学术报告报告题目：Uniform random covering problems
报 告 人：廖灵敏 副教授(法国东巴黎大学)
报告时间：2020年12月28日（星期一）下午16:00-17:00
报告地点：腾讯会议， ID：364 269 623 会议密码：654321
邀 请 人：李兵 教授
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数学学院
2020年12月25 日
报告摘要：
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.