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

Abstract

We study the distributions Fθ,p of the random sums ∑ 1 ∞ ε n θ n where ε1, ε2, … 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, Fθ,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.