\begin{theorem}
\label{thm-main}
	 For $k\geq 1$, the Bergman kernel $\rho_k$ of $H^0(X,L^k)$ at each $p\in X$ satisfies
	 \[\rho_k(p)=(\frac{k}{2\pi})^n\left(1+\sum_{v\in \Lambda\setminus \{0\}}e^{-\frac{k}{4}(|v|_{H})^2}\cos\big(2\pi k\alpha_v(p)\big)\right). \]

\end{theorem}

\begin{proposition}\label{prop-surj}
	 $\Phi_\vcal$ is surjective.
\end{proposition}

Let $(M,\omega)$ be a K\"{a}hler  manifold of complex dimension $n$, and let $L\to M$ be a holomorphic line bundle equipped with a Hermitian metric $h$ whose Chern curvature equals $-i\omega$. The Bergman space $\hcal_k$ consists of holomorphic sections of $L^k$ that are square integrable:
\[
\int_M |s|_{h}^2\,\frac{\omega^n}{n!}<\infty.
\]
With the $L^2$--inner product induced by $h$ and $\omega$, $\hcal_k$ is a closed subspace of $L^2(M,L^k)$. The Bergman kernel $K_k(x,y)$ is the integral kernel of the orthogonal projection $L^2(M,L^k)\to\hcal_k$, characterized by
\[
s(y)=\int_M \langle s(x),K_k(x,y)\rangle_{h^k}\,\frac{\omega^n(x)}{n!}\qquad(\forall s\in\hcal_k).
\]
The diagonal density (or Bergman kernel function) is the pointwise trace of $K_k$,
\[
\rho_k(x)=|K_k(x,x)|_{h^k}.
\]

For trivial bundles over domains in $\mathbb C^n$ one uses the analogous definition for holomorphic $L^2$--functions.

The Bergman kernel has been intensively studied for domains in $\mathbb C^n$ and for K\"{a}hler  manifolds (see, e.g., \cite{Fefferman1974,boutet1975singularite,kerzman1978cauchy,donaldson2001,Sun2011Expected,Donaldson2014Gromov,berman2011fekete,Donaldson15, dsz1,dsz2,dsz3,Shiffman2008NV,sz1,bsz0,bsz1}). Except in very symmetric cases (for instance complex space forms), explicit formulas are rare. A fundamental substitute is the Tian-Catlin-Zelditch expansion (\cite{Tian1990On, Zelditch2000Szego, Lu2000On, Catlin}): when $M$ is compact,
\[
\rho_k(x)\sim\Big(\frac{k}{2\pi}\Big)^n\Big(1+\frac{b_1(x)}{k}+\frac{b_2(x)}{k^2}+\cdots\Big),
\]
where each $b_j(x)$ is a universal polynomial in the curvature of the Riemannian metric corresponding to $\omega$ and its covariant derivatives. There are extensions of this expansion to symplectic manifolds and orbifolds (see \cite{MM,dailiuma}).

In \cite{sun-rem-hyp}, the author made a progress by proving an explicit formula for the Bergman kernel function for positive line bundles on hyperbolic Riemann surfaces. The formula involves summation over all geodesic loops based at the point where the Bergman kernel is evaluated.
 In this paper, inspired by this work, we continue to study the Bergman kernel of abelian varieties.

\

Let $X=\mathbb C^n/\Lambda$ be an abelian variety over $\mathbb C$. Here, $\Lambda$ is a lattice in $\mathbb C^n$. Let $L\to X$ be a positive line bundle over $X$, whose first Chern class is determined by a positive definite Hermitian form $H=(H_{ij})$ on $\mathbb C^n$ such that $\Im H(\Lambda,\Lambda)\subset \mathbb Z$. For each vector $v\in \mathbb C^n$, we denote by $|v|_H$ the norm of $v$ defined by $H$.
Let $\omega=\pi \sqrt{-1} \sum H_{ij} dz^i\wedge d\bar{z}^j \in 2\pi c_1(L)$ be a flat K\"ahler metric on $X$. There exist a Hermitian metric $h$ on $L$ with curvature $- i \omega$.

For each $p\in X$, $\exists \tilde{p}\in\mathbb C^n$ such that $p=\pi(\tilde{p})$. Such a $\tilde{p}$ is called a lift of $p$.
For each $v\in \Lambda$, let $\gamma_{p,v}\subset X$ be the smooth geodesic loop defined by $v$ starting at $p$ with orientation defined by $\frac{\partial}{\partial a}$, where $a$ is the parameter of the line segment $\tilde{p}+av$, $a\in[0,1]$. It is easy to see that $\gamma_{p,v}$ does not depend on the choice of $\tilde{p}$, and that the length of $\gamma_{p,v}$ is $|v|_H$. Let $\nabla$ be the Chern connection of $(L,h)$.
 Let $e^{2\pi i\alpha_v(p)}=\text{Hol}_{L}(\gamma_{p,v})$ be the holonomy of $\nabla$ along $\gamma_{p,v}$.
Our first result is the following formula:

Let $(M,\omega)$ be a K\"{a}hler manifold of complex dimension $n$, and let $L\to M$ be a holomorphic line bundle equipped with a Hermitian metric $h$ whose Chern curvature equals $-i\omega$. The Bergman space $\hcal_k$ consists of holomorphic sections of $L^k$ that are square integrable:
\[
\int_M |s|_{h}^2\,\frac{\omega^n}{n!}<\infty.
\]
With the $L^2$--inner product induced by $h$ and $\omega$, $\hcal_k$ is a closed subspace of $L^2(M,L^k)$. The Bergman kernel $K_k(x,y)$ is the integral kernel of the orthogonal projection $L^2(M,L^k)\to\hcal_k$, characterized by
\[
s(y)=\int_M \langle s(x),K_k(x,y)\rangle_{h^k}\,\frac{\omega^n(x)}{n!}\qquad(\forall s\in\hcal_k).
\]
The diagonal density (or Bergman kernel function) is the pointwise trace of $K_k$,
\[
\rho_k(x)=|K_k(x,x)|_{h^k}.
\]

For trivial bundles over domains in $\mathbb C^n$ one uses the analogous definition for holomorphic $L^2$--functions.

The Bergman kernel has been intensively studied for domains in $\mathbb C^n$ and for K\"{a}hler manifolds (see, e.g., \cite{Fefferman1974,boutet1975singularite,kerzman1978cauchy,donaldson2001,Sun2011Expected,Donaldson2014Gromov,berman2011fekete,Donaldson15, dsz1,dsz2,dsz3,Shiffman2008NV,sz1,bsz0,bsz1}). Except in very symmetric cases (for instance complex space forms), explicit formulas are rare. A fundamental substitute is the Tian-Catlin-Zelditch expansion (\cite{Tian1990On, Zelditch2000Szego, Lu2000On, Catlin}): when $M$ is compact,
\[
\rho_k(x)\sim\Big(\frac{k}{2\pi}\Big)^n\Big(1+\frac{b_1(x)}{k}+\frac{b_2(x)}{k^2}+\cdots\Big),
\]
where each $b_j(x)$ is a universal polynomial in the curvature of the Riemannian metric corresponding to $\omega$ and its covariant derivatives. There are extensions of this expansion to symplectic manifolds and orbifolds (see \cite{MM,dailiuma}).

In \cite{sun-rem-hyp}, the author made a progress by proving an explicit formula for the Bergman kernel function for positive line bundles on hyperbolic Riemann surfaces. The formula involves summation over all geodesic loops based at the point where the Bergman kernel is evaluated.
 In this paper, inspired by this work, we continue to study the Bergman kernel of abelian varieties.

Let $X=\mathbb C^n/\Lambda$ be an abelian variety over $\mathbb C$. Here, $\Lambda$ is a lattice in $\mathbb C^n$. Let $L\to X$ be a positive line bundle over $X$, whose first Chern class is determined by a positive definite Hermitian form $H=(H_{ij})$ on $\mathbb C^n$ such that $\Im H(\Lambda,\Lambda)\subset \mathbb Z$. For each vector $v\in \mathbb C^n$, we denote by $|v|_H$ the norm of $v$ defined by $H$.
Let $\omega=\pi \sqrt{-1} \sum H_{ij} dz^i\wedge d\bar{z}^j \in 2\pi c_1(L)$ be a flat K\"ahler metric on $X$. There exist a Hermitian metric $h$ on $L$ with curvature $- i \omega$.

For each $p\in X$, $\exists \tilde{p}\in\mathbb C^n$ such that $p=\pi(\tilde{p})$. Such a $\tilde{p}$ is called a lift of $p$.
For each $v\in \Lambda$, let $\gamma_{p,v}\subset X$ be the smooth geodesic loop defined by $v$ starting at $p$ with orientation defined by $\frac{\partial}{\partial a}$, where $a$ is the parameter of the line segment $\tilde{p}+av$, $a\in[0,1]$. It is easy to see that $\gamma_{p,v}$ does not depend on the choice of $\tilde{p}$, and that the length of $\gamma_{p,v}$ is $|v|_H$. Let $\nabla$ be the Chern connection of $(L,h)$.
 Let $e^{2\pi i\alpha_v(p)}=\text{Hol}_{L}(\gamma_{p,v})$ be the holonomy of $\nabla$ along $\gamma_{p,v}$.
Our first result is the following formula:

Lastly, we state an off-diagonal decay estimate for the Bergman kernel. For $x,y\in X$ let $\mathfrak{G}_{x,y}$ denote the set of geodesic segments joining $x$ to $y$ (parameterized by arc length).
\begin{theorem}\label{thm-off-diag}
 Let $(X,\omega,L,h)$ be a polarized abelian variety as in Theorem \ref{thm-main}. For every $k\ge 1$ and every $x\neq y$,
 \[
 |K_k(x,y)|_{h^k}\;\le\;\Big(\frac{k}{2\pi}\Big)^{\!n}\sum_{\gamma\in\mathfrak{G}_{x,y}} e^{-\frac{k}{4}\,\ell(\gamma)^2},
 \]
 where $\ell(\gamma)$ is the length of $\gamma$.
\end{theorem}

