Dimensions of subsets of cantor-type sets

Let I = [0,1] be the unit interval on the real line andm> 1 be an integer. Let J = {0,1, . . . , m− 1}. For every point x ∈ I , there is a unique base-m representation x = Σk=1 jkm−k with jk ∈ J except for countable many points. Since countable sets do not interfere with our work, we neglect them here. For each j ∈ J , x ∈ [0,1], and n ∈ N, let τj(x,n) = {k : ik = j, 1 ≤ k ≤ n}, then the limit τj(x)= limn→∞(1/n)τj(x,n) is called the frequency of number j in the base-m representation of x. Here and in the following context, the notation “ A” denotes the number of elements in set A. A classical result of Borel [3] says that for Lebesgue almost every x ∈ [0,1], we have τj(x)= 1/m. As for another problem, for a given probability vector p= (p0, p1, . . . , pm−1) such that Σ j∈J p j = 1, consider the set