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Aequationes mathematicae

, Volume 77, Issue 1–2, pp 171–185 | Cite as

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

  • Wenxia Li
  • Lars Olsen
  • Zhiying Wen
Article

Summary.

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 dim 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 

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Copyright information

© Birkhäuser Verlag, Basel 2009

Authors and Affiliations

  1. 1.Department of MathematicsEast China Normal UniversityShanghaiP. R. China
  2. 2.Department of MathematicsUniversity of St. AndrewsSt. AndrewsScotland
  3. 3.Department of Mathematical SciencesTsinghua UniversityBeijingP. R. China

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