\label{thm:gawron-generalization}
	If $z\in \mathcal S$, then $|z| > \tfrac 14$.

\begin{thm}\label{thm:MO-ineq}
	For all real numbers $t$ and all positive integers $n$,
	\begin{equation}\label{eqn:tao-ineq}
		\abs{\left(1+\frac{it}n\right)^n - 1} \geq \big|e^{it} - 1\big| = 2\sin\frac t2.
	\end{equation}
\end{thm}

\begin{thm}\label{thm:roots-disk}
	All roots of $\stern n\lm$ lie outside the disk $\{\abs{z-2}\leq 1\}\subseteq\C$.
\end{thm}

\begin{thm}\label{thm:gawron-generalization}
	If $z\in \calS$, then $|z| > \tfrac 14$.
\end{thm}

\begin{conj}\label{conj:location-roots}
	All elements of $\calS$ lie in the half-plane $\{\Re w < 1\}$.
\end{conj}

\begin{figure}[ht]
	\centering
	\captionsetup{width=0.75\linewidth}
	\includegraphics[scale=0.75]{sternpoly-roots-deg-21-BW}
	\caption{\small The zeros of $\stern n\lm$ in the range $\{a+bi:-4\leq a\leq 1, |b|\leq 3\}$, where $1\leq n<2^{21}$ is odd.}
	\label{fig:stern-roots}
\end{figure}

\begin{corr}\label{thm:prime-poly-irred}
	For each prime number $p$, the Stern polynomial $\stern p \lm$ is irreducible in $\Q$.
\end{corr}

The Stern sequence $(s_n)_{n\geq 0}$ of positive integers, named after Moritz Abraham Stern, is given by $s_0 = 0$, $s_1 = 1$, and
\begin{equation}\label{eqn:stern-seq}
 s_{2n} = s_n,\quad s_{2n+1} = s_n + s_{n+1}.
\end{equation}
There is a vast body of literature (e.g. \cite{Coons2014,Northshield2010,Reznick2008,Stanley2019} and the references therein) about the Stern sequence. Perhaps the most striking property of the Stern sequence is that every positive rational number appears exactly once in the sequence $(s_{n+1}/s_n)_{n\geq 1}$, giving an explicit enumeration of $\mathbb Q^+$. Stern proved this fact 15 years before Cantor introduced the notion of a countable set!

\par In \cite{Klavzar2007}, Klav\v{z}ar, Milutinovi\'{c}, and Petr define a polynomial analogue $\stern n\lambda$ of the Stern sequence via $\stern 0\lambda = 0$, $\stern 1\lambda = 1$, and
\begin{equation}\label{eqn:stern-def}
 \stern{2n}\lambda = \lambda \stern n\lambda,\quad \stern{2n+1}\lambda = \stern n\lambda + \stern{n+1}\lambda.
\end{equation}
The first $16$ terms of the sequence $\stern n\lambda$ are given in Table \ref{table:stern-16}. This table suggests many patterns, several of which are recorded in Proposition \ref{prop:basic-patterns}.

\begin{table}[ht]
 \input{table-stern-16}
 \caption{\small The first $16$ terms of the sequence $\stern n\lambda$.}
 \label{table:stern-16}
\end{table}

While there are several papers which analyze the sequence $\stern n\lambda$ (see \cite{Dilcher2017,Dilcher2018,Gawron2014,Ulas2011} and the references therein), not much is known about the zeros of Stern polynomials. Because $\stern{2n}\lambda = \lambda \stern n \lambda$ for all $n\geq 1$, it suffices to examine the roots of $\stern n\lambda$ only when $n$ is odd. Define
\[
\mathcal S\coloneqq \{z\in\mathbb C: \stern n z = 0\text{ for some odd }n\geq 1\}.
\]
In \cite[Theorem 2.1]{Gawron2014}, Gawron proves that $0$, ${-}1$, ${-}\tfrac12$, and $-\tfrac13$ are the only rational zeros of any Stern polynomial by showing, more generally, that any real number in the closed interval $[-\tfrac14,\tfrac14]$ cannot be in $\mathcal S$. His methods extend easily to show that any complex number $z$ with $|z| \leq \tfrac14$ cannot be in $\mathcal S$. For completeness, we record the proof here.

\begin{prop}\label{prop:basic-patterns}
	Let $n$ be a nonnegative integer.

	\vspace{-3pt}

	\begin{enumerate}
		\item $\stern{n}{0} = 0$ when $n$ is even, and $\stern n 0 = 1$ when $n$ is odd.
		\item $\stern{n}{1} = s_n$, where $s_n$ is the Stern sequence \eqref{eqn:stern-seq}.
		\item $\stern{n}{2} = n$.
		\item When $n = 2^r$ for a nonnegative integer $r$, $\stern n \lm = \lm^r$.
		\item When $n = 2^r - 1$ for a nonnegative integer $r$, $\stern n\lm = (\lm)_r$.
	\end{enumerate}
\end{prop}

\begin{table}[ht]
	\input{table-stern-16}
	\caption{\small The first $16$ terms of the sequence $\stern n\lm$.}
	\label{table:stern-16}
\end{table}

While there are several papers which analyze the sequence $\stern n\lm$ (see \cite{Dilcher2017,Dilcher2018,Gawron2014,Ulas2011} and the references therein), not much is known about the zeros of Stern polynomials. Because $\stern{2n}\lm = \lm \stern n \lm$ for all $n\geq 1$, it suffices to examine the roots of $\stern n\lm$ only when $n$ is odd. Define
\[
\calS\coloneqq \{z\in\C: \stern n z = 0\text{ for some odd }n\geq 1\}.
\]
In \cite[Theorem 2.1]{Gawron2014}, Gawron proves that $0$, ${-}1$, ${-}\tfrac12$, and $-\tfrac13$ are the only rational zeros of any Stern polynomial by showing, more generally, that any real number in the closed interval $[-\tfrac14,\tfrac14]$ cannot be in $\calS$. His methods extend easily to show that any complex number $z$ with $|z| \leq \tfrac14$ cannot be in $\calS$. For completeness, we record the proof here.

\begin{proof}
 Let $z$ be any complex number with $|z| \leq \tfrac 14$. Let $b_n\coloneqq |\stern n z|$ for each positive integer $n$. We show more strongly that
 \begin{equation}\label{eqn:stronger-statement}
 b_{2n+1} > \tfrac12\max\{b_n,b_{n+1}\} > 0
 \end{equation}
 for all $n\geq 1$.

\begin{thm}[Parabola Theorem, {\cite[Theorem 3.43]{Lorentzen2008}}]\label{thm:parabola}
 For fixed $|\alpha| < \tfrac\pi 2$, let
 \[
 V_\alpha\coloneqq -\tfrac12 + e^{i\alpha}\overline{\mathbb H} = \{w\in\C: \Re(we^{- i\alpha})\geq -\tfrac12\cos\alpha\}
 \]
 and
 \[
 E_\alpha \coloneqq \{a\in\C: |a| - \Re(ae^{-2i\alpha})\leq \tfrac12\cos^2\alpha\}.
 \]
 Then $E_\alpha$ is the element set for continued fractions $\boldsymbol{K}(a_n | 1)$ corresponding to the value set $V_\alpha$.
\end{thm}
The region $V_\alpha$ is a half-plane whose boundary is a line intersecting the real axis at $z = -\tfrac12$. In particular, because $|\alpha| < \tfrac\pi 2$, $V_\alpha$ contains the half-line $[-\tfrac12,\infty)$. The boundary of $E_\alpha$ is a parabola with focus at the origin and vertex at $-\tfrac14 e^{2i\alpha}\cos^2\alpha$. This parabola intersects the real axis at $z = -\tfrac14$.

We now tie the continued fraction theory back to Stern polynomials.
Recall that Corollary \ref{thm:stern-cont-frac} expresses the ratio
of two Stern polynomials as a continued fraction with elements in $\C$.
This form on its own is difficult to work with. We instead note (e.g. \cite[Corollary 2.15]{Lorentzen2008}) that any continued fraction $\boldsymbol{K}(a_n|b_n)$ with $b_j\neq 0$ is equivalent to the continued fraction $\boldsymbol{K}(c_n|1)$, where $c_j = \tfrac{a_j}{b_jb_{j+1}}$. As an explicit example,
\begin{equation}\label{eqn:cfrac-example}
 \frac{\stern{[[2,3,5]]}\lm}{\stern{[[3,5]]}\lm} = (\lm)_2 + \cfrac{\lm^2}{(\lm)_3 + \cfrac{\lm^3}{(\lm)_5}}
 = (\lm)_2 \left[1 + \cfrac{\dfrac{\lm^2}{(\lm)_2(\lm)_3}}{1 + \cfrac{\lm^3}{(\lm)_3(\lm)_5}}\right].
\end{equation}
\begin{remark}
 While continued fractions of the form $\boldsymbol{K}(1|d_n)$ are more common, the coefficients $d_n$ are significantly messier than the coefficients $c_n$ (see \cite[Corollary 2.15]{Lorentzen2008}). We opt to use the less-common form $\boldsymbol{K}(c_n|1)$ to simplify our analysis.
\end{remark}
Equation \eqref{eqn:cfrac-example} suggests that the ratios $\tfrac{z^a}{(z)_a(z)_b}$, where $a$ and $b$ are any positive integers, may play some importance.
For this reason, we define $z_{a,b}\coloneqq \tfrac{z^a}{(z)_a(z)_b}$ and
\[
\calA_z\coloneqq \left\{z_{a,b}:(a,b)\in\N^2\right\}.
\]
Suppose $\calA_z\subseteq E_\alpha$ for some angle $\alpha$ with $|\alpha| < \tfrac{\pi}2$.
Then $V_\alpha$ is a value set for the continued fraction \eqref{eqn:stern-cont-frac} at $\lm = z$. In particular, $\infty$ is not a possible value for this fraction, so $\stern n z \neq 0$ for any $z$.

\begin{itemize}
 \item First suppose $x\in[0,\pi]$. Because both $\sin x$ and $x - \tfrac{x^2}{\pi}$ are symmetric about the axis $x = \tfrac\pi 2$, it suffices to prove the inequality for $x\in[0,\tfrac\pi 2]$. Within this smaller interval, cosine is concave down with $\cos 0 = 1$ and $\cos\tfrac \pi 2 = 0$, so
 \begin{equation}\label{eqn:cos-ineq}
 \cos x \geq 1 - \frac{2x}{\pi}.
 \end{equation}
 Integrating both sides of \eqref{eqn:cos-ineq} yields \eqref{ineq-sinc} in this case.
 \item Now suppose $x\notin[0,\pi]$. By symmetry, it suffices to prove the inequality for $x\geq \pi$. In this interval, $1 - \tfrac{2x}{\pi} \leq -1 \leq \cos x$. It follows that \eqref{eqn:cos-ineq}, and thus \eqref{ineq-sinc}, holds as well.
 \end{itemize}
\end{proof}

\begin{thm}\label{thm:min-geom-series}
 Let $n$ be a positive integer, and suppose $z\in\calB$. Then $|(z)_n| \geq \min(n,\tfrac{11}2)$.
\end{thm}

Now let
 \[
 M\coloneqq 2\pi\left(1 - \frac{5.5}{n}\right) = \pi\left(2 - \frac{11}n\right).
 \]
 There are two cases to consider. First, suppose $t\leq M$. Then
 \[
 n\left(1 - \frac{t}{2\pi}\right) \geq n\left(1 - \frac{M}{2\pi}\right) = \frac{11}2.
 \]
 Now suppose $t\geq M$. Because $n\geq 15$, the coefficient $\tfrac{1}{2n} - \tfrac{11}{8n^2}$ is positive, so
 \begin{equation}\label{eqn:asymp-w-tn}
 t + t^3\left(\frac{1}{2n} - \frac{11}{8n^2}\right) \geq \pi\left(2 - \frac{11}n\right) + \pi^3\left(2 - \frac{11}n\right)^3\left(\frac{1}{2n} - \frac{11}{8n^2}\right).
 \end{equation}
 To estimate \eqref{eqn:asymp-w-tn}, let $x = \tfrac 1n$, so it suffices to analyze the polynomial
 \[
 P(x) := \pi(2-11x) + \pi^3(2-11x)^3(\tfrac 12x - \tfrac{11}{8}x^2).
 \]
 The polynomial $P$ has exactly one critical point in the interval $[0,\tfrac 1{15}]$, occurring at $x = x_0 \approx 0.024$. In particular, $P$ is increasing on $[0,x_0]$ and decreasing on $[x_0,\tfrac1{15}]$. Compute
 \[
 P(0) = 2\pi\quad\text{and}\quad P\left(\frac 1{15}\right) = \frac{19 \pi }{15}+\frac{336091 \pi ^3}{6075000} \approx 5.695 > \frac{11}2.
 \]
 It follows that $P(x) > \tfrac{11}2$ for all $x\in[0,\tfrac{1}{15}]$, and thus $|w_{n,t}^n - 1| \geq M|w_{n,t}-1|$ for all $t\in[0,2\pi]$ and $n\geq 15$.
\end{proof}

\begin{prop}\label{prop:silli}
 Let $z\in\calB^+$. Then $0\leq \arg z \leq \tfrac\pi 6$ and
 \begin{equation}\label{eqn:re-ineq}
 |\Re z^{-4}|\leq |\Re z^{-2}|.
 \end{equation}
\end{prop}

\begin{corr}[{\cite[Theorem 1]{Schinzel2014}}]\label{thm:stern-cont-frac}
	Let $a_1,\ldots, a_t$ be positive integers, where $t\geq 2$.  Then
	\begin{equation}\label{eqn:stern-cont-frac}
		\frac{\stern{[[a_1,\ldots, a_t]]}{\lm}}{\stern{[[a_2,\ldots, a_t]]}{\lm}}
		=
		(\lm)_{a_1} + \cfrac{\lm^{a_1}}{(\lm)_{a_2} + \cfrac{\lm^{a_2}}{\cdots + \cfrac{\cdots}{(\lm)_{a_{t-1}} + \cfrac{\lm^{a_{t-1}}}{(\lm)_{a_t}}}}}\,.
	\end{equation}
\end{corr}

\begin{proof}
	Let $z$ be any complex number with $|z| \leq \tfrac 14$.  Let $b_n\coloneqq |\stern n z|$ for each positive integer $n$.  We show more strongly that
	\begin{equation}\label{eqn:stronger-statement}
		b_{2n+1} > \tfrac12\max\{b_n,b_{n+1}\} > 0
	\end{equation}
	for all $n\geq 1$.

