Hitting Probabilities and the Ekstr{\"o}m-Persson conjecture

TL;DR

This paper advances the understanding of the Ekstr{"o}m-Persson conjecture by developing a new approach that establishes the conjectured Hausdorff dimension as a lower bound for random covering sets, with implications for hitting probabilities.

Contribution

The paper introduces a novel method to prove the lower bound of the Hausdorff dimension in the Ekstr{"o}m-Persson conjecture, extending results to more general limsup sets and hitting probability questions.

Findings

Confirmed the conjectured Hausdorff dimension as a lower bound for random covering sets.

Extended the analysis to more general limsup sets.

Determined conditions under which deterministic sets are hit by random coverings.

Abstract

We consider the Ekst\''om-Persson conjecture concerning the value of the Hausdorff dimension of random covering sets formed by balls with radii $(k^{-\alpha})_{k=1}^\infty$ and centres chosen independently at random according to an arbitrary Borel probability measure $\mu$ on $\mathbb{R}^d$. The conjecture has been solved positively in the case $\frac 1\alpha\le \overline{\dim}_H \mu$, where $\overline{\dim}_H \mu$ stands for the upper Hausdorff dimension of $\mu$. In this paper, we develop a new approach in order to answer the full conjecture, proving in particular that the conjectured value is only a lower bound for the dimension. Our approach opens the way to study more general limsup sets, and has consequences on the so-called hitting probability questions. For instance, we are able to determine whether and what part of a deterministic analytic set can be hit by random covering sets formed by open sets.