A lower bound for the dimension of Bernoulli convolutions

TL;DR

This paper establishes a universal lower bound of 0.82 for the Hausdorff dimension of Bernoulli convolutions associated with parameters in (1,2), and identifies conditions under which the dimension equals 1, especially for certain algebraic numbers.

Contribution

It provides a new lower bound for the dimension of Bernoulli convolutions and characterizes algebraic parameters with full dimension.

Findings

Dimension of Bernoulli convolutions is at least 0.82 for all parameters in (1,2).

Certain algebraic numbers have Bernoulli convolutions with full dimension 1.

Improves previous bounds using entropy and algebraic number properties.

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-Varj\'u, this yields $\dim (\mu_\beta)\ge0.82$ 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 $k \geq 2$, then $\dim(\mu_\beta)=1$. Such is, for instance, any root of a Pisot number which is not a Pisot number itself.