arXiv subjects
Publications and source records attributed to Leiye Xu.
We give an affirmative answer to Owings's sumset question: for any $2$-coloring of natural numbers, there is an infinite $B\subseteq\mathbb{N}$ such that $B+B$ is monochromatic. More generally, for every $m,\ell\in\mathbb{N}$ and every $2$-coloring of $\mathbb{N}$, there is an infinite $B\subseteq\mathbb{N}$ such that $$ (m+\ell)B\cup\{mx+\ell y:x,y\in B,\ x<y\} $$ is monochromatic.
Existing LLM-based theorem provers have achieved impressive results on formal mathematics benchmarks, yet they remain confined to acting as autonomous agents that prove a stated proposition. In this paper, we propose MathCoPilot, a human-in-the-loop system that embodies a new human--AI symbiotic paradigm for mathematical research, in which the mathematician steers the high-level mathematical direction while AI agents carry out the detailed formalization and proof work under continuous human guidance. MathCoPilot unifies three core capabilities: (1) an interactive workbench where the mathematician and AI agents collaborate through a living proof blueprint that decomposes a proof into navigable steps the human can directly inspect, direct, and refine; (2) automated proving skill orchestration with adaptive knowledge base search and Lean-integrated iterative verification; and (3) topic-driven paper retrieval and automated formalization into a verified Lean knowledge base. Using MathCoPilot, we systematically compare four state-of-the-art LLMs, including Gemini~3.1~Pro, GPT-5.4, and Claude~Opus~4.7, on a FormalMATH subset and on two real PDE theorems requiring deep domain expertise, evaluating their ability to produce verified Lean~4 proofs and to identify errors in deliberately incorrect proofs. Our results show that while current models can handle undergraduate-level problems with high success rates under favorable autoformalization conditions, substantial challenges remain for domain-specific theorems requiring genuine mathematical understanding.
We prove that every free ergodic measure-preserving action of a countably infinite amenable group with positive Rokhlin entropy has infinite complex $L^1$-orbit multiplicity, both on $L^1$ and on its mean-zero subspace $L^1_0$. This gives a negative answer to a question of J.-P. Thouvenot recorded by Iwanik and establishes the corresponding endpoint statement at $p=1$ of Iwanik's theorem that positive entropy implies infinite $L^p$-multiplicity for every $p>1$. The proof combines Malykhin's rigidity theorem for independent random variables, Seward's Bernoulli factor theorem, and a F{\o}lner set argument.
We construct Bohr obstructions to multiple recurrence along rounded Hardy-field sequences, showing that the real derivative-span criterion of Bergelson, Moreira, and Richter is essentially sharp and answering two of their questions. For $E\subseteq\mathbb N$ and $u:\mathbb N\to\mathbb Z$, set $R_{u}(E):=\{n\in\mathbb N:E\cap(E-u(n))\neq\varnothing\}$. We prove that, if $f_1,\dots,f_k$ are functions of polynomial growth from a Hardy field and some real linear combination of $f_1,\dots,f_k$ and their derivatives has a nonzero finite limit, then there exist $M\in\mathbb N$ and a basic Bohr set $E\subseteq\mathbb N$ such that $\bigcap_{i=1}^k R_{[Mf_i]}(E)$ is not thick. In particular, for some Bohr set $E$, the set $R_{[t^{3/2}]}(E)\cap R_{[\sqrt{2}t^{3/2}+t]}(E)$ is piecewise syndetic but not thick. We also prove that, if for some $\lambda_1,\dots,\lambda_k\in\mathbb{R}$ we have $$\inf_{x\geq 1}\left\|\sum_{i}\lambda_if_i(x)\right\|_{\mathbb{T}}>\frac{1}{2} \sum_i|\lambda_i|,$$ then $\bigcap_i R_{[f_i]}(E)=\varnothing$ for some basic Bohr set $E$. More generally, our results apply with $[\cdot]$ replaced by any rounding function $\rho:\mathbb{R}\to\mathbb{Z}$ satisfying $\sup_{x\in\mathbb{R}}|\rho(x)-x|<\infty$.
