A note on uniform random covering problems in metric spaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908857130811392 |
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| author | Hu, Zhang-nan Li, Bing Wang, YiJing |
| author_facet | Hu, Zhang-nan Li, Bing Wang, YiJing |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00499 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A note on uniform random covering problems in metric spaces Hu, Zhang-nan Li, Bing Wang, YiJing Probability 60D05, 28A80 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. |
| title | A note on uniform random covering problems in metric spaces |
| topic | Probability 60D05, 28A80 |
| url | https://arxiv.org/abs/2603.00499 |