	\par The proof of \eqref{eqn:stronger-statement} proceeds by induction on $n$.  The base case, $n = 1$, follows because
	\[
	b_3 = |z + 1| \geq \tfrac 34 > \tfrac 12
	= \tfrac 12\max\{1, |z|\} = \tfrac 12\max\{b_1,b_2\}.
	\]

	\par There are two cases to consider.  First assume $n=2k$ is even.  Then
	\begin{align*}
		b_{4k+1} &= |\stern{4k+1}z| = |\stern{2k+1} z + \stern{2k} z|\\
		&= |z \stern kz + \stern{2k+1} z| \geq b_{2k+1} - \tfrac14 b_k\\
		&\geq b_{2k+1} - \tfrac12 b_{2k+1} = \tfrac12 b_{2k+1}.
	\end{align*}
	Moreover, $b_{2k+1} \geq \frac12 b_k$, and thus $\max\{b_{2k},b_{2k+1}\} = b_{2k+1} > 0$.  In this case, our inequality is proved.

	\par Now assume $n = 2k + 1$ is odd.  Then
	\begin{align*}
		b_{4k+3} &= |\stern{4k+3}z| = |\stern{2k+1} z + \stern{2k+2} z|\\
		&= |\stern{2k+1} z + z\stern{k+1} z| \geq b_{2k+1} - \frac 14 b_{k+1}\\
		&\geq b_{2k+1} - \tfrac12 b_{2k+1} = \tfrac12 b_{2k+1}.
	\end{align*}
	Moreover, $b_{2k+1} \geq \frac12 b_{k+1}$, and so in this case we also have $\max\{b_{2k},b_{2k+1}\} = b_{2k+1} > 0$.  We have exhausted both cases, completing the proof of Theorem \ref{thm:gawron-generalization}.
\end{proof}

In \cite{Dilcher2017}, Dilcher et. al. focus more specifically on the complex roots of $\stern n\lambda$.  Their paper makes the following conjecture.

\begin{conj}\label{conj:location-roots}
	All elements of $\mathcal S$ lie in the half-plane $\{\operatorname{Re} w < 1\}$.
\end{conj}

By generalizing the Enestrom-Kakeya theorem, they prove Conjecture \ref{conj:location-roots} for several classes of positive integers $n$ taking the form $2^n\pm k$, where $k$ is fixed and $2^n \geq k$.  These are the only two papers the author could find which discuss the complex zeros of $\stern n\lambda$.

\par Figure \ref{fig:stern-roots} shows a snapshot of $\mathcal S$.  One striking feature of this figure is the contrasting behavior of these roots within the half-planes $\{\operatorname{Re} w\geq 0\}$ and $\{\operatorname{Re} w < 0\}$.  These differences present difficulties in fully characterizing the geometry of $\mathcal S$.

\begin{figure}[ht]
	\centering
	\captionsetup{width=0.75\linewidth}
	\includegraphics[scale=0.75]{sternpoly-roots-deg-21-BW}
	\caption{\small The zeros of $\stern n\lambda$ in the range $\{a+bi:-4\leq a\leq 1, |b|\leq 3\}$, where $1\leq n<2^{21}$ is odd.}
	\label{fig:stern-roots}
\end{figure}

In this paper, we partially resolve Conjecture \ref{conj:location-roots} by establishing the following result.

\begin{thm}\label{thm:roots-disk}
	All roots of $\stern n\lambda$ lie outside the disk $\{\left|z-2\right|\leq 1\}\subseteq\mathbb C$.
\end{thm}

To prove Theorem \ref{thm:roots-disk}, we use a continued fraction representation for a ratio of Stern polynomials (Theorem \ref{thm:stern-cont-frac}) independently discovered by Reznick \cite{Reznick2008} and Schinzel \cite{Schinzel2014}.  This allows us to use the Parabola Theorem (Theorem \ref{thm:parabola}) to show that, for certain values of $z\in\mathbb C$, the denominators of these continued fractions can never be zero.  Along the way, we establish inequalities in $\mathbb C$ relating to the sums $1 + z + \cdots + z^{n-1}$ which may be of independent interest.  The most notable of these inequalities is Theorems \ref{thm:MO-ineq}, which proves a lower bound for this geometric series whenever $\operatorname{Re} z \geq 1$.

As a corollary, we obtain the following surprising fact, resolving a conjecture of Ulas and Ulas (\cite{Ulas2011}).

\begin{corr}\label{thm:prime-poly-irred}
	For each prime number $p$, the Stern polynomial $\stern p \lambda$ is irreducible in $\mathbb Q$.
\end{corr}
Ulas and Ulas had verified this conjecture computationally for the first million primes $p$.  Additionally, Schinzel in \cite{Schinzel2011} proved Corollary \ref{thm:prime-poly-irred} for all primes $p < 2017$ by using finite differences to bound the leading coefficient of any proper divisor of $\stern n\lambda$.  However, these previous attempts to prove the conjecture were algebraic in nature, whereas our proof depends on the analytic properties of $\mathcal S$.

\paragraph{Acknowledgments.} This paper is adapted from the author's PhD dissertation \cite{Altizio2025}.  The author thanks his advisor, Dr. Bruce Reznick, for helpful correspondence.

Let $z$ be any complex number with $|z|\leq \tfrac14$ and set $b_n\coloneqq |\stern n z|$. The proof shows the stronger claim
\begin{equation}\label{eqn:stronger-statement}
 b_{2n+1} > \tfrac12\max\{b_n,b_{n+1}\} > 0
\end{equation}
for all $n\ge1$, and proceeds by induction on $n$.

Base case $n=1$:
\[
 b_3=|z+1|\ge \tfrac34>\tfrac12=\tfrac12\max\{1,|z|\}=\tfrac12\max\{b_1,b_2\}.
\]

Inductive step splits into two cases.

(i) If $n=2k$ is even, then
\[
 b_{4k+1}=|\stern{4k+1}z|=|\stern{2k+1}z+\stern{2k}z|=|z\,\stern k z+\stern{2k+1}z|
 \ge b_{2k+1}-\tfrac14 b_k\ge b_{2k+1}-\tfrac12 b_{2k+1}=\tfrac12 b_{2k+1}.
\]
Moreover, $b_{2k+1}\ge \tfrac12 b_k$, hence $\max\{b_{2k},b_{2k+1}\}=b_{2k+1}>0$, giving \eqref{eqn:stronger-statement}.

(ii) If $n=2k+1$ is odd, then
\[
 b_{4k+3}=|\stern{4k+3}z|=|\stern{2k+1}z+\stern{2k+2}z|=|\stern{2k+1}z+z\,\stern{k+1}z|
 \ge b_{2k+1}-\tfrac14 b_{k+1}\ge b_{2k+1}-\tfrac12 b_{2k+1}=\tfrac12 b_{2k+1}.
\]
Moreover, $b_{2k+1}\ge \tfrac12 b_{k+1}$, so again $\max\{b_{2k},b_{2k+1}\}=b_{2k+1}>0$. These two cases exhaust the induction, completing the proof of Theorem~\ref{thm:gawron-generalization}.

Question:
For the Stern polynomials \(\operatorname{St}_n(\lambda)\) defined by \(\operatorname{St}_0(\lambda)=0\), \(\operatorname{St}_1(\lambda)=1\), and for all integers \(n\ge 0\),
\[
\operatorname{St}_{2n}(\lambda)=\lambda\,\operatorname{St}_n(\lambda),\qquad
\operatorname{St}_{2n+1}(\lambda)=\operatorname{St}_n(\lambda)+\operatorname{St}_{n+1}(\lambda),
\]
define
\[
\mathcal S:=\{z\in\mathbb C: \operatorname{St}_n(z)=0\text{ for some odd }n\ge 1\}.
\]
Which quantitative estimate holds for every \(z\in\mathcal S\)?

Every \(z\in\mathcal S\) satisfies \(|z|>\tfrac14\).

Choices:
B. Every \(z\in\mathcal S\) satisfies \(|z|\ge \tfrac14\).
C. Every \(z\in\mathcal S\) satisfies \(|z|>0\).
D. Every \(z\in\mathcal S\) satisfies \(\Re(z)>-\tfrac14\).
E. Every \(z\in\mathcal S\) satisfies \(|z|>\tfrac13\).

The stem defines the Stern polynomials and the set \(\mathcal S\), but it does not explicitly state or strongly hint at the bound \(|z|>\tfrac14\). The correct estimate is not leaked by wording alone.

This is close to a theorem-recall item: after defining \(\mathcal S\), it asks directly which universal estimate holds. The multiple choices introduce some comparison, but the task is still largely a reformulation of the target conclusion rather than a genuinely new inference problem.

There is some reasoning pressure because the student must distinguish between a sharp strict bound, a weak true statement, and stronger or unrelated false variants. However, it mainly tests recognition of the known result rather than substantial derivation or synthesis.

The distractors are well designed: \(\ge \tfrac14\) probes strict vs. non-strict misunderstanding, \(|z|>0\) is a weaker true statement, \(|z|>\tfrac13\) is an overstrengthening trap, and \(\Re(z)>-\tfrac14\) is a distinct but irrelevant-looking alternative. They are plausible and target common mathematical failure modes.

A solid MCQ with strong distractors and no answer leakage, but it is somewhat theorem-recall based and only moderately tests generative reasoning.


\begin{definition}[Strong non-integrability condition]    \label{def:strong non-integrability condition}
    Let $f \colon S^2 \mapping S^2$ be an expanding Thurston map and $d$ be a visual metric on $S^2$ for $f$.
    Fix $\holderexp \in (0, 1]$.
    Let $\potential \in \holderspacesphere$ be a real-valued \holder continuous function with an exponent $\holderexp$.
    \begin{enumerate}[label=\rm{(\arabic*)}]
        \smallskip
        \item We say that $\potential$ satisfies the \emph{$(\mathcal{C}, \holderexp)$-strong non-integrability condition} (with respect to $f$ and $d$), for a Jordan curve $\mathcal{C} \subseteq S^2$ with $\post{f} \subseteq \mathcal{C}$, if there exist
        \begin{enumerate}[label=\rm{(\alph*)}]
            \smallskip
            \item numbers $\juxtapose{N}{M} \in \n$, $\varepsilon \in (0, 1)$,
            \smallskip
            \item $M$-tiles $Y^{M}_{\black} \in \cTile{M}{\black}$, $Y^{M}_{\white} \in \cTile{M}{\white}$
        \end{enumerate}
        such that for each $\colour \in \colours$, each integer $m \geqslant M$, and each $m$-tile $X \in \Tile{m}$ with $X \subseteq Y^{M}_{\colour}$, there exist two points $\juxtapose{x_{1}}{x_{2}} \in X$ with the following properties:
        \begin{enumerate}
            \smallskip
            \item $\min \set[\big]{ d \parentheses[\big]{ x_{1}, S^2 \mysetminus X }, d \parentheses[\big]{ x_{2}, S^2 \mysetminus X }, d(x_{1}, x_{2}) } \geqslant \diameter{d}{X}$, and
            \smallskip
            \item for each integer $n \geqslant N$, there exist two $(n + M)$-tiles $\juxtapose{X^{n + M}_{\colour, 1}}{X^{n + M}_{\colour, 2}} \in \Tile{n + M}$ such that $Y^{M}_{\colour} = f^{n} \parentheses[\big]{ X^{n + M}_{\colour, 1} } = f^{n} \parentheses[\big]{ X^{n + M}_{\colour, 2} }$ and
            \[
                \abs{ S_{n}\potential(\varsigma_{1}(x_{1})) - S_{n}\potential(\varsigma_{2}(x_{1})) - S_{n}\potential(\varsigma_{1}(x_{2})) + S_{n}\potential(\varsigma_{2}(x_{2})) } \geqslant \varepsilon d( x_{1}, x_{2})^{\holderexp},
            \]
            where we write $\varsigma_{i} \define \parentheses[\big]{ f^{n}|_{X^{n + M}_{\colour, i}} }^{-1}$ for each $i \in \set{1, 2}$.
        \end{enumerate}
        \smallskip

        \item We say that $\potential$ satisfies the \emph{$\holderexp$-strong non-integrability condition} (with respect to $f$ and $d$) if $\potential$ satisfies the $(\mathcal{C}, \holderexp)$-strong non-integrability condition with respect to $f$ and $d$ for some Jordan curve $\mathcal{C} \subseteq S^2$ with $\post{f} \subseteq \mathcal{C}$.

