On the dimension of Bernoulli convolutions
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Authors
Breuillard, E
Varjú, PP
Publication Date
2019Journal Title
Annals of Probability
ISSN
0091-1798
Publisher
Institute of Mathematical Statistics
Volume
47
Issue
4
Pages
2582-2617
Type
Article
This Version
VoR
Metadata
Show full item recordCitation
Breuillard, E., & Varjú, P. (2019). On the dimension of Bernoulli convolutions. Annals of Probability, 47 (4), 2582-2617. https://doi.org/10.1214/18-AOP1324
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).
Keywords
Bernoulli convolution, self-similar measure, dimension, entropy, convolution, transcendence measure, Lehmer's conjecture
Sponsorship
Royal Society (UF140146)
Identifiers
External DOI: https://doi.org/10.1214/18-AOP1324
This record's URL: https://www.repository.cam.ac.uk/handle/1810/297783
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