\Phi(H(F))$ gives \na nilpotent orbit \n\\[\nd_{H^\\vee}(\\mathrm{Ad}(H^\\vee) N_\\phi) \\in \\mathcal{N}(\\mathfrak{h}(\\overline{F})). \n\\]\n\n\\subsection{$L$-packets and $A$-packets}\nNow suppose that $H$ is a split classical group over $F$. \nNamely, $H$ is a symplectic group or a special orthogonal group.\nLet $\\mathrm{Irr}(H(F))$ be the set of equivalence classes of irreducible representations of $H(F)$, \nand let $\\Irr_\\mathrm{temp}(H(F))$ be its subset consisting of tempered representations. \nThe local Langlands conjecture proven by Arthur \\cite[Theorem 2.2.1]{Ar} gives a canonical surjective map\n\\[\n\\mathrm{Irr}(H(F)) \\rightarrow \\Phi(H(F))\n\\]\nwith finite fibers, and preserving the temperedness. \nThe fiber of $\\phi \\in \\Phi(H(F))$ is denoted by $\\Pi_\\phi$ \nand is called the \\emph{$L$-packet} associated to $\\phi$.\n\\par\n\nWe say that an irreducible representation $\\pi$ of $H(F)$ is \\emph{co-tempered} \nif its Zelevinsky--Aubert dual $\\hat\\pi$ is tempered. \nFor $\\phi \\in \\Phi_\\mathrm{temp}(H(F))$, its \\emph{dual $A$-parameter} \n\\[\n\\psi = \\widehat\\phi \\colon W_F \\times \\SL_2(\\mathbb{C}) \\times \\SL_2(\\mathbb{C}) \\rightarrow H^\\vee\n\\]\nis defined by $\\psi(w,g_1,g_2) = \\phi(w,g_2)$. \nIts \\emph{$A$-packet} is given by \n\\[\n\\Pi_\\psi = \\{\\pi \\in \\mathrm{Irr}(H(F)) \\;|\\; \\hat\\pi \\in \\Pi_\\phi\\}. \n\\]\n\n\\subsection{Conjecture and the main result}\nThe following conjecture was independently formulated by Kim and the second author \\cite[Conjectures 1.1, 1.9]{CK}\nand Hazeltine--Liu--Lo--Shahidi \\cite[Conjecture 1.1]{HLLS}. \n\n\\begin{conj}\\label{upper}\nLet $\\phi \\in \\Phi(H(F))$. \nThen for any $\\pi \\in \\Pi_\\phi$, we have\n\\[\n\\bar\\mathcal{N}(\\hat\\pi)^{\\max} \\leq d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi).\n\\]\nMoreover, if $\\phi$ is tempered, \nthen the equality holds for some $\\pi \\in \\Pi_\\phi$. \n\\end{conj}\n\nThis is relevant for any connected reductive group $H$ \nprovided the local Langlands correspondence for $H(F)$ is assumed. \nThis conjecture is known in the following cases. \n\n\\begin{itemize}\n\\item\n$H=\\GL_m$, \\cite{MW}, see \\cite[Theorem 1.4]{CK}. \n\n\\item\n$\\pi$ is any depth-zero simple supercuspidal representation of a classical group (\\cite[Theorem 1.8]{CK}). \n\n\\item\n{$H$ is an inner form of a split group and} \n$\\pi$ is any unipotent representation of $H(F)$ with real infinitesimal character\n(\\cite[Theorem 1.4.1]{CMBO2}).\n\n\\item\n$H$ is an exceptional group $G_2$ (\\cite[Theorem 1.10]{CK}).\n\\end{itemize}\n\nAs explained in \\cite[Theorems 1.5, 1.8]{HLLS}, \nConjecture \\ref{upper} implies\na part of Jiang's conjecture and a part of the Generalized Shahidi Conjecture on local ABV packets \n(see \\cite[Conjectures 1.4, 1.7]{HLLS}). \n\n\\begin{rem}\nAn analogue of Conjecture \\ref{upper} for covering groups \nhas been formulated by Gao--Liu--Lo--Shahidi \\cite[Conjecture 1.2]{GLLS}, and proved for the Kazhdan--Patterson cover of $\\GL_m$ in \\cite[Theorem 1.3]{GLLS}.\n\\end{rem}\n\nIn this paper, we give a result toward Conjecture \\ref{upper} for split classical groups under some conditions. \nWe focus on the case where $\\phi$ is tempered \nsince Conjecture \\ref{upper} is reduced to this case by \\cite[Proposition 2.2]{CK} or \\cite[Theorem 1.2]{HLLS}.\nFor $\\phi \\in \\Phi_\\mathrm{temp}(H(F))$, we consider the following condition:\n\\begin{description}\n\\item[($\\star$)]\nfor any irreducible self-dual subrepresentation $\\phi_0 \\subset \\phi$, \nits determinant $\\det(\\phi_0)$ is trivial on the inertia subgroup $I_F$ of $W_F$.\n\\end{description}\nOur main result is stated as follows.", + "context": "\\subsection{Wavefront set}\nLet $H$ be a connected reductive group over $F$ with the Lie algebra $\\mathfrak{h}$.\nWe denote by $\\mathcal{N}(\\mathfrak{h}(F))$ (resp.~ $\\mathcal{N}(\\mathfrak{h}(\\overline{F}))$) \nthe set of nilpotent $\\mathrm{Ad}(H(F))$-orbits (resp.~ $\\mathrm{Ad}(H(\\overline{F}))$-orbits) \nin $\\mathfrak{h}(F)$ (resp.~ $\\mathfrak{h}(\\overline{F})$).\nFor an irreducible admissible representation $\\pi$ of $H(F)$, \nwe denote its distribution character by $\\Theta_\\pi$. \nThe Harish-Chandra--Howe local character expansion (\\cite[Theorem 16.2]{HC}) states that \n\\[\n\\Theta_\\pi(\\exp(X)) = \\sum_{\\oo_H \\in \\mathcal{N}(\\mathfrak{h}(F))} c_{\\oo_H}(\\pi) \\hat\\mu_{\\oo_H}(\\exp(X))\n\\]\nfor $X \\in \\mathfrak{h}(F)$ sufficiently near to $0$, \nwhere $\\hat\\mu_{\\oo_H}$ is the Fourier transform of the orbital integral, \nand $c_{\\oo_H}(\\pi)$ is a constant. \nSet $\\mathcal{N}(\\pi) = \\{ \\oo_H \\in \\mathcal{N}(\\mathfrak{h}(F)) \\;|\\; c_{\\oo_H}(\\pi)\\not= 0\\}$ \nand let $\\bar\\mathcal{N}(\\pi)$ be the set of $\\oo_H^\\mathrm{st} \\in \\mathcal{N}(\\mathfrak{h}(\\overline{F}))$\nsuch that $\\oo_H^\\mathrm{st}$ contains a rational orbit $\\oo_H$ belonging to $\\mathcal{N}(\\pi)$.\nWe denote by $\\bar\\mathcal{N}(\\pi)^{\\max}$ the subset of $\\bar\\mathcal{N}(\\pi)$ consisting of maximal orbits\nwith respect to the closure ordering\n\\[\n\\oo_H^\\mathrm{st} \\leq \\oo_H'^\\mathrm{st} \\overset{\\text{def}}{\\iff} \\oo_H^\\mathrm{st} \\subset \\overline{\\oo_H'^\\mathrm{st}}.\n\\]\nWe call $\\bar\\mathcal{N}(\\pi)^{\\max}$ the \\emph{(geometric) wavefront set} of $\\pi$.\n\n\\subsection{$L$-parameters}\nDenote by $H^\\vee$ the complex Langlands dual group, \nand by $\\mathfrak{h}^\\vee$ the Lie algebra of $H^\\vee$. \nLet $W_F$ be the Weil group of $F$. \nAn \\emph{$L$-parameter} for $H(F)$ is a homomorphism \n\\[\n\\phi \\colon W_F \\times \\SL_2(\\mathbb{C}) \\rightarrow H^\\vee\n\\]\nsuch that $\\phi(W_F)$ consists of semisimple elements, $\\phi|_{W_F}$ is smooth and $\\phi|_{\\SL_2(\\mathbb{C})}$ is algebraic. \nIt is called \\emph{tempered} if $\\phi(W_F)$ is bounded. \nLet $\\Phi(H(F))$ be the set of $H^\\vee$-conjugacy classes of $L$-parameters for $H(F)$, \nand let $\\Phi_\\mathrm{temp}(H(F))$ be its subset consisting of tempered $L$-parameters. \nFor $\\phi \\in \\Phi(H(F))$, by the derivative of $\\phi|_{\\SL_2(\\mathbb{C})}$, \nwe have a linear map\n\\[\nd\\phi|_{\\SL_2(\\mathbb{C})} \\colon \\sl_2(\\mathbb{C}) \\rightarrow \\mathfrak{h}^\\vee.\n\\]\nWe denote by $N_{\\phi} \\in \\mathfrak{h}^\\vee$ the image of \n$\\begin{pmatrix}\n0 & 1 \\\\ 0 & 0\n\\end{pmatrix}$ under this map.\nThe $\\mathrm{Ad}(H^\\vee)$-orbit of $N_\\phi$ is denoted by $\\mathrm{Ad}(H^\\vee) N_\\phi$.\n\\par\n\nLet $\\mathcal{N}(\\mathfrak{h}^\\vee)$ be the set of nilpotent $\\mathrm{Ad}(H^\\vee)$-orbits in $\\mathfrak{h}^\\vee$.\nWe denote by \n\\[\nd_{H} \\colon \\mathcal{N}(\\mathfrak{h}(\\overline{F})) \\rightarrow \\mathcal{N}(\\mathfrak{h}^\\vee), \\quad\nd_{H^\\vee} \\colon \\mathcal{N}(\\mathfrak{h}^\\vee) \\rightarrow \\mathcal{N}(\\mathfrak{h}(\\overline{F}))\n\\]\nthe Spaltenstein duality maps (\\cite[Section 10]{Sp}). \nThe images of these maps are the sets of \\emph{special nilpotent conjugacy classes}.\nMoreover, $\\oo_H^\\mathrm{st} \\leq d_{H^\\vee}(d_H(\\oo_H^\\mathrm{st}))$, \nand the equality holds if and only if $\\oo_H^\\mathrm{st}$ is special. \nSee \\cite[Appendix A]{BV} for more details.\nAn $L$-parameter $\\phi \\in \\Phi(H(F))$ gives \na nilpotent orbit \n\\[\nd_{H^\\vee}(\\mathrm{Ad}(H^\\vee) N_\\phi) \\in \\mathcal{N}(\\mathfrak{h}(\\overline{F})). \n\\]\n\n\\begin{conj}\\label{upper}\nLet $\\phi \\in \\Phi(H(F))$. \nThen for any $\\pi \\in \\Pi_\\phi$, we have\n\\[\n\\bar\\mathcal{N}(\\hat\\pi)^{\\max} \\leq d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi).\n\\]\nMoreover, if $\\phi$ is tempered, \nthen the equality holds for some $\\pi \\in \\Pi_\\phi$. \n\\end{conj}\n\n\\begin{rem}\nAn analogue of Conjecture \\ref{upper} for covering groups \nhas been formulated by Gao--Liu--Lo--Shahidi \\cite[Conjecture 1.2]{GLLS}, and proved for the Kazhdan--Patterson cover of $\\GL_m$ in \\cite[Theorem 1.3]{GLLS}.\n\\end{rem}\n\nIn this paper, we give a result toward Conjecture \\ref{upper} for split classical groups under some conditions. \nWe focus on the case where $\\phi$ is tempered \nsince Conjecture \\ref{upper} is reduced to this case by \\cite[Proposition 2.2]{CK} or \\cite[Theorem 1.2]{HLLS}.\nFor $\\phi \\in \\Phi_\\mathrm{temp}(H(F))$, we consider the following condition:\n\\begin{description}\n\\item[($\\star$)]\nfor any irreducible self-dual subrepresentation $\\phi_0 \\subset \\phi$, \nits determinant $\\det(\\phi_0)$ is trivial on the inertia subgroup $I_F$ of $W_F$.\n\\end{description}\nOur main result is stated as follows.\n\n\\begin{hyp}\\label{hypo}\nLet $\\oo_H^\\st \\in \\NN(\\h(\\overline{F}))$ be a special nilpotent orbit.\nDefine $\\oo_G^\\st \\in \\NN(\\g^\\theta(\\overline{F}))$ by \n\\[\n\\oo_G^\\st = d_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st)). \n\\]\nThen for any constants $\\{c_{\\oo_H} \\;|\\; \\oo_H \\subset \\oo_H^\\st\\}$ such that \n$\\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H} \\hat\\mu_{\\oo_H}$ is stable, \nthere exist constants $\\{a_{\\oo_G} \\;|\\; \\oo_G \\subset \\oo_G^\\st\\}$ such that \n\\[\n\\sum_{\\oo_G \\subset \\oo_G^\\st} a_{\\oo_G} \\hat\\mu_{\\oo_G}\n\\quad\\text{is equal to the transfer of}\\quad\n\\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H} \\hat\\mu_{\\oo_H} \n\\]\non a small enough neighborhood of $\\theta \\in G(F) \\rtimes \\theta$.\n\\end{hyp}\n\n\\begin{conj}\\label{upper}\nLet $\\phi \\in \\Phi(H(F))$. \nThen for any $\\pi \\in \\Pi_\\phi$, we have\n\\[\n\\bar\\NN(\\hat\\pi)^{\\max} \\leq d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi).\n\\]\nMoreover, if $\\phi$ is tempered, \nthen the equality holds for some $\\pi \\in \\Pi_\\phi$. \n\\end{conj}", + "full_context": "\\subsection{Wavefront set}\nLet $H$ be a connected reductive group over $F$ with the Lie algebra $\\mathfrak{h}$.\nWe denote by $\\mathcal{N}(\\mathfrak{h}(F))$ (resp.~ $\\mathcal{N}(\\mathfrak{h}(\\overline{F}))$) \nthe set of nilpotent $\\mathrm{Ad}(H(F))$-orbits (resp.~ $\\mathrm{Ad}(H(\\overline{F}))$-orbits) \nin $\\mathfrak{h}(F)$ (resp.~ $\\mathfrak{h}(\\overline{F})$).\nFor an irreducible admissible representation $\\pi$ of $H(F)$, \nwe denote its distribution character by $\\Theta_\\pi$. \nThe Harish-Chandra--Howe local character expansion (\\cite[Theorem 16.2]{HC}) states that \n\\[\n\\Theta_\\pi(\\exp(X)) = \\sum_{\\oo_H \\in \\mathcal{N}(\\mathfrak{h}(F))} c_{\\oo_H}(\\pi) \\hat\\mu_{\\oo_H}(\\exp(X))\n\\]\nfor $X \\in \\mathfrak{h}(F)$ sufficiently near to $0$, \nwhere $\\hat\\mu_{\\oo_H}$ is the Fourier transform of the orbital integral, \nand $c_{\\oo_H}(\\pi)$ is a constant. \nSet $\\mathcal{N}(\\pi) = \\{ \\oo_H \\in \\mathcal{N}(\\mathfrak{h}(F)) \\;|\\; c_{\\oo_H}(\\pi)\\not= 0\\}$ \nand let $\\bar\\mathcal{N}(\\pi)$ be the set of $\\oo_H^\\mathrm{st} \\in \\mathcal{N}(\\mathfrak{h}(\\overline{F}))$\nsuch that $\\oo_H^\\mathrm{st}$ contains a rational orbit $\\oo_H$ belonging to $\\mathcal{N}(\\pi)$.\nWe denote by $\\bar\\mathcal{N}(\\pi)^{\\max}$ the subset of $\\bar\\mathcal{N}(\\pi)$ consisting of maximal orbits\nwith respect to the closure ordering\n\\[\n\\oo_H^\\mathrm{st} \\leq \\oo_H'^\\mathrm{st} \\overset{\\text{def}}{\\iff} \\oo_H^\\mathrm{st} \\subset \\overline{\\oo_H'^\\mathrm{st}}.\n\\]\nWe call $\\bar\\mathcal{N}(\\pi)^{\\max}$ the \\emph{(geometric) wavefront set} of $\\pi$.\n\n\\subsection{$L$-parameters}\nDenote by $H^\\vee$ the complex Langlands dual group, \nand by $\\mathfrak{h}^\\vee$ the Lie algebra of $H^\\vee$. \nLet $W_F$ be the Weil group of $F$. \nAn \\emph{$L$-parameter} for $H(F)$ is a homomorphism \n\\[\n\\phi \\colon W_F \\times \\SL_2(\\mathbb{C}) \\rightarrow H^\\vee\n\\]\nsuch that $\\phi(W_F)$ consists of semisimple elements, $\\phi|_{W_F}$ is smooth and $\\phi|_{\\SL_2(\\mathbb{C})}$ is algebraic. \nIt is called \\emph{tempered} if $\\phi(W_F)$ is bounded. \nLet $\\Phi(H(F))$ be the set of $H^\\vee$-conjugacy classes of $L$-parameters for $H(F)$, \nand let $\\Phi_\\mathrm{temp}(H(F))$ be its subset consisting of tempered $L$-parameters. \nFor $\\phi \\in \\Phi(H(F))$, by the derivative of $\\phi|_{\\SL_2(\\mathbb{C})}$, \nwe have a linear map\n\\[\nd\\phi|_{\\SL_2(\\mathbb{C})} \\colon \\sl_2(\\mathbb{C}) \\rightarrow \\mathfrak{h}^\\vee.\n\\]\nWe denote by $N_{\\phi} \\in \\mathfrak{h}^\\vee$ the image of \n$\\begin{pmatrix}\n0 & 1 \\\\ 0 & 0\n\\end{pmatrix}$ under this map.\nThe $\\mathrm{Ad}(H^\\vee)$-orbit of $N_\\phi$ is denoted by $\\mathrm{Ad}(H^\\vee) N_\\phi$.\n\\par\n\nLet $\\mathcal{N}(\\mathfrak{h}^\\vee)$ be the set of nilpotent $\\mathrm{Ad}(H^\\vee)$-orbits in $\\mathfrak{h}^\\vee$.\nWe denote by \n\\[\nd_{H} \\colon \\mathcal{N}(\\mathfrak{h}(\\overline{F})) \\rightarrow \\mathcal{N}(\\mathfrak{h}^\\vee), \\quad\nd_{H^\\vee} \\colon \\mathcal{N}(\\mathfrak{h}^\\vee) \\rightarrow \\mathcal{N}(\\mathfrak{h}(\\overline{F}))\n\\]\nthe Spaltenstein duality maps (\\cite[Section 10]{Sp}). \nThe images of these maps are the sets of \\emph{special nilpotent conjugacy classes}.\nMoreover, $\\oo_H^\\mathrm{st} \\leq d_{H^\\vee}(d_H(\\oo_H^\\mathrm{st}))$, \nand the equality holds if and only if $\\oo_H^\\mathrm{st}$ is special. \nSee \\cite[Appendix A]{BV} for more details.\nAn $L$-parameter $\\phi \\in \\Phi(H(F))$ gives \na nilpotent orbit \n\\[\nd_{H^\\vee}(\\mathrm{Ad}(H^\\vee) N_\\phi) \\in \\mathcal{N}(\\mathfrak{h}(\\overline{F})). \n\\]\n\n\\begin{conj}\\label{upper}\nLet $\\phi \\in \\Phi(H(F))$. \nThen for any $\\pi \\in \\Pi_\\phi$, we have\n\\[\n\\bar\\mathcal{N}(\\hat\\pi)^{\\max} \\leq d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi).\n\\]\nMoreover, if $\\phi$ is tempered, \nthen the equality holds for some $\\pi \\in \\Pi_\\phi$. \n\\end{conj}\n\n\\begin{rem}\nAn analogue of Conjecture \\ref{upper} for covering groups \nhas been formulated by Gao--Liu--Lo--Shahidi \\cite[Conjecture 1.2]{GLLS}, and proved for the Kazhdan--Patterson cover of $\\GL_m$ in \\cite[Theorem 1.3]{GLLS}.\n\\end{rem}\n\nIn this paper, we give a result toward Conjecture \\ref{upper} for split classical groups under some conditions. \nWe focus on the case where $\\phi$ is tempered \nsince Conjecture \\ref{upper} is reduced to this case by \\cite[Proposition 2.2]{CK} or \\cite[Theorem 1.2]{HLLS}.\nFor $\\phi \\in \\Phi_\\mathrm{temp}(H(F))$, we consider the following condition:\n\\begin{description}\n\\item[($\\star$)]\nfor any irreducible self-dual subrepresentation $\\phi_0 \\subset \\phi$, \nits determinant $\\det(\\phi_0)$ is trivial on the inertia subgroup $I_F$ of $W_F$.\n\\end{description}\nOur main result is stated as follows.\n\n\\begin{hyp}\\label{hypo}\nLet $\\oo_H^\\st \\in \\NN(\\h(\\overline{F}))$ be a special nilpotent orbit.\nDefine $\\oo_G^\\st \\in \\NN(\\g^\\theta(\\overline{F}))$ by \n\\[\n\\oo_G^\\st = d_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st)). \n\\]\nThen for any constants $\\{c_{\\oo_H} \\;|\\; \\oo_H \\subset \\oo_H^\\st\\}$ such that \n$\\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H} \\hat\\mu_{\\oo_H}$ is stable, \nthere exist constants $\\{a_{\\oo_G} \\;|\\; \\oo_G \\subset \\oo_G^\\st\\}$ such that \n\\[\n\\sum_{\\oo_G \\subset \\oo_G^\\st} a_{\\oo_G} \\hat\\mu_{\\oo_G}\n\\quad\\text{is equal to the transfer of}\\quad\n\\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H} \\hat\\mu_{\\oo_H} \n\\]\non a small enough neighborhood of $\\theta \\in G(F) \\rtimes \\theta$.\n\\end{hyp}\n\n\\begin{conj}\\label{upper}\nLet $\\phi \\in \\Phi(H(F))$. \nThen for any $\\pi \\in \\Pi_\\phi$, we have\n\\[\n\\bar\\NN(\\hat\\pi)^{\\max} \\leq d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi).\n\\]\nMoreover, if $\\phi$ is tempered, \nthen the equality holds for some $\\pi \\in \\Pi_\\phi$. \n\\end{conj}\n\n\\begin{thm}\\label{twist}\nLet $\\phi \\colon W_F \\times \\SL_2(\\C) \\rightarrow G^\\vee$ \nbe a self-dual tempered $L$-parameter for $G(F) = \\GL_m(F)$, \nand let $\\pi_\\psi$ be the Zelevinsky dual of the irreducible tempered representation \n$\\pi_\\phi$ corresponding to $\\phi$. \nThen \n\\[\n\\bar\\NN_\\theta(\\tl\\pi_\\psi)^{\\max} = \\{d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi)\\}.\n\\]\nMoreover, $c_{\\oo_G}(\\tl\\pi_\\psi) = c_{\\oo_G'}(\\tl\\pi_\\psi)$ \nfor any rational orbits $\\oo_G$ and $\\oo_G'$ contained in $d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi)$.\n\\end{thm}\n\\begin{proof}\nRecall that $\\bar\\NN(\\pi_\\psi)^{\\max} = \\{d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi)\\}$ \nby M{\\oe}glin--Waldspurger \\cite{MW} as explained in \\cite[Theorem 1.4]{CK}.\nThen Varma showed in \\cite[Lemma 5.29]{V} that \nfor any $\\oo_G^\\st \\in \\bar\\NN_\\theta(\\tl\\pi_\\psi)^{\\max}$, \nthe corresponding degenerate Whittaker model is nonzero. \nHence we have\n\\[\n\\oo_G^\\st \\leq d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi).\n\\]\nOn the other hand, by Konno \\cite[Theorem 4.1]{K}\ntogether with the multiplicity one result of degenerate Whittaker models by M{\\oe}glin--Waldspurger \\cite{MW}\n(cf., see \\cite[Proposition 4.4]{K}),\nwe have\n\\[\nd_{G^\\vee}(\\Ad(G^\\vee) N_\\phi) \\in \\bar\\NN_\\theta(\\tl\\pi_\\psi)^{\\max}. \n\\]\nHence we obtain the first assertion.\nThe last assertion follows from \\cite[Theorem 4.1(2), Remark 4.2]{K}.\n\\end{proof}\n\nAs a consequence of Theorem \\ref{transfer}, we obtain the following. \n\\begin{cor}\\label{cor1}\nWe have\n\\[\nd_{H^\\vee}(\\Ad(H^\\vee) N_\\phi) \\in \\bar\\NN(\\psi).\n\\]\nMoreover, for $\\oo_H^\\st \\in \\bar\\NN(\\psi)$, \nif $\\oo_H^\\st \\not= d_{H^\\vee}(\\Ad(H^\\vee) N_\\phi)$, then \n\\[\n\\dim(\\oo_H^\\st) < \\dim(d_{H^\\vee}(\\Ad(H^\\vee) N_\\phi)).\n\\]\n\\end{cor}\n\\begin{proof}\nSet $d = \\dim(d_{H^\\vee}(\\Ad(H^\\vee) N_\\phi))$. \nAs in the proof of Theorem \\ref{transfer}, \nconsider the unique $L$-parameter $\\phi_0 \\in \\Phi_\\temp(H(F))$\nsuch that $\\phi_0$ is trivial on $W_F$ and \n\\[\n\\Ad(H^\\vee) N_{\\phi_0} = \\Ad(H^\\vee) N_\\phi. \n\\]\nSet $\\psi_0 = \\widehat\\phi_0$.\nFor a constant $e \\in \\C^\\times$\nthe transfer of $S\\Theta_\\psi - eS\\Theta_{\\psi_0}$ is \n\\[\n\\Theta_{\\tl\\pi_\\psi}-e\\Theta_{\\tl\\pi_{\\psi_0}}\n= \\sum_{\\oo_G \\in \\NN(\\g^\\theta(F))} \\left( c_{\\oo_G}(\\tl\\pi_\\psi)-e c_{\\oo_G}(\\tl\\pi_{\\psi_0}) \\right) \\hat\\mu_{\\oo_G}.