        \smallskip

        \item We say that $\potential$ satisfies the \emph{strong non-integrability condition} (with respect to $f$ and $d$) if $\potential$ satisfies the $\holderexp'$-strong non-integrability condition with respect to $f$ and $d$ for some $\holderexp' \in (0, \holderexp]$.
    \end{enumerate}
\end{definition}

\begin{definition}[Eventually positive function]    \label{def:eventually positive functions}
    Let $g \colon X \mapping X$ be a map on a set $X$, and $\varphi \colon X \mapping \real$ be a real-valued function on $X$.
    Then $\varphi$ is \emph{eventually positive} if there exists $N \in \n$ such that $S_n \varphi(x) > 0$ for each $x \in X$ and each $n \in \n$ with $n \geqslant N$.
\end{definition}

\begin{equation}  \label{eq:Variational Principle for entropy}
    h_{\operatorname{top}}(g) = \sup\{h_{\mu}(g) \describe\mu \in \mathcal{M}(X, g)\}.
\end{equation}
A measure $\mu$ that attains the supremum in \eqref{eq:Variational Principle for pressure} is called an \emph{equilibrium state} for the map $g$ and the potential $\psi$. A measure $\mu$ that attains the supremum in \eqref{eq:Variational Principle for entropy} is called a \emph{measure of maximal entropy} of $g$.

Let $\widetilde{X}$ be another compact metric space. If $\mu$ is a measure on $X$ and the map $\pi \colon X \mapping \widetilde{X}$ is continuous, then the \emph{push-forward} $\pi_{*} \mu$ of $\mu$ by $\pi$ is the measure given by $\pi_{*}\mu(A) \define \mu \parentheses[\big]{ \pi^{-1}(A) }$ for all Borel sets $A \subseteq \widetilde{X}$.

\subsection{Thurston maps}\label{sub:Thurston_maps}

In this subsection, we go over some key concepts and results on Thurston maps, and expanding Thurston maps in particular.
For a more thorough treatment of the subject, we refer to \cite{bonk2017expanding,li2017ergodic}.

Let $S^2$ denote an oriented topological $2$-sphere and $f \colon S^2 \mapping S^2$ be a branched covering map.
We denote by $\deg_f(x)$ the local degree of $f$ at $x \in S^2$.
The \emph{degree} of $f$ is $\deg{f} = \sum_{x\in f^{-1}(y)} \deg_{f}(x)$ for $y\in S^2$ and is independent of $y$.

A point  $x\in S^2$ is a \emph{critical point} of $f$ if $\deg_f(x) \geqslant 2$.
The set of critical points of $f$ is denoted by $\crit{f}$. A point $y\in S^2$ is a \emph{postcritical point} of $f$ if $y = f^n(x)$ for some $x \in \crit{f}$ and $n\in \n$.
The set of postcritical points of $f$ is denoted by $\post{f}$.
We observe that $\post{f} = \post{f^{n}}$ for all $n \in \n$.

\begin{definition}[Thurston maps]
    A Thurston map is a branched covering map $f \colon S^2 \mapping S^2$ on $S^2$ with $\deg f \geqslant 2$ and $\card{ \post{f} }  < +\infty$.
\end{definition}

We now recall the notation for cell decompositions of $S^2$ as used in \cite{bonk2017expanding} and \cite{li2017ergodic}. A \emph{cell of dimension $n$} in $S^2$, $n \in \{1, \, 2\}$, is a subset $c \subseteq S^2$ that is homeomorphic to the closed unit ball $\overline{\mathbb{B}^n}$ in $\real^n$, where $\mathbb{B}^{n}$ is the open unit ball in $\real^{n}$. We define the \emph{boundary of $c$}, denoted by $\partial c$, to be the set of points corresponding to $\partial \mathbb{B}^n$ under such a homeomorphism between $c$ and $\overline{\mathbb{B}^n}$. The \emph{interior of $c$} is defined to be $\inte{c} = c \mysetminus \partial c$. For each point $x\in S^2$, the set $\{x\}$ is considered as a \emph{cell of dimension $0$} in $S^2$. For a cell $c$ of dimension $0$, we adopt the convention that $\partial c = \emptyset$ and $\inte{c} = c$.

Let $f \colon S^2 \mapping S^2$ be a Thurston map, and $\mathcal{C}\subseteq S^2$ be a Jordan curve containing $\post{f}$.
Then the pair $f$ and $\mathcal{C}$ induces natural cell decompositions $\mathbf{D}^n(f,\mathcal{C})$ of $S^2$, for each $n \in \n_0$, in the following way:

By the Jordan curve theorem, the set $S^2 \mysetminus \mathcal{C}$ has two connected components. We call the closure of one of them the \emph{white $0$-tile} for $(f,\mathcal{C})$, denoted by $X^0_{\white}$, and the closure of the other one the \emph{black $0$-tile} for $(f,\mathcal{C})$, denoted be $X^0_{\black}$.
The set of $0$-\emph{tiles} is $\mathbf{X}^0(f, \mathcal{C}) \define \set[\big]{ X^0_{\black}, \, X^0_{\white} }$.
The set of $0$-\emph{vertices} is $\mathbf{V}^0(f, \mathcal{C}) \define \post{f}$.
We set $\overline{\mathbf{V}}^0(f, \mathcal{C}) \define \set[\big]{ \{x\} \describe x\in \mathbf{V}^0(f,\mathcal{C}) }$.
The set of $0$-\emph{edges} $\mathbf{E}^0(f,\mathcal{C})$ consists of the closures of the connected components of $\mathcal{C} \mysetminus \post{f}$.
Then we get a cell decomposition\[
    \mathbf{D}^0(f,\mathcal{C}) \define \mathbf{X}^0(f, \mathcal{C}) \cup \mathbf{E}^0(f,\mathcal{C}) \cup \overline{\mathbf{V}}^0(f,\mathcal{C})
\]
of $S^2$ consisting of \emph{cells of level }$0$, or $0$-\emph{cells}.

We can recursively define the unique cell decomposition $\mathbf{D}^n(f,\mathcal{C})$ for $n\in \n$, consisting of $n$-\emph{cells}, such that $f$ is cellular for $\parentheses[\big]{ \mathbf{D}^{n + 1}(f,\mathcal{C}), \mathbf{D}^n(f,\mathcal{C}) }$.
See \cite[Lemma~5.12]{bonk2017expanding} for details.
We denote by $\mathbf{X}^n(f,\mathcal{C})$ the set of $n$-cells of dimension 2, called $n$-\emph{tiles}; by $\mathbf{E}^n(f,\mathcal{C})$ the set of $n$-cells of dimension $1$, called $n$-\emph{edges}; by $\overline{\mathbf{V}}^n(f,\mathcal{C})$ the set of $n$-cells of dimension $0$; and by $\mathbf{V}^n(f,\mathcal{C})$ the set $\set[\big]{x \describe \{x\} \in \overline{\mathbf{V}}^n(f,\mathcal{C}) }$, called the set of $n$-\emph{vertices}.

\smallskip

For $n\in \n_0$, we define the \emph{set of black $n$-tiles} as\[
    \mathbf{X}^n_{\black}(f,\mathcal{C}) \define \left\{ X \in \mathbf{X}^n (f,\mathcal{C}) \describe f^n(X) = X^0_{\black} \right\},
\]
and the \emph{set of white $n$-tiles} as\[
    \mathbf{X}^n_{\white}(f,\mathcal{C}) \define \left\{X\in \mathbf{X}^n(f,\mathcal{C}) \describe f^n(X) = X^0_{\white}\right\}.
\]

From now on, if the map $f$ and the Jordan curve $\mathcal{C}$ are clear from the context, we will sometimes omit $(f,\mathcal{C})$ in the notation above.

We can now give a definition of expanding Thurston maps.

\begin{definition}[Expansion]     \label{def:expanding_Thurston_maps}
    A Thurston map $f \colon S^2 \mapping S^2$ is called \emph{expanding} if there exists a metric $d$ on $S^2$ that induces the standard topology on $S^2$ and a Jordan curve $\mathcal{C} \subseteq S^2$ containing $\post{f}$ such that
    \begin{equation}    \label{eq:definition of expansion}
        \lim_{n \to +\infty} \max\{ \diameter{d}{X} \describe X \in \mathbf{X}^n(f,\mathcal{C}) \} = 0.
    \end{equation}

\begin{theorem} \label{thm:main theorem}
    Let $f \colon S^2 \to S^2$ be an expanding Thurston map and $d$ be a visual metric on $S^2$ for $f$.
    Let $\beta \in (0, 1]$ and $\phi \in C^{0, \beta}(S^2, d)$ be an eventually positive real-valued \holder continuous function satisfying the $\beta$-strong non-integrability condition (with respect to $f$ and $d$).
    Then there exists a unique positive number $\rootpressure > 0$ with topological pressure $P(f, -\rootpressure \phi) = 0$ and there exists $N_{f} \in \n$ depending only on $f$ such that for each $N \in \n$ with $N \geqslant N_{f}$, the following statement holds for the iterate $F \coloneqq f^{N}$ and the potential $\Phi \coloneqq \sum_{i = 0}^{N - 1} \phi \circ f^{i}$:

    Denote $\alpha \coloneqq \frac{\mathrm{d}}{\mathrm{d} t} P(F, t \Phi) |_{t = -\rootpressure}$ and $\sigma \coloneqq \sqrt{ \frac{\mathrm{d}^2}{\mathrm{d} t^2} P(F, t \Phi) |_{t = -\rootpressure } }$.
    Let $\sequen{I_{n}}$ be a sequence of intervals contained in a compact set $K \subseteq \real$ with $\sequen[\big]{|I_{n}|^{-1}}$ having sub-exponential growth.
    Then
    \[
        \pi_{F, \Phi}(n; \alpha, I_n) \sim \frac{ \int_{I_{n}} e^{\rootpressure t} \, \mathrm{d}t }{ \sqrt{2\pi} \, \sigma } \, \frac{e^{\rootpressure \alpha n}}{n^{3/2}} \qquad \text{as } n \to +\infty.
    \]
\end{theorem}

\begin{remark}\label{rem:chordal metric visual metric qs equiv}
    If $f \colon \ccx \mapping \ccx$ is a rational expanding Thurston map, then a visual metric is quasisymmetrically equivalent to the chordal metric on the Riemann sphere $\ccx$ (see \cite[Theorem~18.1~(ii)]{bonk2017expanding}).
    Here the chordal metric $\sigma$ on $\ccx$ is given by $\sigma (z, w) \define \frac{2\abs{z - w}}{\sqrt{1 + \abs{z}^2} \sqrt{1 + \abs{w}^2}}$ for all $\juxtapose{z}{w} \in \cx$, and $\sigma(\infty, z) = \sigma(z, \infty) \define \frac{2}{\sqrt{1 + \abs{z}^2}}$ for all $z \in \cx$.
    Quasisymmetric embeddings of bounded connected metric spaces are \holder continuous (see \cite[Section~11.1 and Corollary~11.5]{heinonen2001lectures}).
    Accordingly, the classes of \holder continuous functions on $\ccx$ equipped with the chordal metric and on $S^2 = \ccx$ equipped with any visual metric for $f$ are the same (up to a change of the \holder exponent).
\end{remark}

\label{sec:Introduction}

Periodic orbits serve as the skeleton of chaotic dynamics, encoding essential information about the system's long-term behavior.
A fundamental objective in this field is counting these orbits, a problem analogous to the Prime Number Theorem in number theory.
Recently, the dynamical counterpart---the Prime Orbit Theorem---was successfully established for expanding Thurston maps \cite{li2024prime:dirichlet, li2024prime:split}, which serve as topological models for postcritically-finite rational maps.
While this answers the question of ``how many'' orbits exist, it leaves open the more subtle question of ``how they are distributed'' with respect to statistical observables.
Understanding this fine-scale distribution is crucial for characterizing the fluctuations inherent in the system.

