Let $(M^4,g)$ be a closed Riemannian manifold and denote by $A_{\min}(M,g)$
the area of a smallest $2$-dimensional stationary integral varifold in $(M,g)$.
In general, the existence of such a minimal object follows from the
Almgren--Pitts min--max theory \cite{pitts2014existence}, and in low
codimension and low dimension one can often upgrade the varifold to a smooth
embedded minimal hypersurface using the regularity theory of
Schoen--Simon--Yau \cite{schoen1975,schoen1981regularity}.

In this article we are interested in bounding $A_{\min}(M,g)$ in terms of
global geometric data in the special case when $(M,g)$ is an Einstein
$4$-manifold subject to natural non-collapsing and diameter bounds.
Throughout we assume that
\begin{equation}\label{eq:Ein-assumptions}
 \Ric_g = \lambda g,\quad |\lambda|\le 3,\quad \Vol(M,g)\ge v>0,\quad
 \diam(M,g)\le D,\quad H_1(M;\mathbb{Z})=0.
\end{equation}
We write $\Ee_1(4,v,D)$ for the class of all closed Einstein $4$-manifolds
satisfying \eqref{eq:Ein-assumptions}.

This paper is a sequel to our previous work \cite{LW_HF1}, where we considered
the larger class with $ |{\rm Ric}_g|\le 3,\ \Vol(M,g)\ge v>0,\
 \diam(M,g)\le D,\ H_1(M;\mathbb{Z})=0, $
and proved a linear upper bound for the first homological filling function
$\HF_1$ of $(M,g)$. More precisely, for every $(M^4,g)\in\Mm_1(4,v,D)$ we
showed that there exist functions $f_1(v,D),f_2(v,D)>0$ such that
\[
 \HF_1(l)\ \le\ f_1(v,D)\,l + f_2(v,D)\qquad\text{for all }l\ge 0.
\]
Combined with an effective version of the Almgren--Pitts min--max theory due to
Nabutovsky--Rotman \cite{nabutovsky2006curvature}, this yields
\[
 A_{\min}(M,g)\ \le\ F(v,D)
\]
for some function $F$ depending only on $v$ and $D$. In that setting, however,
the dependence of $f_1,f_2$ and $F$ on the geometric parameters was not made
explicit, since the constants coming from the Cheeger--Naber bubble--tree
decomposition \cite{cheeger2014regularity} were only known to exist abstractly.

In the present work we revisit this question in the more rigid Einstein setting.
For $(M^4,g)\in\Ee_1(4,v,D)$ the curvature solves an elliptic system, and
global Sobolev inequalities can be used, following Anderson
\cite{anderson1989ricci,anderson1992thel}, to derive quantitative $L^2$–curvature
bounds and $\varepsilon$–regularity with constants depending only on a Sobolev
constant. This analytic input, combined with the Einstein specialization of the
Cheeger--Naber bubble--tree decomposition and an improved combinatorial filling
estimate, allows us to obtain an explicit upper bound for $A_{\min}(M,g)$ in
terms of $(v,D)$ and the Sobolev constants.

Our first main result is the following quantitative bound for minimal
$2$-varifolds in Einstein $4$-manifolds.

\begin{equation}\label{eq:Ein-assumptions}
 \Ric_g = \lambda g,\quad |\lambda|\le 3,\quad \Vol(M,g)\ge v>0,\quad
 \diam(M,g)\le D,\quad H_1(M;\Z)=0.
\end{equation}

\begin{abstract}
We study the smallest area $A_{\min}(M,g)$ of a $2$-dimensional stationary integral varifold in a closed Einstein $4$-manifold $(M^4,g)$ with
\[
\Ric_g = \lambda g,\quad |\lambda|\le 3,\quad \Vol(M,g)\ge v>0,\quad \diam(M,g)\le D,\quad H_1(M;\Z)=0.
\]
Building on the previous work \cite{LW_HF1} on homological filling functions, we show that for every $(M^4,g)$ in this Einstein class there is an upper bound
\[
A_{\min}(M,g)\le F_{\Ein}(v,D),
\]
where $F_{\Ein}$ depends only on $(v,D)$ and on quantitative Sobolev and $\varepsilon$–regularity constants for Einstein metrics.
\end{abstract}

In the present work we revisit this question in the more rigid Einstein setting.
For $(M^4,g)\in\Ee_1(4,v,D)$ the curvature solves an elliptic system, and
global Sobolev inequalities can be used, following Anderson
\cite{anderson1989ricci,anderson1992thel}, to derive quantitative $L^2$–curvature
bounds and $\varepsilon$–regularity with constants depending only on a Sobolev
constant. This analytic input, combined with the Einstein specialization of the
Cheeger--Naber bubble--tree decomposition and an improved combinatorial filling
estimate, allows us to obtain an explicit upper bound for $A_{\min}(M,g)$ in
terms of $(v,D)$ and the Sobolev constants.

Our first main result is the following quantitative bound for minimal
$2$-varifolds in Einstein $4$-manifolds.

As in \cite{LW_HF1}, the proof of Theorem~\ref{thm:main} proceeds via a
homological filling inequality expressed in terms of the first homological
filling function $\HF_1$ of $(M,g)$.

\begin{theorem}\label{thm:Ein-fill}
For every $v,D>0$ there exist functions $$f_1^{\Ein}(v,D),f_2^{\Ein}(v,D)>0, $$
with the following property. If $(M^4,g)\in \Ee_1(4,v,D)$ and
$C\in\Zz_1(M;\Z)$ is a singular Lipschitz $1$-cycle, then $C$ bounds a
singular $2$-chain $E\in\Cc_2(M;\Z)$ such that
\[
 \mass_2(E)\ \le\ f_1^{\Ein}(v,D)\,\mass_1(C) + f_2^{\Ein}(v,D).
\]
Equivalently, the first homological filling function of $(M,g)$ satisfies
\[
 \HF_1(l)\ \le\ f_1^{\Ein}(v,D)\,l + f_2^{\Ein}(v,D)
 \qquad\text{for all }l\ge 0.
\]
\end{theorem}

Theorem~\ref{thm:main} is then an immediate consequence of
Theorem~\ref{thm:Ein-fill} and the min–max theorem of Nabutovsky--Rotman
\cite{nabutovsky2006curvature}, which relates $A_{\min}(M,g)$ to $\HF_1$.
In the Einstein setting we can track all constants arising from the Sobolev
inequality, the $L^2$–curvature bound and the bubble–tree decomposition,
thus making $F_{\Ein}(v,D)$ completely explicit once the analytic data are fixed.

\begin{theorem}[Einstein $\varepsilon$-regularity via Sobolev]\label{thm:eps_reg_Einstein}
There exist dimensional constants $\varepsilon_{\mathrm{Ein}}>0$ and $c_{\mathrm{Ein}}>0$,
depending only on $C_S(v,D)$, such that the following holds.

\begin{theorem}[Nabutovsky--Rotman~{\cite{nabutovsky2006curvature}}]\label{thm:NR}
Let $M$ be a closed $n$-dimensional Riemannian manifold with $H_1(M;\Z)=0$
and diameter $\diam(M)=D$. Let $A_{\min}(M)$ denote the area of a smallest
$2$-dimensional stationary integral varifold in $M$. Then
\[
 A_{\min}(M)\;\le\;\frac{(n+1)!}{2}\,\HF_1(2D).
\]
In particular, for $n=4$,
\[
 A_{\min}(M)\;\le\;60\,\HF_1(2D).
\]
\end{theorem}

\begin{theorem}\label{thm:Ein-fill}
For every $v,D>0$ there exist functions $$f_1^{\Ein}(v,D),f_2^{\Ein}(v,D)>0, $$
with the following property.  If $(M^4,g)\in \Ee_1(4,v,D)$ and
$C\in\Zz_1(M;\Z)$ is a singular Lipschitz $1$-cycle, then $C$ bounds a
singular $2$-chain $E\in\Cc_2(M;\Z)$ such that
\[
 \mass_2(E)\ \le\ f_1^{\Ein}(v,D)\,\mass_1(C) + f_2^{\Ein}(v,D).
\]
Equivalently, the first homological filling function of $(M,g)$ satisfies
\[
 \HF_1(l)\ \le\ f_1^{\Ein}(v,D)\,l + f_2^{\Ein}(v,D)
 \qquad\text{for all }l\ge 0.
\]
\end{theorem}

\begin{theorem}\label{thm:main}
For every $v,D>0$ there exists a constant $F_{\Ein}(v,D)>0$ with the following
property.  If $(M^4,g)\in \Ee_1(4,v,D)$, then the area $A_{\min}(M,g)$ of a
smallest $2$-dimensional stationary integral varifold in $(M,g)$ satisfies
\[
 A_{\min}(M,g)\ \le\ F_{\Ein}(v,D).
\]
Moreover $F_{\Ein}(v,D)$ can be written explicitly in terms of the Sobolev
constant $C_S(v,D)$, the global $L^2$–curvature bound, and the quantitative
Einstein $\varepsilon$–regularity constants.
\end{theorem}

As in \cite{LW_HF1}, the proof of Theorem~\ref{thm:main} proceeds via a
homological filling inequality expressed in terms of the first homological
filling function $\HF_1$ of $(M,g)$.

\begin{theorem}\label{thm:Ein-fill}
For every $v,D>0$ there exist functions $$f_1^{\Ein}(v,D),f_2^{\Ein}(v,D)>0, $$
with the following property.  If $(M^4,g)\in \Ee_1(4,v,D)$ and
$C\in\Zz_1(M;\mathbb{Z})$ is a singular Lipschitz $1$-cycle, then $C$ bounds a
singular $2$-chain $E\in\Cc_2(M;\mathbb{Z})$ such that
\[
 \mass_2(E)\ \le\ f_1^{\Ein}(v,D)\,\mass_1(C) + f_2^{\Ein}(v,D).
\]
Equivalently, the first homological filling function of $(M,g)$ satisfies
\[
 \HF_1(l)\ \le\ f_1^{\Ein}(v,D)\,l + f_2^{\Ein}(v,D)
 \qquad\text{for all }l\ge 0.
\]
\end{theorem}

Theorem~\ref{thm:main} is then an immediate consequence of
Theorem~\ref{thm:Ein-fill} and the min–max theorem of Nabutovsky--Rotman
\cite{nabutovsky2006curvature}, which relates $A_{\min}(M,g)$ to $\HF_1$.
In the Einstein setting we can track all constants arising from the Sobolev
inequality, the $L^2$–curvature bound and the bubble–tree decomposition,
thus making $F_{\Ein}(v,D)$ completely explicit once the analytic data are fixed.

As in \cite{LW_HF1}, the proof of Theorem~\ref{thm:main} "proceeds via a homological filling inequality expressed in terms of the first homological filling function $\HF_1$ of $(M,g)$." One first proves Theorem~\ref{thm:Ein-fill}, i.e. a linear filling estimate for every singular Lipschitz $1$-cycle $C$ by a singular $2$-chain $E$ with \(\mass_2(E)\le f_1^{\Ein}(v,D)\,\mass_1(C)+f_2^{\Ein}(v,D)\), equivalently \(\HF_1(l)\le f_1^{\Ein}(v,D)\,l+f_2^{\Ein}(v,D)\). Then Theorem~\ref{thm:main} is "an immediate consequence" of Theorem~\ref{thm:Ein-fill} together with the min--max theorem of Nabutovsky--Rotman \cite{nabutovsky2006curvature}, "which relates $A_{\min}(M,g)$ to $\HF_1$." In the Einstein setting, the constants are tracked using "the Sobolev inequality, the $L^2$--curvature bound and the bubble--tree decomposition," making $F_{\Ein}(v,D)$ explicit once the analytic data are fixed.

As in Larry Guth and David W. Rotman, \emph{The first homology of a Riemannian manifold and the first homological filling function}, the proof of the main theorem “proceeds via a homological filling inequality expressed in terms of the first homological filling function $\HF_1$ of $(M,g)$.” We first prove the following theorem.

\begin{theorem}\label{thm:Ein-fill}
For every $v,D>0$ there exist functions $$f_1^{\Ein}(v,D),f_2^{\Ein}(v,D)>0, $$
with the following property.  If $(M^4,g)\in \Ee_1(4,v,D)$ and
$C\in\Zz_1(M;\Z)$ is a singular Lipschitz $1$-cycle, then $C$ bounds a
singular $2$-chain $E\in\Cc_2(M;\Z)$ such that
\[
 \mass_2(E)\ \le\ f_1^{\Ein}(v,D)\,\mass_1(C) + f_2^{\Ein}(v,D).
\]
Equivalently, the first homological filling function of $(M,g)$ satisfies
\[
 \HF_1(l)\ \le\ f_1^{\Ein}(v,D)\,l + f_2^{\Ein}(v,D)
 \qquad\text{for all }l\ge 0.
\]
\end{theorem}

To prove the main theorem, we then combine this filling estimate with the min--max theorem of Nabutovsky--Rotman (Alexander Nabutovsky and Regina Rotman, \emph{Curvature-Free Bounds for the Least Area of a Surface}, 2006), which “relates $A_{\min}(M,g)$ to $\HF_1$.” In the Einstein setting, the constants are tracked using “the Sobolev inequality, the $L^2$--curvature bound and the bubble--tree decomposition,” making $F_{\Ein}(v,D)$ explicit once the analytic data are fixed.

\label{thm:main}
For every $v,D>0$ there exists a constant $F_{\Ein}(v,D)>0$ with the following
property.  If $(M^4,g)\in \Ee_1(4,v,D)$, then the area $A_{\min}(M,g)$ of a
smallest $2$-dimensional stationary integral varifold in $(M,g)$ satisfies
\[
 A_{\min}(M,g)\ \le\ F_{\Ein}(v,D).
\]
Moreover $F_{\Ein}(v,D)$ can be written explicitly in terms of the Sobolev
constant $C_S(v,D)$, the global $L^2$–curvature bound, and the quantitative
Einstein $\varepsilon$–regularity constants.,

Question:
Let $v,D>0$. Suppose $(M^4,g)$ is a closed Einstein $4$-manifold in the class $\Ee_1(4,v,D)$, meaning that
\[
\Ric_g=\lambda g,\qquad |\lambda|\le 3,\qquad \Vol(M,g)\ge v,\qquad \diam(M,g)\le D,\qquad H_1(M;\mathbb Z)=0.
\]
Let $A_{\min}(M,g)$ denote the area of a smallest $2$-dimensional stationary integral varifold in $(M,g)$. Which conclusion about $A_{\min}(M,g)$ is valid under these hypotheses?

