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Periodicity of Klein Polyhedra. Generalization of Lagrange’s Theorem

  • Oleg Karpenkov
Chapter
  • 2.2k Downloads
Part of the Algorithms and Computation in Mathematics book series (AACIM, volume 26)

Abstract

The sails of algebraic multidimensional continued fractions possess combinatorial periodicity due to the action of the positive Dirichlet group on the sails. In case of one-dimensional geometric continued fractions this periodicity is completely described by the periodicity of the corresponding LLS sequences. The questions related to periodicity of multidimensional algebraic sails are important in algebraic number theory, since they are in correspondence with algebraic irrationalities. In particular, periods of algebraic sails characterize the groups of units in the corresponding orders.

This chapter is dedicated to such problems. First, we associate to any matrix with real distinct eigenvalues a multidimensional continued fraction. Second, we discuss periodicity of associated sails in the algebraic case. Further, we give examples of periods of two-dimensional algebraic continued fractions and formulate several questions arising in that context. Then we state and prove a version of Lagrange’s theorem for multidimensional continued fractions. Finally, we say a few words about the relation of the Littlewood and Oppenheim conjectures to periodic sails.

Keywords

Continue Fraction Infinite Series Golden Ratio Polygonal Line Algebraic Number Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 27.
    A.D. Bryuno, V.I. Parusnikov, Klein polyhedra for two Davenport cubic forms. Math. Notes - Ross. Akad. 56(3–4), 994–1007 (1995). Russian version: Mat. Zametki 56(4), 9–27 (1994) MathSciNetGoogle Scholar
  2. 34.
    J.W.S. Cassels, H.P.F. Swinnerton-Dyer, On the product of three homogeneous linear forms and the indefinite ternary quadratic forms. Philos. Trans. R. Soc. Lond. Ser. A 248, 73–96 (1955) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 52.
    B.N. Delone, D.K. Faddeev, The Theory of Irrationalities of the Third Degree. Translations of Mathematical Monographs, vol. 10 (Am. Math. Soc., Providence, 1964) zbMATHGoogle Scholar
  4. 68.
    O.N. German, Sails and norm minima of lattices. Sb. Math. 196(3), 337–367 (2005). Russian version: Mat. Sb. 196(3), 31–60 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 69.
    O.N. German, Klein polyhedra and norm minima of lattices. Dokl. Akad. Nauk, Ross. Akad. Nauk 406(3), 298–302 (2006) MathSciNetGoogle Scholar
  6. 70.
    O.N. German, E.L. Lakshtanov, On a multidimensional generalization of Lagrange’s theorem for continued fractions. Izv. Math. 72(1), 47–61 (2008). Russian version: Izv. Ross. Akad. Nauk, Ser. Mat. 72(1), 51–66 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 89.
    O. Karpenkov, On the triangulations of tori associated with two-dimensional continued fractions of cubic irrationalities. Funct. Anal. Appl. 38(2), 102–110 (2004). Russian version: Funkc. Anal. Prilozh. 38(2), 28–37 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 90.
    O. Karpenkov, On two-dimensional continued fractions of hyperbolic integer matrices with small norm. Russ. Math. Surv. 59(5), 959–960 (2004). Russian version: Usp. Mat. Nauk 59(5), 149–150 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 92.
    O. Karpenkov, Three examples of three-dimensional continued fractions in the sense of Klein. C. R. Math. Acad. Sci. Paris 343(1), 5–7 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 116.
    E. Korkina, La périodicité des fractions continues multidimensionnelles. C. R. Acad. Sci., Sér. 1 Math. 319(8), 777–780 (1994) MathSciNetzbMATHGoogle Scholar
  11. 118.
    E.I. Korkina, Two-dimensional continued fractions. The simplest examples. Tr. Mat. Inst. Steklova 209, 143–166 (1995) MathSciNetGoogle Scholar
  12. 123.
    G. Lachaud, Polyèdre d’Arnol’d et voile d’un cône simplicial: analogues du théorème de Lagrange. C. R. Acad. Sci., Sér. 1 Math. 317(8), 711–716 (1993) MathSciNetzbMATHGoogle Scholar
  13. 124.
    G. Lachaud, Sails and Klein polyhedra, in Number Theory, Tiruchirapalli, 1996. Contemp. Math., vol. 210 (Am. Math. Soc., Providence, 1998), pp. 373–385 CrossRefGoogle Scholar
  14. 125.
    G. Lachaud, Voiles et Polyhedres de Klein (Act. Sci. Ind., Hermann, 2002) Google Scholar
  15. 159.
    V.I. Parusnikov, Klein’s polyhedra for the third extremal ternary cubic form. Technical report, preprint 137, Keldysh Institute of the RAS, Moscow (1995) Google Scholar
  16. 160.
    V.I. Parusnikov, Klein’s polyhedra for the fifth extremal cubic form. Technical report, preprint 69, Keldysh Institute of the RAS, Moscow (1998) Google Scholar
  17. 161.
    V.I. Parusnikov, Klein’s polyhedra for the seventh extremal cubic form. Technical report, preprint 79, Keldysh Institute of the RAS, Moscow (1999) Google Scholar
  18. 162.
    V.I. Parusnikov, Klein polyhedra for the fourth extremal cubic form. Math. Notes - Ross. Akad. 67(1–2), 87–102 (2000). Russian version: Mat. Zametki 67(1), 110–128 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 191.
    B.F. Skubenko, Minima of a decomposable cubic form of three variables. Zapiski nauch. sem. LOMI (1988) Google Scholar
  20. 192.
    B.F. Skubenko, Minima of decomposable forms of degree n of n variables for n≥3. Zapiski nauch. sem. LOMI (1990) Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Oleg Karpenkov
    • 1
  1. 1.Dept. of Mathematical SciencesUniversity of LiverpoolLiverpoolUK

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