Uniform lower bounds on the dimension of Bernoulli convolutions

In this note we present an algorithm to obtain a uniform lower bound on Hausdorff dimension of the stationary measure of an affine iterated function scheme with similarities, the best known example of which is Bernoulli convolution. The Bernoulli convolution measure mu(lambda) is the probability measure corresponding to the law of the random variable xi =Sigma(infinity)(k=0) xi(k)lambda(k), where xi(k) are i.i.d. random variables assuming values -1 and 1 with equal probability and 1/2 < lambda < 1. In particular, for Bernoulli convolutions we give a uniform lower bound dim(H)(mu(lambda)) >= 0.96399 for all 1/2 < lambda < 1. (c) 2021 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).