On the Hausdorff Dimension of Bernoulli Convolutions

Abstract

We give an expression for the Garsia entropy of Bernoulli convolutions in terms of products of matrices. This gives an explicit rate of convergence of the Garsia entropy and shows that one can calculate the Hausdorff dimension of the Bernoulli convolution |$\nu _{\beta }$| to arbitrary given accuracy whenever |$\beta $| is algebraic. In particular, if the Garsia entropy |$H(\beta )$| is not equal to |$\log (\beta )$| then we have a finite time algorithm to determine whether or not |$\operatorname{dim_H} (\nu _{\beta })=1$|⁠.