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.