A note on the hitting probabilities of random covering sets

TL;DR

This paper investigates the probability that a random covering set on a torus intersects with certain sets, establishing conditions based on Hausdorff dimension and providing counterexamples involving packing dimension.

Contribution

It proves a new almost sure intersection criterion for random covering sets on the torus based on Hausdorff dimension, and shows limitations of this criterion with respect to packing dimension.

Findings

Almost sure intersection occurs if Hausdorff dimension exceeds a threshold.

Counterexamples demonstrate the condition cannot be replaced by packing dimension.

Provides a detailed analysis of the Hausdorff dimension of random covering sets.

Abstract

Let $E=\limsup\limits_{n\to\infty}(g_n+\xi_n)$ 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 $E\cap F\neq\emptyset$ 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 $F$.