There exists a constant $F_{\Ein}(v,D)>0$, depending only on $v$ and $D$, such that
\[
A_{\min}(M,g)\le F_{\Ein}(v,D).
\]
Moreover, $F_{\Ein}(v,D)$ can be written explicitly in terms of the Sobolev constant $C_S(v,D)$, the global $L^2$-curvature bound, and the quantitative Einstein $\varepsilon$-regularity constants.

Choices:
B.
There exists a universal constant $F_{\Ein}>0$, independent of $v$ and $D$, such that for every $(M^4,g)\in \Ee_1(4,v,D)$ one has
\[
A_{\min}(M,g)\le F_{\Ein}.
\]
Moreover, this bound can be written explicitly in terms of dimensional Einstein $\varepsilon$-regularity constants alone.
C.
There exists a constant $F_{\Ein}(v,D)>0$ such that
\[
A_{\min}(M,g)\le F_{\Ein}(v,D)
\]
for every $(M^4,g)\in \Ee_1(4,v,D)$.
D. For every $v,D>0$ there exists a constant $F_{\Ein}(v,D)>0$ with the following property: if $(M^4,g)\in \Ee_1(4,v,D)$, then every $2$-dimensional stationary integral varifold in $(M,g)$ has area at most $F_{\Ein}(v,D)$. Moreover $F_{\Ein}(v,D)$ can be written explicitly in terms of the Sobolev constant $C_S(v,D)$, the global $L^2$-curvature bound, and the quantitative Einstein $\varepsilon$-regularity constants.
E.
For every $v,D>0$ there exists a constant $F_{\Ein}(v,D)>0$ with the following property. If $(M^4,g)\in \Ee_1(4,v,D)$, then the area $A_{\min}(M,g)$ of a smallest $2$-dimensional stationary integral varifold in $(M,g)$ satisfies
\[
A_{\min}(M,g)\le F_{\Ein}(v,D),
\]
and one may choose $F_{\Ein}(v,D)$ to depend only on $v$, $D$, and the diameter term appearing in the Nabutovsky--Rotman estimate, without any use of the Sobolev constant, the global $L^2$-curvature bound, or Einstein $\varepsilon$-regularity constants.

dependence of the bound on v and D and analytic data

explicit description of how F_{\Ein}(v,D) is written in terms of Sobolev/L^2-curvature/\varepsilon-regularity data

smallest varifold versus arbitrary stationary varifold

necessity of Sobolev, L^2-curvature, and bubble-tree/\varepsilon-regularity input for explicit constants


Question:
Let \(\Ee_1(4,v,D)\) denote the class of closed Einstein \(4\)-manifolds \((M^4,g)\) satisfying
\[
\operatorname{Ric}_g=\lambda g,\qquad |\lambda|\le 3,\qquad \operatorname{Vol}(M,g)\ge v>0,\qquad \operatorname{diam}(M,g)\le D,\qquad H_1(M;\mathbb Z)=0.
\]
For such a manifold, let \(A_{\min}(M,g)\) be the area of a smallest \(2\)-dimensional stationary integral varifold in \((M,g)\). Which explicit uniform existence statement holds for this class of manifolds?

For every \(v,D>0\), there exists a constant \(F_{\mathrm{Ein}}(v,D)>0\) such that for every \((M^4,g)\in \Ee_1(4,v,D)\),
\[
A_{\min}(M,g)\le F_{\mathrm{Ein}}(v,D).
\]
Moreover, \(F_{\mathrm{Ein}}(v,D)\) can be written explicitly in terms of the Sobolev constant \(C_S(v,D)\), the global \(L^2\)-curvature bound, and the quantitative Einstein \(\varepsilon\)-regularity constants.

Choices:
B.
For every \(v,D>0\), there exists a constant \(F_{\mathrm{Ein}}(v,D)>0\) such that for every closed Einstein \(4\)-manifold \((M^4,g)\) with
\[
\operatorname{Ric}_g=\lambda g,\qquad |\lambda|\le 3,\qquad \operatorname{Vol}(M,g)\ge v>0,\qquad \operatorname{diam}(M,g)\le D,
\]
one has
\[
A_{\min}(M,g)\le F_{\mathrm{Ein}}(v,D).
\]
Moreover, \(F_{\mathrm{Ein}}(v,D)\) can be written explicitly in terms of the Sobolev constant \(C_S(v,D)\), the global \(L^2\)-curvature bound, and the quantitative Einstein \(\varepsilon\)-regularity constants.
C.
For every \(v,D>0\), there exists a constant \(F_{\mathrm{Ein}}(v,D)>0\) such that for every \((M^4,g)\in \Ee_1(4,v,D)\),
\[
A_{\min}(M,g)\le F_{\mathrm{Ein}}(v,D).
\]
D.
For every \(v,D>0\), there exists a constant \(F_{\mathrm{Ein}}(v,D)>0\), independent of any Sobolev or curvature data, such that for every \((M^4,g)\in \Ee_1(4,v,D)\),
\[
A_{\min}(M,g)\le F_{\mathrm{Ein}}(v,D).
\]
Moreover, \(F_{\mathrm{Ein}}(v,D)\) is determined solely by the volume lower bound \(v\) and the diameter bound \(D\).
E.
For every \(v,D>0\), there exists a constant \(F_{\mathrm{Ein}}(v,D)>0\) such that for every \((M^4,g)\in \Ee_1(4,v,D)\),
\[
A_{\min}(M,g)\le 60\,F_{\mathrm{Ein}}(v,D).
\]
Moreover, \(F_{\mathrm{Ein}}(v,D)\) can be written explicitly in terms of the Sobolev constant \(C_S(v,D)\), the global \(L^2\)-curvature bound, and the quantitative Einstein \(\varepsilon\)-regularity constants.


The study of \hk \ manifolds is a central topic in the theory of compact Kähler manifolds: in virtue of the Beauville--Bogomolov decomposition theorem \cite[Thm.\ 1]{beauville}, they represent one of three ``fundamental" types of compact Kähler manifolds with \(c_1(K_X)=0\), together with abelian varieties and strict Calabi-Yau manifolds.
Their topology is extremely interesting: on the one hand it is severely restricted compared to that of a general compact \Kah \ manifold, for example by the Fujiki relations, \cite{Fujiki}. Moreover, endowed with the usual Hodge structure, the cohomology of \hkm s largely determines their geometry by the Torelli Theorem, see for example \cite{Markman_torelli}. On the other hand, the topology of \hkm s remains very mysterious, with a lot of basic open questions. One of these questions is the following:

\begin{question}\label{que:hk_torsion}
Let $X$ be a \hk \ manifold. Is $H^*(X, \mathbb Z)$ torsion-free?
\end{question}
For \hk \ manifolds of $K3^{[n]}$-type, the answer is positive by \cite{Markman2007_integral}, which has then been further generalized by \cite{Totaro_2020_integral} to the Hilbert scheme of points on any smooth projective surface with torsion-free cohomology. Already for the next most understood \hk \ deformation type, that of \textit{generalized Kummer varieties} (or \textit{Kummer manifolds}), denoted as \(\Kum_n(A)\), this question is open in its full generality. In \cite{kapfermenet}, however, Kapfer and Menet give a positive answer in the case of Kummer fourfolds.

The abelian group $H^3(X, \mathbb Z)_{\tors}$ is of particular interest as it sits in the short exact sequence
$$0 \to H^2(X, \mathbb Z) / \NS(X, \mathbb Z) \otimes_\Z \Q/\Z \to \Br(X) \to H^3(X, \mathbb Z)_{\tors} \to 0.$$
For this reason it is often called the \textit{topological Brauer group}, see also \cite[Sec.\ 1]{grothendieck}, and it is the lowest degree cohomology group whose torsion part is not known for any \hk \ manifold other than those of K3\(^{[n]}\)-type and Kummer fourfolds.

The aim of this note is to prove the following result:

\begin{question}\label{que:hk_torsion}
Let $X$ be a \hk \ manifold. Is $H^*(X, \mathbb Z)$ torsion-free?
\end{question}
For \hk \ manifolds of $K3^{[n]}$-type, the answer is positive by \cite{Markman2007_integral}, which has then been further generalized by \cite{Totaro_2020_integral} to the Hilbert scheme of points on any smooth projective surface with torsion-free cohomology. Already for the next most understood \hk \ deformation type, that of \textit{generalized Kummer varieties} (or \textit{Kummer manifolds}), denoted as \(\Kum_n(A)\), this question is open in its full generality. In \cite{kapfermenet}, however, Kapfer and Menet give a positive answer in the case of Kummer fourfolds.

The abelian group $H^3(X, \mathbb Z)_{\tors}$ is of particular interest as it sits in the short exact sequence
$$0 \to H^2(X, \mathbb Z) / \NS(X, \mathbb Z) \otimes_\Z \Q/\Z \to \Br(X) \to H^3(X, \mathbb Z)_{\tors} \to 0.$$
For this reason it is often called the \textit{topological Brauer group}, see also \cite[Sec.\ 1]{grothendieck}, and it is the lowest degree cohomology group whose torsion part is not known for any \hk \ manifold other than those of K3\(^{[n]}\)-type and Kummer fourfolds.

The strategy of the proof is inspired by and generalizes the approach of \cite{kapfermenet}:
we first consider a rational cover of degree \(n\) by a smooth projective variety with no torsion in degree three cohomology, which forces \(H^3(\Kum_{n-1}(A),\Z)_{\tors}\) to be entirely of \(n\)-torsion. By considering a double cover of a suitable open of \(\Kum_{n-1}(A)\), we then show that \(H^3(\Kum_{n-1}(A),\Z)_{\tors}\) is a $2$-group, via an analysis of the cohomology of the alternating group and the Cartan--Leray spectral sequence. When \(n\) is odd, we obtain Corollary \ref{cor:nice_statement_even}.
We are then left with asking ourselves the following

\end{lemma}
\begin{proof}
Since \(H^3(-,\Z)_{\tors}\) is a birational invariant, by passing to a resolution of indeterminacies we may assume that \(f\) is everywhere defined. The result then follows from the projection formula.
\end{proof}
\begin{corollary}\label{cor:n-tors_of_Kum}
 The abelian group $H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors}$ is $n$-torsion.
\end{corollary}
\begin{proof}
 The map \(K_{n-1}(A) \dashrightarrow \Kum_{n-1}(A)\) factors through the quotient by \(\mathfrak{S}_{n-1}\) acting on the first \(n-1\) entries of \(A^n\), yielding a map of degree \(n\).
 The quotient \(K_{n-1}(A)/\mathfrak{S}_{n-1}\) is birational to \(A^{[n-1]}\), hence we have a generically finite rational map of degree \(n\)
 \[A^{[n-1]} \dashrightarrow \Kum_{n-1}(A).\]
By \cite{Markman2007_integral} the group $H^3(A^{[n-1]},\Z)$ is torsion-free. The result then follows from Lemma \ref{lem:dn_tors} with \(N=1\).
\end{proof}
\section{Quotient by the alternating group}
Ideally, one would like to apply the argument of the previous section to the rational double cover $$A^{n-1} / \mathfrak A_n \dashrightarrow \Kum_{n-1}(A).$$ However, there is no smooth projective model \(Z\) of $A^{n-1} / \alt_n$ for which \(H^3(Z, \Z)_{\tors}\) is known.

Let $U \coloneqq K_{n-1}(A) \cap (A^n\setminus \Delta_2) \subset A^n$ and $V \coloneqq \Kum_{n-1}(A) \cap (A^{[n]} \setminus D_2) \subset A^{[n]}$.
For codimension reasons, avoiding the smaller diagonals does not change $H^3(-, \mathbb Z)_{\tors}$:
\begin{lemma}\label{lem:torsvkum}
 We have
 $H^3(V, \mathbb Z)_{\tors} \simeq H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors}.$
\end{lemma}
\begin{proof}
We consider the stratification
\[D_{2}= Z_0 \supset Z_1\supset \cdots \]
where \(Z_{i+1}\) is the singular locus of \(Z_i\). By the long exact sequence of relative cohomology and the Thom isomorphism, we have
\[H^3(\Kum_{n-1}\setminus Z_{1},\Z)\simeq H^3(\Kum_{n-1}\setminus Z_{2},\Z) \simeq \dots \simeq H^3(\Kum_{n-1}(A), \mathbb Z),\]
see \cite[Lemma 11.13]{Voisin_book_I}. By the same argument we also have the exact sequence
$$ 0\to H^3(\Kum_{n-1} \setminus Z_1,\mathbb Z) \to H^3( \Kum_{n-1}(A) \setminus Z_0, \mathbb Z) \to \mathbb Z^{\pi_0(Z_0 \setminus Z_1)}.$$
Thus
\[H^3(\Kum_{n-1}(A),\Z)_{\tors}\simeq H^3(\Kum_{n-1}(A)\setminus Z_0,\Z)_{\tors}.\]
Since $V = \Kum_{n-1}(A) \setminus Z_0$, we conclude.
\end{proof}
Since the alternating group acts freely on $U$, the relation between the quotient $U / \alt_n$ and the open subset $V \subset \Kum_{n-1}(A)$ can be made precise as follows:
\begin{lemma}\label{lem:doublecover_V}
There exists a ramified double cover
 $$\pi \colon \Bl_{\overline{\Delta}} (U/\alt_n) \to V,$$
 where \(\overline{\Delta}\) is the image in \(U/\alt_n\) of \(\Delta\subset U\).
\end{lemma}
\begin{proof}
By \cite[Sec.\ 8, p.\ 770]{beauville}, we have
\[(\Bl_{\Delta} U )/ \mathfrak S_n \simeq V\]
Moreover, since $\mathfrak A_n$ acts freely on $U$ and stabilizes $\Delta$, we have
$$\Bl_{\Delta}(U) / \mathfrak A_n \simeq \Bl_{\overline{\Delta}}(U / \mathfrak A_n).$$
The action of \(\Z/2\Z\simeq\mathfrak{S}_n/\alt_n\) induced onto the quotient \(\Bl_{\Delta}(U) / \mathfrak A_n \) produces a double cover
 $$\Bl_{\overline\Delta}(U / \mathfrak A_n) \simeq (\Bl_{\Delta} U ) / \mathfrak A_n \to (\Bl_{\Delta} U) / \mathfrak S_n \simeq V,$$
 which is what we wanted.
 \end{proof}

