Exponents of diophantine approximation in dimension 2 for numbers of Sturmian type

Abstract

We generalize the construction of Roy’s Fibonacci type numbers to the case of a Sturmian recurrence and we determine the classical exponents of approximation \(\omega _2(\xi )\), \({\widehat{\omega }}_2(\xi )\), \(\lambda _2(\xi )\), \({\widehat{\lambda }}_2(\xi )\) associated with these real numbers. This also extends similar results established by Bugeaud and Laurent in the case of Sturmian continued fractions. More generally we provide an almost complete description of the combined graph of parametric successive minima functions defined by Schmidt and Summerer in dimension two for such Sturmian type numbers. As a side result we obtain new information on the joint spectra of the above exponents as well as a new family of numbers for which it is possible to construct the sequence of the best rational approximations.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3

Notes

  1. 1.

    Note that there is also a construction which gives for every \(\omega \ge 3\) a number \(\xi \) for which \(\omega _2(\xi )=\omega \), but for such \(\xi \) we always have \({\widehat{\omega }}_2(\xi ) = 2\) (see the proof of Theorem 5.1 and Theorem 5.5 of [5]).

References

  1. 1.

    Allouche, J.-P., Davison, J., Queffélec, M., Zamboni, L.: Transcendence of Sturmian or morphic continued fractions. J. Number Theory 91(1), 39–66 (2001)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Arbour, B., Roy, D.: A Gel’fond type criterion in degree two. Acta Arith. 111(1), 97–103 (2004)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Beresnevich, V., Dickinson, D., Velani, S.: Diophantine approximation on planar curves and the distribution of rational points. Ann. Math. 166(2), 367–426 (2007)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bernik, V.: Use of Hausdorff dimension in the theory of Diophantine approximations. Acta Arith. 42(3), 219–253 (1983)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bugeaud, Y.: Exponents of diophantine approximation. In: Badziahin, D., Gorodnik, A., Peyerimhoff, N. (eds.) Dynamics and Analytic Number Theory, volume 437 of London Math. Soc. Lecture Note Ser., chap. 2, pp. 96–135. Cambridge University Press, Cambridge (2016)

  6. 6.

    Bugeaud, Y., Laurent, M.: Exponents of diophantine approximation and Sturmian continued fractions. Ann. Inst. Fourier 55(3), 773–804 (2005)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Cassaigne, J.: Limit values of the recurrence quotient of Sturmian sequences. Theoret. Comput. Sci. 218(1), 3–12 (1999)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Davenport, H., Schmidt, W.: Approximation to real numbers by algebraic integers. Acta Arith. 15(4), 393–416 (1969)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Fischler, S.: Palindromic prefixes and episturmian words. J. Comb. Theory, Ser. A 113(7), 1281–1304 (2006)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Fischler, S.: Palindromic prefixes and diophantine approximation. Monatsh. Math. 151(1), 11–37 (2007)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Jarník, V.: Zum Khintchineschen” Übertragungssatz”. Trudy Tbilisskogo mathematicheskogo instituta im. A. M. Razmadze = Travaux de l’Institut mathématique de Tbilissi 3, 193–212 (1938)

    MATH  Google Scholar 

  12. 12.

    Roy, D.: Approximation simultanée d’un nombre et de son carré. C. R. Acad. Sci. Paris 336(1), 1–6 (2003)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Roy, D.: Approximation to real numbers by cubic algebraic integers I. Proc. Lond. Math. Soc. 88(1), 42–62 (2004)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Roy, D.: On two exponents of approximation related to a real number and its square. Canad. J. Math 59(1), 211–224 (2007)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Roy, D.: On Schmidt and Summerer parametric geometry of numbers. Ann. Math. 182, 739–786 (2015)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Roy, D.: Spectrum of the exponents of best rational approximation. Math. Z. 283(1–2), 143–155 (2016)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Schleischitz, J.: Approximation to an extremal number, its square and its cube. Pac. J. Math. 287(2), 485–510 (2017)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Schleischitz, J.: Cubic approximation to Sturmian continued fractions. J. Number Theory 184, 270–299 (2018)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Schmidt, W.M.: Diophantine approximation, volume 785 of Lecture Notes in Math. Springer, Berlin(1980)

  20. 20.

    Schmidt, W.M., Summerer, L.: Parametric geometry of numbers and applications. Acta Arith. 140, 67–91 (2009)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Schmidt, W.M., Summerer, L.: Diophantine approximation and parametric geometry of numbers. Monatsh. Math. 169, 51–104 (2013)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Vaughan, R.C., Velani, S.: Diophantine approximation on planar curves: the convergence theory. Invent. Math. 166(1), 103–124 (2006)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

I am very grateful to Stéphane Fischler and Damien Roy for giving me a lot of feedback on this paper. I also thank the anonymous referees for their work.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Anthony Poëls.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Poëls, A. Exponents of diophantine approximation in dimension 2 for numbers of Sturmian type. Math. Z. 294, 951–993 (2020). https://doi.org/10.1007/s00209-019-02280-2

Download citation

Keywords

  • Diophantine approximation
  • Geometry of numbers
  • Sturmian sequence
  • Simultaneous approximation

Mathematics Subject Classification

  • Primary 11J13
  • Secondary 11H06
  • 11J82