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Journal of Mathematical Sciences

, Volume 234, Issue 5, pp 640–658 | Cite as

Linear-Fractional Invariance of the Simplex-Module Algorithm for Expanding Algebraic Numbers in Multidimensional Continued Fractions

  • V. G. Zhuravlev
Article
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The paper establishes the invariance of the simplex-module algorithm for expanding real numbers α = (α1, …, αd) in multidimensional continued fractions under linear-fractional transformations \( {\alpha}^{\prime }=\left({\alpha}_1^{\prime },\dots, {\alpha}_d^1\right)=U\left\langle \alpha \right\rangle \) with matrices U from the unimodular group GLd+1(ℤ). It is shown that the convergents of the transformed collections of numbers α satisfy the same recurrence relation and have the same approximation order.

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References

  1. 1.
    V. G. Zhuravlev, “The simplex-module algorithm for expansion of algebraic numbers in multidimensional continued fractions,” Zap. Nauchn. Semin. POMI, 449, 168–195 (2016).Google Scholar
  2. 2.
    V. G. Zhuravlev, “Localized Pisot matrices and joint approximation of algebraic numbers,” Zap. Nauchn. Semin. POMI, 458, 104–134 (2017).Google Scholar
  3. 3.
    J. W. S. Cassels, An Introduction to Diophantine Approximations [Russian translation], Moscow (1961).Google Scholar
  4. 4.
    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
  5. 5.
    E. S. Selmer, “Continued fractions in several dimensions,” Nordisk Nat. Tidskr., 9, 37–43 (1961).MathSciNetzbMATHGoogle Scholar
  6. 6.
    A. Nogueira, “The three-dimensional Poincare continued fraction algorithm,” Isr. J. Math., 90, No. 1–3, 373–401 (1995).MathSciNetCrossRefGoogle Scholar
  7. 7.
    F. Schweiger, Multidimensional Continued Fractions, Oxford Univ. Press, New York (2000).zbMATHGoogle Scholar
  8. 8.
    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
  9. 9.
    P. Arnoux and S. Labbe, “On some symmetric multidimensional continued fraction algorithms,” arXiv:1508.07814, August 2015.Google Scholar
  10. 10.
    J. Cassaigne, “Un algorithme de fractions continues de complexité linéaire,” DynA3S meeting, LIAFA, Paris, October 12th, 2015.Google Scholar
  11. 11.
    V. G. Zhuravlev, “Two-dimensional approximations by the method of dividing toric tilings,” Zap. Nauchn. Semin. POMI, 440, 81–98 (2015).Google Scholar
  12. 12.
    R. Mönkemeyer, “Über Fareynetze in n Dimensionen,” Math. Nachr., 11, 321–344 (1963).CrossRefGoogle Scholar
  13. 13.
    D. Grabiner, “Farey nets and multidimensional continued fractions,” Monatsh. Math., 114, No. 1, 35–61 (1992).MathSciNetCrossRefGoogle Scholar
  14. 14.
    V. G. Zhuravlev, “Differentiation of induced toric tilings and multidimensional approximations of algebraic numbers,” Zap. Nauchn. Semin. POMI, 445, 33–92 (2016).Google Scholar
  15. 15.
    V. G. Zhuravlev, “The simplex-karyon algorithm for expansion in multidimensional continued fractions,” Tr. MIAN, 299, 283–303 (2017).Google Scholar
  16. 16.
    A. Yu. Khinchin, Continued Fractions [in Russian], fourth ed., Nauka, Moscow (1978).Google Scholar
  17. 17.
    J. Lagarias, “Best simultaneous Diophantine approximations. I. Growth rates of best approximation denomimators,” Trans. Amer. Math. Soc., 272, No. 2, 545–554 (1982).MathSciNetzbMATHGoogle Scholar
  18. 18.
    Z. I. Borevich and I. R. Shafarevich, Number Theory [in Russian], second ed., Nauka, Moscow (1972).Google Scholar
  19. 19.
    I. M. Vinogradov, Elements of Number Theory [in Russian], sixth ed., Nauka, Moscow (1953).Google Scholar
  20. 20.
    V. G. Zhuravlev, “Linear-fractional invariance of multidimensional continued fractions,” Zap. Nauchn. Semin. POMI, 458, 42–76 (2017).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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