\n\\]\nSince $d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi) = d_{G^\\vee}(\\Ad(G^\\vee) N_{\\phi_0})$,\nby Theorem \\ref{twist}, we can choose $e \\in \\C^\\times$ such that \nfor $\\oo_G^\\st \\in \\NN(\\g^\\theta(\\overline{F}))$, \nif $c_{\\oo_G}(\\tl\\pi_\\psi)-e c_{\\oo_G}(\\tl\\pi_{\\psi_0}) \\not= 0$ for some $\\oo_G \\subset \\oo_G^\\st$, \nthen $\\oo_G^\\st < d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi)$.\nThis together with Theorem \\ref{transfer} implies that \n\\[\n\\sum_{\\substack{\\oo_H^\\st \\in \\NN(\\h(\\overline{F})) \\\\ \\dim(\\oo_H^\\st) \\geq d}} \n\\left( \\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H}(\\psi) \\hat\\mu_{\\oo_H}\\right)\n=\ne\\sum_{\\substack{\\oo_H^\\st \\in \\NN(\\h(\\overline{F})) \\\\ \\dim(\\oo_H^\\st) \\geq d}} \n\\left( \\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H}(\\psi_0) \\hat\\mu_{\\oo_H}\\right)\n\\]\nBy \\cite[Theorem 1.4.1]{CMBO2}, \nin the right-hand side, only $\\oo_H^\\st = d_{H^\\vee}(\\Ad(H^\\vee) N_\\phi)$ can contribute. \nBy the linear independence of $\\hat\\mu_{\\oo_H}$ (\\cite[Theorem 5.11]{HC}), \nthe same holds for the left-hand side.\n\\end{proof}\n\n\\begin{prop}\\label{refine}\nAssume Hypothesis \\ref{hypo}.\nThen for any $\\oo_H^\\st \\in \\bar\\NN(\\psi)$, \nwe have \n\\[\n\\oo_H^\\st \\leq d_{H^\\vee}(\\Ad(H^\\vee) N_\\phi).\n\\]\n\\end{prop}\n\\begin{proof}\nFor each special orbit $\\oo_H^\\st \\in \\NN(\\h(\\overline{F}))$, \nlet $\\{a_{\\oo_G}(\\psi) \\;|\\; \\oo_G \\subset d_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st))\\}$ \nbe the constants in Hypothesis \\ref{hypo}\ngiven by $\\{c_{\\oo_H}(\\psi) \\;|\\; \\oo_H \\subset \\oo_H^\\st\\}$.\nThen by considering the transfer of $S\\Theta_\\psi$, \nwe see that the local character expansion of $\\Theta_{\\tl\\pi_\\psi}$ is given by\n\\[\n\\Theta_{\\tl\\pi_\\psi} = \\sum_{\\oo_H^\\st} \n\\sum_{\\oo_G \\subset \\oo_G^\\st} a_{\\oo_G}(\\psi) \\hat\\mu_{\\oo_G}, \n\\]\nwhere $\\oo_H^\\st$ runs over special nilpotent orbits in $\\NN(\\h(\\overline{F}))$\nand $\\oo_G$ runs over rational orbits contained in\n$\\oo_G^\\st = d_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st))$.\nIn particular, if $\\oo_H^\\st \\in \\bar\\NN(\\psi)$, \nthen $d_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st)) \\in \\bar\\NN_\\theta(\\tl\\pi_\\psi)$. \nBy Theorem \\ref{twist}, we have\n\\[\nd_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st)) \\leq d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi). \n\\]\nHence $\\Ad(G^\\vee)d_{H}(\\oo_H^\\st) \\geq \\Ad(G^\\vee) N_\\phi$.\nNote that \n\\[\n\\Ad(G^\\vee)d_{H}(\\oo_H^\\st) \\cap H^\\vee = d_{H}(\\oo_H^\\st), \n\\quad\n\\Ad(G^\\vee) N_\\phi \\cap H^\\vee = \\Ad(H^\\vee) N_\\phi. \n\\]\n(Here, if $H=\\SO_{2n}$, we consider $\\O_{2n}(\\C)$-orbits actually. See Remark \\ref{OvsSO}.)\nHence $d_{H}(\\oo_H^\\st) \\geq \\Ad(H^\\vee) N_\\phi$, \nand we conclude that \n$\\oo_H^\\st \\leq d_{H^\\vee}(d_{H}(\\oo_H^\\st)) \\leq d_{H^\\vee}(\\Ad(H^\\vee) N_\\phi)$.\n\\end{proof}\n\nNow we assume that we have\n\\begin{itemize}\n\\item\n$\\oo_H^\\st \\in \\NN(\\h(\\overline{F}))$ \nwith $\\dim(\\oo_H^\\st) \\geq \\dim(d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi))$; \n\\item\n$\\oo_H \\in \\NN(\\h(F))$ with $\\oo_H \\subset \\oo_H^\\st$; \n\\item\n$\\pi_0 \\in \\Pi_\\psi$ with $\\psi = \\widehat\\phi$\n\\end{itemize}\nsuch that $c_{\\oo_H}(\\pi_0) \\not= 0$. \nThe goal is to show that $\\oo_H^\\st = d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi)$.\n\\par\n\nBy the induction hypothesis, we have \n$\\dim(\\oo_{H_i}^\\st) \\leq \\dim (d_{H_i^\\vee}(\\Ad(H_i^\\vee)N_{\\phi_i}))$ for $i \\in \\{1,2\\}$.\nHence by Proposition \\ref{WS} and Lemma \\ref{dimW}, \nwe have \n\\begin{align*}\n\\dim(\\oo_H^\\st) &= \\dim\\left(W(\\oo_{H_1}^\\st, \\oo_{H_2}^\\st)\\right)\n\\\\&\\leq \\dim\\left(\nW(d_{H_1^\\vee}(\\Ad(H_1^\\vee)N_{\\phi_1}), d_{H_2^\\vee}(\\Ad(H_2^\\vee)N_{\\phi_2}))\n\\right)\n\\\\&\\leq \n\\dim\\left(\nd_{H^\\vee}(\\Ad(H^\\vee)\\xi(\nd_{H_1}(d_{H_1^\\vee}(\\Ad(H_1^\\vee)N_{\\phi_1})), d_{H_2}(d_{H_2^\\vee}(\\Ad(H_2^\\vee)N_{\\phi_2}))\n))\n\\right)\n\\\\&\\leq\n\\dim\\left(\nd_{H^\\vee}(\\Ad(H^\\vee)\\xi(\\Ad(H_1^\\vee)N_{\\phi_1}, \\Ad(H_2^\\vee)N_{\\phi_2}))\n\\right)\n\\\\&= \n\\dim\\left(d_{H^\\vee}(\\Ad(H^\\vee)N_{\\phi})\\right).\n\\end{align*}\nHere, we used the facts \nthat $d_{H_i}(d_{H_i^\\vee}(\\Ad(H_i^\\vee)N_{\\phi_i})) \\geq \\Ad(H_i^\\vee)N_{\\phi_i}$\nand that $d_{H^\\vee}$ is an order-reversing map.\nSince we assume that $\\dim(\\oo_H^\\st) \\geq \\dim\\left(d_{H^\\vee}(\\Ad(H^\\vee)N_{\\phi})\\right)$, \nthis must be an equality.\nMoreover, it together with Lemma \\ref{dimW} implies that \n$\\oo_{H_i}^\\st = d_{H_i^\\vee}(\\Ad(H_i^\\vee)N_{\\phi_i})$ for $i \\in \\{1,2\\}$. \nThen by proposition \\ref{WS}, we have\n\\begin{align*}\n\\oo_{H^\\st} \n&\\leq d_{H^\\vee}(\\Ad(H^\\vee)\\xi(d_{H_1}(\\oo_{H_1}^\\st), d_{H_2}(\\oo_{H_2}^\\st)))\n\\\\&\\leq d_{H^\\vee}(\\Ad(H^\\vee)\\xi(\\Ad(H_1^\\vee)N_{\\phi_1}, \\Ad(H_2^\\vee)N_{\\phi_2}))\n\\\\&= d_{H^\\vee}(\\Ad(H^\\vee)N_{\\phi}).\n\\end{align*}\nSince $\\dim(\\oo_H^\\st) = \\dim\\left(d_{H^\\vee}(\\Ad(H^\\vee)N_{\\phi})\\right)$, \nwe conclude that $\\oo_H^\\st = d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi)$, \nas desired.\n\\par\n\nThe proof of Theorem \\ref{main} (3) is similar. \nSuppose that $\\oo_H^\\st \\in \\NN(\\h(\\overline{F}))$ contains a rational orbit $\\oo_H$ \nsuch that $c_{\\oo_H}(\\pi_0) \\not= 0$ for some $\\pi_0 \\in \\Pi_\\psi$. \nConsider $s \\in \\Sc_\\psi$ satisfying ($\\ast$).\nIf $s = 1$, then Proposition \\ref{refine} implies that $\\oo_H^\\st \\leq d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi)$. \nOtherwise, we can write $\\oo_H^\\st = W(\\oo_{H_1}^\\st, \\oo_{H_2}^\\st)$ as above. \nThen $\\oo_{H_i}^\\st \\leq d_{H_i^\\vee}(\\Ad(H_i^\\vee)N_{\\phi_i})$ by the induction hypothesis.\nBy Proposition \\ref{WS} and Lemma \\ref{Worder} below, we have \n\\begin{align*}\n\\oo_H^\\st &= W(\\oo_{H_1}^\\st, \\oo_{H_2}^\\st)\n\\\\&\\leq W(d_{H_1^\\vee}(\\Ad(H_1^\\vee)N_{\\phi_1}), d_{H_2^\\vee}(\\Ad(H_2^\\vee)N_{\\phi_2}))\n\\\\&\\leq \nd_{H^\\vee}(\\Ad(H^\\vee)\\xi(\nd_{H_1}(d_{H_1^\\vee}(\\Ad(H_1^\\vee)N_{\\phi_1})), d_{H_2}(d_{H_2^\\vee}(\\Ad(H_2^\\vee)N_{\\phi_2}))\n))\n\\\\&\\leq\nd_{H^\\vee}(\\Ad(H^\\vee)\\xi(\\Ad(H_1^\\vee)N_{\\phi_1}, \\Ad(H_2^\\vee)N_{\\phi_2}))\n\\\\&= \nd_{H^\\vee}(\\Ad(H^\\vee)N_{\\phi}).\n\\end{align*}\nThis completes the proof of Theorem \\ref{main}.\n\\end{proof}", + "post_theorem_intro_text_len": 6376, + "post_theorem_intro_text": "\\subsection{Outline of the proof}\\label{sec.outline}\nThroughout the paper, we assume that $p \\gg 0$ as in Theorem \\ref{main}.\nThe proof of Theorem \\ref{main} is similar to the proofs of the Generic Packet Conjecture \nby Konno \\cite{K} and Varma \\cite{V}. \nThe main idea is to compare local character expansions via \n\\emph{endoscopic character identities}.\n\\par\n\nFix $\\phi \\in \\Phi_\\mathrm{temp}(H(F))$ and set $\\psi = \\widehat\\phi$. \nSet $G = \\GL_m$, where $m$ is the dimension of the standard representation $H^\\vee \\rightarrow G^\\vee = \\GL_m(\\mathbb{C})$.\nBy composing with this map, we can regard $\\psi$ as an $A$-parameter for $G(F)$, \nand obtain an irreducible self-dual representation $\\pi_\\psi$ of $G(F)$. \nDefine an involution $\\theta$ on $G(F)$ by \n\\[\n\\theta(g) = J{}^tg^{-1}J^{-1}, \n\\quad\nJ = \\begin{pmatrix}\n&&&1 \\\\\n&&-1&\\\\\n&\\iddots&& \\\\\n(-1)^{m-1}&&\n\\end{pmatrix} \\in G(F).\n\\]\nWe can extend $\\pi_\\psi$ to an irreducible representation $\\tl\\pi_\\psi$ of $G \\rtimes \\langle \\theta \\rangle$.\nThe twisted character of $\\tl\\pi_\\psi$ is denoted by $\\Theta_{\\tl\\pi_\\psi}$. \n\\par\n\nLet $\\Sc_\\psi$ be the component group for $\\psi$, \nand let $\\widehat\\Sc_\\psi$ be its Pontryagin dual. \nRecall that Arthur defined a map\n\\[\n\\Pi_\\psi \\rightarrow \\widehat\\Sc_\\psi, \\;\n\\pi \\mapsto \\langle \\cdot, \\pi \\rangle_\\psi. \n\\]\nNotice that this is bijective since $\\psi = \\widehat\\phi$.\nSet $s_\\psi = \\psi(\\1_{W_F}, \\mathbf{1}, -\\mathbf{1}) = \\phi(\\1_{W_F}, -\\mathbf{1})$.\nIn this setting, \\cite[Theorem 2.2.1]{Ar} asserts that \n\\begin{description}\n\\item[(T1)]\nthe distribution\n\\[\nS\\Theta_{\\psi} = \\sum_{\\pi \\in \\Pi_\\psi} \\langle s_\\psi, \\pi \\rangle_\\psi \\Theta_\\pi\n\\]\nis stable, and $\\Theta_{\\tl\\pi_\\psi}$ is the transfer of $S\\Theta_\\psi$; \n\\item[(T2)]\nEach $s \\in \\Sc_\\psi$ with $s \\not= 1$ gives an elliptic endoscopic group $H_1 \\times H_2$ of $H$\nand an $A$-parameter $\\psi_i$ for $H_i$ \n(of the form $\\psi_i = \\widehat\\phi_i$ for some $\\phi_i \\in \\Phi_\\mathrm{temp}(H_i)$) such that \n\\[\nS\\Theta_{\\psi, s} = \\sum_{\\pi \\in \\Pi_\\psi} \\langle s \\cdot s_\\psi, \\pi \\rangle_\\psi \\Theta_\\pi\n\\]\nis the transfer of $S\\Theta_{\\psi_1} \\otimes S\\Theta_{\\psi_2}$. \n\\end{description}\n\\par\n\nThe first step of the proof of Theorem \\ref{main} is to establish an analogue for the twisted $\\GL_m$ case.\nBy the twisted local character expansion established by Clozel \\cite[Theorem 3]{C}, \nwe can expand\n\\[\n\\Theta_{\\tl\\pi_\\psi} \n= \\sum_{\\oo_G \\in \\mathcal{N}(\\mathfrak{g}^\\theta(F))} c_{\\oo_G}(\\tl\\pi_\\psi) \\hat\\mu_{\\oo_G}. \n\\]\nSee Section \\ref{sec.TLCE} below for the notations. \nWe define $\\NN_\\theta(\\tl\\pi_\\psi) \\subset \\mathcal{N}(\\mathfrak{g}^\\theta(F))$ \nand $\\bar\\NN_\\theta(\\tl\\pi_\\psi) \\subset \\mathcal{N}(\\mathfrak{g}^\\theta(\\overline{F}))$\nsimilar to $\\mathcal{N}(\\pi)$ and $\\bar\\mathcal{N}(\\pi)$, respectively. \nIn Theorem \\ref{twist} below, we will show that \n\\[\n\\bar\\NN_\\theta(\\tl\\pi_\\psi)^{\\max} = \\{d_{G^\\vee}(\\mathrm{Ad}(G^\\vee) N_\\phi)\\}.\n\\]\nThis is an immediate consequence of \nresults of Konno \\cite{K}, Varma \\cite{V}, and M{\\oe}glin--Waldspurger \\cite{MW}.\nMoreover, Konno's result \\cite[Theorem 4.1(2)]{K} says that \nthe constant $c_{\\oo_G}(\\tl\\pi_\\psi)$ for $\\oo_G \\subset d_{G^\\vee}(\\mathrm{Ad}(G^\\vee) N_\\phi)$ \ndoes not depend on the choice of the rational orbit.\n\\par\n\nSet $\\oo_H^\\mathrm{st} = d_{H^\\vee}(\\mathrm{Ad}(H^\\vee) N_\\phi)$ and $\\oo_G^\\mathrm{st} = d_{G^\\vee}(\\mathrm{Ad}(G^\\vee) N_\\phi)$.\nThe second step of the proof of Theorem \\ref{main} is to show that \nthere exists a constant $\\gamma_{\\oo_H}$ for each rational orbit $\\oo_H \\subset \\oo_H^\\mathrm{st}$ such that \n\\[\n\\sum_{\\oo_G \\subset \\oo_G^\\mathrm{st}} \\hat\\mu_{\\oo_G}\n\\quad\n\\text{is equal to the transfer of} \n\\quad\n\\sum_{\\oo_H \\subset \\oo_H^\\mathrm{st}} \\gamma_{\\oo_H}\\hat\\mu_{\\oo_H}\n\\]\non a small enough neighborhood of $\\theta \\in G(F) \\rtimes \\theta$.\nSee Theorem \\ref{transfer} below. \nHypothesis \\ref{hypo} is an analogue of this result. \nThese claims could be proven by similar arguments to the standard case explained in the next paragraph, \nbut there is no explicit reference in the literature.\nThus we will prove Theorem \\ref{transfer} in this paper. \nThe proof uses Conjecture \\ref{upper} in the case established in \\cite{CMBO2},\nwhich requires that $H$ is split over $F$.\n\\par\n\nAn analogous result for the standard endoscopy with unramified endoscopic groups\nis known by Waldspurger \\cite{W_pave}. \nLet $H_1 \\times H_2$ be an unramified endoscopic group of $H$. \nFor a special nilpotent orbit $\\oo_{H_i}^\\mathrm{st} \\in \\mathcal{N}(\\h_i(\\overline{F}))$, \nWaldspurger explicitly defined a nilpotent orbit\n\\[\n\\oo_H^\\mathrm{st} = W(\\oo_{H_1}^\\mathrm{st}, \\oo_{H_2}^\\mathrm{st}) \\in \\mathcal{N}(\\mathfrak{h}(\\overline{F})),\n\\]\nand proved a certain transfer result in \\cite[XII.9 Th\\'eor\\`eme]{W_pave}.\nThe third step of the proof of Theorem \\ref{main} is to relate the Waldspurger map $W(\\oo_{H_1}^\\mathrm{st}, \\oo_{H_2}^\\mathrm{st})$\nand the Spaltenstein duality (Proposition \\ref{WS}). \n\\par\n\nCombining the two transfer results, we can relate the local character expansion of $S\\Theta_\\psi$ (resp.~ $S\\Theta_{\\psi,s}$)\nwith the one of $\\Theta_{\\tl\\pi_\\psi}$ (resp.~ $S\\Theta_{\\psi_1} \\otimes S\\Theta_{\\psi_2}$).\nThen by induction on $m$, \nwe can deduce Theorem \\ref{main} in Section \\ref{sec.proof}. \nTo use \\cite[XII.9 Th\\'eor\\`eme]{W_pave}, we need the condition ($\\star$) in Cases (C) and (D), \nwhich is equivalent to saying that all endoscopic groups $H_1 \\times H_2$ arising from $s \\in \\Sc_\\psi$ are unramified. \n\n\\begin{rem}\\label{OvsSO}\nThe local Langlands correspondence for $H = \\SO_{2n}$ as proven by Arthur, \nis given up to $\\O_{2n}(F)$-conjugation. \nThus, in turn, we should replace $\\mathrm{Ad}(H^\\vee)N_\\phi$ by $\\mathrm{Ad}(\\O_{2n}(\\mathbb{C}))N_\\phi$ in our results. \nWe will ignore this discrepancy since it is not an important difference for the geometric wavefront set results.\nBy replacing $\\SO_{2n}$ with $\\O_{2n}$, we can use the following properties:\n\\begin{enumerate}\n\\item\nthe distributions on $\\SO_{2n}(F)$ considered in the paper are all $\\O_{2n}(F)$-invariant;\n\\item\nthe transfer of stable distributions on $H(F)$ to $G(F) \\rtimes \\theta$ is injective;\n\\item\n$\\mathrm{Ad}(G^\\vee)\\oo_{H^\\vee} \\cap H^\\vee = \\oo_{H^\\vee}$ for $\\oo_{H^\\vee} \\in \\mathcal{N}(\\mathfrak{h}^\\vee)$.\n\\end{enumerate}\nNote that (2) and (3) hold when $H = \\SO_{2n+1}$ or $H = \\Sp_{2n}$. \n\\end{rem}", + "sketch": "The proof of Theorem~\\ref{main} follows Konno~\\cite{K} and Varma~\\cite{V}: the \"main idea is to compare local character expansions via \\emph{endoscopic character identities}.\" Fix $\\phi\\in\\Phi_\\mathrm{temp}(H(F))$, set $\\psi=\\widehat\\phi$, and embed into $G=\\GL_m$ via the standard map $H^\\vee\\to G^\\vee$. This yields a self-dual $\\pi_\\psi$ of $G(F)$, extended to $\\tilde\\pi_\\psi$ on $G\\rtimes\\langle\\theta\\rangle$, whose twisted character $\\Theta_{\\tilde\\pi_\\psi}$ matches Arthur’s stable distribution\n\\[\nS\\Theta_{\\psi}=\\sum_{\\pi\\in\\Pi_\\psi}\\langle s_\\psi,\\pi\\rangle_\\psi\\,\\Theta_\\pi\n\\]\nby endoscopic character identities (Arthur’s (T1)), and similarly for $S\\Theta_{\\psi,s}$ (Arthur’s (T2)).\n\n(1) \"The first step\" is an analogue in the twisted $\\GL_m$ case: using Clozel’s twisted local character expansion\n\\[\n\\Theta_{\\tilde\\pi_\\psi}=\\sum_{\\oo_G\\in\\mathcal N(\\mathfrak g^\\theta(F))} c_{\\oo_G}(\\tilde\\pi_\\psi)\\,\\hat\\mu_{\\oo_G},\n\\]\none defines $\\bar\\mathcal N_\\theta(\\tilde\\pi_\\psi)^{\\max}$ and proves (Theorem~\\ref{twist})\n\\[\n\\bar\\mathcal N_\\theta(\\tilde\\pi_\\psi)^{\\max}=\\{d_{G^\\vee}(\\mathrm{Ad}(G^\\vee)N_\\phi)\\},\n\\]\nusing results of Konno, Varma, and M\\oe glin--Waldspurger; additionally, Konno’s result implies the constants $c_{\\oo_G}(\\tilde\\pi_\\psi)$ for $\\oo_G\\subset d_{G^\\vee}(\\mathrm{Ad}(G^\\vee)N_\\phi)$ do not depend on the rational orbit.\n\n(2) Setting $\\oo_H^{\\mathrm{st}}=d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)$ and $\\oo_G^{\\mathrm{st}}=d_{G^\\vee}(\\mathrm{Ad}(G^\\vee)N_\\phi)$, \"the second step\" is a transfer statement (Theorem~\\ref{transfer}): for each rational orbit $\\oo_H\\subset\\oo_H^{\\mathrm{st}}$ there is a constant $\\gamma_{\\oo_H}$ such that on a small neighborhood of $\\theta$,\n\\[\n\\sum_{\\oo_G\\subset\\oo_G^{\\mathrm{st}}}\\hat\\mu_{\\oo_G}\\quad\\text{equals the transfer of}\\quad\\sum_{\\oo_H\\subset\\oo_H^{\\mathrm{st}}}\\gamma_{\\oo_H}\\hat\\mu_{\\oo_H}.\n\\]\nHypothesis~\\ref{hypo} is described as an analogue of this transfer result.\n\n(3) \"The third step\" relates Waldspurger’s explicit map for standard endoscopy,\n\\[\n\\oo_H^{\\mathrm{st}}=W(\\oo_{H_1}^{\\mathrm{st}},\\oo_{H_2}^{\\mathrm{st}}),\n\\]\nto Spaltenstein duality (Proposition~\\ref{WS}).\n\nFinally, \"combining the two transfer results,\" one relates the local character expansions of $S\\Theta_\\psi$ and $S\\Theta_{\\psi,s}$ to those of $\\Theta_{\\tilde\\pi_\\psi}$ and $S\\Theta_{\\psi_1}\\otimes S\\Theta_{\\psi_2}$, and then \"by induction on $m$\" deduces Theorem~\\ref{main} (in Section~\\ref{sec.proof}). The condition $(\\star)$ in Cases (C) and (D) is needed to apply Waldspurger’s theorem, equivalently ensuring the relevant endoscopic groups $H_1\\times H_2$ are unramified.", + "expanded_sketch": "The proof of Theorem~\\ref{main} follows Konno~\\cite{K} and Varma~\\cite{V}: the \"main idea is to compare local character expansions via \\emph{endoscopic character identities}.\" Fix $\\phi\\in\\Phi_\\mathrm{temp}(H(F))$, set $\\psi=\\widehat\\phi$, and embed into $G=\\GL_m$ via the standard map $H^\\vee\\to G^\\vee$. This yields a self-dual $\\pi_\\psi$ of $G(F)$, extended to $\\tilde\\pi_\\psi$ on $G\\rtimes\\langle\\theta\\rangle$, whose twisted character $\\Theta_{\\tilde\\pi_\\psi}$ matches Arthur’s stable distribution\n\\[\nS\\Theta_{\\psi}=\\sum_{\\pi\\in\\Pi_\\psi}\\langle s_\\psi,\\pi\\rangle_\\psi\\,\\Theta_\\pi\n\\]\nby endoscopic character identities (Arthur’s (T1)), and similarly for $S\\Theta_{\\psi,s}$ (Arthur’s (T2)).\n\n(1) \"The first step\" is an analogue in the twisted $\\GL_m$ case: using Clozel’s twisted local character expansion\n\\[\n\\Theta_{\\tilde\\pi_\\psi}=\\sum_{\\oo_G\\in\\mathcal N(\\mathfrak g^\\theta(F))} c_{\\oo_G}(\\tilde\\pi_\\psi)\\,\\hat\\mu_{\\oo_G},\n\\]\none defines $\\bar\\mathcal N_\\theta(\\tilde\\pi_\\psi)^{\\max}$ and proves the following theorem.\n\\begin{thm}\\label{twist}\nLet $\\phi \\colon W_F \\times \\SL_2(\\C) \\rightarrow G^\\vee$ \nbe a self-dual tempered $L$-parameter for $G(F) = \\GL_m(F)$, \nand let $\\pi_\\psi$ be the Zelevinsky dual of the irreducible tempered representation \n$\\pi_\\phi$ corresponding to $\\phi$. \nThen \n\\[\n\\bar\\NN_\\theta(\\tl\\pi_\\psi)^{\\max} = \\{d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi)\\}.\n\\]\nMoreover, $c_{\\oo_G}(\\tl\\pi_\\psi) = c_{\\oo_G'}(\\tl\\pi_\\psi)$ \nfor any rational orbits $\\oo_G$ and $\\oo_G'$ contained in $d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi)$.\n\\end{thm}\nThis uses results of Konno, Varma, and M\\oe glin--Waldspurger.\n\n(2) Setting $\\oo_H^{\\mathrm{st}}=d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)$ and $\\oo_G^{\\mathrm{st}}=d_{G^\\vee}(\\mathrm{Ad}(G^\\vee)N_\\phi)$, \"the second step\" is the following transfer statement.\n\\begin{thm}\\label{transfer}\nLet $\\oo_H^\\st \\in \\NN(\\h(\\overline{F}))$ be a special nilpotent orbit.