In this paper, we investigate the asymptotic distribution of primitive periodic orbits restricted to shrinking intervals for expanding Thurston maps.
We count orbits whose Birkhoff sums for a given potential lie within a prescribed family of shrinking intervals.
we obtain a precise asymptotic formula for the number of these constrained orbits, which resemble a local central limit theorem.
Unlike the central limit theorem, which describes the distribution of Birkhoff sums on the scale of $\sqrt{n}$, the local central limit theorem can probe the density at a given point.
Our main result refines the coarse counting of the Prime Orbit Theorem \cite{li2024prime:dirichlet} for expanding Thurston maps.

The problem of counting orbits in shrinking intervals has been successfully addressed for uniformly expanding systems.
For instance, Petkov and Stoyanov \cite{petkovDistributionPeriodsClosed2012} investigated the distribution of closed orbits for hyperbolic flows, and Sharp and Stylianou \cite{sharpStatisticsMultipliersHyperbolic2022} studied the multipliers and holonomies for hyperbolic rational maps.
However, these results rely heavily on the hyperbolicity and smoothness of the underlying systems.
The context of non-uniformly expanding dynamics, particularly for branched covering maps, remains largely unexplored.
To the best of our knowledge, our work is the first to address this problem in such a setting.

\subsection{Main results}\label{sub:Main results}

Let $f \colon S^2 \to S^2$ be an expanding Thurston map and $\phi \colon S^2 \to \mathbb{R}$ be a real-valued H\"{o}lder  continuous function.
The topological $2$-sphere $S^2$ is equipped with a \emph{visual metric} $d$ (see Section~\ref{sub:Thurston_maps} for details).
A periodic orbit $\tau = \set[\big]{x, f(x), \dots, f^{n-1}(x)}$ (where $f^n(x) = x$) is called \emph{primitive} if $f^m(x) \ne x$ for each integer $m$ with $1 \leqslant m < n$.
We denote the set of primitive periodic orbits of $f$ by $\mathfrak{P}(f)$.
For each $\tau \in \mathfrak{P}(f)$, we write $\phi(\tau) \coloneqq \sum_{x \in \tau} \phi(x)$.

In this article, we investigate the asymptotic distribution of primitive periodic orbits subject to constraints on their Birkhoff sums.
Specifically, for a given number $\alpha \in \mathbb{R}$ and a sequence $\sequen{I_{n}}$ of intervals contained in a compact set $K \subseteq \mathbb{R}$, we study the asymptotic behavior of
\[
    \pi_{f, \phi}(n; \alpha, I_n) \coloneqq \operatorname{card} \set[\big]{ \tau \in \mathfrak{P}_n(f) : \phi(\tau) - n \alpha \in I_{n}}
\]
as $n \to +\infty$, where $\mathfrak{P}_n(f) \coloneqq \set[\big]{\tau \in \mathfrak{P}(f) : |\tau| = n}$.

To obtain precise estimates, we impose specific conditions on the potential $\phi$ and the sequence $\sequen{I_{n}}$.
We assume that $\phi$ is \emph{eventually positive} and satisfies the \emph{strong non-integrability condition} (see Definitions~\ref{def:eventually positive functions} and~\ref{def:strong non-integrability condition}).
Furthermore, denoting the length of $I_{n}$ by $|I_n|$, we assume that the sequence $\sequen[\big]{|I_{n}|^{-1}}$ exhibits sub-exponential growth, i.e., $\limsup_{n \to +\infty} \frac{1}{n} \log \parentheses[\big]{ |I_{n}|^{-1} } = 0$.

We write $A(n) \sim B(n)$ as $n \to +\infty$ if $\lim_{n \to +\infty} A(n) / B(n) = 1$.

Periodic orbits serve as the skeleton of chaotic dynamics, encoding essential information about the system's long-term behavior.
A fundamental objective in this field is counting these orbits, a problem analogous to the Prime Number Theorem in number theory.
Recently, the dynamical counterpart---the Prime Orbit Theorem---was successfully established for expanding Thurston maps \cite{li2024prime:dirichlet, li2024prime:split}, which serve as topological models for postcritically-finite rational maps.
While this answers the question of ``how many'' orbits exist, it leaves open the more subtle question of ``how they are distributed'' with respect to statistical observables.
Understanding this fine-scale distribution is crucial for characterizing the fluctuations inherent in the system.

In this paper, we investigate the asymptotic distribution of primitive periodic orbits restricted to shrinking intervals for expanding Thurston maps.
We count orbits whose Birkhoff sums for a given potential lie within a prescribed family of shrinking intervals.
we obtain a precise asymptotic formula for the number of these constrained orbits, which resemble a local central limit theorem.
Unlike the central limit theorem, which describes the distribution of Birkhoff sums on the scale of $\sqrt{n}$, the local central limit theorem can probe the density at a given point.
Our main result refines the coarse counting of the Prime Orbit Theorem \cite{li2024prime:dirichlet} for expanding Thurston maps.

Let $f \colon S^2 \to S^2$ be an expanding Thurston map and $\phi \colon S^2 \to \mathbb{R}$ be a real-valued H\"{o}lder continuous function.
The topological $2$-sphere $S^2$ is equipped with a \emph{visual metric} $d$ (see Section~\ref{sub:Thurston_maps} for details).
A periodic orbit $\tau = \set[\big]{x, f(x), \dots, f^{n-1}(x)}$ (where $f^n(x) = x$) is called \emph{primitive} if $f^m(x) \ne x$ for each integer $m$ with $1 \leqslant m < n$.
We denote the set of primitive periodic orbits of $f$ by $\mathfrak{P}(f)$.
For each $\tau \in \mathfrak{P}(f)$, we write $\phi(\tau) \coloneqq \sum_{x \in \tau} \phi(x)$.

In this article, we investigate the asymptotic distribution of primitive periodic orbits subject to constraints on their Birkhoff sums.
Specifically, for a given number $\alpha \in \mathbb{R}$ and a sequence $\sequen{I_{n}}$ of intervals contained in a compact set $K \subseteq \mathbb{R}$, we study the asymptotic behavior of
\[
 \pi_{f, \phi}(n; \alpha, I_n) \coloneqq \operatorname{card} \set[\big]{ \tau \in \mathfrak{P}_n(f) : \phi(\tau) - n \alpha \in I_{n}}
\]
as $n \to +\infty$, where $\mathfrak{P}_n(f) \coloneqq \set[\big]{\tau \in \mathfrak{P}(f) : |\tau| = n}$.

To obtain precise estimates, we impose specific conditions on the potential $\phi$ and the sequence $\sequen{I_{n}}$.
We assume that $\phi$ is \emph{eventually positive} and satisfies the \emph{strong non-integrability condition} (see Definitions~\ref{def:eventually positive functions} and~\ref{def:strong non-integrability condition}).
Furthermore, denoting the length of $I_{n}$ by $|I_n|$, we assume that the sequence $\sequen[\big]{|I_{n}|^{-1}}$ exhibits sub-exponential growth, i.e., $\limsup_{n \to +\infty} \frac{1}{n} \log \parentheses[\big]{ |I_{n}|^{-1} } = 0$.

We write $A(n) \sim B(n)$ as $n \to +\infty$ if $\lim_{n \to +\infty} A(n) / B(n) = 1$.

We write $A(n) \sim B(n)$ as $n \to +\infty$ if $\lim_{n \to +\infty} A(n) / B(n) = 1$.

Recall that a postcritically-finite rational map is expanding if and only if it has no periodic critical points (see \cite[Proposition~2.3]{bonk2017expanding}).
Therefore, when we restrict our attention to rational maps, we obtain the following corollary of Theorem~\ref{thm:main theorem} and Remark~\ref{rem:chordal metric visual metric qs equiv}.

\begin{corollary}\label{coro:main theorem for postcritically-finite rational maps}
 Let $f \colon \ccx \mapping \ccx$ be a postcritically-finite rational map without periodic critical points.
 Let $\sigma$ be the chordal metric or the spherical metric on the Riemann sphere $\ccx$, and $\phi \in C^{0, \holderexp} \parentheses[\big]{ \ccx, \sigma }$ be an eventually positive real-valued \holder continuous function with exponent $\holderexp \in (0, 1]$ satisfying the $\holderexp$-strong non-integrability condition (with respect to $f$ and a visual metric).
 Then there exists a unique positive number $\rootpressure > 0$ with topological pressure $P(f, -\rootpressure \phi) = 0$ and there exists $N_f \in \n$ depending only on $f$ such that for each $N \in \n$ with $N \geqslant N_f$, the following statements hold for $F \define f^N$ and $\Phi \define \sum_{i=0}^{N-1} \phi \circ f^i$:

Denote $\alpha \coloneqq \frac{\mathrm{d}}{\mathrm{d} t} P(F, t \Phi) |_{t = -\rootpressure}$ and $\sigma \coloneqq \sqrt{ \frac{\mathrm{d}^2}{\mathrm{d} t^2} P(F, t \Phi) |_{t = -\rootpressure } }$.
 Let $\sequen{I_{n}}$ be a sequence of intervals contained in a compact set $K \subseteq \real$ with $\sequen[\big]{|I_{n}|^{-1}}$ having sub-exponential growth.
 Then
 \[
 \pi_{F, \Phi}(n; \alpha, I_n) \sim \frac{ \int_{I_{n}} e^{\rootpressure t} \, \mathrm{d}t }{ \sqrt{2\pi} \, \sigma } \, \frac{e^{\rootpressure \alpha n}}{n^{3/2}} \qquad \text{as } n \to +\infty.
 \]
\end{corollary}

We now consider the following three quantities:
 \begin{align*}
 A_1(n) &\define \int_{ \abs{t} < \varepsilon \sigma \sqrt{n} } \, \abs[\bigg]{ \frac{\ell_{n}^{-1}}{e^{P(f, -\rootpressure \normpotential)n}} \widehat{\psi}_{n}\parentheses[\Big]{ \frac{t}{2 \pi \sigma \sqrt{n}} } \normpartifun[\Big]{-\rootpressure + \frac{\imaginary t}{\sigma \sqrt{n}}} \, - \, \ell_{n}^{-1} e^{-\frac{t^{2}}{2}} \! \int_{\real} \! \psi_{n}(x) \,\mathrm{d}x } \,\mathrm{d}t, \\
 A_2(n) &\define \int_{ \abs{t} \geqslant \varepsilon \sigma \sqrt{n} } \, \abs[\bigg]{ \frac{\ell_{n}^{-1}}{e^{P(f, -\rootpressure \normpotential)n}} \widehat{\psi}_{n}\parentheses[\Big]{ \frac{t}{2 \pi \sigma \sqrt{n}} } \normpartifun[\Big]{-\rootpressure + \frac{\imaginary t}{\sigma \sqrt{n}}} } \,\mathrm{d}t, \\
 A_3(n) &\define \int_{ \abs{t} \geqslant \varepsilon \sigma \sqrt{n} } \, \abs[\bigg]{ \ell_{n}^{-1} e^{-\frac{t^{2}}{2}} \! \int_{\real} \! \psi_{n}(x) \,\mathrm{d}x } \,\mathrm{d}t.
 \end{align*}
 Here $\varepsilon \in (0, 1)$ is chosen to be smaller than $\min\set[\Big]{ \delta , \frac{\sigma^{2}}{4 C_{\delta}}, t_0 }$, where the constants $\delta$ and $C_{\delta}$ are given by Lemma~\ref{lem:Taylor extension of pressure function in imaginary part}, and the constant $t_0$ is given by Lemma~\ref{lem:partition function estimate:expanding Thurston map:bounded imaginary bounded cases}~\ref{item:lem:partition function estimate:expanding Thurston map:bounded imaginary bounded cases:local part}.
 It follows from Claim~1 that
 \[
 A(n) \leqslant \frac{1}{\sqrt{2\pi}} \parentheses[\big]{ A_1(n) + A_2(n) + A_3(n)} \qquad \text{for each } n \in \n.
 \]
 Thus it suffices to show that $\lim_{n \to +\infty} A_{i}(n) = 0$ for each $i \in \{1, 2, 3\}$.

