Regularity dimensions: quantifying doubling and uniform perfectness

TL;DR

This paper investigates the regularity dimensions related to doubling and uniform perfectness, computes these for measures under Brownian motion graphs and quasisymmetric maps, and applies findings to Diophantine approximation in Kleinian groups.

Contribution

It introduces a detailed analysis of regularity dimensions, computes them for specific measure transformations, and connects these concepts to Diophantine approximation in Kleinian groups.

Findings

Regularity dimensions of measures on Brownian motion graphs are computed.

Regularity dimensions under quasisymmetric homeomorphisms are analyzed.

Application to Diophantine approximation in Kleinian groups is demonstrated.

Abstract

We study the upper and lower regularity dimensions in relation to the notions of doubling and uniformly perfect. These two regularity properties are closely related which is quantified thanks to the regularity dimensions. The regularity dimensions of pushforward measures onto graphs of Brownian motion are calculated, similarly for pushforwards with respect to quasisymmetric homeomorphisms. We finish by introducing an application to Diophantine approximation in the setting of Kleinian groups.