Davit Karagulyan: On random covering of the circle by arcs with non-uniformly distributed centers

Abstract: The classical Dvoretzky covering problem asks for conditions on the sequence of lengths $\{\ell_n\}_{n\in \mathbb{N}}$ so that the random intervals$I_n : = (\omega_n -(\ell_n/2), \omega_n +(\ell_n/2))$ where $\omega_n$ is a sequence of i.i.d. uniformly distributed random variables, covers any point on the circle $\mathbb{T}$ infinitely often. We consider the case when the distribution of $\omega_n$ is absolutely continuous with a density function $f$. When the set $K_f$ of essential infimum points of $f$ has an upper box-counting dimension strictly less than 1, we give a necessary and sufficient condition for Dvoretzky covering. Under more restrictive assumptions on the sequence $\{\ell_n\}$ the above result is true if $\dim_H K_f<1$. We next show that as long as $\{\ell_n\}_{n\in \mathbb{N}}$ and $f$ satisfy the assumption of the above theorem and $|K_f|=0$, then a Menshov type result holds, i.e. Dvoretzky covering can be achieved by changing $f$ on a set of arbitrarily small Lebesgue measure.
This is joint work with M. Hirayama.