Let $K_k(z,w)$ be the Bergman kernel of $\hcal_{\C^n,k}$. Then we can write \[K_k(z,w)=\sum_{i=1}^{\infty}f_i(z)\otimes \bar{f}_i(w),\] where $\{f_i\}_{i=1}^\infty$ is an orthonormal basis of $\hcal_{\C^n,k}$.
For each $v\in \Lambda$, we define \[K^v_k(z,w)=\sum_{i=1}^{\infty}A_v^*f_i(z)\otimes \bar{f}_i(w).\]
\begin{proposition}
 $K^v_k(z,w)=\sum_{i=1}^{\infty}A_v^*f_i(z)\otimes \bar{f}_i(w)$ is independent of the choice of orthonormal basis $\{f_i\}_{i=1}^\infty$ of $\hcal_{\C^n,k}$.
\end{proposition}
\begin{proof}
 For any $f\in \hcal_{\C^n,k}$, we have $f=\sum_{i=1}^{\infty}c_i A_v^*f_i$, since $\{A_v^*f_i\}$ is also an orthonormal basis of $\hcal_{\C^n,k}$. Then for any fixed $w$, \[\int_{\C^n} (f,\sum_{i=1}^{\infty}A_v^*f_i(z)\otimes \bar{f}_i(w))_{\pi^*h}\frac{w_0^n}{n!}=\sum_{i=1}^{\infty}c_i f_i(w)=(A_v^{-1})^*f(w), \]
 namely, $K^v_k(z,w)$ is the representation element for the functional $\val_w\circ A_{-v}^*$, hence independent of the choice of the orthonormal basis.
\end{proof}
\begin{defandthm}\label{def-and-thm-k-gamma}
 We define
\[K_{k}^\Lambda(z,w)\eqd \sum_{v\in \Lambda}K^v_k(z,w). \]
Then for $k>0$, and for each fixed $w$, it converges locally uniformly and absolutely. Moreover, we have that for $k>0$, $\sum_{v\in \Lambda}K^v_k(z,w)\in W_k$ for each fixed $w$.
\end{defandthm}
\begin{proof}
 Since $K^v_k(z,w)$ is independent of the choice of orthonormal basis, for each fixed $w$, we can choose the orthonormal basis $\{f_i\}$ such that $f_1=s_w$, a peak section at $w$. So $f_i(w)=0$ for $i>1$. So we have \[K^v_k(z,w)=A_v^*s_w(z)\otimes \bar{s}_w(w).\]
 So by Lemma \ref{lem-sum-g-f-0}, we have $\sum_{v\in \Lambda}K^v_k(z,w)$ converges locally uniformly and absolutely. And by Lemma \ref{lem-sum-g-f-0-in-v-k}, $\sum_{v\in \Lambda}K^v_k(z,w)\in W_k$ for each fixed $w$.
\end{proof}
\begin{lemma}\label{lem-int-s-sum-g-s-p}
 For any $s\in W_k$, we have
 \[\int_{F_\Lambda}(s,\sum_{v\in \Lambda}A_v^*s_p)_{\pi^*h}\frac{\omega_0^n}{n!}=(\frac{2\pi}{k})^{n/2}\frac{s(p)}{s_p(p)}.\]
\end{lemma}
\begin{proof}
 It suffices to show this for $p=0$. By Lemmas \ref{lem-sum-g-f-0} and \ref{lem-sum-g-f-0-in-v-k},
 \[ \int_{F_\Lambda}(s,\sum_{v\in \Lambda}A_v^*s_0)_{\pi^*h}\frac{\omega_0^n}{n!}=\sum_{v\in \Lambda}\int_{F_\Lambda}(s,A_v^*s_0)_{\pi^*h}\frac{\omega_0^n}{n!}.\]
 On the other hand, since $s_0=(\frac{k}{2\pi})^{n/2}\bm{e}_k$ is $L_1$ integrable on $\C^n$, and $s$ is bounded on $\C^n$, we have $(s,s_0)_{\pi^*h}\in L^1(\C^n,\frac{\omega_0^n}{n!})$. So \[\sum_{v\in \Lambda}\int_{F_\Lambda}(s,A_v^*s_0)_{\pi^*h}\frac{\omega_0^n}{n!}=\int_{\C^n} (s,s_0)_{\pi^*h}\frac{\omega_0^n}{n!}.\]
 Then by taking the Taylor expansion of $\frac{s}{\bm{e}_k}$, we get the right hand side equals $(\frac{2\pi}{k})^{n/2}\frac{s(0)}{\bm{e}_k(0)}=\frac{s(0)}{s_0(0)}$.
\end{proof}
\begin{theorem}\label{thm-k-gamma-bergman}
 For $k>0$, $K_{k}^\Lambda(z,w)$ is the Bergman kernel of $W_k$.
\end{theorem}
\begin{proof}
 For fixed $w$, we have \[K_{k}^\Lambda(z,w)=(\sum_{v\in \Lambda}A_v^*s_w(z))\otimes \bar{s}_w(w).\]
 So for each $S\in \hcal_{X,k}$, we have \[\int_{F_\Lambda} (S,K_{k}^\Lambda(z,w))_{\pi^*h}\frac{\omega_0^n}{n!}=\frac{S(w)}{s_w(w)}s_w(w)=S(w).\]
 We have proved the theorem.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm-main}]
 For a point \(p\in X\) choose a lift \(\tilde p\in\C^n\); without loss of generality we may take \(\tilde p=0\). By Theorem \ref{thm-k-gamma-bergman} we have the identity
 \[
 \rho_{X,k}(0)=K_{k}^\Lambda(0,0)
 = (\frac{k}{2\pi})^n \;+\; \sum_{v\in\Lambda\setminus\{0\}} K^v_k(0,0).
 \]
 Recall that an element $v\in \Lambda$ is primitive if it can not be written as $v=mu$ for some $u\in\Lambda$ and $m>1$.
 Let \(\mathcal P\) denote the set of primitive elements of \(\Lambda\). Writing \(\langle v\rangle=\{mv:\,m\in\Z \}\) for the cyclic subgroup generated by \(v\in\mathcal P\), we may reorganise the sum as
 \[
 \sum_{v\in\Lambda\setminus\{0\}} K^v_k(0,0)
 = \frac12\sum_{v\in\mathcal P}\sum_{u\in\langle v\rangle\setminus\{0\}} K^{u}_k(0,0).
 \]

Applying Theorem \ref{thm-cylinder-main} to the quotient \(\C^n/\langle v\rangle\) yields, for each \(v\in\mathcal P\),
 \[
 \sum_{u\in\langle v\rangle\setminus\{1\}} K^{u}_k(0,0)
 = (\frac{k}{2\pi})^n\sum_{u\in\langle u\rangle\setminus\{0\}}
 e^{-k(\frac{|u|}{2})^2}\cos\!\big(2\pi k\alpha_{u}(p)\big).
 \]
 Summing over all primitive elements \(g\in\mathcal P\) gives the formula in Theorem \ref{thm-main}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm-off-diag}]
 By Theorem \ref{thm-k-gamma-bergman}, we have
 \[\pi^*K_{X,k}(z,w)=\sum_{v\in \Lambda}(A_v^*s_w(z))\otimes \bar{s}_w(w).\]
 We can assume that $w=0$. Then since $|A_v^*s_0(z)|=(\frac{k}{2\pi})^{n/2}e^{-\frac{k}{4}d^2(-v,z)}$. We have \[|\pi^*K_{X,k}(z,0)|_h\leq (\frac{k}{2\pi})^{n}\sum_{v\in \Lambda}e^{-\frac{k}{4}d^2(-v,z)}.\]
 Since $d(-v,z)$ can be identified with the length of the corresponding geodesic segment in $X$ joining $\pi(z)$ and $\pi(0)$, the theorem follows.
\end{proof}
\subsection{Proof of Theorem \ref{thm-second}}
For a fixed line bundle $L$ on $X$ with $\omega\in 2\pi c_1(L)$, let $h$ be a Hermitian metric on $L$ such that $\Theta_h=-i\omega$. Let $F$ denote the underlying $C^\infty$ complex line bundle of $L$. Let $\ucal$ be the space of unitary connections on $(F,h)$ whose curvatures satisfy the condition that the $(0,2)$-part is $0$. Let $\nabla_0\in \ucal$ be the Chern connection of $(L,h)$.

For $L'$, we let $\text{Hol}_{L'}(\gamma_{p,v})=e^{2\pi i \alpha'_v(p)}$ and $\phi'_p=\alpha'_{v_1}(p)$. Then we also have \[(P_*(\frac{(2\pi)^n}{k^n}\rho_{L',k}-1))(x)=2\nu \sum_{m\geq 1} e^{-\frac{k}{4}|m|^2|v_1|_H^2}\cos(2\pi m (\lambda t+\phi'_p)). \]
Therefore, if $\rho_{L',k}=\rho_{L,k}$, we get \[\sum_{m\geq 1} e^{-\frac{k}{2}|m|^2|v_1|_H^2}\cos(2\pi m (\lambda t+\phi'_p))= \sum_{m\geq 1} e^{-\frac{k}{2}|m|^2\|v_1\|^2}\cos(2\pi m (\lambda t+\phi_p)), \] for $t\in [0,1]$. So we get $\phi_p'=\phi_p, \mod 1$, namely $\text{Hol}_{L'}(\gamma_{p,v_1})=\text{Hol}_{L'}(\gamma_{p,v_1})$. So we have proved the theorem.

\end{proof}
\begin{proof}[Proof of Theorem \ref{thm-max-min}]
 Part (a) follows directly from Theorem \ref{thm-main}. For part (b) and (c), we have that $\exists C_1>0$, $k_0>0$ such that for $k\geq k_0$, \[\sum_{v\in \Lambda, |v|_H>l_1}e^{-\frac{k}{4}|v|_H^2}\leq C_1 e^{-\frac{k}{4}l_2^2}.\]
 Let $p$ be a point satisfying the condition that $\text{Hol}_{L}(\gamma_{p,v})=1$ for all $v\in S_1$. Suppose that $k\geq k_0$ and $\rho_k(q)>\rho_k(p)$, then we have \[\sum_{v\in S_1}e^{-\frac{k}{4}l_1^2}(1-\cos(2\pi k\alpha_v(q)))\leq 2C_1 e^{-\frac{k}{4}l_2^2}, \]namely \[\sum_{v\in S_1}(1-\cos(2\pi k\alpha_v(q)))\leq 2C_1 e^{-\frac{k}{4}(l_2^2-l_1^2)}.\]
 If we require that $\alpha_v(q)\in (-\pi,\pi]$ for all $v$, then this implies that $|\alpha_v(q)|$ is small for $v\in S_1$.
Let $\{v_1, v_2,\cdots, v_m\}$ be a maximally linearly independent subset of $S_1$.
 From the proof of Proposition \ref{prop-surj}, it is easy to see that $\exists C_2>0$, independent of $k$, such that when $\epsilon$ is small enough, for any $q'\in X$, if $1-\cos(2\pi k\alpha_{v_j}(q'))<\epsilon$, $1\leq j\leq m$, then $\exists p'\in X$ such that $d(p',q')<C_2\epsilon$ and $e^{2\pi k\alpha_{v_j}(p')\sqrt{-1}}=1$, $1\leq j\leq m$. Let $q'=q$, we get $p'$ such that $d(p',q)<2C_2C_1 e^{-\frac{k}{4}(l_2^2-l_1^2)}$ and $e^{2\pi k\alpha_{v_j}(p')\sqrt{-1}}=1$, $1\leq j\leq m$.

\begin{remark}
\begin{itemize}
\item This formula is closely parallel to Theorem~1 of \cite{sun-rem-hyp}, and the proof follows the same strategy. Equivalently, the sum may be written over all geodesic loops based at \(p\), as in \cite{sun-rem-hyp}. It is instructive to compare the two settings: in the hyperbolic case the contribution of a loop of length \(l(\gamma)\) is essentially \(\cosh^{-2k}\!\big(\tfrac{l(\gamma)}{2}\big)\), while in the flat (abelian) case it is the Gaussian factor \(e^{-\frac{k}{2}(|v|_{H})^2}\). This contrast reflects the difference between negative curvature and zero curvature geometries.
\item The formula is also valid when the rank of $\Lambda$ is not maximal, i.e., for polarized complex tori that are not necessarily compact. For simplicity, we only state the compact case here, as we think people care most about the compact case.
\end{itemize}
\end{remark}

For a general polarized manifold \((X,L)\) one may ask:
\begin{question}
Given another Hermitian polarization \((L',h')\) with \(\Theta_{h'}=\Theta_h\) (so \(c_1(L')=c_1(L)\)), under what conditions can one detect that the Bergman kernels \(\rho_{L',k}\) and \(\rho_{L,k}\) are different?
\end{question}
Since the Tian--Catlin--Zelditch expansion depends only on local curvature data, it cannot distinguish such global differences. As an application of Theorem \ref{thm-main} we obtain the following rigidity statement.

\begin{theorem}\label{thm-second}
	Let $(X,L)$ be a polarized abelian variety. Let $L'$ be another polarization of $X$ with respect to a Hermitian metric $h'$ on $L'$ such that $\Theta_{h'}=\Theta_h$. Suppose that $$\rho_{L',k}=\rho_{L,k}$$ for some $k\geq 1$. Then $kL'$ is isomorphic to $kL$.
\end{theorem}
Similar to \cite{sun-rem-hyp}, another application of Theorem \ref{thm-main} concerns the locations of the global maxima and minima of $\rho_k$. Set
\[
l_1=\min_{v\in \Lambda\setminus\{0\}} |v|_{H}, \qquad
l_2=\min_{\substack{v\in \Lambda\\ |v|_{H}>l_1}} |v|_{H},
\]
and write $S_1=\{v\in\Lambda:\;|v|_H=l_1\}$.

