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

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## Summary.

We compute the Hausdorff dimension and the packing dimension of subsets of Moran fractals with prescribed , where dim

*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$$ {\rm dim_{\sf H}} E = {\rm dim_{\sf P}} E = - {\frac {1} {log 10}} log \left( {\frac {t^{t_1}_{1} t^{t_2}_{2} (1 - t_1)^{1-t_1} (1 - t_2)^{1-t_2}} {2^{t_1}3^{1-t_1}}}\right) $$

_{ H }denotes the Hausdorff dimension and dim_{ P }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.## Mathematics Subject Classification (2000).

28A80## Keywords.

Hausdorff dimension packing dimension group frequencies of digits mixed group frequencies of digits## Preview

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© Birkhäuser Verlag, Basel 2009