Functiones et Approximatio Commentarii Mathematici

New generalization of continued fraction, I

Alexander D. Bruno

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Abstract

Let three homogeneous real linear forms be given in a three-dimensional real space. Their moduli give a mapping of the space into another space. In the second space, we consider the convex hull of images of all integer points of the first space except its origin. This convex hull is called the modular polyhedron. The best integer approximations to the root subspaces of these forms are given by the integer points whose images lie on the boundary of the modular polyhedron. Here we study the properties of the modular polyhedron and use them for the construction of an algorithm generalizing continued fraction. The algorithm gives the best approximations, and it is periodic for cubic irrationalities with positive discriminant. Attempts to generalize continued fraction were made by Euler, Jacobi, Dirichlet, Hermite, Poincare, Hurwitz, Klein, Minkowski, Voronoi, and by many others.

Article information

Source
Funct. Approx. Comment. Math., Volume 43, Number 1 (2010), 55-104.

Dates
First available in Project Euclid: 28 September 2010

Permanent link to this document
https://projecteuclid.org/euclid.facm/1285679146

Digital Object Identifier
doi:10.7169/facm/1285679146

Mathematical Reviews number (MathSciNet)
MR2683574

Zentralblatt MATH identifier
1213.11145

Subjects
Primary: 11J70: Continued fractions and generalizations [See also 11A55, 11K50]
Secondary: 11K60: Diophantine approximation [See also 11Jxx]

Keywords
generalized continued fraction lattice modular polyhedron face

Citation

Bruno, Alexander D. New generalization of continued fraction, I. Funct. Approx. Comment. Math. 43 (2010), no. 1, 55--104. doi:10.7169/facm/1285679146. https://projecteuclid.org/euclid.facm/1285679146


