On normal numbers and self-similar measures

TL;DR

This paper proves that for a class of self-similar measures generated by equicontractive affine maps, almost all points are normal in any base when a certain irrationality condition on the contraction ratio is met.

Contribution

It establishes the normality of almost every point in self-similar measures under specific irrationality conditions on the contraction ratio.

Findings

Almost every point is normal in base b.

Normality holds for non-atomic self-similar measures.

Condition involves irrationality of log b over log |λ|.

Abstract

In this paper we prove that if $\{\varphi_i(x)=\lambda x+t_i\}$ is an equicontractive iterated function system and $b$ is a positive integer satisfying $\frac{\log b}{\log |\lambda|}\notin\mathbb{Q},$ then almost every $x$ is normal in base $b$ for any non-atomic self-similar measure of $\{\varphi_i\}$.