\begin{proposition}\label{prop:spectral_sequence_argument}
 If there are integers $N_p$ for $1 \leq p \leq 3$ such that $$N_p \cdot H^p(\mathfrak A_n, H^{3-p}(K_{n-1}(A), \mathbb Z)) = 0,$$ then we have
 $$2N_1N_2N_3 \cdot H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors} = 0.$$
\end{proposition}
\begin{proof}
Since the action of $\mathfrak A_n$ on $U$ is free, there is the Cartan--Leray spectral sequence
$$H^p(\mathfrak A_n, H^q(U, \mathbb Z)) \Rightarrow H^{p+q}(U / \mathfrak A_n, \mathbb Z).$$
Note that for $p \leq 6$, by the same argument as in Lemma \ref{lem:torsvkum} we have $H^p(U, \mathbb Z) \simeq H^p(K_{n-1}(A), \mathbb Z)$, as the complement of $U \subset K_{n-1}(A)$ is of codimension four. In particular, $H^0(\mathfrak A_n, H^3(U, \mathbb Z)) \simeq H^3(K_{n-1}(A))^{\mathfrak A_n}$ is torsion-free.
It then follows that $H^3(U / \mathfrak A_n, \mathbb Z)_{\tors}$ is an extension of subquotients of $H^{p}(\mathfrak A_n, H^{3-p}(K_{n-1}(A, \mathbb Z)))$ for $1 \leq p \leq 3$, which implies that \(N_1 N_2 N_3 \cdot H^3(U / \mathfrak A_n, \mathbb Z)_{\tors} = 0\) .
As blowing-up smooth subvarieties leaves $H^3(-, \mathbb Z)_{\tors}$ invariant, we have
$$H^3(\Bl_{\overline{\Delta}}(U / \mathfrak A_n), \mathbb Z)_{\tors} \simeq H^3(U / \mathfrak A_n, \mathbb Z)_{\tors}.$$
By applying \cite[Thm.\ 5.4]{aguilarprieto} to the double cover of Lemma \ref{lem:doublecover_V} $$\pi \colon \Bl_{\overline\Delta}(U / \mathfrak A_n) \to V,$$
we have that \(\pi_* \pi^* \alpha =2 \alpha\) for any \(\alpha \in H^*(V,\Z)\), which allows us to conclude by Lemma \ref{lem:torsvkum}.
\end{proof}

\begin{theorem*}[Theorem \ref{thm:main}]
If \(n \neq 6\), then
 $$ \gcd(2^3, n) \cdot H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors} = 0.$$
If instead \(n=6\), then
$$6 \cdot H^3(\Kum_{5}(A), \mathbb Z)_{\tors} = 0.$$
\end{theorem*}

\begin{proof}
On the one hand, we have $n \cdot H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors} = 0$ by Corollary \ref{cor:n-tors_of_Kum}. Note that this already implies the result for $n \in \{4, 6, 8\}$. On the other hand, by combining Proposition \ref{prop:spectral_sequence_argument} and Proposition \ref{prop:spectralval}, we have $2^3 \cdot H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors} = 0$ for $n = 3$ or $n \geq 8$ and $2 \cdot 12^3 \cdot H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors} = 0$ for $n=5,7$. The claim then follows by taking the greatest common divisor of $n$ and $2^3$ (respectively \ $2 \cdot 12^3$).
\end{proof}

\begin{corollary}\label{cor:nice_statement_even}
    The third integral cohomology group of a generalized Kummer variety of dimension divisible by four is torsion-free.
\end{corollary}

For odd values of \(n\), this amounts to the following:
\begin{corollary}\label{cor:nice_statement_even}
    The third integral cohomology group of a generalized Kummer variety of dimension divisible by four is torsion-free.
\end{corollary}
As already mentioned, the case \(n=2\) follows from \cite{kapfermenet}. This result confirms folklore expectations, as stated for example in \cite[Rem.\ 2.1.(ii)]{huyb2024}

The strategy of the proof is inspired by and generalizes the approach of \cite{kapfermenet}:
we first consider a rational cover of degree \(n\) by a smooth projective variety with no torsion in degree three cohomology, which forces \(H^3(\Kum_{n-1}(A),\Z)_{\tors}\) to be entirely of \(n\)-torsion. By considering a double cover of a suitable open of \(\Kum_{n-1}(A)\), we then show that \(H^3(\Kum_{n-1}(A),\Z)_{\tors}\) is a $2$-group, via an analysis of the cohomology of the alternating group and the Cartan--Leray spectral sequence. When \(n\) is odd, we obtain Corollary \ref{cor:nice_statement_even}.
We are then left with asking ourselves the following

\begin{question}
    Does \(H^3(\Kum_{2n+1}(A),\Z)_{\tors}\) vanish for \(n \geq 1\) as well?
\end{question}

\subsection{Acknowledgements}
The authors would like to thank Daniel Huybrechts and Emanuele Macrì for their interest in the project, as well as Nick Addington for sharing his computations and suggesting that earlier bounds could be improved. The first author would like to thank Claire Voisin for the invitation to Paris, where this project was initiated.
Many thanks to Daniel Huybrechts and Paolo Stellari, respectively, for bringing up Question \ref{que:hk_torsion} at the ``K3 surfaces and friends"\footnote{Lorentz Center, Leiden, June 2025  (\href{https://www.lorentzcenter.nl/k3-surfaces-en-friends-brauer-groups-and-moduli.html}{\underline{link}}) }
and PRAGMATIC\footnote{University of Catania, Italy, September 2025 (\href{https://www.dmi.unict.it/pragmatic/docs/Pragmatic2025.html}{\underline{link}})} summer schools. We thank the organizers of said events for the wonderful research environment.
Both authors were supported by the ERC Synergy Grant 854361 HyperK.
The first author is grateful for the support provided by the International Max Planck Research School on Moduli Spaces at the Max Planck Institute for Mathematics in Bonn.

The strategy of the proof (inspired by and generalizing \,\cite{kapfermenet}) is as follows. To prove Theorem~\ref{thm:main}, one first considers “a rational cover of degree \(n\) by a smooth projective variety with no torsion in degree three cohomology,” which “forces \(H^3(\Kum_{n-1}(A),\Z)_{\tors}\) to be entirely of \(n\)-torsion.” Then, “by considering a double cover of a suitable open of \(\Kum_{n-1}(A)\),” one shows that \(H^3(\Kum_{n-1}(A),\Z)_{\tors}\) “is a \(2\)-group,” using “an analysis of the cohomology of the alternating group and the Cartan--Leray spectral sequence.” When \(n\) is odd, combining these conclusions yields Corollary~\ref{cor:nice_statement_even}.

\label{thm:main}
If \(n \neq 6\), then
    $$ \gcd(2^3, n) \cdot H^3(\Kum_{n-1}(A), \mathbb Z)_{\tors} = 0.$$
If instead \(n=6\), then
$$6 \cdot H^3(\Kum_{5}(A), \mathbb Z)_{\tors} = 0.$$,

Question: Let \(A\) be an abelian surface, and for an integer \(n\ge 2\) let \(\Kum_{n-1}(A)\) denote the corresponding generalized Kummer variety. Write \(H^3(\Kum_{n-1}(A),\mathbb Z)_{\tors}\) for the torsion subgroup of its third integral cohomology. Which annihilation statement holds for this torsion subgroup?

If \(n\neq 6\), then \[\gcd(2^3,n)\cdot H^3(\Kum_{n-1}(A),\mathbb Z)_{\tors}=0.\] If \(n=6\), then \[6\cdot H^3(\Kum_{5}(A),\mathbb Z)_{\tors}=0.\]

Choices:
B.
For every integer \(n\ge 2\), one has
\[2^3\cdot H^3(\Kum_{n-1}(A),\mathbb Z)_{\tors}=0.\]
C.
If \(n\) is odd, then
\[H^3(\Kum_{n-1}(A),\mathbb Z)_{\tors}=0.\]
D.
If \(n\neq 6\), then
\[\operatorname{lcm}(2^3,n)\cdot H^3(\Kum_{n-1}(A),\mathbb Z)_{\tors}=0.\]
If \(n=6\), then
\[6\cdot H^3(\Kum_{5}(A),\mathbb Z)_{\tors}=0.\]
E.
If \(n\neq 6\), then
\[\gcd(2^2,n)\cdot H^3(\Kum_{n-1}(A),\mathbb Z)_{\tors}=0.\]
If \(n=6\), then
\[2\cdot H^3(\Kum_{5}(A),\mathbb Z)_{\tors}=0.\]


\label{thm:Cd-main}
For every $d \geq 1$ and every finite set $A\subset \mathbb{R}^m$ with exponential Freiman dimension $d$, we have
\begin{equation} \label{eq:main-tight}
  |A+A|\geq C_d(|A|-2^d)+3^d,
\end{equation}
where
\[
  C_d \;=\;
  \begin{cases}
    2\bigl(\tfrac32\bigr)^{d-1}, & d \le 5,\\[4pt]
    2^{d/2+1} - 1, & d \ge 6 \text{ even},\\[4pt]
    3\cdot 2^{(d-1)/2} - \tfrac32, & d \ge 7 \text{ odd}.
  \end{cases}
\]

\begin{theorem}\label{thm:Cd-main}
For every $d \geq 1$ and every finite set $A\subset \R^m$ with exponential Freiman dimension $d$, we have
\begin{equation} \label{eq:main-tight}
  |A+A|\geq C_d(|A|-2^d)+3^d,
\end{equation}
where
\[
  C_d \;=\;
  \begin{cases}
    2\bigl(\tfrac32\bigr)^{d-1}, & d \le 5,\\[4pt]
    2^{d/2+1} - 1, & d \ge 6 \text{ even},\\[4pt]
    3\cdot 2^{(d-1)/2} - \tfrac32, & d \ge 7 \text{ odd}.
  \end{cases}
\]
\end{theorem}

A central theme in additive combinatorics suggests that in any abelian group, finite sets $A$ with small doubling
\[
 \sigma(A) \;:=\; \frac{|A+A|}{|A|}
\]
must exhibit strong additive structure. Here, $A+ A = \{ a+ a' : a,a' \in A\}$ denotes the sumset of $A$. Classical results of Freiman and their later refinements show that such sets are efficiently contained in
generalized arithmetic progressions or subspaces with bounded dimension; see, for example, Freiman's monograph
\cite{FreimanBook} and the book of Tao and Vu~\cite{TaoVu}.

When the abelian group above has no torsion, these questions are often related to studying the following problem: given a finite set $A \subset \mathbb{R}^m$, what type of geometric conditions must $A$ satisfy so as to ensure that the doubling $\sigma(A)$ grows rapidly, see \cite{Bilu, Chang}.
A basic instance of this is \emph{Freiman's lemma}, which states that if a finite set \(A\subset\mathbb{R}^m\) contains the vertices of a non-degenerate \(d\)-simplex, then
\[
 |A+A| \geq (d+1)|A| - \tfrac12 d(d+1) \geq d|A|/2.
\]
A non-degenerate \(d\)-simplex is a set $P \;=\; \{v_0,\,v_0+v_1,\dots,v_0+v_d\}$
with \(v_1,\dots,v_d\) linearly independent. So the mere presence of a simplex of dimension \(d\) forces the additive doubling $\sigma(A)$ to grow linearly in $d$.

This can be compared with the continuous setting where given a non-empty, compact set $\mathcal{A} \subset \mathbb{R}^d$, one can apply the Brunn--Minkowski inequality to deduce that $\mu(\mathcal{A} + \mathcal{A}) \geq 2^d \mu(\mathcal{A})$, with $\mu$ denoting the Lebesgue measure in $\mathbb{R}^d$. Thus it is natural to ask what local conditions can be prescribed for a finite set $A \subseteq \mathbb{R}^d$ to ensure that $\sigma(A)$ grows exponentially in $d$.

In this direction, Green and Tao \cite{GTcompressions} introduced the notion of \emph{exponential Freiman dimension}. Thus, given a finite, non-empty set $A \subset \mathbb{R}^m$, we define its exponential Freiman dimension to be the the largest positive integer $d$ for which $A$ contains the vertices of a non-degenerate $d$-dimensional parallelepiped
\[
 P \;=\; v_0 + \{0,1\} \cdot v_1 + \cdots + \{0,1\} \cdot v_d
\]
with \(v_1,\dots,v_d\) linearly independent. Green and Tao \cite{GTcompressions} proved that if $A \subset \mathbb{R}^m$ has exponential Freiman dimension $d$, then
\begin{equation} \label{sqrt2}
 |A+A| \geq (\sqrt{2})^{d} \,|A|.
 \end{equation}
Such inequalities, while being of independent interest, have also had some exciting recent applications to sum-product theory, see work of P\'{a}lv\"{o}lgyi--Zhelezov ~ \cite{PZ2020}. This circle of ideas has been significantly generalized in \cite{EntropyPFR}, the latter leading to the breakthrough work of Gowers--Green--Manners--Tao \cite{PFR} on the polynomial Freiman--Ruzsa conjecture over $\mathbb{F}_2^n$, and subsequently yielding previously inaccessible sum-product estimates over $\mathbb{R}$.

In \cite{GTcompressions}, Green and Tao noted that the inequality \eqref{sqrt2} is somewhat close to sharp: for example, if $A = \{0,1\}^d\subset\mathbb{R}^d$ one has $|A+A| = 3^d = (3/2)^d|A|$. More specifically, given a positive integer $d$, if we denote by $C_d>0$ the largest constant such that
\begin{equation}\label{eq:Cd-def}
 |A+A| \;\ge\; C_d\,|A| - O_d(1)
\end{equation}
holds for all finite sets $A \subset \mathbb{R}^{m}$ with exponential Freiman dimension $d$, then Green and Tao established that
$$(\sqrt{2})^{d} \leq C_{d} \leq 2(3/2)^{d-1},$$
with the upper bound following from considering the set $A = \{0,1\}^{d-1} \times \{0,1,\dots, N\}$.

Our main result closes the above exponential gap by determining the exact value of \(C_d\) for every $d \geq 1$.

\begin{equation} \label{sqrt2}
  |A+A| \geq (\sqrt{2})^{d} \,|A|.
  \end{equation}

In this direction, Green and Tao \cite{GTcompressions} introduced the notion of \emph{exponential Freiman dimension}. Thus, given a finite, non-empty set $A \subset \mathbb{R}^m$, we define its exponential Freiman dimension to be the the largest positive integer $d$ for which $A$ contains the vertices of a non-degenerate $d$-dimensional parallelepiped
\[
 P \;=\; v_0 + \{0,1\} \cdot v_1 + \cdots + \{0,1\} \cdot v_d
\]
with \(v_1,\dots,v_d\) linearly independent. Green and Tao \cite{GTcompressions} proved that if $A \subset \mathbb{R}^m$ has exponential Freiman dimension $d$, then
\begin{equation} \label{sqrt2}
 |A+A| \geq (\sqrt{2})^{d} \,|A|.
 \end{equation}
Such inequalities, while being of independent interest, have also had some exciting recent applications to sum-product theory, see work of P\'{a}lv\"{o}lgyi--Zhelezov ~ \cite{PZ2020}. This circle of ideas has been significantly generalized in \cite{EntropyPFR}, the latter leading to the breakthrough work of Gowers--Green--Manners--Tao \cite{PFR} on the polynomial Freiman--Ruzsa conjecture over $\mathbb{F}_2^n$, and subsequently yielding previously inaccessible sum-product estimates over $\mathbb{R}$.

