On μ-Dvoretzky random covering of the circle

Abstract

In this paper, we study the Dvoretzky covering problem with non-uniformly distributed centers. When the probability law of the centers is absolutely continuous w.r.t. Lebesgue measure and satisfies a regularity 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 ${\ell }_{\mathit{n}}=\frac{\mathit{c}}{\mathit{n}}$, we give a necessary and sufficient condition for covering the circle, without imposing any regularity on the density function.