Uniform Random Covering Problems. (1st October 2021) - Koivusalo, Henna; Liao, Lingmin; Persson, Tomas
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 ω =(ω _n)_n≥ 1 uniformly distributed on the unit circle T and a sequence (r_n)_n≥ 1 of positive real numbers with limit 0 . We investigate the size of the random set align* & U (ω):={yT: N1, n ≤ N, s.t. | ω_n -y | < r_N }. align* Some sufficient conditions for U(ω ) to be almost surely the whole space, of full Lebesgue measure, or countable, are given. In the case that U(ω ) is a Lebesgue null measure set, we provide some estimations for the upper and lower bounds of Hausdorff dimension.