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Other Generalizations of Continued Fractions

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

Abstract

In this chapter we present some other generalizations of regular continued fractions to the multidimensional case. The main goal for us here is to give different geometric constructions related to such continued fractions (whenever possible). We say a few words about Minkowski–Voronoi continued fractions, triangle sequences related to Farey addition, O’Hara’s algorithm related to decomposition of rectangular parallelepipeds, geometric continued fractions, and determinant generalizations of continued fractions. Finally, we describe the relation of regular continued fractions to rational knots and links.

We do not pretend to give a complete list of generalizations of continued fractions. The idea is to show the diversity of generalizations.

Keywords

Continue Fraction Voronoi Tessellation Multidimensional Case Relative Minimum Euclidean Algorithm 
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. 1.
    F. Aicardi, Symmetries of quadratic form classes and of quadratic surd continued fractions. I. A Poincaré tiling of the de Sitter world. Bull. Braz. Math. Soc. (N.S.) 40(3), 301–340 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    F. Aicardi, Symmetries of quadratic form classes and of quadratic surd continued fractions. II. Classification of the periods’ palindromes. Bull. Braz. Math. Soc. (N.S.) 41(1), 83–124 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    G.E. Andrews, The Theory of Partitions. Cambridge Mathematical Library (Cambridge University Press, Cambridge, 1998). Reprint of the 1976 original zbMATHGoogle Scholar
  4. 5.
    V. Arnold, Arithmetics of binary quadratic forms, symmetry of their continued fractions and geometry of their de Sitter world. Bull. Braz. Math. Soc. (N.S.) 34(1), 1–42 (2003). Dedicated to the 50th anniversary of IMPA MathSciNetzbMATHCrossRefGoogle Scholar
  5. 12.
    S. Assaf, L.-C. Chen, T. Cheslack-Postava, B. Cooper, A. Diesl, T. Garrity, M. Lepinski, A. Schuyler, A dual approach to triangle sequences: a multidimensional continued fraction algorithm. Integers 5(1), A8 (2005) MathSciNetGoogle Scholar
  6. 17.
    O.R. Beaver, T. Garrity, A two-dimensional Minkowski ?(x) function. J. Number Theory 107(1), 105–134 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 24.
    A.J. Brentjes, Multidimensional Continued Fraction Algorithms. Mathematical Centre Tracts, vol. 145 (Mathematisch Centrum, Amsterdam, 1981) Google Scholar
  8. 26.
    V. Brun, Algorithmes euclidiens pour trois et quatre nombres, in Treizième congrès des mathématiciens scandinaves, tenu à Helsinki 18–23 août 1957 (Mercators Tryckeri, Helsinki, 1958), pp. 45–64 Google Scholar
  9. 30.
    G. Bullig, Zur Kettenbruchtheorie im Dreidimensionalen (Z 1). Abh. Math. Semin. Hansische Univ. 13, 321–343 (1940) MathSciNetCrossRefGoogle Scholar
  10. 32.
    V.A. Bykovskiĭ, Connectivity of Minkowskii Graph for Multidimensional Complete Lattices (Dalnauka, Vladivostok, 2000) Google Scholar
  11. 33.
    V.A. Bykovskiĭ, O.A. Gorkusha, Minimal bases of three-dimensional complete lattices. Sb. Math. 192(1–2), 215–223 (2001). Russian version: Mat. Sb. 192(2), 57–66 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 36.
    J.H. Conway, An enumeration of knots and links, and some of their algebraic properties, in Computational Problems in Abstract Algebra, Oxford, 1967 (Pergamon, Oxford, 1970), pp. 329–358 Google Scholar
  13. 44.
    K. Dasaratha, L. Flapan, T. Garrity, Ch. Lee, C. Mihaila, N. Neumann-Chun, S. Peluse, M. Stroffregen, Cubic irrationals and periodicity via a family of multi-dimensional continued fraction algorithms (2012). arXiv:1208.4244
  14. 45.
    K. Dasaratha, L. Flapan, T. Garrity, Ch. Lee, C. Mihaila, N. Neumann-Chun, S. Peluse, M. Stroffregen, A generalized family of multidimensional continued fractions: TRIP maps (2012). arXiv:1206.7077
  15. 56.
    J. Fiala, P. Kleban, Generalized number theoretic spin chain-connections to dynamical systems and expectation values. J. Stat. Phys. 121(3–4), 553–577 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 58.
    C. Freiling, M. Laczkovich, D. Rinne, Rectangling a rectangle. Discrete Comput. Geom. 17(2), 217–225 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 60.
    T. Garrity, On periodic sequences for algebraic numbers. J. Number Theory 88(1), 86–103 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 61.