Let $T > 1$ be the constant given by Proposition~\ref{prop:partition function estimate:expanding Thurston map:unbounded imaginary}.
 By Lemma~\ref{lem:partition function estimate:expanding Thurston map:bounded imaginary bounded cases}~\ref{item:lem:partition function estimate:expanding Thurston map:bounded imaginary bounded cases:intermediate part}, there exists $\vartheta \in (0, 1)$ such that
 \begin{equation} \label{eq:temp:prop:estimate of partition function with periodic orbits:Claim3:medium case}
 \int_{ \varepsilon \leqslant \frac{ \abs{t}}{\sigma \sqrt{n} } \leqslant T } \abs[\bigg]{ \frac{\ell_{n}^{-1}}{e^{P(f, -\rootpressure \normpotential)n}} \widehat{\psi}_{n}\parentheses[\Big]{ \frac{t}{2 \pi \sigma \sqrt{n}} } \normpartifun[\Big]{-\rootpressure + \frac{\imaginary t}{\sigma \sqrt{n}}} } \,\mathrm{d}t
 = \mathcal{O}\parentheses[\big]{ \sigma \sqrt{n} \, \ell_{n}^{-1} \uniformnorm{\widehat{\psi}_{n}} \vartheta^{n} }
 \end{equation}
 as $n \to +\infty$, where $\ell_{n}^{-1} \uniformnorm{\widehat{\psi}_{n}} \leqslant \ell_{n}^{-1} \int_{\real} \! \psi_{n}(x) \,\mathrm{d}x = \int_{\real} \! \psi(y) e^{\rootpressure \ell_{n} y} \,\mathrm{d}y$ is uniformly bounded for $n \in \n$ since $\psi$ has compact support.
 Since $\psi_n \in C^{4}(\real, \real)$, we have that for all $n \in \n$ and $t \in \real$,
 \[
 (2 \pi \imaginary t)^{4} \widehat{\psi}_{n}(t) = \widehat{\psi_{n}^{(4)}}(t).
 \]
 Moreover, since $\psi$ has compact support, we have that
 \[
 \begin{split}
 \ell_{n}^{3} \, \uniformnorm[\Big]{ \widehat{\psi_{n}^{(4)}} }
 \leqslant \ell_{n}^{3} \! \int_{\real} \abs[\big]{ \psi_{n}^{(4)}(x) } \,\mathrm{d}x
 &= \int_{\real} \, \abs[\bigg]{ \sum_{k=0}^{4} \binom{4}{k} \ell_{n}^{k} \, \psi^{(4 - k)}(y) (\rootpressure )^{k} e^{\rootpressure \ell_{n} y} } \,\mathrm{d}y \leqslant C'
 \end{split}
 \]
 for some constant $C' \geqslant 0$, which is independent of $n$.
 Then, by Proposition~\ref{prop:partition function estimate:expanding Thurston map:unbounded imaginary}, there exist $C > 0$ and $\rho \in (0, 1)$ such that for each integer $n \geqslant 2$,
 \begin{align*}
 &\int_{ \abs{t} > T \sigma \sqrt{n} } \, \abs[\bigg]{ \frac{\ell_{n}^{-1}}{e^{P(f, -\rootpressure \normpotential)n}} \widehat{\psi}_{n}\parentheses[\Big]{ \frac{t}{2 \pi \sigma \sqrt{n}} } \normpartifun[\Big]{-\rootpressure + \frac{\imaginary t}{\sigma \sqrt{n}}} } \,\mathrm{d}t \\
 &\qquad \leqslant \int_{ \abs{t} > T \sigma \sqrt{n} } \, \abs[\bigg]{ \ell_{n}^{-1} \widehat{\psi}_{n}\parentheses[\Big]{ \frac{t}{2 \pi \sigma \sqrt{n}} } C \abs[\Big]{ \frac{t}{\sigma\sqrt{n}} }^{2 + \varepsilon} \rho^{n} } \,\mathrm{d}t\\
 &\qquad = \int_{ \abs{t} > T } C \sigma \sqrt{n} \, \ell_{n}^{-1} \rho^{n} \abs{t}^{2 + \varepsilon} \abs[\Big]{ \widehat{\psi}_{n}\parentheses[\Big]{ \frac{t}{2 \pi} } } \,\mathrm{d}t \\
 &\qquad = C \sigma \sqrt{n} \, \ell_{n}^{-4} \rho^{n} \int_{ \abs{t} > T } \abs{t}^{-2 + \varepsilon} \, \ell_{n}^{3} \abs[\Big]{ \widehat{\psi^{(4)}_{n}} \parentheses[\Big]{ \frac{t}{2 \pi} } } \,\mathrm{d}t \\
 &\qquad \leqslant C C' \sigma \sqrt{n} \, \ell_{n}^{-4} \rho^{n} \int_{ \abs{t} > T } \abs{t}^{-2 + \varepsilon} \,\mathrm{d}t.
 \end{align*}
 Combining this with \eqref{eq:temp:prop:estimate of partition function with periodic orbits:Claim3:medium case} and recalling that the sequences $\sequen{\ell_{n}^{-1}}$ are of sub-exponential growth, we establish Claim~3.

Recall that a postcritically-finite rational map is expanding if and only if it has no periodic critical points (see \cite[Proposition~2.3]{bonk2017expanding}).
Therefore, when we restrict our attention to rational maps, we obtain the following corollary of Theorem~\ref{thm:main theorem} and Remark~\ref{rem:chordal metric visual metric qs equiv}.

\begin{corollary}\label{coro:main theorem for postcritically-finite rational maps}
    Let $f \colon \ccx \rightarrow \ccx$ be a postcritically-finite rational map without periodic critical points.
    Let $\sigma$ be the chordal metric or the spherical metric on the Riemann sphere $\ccx$, and $\phi \in C^{0, \beta} \parentheses[\big]{ \ccx, \sigma }$ be an eventually positive real-valued H\"{o}lder  continuous function with exponent $\beta \in (0, 1]$ satisfying the $\beta$-strong non-integrability condition (with respect to $f$ and a visual metric).
    Then there exists a unique positive number $\rootpressure > 0$ with topological pressure $P(f, -\rootpressure \phi) = 0$ and there exists $N_f \in \mathbb{N}$ depending only on $f$ such that for each $N \in \mathbb{N}$ with $N \geqslant N_f$, the following statements hold for $F \coloneqq f^N$ and $\Phi \coloneqq \sum_{i=0}^{N-1} \phi \circ f^i$:

    Denote $\alpha \coloneqq \frac{\mathrm{d}}{\mathrm{d} t} P(F, t \Phi) |_{t = -\rootpressure}$ and $\sigma \coloneqq \sqrt{ \frac{\mathrm{d}^2}{\mathrm{d} t^2} P(F, t \Phi) |_{t = -\rootpressure } }$.
    Let $\sequen{I_{n}}$ be a sequence of intervals contained in a compact set $K \subseteq \mathbb{R}$ with $\sequen[\big]{|I_{n}|^{-1}}$ having sub-exponential growth.
    Then
    \[
        \pi_{F, \Phi}(n; \alpha, I_n) \sim \frac{ \int_{I_{n}} e^{\rootpressure t} \, \mathrm{d}t }{ \sqrt{2\pi} \, \sigma } \, \frac{e^{\rootpressure \alpha n}}{n^{3/2}} \qquad \text{as } n \to +\infty.
    \]
\end{corollary}

\subsection{Strategy and organization}
Our approach relies on a combination of thermodynamic formalism and operator theory, specifically adapted to the branched covering setting.

The main technical obstacle in studying Thurston maps is the presence of critical points, which disrupts the functional analytic properties of the standard Ruelle transfer operator.
To overcome this, we employ the \emph{split Ruelle operators} introduced in \cite{li2024prime:split}.
The idea is to decompose the sphere into ``black'' and ``white'' tiles (based on a checkerboard coloring induced by an invariant Jordan curve) and define a pair of operators acting on functions supported on these tiles.
This construction effectively ``unfolds'' the singularities, allowing us to recover good spectral properties.

To obtain the precise asymptotics required for the local central limit theorem, we need to control the decay of the characteristic function of the Birkhoff sums.
In terms of operator theory, this translates to bounding the spectral radius of the twisted transfer operator $\mathbb{L}_{s\phi}$ as the complex parameter $s$ moves along the imaginary axis.
The detailed estimates are separated into three parts: the unbounded part, the bounded part, and the local part.
For the unbounded part, we employ Dolgopyat-type estimates for the split Ruelle operators established in \cite{li2024prime:split}.
This requires checking a strong non-integrability condition (Definition~\ref{def:strong non-integrability condition}), which in particular implies that the potential is not cohomologous to a constant.
For the bounded part, we employ Ruelle's estimate (see Appendix~\ref{sec:Appendix:Ruelle lemma}).
For the local part, we employ the complex Ruelle--Perron--Frobenius theorem \cite[Theorem~2]{pollicott1984complex} and arguments in \cite{pollicottZetaFunctionsPeriodic1990}.

Finally, to count orbits in intervals $I_n$, we approximate the indicator function $\mathbbm{1}_{I_n}$ by smooth test functions.
We then apply Fourier transforms to relate the smoothed count to partition functions, which allows us to apply the established decay estimates.

\smallskip

The paper is organized as follows.
In Section~\ref{sec:Preliminaries}, we fix our notation, review fundamental concepts from thermodynamic formalism, and recall key results from the theory of expanding Thurston maps.
In Section~\ref{sec:The Assumptions}, we collect the main assumptions used throughout the paper.
The technical core of the paper is Section~\ref{sec:Pressure function and partition function estimates}, where we employ previous results and derive crucial decay estimates for associated partition functions.
These estimates are then used in Section~\ref{sec:Proof of the main theorem} to prove Theorem~\ref{thm:main theorem}.
section/Proof.tex0000664000000000000000000007214415112046574013046 0ustar  rootroot

To prove Theorem~\ref{thm:main theorem}, the approach uses “a combination of thermodynamic formalism and operator theory, specifically adapted to the branched covering setting.” The “main technical obstacle” is that “critical points … disrupt the functional analytic properties of the standard Ruelle transfer operator,” so the argument “employ[s] the \emph{split Ruelle operators}” by “decompos[ing] the sphere into ‘black’ and ‘white’ tiles … and defin[ing] a pair of operators acting on functions supported on these tiles,” which “unfolds the singularities” and restores “good spectral properties.”

For the local central limit theorem asymptotics, one “control[s] the decay of the characteristic function of the Birkhoff sums,” i.e. “bound[s] the spectral radius of the twisted transfer operator $\mathbb{L}_{s\phi}$ as the complex parameter $s$ moves along the imaginary axis.” The needed estimates are split into “three parts: the unbounded part, the bounded part, and the local part”: for the unbounded part use “Dolgopyat-type estimates for the split Ruelle operators,” which “requires checking a strong non-integrability condition … (in particular [implying] that the potential is not cohomologous to a constant)”; for the bounded part use “Ruelle’s estimate”; and for the local part use “the complex Ruelle--Perron--Frobenius theorem … and arguments in \cite{pollicottZetaFunctionsPeriodic1990}.”

Finally, “to count orbits in intervals $I_n$,” one “approximate[s] the indicator function $\mathbbm{1}_{I_n}$ by smooth test functions,” then “apply Fourier transforms to relate the smoothed count to partition functions,” so that the previously established decay estimates can be applied.

To prove the main theorem, the approach uses “a combination of thermodynamic formalism and operator theory, specifically adapted to the branched covering setting.” The “main technical obstacle” is that “critical points … disrupt the functional analytic properties of the standard Ruelle transfer operator,” so the argument “employ[s] the \emph{split Ruelle operators}” by “decompos[ing] the sphere into ‘black’ and ‘white’ tiles … and defin[ing] a pair of operators acting on functions supported on these tiles,” which “unfolds the singularities” and restores “good spectral properties.”

For the local central limit theorem asymptotics, one “control[s] the decay of the characteristic function of the Birkhoff sums,” i.e. “bound[s] the spectral radius of the twisted transfer operator $\mathbb{L}_{s\phi}$ as the complex parameter $s$ moves along the imaginary axis.” The needed estimates are split into “three parts: the unbounded part, the bounded part, and the local part”: for the unbounded part use “Dolgopyat-type estimates for the split Ruelle operators,” which “requires checking a strong non-integrability condition … (in particular [implying] that the potential is not cohomologous to a constant)”; for the bounded part use “Ruelle’s estimate”; and for the local part use “the complex Ruelle--Perron--Frobenius theorem … and arguments in Mark Pollicott, \emph{Zeta functions, periodic points and oscillations of the prime orbit theorem} (1990).”

Finally, “to count orbits in intervals $I_n$,” one “approximate[s] the indicator function $\mathbbm{1}_{I_n}$ by smooth test functions,” then “apply Fourier transforms to relate the smoothed count to partition functions,” so that the previously established decay estimates can be applied.

