The Annals of Probability

On the dimension of Bernoulli convolutions

Emmanuel Breuillard and Péter P. Varjú

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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

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

Received: October 2016
Revised: November 2017
First available in Project Euclid: 4 July 2019

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Mathematical Reviews number (MathSciNet)

Primary: 28A80: Fractals [See also 37Fxx] 42A85: Convolution, factorization

Bernoulli convolution self-similar measure dimension entropy convolution transcendence measure Lehmer’s conjecture


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

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