On the dimension of Bernoulli convolutions for all transcendental parameters

Abstract

The Bernoulli convolution $\nu_\lambda$ with parameter $\lambda \in (0,1)$ is the probability measure supported on $\mathbf{R}$ that is the law of the random variable $\sum\pm\lambda^n$, where the $\pm$ are independent fair coin-tosses. We prove that $\mathrm{dim}\ \nu_\lambda = 1$ for all transcendental $\lambda = (1/2,1)$.