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Ukrainian Mathematical Journal

, Volume 57, Issue 9, pp 1361–1370 | Cite as

Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits

  • S. Albeverio
  • M. Prats'ovytyi
  • G. Torbin
Article

Abstract

Properties of the set T s of “particularly nonnormal numbers” of the unit interval are studied in detail (T s consists of real numbers x some of whose s-adic digits have the asymptotic frequencies in the nonterminating s-adic expansion of x, and some do not). It is proved that the set T s is residual in the topological sense (i.e., it is of the first Baire category) and is generic in the sense of fractal geometry (T s is a superfractal set, i.e., its Hausdorff-Besicovitch dimension is equal to 1). A topological and fractal classification of sets of real numbers via analysis of asymptotic frequencies of digits in their s-adic expansions is presented.

Keywords

Probability Distribution Real Number Fractal Property Unit Interval Fractal Geometry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • S. Albeverio
    • 1
  • M. Prats'ovytyi
    • 2
  • G. Torbin
    • 2
    • 3
  1. 1.Institut fur Angewandte MathematikUniversitat BonnGermany
  2. 2.National Pedagogic UniversityKyivUkraine
  3. 3.Institute of MathematicsNational Academy of Sciences of UkraineKyivUkraine

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