Equidistribution from Fractals

TL;DR

This paper establishes a fractal-geometric condition for measures on [0,1] to support points that are normal in base n, with robustness under coordinate changes and applications to fractal sets and measure rigidity.

Contribution

It introduces a new fractal-geometric criterion for normality support, extending to Pisot numbers and beta-maps, and advances measure rigidity results.

Findings

Supports normality in fractal sets.

Strengthens measure rigidity theorems.

Applies to Pisot numbers and beta transformations.

Abstract

We give a fractal-geometric condition for a measure on [0,1] to be supported on points x that are normal in base n, i.e. such that the sequence x,nx,n^2 x,... equidistributes modulo 1. This condition is robust under C^1 coordinate changes, and it applies also when n is a Pisot number and equidistribution is understood with respect to the beta-map and Parry measure. As applications we obtain new results (and strengthen old ones) about the prevalence of normal numbers in fractal sets, and new results on measure rigidity, specifically completing Host's theorem to multiplicatively independent integers and proving a Rudolph-Johnson-type theorem for certain pairs of beta transformations.