\begin{theorem}\label{thm-max-min}
\begin{enumerate}
\item[(a)] Assume $\Im H(u,v)\in 2\mathbb{Z}$ for all $u,v\in\Lambda$. Then for every $k\ge1$ the Bergman density $\rho_k$ attains its maximum precisely at those points $p\in X$ for which
\[
\mathrm{Hol}_L(\gamma_{p,v})=1\quad\text{for all }v\in\Lambda.
\]

\item[(b)] Assume $\Im H(u,v)\in 2\mathbb{Z}$ for all $u,v\in S_1$. There exist constants $C>0$ and $k_0\in\mathbb{N}$ such that for every $k\ge k_0$ every global maximum of $\rho_k$ lies within a neighborhood of radius
\[
C\exp\!\Big(\tfrac{k}{4}(l_1^2-l_2^2)\Big)
\]
of some point $p$ satisfying
\[
\mathrm{Hol}_{kL}(\gamma_{p,v})=1\quad\text{for all }v\in S_1.
\]

\item[(c)] Let $\{v_1,\dots,v_m\}$ be a maximally linearly independent subset of $S_1$ and suppose $\#S_1=2m$ (so $S_1=\{\pm v_1,\dots,\pm v_m\}$). Then the statement of (b) holds. Moreover, there exists $C'>0$ and $k_0'$ such that for all $k\ge k_0'$ every global minimum of $\rho_k$ lies within a neighborhood of radius
\[
C'\exp\!\Big(\tfrac{k}{4}(l_1^2-l_2^2)\Big)
\]
of some point $p$ with
\[
\mathrm{Hol}_{kL}(\gamma_{p,v})=-1\quad\text{for all }v\in S_1.
\]
\end{enumerate}
\end{theorem}

\begin{remark}
\begin{itemize}
\item If $(v_1,\dots,v_{2n})$ is any basis of $\Lambda$, Proposition \ref{prop-surj} implies there exist points $p$ with $\mathrm{Hol}_L(\gamma_{p,v_j})=1$ for all $j$. Hence the hypothesis $\Im H(u,v)\in2\mathbb{Z}$ on $\Lambda\times\Lambda$ ensures existence of points where $\mathrm{Hol}_L(\gamma_{p,v})=1$ for every $v\in\Lambda$.
\item In part (b) it is enough to verify the holonomy condition $\mathrm{Hol}_{kL}(\gamma_{p,v_j})=1$ for a maximally independent subset $\{v_j\}_{j=1}^m\subset S_1$.
\end{itemize}
\end{remark}
Similar to Conjecture 1.6 in \cite{sun-rem-hyp}, we propose the following rigidity conjecture for polarized abelian varieties:

\begin{conjecture}
	Let $(X,L)$ and $(X',L')$ be polarized abelian varieties of dimension $n$. Suppose there is a diffeomorphism $\Phi:X\to X'$ and an integer $k\geq 1$ such that
	\[
	\rho_{L,k}=\Phi^*\rho_{L',k}.
	\]
	Then $\Phi$ is either a biholomorphism or an anti-biholomorphism, and moreover
	\[
	\Phi^*((L')^{\!k})\cong L^{\!k}\qquad\text{or}\qquad \Phi^*((L')^{\!k})\cong\overline{L^{\!k}},
	\]
	respectively. Here $\overline{L}$ denotes the complex-conjugate line bundle of $L$ (transition functions given by complex conjugates).
\end{conjecture}

Theorem \ref{thm-second} gives partial evidence for this conjecture.

Lastly, we state an off-diagonal decay estimate for the Bergman kernel. For $x,y\in X$ let $\mathfrak{G}_{x,y}$ denote the set of geodesic segments joining $x$ to $y$ (parameterized by arc length).
\begin{theorem}\label{thm-off-diag}
	Let $(X,\omega,L,h)$ be a polarized abelian variety as in Theorem \ref{thm-main}. For every $k\ge 1$ and every $x\neq y$,
	\[
	|K_k(x,y)|_{h^k}\;\le\;\Big(\frac{k}{2\pi}\Big)^{\!n}\sum_{\gamma\in\mathfrak{G}_{x,y}} e^{-\frac{k}{4}\,\ell(\gamma)^2},
	\]
	where $\ell(\gamma)$ is the length of $\gamma$.
\end{theorem}

The corresponding result in \cite{sun-rem-hyp} was the first to relate the global off-diagonal decay to the geodesic distance (compared to many local results, see the references therein); the theorem above provides a second instance, now in the flat setting.

\medskip
We briefly outline the proofs. The argument for Theorem \ref{thm-main} follows the same two-step strategy as in \cite{sun-rem-hyp}: first establish the formula for the cylinder (quotients of $\mathbb C^n$ by a cyclic subgroup), and then lift the global Bergman kernel to the universal cover and express it as a summation over the lattice $\Lambda$. This strategy parallels the construction behind the Selberg trace formula.

Theorem \ref{thm-second} is obtained by integrating the kernel along complementary subtori to isolate the contributions of a chosen lattice direction; comparison of these averaged formulas recovers the holonomy data. Theorems \ref{thm-max-min} and \ref{thm-off-diag} are direct consequences of Theorem \ref{thm-main}, while the off-diagonal decay is obtained by estimating the contribution of peak sections and summing over geodesic segments joining two points.

\medskip

\noindent\textbf{Organization of the paper.}
In Section~2 we collect background material and derive the one-dimensional cylinder formula. Section~\ref{section-cylinder} treats the higher-dimensional cylinder and proves the cylinder case of Theorem~\ref{thm-main}. Section~\ref{sec-proof-thm-main} carries out the lattice summation on the universal cover and completes the proof of Theorem~\ref{thm-main}; the remaining theorems are then proved as applications.

\medskip

\noindent\textbf{Acknowledgements.} The author would like to thank Professor Song Sun for many very helpful discussions.

We briefly outline the proofs. The argument for the main theorem follows the same two-step strategy as in Sun, “Removable singularities in hyperbolic geometry”: first establish the formula for the cylinder (quotients of $\mathbb C^n$ by a cyclic subgroup), and then lift the global Bergman kernel to the universal cover and express it as a summation over the lattice $\Lambda$. This strategy parallels the construction behind the Selberg trace formula.,

Question: Let \(X=\mathbb C^n/\Lambda\) be an abelian variety, where \(\Lambda\subset \mathbb C^n\) is a lattice. Let \(L\to X\) be a positive holomorphic line bundle whose first Chern class is determined by a positive definite Hermitian form \(H\) on \(\mathbb C^n\) with \(\operatorname{Im} H(\Lambda,\Lambda)\subset \mathbb Z\). Write \(|v|_H\) for the norm induced by \(H\), and let \(\omega=\pi i\sum H_{ij}\,dz^i\wedge d\bar z^j\in 2\pi c_1(L)\). Choose a Hermitian metric \(h\) on \(L\) with Chern curvature \(-i\omega\). For each \(p\in X\) and \(v\in\Lambda\), let \(\gamma_{p,v}\) be the geodesic loop based at \(p\) determined by \(v\), and define \(\alpha_v(p)\in \mathbb R/\mathbb Z\) by \(e^{2\pi i\alpha_v(p)}=\operatorname{Hol}_L(\gamma_{p,v})\). For each integer \(k\ge 1\), let \(\rho_k(p)\) denote the diagonal Bergman kernel of \(H^0(X,L^k)\) at \(p\). Which statement holds for every such integer \(k\ge 1\) and every point \(p\in X\)?

\[\rho_k(p)=\left(\frac{k}{2\pi}\right)^n\left(1+\sum_{v\in \Lambda\setminus\{0\}} e^{-\frac{k}{4}|v|_H^2}\cos\big(2\pi k\alpha_v(p)\big)\right).\]

Choices:
B. \[\rho_k(p)=\left(\frac{k}{2\pi}\right)^n\left(1+\sum_{v\in \Lambda} e^{-\frac{k}{4}|v|_H^2}\cos\big(2\pi k\alpha_v(p)\big)\right).\]
C. \[\rho_k(p)=\left(\frac{k}{2\pi}\right)^n\left(1+\sum_{v\in \Lambda\setminus\{0\}} e^{-\frac{k}{4}|v|_H^2}c_v(p)\right),\qquad |c_v(p)|\le 1\ \text{for all }v\in\Lambda\setminus\{0\}.\]
D. \[\rho_k(p)=\left(\frac{k}{2\pi}\right)^n\left(1+\sum_{v\in \Lambda\setminus\{0\}} e^{-\frac{k}{2}|v|_H^2}\cos\big(2\pi k\alpha_v(p)\big)\right).\]
E. \[\rho_k(p)=\left(\frac{k}{2\pi}\right)^n\left(1+\sum_{v\in \Lambda\setminus\{0\}} e^{-\frac{k}{4}|v|_H^2}\,e^{2\pi i k\alpha_v(p)}\right).\]


\begin{thm}\label{mainthm}
    Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.
\end{thm}

\par In slightly more detail,
given a boundaryless surface with negative Euler characteristic
and a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\colon S \to S$, one can construct the veering triangulation of the mapping torus of $f$, whose
flow graph encodes the transition matrix for this train track. Using the work of \cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.

\subsection{Connections to literature}\label{discuss}

\par Work of Penner-Papadopoulos \cite[Theorem 4.1]{PP} is sometimes cited as producing $f$-invariant generic train tracks with irreducible transition matrix for a pseudo-Anosov $f$, however it was pointed out to us by Chris Leininger that this construction seems to only produce train tracks which are $f^n$-invariant with irreducible transition matrix for some $n$ possibly greater than one.
\par In the setting of \cite[Theorem 4.1]{PP},
the support map $\sigma$ produced for the $f$-invariant train track $\tau$ is not required to map switches to switches, rather the transition matrix is defined by choosing test points in the interiors of branches and counting the number of times the image of each branch hits each test point. Since switches don't have to map to switches, a branch $b$ of $\tau$ could e.g. be mapped by $\sigma \circ f$ over itself and a small piece of an adjacent branch. Then the transition matrix for $f$ will record that $b$ only maps over itself. However, for some $n$, $(\sigma \circ f)^n(b)$ will be mapped over all of the adjacent branch, so the transition matrix for $(\sigma \circ f)^n$ will record that $b$ maps over itself and the adjacent branch. Hence, it's possible that $f^n(\tau) \prec \tau$ has irreducible transition matrix while $f(\tau) \prec \tau$ does not.

\par Returning to discussion of the present work, we note that much of the theory of train tracks assumes that the tracks considered are \textit{\textbf{generic}}, i.e. trivalent, since this simplification can often be made without loss of generality. In particular, the train tracks associated to veering triangulations are always generic. However, if the flow graph for a pseudo-Anosov $f$ is not irreducible, \Cref{mainthm} necessarily produces train tracks which are not generic.
This motivates the following question:

\begin{qn}
    For all $f\colon S \to S$ pseudo-Anosov, does there exist a \textit{generic} invariant train track for $f$ with irreducible transition matrix?
\end{qn}

\par \textbf{Acknowledgements.} I would like to thank Sam Taylor for his generous guidance and feedback. I would also like to thank Chi Cheuk Tsang for helpful conversations and comments on a draft of this paper.

