Advertisement

Rational approximation of maximal commutative subgroups of \({GL(n,\mathbb{R})}\)

  • Oleg N. Karpenkov
  • Anatoly M. VershikEmail author
Article

Abstract

How to find “best rational approximations” of maximal commutative subgroups of \({GL(n,\mathbb{R})}\)? In this paper we specify this problem and make first steps in its study. It contains the classical problems of Diophantine and simultaneous approximation as particular subcases but in general is much wider. We prove estimates for n = 2 for both totally real and complex cases and give an algorithm to construct best approximations of a fixed size. In addition we introduce a relation between best approximations and sails of cones and interpret the result for totally real subgroups in geometric terms of sails.

Mathematics Subject Classification (2010)

11J13 11K60 11J70 

Keywords

Maximal commutative subgroups centralizers Diophantine approximation Markov–Davenport forms sails of simplicial cones 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. L. Adler and L. Flatto, Cross section maps for geodesic flows. I. The modular surface. In: Ergodic Theory and Dynamical Systems, II (College Park, MD, 1979/1980), Progr. Math. 21, Birkhäuser, Boston, 1982, 103–161.Google Scholar
  2. 2.
    Arnold V.I.: A-graded algebras and continued fractions. Comm. Pure Appl. Math. 142, 993–1000 (1989)CrossRefGoogle Scholar
  3. 3.
    Arnold V.I.: Higher dimensional continued fractions. Regul. Chaotic Dynam. 3, 10–17 (1998)zbMATHCrossRefGoogle Scholar
  4. 4.
    V. I. Arnold, Continued Fractions. Moscow Center of Continuous Mathematical Education, 2002.Google Scholar
  5. 5.
    M. O. Avdeeva, On statistics of incomplete quotients of finite continued fractions. Funktsional. Anal. i Prilozhen. 38 (2004), no. 2, 1–11 (in Russian).Google Scholar
  6. 6.
    M. O. Avdeeva and V. A. Bykovskiĭ, Solution of Arnold’s problem on Gauss–Kuzmin statistics. Preprint, Vladivostok, 2002.Google Scholar
  7. 7.
    A. D. Bryuno and V. I. Parusnikov, Klein polyhedrals for two cubic Davenport forms. Mat. Zametki 56 (1994), no. 4, 9–27 (in Russian).Google Scholar
  8. 8.
    Buchmann J.A.: A generalization of Voronoi’s algorithm I, II. J. Number Theory 20, 177–209 (1985)CrossRefMathSciNetGoogle Scholar
  9. 9.
    V. A. Bykovskiĭ, Relative minima of lattices, and vertices of Klein polyhedra. Funktsional. Anal. i Prilozhen. 40 (2006), no. 1, 56–57 (in Russian)Google Scholar
  10. 10.
    Davenport H.: On the product of three homogeneous linear forms, I. Proc. London Math. Soc. 13, 139–145 (1938)zbMATHCrossRefGoogle Scholar
  11. 11.
    Davenport H.: Note on the product of three homogeneous linear forms. J. London Math. Soc. 16, 98–101 (1941)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Davenport H.: On the product of three homogeneous linear forms. IV. Proc. Cambridge Philos. Soc. 39, 1–21 (1943)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    O. N. German and E. L. Lakshtanov, On a multidimensional generalization of Lagrange’s theorem for continued fractions. Izv. Ross. Akad. Nauk Ser. Mat. 2008, no. 1, 51–66 (in Russian).Google Scholar
  14. 14.
    O. Karpenkov, On tori decompositions associated with two-dimensional continued fractions of cubic irrationalities. Funktsional. Anal. i Prilozhen. 38 (2004), no. 2, 28–37 (in Russian).Google Scholar
  15. 15.
    O. Karpenkov, On two-dimensional continued fractions for integer hyperbolic matrices with small norm. Uspekhi Mat. Nauk 59 (2004), no. 5, 149–150 (in Russian).Google Scholar
  16. 16.