    T. Garrity, A thermodynamic classification of real numbers. J. Number Theory 130(7), 1537–1559 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 62.
    T. Garrity, A thermodynamic classification of pairs of real numbers via the triangle multi-dimensional continued fraction (2012). arXiv:1205.5663
  20. 64.
    I. Gelfand, V. Retakh, Quasideterminants. I. Sel. Math. New Ser. 3(4), 517–546 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 65.
    I.M. Gelfand, V.S. Retakh, Determinants of matrices over noncommutative rings. Funct. Anal. Appl. 25(2), 91–102 (1991). Russian version: Funkc. Anal. Prilozh. 25(2), 13–25 (1991) MathSciNetCrossRefGoogle Scholar
  22. 66.
    I.M. Gelfand, V.S. Retakh, Theory of noncommutative determinants, and characteristic functions of graphs. Funct. Anal. Appl. 26(4), 231–246 (1993). Russian version: Funkc. Anal. Prilozh. 26(4), 1–20 (1992) MathSciNetCrossRefGoogle Scholar
  23. 71.
    O.A. Gorkusha, Minimal bases of three-dimensional complete lattices. Math. Notes - Ross. Akad. 69(3–4), 320–328 (2001). Russian version: Mat. Zametki 69(3), 353–362 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 74.
    J. Hančl, A. Jaššová, P. Lertchoosakul, R. Nair, On the metric theory of p-adic continued fractions. Indag. Math. (N.S.) 24(1), 42–56 (2013) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 75.
    H. Hancock, Development of the Minkowski Geometry of Numbers, vol. 1 (Dover, New York, 1964) zbMATHGoogle Scholar
  26. 76.
    H. Hancock, Development of the Minkowski Geometry of Numbers, vol. 2 (Dover, New York, 1964) zbMATHGoogle Scholar
  27. 80.
    J. Hirsh, L.C. Washington, p-adic continued fractions. Ramanujan J. 25(3), 389–403 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 82.
    A.A. Illarionov, Estimates for the number of relative minima of lattices. Math. Notes - Ross. Akad. 89(1–2), 245–254 (2011). Russian version: Mat. Zametki 89(2), 249–259 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 83.
    A.A. Illarionov, The average number of relative minima of three-dimensional integer lattices. St. Petersburg Math. J. 23(3), 551–570 (2012). Russian version: Algebra Anal. 23(3), 189–215 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  30. 85.
    C.G.J. Jacobi, Allgemeine Theorie der Kettenbruchähnlichen Algorithmen, in welchen jede Zahl aus drei vorhergehenden gebildet wird (Aus den hinterlassenen Papieren von C. G. J. Jacobi mitgetheilt durch Herrn E. Heine). J. Reine Angew. Math. 69(1), 29–64 (1868) zbMATHCrossRefGoogle Scholar
  31. 87.
    V.F. Kagan, Origins of Determinant Theory (Gos. Izd. Ukraine, Odessa, 1922) (in Russian) Google Scholar
  32. 100.
    O.N. Karpenkov, A.V. Ustinov, On construction of the Minkowski-Voronoi complex. Preprint (2013) Google Scholar
  33. 103.
    L.H. Kauffman, S. Lambropoulou, On the classification of rational knots. Enseign. Math. (2) 49(3–4), 357–410 (2003) MathSciNetzbMATHGoogle Scholar
  34. 104.
    L.H. Kauffman, S. Lambropoulou, On the classification of rational tangles. Adv. Appl. Math. 33(2), 199–237 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 110.
    A. Knauf, On a ferromagnetic spin chain. Commun. Math. Phys. 153(1), 77–115 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  36. 111.
    A. Knauf, On a ferromagnetic spin chain. II. Thermodynamic limit. J. Math. Phys. 35(1), 228–236 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  37. 112.
    A. Knauf, Number theory, dynamical systems and statistical mechanics. Rev. Math. Phys. 11(8), 1027–1060 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  38. 114.
    M. Konvalinka, I. Pak, Geometry and complexity of O’Hara’s algorithm. Adv. Appl. Math. 42(2), 157–175 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  39. 120.
    C. Kraaikamp, I. Smeets, Approximation results for α-Rosen fractions. Unif. Distrib. Theory 5(2), 15–53 (2010) MathSciNetzbMATHGoogle Scholar
  40. 126.
    M. Laczkovich, G. Szekeres, Tilings of the square with similar rectangles. Discrete Comput. Geom. 13(3–4), 569–572 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  41. 128.
    J.C. Lagarias, Geodesic multidimensional continued fractions. Proc. Lond. Math. Soc. (3) 69(3), 464–488 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  42. 133.
