On $\mu$-Dvoretzky random covering of the circle

TL;DR

This paper investigates the conditions under which a circle can be covered by random intervals centered at points with non-uniform distributions, providing necessary and sufficient criteria for coverage in various cases.

Contribution

It introduces new criteria for circle coverage with non-uniformly distributed centers, including cases with regular densities and specific interval lengths.

Findings

Established necessary and sufficient conditions for coverage with regular densities.

Derived criteria for coverage when interval lengths are proportional to 1/n.

Extended results to cases without regularity assumptions on the density.

Abstract

In this paper, we study the Dvoretzky covering problem with non-uniformly distributed centers. When the probability law of the centers admits an absolutely continuous density which satisfies a regular condition on the set of essential infimum points, we give a necessary and sufficient condition for covering the circle. When the lengths of covering intervals are of the form $\ell_n = \frac{c}{n}$, we give a necessary and sufficient condition for covering the circle, without imposing any regularity on the density function.