On the dimension of Bernoulli convolutions


Type
Article
Change log
Authors
Breuillard, E 
Varjú, PP 
Abstract

The Bernoulli convolution with parameter λ ∈ (0, 1) is the probability measure μλ that is the law of the random variable σn ≥ 0 ±λn, where the signs are independent unbiased coin tosses. We prove that each parameter λ ∈ (1/2, 1) with dimμλ < 1 can be approximated by algebraic parameters η ∈ (1/2, 1) within an error of order exp(-deg(η)A) such that dimμη < 1, for any number A. As a corollary, we conclude that dimμλ = 1 for each of λ = ln 2, e-1/2,π/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 dimμλ = 1 for all λ ∈ (a, 1).

Description
Keywords
Bernoulli convolution, self-similar measure, dimension, entropy, convolution, transcendence measure, Lehmer's conjecture
Journal Title
Annals of Probability
Conference Name
Journal ISSN
0091-1798
2168-894X
Volume Title
47
Publisher
Institute of Mathematical Statistics
Rights
All rights reserved
Sponsorship
Royal Society (UF140146)