    Karpenkov O.: Three examples of three-dimensional continued fractions in the sense of Klein. C. R. Math. Acad. Sci. Paris 343, 5–7 (2006)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Karpenkov O.: Completely empty pyramids on integer lattices and twodimensional faces of multidimensional continued fractions. Monatsh. Math. 152, 217–249 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    O. Karpenkov, On invariant Möbius measure and Gauss–Kuzmin face distribution. Proc. Steklov Inst. Math. 258 (2007), 74–86; http://arxiv.org/abs/math.NT/0610042.
  19. 19.
    Karpenkov O.: Elementary notions of lattice trigonometry. Math. Scand. 102, 161–205 (2008)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Karpenkov O.N.: Constructing multidimensional periodic continued fractions in the sense of Klein. Math. Comp. 78, 1687–1711 (2009)CrossRefMathSciNetGoogle Scholar
  21. 21.
    O. Karpenkov, Integer conjugacy classes of \({SL(3, \mathbb{Z})}\) and Hessenberg matrices. 2007; http://arxiv.org/abs/0711.0830.
  22. 22.
    A. Ya. Khinchin, Continued Fractions. 4-th ed., Nauka, Moscow, 1978 (in Russian); English transl., Dover Publ., Mineola, NY, 1997.Google Scholar
  23. 23.
    F. Klein, Ueber eine geometrische Auffassung der gewöhnliche Kettenbruchentwickelung. Nachr. Ges. Wiss. Göttingen Math.-Phys. Kl. 1895, 357–359.Google Scholar
  24. 24.
    Kontsevich M.L., Sukhov Yu.M.: Statistics of Klein polyhedra and multidimensional continued fractions. Amer. Math. Soc. Transl. 197(2), 9–27 (1999)MathSciNetGoogle Scholar
  25. 25.
    E. I. Korkina, La périodicité des fractions continues multidimensionnelles. C. R. Acad. Sci. Paris 319 (1994), 778–730.Google Scholar
  26. 26.
    E. I. Korkina, Two-dimensional continued fractions. The simplest examples. Trudy Mat. Inst. Steklov. 209 (1995), 143–166 (in Russian).Google Scholar
  27. 27.
    Korkina E.I.: The simplest 2-dimensional continued fraction. J. Math. Sci. 82, 3680–3685 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    R. O. Kuz’min, On a problem of Gauss. Dokl. Akad. Nauk SSSR Ser. A (1928), 375–380.Google Scholar
  29. 29.
    G. Lachaud, Voiles et polyèdres de Klein. Act. Sci. Ind., Hermann, 2002.Google Scholar
  30. 30.
    Lagarias J.C.: Best simultaneous Diophantine approximations. I. Growth rates of best approximation denominators. Trans. Amer. Math. Soc. 272, 545–554 (1982)zbMATHMathSciNetGoogle Scholar
  31. 31.
    Markoff A.: Sur les formes quadratiques binaires indéfinies. Math. Ann. 15, 381–409 (1879)CrossRefGoogle Scholar
  32. 32.
    Minkowski H.: Généralisation de la théorie des fractions continues. Ann. Sci. École Norm. Sup. (3) 33, 1057–1070 (1896)Google Scholar
  33. 33.
    J.-O. Moussafir, Voiles et polyèdres de Klein: géométrie, algorithmes et statistiques. Docteur ès sciences thése, Université Paris IX, 2000; http://www.ceremade.dauphige.fr/~msfr/
  34. 34.
    Perron O.: Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus. Math. Ann. 64, 1–76 (1907)zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Schweiger F.: Multidimensional Continued Fractions. Oxford Sci. Publ., Oxford Univ. Press, Oxford (2000)zbMATHGoogle Scholar
  36. 36.
    Series C.: The modular surface and continued fractions. J. London Math. Soc. (2) 31, 69–80 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Tsuchihashi H.: Higher dimensional analogues of periodic continued fractions and cusp singularities. Tohoku Math. J. 35, 176–193 (1983)CrossRefMathSciNetGoogle Scholar
  38. 38.
    A. Vershik. Statistical mechanics of combinatorial partitions, and their limit configurations. Funktsional. Anal. i Prilozhen. 30, no. 2, 90–105 (1996) (in Russian).Google Scholar
  39. 39.
    G. F. Voronoy, On a generalization of the algorithm of continued fractions. Izd. Varsh. Univ., Warszawa, 1896; also in: Collected Works in 3 Volumes, Vol. 1, Izd. Akad. Nauk Ukrain. SSSR, Kiev, 1952 (in Russian).Google Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.TU GrazGrazAustria
  2. 2.St. Petersburg Department of Steklov Institute of MathematicsSt. PetersburgRussia

Personalised recommendations