\label{thm:main theorem}
    Let $f \colon S^2 \to S^2$ be an expanding Thurston map and $d$ be a visual metric on $S^2$ for $f$.
    Let $\beta \in (0, 1]$ and $\phi \in C^{0, \beta}(S^2, d)$ be an eventually positive real-valued H\"{o}lder  continuous function satisfying the $\beta$-strong non-integrability condition (with respect to $f$ and $d$).
    Then there exists a unique positive number $\rootpressure > 0$ with topological pressure $P(f, -\rootpressure \phi) = 0$ and there exists $N_{f} \in \mathbb{N}$ depending only on $f$ such that for each $N \in \mathbb{N}$ with $N \geqslant N_{f}$, the following statement holds for the iterate $F \coloneqq f^{N}$ and the potential $\Phi \coloneqq \sum_{i = 0}^{N - 1} \phi \circ f^{i}$:

    Denote $\alpha \coloneqq \frac{\mathrm{d}}{\mathrm{d} t} P(F, t \Phi) |_{t = -\rootpressure}$ and $\sigma \coloneqq \sqrt{ \frac{\mathrm{d}^2}{\mathrm{d} t^2} P(F, t \Phi) |_{t = -\rootpressure } }$.
    Let $\sequen{I_{n}}$ be a sequence of intervals contained in a compact set $K \subseteq \mathbb{R}$ with $\sequen[\big]{|I_{n}|^{-1}}$ having sub-exponential growth.
    Then
    \[
        \pi_{F, \Phi}(n; \alpha, I_n) \sim \frac{ \int_{I_{n}} e^{\rootpressure t} \, \mathrm{d}t }{ \sqrt{2\pi} \, \sigma } \, \frac{e^{\rootpressure \alpha n}}{n^{3/2}} \qquad \text{as } n \to +\infty.
    \],

Question:
Let $f\colon S^2\to S^2$ be an expanding Thurston map, let $d$ be a visual metric on $S^2$ for $f$, let $\beta\in(0,1]$, and let $\phi\in C^{0,\beta}(S^2,d)$ be an eventually positive real-valued H\"older continuous function satisfying the $\beta$-strong non-integrability condition with respect to $f$ and $d$. For $N\in\mathbb N$, set $F:=f^N$ and $\Phi:=\sum_{i=0}^{N-1}\phi\circ f^i$. If $\tau=\{x,F(x),\dots,F^{n-1}(x)\}$ is a primitive periodic orbit of $F$ (meaning $F^n(x)=x$ and $F^m(x)\neq x$ for $1\le m<n$), write $|\tau|=n$, let $\mathfrak P_n(F)$ be the set of primitive periodic orbits of $F$ of length $n$, and define $\Phi(\tau):=\sum_{y\in\tau}\Phi(y)$. For $\alpha\in\mathbb R$ and intervals $I_n\subseteq K$ with $K\subseteq\mathbb R$ compact, define
\[
\pi_{F,\Phi}(n;\alpha,I_n):=\operatorname{card}\{\tau\in\mathfrak P_n(F):\Phi(\tau)-n\alpha\in I_n\}.
\]
Which statement about existence, uniqueness, and the asymptotic formula for $\pi_{F,\Phi}(n;\alpha,I_n)$ is valid?

There exists a unique positive number $s_0>0$ such that $P(f,-s_0\phi)=0$, and there exists $N_f\in\mathbb N$, depending only on $f$, such that for every $N\ge N_f$, if $F=f^N$ and $\Phi=\sum_{i=0}^{N-1}\phi\circ f^i$, then with
\[
\alpha:=\left.\frac{d}{dt}P(F,t\Phi)\right|_{t=-s_0},\qquad
\sigma:=\sqrt{\left.\frac{d^2}{dt^2}P(F,t\Phi)\right|_{t=-s_0}},
\]
for every sequence of intervals $(I_n)$ contained in a compact set $K\subseteq\mathbb R$ such that $(|I_n|^{-1})$ has sub-exponential growth, one has
\[
\pi_{F,\Phi}(n;\alpha,I_n)\sim \frac{\int_{I_n} e^{s_0 t}\,dt}{\sqrt{2\pi}\,\sigma}\,\frac{e^{s_0\alpha n}}{n^{3/2}}
\qquad\text{as } n\to+\infty.
\]"

Choices:
B.
There exists a unique positive number $s_0>0$ such that $P(f,-s_0\phi)=0$, and for every $N\in\mathbb N$, if $F=f^N$ and $\Phi=\sum_{i=0}^{N-1}\phi\circ f^i$, then with
\[
\alpha:=\left.\frac{d}{dt}P(F,t\Phi)\right|_{t=-s_0},\qquad
\sigma:=\sqrt{\left.\frac{d^2}{dt^2}P(F,t\Phi)\right|_{t=-s_0}},
\]
for every sequence of intervals $(I_n)$ contained in a compact set $K\subseteq\mathbb R$ such that $(|I_n|^{-1})$ has sub-exponential growth, one has
\[
\pi_{F,\Phi}(n;\alpha,I_n)\sim \frac{\int_{I_n} e^{s_0 t}\,dt}{\sqrt{2\pi}\,\sigma}\,\frac{e^{s_0\alpha n}}{n^{3/2}}
\qquad\text{as } n\to+\infty.
\]
C.
There exists a unique positive number $s_0>0$ such that $P(f,-s_0\phi)=0$, and there exists $N_f\in\mathbb N$, depending only on $f$, such that for every $N\ge N_f$, if $F=f^N$ and $\Phi=\sum_{i=0}^{N-1}\phi\circ f^i$, then with
\[
\alpha:=\left.\frac{d}{dt}P(F,t\Phi)\right|_{t=-s_0},
\]
for every sequence of intervals $(I_n)$ contained in a compact set $K\subseteq\mathbb R$ such that $(|I_n|^{-1})$ has sub-exponential growth, one has
\[
\pi_{F,\Phi}(n;\alpha,I_n)=\mathcal O\!\left(\frac{e^{s_0\alpha n}}{n^{3/2}}\right)
\qquad\text{as } n\to+\infty.
\]
D.
There exists a unique positive number $s_0>0$ such that $P(f,-s_0\phi)=0$, and there exists $N_f\in\mathbb N$, depending only on $f$, such that for every $N\ge N_f$, if $F=f^N$ and $\Phi=\sum_{i=0}^{N-1}\phi\circ f^i$, then with
\[
\alpha:=\left.\frac{d}{dt}P(F,t\Phi)\right|_{t=-s_0},\qquad
\sigma:=\sqrt{\left.\frac{d^2}{dt^2}P(F,t\Phi)\right|_{t=-s_0}},
\]
for every sequence of intervals $(I_n)$ contained in a compact set $K\subseteq\mathbb R$, one has
\[
\pi_{F,\Phi}(n;\alpha,I_n)\sim \frac{|I_n|}{\sqrt{2\pi}\,\sigma}\,\frac{e^{s_0\alpha n}}{n^{3/2}}
\qquad\text{as } n\to+\infty.
\]
E.
There exists a unique positive number $s_0>0$ such that $P(f,-s_0\phi)=0$, and there exists $N_f\in\mathbb N$, depending only on $f$, such that for every $N\ge N_f$, if $F=f^N$ and $\Phi=\sum_{i=0}^{N-1}\phi\circ f^i$, then with
\[
\alpha:=\left.\frac{d}{dt}P(F,t\Phi)\right|_{t=-s_0},\qquad
\sigma:=\sqrt{\left.\frac{d^2}{dt^2}P(F,t\Phi)\right|_{t=-s_0}},
\]
for every sequence of intervals $(I_n)$ contained in a compact set $K\subseteq\mathbb R$ such that $(|I_n|^{-1})$ has sub-exponential growth, one has
\[
\pi_{F,\Phi}(n;\alpha,I_n)\sim \frac{\int_{I_n} e^{s_0 t}\,dt}{\sqrt{2\pi}\,\sigma}\,\frac{e^{s_0\alpha n}}{n^{1/2}}
\qquad\text{as } n\to+\infty.
\]

threshold iterate N\ge N_f replaced by all N

dropped the sharp asymptotic constant and equivalence, retaining only a coarse upper bound of the same order

smooth approximation/Fourier-weighted interval term replaced by bare length and sub-exponential hypothesis removed

prime-orbit scale n^{-3/2} replaced by CLT-like n^{-1/2}

The stem itself does not state the conclusion or uniquely reveal the correct option; it only gives the hypotheses and asks which resulting theorem statement is valid.

The item is essentially a theorem-recognition question: under a full list of hypotheses, the student must pick the exact conclusion. It is not a verbatim restatement in the stem, but it remains close to selecting the correct theorem statement.

There is some reasoning pressure because the options differ in subtle but meaningful ways (quantifier on N, sharp asymptotic vs bound, interval factor, power of n). However, it mainly tests precise recall/discrimination rather than constructing a conclusion from first principles.

The distractors are mathematically plausible and target realistic failure modes: overgeneralizing to all N, weakening an asymptotic to an O-bound, dropping the weighted interval term/sub-exponential condition, and confusing n^{-3/2} with n^{-1/2}.

A solid high-level theorem-discrimination MCQ with little stem leakage and strong distractors, but it primarily assesses recognition of a precise result rather than genuinely generative mathematical reasoning.


\begin{Definition}\label{def:phi homogenization}
Fix $\lambda = \lambda(n) > 0$ sufficiently small such that $\sigma = C_1(n)\lambda$ satisfies $4^{n+1}\sigma < c(n)$, where $C_1(n)$ is the one in Lemma \ref{lem:covering Lemma} and $c(n)$ is the one in Proposition \ref{prop:basic}. For $\Lambda\ge \max\{\Lambda_0,\Lambda_1,\Lambda_2\}$, and any nonnegative functions $v \in C(\Omega)$, we define its $\phi$-homogenization $M_{\phi,\Lambda}v(x):=M_{\phi,\lambda,\Lambda}v(x)$ in $\frac{7}{8}\Omega$ through the following maximal averaging type operator:
\begin{equation}\label{eq:phi homogenization}
\begin{split}
M_{\phi,\Lambda}v(x)=  \sup\left\{ t\in\R:\; \exists \ \check{h}(x)<h<  \hat{h}(x),\ s.t., \
\frac{\mu ( \{v> t\} \cap S_h (x)  )}{\mu(S_h (x))} > \lambda \right\}.
\end{split}
\end{equation}
\end{Definition}

\begin{Theorem}\label{thm:uniqueness within periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying \eqref{eq:periodic f}, and let $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ be a quadratic function with $A\in \mathcal{S}_+^{n\times n}$ satisfying \eqref{eq:A compatible}. Then there exists a unique periodic solution (up to addition of constants) to \eqref{eq:periodic equation ma} with quadratic component $P$.
\end{Theorem}

In this paper, we study convex solutions to the Monge-Amp\`ere equation
\begin{equation}\label{eq:periodic equation ma}
\det D^2 u = \mu \quad \text{in } \mathbb{R}^n,
\end{equation}
where $\mu\not\equiv 0$ is a nonnegative locally finite Borel measure on $\mathbb{R}^n$ and is periodic in $n$ linearly independent directions. By exploiting the affine invariance of the equation, we may, without loss of generality, restrict our analysis to the case where $\mu $ is periodic with respect to the integer lattice $\mathbb{Z}^n := \{(k_1,\dots,k_n) \mid k_i \in \mathbb{Z}\} \subset \mathbb{R}^n$, i.e., for any Borel set $E \subset \mathbb{R}^n$,
\begin{equation}\label{eq:periodic f}
\mu (E+z)=\mu (E),\quad \forall \ z \in \mathbb{Z}^n.
\end{equation}

Let $\mathcal{S}_+^{n\times n}$ denote the set of positive definite symmetric $n\times n$ matrices.
\begin{Definition}[Periodic solution]\label{def:periodic solution}
A \emph{periodic solution} to \eqref{eq:periodic equation ma} is a convex function $u \in C(\mathbb{R}^n)$ satisfying \eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as
\begin{equation}\label{eq:periodic solution}
u(x) = v(x) +P(x),
\end{equation}
where $v$ is periodic with respect to $\mathbb{Z}^n$ and $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ is a quadratic function with $A\in \mathcal{S}_+^{n\times n}$, $b\in\mathbb{R}^n$ and $c\in\mathbb{R}$.
\end{Definition}

Note that if $u$ is a periodic solution to \eqref{eq:periodic equation ma} and is expressed as \eqref{eq:periodic solution}, then the quadratic component $P(x)$ always satisfies the compatibility condition
\begin{equation}\label{eq:A compatible}
 \det A =\mu (\mathbb{T}^n),
\end{equation}
where $\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$.

Our first result is the uniqueness of periodic solutions.

\begin{equation}\label{eq:periodic equation ma}
\det D^2 u = \mu \quad \text{in } \R^n,
\end{equation}

Let $\mu \not\equiv 0$ be a nonnegative locally finite periodic Borel measure on $\R^n$. We show that any convex solution to the Monge-Amp\`ere equation
\[
\det D^2 u = \mu \quad \text{in } \R^n
\]
admits a unique decomposition (up to addition of constants) as the sum of a quadratic polynomial and a periodic function. This result extends,
in full generality, the earlier works for the case $\mu=f(x)\,\ud x$: when $\log f \in C^\alpha$, it was established by Caffarelli and Li; and
when $\log f$ is merely bounded, it was proved by Li and Lu. Our result thus answers a question raised by Li and Lu. A key ingredient in the proof is a new dichotomous Harnack-type inequality for linearized Monge-Amp\`ere equations with nonnegative periodic measures.

