Journal of Mathematical Sciences

, Volume 234, Issue 5, pp 659–679 | Cite as

Localized Pisot Matrices and Joint Approximations of Algebraic Numbers

  • V. G. ZhuravlevEmail author

A development of the simplex-module algorithm for expansion of algebraic numbers in multidimensional continued fractions is proposed. To this end, localized Pisot matrices are constructed, whose eigenvalues with moduli less than one are contained in an interval of small length. Such Pisot matrices generate continued fractions whose convergents are arbitrarily close to the best approximations.


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  1. 1.
    V. G. Zhuravlev, “Simplex-module algorithm for expansion of algebraic numbers in multidimensional continued fractions,” Zap. Nauchn. Semin. POMI, 449, 130–167 (2016).Google Scholar
  2. 2.
    V. Brun, Algorithmes euclidiens pour trois et quatre nombres, in: Treizième Congrès des Mathèmaticiens Scandinaves (Helsinki, 18–23 août 1957), Mercators Tryckeri, Helsinki (1958), pp. 45–64.Google Scholar
  3. 3.
    E. S. Selmer, “Continued fractions in several dimensions,” Nordisk Nat. Tidskr., 9, 37–43 (1961).MathSciNetzbMATHGoogle Scholar
  4. 4.
    A. Nogueira, “The three-dimensional Poincare continued fraction algorithm,” Isr. J. Math., 90, No. 1–3, 373–401 (1995).MathSciNetCrossRefGoogle Scholar
  5. 5.
    F. Schweiger, Multidimensional Continued Fractions, Oxford Univ. Press, New York (2000).zbMATHGoogle Scholar
  6. 6.
    V. Berthe and S. Labbe, “Factor complexity of S-adic words generated by the Arnoux–Rauzy–Poincaré algorithm,” Adv. Appl. Math., 63, 90–130 (2015).MathSciNetCrossRefGoogle Scholar
  7. 7.
    P. Arnoux and S. Labbe, On some symmetric multidimensional continued fraction algorithms, arXiv:1508.07814, August 2015.Google Scholar
  8. 8.
    J. Cassaigne, Un algorithme de fractions continues de complexité linéaire, DynA3S meeting, LIAFA, Paris, October 12th, 2015.Google Scholar
  9. 9.
    V. G. Zhuravlev, “Simplex-karyon algorithm for expansion in multidimensional continued fractions,” Tr. MIAN, 299, 283–303 (2017).Google Scholar
  10. 10.
    J. W. S. Cassels, An Introduction to Diophantine Approximation [Russian translation], Moscow (1961).Google Scholar
  11. 11.
    A. Ya. Khinchin, Continued Fractions [in Russian], 4th ed., Moscow (1978).Google Scholar
  12. 12.
    J. Lagarias, “Best simultaneous Diophantine approximations. I. Growth rates of best approximation denominators,” Trans. Amer. Math. Soc., 272, No. 2, 545–554 (1982).MathSciNetzbMATHGoogle Scholar
  13. 13.
    N. G. Moshchevitin, On some open problems in Diophantine approximation, arXiv:1202.4539v5 [math.NT], 22 Dec. 2012, pp. 1–42.Google Scholar
  14. 14.
    N. Chevallier, “Best simultaneous Diophantine approximations and multidimensional continued fraction expansions,” Moscow J. Comb. Number Theory, 3, No. 3, 3–56 (2013).MathSciNetzbMATHGoogle Scholar
  15. 15.
    Z. I. Borevich and I. R. Schafarevich, Number Theory [in Russian], 2nd ed., Moscow (1972).Google Scholar
  16. 16.
    I. M. Vinogradov, Elements of Number Theory [in Russian], 5th ed., Moscow (1972).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Vladimir State UniversityVladimirRussia

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