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 $\ddot{I}‰=(\ddot{I}{‰}_n{)}_{n\in \mathbb{N}}$ be a sequence of independent identically distributed random variables on $(X,\hat{I}¼)$, 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 $\mathcal{U}(\ddot{I}‰,\ell )$, its measure and the event $\mathcal{U}(\ddot{I}‰,\ell )=X$. Some sufficient conditions are provided for $\mathcal{U}(\ddot{I}‰,\ell )$ to have full measure or be countable almost surely. Furthermore, we employ the local dimension of $\hat{I}¼$ to estimate the Hausdorff dimension of $\mathcal{U}(\ddot{I}‰,\ell )$. 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.