arXiv subjects

Zhang-nan Hu

Publications and source records attributed to Zhang-nan Hu.

8 recordsLinked to original sources

arXiv2026

A note on uniform random covering problems in metric spaces

In this paper, we study the uniform random covering problem in general metric space $(X,d)$. Let $ω=(ω_n)_{n\in\mathbb N}$ be a sequence of independent identically distributed random variables on $(X,μ)$, and $\ell=(\ell_n)_{n\in\mathbb N}$ a sequence of positive real numbers. We analyze the size of the set \[\mathcal{U}(ω,\ell)=\left\{y\in X\colon \forall N\gg1,~\exists 1\le n\le N,~s.t. ~d(ω_n,y)<\ell_N\right\},\] and establish the 0-1 law for the Hausdorff dimension of $\mathcal{U}(ω,\ell)$, its measure and the event $\mathcal{U}(ω,\ell)=X$. Some sufficient conditions are provided for $\mathcal{U}(ω,\ell)$ to have full measure or be countable almost surely. Furthermore, we employ the local dimension of $μ$ to estimate the Hausdorff dimension of $\mathcal{U}(ω,\ell)$. While prior work by Koivusalo, Liao and Persson ( Int. Math. Res. Not. 2023) addressed the case of the torus $\mathbb{T}$, we apply our results to the $d$-dimensional torus $\mathbb{T}^d$, and explicit analysis of the Hausdorff dimension in a critical case is given.

math.PR↗

arXiv2025

Dimension theory of inhomogeneous Diophantine approximation with matrix sequences

In this paper, we investigate the Hausdorff dimension of naturally occurring sets of inhomogeneous well-approximable points with a sequence of real invertible matrices $\mathcal{A}=(A_n)_{n\in\mathbb{N}}$. Specifically, for a given point $\mathbf{y}\in [0,1)^d$ and a function $ψ: \mathbb{N} \to \mathbb{R}^+$, we study the limsup set \[ W\big(\mathcal{A},ψ,{\bf y}\big) =\Big\{\mathbf{x}\in [0,1)^d\colon A_n\mathbf{x}~(\bmod~1)\in B\big(\mathbf{y}, ψ(n)\big) {\rm ~ for~ infinitely ~many}~n\in\mathbb{N}\Big\}.\] The upper and lower bounds on the Hausdorff dimension of $W\big(\mathcal{A},ψ,{\bf y}\big)$ are determined by involving the singular values of $A_n$ and the successive minima of the lattice $A_n^{-1}\mathbb{Z}^d$, and both bounds are shown to be attainable for some matrices. Within this framework, we unify the problem of shrinking target sets and recurrence sets, establishing the Hausdorff dimensions for such limsup sets. As applications, our corresponding upper bounds for shrinking target and recurrence sets essentially improve those appearing in the present literature. Furthermore, explicit Hausdorff dimension formulas are derived for shrinking targets and recurrence sets associated with concrete classes of matrices. We extend the Mass Transference Principle for rectangles of Li-Liao-Velani-Wang-Zorin (Adv. Math., 2025) to rectangles under local isometries. This generalization yields a general lower bound for the Hausdorff dimension of $W\big(\mathcal{A},ψ,{\bf y}\big)$.

math.NT↗

arXiv2024

On shrinking targets for linear expanding and hyperbolic toral endomorphisms

Let $A$ be an invertible $d\times d$ matrix with integer elements. Then $A$ determines a self-map $T$ of the $d$-dimensional torus $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$. Given a real number $τ>0$, and a sequence $\{z_n\}$ of points in $\mathbb{T}^d$, let $W_τ$ be the set of points $x\in\mathbb{T}^d$ such that $T^n(x)\in B(z_n,e^{-nτ})$ for infinitely many $n\in\mathbb{N}$. The Hausdorff dimension of $W_τ$ has previously been studied by Hill--Velani and Li--Liao--Velani--Zorin. We provide complete results on the Hausdorff dimension of $W_τ$ for any expanding matrix. For hyperbolic matrices, we compute the dimension of $W_τ$ only when $A$ is a $2 \times 2$ matrix. We give counterexamples to a natural candidate for a dimension formula for general dimension $d$.

