Uniform random covering problems

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 $\omega=(\omega_n)_{n\geq 1}$ 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 \[ \mathcal U (\omega):=\{y\in \mathbb{T}: \ \forall N\gg 1, \ \exists n \leq N, \ \text{s.t.} \ \| \omega_n -y \| < r_N \}. \] Some sufficient conditions for $\mathcal U(\omega)$ to be almost surely the whole space, of full Lebesgue measure, or countable, are given. In the case that $\mathcal U(\omega)$ is a Lebesgue null measure set, we provide some estimations for the upper and lower bounds of Hausdorff dimension.