We develop a new theory of maximizing sets in dynamical systems, for the study of ergodic optimization in systems with weak hyperbolicity but where the Mañé cohomology lemma does not hold. This leads to new solutions of the Typical Periodic Optimization problem in the Lipschitz category: existence of an open dense set of Lipschitz functions such that each member has a unique maximizing measure and this measure is periodic (an equi-distribution on a single periodic orbit). The theory yields a structural theorem, that isolates the part of the system responsible for any robust non-periodic optimization. The structural theorem is developed further in the setting of symbolic dynamics: given any shift space, for typical Lipschitz functions the maximizing measure is shown to be either periodic or supported on the Markov boundary of the shift space. It follows that Contreras' Typical Periodic Optimization theorem for shifts of finite type can be extended to a wide class of shift spaces, including every sofic shift. The structural theorem is used to provide the first known example of a shift space where Typical Periodic Optimization fails despite periodic measures being dense in the set of all invariant measures.
In this paper, we introduce topological dynamical systems with almost countable spectrum. We prove that the Logarithmic Sarnak Conjecture holds for zero-entropy topological dynamical systems whose spectrum is almost countable. This class includes Anzai skew product on $\mathbb{T}^2$ over a rotation of $\mathbb{T}^1$, time-one maps of continuous suspension flows over rotations, systems with finite maximal pattern entropy, and bounded tame systems.
In this paper, we mainly study the relation between regularity, independence and mean sensitivity for minimal systems. In the first part, we show that if a minimal system is incontractible, or local Bronstein with an invariant Borel probability measure, then the regularity is strictly bounded by the infinite independence. In particular, the following two types of minimal systems are applicable to our result: (1) The acting group of the minimal system is a virtually nilpotent group. (2) The minimal system is a proximal extension of its maximal equicontinuous factor and admits an invariant Borel probability measure. Items (1) and (2) correspond to Conjectures 1 and 2 from Huang, Lian, Shao, and Ye (J. Funct. Anal., 2021); item (1) verifies Conjecture 1 in the virtually nilpotent case, and item (2) gives an affirmative answer to Conjecture 2. In the second part, for a minimal system acting by an amenable group, under the local Bronstein condition, we establish parallel results regarding weak mean sensitivity and establish that every mean-sensitive tuple is an IT-tuple.
We investigate the connections between independence, sequence entropy, and mean sensitivity for a measure preserving system under the action of a countable infinite discrete group. We establish that every sequence entropy tuple for an invariant measure is an IT tuple. Furthermore, if the acting group is amenable, we show that for an ergodic measure, the sequence entropy tuples, the mean sensitive tuples along some tempered Følner sequence, and the sensitive in the mean tuples along some tempered Følner sequence coincide.
In this paper, we investigate the discrete spectrum of probability measures for actions of locally compact groups. We establish that a probability measure has a discrete spectrum if and only if it has bounded measure-max-mean-complexity. As applications: 1) An invariant measure for a locally compact amenable group action has a discrete spectrum if and only if it has bounded mean-complexity along Følner sequences; 2) An invariant measure for a locally compact amenable group action has a discrete spectrum if and only if it is mean equicontinuous along a tempered Følner sequence, or equicontinuous in the mean along a tempered Følner sequence.
In this paper, we introduce the directional Pinsker algebra, and construct a skew product to study it. As applications, we show that 1. if a $\mathbb{Z}^2$-system with positive directional measure-theoretic entropy then it is multivariant directional mean Li-Yorke chaotic along the corresponding direction; 2. for any ergodic measure on a $\mathbb{Z}^2$-system, the intersection of the set of directional measure-theoretic entropy tuples with the set of directional asymptotic tuples is dense in the set of directional measure-theoretic entropy tuples.
We show that a minimal toplogical dynamical system that is frequently stable if and only if it is almost automorphic.
Let $G$ be an infinite discrete countable group and $(X,G)$ be a minimal $G$-system. In this paper, we prove the supremum of topological sequence entropy of $(X,G)$ is not less than $\log(\sum_{μ\in\mathcal{M}^e(X,G)}e^{h_μ^*(X,G)})$. If additionally $G$ is abelian then there is a constant $K\in\mathbb{N}\cup\{\infty\}$ with $\log K\le h_{top}^*(X,G)$ such that $ν(\{y\in H:|π^{-1}(y)|=K\})=1$ where $(H,G)$ is the maximal equicontinuous factor of $(X,G)$, $π:(X,G)\to (H,G)$ is the factor map and $ν$ is the Haar measure of $H$.