\section{Background}\label{background}

\par A \textit{\textbf{strongly connected component}} of a directed graph $G$ is a maximal subset $C$ of the vertices of $G$ such that for any two vertices in $C$, there is a path in $G$ from the first vertex to the second.
If $G$ has only one strongly connected component, it is called \textit{\textbf{strongly connected}}.
A nonnegative square matrix $A$ is called \textit{\textbf{irreducible}} if for every pair of indices $i$, $j$ there is an $n>0$ such that the $(i,j)$-th entry of $A^n$ is positive. $A$ is irreducible if and only if it is the adjacency matrix of a strongly connected graph.
\par
    Given an orientable finite-type surface $S$ without boundary and negative Euler characteristic, a homeomorphism $f\colon S \to S$ is called \textit{\textbf{pseudo-Anosov}} if there is a pair of transverse measured singular foliations $(\Lambda^u,\mu_u)$ and $(\Lambda^s,\mu_s)$, called the \textit{\textbf{unstable}} and \textit{\textbf{stable}} foliations, and a constant $\lambda > 1$, called the \textit{\textbf{stretch factor}}, such that $f(\Lambda^u)=\Lambda^u$, $f(\Lambda^s)=\Lambda^s$ and $f(\mu^u) = \lambda \mu^u$, $f(\mu^s) = \lambda^{-1} \mu^s$. No nontrivial power of a pseudo-Anosov map fixes the homotopy class of any essential closed curve in $S$. A pseudo-Anosov map is a diffeomorphism except at the singular points of $\Lambda^u$.

    \par The \textit{\textbf{mapping torus of $f$}} is the 3-manifold
    $$ M_f = \faktor{(S \times [0,1])}{ (f(p),0) \sim (p,1)} $$ which fibers over the circle with fiber $S$ and monodromy $f$.

    \par If all of the singularities of the foliations are at the punctures of $S$, we say $f$ is \textit{\textbf{fully punctured}}. We will study general pseudo-Anosovs by deleting the singularities of the foliations to obtain a new surface $S^\circ$ on which the restriction of $f$ is fully punctured.
For full background on pseudo-Anosov homeomorphisms, see Fathi--Laudenbach--Poénaru \cite{FLP}.

\subsection{Train tracks}\label{tts}
\par A \textit{\textbf{train track}} $\tau$ is a closed 1-complex embedded in a surface $S$ with a ``smoothing" at each vertex so that $\tau$ has a well-defined tangent space at each vertex, and $S \setminus \tau$ contains no nullgons, unpunctured monogons or unpunctured bigons. See Penner-Harer \cite{PH}. The vertices of $\tau$ are called \textit{\textbf{switches}} and the edges are called \textit{\textbf{branches}}. If the switches of $\tau$ all have degree three, $\tau$ is called \textit{\textbf{generic}}.

\par Given two train tracks $\tau_1$ and $\tau_2$ on $S$, $\tau_2$ is said to \textit{\textbf{carry}} $\tau_1$ (or $\tau_1$ \textit{\textbf{is carried by}} $\tau_2$), denoted $\tau_1 \prec \tau_2$, if there is a $C^1$ map $\sigma\colon S \to S$ called the \textit{\textbf{support map}}
such that
    \begin{enumerate}
        \item $\sigma$ is homotopic to the identity,
        \item $\sigma(\tau_1) \subseteq \tau_2$,
        \item for all points $x$ in $\tau_1$,
    $d_x \sigma_{|\tau_1}\colon T_x \tau_1 \to T_{h(x)}\tau_2$ is an isomorphism of tangent spaces,
    \end{enumerate}
Note that carrying is transitive: if $\tau_1 \prec \tau_2$ and $\tau_2 \prec \tau_3$, then
$\tau_1 \prec \tau_3$.
We also write $\tau_1 \prec_\sigma \tau_2$ to indicate that $\sigma$ is the support map for $\tau_1 \prec \tau_2$. A carrying map $\sigma$ for $\tau_1 \prec \tau_2$  is called \textit{\textbf{combinatorial}} if for every switch $v$ in $\tau_1$, $\sigma(v)$ is a switch in $\tau_2$.
    \par
    Given $\tau_1$ with branches $\{b^1_j\}_{j=1\ldots n}$ and $\tau_2$ with branches $\{b^2_i\}_{i=1\ldots m}$ such that $\tau_1$ is carried by $\tau_2$ with combinatorial support map $\sigma$,
    the \textit{\textbf{transition matrix}} is the $m \times n$ matrix $T = [t_{i,j}]$ where
    $t_{i,j}$ is the number of times the image of $b^1_j$ under $\sigma$ passes over $b^2_i$.
\par
    A \textit{\textbf{train path}} is a $C^1$ immersion $p\colon [0,1] \to \tau$ so that the endpoints $\{0,1\} = \partial[0,1]$ are mapped to switches. We will sometimes conflate $p$ with its image in $\tau$. For $t \in (0,1)$ such that $p(t)$ is a switch, the sub-paths $p_0 = p_{|[0,t]}\colon [0,t] \to \tau$, $p_1 = p_{|[t,1]}\colon [t,1] \to \tau$ are also train paths (after reparametrizing). Similarly, if $p_0$ and $p_1$ are train paths such that $p_0(1)=p_1(0)$ and the differentials $d_1 p_0(\partial_t)$ and $d_0 p_1(\partial_t)$ are either both positive or both negative. The concatenation $(p_0 p_1)\colon [0,2] \to \tau$ is a train path after reparametrizing, hence any train path can be written as $(p_0\ldots p_{n-1})$, where $p_i\colon\left[\frac{i}{n},\frac{i+1}{n}\right] \to \tau$ has image consisting of a single branch. If $p$ is injective we say the train path is \textit{\textbf{embedded}}.
\par
    A \textit{\textbf{fold}} is a particular combinatorial support map $\phi$ for $\tau_1 \prec \tau_2$ which induces a bijection between the branches of $\tau_1$ and $\tau_2$
    except on a particular connected subgraph where it is defined according to either of the two pictures in \Cref{fig:fold}. The three branches of $\tau_1$ which are mapped over the small branch $e'$ in the domain of \Cref{fig:fold} are said to \textit{\textbf{fold to}} $e'$.
    If there is a $\tau_1 \prec_\phi \tau_2$
    we say $\tau_1$ \textit{\textbf{folds to}} $\tau_2$, denoted $\tau_1 \leftharpoonup \tau_2$. If $\tau_1$ folds to $\tau_2$, $\tau_1$ is carried by $\tau_2$. If $\tau_1 \prec \tau_2$ with support map given by a sequence of folds we also say $\tau_1$ folds to $\tau_2$.
    \begin{figure}[h]
    \centering
    \includegraphics[width=0.66\textwidth]{foldgood.pdf}
    \caption{The two possible pictures of a fold. The fold on the left is a left fold and the one on the right is a right fold. Colors indicate the image of each branch on the top after folding.}
    \label{fig:fold}
\end{figure}
    \par
    A train track $\tau$ is an \textit{\textbf{invariant}} track for a map $f: S \to S$ with support map $\sigma$ if $f(\tau)$ is carried by $\tau$ with support map $\sigma$.
    If the transition matrix for $f(\tau) \prec_\sigma \tau$ is irreducible, we also say $\tau$ is \textit{\textbf{irreducible}}.
By smoothly isotoping $\tau$ if necessary, we may assume it is disjoint from the singularities of $\Lambda^u$. This will be convenient later when we remove the singularities of $\Lambda^u$. The following theorem due to Agol is the starting point of the construction of the veering triangulation of the mapping torus of a pseudo-Anosov $f$:

\begin{thm}[\protect{\cite[Theorem 3.5]{A}}]\label{ttfs} Given $f\colon S \to S$ pseudo-Anosov, there exists a generic invariant train track $\tau$ for $f$ such that $f(\tau)$ folds to $\tau$, i.e. there exist a finite sequence of train tracks
$$f(\tau) = \tau_0 \leftharpoonup \tau_1 \leftharpoonup \ldots \leftharpoonup \tau_n = \tau,$$
where $\tau_i$ is carried by $\tau_{i+1}$ with support map given by a single fold.
\end{thm}

\label{intro}
Given a pseudo-Anosov homeomorphism $f$, an invariant train track provides a combinatorial way of recording how curves and other objects behave under iteration of $f$.
The existence of invariant train tracks with irreducible transition matrix
allows the application of Perron-Frobenius theory to the study of the properties of these maps.
For example, when an invariant train track is irreducible it can be used to build a Markov partition for the map.
The existence of irreducible invariant train tracks is often taken for granted
(See e.g. \cite{FR}, \cite{F}, \cite{J}, \cite{EF}, \cite{M}, \cite{Pe})
but to our knowledge no complete proof is currently in the literature. See \Cref{discuss} for more details.

\par The veering triangulation associated to $f$ provides a unified way to study both the topology of its mapping torus and the dynamics of its invariant train tracks.
One important datum attached to a veering triangulation is its \textit{flow graph}, which encodes a Markov partition of a pseudo-Anosov map or flow. Flow graphs have been used by Landry--Minsky--Taylor used to count periodic points of pseudo-Anosov maps, and to define polynomial invariants of pseudo-Anosov flows without perfect fits \cite{LMT1}.
When the flow graph associated to the veering triangulation of a pseudo-Anosov mapping torus is strongly connected, the triangulation can be used to obtain invariant train tracks with
irreducible transition matrices.
\par For general veering triangulations, Agol--Tsang have characterized the strongly connected components of the flow graph, which consist of one ``reduced component" from which there are paths to every vertex, and some number of ``infinitesimal cycles" which do not have any paths back to the rest of the graph \cite{AT}.
By studying the interaction between these infinitesimal components and the invariant train tracks arising from the triangulation, we are able to produce new train tracks where the obstructions to irreducibility coming from the infinitesimal components have been bypassed. This allows us to prove the following:

\par In slightly more detail,
given a boundaryless surface with negative Euler characteristic
and a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\colon S \to S$, one can construct the veering triangulation of the mapping torus of $f$, whose
flow graph encodes the transition matrix for this train track. Using the work of \cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.

\begin{qn}
 For all $f\colon S \to S$ pseudo-Anosov, does there exist a \textit{generic} invariant train track for $f$ with irreducible transition matrix?
\end{qn}