Let $\mathcal{S}_+^{n\times n}$ denote the set of positive definite symmetric $n\times n$ matrices.
\begin{Definition}[Periodic solution]\label{def:periodic solution}
A \emph{periodic solution} to \eqref{eq:periodic equation ma} is a convex function $u \in C(\mathbb{R}^n)$ satisfying \eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as
\begin{equation}\label{eq:periodic solution}
u(x) = v(x) +P(x),
\end{equation}
where $v$ is periodic with respect to $\mathbb{Z}^n$ and $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ is a quadratic function with $A\in \mathcal{S}_+^{n\times n}$, $b\in\R^n$ and $c\in\R$.
\end{Definition}

Our first result is the uniqueness of periodic solutions.

The existence of periodic solutions follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof. For the uniqueness, when $\mu=f(x)\,\ud x$ is a positive measure, the case $\log f \in C^\alpha$ was proved by Li \cite{li1990existence}, and the case $\log f\in L^\infty$ was shown by Li-Lu \cite{li2019monge}. Our theorem extends these results to general periodic Borel measures.

\begin{Theorem}\label{thm:solutions are periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying \eqref{eq:periodic f}, and let $u$ be a convex solution of \eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \ref{def:periodic solution}.
\end{Theorem}

\begin{Proposition}\label{prop:semiconcave}
Let $u$ be a convex solution to \eqref{eq:periodic equation ma} with $\mu$ being a $\mathbb{Z}^n/8n$-periodic measure (where $\mathbb{Z}^n := \Gamma_{\{e_1,\cdots,e_n\}}$) satisfying $\mu(\mathbb{Z}^n) = 1$.
Under the nondegeneracy condition
\begin{equation}\label{eq:nondegenerate}
\Delta^2_{i} u(0):=\Delta^2_{e_i} u(0) > 0 \quad \text{for all } 1 \leq i \leq n,
\end{equation}
there exists a unique compatible matrix $A$ satisfying:
\[
\sup_{ \R^n}\Delta_z^2 u=z^{\top}Az ,\quad \forall z \in \mathbb{Z}^n.
\]
\end{Proposition}
\begin{proof}
Without loss of generality, we may assume $u(0) = 0$ and $u \geq 0$.
Let us define $S_h := \{x \in \mathbb{R}^n : u(x) \leq h\}$, and select $\Lambda_u>0$ such that the section $S_{\Lambda_u}$ satisfies $\kappa_{\mathbb{Z}^n}(S_{\Lambda_u}) \geq \Lambda_0^2$.
Let us now normalize $S_h$ such that $B_1(0) \subset T_h^{-1} S_h \subset B_n(0)$ and consider the normalized function
\[
u_{h}(x) =\frac{u(T_h x)}{(\det T_h)^{\frac{2}{n}}} \quad \text{with } \det D^2 u_h= \mu_h=\mu \circ T_h .
\]
Due to the periodicity of $\mu$ and large $h$, we know that
\[
c(n)h^{\frac{n}{2}} \leq \det T_h \leq C(n)h^{\frac{n}{2}},
\]
and thus, $c(n)h^{\frac{n}{2}} \leq |S_h| \leq C(n)h^{\frac{n}{2}}$. Applying Proposition \ref{prop:basic} to $\phi=u_h-h(\det T_h)^{-\frac{2}{n}}$, we obtain that for sufficiently large $h$,
\[
c(n) (h/\Lambda_u)^{\frac{1}{3}} S_{\Lambda_u} \subset S_h \subset C(n) (h/\Lambda_u)^{\frac{2}{3}} S_{\Lambda_u},
\]
and consequently, $B_{c(n) h^{\frac{1}{4}}}\subset S_h\subset B_{C(n) h^{\frac{3}{4}}}$ for all large $h$. That is, for sufficiently large $h$, we now have
\[
\quad c(n)h^{\frac{1}{4}} I\leq T_h \leq C(n)h^{\frac{3}{4}} I.
\]
The transformation law \eqref{eq:transformation law} then yields the following estimates at corresponding points
\begin{equation}\label{eq:transformh}
c(n)h^{-\frac{1}{2}}\Delta^2_{e_i} u \leq \Delta^2_{T_h^{-1} e_i} u_h \leq C(n)h^{\frac{1}{2}}\Delta^2_{e_i} u.
\end{equation}

\textbf{Step 4.} Suppose for some sequence $h \to \infty$, the rescaled functions $\tilde{u}_h$ locally converge to a quadratic form $P_A(x) = \frac{1}{2}x^{\top}A x$ for $A \in \mathcal{S}_+^{n\times n}$ with $\det A = 1$.
For each fixed direction $i$, we denote
\[
\alpha=\sup_{\R^n}\Delta^2_{e_i} u,\quad \beta= e_i^{\top} A e_i.
\]
Note that the estimate \eqref{eq:w21 estimate} continues to hold for $\tilde{u}_{h}$, from which we derive $\alpha \geq \beta$.
We claim
\begin{equation}\label{eq:w2infty sup}
\alpha=\beta.
\end{equation}
Assume by contradiction that $\alpha = \beta + 4s$ for some $s > 0$.
Through rescaling from infinity, we may assume without loss of generality that there exist points $x_h \in B_{1/8}$ satisfying
\[
\alpha- \Delta^2_{h^{\frac{1}{2}}e_i} \tilde{u}_{h}(x_h)= \inf_{B_{1/8}} (\alpha- \Delta^2_{h^{\frac{1}{2}} e_i} \tilde{u}_{h} ) \leq a_h,
\]
where $a_h \to 0 $ as $h \to \infty$. Define the normalized function $v = (\alpha - \Delta^2_{h^{\frac{1}{2}}e_i} \tilde{u}_{h})/a_h$ and set $\delta = s/[C(\beta + 2s)]$. Applying Lemma \ref{lem:Theorem 7.3.1 modify} yields with $\phi=\tilde{u}_h-1$ and $\Omega=S_{1,h}=\{\tilde{u}_h \leq 1\} $, we obtain that
\[
\tilde{\mu}_h ( \{v\leq M_1(n,\delta) \} \cap S_{1,h} ) \geq (1-\delta) \tilde{\mu}_h(S_{1,h} ),
\]
where $\tilde{\mu}_h=\tilde{f}_{h}\,\ud x$. For sufficiently small $a_h$, we consequently obtain
\[
\tilde{\mu}_h\left(\left\{\Delta^2_{h^{\frac{1}{2}} e_i} \tilde{u}_h>\beta +2 s\right\} \cap S_{1,h}\right)= \tilde{\mu}_h\left(\left\{v <\frac{2s}{a_h}\right\} \cap S_{1,h}\right)\geq (1-\delta) \tilde{\mu}_h (S_{1,h}).
\]
This leads to the integral lower bound
\[
\int_{S_{1,h}} \Delta^2_{h^{\frac{1}{2}} e_i} \tilde{u}_h \ud \tilde{\mu}_h \geq (1-\delta) (\beta+2s) \tilde{\mu}_h (S_{1,h})\geq
(\beta+s) \tilde{\mu}_h (S_{1,h}).
\]
This contradicts the uniform estimate \eqref{eq:w21 estimate} after performing a suitable rescaling.

\begin{proof}[Proof of Theorems \ref{thm:uniqueness within periodic} and \ref{thm:solutions are periodic}]
Let $u$ be a global solution of \eqref{eq:periodic equation ma}.
Clearly, $u$ cannot be linear on any ray segment. Indeed, if it were linear on a ray, after subtracting its supporting hyperplane on this ray and by using the convexity of $u$, $u$ would be bounded within any cylinder having that ray as its axis. This would imply that the associated measure $\mu$ is finite on any strictly smaller cylinder, contradicting the assumptions that $\mu$ is periodic and $\mu \not\equiv 0$. Therefore, there exists a sufficiently large integer $R$ such that $S_1(0) \subset B_R(0)$.
We now rescale $u$ by defining $u(8nRx)/64n^2R^2$. For simplicity, we continue to denote this rescaled function as $u$. Under this scaling, one can verify that condition \eqref{eq:nondegenerate} in Proposition \ref{prop:semiconcave} is satisfied. Hence, by Proposition \ref{prop:semiconcave}, there exists a compatible matrix $A \in \mathcal{S}_+^{n \times n}$ such that
\[
\sup_{ \R^n}\Delta_z^2 u=z^{\top}Az,\quad \forall z \in \mathbb{Z}^n.
\]
Let $w$ be a periodic solution to \eqref{eq:periodic equation ma} in the sense of Definition \ref{def:periodic solution} with quadratic part $\frac{1}{2}z^{\top} A z $. Then $\Delta_z^2 w=z^{\top}Az$. Let
\[
v(x)=u(x)-w(x).
\]
By subtracting a suitable linear function, we now assume that $v(0)=0$ and
\[
v(e_i)=v(-e_i),\quad 1\leq i\leq n.
\]
Noting that $\sup_{\R^n} \Delta_z^2 v = \sup_{\R^n} \Delta_z^2 (u - w) = 0$, we then find that
\[
v(\pm ke_i)\leq 0 \quad \text{for all } k\in \mathbb{Z},\ 1\leq i\leq n.
\]

The existence of periodic solutions follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof.  For the uniqueness, when $\mu=f(x)\,\mathrm{d} x$ is a positive measure, the case $\log f \in C^\alpha$ was proved by Li \cite{li1990existence}, and the case  $\log f\in L^\infty$ was shown by Li-Lu \cite{li2019monge}. Our theorem extends these results to general periodic Borel measures.

Our second result is the classification below, which extends the main results  of Caffarelli-Li \cite{caffarelli2004liouville} and Li-Lu \cite{li2019monge} to the full generality. In particular, it provides a complete answer to Question 1.1 in \cite{li2019monge}.

\begin{Theorem}\label{thm:solutions are periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying \eqref{eq:periodic f}, and let $u$ be a convex solution of \eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \ref{def:periodic solution}.
\end{Theorem}

Caffarelli-Li \cite{caffarelli2004liouville} first proved Theorem \ref{thm:solutions are periodic} for positive measures $\mu=f(x)\,\mathrm{d} x$ with $\log f \in C^\alpha$, and conjectured the extension to merely bounded $\log f$. This conjecture was resolved by Li-Lu \cite{li2019monge}, who further asked whether the result holds for the nonnegative case $f \geq 0$ with $f \not\equiv 0$. Theorem \ref{thm:solutions are periodic} affirmatively answers this question -- and indeed establishes it in full generality.

\begin{Remark}
When $\mu \equiv 0$, equation \eqref{eq:periodic equation ma} reduces to the homogeneous Monge-Amp\`ere equation $\det D^2u = 0$. The solutions are precisely those functions that are linear along certain directions (see, e.g., Caffarelli-Nirenberg-Spruck \cite{caffarelli1986dirichlet}).
In particular, writing $x = (x',x_n) \in \mathbb{R}^{n-1} \times \mathbb{R}$, any convex function of the form $u(x) = w(x')$ is a solution, and such solutions are not necessarily periodic.
\end{Remark}

Since $\mu$ is assumed to be nonnegative,  equation \eqref{eq:periodic equation ma} represents a possibly degenerate Monge-Amp\`ere equation, for which many arguments in  \cite{caffarelli2004liouville,li2019monge} do not apply.  The principal challenge stems from the behavior of $u$ on small sections: the associated measure $\mu$ may not satisfy the doubling condition. This failure implies that sections lose the engulfing property, consequently invalidating the Vitali or  Besicovitch covering lemma which is essential for the Calder\'on-Zygmund decomposition on sections.
Therefore, we cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\'errez \cite{caffarelli1997properties} for subsolutions or supersolutions of linearized Monge-Amp\`ere equations.

Nonetheless, by exploiting the equation's periodic structure, we derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator (see Definition \ref{def:phi homogenization}). This constitutes one of the key innovations of the present work. For supersolutions, we address the challenge by excluding contributions from small sections. Theorem \ref{thm:harnack super} establishes decay estimates for homogenized level-sets of supersolutions within our framework. For subsolutions, for which standard $L^{\infty}$ bounds fail, Theorem \ref{thm:harnack sub} establishes an alternative dichotomy regarding their growth behavior: either the $L^{\infty}$ bound holds or the $L^{\infty}$ norm grows exponentially. Finally, we obtain  in Theorem \ref{thm:harnack} a Harnack inequality with a quantitatively controlled error term.

