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 $\ell=(\ell_n)_{n\in\mathbb N}$ 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.