Uniform random covering problems

Motivated by the random covering problem and the study of Dirichlet uniform approximable numbers, we investigate the uniform random covering problem. Precisely, consider an i.i.d. sequence $ω=(ω_n)_{n\geq 1}$ uniformly distributed on the unit circle $\mathbb{T}$ and a sequence $(r_n)_{n\geq 1}$ of positive real numbers with limit $0$. We investigate the size of the random set $\mathcal U (ω):=\{y\in \mathbb{T}: \ \forall N\gg 1, \ \exists n \leq N, \ \text{s.t.} \ \| ω_n -y \| < r_N \}.$ Some sufficient conditions for $\mathcal U(ω)$ to be almost surely the whole space, of full Lebesgue measure, or countable, are given. In the case that $\mathcal U(ω)$ is a Lebesgue null measure set, we provide some estimations for the upper and lower bounds of Hausdorff dimension.

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