\nDefine $\\oo_G^\\st \\in \\NN(\\g^\\theta(\\overline{F}))$ by \n\\[\n\\oo_G^\\st = d_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st)). \n\\]\nThen for each rational orbit $\\oo_H \\subset \\oo_H^\\st$, \nthere exists a constant $\\gamma_{\\oo_H} \\in \\C$ such that \n\\[\n\\sum_{\\oo_G \\subset \\oo_G^\\st} \\hat\\mu_{\\oo_G}\n\\quad\\text{is equal to the transfer of }\\quad\n\\sum_{\\oo_H \\subset \\oo_H^\\st} \\gamma_{\\oo_H} \\hat\\mu_{\\oo_H}\n\\]\non a small enough neighborhood of $\\theta \\in G(F) \\rtimes \\theta$.\n\\end{thm}\nThe following hypothesis is described as an analogue of this transfer result.\n\\begin{hyp}\\label{hypo}\nLet $\\oo_H^\\st \\in \\NN(\\h(\\overline{F}))$ be a special nilpotent orbit.\nDefine $\\oo_G^\\st \\in \\NN(\\g^\\theta(\\overline{F}))$ by \n\\[\n\\oo_G^\\st = d_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st)). \n\\]\nThen for any constants $\\{c_{\\oo_H} \\;|\\; \\oo_H \\subset \\oo_H^\\st\\}$ such that \n$\\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H} \\hat\\mu_{\\oo_H}$ is stable, \nthere exist constants $\\{a_{\\oo_G} \\;|\\; \\oo_G \\subset \\oo_G^\\st\\}$ such that \n\\[\n\\sum_{\\oo_G \\subset \\oo_G^\\st} a_{\\oo_G} \\hat\\mu_{\\oo_G}\n\\quad\\text{is equal to the transfer of}\\quad\n\\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H} \\hat\\mu_{\\oo_H} \n\\]\non a small enough neighborhood of $\\theta \\in G(F) \\rtimes \\theta$.\n\\end{hyp}\n\n(3) \"The third step\" relates Waldspurger’s explicit map for standard endoscopy,\n\\[\n\\oo_H^{\\mathrm{st}}=W(\\oo_{H_1}^{\\mathrm{st}},\\oo_{H_2}^{\\mathrm{st}}),\n\\]\nto Spaltenstein duality using the proposition below.\n\\begin{prop}\\label{WS}\nFor $i \\in \\{1,2\\}$, let $\\oo_{H_i}^\\st \\in \\NN(\\h_i(\\overline{F}))$ be a special nilpotent orbit.\nThen \n\\[\nW(\\oo_{H_1}^\\st, \\oo_{H_2}^\\st) \n\\leq d_{H^\\vee}(\\Ad(H^\\vee)\\xi(d_{H_1}(\\oo_{H_1}^\\st), d_{H_2}(\\oo_{H_2}^\\st))). \n\\]\n\\end{prop}\n\nFinally, \"combining the two transfer results,\" one relates the local character expansions of $S\\Theta_\\psi$ and $S\\Theta_{\\psi,s}$ to those of $\\Theta_{\\tilde\\pi_\\psi}$ and $S\\Theta_{\\psi_1}\\otimes S\\Theta_{\\psi_2}$, and then \"by induction on $m$\" establishes the main theorem (later in the paper). The condition $(\\star)$ in Cases (C) and (D) is needed to apply Waldspurger’s theorem, equivalently ensuring the relevant endoscopic groups $H_1\\times H_2$ are unramified.", + "expanded_theorem": "\\label{main}\nLet $H$ be a split classical group over $F$, \nand let $\\phi \\in \\Phi_\\mathrm{temp}(H(F))$. \nSuppose that one of the following holds: \n\\begin{description}\n\\item[(B)]\n$H = \\SO_{2n+1}$, $p > 6n+3$ and $\\phi$ is arbitrary; \n\\item[(C)]\n$H = \\Sp_{2n}$, $p > 6n$ and $\\phi$ satisfies $(\\star)$; \n\\item[(D)]\n$H = \\SO_{2n}$, $p > 6n$ and $\\phi$ satisfies $(\\star)$. \n\\end{description}\nSet $\\psi = \\widehat\\phi$. \n\n\\begin{enumerate}\n\\item\nThere exists $\\pi \\in \\Pi_\\psi$ such that \n\\[\nd_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi) \\in \\bar\\mathcal{N}(\\pi)^{\\max}.\n\\]\n\\item\nFor $\\pi \\in \\Pi_\\psi$ and for $\\oo_H^\\mathrm{st} \\in \\bar\\mathcal{N}(\\pi)$, \nif $\\oo_H^\\mathrm{st} \\not= d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)$, then \n\\[\n\\dim(\\oo_H^\\mathrm{st}) < \\dim(d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)).\n\\]\nHere, $\\dim(\\oo_H^\\mathrm{st})$ is the Zariski dimension of the orbit as an algebraic variety.\n\\item\nIf we further assume the following hypothesis, then the corresponding upper-bound conjecture holds for $\\phi$, i.e., \n\\[\n\\bar\\mathcal{N}(\\pi)^{\\max} \\leq d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\n\\]\nfor any $\\pi \\in \\Pi_\\psi$.\n\n\\begin{hyp}\\label{hypo}\nLet $\\oo_H^\\st \\in \\NN(\\h(\\overline{F}))$ be a special nilpotent orbit.\nDefine $\\oo_G^\\st \\in \\NN(\\g^\\theta(\\overline{F}))$ by \n\\[\n\\oo_G^\\st = d_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st)). \n\\]\nThen for any constants $\\{c_{\\oo_H} \\;|\\; \\oo_H \\subset \\oo_H^\\st\\}$ such that \n$\\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H} \\hat\\mu_{\\oo_H}$ is stable, \nthere exist constants $\\{a_{\\oo_G} \\;|\\; \\oo_G \\subset \\oo_G^\\st\\}$ such that \n\\[\n\\sum_{\\oo_G \\subset \\oo_G^\\st} a_{\\oo_G} \\hat\\mu_{\\oo_G}\n\\quad\\text{is equal to the transfer of}\\quad\n\\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H} \\hat\\mu_{\\oo_H} \n\\]\non a small enough neighborhood of $\\theta \\in G(F) \\rtimes \\theta$.\n\\end{hyp}\n\n\\begin{conj}\\label{upper}\nLet $\\phi \\in \\Phi(H(F))$. \nThen for any $\\pi \\in \\Pi_\\phi$, we have\n\\[\n\\bar\\NN(\\hat\\pi)^{\\max} \\leq d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi).\n\\]\nMoreover, if $\\phi$ is tempered, \nthen the equality holds for some $\\pi \\in \\Pi_\\phi$. \n\\end{conj}\n\\end{enumerate}", + "theorem_type": [ + "Existence", + "Implication" + ], + "mcq": { + "question": "Let $H$ be a split classical group over $F$ with Lie algebra $\\mathfrak h$, and let\n$\\phi: W_F\\times \\mathrm{SL}_2(\\mathbb C)\\to H^\\vee$ be a tempered $L$-parameter. Assume one of the following:\n(B) $H=\\mathrm{SO}_{2n+1}$, $p>6n+3$, and $\\phi$ is arbitrary;\n(C) $H=\\mathrm{Sp}_{2n}$, $p>6n$, and for every irreducible self-dual subrepresentation $\\phi_0\\subset \\phi$, one has $\\det(\\phi_0)|_{I_F}=1$;\n(D) $H=\\mathrm{SO}_{2n}$, $p>6n$, and for every irreducible self-dual subrepresentation $\\phi_0\\subset \\phi$, one has $\\det(\\phi_0)|_{I_F}=1$.\nSet $\\psi=\\widehat\\phi$, and let $\\Pi_\\psi$ be the packet attached to $\\psi$. Let $N_\\phi\\in \\mathfrak h^\\vee$ be the image of $\\begin{pmatrix}0&1\\\\0&0\\end{pmatrix}$ under $d\\phi|_{\\mathrm{SL}_2(\\mathbb C)}$, and let $d_H:\\mathcal N(\\mathfrak h(\\overline F))\\to \\mathcal N(\\mathfrak h^\\vee)$ and $d_{H^\\vee}:\\mathcal N(\\mathfrak h^\\vee)\\to \\mathcal N(\\mathfrak h(\\overline F))$ be the Spaltenstein duality maps. For an irreducible admissible representation $\\pi$ of $H(F)$, write\n$\\Theta_\\pi(\\exp X)=\\sum_{\\mathcal O_H} c_{\\mathcal O_H}(\\pi)\\,\\widehat\\mu_{\\mathcal O_H}(\\exp X)$\nnear $0$, define $\\mathcal N(\\pi)=\\{\\mathcal O_H: c_{\\mathcal O_H}(\\pi)\\neq 0\\}$, let $\\bar{\\mathcal N}(\\pi)$ be the set of nilpotent $\\mathrm{Ad}(H(\\overline F))$-orbits containing some rational orbit in $\\mathcal N(\\pi)$, and let $\\bar{\\mathcal N}(\\pi)^{\\max}$ be the maximal elements of $\\bar{\\mathcal N}(\\pi)$ for the closure order $\\mathcal O\\le \\mathcal O'$ iff $\\mathcal O\\subset \\overline{\\mathcal O'}$. Under these assumptions, which statement is valid?", + "correct_choice": { + "label": "A", + "text": "All of the following hold:\n(1) There exists $\\pi\\in \\Pi_\\psi$ such that\n\\[\nd_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\\in \\bar{\\mathcal N}(\\pi)^{\\max}.\n\\]\n(2) For every $\\pi\\in \\Pi_\\psi$ and every $\\mathcal O_H^{\\mathrm{st}}\\in \\bar{\\mathcal N}(\\pi)$, if\n$\\mathcal O_H^{\\mathrm{st}}\\neq d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)$, then\n\\[\n\\dim(\\mathcal O_H^{\\mathrm{st}})<\\dim\\bigl(d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\\bigr),\n\\]\nwhere $\\dim$ is the Zariski dimension of the orbit.\n(3) If one further assumes the following transfer hypothesis: for every special nilpotent orbit $\\mathcal O_H^{\\mathrm{st}}\\in \\mathcal N(\\mathfrak h(\\overline F))$, after defining\n\\[\n\\mathcal O_G^{\\mathrm{st}}=d_{G^\\vee}(\\mathrm{Ad}(G^\\vee)d_H(\\mathcal O_H^{\\mathrm{st}})),\n\\]\nany stable distribution of the form $\\sum_{\\mathcal O_H\\subset \\mathcal O_H^{\\mathrm{st}}} c_{\\mathcal O_H}\\widehat\\mu_{\\mathcal O_H}$ is transferred, on a sufficiently small neighborhood of $\\theta\\in G(F)\\rtimes \\theta$, to some distribution $\\sum_{\\mathcal O_G\\subset \\mathcal O_G^{\\mathrm{st}}} a_{\\mathcal O_G}\\widehat\\mu_{\\mathcal O_G}$, then for every $\\pi\\in \\Pi_\\psi$ one has\n\\[\n\\bar{\\mathcal N}(\\pi)^{\\max}\\le d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi).\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "All of the following hold:\n(1) For every $\\pi\\in \\Pi_\\psi$ such that $d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\\in \\bar{\\mathcal N}(\\pi)$, one has\n\\[\nd_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\\in \\bar{\\mathcal N}(\\pi)^{\\max}.\n\\]\n(2) For every $\\pi\\in \\Pi_\\psi$ and every $\\mathcal O_H^{\\mathrm{st}}\\in \\bar{\\mathcal N}(\\pi)$, if\n$\\mathcal O_H^{\\mathrm{st}}\\neq d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)$, then\n\\[\n\\dim(\\mathcal O_H^{\\mathrm{st}})\\le \\dim\\bigl(d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\\bigr),\n\\]\nwhere $\\dim$ is the Zariski dimension of the orbit.\n(3) If one further assumes the transfer hypothesis stated in the theorem, then for every $\\pi\\in \\Pi_\\psi$ one has\n\\[\n\\bar{\\mathcal N}(\\pi)\\le d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi).\n\\]" + }, + { + "label": "C", + "text": "Both of the following hold:\n(1) There exists $\\pi\\in \\Pi_\\psi$ such that\n\\[\nd_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\\in \\bar{\\mathcal N}(\\pi)^{\\max}.\n\\]\n(2) For every $\\pi\\in \\Pi_\\psi$ and every $\\mathcal O_H^{\\mathrm{st}}\\in \\bar{\\mathcal N}(\\pi)$, if\n$\\mathcal O_H^{\\mathrm{st}}\\neq d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)$, then\n\\[\n\\dim(\\mathcal O_H^{\\mathrm{st}})<\\dim\\bigl(d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\\bigr).\n\\]" + }, + { + "label": "D", + "text": "All of the following hold:\n(1) There exists $\\pi\\in \\Pi_\\psi$ such that\n\\[\nd_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)= \\bar{\\mathcal N}(\\pi)^{\\max}.\n\\]\n(2) For every $\\pi\\in \\Pi_\\psi$ and every $\\mathcal O_H^{\\mathrm{st}}\\in \\bar{\\mathcal N}(\\pi)$, one has\n\\[\n\\mathcal O_H^{\\mathrm{st}}\\le d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi).\n\\]\n(3) In particular, without any additional transfer hypothesis, for every $\\pi\\in \\Pi_\\psi$ one has\n\\[\n\\bar{\\mathcal N}(\\pi)^{\\max}\\le d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi).\n\\]" + }, + { + "label": "E", + "text": "All of the following hold:\n(1) There exists $\\pi\\in \\Pi_\\psi$ such that\n\\[\nd_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\\in \\bar{\\mathcal N}(\\pi)^{\\max}.\n\\]\n(2) For every $\\pi\\in \\Pi_\\psi$ and every $\\mathcal O_H^{\\mathrm{st}}\\in \\bar{\\mathcal N}(\\pi)$, if\n$\\mathcal O_H^{\\mathrm{st}}\\neq d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)$, then\n\\[\n\\dim(\\mathcal O_H^{\\mathrm{st}})<\\dim\\bigl(d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\\bigr).\n\\]\n(3) If one further assumes the following transfer hypothesis: for every special nilpotent orbit $\\mathcal O_H^{\\mathrm{st}}\\in \\mathcal N(\\mathfrak h(\\overline F))$, after defining\n\\[\n\\mathcal O_G^{\\mathrm{st}}=d_{G^\\vee}(\\mathrm{Ad}(G^\\vee)d_H(\\mathcal O_H^{\\mathrm{st}})),\n\\]\nthere exist constants $a_{\\mathcal O_G}$ such that the specific distribution\n\\[\n\\sum_{\\mathcal O_G\\subset \\mathcal O_G^{\\mathrm{st}}}\\widehat\\mu_{\\mathcal O_G}\n\\]\nis equal, on a sufficiently small neighborhood of $\\theta\\in G(F)\\rtimes \\theta$, to the transfer of every stable distribution of the form $\\sum_{\\mathcal O_H\\subset \\mathcal O_H^{\\mathrm{st}}} c_{\\mathcal O_H}\\widehat\\mu_{\\mathcal O_H}$; then for every $\\pi\\in \\Pi_\\psi$ one has\n\\[\n\\bar{\\mathcal N}(\\pi)^{\\max}\\le d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi).\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": "maximal-elements conclusion strengthened to all orbits and strict inequality weakened to non-strict", + "template_used": "stronger_trap" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped the additional upper-bound conclusion under the extra transfer hypothesis", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "removed dependence on the extra transfer hypothesis and replaced existence/membership by equality of maximal set", + "template_used": "uniformity_effectivity" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "quantification in the transfer hypothesis replaced by a single fixed transferred distribution independent of the stable coefficients", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem itself does not reveal the correct option; the correct statement must be identified by comparing nuanced alternatives. Although the source JSON contains an annotated correct choice, that is external metadata rather than leakage in the posed MCQ." + }, + "TAS": { + "score": 1, + "justification": "The item is very close to theorem recall: one option essentially reproduces the intended theorem, while others are slight quantifier or hypothesis modifications. It is not purely tautological, but it is only a mild reformulation rather than a genuinely new application." + }, + "GPS": { + "score": 1, + "justification": "Selecting the answer requires some careful reasoning about logical strength, strict vs. non-strict inequalities, maximal elements vs. all elements, and the role of the extra transfer hypothesis. Still, the task mainly tests precise recognition of a theorem statement rather than substantial generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong: they are plausible theorem-like variants involving common failure modes such as overstrengthening conclusions, weakening hypotheses, or altering quantification. They are distinct and mathematically aligned with realistic misunderstandings." + }, + "total_score": 6, + "overall_assessment": "A well-constructed theorem-discrimination MCQ with strong distractors and no stem-level answer leakage, but it primarily tests precise recall/logical parsing of a theorem rather than deeper generative reasoning." + } + }, + { + "id": "2602.22912v1", + "paper_link": "http://arxiv.org/abs/2602.22912v1", + "theorems_cnt": 4, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm:main:intro:ReductionToOneEnd:doubleRays}\n The consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.", + "start_pos": 7951, + "end_pos": 8165, + "label": "thm:main:intro:ReductionToOneEnd:doubleRays" + }, + "ref_dict": { + "prob:ubiquity:antiraysRays": "\\begin{problem}\\label{prob:ubiquity:antiraysRays}\n Is the disjoint union of a consistently forward and a consistently backward oriented ray ubiquitous?\n\\end{problem}", + "thm:ubiquity:antiraysRays": "\\begin{thm}\\label{thm:ubiquity:antiraysRays}\n Let $D$ be a digraph.\n If there is a sequence $(\\cX_n)_{n\\in\\N}$ such that, for every $n\\in\\N$, the set $\\cX_n$ consists of $n$ rays and $n$ anti-rays of~$D$ that are all pairwise disjoint and such that the sequence $(\\cX_n)_{n\\in\\N}$ does not concentrate in any end of~$D$, then $D$ contains a set of infinitely many rays and infinitely many anti-rays every two of which are disjoint.\n\\end{thm}", + "thm:main:intro:ReductionToOneEnd:doubleRays": "\\begin{theorem}\\label{thm:main:intro:ReductionToOneEnd:doubleRays}\n The consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.\n\\end{theorem}", + "prob:ubiquity:doubleRays": "\\begin{problem}\\cite{GKR2024}*{Problem 1.3}\\label{prob:ubiquity:doubleRays}\n Is the consistently oriented double ray ubiquitous?\n\\end{problem}", + "thm:main:intro:ReductionToOneEnd:RaysAntiRays": "\\begin{theorem}\\label{thm:main:intro:ReductionToOneEnd:RaysAntiRays}\n The disjoint union of a consistently forward and a consistently backward oriented ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.\n\\end{theorem}", + "thm:ubiquity:doubleRays": "\\begin{thm}\\label{thm:ubiquity:doubleRays}\n Let $D$ be a digraph.\n If there is a valid sequence $(\\cR_n)_{n\\in\\N}$ that does not concentrate in finitely many ends, then $D$ contains infinitely many pairwise disjoint double rays.\n\\end{thm}" + }, + "pre_theorem_intro_text_len": 2640, + "pre_theorem_intro_text": "\\label{sec:intro}\n\nThe property of a graph $H$ to be \\emph{ubiquitous} describes the concept that the existence of $n$ pairwise disjoint copies for all $n\\in\\ensuremath{\\mathbb N}$ of a graph $H$ in a graph $G$ implies that $G$ already contains $\\aleph_0$ many pairwise disjoint copies of~$H$.\nOriginally, this was considered with respect to the subgraph relation and Halin showed that the ray is ubiquitous~\\cite{Halin65}*{Satz 1} and that the double ray is ubiquitous~\\cite{Halin70}*{Satz 3}.\nLater, counterexamples \\cites{A77, A02} regarding being ubiquitous were found, which even include trees.\nHence, ubiquity was also considered with respect to other relations such as the minor and topological minor relation, and we refer to \\cites{A79,A80,BEEGHPTI,BEEGHPTII,BEEGHPTIII} for ubiquity results regarding these notions.\nMany of these results leverage that finite graphs (resp.~trees) are well-quasi ordered under the minor (resp.~topological minor) relation.\n\nIt should be noted that Andreae~\\cite{A02} posed the major and still open conjecture whether all locally finite connected graphs are ubiquitous under the minor relation. In the same paper he also provided an example of an uncountable non-ubiquitous graph under the minor relation, whose construction exploits that uncountable graphs are not well-quasi ordered under minors.\n\nFurther modified questions regarding ubiquity have been raised, e.g.~where disjointness is replaced by edge-disjointness.\nIn this setting it turned out to be complicated to even verify that double rays are ubiquitous regarding the subgraph relation~\\cite{BCP15}.\n\nFor digraphs, much less is known, even for the subdigraph relation, and for the rest of this article, we shall only consider the property of being ubiquitous under that relation.\nWhile Zuther~\\cite{Z1997}*{Theorem 2.17} proved that the consistently oriented ray is ubiquitous, Gut et al.~\\cite{GKR2024}*{Theorem 1.1} proved the same for all possible orientations of the undirected ray.\nLater, Gut et al.~\\cite{GKR2025}*{Theorem 1.3} classified almost all orientations of the undirected double ray that are ubiquitous.\nThe only orientation they left out was the consistently oriented one.\nDue to that, they posed the following problem.\n\n\\begin{problem}\\cite{GKR2024}*{Problem 1.3}\\label{prob:ubiquity:doubleRays}\n Is the consistently oriented double ray ubiquitous?\n\\end{problem}\n\nOur main result (Theorem~\\ref{thm:main:intro:ReductionToOneEnd:doubleRays}) states that it suffices to verify Problem~\\ref{prob:ubiquity:doubleRays} for one-ended digraphs.