## The Annals of Probability

### 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 $\lambda\in(a,1)$.

#### Article information

Source
Ann. Probab., Volume 47, Number 4 (2019), 2582-2617.

Dates
Revised: November 2017
First available in Project Euclid: 4 July 2019

https://projecteuclid.org/euclid.aop/1562205717

Digital Object Identifier
doi:10.1214/18-AOP1324

Mathematical Reviews number (MathSciNet)
MR3980929

#### Citation

Breuillard, Emmanuel; Varjú, Péter P. On the dimension of Bernoulli convolutions. Ann. Probab. 47 (2019), no. 4, 2582--2617. doi:10.1214/18-AOP1324. https://projecteuclid.org/euclid.aop/1562205717

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