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 who se conjugates all have moduli less than 1) between 1 and 2. It is known tha t, when p = .5, F-theta,F-p is a singular measure with exact Hausdorff dime nsion 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, a nd provide an algorithm for the construction of these matrices. We show tha t for certain beta of small degree, simulation gives the Hausdorff dimensio n to several decimal places.