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

We study the distributions F θ, p of the random sums∑ 1 ∞ε n θ nwhere ε 1 , ε 2 , … are i.i.d. Bernoulli‐ p and θ is the inverse of a Pisot number (an algebraic integer β whose conjugates all have moduli less than 1) between 1 and 2. It is known that, when p =.5, F θ, 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 β of small degree, simulation gives the Hausdorff dimension to several decimal places.