In \cite{GTcompressions}, Green and Tao noted that the inequality \eqref{sqrt2} is somewhat close to sharp: for example, if $A = \{0,1\}^d\subset\R^d$ one has $|A+A| = 3^d = (3/2)^d|A|$. More specifically, given a positive integer $d$, if we denote by $C_d>0$ the largest constant such that
\begin{equation}\label{eq:Cd-def}
 |A+A| \;\ge\; C_d\,|A| - O_d(1)
\end{equation}
holds for all finite sets $A \subset \mathbb{R}^{m}$ with exponential Freiman dimension $d$, then Green and Tao established that
$$(\sqrt{2})^{d} \leq C_{d} \leq 2(3/2)^{d-1},$$
with the upper bound following from considering the set $A = \{0,1\}^{d-1} \times \{0,1,\dots, N\}$.

A perhaps surprising aspect of Theorem~\ref{thm:Cd-main} is that the natural upper bound example $A = \{0,1\}^{d-1} \times \{0,1,2,\dots, N\}$ which satisfies $|A+A| = 2 (3/2)^{d-1}|A| + O_d(1)$ is only optimal when $d \leq 5$. When $d >5$, there is a significantly different family of examples that extremize the doubling $\sigma(A)$.
Writing \([0,N] := \{0,1,\dots,N\}\)
for any \(N\in\N\), we describe these examples below.
\begin{align}
 A &= \{0,1\}^{d-1} \times [0,N], \label{eq:example-0.1}\\
 A &= \{0,1\}^d \,\cup\, \bigl(\{0\}^{d/2} \times [0,N]^{d/2}\bigr)
 &&\text{for \(d\) even}, \label{eq:example-0.2}\\
 A &= \{0,1\}^d \,\cup\, \bigl(\{0\}^{(d-1)/2} \times \{0,1\}\times [0,N]^{(d-1)/2}\bigr)
 &&\text{for \(d\) odd}. \label{eq:example-0.3}
\end{align}
Example~\eqref{eq:example-0.1} satisfies $|A| = 2^{d-1}(N+1)$, and $A+A = \{0,1,2\}^{\,d-1}\times[0,2N]$, so
$$|A+A| = 3^{\,d-1}(2N+1)= 2(3/2)^{d-1}(|A|-2^d)+3^d.$$
Example~\eqref{eq:example-0.2} satisfies $|A| = (N+1)^{d/2} + 2^d - 2^{d/2}$ and
\[ A+ A = \{0,1,2\}^d \cup (\{0,1\}^{d/2} \times [0,N+1]^{d/2}) \cup (\{0\}^{d/2} \times [0, 2N]^{d/2}), \]
whence,
\[ \frac{|A+A|}{|A|} = \frac{2^{d/2}(N+2)^{d/2} + (2N+1)^{d/2} - (N+2)^{d/2} + O_d(1)}{(N+1)^{d/2} + O_d(1)} \rightarrow 2 \cdot 2^{d/2}-1 \]
as $N\rightarrow\infty$.
Finally, Example~\eqref{eq:example-0.3} satisfies $|A| = 2(N+1)^{(d-1)/2} + 2^d - 2^{(d+1)/2}$ and
\begin{align*}
 A+A &= \{0,1,2\}^d \cup (\{0,1\}^{(d-1)/2}\times \{0,1,2\}\times [0,N+1]^{(d-1)/2}) \\
 & \quad \cup (\{0\}^{(d-1)/2}\times \{0,1,2\}\times [0,2N]^{(d-1)/2}),
\end{align*}
and so, we get that
\begin{align*}
 \frac{|A+A|}{|A|} &= \frac{3\cdot 2^{(d-1)/2}(N+2)^{(d-1)/2} + 3\cdot (2N+1)^{(d-1)/2} - 3\cdot (N+2)^{(d-1)/2} + O_d(1)}{2(N+1)^{(d-1)/2} + O_d(1)}\\
 &\rightarrow 3\cdot 2^{(d-1)/2} - \tfrac32
\end{align*}
as $N\rightarrow\infty$.

\begin{theorem} \label{lem:mainineq}
 Let $0\leq k<d$ be integers. Let $\{x_u: u\in \{0,1\}^k\}$ and $\{y_u: u\in \{0,1,2\}^k\}$ be sets of non-negative real numbers such that $y_{u+v}\geq x_u+x_v$ for each $u,v\in \{0,1\}^k$. Assume further that $x_u\geq x_v$ if $v\succeq u$. Then we have
 \begin{equation} \label{anineq}
 \sum_{u\in \{0,1,2\}^k} y_u^{d-k}\geq C_d \sum_{u\in \{0,1\}^k} x_u^{d-k},
 \end{equation}
 where $C_d$ is as in Theorem \ref{thm:Cd-main}, namely
 \[
 C_d \;=\;
 \begin{cases}
 2\bigl(\tfrac32\bigr)^{d-1}, & d \le 5,\\[4pt]
 2^{d/2+1} - 1, & d \ge 6 \text{ even},\\[4pt]
 3\cdot 2^{(d-1)/2} - \tfrac32, & d \ge 7 \text{ odd}.
 \end{cases}
\]
\end{theorem}

\begin{definition}[$i$-compression]
Let $A \subset \R^{d}$ be finite and fix $1 \le i \le d$. For each $\mathbf{u}\in\R^{d}$, set
\[
m(\mathbf{u}) := |A \cap H_{\mathbf{u}}|.
\]
We define the \emph{$i$-compression} $C_i(A)$ of $A$ by
\[
C_i(A)
 := \Bigl\{
(u_1,\dots,u_{i-1},t,u_{i+1},\dots,u_d) :
t\in \N_0, 0 \le t < m(\mathbf{u}) \text{ for some } \mathbf{u}
\Bigr\},
\]
that is, in each fiber $H_{\mathbf{u}}$, we replace $A \cap H_{\mathbf{u}}$ by the initial segment
$\{0,1,\dots,m(\mathbf{u})-1\}$.
\end{definition}

We begin by recording some useful notations. First, for vectors $u = (u_1, \dots, u_d)$ and $v = (v_1, \dots, v_d)$ in $\{0,1\}^d$, we define the Hamming distance $d(u,v)$ to be
\[ d(u,v) = \sum_{1 \leq i \leq d: \ u_i \neq v_i} 1 = \sum_{1 \leq i \leq d} |u_i - v_i|. \]
Moreover, we define
\[ |u| = d(u,0) = u_1 + \dots + u_d. \]
Next, recall that
\[C_d \;=\;
 \begin{cases}
 2\bigl(\tfrac32\bigr)^{d-1}, & d \le 5,\\[4pt]
 2^{d/2+1} - 1, & d \ge 6 \text{ even},\\[4pt]
 3\cdot 2^{(d-1)/2} - \tfrac32, & d \ge 7 \text{ odd}.
 \end{cases}
\]

Just as before, we will redistribute some of the weight from $D_{k-m-2}$ to $D_{k-m-1}$ and $D_{k-m}$. By \nameref{lem:redistribution} twice, we have
\[X_{k-m-2} \geq \frac{k-m-1}{m+2} X_{k-m-1}\geq \frac{(k-m-1)(k-m)}{(m+2)(m+1)} X_{k-m}.\]
Thus we will be done if we can show that
\[\frac{(m+2)(m+1)}{(k-m-1)(k-m)}(C_d-D_{k-m})+\frac{m+1}{k-m}(C_d-D_{k-m-1})\leq D_{k-m-2} - C_d,\]
or equivalently,
\begin{equation} \label{eq:doddweight}
\frac{(m+2)(m+1)}{(k-m-1)(k-m)}\left(2^{k-m}-\frac{3}{2}\right)+\frac{m+1}{k-m}\left(2^{k-m-1}-\frac{3}{2}\right)\leq 3\cdot 2^{m-1}-2^{k-m-2} + \frac{3}{2}.
\end{equation}
Writing
\begin{align*}
 f_1(k,m) &= (m+2)(m+1)\left(2^{k-m}-\frac{3}{2}\right)+(m+1)(k-m-1)\left(2^{k-m-1}-\frac{3}{2}\right)\\
 &\quad - (k-m-1)(k-m)\left(3\cdot 2^{m-1}-2^{k-m-2}+\frac{3}{2}\right)
\end{align*}
for all $k,m \in \mathbb{Z}$, we see that \eqref{eq:doddweight} follows for $k,m$ if $f_1(k,m)\leq 0$.

The permutohedron $P_{d}$ lives inside the hyperplane $\left\{x \in \mathbb{R}^{d}:\ x_1+\ldots+x_d = \binom{d+1}{2}\right\}$, and has a number of remarkable properties (see for example \cite{Postnikov} or \cite{Stanley}).
For instance, when $d$ is even, the vertex set of $P_d$ contains $d/2$-dimensional affine cubes. Indeed, consider the $d/2$ disjoint adjacent transpositions
\[
t_j = (2j-1\;\;2j) \in S_{d}, \qquad j = 1,\dots,d/2.
\]
These transpositions commute and each of them has order two. Thereby, for each
$\varepsilon = (\varepsilon_1,\dots,\varepsilon_{d/2})\in\{0,1\}^{d/2}$, we can define the permutation
\[
\sigma_\varepsilon := t_1^{\varepsilon_1} \circ t_2^{\varepsilon_2} \cdots \circ t_{d/2}^{\varepsilon_{d/2}} \in S_{d}.
\]
Then the $2^{d/2}$ vertices of $P_{d}$ defined by
\[
\Bigl\{\,\bigl(\sigma_\varepsilon(1),\dots,\sigma_\varepsilon(d)\bigr)
 : \varepsilon\in\{0,1\}^{d/2} \Bigr\}
\]
form an affine copy of the ${d/2}$-dimensional cube $\{0,1\}^{d/2}$. Consequently, if a finite set \(A \subset \mathbb{Z}^d\) contains the vertex set of the permutohedron, then \(A\) automatically contains
an affine copy of \(\{0,1\}^{\lfloor d/2 \rfloor}\).
Our Theorem \ref{thm:Cd-main} therefore implies
\begin{equation} \label{cubeinP}
|A+A|\geq C_{d/2}(|A|-2^{d/2})+3^{d/2}.
\end{equation}

A perhaps surprising aspect of Theorem~\ref{thm:Cd-main} is  that the natural upper bound example $A = \{0,1\}^{d-1} \times \{0,1,2,\dots, N\}$ which satisfies $|A+A| = 2 (3/2)^{d-1}|A| + O_d(1)$ is only optimal when $d \leq 5$. When $d >5$, there is a significantly different family of examples that extremize the doubling $\sigma(A)$.
Writing \([0,N] := \{0,1,\dots,N\}\)
for any \(N\in\mathbb{N}\), we describe these examples below.
\begin{align}
  A &= \{0,1\}^{d-1} \times [0,N], \label{eq:example-0.1}\\
  A &= \{0,1\}^d \,\cup\, \bigl(\{0\}^{d/2} \times [0,N]^{d/2}\bigr)
      &&\text{for \(d\) even}, \label{eq:example-0.2}\\
  A &= \{0,1\}^d \,\cup\, \bigl(\{0\}^{(d-1)/2} \times \{0,1\}\times [0,N]^{(d-1)/2}\bigr)
      &&\text{for \(d\) odd}. \label{eq:example-0.3}
\end{align}
Example~\eqref{eq:example-0.1} satisfies $|A| = 2^{d-1}(N+1)$, and $A+A = \{0,1,2\}^{\,d-1}\times[0,2N]$, so
$$|A+A| = 3^{\,d-1}(2N+1)= 2(3/2)^{d-1}(|A|-2^d)+3^d.$$
Example~\eqref{eq:example-0.2} satisfies $|A| = (N+1)^{d/2} + 2^d - 2^{d/2}$ and
\[ A+ A = \{0,1,2\}^d \cup (\{0,1\}^{d/2} \times [0,N+1]^{d/2}) \cup (\{0\}^{d/2} \times [0, 2N]^{d/2}),  \]
whence,
\[ \frac{|A+A|}{|A|} = \frac{2^{d/2}(N+2)^{d/2} + (2N+1)^{d/2} - (N+2)^{d/2} + O_d(1)}{(N+1)^{d/2} + O_d(1)} \rightarrow 2 \cdot 2^{d/2}-1 \]
as $N\rightarrow\infty$.
Finally, Example~\eqref{eq:example-0.3} satisfies  $|A| = 2(N+1)^{(d-1)/2} + 2^d - 2^{(d+1)/2}$ and
\begin{align*}
    A+A &= \{0,1,2\}^d \cup (\{0,1\}^{(d-1)/2}\times \{0,1,2\}\times [0,N+1]^{(d-1)/2}) \\
    & \quad \cup (\{0\}^{(d-1)/2}\times \{0,1,2\}\times [0,2N]^{(d-1)/2}),
\end{align*}
and so, we get that
\begin{align*}
    \frac{|A+A|}{|A|} &= \frac{3\cdot 2^{(d-1)/2}(N+2)^{(d-1)/2} + 3\cdot (2N+1)^{(d-1)/2} - 3\cdot (N+2)^{(d-1)/2} + O_d(1)}{2(N+1)^{(d-1)/2} + O_d(1)}\\
    &\rightarrow 3\cdot 2^{(d-1)/2} - \tfrac32
\end{align*}
as $N\rightarrow\infty$.

Taking the minimum of these three upper bounds for each \(d\) recovers exactly
the values in Theorem~\ref{thm:Cd-main}. Furthermore, \eqref{eq:main-tight} is exactly tight for $d\leq 5$. Interestingly, for sufficiently large $d$, since $C_d<(3/2)^d$, Theorem \ref{thm:Cd-main} yields the bound
\[ |A+A|\geq C_d|A|+(3^d-2^dC_d), \]
wherein the error term is actually positive.

The main difficulty will be to show that $C_{d}$ is at least the indicated values above. Before we move on to the proof of Theorem \ref{thm:Cd-main} in \S2, let us start by discussing some of the high-level ideas behind it.

\medskip

{\bf{Proof ideas}}. By applying a suitable linear transformation and translation we may assume
that $P=\left\{0,1\right\}^{d} \subseteq A \subset \mathbb{R}^{d}$. By further applying translations
and coordinate-wise compressions, we may also assume that $A$ is a \emph{down-set} in $\N_0^d$, that is, for any \(u\in A\) and any $v \in \mathbb{Z}^d$ satisfying the inequality \(0\le v\le u\) coordinate-wise, we have \(v\in A\).  In particular, the
assumption \(\{0,1\}^d \subseteq A\) says that the ``bottom layer'' of \(A\) is a
complete discrete hypercube.  We then partition \(A\) according to which
coordinates lie below or above a fixed threshold, and obtain a parallel
partition of the sumset \(B := A+A\).  We will then employ a discrete Brunn--Minkowski type inequality, due to Green and Tao \cite{GTcompressions}, to get lower bounds on the sizes of the blocks of \(B\)
in terms of the blocks of \(A\).

