On the dimension of Bernoulli convolutions

Abstract

The Bernoulli convolution with parameter $\lambda \in (0,1)$ is the probability measure ${\mu }_{\lambda }$ that is the law of the random variable ${\sum }_{n\ge 0}\pm {\lambda }^n$, 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 $\xi \in (1/2,1)$ within an error of order $\exp (-deg(\xi {)}^A)$ for any number $A$, such that $\dim {\mu }_{\xi }<1$. As a corollary, we conclude that $\dim {\mu }_{\lambda }=1$ for each of $\lambda =\ln 2,e^{-1/2},\pi /4$. These are the first explicit examples of such transcendental parameters. Moreover, we show that Lehmer's conjecture implies the existence of a constant $a<1$ such that $\dim {\mu }_{\lambda }=1$ for all $\lambda \in (a,1)$.

34 pages; version accepted for publication in Ann. Probab.; two typos corrected; results and proofs are unchanged

References in corpus (2)

• Absolute continuity of Bernoulli convolutions for algebraic parameters
• Entropy of Bernoulli convolutions and uniform exponential growth for linear groups

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