Skip to main content
SpringerLink
Log in

Simplex—Karyon Algorithm of Multidimensional Continued Fraction Expansion

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

A simplex–karyon algorithm for expanding real numbers α = (α1,..., α d ) in multidimensional continued fractions is considered. The algorithm is based on a (d + 1)-dimensional superspace S with embedded hyperplanes: a karyon hyperplane K and a Farey hyperplane F. The approximation of numbers α by continued fractions is performed on the hyperplane F, and the degree of approximation is controlled on the hyperplane K. A local ℘(r)-strategy for constructing convergents is chosen, with a free objective function ℘(r) on the hyperplane K.

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. Arnoux and S. Labbé, “On some symmetric multidimensional continued fraction algorithms,” arXiv: 1508.07814 [math.DS].

  2. V. Berthé and S. Labbé, “Factor complexity of S-adic words generated by the Arnoux–Rauzy–Poincaré algorithm,” Adv. Appl. Math. 63, 90–130 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  3. V. Brun, “Algorithmes euclidiens pour trois et quatre nombres,” in Kolmastoista Skandinaavinen matemaatikkokongressi, Helsingissä, 1957 (Treizième Congrès des mathématiciens scandinaves, Helsinki. 1957) (Mercators Tryckeri, Helsinki, 1958), pp. 45–64.

    Google Scholar 

  4. J. Cassaigne, “Un algorithme de fractions continues de complexité linéaire,” Talk at the Dyna3S Meeting, Paris (Oct. 12, 2015).

    Google Scholar 

  5. D. J. Grabiner, “Farey nets and multidimensional continued fractions,” Monatsh. Math. 114 (1), 35–61 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Mönkemeyer, “Über Fareynetze in n Dimensionen,” Math. Nachr. 11, 321–344 (1954).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Nogueira, “The three-dimensional Poincaré continued fraction algorithm,” Isr. J. Math. 90 (1–3), 373–401 (1995).

    Article  MATH  Google Scholar 

  8. W. M. Schmidt, Diophantine Approximation (Springer, Berlin, 1980), Lect. Notes Math. 785.

    MATH  Google Scholar 

  9. F. Schweiger, Multidimensional Continued Fractions (Oxford Univ. Press, Oxford, 2000).

    MATH  Google Scholar 

  10. E. S. Selmer, “Continued fractions in several dimensions,” Nordisk Mat. Tidskr. 9, 37–43 (1961).

    MathSciNet  MATH  Google Scholar 

  11. V. G. Zhuravlev, “Two-dimensional approximations by the method of dividing toric tilings,” Zap. Nauchn. Semin. POMI 440, 81–98 (2015) [J. Math. Sci. 217 (1), 54–64 (2016)].

    Google Scholar 

  12. V. G. Zhuravlev, “Differentiation of induced toric tiling and multidimensional approximations of algebraic numbers,” Zap. Nauchn. Semin. POMI 445, 33–92 (2016) [J. Math. Sci. 222 (5), 544–584 (2017)].

    Google Scholar 

  13. V. G. Zhuravlev, “Simplex-module algorithm for expansion of algebraic numbers in multidimensional continued fractions,” Zap. Nauchn. Semin. POMI 449, 130–167 (2016) [J. Math. Sci. 225 (6), 924–949 (2017)].

    Google Scholar 

  14. V. G. Zhuravlev, “Periodic karyon expansions of cubic irrationals in continued fractions,” Sovrem. Probl. Mat. 23, 43–68 (2016) [Proc. Steklov Inst. Math. 296 (Suppl. 2), 36–60 (2017)].

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Vladimir State University Named after Alexander and Nikolay Stoletovs, ul. Gor’kogo 87, Vladimir, 600000, Russia

    V. G. Zhuravlev

Authors
  1. V. G. Zhuravlev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. G. Zhuravlev.

Additional information

Original Russian Text © V.G. Zhuravlev, 2017, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2017, Vol. 299, pp. 283–303.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhuravlev, V.G. Simplex—Karyon Algorithm of Multidimensional Continued Fraction Expansion. Proc. Steklov Inst. Math. 299, 268–287 (2017). https://doi.org/10.1134/S008154381708017X

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S008154381708017X

Search

Navigation