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 $\dim {\nu }_{\lambda }=1$ for all transcendental $\lambda \in (1/2,1)$.

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 unchanged

References in corpus (3)

• Absolute continuity of Bernoulli convolutions for algebraic parameters
• On the dimension of Bernoulli convolutions
• On the Hausdorff Dimension of Bernoulli Convolutions