Monatshefte für Mathematik

, Volume 162, Issue 4, pp 409–427 | Cite as

Simultaneously non-convergent frequencies of words in different expansions

  • David FärmEmail author


We consider expanding maps such that the unit interval can be represented as a full symbolic shift space with bounded distortion. There are already theorems about the Hausdorff dimension for sets defined by the set of accumulation points for the frequencies of words in one symbolic space at a time. We show that the dimension is preserved when such sets defined using different maps are intersected. More precisely, it is proven that the dimension of any countable intersection of sets defined by their sets of accumulation for frequencies of words in different expansions, has dimension equal to the infimum of the dimensions of the sets that are intersected. As a consequence, the set of numbers for which the frequencies do not exist has full dimension even after countable intersections. We also prove that this holds for a dense set of non-integer base expansions.


Interval map Non-typical point Hausdorff dimension Beta shift 

Mathematics Subject Classification (2000)

37E05 37C45 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Barreira L., Saussol B., Schmeling J.: Distribution of frequencies of digits via multifractal analysis. J. Number Theory 97, 410–438 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Barreira L., Schmeling J.: Sets of ‘non-typical’ points have full topological entropy and full Hausdorff dimension. Israel J. Math. 116, 29–70 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Barreira L., Saussol B., Schmeling J.: Higher-dimensional multifractal analysis. J. Math. Pures Appl. (9) 81(1), 67–91 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Billingsley P.: Ergodic Theory and Information. Wiley, New York (1965)zbMATHGoogle Scholar
  5. 5.
    Borel E.: Les probabilités dénombrables et leurs applications arithmétiques. Supplemento di Rend. Circ. Mat. Palermo 27, 247–271 (1909)CrossRefzbMATHGoogle Scholar
  6. 6.
    Eggleston H.: The fractional dimension of a set defined by decimal properties. Q. J. Math. Oxford Ser. 20, 31–36 (1949)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Falconer K.: Sets with large intersection properties. J. Lond. Math. Soc. (2) 49(2), 267–280 (1994)MathSciNetGoogle Scholar
  8. 8.
    Olsen L.: Extremely non-normal numbers. Math. Proc. Camb. Philos. Soc. 137(1), 43–53 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Parry W.: On the β-expansion of real numbers. Acta Mathematica Academiae Scientiarum Hungaricae 11, 401–416 (1960)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Pollington A.D.: The Hausdorff dimension of a set of normal numbers. Pac. J. Math 95(1), 193–204 (1981)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Rényi A.: Representations for real numbers and their ergodic properties. Acta Mathematica Academiae Scientiarum Hungaricae 8, 477–493 (1957)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Schmidt W.M.: Über die Normalität von Zahlen zu verschiedenen Basen. Acta Arith. 7, 299–309 (1961/1962)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Centre for Mathematical SciencesLund UniversityLundSweden

Personalised recommendations