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


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.


Probability Distribution Real Number Fractal Property Unit Interval Fractal Geometry 
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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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