Uniform Random Covering Problems

Henna Koivusalo, Lingmin Liao, Tomas Persson

Research output: Contribution to journalArticlepeer-review

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 = (ωn)n≥1 uniformly distributed on the unit circle &x1D54B; and a sequence (rn)n≥1 of positive real numbers with limit 0. We investigate the size of the random set U(ω) := {y ∈ &x1D54B;: ∀N ≫ 1, ∃n ≤ N, s.t. |ωn - y| < rN}. 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.

Original languageEnglish
Pages (from-to)455-481
Number of pages27
JournalInternational Mathematics Research Notices
Volume2023
Issue number1
DOIs
Publication statusPublished - 2023 Jan 1

Subject classification (UKÄ)

  • Mathematical Analysis

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