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July 2019 On the dimension of Bernoulli convolutions
Emmanuel Breuillard, Péter P. Varjú
Ann. Probab. 47(4): 2582-2617 (July 2019). DOI: 10.1214/18-AOP1324


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


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Emmanuel Breuillard. Péter P. Varjú. "On the dimension of Bernoulli convolutions." Ann. Probab. 47 (4) 2582 - 2617, July 2019.


Received: 1 October 2016; Revised: 1 November 2017; Published: July 2019
First available in Project Euclid: 4 July 2019

zbMATH: 07114725
MathSciNet: MR3980929
Digital Object Identifier: 10.1214/18-AOP1324

Primary: 28A80 , 42A85‎

Keywords: Bernoulli convolution , convolution , dimension , Entropy , Lehmer’s conjecture , Self-similar measure , transcendence measure

Rights: Copyright © 2019 Institute of Mathematical Statistics


Vol.47 • No. 4 • July 2019
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