Skip to main content
SpringerLink
Log in

Schneider’s p-adic continued fractions

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

We study Schneider's version of p-adic continued fractions. We are interested in the finiteness of rational number expansion, the quality of approximation by convergents, the irrationality exponent of a number with a given continued fraction expansion, and the convergence of Schneider's continued fractions in the field of real numbers. The main requirement for all of these problems is a good estimate of growth for the sequences of numerators and denominators of convergents.

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. P.-G. Becker, Periodizitätseigenschaften p-adischer Kettenbrüche, Elem. Math., 45 (1990), 1–8.

  2. R. Belhadef and H. A. Esbelin, On the limits of some p-adic Schneider continued fractions, Adv. Math. Sci. J., 10, (2021), 2581–2591.

  3. Y. Bugeaud, Approximation by Algebraic Numbers, Cambridge Tracts in Mathematics, 160, Cambridge University Press (Cambridge, 2004).

  4. Y. Bugeaud, On simultaneous uniform approximation to a p-adic number and its square, Proc. Amer. Math. Soc., 138 (2010), 3821–3826.

  5. Y. Bugeaud, Distribution Modulo One and Diophantine Approximation, Cambridge Tracts in Mathematics, 193, Cambridge University Press (Cambridge, 2012).

  6. Y. Bugeaud, N. Budarina, D. Dickinson, and H. O'Donnell, On simultaneous rational approximation to a p-adic number and its integral powers, Proc. Edinb. Math. Soc., 54 (2011), 599–612.

  7. Y. Bugeaud and T. Pejković, Quadratic approximation in \(\mathbb{Q}_p\), Int. J. Number Theory, 11 (2015), 193–209.

  8. Y. Bugeaud and T. Pejković, Explicit examples of p-adic numbers with prescribed irrationality exponent, Integers, 18A (2018), Paper No. A5, 15 pp.

  9. Y. Bugeaud and J. Schleischitz, Classical and uniform exponents of multiplicative p-adic approximation, Publ. Mat., to appear.

  10. P. Bundschuh, p-adische Kettenbrüche und Irrationalität p-adischer Zahlen, Elem. Math., 32 (1977), 36–40.

  11. A. A. Deanin, Periodicity of p-adic continued fraction expansions, J. Number Theory, 23 (1986), 367–387.

  12. B. M. M. de Weger, Periodicity of p-adic continued fractions, Elem. Math., 43 (1988), 112–116.

  13. A. Dujella, Number Theory, Školska Knjiga (Zagreb, 2021).

  14. A. Haddley and R. Nair, On Schneider's continued fraction map on a complete non-Archimedean field, Arnold Math. J., 8 (2022), 19–38.

  15. J. Haněl, A. Jaššová, P. Lertchoosakul, and R. Nair, On the metric theory of p-adic continued fractions, Indag. Math. (N.S.), 24 (2013), 42–56.

  16. J. Hirsh and L. C. Washington, p-adic continued fractions, Ramanujan J., 25 (2011), 389–403.

  17. H. Hu, Y. Yu, and Y. Zhao, On the digits of Schneider's p-adic continued fractions, J. Number Theory, 187 (2018), 372–390.

  18. N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-functions, 2nd ed., Graduate Texts in Mathematics, 58, Springer-Verlag (New York, 1984).

  19. J. Miller, On p-adic Continued Fractions and Quadratic Irrationals, PhD Thesis, University of Arizona (Tucson, 2007).

  20. Th. Schneider, Über p-adische Kettenbrüche, in: Symposia Mathematica, vol. IV (INDAM, Rome, 1968/69), Academic Press (London), pp. 181–189.

  21. F. Tilborghs, Periodic p-adic continued fractions, Simon Stevin, 64 (1990), 383–390.

  22. A. J. van der Poorten, Schneider's continued fraction, in: Number Theory with an Emphasis on the Markoff Spectrum (Provo, UT, 1991), Lecture Notes in Pure and Appl. Math., 147, Dekker (New York, 1993), pp. 271–281.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000, Zagreb, Croatia

    T. Pejković

Authors
  1. T. Pejković
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Pejković.

Additional information

The author was supported by the Croatian Science Foundation under the project no. IP- 2018-01-1313 and the QuantiXLie Center of Excellence, a project co-financed by the Croatian Government and European Union through the European Regional Development Fund – the Competitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pejković, . Schneider’s p-adic continued fractions. Acta Math. Hungar. 169, 191–215 (2023). https://doi.org/10.1007/s10474-023-01306-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-023-01306-w

Key words and phrases

Mathematics Subject Classification

Search

Navigation