Extraction of optimal subsequences of sequence of balls, and application to optimality estimates of mass transference principles

TL;DR

This paper proves the extraction of well-distributed subsequences from sequences of balls with full measure, establishing optimal bounds for Hausdorff dimensions of limsup sets and introducing a new Borel-Cantelli divergence lemma applicable without doubling measure assumptions.

Contribution

It introduces a method to extract well-distributed subsequences from sequences of balls with full measure, and proves the optimality of lower bounds for Hausdorff dimensions in mass transference principles.

Findings

Extraction of well-distributed subsequences from sequences with full measure.

Optimal lower bounds for Hausdorff dimension of limsup sets.

A new Borel-Cantelli divergence lemma without doubling measure assumption.

Abstract

In this article, we prove that from any sequence of balls whose associated limsup set has full $\mu$-measure, one can extract a well-distributed subsequence of balls. From this, we deduce the optimality of various lower bounds for the Hausdorff dimension of limsup sets of balls obtained by mass transference principles. We also establish a version of Borel-Cantelli divergence lemma particulary suited for limsup set generated by balls. This lemma is very similar to the one proved by Bersnevich and Velani but the measure is not assumed ot be doubling.