Computing Garsia Entropy for Bernoulli Convolutions with Algebraic Parameters

Abstract

We introduce a parameter space containing all algebraic integers $\beta\in(1,2]$ that are not Pisot or Salem numbers, and a sequence of increasing piecewise continuous function on this parameter space which gives a lower bound for the Garsia entropy of the Bernoulli convolution $\nu_{\beta}$. This allows us to show that $\mathrm{dim}_\mathrm{H} (\nu_{\beta})=1$ for all $\beta$ with representations in certain open regions of the parameter space.