After a suitable renormalisation, this reduces Theorem~\ref{thm:Cd-main} to a
purely analytic inequality for families of non-negative real numbers
\(\{x_u\}_{u\in\{0,1\}^k}\) and \(\{y_w\}_{w\in\{0,1,2\}^k}\), indexed by slices
of the discrete (Hamming) hypercube and hypergrid, subject to two constraints:
a collection of inequalities of the form \(y_{u+v}\ge x_u+x_v\) arising from the sumset structure, and a monotonicity condition reflecting the downset
structure. The monotonicity condition is not strictly necessary, but it is convenient to have. In order to state this, we will require one further piece of notation, and so, given vectors $u=(u_1, \dots, u_d)$ and $v= (v_1, \dots, v_d)$ in $\mathbb{R}^d$, we say that $v\succeq u$ if $v_i \geq u_i$ for every $1 \leq i \leq d$.

\begin{theorem} \label{lem:mainineq}
    Let $0\leq k<d$ be integers. Let $\{x_u: u\in \{0,1\}^k\}$ and $\{y_u: u\in \{0,1,2\}^k\}$ be sets of non-negative real numbers such that $y_{u+v}\geq x_u+x_v$ for each $u,v\in \{0,1\}^k$. Assume further that $x_u\geq x_v$ if $v\succeq u$. Then we have
    \begin{equation} \label{anineq}
        \sum_{u\in \{0,1,2\}^k} y_u^{d-k}\geq C_d \sum_{u\in \{0,1\}^k} x_u^{d-k},
    \end{equation}
    where $C_d$ is as in Theorem \ref{thm:Cd-main}, namely
    \[
  C_d \;=\;
  \begin{cases}
    2\bigl(\tfrac32\bigr)^{d-1}, & d \le 5,\\[4pt]
    2^{d/2+1} - 1, & d \ge 6 \text{ even},\\[4pt]
    3\cdot 2^{(d-1)/2} - \tfrac32, & d \ge 7 \text{ odd}.
  \end{cases}
\]
\end{theorem}

We remark that the constant $C_d$ in Theorem~\ref{thm:Cd-main} is tight for at least one value of $k$ for each $d$. In particular, for $d\leq 5$, tightness occurs at $k=d-1$, with $x_u=1$ for all $u$. For even $d\geq 6$, tightness occurs at $k=d/2$ with $x_0=1$ and $x_u=0$ for all $u\neq 0$. Finally, for odd $d\geq 7$, tightness occurs at $k=(d+1)/2$ with $x_0=x_{e_1}=1$ and $x_u=0$ for all other $u$. These equality cases naturally result in the constructions given in \eqref{eq:example-0.1}, \eqref{eq:example-0.2} and \eqref{eq:example-0.3}.

Theorem \ref{lem:mainineq} is proved by a
combination of averaging arguments, case distinctions in small dimensions, and compression properties that allow us to redistribute weight from parts with excess weight to parts with less weight.  The
dimension-dependent constants that arise along the way match precisely the
values of \(C_d\) asserted in Theorem~\ref{thm:Cd-main}.

\medskip

{\bf{Organization of the paper}}. In \S2 we discuss various prerequisites concerning compressions and prove how  Theorem~\ref{thm:Cd-main} follows from Theorem \ref{lem:mainineq}. We lay some preliminary brickwork for proving our main analytic inequality in \S3, which already turns out to be sufficient to deal with the small-dimensional cases \(d \in \{3,4,5\}\). The case when $d \geq 6$ is even is resolved in \S4, while the case when $d \geq 7$ is odd is analysed in \S5.  The odd case is slightly more involved compared to the even case, since the corresponding doubling constant $C_d$ is larger and the construction \eqref{eq:example-0.3} attaining ``equality'' is more complicated. In \S6, we will discuss a few natural future directions.

\medskip

{\bf{Acknowledgments.}} The authors would like to thank Ben Green for useful discussions. J.L. was supported by ERC Advanced Grant 883810. A.M. was supported by Leverhulme early career fellowship ECF-2025-148. C.P. was supported by NSF grant DMS-2246659. X.S. was supported by NSF grant DMS-2452462.

\medskip

{\bf{Proof ideas}}. By applying a suitable linear transformation and translation we may assume that $P=\left\{0,1\right\}^{d} \subseteq A \subset \mathbb{R}^{d}$. By further applying translations and coordinate-wise compressions, we may also assume that $A$ is a \emph{down-set} in $\N_0^d$. We then partition $A$ according to which coordinates lie below or above a fixed threshold, and obtain a parallel partition of the sumset $B := A+A$. We will then employ a discrete Brunn--Minkowski type inequality, due to Green and Tao \cite{GTcompressions}, to get lower bounds on the sizes of the blocks of $B$ in terms of the blocks of $A$.

After a suitable renormalisation, this reduces Theorem~\ref{thm:Cd-main} to a purely analytic inequality for families of non-negative real numbers $\{x_u\}_{u\in\{0,1\}^k}$ and $\{y_w\}_{w\in\{0,1,2\}^k}$, indexed by slices of the discrete (Hamming) hypercube and hypergrid, subject to two constraints: inequalities of the form $y_{u+v}\ge x_u+x_v$ (from the sumset structure), and a monotonicity condition reflecting the downset structure. This analytic inequality is stated as Theorem~\ref{lem:mainineq}.

Finally, Theorem~\ref{lem:mainineq} is proved by a combination of averaging arguments, case distinctions in small dimensions, and compression properties that allow us to redistribute weight from parts with excess weight to parts with less weight; the resulting dimension-dependent constants match precisely the values of $C_d$ in Theorem~\ref{thm:Cd-main}.

{\bf{Proof ideas}}. By applying a suitable linear transformation and translation we may assume that $P=\left\{0,1\right\}^{d} \subseteq A \subset \mathbb{R}^{d}$. By further applying translations and coordinate-wise compressions, we may also assume that $A$ is a \emph{down-set} in $\N_0^d$. We then partition $A$ according to which coordinates lie below or above a fixed threshold, and obtain a parallel partition of the sumset $B := A+A$. We will then employ a discrete Brunn--Minkowski type inequality, due to Green and Tao \cite{GTcompressions}, to get lower bounds on the sizes of the blocks of $B$ in terms of the blocks of $A$.

After a suitable renormalisation, to prove the main theorem we reduce to a purely analytic inequality for families of non-negative real numbers $\{x_u\}_{u\in\{0,1\}^k}$ and $\{y_w\}_{w\in\{0,1,2\}^k}$, indexed by slices of the discrete (Hamming) hypercube and hypergrid, subject to two constraints: inequalities of the form $y_{u+v}\ge x_u+x_v$ (from the sumset structure), and a monotonicity condition reflecting the downset structure. We first prove the following theorem.

\begin{theorem} \label{lem:mainineq}
    Let $0\leq k<d$ be integers. Let $\{x_u: u\in \{0,1\}^k\}$ and $\{y_u: u\in \{0,1,2\}^k\}$ be sets of non-negative real numbers such that $y_{u+v}\geq x_u+x_v$ for each $u,v\in \{0,1\}^k$. Assume further that $x_u\geq x_v$ if $v\succeq u$. Then we have
    \begin{equation} \label{anineq}
        \sum_{u\in \{0,1,2\}^k} y_u^{d-k}\geq C_d \sum_{u\in \{0,1\}^k} x_u^{d-k},
    \end{equation}
    where $C_d$ is as in Theorem \ref{thm:Cd-main}, namely
    \[
  C_d \;=\;
  \begin{cases}
    2\bigl(\tfrac32\bigr)^{d-1}, & d \le 5,\\[4pt]
    2^{d/2+1} - 1, & d \ge 6 \text{ even},\\[4pt]
    3\cdot 2^{(d-1)/2} - \tfrac32, & d \ge 7 \text{ odd}.
  \end{cases}
\]
\end{theorem}

Finally, the theorem above is proved by a combination of averaging arguments, case distinctions in small dimensions, and compression properties that allow us to redistribute weight from parts with excess weight to parts with less weight; the resulting dimension-dependent constants match precisely the values of $C_d$ in the main theorem.

Question:
Let \(A\subset \mathbb{R}^m\) be a finite set for some \(m\), and write \(A+A=\{a+a':a,a'\in A\}\). The exponential Freiman dimension of \(A\) is the largest positive integer \(d\) such that \(A\) contains the vertices of a non-degenerate \(d\)-dimensional parallelepiped
\[
v_0+\{0,1\}\cdot v_1+\cdots+\{0,1\}\cdot v_d,
\]
where \(v_1,\dots,v_d\) are linearly independent. Define
\[
C_d=
\begin{cases}
2\left(\tfrac32\right)^{d-1}, & d\le 5,\\[4pt]
2^{d/2+1}-1, & d\ge 6 \text{ even},\\[4pt]
3\cdot 2^{(d-1)/2}-\tfrac32, & d\ge 7 \text{ odd}.
\end{cases}
\]
Which quantitative estimate holds for every integer \(d\ge 1\) and every finite set \(A\subset \mathbb{R}^m\) with exponential Freiman dimension \(d\)?

\[
|A+A|\geq C_d\bigl(|A|-2^d\bigr)+3^d.
\]

Choices:
B.
\[
|A+A|\geq C_d\bigl(|A|-2^d\bigr)+2^d.
\]
C.
\[
|A+A|\geq C_d\bigl(|A|-2^d\bigr).
\]
D.
\[
|A+A|\geq C_d\,|A|.
\]
E.
\[
|A+A|\geq C_{d+1}\bigl(|A|-2^d\bigr)+3^d.
\]

additive offset from compression/renormalisation


Two amplitudes in the spectral action matrix model are
\begin{align*}
	\raisebox{-20pt}{
\begin{tikzpicture}[thick]
\draw (0.45,-0.7) to (1,0);
\draw (1.55,-0.7) to (1,0);
\draw (1,0) arc (-90:270:0.5cm);
\node at (1,-0.55) {$i$};
\node at (0.2,0.8) {$j$};
lldraw (1,0) circle (4pt);
lldraw[white] (1,0) circle (2pt)
	;
\end{tikzpicture}
}
\quad=\quad\sum_{k=1}^N
	\raisebox{-20pt}{
\begin{tikzpicture}[thick]
	\draw (0.45,-.7) to (1,0);
	\draw (1.55,-.7) to (1,0);
	\draw (1,0) arc (-90:270:0.5cm);
	\node at (1,-.55) {$i$};
	\node at (0.2,0.8) {$j$};
	lldraw (1,0) circle (4pt);
lldraw[white] (1,0) circle (2pt)
	;
        \node at (1,.575) {$k$};
\end{tikzpicture}
}\quad=\quad\sum_{k=1}^N\frac{f'[\lambda_j,\lambda_k,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_k]},
\end{align*}
and
\begin{align*}
	\raisebox{-20pt}{
\begin{tikzpicture}[thick]
	\draw (0.45,-.7) to (1,0);
	\draw (1.55,-.7) to (1,0);
	\draw (1,0.054) arc (-90:270:0.3cm);
	\draw (1,-0.035) arc (-90:270:.7cm);
	\node at (1,-.55) {$i$};
	\node at (0,1) {$j$};
	lldraw (1,0) circle (4pt);
lldraw[white] (1,0) circle (2pt)
	;
\end{tikzpicture}
}
\quad=\quad\sum_{k,l=1}^N
	\raisebox{-20pt}{
\begin{tikzpicture}[thick]
	\draw (0.45,-.7) to (1,0);
	\draw (1.55,-.7) to (1,0);
	\draw (1,0.054) arc (-90:270:0.3cm);
	\draw (1,-0.035) arc (-90:270:.7cm);
	\node at (1,-.55) {$i$};
	\node at (0,1) {$j$};
	\node at (1,1) {$k$};
	\node at (1,0.4) {$l$};
	lldraw (1,0) circle (4pt);
lldraw[white] (1,0) circle (2pt)
	;
\end{tikzpicture}
}\quad=\quad\sum_{k,l=1}^N\frac{f'[\lambda_j,\lambda_k,\lambda_l,\lambda_k,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_k]f'[\lambda_k,\lambda_l]}.
\end{align*}

\begin{tikzpicture}[thick]
\draw (-1.5,0) to (-1,0);
\draw (-1,0) to[out=90,in=90] (0,0);
\draw (-1,0) to[out=-90,in=-90] (0,0);
\draw (1,0) to[out=90,in=90] (0,0);
\draw (1,0) to[out=-90,in=-90] (0,0);
\draw (1,0) to (1.5,0);
\vertex{-1,0};
\vertex{0,0};
\vertex{1,0};
\node at (-1.65,0) {\footnotesize $1$};
\node at (1.65,0) {\footnotesize$2$};
\end{tikzpicture}$$
does not appear, because its two unbroken faces do not share an edge.
\end{rem}

\section{Power counting when allowing eigenvalues with vanishing derivative}\label{sct:main2}
In our first main theorem, Theorem \ref{thm:main}, we assumed that $f'[\lambda_{k_1},\ldots,\lambda_{k_n}]\neq0$. With $\{\lambda_k\}_{k\geq1}$ in general position, this assumption is typically satisfied. However, there will be at least one~${x\in\R}$ such that $f'(x)=0$, which means that the assumption of nonvanishing divided differences is sensitive to infinitesimal changes in $\{\lambda_k\}_{k\geq1}$. In fact, we shall show that the actual order of divergence of many graphs is sensitive to the question of whether or not $f'$ vanishes on $\{\lambda_k\}_{k\geq1}$.