\n(We refer to Section~\\ref{sec:prelims} for the definition of ends of digraphs.)", + "context": "\\label{sec:intro}\n\nThe property of a graph $H$ to be \\emph{ubiquitous} describes the concept that the existence of $n$ pairwise disjoint copies for all $n\\in\\ensuremath{\\mathbb N}$ of a graph $H$ in a graph $G$ implies that $G$ already contains $\\aleph_0$ many pairwise disjoint copies of~$H$.\nOriginally, this was considered with respect to the subgraph relation and Halin showed that the ray is ubiquitous~\\cite{Halin65}*{Satz 1} and that the double ray is ubiquitous~\\cite{Halin70}*{Satz 3}.\nLater, counterexamples \\cites{A77, A02} regarding being ubiquitous were found, which even include trees.\nHence, ubiquity was also considered with respect to other relations such as the minor and topological minor relation, and we refer to \\cites{A79,A80,BEEGHPTI,BEEGHPTII,BEEGHPTIII} for ubiquity results regarding these notions.\nMany of these results leverage that finite graphs (resp.~trees) are well-quasi ordered under the minor (resp.~topological minor) relation.\n\nIt should be noted that Andreae~\\cite{A02} posed the major and still open conjecture whether all locally finite connected graphs are ubiquitous under the minor relation. In the same paper he also provided an example of an uncountable non-ubiquitous graph under the minor relation, whose construction exploits that uncountable graphs are not well-quasi ordered under minors.\n\nFor digraphs, much less is known, even for the subdigraph relation, and for the rest of this article, we shall only consider the property of being ubiquitous under that relation.\nWhile Zuther~\\cite{Z1997}*{Theorem 2.17} proved that the consistently oriented ray is ubiquitous, Gut et al.~\\cite{GKR2024}*{Theorem 1.1} proved the same for all possible orientations of the undirected ray.\nLater, Gut et al.~\\cite{GKR2025}*{Theorem 1.3} classified almost all orientations of the undirected double ray that are ubiquitous.\nThe only orientation they left out was the consistently oriented one.\nDue to that, they posed the following problem.\n\n\\begin{problem}\\cite{GKR2024}*{Problem 1.3}\\label{prob:ubiquity:doubleRays}\n Is the consistently oriented double ray ubiquitous?\n\\end{problem}\n\nOur main result (Theorem~\\ref{thm:main:intro:ReductionToOneEnd:doubleRays}) states that it suffices to verify Problem~\\ref{prob:ubiquity:doubleRays} for one-ended digraphs.\n(We refer to Section~\\ref{sec:prelims} for the definition of ends of digraphs.)\n\n\\begin{problem}\\cite{GKR2024}*{Problem 1.3}\\label{prob:ubiquity:doubleRays}\n Is the consistently oriented double ray ubiquitous?\n\\end{problem}\n\n\\begin{theorem}\\label{thm:main:intro:ReductionToOneEnd:doubleRays}\n The consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.\n\\end{theorem}", + "full_context": "\\label{sec:intro}\n\nThe property of a graph $H$ to be \\emph{ubiquitous} describes the concept that the existence of $n$ pairwise disjoint copies for all $n\\in\\ensuremath{\\mathbb N}$ of a graph $H$ in a graph $G$ implies that $G$ already contains $\\aleph_0$ many pairwise disjoint copies of~$H$.\nOriginally, this was considered with respect to the subgraph relation and Halin showed that the ray is ubiquitous~\\cite{Halin65}*{Satz 1} and that the double ray is ubiquitous~\\cite{Halin70}*{Satz 3}.\nLater, counterexamples \\cites{A77, A02} regarding being ubiquitous were found, which even include trees.\nHence, ubiquity was also considered with respect to other relations such as the minor and topological minor relation, and we refer to \\cites{A79,A80,BEEGHPTI,BEEGHPTII,BEEGHPTIII} for ubiquity results regarding these notions.\nMany of these results leverage that finite graphs (resp.~trees) are well-quasi ordered under the minor (resp.~topological minor) relation.\n\nIt should be noted that Andreae~\\cite{A02} posed the major and still open conjecture whether all locally finite connected graphs are ubiquitous under the minor relation. In the same paper he also provided an example of an uncountable non-ubiquitous graph under the minor relation, whose construction exploits that uncountable graphs are not well-quasi ordered under minors.\n\nFor digraphs, much less is known, even for the subdigraph relation, and for the rest of this article, we shall only consider the property of being ubiquitous under that relation.\nWhile Zuther~\\cite{Z1997}*{Theorem 2.17} proved that the consistently oriented ray is ubiquitous, Gut et al.~\\cite{GKR2024}*{Theorem 1.1} proved the same for all possible orientations of the undirected ray.\nLater, Gut et al.~\\cite{GKR2025}*{Theorem 1.3} classified almost all orientations of the undirected double ray that are ubiquitous.\nThe only orientation they left out was the consistently oriented one.\nDue to that, they posed the following problem.\n\n\\begin{problem}\\cite{GKR2024}*{Problem 1.3}\\label{prob:ubiquity:doubleRays}\n Is the consistently oriented double ray ubiquitous?\n\\end{problem}\n\nOur main result (Theorem~\\ref{thm:main:intro:ReductionToOneEnd:doubleRays}) states that it suffices to verify Problem~\\ref{prob:ubiquity:doubleRays} for one-ended digraphs.\n(We refer to Section~\\ref{sec:prelims} for the definition of ends of digraphs.)\n\n\\begin{problem}\\cite{GKR2024}*{Problem 1.3}\\label{prob:ubiquity:doubleRays}\n Is the consistently oriented double ray ubiquitous?\n\\end{problem}\n\n\\begin{theorem}\\label{thm:main:intro:ReductionToOneEnd:doubleRays}\n The consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.\n\\end{theorem}\n\n\\begin{abstract}\nWe prove that the consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.\nAdditionally, we prove the same equivalence for the disjoint union of a consistently forward and a consistently backward oriented ray. \nFurthermore, we discuss the connection between these two ubiquity problems.\n\\end{abstract}\n\nFor digraphs, much less is known, even for the subdigraph relation, and for the rest of this article, we shall only consider the property of being ubiquitous under that relation.\nWhile Zuther~\\cite{Z1997}*{Theorem 2.17} proved that the consistently oriented ray is ubiquitous, Gut et al.~\\cite{GKR2024}*{Theorem 1.1} proved the same for all possible orientations of the undirected ray.\nLater, Gut et al.~\\cite{GKR2025}*{Theorem 1.3} classified almost all orientations of the undirected double ray that are ubiquitous.\nThe only orientation they left out was the consistently oriented one.\nDue to that, they posed the following problem.\n\n\\begin{problem}\\cite{GKR2024}*{Problem 1.3}\\label{prob:ubiquity:doubleRays}\n Is the consistently oriented double ray ubiquitous?\n\\end{problem}\n\nOur main result (Theorem~\\ref{thm:main:intro:ReductionToOneEnd:doubleRays}) states that it suffices to verify Problem~\\ref{prob:ubiquity:doubleRays} for one-ended digraphs.\n(We refer to Section~\\ref{sec:prelims} for the definition of ends of digraphs.)\n\nAs a major step in the proof of Theorem~\\ref{thm:main:intro:ReductionToOneEnd:doubleRays}, we verify Problem~\\ref{prob:ubiquity:doubleRays} for a large, but technical, class of digraphs, see Theorem~\\ref{thm:ubiquity:doubleRays}.\n\n\\begin{problem}\\label{prob:ubiquity:antiraysRays}\n Is the disjoint union of a consistently forward and a consistently backward oriented ray ubiquitous?\n\\end{problem}\n\n\\begin{theorem}\\label{thm:main:intro:ReductionToOneEnd:RaysAntiRays}\n The disjoint union of a consistently forward and a consistently backward oriented ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.\n\\end{theorem}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:main:intro:ReductionToOneEnd:doubleRays}.]\n Let us assume that Problem~\\ref{prob:ubiquity:doubleRays} holds for the class of one-ended digraphs and let $D$ be a digraphs and $(\\cR_n)_{n\\in\\N}$ be a valid sequence in~~$D$.\n By Theorem~\\ref{thm:ubiquity:doubleRays}, we may assume that $(\\cR_n)_{n\\in\\N}$ concentrates in finitely many ends.\n Then there is a valid sequence $(\\cQ_n)_{n\\in\\N}$ induced by $(\\cR_n)_{n\\in\\N}$ such that all tails from all elements of $\\bigcup_{n\\in\\N}\\cQ_n$ are equivalent.\n By Lemma~\\ref{lem:reduction:doubleRays:oneEnd}, there exists a one-ended subdigraph $D'$ of~$D$ and a valid sequence $(\\cP_n)_{n\\in\\N}$ in~$D'$ induced by $(\\cQ_n)_{n\\in\\N}$.\n Thus, there are infinitely many pairwise disjoint double rays in~$D'$ and hence in~$D$.\n\\end{proof}\n\n\\begin{problem}\\cite{GKR2024}*{Problem 1.3}\\label{prob:ubiquity:doubleRays}\n Is the consistently oriented double ray ubiquitous?\n\\end{problem}\n\n\\begin{theorem}\\label{thm:main:intro:ReductionToOneEnd:doubleRays}\n The consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.\n\\end{theorem}\n\n\\begin{thm}\\label{thm:ubiquity:doubleRays}\n Let $D$ be a digraph.\n If there is a valid sequence $(\\cR_n)_{n\\in\\N}$ that does not concentrate in finitely many ends, then $D$ contains infinitely many pairwise disjoint double rays.\n\\end{thm}", + "post_theorem_intro_text_len": 1487, + "post_theorem_intro_text": "As a major step in the proof of Theorem~\\ref{thm:main:intro:ReductionToOneEnd:doubleRays}, we verify Problem~\\ref{prob:ubiquity:doubleRays} for a large, but technical, class of digraphs, see Theorem~\\ref{thm:ubiquity:doubleRays}.\n\nAdditionally, we consider the following more general version of \\cite{HH2024+}*{Problem 4.2}.\n\n\\begin{problem}\\label{prob:ubiquity:antiraysRays}\n Is the disjoint union of a consistently forward and a consistently backward oriented ray ubiquitous?\n\\end{problem}\n\nWe also verify Problem~\\ref{prob:ubiquity:antiraysRays} for a large, but technical, class of digraphs (Theorem~\\ref{thm:ubiquity:antiraysRays}) and show that Problem~\\ref{prob:ubiquity:antiraysRays} holds if it holds for all one-ended digraphs.\n\n\\begin{theorem}\\label{thm:main:intro:ReductionToOneEnd:RaysAntiRays}\n The disjoint union of a consistently forward and a consistently backward oriented ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.\n\\end{theorem}\n\n\\medskip\n\nThis paper is structured as follows. \nAfter introducing some terminology in Section~\\ref{sec:prelims}, we prove Theorem~\\ref{thm:main:intro:ReductionToOneEnd:RaysAntiRays} in Section~\\ref{sec:ubiquity:antiraysRays}.\nIn Section~\\ref{sec:ubiquity:DoubleRays}, we prove Theorem~\\ref{thm:main:intro:ReductionToOneEnd:doubleRays}.\nIn Section~\\ref{sec:equivalence}, we discuss connections between Problems~\\ref{prob:ubiquity:doubleRays} and \\ref{prob:ubiquity:antiraysRays}.", + "sketch": "As a major step in the proof of Theorem~\\ref{thm:main:intro:ReductionToOneEnd:doubleRays}, the authors \"verify Problem~\\ref{prob:ubiquity:doubleRays} for a large, but technical, class of digraphs\" (see Theorem~\\ref{thm:ubiquity:doubleRays}).", + "expanded_sketch": "No expanded sketch found.", + "expanded_theorem": "\\label{thm:main:intro:ReductionToOneEnd:doubleRays}\n The consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.", + "theorem_type": [ + "Biconditional or Equivalence" + ], + "mcq": { + "question": "A digraph $H$ is called ubiquitous under the subdigraph relation if, whenever a digraph $G$ contains $n$ pairwise disjoint copies of $H$ for every $n\\in\\mathbb{N}$, the digraph $G$ already contains countably infinitely many pairwise disjoint copies of $H$. Let $C$ be the consistently oriented double ray, that is, the two-way infinite directed path $\\cdots \\to v_{-1} \\to v_0 \\to v_1 \\to \\cdots$. Which of the following statements is equivalent to saying that $C$ is ubiquitous under the subdigraph relation?", + "correct_choice": { + "label": "A", + "text": "The consistently oriented double ray $C$ is ubiquitous when restricted to one-ended digraphs; that is, for every one-ended digraph $G$ (a digraph with exactly one end), if $G$ contains $n$ pairwise disjoint copies of $C$ for every $n\\in\\mathbb{N}$, then $G$ contains countably infinitely many pairwise disjoint copies of $C$." + }, + "choices": [ + { + "label": "B", + "text": "The consistently oriented double ray $C$ is ubiquitous when restricted to digraphs with at most one end; that is, for every digraph $G$ with at most one end, if $G$ contains $n$ pairwise disjoint copies of $C$ for every $n\\in\\mathbb{N}$, then $G$ contains countably infinitely many pairwise disjoint copies of $C$." + }, + { + "label": "C", + "text": "For every one-ended digraph $G$, if $G$ contains countably infinitely many pairwise disjoint copies of $C$, then in particular $G$ contains $n$ pairwise disjoint copies of $C$ for every $n\\in\\mathbb{N}$." + }, + { + "label": "D", + "text": "There exists a one-ended digraph $G$ such that, whenever $G$ contains $n$ pairwise disjoint copies of $C$ for every $n\\in\\mathbb{N}$, every digraph containing $G$ as a subdigraph contains countably infinitely many pairwise disjoint copies of $C$." + }, + { + "label": "E", + "text": "The consistently oriented double ray $C$ is ubiquitous when restricted to finitely-ended digraphs; that is, for every digraph $G$ with finitely many ends, if $G$ contains $n$ pairwise disjoint copies of $C$ for every $n\\in\\mathbb{N}$, then $G$ contains countably infinitely many pairwise disjoint copies of $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": "one-ended_exactness_replaced_by_at_most_one_end", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "ubiquity_implication_replaced_by_trivial_converse_on_one-ended_digraphs", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "for_every_one-ended_digraph_replaced_by_there_exists_one-ended_digraph", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "restriction_class_changed_from_one-ended_to_finitely-ended", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines ubiquity and the digraph C but does not explicitly or implicitly point to the correct restriction class ('one-ended digraphs'). The correct answer is not leaked by wording in the prompt." + }, + "TAS": { + "score": 2, + "justification": "The item is not a direct restatement of the definition of ubiquity. It asks for an equivalent reformulation involving a nontrivial restriction to a specific class of digraphs, so the student must distinguish among genuinely different statements." + }, + "GPS": { + "score": 1, + "justification": "The question requires some reasoning or theorem recognition to separate exact equivalence from nearby variants (at most one end, finitely many ends, converse implication, existential quantifier). However, it mainly tests precise recall/recognition rather than deep generative derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically meaningful: they vary boundary conditions, weaken/alter quantifiers, or reverse implication. These reflect common failure modes in reading equivalence statements and theorem hypotheses." + }, + "total_score": 7, + "overall_assessment": "A strong MCQ: it avoids answer leakage, is not tautological, and uses high-quality distractors. Its main limitation is that it leans more toward theorem recognition than substantial generative reasoning." + } + }, + { + "id": "2602.23137v1", + "paper_link": "http://arxiv.org/abs/2602.23137v1", + "theorems_cnt": 5, + "theorem": { + "env_name": "theorem", + "content": "\\label{ergodic-th}\nIf Assumption B holds, then $\\{u(t,x)\\}_{x\\in \\mathbb{R}}$ is ergodic, for any $t>0$. Consequently, by the mean ergodic theorem,\n\\[\n\\frac{1}{R}F_R(t) \\to 0 \\quad \\mbox{a.s and in $L^2(\\Omega)$}, \\quad \\mbox{as $R \\to \\infty$}.\n\\]", + "start_pos": 15922, + "end_pos": 16193, + "label": "ergodic-th" + }, + "ref_dict": { + "FCLT": "\\begin{theorem}[Functional Central Limit Theorem]\n\\label{FCLT}\nUnder the hypotheses of Theorem \\ref{QCLT}, for any $R>0$,\nthe process $\\{F_R(t)\\}_{t\\geq 0}$ has a $\\gamma$-H\\\"older continuous modification (denoted also $F_R$), for any $\\gamma \\in (0,\\frac{\\beta}{2})$, where $\\beta$ is given by \\eqref{def-beta}.\nMoreover,\n\\[\n\\frac{1}{R^{\\beta/2}}F_R(\\cdot) \\stackrel{d}{\\to} \\cG(\\cdot) \\quad \\mbox{in $C[0,\\infty)$ as $R \\to \\infty$},\n\\]\nwhere $\\{\\cG(t)\\}_{t\\geq 0}$ is a zero-mean Gaussian process with covariance \n$\\bE[\\cG(t) \\cG(s)]=K(t,s)$, and $K(t,s)$ given by Theorem \\ref{cov-th}. Here $\\stackrel{d}{\\to}$ denotes the convergence in distribution, and $C[0,\\infty)$ is equipped with the uniform convergence on compact sets.\n\\end{theorem}", + "ergodic-th": "\\begin{theorem}\n\\label{ergodic-th}\nIf Assumption B holds, then $\\{u(t,x)\\}_{x\\in \\bR}$ is ergodic, for any $t>0$. Consequently, by the mean ergodic theorem,\n\\[\n\\frac{1}{R}F_R(t) \\to 0 \\quad \\mbox{a.s and in $L^2(\\Omega)$}, \\quad \\mbox{as $R \\to \\infty$}.\n\\]\n\\end{theorem}", + "key-D": "\\begin{equation}\n\\label{key-D}\n\\|D_{r,y,z}u(t,x)\\|_p \\leq C'_{T,p,\\nu,\\k}|z| \\int_{\\bR}G_{t-r}(x-y')\\k(y-y')dy',\n\\end{equation}", + "def-Mp": "\\begin{equation}\n\\label{def-Mp}\n\\cM_p(t):=\\int_0^t \\int_{\\bR} \\big|(G_{t-s}*\\k)(x)\\big|^p dx ds<\\infty \\quad \\mbox{for all} \\quad t>0,\n\\end{equation}", + "Gauss-color": "\\begin{equation}\n\\label{Gauss-color}\nW_t(\\varphi)=B_t(\\varphi*\\k) \\quad \\mbox{for all $t>0$ and $\\varphi \\in \\cS(\\bR^d)$},\n\\end{equation}", + "def-cov-W": "\\begin{equation}\n\\label{def-cov-W}\n\\bE[W_t(\\varphi) W_s(\\psi)]=(t \\wedge s) \\int_{\\bR^d} \\int_{\\bR^d}\\varphi(x)\\psi(y) f(x-y)dxdy,\n\\end{equation}", + "appB": "\\begin{proof}\nBy direct calculation, we see that $\\tilde{f} \\star_{k}^{1} \\tilde{g} =0$ implies that $\\tilde{f} \\star_{k}^{\\ell} \\tilde{g} =0$ for all $\\ell=2,\\ldots,k$. Hence, all the terms in the product formula are zero, except those corresponding to $\\ell=0$ and $\\ell=1$:\n\\begin{align*}\nI_n(f) I_m(g) &=I_n(\\tilde{f}) I_m(\\tilde{g})=I_{n+m}(\\tilde{f} \\otimes \\tilde{g})+\n\\sum_{k=1}^{n\\wedge m}k! \\binom{n}{k}\\binom{m}{k}I_{n+m-k}(\\tilde{f} \\star_{k}^{0} \\tilde{g} ).\n\\end{align*}\nNote that $I_{n+m}(\\tilde{f} \\otimes \\tilde{g})=I_{n+m}(f \\otimes g)$ since $\\tilde{f} \\otimes \\tilde{g}$ and $f \\otimes g$ have the same symmetrization.\n\\end{proof}\n\n\\section{Inequalities for Riesz potentials}\n\\label{appB}\n\nIn this section, we include some inequalities for Riesz potentials which were used for the proof of the QCLT in the case when $\\k$ is the Riesz kernel.