Non-differentiability of devil's staircases and dimensions of subsets of Moran sets

Let C be the homogeneous Cantor set invariant for x → ax and x → 1 - a + ax. It has been shown by Darst that the Hausdorff dimension of the set of non- differentiability points of the distribution function of uniform measure on C equals (dim_HC)² = (log 2/log a)². In this paper we generalize the essential ingredient of the proof of this result. Let Ω = {0, 1, ..., r}. Let F be a Moran set associated with {0 < a _i < 1, i ∈ Ω} and Ω^w= Ω × Ω × .Let ø be the associated coding map from Ω^w onto F. Fix a non-empty set ⌈ ⊆ Ω with ⌈^c ≠ ∅ and let z(σ, n) denote the position of the nth occurrence of the elements of ⌈ in σ ∈ Ω^w. For given 0 ≤ ξ ≤ 1, let and A = {σ ∈ Ω^w: lim sup z(σ, n+1) = ξ^-1}, F _ξ =ø(∧) and A = ^* = {σ ∈ Ω^w: lim supz(σ, n+1) ≥ ξ^-1}, F ^*_ξø(∧^*). We show that dim_P F_ξ = dim_P F^* _ξ = dim_B F_ξ = dim_B F ^*_ξ = s with ∑j∈Ω a _i^s = 1, and dim_H F_ξ = dim _H F^*_ξ = η] where η is such that F^*_ξ ∑ aⁿ_{j j εΩ} + (1-ξ)log ∑ aⁿ_{j j ε Γc} = 0.