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 Mat. Zametki, 2012, Volume 92, Issue 3, Pages 343–360 (Mi mz9020)

A Multidimensional Generalization of Lagrange's Theorem on Continued Fractions

A. V. Bykovskaya

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: A multidimensional geometric analog of Lagrange's theorem on continued fractions is proposed. The multidimensional generalization of the geometric interpretation of a continued fraction uses the notion of a Klein polyhedron, that is, the convex hull of the set of nonzero points in the lattice $\mathbb Z^n$ contained inside some $n$-dimensional simplicial cone with vertex at the origin. A criterion for the semiperiodicity of the boundary of a Klein polyhedron is obtained, and a statement about the nonempty intersection of the boundaries of the Klein polyhedra corresponding to a given simplicial cone and to a certain modification of this cone is proved.

Keywords: Lagrange's theorem on continued fractions, Klein polyhedron, simplicial cone, sail, hyperbolic operator, eigenbasis, eigencone, integer lattice, semiperiodic boundary

DOI: https://doi.org/10.4213/mzm9020

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English version:
Mathematical Notes, 2012, 92:3, 312–326

Bibliographic databases:

UDC: 511.9+511.48
Revised: 04.04.2011

Citation: A. V. Bykovskaya, “A Multidimensional Generalization of Lagrange's Theorem on Continued Fractions”, Mat. Zametki, 92:3 (2012), 343–360; Math. Notes, 92:3 (2012), 312–326

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz9020
• https://doi.org/10.4213/mzm9020
• http://mi.mathnet.ru/eng/mz/v92/i3/p343

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This publication is cited in the following articles:
1. A. A. Illarionov, “Some properties of three-dimensional Klein polyhedra”, Sb. Math., 206:4 (2015), 510–539
2. A. A. Illarionov, “Distribution of facets of higher-dimensional Klein polyhedra”, Sb. Math., 209:1 (2018), 56–70
3. A. A. Illarionov, “The statistical properties of 3D Klein polyhedra”, Sb. Math., 211:5 (2020), 689–708
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