Applications of multifractal divergence points to sets of numbers defined by their $N$-adic expansion

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Abstract

In this paper we apply the techniques and results from the theory of multifractal divergence points to give a systematic and detailed account of the Hausdorff dimensions of sets of numbers defined in terms of the asymptotic behaviour of the frequencies of the digits in their N-adic expansion. Using earlier methods and results we investigate and compute the Hausdorff dimension of several new sets of numbers. In particular, we compute the Hausdorff dimension of a large class of sets of numbers for which the limiting frequencies of the digits in their N-adic expansion do not exist. Such sets have only very rarely been studied. In addition, our techniques provide simple proofs of (substantial generalizations of) known results, by Cajar and Drobot and Turner and others, on the Hausdorff dimension of sets of normal and non-normal numbers.

Original languageEnglish
Pages (from-to)139-165
Number of pages27
JournalMathematical Proceedings of the Cambridge Philosophical Society
Volume136
Issue number1
DOIs
Publication statusPublished - Jan 2004

Keywords

  • PERRON-FROBENIUS THEORY
  • HAUSDORFF DIMENSION

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