Hausdorff dimension of subsets of Moran fractals with prescribed group frequency of their codings

A well known result states that the set of numbers in base r in which the digits i occur with relative frequency p_i for i = 0,..., r - 1 is a set of Hausdorff dimension -(1/log r) ∑_i=0^r-1 p _i log p_i. For instance, decimal numbers in which only the digits 1 and 6 occur, both with relative frequencies 1/2, have Hausdorff dimension log 2/log 10. In this paper we generalize this result to the situation where one prescribes the relative frequencies of groups of digits in the expansion. For example, suppose we require that in the decimal expansion digits from {0, 1, 2} occur with relative frequency 1/2, and also that digits from {3, 4,..., 9} occur with this relative frequency. Our result shows that the Hausdorff dimension of this set is (log 2 + 1/2 log 3 + 1/2 log 7)/log 10. Actually, we take a much more general geometric viewpoint, considering subsets of Moran fractals specified by prescribing the relative frequencies of groups of symbols in their codings. We determine the Hausdorff dimension of such sets, and moreover give necessary and sufficient conditions for such a set to have positive Hausdorff measure in its dimension.