Hausdorff dimension of random limsup sets

TL;DR

This paper establishes bounds for the Hausdorff dimension of random limsup sets formed by balls with random centers in Euclidean space, linking the dimension to ball radii decay and measure multifractality.

Contribution

It generalizes existing formulas for Hausdorff dimension of limsup sets to broader classes of measures with random centers in Euclidean space.

Findings

Derived bounds for Hausdorff dimension of random limsup sets.

Connected dimension estimates to measure decay rates and multifractal properties.

Extended known formulas to more general measure classes.

Abstract

We prove bounds for the almost sure value of the Hausdorff dimension of the limsup set of a sequence of balls in $\mathbf{R}^d$ whose centres are independent, identically distributed random variables. The formulas obtained involve the rate of decrease of the radii of the balls and multifractal properties of the measure according to which the balls are distributed, and generalise formulas that are known to hold for particular classes of measures.