Skip to main content
Log in

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

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript


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 via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others


  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]).


  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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references


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

Authors and Affiliations


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

Check for updates. 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).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification