## 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 &x1D54B; and a sequence (r_{n})_{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. |ω_{n} - y| < r_{N}}. 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 language | English |
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Pages (from-to) | 455-481 |

Number of pages | 27 |

Journal | International Mathematics Research Notices |

Volume | 2023 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2023 Jan 1 |

## Subject classification (UKÄ)

- Mathematical Analysis