\par Given two train tracks $\tau_1$ and $\tau_2$ on $S$, $\tau_2$ is said to \textit{\textbf{carry}} $\tau_1$ (or $\tau_1$ \textit{\textbf{is carried by}} $\tau_2$), denoted $\tau_1 \prec \tau_2$, if there is a $C^1$ map $\sigma\colon S \to S$ called the \textit{\textbf{support map}}
such that
 \begin{enumerate}
 \item $\sigma$ is homotopic to the identity,
 \item $\sigma(\tau_1) \subseteq \tau_2$,
 \item for all points $x$ in $\tau_1$,
 $d_x \sigma_{|\tau_1}\colon T_x \tau_1 \to T_{h(x)}\tau_2$ is an isomorphism of tangent spaces,
 \end{enumerate}
Note that carrying is transitive: if $\tau_1 \prec \tau_2$ and $\tau_2 \prec \tau_3$, then
$\tau_1 \prec \tau_3$.
We also write $\tau_1 \prec_\sigma \tau_2$ to indicate that $\sigma$ is the support map for $\tau_1 \prec \tau_2$. A carrying map $\sigma$ for $\tau_1 \prec \tau_2$ is called \textit{\textbf{combinatorial}} if for every switch $v$ in $\tau_1$, $\sigma(v)$ is a switch in $\tau_2$.
 \par
 Given $\tau_1$ with branches $\{b^1_j\}_{j=1\ldots n}$ and $\tau_2$ with branches $\{b^2_i\}_{i=1\ldots m}$ such that $\tau_1$ is carried by $\tau_2$ with combinatorial support map $\sigma$,
 the \textit{\textbf{transition matrix}} is the $m \times n$ matrix $T = [t_{i,j}]$ where
 $t_{i,j}$ is the number of times the image of $b^1_j$ under $\sigma$ passes over $b^2_i$.
\par
 A \textit{\textbf{train path}} is a $C^1$ immersion $p\colon [0,1] \to \tau$ so that the endpoints $\{0,1\} = \partial[0,1]$ are mapped to switches. We will sometimes conflate $p$ with its image in $\tau$. For $t \in (0,1)$ such that $p(t)$ is a switch, the sub-paths $p_0 = p_{|[0,t]}\colon [0,t] \to \tau$, $p_1 = p_{|[t,1]}\colon [t,1] \to \tau$ are also train paths (after reparametrizing). Similarly, if $p_0$ and $p_1$ are train paths such that $p_0(1)=p_1(0)$ and the differentials $d_1 p_0(\partial_t)$ and $d_0 p_1(\partial_t)$ are either both positive or both negative. The concatenation $(p_0 p_1)\colon [0,2] \to \tau$ is a train path after reparametrizing, hence any train path can be written as $(p_0\ldots p_{n-1})$, where $p_i\colon\left[\frac{i}{n},\frac{i+1}{n}\right] \to \tau$ has image consisting of a single branch. If $p$ is injective we say the train path is \textit{\textbf{embedded}}.
\par
 A \textit{\textbf{fold}} is a particular combinatorial support map $\phi$ for $\tau_1 \prec \tau_2$ which induces a bijection between the branches of $\tau_1$ and $\tau_2$
 except on a particular connected subgraph where it is defined according to either of the two pictures in \Cref{fig:fold}. The three branches of $\tau_1$ which are mapped over the small branch $e'$ in the domain of \Cref{fig:fold} are said to \textit{\textbf{fold to}} $e'$.
 If there is a $\tau_1 \prec_\phi \tau_2$
 we say $\tau_1$ \textit{\textbf{folds to}} $\tau_2$, denoted $\tau_1 \leftharpoonup \tau_2$. If $\tau_1$ folds to $\tau_2$, $\tau_1$ is carried by $\tau_2$. If $\tau_1 \prec \tau_2$ with support map given by a sequence of folds we also say $\tau_1$ folds to $\tau_2$.
 \begin{figure}[h]
 \centering
 \includegraphics[width=0.66\textwidth]{foldgood.pdf}
 \caption{The two possible pictures of a fold. The fold on the left is a left fold and the one on the right is a right fold. Colors indicate the image of each branch on the top after folding.}
 \label{fig:fold}
\end{figure}
 \par
 A train track $\tau$ is an \textit{\textbf{invariant}} track for a map $f: S \to S$ with support map $\sigma$ if $f(\tau)$ is carried by $\tau$ with support map $\sigma$.
 If the transition matrix for $f(\tau) \prec_\sigma \tau$ is irreducible, we also say $\tau$ is \textit{\textbf{irreducible}}.
By smoothly isotoping $\tau$ if necessary, we may assume it is disjoint from the singularities of $\Lambda^u$. This will be convenient later when we remove the singularities of $\Lambda^u$. The following theorem due to Agol is the starting point of the construction of the veering triangulation of the mapping torus of a pseudo-Anosov $f$:

\par Note that the referenced theorem is stated in terms of \textit{splits} rather than folds, which are the combinatorial inverse of folds.
 Additionally, the referenced theorem is stronger than what we state here because the splitting sequence is shown to be canonical up to ``commuting maximal splits": at each step in the sequence, the train track is split at a branch carrying maximal measure of the invariant lamination, and the splits of maximal branches commute if there is not a unique branch carrying maximal measure.
 This theorem motivates the following definition:
\begin{defn}
 An invariant train track $\tau$ for a pseudo-Anosov map $f$ is called \textit{\textbf{veering}} if it
 is generic and $\tau$ can be obtained from $f(\tau)$ by a sequence of folds, in which case there is a sequence of train tracks
 $$f(\tau) = \tau_0 \leftharpoonup \tau_1 \leftharpoonup \ldots \leftharpoonup \tau_n = \tau$$
 where $\tau_i$ is carried by $\tau_{i+1}$ with support map consisting of a single fold.
\end{defn}
\par In \Cref{clvts} we will see that a veering train track as defined here is sufficient to produce the veering triangulation of the fully punctured mapping torus of $f$. While veering train tracks are not necessarily irreducible, they will be the beginning of our construction of irreducible tracks.
\par
 Given two train tracks $\tau_1$, $\tau_2$ on $S$, a (surface) \textit{\textbf{train track map}} is a map $t: S \to S$ so that $t(\tau_1)$ is contained in $\tau_2$, and the restriction of $t$ to $\tau_1$ is
 such that for any train path $p\colon [0,1] \to \tau_1$, $t \circ p\colon [0,1] \to \tau_2$ is also a train path. In other words, a train track map is like a combinatorial support map except that it does not have to be homotopic to the identity map.
 \par The following lemma will be useful for producing support maps:
\begin{lemma}\label{supplemma}
 Let $t\colon S \to S$ be a train track map taking $\tau_1$ to $\tau_2$, and $g\colon S \to S$ be a diffeomorphism.
 If $t$ is homotopic to $g$, then $g(\tau_1) \prec \tau_2$ with support map given by $t \circ g^{-1}$.
\end{lemma}
\begin{proof}
 Since $t$ and $g$ are homotopic, $t \circ g^{-1}$ is homotopic to the identity. Since $t(\tau_1) \subseteq \tau_2$, $(t \circ g^{-1})(g(\tau_1)) = t(\tau_1) \subseteq \tau_2$. Let $p \in g(\tau_1)$. Since $g$ is a diffeomorphism, the restriction of the differential
 $$d_p g^{-1}_{|g(\tau_1)}\colon T_p g(\tau_1) \to T_{g^{-1}(p)} \tau_1$$
 is an isomorphism. Since $t$ is a train track map, $d_{g^{-1}(p)} t_{| \tau_1}$ is an isomorphism, so the composition
 $$d_p (t \circ g^{-1})_{|g(\tau_1)} = (d_{g^{-1}(p)} t_{| \tau_1}) \circ (d_p g^{-1}_{|g(\tau_1)})\colon T_p g(\tau_1) \to T_{(t \circ g^{-1})(p)} \tau_2$$
 is also an isomorphism.
\end{proof}

Given a boundaryless surface with negative Euler characteristic and a train track folding (splitting) sequence for a pseudo-Anosov $f\colon S\to S$, one constructs the veering triangulation of the mapping torus of $f$, whose flow graph encodes the transition matrix for this train track. Using the work of \cite{AT}, one shows that the only obstructions to irreducibility are branches dual to infinitesimal cycles of the flow graph, and characterizes the local structure of veering tracks near these branches. Using this local picture, one modifies the track by contracting the obstructing branches, and recovers a support map for the modified train track with irreducible transition matrix.

\label{mainthm}
    Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.,

Question: Let $f\colon S\to S$ be a pseudo-Anosov homeomorphism of a surface. A train track $\tau$ is called invariant for $f$ if $f(\tau)$ is carried by $\tau$. If $f(\tau)$ has branches $\{b_j^1\}$, $\tau$ has branches $\{b_i^2\}$, and the carrying is given by a combinatorial support map, the associated transition matrix is the matrix $T=[t_{i,j}]$ where $t_{i,j}$ is the number of times the image of $b_j^1$ passes over $b_i^2$. A square nonnegative matrix is irreducible if for every pair of indices $i,j$, some power of the matrix has positive $(i,j)$-entry. Which statement holds for every pseudo-Anosov homeomorphism $f$?

There exists an invariant train track $\tau$ for $f$ such that the associated transition matrix is irreducible.

Choices:
B. There exists a generic invariant train track $\tau$ for $f$ such that the associated transition matrix is irreducible.
C. There exists an invariant train track $\tau$ for $f$ such that the associated transition matrix is nonzero.
D. For every invariant train track $\tau$ for $f$, the associated transition matrix is irreducible.
E. There exists an invariant train track $\tau$ for $f$ such that, after replacing $f$ by some iterate $f^n$ with $n\ge 1$, the associated transition matrix is irreducible for $f^n$.

existential construction versus universal claim

need for irreducibility for $f$ itself, not merely an iterate


\label{main-2}  Let $R$ be a finite family of slopes containing $0,1,\infty$ of some cardinality $k+3$, and let $s$ be a slope not lying in $R$.
\begin{itemize}
    \item[(i)] (Three slopes) If $k=0$, then
\begin{equation}\label{2ab} 2 - \frac{c_2}{D(R;s)} \leq \SD\left(R; s\right) \leq 2 - \frac{c_1}{D(R;s)}
\end{equation}
    for some absolute constants $c_2 > c_1 > 0$.
    \item[(ii)]  (Many slopes) In general, one has
    \begin{equation}\label{sam} 2 - \frac{C_k \log(2+D(R;s))}{D(R;s)} \leq \SD(R; s) \leq 2 - \frac{c_k}{D(R;s)^{k+1}}
    \end{equation}
    for some absolute constants $c_k, C_k > 0$.
\end{itemize}

\begin{equation}\label{dxy}
d[X;Y], d[Y;Y] \ll \log K.
\end{equation}

For $a \in \Q$, define $g(a) \coloneqq \H[Y-aY'] - \H[Y]$, where $Y'$ is an independent copy of $Y$. From \eqref{hx}, \eqref{ruzsa}, \eqref{ruzsa-diff} we have $g(r) \ll \log K$ for $r \in R \backslash \{\infty\}$.  By Proposition \ref{dilate}(iv), we conclude that $g(s) \ll D^{k+1} \log K$, thus $d[Y; sY] \ll D^{k+1} \log K$.  Combining with \eqref{dxy} and \eqref{ruzsa}, \eqref{ruzsa-diff}, we obtain \eqref{dxsy} as required.

\section{A continuous limit}\label{cts-sec}

In this section we establish \Cref{cts-limit}.  Let $\alpha$, $a$, $b$, $f$, $f_0, f_1, f_\infty$ be as in that theorem.  We allow implied constants in the asymptotic notation to depend on $\alpha, f$. We will establish the lower bound
\begin{equation}\label{sd-targ}
 \SD\left(\{0,1,\infty\},\frac{a}{b}\right) \geq 2 - \frac{h_2(f_\alpha) - \max(h(f_0), h(f_\infty), h(f_1))+o(1)}{\log b}.
 \end{equation}

\subsection{The arithmetic Kakeya conjecture}

Define a \emph{slope} to be an element $r$ of the projective rational\footnote{One could also formulate the arithmetic Kakeya conjecture in other fields than the rationals, and in fact most of the results here apply to arbitrary infinite fields, with minor modifications in the case where the field has positive characteristic.  But in this paper we shall restrict attention for simplicity to the rational case, which is the case of interest for applications to the Kakeya problem.} line $\Q \cup \{\infty\}$.  We then define the projection operators $\pi_r \colon \Q \times \Q \to \Q$ by setting
$$ \pi_r(x,y) \coloneqq x + ry$$
for $r \neq \{\infty\}$, and
$$ \pi_\infty(x,y) \coloneqq y.$$
Given a finite set $R$ of slopes and a further slope $s$ not in $R$, we define the \emph{sum-difference constant} $\SD(R;s)$ to be the least exponent such that the bound
\begin{equation}\label{sdef}
\H[ \pi_s(X,Y) ] \leq \SD(R;s) \max_{r \in R} \H[\pi_r(X,Y)]
\end{equation}
holds for $\Q$-valued random variables\footnote{In this paper, all random variables are understood to be discrete, and in fact take only finitely many values.} $X,Y$ (not necessarily independent), where
$$ \H[X] \coloneqq \sum_{x} \h(\P(X=x))$$
is the Shannon entropy of a random variable $X$, with $\h(t) \coloneqq t \log \frac{1}{t}$ using the convention $\h(0) \coloneqq 0$.  Thus for instance
$$\H[ X-Y ] \leq \SD(\{0,1,\infty\};-1) \max\left( \H[X], \H[X+Y], \H[Y] \right)$$
whenever $X,Y$ are (possibly dependent) $\Q$-valued random variables.

