Skip to main content
SpringerLink
Log in

On the quadratic Lagrange spectrum

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We study the quadratic Lagrange spectrum defined by Parkkonen and Paulin by considering the approximation by elements of the orbit of a given real quadratic irrational number for the action by homographies and anti-homographies of \(PSL_2(\mathbf{Z})\) on \(\mathbf{R}\cup \{\infty \}\). Our approach is based on the theory of continued fractions.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adamczewski, B., Bugeaud, Y.: On the Littlewood conjecture in simultaneous Diophantine approximation. J. Lond. Math. Soc. 73, 355–366 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Adamczewski, B., Bugeaud, Y.: On the Maillet–Baker continued fractions. J. Reine Angew. Math. 606, 105–121 (2007)

    MATH  MathSciNet  Google Scholar 

  3. Adamczewski, B., Bugeaud, Y.: Transcendence measures for continued fractions involving repetitive or symmetric patterns. J. Eur. Math. Soc. 12, 883–914 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Amou, M., Bugeaud, Y.: Expansions in integer bases and exponents of Diophantine approximation. J. Lond. Math. Soc. 81, 297–316 (2010)

    Google Scholar 

  5. Bugeaud, Y.: Approximation by Algebraic Numbers. Cambridge Tracts in Mathematics 160, Cambridge (2004)

  6. Bugeaud, Y. : Automatic continued fractions are transcendental or quadratic. Ann. Sci. École Norm. Sup. (to appear)

  7. Bugeaud, Y.: Continued fractions with low complexity: transcendence measures and quadratic approximation. Compos. Math. 148, 718–750 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Clarendon Press, Oxford (1979)

    MATH  Google Scholar 

  9. Parkkonen, J., Paulin, F.: Erratum to Spiraling spectra of geodesic lines in negatively curved manifolds. Math. Z. (to appear)

  10. Parkkonen, J., Paulin, F.: Spiraling spectra of geodesic lines in negatively curved manifolds. Math. Z. 268, 101–142 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Parkkonen, J., Paulin, F.: Équidistribution, comptage et approximation par des irrationnels quadratiques. J. Mod. Dyn. 6, 1–40 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  12. Perron, O.: Die Lehre von den Ketterbrüchen. Teubner, Leipzig (1929)

    Google Scholar 

  13. Schmidt, W.M.: Diophantine Approximation, Lecture Notes in Math., vol. 785. Springer, Berlin (1980)

  14. Schmidt, W.M.: On badly approximable numbers and certain games. Trans. Am. Math. Soc. 123, 178–199 (1966)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

I am very pleased to thank Jouni Parkkonen and Frédéric Paulin for having encouraged me to write this note and for fruitful correspondence. Many thanks to the referee for a very careful reading.

Author information

Authors and Affiliations

  1. Université de Strasbourg, Mathématiques, 7, rue René Descartes, 67084 , Strasbourg, France

    Yann Bugeaud

Authors
  1. Yann Bugeaud
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yann Bugeaud.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bugeaud, Y. On the quadratic Lagrange spectrum. Math. Z. 276, 985–999 (2014). https://doi.org/10.1007/s00209-013-1230-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-013-1230-1

Keywords

Mathematics Subject Classification (2000)

Search

Navigation