math.DS↗

arXiv2023

Hausdorff dimension of recurrence sets

We consider linear mappings on the $d$-dimensional torus, defined by $T(x) = Ax \pmod 1$, where $A$ is an invertible $d \times d$ integer matrix, with no eigenvalues on the unit circle. In the case $d = 2$ and $\det A = \pm 1$, we give a formula for the Hausdorff dimension of the set \[ \{ \, x \in \mathbb{T}^d : d (T^n (x), x) < e^{- αn} \text{ for infinitely many } n \, \}. \]

math.DS↗

arXiv2022

Large intersection property for limsup sets in metric space

We show that limsup sets generated by a sequence of open sets in compact Ahlfors $s$-regular space $(X,\mathscr{B},μ,ρ)$ belong to the classes of sets with large intersections with index $λ$, denoted by $\mathcal{G}^λ(X)$, under some conditions. In particular, this provides a lower bound on Hausdorff dimension of such sets. These results are applied to obtain that limsup random fractals with indices $γ_2$ and $δ$ belong to $\mathcal{G}^{s-δ-γ_2}(X)$ almost surely, and random covering sets with exponentially mixing property belong to $\mathcal{G}^{s_0}(X)$ almost surely, where $s_0$ equals to the corresponding Hausdorff dimension of covering sets almost surely. We also investigate the large intersection property of limsup sets generated by rectangles in metric space.

math.MG↗

arXiv2021

On the Intersection of Dynamical Covering Sets with Fractals

Let $(X,\mathscr{B}, μ,T,d)$ be a measure-preserving dynamical system with exponentially mixing property, and let $μ$ be an Ahlfors $s$-regular probability measure. The dynamical covering problem concerns the set $E(x)$ of points which are covered by the orbits of $x\in X$ infinitely many times. We prove that the Hausdorff dimension of the intersection of $E(x)$ and any regular fractal $G$ equals $\dim_{\rm H}G+α-s$, where $α=\dim_{\rm H}E(x)$ $μ$--a.e. Moreover, we obtain the packing dimension of $E(x)\cap G$ and an estimate for $\dim_{\rm H}(E(x)\cap G)$ for any analytic set $G$.

math.DS↗

arXiv2021

On the hitting probabilities of limsup random fractals

Let $A$ be a limsup random fractal with indices $γ_1, ~γ_2 ~$and $δ$ on $[0,1]^d$. We determine the hitting probability $\mathbb{P}(A\cap G)$ for any analytic set $G$ with the condition $(\star)$$\colon$ $\dim_{\rm H}(G)>γ_2+δ$, where $\dim_{\rm H}$ denotes the Hausdorff dimension. This extends the correspondence of Khoshnevisan, Peres and Xiao [10] by relaxing the condition that the probability $P_n$ of choosing each dyadic hyper-cube is homogeneous and $\lim\limits_{n\to\infty}\frac{\log_2P_n}{n}$ exists. We also present some counterexamples to show the Hausdorff dimension in condition $(\star)$ can not be replaced by the packing dimension.

math.PR↗

arXiv2020

Random Covering Sets in Metric Space with Exponentially Mixing Property

Let $\{B(ξ_n,r_n)\}_{n\ge1}$ be a sequence of random balls whose centers $\{ξ_n\}_{n\ge1}$ is a stationary process, and $\{r_n\}_{n\ge1}$ is a sequence of positive numbers decreasing to 0. Our object is the random covering set $E=\limsup\limits_{n\to\infty}B(ξ_n,r_n)$, that is, the points covered by $B(ξ_n,r_n)$ infinitely often. The sizes of $E$ are investigated from the viewpoint of measure, dimension and topology.

math.PR↗