Let $G$ be an infinite countable discrete amenable group. For any $G$-action on a compact metric space $X$, it is proved that for any sequence $(G_n)_{n\ge 1}$ consisting of non-empty finite subsets of $G$ with $\lim_{n\to \infty}|G_n|=\infty$, Pinsker $σ$-algebra is a characteristic factor for $(G_n)_{n\ge 1}$. As a consequence, for a class of $G$-topological dynamical systems, positive topological entropy implies mean Li-Yorke chaos along a class of sequences consisting of non-empty finite subsets of $G$.
Given $q\in\mathbb{N}$, let $(X,T)$ be a $\mathbb{Z}^q$-system, $\vec{v}\in\mathbb{R}^q\setminus\{\vec{0}\}$ be a direction vector and $\textbf{b}\in\mathbb{R}_+^{q-1}$. We study $(X,T)$ that has bounded complexity with respect to three kinds of metrics defined along direction $\vec{v}$: the directional Bowen metric $d_k^{\vec{v},\textbf{b}}$, the directional max-mean metric $\hat{d}_k^{\vec{v},\textbf{b}}$ and the directional mean metric $\bar{d}_k^{\vec{v},\textbf{b}}$. It is shown that $(X,T)$ has bounded topological complexity with respect to $\{d_k^{\vec{v},\textbf{b}}\}_{k=1}^{\infty}$ (resp. $\{\hat{d}_k^{\vec{v},\textbf{b}}\}_{k=1}^{\infty}$) if and only if $T$ is $(\vec{v},\textbf{b})$-equicontinuous (resp. $(\vec{v},\textbf{b})$-equicontinuous in the mean). Meanwhile, it turns out that an invariant Borel probability measure $μ$ on $X$ has bounded complexity with respect to $\{d_k^{\vec{v},\textbf{b}}\}_{k=1}^{\infty}$ if and only if $T$ is $(μ,\vec{v},\textbf{b})$-equicontinuous. Moreover, it is shown that $μ$ has bounded complexity with respect to $\{\bar{d}_k^{\vec{v},\textbf{b}}\}_{k=1}^{\infty}$ if and only if $μ$ has bounded complexity with respect to $\{\hat{d}_k^{\vec{v},\textbf{b}}\}_{k=1}^{\infty}$ if and only if $T$ is $(μ,\vec{v},\textbf{b})$-mean equicontinuous if and only if $T$ is $(μ,\vec{v},\textbf{b})$-equicontinuous in the mean if and only if $μ$ has $\vec{v}$-discrete spectrum.
In this paper, directional sequence entropy and directional Kronecker algebra for $\mathbb{Z}^q$-systems are introduced. The relation between sequence entropy and directional sequence entropy are established. Meanwhile, direcitonal discrete spectrum systems and directional null systems are defined. It is shown that a $\mathbb{Z}^q$-system has directional discrete spectrum if and only if it is directional null. Moreover, it turns out that a $\mathbb{Z}^q$-system has directional discrete spectrum along $q$ linearly independent directions if and only if it has discrete spectrum.
We prove that there exists an open and dense subset $\mathcal{U}$ in the space of $C^{2}$ expanding self-maps of the circle $\mathbb{T}$ such that the Lyapunov minimizing measures of any $T\in{\mathcal U}$ are uniquely supported on a periodic orbit.This answers a conjecture of Jenkinson-Morris in the $C^2$ topology.
The weak mean equicontinuous properties for a countable discrete amenable group $G$ acting continuously on a compact metrizable space $X$ are studied. It is shown that the weak mean equicontinuity of $(X \times X,G)$ is equivalent to the mean equicontinuity of $(X,G)$. Moreover, when $(X,G)$ has full measure center or $G$ is abelian, it is shown that $(X,G)$ is weak mean equicontinuous if and only if all points in $X$ are uniquely ergodic points and the map $x \to μ_x^G$ is continuous, where $μ_x^G$ is the unique ergodic measure on $\{\ol{Orb(x)}, G\}$.
In this paper, we consider measure-theoretical restricted sensitivity and topological restricted sensitivities by restricting the first sensitive time. For a given topological dynamical system, we define measure-theoretical restricted asymptotic rate with respect to sensitivity, and obtain that it equal to the reciprocal of the Brin-Katok local entropy for almost every point. For topological version we have similar definitions and conclusions.