On the dimension of Bernoulli convolutions for all transcendental parameters

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 $\dim\nu_\lambda=1$ for all transcendental $\lambda\in(1/2,1)$.Comment: 11 pages; version accepted for publication in Ann. of Math.; dedicated to the memory of Jean Bourgain; minor corrections based on referee's report; results and proofs are unchange

Recent progress on Bernoulli convolutions

The Bernoulli convolution with parameter $\lambda\in(0,1)$ is the measure on $\bf R$ that is the distribution of the random power series $\sum\pm\lambda^n$, where $\pm$ are independent fair coin-tosses. This paper surveys recent progress on our understanding of the regularity properties of these measures.Support received from the Royal Society