Title: Uniform random covering problems

Speaker:  Prof. Lingming Liao  ( University of Paris East, France )

Time: Mon, Dec.28 2020, PM: 16:00-17:00

Location: Tencent Conference

Meeting  Number: 364 269 623

Password: 654321

Inviter: Prof. Bing Li

Abstract:

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