\n\nWe recall that the Riesz kernel of order $\\alpha \\in (0,d)$ is defined by \n\\[\nR_{d,\\alpha}(x)=C_{d,\\alpha}|x|^{-(d-\\alpha)} \\quad \\mbox{with} \\quad C_{d,\\alpha}=\\pi^{-\\frac{d}{2}}2^{-\\alpha} \\frac{\\Gamma(\\frac{d-\\alpha}{2})}{\\Gamma(\\frac{\\alpha}{2})}.\n\\]\nIts Fourier transform (in the sense of distributions) is $\\cF R_{d,\\alpha}(\\xi)=|\\xi|^{-\\alpha}$, since\n\\[\n\\int_{\\bR^d}\\varphi(x)|x|^{-\\alpha}dx=C_{d,\\alpha} \\int_{\\bR^d}\\cF \\varphi(\\xi)|\\xi|^{-(d-\\alpha)}d\\xi \\quad \n\\mbox{for any $\\varphi \\in \\cS(\\bR^d)$},\n\\]\nwhere $\\cS(\\bR^d)$ is the set of rapidly decreasing functions on $\\bR^d$.\n\n\\medskip\n\nFor the next result, we refer to Theorem 1, p.119 of \\cite{stein70}.\n\n\\begin{theorem}[Hardy-Littlewood-Sobolov inequality]\n\\label{HLS}\nFor any $\\alpha \\in (0,d)$ and $p>\\frac{d}{d-\\alpha}$, \n\\[\n\\| \\varphi *R_{d,\\alpha}\\|_{L^p(\\bR^d)} \\leq A_{d.\\alpha,p}\\|\\varphi\\|_{L^{q}(\\bR^d)} \\quad \\mbox{for any $\\varphi \\in L^q(\\bR^d)$},\n\\]\nwhere $A_{d,\\alpha,p}>0$ is a constant depending on $(d,\\alpha,p)$, and $q$ is defined by:\n\\[\n\\frac{1}{q}=\\frac{1}{p}+\\frac{\\alpha}{d}.\n\\]\n(Condition $p>\\frac{d}{d-\\alpha}$ is equivalent to $q>1$.)\n\\end{theorem}\n\nAs a consequence of Theorem \\ref{HLS}, we obtain the following lemma.\n\n\\begin{lemma}\n\\label{Hol-HLS}\nFor any $\\alpha \\in (0,d)$, $q \\in (1,\\frac{d}{\\alpha})$, $\\varphi \\in L^q(\\bR^d)$ and $\\psi \\in L^{q'}(\\bR^d)$,\n\\[\n\\left|\\int_{\\bR^d}\\int_{\\bR^d}\\varphi(x)\\psi(y) R_{d,\\alpha}(x-y) dxdy \\right| \\leq A_{d,\\alpha,p} \\|\\varphi\\|_{L^q(\\bR^d)} \\|\\psi\\|_{L^{q'}(\\bR^d)},\n\\]\nwhere $A_{d,\\alpha,p}$ is the constant from Theorem \\ref{HLS}, and $(p,q')$ are defined by\n\\[\n\\frac{1}{p}+\\frac{1}{q}=1 \\quad \\mbox{and} \\quad \\frac{1}{q}+\\frac{1}{q'}=1+\\frac{\\alpha}{d}.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nWe apply H\\\"older's inequality, letting $\\frac{1}{p}+\\frac{1}{q}=1$, and obtain that\n\\begin{align*}\n\\left|\\int_{\\bR^d}\\int_{\\bR^d}\\varphi(x)\\psi(y) R_{d,\\alpha}(x-y) dxdy \\right|\n& =\\left| \\int_{\\bR^d}\\varphi(x) \\big(\\psi * R_{d,\\alpha} \\big) dx\\right| \\leq \\|\\varphi\\|_{L^q(\\bR^d)} \\|\\psi * R_{d,\\alpha}\\|_{L^p(\\bR^d)}.\n\\end{align*}\nNote that $q<\\frac{d}{\\alpha}$ is equivalent to $p>\\frac{d}{d-\\alpha}$.\nTherefore, by Theorem \\ref{HLS} \n\\[\n \\|\\psi * R_{d,\\alpha}\\|_{L^p(\\bR^d)} \\leq A_{d,\\alpha,p} \\|\\psi \\|_{L^{q'}(\\bR^d)},\n\\]\nwhere $\\frac{1}{q'}=\\frac{1}{p}+\\frac{\\alpha}{d}$. Finally, we note that $\n1-\\frac{1}{q}=\\frac{1}{p}=\\frac{1}{q'}-\\frac{\\alpha}{d}$, and hence, $\\frac{1}{q}+\\frac{1}{q'}=1+\\frac{\\alpha}{d}$.\n\n\\end{proof}", + "mom-u": "\\begin{equation}\n\\label{mom-u}\nK_p(T):=\\sup_{(t,x)\\in [0,T] \\times \\bR}\\|u(t,x)\\|_p<\\infty \\quad \\mbox{for all} \\quad T>0,\n\\end{equation}", + "cov-th": "\\begin{theorem}[Limiting covariance]\n\\label{cov-th}\nIf Assumption B holds, then\nfor any $t,s>0$,\n\\[\n\\lim_{R \\to \\infty}\\frac{1}{R^{\\beta}}\\bE[F_R(t) F_R(s)]=K(t,s) \\quad \\mbox{is finite},\n\\] \nwhere \n\\begin{equation}\n\\label{def-beta}\n\\beta:=\\left\\{\n\\begin{array}{ll} 1 & \\mbox{if $\\k \\in L^1(\\bR)$,} \\\\\n\\alpha+1 & \\mbox{if $\\k=R_{1,\\alpha/2}$ for some $\\alpha \\in (0,1)$. }\n\\end{array} \\right.\n\\end{equation}\nIn particular, $R^{-\\beta} \\sigma_R^2(t) \\to K(t,t)$ as $R \\to \\infty$, for any $t>0$.\n\\end{theorem}", + "QCLT": "\\begin{theorem}[Quantitative Central Limit Theorem]\n\\label{QCLT}\nSuppose that Assumption B holds, and there exists $p \\in (1,2]$ such that \n\\begin{equation}\n\\label{mp-m2p}\nm_{p}<\\infty \\quad \\mbox{and} \\quad m_{2p}<\\infty.\n\\end{equation}\n\nIf $\\k \\in L^1(\\bR)$, then\nfor any $t>0$,\n\\[\n{\\rm dist}\\left( \\frac{F_R(t)}{\\sigma_R(t)},Z\\right) \\leq C_t R^{-(1-\\frac{1}{p})}\n\\]\n\nIf $\\k=R_{1,\\alpha/2}$ for some $\\alpha \\in (0,1)$, and $p>\\frac{2}{2-\\alpha}$, then\n\\[\n{\\rm dist}\\left( \\frac{F_R(t)}{\\sigma_R(t)},Z\\right) \\leq\nC_t R^{-\\e} \\quad \\mbox{for any $\\varepsilon \\in \\big(0, 1-\\frac{1}{p}-\\frac{\\alpha}{2}\\big)$}.\n\\]\n\nHere $C_t>0$ is a constant depending on $t$, and ${\\rm dist}$ is the Fortet-Mourier distance, the 1-Wasserstein distance, or the Kolmogorov distance.\n\\end{theorem}", + "mp-m2p": "\\begin{equation}\n\\label{mp-m2p}\nm_{p}<\\infty \\quad \\mbox{and} \\quad m_{2p}<\\infty.\n\\end{equation}", + "def-G": "\\begin{equation}\n\\label{def-G}\nG_t(x)=\\frac{1}{2}1_{\\{|x|0$ and $x \\in \\bR$.}\n\\end{equation}", + "appA": "\\label{appA}\n\nIn this appendix section, we include some auxiliary results from Malliavin calculus which were used in the sequel. \n\\medskip\n\nLet $(Z,\\cZ,\\fm)$ be a $\\sigma$-finite measure space, and $N", + "key-D2": "\\begin{equation}\n\\label{key-D2}\n\\|D_{(r_1,y_1,z_1),(r_2,y_2,z_2)}^2 u(t,x) \\|_p \\leq C_{T,p,\\nu,\\k}'' |z_1 z_2| \\int_{\\bR^2} \\widetilde{f}_2(r_1,y_1',r_2,y_2',t,x) \\k(y_1-y_1')\\k(y_2-y_2') dy_1'dy_2',\n\\end{equation}", + "HAM": "\\begin{align}\n\\label{HAM}\n\t\\begin{cases}\n\t\t\\dfrac{\\partial^2 u}{\\partial^2 t} (t,x)\n\t\t= \\dfrac{\\partial^2 u}{\\partial x^2} (t,x)+u(t,x) \\dot{X}(t,x), \\\n\t\tt>0, \\ x \\in \\bR, \\\\\n\t\tu(0,x) = 1, \\dfrac{\\partial u}{\\partial t} (0,x)=0, \\quad x \\in \\bR,\n\t\\end{cases}\n\\end{align}", + "def-beta": "\\begin{equation}\n\\label{def-beta}\n\\beta:=\\left\\{\n\\begin{array}{ll} 1 & \\mbox{if $\\k \\in L^1(\\bR)$,} \\\\\n\\alpha+1 & \\mbox{if $\\k=R_{1,\\alpha/2}$ for some $\\alpha \\in (0,1)$. }\n\\end{array} \\right.\n\\end{equation}", + "lem-key-Z": "\\begin{lemma}\n\\label{lem-key-Z}\nSuppose that Assumption B holds.\nLet $\\{Z(t,x);t\\geq 0,x\\in \\bR\\}$ be an adapted random field such that\n$\\{Z(t,x)\\}_{x\\in \\bR^d}$ is strictly stationary and $Z(t,x) \\in {\\rm dom}(D)$ for any $t\\geq 0$ and $x \\in \\bR$. Assume that for any $00$ is a constant depending on $t$. Then $\\{Z(t,x)\\}_{x\\in \\bR}$ is ergodic.\n\\end{lemma}", + "HLS": "\\begin{theorem}[Hardy-Littlewood-Sobolov inequality]\n\\label{HLS}\nFor any $\\alpha \\in (0,d)$ and $p>\\frac{d}{d-\\alpha}$, \n\\[\n\\| \\varphi *R_{d,\\alpha}\\|_{L^p(\\bR^d)} \\leq A_{d.\\alpha,p}\\|\\varphi\\|_{L^{q}(\\bR^d)} \\quad \\mbox{for any $\\varphi \\in L^q(\\bR^d)$},\n\\]\nwhere $A_{d,\\alpha,p}>0$ is a constant depending on $(d,\\alpha,p)$, and $q$ is defined by:\n\\[\n\\frac{1}{q}=\\frac{1}{p}+\\frac{\\alpha}{d}.\n\\]\n(Condition $p>\\frac{d}{d-\\alpha}$ is equivalent to $q>1$.)\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 13063, + "pre_theorem_intro_text": "The theory of stochastic partial differential equations (SPDEs) has grown consistently in the last three decades, and now constitutes a central area of stochastic analysis. SPDEs model the evolution of random systems across multiple scales, ranging from microscopic particle dynamics to macroscopic continuum limits, and arise in diverse contexts, such as interface growth, disordered media, turbulence, and transport in random environments. They provide a rigorous mathematical framework for universality phenomena observed in physics. Motivated by these physical models and mathematical challenges, systematic theories for SPDEs began to emerge in the late 1980s and 1990s. Prominent developments include Walsh's \\cite{walsh86} probabilistic approach based on martingale measures and random fields, and the semigroup approach introduced by Da Prato and Zabczyk \\cite{DZ92}. Basic equations which have been studied are the stochastic heat equation and the stochastic wave equation, and their particular cases, the parabolic Anderson model (pAm), respectively the hyperbolic Anderson model (hAm). However, comprehensive solution theories for numerous other SPDEs\nremained beyond the grasp of mathematical analysis, until the early 21st century. These challenges\nstemmed from the singular nature of such equations, a singularity intricately linked to the irregularities inherent in the random data involved. A significant breakthrough arrived in 2014 when Hairer \\cite{hairer14}\nconstructed a well-posedness framework for the Kardar-Parisi-Zhang (KPZ) equation, using concepts from Lyons' rough paths theory \\cite{lyons98}. In dimension 1, the solution of the KPZ equation is directly related to the solution of (pAm\n) via the Cole-Hopf transformation.\n\n\\medskip\n\nTraditionally, SPDEs are perturbed by a space-time Gaussian white noise in dimension $d=1$, which is a zero-mean Gaussian process $\\{B_t(\\varphi);t> 0,\\varphi \\in L^2(\\mathbb{R})\\}$ with covariance:\n\\[\n\\mathbb{E}[B_t(\\varphi) B_s(\\psi)]=(t \\wedge s) \\langle \\varphi,\\psi \\rangle_{L^2(\\mathbb{R})}.\n\\]\nIn higher dimensions, Dalang introduced in \\cite{dalang99}, a spatially-homogeneous (or colored) noise, as a zero-mean Gaussian process $\\{W_t(\\varphi);t\\geq 0,\\varphi \\in \\mathcal{S}(\\mathbb{R})\\}$ with covariance:\n\\begin{equation}\n\\label{def-cov-W}\n\\mathbb{E}[W_t(\\varphi) W_s(\\psi)]=(t \\wedge s) \\int_{\\mathbb{R}^d} \\int_{\\mathbb{R}^d}\\varphi(x)\\psi(y) f(x-y)dxdy,\n\\end{equation}\nwhere $\\mathcal{S}(\\mathbb{R}^d)$ is the set of all rapidly decreasing functions on $\\mathbb{R}^d$, and $f:\\mathbb{R}^d \\to [0,\\infty]$ is the Fourier transform of a tempered measure $\\mu$. This noise can also be defined as:\n\\begin{equation}\n\\label{Gauss-color}\nW_t(\\varphi)=B_t(\\varphi*\\kappa) \\quad \\mbox{for all $t>0$ and $\\varphi \\in \\mathcal{S}(\\mathbb{R}^d)$},\n\\end{equation}\nwhere the kernel $\\kappa$ is chosen such that $k * \\widetilde{k}=f$, and $\\widetilde{k}(x):=k(-x)$ for all $x \\in \\mathbb{R}^d$.\n\n\\medskip\n\nOne of the problems which has received a lot of interest in the literature in the recent years is the study of the asymptotic behaviour as $R \\to \\infty$ of the spatial integral \n\\[\nF_R(t)=\\int_{|x|0, \\ x \\in \\mathbb{R}, \\\\\n\t\tu(0,x) = 1, \\dfrac{\\partial u}{\\partial t} (0,x)=0, \\quad x \\in \\mathbb{R},\n\t\\end{cases}\n\\end{align}\nwhere $X=\\{X_t(\\varphi);t>0,\\varphi \\in \\mathcal{S}(\\mathbb{R})\\}$ is the L\\'evy colored noise, given by:\n\\[\nX_t(\\varphi)=L_t(\\varphi *\\kappa).\n\\]\nwith $L_t(\\varphi)=L(1_{[0,t]} \\varphi)$ for all $\\varphi \\in L^2(\\mathbb{R})$, and $L=\\{L(\\varphi);\\varphi \\in L^2(\\bR_{+} \\times \\mathbb{R})\\}$ being the {\\em L\\'evy white noise}:\n\\[\nL(\\varphi)=\\int_{\\bR_{+} \\times \\mathbb{R} \\times \\bR_0} \\varphi(t,x)z \\widehat{N}(dt,dx,dz) \\quad \\mbox{for all} \\quad \\varphi \\in L^2(\\bR_{+} \\times \\mathbb{R}).\n\\]\n\nHere $N$ is a Poisson random measure on $Z=\\bR_{+}\\times \\mathbb{R} \\times \\bR_0$ with intensity \n\\[\n\\mathfrak{m}(dt,dx,dz)=dt dx \\nu(dz),\n\\] \ndefined on a complete probability space $(\\Omega,\\mathcal{F},\\mathbb{P})$, and $\\widehat{N}(A)=N(A)-\\mathfrak{m}(A)$ is the compensated version of $N$.\nThe space $\\bR_0=\\mathbb{R} \\verb2\\2 \\{0\\}$ is equipped with the distance $d(x,y)=|x^{-1}-y^{-1}|$, so that the bounded subsets of $\\bR_0$ are those that are bounded away from 0.\nWe assume that the measure $\\nu$ satisfies the following condition:\n\\[\nm_2:=\\int_{\\bR_0}|z|^2 \\nu(dz)<\\infty.\n\\]\nUnder this condition, the following isometry property holds: for any $\\varphi,\\psi \\in L^2(\\bR_{+}\\times \\mathbb{R})$,\n\\[\n\\mathbb{E}[L(\\varphi)L(\\psi)]=m_2 \\int_{\\bR_+ \\times \\mathbb{R}} \\varphi(t,x)\\psi(t,x)dtdx.\n\\]\n\nWe denote by $\\mathcal{S}'(\\mathbb{R})$ the set of tempered distributions on $\\mathbb{R}$.\nAs in \\cite{BJ25}, we assume that the kernel $\\kappa$ satisfies the following assumption:\n\n\\medskip\n\n\\noindent {\\bf Assumption A.}\n$\\kappa:\\mathbb{R} \\to [0,\\infty]$ is a continuous, symmetric, and tempered function such that:\\\\\n(a) $\\mathcal{F}\\kappa=h$ in $\\mathcal{S}'(\\mathbb{R})$ is a tempered non-negative function, and $h^2$ is tempered;\\\\\n(b) $f=\\kappa* \\widetilde{\\kappa}$ is a continuous, symmetric, and tempered function ($\\mathcal{F} f=h^2$ in $\\mathcal{S}'(\\mathbb{R}))$;\\\\\n(c) $\\kappa(x)<\\infty$ for all $x\\not=0$, and $f(x)<\\infty$ for all $x\\not=0$.\n\n\\medskip\n\nWe will impose Assumption A throughout this article.\nUnder this assumption, the noise $X$ is well-defined, since $\\varphi*\\kappa \\in \\L^2(\\mathbb{R})$ for any $\\varphi \\in \\mathcal{S}(\\mathbb{R})$. Moreover, for any $\\varphi,\\psi \\in \\mathcal{S}(\\mathbb{R})$,\n\\[\n\\int_{\\mathbb{R}} \\int_{\\mathbb{R}}\\varphi(x)\\psi(y) f(x-y)dxdy=\\int_{\\mathbb{R}} \\mathcal{F} \\varphi(\\xi) \\overline{\\mathcal{F} \\psi(\\xi)} \\mu(d\\xi),\n\\]\nwhere $\\mathcal{F} \\varphi(\\xi)=\\int_{\\mathbb{R}}e^{-i \\xi x} \\varphi(x) dx$ is the Fourier transform of $\\varphi$, and\n\\begin{equation}\n\\label{def-mu}\n\\mu(d\\xi)=\\frac{1}{2\\pi} |\\mathcal{F} \\kappa(\\xi)|^2 d\\xi.\n\\end{equation}\n\nSince $d=1$, $\\mu$ satisfies {\\em Dalang's condition}: (see Remark 10.(b) of \\cite{dalang99})\n\\[\n\\int_{\\mathbb{R}}\\frac{1}{1+|\\xi|^2}\\mu(d\\xi)<\\infty.\n\\]\n\n\\medskip\n\nThe stochastic integral with respect to $X$ is defined using Walsh's theory \\cite{walsh86}, \nsince $X$ induces a martingale measure. This integral shares many properties with the integral defined in \\cite{dalang99} for the Gaussian colored noise, except that its moments are estimated in a different way.\n\nLet $G$ be the fundamental solution of the wave equation in dimension $d=1$, given by:\n\\begin{equation}\n\\label{def-G}\nG_t(x)=\\frac{1}{2}1_{\\{|x|0$ and $x \\in \\mathbb{R}$.}\n\\end{equation}\n\nInspired by D'Alembert formula, we introduce the following definition. \n\n\\begin{definition}\n{\\rm A predictable process $\\{u(t,x);t\\geq 0,x\\in \\mathbb{R}\\}$ is called a {\\em (mild) solution} of \\eqref{HAM}\nif it satisfies the integral equation:\n\\begin{equation}\n\\label{def-sol}\nu(t,x)=1+\\int_0^t \\int_{\\mathbb{R}}G_{t-s}(x-y)u(s,y)X(ds,dy).\n\\end{equation}\n}\n\\end{definition}\n\nWe recall that a random field $\\{\\Phi(t,x);t\\geq 0,x\\in \\mathbb{R}\\}$ is {\\em predictable} if it is measurable with respect to the predictable $\\sigma$-field on $\\Omega \\times \\bR_{+} \\times \\mathbb{R}$, which is the minimal $\\sigma$-field with respect to which all elementary processes are measurable. An {\\em elementary process} is a linear combination of processes of the form\n\\[\n\\Phi(t,x)=Y 1_{(a,b]}(t) 1_{A}(x)\n\\]\nwhere $0\\leq a0.\n\\]\n\nBy Lemma 5.5 of \\cite{BJ25}, the process $\\{u(t,x)\\}_{x\\in \\mathbb{R}}$ is strictly stationary, for any $t>0$.\nMoreover, by Theorem 5.6 of \\cite{BJ25}, if for some $p\\geq 2$, we have\n\\begin{equation}\n\\label{def-mp}\nm_p:=\\int_{\\bR_0}|z|^p \\nu(dz)<\\infty,\n\\end{equation}\nand\n\\begin{equation}\n\\label{def-Mp}\n\\cM_p(t):=\\int_0^t \\int_{\\mathbb{R}} \\big|(G_{t-s}*\\kappa)(x)\\big|^p dx ds<\\infty \\quad \\mbox{for all} \\quad t>0,\n\\end{equation}\nthen\n\\begin{equation}\n\\label{mom-u}\nK_p(T):=\\sup_{(t,x)\\in [0,T] \\times \\mathbb{R}}\\|u(t,x)\\|_p<\\infty \\quad \\mbox{for all} \\quad T>0,\n\\end{equation}\nwhere $\\|\\cdot\\|_p$ denotes the norm in $L^p(\\Omega)$. The goal of the present paper is to complement this analysis by showing that the {\\em spatial average} of the (centered) solution:\n\\[\nF_R(t)=\\int_{-R}^{R}\\big(u(t,x)-1 \\big)dx.\n\\]\nhas asymptotic Gaussian fluctuations when $R \\to \\infty$, with a precise rate of convergence which depends on the noise. We denote $\\sigma_R^2(t)={\\rm Var}(F_R(t))$. \n\n\\medskip\n\n\\medskip\n\nRecall that the {\\em Riesz kernel of order $\\alpha \\in (0,1)$} is given by:\n\\[\nR_{1,\\alpha}(x)=C_{1,\\alpha}|x|^{-(1-\\alpha)} \\quad \\mbox{where} \\quad C_{1,\\alpha}=\\pi^{-1/2}2^{-\\alpha}\\frac{\\Gamma(\\frac{1-\\alpha}{2})}{\\Gamma(\\frac{\\alpha}{2})}.\n\\]\nMoreover, $\\mathcal{F} R_{1,\\alpha}(\\xi)=|\\xi|^{-\\alpha}d\\xi$, and $R_{1,\\alpha}*R_{1,\\beta}=R_{1,\\alpha+\\beta}$ for any $\\alpha,\\beta>0$ with $\\alpha+\\beta<1$.\n\n\\medskip\n\nWe introduce the following assumption.\n\n\\medskip\n\n\\noindent {\\bf Assumption B.} \n(i) $\\kappa\\in L^1(\\mathbb{R})$; or (ii) $\\kappa=R_{1,\\alpha/2}$ for some $\\alpha \\in (0,1)$.\n\n\\begin{remark}\n\\label{rem-M}\n{\\rm Suppose that Assumption B holds. Applying Young's inequality if $\\kappa \\in L^1(\\mathbb{R})$, and Theorem \\ref{HLS} if $\\kappa=R_{1,\\alpha/2}$, we see that:\n\\[\n\\|G_t* \\kappa\\|_{L^p(\\mathbb{R})} \\leq\n\\left\\{\n\\begin{array}{ll} \n\\|\\kappa\\|_{L^1(\\mathbb{R})} \\|G_t\\|_{L^p(\\mathbb{R})} & \\mbox{if $\\kappa \\in L^1(\\mathbb{R})$ and $p\\geq 1$} \\\\\nA_{1,\\alpha,p}\\|G_t\\|_{L^q(\\mathbb{R})} & \\mbox{if $\\kappa=R_{1,\\alpha/2}$, $\\alpha \\in (0,1)$, $p>\\frac{2}{2-\\alpha}$ and $\\frac{1}{q}=\\frac{1}{p}+\\frac{\\alpha}{2}$}\n\\end{array} \\right.\n\\]\nSince $\\|G_t\\|_{L^p(\\mathbb{R})}^p=2^{1-p}t$ for any $p>0$, we infer that condition \\eqref{def-Mp} holds for any $p\\geq 2$. Therefore, \\eqref{mom-u} holds for any $p \\geq 2$ such that $m_p<\\infty$.\n}\n\\end{remark}\n\nWe recall the following definitions. Let $X$ and $Y$ be random variables defined on the same probability space.\nThe {\\em $1$-Wasserstein distance} between $X$ and $Y$ is:\n\\[\nd_{W}(X,Y)=\\sup_{{\\rm Lip}(h) \\leq 1} \\big|\\mathbb{E}[h(X)]-\\mathbb{E}[h(Y)]\\big|, \\quad\n\\mbox{where} \\quad\n{\\rm Lip}(h):=\\sup_{x \\not=y}\\frac{|h(x)-h(y)|}{|x-y|}.\n\\]\nThe {\\em Fortet-Mourier distance} between $X$ and $Y$ is:\n\\[\nd_{FM}(X,Y)=\\sup_{\\|h\\|_{\\infty}+{\\rm Lip}(h) \\leq 1} \\big|\\mathbb{E}[h(X)]-\\mathbb{E}[h(Y)]\\big|,\n\\]\nThe {\\em Kolmogorov distance} between $X$ and $Y$ is:\n\\[\nd_{K}(X,Y)=\\sup_{x \\in \\mathbb{E}} \\big|P(X \\leq x)-P(Y \\leq x)\\big|.\n\\]\n\nWe are now ready to state the main results of this article.", + "context": "Traditionally, SPDEs are perturbed by a space-time Gaussian white noise in dimension $d=1$, which is a zero-mean Gaussian process $\\{B_t(\\varphi);t> 0,\\varphi \\in L^2(\\mathbb{R})\\}$ with covariance:\n\\[\n\\mathbb{E}[B_t(\\varphi) B_s(\\psi)]=(t \\wedge s) \\langle \\varphi,\\psi \\rangle_{L^2(\\mathbb{R})}.