Equidistribution Results for Self-Similar Measures

Simon Baker

doi:10.1093/imrn/rnab056

Abstract

Abstract A well-known theorem due to Koksma states that for Lebesgue almost every $x&gt;1$ 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.

Highlights

A sequence∞ n=1 of real numbers is said to be uniformly distributed modulo one if for every pair of real numbers u, v with 0 ≤ u < v ≤ 1 we have lim N →∞ #{1 ≤ n N : xn N mod ∈ [u, v]} = v
Some other important contributions in this area include the papers by Kaufman [20], and Queffelec and Ramare [27], who constructed Borel probability measures supported on subsets of the badly approximable numbers whose Fourier transform converges to zero polynomially fast
+t coincides with the self-similar measure corresponding to the probability vector2i=1 = (1/2, 1/2), we see that Theorem 2.1 immediately implies the following corollary2

Summary

Introduction

A sequence (xn)∞ n=1 of real numbers is said to be uniformly distributed modulo one if for every pair of real numbers u, v with 0 ≤ u < v ≤ 1 we have lim. Suppose μ is a Borel probability measure supported on [1, ∞) that is defined “independently” from the family of maps {fn(x) = xn}∞ n=1, we are interested in determining whether for μ almost every x the sequence (xn)∞ n=1 is uniformly distributed modulo one. Some other important contributions in this area include the papers by Kaufman [20], and Queffelec and Ramare [27], who constructed Borel probability measures supported on subsets of the badly approximable numbers whose Fourier transform converges to zero polynomially fast. By a recent result of Pollington et al [26], if the Fourier transform of a Borel probability measure converges to zero sufficiently fast, for almost every x, (1.1) holds for the sequence (bnx)∞ n=1 with an explicit error term.

Statement of results

Fractal Geometry

Uniform distribution