On Minkowski type question mark functions associated with even or odd continued fractions

Abstract

We study analogues of Minkowski’s question mark function ?(x) related to continued fractions with even or with odd partial quotients. We prove that these functions are Hölder continuous with precise exponents, and that they linearize the appropriate versions of the Gauss and Farey maps.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

References

  1. 1.

    Aaronson, J., Denker, M.: The Poincaré series of \({\mathbb{C}} \setminus {\mathbb{Z}}\). Ergod. Theory Dyn. Syst. 19, 1–20 (1999)

    Article  MATH  Google Scholar 

  2. 2.

    Alkauskas, G.: Integral transforms of the Minkowski question mark function. Ph.D. thesis, University of Nottingham (2008)

  3. 3.

    Beaver, O., Garrity, T.: A two-dimensional Minkowski \(?(x)\) function. J. Number Theory 107, 105–134 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Bonnano, C., Isola, S.: Orderings of the rationals and dynamical systems. Colloq. Math. 116, 165–189 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Denjoy, A.: Sur une fonction réele de Minkowski. J. Math. Pure Appl. 17, 106–151 (1938)

    Google Scholar 

  6. 6.

    Dilcher, K., Stolarsky, K.: A polynomial analogue to the Stern sequence. Int. J. Number Theory 3, 85–103 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Dresse, Z., Van Assche, W.: Orthogonal polynomials for Minkowski’s question mark function. J. Comput. Appl. Math. 284, 171–183 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Dushistova, A., Kan, I.D., Moshchevitin, N.G.: Differentiability of the Minkowski question mark function. J. Math. Anal. Appl. 401, 774–794 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Grabner, P.J., Kirschenhofer, P., Tichy, R.F.: Combinatorial and arithmetical properties of linear numeration systems. Combinatorica 22, 245–267 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Jordan, T., Sahlsten, T.: Fourier transforms of Gibbs measures for the Gauss map. Math. Ann. 364, 983–1023 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Kesseböhmer, M., Stratmann, B.O.: Fractal analysis for sets of non-differentiability of Minkowski’s question mark function. J. Number Theory 128, 2663–2686 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Kesseböhmer, M., Munday, S., Stratmann, B.O.: Infinite Ergodic Theory of Numbers. De Gruyter Graduate, Berlin (2016)

    MATH  Google Scholar 

  13. 13.

    Kinney, J.R.: Note on a singular function of Minkowski. Proc. Am. Math. Soc. 11, 788–789 (1960)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Kraaikamp, C.: A new class of continued fraction expansions. Acta Arith. 57, 1–39 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Kraaikamp, C., Nakada, H.: On normal numbers for continued fractions. Ergod. Theory Dyn. Syst. 20, 1405–1421 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Masarotto, V.: Metric and arithmetic properties of a new class of continued fraction expansions. Master Thesis, Universitá di Padova and Leiden University (2008/2009)

  17. 17.

    Minkowski, H.: Zur Geometrie der Zahlen, Verhandlungen des Internationalen Mathematiker-Kongresous, Heildelberg, pp. 164–173 1904 (see also Gesammelte Abhandlungen 2, B. G. Teubner, Leipzig und Berlin 1911, pp. 43–52)

  18. 18.

    Northshield, S.: An analogue of Stern’s sequence for \(\mathbb{Z}[\sqrt{2}\)]. J. Integer Seq. 18(11) (2015). Article 15.11.6

  19. 19.

    OEIS: The on-line encyclopedia of integer sequences. https://oeis.org/. Accessed 9 June 2018

  20. 20.

    Panti, G.: Multidimensional continued fractions and a Minkowski function. Mon. Math. 154, 247–264 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Paradis, J., Viader, P., Bibiloni, L.: The derivative of Minkowski’s \(?(x)\) function. J. Math. Anal. Appl. 253, 107–125 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Rieger, G. J.: On the metrical theory of continued fractions with odd partial quotients. In: Colloquia Mathematica Societatis János Bolyai 34, Topics in Classical Number Theory, Budapest, pp. 1371–1418 (1981)

  23. 23.

    Romik, D.: The dynamics of Pythagorean triples. Trans. Am. Math. Soc. 360, 6045–6064 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Salem, R.: On some singular monotonic functions which are strictly increasing. Trans. Am. Math. Soc. 53, 427–439 (1943)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Schweiger, F.: Continued fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universität Salzburg 4, 59–70 (1982)

    Google Scholar 

  26. 26.

    Schweiger, F.: On the approximation by continued fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universität Salzburg 1–2, 105–114 (1984)

    Google Scholar 

  27. 27.

    Zhabitskaya, E.N.: Continued fractions with odd partial quotients. Int. J. Number Theory 8, 1541–1556 (2012)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

We are grateful to the referees for their valuable input that contributed to a number of clarifications and improved the presentation of the paper. Florin P. Boca would like to acknowledge partial support during his visits to IMAR Bucharest by a grant from Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, Project PN-III-P4-ID-PCE-2016-0823, within PNCDI III.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Florin P. Boca.

Additional information

Communicated by A. Constantin.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Boca, F.P., Linden, C. On Minkowski type question mark functions associated with even or odd continued fractions. Monatsh Math 187, 35–57 (2018). https://doi.org/10.1007/s00605-018-1205-8

Download citation

Keywords

  • Question mark function
  • Even continued fraction
  • Odd continued fraction
  • Hölder exponent
  • Gauss map
  • Farey map

Mathematics Subject Classification

  • 26A30
  • 11A55
  • 11B57
  • 11B83
  • 37B10
  • 37E25