On genericity of non-uniform Dvoretzky coverings of the circle

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 variable, covers any point on the circle $\mathbb{T}$ infinitely often. We consider the case when ${\omega }_n$ are absolutely continuous with a density function $f$. When $m_f=essin{f}_{\mathbb{T}}f>0$ and the set $K_f$ of its essential infimum points satisfies ${\overline{\dim }}_{\mathrm{B}}{K}_f<1$, where ${\overline{\dim }}_{\mathrm{B}}$ is the upper box-counting dimension, we show that the following condition is necessary and sufficient for $\mathbb{T}$ to be ${\mu }_f$-Dvoretzky covered \[ \limsup_{n \rightarrow \infty} \left(\frac{\ell_1 + \dots + \ell_n}{\ln n}\right)\geq \frac{1}{m_f}. \] Under more restrictive assumptions on $\{{\ell }_n\}$ 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 above condition 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, however, is not true for the uniform density.