The quantity $\SD(R;s)$ can equivalently be defined as the least exponent such that the bound
\begin{equation}\label{pise}
|\pi_{s}(E)| \leq (\max_{r \in R} |\pi_s(E)|)^{\SD(R;s)}
\end{equation}
for all finite non-empty $E \subset \Q$, where $|A|$ denotes the cardinality of a finite set $A$; see \cite{green}.  However, it will be convenient in this paper to work with the entropy formulation, in order to take advantage of the ``entropic Pl\"unnecke--Ruzsa calculus'' that are founded on the Shannon entropy inequalities.

It is easy to see that one has the projective invariance
$$ \SD(\phi(R); \phi(s)) = \SD(R; s)$$
for any projective transformation $\phi \colon \Q \cup \{\infty\} \to \Q \cup \{\infty\}$, that is to say a map of the form $\phi(r) \coloneqq \frac{ar+b}{cr+d}$ for some $a,b,c,d \in \Q$ with $ad-bc \neq 0$, with the usual conventions when $r$ is infinite or $cr+d$ vanishes.  For instance, by using a dilation transformation, we have
$$ \SD(\{0,1,\infty\}; s) = \SD(\{0,-1/s,\infty\}; -1)$$
for any slope $s$ other than $0,1,\infty$.
In the literature it is conventional to use this $3$-transitive projective symmetry to normalize $s = -1$ and $0, \infty \in R$ (assuming that $|R|\geq 2$ of course), though in this paper it will be more convenient to adopt the normalization $0,1,\infty \in R$ (assuming $|R| \geq 3$).

It is easy to see that $\SD(R;s)=\infty$ when $|R|<2$.
From the entropy inequality $\H[X-Y] \leq \H[X,Y] \leq \H[X]+\H[Y]$ we see that $\SD(R;-1) \leq 2$ when $0,\infty \in S$, and it is easy to see (using the uniform distribution on a long arithmetic progression) that we have equality when $R = \{0,\infty\}$.  By projective invariance, this implies that $\SD(R;s)=2$ whenever $|R|=2$.  As these quantities are clearly non-decreasing in $R$, we then have the trivial bound
\begin{equation}\label{sdf-upper}
\SD(R;s) \leq 2
\end{equation}
for $|R| \geq 2$.

Improvements upon \eqref{sdf-upper} directly lead to improved upper bounds on the dimension of Kakeya and Nikodym sets in high dimensions.  Indeed, in \cite{bourgain} it was observed that a bound of the form $\SD(R;-1) \leq \alpha$ implies that Kakeya and Nikodym sets in dimension $d$ have (upper) Minkowski dimension\footnote{It was recently observed by Thomas Bloom (private communication) that, by combining Bourgain's arguments with the recent quantitative progress on Szemer\'edi's theorem by Leng, Sawhney, and Sah \cite{lss}, that one also obtains this bound for the Hausdorff dimension as well.  For the corresponding results for packing dimension, see \cite{cowen}.} at most $\frac{d-1}{\alpha}+1$.  In particular, if one can establish the \emph{arithmetic Kakeya conjecture}
\begin{equation}\label{infr}
\inf_R \SD(R;-1) = 1
\end{equation}
then this would imply that Kakeya and Nikodym sets in $\R^d$ have full Minkowski and Hausdorff dimension for all $d$.  This is currently only known for $d \leq 3$ \cite{wang}. We refer the reader to \cite{green}, \cite{cowen}, \cite{pohoata} for a discussion of this conjecture (and other equivalent forms of it), and its connection with other variants of the Kakeya conjecture.

Nontrivial progress towards the arithmetic Kakeya conjecture was first obtained in  \cite{bourgain}, who in our notation showed that
$$ \SD(\{0,1,\infty\};-1)  \leq 2 - \frac{1}{13} = 1.923\dots,$$
and used this to obtain new bounds on the Kakeya conjecture in high dimensions.
Further improvements were then obtained in \cite{katz-tao}, \cite{katz-tao-new}. For instance, it is known that
$$ 1.77898 \leq \SD(\{0,1,\infty\};-1) \leq 2 - \frac{1}{6} = 1.833\dots$$
and
$$ 1.668 \leq \SD(\{0,1,2,\infty\};-1) \leq 2 - \frac{1}{4} = 1.75,$$
with the upper bounds established in \cite{katz-tao}, and the lower bounds in \cite{lemm}, \cite{gdm} respectively.
At present, the best upper bound known towards \eqref{infr} is
$$\inf_R \SD(R;-1) \leq 1.67513\dots;$$
see \cite{katz-tao-new}.

\subsection{Asymptotic behavior}

Informally, the arithmetic Kakeya conjecture asserts that in the asymptotic regime where the number of slopes $R$ is large, the constants $\SD(R;s)$ converge to $1$.  Here we consider a complementary regime, in which the number of slopes $R$ is fixed, but we instead let the elements of $R$ (or $s$) vary.  Our main results assert, roughly speaking, that the behavior of these constants is determined by the \emph{rational complexity} of $s$ relative to $R$, with the constants approaching $2$ as it becomes harder to express $s$ in terms of a rational expression of the $R$.  We give a (slightly artificial) definition of this quantity, restricting attention to the normalized setting $\{0,1,\infty\} \subset R$ for simplicity.

\begin{definition}[Rational complexity] Given a family of slopes $R = \{0,1,\infty,r_1,\dots,r_k\}$ containing $0,1,\infty$ and a further slope $s$ not in $R$, we define the \emph{rational complexity} $D = D(R;s)$ of $s$ relative to $R$ to be the least natural number $D$ for which one has a representation of the form
\begin{equation}\label{rat}
 s = \frac{P(r_1,\dots,r_k)}{Q(r_1,\dots,r_k)}
 \end{equation}
where $P,Q$ are polynomials of degree at most $D$ with integer coefficients of magnitude at most $2^D$, with $Q(r_1,\dots,r_k)$ non-zero; this complexity is finite since $s$ is rational.
\end{definition}

Informally, if the complexity of $s$ with respect to $R$ is equal to $D$, then $s$ can be expressed in terms of the slopes in $R$ by a rational expression whose length (when expressed as string of characters) is comparable to $D$.  The rational complexity is reminiscent of the \emph{arithmetic circuit complexity} of $s$ in terms of $R$, but with the key difference that the circuit must take the specific rational form \eqref{rat}.

\begin{example}  $D(\{0,1,\infty\};s)$ is the least natural number for which one can express $s$ as $a/b$ where $a,b$ are integers of magnitude at most $2^D$.  In particular, if $a,b$ are coprime then
$$D \left( \{0,1,\infty\};\frac{a}{b} \right) \asymp \log(2+|a|+|b|).$$
\end{example}

Our main result, proven in Section \ref{main-sec}, is then as follows.

Define a \emph{slope} to be an element $r$ of the projective rational\footnote{One could also formulate the arithmetic Kakeya conjecture in other fields than the rationals, and in fact most of the results here apply to arbitrary infinite fields, with minor modifications in the case where the field has positive characteristic. But in this paper we shall restrict attention for simplicity to the rational case, which is the case of interest for applications to the Kakeya problem.} line $\Q \cup \{\infty\}$. We then define the projection operators $\pi_r \colon \Q \times \Q \to \Q$ by setting
$$ \pi_r(x,y) \coloneqq x + ry$$
for $r \neq \{\infty\}$, and
$$ \pi_\infty(x,y) \coloneqq y.$$
Given a finite set $R$ of slopes and a further slope $s$ not in $R$, we define the \emph{sum-difference constant} $\SD(R;s)$ to be the least exponent such that the bound
\begin{equation}\label{sdef}
\H[ \pi_s(X,Y) ] \leq \SD(R;s) \max_{r \in R} \H[\pi_r(X,Y)]
\end{equation}
holds for $\Q$-valued random variables\footnote{In this paper, all random variables are understood to be discrete, and in fact take only finitely many values.} $X,Y$ (not necessarily independent), where
$$ \H[X] \coloneqq \sum_{x} \h(\P(X=x))$$
is the Shannon entropy of a random variable $X$, with $\h(t) \coloneqq t \log \frac{1}{t}$ using the convention $\h(0) \coloneqq 0$. Thus for instance
$$\H[ X-Y ] \leq \SD(\{0,1,\infty\};-1) \max\left( \H[X], \H[X+Y], \H[Y] \right)$$
whenever $X,Y$ are (possibly dependent) $\Q$-valued random variables.

It is easy to see that $\SD(R;s)=\infty$ when $|R|<2$.
From the entropy inequality $\H[X-Y] \leq \H[X,Y] \leq \H[X]+\H[Y]$ we see that $\SD(R;-1) \leq 2$ when $0,\infty \in S$, and it is easy to see (using the uniform distribution on a long arithmetic progression) that we have equality when $R = \{0,\infty\}$. By projective invariance, this implies that $\SD(R;s)=2$ whenever $|R|=2$. As these quantities are clearly non-decreasing in $R$, we then have the trivial bound
\begin{equation}\label{sdf-upper}
\SD(R;s) \leq 2
\end{equation}
for $|R| \geq 2$.

Improvements upon \eqref{sdf-upper} directly lead to improved upper bounds on the dimension of Kakeya and Nikodym sets in high dimensions. Indeed, in \cite{bourgain} it was observed that a bound of the form $\SD(R;-1) \leq \alpha$ implies that Kakeya and Nikodym sets in dimension $d$ have (upper) Minkowski dimension\footnote{It was recently observed by Thomas Bloom (private communication) that, by combining Bourgain's arguments with the recent quantitative progress on Szemer\'edi's theorem by Leng, Sawhney, and Sah \cite{lss}, that one also obtains this bound for the Hausdorff dimension as well. For the corresponding results for packing dimension, see \cite{cowen}.} at most $\frac{d-1}{\alpha}+1$. In particular, if one can establish the \emph{arithmetic Kakeya conjecture}
\begin{equation}\label{infr}
\inf_R \SD(R;-1) = 1
\end{equation}
then this would imply that Kakeya and Nikodym sets in $\R^d$ have full Minkowski and Hausdorff dimension for all $d$. This is currently only known for $d \leq 3$ \cite{wang}. We refer the reader to \cite{green}, \cite{cowen}, \cite{pohoata} for a discussion of this conjecture (and other equivalent forms of it), and its connection with other variants of the Kakeya conjecture.

\begin{definition}[Rational complexity] Given a family of slopes $R = \{0,1,\infty,r_1,\dots,r_k\}$ containing $0,1,\infty$ and a further slope $s$ not in $R$, we define the \emph{rational complexity} $D = D(R;s)$ of $s$ relative to $R$ to be the least natural number $D$ for which one has a representation of the form
\begin{equation}\label{rat}
 s = \frac{P(r_1,\dots,r_k)}{Q(r_1,\dots,r_k)}
 \end{equation}
where $P,Q$ are polynomials of degree at most $D$ with integer coefficients of magnitude at most $2^D$, with $Q(r_1,\dots,r_k)$ non-zero; this complexity is finite since $s$ is rational.
\end{definition}

\begin{example} $D(\{0,1,\infty\};s)$ is the least natural number for which one can express $s$ as $a/b$ where $a,b$ are integers of magnitude at most $2^D$. In particular, if $a,b$ are coprime then
$$D \left( \{0,1,\infty\};\frac{a}{b} \right) \asymp \log(2+|a|+|b|).$$
\end{example}

Our main result, proven in Section \ref{main-sec}, is then as follows.