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References

  • V.I. Arnold, Higher dimensional continued fractions, Regular and Chaotic Dynamics, v. 3, no. 3 (1998), 10--17.
  • L. Bernstein, The Jacobi--Perron algorithm -- its theory and application, LNM 207. Berlin/Heidelberg/New York: Springer Verlag, 1971.
  • Z.I. Borevich and I.R. Shafarevich, Number Theory, Moscow, Nauka, 1972 (Russian) = Academic Press, 1966 (English).
  • A.J. Brentjes, Multi-dimensional Continued Fraction Algorithms, Mathematical Centre Tracts 145, Amsterdam: Mathematisch Centrum, 1981.
  • K. Briggs, Klein polyhedra http://www.btexact.com/people/briggsk2/klein-polyhedra.html
  • V. Brun, En generalisation av Kjedebroken, Skrifter utgit av Videnskapsselskapeti Kristiania. I. Matematisk-Naturvidenskabelig Klasse 1919. N 6; 1920.
  • A.D. Bruno (Bryuno), Continued fraction expansion of algebraic numbers, Zhurnal Vychisl. Matem. i Matem. Fiziki 4:2 (1964), 211--221 (Russian) = USSR Comput. Math. and Math. Phys. 4:2 (1964), 1--15 (English).
  • A.D. Bruno (Brjuno), Convergence of transformations of differential equations to normal form, Dokl. Akad. Nauk SSSR 165:5 (1965), 987-989 (R) = Soviet Math. Dokl. 6 (1965), 1536--1538 (E).
  • A.D. Bruno, Small denominators, Mat. Encyclopedia. Moscow, 1982, v. 3, 505--507 (R) = Encyclopaedia of Mathematics (in 10 vols). Kluver Acad. Publ.: Dordrecht, 1993, v. 8, 366--368 (E).
  • A.D. Bruno, The correct generalization of the continued fraction, Preprint no. 86 of the Keldysh Inst. of Applied Math.: Moscow, 2003 (in Russian).
  • A.D. Bruno, On generalizations of the continued fractions, Preprint no. 10 of the Keldysh Inst. of Applied Math.; Moscow, 2004. 31 p (Russian).
  • A.D. Bruno, Algorithm of the generalized continued fraction, Preprint no. 45 of the Keldysh Inst. of Applied Math.: Moscow, 2004 (in Russian).
  • A.D. Bruno, Structure of the best Diophantine approximations, Doklady Akademii Nauk 402, no. 4 (2005), 439--444 (Russian). = Doklady Mathematics 71, no. 3 (2005), 396--400 (English).
  • A.D. Bruno, Generalized continued fraction algorithm, Doklady Akademii Nauk 402, no. 6 (2005), 732--736 (Russian). = Doklady Mathematics 71, no. 3 (2005), 446--450 (English).
  • A.D. Bruno, Properties of the modular polyhedron, Preprint no. 72 of the Keldysh Inst. of Applied Math.; Moscow, 2005. 31 p (Russian).
  • A.D. Bruno (Bryuno) and V.I. Parusnikov, Klein polyhedrals for two cubic Davenport forms, Matem. Zametki 56:4 (1994), 9--27 (Russian); Math. Notes 56:3--4 (1994), 994--1007 (English).
  • A.D. Bruno (Bryuno) and V.I. Parusnikov, Comparison of various generalizations of continued fractions, Matem. Zametki 61:3 (1997), 339--348 (Russian) Math. Notes 61:3 (1997), 278--286 (English).
  • A.D. Bruno and V.I. Parusnikov, Polyhedra of absolute values for triples of linear forms, Preprint no. 93 of the Keldysh Inst. of Applied Math.: Moscow, 2003 (in Russian).
  • A.D. Bruno and V.I. Parusnikov, New generalizations of the continued fraction, Preprint no. 52, Keldysh Inst. of Applied Math.; Moscow, 2005. 20 p (English).
  • J. Buchmann, On the period length of the generalized Lagrange algorithm, J. Number Theory 26 (1987), 8--37.
  • V.A. Bykovsky, The Valen's theorem for two-dimensional convergent fraction, Matem. Zametki 66:1 (1999), 30--37 (Russian) = Math. Notes 66:1 (1999) (English).
  • J.W.S Cassels, An Introduction to the Geometry of Numbers, Springer-Verlag, Berlin 1959 (English) = Mir, Moscow 1965 (Russian).
  • B.N. Delone, The Petersburg's School of Mathematics, Moscow-Leningrad, AN SSSR, 1947 (Russian).
  • B.N. Delone and D.K. Faddeev, The theory of irrationalities of the third degree, Trudy Matem. Inst. Akad. Nauk 11, Moscow-Leningrad, AN SSSR, (1947) (Russian) = Am. Math. Soc. Transl. of Math. Monographs 10 (1964) (English).
  • L. Euler, De fractinibus continuis, Comm. Acad. Sci. Imper. Petropol. 9 (1737).
  • L. Euler, De relatione inter ternas pluresve quantitates instituenda, Petersburger Akademie Notiz. Exhib. August 14, 1775, Commentationes arithmeticae collectae. V. II. St. Petersburg, 1849, 99--104.
  • Ch. Hermite, Lettres de M. Ch. Hermite á M. Jacobi sur differents objets de la theorie des nombres, J. Reine Angew. Math. 40 (1850), 261--315; Oeuvres, T. I, Paris: Gauther--Villares, 1905, 100--163, Opuscule Mathematica de Jacobi, v. II.
  • Ch. Hermite, Correspondance d$'$Hermite et de Stieltjes. T. II, lettres 232, 238, 408. Gauthier--Villars, Paris, 1905.
  • A. Hurwitz, Ueber die angenäherte Darstellung der Zahlen durch rationale Brüche, Math. Ann. 44 (1894), 417--436.
  • A. Hurwitz, Ueber eine besondere Art der Kettenbruch-Entwiklung reller Grössen, Acta math. 12 (1889), 367--405.
  • C.G.J Jacobi, Ueber die Auflösung der Gleichung $a_1x_1+a_2x_2+\ldots+a_nx_n=f_n$, J. Reine Angew. Math. 69 (1868), 21--28.
  • C.G.J. Jacobi, Allgemeine Theorie der Kettenbruchänlichen Algorithmen, in welchen jede Zahl aus drei vorhergehenden gebildet wird, J. Reine Angew. Math. 69 (1868), 29--64; Gesammelte Werke, Bd. IV. Berlin: Reimer, 1891, 385--426.
