Monatshefte für Mathematik

, Volume 185, Issue 2, pp 167–188 | Cite as

Irrationality exponent, Hausdorff dimension and effectivization

  • Verónica Becher
  • Jan Reimann
  • Theodore A. Slaman
Article
  • 18 Downloads

Abstract

We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension and show that the two notions are independent. For any real number a greater than or equal to 2 and any non-negative real b be less than or equal to 2 / a, we show that there is a Cantor-like set with Hausdorff dimension equal to b such that, with respect to its uniform measure, almost all real numbers have irrationality exponent equal to a. We give an analogous result relating the irrationality exponent and the effective Hausdorff dimension of individual real numbers. We prove that there is a Cantor-like set such that, with respect to its uniform measure, almost all elements in the set have effective Hausdorff dimension equal to b and irrationality exponent equal to a. In each case, we obtain the desired set as a distinguished path in a tree of Cantor sets.

Keywords

Diophantine approximation Cantor sets Effective Hausdorff dimension 

Mathematics Subject Classification

11J82 11J83 03D32 

Notes

Acknowledgements

The authors thank two anonymous referees for helpful comments and remarks.

References

  1. 1.
    Becher, V., Bugeaud, Y., Slaman, T.A.: The irrationality exponents of computable numbers. Proc. Am. Math. Soc. 144(4), 1509–1521 (2016)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Beresnevich, V., Dickinson, D., Velani, S.: Sets of exact ‘logarithmic’ order in the theory of diophantine approximation. Math. Ann. 321(2), 253–273 (2001)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Besicovitch, A.S.: Sets of fractal dimensions (IV): on rational approximation to real numbers. J. Lond. Math. Soc. 9, 126–131 (1934)CrossRefMATHGoogle Scholar
  4. 4.
    Besicovitch, A.S.: On existence of subsets of finite measure of sets of infinite measure. Nederl. Akad. Wetensch. Proc. Ser. A 14, 339–344 (1952)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bugeaud, Y.: Sets of exact approximation order by rational numbers. Math. Ann. 327(1), 171–190 (2003)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bugeaud, Y.: Approximation by Algebraic Numbers, Volume 160 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  7. 7.
    Cai, J.-Y., Hartmanis, J.: On Hausdorff and topological dimensions of the Kolmogorov complexity of the real line. J. Comput. Syst. Sci. 49(3), 605–619 (1994)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Calude, C., Staiger, L.: Liouville numbers, Borel normality and Martin–Löf randomness. Technical report, Centre for Discrete Mathematics and Theoretical Computer Science, University of Auckland (2013)Google Scholar
  9. 9.
    Downey, R., Hirschfeldt, D.: Algorithmic Randomness and Complexity. Springer, New York (2010)CrossRefMATHGoogle Scholar
  10. 10.
    Falconer, K.J.: The Geometry of Fractal Sets, vol. 85. Cambridge University Press, Cambridge (1986)MATHGoogle Scholar
  11. 11.
    Falconer, K.J.: Fractal Geometry, 2nd edn. Wiley, Hoboken (2003)CrossRefMATHGoogle Scholar
  12. 12.
    Güting, R.: On Mahler’s function \(\theta _{1}\). Mich. Math. J. 10, 161–179 (1963)CrossRefMATHGoogle Scholar
  13. 13.
    Jarník, V.: Zur metrischen Theorie der diophantischen Approximation. Prace Mat.-Fiz. 36, 91–106 (1929)MATHGoogle Scholar
  14. 14.
    Jarník, V.: Diophantische Approximationen und Hausdorffsches Mass. Rec. Math. Moscou 36, 371–382 (1929)MATHGoogle Scholar
  15. 15.
    Khintchine, A.: Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der diophantischen Approximationen. Math. Ann. 92(1), 115–125 (1924)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kjos-Hanssen, B., Reimann, J.: The strength of the Besicovitch–Davies theorem. In: Computability in Europe, Volume 6158 of Lecture Notes in Computer Science, pp. 229–238. Springer (2010)Google Scholar
  17. 17.
    Lutz, J.H.: Gales and the constructive dimension of individual sequences. In: Automata, languages and programming (Geneva, 2000), Volume 1853 of Lecture Notes in Computer Science, pp. 902–913. Springer, Berlin (2000)Google Scholar
  18. 18.
    Reimann, J., Stephan, F.: Effective Hausdorff dimension. In: Logic Colloquium ’01, Volume 20 of Lecture Notes in Logic, pp. 369–385. Association for Symbolic Logic, Urbana (2005)Google Scholar
  19. 19.
    Staiger, L.: The Kolmogorov complexity of real numbers. Theor. Comput. Sci. 284(2), 455–466 (2002)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria 2017

Authors and Affiliations

  1. 1.Departamento de Computación, Facultad de Ciencias Exactas y NaturalesUniversidad de Buenos AiresBuenos AiresArgentina
  2. 2.CONICETBuenos AiresArgentina
  3. 3.Department of MathematicsPenn State UniversityUniversity ParkUSA
  4. 4.Department of MathematicsUniversity of California, BerkeleyBerkeleyUSA

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