On the Hausdorff Dimension of Bernoulli Convolutions

TL;DR

This paper provides a method to compute the Hausdorff dimension of Bernoulli convolutions using Garsia entropy, enabling precise calculations for algebraic parameters and an algorithm to determine when the dimension equals one.

Contribution

It introduces an explicit matrix product expression for Garsia entropy, allowing accurate Hausdorff dimension computation and an algorithm for algebraic parameters.

Findings

Explicit rate of convergence for Garsia entropy

Algorithm to determine if Hausdorff dimension equals one

Calculation of Hausdorff dimension for algebraic $eta$

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 $\mathrm{dim}_\mathrm{H} (\nu_\beta)=1$.