  • A.Ya. Khinchin, Continued fractions, Moscow, Fizmatgiz, 1961 (in Russian) = Noordhoff, Groningen 1963, or Univ. of Chicago Press 1964. (in English)
  • F. Klein, Über eine geometrische Auffassung der gewöhnlichen Kettenbruchentwicklung, Nachr. Ges. Wiss. Göttingen Math.-Phys. Kl. 3 (1895), 357--359.
  • F. Klein, Sur une representation geometrique du developpement en fraction continue ordinare, Nouv. Ann. Math. (3) 15 (1896), 327--331.
  • F. Klein, Ausgewählte Kapitel der Zahlentheorie, Bd. I, Einleitung. Göttingen, 1896, 16--50.
  • J.F. Koksma, Diophantische Approximationen, Berlin: Julius Springer, 1936.
  • M.L. Kontsevich and Yu.M. Suhov, Statistics of Klein polyhedra and multidimensional continued fractions, Amer. Math. Soc. Transl. (2) 197 (1999), 9--27.
  • E.I. Korkina, Two--dimensional convergent fractions. The simplest examples, Trudy Matem. Inst. im V.A. Steklova. 209 (1995), 143--166 (Russian) = Proceedings of the Steklov Inst. of Math. 209 (1995), 124--144 (English).
  • G. Lachaud, Polyèdre d$'$Arnol$'$d et voile d$'$un cône simplicial: analogues du théorème de Lagrange, C.R. Acad. Sci. Ser. 1. 317 (1993), 711--716.
  • G. Lachaud, Polyèdre d$'$Arnol$'$d et voile d$'$un cône simplicial, analogues du théorème de Lagrange pour les irrationnels de degré quelconque, Prétirage N 93--17. Marseille: Laboratoire de Mathématiques Discretes du C.N.R.S., 1993.
  • J.L. Lagrange, Complement chez Elements d$'$algebre etc. par M.L. Euler, t. III, 1774.
  • S. Lang and H. Trotter, Continued fractions of some algebraic numbers, J. Reine Angew. Math. 252 (1972), 112--134.
  • G.P. Lejeune Dirichlet, Verallgemeinerung eines Satzes aus der Lehre von den Kettenbrüchen nebst einigen Anwendungen auf die Theorie der Zahlen, S.--B. Press. Akad. Wiss. 1842, 93--95; Werke. Bd. I. Berlin: Reimer, 1889, 635--638.
  • H. Minkowski, Généralisation de le théorie des fractions continues, Ann. Sci. Ec. Norm. Super. ser III, 13 (1896), 41--60. Also in: Gesamm. Abh. I, 278--292.
  • H. Minkowski, Über die Annäherung an eine reele Größ e durch rationale Zahlen, Math. Annalen 54 (1901), 91--124; also in: Gesamm. Abh. I, 320--352.
  • V.I. Nechaev, Diagonal continued fraction, Matem. Encyclop. 2(1979), Moscow, p. 123 (Russian) = Math. Encyclop., Kluwer Acad. Publ. (English).
  • V.I. Parusnikov, Klein$'$s polyhedra with big faces, Preprint no. 93 of the Keldysh Inst. of Applied Math.: Moscow, 1997 (in Russian).
  • V.I. Parusnikov, Klein polyhedra for complete decomposable forms, Number theory. Diophantine, Computational and Algebraic Aspects. Editors: K. Győry, A. Pethő and V.T. Sós. De Gruyter. Berlin, New York. 1998, 453--463.
  • V.I. Parusnikov, Klein$'$s polyhedra for the fifth extremal cubic form, Preprint no. 69 of the Keldysh Inst. of Applied Math.: Moscow, 1998 (in Russian).
  • V.I. Parusnikov, Klein$'$s polyhedra for the sixth extremal cubic form, Preprint no. 69 of the Keldysh Inst. of Applied Math.: Moscow, 1999 (in Russian).
  • V.I. Parusnikov, Klein$'$s polyhedra for the seventh extremal cubic form, Preprint no. 79 of the Keldysh Inst. of Applied Math.: Moscow, 1999 (in Russian).
  • V.I. Parusnikov, Klein polyhedra for the fourth extremal cubic form, Math. Notes 67:1 (2000), 87--102.
  • V.I. Parusnikov, Klein$'$s polyhedra for three extremal forms, Matem. Zametki 77:4 (2005), 566-583 (Russian) = Math. Notes 77:4 (2005), 523--538 (English).
  • V.I. Parusnikov, Comparison of several generalizations of the continued fraction, Chebysevsky Sbornik 5 (2005), no. 4, Tula, 180--188.
  • O. Perron, Grundlagen für eine Theorie des Jacobischen Ketten-bruchalgorithmus, Math. Ann. 64 (1907), 1--76.
  • O. Perron, Die Lehre von den Kettenbrüchen, Teubner Verlag, Leipzig, 1913; Stuttgart, 1954, 1977.
  • N. Pipping, Zur Theorie der Diagonalkettenbrüche, Acta Acad. Aboens. 3 (1924), p. 22.
  • H. Poincaré, Sur une généralisation des fractions continues, C.R. Acad. Sci. Paris. Ser. 1 99 (1884), 1014--1016.
  • L.D. Pustylnikov, Generalized continued fractions and the ergodic theory, Uspekhi Matem. Nauk 58:1 (2003), 113-164 (Russian) = Russian Math.-Surveys 58:1 (2003) (English).
  • F. Schweiger, Multidimensional Continued Fractions, Oxford Univ. Press: New York, 2000.
  • B.F. Skubenko, Minimum of a decomposable cubic form of three variables, J. Sov. Math. 53(3) (1991), 302--321.
  • H.M. Stark, An explanation of some exotic continued fractions found by Brillhart, Computers in Number Theory, Academic Press, London and New-York, 1971, 21--35.
  • H.P.F Swinnerton-Dyer, On the product of three homogeneous linear forms, Acta Arithmetica 18 (1971), 371--385.
  • B.A. Venkov, Elementary theory of numbers, Moscow-Leningrad, ONTI, 1937 (in Russian).
  • G.F. Voronoi, On Generalization of the Algorithm of Continued Fractions, Warszawa University, 1896; also in: Collected Works in 3 Volumes, Vol. 1, Izdat. Akad. Nauk USSR, Kiev, 1952 (in Russian).
  • J.A. Wallis, Arithmetica infinitorum, 1655.