    K. Mahler, Lectures on Diophantine Approximations. Part I: g-adic Numbers and Roth’s Theorem (University of Notre Dame Press, Notre Dame, 1961). Prepared from the notes by R.P. Bambah of my lectures given at the University of Notre Dame in the Fall of 1957 zbMATHGoogle Scholar
  43. 138.
    A. Messaoudi, A. Nogueira, F. Schweiger, Ergodic properties of triangle partitions. Monatshefte Math. 157(3), 283–299 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  44. 139.
    H. Minkowski, Généralisation de la théorie des fractions continues. Ann. Sci. Éc. Norm. Sup. (3) 13, 41–60 (1896) MathSciNetzbMATHGoogle Scholar
  45. 140.
    H. Minkowski, Gesammelte Abhandlungen (Am. Math. Soc., Chelsea, 1967), pp. 293–315 Google Scholar
  46. 143.
    V.V. Nikulin, On the classification of arithmetic groups generated by reflections in Lobachevskiĭ spaces. Izv. Akad. Nauk SSSR, Ser. Mat. 45(1), 113–142 (1981) MathSciNetGoogle Scholar
  47. 144.
    V.V. Nikulin, On the classification of arithmetic groups generated by reflections in Lobachevskiĭ spaces. Izv. Akad. Nauk SSSR, Ser. Mat. 45(1), 240 (1981) MathSciNetGoogle Scholar
  48. 145.
    V.V. Nikulin, Quotient-groups of groups of automorphisms of hyperbolic forms by subgroups generated by 2-reflections. Algebro-geometric applications, in Current Problems in Mathematics, vol. 18 (Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1981), pp. 3–114 Google Scholar
  49. 146.
    V.V. Nikulin, Discrete reflection groups in Lobachevsky spaces and algebraic surfaces, in Proceedings of the International Congress of Mathematicians, Berkeley, CA, 1986, vol. 1 (Am. Math. Soc., Providence, 1987), pp. 654–671 Google Scholar
  50. 147.
    V.V. Nikulin, Finiteness of the number of arithmetic groups generated by reflections in Lobachevskiĭ spaces. Izv. Math. 71(1), 53–56 (2007). Russian version: Izv. Ross. Akad. Nauk, Ser. Mat. 71(1), 55–60 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  51. 148.
    S. Northshield, Stern’s diatomic sequence 0,1,1,2,1,3,2,3,1,4,… . Am. Math. Mon. 117(7), 581–598 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  52. 151.
    K.M. O’Hara, Structure and Complexity of the Involution Principle for Partitions (ProQuest LLC, Ann Arbor, 1984). Ph.D. thesis, University of California, Berkeley Google Scholar
  53. 152.
    K.M. O’Hara, Bijections for partition identities. J. Comb. Theory, Ser. A 49(1), 13–25 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  54. 155.
    I. Pak, Partition identities and geometric bijections. Proc. Am. Math. Soc. 132(12), 3457–3462 (2004) (electronic) zbMATHCrossRefGoogle Scholar
  55. 156.
    I. Pak, Partition bijections, a survey. Ramanujan J. 12(1), 5–75 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  56. 157.
    I. Pak, A. Postnikov, V. Retakh, Noncommutative Lagrange theorem and inversion polynomials. Preprint (1995). www.math.ucla.edu/~pak/papers
  57. 158.
    G. Panti, Multidimensional continued fractions and a Minkowski function. Monatshefte Math. 154(3), 247–264 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  58. 164.
    O. Perron, Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus. Math. Ann. 64(1), 1–76 (1907) MathSciNetzbMATHCrossRefGoogle Scholar
  59. 165.
    O. Perron, Erweiterung eines Markoffschen Satzes über die Konvergenz gewisser Kettenbrüche. Math. Ann. 74(4), 545–554 (1913) MathSciNetzbMATHCrossRefGoogle Scholar
  60. 170.
    M. Prasolov, M. Skopenkov, Tiling by rectangles and alternating current. J. Comb. Theory, Ser. A 118(3), 920–937 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  61. 172.
    D. Rosen, A class of continued fractions associated with certain properly discontinuous groups. Duke Math. J. 21, 549–563 (1954) MathSciNetzbMATHCrossRefGoogle Scholar
  62. 173.
    A.A. Ruban, Certain metric properties of the p-adic numbers. Sib. Mat. Zh. 11, 222–227 (1970) MathSciNetzbMATHCrossRefGoogle Scholar
  63. 174.
    A.L. Schmidt, Ergodic theory of complex continued fractions, in Number Theory with an Emphasis on the Markoff Spectrum, Provo, UT, 1991. Lecture Notes in Pure and Appl. Math., vol. 147 (Dekker, New York, 1993), pp. 215–226 Google Scholar
  64. 175.
