The exact Hausdorff dimension functions of some Cantor subsets

Let Q : [1, infinity) --> [1, infinity) be a strictly increasing function with n less than or equal to Q(n) for all sufficiently large integers n. Fix a positive integer N with N less than or equal to [10(Q(n))/10(Q(n-1))] for all sufficiently large integers n (here [x] denotes the integer part of x), and define M by

M = {Sigma(n=1)(infinity) x(n)/10(Q(n)) \ x(n) = 0, 1,..., min ([10(Q(n))/10(Q(n-1))], N) - 1 for all n}.

We determine the exact Hausdorff dimension function of M for a large class of functions Q including Q(t) = Gamma(t + 1) for t greater than or equal to 1. As an application of our results we exhibit a large class of dimension functions h for which the h-dimensional Hausdorff measure H-h (L) of the set L of Liouville numbers is positive, i.e., such that 0 < H-h (L).