Hitting probabilities of random covering sets in tori and metric spaces

TL;DR

This paper establishes precise bounds for the Hausdorff dimension of intersections between random covering sets and fixed analytic sets in both metric spaces and tori, advancing understanding of geometric measure theory.

Contribution

It provides sharp bounds for the Hausdorff dimension of intersections in Ahlfors regular metric spaces and tori, extending previous results to more general settings.

Findings

Derived sharp bounds for Hausdorff dimensions of intersections

Applied results to both metric spaces and tori

Extended understanding of random covering sets

Abstract

We provide sharp lower and upper bounds for the Hausdorff dimension of the intersection of a typical random covering set with a fixed analytic set both in Ahlfors regular metric spaces and in the $d$-dimensional torus. In metric spaces, we consider covering sets generated by balls and, in the torus, we deal with general analytic generating sets.