A note on uniform random covering problems in metric spaces

Abstract

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.

Theorems & Definitions (34)

• Theorem 1.1: Dirichlet's theorem
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 3.1
• proof : Proof of Theorem \ref{['theorem1']}
• Theorem 4.1: Theorem 4.3.1 of galambos
• proof : Proof of Theorem \ref{['theorem2']}
• proof : Proof of Theorem \ref{['theorem3']}