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Aihua Fan

Publications and source records attributed to Aihua Fan.

At least 19 recordsLinked to original sources
arXiv2026

A Weyl-type theorem for Diophantine approximations driven by LCA groups and applications

We investigate actions of locally compact Abelian (LCA) groups on the torus $\mathbb{T}^n$, motivated by their close connection with Diophantine approximation. While Kronecker's theorem yields a classical density result, we prove a stronger equidistribution theorem of Weyl type: every such action admits a decomposition into uniquely ergodic subsystems. The proof of this result is based on a characterization of unique ergodicity for actions of amenable groups on compact metric spaces. As consequences, we establish several foundational results for LCA groups, including the Bohr orthogonality of characters along arbitrary Folner sequences, a Bohr mean formula for almost periodic functions, and a Wiener-type theorem on LCA groups characterizing the discrete part of a Borel probability measure through its Fourier transform. An application to numerical analysis is also discussed.

math.DS↗
arXiv2026

Khintchin conjecture and related topics

Motivated by Khintchin's 1923 conjecture, refuted by Marstrand in 1970, we study the Khintchin class of functions associated to a given increasing sequence of integers. When the Khintchin class contains L^p(\mathbb{T}), we call the sequence a L^p-Khintchin sequence. We establish basic properties of Khintchin sequences, provide several constructions, and propose open problems for further research. We also initiate the study of Khintchin sequences of group endomorphisms on compact abelian groups. Under a Fourier-tightness assumption, we show that ergodicity (respectively, weakly mixing or strongly mixing) of a skew product of endomorphisms is equivalent to the corresponding property of the base system, supporting the idea that typical fiber orbits in such skew products should form Khintchin sequences.

math.DS↗
arXiv2025

Representation of quasi-periodic functions and Hausdorff-Young inequalities for Besicovitch almost periodic functions

For a class of $\mathbb{R}^d$-ations and $\mathbb{Z}^d$-actions on the $n$-dimensional torus $\mathbb{T}^n$, we characterize their unique ergodicity and establish a theorem of Weyl type. This result allows us to establish an isomorphism between the Banach algebra of quasi-periodic functions with spectrum in a given $\mathbb{Z}$-module and the Banach algebra of periodic functions on a torus. This, in return, allows us to give a very simple proof of Hausdorff-Young inequalities for Besicovitch almost periodic functions. The regularity of the parent function of a quasi-periodic function is also studied.

math.CA↗
arXiv2025

Infinite Dimensional Multifractal Analysis of the Wiener measure

We present a multifractal formalism for measures on infinite dimensional metric spaces, in terms of scales instead of dimensions in the classical multifractal analysis. We prove a multifractal formalism with a suitable scaling, called order, for the Wiener measure, which is the probability law of the standard Brownian motion. We also prove the fundamental Frostman Lemma on a large class of Polish spaces, for which the increasing sets lemma holds.

math.PR↗
arXiv2025

Non-Archimedean Koksma Theorems and Dimensions of Exceptional Sets

We establish a non-Archimedean analogue of Koksma's theorem. For a local field F of characteristic zero, we prove that the sequence ([{\alpha}x^n]) is uniformly distributed in the valuation ring O for almost every x with |x|_p>1. In the case of positive characteristic, ([x^n]) fails to be uniformly distributed, but it becomes {\mu}*-uniformly distributed for some weighted measure {\mu}*. These results are derived from a general metric theorem for sequences generated by expanding scaling maps. On the other hand, we demonstrate that the exceptional set of parameters x for which these sequences are not uniformly distributed is large (i.e. having full Hausdorff dimension) and share a rich q-homogeneous fractal structure.

