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 $\omega =({\omega }_n{)}_{n\in \mathbb{N}}$ be a sequence of independent identically distributed random variables on $(X,\mu )$, and $\mathrm{\ell }=({\mathrm{\ell }}_n{)}_{n\in \mathbb{N}}$ a sequence of positive real numbers. We analyze the size of the set $$\mathcal{U}(\omega ,\mathrm{\ell })=\{y\in X\text{\hspace{0.17em}⁣}:\mathrm{\forall }N\gg 1,\mathrm{\exists }1\le n\le N,s\mathrm{.}t\mathrm{.}d({\omega }_n,y)<{\mathrm{\ell }}_N\},$$ and establish the 0-1 law for the Hausdorff dimension of $\mathcal{U}(\omega ,\mathrm{\ell })$, its measure and the event $\mathcal{U}(\omega ,\mathrm{\ell })=X$. Some sufficient conditions are provided for $\mathcal{U}(\omega ,\mathrm{\ell })$ to have full measure or be countable almost surely. Furthermore, we employ the local dimension of $\mu$ to estimate the Hausdorff dimension of $\mathcal{U}(\omega ,\mathrm{\ell })$. 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.