We will refer to faces with an index $i_0$ such that $f'(\lambda_{i_0})=0$ as \emph{0-faces} or \emph{singular faces}, and similarly for indices and eigenvalues.
As in Section~\ref{sect: lower}, given a subset $\bfr\subseteq\mathcal U$ of the set of unbroken faces of a Feynman ribbon graph~$G$, we let~$G_{\bfr}$ be the graph obtained from $G$ by artificially declaring the faces in $\mathfrak{b}$ to be broken. Elements of $\mathfrak{b}$ are called 0-faces.\begin{thm}\label{thm:main2}
Let $f\in C^{\infty}(\mathbb{R})_\R$ be an even function with $f'$ of precise order $-p-1$ for some $p \in\mathbb{R}_{\ge0}$.
Let~$\{\lambda_k\}_{k=1}^\infty$ be a sequence of real numbers with constants $K,c_1,c_2>0$ such that $c_1k^{1/d}\leq |\lambda_k|\leq c_2k^{1/d}$ for all $k\geq K$. Assume that for all $k,l\in\N_{\geq1}$ we have $f'[\lambda_{k},\lambda_{l}]\neq0$. Then, for any Feynman ribbon graph $G=(G^0,n,G^1)$ whose vertices have valence $\geq3$, for all external indices $i_1,\ldots,i_n\in\N_{\geq1}$, there exist $M,c_4>0$ such that for all~$N\geq M$,
$$|\Ampl_{N,i_1,\ldots,i_n}(G)|\leq c_4 N^{\tilde\omega(G)},$$
$$\tilde \omega(G):=\max_{\bfr\subseteq\mathcal U}\omega_{\bfr}(G_{\bfr}):=\max_{\bfr\subseteq\mathcal U}(U^{\bfr}+\frac{p}{d}(E^{\bfr}_{\textnormal{fi}}-V^{\bfr}_{\textnormal{fi}})+\frac{p+1}{d}(E^{\bfr}_{10}-V^{\bfr}_{10})),$$
where $U^{\bfr}$ is the number of unbroken faces of $G_{\bfr}$, $E^{\bfr}_{\textnormal{fi}}$ is the number of fully internal edges of $G_{\bfr}$ (propagators bordered on both sides by unbroken faces) and $V^{\bfr}_{\textnormal{fi}}$ is the number of fully internal vertices of $G_{\bfr}$ (vertices bordered on all sides by unbroken faces). Respectively, $E^{\bfr}_{10}$ and $V^{\bfr}_{10}$ are the number of edges and vertices of~$G_{\bfr}$ that border exactly one 0-face and for the rest unbroken faces.
\end{thm}
\begin{proof}
Using the precise order of $f'$, there exists $R>0$ such that, for all $x\in\mathbb{R}\setminus[-R,R]$ and all~$0\le k\le \max_{v\in G^{0}}\;\deg(v)$, equation \eqref{eqn: precise_order} is satisfied and the conclusion of Lemma \ref{lem:zero-index bounds} holds. Let~$i_R$ be the lowest index such that $|\lambda_i| > R$ for $i\ge i_R$. For brevity we assume -- without loss of generality -- that $f'(\lambda_0)=0$ and $f'(\lambda_k)\neq0$ for~${k\geq 1}$. As in the proof of Lemma~\ref{lem: restricted_lower_sufficient}, a sum splitting argument yields
\begin{equation*}
    \Ampl_{N,i_1,\ldots,i_n}(G) = \sum_{\bfr\subseteq \mathcal{U}}\sum_{\gamma:\bfr\to\{0,\dotsc,i_R-1\}}\Ampl^{\ge i_R}_{N,{i_1},\dotsc,i_n,\gamma}(G_{\bfr}).
\end{equation*}
We claim that it suffices to show that \begin{align}\label{eqn: ampl_Gbgamma}
|\Ampl^{\ge i_R}_{N,{i_1},\dotsc,i_n,\gamma}(G_{\bfr})|&\lesssim N^{\omega_{\bfr,\gamma}(G_{\bfr})},\\
\omega_{\bfr,\gamma}(G_{\bfr})&:=(U^{\bfr}+\frac{p}{d}(E_{\textnormal{fi}}^b-V_{\textnormal{fi}}^b)+\frac{p+1}{d}(E_{10}^{\bfr,\gamma}-V_{10}^{\bfr,\gamma})),\nonumber
\end{align}
as the following graph-theoretical argument shows that $\omega_{\bfr,\gamma}(G_{\bfr})\le \omega_{\bfr}(G_{\bfr})$ for all such $\gamma$. In particular, if we let $\gamma_0$ denote the map that sends all artificially broken faces to the zero index, then $\omega_{\bfr,\gamma_0}(G_{\bfr})= \omega_{\bfr}(G_{\bfr})$, hence this bound is sharp.

More generally, we have $\omega_{\bfr}(G_{\bfr,\gamma})\leq\omega_{\bfr}(G_{\bfr,\gamma'})$ whenever $\gamma'$ is obtained from $\gamma$ by setting \begin{equation*}
    \gamma'(b)=\begin{cases}
        0,&b=b', \\
        \gamma(b),&\text{else},
    \end{cases}
\end{equation*}
for some $b'$ with $\gamma(b')\neq0$. Indeed, as the values $U^{\bfr}$, $\Efi^b$, and $\Vfi^b$ do not depend on $\gamma$, it suffices to show \begin{equation*}
E_{10}^{\bfr,\gamma}-V_{10}^{\bfr,\gamma}\leq E_{10}^{\bfr,\gamma'}-V_{10}^{\bfr,\gamma'}.
\end{equation*}
Towards this, first note that $E_{10}^{\bfr,\gamma'}-E_{10}^{\bfr,\gamma}$ equals the number of edges that border the face $b'$ on the one side and border an unbroken face (i.e.\ an element of $\mathcal{U}\setminus\bfr$) on the other. Moreover, $V_{10}^{\bfr,\gamma'}-V_{10}^{\bfr,\gamma}$ equals the number of vertices that border $b'$ and for the rest only unbroken faces. For each distinct vertex of the latter kind there is at least one distinct edge of the former kind, implying that $$ V_{10}^{\bfr,\gamma'}-V_{10}^{\bfr,\gamma}\leq E_{10}^{\bfr,\gamma'}-E_{10}^{\bfr,\gamma},$$which concludes the argument.

It remains to prove~\eqref{eqn: ampl_Gbgamma}. The strategy is similar to the proof of Theorem \ref{thm:upper}, but with different bounds for the edges and vertices, in which we distinguish three different types of indices and their corresponding eigenvalues.
We distinguish between (i) eigenvalues $\lambda_0$ such that $f'(\lambda_0)=0$, (ii) eigenvalues $\lambda_i$ for which $f'(\lambda_i)\neq0$ and the index $i<R$ is fixed, and (iii) eigenvalues $\lambda_k$ for which $f'(\lambda_k)\neq0$ and the index $k\geq R$ is running. Analogously to the proof of Theorem \ref{thm:upper}, noting Remark \ref{rem:apparent shortcut} and its resolution, the above estimate follows from the bounds on edges and vertices that are summarized and proved in Lemma \ref{lem:final bounds} below, where we stress that all constants that arise are independent of the running indices.
\end{proof}

\begin{lem}\label{lem:final bounds}
Let $n\geq3$. Let $f,\{\lambda_k\}_{k=1}^\infty$ be as in Theorem \ref{thm:main2}. Let $\lambda_0\in\R$ be such that $f'(\lambda_0)=0$. There exist $c_1,c_2>0$ such that we have
\begin{multicols}{2}
\begin{enumerate}
\item $|f'\{\lambda_{k_1},\lambda_{k_2}\}|>c_1|\lambda_{k_1}|^{-p}$,
\item $|f'\{\lambda_{i},\lambda_{k}\}|>c_1$,
\item $|f'\{\lambda_{i_1},\lambda_{i_2}\}|>c_1$,
\item $|f'\{\lambda_0,\lambda_k\}|> c_1|\lambda_k|^{-p-1}$,
\item $|f'\{\lambda_0,\lambda_i\}|>c_1$,
\item $|f'\{\lambda_0,\lambda_0\}|>c_1$,
\item $|f'\{\lambda_{k_1},\ldots,\lambda_{k_n}\}|\leq c_2|\lambda_{k_1}|^{-p}$,
\item $|f'\{\lambda_0,\lambda_{k_2},\ldots,\lambda_{k_n}\}|\leq c_2|\lambda_{k_2}|^{-p-1}$,
\item $|f'\{\lambda_{j_1},\ldots,\lambda_{j_n}\}|\leq c_2$,
ll}]
ll}]
ll}]
\end{enumerate}
\end{multicols}
\noindent
for all $i,i_1,i_2\leq i_R$, $k,k_1,\ldots,k_n\geq i_R$ such that $f'(\lambda_j)\neq0$ for $j\in\{i,i_1,i_2,k,k_1,\ldots,k_n\}$ and such that $|\lambda_{k_1}|\leq\cdots\leq|\lambda_{k_n}|$, and for all $j_1,\ldots,j_n\in\N$.
\end{lem}

\begin{proof}
The estimates 1.\ and 7.\ follow from Theorem~\ref{thm:weighted_divdiff_bound}, 4.\ and 8.\ follow from Lemma~\ref{lem:zero-index bounds}, and 9.\ was shown in Corollary~\ref{cor:wdivdifs are bounded}. Estimates 3., 5., and 6.\ follow from the fact that there is a finite amount of indices $i<i_R$, and for each $i_1,i_2<i_R$ the (weighted) divided difference is nonzero. {Estimate 2.\ follows from Theorem~\ref{thm:no_vanishing_divdifs}.}
\end{proof}
We conclude this section with some remarks and examples illustrating the influence of 0-faces on the behaviour of amplitudes.

\begin{rem}
The following explains the absence of a lower bound in Theorem \ref{thm:main2}. Suppose $i_1,i_2,i_3$ are such that $f'[i_1,i_2,i_3]=0$, $f'(\lambda_{i_1})\neq0$, and $f'(\lambda_{i_3})\neq0$. We then compute the amplitude
\begin{align*}
\raisebox{-19.5pt}{
\begin{tikzpicture}[thick]
\draw (-0.5,0.5) to (0,0);
\draw (-0.5,-.5) to (0,0);
\draw (0,0) to[out=60,in=120] (1,0);
\draw (0,0) to[out=-60,in=-120] (1,0);
\draw (1,0) to (1.5,0);
\node at (0,0.5) {$i_1$};
\node at (-.5,0) {$i_2$};
\node at (0,-.5) {$i_3$};
\vertex{0,0};
\vertex{1,0};
\end{tikzpicture}

\begin{align}\label{eq:power counting formula 2}
\tilde \omega(G)=\max(U^{\bfr}+\frac{p}{d}(E^{\bfr}_{\textnormal{fi}}-V^{\bfr}_{\textnormal{fi}})+\frac{p+1}{d}(E^{\bfr}_{10}-V^{\bfr}_{10})),
\end{align}
in which the maximum is taken over all subsets $\bfr$ of unbroken faces, $U^{\bfr},\Efi^b,\Vfi^b$ are as before when designating the elements of $\bfr$ as broken, and $V^{\bfr}_{10}/E^{\bfr}_{10}$ denotes the number of vertices/edges bordering only unbroken faces except for either exactly one element of $\bfr$ or exactly one external index $i_0$ satisfying~${f'(\lambda_{i_0})=0}$.

\paragraph{Consequences of \eqref{eq:power counting formula}}
Note that $\Efi-\Vfi\geq 0$ for any graph. A remarkable result is that a function~$f$ with \textit{faster} decay results in a \textit{higher} degree of divergence of the graph. This is not obvious from the definition of the Feynman rules, even for simple graphs like in Example 1, and even for concrete functions like $f(x)=x^{-p}$.
Because divided differences of such functions may well be negative, one might \textit{a priori} expect cancellations of terms damping the degree of divergence, but this does not happen.

Another corollary of our power counting formula is the following observation. Among the connected graphs with~$n\geq1$ external edges and $L$ loops, the maximal value of $U$ is $L-1$, and the maximal value of $\Efi-\Vfi$ is $L-1$. The maximal value $\omega(G)=L-1+\frac{p}{d}(L-1)$ is attained by a nonempty set of diagrams, by virtue of our lower bound. These diagrams with maximal divergence are precisely the planar diagrams that cannot be split into two connected components by removing one vertex and all external edges, and moreover have only one unbroken face -- cf.\ Remark \ref{rem:2-point 2-loop}.
\paragraph{Consequences of \eqref{eq:power counting formula 2}}
The situation where divided differences of $f'$ may vanish is more complicated (cf. Section \ref{sct:main2}), but \eqref{eq:power counting formula 2} leads us to conjecture that the diagrams with maximal divergence are of the same planar form as before -- cf.\ Remark \ref{rem:conjectures}. As we discuss in Remark \ref{rem:UV/IR}, the influence of modes $\lambda_i$ with $f'(\lambda_i)=0$ is reminiscent of the UV/IR behavior of scalar field theories on noncommutative spacetime, which underlines the need for a rigorous renormalization analysis of the spectral action beyond the weak field approximation.

\paragraph{Techniques} To prove the above power counting formulas, we introduce several techniques in the asymptotic analysis of divided differences, and prove upper and lower bounds for them, which appear to be novel. A pivotal concept throughout the proof is the weighted divided difference
$$f'\{x_1,\ldots,x_n\}:=(-1)^n x_1\cdots x_n f'[x_1,\ldots,x_n].$$
Its asymptotic behavior is more easily understood, because sending any of its variables to infinity yields another weighted divided difference.
Furthermore, a key result is that weighted divided differences are positive on large enough subsets of $\R^n$ for the functions $f$ we are considering. We thus generalize and give a new proof for Hunter's positivity theorem \cite{Hunter1977}, which is recovered by taking $f(x)=x^{-p}$ ($p\in 2\N$). Moreover, we show that for functions bounded above or below by $x^{-p}$, with similar bounds on their derivatives, the asymptotic behavior of the weighed divided differences away from the origin is determined by the variable with the smallest modulus, which is crucial in the proof of our main theorem.

Our main theorem is proved by combining this positivity and the mentioned upper and lower bounds with an interesting graph-theoretical argument.

\paragraph{Acknowledgements}
We are grateful to Martijn Caspers, Séverin Charbonnier, Harald Grosse, and Walter van Suijlekom for useful discussions. TvN thanks the Erwin Schr\"odinger Institute and the organizers and participants of the April 2023 conference 'Non-commutative Geometry meets Topological Recursion', where the motivation for this paper originated.
TvN was supported by NWO project ‘Noncommutative multi-linear harmonic analysis and higher order spectral shift’,
OCENW.M.22.070. EMH thanks the Max Planck Institute for Mathematics in Bonn for its financial support.

\section{Definitions}
\label{sct:Definitions}

Throughout we fix a positive real number $d\in\R_{>0}$, and a sequence $\{\lambda_k\}_{k=1}^\infty\subseteq\R$ with the property that there exist numbers $K\in\N$, $c_1,c_2\in\R_{>0}$ such that
\begin{align}\label{eq:eigenvalues asymptotics assumption}
    c_1 k^{1/d}\leq|\lambda_k|\leq c_2 k^{1/d}
\end{align}

\begin{align}\label{eq:power counting formula}
\omega(G)= U+\frac{p}{d}(E_{\f}-V_{\f}).
\end{align}

\begin{rem}\label{rem:UV/IR}
Smooth even functions $f$ satisfy $f'(0)=0$, and if $\{\lambda_k\}$ is the spectrum of a typical Dirac operator, these eigenvalues correspond to modes of zero momentum. The fact that UV-divergences can become higher as external momenta vanish is reminiscent of noncommutative quantum field theory, cf.\ \cite{MRS2000}. For instance, in the naive version of noncommutative $\phi^4_4$, such behavior was shown to lead to UV/IR mixing and, consequently, nonrenormalizability \cite{CR2000,CR2001,GW2005a}. This prompted the Grosse--Wulkenhaar model which solved the UV/IR problem \cite{GW2005b} and proved incredibly successful \cite{DGMR2007,GW2014}.

