A note on the hitting probabilities of random covering sets

Abstract

Let $E=\underset{n\to \infty }{\mathrm{limsup}}(g_n+{\xi }_n)$ be the random covering set on the torus ${\mathbb{T}}^d$, where $\{g_n\}$ is a sequence of ball-like sets and ${\xi }_n$ is a sequence of independent random variables uniformly distributed on $\T^d$. We prove that $E\cap F\ne \varnothing$ almost surely whenever $F\subset {\mathbb{T}}^d$ is an analytic set with Hausdorff dimension, ${\dim }_H(F)>d-\alpha$, where $\alpha$ is the almost sure Hausdorff dimension of $E$. Moreover, examples are given to show that the condition on ${\dim }_H(F)$ cannot be replaced by the packing dimension of $F$.

11 pages