Equidistribution from fractal measures

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 (Formula presented.) equidistributes modulo 1. This condition is robust under C1 coordinate changes, and it applies also when n is a Pisot number rather than an integer. 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.