Dimension of a Family of Singular Bernoulli Convolutions

Let {Xn}∞n= 0 be a sequence of i.i.d. Bernoulli random variables (i.e., Xn takes values {0, 1} with probability 12 each), let X = ∑∞n= 0ρnXn and let μ be the corresponding probability measure. Erdös-Salem proved that if 12 < ρ < 1, and if ρ−1 is a P.V. number, then μ is singular. In this paper, we study the algebraic structure of ρ and the singularity of the correspondent μ in more detail. We introduce a new class of algebraic numbers containing the P.V. numbers, and make use of the self-similar property determined by such numbers to calculate the exact mean-quadratictariation dimension of μ. This dimension is most relevant to Strichartz′s recent work on Fourier asymptotics of fractal measures.