    H. Schubert, Knoten mit zwei Brücken. Math. Z. 65, 133–170 (1956) MathSciNetzbMATHCrossRefGoogle Scholar
  65. 176.
    F. Schweiger, The Metrical Theory of Jacobi-Perron Algorithm. Lecture Notes in Mathematics, vol. 334 (Springer, Berlin, 1973) zbMATHGoogle Scholar
  66. 177.
    F. Schweiger, Ergodic properties of multi-dimensional subtractive algorithms, in New Trends in Probability and Statistics, Palanga, 1991, vol. 2 (VSP, Utrecht, 1992), pp. 91–100 Google Scholar
  67. 178.
    F. Schweiger, Invariant measures for fully subtractive algorithms. Anz. Österreich. Akad. Wiss. Math.-Natur. Kl. 131, 25–30 (1995) MathSciNetGoogle Scholar
  68. 179.
    F. Schweiger, Fully subtractive algorithms. Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 204, 23–32 (1996) MathSciNetGoogle Scholar
  69. 180.
    F. Schweiger, Multidimensional Continued Fractions. Oxford Science Publications (Oxford University Press, Oxford, 2000) zbMATHGoogle Scholar
  70. 181.
    F. Schweiger, Multidimensional continued fractions—new results and old problems. Preprint (2008). http://www.cirm.univ-mrs.fr/videos/2008/exposes/287w2/Schweiger.pdf
  71. 182.
    H. Seifert, W. Threlfall, Seifert and Threlfall: A Textbook of Topology. Pure and Applied Mathematics, vol. 89 (Academic Press, New York, 1980). Translated from the German edition of 1934 by Michael A. Goldman, with a preface by Joan S. Birman, with “Topology of 3-dimensional fibered spaces” by Seifert, translated from the German by Wolfgang Heil zbMATHGoogle Scholar
  72. 183.
    E.S. Selmer, Continued fractions in several dimensions. Nord. Mat. Tidskr. 9, 37–43 (1961) MathSciNetzbMATHGoogle Scholar
  73. 184.
    E.S. Selmer, Continued fractions in several dimensions. Nord. Mat. Tidskr. 9, 95 (1961) MathSciNetGoogle Scholar
  74. 186.
    V.I. Shmoĭlov, Nepreryvnye Drobi. Tom I. Periodicheskie Nepreryvnye Drobi (Merkator, Lviv, 2004) Google Scholar
  75. 187.
    V.I. Shmoĭlov, Nepreryvnye Drobi. Tom II. Raskhodyashchiesya Nepreryvnye Drobi (Merkator, Lviv, 2004) Google Scholar
  76. 188.
    V.I. Shmoĭlov, Nepreryvnye Drobi. Tom III. Iz Istorii Nepreryvnykh Drobei (Merkator, Lviv, 2004) Google Scholar
  77. 189.
    V.Ya. Skorobogatko, Ideas and results of the theory of branching continued fractions and their application to the solution of differential equations, in General Theory of Boundary Value Problems (Naukova Dumka, Kiev, 1983), pp. 187–197 Google Scholar
  78. 190.
    V.Ya. Skorobogatko, Teoriya Vetvyashchikhsya Tsepnykh Drobei i Ee Primenenie v Vychislitelnoi Matematike (Nauka, Moscow, 1983) Google Scholar
  79. 195.
    G. Szekeres, Multidimensional continued fractions. Ann. Univ. Sci. Bp. Rolando Eötvös Nomin., Sect. Math. 13, 113–140 (1970) MathSciNetGoogle Scholar
  80. 197.
    R.F. Tichy, J. Uitz, An extension of Minkowski’s singular function. Appl. Math. Lett. 8(5), 39–46 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  81. 201.
    A.V. Ustinov, Minimal vector systems in 3-dimensional lattices and analog of Vahlen’s theorem for 3-dimensional Minkowski’s continued fractions. Sovrem. Probl. Mat. 16, 103–128 (2012) CrossRefGoogle Scholar
  82. 203.
    P. Viader, J. Paradís, L. Bibiloni, A new light on Minkowski’s ?(x) function. J. Number Theory 73(2), 212–227 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  83. 204.
    G. Voronoĭ, A generalization of the algorithm of continued fractions. Ph.D. thesis, Warsaw, 1896 (in Russian) Google Scholar
  84. 205.
    G.F. Voronoĭ, On a Generalization of the Algorithm of Continued Fraction. Collected Works in Three Volumes (USSR Ac. Sci, Kiev, 1952) (in Russian) Google Scholar
  85. 210.
    C. Yannopoulos, Zur Kettenbruchtheorie im Dreidimensionalen. Math. Z. 47, 105–110 (1940) MathSciNetCrossRefGoogle 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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