Skip to main content
SpringerLink
Log in

Periodic Karyon Expansions of Algebraic Units in Multidimensional Continued Fractions

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Periodic expansions of algebraic numbers in multidimensional continued fractions are obtained by using multidimensional backward maps and a method for differentiating induced toric tilings.

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. V. G. Zhuravlev, “Differentiation of induced toric tilings and multidimensional approximation of algebraic numbers,” Zap. Nauchn. Semin. POMI, 445, 33–92 (2016).

    MathSciNet  Google Scholar 

  2. A. Ya. Khinchin, Continued Fractions [in Russian], 4th ed., Moscow (1978).

  3. V. G. Zhuravlev, “Two-dimensional approximations by the method of tilings,” Zap. Nauchn. Semin. POMI, 440, 81–98 (2015).

    Google Scholar 

  4. V. G. Zhuravlev, “Dividing toric tilings and bounded remainder sets,” Zap. Nauchn. Semin. POMI, 440, 99–122 (2015).

    Google Scholar 

  5. V. G. Zhuravlev, “Periodic karyon continued fraction expansions of cubic irrationalities,” Sovrem. Probl. Matem., 23, 41–66 (2016).

    Google Scholar 

  6. Z. Coelho, A. Lopes, and L. F. Da Rocha, “Absolutely continuous invariant measures for a class of affine interval exchange maps,” Proc. Amer. Math. Soc., 123, No. 11, 3533–3542 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  7. V. G. Zhuravlev and A. V. Shutov, “Derivatives of circle rotations and similarity of orbits,” Max Planck Inst. Math., Preprint Ser., 62, 1–11 (2004).

  8. M. Furukado, Sh. Ito, A. Saito, J. Tamura, and Sh. Yasutomi, “A new multidimensional slow continued fraction algorithm and stepped surface,” Exper. Math., 23, No. 4, 390–410 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Abrate, S. Barbero, U. Cerruti, and N. Murru, “Periodic representations for cubic irrationalities,” Fibonacci Quart., 50, No. 3, 252–264 (2012).

    MathSciNet  MATH  Google Scholar 

  10. N. Murru, “On the periodic writing of cubic irrationals and a generalization of Rédei functions,” Int. J. Number Theory, 11, 779–799 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  11. Sh. Ito, J. Fujii, H. Higashino, and Sh. Yasutomi, “On simultaneous approximation to (α, α 2) with α 3 + kα − 1 = 0,” J. Number Theory, 99, 255–283 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  12. Q. Wang, K. Wang, and Z. Dai, “On optimal simultaneous rational approximation to (ω,ω 2)τ with ω being some kind of cubic algebraic function,” J. Approx. Theory, 148, 194–210 (2007).

  13. P. Hubert and A. Messaoudi, “Best simultaneous Diophantine approximations of Pisot numbers and Rauzy fractals,” Acta Arithm., 124, No. 1, 1–15 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  14. N. Chevallier, “Best simultaneous Diophantine approximations of some cubic numbers,” J. Théor. Nombres Bordeaux, 14, No. 2, 403–414 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  15. N. Chevallier, “Best simultaneous Diophantine approximations and multidimensional continued fraction expansions,” Moscow J. Comb. Number Theory, 1, No. 1, 3–56 (2013).

  16. V. Brun, “Algorithmes euclidiens pour trois et quatre nombres,” in: Treiziéme congres des mathématiciens scandinaves (Helsinki 18–23 août, 1957), Mercators Tryckeri, Helsinki (1958), pp. 45–64.

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

    MathSciNet  MATH  Google Scholar 

  18. A. Nogueira, “The three-dimensional Poincare continued fraction algorithm,” Isr. J. Math., 90, No. 1–3, 373–401 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  19. F. Schweiger, Multidimensional Continued Fractions, Oxford Univ. Press, New York (2000).

  20. V. Berthe and S. Labbe, “Factor complexity of S-adic words generated by the Arnoux–Rauzy–Poincare algorithm,” Adv. Appl. Math., 63, 90–130 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  21. P. Arnoux and S. Labbe, “On some symmetric multidimensional continued fraction algorithms,” arXiv:1508.07814 (2015).

  22. J. Cassaigne, “Un algorithme de fractions continues de complexité linéaire,” Oct. 2015. Dyna3S Meeting, LIAFA, Paris (2015).

  23. J. Lagarias, “Best simultaneous Diophantine approximations. I. Growth rates of best approximation denomimators,” Trans. Amer. Math. Soc., 272, No. 2, 545–554 (1982).

  24. V. G. Zhuravlev, “Bounded remainder polyhedra,” Sovrem. Probl. Matem., 16, 82-102 (2012).

    Article  Google Scholar 

  25. V. G. Zhuravlev, “Exchanged toric developments and bounded remainder sets,” Zap. Nauchn. Semin. POMI, 392, 95-145 (2011).

    Google Scholar 

  26. E. S. Fedorov, The Elements of the Study of Figures [in Russian], Moscow (1953).

  27. G. F. Voronoi, Collected Works [in Russian], Vol. 2, Kiev (1952).

Download references

Author information

Authors and Affiliations

  1. Vladimir State University, Vladimir, 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

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 449, 2016, pp. 84–129.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhuravlev, V.G. Periodic Karyon Expansions of Algebraic Units in Multidimensional Continued Fractions. J Math Sci 225, 893–923 (2017). https://doi.org/10.1007/s10958-017-3505-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-017-3505-2

Search

Navigation