The dimension of subsets of Moran sets determined by the success run behaviour of their codings

Let F ⊂ Rⁿ be a Moran set associated with the set {0 < a_j 1, j = 0, 1,...,r}. Let Γ be a non-empty subset of {0, 1, 2,...,r} with non-empty complement. Associated with the behaviour of success run of symbols from Γ in the coding space {0, 1,...,r}^N is a decomposition of F such that F = ∪ F_t. t∈[0,+ ∞] Depending on F this might be a partition of F or almost a partition of F in the sense that sup _x∈F#{t : x ∈ F_t} < + ∞ We prove that each F_t is dense in F, and dim_HF_t = dim_PF_t = dim_BF_t = dim_HF = dim_PF = dim_BF = s with ∑^r_j=0a^s_j; = 1. For ℒ¹-a.e. t ∈ [0, + ∞], Script H sign^s.«(F_t) = 0 and F_t is an s-set when t =-(log∑_j∈Γa^s_j)^-1. Moreover, associated with this decomposition {Ft : t ∈ [0, + ∞]} of F is a measurable function Y such that each F_t is a level set of Y. The fractal dimensions of the graph of Y are also determined.