2021/03/02 by Koivusalo, Henna, Liao, Lingmin, Persson, Tomas
#28A78 #60D05 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2103.01595
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=(ωn)n≥ 1 uniformly distributed on the unit circle \mathbbT and a sequence (rn)n≥ 1 of positive real numbers with limit 0. We investigate the size of the random set \mathcal U (ω):=\y∈ \mathbbT: ∀ N≫ 1, ∃ n ≤ N, s.t. ‖ ωn -y ‖ lt; rN \. Some sufficient conditions for \mathcal U(ω) to be almost surely the whole space, of full Lebesgue measure, or countable, are given. In the case that \mathcal U(ω) is a Lebesgue null measure set, we provide some estimations for the upper and lower bounds of Hausdorff dimension.