\n\\]\nIn higher dimensions, Dalang introduced in \\cite{dalang99}, a spatially-homogeneous (or colored) noise, as a zero-mean Gaussian process $\\{W_t(\\varphi);t\\geq 0,\\varphi \\in \\mathcal{S}(\\mathbb{R})\\}$ with covariance:\n\\begin{equation}\n\\label{def-cov-W}\n\\mathbb{E}[W_t(\\varphi) W_s(\\psi)]=(t \\wedge s) \\int_{\\mathbb{R}^d} \\int_{\\mathbb{R}^d}\\varphi(x)\\psi(y) f(x-y)dxdy,\n\\end{equation}\nwhere $\\mathcal{S}(\\mathbb{R}^d)$ is the set of all rapidly decreasing functions on $\\mathbb{R}^d$, and $f:\\mathbb{R}^d \\to [0,\\infty]$ is the Fourier transform of a tempered measure $\\mu$. This noise can also be defined as:\n\\begin{equation}\n\\label{Gauss-color}\nW_t(\\varphi)=B_t(\\varphi*\\kappa) \\quad \\mbox{for all $t>0$ and $\\varphi \\in \\mathcal{S}(\\mathbb{R}^d)$},\n\\end{equation}\nwhere the kernel $\\kappa$ is chosen such that $k * \\widetilde{k}=f$, and $\\widetilde{k}(x):=k(-x)$ for all $x \\in \\mathbb{R}^d$.\n\nLet $G$ be the fundamental solution of the wave equation in dimension $d=1$, given by:\n\\begin{equation}\n\\label{def-G}\nG_t(x)=\\frac{1}{2}1_{\\{|x|0$ and $x \\in \\mathbb{R}$.}\n\\end{equation}\n\nBy Lemma 5.5 of \\cite{BJ25}, the process $\\{u(t,x)\\}_{x\\in \\mathbb{R}}$ is strictly stationary, for any $t>0$.\nMoreover, by Theorem 5.6 of \\cite{BJ25}, if for some $p\\geq 2$, we have\n\\begin{equation}\n\\label{def-mp}\nm_p:=\\int_{\\bR_0}|z|^p \\nu(dz)<\\infty,\n\\end{equation}\nand\n\\begin{equation}\n\\label{def-Mp}\n\\cM_p(t):=\\int_0^t \\int_{\\mathbb{R}} \\big|(G_{t-s}*\\kappa)(x)\\big|^p dx ds<\\infty \\quad \\mbox{for all} \\quad t>0,\n\\end{equation}\nthen\n\\begin{equation}\n\\label{mom-u}\nK_p(T):=\\sup_{(t,x)\\in [0,T] \\times \\mathbb{R}}\\|u(t,x)\\|_p<\\infty \\quad \\mbox{for all} \\quad T>0,\n\\end{equation}\nwhere $\\|\\cdot\\|_p$ denotes the norm in $L^p(\\Omega)$. The goal of the present paper is to complement this analysis by showing that the {\\em spatial average} of the (centered) solution:\n\\[\nF_R(t)=\\int_{-R}^{R}\\big(u(t,x)-1 \\big)dx.\n\\]\nhas asymptotic Gaussian fluctuations when $R \\to \\infty$, with a precise rate of convergence which depends on the noise. We denote $\\sigma_R^2(t)={\\rm Var}(F_R(t))$.\n\n\\begin{remark}\n\\label{rem-M}\n{\\rm Suppose that Assumption B holds. Applying Young's inequality if $\\kappa \\in L^1(\\mathbb{R})$, and Theorem \\ref{HLS} if $\\kappa=R_{1,\\alpha/2}$, we see that:\n\\[\n\\|G_t* \\kappa\\|_{L^p(\\mathbb{R})} \\leq\n\\left\\{\n\\begin{array}{ll} \n\\|\\kappa\\|_{L^1(\\mathbb{R})} \\|G_t\\|_{L^p(\\mathbb{R})} & \\mbox{if $\\kappa \\in L^1(\\mathbb{R})$ and $p\\geq 1$} \\\\\nA_{1,\\alpha,p}\\|G_t\\|_{L^q(\\mathbb{R})} & \\mbox{if $\\kappa=R_{1,\\alpha/2}$, $\\alpha \\in (0,1)$, $p>\\frac{2}{2-\\alpha}$ and $\\frac{1}{q}=\\frac{1}{p}+\\frac{\\alpha}{2}$}\n\\end{array} \\right.\n\\]\nSince $\\|G_t\\|_{L^p(\\mathbb{R})}^p=2^{1-p}t$ for any $p>0$, we infer that condition \\eqref{def-Mp} holds for any $p\\geq 2$. Therefore, \\eqref{mom-u} holds for any $p \\geq 2$ such that $m_p<\\infty$.\n}\n\\end{remark}\n\nWe recall the following definitions. Let $X$ and $Y$ be random variables defined on the same probability space.\nThe {\\em $1$-Wasserstein distance} between $X$ and $Y$ is:\n\\[\nd_{W}(X,Y)=\\sup_{{\\rm Lip}(h) \\leq 1} \\big|\\mathbb{E}[h(X)]-\\mathbb{E}[h(Y)]\\big|, \\quad\n\\mbox{where} \\quad\n{\\rm Lip}(h):=\\sup_{x \\not=y}\\frac{|h(x)-h(y)|}{|x-y|}.\n\\]\nThe {\\em Fortet-Mourier distance} between $X$ and $Y$ is:\n\\[\nd_{FM}(X,Y)=\\sup_{\\|h\\|_{\\infty}+{\\rm Lip}(h) \\leq 1} \\big|\\mathbb{E}[h(X)]-\\mathbb{E}[h(Y)]\\big|,\n\\]\nThe {\\em Kolmogorov distance} between $X$ and $Y$ is:\n\\[\nd_{K}(X,Y)=\\sup_{x \\in \\mathbb{E}} \\big|P(X \\leq x)-P(Y \\leq x)\\big|.\n\\]\n\nWe are now ready to state the main results of this article.\n\n\\begin{theorem}[Hardy-Littlewood-Sobolov inequality]\n\\label{HLS}\nFor any $\\alpha \\in (0,d)$ and $p>\\frac{d}{d-\\alpha}$, \n\\[\n\\| \\varphi *R_{d,\\alpha}\\|_{L^p(\\bR^d)} \\leq A_{d.\\alpha,p}\\|\\varphi\\|_{L^{q}(\\bR^d)} \\quad \\mbox{for any $\\varphi \\in L^q(\\bR^d)$},\n\\]\nwhere $A_{d,\\alpha,p}>0$ is a constant depending on $(d,\\alpha,p)$, and $q$ is defined by:\n\\[\n\\frac{1}{q}=\\frac{1}{p}+\\frac{\\alpha}{d}.\n\\]\n(Condition $p>\\frac{d}{d-\\alpha}$ is equivalent to $q>1$.)\n\\end{theorem}\n\n\\begin{equation}\n\\label{def-Mp}\n\\cM_p(t):=\\int_0^t \\int_{\\bR} \\big|(G_{t-s}*\\k)(x)\\big|^p dx ds<\\infty \\quad \\mbox{for all} \\quad t>0,\n\\end{equation}\n\n\\begin{equation}\n\\label{mom-u}\nK_p(T):=\\sup_{(t,x)\\in [0,T] \\times \\bR}\\|u(t,x)\\|_p<\\infty \\quad \\mbox{for all} \\quad T>0,\n\\end{equation}", + "full_context": "Traditionally, SPDEs are perturbed by a space-time Gaussian white noise in dimension $d=1$, which is a zero-mean Gaussian process $\\{B_t(\\varphi);t> 0,\\varphi \\in L^2(\\mathbb{R})\\}$ with covariance:\n\\[\n\\mathbb{E}[B_t(\\varphi) B_s(\\psi)]=(t \\wedge s) \\langle \\varphi,\\psi \\rangle_{L^2(\\mathbb{R})}.\n\\]\nIn higher dimensions, Dalang introduced in \\cite{dalang99}, a spatially-homogeneous (or colored) noise, as a zero-mean Gaussian process $\\{W_t(\\varphi);t\\geq 0,\\varphi \\in \\mathcal{S}(\\mathbb{R})\\}$ with covariance:\n\\begin{equation}\n\\label{def-cov-W}\n\\mathbb{E}[W_t(\\varphi) W_s(\\psi)]=(t \\wedge s) \\int_{\\mathbb{R}^d} \\int_{\\mathbb{R}^d}\\varphi(x)\\psi(y) f(x-y)dxdy,\n\\end{equation}\nwhere $\\mathcal{S}(\\mathbb{R}^d)$ is the set of all rapidly decreasing functions on $\\mathbb{R}^d$, and $f:\\mathbb{R}^d \\to [0,\\infty]$ is the Fourier transform of a tempered measure $\\mu$. This noise can also be defined as:\n\\begin{equation}\n\\label{Gauss-color}\nW_t(\\varphi)=B_t(\\varphi*\\kappa) \\quad \\mbox{for all $t>0$ and $\\varphi \\in \\mathcal{S}(\\mathbb{R}^d)$},\n\\end{equation}\nwhere the kernel $\\kappa$ is chosen such that $k * \\widetilde{k}=f$, and $\\widetilde{k}(x):=k(-x)$ for all $x \\in \\mathbb{R}^d$.\n\nLet $G$ be the fundamental solution of the wave equation in dimension $d=1$, given by:\n\\begin{equation}\n\\label{def-G}\nG_t(x)=\\frac{1}{2}1_{\\{|x|0$ and $x \\in \\mathbb{R}$.}\n\\end{equation}\n\nBy Lemma 5.5 of \\cite{BJ25}, the process $\\{u(t,x)\\}_{x\\in \\mathbb{R}}$ is strictly stationary, for any $t>0$.\nMoreover, by Theorem 5.6 of \\cite{BJ25}, if for some $p\\geq 2$, we have\n\\begin{equation}\n\\label{def-mp}\nm_p:=\\int_{\\bR_0}|z|^p \\nu(dz)<\\infty,\n\\end{equation}\nand\n\\begin{equation}\n\\label{def-Mp}\n\\cM_p(t):=\\int_0^t \\int_{\\mathbb{R}} \\big|(G_{t-s}*\\kappa)(x)\\big|^p dx ds<\\infty \\quad \\mbox{for all} \\quad t>0,\n\\end{equation}\nthen\n\\begin{equation}\n\\label{mom-u}\nK_p(T):=\\sup_{(t,x)\\in [0,T] \\times \\mathbb{R}}\\|u(t,x)\\|_p<\\infty \\quad \\mbox{for all} \\quad T>0,\n\\end{equation}\nwhere $\\|\\cdot\\|_p$ denotes the norm in $L^p(\\Omega)$. The goal of the present paper is to complement this analysis by showing that the {\\em spatial average} of the (centered) solution:\n\\[\nF_R(t)=\\int_{-R}^{R}\\big(u(t,x)-1 \\big)dx.\n\\]\nhas asymptotic Gaussian fluctuations when $R \\to \\infty$, with a precise rate of convergence which depends on the noise. We denote $\\sigma_R^2(t)={\\rm Var}(F_R(t))$.\n\n\\begin{remark}\n\\label{rem-M}\n{\\rm Suppose that Assumption B holds. Applying Young's inequality if $\\kappa \\in L^1(\\mathbb{R})$, and Theorem \\ref{HLS} if $\\kappa=R_{1,\\alpha/2}$, we see that:\n\\[\n\\|G_t* \\kappa\\|_{L^p(\\mathbb{R})} \\leq\n\\left\\{\n\\begin{array}{ll} \n\\|\\kappa\\|_{L^1(\\mathbb{R})} \\|G_t\\|_{L^p(\\mathbb{R})} & \\mbox{if $\\kappa \\in L^1(\\mathbb{R})$ and $p\\geq 1$} \\\\\nA_{1,\\alpha,p}\\|G_t\\|_{L^q(\\mathbb{R})} & \\mbox{if $\\kappa=R_{1,\\alpha/2}$, $\\alpha \\in (0,1)$, $p>\\frac{2}{2-\\alpha}$ and $\\frac{1}{q}=\\frac{1}{p}+\\frac{\\alpha}{2}$}\n\\end{array} \\right.\n\\]\nSince $\\|G_t\\|_{L^p(\\mathbb{R})}^p=2^{1-p}t$ for any $p>0$, we infer that condition \\eqref{def-Mp} holds for any $p\\geq 2$. Therefore, \\eqref{mom-u} holds for any $p \\geq 2$ such that $m_p<\\infty$.\n}\n\\end{remark}\n\nWe recall the following definitions. Let $X$ and $Y$ be random variables defined on the same probability space.\nThe {\\em $1$-Wasserstein distance} between $X$ and $Y$ is:\n\\[\nd_{W}(X,Y)=\\sup_{{\\rm Lip}(h) \\leq 1} \\big|\\mathbb{E}[h(X)]-\\mathbb{E}[h(Y)]\\big|, \\quad\n\\mbox{where} \\quad\n{\\rm Lip}(h):=\\sup_{x \\not=y}\\frac{|h(x)-h(y)|}{|x-y|}.\n\\]\nThe {\\em Fortet-Mourier distance} between $X$ and $Y$ is:\n\\[\nd_{FM}(X,Y)=\\sup_{\\|h\\|_{\\infty}+{\\rm Lip}(h) \\leq 1} \\big|\\mathbb{E}[h(X)]-\\mathbb{E}[h(Y)]\\big|,\n\\]\nThe {\\em Kolmogorov distance} between $X$ and $Y$ is:\n\\[\nd_{K}(X,Y)=\\sup_{x \\in \\mathbb{E}} \\big|P(X \\leq x)-P(Y \\leq x)\\big|.\n\\]\n\nWe are now ready to state the main results of this article.\n\n\\begin{theorem}[Hardy-Littlewood-Sobolov inequality]\n\\label{HLS}\nFor any $\\alpha \\in (0,d)$ and $p>\\frac{d}{d-\\alpha}$, \n\\[\n\\| \\varphi *R_{d,\\alpha}\\|_{L^p(\\bR^d)} \\leq A_{d.\\alpha,p}\\|\\varphi\\|_{L^{q}(\\bR^d)} \\quad \\mbox{for any $\\varphi \\in L^q(\\bR^d)$},\n\\]\nwhere $A_{d,\\alpha,p}>0$ is a constant depending on $(d,\\alpha,p)$, and $q$ is defined by:\n\\[\n\\frac{1}{q}=\\frac{1}{p}+\\frac{\\alpha}{d}.\n\\]\n(Condition $p>\\frac{d}{d-\\alpha}$ is equivalent to $q>1$.)\n\\end{theorem}\n\n\\begin{equation}\n\\label{def-Mp}\n\\cM_p(t):=\\int_0^t \\int_{\\bR} \\big|(G_{t-s}*\\k)(x)\\big|^p dx ds<\\infty \\quad \\mbox{for all} \\quad t>0,\n\\end{equation}\n\n\\begin{equation}\n\\label{mom-u}\nK_p(T):=\\sup_{(t,x)\\in [0,T] \\times \\bR}\\|u(t,x)\\|_p<\\infty \\quad \\mbox{for all} \\quad T>0,\n\\end{equation}\n\nBy Lemma 5.5 of \\cite{BJ25}, the process $\\{u(t,x)\\}_{x\\in \\bR}$ is strictly stationary, for any $t>0$.\nMoreover, by Theorem 5.6 of \\cite{BJ25}, if for some $p\\geq 2$, we have\n\\begin{equation}\n\\label{def-mp}\nm_p:=\\int_{\\bR_0}|z|^p \\nu(dz)<\\infty,\n\\end{equation}\nand\n\\begin{equation}\n\\label{def-Mp}\n\\cM_p(t):=\\int_0^t \\int_{\\bR} \\big|(G_{t-s}*\\k)(x)\\big|^p dx ds<\\infty \\quad \\mbox{for all} \\quad t>0,\n\\end{equation}\nthen\n\\begin{equation}\n\\label{mom-u}\nK_p(T):=\\sup_{(t,x)\\in [0,T] \\times \\bR}\\|u(t,x)\\|_p<\\infty \\quad \\mbox{for all} \\quad T>0,\n\\end{equation}\nwhere $\\|\\cdot\\|_p$ denotes the norm in $L^p(\\Omega)$. The goal of the present paper is to complement this analysis by showing that the {\\em spatial average} of the (centered) solution:\n\\[\nF_R(t)=\\int_{-R}^{R}\\big(u(t,x)-1 \\big)dx.\n\\]\nhas asymptotic Gaussian fluctuations when $R \\to \\infty$, with a precise rate of convergence which depends on the noise. We denote $\\sigma_R^2(t)={\\rm Var}(F_R(t))$.\n\n\\begin{remark}\n\\label{rem-M}\n{\\rm Suppose that Assumption B holds. Applying Young's inequality if $\\k \\in L^1(\\bR)$, and Theorem \\ref{HLS} if $\\k=R_{1,\\alpha/2}$, we see that:\n\\[\n\\|G_t* \\k\\|_{L^p(\\bR)} \\leq\n\\left\\{\n\\begin{array}{ll} \n\\|\\k\\|_{L^1(\\bR)} \\|G_t\\|_{L^p(\\bR)} & \\mbox{if $\\k \\in L^1(\\bR)$ and $p\\geq 1$} \\\\\nA_{1,\\alpha,p}\\|G_t\\|_{L^q(\\bR)} & \\mbox{if $\\k=R_{1,\\alpha/2}$, $\\alpha \\in (0,1)$, $p>\\frac{2}{2-\\alpha}$ and $\\frac{1}{q}=\\frac{1}{p}+\\frac{\\alpha}{2}$}\n\\end{array} \\right.\n\\]\nSince $\\|G_t\\|_{L^p(\\bR)}^p=2^{1-p}t$ for any $p>0$, we infer that condition \\eqref{def-Mp} holds for any $p\\geq 2$. Therefore, \\eqref{mom-u} holds for any $p \\geq 2$ such that $m_p<\\infty$.\n}\n\\end{remark}\n\nWe are now ready to state the main results of this article.\n\n\\begin{theorem}[Limiting covariance]\n\\label{cov-th}\nIf Assumption B holds, then\nfor any $t,s>0$,\n\\[\n\\lim_{R \\to \\infty}\\frac{1}{R^{\\beta}}\\bE[F_R(t) F_R(s)]=K(t,s) \\quad \\mbox{is finite},\n\\] \nwhere \n\\begin{equation}\n\\label{def-beta}\n\\beta:=\\left\\{\n\\begin{array}{ll} 1 & \\mbox{if $\\k \\in L^1(\\bR)$,} \\\\\n\\alpha+1 & \\mbox{if $\\k=R_{1,\\alpha/2}$ for some $\\alpha \\in (0,1)$. }\n\\end{array} \\right.\n\\end{equation}\nIn particular, $R^{-\\beta} \\sigma_R^2(t) \\to K(t,t)$ as $R \\to \\infty$, for any $t>0$.\n\\end{theorem}\n\n\\begin{theorem}[Quantitative Central Limit Theorem]\n\\label{QCLT}\nSuppose that Assumption B holds, and there exists $p \\in (1,2]$ such that \n\\begin{equation}\n\\label{mp-m2p}\nm_{p}<\\infty \\quad \\mbox{and} \\quad m_{2p}<\\infty.\n\\end{equation}\n\n\\begin{theorem}[Functional Central Limit Theorem]\n\\label{FCLT}\nUnder the hypotheses of Theorem \\ref{QCLT}, for any $R>0$,\nthe process $\\{F_R(t)\\}_{t\\geq 0}$ has a $\\gamma$-H\\\"older continuous modification (denoted also $F_R$), for any $\\gamma \\in (0,\\frac{\\beta}{2})$, where $\\beta$ is given by \\eqref{def-beta}.\nMoreover,\n\\[\n\\frac{1}{R^{\\beta/2}}F_R(\\cdot) \\stackrel{d}{\\to} \\cG(\\cdot) \\quad \\mbox{in $C[0,\\infty)$ as $R \\to \\infty$},\n\\]\nwhere $\\{\\cG(t)\\}_{t\\geq 0}$ is a zero-mean Gaussian process with covariance \n$\\bE[\\cG(t) \\cG(s)]=K(t,s)$, and $K(t,s)$ given by Theorem \\ref{cov-th}. Here $\\stackrel{d}{\\to}$ denotes the convergence in distribution, and $C[0,\\infty)$ is equipped with the uniform convergence on compact sets.\n\\end{theorem}\n\n$\\bullet$\nAny random variable $F \\in L^2(\\Omega)$ which is $\\cF^N$-measurable has the {\\em Poisson-chaos expansion}:\n\\begin{equation}\n\\label{Poisson-chaos}\nF=\\bE(F)+\\sum_{n\\geq 1}I_n(f_n), \\quad \\mbox{for some $f_n \\in \\cH^{\\odot n}$},\n\\end{equation}\nand the series is orthogonal in $L^2(\\Omega)$. Here $I_n$ is the multiple integral with respect to $\\widehat{N}$ and $\\cH^{\\odot n}$ is the set of symmetric functions in $\\cH^{\\otimes n}$.\nFor any $f \\in \\cH^{\\otimes n}$,\n\\[\n\\bE[I_n(f)]=0 \\quad \\mbox{and} \\quad \\bE|I_n(f)|^2=n! \\|\\widetilde{f}\\|_{\\cH^{\\otimes n}}^2,\n\\]\nwhere $\\widetilde{f}$ is the symmetrization of $f$:\n\\[\n\\widetilde{f}(\\xi_1,\\ldots,\\xi_n)=\\frac{1}{n!}\\sum_{\\rho \\in S_n}f(\\xi_{\\rho(1)},\\ldots,\\xi_{\\rho(n)}),\n\\]\n$S_n$ being the set of all permutations of $1,\\ldots,n$.\nMoreover, $I_n(f)=I_n(\\widetilde{f})$ for any $f \\in \\cH^{\\otimes n}$.\n\n{\\em Step 1.} (finite-dimensional convergence) We will prove that for any integer $m\\geq 1$ and form any $t_1,\\ldots,t_m \\in [0,T]$\n\\[\n\\Big(\\frac{1}{R^{\\beta/2}} F_R(t_1),\\ldots, \\frac{1}{R^{\\beta/2}}F_R(t_m) \\Big)\n\\stackrel{d}{\\to} \\Big(\\cG(t_1),\\ldots, \\cG(t_m) \\Big) \\quad \\mbox{as $R \\to \\infty$}.\n\\]\nBy Cram\\'er-Wold theorem, this is equivalent to showing that for any $b_1,\\ldots,b_m \\in \\bR$,\n\\[\nX_R:=\\frac{1}{R^{\\beta/2}}\\sum_{j=1}^{m}b_jF_R(t_j) \\stackrel{d}{\\to}\\sum_{j=1}^{m}b_j \\cG(t_j) \n\\quad \\mbox{as $R \\to \\infty$}.\n\\]\nUsing the same argument as on page 4215 of \\cite{BZ24}, it is enough to prove that $X_R/\\tau_R \\stackrel{d}{\\to} Z$ as $R \\to \\infty$, where $\\tau_R^2={\\rm Var}(X_R)$ and $Z \\sim N(0,1)$. By Proposition \\ref{tara}, \n\\[\nd_{W}\\left(\\frac{X_R}{\\tau_R},Z \\right) \\leq \\gamma_1+\\gamma_2+\\gamma_3,\n\\]\nwhere $\\gamma_1,\\gamma_2$ and $\\gamma_3$ are defined as in \\eqref{gamma17} with $F=X_R$. Using the same argument as in the proof of Theorem \\ref{QCLT}, we infer that \n\\[\nd_{W}\\left(\\frac{X_R}{\\tau_R},Z \\right) \\les R^{1-p} \\quad \\mbox{if $\\k \\in L^1(\\bR)$,}\n\\]\nand\n\\[\nd_{W}\\left(\\frac{X_R}{\\tau_R},Z \\right) \\les R^{-\\varepsilon} \\quad \\mbox{for any $\\varepsilon \\in \\big(0,1-\\frac{1}{p}-\\frac{\\alpha}{2}\\big)$, \\quad if $\\k =R_{1,\\alpha/2}$.}\n\\]\n\n\\begin{theorem}[Hardy-Littlewood-Sobolov inequality]\n\\label{HLS}\nFor any $\\alpha \\in (0,d)$ and $p>\\frac{d}{d-\\alpha}$, \n\\[\n\\| \\varphi *R_{d,\\alpha}\\|_{L^p(\\bR^d)} \\leq A_{d.\\alpha,p}\\|\\varphi\\|_{L^{q}(\\bR^d)} \\quad \\mbox{for any $\\varphi \\in L^q(\\bR^d)$},\n\\]\nwhere $A_{d,\\alpha,p}>0$ is a constant depending on $(d,\\alpha,p)$, and $q$ is defined by:\n\\[\n\\frac{1}{q}=\\frac{1}{p}+\\frac{\\alpha}{d}.\n\\]\n(Condition $p>\\frac{d}{d-\\alpha}$ is equivalent to $q>1$.)\n\\end{theorem}\n\n\\begin{theorem}[Quantitative Central Limit Theorem]\n\\label{QCLT}\nSuppose that Assumption B holds, and there exists $p \\in (1,2]$ such that \n\\begin{equation}\n\\label{mp-m2p}\nm_{p}<\\infty \\quad \\mbox{and} \\quad m_{2p}<\\infty.\n\\end{equation}\n\nIf $\\k \\in L^1(\\bR)$, then\nfor any $t>0$,\n\\[\n{\\rm dist}\\left( \\frac{F_R(t)}{\\sigma_R(t)},Z\\right) \\leq C_t R^{-(1-\\frac{1}{p})}\n\\]\n\nIf $\\k=R_{1,\\alpha/2}$ for some $\\alpha \\in (0,1)$, and $p>\\frac{2}{2-\\alpha}$, then\n\\[\n{\\rm dist}\\left( \\frac{F_R(t)}{\\sigma_R(t)},Z\\right) \\leq\nC_t R^{-\\e} \\quad \\mbox{for any $\\varepsilon \\in \\big(0, 1-\\frac{1}{p}-\\frac{\\alpha}{2}\\big)$}.\n\\]\n\nHere $C_t>0$ is a constant depending on $t$, and ${\\rm dist}$ is the Fortet-Mourier distance, the 1-Wasserstein distance, or the Kolmogorov distance.\n\\end{theorem}", + "post_theorem_intro_text_len": 6581, + "post_theorem_intro_text": "\\begin{theorem}[Limiting covariance]\n\\label{cov-th}\nIf Assumption B holds, then\nfor any $t,s>0$,\n\\[\n\\lim_{R \\to \\infty}\\frac{1}{R^{\\beta}}\\mathbb{E}[F_R(t) F_R(s)]=K(t,s) \\quad \\mbox{is finite},\n\\] \nwhere \n\\begin{equation}\n\\label{def-beta}\n\\beta:=\\left\\{\n\\begin{array}{ll} 1 & \\mbox{if $\\kappa \\in L^1(\\mathbb{R})$,} \\\\\n\\alpha+1 & \\mbox{if $\\kappa=R_{1,\\alpha/2}$ for some $\\alpha \\in (0,1)$. }\n\\end{array} \\right.\n\\end{equation}\nIn particular, $R^{-\\beta} \\sigma_R^2(t) \\to K(t,t)$ as $R \\to \\infty$, for any $t>0$.\n\\end{theorem}\n\n\\begin{theorem}[Quantitative Central Limit Theorem]\n\\label{QCLT}\nSuppose that Assumption B holds, and there exists $p \\in (1,2]$ such that \n\\begin{equation}\n\\label{mp-m2p}\nm_{p}<\\infty \\quad \\mbox{and} \\quad m_{2p}<\\infty.\n\\end{equation}\n\nIf $\\kappa \\in L^1(\\mathbb{R})$, then\nfor any $t>0$,\n\\[\n{\\rm dist}\\left( \\frac{F_R(t)}{\\sigma_R(t)},Z\\right) \\leq C_t R^{-(1-\\frac{1}{p})}\n\\]\n\nIf $\\kappa=R_{1,\\alpha/2}$ for some $\\alpha \\in (0,1)$, and $p>\\frac{2}{2-\\alpha}$, then\n\\[\n{\\rm dist}\\left( \\frac{F_R(t)}{\\sigma_R(t)},Z\\right) \\leq\nC_t R^{-\\varepsilon} \\quad \\mbox{for any $\\varepsilon \\in \\big(0, 1-\\frac{1}{p}-\\frac{\\alpha}{2}\\big)$}.\n\\]\n\nHere $C_t>0$ is a constant depending on $t$, and ${\\rm dist}$ is the Fortet-Mourier distance, the 1-Wasserstein distance, or the Kolmogorov distance.\n\\end{theorem}\n\n\\begin{theorem}[Functional Central Limit Theorem]\n\\label{FCLT}\nUnder the hypotheses of Theorem \\ref{QCLT}, for any $R>0$,\nthe process $\\{F_R(t)\\}_{t\\geq 0}$ has a $\\gamma$-H\\\"older continuous modification (denoted also $F_R$), for any $\\gamma \\in (0,\\frac{\\beta}{2})$, where $\\beta$ is given by \\eqref{def-beta}.