A lower bound for the dimension of Bernoulli convolutions

Abstract

Let $\beta \in (1,2)$ and let $H_{\beta }$ denote Garsia's entropy for the Bernoulli convolution ${\mu }_{\beta }$ associated with $\beta$. In the present paper we show that $H_{\beta }>0.82$ for all $\beta \in (1,2)$ and improve this bound for certain ranges. Combined with recent results by Hochman and Breuillard-Varjú, this yields $\dim ({\mu }_{\beta })\ge 0.82$ for all $\beta \in (1,2)$. In addition, we show that if an algebraic $\beta$ is such that $[\mathbb{Q}(\beta ):\mathbb{Q}({\beta }^k)]=k$ for some $k\ge 2$, then $\dim ({\mu }_{\beta })=1$. Such is, for instance, any root of a Pisot number which is not a Pisot number itself.

8 pages, no figures

References in corpus (3)

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

Cited by in corpus (3)

• Estimates on the dimension of self-similar measures with overlaps
• Computing Garsia Entropy for Bernoulli Convolutions with Algebraic Parameters
• Measures on the Spectra of Algebraic Integers