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 $\xi\in(1/2,1)$ within an error of order $\exp(-deg(\xi)^{A})$ for any number $A$, such that $\dim\mu_\xi<1$. As a corollary, we conclude that $\dim\mu_\lambda=1$ for each of $\lambda=\ln 2, e^{-1/2}, \pi/4$. 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 $\lambda\in(a,1)$.

Locations

• The Annals of Probability - View - PDF
• arXiv (Cornell University) - View - PDF
• Apollo (University of Cambridge) - View - PDF
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