Normal numbers from infinite convolution measures

Abstract

Let r, s be natural numbers, r, s \geq 2, such that {\rm log}\,r/{\rm log}\,s is irrational. Let \mu_j = \sum_{k=0}^{s-1} p^{(j)}_{k}\delta_{k/s^j}, where p^{(j)}_k \in [0,1), \sum_{k=0}^{s-1} p^{(j)}_k =1, j = 1, 2, \dotsc and {\delta}_{x} denotes the probability atom at x. We prove that if there exists k_0\in \{0,\dotsc , s-2\}, such that \inf \{ p^{(j)}_{k_0}, p^{(j)}_{k_0+1} : j\in \mathbb N \} > 0 and \mu is the infinite convolution measure \mu = *^{\infty}_{j=1} {\mu_{j} }, where the convergence is in the weak* sense, then \mu-almost all numbers are normal to base r. This generalizes some earlier results of Feldman, Smorodinsky and Pushkin.