Geodesic
A Multidimensional Generalization of Lagrange's Theorem on Continued Fractions
Matematičeskie zametki, Tome 92 (2012) no. 3, pp. 343-360

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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, hyperbolic operator, eigenbasis, integer lattice, semiperiodic boundary.
Mots-clés : sail, eigencone 
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     author = {A. V. Bykovskaya},
     title = {A {Multidimensional} {Generalization} of {Lagrange's} {Theorem} on {Continued} {Fractions}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {343--360},
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     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2012_92_3_a2/}
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A. V. Bykovskaya. A Multidimensional Generalization of Lagrange's Theorem on Continued Fractions. Matematičeskie zametki, Tome 92 (2012) no. 3, pp. 343-360. http://geodesic.mathdoc.fr/item/MZM_2012_92_3_a2/