In light of this, it seems instructive to determine whether UV/IR-mixing is present in the spectral action matrix model and its relatives. This is a difficult question to answer,
requiring a careful renormalization analysis and the passage to continuous spectrum.
\end{rem}

\label{sect: intro}
Motivated by the quest for a QFT description of the spectral action~\cite{ChamseddineConnes1997} we would like to analyze the large $N$ behavior of the correlation functions
\begin{align}\label{eq:path integral}
\frac{\int_{H_N}V_{i_1j_1}\cdots V_{i_mj_m}e^{-\hbar^{-1}\Tr(f(D+V)-f(D))}d V}{\int_{H_N}e^{-\hbar^{-1}\Tr(f(D+V)-f(D))}d V},
\end{align}
where $\hbar$ is a formal parameter, $f:\mathbb{R}\to\mathbb{R}$ is a sufficiently regular function,
$D=\operatorname{diag}(\lambda_k)_{k=1}^\infty$ is a self-adjoint operator diagonalized with respect to some countable orthonormal basis, and $H_N$ is the space of hermitian matrices acting on the first $N$ basis elements\footnote{$H_N$ is equipped with the canonical measure coming from the real and imaginary parts of the components of $V$. At this point the integral is only formal.}.

Following \cite{vNvS21,vNvS22b,vNvS23}, we write
$$\Tr(f(D+V)-f(D))=\sum_{n=1}^\infty\sum_{i_1,\ldots,i_n=1}^N \frac{1}{n}f'[\lambda_{i_1},\ldots,\lambda_{i_n}]V_{i_1i_2}\cdots V_{i_n i_1},$$ in terms of the divided differences of the derivative of $f$, defined inductively
by $f'[x]:=f'(x)$, $f'[x,y]:=\frac{f'(x)-f'(y)}{x-y}$, $f'[x,y,z]:=\frac{f'[x,z]-f'[y,z]}{x-y}=\frac{f'(x)-f'(z)}{(x-y)(x-z)}-\frac{f'(y)-f'(z)}{(x-y)(y-z)}$, \textit{et cetera}. By designating the second order term $\frac12\sum_{k,l=1}^Nf'[\lambda_k,\lambda_l]|V_{kl}|^2$ as the free theory, we can apply standard methods of Gaussian integration in order to express \eqref{eq:path integral} as a combinatorial series whose terms are ribbon graph amplitudes \cite{Eynard2016,vNvS23}. The respective ribbon graphs are generalized Kontsevich graphs in the terminology of~\cite{BCEG}. Although \cite{BCEG} considers polynomials $f$ (the order of which limits the valency of the vertices), the corresponding amplitudes turn out to be formally the same as in \cite{vNvS22b}: these amplitudes are certain fractions of divided differences of $f'$.

A stunning feature of these graph amplitudes, noted independently by \cite{BCEG,vNvS22b}, is the Ward--Takahashi identity, which \cite{vNvS22b} showed to imply one-loop renormalizability of the spectral action matrix model in the Gomis--Weinberg sense, raising the question what algebraic relations govern the graphs at arbitrary loop order, and if their renormalization flow can be understood. These questions, besides being inherently interesting \cite{Azarfar2024,BG2016,HKPV2022,Perez2022,Perez2025,Steinacker2010,tHooft1982}, prepare for replacing the integration space~$H_N$ in \eqref{eq:path integral} by a space of fields incorporating physical content such as the spectral Standard Model \cite{ChamseddineConnes1997,CC2012,CIS2020}, and beyond \cite{AMST2016,CCM2015,CCS2013,DM2017,Suijlekom2015}. Moreover, their positive answer would bring the spectral action more in line with noncommutative quantum field theory \cite{GW2005a,GW2005b,GW2014,Riv2007}.

\paragraph{Feynman rules}
We summarize the Feynman rules referred to above --
precise definitions are in Section \ref{sct:Definitions}. The Feynman diagrams are ribbon graphs for which not the edges but the faces are labeled by indices.
For each vertex bordered by faces with indices $i_1,\ldots,i_n$, we multiply by a factor $f'[\lambda_{i_1},\ldots,\lambda_{i_n}]$, e.g.,
\begin{align*}
\raisebox{-18pt}{
\begin{tikzpicture}[thick]
 \draw (-0.5,-0.5) to (0,0);
 \draw (-0.5,0.5) to (0,0);
 \draw (0.5,0.5) to (0,0);
 \draw (0.5,-0.5) to (0,0);
 \node at (-0.5,0) {$i$};
 \node at (0,0.5) {$j$};
 \node at (0.5,0) {$k$};
 \node at (0,-0.5) {$l$};
 lldraw (0,0) circle (4pt);
lldraw[white] (0,0) circle (2pt)
 ;
\end{tikzpicture}}
\,\quad&=\quad f'[\lambda_i,\lambda_j,\lambda_k,\lambda_l],\\
\intertext{for each internal edge bordered by $i$ and $j$, we divide by a factor $f'[\lambda_i,\lambda_j]$, i.e.,}
\raisebox{-14pt}{
\begin{tikzpicture}[thick]
 \draw (-0.5,-0.5) to (0.5,0.5);
 \node at (-0.25,0.25) {$i$};
 \node at (0.25,-0.25) {$j$};
\end{tikzpicture}}
\,\quad&=\quad\frac{1}{f'[\lambda_i,\lambda_j]},\\
\intertext{and, finally, we sum over each unbroken face (face without external edges), e.g.,}
\raisebox{-18pt}{
\begin{tikzpicture}[thick]
\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);
\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);
\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);
\draw (-0.5,0) to (-0.8,0);
\draw (0.5,0.5) to (0.7,0.7);
\draw (0.5,-0.5) to (0.7,-0.7);
lldraw (-0.5,0) circle (4pt);
lldraw[white] (-0.5,0) circle (2pt)
 ;
lldraw (0.5,0.5) circle (4pt);
lldraw[white] (0.5,0.5) circle (2pt)
 ;
lldraw (0.5,-0.5) circle (4pt);
lldraw[white] (0.5,-0.5) circle (2pt)
 ;
\end{tikzpicture}}
\,\quad&=\quad\sum_{k=1}^N
\raisebox{-18pt}{
\begin{tikzpicture}[thick]
\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);
\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);
\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);
\draw (-0.5,0) to (-0.8,0);
\draw (0.5,0.5) to (0.7,0.7);
\draw (0.5,-0.5) to (0.7,-0.7);
\node at (0.15,0) {$k$};
lldraw (-0.5,0) circle (4pt);
lldraw[white] (-0.5,0) circle (2pt)
 ;
lldraw (0.5,0.5) circle (4pt);
lldraw[white] (0.5,0.5) circle (2pt)
 ;
lldraw (0.5,-0.5) circle (4pt);
lldraw[white] (0.5,-0.5) circle (2pt)
 ;
\end{tikzpicture}}\,\,.
\end{align*}

For renormalization purposes, it is relevant \cite{CR2000,CR2001,GW2005a,GW2005b,ILV2012,KLV2014,LOR2015,Riv2007,RVW} to know the asymptotic behavior of these amplitudes as $N\to\infty$. The main objective of this paper is to prove the following formulas describing this asymptotic behavior.

\begin{rem}\label{rem:apparent shortcut}
We explain here a problem with an apparent shortcut to the above proof. Indeed, given positive~${n_1,\ldots,n_U}$ it is not hard to show (using $\lambda_j\sim j^{1/d}$) that
\begin{align}\label{eq:sums like integrals}
 \sum_{j_1=1}^N\cdots\sum_{j_U=1}^N \lambda_{j_1}^{n_1}\cdots\lambda_{j_U}^{n_U}=\O(N^{U+(n_1+\ldots+n_U)/d}),
\end{align}
exactly as in the case with integrals instead of sums. If one skips the graph-theoretical Lemma \ref{lem:injection fi vertices to fi edges} and applies the estimates of Theorem \ref{thm:weighted_divdiff_bound} for arbitrary bordering indices of the fully internal vertices and edges, one can estimate the amplitude by the sum on the left-hand side of \eqref{eq:sums like integrals} where indeed the powers automatically add up to $n_1+\ldots+n_U=p(\Efi-\Vfi)$ as required! However, the $n_1,\ldots,n_U$ may not be positive, which invalidates \eqref{eq:sums like integrals}. The following example shows that such an estimate of the amplitude by \eqref{eq:sums like integrals} really is too coarse in general. We surely have
\begin{align}\label{eq:example diagram}
\raisebox{-20pt}{
\begin{tikzpicture}[thick]
 \draw (0.45,-.7) to (1,0);
 \draw (1.55,-.7) to (1,0);
 \draw (1,0.054) arc (-90:270:0.3cm);
 \draw (1,-0.035) arc (-90:270:.7cm);
 \node at (1,-.55) {$i_1$};
 \node at (0,1) {$i_2$};
 \vertex{1,0};
\end{tikzpicture}
}
=\lambda_{i_1}^{-1}\lambda_{i_2}^{-1}\sum_{k=1}^N\sum_{l=1}^N\frac{f'\{i_1,k,l,k,i_1,i_2\}}{f'\{i_1,k\}f'\{k,l\}}\lesssim
\sum_{k=1}^N\sum_{l=1}^N\frac{l^{-p/d}}{k^{-p/d}k^{-p/d}}=\sum_{k=1}^N k^{2p/d}\sum_{l=1}^N l^{-p/d}.
\end{align}
Naively adding up the orders gives the correct result, $\O(N^{2+\frac{p}{d}})$. But because $-p/d$ is negative, we cannot add up the powers: Assuming $p/d>1$, the sequence $(\sum_{l=1}^N l^{-p/d})_{N\in\N}$ is convergent with nonzero limit. Hence, $\sum_{l=1}^N l^{-p/d}$ is of order $N^0$, not of order $N^{1-p/d}$. The right-hand side of \eqref{eq:example diagram} is therefore $\O(N^{1+2p/d})$, and as we now know we have a better bound for the left-hand side. The trick is simply to choose the indices so that each vertex contribution is canceled by an edge contribution.
\end{rem}

\begin{rem}
The following explains the absence of a lower bound in Theorem \ref{thm:main2}. Suppose $i_1,i_2,i_3$ are such that $f'[i_1,i_2,i_3]=0$, $f'(\lambda_{i_1})\neq0$, and $f'(\lambda_{i_3})\neq0$. We then compute the amplitude
\begin{align*}
\raisebox{-19.5pt}{
\begin{tikzpicture}[thick]
\draw (-0.5,0.5) to (0,0);
\draw (-0.5,-.5) to (0,0);
\draw (0,0) to[out=60,in=120] (1,0);
\draw (0,0) to[out=-60,in=-120] (1,0);
\draw (1,0) to (1.5,0);
\node at (0,0.5) {$i_1$};
\node at (-.5,0) {$i_2$};
\node at (0,-.5) {$i_3$};
\vertex{0,0};
\vertex{1,0};
\end{tikzpicture}
}
=-\lambda_{i_1}^{-1}\lambda_{i_2}^{-1}\lambda_{i_3}^{-1}\sum_{k=1}^N\frac{f'\{i_1,i_2,i_3,k\}f'\{i_1,i_3,k\}}{f'\{i_1,k\}f'\{i_3,k\}}.
\end{align*}
Contrary to the situation where $f'[i_1,i_2,i_3]\neq0$, the factor $|f'\{i_1,i_2,i_3,k\}|$ obtained from the 4-vertex is proportional to $|\lambda_k|^{-1}$ as $k\to\infty$.
The other factors are proportional to $1$ as usual. We obtain
\begin{align*}
\raisebox{-19.5pt}{
\begin{tikzpicture}[thick]
\draw (-0.5,0.5) to (0,0);
\draw (-0.5,-.5) to (0,0);
\draw (0,0) to[out=60,in=120] (1,0);
\draw (0,0) to[out=-60,in=-120] (1,0);
\draw (1,0) to (1.5,0);
\node at (0,0.5) {$i_1$};
\node at (-.5,0) {$i_2$};
\node at (0,-.5) {$i_3$};
\vertex{0,0};
\vertex{1,0};
\end{tikzpicture}}
\sim\sum_{k=1}^N|\lambda_k|^{-1}\sim\sum_{k=1}^Nk^{-\frac{1}{d}}<\O(N).
\end{align*}
The order depends on $d$ but (since $d\geq0$) at least it is smaller than $\O(N)$ in the sense that there exists no $c$ such that the lower bound is $\geq cN$. For $d<1$ the graph is in fact finite.
\end{rem}

