Uniform random covering problems

TL;DR

This paper studies the size and measure of a random set on the circle formed by points close to a sequence of random points, providing conditions for it to be full, null, or countable, and estimating its Hausdorff dimension.

Contribution

It introduces new conditions determining when the random covering set is full measure, null measure, or countable, and estimates its Hausdorff dimension in the null case.

Findings

Conditions for full Lebesgue measure of the set.

Conditions for the set to be countable.

Bounds on Hausdorff dimension when the set is null.

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.