Sets of beta-expansions and the Hausdorff Measure of Slices through Fractals

TL;DR

This paper investigates measures on beta-expansions and fractal slices, enhancing understanding of Bernoulli convolutions and Hausdorff measures through new measure disintegration techniques.

Contribution

It introduces novel measures on beta-expansions and fractal slices, providing new insights into measure disintegration and generalizing results on Hausdorff dimensions.

Findings

Better understanding of the measure of maximal entropy for the beta-transformation

Reinterpretation of Lindenstrauss, Peres, and Schlag's results in terms of equidistribution

Conditions for positive Hausdorff measure in almost every slice through self-similar sets

Abstract

We study natural measures on sets of beta-expansions and on slices through self similar sets. In the setting of beta-expansions, these allow us to better understand the measure of maximal entropy for the random beta-transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing, leading to conditions under which almost every slice through a self similar set has positive Hausdorff measure, generalising long known results about almost everywhere values of the Hausdorff dimension.