math.NT↗
arXiv2025

Computation of Lyapunov exponents of matrix products

For $m$ given square matrices $A_0, A_1, \cdots, A_{m-1}$ ($m\ge 2$), one of which is assumed to be of rank $1$, and for a given sequence $(\omega_n)$ in $\{0,1, \cdots, m-1\}^\mathbb{N}$, the following limit, if it exists, $$L(\omega):=\lim_{n\to \infty} \frac 1n \log \|A_{\omega_0} A_{\omega_2}\cdots A_{\omega_{n-1}}\|$$ defines the Lyapunov exponent of the sequence of matrices $(A_{\omega_n})_{n\ge 0}$. It is proved that the Lyapunov exponent $L(\omega)$ has a closed-form expression under certain conditions. One special case arises when $A_j$'s are non-negative and $\omega$ is generic with respect to some shift-invariant measure; a second special case occurs when $A_j$'s (for $1\le j<m$) are invertible and $\omega$ is a typical point with respect to some shift-ergodic measure. Substitutive sequences and characteristic sequences of $\mathcal{B}$-free integers are considered as examples. An application is presented for the computation of multifractal spectrum of weighted Birkhoff averages.

math.DS↗
arXiv2024

$p$-adic rational maps having empty Fatou set

On any finite algebraic extension $K$ of the field $\Q_p$ of $p$-adic numbers, there exist rational maps $\phi\in K(z)$ such that dynamical system $(\mathbb{P}^{1}(K),\phi)$ has empty Fatou set, i.e. the iteration family $\{\phi^n: n\geq 0\}$ is nowhere equicontinuous.

math.DS↗
arXiv2023

Old and new results on the Furstenberg sets

This paper is a complement to our previous paper [21]. It surveys the works on the Furstenberg set $S=\{2^{m}3^{n}: n\ge 0, m\ge 0\}$ and its random version $T$. We also present some new results. For example, it is proved that $T$ almost surely contains a subset of positive lower density which is $\frac{4}{3}$-Rider. It is also proved that a class of random sets of integers are Sidon sets when Bourgain's condition is not satisfied; this generalizes a result of Kahane-Katznelson. Some open questions about $S$ and $T$ are listed at the end of the paper.

math.FA↗
arXiv2022

Multifractal Analysis of generalized Thue-Morse trigonometric polynomials

We consider the generalized Thue-Morse sequences $(t_n^{(c)})_{n\ge 0}$ ($c \in [0,1)$ being a parameter) defined by $t_n^{(c)} = e^{2πi c s_2(n)}$, where $s_2(n)$ is the sum of digits of the binary expansion of $n$. For the polynomials $σ_{N}^{(c)} (x) := \sum_{n=0}^{N-1} t_n^{(c)} e^{2πi n x}$, we have proved in [18] that the uniform norm $\|σ_N^{(c)}\|_\infty$ behaves like $N^{γ(c)}$ and the best exponent $γ(c)$ is computed. In this paper, we study the pointwise behavior and give a complete multifractal analysis of the limit $\lim_{n\to\infty}n^{-1}\log |σ_{2^n}^{(c)}(x)|$.

math.DS↗
arXiv2022

A topological version of Furstenberg-Kesten theorem

Let $A(x): =(A_{i, j}(x))$ be a continuous function defined on some subshift of $Ω:= \{0,1, \cdots, m-1\}^\mathbb{N}$, taking $d\times d$ non-negative matrices as values and let $ν$ be an ergodic $σ$-invariant measure on the subshift where $σ$ is the shift map. Under the condition that $ A(x)A(σx)\cdots A(σ^{\ell-1} x)$ is a positive matrix for some point $ x$ in the support of $ν$ and some integer $\ell\ge 1$ and that every entry function $A_{i,j}(\cdot)$ is either identically zero or bounded from below by a positive number which is independent of $i$ and $j$, it is proved that for any $ν$-generic point $ω\in Ω$, the limit defining the Lyapunov exponent $\lim_{n\to \infty} n^{-1} \log \|A(ω) A(σω)\cdots A(σ^{n-1}ω)\|$ exists.

math.DS↗
arXiv2022

The Furstenberg set and its random version

We study some number-theoretic, ergodic and harmonic analysis properties of the Furstenberg set of integers $S=\{2^{m}3^{n}\}$ and compare them to those of its random analogue $T$. In this half-expository work, we show for example that $S$ is "Khinchin distributed", is far from being Hartman-distributed while $T$ is, and that $S$ is a $Λ(p)$ set for all $2<p<\infty$ and that $T$ is a $p$-Rider set for all $p$ such that $4/3<p<2$. Measure-theoretic and probabilistic techniques, notably martingales, play an important role in this work.

