Uniform Random Covering Problems. (1st October 2021)

• Record Type: Journal Article
• Authors: Koivusalo, Henna; Liao, Lingmin; Persson, Tomas
• Is Part Of: International mathematics research notices. Volume 2023:Number 1(2023)
• Page Start: 455
• Page End: 481
• Publication Date: 2021-10-01
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnab272
• Languages: English
• ISSNs: 1073-7928

Abstract

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