Equidistribution Results for Self-Similar Measures. (23rd April 2021)

• Record Type: Journal Article
• Authors: Baker, Simon
• Is Part Of: International mathematics research notices. Volume 2022:Number 16(2022)
• Page Start: 12378
• Page End: 12401
• Publication Date: 2021-04-23
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnab056
• Languages: English
• ISSNs: 1073-7928

Abstract

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 a self-similar measure. 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 some nonlinearity conditions. As a corollary of our main result, we show that if $C$ is equal to the middle 3rd Cantor set and $t\geq 1$, then with respect to the natural measure on $C+t, $ for almost every $x$, the sequence $(x^n)_{n=1}^{\infty }$ is uniformly distributed modulo one.