Random Covering Sets in Metric Space with Exponentially Mixing Property

TL;DR

This paper investigates the measure, dimension, and topology of random covering sets in metric spaces with exponentially mixing properties, focusing on the behavior of points covered infinitely often by decreasing-radius random balls.

Contribution

It introduces new results on the size and structure of random covering sets in metric spaces with exponential mixing, extending previous work to more general settings.

Findings

Determines conditions for full measure of the covering set

Establishes dimension results for the covering set

Analyzes topological properties of the covering set

Abstract

Let $\{B(\xi_n,r_n)\}_{n\ge1}$ be a sequence of random balls whose centers $\{\xi_n\}_{n\ge1}$ is a stationary process, and $\{r_n\}_{n\ge1}$ is a sequence of positive numbers decreasing to 0. Our object is the random covering set $E=\limsup\limits_{n\to\infty}B(\xi_n,r_n)$, that is, the points covered by $B(\xi_n,r_n)$ infinitely often. The sizes of $E$ are investigated from the viewpoint of measure, dimension and topology.