A lower bound for Garsia's entropy for certain Bernoulli convolutions

Let $\beta \in (1,2)$ be a Pisot number and let $H_{\beta }$ denote Garsia’s entropy for the Bernoulli convolution associated with $\beta$. Garsia, in 1963 showed that $H_{\beta }<1$ for any Pisot $\beta$. For the Pisot numbers which satisfy $x^m=x^{m-1}+x^{m-2}+\mathrm{.}\mathrm{.}\mathrm{.}+x+1$ (with $m\ge 2$) Garsia’s entropy has been evaluated with high precision by Alexander and Zagier and later improved by Grabner, Kirschenhofer and Tichy, and it proves to be close to 1. No other numerical values for $H_{\beta }$ are known. In the present paper we show that $H_{\beta }>0.81$ for all Pisot $\beta$, and improve this lower bound for certain ranges of $\beta$. Our method is computational in nature.