Hausdorff and packing dimensions of subsets of Moran fractals with prescribed mixed group frequency of their codings

Abstract

We compute the Hausdorff dimension and the packing dimension of subsets of Moran fractals with prescribed mixed group frequencies. For example, if E denotes the set of real numbers x in [0, 1] for which the group of digits {1, 2, 3, 4} in the decimal expansion of x occurs with relative frequency \(t_1 \in [0, 1]\) and the group of digits {0, 1, 2, 8, 9} in the decimal expansion of x occurs with relative frequency \(t_2 \in [0, 1]\), then our results shows that $$ {\rm dim_{\sf H}} E = {\rm dim_{\sf P}} E = - {\frac {1} {log 10}} log \left( {\frac {t^{t_1}_{1} t^{t_2}_{2} (1 - t_1)^{1-t_1} (1 - t_2)^{1-t_2}} {2^{t_1}3^{1-t_1}}}\right) $$ , where dimH denotes the Hausdorff dimension and dimP denotes the packing dimension. Observe that the two groups of digits with prescribed frequencies, namely {1, 2, 3, 4} and {0, 1, 2, 8, 9}, are mixed, i.e. they are not disjoint. Previous work [LD, O1, V] has investigated the non-mixed case. In this paper we investigate the more difficult problem of finding the Hausdorff dimension and packing dimension of subsets of Moran fractals with prescribed mixed group frequencies.