On the dimension of Bernoulli convolutions

Abstract

The Bernoulli convolution with parameter $\lambda\in(0,1)$ is the probability measure $\mu_{\lambda}$ that is the law of the random variable $\sum_{n\ge0}\pm\lambda^{n}$, where the signs are independent unbiased coin tosses.

We prove that each parameter $\lambda\in(1/2,1)$ with $\dim\mu_{\lambda}<1$ can be approximated by algebraic parameters $\eta\in(1/2,1)$ within an error of order $\exp(-\deg(\eta)^{A})$ such that $\dim\mu_{\eta}<1$, for any number $A$. As a corollary, we conclude that $\dim\mu_{\lambda}=1$ for each of $\lambda=\ln2,e^{-1/2},\pi/4$. These are the first explicit examples of such transcendental parameters. Moreover, we show that Lehmer’s conjecture implies the existence of a constant $a<1$ such that $\dim\mu_{\lambda}=1$ for all $\lambda\in(a,1)$.