V. I. Arnold, “Higher dimensional continued fractions”, Regul. Chaotic Dyn., 3:3 (1998), 10

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Regular and Chaotic Dynamics, 1998, Volume 3, Issue 3, Pages 10–17
DOI: https://doi.org/10.1070/RD1998v003n03ABEH000076 (Mi rcd944)

This article is cited in 28 scientific papers (total in 29 papers)

On the 70th birthday of J.Moser

Higher dimensional continued fractions

V. I. Arnold

Steklov Mathematical Institute, ul. Gublcina 8, Moscow 117966, Russia

Citations (29)

DOI: https://doi.org/10.1070/RD1998v003n03ABEH000076

Abstract: The higher-dimensional analogue of a continuous fraction is the polyhedral surface, bounding the convex hull of the semigroup of the integer points in a simplicial cone of the euclidian space. The article describes some conjectures and theorems, extending to such higher-dimensional continouos fraction the Lagrange theorem on quadraticirrationals and the Gauss–Kuzmin statistics.

Received: 03.09.1998

Bibliographic databases:

Document Type: Article

MSC: 11B37, 13A99, 22A20, 41.'VG3, 52B70, C2L12

Language: English

Citation: V. I. Arnold, “Higher dimensional continued fractions”, Regul. Chaotic Dyn., 3:3 (1998), 10–17

Citation in format AMSBIB

\Bibitem{Arn98}

\by V. I. Arnold

\paper Higher dimensional continued fractions

\jour Regul. Chaotic Dyn.

\yr 1998

\vol 3

\issue 3

\pages 10--17

\mathnet{http://mi.mathnet.ru/rcd944}

\crossref{https://doi.org/10.1070/RD1998v003n03ABEH000076}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1704965}

\zmath{https://zbmath.org/?q=an:1044.11596}

Linking options:

https://www.mathnet.ru/eng/rcd944

https://www.mathnet.ru/eng/rcd/v3/i3/p10

This publication is cited in the following 29 articles:

V. Berthé, K. Dajani, C. Kalle, E. Krawczyk, H. Kuru, A. Thevis, Association for Women in Mathematics Series, 32, Women in Numbers Europe IV, 2024, 111
Roland Bacher, “Euclid meets Popeye: The Euclidean Algorithm for 2×2 Matrices”, Comptes Rendus. Mathématique, 361:G5 (2023), 889
O. N. German, “Badly Approximable Matrices and Diophantine Exponents”, Dokl. Math., 106:S2 (2022), S201
Mikhail Geraskin, Studies in Systems, Decision and Control, 338, Cyber-Physical Systems: Modelling and Intelligent Control, 2021, 309
A. A. Illarionov, “The statistical properties of 3D Klein polyhedra”, Sb. Math., 211:5 (2020), 689–708
A. A. Illarionov, “Distribution of facets of higher-dimensional Klein polyhedra”, Sb. Math., 209:1 (2018), 56–70
Gerardo Gonzalez Sprinberg, Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics, 2018, 593
K. Spalding, A. P. Veselov, “Conway River and Arnold Sail”, Arnold Math J., 4:2 (2018), 169
A. A. Illarionov, “Some properties of three-dimensional Klein polyhedra”, Sb. Math., 206:4 (2015), 510–539
A. D. Bryuno, “Universalnoe obobschenie algoritma tsepnoi drobi”, Chebyshevskii sb., 16:2 (2015), 35–65
A. A. Illarionov, “On the statistical properties of Klein polyhedra in three-dimensional lattices”, Sb. Math., 204:6 (2013), 801–823
Oleg Karpenkov, Algorithms and Computation in Mathematics, 26, Geometry of Continued Fractions, 2013, 271
Oleg Karpenkov, Algorithms and Computation in Mathematics, 26, Geometry of Continued Fractions, 2013, 215
Kiumars Kaveh, Askold Khovanskii, “Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory”, Ann. Math., 176:2 (2012), 925
V. Berthé, “Numeration and discrete dynamical systems”, Computing, 94:2-4 (2012), 369
Émilie Charrier, Fabien Feschet, Lilian Buzer, “Computing efficiently the lattice width in any dimension”, Theoretical Computer Science, 412:36 (2011), 4814
A. D. Bruno, “Structure of the best diophantine approximations and multidimensional generalizations of the continued fraction”, Chebyshevskii sb., 11:1 (2010), 68–73
Oleg N. Karpenkov, Anatoly M. Vershik, “Rational approximation of maximal commutative subgroups of ${GL(n,\mathbb{R})}$ ”, J. Fixed Point Theory Appl., 7:1 (2010), 241
Émilie Charrier, Lilian Buzer, Fabien Feschet, Lecture Notes in Computer Science, 5810, Discrete Geometry for Computer Imagery, 2009, 46
O. N. German, E. L. Lakshtanov, “On a multidimensional generalization of Lagrange's theorem on continued fractions”, Izv. Math., 72:1 (2008), 47–61

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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