Equidistribution results for self-similar measures

Abstract

A well known theorem due to Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n{)}_{n=1}^{\infty }$ is uniformly distributed modulo one. In this paper we give sufficient conditions for an analogue of this theorem to hold for self-similar measures. Our approach applies more generally to sequences of the form $(f_n(x){)}_{n=1}^{\infty }$ where $(f_n{)}_{n=1}^{\infty }$ is a sequence of sufficiently smooth real valued functions satisfying a nonlinearity assumption. As a corollary of our main result, we show that if $C$ is equal to the middle third Cantor set and $t\ge 1$, then with respect to the Cantor-Lebesgue measure on $C+t$ the sequence $(x^n{)}_{n=1}^{\infty }$ is uniformly distributed for almost every $x$.