\nMoreover,\n\\[\n\\frac{1}{R^{\\beta/2}}F_R(\\cdot) \\stackrel{d}{\\to} \\mathcal{G}(\\cdot) \\quad \\mbox{in $C[0,\\infty)$ as $R \\to \\infty$},\n\\]\nwhere $\\{\\mathcal{G}(t)\\}_{t\\geq 0}$ is a zero-mean Gaussian process with covariance \n$\\mathbb{E}[\\mathcal{G}(t) \\mathcal{G}(s)]=K(t,s)$, and $K(t,s)$ given by Theorem \\ref{cov-th}. Here $\\stackrel{d}{\\to}$ denotes the convergence in distribution, and $C[0,\\infty)$ is equipped with the uniform convergence on compact sets.\n\\end{theorem}\n\nThe proofs of these theorems are based on a key estimate which shows that the moments of the Malliavin derivative of the solution of \\eqref{HAM} can be bounded, up to a constant, by a deterministic function (see relation \\eqref{key-D} below). Similar estimates appear in all references dedicated to QCLT, where they play a crucial role. To prove this key estimate, we proceed as in \\cite{BZ24}, developing a connection to (hAm) with delta initial velocity by using the form \\eqref{def-G} of the fundamental solution $G$.\n\nTheorem \\ref{ergodic-th} follows from the key estimate \\eqref{key-D} and a criterion which may be of independent interest (Lemma \\ref{lem-key-Z} below). \nTo prove Theorem \\ref{cov-th} we show that the covariance of the solution of \\eqref{HAM} coincides with the covariance of the solution of (hAm) driven by $\\sqrt{m_2}W$, where $W$ is the Gaussian colored noise with covariance \\eqref{def-cov-W}.\n\nTo prove Theorem \\ref{QCLT}, we apply the recent result of Trauthwein \\cite{trauthwein25}, which gives the optimal rates for the Wasserstein and Kolmogorov distances between $F/\\sqrt{{\\rm Var}(F)}$ and $Z \\sim N(0,1)$, for a centered random variable $F$ with finite variance, which is Malliavin differentiable with respect to a compensated Poisson random measure. These estimates involve the first and second order Malliavin derivatives. Therefore, to apply the result in \\cite{trauthwein25}, we develop a similar key estimate for the second Malliavin derivative of the solution (see relation \\eqref{key-D2} below). Different techniques are used for estimating the 7 quantities $\\gamma_1,\\ldots,\\gamma_7$ which appear in the result of \\cite{trauthwein25}, which rely on Young's inequality in the case when $k$ is integrable, respectively on Hardy-Littlewood-Sobolev inequality when $\\kappa$ is the Riesz kernel. Finally, Theorem \\ref{FCLT} follows by the classical method of finite dimensional convergence and tightness.\n\n\\begin{remark}\n{\\rm a) If $\\kappa \\in L^1(\\mathbb{R})$, then $f=\\kappa * \\tilde{\\kappa} \\in L^1(\\mathbb{R})$ by Young's inequality. In this case, the variance $\\sigma_R^2(t)$ decays with rate $R$, as in the L\\'evy white noise case and the Gaussian colored noise case with integrable kernel $f$ (see Theorem 1.2 of \\cite{NZ22}). The decay rate $R^{-(1-\\frac{1}{p})}$ in the QCLT (in the $d_{W}$, $d_{FM}$ or $d_K$ distances) is the same as in the L\\'evy white noise case (see Theorem 1.1.(iii) of \\cite{BZ24}), and depends on the parameter $p$ from \\eqref{mp-m2p}. This can be explained since formally, the L\\'evy white noise can be viewed as a L\\'evy colored noise with integrable kernel $\\kappa=\\delta_0$. When $p=2$ (i.e. $m_4<\\infty)$, this rate coincides with the rate $R^{-1/2}$ obtained in the case of the Gaussian colored noise case with integrable kernel $f$, for the QCLT in the total variation distance (see Theorem 1.2 of \\cite{NZ22}).\n\n\\medskip\n\nb) If $\\kappa=R_{1,\\alpha/2}$ for some $\\alpha \\in (0,1)$, then $f =R_{1,\\alpha}$. In this case, using the parametrization $\\alpha=2H-1$ with $H \\in (\\frac{1}{2},1)$, we see that the decay rate $R^{2H}$ of $\\sigma_R^2(t)$ is the same as in the Gaussian colored noise case (see Proposition 3.3 of \\cite{DNZ20}). In this Gaussian case, Theorem 1.1 {\\em ibid.} gives the rate $R^{-(1-H)}=R^{-\\frac{1-\\alpha}{2}}$ for the QCLT in the total variation distance, which is almost the same as the rate $R^{-\\varepsilon}$ with $\\varepsilon \\in (0,\\frac{1-\\alpha}{2})$ obtained in Theorem \\ref{QCLT} above when $p=2$ (i.e. $m_4<\\infty$).\n}\n\\end{remark}\n\nThis article is organized as follows. In Section \\ref{section-prelim}, we include some preliminaries about Poisson-Malliavin calculus and moment inequalities with respect to noises $L$ and $X$. In Sections \\ref{section-Mall1} and \\ref{section-Mall2}, we prove some key estimates for the first, respectively the second, Malliavin derivative of the solution. In Section \\ref{section-proofs}, we present the proofs of Theorems \\ref{ergodic-th}, \\ref{cov-th}, \\ref{QCLT} and \\ref{FCLT}. Appendix \\ref{appA} contains some auxiliary results about Poisson-Malliavin calculus. Appendix \\ref{appB} presents some inequalities for Riesz potentials.\n\n\\medskip\n\nWe conclude the introduction with few words about the notation. We write $a\\lesssim b$ to indicate that $a \\leq C b$ for some positive constant $C>0$ that does not depend on $(a,b)$. \nFor any $p\\geq 1$, we denote by $\\|\\cdot \\|_p$ the norm in $L^p(\\Omega)$.", + "sketch": "The post-theorem introduction outlines the following proof strategy (centered around Theorem~\\ref{ergodic-th}):\n\n- The proofs are \"based on a key estimate\" showing that \"the moments of the Malliavin derivative of the solution of \\eqref{HAM} can be bounded, up to a constant, by a deterministic function\" (relation \\eqref{key-D}).\n\n- To prove this key estimate, the authors \"proceed as in \\cite{BZ24}, developing a connection to (hAm) with delta initial velocity by using the form \\eqref{def-G} of the fundamental solution $G$.\"\n\n- Then, explicitly for the main theorem: \"Theorem \\ref{ergodic-th} follows from the key estimate \\eqref{key-D} and a criterion which may be of independent interest (Lemma \\ref{lem-key-Z} below).\"", + "expanded_sketch": "The post-theorem introduction outlines the following proof strategy (centered around the main theorem):\n\n- The proofs are based on a key estimate showing that the moments of the Malliavin derivative of the solution of\n\\begin{align}\n\\label{HAM}\n\t\\begin{cases}\n\t\t\\dfrac{\\partial^2 u}{\\partial^2 t} (t,x)\n\t\t= \\dfrac{\\partial^2 u}{\\partial x^2} (t,x)+u(t,x) \\dot{X}(t,x), \\\\\n\t\tt>0, \\ x \\in \\bR, \\\\\n\t\tu(0,x) = 1, \\dfrac{\\partial u}{\\partial t} (0,x)=0, \\quad x \\in \\bR,\n\t\\end{cases}\n\\end{align}\ncan be bounded, up to a constant, by a deterministic function, namely\n\\begin{equation}\n\\label{key-D}\n\\|D_{r,y,z}u(t,x)\\|_p \\leq C'_{T,p,\\nu,\\k}|z| \\int_{\\bR}G_{t-r}(x-y')\\k(y-y')dy',\n\\end{equation}\n\n- To prove this key estimate, the authors “proceed as in \\cite{BZ24}”, developing a connection to (hAm) with delta initial velocity by using the form\n\\begin{equation}\n\\label{def-G}\nG_t(x)=\\frac{1}{2}1_{\\{|x|0$ and $x \\in \\bR$.}\n\\end{equation}\nof the fundamental solution $G$.\n\n- Then, in establishing the main theorem: it follows from the key estimate (the equation above) and a criterion which may be of independent interest. We first record that criterion.\n\\begin{lemma}\n\\label{lem-key-Z}\nSuppose that Assumption B holds.\nLet $\\{Z(t,x);t\\geq 0,x\\in \\bR\\}$ be an adapted random field such that\n$\\{Z(t,x)\\}_{x\\in \\bR^d}$ is strictly stationary and $Z(t,x) \\in {\\rm dom}(D)$ for any $t\\geq 0$ and $x \\in \\bR$. Assume that for any $00$ is a constant depending on $t$. Then $\\{Z(t,x)\\}_{x\\in \\bR}$ is ergodic.\n\\end{lemma}", + "expanded_theorem": "\\label{ergodic-th}\nIf Assumption B holds, then $\\{u(t,x)\\}_{x\\in \\mathbb{R}}$ is ergodic, for any $t>0$. Consequently, by the mean ergodic theorem,\n\\[\n\\frac{1}{R}F_R(t) \\to 0 \\quad \\mbox{a.s and in $L^2(\\Omega)$}, \\quad \\mbox{as $R \\to \\infty$}.\n\\]", + "theorem_type": [ + "Implication", + "Asymptotic or Limit" + ], + "mcq": { + "question": "Let \\(u(t,x)\\) be the one-dimensional random field under Assumption B, and for each fixed \\(t>0\\) define the centered spatial average over \\([-R,R]\\) by\n\\[\nF_R(t)=\\int_{-R}^{R}(u(t,x)-1)\\,dx.\n\\]\nUnder Assumption B, which statement about the spatial process \\(\\{u(t,x)\\}_{x\\in\\mathbb{R}}\\) and the asymptotic behavior of \\(F_R(t)\\) is valid for every \\(t>0\\)?", + "correct_choice": { + "label": "A", + "text": "For every \\(t>0\\), the spatial process \\(\\{u(t,x)\\}_{x\\in\\mathbb{R}}\\) is ergodic. Consequently,\n\\[\n\\frac{1}{R}F_R(t)=\\frac{1}{R}\\int_{-R}^{R}(u(t,x)-1)\\,dx \\to 0\n\\quad \\text{as } R\\to\\infty,\n\\]\nboth almost surely and in \\(L^2(\\Omega)\\)." + }, + "choices": [ + { + "label": "B", + "text": "For every \\(t>0\\), the spatial process \\(\\{u(t,x)\\}_{x\\in\\mathbb{R}}\\) is mixing. Consequently,\n\\[\n\\frac{1}{R}F_R(t)=\\frac{1}{R}\\int_{-R}^{R}(u(t,x)-1)\\,dx \\to 0\n\\quad \\text{as } R\\to\\infty,\n\\]\nboth almost surely and in \\(L^2(\\Omega)\\)." + }, + { + "label": "C", + "text": "For every \\(t>0\\),\n\\[\n\\frac{1}{R}F_R(t)=\\frac{1}{R}\\int_{-R}^{R}(u(t,x)-1)\\,dx \\to 0\n\\quad \\text{as } R\\to\\infty,\n\\]\nin \\(L^2(\\Omega)\\)." + }, + { + "label": "D", + "text": "For every \\(t>0\\), the spatial process \\(\\{u(t,x)\\}_{x\\in\\mathbb{R}}\\) is ergodic. Consequently,\n\\[\n\\frac{1}{\\sqrt{R}}F_R(t)=\\frac{1}{\\sqrt{R}}\\int_{-R}^{R}(u(t,x)-1)\\,dx \\to 0\n\\quad \\text{as } R\\to\\infty,\n\\]\nboth almost surely and in \\(L^2(\\Omega)\\)." + }, + { + "label": "E", + "text": "For every \\(t>0\\), the spatial process \\(\\{u(t,x)\\}_{x\\in\\mathbb{R}}\\) is ergodic. Consequently,\n\\[\n\\frac{1}{R}F_R(t)=\\frac{1}{R}\\int_{-R}^{R}(u(t,x)-1)\\,dx \\to 0\n\\quad \\text{as } R\\to\\infty,\n\\]\nalmost surely, and the convergence is uniform in \\(t\\) on compact time intervals in \\(L^2(\\Omega)\\)." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "B" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "regularity", + "tampered_component": "ergodicity_vs_stronger_mixing", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "finiteness", + "tampered_component": "dropped_ergodicity_and_a.s._convergence", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "normalization_scale_1_over_R_replaced_by_1_over_sqrtR", + "template_used": "boundary_range" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "pointwise_in_t_conclusion_replaced_by_uniform_on_compacts", + "template_used": "quantifier_dependence" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines the averaged quantity and asks for the correct limiting statement, but it does not explicitly reveal ergodicity or the exact convergence mode. There is no direct answer leakage from the wording." + }, + "TAS": { + "score": 1, + "justification": "The correct option appears to be very close to a theorem-style statement under Assumption B, so the item risks being a near restatement of a known result. However, it is not completely tautological because the alternatives vary meaningfully in strength, mode of convergence, and scaling." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is required to distinguish the strongest valid conclusion from weaker true or stronger false variants, especially between A and C and among different convergence claims. Still, for someone who knows the source result, the correct answer is likely recognized rather than generated." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically well targeted: B overstates ergodicity to mixing and adds a rate, C is a weaker true statement, D adds unjustified uniformity in time, and E uses the wrong normalization scale. These reflect realistic failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with good distractors and little answer leakage, but it remains somewhat theorem-recall driven rather than deeply generative." + } + }, + { + "id": "2602.23597v1", + "paper_link": "http://arxiv.org/abs/2602.23597v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm:general}\n Let $\\beta$ be an algebraic number with $|\\beta|=1$ which is not a root of unity. Then $\\frac{\\operatorname{Arg}(\\beta)}{2\\pi} \\in \\left(-\\frac{1}{2},\\frac{1}{2}\\right]$ is a Diophantine number.", + "start_pos": 8080, + "end_pos": 8313, + "label": "thm:general" + }, + "ref_dict": { + "thm:general": "\\begin{theorem}\\label{thm:general}\n Let $\\beta$ be an algebraic number with $|\\beta|=1$ which is not a root of unity. Then $\\frac{\\Arg(\\beta)}{2\\pi} \\in \\left(-\\frac{1}{2},\\frac{1}{2}\\right]$ is a Diophantine number.\n\\end{theorem}", + "ineq:dioph": "\\begin{equation}\\label{ineq:dioph}\n \\left| \\theta - \\frac{p}{q} \\right| \\ge \\frac{c}{q^\\tau}.\n\\end{equation}", + "eq:main": "\\begin{equation}\\label{eq:main}\n n \\tan \\alpha = \\tan(n \\alpha).\n\\end{equation}", + "thm:3props": "\\begin{theorem}\\label{thm:3props}\n Let $n \\ge 2$ be an integer and let $\\alpha \\in \\left( 0, \\frac{\\pi}{2} \\right)$ be a solution of \\eqref{eq:main}. Then the following statements hold.\n \\begin{enumerate}[label=(\\alph*)]\n \\item $\\frac{\\alpha}{2\\pi}$ is a Diophantine number.\n \\item $\\frac{\\alpha}{2\\pi}$ is transcendental.\n \\item $\\alpha$ is transcendental.\n \\end{enumerate}\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 5205, + "pre_theorem_intro_text": "Let $\\theta$ be a real number. The {\\em irrationality exponent} of $\\theta$ is \n$$\\mu(\\theta) = \\sup\\left\\{ \\mu > 0 \\colon \\left| \\theta - \\frac{p}{q} \\right| < \\frac{1}{q^\\mu} \\text{ holds for infinitely many pairs } (p,q) \\in \\mathbb{Z} \\times \\mathbb{N} \\right\\}.$$\nWe say that $\\theta$ is a {\\em Liouville number} whenever $\\mu(\\theta) = \\infty$. On the other hand, $\\theta$ is called {\\em Diophantine} if there exist $c > 0$ and $\\tau < \\infty$ such that, for every $(p,q) \\in \\mathbb{Z} \\times \\mathbb{N}$, \n\\begin{equation}\\label{ineq:dioph}\n \\left| \\theta - \\frac{p}{q} \\right| \\ge \\frac{c}{q^\\tau}.\n\\end{equation}\nLet $\\mathcal L$ be the set of Liouville numbers and let $\\mathcal D$ be the set of Diophantine numbers.\n\nWe observe that if $\\theta = \\frac{a}{b}$ is rational, then $\\mu(\\theta) = 1$, and one may choose $\\frac{p}{q} = \\frac{a}{b}$ so that \\eqref{ineq:dioph} fails to hold. Hence, no rational number is either Diophantine or Liouville. Among irrational numbers, the sets $\\mathcal L$ and $\\mathcal D$ form a partition.\n\nA very strong result due to Roth \\cite{Roth1955} states that $\\mu(\\theta) = 2$ for every algebraic number $\\theta$. In particular, every Liouville number must be transcendental, or equivalently, every algebraic irrational number is Diophantine.\n\nArithmetic properties such as rationality or irrationality, algebraicity or transcendence,\nand Diophantine or Liouville behavior play a fundamental role in several areas of mathematics. They arise naturally in Diophantine approximation, transcendence theory, and number theory, but also have significant implications in dynamics, geometry, and analysis.\n\nIn many contexts, good arithmetic properties govern rigidity versus flexibility phenomena and the qualitative behavior of dynamical systems. In what follows, we introduce a simple trigonometric equation whose solutions exhibit\nremarkable arithmetic features and give rise to several geometric and dynamical implications,\nwhich will be discussed in the subsequent paragraphs.\n\nLet $n \\in \\mathbb{Z}$ and consider the equation \n\\begin{equation}\\label{eq:main}\n n \\tan \\alpha = \\tan(n \\alpha).\n\\end{equation}\nIf $n\\in \\{0,1\\}$, then every $\\alpha \\in \\mathbb{R}$ is a solution. Furthermore, $\\alpha = 0$ is a solution for every $n \\in \\mathbb{Z}$. Since the tangent function is odd and $\\pi$-periodic, we may restrict attention to the case $n \\ge 2$ and $\\alpha \\in \\left(0,\\frac{\\pi}{2}\\right)$. In \\cite{Cyr2012}, Cyr proved that $\\frac{\\alpha}{2\\pi}$ is irrational.\n\nThis equation appears naturally in several topics involving plane curves and billiard dynamics. Despite its elementary appearance, its solutions encode subtle rigidity phenomena with both geometric and dynamical consequences.\n\nOne prominent context in which this equation appears is the {\\em theory of bicycle curves}. In an effort to classify noncircular bicycle curves, Tabachnikov \\cite{Tabachnikov2006} proved that the circle admits a nontrivial infinitesimal deformation as a smooth plane bicycle curve with rotation number $\\frac{\\alpha}{\\pi}$ if and only if \\eqref{eq:main} holds for some integer $n \\ge 2$. Thus, the solutions of this equation characterize the values of the rotation number for which the circle fails to be rigid in this geometric setting. In light of the Cyr theorem, it follows that the circle is rigid as a bicycle curve for any rational rotation number, except in the case $\\alpha = \\frac{\\pi}{2}$, where any curve of constant width is a bicycle curve.\n\nThe same equation arises independently in the context of {\\em mathematical billiards}. In his study of the dynamics of the billiard map inside the cross section of an infinite cylinder that floats in neutral equilibrium in any orientation, Gutkin \\cite{Gutkin2010} showed that for a regular, noncircular billiard table with boundary $\\Gamma$, the existence of a constant-angle caustic imposes the arithmetic constraint that there exist an integer $n \\ge 2$ and a real parameter $\\alpha$ satisfying \\eqref{eq:main}. Moreover, under additional assumptions, the arithmetic nature of $\\frac{\\alpha}{\\pi}$ plays a decisive role in determining whether the induced billiard dynamics on the caustic is rational or irrational.\n\nMore recently, equations of the form \\eqref{eq:main} have also appeared in the study of {\\em self-Bäcklund curves}. In this setting, the existence of non-trivial infinitesimal deformations of central conics as a self-Bäcklund is again governed by the solvability of \\eqref{eq:main} for certain values of $n \\ge 4$ (see \\cite[Section~3]{BialyBorTabachnikov2022}).\n\nThe examples discussed above show that arithmetic properties of parameters appearing in geometric and dynamical problems can strongly influence their behavior. This perspective motivates the present work, which focuses on the arithmetic properties of real numbers arising in such contexts.\n\nBefore stating the main results of this paper, we introduce some additional notation. Let $z \\in \\mathbb{C} \\setminus \\{0\\}$. We denote by $\\operatorname{Log} z = \\ln|z| + i\\operatorname{Arg}(z)$ the principal branch of the logarithm, where $\\operatorname{Arg}(z) \\in (-\\pi,\\pi]$. \nThe main results of this paper are the following.", + "context": "Let $\\theta$ be a real number. The {\\em irrationality exponent} of $\\theta$ is \n$$\\mu(\\theta) = \\sup\\left\\{ \\mu > 0 \\colon \\left| \\theta - \\frac{p}{q} \\right| < \\frac{1}{q^\\mu} \\text{ holds for infinitely many pairs } (p,q) \\in \\mathbb{Z} \\times \\mathbb{N} \\right\\}.