Roope Anttila

Dvoretzky covering problem for general measures

A random covering set is a limsup set generated by a collection of balls with deterministic radii and centres chosen at random with respect to a fixed probability measure. In 1956, in the context of the Lebesgue measure on the unit circle, Aryeh Dvoretzky asked the following question: When does the random covering set fully cover the support of the measure almost surely? For the Lebesgue measure, this question was answered in 1972 by Shepp, who showed that a necessary and sufficient condition for full covering is given by the divergence of a certain series which only depends on the sequence of radii. Shepp's result was later generalised by Kahane, who gave a potential theoretic characterisation for covering an arbitrary compact subset of the circle. In this talk, I will discuss recent joint work with Markus Myllyoja, where we show that a generalisation of Kahane's potential theoretic approach characterises the covering of analytic sets by random covering sets driven by arbitrary Borel probability measures on the real line.