\label{main-sec}

We now prove Theorem \ref{main-2}.  We begin with part (i).  Write $s=a/b$, where $b \geq 1$ and $a \neq 0,b$ is coprime to $b$, then $D(R;s) \asymp \log(2+|a|+|b|)$.  Let $N$ be the

\begin{definition}[Rational complexity] Given a family of slopes $R = \{0,1,\infty,r_1,\dots,r_k\}$ containing $0,1,\infty$ and a further slope $s$ not in $R$, we define the \emph{rational complexity} $D = D(R;s)$ of $s$ relative to $R$ to be the least natural number $D$ for which one has a representation of the form
\begin{equation}\label{rat}
 s = \frac{P(r_1,\dots,r_k)}{Q(r_1,\dots,r_k)}
 \end{equation}
where $P,Q$ are polynomials of degree at most $D$ with integer coefficients of magnitude at most $2^D$, with $Q(r_1,\dots,r_k)$ non-zero; this complexity is finite since $s$ is rational.
\end{definition}

Our main result, proven in Section \ref{main-sec}, is then as follows.

\begin{figure}
 \centering
\centerline{\includegraphics[width=\linewidth]{best.png}}
 \caption{Lower bounds on $\SD(\{0,1,\infty\}; s)$ obtained by \texttt{AlphaEvolve} for various $s=a/b$, plotted against $2 - 0.16 / \log(|a|+b|)$. The horizontal axis is $|a|+|b|$ (so in some cases, multiple fractions $a/b$ are plotted on a single vertical line).}
 \label{fig:kak}
\end{figure}

\begin{theorem}[Continuous limit]\label{cts-limit} Let $\alpha$ be a real number, and let $a,b$ be coprime integer parameters with $b \to \infty$ and $a/b \to \alpha$. Then
\begin{equation}\label{up}
 \SD\left(\{0,1,\infty\};\frac{a}{b}\right) \geq 2 - \frac{c_\alpha+o(1)}{\log b}
 \end{equation}
 as $b\to \infty$ and $a/b \to \alpha$, where $0 < c_\alpha < \infty$ is the infimum of all quantities
\begin{equation}\label{hf}
 h_2(f_{(\alpha)}) - \max(h(f_0), h(f_\infty), h(f_\alpha))
 \end{equation}
where $f \colon \R^2 \to \R^+$ ranges over smooth compactly supported functions with total mass $\int_{\R^2} f(x,y)\ dx dy$ equal to $1$, $f_{(\alpha)} \colon \R \times [0,1) \to \R^+$ is a ``folded'' version
\begin{equation}\label{falpha-def}
 f_{(\alpha)}(x,y) \coloneqq \sum_{j \in \Z} f(x-\alpha j, y+j)
 \end{equation}
of $f$, $f_0, f_\infty, f_1 \colon \R \to \R^+$ are the projections
\begin{align}
f_0(x) &\coloneqq \int_\R f(x,y)\ dy\label{f0-def}\\
f_\infty(y) &\coloneqq \int_\R f(x,y)\ dx\label{finfty-def}\\
f_1(z) &\coloneqq \int_\R f(x,z- x)\ dx,\label{f1-def}
\end{align}
the two-dimensional differential entropy $h_2(f_{(\alpha)})$ of $f_{(\alpha)}$ is defined as
$$h_2(f_{(\alpha)}) \coloneqq \int_0^1 \int_\R \h(f_{(\alpha)}(x,y))\ dx dy,$$
and the one-dimensional differential entropies $h(f_r)$ for $r=0,1,\infty$ is defined as
$$ h(f_r) \coloneqq \int_\R \h(f_r(x))\ dx.$$
\end{theorem}

\begin{proposition}\label{dilate} Let $X$ be an $\Q$-valued random variable, and for any $a \in \Q$, let $g(a)$ denote the quantity $g(a) \coloneqq \H[X-aX'] - \H[X]$, where $X'$ is an independent copy of $X$.
\begin{itemize}
 \item[(i)] $g(0) = 0$ and $g(1)=d[X;X]$.
 \item[(ii)] For any $a \in \Q$, we have $g(-a) \leq 3g(a)$, and if $a$ is non-zero, $g(a^{-1}) = g(a)$.
 \item[(iii)] For any $k \geq 1$ and $a_1,\dots,a_k \in \Q$, we have $g(a_1 \dots a_k) \leq g(a_1)+ \dots + g(a_k)$ and $g(a_1 + \dots + a_k) \leq g(a_1)+\dots+g(a_k)+(k-1) g(1)$.
 \item[(iv)] If $a$ is a non-zero integer, then $g(a) \leq (4 + 10 \lfloor \log_2 |a| \rfloor) g(1)$.
 \item[(v)] If $R$ is a finite set of slopes containing $0,1,\infty$ of cardinality $k+3$, then
 $$ g(a) \ll D(R;a)^{k+1} \max_{r \in R \backslash \{\infty\}} g(r).$$
\end{itemize}
\end{proposition}

In this section we establish \Cref{cts-limit}. Let $\alpha$, $a$, $b$, $f$, $f_0, f_1, f_\infty$ be as in that theorem. We allow implied constants in the asymptotic notation to depend on $\alpha, f$. We will establish the lower bound
\begin{equation}\label{sd-targ}
 \SD\left(\{0,1,\infty\},\frac{a}{b}\right) \geq 2 - \frac{h_2(f_\alpha) - \max(h(f_0), h(f_\infty), h(f_1))+o(1)}{\log b}.
 \end{equation}
Comparing this with Theorem \ref{main-2}, we conclude that the expression $h_2(f_\alpha) - \max(h(f_0), h(f_\infty), h(f_1))$ is bounded from below, hence $c_\alpha > 0$; by taking an arbitrary test function for $f$ we also see that $c_\alpha < \infty$. Taking $h_2(f_\alpha) - \max(h(f_0), h(f_\infty), h(f_1))$ arbitrarily close to $c_\alpha$, we obtain the desired claim \eqref{up}.

It remains to establish \eqref{sd-targ}.
By the Poisson summation formula, the rapid decrease of the Fourier transform of the smooth compactly supported $f$, and the mass one hypothesis, we have
\begin{equation}\label{bib}
 \frac{1}{b^2} \sum_{n,m \in \Z} f\left(\frac{n}{b}, \frac{m}{b}\right) = 1 + \eps
 \end{equation}
for some $\eps = O(1/b)$; in fact one can get much better decay than this, but for our purposes any decay faster than $1/\log b$ will suffice. We then take $X,Y$ to be supported on the grid $\Z^2$ with probability distribution
$$ \P((X,Y) = (n,m)) \coloneqq \frac{1}{(1+\eps) b^2} \left(\frac{n}{b}, \frac{m}{b}\right)$$
thus $(X,Y)$ takes values in a ball of radius $O(b)$ and
\begin{equation}\label{pnx}
\P((X,Y) = (n,m)) = \frac{1}{b^2} f\left(\frac{n}{b}, \frac{m}{b}\right) + O\left( \frac{1}{b^3} \right).
\end{equation}
By Bezout's theorem, any integer can be uniquely written in the form $bn+am$ for some $0 \leq m < b$, and any other representation of the form $bn'+am'$ takes the form $b(n-aj) + a(m+bj)$. Thus
$$ \P(bX+aY = bn+am) = \sum_{j \in\Z} \P( (X,Y) = (n-aj, m+bj) ).$$
There are only $O(1)$ values of $j$ for which this sum is non-zero, so from \eqref{pnx} one has
$$ \P(bX+aY = bn+m) = \frac{1}{b^2} f_{(a/b)}\left(\frac{n}{b}, \frac{m}{b}\right) + O\left( \frac{1}{b^3} \right).$$
where $f_{(a,b)}: \R \times [0,1) \to \R^+$ is the function
$$f_{(a/b)}(x,y) \coloneqq \sum_{j \in \Z} f\left(x-\frac{a}{b} j, y+j\right).$$
Applying the entropy function $\h$, we conclude that
$$ \h(\P(bX+aY = bn+am)) = \frac{2 \log b}{b^2} f_{(a/b)}\left(\frac{n}{b}, \frac{m}{b}\right) +
\frac{1}{b^2} \h\left(f_{(a/b)}\left(\frac{n}{b}, \frac{m}{b}\right)\right) + O\left( \frac{\log b}{b^3} \right).$$
From \eqref{bib} one has
$$ \frac{1}{b^2} \sum_{n \in\Z; 0 \leq m < b} f_{(a/b)}\left(\frac{n}{b}, \frac{m}{b}\right) = 1 + \eps = 1 + O\left(\frac{1}{b} \right)$$
and from (uniform) Riemann integrability of the $f_{(a/b)}$ one has
$$ \frac{1}{b^2} \sum_{n \in\Z; 0 \leq m < b} \h\left(f_{(a/b)}\left(\frac{n}{b}, \frac{m}{b}\right)\right) = h_2(f_{(a/b)}) + o(1).$$
Finally, from dominated convergence one has
$$ h_2(f_{(a/b)}) = h_2(f_{(\alpha)})+o(1)$$
so we conclude that
$$ \H\left[\pi_{a/b}(X,Y)\right] = \H[bX+aY] = 2 \log b + h_2(f_{(\alpha)}) + o(1).$$
In a similar vein, from another application of Poisson summation and \eqref{f0-def} we see that
$$ \P(X=n) = \frac{1}{b} f_0\left(\frac{n}{b}\right) + O\left(\frac{1}{b^2}\right)$$
for any integer $n$, hence
$$ \h(\P(X=n)) = \frac{\log b}{b} f_0\left(\frac{n}{b}\right) + \frac{1}{b} \h\left(f_0 \left(\frac{n}{b}\right)\right) + O\left(\frac{\log b}{b^2}\right).$$
Since $X = O(b)$, one can sum using Riemann integrability and \eqref{bib} to conclude that
$$ \H[\pi_0(X,Y)] = \H[X]= \log b + h(f_0) + o(1).$$
Similar arguments give
$$ \H[\pi_\infty(X,Y)] = \H[Y] = \log b + h(f_\infty) + o(1)$$
and
$$ \H[\pi_1(X,Y)] = \H[X+Y] = \log b + h(f_1) + o(1).$$
Comparing this with \eqref{sdef}, we obtain \eqref{sd-targ}.

\begin{figure}
    \centering
\centerline{\includegraphics[width=\linewidth]{best.png}}
    \caption{Lower bounds on $\SD(\{0,1,\infty\}; s)$ obtained by \texttt{AlphaEvolve} for various $s=a/b$, plotted against $2 - 0.16 / \log(|a|+b|)$.  The horizontal axis is $|a|+|b|$ (so in some cases, multiple fractions $a/b$ are plotted on a single vertical line).}
    \label{fig:kak}
\end{figure}

The logarithmic convergence in \eqref{2ab} was suggested to us by experiments in \cite{gdm} using \texttt{AlphaEvolve} to obtain lower bounds for $\SD(\{0,1,\infty\}; s)$ for various slopes $s$.  This data was of low accuracy, as \texttt{AlphaEvolve} could only provide lower bounds and not upper bounds for these quantities; nevertheless, a logarithmic decay was numerically evident (see Figure \ref{fig:kak}), and furthermore the approximate discrete gaussian shape of the joint distributions of the random variables $X,Y$ obtained by this tool (see \cite[Figure 18]{gdm}) suggested an approach to make the lower bound in \eqref{2ab} rigorous.  Once this was accomplished, the author was also able to obtain the matching upper bound in \eqref{2ab} using the entropic Pl\"unnecke--Ruzsa calculus.   A modification of these arguments, with some inefficiencies, then gives \eqref{sam}. We tentatively conjecture that the bounds in \eqref{sam} can be improved to be of the form \eqref{2ab} for all $k$, not just $k=0$ (possibly after some slight adjustments to the definition of rational complexity).

