Uniform Random Covering Problems. (1st October 2021) - Koivusalo, Henna; Liao, Lingmin; Persson, Tomas
Abstract: 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 $ω =(ω _n)_{n≥ 1}$ uniformly distributed on the unit circle $\mathbb{T}$ and a sequence $(r_n)_{n≥ 1}$ of positive real numbers with limit $0$ . We investigate the size of the random set $$\begin{align*} & {\operatorname{{{\mathcal{U}}}}} (ω):=\{y\in \mathbb{T}: \ \forall N\gg 1, \ \exists n ≤ N, \ \text{s.t.} \ | ω_n -y | < r_N \}. \end{align*}$$ Some sufficient conditions for ${\operatorname{{{\mathcal{U}}}}}(ω )$ to be almost surely the whole space, of full Lebesgue measure, or countable, are given. In the case that ${\operatorname{{{\mathcal{U}}}}}(ω )$ is a Lebesgue null measure set, we provide some estimations for the upper and lower bounds of Hausdorff dimension.