Random Series in Powers of Algebraic Integers: Hausdorff Dimension of the Limit Distribution

We study the distributions F_,p of the random sums where ₁, ₂, ... are i.i.d. Bernoulli-p and is theinverse of a Pisot number (an algebraic integer ßwhose conjugates all have moduli less than 1) between 1 and2. It is known that, when p=.5, F_,p is a singular measure withexact Hausdorff dimension less than 1. We show that in all casesthe Hausdorff dimension can be expressed as the top Lyapunovexponent of a sequence of random matrices, and provide an algorithmfor the construction of these matrices. We show that for certainß of small degree, simulation gives the Hausdorffdimension to several decimal places.