Let { n } ∞ n = 1 be i.i.d. Bernoulli random variables. For 12<ρ<1, let X = ∞ n = 1 ρ n n be the discount sum and let μ ρ be the distribution measure. It is known that if ρ - 1 is a P.V. number, then μ ρ is continuously singular. In this paper we use a Markov chain technique to obtain the precise L 2 -dimension of such measures. In particular for ρ = (5 - 1)2, we use a device of Strichartz et al. and the renewal equation to derive a formula for the L p -dimension and the entropy dimension of the corresponding μ ρ .