Equidistribution for measures defined by digit restrictions

Abstract

Abstract In this paper, we study the pointwise equidistribution properties of measures μ p defined by digit restrictions on the b-adic expansion, where b ⩾ 2 is an integer. We prove that, if a sequence ( α n ) n ⩾ 1 satisfies a certain b-adic diversity condition, then the sequence ( α n x ) n ⩾ 1 is uniformly distributed modulo one for μ p -a.e. x. We also find some sufficient conditions to ensure the b-adic diversity. Moreover, we apply these results to establish the b-adic diversity for the sequences that can be written as certain combination of polynomial and exponential functions.