In the limit as $a/b$ converges to some real number $\alpha$ and $b \to \infty$, we can obtain a more precise lower bound (also suggested by the aforementioned \texttt{AlphaEvolve} numerics) as follows.

\begin{theorem}[Continuous limit]\label{cts-limit}  Let $\alpha$ be a real number, and let $a,b$ be coprime integer parameters with $b \to \infty$ and $a/b \to \alpha$.  Then
\begin{equation}\label{up}
 \SD\left(\{0,1,\infty\};\frac{a}{b}\right) \geq 2 - \frac{c_\alpha+o(1)}{\log b}
 \end{equation}
 as $b\to \infty$ and $a/b \to \alpha$, where $0 < c_\alpha < \infty$ is the infimum of all quantities
\begin{equation}\label{hf}
 h_2(f_{(\alpha)}) - \max(h(f_0), h(f_\infty), h(f_\alpha))
 \end{equation}
where $f \colon \R^2 \to \R^+$ ranges over smooth compactly supported functions with total mass $\int_{\R^2} f(x,y)\ dx dy$ equal to $1$, $f_{(\alpha)} \colon \R \times [0,1) \to \R^+$ is a ``folded'' version
\begin{equation}\label{falpha-def}
 f_{(\alpha)}(x,y) \coloneqq \sum_{j \in \Z} f(x-\alpha j, y+j)
 \end{equation}
of $f$, $f_0, f_\infty, f_1 \colon \R \to \R^+$ are the projections
\begin{align}
f_0(x) &\coloneqq \int_\R f(x,y)\ dy\label{f0-def}\\
f_\infty(y) &\coloneqq \int_\R f(x,y)\ dx\label{finfty-def}\\
f_1(z) &\coloneqq \int_\R f(x,z- x)\ dx,\label{f1-def}
\end{align}
the two-dimensional differential entropy $h_2(f_{(\alpha)})$  of $f_{(\alpha)}$ is defined as
$$h_2(f_{(\alpha)}) \coloneqq \int_0^1 \int_\R \h(f_{(\alpha)}(x,y))\ dx dy,$$
and the one-dimensional differential entropies $h(f_r)$ for $r=0,1,\infty$ is defined as
$$ h(f_r) \coloneqq \int_\R \h(f_r(x))\ dx.$$
\end{theorem}

We establish this result in \Cref{cts-sec}.
We tentatively conjecture that the upper bound in \eqref{up} is in fact an asymptotic equality, so that the asymptotic behavior of
$ \SD\left(\{0,1,\infty\}, \frac{a}{b}\right)$ is controlled not only by the rational complexity (as represented by the $\log b$ denominator), but also by the variational quantity $c_\alpha$ appearing in the numerator.  It is not clear whether this quantity $c_\alpha$ can be computed exactly; numerically, two-dimensional gaussians are reasonably good candidates for $f$, but in practice they do not extremize the functional \eqref{hf} precisely.

\subsection{Notation}

We use the asymptotic notation $X = O(Y)$, $X \ll Y$, or $Y \gg X$ to denote the assertion that $|X| \leq CY$ for some absolute constant $C$; if we need this implied constant $C$ to depend on some fixed quantities (such as the number $k$ of slopes), we will indicate this in the text.

When there is no possibility of ambiguity, we omit parentheses from pairs $(X,Y)$ of random variables, for instance abbreviating $\H[(X,Y)]$ as $\H[X,Y]$.  Given a random variable $Y$ taking values in some set $S$ and some function $f \colon S \to \R$, we define the expectation
$$ \E_{Y=y} f(y) \coloneqq \sum_y \P(Y=y) f(y)$$
where $y$ ranges over the essential range of $Y$.  While this expression could also be abbreviated as $\E f(Y)$, it will be notationally useful to distinguish between the random variable $Y$ and the possible values $y$ that this variable could take.  For instance, with this notation, the \emph{conditional entropy} $\H[X|Y]$ of one random variable $X$ with respect to another $Y$ can now be defined by the formula
$$ \H[X|Y] \coloneqq \E_{Y=y} \H[ X | Y = y ]$$
where $(X|Y=y)$ is $X$ conditioned to the event $Y=y$ (again, we omit parentheses when there is no possibility of ambiguity).  The chain rule asserts that $\H[X|Y]$ can also be expressed by the formula
$$ \H[X|Y] = \H[X,Y] - \H[Y].$$
The \emph{mutual information} $\I(X:Y)$ between two random variables is given by the formula
$$ \I(X:Y) = \H[X] - \H[X|Y] = \H[Y] - \H(Y|X) = \H[X] + \H[Y] - \H[X,Y].$$
As is well known, $\I(X:Y)$ is non-negative, and vanishes precisely when $X,Y$ are independent.  Equivalently, one has the subadditivity property
$$ \H[X,Y] \leq \H[X] + \H[Y]$$
with equality precisely when $X,Y$ are independent.

We also define the conditional mutual information
$$ \I[X:Y|Z] \coloneqq \E_{Z=z} \I[(X|Z=z):(Y|Z=z)].$$
Clearly, $\I[X:Y|Z]$ is non-negative, and vanishes precisely when $X,Y$ are independent conditionally on $Z$.
From the chain rule we have
\begin{equation}\label{i-split}
\begin{split}
\I[X:Y|Z] &= \H[X|Z] - \H[X|Y,Z] \\
&= \H[Y|Z] - \H[Y|X,Z] \\
&= \H[X|Z] + \H[Y|Z] - \H[X,Y|Z].
\end{split}
\end{equation}

\subsection{Acknowledgments}

The author was supported by the James and Carol Collins Chair, the Mathematical Analysis \& Application Research Fund, and by NSF grants DMS-2347850, and is particularly grateful to recent donors to the Research Fund.  He particularly thanks his coauthors Bogdan Georgiev, Javier G\'omez-Serrano, and Adam Zsolt Wagner for the highly productive and enjoyable collaboration \cite{gdm}, and for generously sharing the outputs of that collaboration for the purposes of writing the current paper.

While some of the results proven here were suggested by the outcome of AI-assisted experiments, the arguments in this paper are completely human-generated.

The post-theorem introduction does not give a step-by-step proof, but it does outline the methods used to establish the bounds in Theorem~\ref{main-2}. For the three-slope case \eqref{2ab}, the “logarithmic convergence” was “suggested… by experiments… using \texttt{AlphaEvolve} to obtain lower bounds,” and the “approximate discrete gaussian shape of the joint distributions of the random variables $X,Y$… suggested an approach to make the lower bound in \eqref{2ab} rigorous.” “Once this was accomplished, the author was also able to obtain the matching upper bound in \eqref{2ab} using the entropic Pl\"unnecke--Ruzsa calculus.” Finally, “a modification of these arguments, with some inefficiencies, then gives \eqref{sam}.”

The post-theorem introduction does not give a step-by-step proof, but it does outline the methods used to establish the bounds in the main theorem. For the three-slope case
\begin{equation}\label{2ab} 2 - \frac{c_2}{D(R;s)} \leq \SD\left(R; s\right) \leq 2 - \frac{c_1}{D(R;s)}
\end{equation}
, the “logarithmic convergence” was “suggested… by experiments… using \texttt{AlphaEvolve} to obtain lower bounds,” and the “approximate discrete gaussian shape of the joint distributions of the random variables $X,Y$… suggested an approach to make the lower bound in the equation above rigorous.” “Once this was accomplished, the author was also able to obtain the matching upper bound in the equation above using the entropic Pl\"unnecke--Ruzsa calculus.” Finally, “a modification of these arguments, with some inefficiencies, then gives
\begin{equation}\label{sam} 2 - \frac{C_k \log(2+D(R;s))}{D(R;s)} \leq \SD(R; s) \leq 2 - \frac{c_k}{D(R;s)^{k+1}}
    \end{equation}
.”

Question:
Let a slope mean an element of \(\mathbb{Q}\cup\{\infty\}\). For each slope \(r\), define \(\pi_r:\mathbb{Q}\times\mathbb{Q}\to\mathbb{Q}\) by \(\pi_r(x,y)=x+ry\) if \(r\neq\infty\), and \(\pi_\infty(x,y)=y\). For a finite family of slopes \(R\) and a slope \(s\notin R\), let \(\mathrm{SD}(R;s)\) be the least exponent such that
\[
H[\pi_s(X,Y)]\le \mathrm{SD}(R;s)\max_{r\in R} H[\pi_r(X,Y)]
\]
for all finitely supported \(\mathbb{Q}\)-valued random variables \(X,Y\) (not necessarily independent), where \(H[Z]=\sum_z h(\mathbb{P}(Z=z))\) is Shannon entropy and \(h(t)=t\log(1/t)\) with \(h(0)=0\). Now let \(R=\{0,1,\infty,r_1,\dots,r_k\}\subset \mathbb{Q}\cup\{\infty\}\) be a finite family of slopes of cardinality \(k+3\), and let \(s\notin R\). Define the rational complexity \(D(R;s)\) to be the least natural number \(D\) for which
\[
s=\frac{P(r_1,\dots,r_k)}{Q(r_1,\dots,r_k)}
\]
for some polynomials \(P,Q\) of degree at most \(D\) with integer coefficients of magnitude at most \(2^D\), with \(Q(r_1,\dots,r_k)\neq 0\). Which statement holds for every such \(R\) and \(s\)?

If \(k=0\) (so \(R=\{0,1,\infty\}\)), then there exist absolute constants \(c_2>c_1>0\) such that
\[
2-\frac{c_2}{D(R;s)}\le \mathrm{SD}(R;s)\le 2-\frac{c_1}{D(R;s)}.
\]
More generally, for arbitrary \(k\), there exist constants \(c_k,C_k>0\) depending only on \(k\) such that
\[
2-\frac{C_k\log(2+D(R;s))}{D(R;s)}\le \mathrm{SD}(R;s)\le 2-\frac{c_k}{D(R;s)^{k+1}}.
\]

Choices:
B.
If \(k=0\) (so \(R=\{0,1,\infty\}\)), then there exist absolute constants \(c_2>c_1>0\) such that
\[
2-\frac{c_2\log(2+D(R;s))}{D(R;s)}\le \mathrm{SD}(R;s)\le 2-\frac{c_1\log(2+D(R;s))}{D(R;s)}.
\]
More generally, for arbitrary \(k\), there exist constants \(c_k,C_k>0\) depending only on \(k\) such that
\[
2-\frac{C_k}{D(R;s)}\le \mathrm{SD}(R;s)\le 2-\frac{c_k}{D(R;s)^{k+1}}.
\]
C.
For arbitrary \(k\), there exist constants \(c_k,C_k>0\) depending only on \(k\) such that
\[
2-\frac{C_k\log(2+D(R;s))}{D(R;s)}\le \mathrm{SD}(R;s)\le 2.
\]
In particular, when \(k=0\), one has \(2-\frac{C_0\log(2+D(R;s))}{D(R;s)}\le \mathrm{SD}(R;s)\le 2\).
D.
If \(k=0\) (so \(R=\{0,1,\infty\}\)), then there exist absolute constants \(c_2>c_1>0\) such that
\[
2-\frac{c_2}{D(R;s)}\le \mathrm{SD}(R;s)\le 2-\frac{c_1}{D(R;s)^2}.
\]
More generally, for arbitrary \(k\), there exist constants \(c_k,C_k>0\) depending only on \(k\) such that
\[
2-\frac{C_k\log(2+D(R;s))}{D(R;s)}\le \mathrm{SD}(R;s)\le 2-\frac{c_k}{D(R;s)^k}.
\]
E.
If \(k=0\) (so \(R=\{0,1,\infty\}\)), then there exist absolute constants \(c_2>c_1>0\) such that
\[
2-\frac{c_2}{D(R;s)}\le \mathrm{SD}(R;s)\le 2-\frac{c_1}{D(R;s)}.
\]
More generally, there exist absolute constants \(c,C>0\), independent of \(k\), \(R\), and \(s\), such that for every \(k\)
\[
2-\frac{C\log(2+D(R;s))}{D(R;s)}\le \mathrm{SD}(R;s)\le 2-\frac{c}{D(R;s)^{k+1}}.
\]

special k=0 logarithmic-vs-linear behavior

nontrivial upper bound improvement below 2

upper-bound exponent in D for many slopes and three-slope matching rate