$$\nWe say that $\\theta$ is a {\\em Liouville number} whenever $\\mu(\\theta) = \\infty$. On the other hand, $\\theta$ is called {\\em Diophantine} if there exist $c > 0$ and $\\tau < \\infty$ such that, for every $(p,q) \\in \\mathbb{Z} \\times \\mathbb{N}$, \n\\begin{equation}\\label{ineq:dioph}\n \\left| \\theta - \\frac{p}{q} \\right| \\ge \\frac{c}{q^\\tau}.\n\\end{equation}\nLet $\\mathcal L$ be the set of Liouville numbers and let $\\mathcal D$ be the set of Diophantine numbers.\n\nWe observe that if $\\theta = \\frac{a}{b}$ is rational, then $\\mu(\\theta) = 1$, and one may choose $\\frac{p}{q} = \\frac{a}{b}$ so that \\eqref{ineq:dioph} fails to hold. Hence, no rational number is either Diophantine or Liouville. Among irrational numbers, the sets $\\mathcal L$ and $\\mathcal D$ form a partition.\n\nLet $n \\in \\mathbb{Z}$ and consider the equation \n\\begin{equation}\\label{eq:main}\n n \\tan \\alpha = \\tan(n \\alpha).\n\\end{equation}\nIf $n\\in \\{0,1\\}$, then every $\\alpha \\in \\mathbb{R}$ is a solution. Furthermore, $\\alpha = 0$ is a solution for every $n \\in \\mathbb{Z}$. Since the tangent function is odd and $\\pi$-periodic, we may restrict attention to the case $n \\ge 2$ and $\\alpha \\in \\left(0,\\frac{\\pi}{2}\\right)$. In \\cite{Cyr2012}, Cyr proved that $\\frac{\\alpha}{2\\pi}$ is irrational.\n\nOne prominent context in which this equation appears is the {\\em theory of bicycle curves}. In an effort to classify noncircular bicycle curves, Tabachnikov \\cite{Tabachnikov2006} proved that the circle admits a nontrivial infinitesimal deformation as a smooth plane bicycle curve with rotation number $\\frac{\\alpha}{\\pi}$ if and only if \\eqref{eq:main} holds for some integer $n \\ge 2$. Thus, the solutions of this equation characterize the values of the rotation number for which the circle fails to be rigid in this geometric setting. In light of the Cyr theorem, it follows that the circle is rigid as a bicycle curve for any rational rotation number, except in the case $\\alpha = \\frac{\\pi}{2}$, where any curve of constant width is a bicycle curve.\n\nThe examples discussed above show that arithmetic properties of parameters appearing in geometric and dynamical problems can strongly influence their behavior. This perspective motivates the present work, which focuses on the arithmetic properties of real numbers arising in such contexts.\n\nBefore stating the main results of this paper, we introduce some additional notation. Let $z \\in \\mathbb{C} \\setminus \\{0\\}$. We denote by $\\operatorname{Log} z = \\ln|z| + i\\operatorname{Arg}(z)$ the principal branch of the logarithm, where $\\operatorname{Arg}(z) \\in (-\\pi,\\pi]$. \nThe main results of this paper are the following.\n\n\\begin{equation}\\label{eq:main}\n n \\tan \\alpha = \\tan(n \\alpha).\n\\end{equation}\n\n\\begin{equation}\\label{ineq:dioph}\n \\left| \\theta - \\frac{p}{q} \\right| \\ge \\frac{c}{q^\\tau}.\n\\end{equation}", + "full_context": "Let $\\theta$ be a real number. The {\\em irrationality exponent} of $\\theta$ is \n$$\\mu(\\theta) = \\sup\\left\\{ \\mu > 0 \\colon \\left| \\theta - \\frac{p}{q} \\right| < \\frac{1}{q^\\mu} \\text{ holds for infinitely many pairs } (p,q) \\in \\mathbb{Z} \\times \\mathbb{N} \\right\\}.$$\nWe say that $\\theta$ is a {\\em Liouville number} whenever $\\mu(\\theta) = \\infty$. On the other hand, $\\theta$ is called {\\em Diophantine} if there exist $c > 0$ and $\\tau < \\infty$ such that, for every $(p,q) \\in \\mathbb{Z} \\times \\mathbb{N}$, \n\\begin{equation}\\label{ineq:dioph}\n \\left| \\theta - \\frac{p}{q} \\right| \\ge \\frac{c}{q^\\tau}.\n\\end{equation}\nLet $\\mathcal L$ be the set of Liouville numbers and let $\\mathcal D$ be the set of Diophantine numbers.\n\nWe observe that if $\\theta = \\frac{a}{b}$ is rational, then $\\mu(\\theta) = 1$, and one may choose $\\frac{p}{q} = \\frac{a}{b}$ so that \\eqref{ineq:dioph} fails to hold. Hence, no rational number is either Diophantine or Liouville. Among irrational numbers, the sets $\\mathcal L$ and $\\mathcal D$ form a partition.\n\nLet $n \\in \\mathbb{Z}$ and consider the equation \n\\begin{equation}\\label{eq:main}\n n \\tan \\alpha = \\tan(n \\alpha).\n\\end{equation}\nIf $n\\in \\{0,1\\}$, then every $\\alpha \\in \\mathbb{R}$ is a solution. Furthermore, $\\alpha = 0$ is a solution for every $n \\in \\mathbb{Z}$. Since the tangent function is odd and $\\pi$-periodic, we may restrict attention to the case $n \\ge 2$ and $\\alpha \\in \\left(0,\\frac{\\pi}{2}\\right)$. In \\cite{Cyr2012}, Cyr proved that $\\frac{\\alpha}{2\\pi}$ is irrational.\n\nOne prominent context in which this equation appears is the {\\em theory of bicycle curves}. In an effort to classify noncircular bicycle curves, Tabachnikov \\cite{Tabachnikov2006} proved that the circle admits a nontrivial infinitesimal deformation as a smooth plane bicycle curve with rotation number $\\frac{\\alpha}{\\pi}$ if and only if \\eqref{eq:main} holds for some integer $n \\ge 2$. Thus, the solutions of this equation characterize the values of the rotation number for which the circle fails to be rigid in this geometric setting. In light of the Cyr theorem, it follows that the circle is rigid as a bicycle curve for any rational rotation number, except in the case $\\alpha = \\frac{\\pi}{2}$, where any curve of constant width is a bicycle curve.\n\nThe examples discussed above show that arithmetic properties of parameters appearing in geometric and dynamical problems can strongly influence their behavior. This perspective motivates the present work, which focuses on the arithmetic properties of real numbers arising in such contexts.\n\nBefore stating the main results of this paper, we introduce some additional notation. Let $z \\in \\mathbb{C} \\setminus \\{0\\}$. We denote by $\\operatorname{Log} z = \\ln|z| + i\\operatorname{Arg}(z)$ the principal branch of the logarithm, where $\\operatorname{Arg}(z) \\in (-\\pi,\\pi]$. \nThe main results of this paper are the following.\n\n\\begin{equation}\\label{eq:main}\n n \\tan \\alpha = \\tan(n \\alpha).\n\\end{equation}\n\n\\begin{equation}\\label{ineq:dioph}\n \\left| \\theta - \\frac{p}{q} \\right| \\ge \\frac{c}{q^\\tau}.\n\\end{equation}\n\n\\begin{abstract}\n An irrational number $\\theta$ is called Diophantine if there exist $c>0$ and $\\tau < \\infty$ such that $\\left| \\theta - \\frac{p}{q} \\right| \\ge \\frac{c}{q^\\tau}$ holds for every $(p,q) \\in \\mathbb{Z} \\times \\mathbb{N}$. \n In this paper, we study Diophantine and transcendence properties of some real numbers. Using lower bounds for linear forms in logarithms, we show that if $\\beta \\in \\mathbb{C}$ is an algebraic number with $|\\beta|=1$ that is not a root of unity, then $\\frac{\\Arg(\\beta)}{2\\pi}$ is Diophantine. We also prove that if $\\beta = e^{i\\alpha}$ is algebraic, then $\\frac{\\alpha}{\\pi}$ is either rational or transcendental.\n\nLet $\\theta$ be a real number. The {\\em irrationality exponent} of $\\theta$ is \n$$\\mu(\\theta) = \\sup\\left\\{ \\mu > 0 \\colon \\left| \\theta - \\frac{p}{q} \\right| < \\frac{1}{q^\\mu} \\text{ holds for infinitely many pairs } (p,q) \\in \\mathbb{Z} \\times \\mathbb{N} \\right\\}.$$\nWe say that $\\theta$ is a {\\em Liouville number} whenever $\\mu(\\theta) = \\infty$. On the other hand, $\\theta$ is called {\\em Diophantine} if there exist $c > 0$ and $\\tau < \\infty$ such that, for every $(p,q) \\in \\mathbb{Z} \\times \\mathbb{N}$, \n\\begin{equation}\\label{ineq:dioph}\n \\left| \\theta - \\frac{p}{q} \\right| \\ge \\frac{c}{q^\\tau}.\n\\end{equation}\nLet $\\mathcal L$ be the set of Liouville numbers and let $\\mathcal D$ be the set of Diophantine numbers.\n\nBefore stating the main results of this paper, we introduce some additional notation. Let $z \\in \\mathbb{C} \\setminus \\{0\\}$. We denote by $\\Log z = \\ln|z| + i\\Arg(z)$ the principal branch of the logarithm, where $\\Arg(z) \\in (-\\pi,\\pi]$. \nThe main results of this paper are the following.\n\nBy applying the previous result to the solutions of equation \\eqref{eq:main}, together with the classical theorems described in Section \\ref{sec:pre}, we obtain the following arithmetic statements.\n\n\\begin{theorem}\\label{thm:3props}\n Let $n \\ge 2$ be an integer and let $\\alpha \\in \\left( 0, \\frac{\\pi}{2} \\right)$ be a solution of \\eqref{eq:main}. Then the following statements hold.\n \\begin{enumerate}[label=(\\alph*)]\n \\item $\\frac{\\alpha}{2\\pi}$ is a Diophantine number.\n \\item $\\frac{\\alpha}{2\\pi}$ is transcendental.\n \\item $\\alpha$ is transcendental.\n \\end{enumerate}\n\\end{theorem}\n\nFor $z_1,z_2 \\in \\mathbb{C}$, $p \\in \\mathbb{Z}$ and $q \\in \\mathbb{N}$, set $L(z_1,z_2) = qz_1 - 2pz_2$ and \n$$\\Lambda(q,p) = L(\\Log(\\beta),\\Log(-1)) = q \\Log(\\beta) - 2p \\Log(-1)$$\nto be a linear form in logarithms with algebraic numbers $\\{\\beta,-1\\}$. Furthermore, we have that\n\\begin{equation}\\label{eq:Lqp}\n \\Lambda(q,p) = i\\left( q\\alpha - 2\\pi p \\right).\n\\end{equation}\nLet $\\theta = \\frac{\\Arg(\\beta)}{2\\pi} = \\frac{\\alpha}{2\\pi} \\in \\left( -\\frac{1}{2},\\frac{1}{2} \\right]$. It follows that \n\\begin{equation}\\label{eq:|Lambda|}\n |\\Lambda(q,p)| = |q\\alpha - 2\\pi p| = 2\\pi q \\left| \\theta - \\frac{p}{q} \\right|.\n\\end{equation}\n\nWe now apply Baker-Wüstholz Theorem (Theorem \\ref{thm:bw}) with $(\\alpha_1, \\alpha_2, b_1, b_2) = (\\beta, -1, q, -2p)$. Let $d = [\\mathbb{Q}(\\beta):\\mathbb{Q}]$. Since \n$$h'(L) = \\max \\left\\{ \\ln(\\max\\{q,|2p|\\}), 1\\right\\} = \\ln(\\max\\{q,|2p|,e\\}) ,$$ \nobtain that \n$$\\ln|\\Lambda(q,p)| > -C_0 \\ln(\\max\\{q,|2p|,e\\})$$\nwith $C_0 = C_0(\\beta) = C(2,d) h'(\\beta) h'(-1) \n> 0$ depending only on $\\beta$. Therefore \n$$|\\Lambda(q,p)| > (\\max\\{q,|2p|,e\\})^{-C_0}.$$\nBy \\eqref{eq:|Lambda|}, we obtain\n\\begin{equation}\\label{ineq:modulotheta}\n \\left| \\theta - \\frac{p}{q} \\right| > \\frac{1}{2\\pi} \\cdot \\frac{1}{q(\\max\\{q,|2p|,e\\})^{C_0}} \\quad \\text{ for every } p \\in \\mathbb{Z}.\n\\end{equation}\n\nNotice that $\\theta$ is irrational; otherwise $\\beta = e^{2\\pi i \\theta}$ would be a root of unity, which is a contradiction. Therefore, there exists a unique integer $p'$ closest to $q\\theta$, that is, $\\left|q\\theta - p'\\right| < \\frac{1}{2}$. Since $|p'| < q|\\theta| + \\frac{1}{2} < \\frac{q+1}{2}$, it follows that \n$$\\max\\{q,2|p'|,e\\} < 3q$$\nfor every $q \\in \\mathbb{N}$. By \\eqref{ineq:modulotheta}, we obtain\n$$\\left| \\theta - \\frac{p'}{q} \\right| > \\frac{1}{2\\pi} \\cdot \\frac{1}{q(3q)^{C_0}} = \\frac{1}{2\\pi \\cdot 3^{C_0}} \\cdot \\frac{1}{q^{C_0+1}}.$$\nLet $c = \\frac{1}{2\\pi \\cdot 3^{C_0}} > 0$ and $\\tau = C_0+1 < \\infty$. Notice that both $c$ and $\\tau$ depend only on $\\beta$, and hence only on $\\theta$. For every $p \\in \\mathbb{Z}$ and $q \\in \\mathbb{N}$, we have that $|q\\theta - p| \\ge |q\\theta - p'|$, therefore\n$$\\left| \\theta - \\frac{p}{q} \\right| \\ge \\left| \\theta - \\frac{p'}{q} \\right| \\ge \\frac{c}{q^\\tau}.$$\nThis is precisely the definition of a Diophantine number.\n\\qed\n\n\\begin{equation}\\label{eq:main}\n n \\tan \\alpha = \\tan(n \\alpha).\n\\end{equation}", + "post_theorem_intro_text_len": 1638, + "post_theorem_intro_text": "By applying the previous result to the solutions of equation \\eqref{eq:main}, together with the classical theorems described in Section \\ref{sec:pre}, we obtain the following arithmetic statements.\n\n\\begin{theorem}\\label{thm:3props}\n Let $n \\ge 2$ be an integer and let $\\alpha \\in \\left( 0, \\frac{\\pi}{2} \\right)$ be a solution of \\eqref{eq:main}. Then the following statements hold.\n \\begin{enumerate}[label=(\\alph*)]\n \\item $\\frac{\\alpha}{2\\pi}$ is a Diophantine number.\n \\item $\\frac{\\alpha}{2\\pi}$ is transcendental.\n \\item $\\alpha$ is transcendental.\n \\end{enumerate}\n\\end{theorem}\n\nWe observe that items $(a)$ and $(b)$ of the previous theorem extend the result of Cyr~\\cite{Cyr2012}, while item $(c)$ provides the transcendence of the solution $\\alpha$ itself.\n\nThe paper is organized as follows. In Section \\ref{sec:pre}, we introduce the main tools used in the proof of Theorem \\ref{thm:general}, including several algebraic ingredients, as well as the Baker-Wüstholz Theorem, which provides a lower bound for a non-zero linear forms in logarithms, the Gelfond-Schneider Theorem and the Hermite-Lindemann Theorem, which are two remarkable results in transcendence theory. Section \\ref{sec:proof} contains the proof of the main result (Theorem \\ref{thm:general}). In Section \\ref{sec:eqtan}, we apply the main result, together with the remarkable theorems presented in Section \\ref{sec:pre}, to the solutions of equation \\eqref{eq:main} in order to prove Theorem \\ref{thm:3props}. Finally, in Section \\ref{sec:conclusion}, we summarize the results obtained in this paper and present an open problem.", + "sketch": "The post-theorem introduction does not give a step-by-step proof sketch of Theorem~\\ref{thm:general}, but it does indicate the proof ingredients and where the proof is carried out: Section~\\ref{sec:pre} “introduce[s] the main tools used in the proof of Theorem~\\ref{thm:general}, including several algebraic ingredients, as well as the Baker-Wüstholz Theorem, which provides a lower bound for a non-zero linear forms in logarithms, the Gelfond-Schneider Theorem and the Hermite-Lindemann Theorem.” It then states that “Section~\\ref{sec:proof} contains the proof of the main result (Theorem~\\ref{thm:general}).”", + "expanded_sketch": "No expanded sketch found.", + "expanded_theorem": "\\label{thm:general}\n Let $\\beta$ be an algebraic number with $|\\beta|=1$ which is not a root of unity. Then $\\frac{\\operatorname{Arg}(\\beta)}{2\\pi} \\in \\left(-\\frac{1}{2},\\frac{1}{2}\\right]$ is a Diophantine number.,", + "theorem_type": [ + "Implication", + "Universal" + ], + "mcq": { + "question": "Let \\(\\Arg(z)\\in(-\\pi,\\pi]\\) denote the principal argument of a nonzero complex number \\(z\\). A real number \\(\\theta\\) is called Diophantine if there exist constants \\(c>0\\) and \\(\\tau<\\infty\\) such that for every \\((p,q)\\in \\mathbb Z\\times \\mathbb N\\),\n\\[\n\\left|\\theta-\\frac pq\\right|\\ge \\frac{c}{q^{\\tau}}.\n\\]\nWhich statement holds for every algebraic number \\(\\beta\\) satisfying \\(|\\beta|=1\\) and such that \\(\\beta\\) is not a root of unity?", + "correct_choice": { + "label": "A", + "text": "The number \\(\\dfrac{\\Arg(\\beta)}{2\\pi}\\in\\left(-\\dfrac12,\\dfrac12\\right]\\) is Diophantine; equivalently, there exist constants \\(c>0\\) and \\(\\tau<\\infty\\) such that for all \\((p,q)\\in\\mathbb Z\\times\\mathbb N\\),\n\\[\n\\left|\\frac{\\Arg(\\beta)}{2\\pi}-\\frac pq\\right|\\ge \\frac{c}{q^{\\tau}}.\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "The number \\(\\dfrac{\\Arg(\\beta)}{2\\pi}\\in\\left(-\\dfrac12,\\dfrac12\\right]\\) is Diophantine; equivalently, there exist constants \\(c>0\\) and \\(\\tau<\\infty\\), depending only on the degree \\([\\mathbb Q(\\beta):\\mathbb Q]\\), such that for all \\((p,q)\\in\\mathbb Z\\times\\mathbb N\\),\n\\[\n\\left|\\frac{\\Arg(\\beta)}{2\\pi}-\\frac pq\\right|\\ge \\frac{c}{q^{\\tau}}.\n\\]" + }, + { + "label": "C", + "text": "The number \\(\\dfrac{\\Arg(\\beta)}{2\\pi}\\in\\left(-\\dfrac12,\\dfrac12\\right]\\) is not a rational number." + }, + { + "label": "D", + "text": "The number \\(\\dfrac{\\Arg(\\beta)}{2\\pi}\\in\\left(-\\dfrac12,\\dfrac12\\right]\\) is Liouville; equivalently, for every \\(\\tau<\\infty\\) there exist infinitely many pairs \\((p,q)\\in\\mathbb Z\\times\\mathbb N\\) such that\n\\[\n\\left|\\frac{\\Arg(\\beta)}{2\\pi}-\\frac pq\\right|< \\frac{1}{q^{\\tau}}.\n\\]" + }, + { + "label": "E", + "text": "For every algebraic number \\(\\beta\\) with \\(|\\beta|=1\\), the number \\(\\dfrac{\\Arg(\\beta)}{2\\pi}\\in\\left(-\\dfrac12,\\dfrac12\\right]\\) is Diophantine; equivalently, there exist constants \\(c>0\\) and \\(\\tau<\\infty\\) such that for all \\((p,q)\\in\\mathbb Z\\times\\mathbb N\\),\n\\[\n\\left|\\frac{\\Arg(\\beta)}{2\\pi}-\\frac pq\\right|\\ge \\frac{c}{q^{\\tau}}.\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": "dependence_of_constants_on_beta", + "template_used": "uniformity_effectivity" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped_Diophantine_conclusion_to_irrationality_only", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "Diophantine_vs_Liouville_property", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "excluded_root_of_unity_hypothesis", + "template_used": "boundary_range" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not reveal the correct conclusion; it only supplies the definition of Diophantine and the hypothesis on algebraic numbers on the unit circle. There is no explicit answer leakage." + }, + "TAS": { + "score": 1, + "justification": "The intended correct option is essentially the theorem itself specialized to the given hypotheses, so the item is close to a direct theorem recall rather than a substantially reformulated problem. The competing options prevent it from being fully tautological, but only mildly." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish the exact Diophantine conclusion from a stronger uniform version, a weaker irrationality-only claim, and false alternatives. However, recognition of the underlying theorem is largely sufficient, so the generative demand is moderate rather than strong." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically meaningful and target common failure modes: confusing exact vs. uniform bounds (B), accepting a weaker true statement (C), reversing Diophantine to Liouville behavior (D), and forgetting the non-root-of-unity hypothesis (E). They are distinct and plausible." + }, + "total_score": 6, + "overall_assessment": "Conceptually a solid theorem-recall MCQ with good distractors and no answer leakage, but it is only mildly non-tautological and does not strongly force generative reasoning. There is also a serious formatting/validity issue: the marked correct answer is labeled A, but A is missing from the displayed choices." + } + } +]