A lower bound for the dimension of Bernoulli convolutions

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.