A note on uniform random covering problems in metric spaces

TL;DR

This paper investigates the properties of a uniform random covering set in general metric spaces, establishing a 0-1 law for its Hausdorff dimension and measure, with applications to the d-dimensional torus.

Contribution

It extends the analysis of random covering problems from the torus to general metric spaces and provides conditions for full measure and explicit Hausdorff dimension estimates.

Findings

Established a 0-1 law for Hausdorff dimension and measure of the covering set.

Provided sufficient conditions for the set to have full measure or be countable.

Applied results to the d-dimensional torus with explicit dimension analysis.

Abstract

In this paper, we study the uniform random covering problem in general metric space $(X,d)$. Let $\omega=(\omega_n)_{n\in\mathbb N}$ be a sequence of independent identically distributed random variables on $(X,\mu)$, and $\ell=(\ell_n)_{n\in\mathbb N}$ a sequence of positive real numbers. We analyze the size of the set \[\mathcal{U}(\omega,\ell)=\left\{y\in X\colon \forall N\gg1,~\exists 1\le n\le N,~s.t. ~d(\omega_n,y)<\ell_N\right\},\] and establish the 0-1 law for the Hausdorff dimension of $\mathcal{U}(\omega,\ell)$, its measure and the event $\mathcal{U}(\omega,\ell)=X$. Some sufficient conditions are provided for $\mathcal{U}(\omega,\ell)$ to have full measure or be countable almost surely. Furthermore, we employ the local dimension of $\mu$ to estimate the Hausdorff dimension of $\mathcal{U}(\omega,\ell)$. 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.