Random series in powers of algebraic integers: Hausdorff dimension of the limit distribution

We study the distributions F-theta,F-p of the random sums Sigma(1)(infinity)epsilon(n)theta(n), where epsilon(1), epsilon(2),... are i.i.d. Bernoulli-p and theta is the inverse of a Pisot number (an algebraic integer beta whose conjugates all have moduli less than 1) between 1 and 2. It is known that, when p = .5, F-theta,F-p is a singular measure with exact Hausdorff dimension less than 1. We show that in all cases the Hausdorff dimension can be expressed as the top Lyapunov exponent of a sequence of random matrices, and provide an algorithm for the construction of these matrices. We show that for certain beta of small degree, simulation gives the Hausdorff dimension to several decimal places.