ON THE DIMENSION OF BERNOULLI CONVOLUTIONS

• 成果类型: Article
• 署名作者: Breuillard, Emmanuel; Varju, Peter P.
• 署名单位: University of Cambridge
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/18-AOP1324
• 发表日期: 2019
• 页码: 2582-2617
• 关键词: entropy; sumset; FAMILY

摘要

The Bernoulli convolution with parameter lambda is an element of (0, 1) is the probability measure mu(lambda) that is the law of the random variable Sigma(n >= 0) +/-lambda(n), where the signs are independent unbiased coin tosses. We prove that each parameter lambda is an element of (1/2, 1) with dim mu(lambda) < 1 can be approximated by algebraic parameters eta is an element of (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 =In 2, 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 alpha < 1 such that dim mu(lambda) = 1 for all lambda is an element of (a, 1).