How smooth is a devil's staircase?

Let the self-similar set C in R be defined by C = ∪_j=0^r h_j(C} with a disjoint union, where the h_j's are similitude mappings with ratios 0 < a_j < 1. Let μ, on C be the self-similar probability measure corresponding to the probability vector (a₀^ζ, a₁^ζ, . . ., a_r^ζ), where ζ = dim_H C is the Hausdorff dimension of C. Let S be the set of points at which the probability distribution function F of μ has no derivative, finite or infinite. We prove that dim_H S = (dim_H C)² and dim_P S = dim_B S = dim_H C.