\begin{rem}
Even though there exist only finitely many eigenmodes $\lambda_k$ with $f'(\lambda_k)=0$, these singular modes can boost the order of divergence not only when occurring as external indices. For instance, assuming non-singular external indices, we have the divergences
\begin{align*}
\raisebox{-20pt}{
\begin{tikzpicture}[thick]
\draw (-1,0) to (1,0);
\draw (-.7,.7) arc (180:0:.7cm);
\draw (0,0) to[out=160,in=-90] (-.7,.7);
\draw (0,0) to[out=20,in=-90] (.7,.7);
\draw (-.3,.6) arc (180:0:.3cm);
\draw (0,0) to[out=110,in=-90] (-.3,.6);
\draw (0,0) to[out=70,in=-90] (.3,.6);
\vertex{0,0};
\end{tikzpicture}}
=\O(N^{2+\frac{p}{d}})
\quad\text{and}\quad
\raisebox{-20pt}{
\begin{tikzpicture}[thick]
\draw (-1,0) to (1,0);
\draw (-.7,.7) arc (180:0:.7cm);
\draw (0,0) to[out=160,in=-90] (-.7,.7);
\draw (0,0) to[out=20,in=-90] (.7,.7);
\draw (-.3,.6) arc (180:0:.3cm);
\draw (0,0) to[out=110,in=-90] (-.3,.6);
\draw (0,0) to[out=70,in=-90] (.3,.6);
\vertex{0,0};
\node at (0,.5) {$0$};
\end{tikzpicture}}
=\O(N^{1+\frac{p+1}{d}}).
\end{align*}
The latter is larger than the former precisely if $d<1$. More generally, at second loop order, the boost of UV-divergence by internal singular indices is not apparent for $d\geq1$. Indeed, if $d\geq1$, then the diagrams of Remark \ref{rem:2-point 2-loop} remain precisely those of maximal order, also in the more general setting of Theorem \ref{thm:main2}.
However, for any $d\in\N$, at loop order $L=d+2$ the maximal diagrams become those where one of the faces is artificially broken by a singular index, such as
\begin{align}
\raisebox{-35pt}{
\begin{tikzpicture}[thick]
\draw (-1.5,0) to (-1,0);
\draw (-1,0) arc (-180:180:1cm);
\draw (-1,0) to (0.5,.866);
\draw (0.5,.866) to (0.5,-.866);
\draw (-1,0) to (0.5,-.866);
\draw (0.5,.866) to (.75,1.3);
\draw (0.5,-.866) to (.75,-1.3);
\vertex{-1,0};
\vertex{0.5,.866};
\vertex{0.5,-.866};
\end{tikzpicture}}
~~
=\O(N^{4+3p/d})\quad\text{and}\quad
\raisebox{-35pt}{
\begin{tikzpicture}[thick]
\draw (-1.5,0) to (-1,0);
\draw (-1,0) arc (-180:180:1cm);
\draw (-1,0) to (0.5,.866);
\draw (0.5,.866) to (0.5,-.866);
\draw (-1,0) to (0.5,-.866);
\draw (0.5,.866) to (.75,1.3);
\draw (0.5,-.866) to (.75,-1.3);
\node at (0,0) {$0$};
\vertex{-1,0};
\vertex{0.5,.866};
\vertex{0.5,-.866};
\end{tikzpicture}}
=\O(N^{3+3(p+1)/d}).\label{eq:3-point 4-loop}
\end{align}
The latter is larger than the former precisely if $d<3$. Graphs similar to the examples above yield divergences increased by breaking faces to singular modes for any $d<L-1$.
\end{rem}
\begin{rem}\label{rem:conjectures}
In the setting of Theorem \ref{thm:main2} and for $d<3$, the second graph of \eqref{eq:3-point 4-loop} has maximal order among the 3-point 4-loop diagrams, but not every 3-point 4-loop diagram that has maximal order in the setting of Theorem \ref{thm:main} has maximal order in the setting of Theorem \ref{thm:main2}. That being said, we conjecture that, among the $n$-points $L$-loop graphs, every graph with maximal order in the setting of Theorem \ref{thm:main2} has maximal order in the setting of Theorem \ref{thm:main}. We moreover conjecture that the Ward identity, in the sense of \cite{vNvS22b}, holds in both cases when restricting to the graphs of maximal order of divergence.
This might help generalize the results of \cite{vNvS22b} to higher loop, which is a pressing open problem.
\end{rem}

For renormalization purposes, it is relevant \cite{CR2000,CR2001,GW2005a,GW2005b,ILV2012,KLV2014,LOR2015,Riv2007,RVW} to know the asymptotic behavior of these amplitudes as $N\to\infty$. The main objective of this paper is to prove the following formulas describing this asymptotic behavior.

\paragraph{Main results}
Suppose that $\{\lambda_k\}_{k=1}^\infty\subseteq\mathbb{R}$ is a sequence of asymptotic behavior $\lambda_k\asymp k^{1/d}$ for $d\in\R_{>0}$. Suppose that $f$ is smooth, even, and satisfies, for $x\to\infty$, $f^{(n)}(x)\asymp (-1)^nx^{-p-n}$ ($n\in\mathbb{N}$) for some~$p>0$. Suppose moreover that the divided differences of $f$ do not vanish on $\{\lambda_k\}_{k=1}^\infty$. For a graph $G$ with $U$ unbroken faces, $E_{\textnormal{fi}}$ edges which do not border a broken face, and $V_{\textnormal{fi}}$ vertices which do not border a broken face, the amplitude of $G$ is bounded from above and below by a constant times $N^{\omega(G)}$, where
\begin{align}\label{eq:power counting formula}
\omega(G)= U+\frac{p}{d}(E_{\textnormal{fi}}-V_{\textnormal{fi}}).
\end{align}

However, when the assumption of vanishing divided differences is not satisfied, the divergences become (for certain graphs \textit{strictly}) larger. In this case, the amplitude is bounded from above by $N^{\tilde\omega(G)}$, where
\begin{align}\label{eq:power counting formula 2}
\tilde \omega(G)=\max(U^{\mathfrak{b}}+\frac{p}{d}(E^{\mathfrak{b}}_{\textnormal{fi}}-V^{\mathfrak{b}}_{\textnormal{fi}})+\frac{p+1}{d}(E^{\mathfrak{b}}_{10}-V^{\mathfrak{b}}_{10})),
\end{align}
in which the maximum is taken over all subsets $\mathfrak{b}$ of unbroken faces, $U^{\mathfrak{b}},E_\textnormal{fi}^b,V_\textnormal{fi}^b$ are as before when designating the elements of $\mathfrak{b}$ as broken, and $V^{\mathfrak{b}}_{10}/E^{\mathfrak{b}}_{10}$ denotes the number of vertices/edges bordering only unbroken faces except for either exactly one element of $\mathfrak{b}$ or exactly one external index $i_0$ satisfying~${f'(\lambda_{i_0})=0}$.

\paragraph{Consequences of \eqref{eq:power counting formula}}
Note that $E_\textnormal{fi}-V_\textnormal{fi}\geq 0$ for any graph. A remarkable result is that a function~$f$ with \textit{faster} decay results in a \textit{higher} degree of divergence of the graph. This is not obvious from the definition of the Feynman rules, even for simple graphs like in Example 1, and even for concrete functions like $f(x)=x^{-p}$.
Because divided differences of such functions may well be negative, one might \textit{a priori} expect cancellations of terms damping the degree of divergence, but this does not happen.

Another corollary of our power counting formula is the following observation. Among the connected graphs with~$n\geq1$ external edges and $L$ loops, the maximal value of $U$ is $L-1$, and the maximal value of $E_\textnormal{fi}-V_\textnormal{fi}$ is $L-1$. The maximal value $\omega(G)=L-1+\frac{p}{d}(L-1)$ is attained by a nonempty set of diagrams, by virtue of our lower bound. These diagrams with maximal divergence are precisely the planar diagrams that cannot be split into two connected components by removing one vertex and all external edges, and moreover have only one unbroken face -- cf.\ Remark \ref{rem:2-point 2-loop}.
\paragraph{Consequences of \eqref{eq:power counting formula 2}}
The situation where divided differences of $f'$ may vanish is more complicated (cf. Section \ref{sct:main2}), but \eqref{eq:power counting formula 2} leads us to conjecture that the diagrams with maximal divergence are of the same planar form as before -- cf.\ Remark \ref{rem:conjectures}. As we discuss in Remark \ref{rem:UV/IR}, the influence of modes $\lambda_i$ with $f'(\lambda_i)=0$ is reminiscent of the UV/IR behavior of scalar field theories on noncommutative spacetime, which underlines the need for a rigorous renormalization analysis of the spectral action beyond the weak field approximation.

\paragraph{Techniques} To prove the above power counting formulas, we introduce several techniques in the asymptotic analysis of divided differences, and prove upper and lower bounds for them, which appear to be novel. A pivotal concept throughout the proof is the weighted divided difference
$$f'\{x_1,\ldots,x_n\}:=(-1)^n x_1\cdots x_n f'[x_1,\ldots,x_n].$$
Its asymptotic behavior is more easily understood, because sending any of its variables to infinity yields another weighted divided difference.
Furthermore, a key result is that weighted divided differences are positive on large enough subsets of $\mathbb{R}^n$ for the functions $f$ we are considering. We thus generalize and give a new proof for Hunter's positivity theorem \cite{Hunter1977}, which is recovered by taking $f(x)=x^{-p}$ ($p\in 2\mathbb{N}$). Moreover, we show that for functions bounded above or below by $x^{-p}$, with similar bounds on their derivatives, the asymptotic behavior of the weighed divided differences away from the origin is determined by the variable with the smallest modulus, which is crucial in the proof of our main theorem.

Our main theorem is proved by combining this positivity and the mentioned upper and lower bounds with an interesting graph-theoretical argument.

\paragraph{Acknowledgements}
We are grateful to Martijn Caspers, Séverin Charbonnier, Harald Grosse, and Walter van Suijlekom for useful discussions. TvN thanks the Erwin Schr\"odinger Institute and the organizers and participants of the April 2023 conference 'Non-commutative Geometry meets Topological Recursion', where the motivation for this paper originated.
TvN was supported by NWO project ‘Noncommutative multi-linear harmonic analysis and higher order spectral shift’,
OCENW.M.22.070. EMH thanks the Max Planck Institute for Mathematics in Bonn for its financial support.

To prove the above power counting formulas, the authors “introduce several techniques in the asymptotic analysis of divided differences, and prove upper and lower bounds for them.” A “pivotal concept throughout the proof is the weighted divided difference
$$f'\{x_1,\ldots,x_n\}:=(-1)^n x_1\cdots x_n f'[x_1,\ldots,x_n],$$
whose asymptotic behavior is “more easily understood, because sending any of its variables to infinity yields another weighted divided difference.”

A “key result” used in the proof is that “weighted divided differences are positive on large enough subsets of $\mathbb{R}^n$ for the functions $f$ we are considering,” which generalizes Hunter’s positivity theorem. Moreover, for functions “bounded above or below by $x^{-p}$, with similar bounds on their derivatives,” they show that “the asymptotic behavior of the weighed divided differences away from the origin is determined by the variable with the smallest modulus,” and this is stated to be “crucial in the proof of our main theorem.”

Finally, “our main theorem is proved by combining this positivity and the mentioned upper and lower bounds with an interesting graph-theoretical argument.”,

Question: Consider the spectral action matrix model determined by a self-adjoint diagonal operator $D=\operatorname{diag}(\lambda_k)_{k=1}^\infty$ and a sufficiently regular function $f:\mathbb R\to\mathbb R$. For ribbon-graph amplitudes, use the Feynman rules from the spectral action model: a vertex bordered by faces with labels $i_1,\dots,i_n$ contributes $f'[\lambda_{i_1},\dots,\lambda_{i_n}]$, an internal edge bordered by face labels $a,b$ contributes $1/f'[\lambda_a,\lambda_b]$, and each unbroken internal face is summed over its label from $1$ to $N$. For the two one-vertex 2-point ribbon graphs with external face labels $j$ and $i$—the first having one internal loop face, and the second having two nested internal loop faces— which statement gives their amplitudes for all choices of the external labels?

Their amplitudes are
\[
\sum_{k=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_k]}
\]
for the graph with one internal loop face, and
\[
\sum_{k,l=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_l,\lambda_k,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_k]f'[\lambda_k,\lambda_l]}
\]
for the graph with two nested internal loop faces.

Choices:
B.
Their amplitudes are
\[
\sum_{k=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_i]}{f'[\lambda_j,\lambda_k]}
\]
for the graph with one internal loop face, and
\[
\sum_{k,l=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_l,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_k]f'[\lambda_k,\lambda_l]}
\]
for the graph with two nested internal loop faces.
C. Their amplitudes are obtained by summing over the internal face labels and taking the product of the vertex factor with the reciprocal edge factors. In particular, the first graph is a single sum over one internal face label and the second graph is a double sum over two internal face labels, with denominators \(f'[\lambda_j,\lambda_k]\) and \(f'[\lambda_j,\lambda_k]f'[\lambda_k,\lambda_l]\), respectively.
D.
Their amplitudes are
\[
\sum_{k=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_i]}
\]
for the graph with one internal loop face, and
\[
\sum_{k,l=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_l,\lambda_k,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_i]f'[\lambda_k,\lambda_l]}
\]
for the graph with two nested internal loop faces.
E.
Their amplitudes are
\[
\sum_{k=1}^N \frac{f'[\lambda_i,\lambda_j,\lambda_k,\lambda_j]}{f'[\lambda_j,\lambda_k]}
\]
for the graph with one internal loop face, and
\[
\sum_{k,l=1}^N \frac{f'[\lambda_i,\lambda_j,\lambda_k,\lambda_l,\lambda_k,\lambda_j]}{f'[\lambda_j,\lambda_k]f'[\lambda_k,\lambda_l]}
\]
for the graph with two nested internal loop faces.

cyclic face-label repetition at the vertex

explicit vertex numerators were dropped while retaining the summation/edge-factor structure

edge denominator depends on adjacent face labels, not the external pair

ordered incidence data of face labels around the vertex


Question: In the spectral action matrix model, let the ribbon-graph amplitude be computed by the rules: a vertex bordered by faces with labels \(i_1,\dots,i_n\) contributes \(f'[\lambda_{i_1},\dots,\lambda_{i_n}]\), an internal edge bordered by faces \(a,b\) contributes \(1/f'[\lambda_a,\lambda_b]\), and each unbroken internal face label is summed from \(1\) to \(N\). Here \(f'[x_1,\dots,x_n]\) denotes the divided difference of \(f'\). What are the explicit amplitudes of the one-vertex 2-point ribbon graphs with external face label \(i\): (1) the graph with one internal loop and outer face label \(j\), and (2) the graph with two nested internal loops and outer face label \(j\)?

They are
\[
\sum_{k=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_k]}
\]
for the one-loop graph, and
\[
\sum_{k,l=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_l,\lambda_k,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_k]f'[\lambda_k,\lambda_l]}
\]
for the graph with two nested loops.

Choices:
B.
They are
\[
\sum_{k=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_i]}
\]
for the one-loop graph, and
\[
\sum_{k,l=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_l,\lambda_k,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_i]f'[\lambda_k,\lambda_l]}
\]
for the graph with two nested loops.
C.
They are obtained by summing a divided-difference numerator over the internal face labels, namely
\[
\sum_{k=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_k]}
\]
for the one-loop graph, and for the graph with two nested loops a double sum over \(k,l\) of a term with numerator
\[
f'[\lambda_j,\lambda_k,\lambda_l,\lambda_k,\lambda_j,\lambda_i],
\]
divided by the product of the edge factors prescribed by the internal edges.
D.
They are
\[
\sum_{k=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_i]}{f'[\lambda_j,\lambda_k]}
\]
for the one-loop graph, and
\[
\sum_{k,l=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_l,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_k]f'[\lambda_k,\lambda_l]}
\]
for the graph with two nested loops.
E.
They are
\[
\sum_{k=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_k]^2}
\]
for the one-loop graph, and
\[
\sum_{k,l=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_l,\lambda_k,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_k]f'[\lambda_j,\lambda_l]}
\]
for the graph with two nested loops.

edge-factor depends on adjacent face labels, not external pair

dropped the fully explicit denominator/product specification while retaining the correct summation and numerator pattern

cyclic repetition of face labels around the vertex

one denominator factor per internal edge with the correct adjacent labels