After overcoming this critical difficulty arising from the degeneracy of equation \eqref{eq:periodic equation ma}, the rest proof focuses on analyzing the second-order difference quotient $\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart. As observed in \cite{caffarelli2004liouville}, these quantities naturally arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\`ere equations. This is another point where the periodicity of the measure plays a crucial role. We then establish the quadratic behavior of solutions near infinity, extending the arguments in \cite{caffarelli2004liouville,li2019monge} for positive measures $\mu=f(x)\mathrm{d} x$ to  the degenerate case considered here.  In this step, we must also overcome additional difficulties due to the degeneracy of
 \eqref{eq:periodic equation ma}. Consequently, when Theorem \ref{thm:harnack} is applied to the difference of two solutions, exponential growth is precluded. Hence, the classical Harnack inequality holds for this difference, and our theorems would follow.

This paper is organized as follows. In Section \ref{sec:harnack}, we study convex functions whose Monge-Amp\`ere measure has small period, establishing a dichotomous Harnack type  inequality for linearized Monge-Amp\`ere equations. Section \ref{sec:semiconvavity} investigates  the uniqueness of compatible quadratic component of each solution in the sense of Proposition \ref{prop:semiconcave}. Finally, in Section \ref{sec:main theorem}, we prove  Theorems \ref{thm:uniqueness within periodic} and \ref{thm:solutions are periodic}.

The post-theorem introduction says: existence of periodic solutions “follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof,” while Theorem~\ref{thm:uniqueness within periodic} “extends these results to general periodic Borel measures.” It then explains the main difficulty in proving the results (including Theorem~\ref{thm:uniqueness within periodic}) in the nonnegative/degenerate setting: “the associated measure $\mu$ may not satisfy the doubling condition,” so “sections lose the engulfing property,” which “invalidat[es] the Vitali or Besicovitch covering lemma … essential for the Calder\'on-Zygmund decomposition on sections,” and therefore one “cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\'errez.”

To overcome this, “by exploiting the equation's periodic structure,” the authors “derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator.” For supersolutions they do so “by excluding contributions from small sections,” obtaining “decay estimates for homogenized level-sets” (Theorem~\ref{thm:harnack super}); for subsolutions (where “standard $L^{\infty}$ bounds fail”), they prove a “dichotomy” (Theorem~\ref{thm:harnack sub}): “either the $L^{\infty}$ bound holds or the $L^{\infty}$ norm grows exponentially”; combining these yields “a Harnack inequality with a quantitatively controlled error term” (Theorem~\ref{thm:harnack}).

“After overcoming this critical difficulty,” the remaining proof “focuses on analyzing the second-order difference quotient $\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart,” which “arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\`ere equations,” with periodicity “play[ing] a crucial role.” They then “establish the quadratic behavior of solutions near infinity,” extending arguments from \cite{caffarelli2004liouville,li2019monge} to the degenerate case. Finally, applying Theorem~\ref{thm:harnack} “to the difference of two solutions,” “exponential growth is precluded,” so “the classical Harnack inequality holds for this difference,” and “our theorems would follow.”

The post-theorem introduction says: existence of periodic solutions “follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof,” while in establishing the main theorem it “extends these results to general periodic Borel measures.” It then explains the main difficulty in proving the results (including the main theorem) in the nonnegative/degenerate setting: “the associated measure $\mu$ may not satisfy the doubling condition,” so “sections lose the engulfing property,” which “invalidat[es] the Vitali or Besicovitch covering lemma … essential for the Calder\'on-Zygmund decomposition on sections,” and therefore one “cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\'errez.”

To overcome this, “by exploiting the equation's periodic structure,” the authors “derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator.” We first use the following theorem for supersolutions:
\begin{Theorem}\label{thm:harnack super}
Assume that $v \in C^2(\Omega)$ is a nonnegative supersolution of \eqref{eq:linearized ma phi} in $\Omega$.
Then there exist positive constants  $\Lambda_6(n)$, $\epsilon(n) $ and $C(n) $ such that if
\[
\inf_{B_{1/4}} v \leq 1,
\]
then for $\Lambda\ge \Lambda_6(n)$, we have
\begin{equation}\label{eq:harnack super}
\mu(\{M_{\phi,\Lambda}v >  t\} \cap B_{1/2})< Ct^{-\epsilon} .
\end{equation}
\end{Theorem}
For subsolutions (where “standard $L^{\infty}$ bounds fail”), they prove the following “dichotomy”:
\begin{Theorem}\label{thm:harnack sub}
Assume that $v \in C^2(\Omega)$ with $v \geq 0$ is a subsolution of \eqref{eq:linearized ma phi} in $\Omega$.
Then, there exist positive constants $\Lambda_7(n)$ and $\beta(n)$ such that for any $\epsilon > 0$, there is a constant $C_2(n, \epsilon)$ with the following property: If $\Lambda\ge \Lambda_7(n)$ and
\[
\min\{\mu(\{v >  t\}  \cap B_{1/2}), \mu(\{M_{\phi,\Lambda}v >  t\} \cap B_{1/2})  \} \leq t^{-\epsilon}   \quad  \forall t>0,
\]
then either
\[
\sup_{B_{1/4} } v  \leq C_2(n,\epsilon),
\]
or for $\kappa =\kappa(B_1)\geq \Lambda^2$ that
\[
\sup_{B_{1/2} } v  \geq e^{\beta \kappa^{\frac{1}{3}} } \sup_{B_{1/4} }v.
\]
\end{Theorem}
Combining these yields “a Harnack inequality with a quantitatively controlled error term,” namely the following theorem:
\begin{Theorem}\label{thm:harnack}
Let $\Omega$ and $\widetilde{\Omega}$ be open convex subsets of $\mathbb{R}^n$ satisfying $B_1\subset \Omega \subset B_n, B_1\subset \widetilde{\Omega} \subset B_n$, $n\geq 2$, and let $\phi\in C^2(\overline{\Omega})$ and $\tilde{\phi}\in C^2(\overline{\widetilde{\Omega}})$ be convex functions satisfying,
\[
\mu_\phi =\mu \quad \text{in } \Omega, \quad \phi=0 \quad \text{on } \partial \Omega,
\]
\[
\mu_{\tilde{\phi}} =\mu \quad \text{in }  \widetilde{\Omega}, \quad \tilde{\phi}=0 \quad \text{on } \partial \widetilde{\Omega},
\]
where $\mu$ is a periodic Borel measure defined on $\R^n$ satisfying \eqref{eq:period gamma},  \eqref{eq:normalized assumption}, and \eqref{eq:small period assumption}. Assume that $v\in C^2(\Omega) \cap C^2(\widetilde{\Omega})$ with $v\geq 0$ satisfies
\[
L_{\phi}v \leq 0   \quad  \text{in } \Omega, \quad L_{\tilde{\phi}}v \geq 0   \quad  \text{in }\widetilde{\Omega}.
\]

Then there exist positive constants $\beta(n)$, $\Lambda_9(n)$,  and $C(n)$ such that if $\Lambda>\Lambda_9(n)$, then either
\[
\sup_{B_{1/4}}v\leq C\inf_{B_{1/4}}v,
\]
or for $\kappa =\kappa(B_1)\geq \Lambda^2$ that
\[
\sup_{B_{1/2}} v  \geq e^{\beta \kappa^{\frac{1}{6}}}\sup_{B_{1/4}}v.
\]
Consequently, we have
\[
\sup_{B_{1/4}}v \leq C\inf_{B_{1/4}}v+e^{-\beta \kappa^{\frac{1}{6}}}\sup_{B_{1/2}} v.
\]
\end{Theorem}

“After overcoming this critical difficulty,” the remaining proof “focuses on analyzing the second-order difference quotient $\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart,” which “arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\`ere equations,” with periodicity “play[ing] a crucial role.” They then “establish the quadratic behavior of solutions near infinity,” extending arguments from \cite{caffarelli2004liouville,li2019monge} to the degenerate case. Finally, applying the preceding theorem (the Harnack inequality with controlled error term) “to the difference of two solutions,” “exponential growth is precluded,” so “the classical Harnack inequality holds for this difference,” and this completes the proof strategy for the main theorem.

\label{thm:uniqueness within periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying
\begin{equation}\label{eq:periodic f}
\mu (E+z)=\mu (E),\quad \forall \ z \in \mathbb{Z}^n.
\end{equation}
and let $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ be a quadratic function with $A\in \mathcal{S}_+^{n\times n}$ satisfying
\begin{equation}\label{eq:A compatible}
 \det A =\mu (\mathbb{T}^n),
\end{equation}
Then, to prove the main theorem, we show that there exists a unique periodic solution (up to addition of constants) to
\begin{equation}\label{eq:periodic equation ma}
\det D^2 u = \mu \quad \text{in } \R^n,
\end{equation}
with quadratic component $P$.

Question: Let \(\mu \not\equiv 0\) be a nonnegative locally finite Borel measure on \(\mathbb{R}^n\) that is periodic with respect to \(\mathbb{Z}^n\), meaning that \(\mu(E+z)=\mu(E)\) for every Borel set \(E\subset \mathbb{R}^n\) and every \(z\in \mathbb{Z}^n\). Let \(P(x)=\frac12 x^{\top}Ax+b\cdot x+c\), where \(A\in \mathcal{S}_+^{n\times n}\) is positive definite symmetric, \(b\in\mathbb{R}^n\), \(c\in\mathbb{R}\), and \(\det A=\mu(\mathbb{T}^n)\) with \(\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n\). A periodic solution with quadratic component \(P\) means a convex function \(u\in C(\mathbb{R}^n)\) satisfying \(\det D^2u=\mu\) in \(\mathbb{R}^n\) in the Alexandrov sense and admitting a decomposition \(u(x)=v(x)+P(x)\), where \(v(x+z)=v(x)\) for all \(z\in\mathbb{Z}^n\). Which statement holds?

There exists a periodic solution with quadratic component \(P\), and it is unique up to addition of constants. Equivalently, there exists a convex \(u\in C(\mathbb{R}^n)\) solving \(\det D^2u=\mu\) in the Alexandrov sense such that \(u(x)=v(x)+P(x)\) for some \(\mathbb{Z}^n\)-periodic function \(v\), and whenever \(u_1=u_1^{\mathrm{per}}+P\) and \(u_2=u_2^{\mathrm{per}}+P\) are two such periodic solutions, \(u_1-u_2\) is constant on \(\mathbb{R}^n\).

Choices:
B. There exists a periodic solution with quadratic component \(P\), and it is unique in the strict sense. Equivalently, there exists a convex \(u\in C(\mathbb{R}^n)\) solving \(\det D^2u=\mu\) in the Alexandrov sense such that \(u(x)=v(x)+P(x)\) for some \(\mathbb{Z}^n\)-periodic function \(v\), and whenever \(u_1=u_1^{\mathrm{per}}+P\) and \(u_2=u_2^{\mathrm{per}}+P\) are two such periodic solutions, one has \(u_1=u_2\) on \(\mathbb{R}^n\).
C. There exists at most one periodic solution with quadratic component \(P\) up to addition of constants. Equivalently, whenever \(u_1=u_1^{\mathrm{per}}+P\) and \(u_2=u_2^{\mathrm{per}}+P\) are two such periodic solutions of \(\det D^2u=\mu\) in the Alexandrov sense, \(u_1-u_2\) is constant on \(\mathbb{R}^n\).
D. For every quadratic function \(P(x)=\frac12 x^{\top}Ax+b\cdot x+c\) with \(A\in \mathcal{S}_+^{n\times n}\), there exists a periodic solution with quadratic component \(P\), and it is unique up to addition of constants; in particular, the compatibility condition \(\det A=\mu(\mathbb{T}^n)\) is not needed for existence or uniqueness.
E. There exists a periodic solution with quadratic component \(P\), and any two such periodic solutions differ by an affine function on \(\mathbb{R}^n\). Equivalently, there exists a convex \(u\in C(\mathbb{R}^n)\) solving \(\det D^2u=\mu\) in the Alexandrov sense such that \(u(x)=v(x)+P(x)\) for some \(\mathbb{Z}^n\)-periodic function \(v\), and whenever \(u_1=u_1^{\mathrm{per}}+P\) and \(u_2=u_2^{\mathrm{per}}+P\) are two such periodic solutions, \(u_1-u_2=\ell\) for some affine function \(\ell\).

The stem gives hypotheses and definitions but does not explicitly state the conclusion. It does not directly reveal that existence plus uniqueness up to constants is the correct option.

The item is close to a theorem-recall question: under the stated assumptions, the student is asked to identify the correct conclusion. However, the alternatives are meaningfully varied, so it is not a pure verbatim restatement.

Some reasoning is needed to distinguish strict uniqueness, uniqueness up to constants, mere uniqueness without existence, and incorrect affine ambiguity. But the task is still mainly recognition of the theorem rather than substantial derivation.

The distractors are strong and mathematically plausible: they test common overstatements and confusions about compatibility conditions and uniqueness classes. They are distinct and well-targeted.

A solid theorem-identification MCQ with strong distractors and little answer leakage, though it leans more toward recall of a known result than deep generative reasoning.