math.FA↗
arXiv2022

Trigonometric multiplicative chaos and Application to random distributions

The random trigonometric series $\sum_{n=1}^\infty ρ_n \cos (nt +ω_n)$ on the circle $\mathbb{T}$ are studied under the conditions $\sum |ρ_n|^2=\infty$ and $ρ_n\to 0$, where $\{ω_n\}$ are iid and uniformly distributed on $\mathbb{T}$. They are almost surely not Fourier-Stieljes series but define pseudo-functions. This leads us to develop the theory of trigonometric multiplicative chaos, which are the limits of the exponentiations of partials sums. which produces a class of random measures. The behaviors of the partial sums of the above series are proved to be multifractal. Our theory holds on the torus $\mathbb{T}^d$ of dimension $d\ge 1$.

math.PR↗
arXiv2021

Bohr chaoticity of topological dynamical systems

We introduce the notion of Bohr chaoticity, which is a topological invariant for topological dynamical systems, and which is opposite to the property required by Sarnak's conjecture. We prove the Bohr chaoticity for all systems which have a horseshoe and for all toral affine dynamical systems of positive entropy, some of which don't have a horseshoe. But uniquely ergodic dynamical systems are not Bohr chaotic.

math.DS↗
arXiv2021

Bohr chaoticity of principal algebraic actions and Riesz product measures

For a continuous $\mathbb{N}^d$ or $\mathbb{Z}^d$ action on a compact space, we introduce the notion of Bohr chaoticity, which is an invariant of topological conjugacy and which is proved stronger than having positive entropy. We prove that all principal algebraic $\mathbb{Z}$ actions of positive entropy are Bohr-chaotic. The same is proved for principal algebraic $\mathbb{Z}^d$ ($d\ge 2$) actions of positive entropy under the condition of existence of summable homoclinic points.

math.DS↗
arXiv2020

Multifractal analysis of weighted ergodic averages

We propose to study the multifractal behavior of weighted ergodic averages. Our study in this paper is concentrated on the symbolic dynamics. We introduce a thermodynamical formalism which leads to a multifractal spectrum. It is proved that this thermodynamical formalism applies to different kinds of dynamically defined weights, including stationary ergodic random weights, uniquely ergodic weights etc. But the validity of the thermodynamical formalism for very irregular weights, like Möbius function, is an unsolved problem. The paper ends with some other unsolved problems.

math.DS↗
arXiv2019

On $μ$-Dvoretzky random covering of the circle

In this paper, we study the Dvoretzky covering problem with non-uniformly distributed centers. When the probability law of the centers admits an absolutely continuous density which satisfies a regular condition on the set of essential infimum points, we give a necessary and sufficient condition for covering the circle. When the lengths of covering intervals are of the form $\ell_n = \frac{c}{n}$, we give a necessary and sufficient condition for covering the circle, without imposing any regularity on the density function.

math.PR↗
arXiv2019

$L^\infty$-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization

Given an integer $q\ge 2$ and a real number $c\in [0,1)$, consider the generalized Thue-Morse sequence $(t_n^{(q;c)})_{n\ge 0}$ defined by $t_n^{(q;c)} = e^{2πi c S_q(n)}$, where $S_q(n)$ is the sum of digits of the $q$-expansion of $n$. We prove that the $L^\infty$-norm of the trigonometric polynomials $σ_{N}^{(q;c)} (x) := \sum_{n=0}^{N-1} t_n^{(q;c)} e^{2πi n x}$, behaves like $N^{γ(q;c)}$, where $γ(q;c)$ is equal to the dynamical maximal value of $\log_q \left|\frac{\sin qπ(x+c)}{\sin π(x+c)}\right|$ relative to the dynamics $x \mapsto qx \mod 1$ and that the maximum value is attained by a $q$-Sturmian measure. Numerical values of $γ(q;c)$ can be computed.

math.DS↗