Frequencies of digits, divergence points, and Schmidt games

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Abstract

Sets of divergence points, i.e. numbers x (or tuples of numbers) for which the limiting frequency of a given string of N-adic digits of x fails to exist, have recently attracted huge interest in the literature. In this paper we consider sets of simultaneous divergence points, i.e. numbers x (or tuples of numbers) for which the limiting frequencies of all strings of N-adic digits of x fail to exist. We show that many natural sets of simultaneous divergence points are (alpha,beta)-wining sets in the sense of the Schmidt game. As all application we obtain lower bounds for the Hausdorff dimension of these sets. (C) 2007 Elsevier Ltd. All rights reserved.

Original languageEnglish
Pages (from-to)2222-2232
Number of pages11
JournalChaos, Solitons and Fractals
Volume40
Issue number5
DOIs
Publication statusPublished - 15 Jun 2009

Keywords

  • BADLY APPROXIMABLE NUMBERS
  • SELF-SIMILAR MEASURES
  • HAUSDORFF DIMENSION
  • SETS
  • FRACTALS
  • AVERAGES
  • ORBITS
  • SPACE

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