A multidimensional generalization of Lagrange’s theorem on continued fractions
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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 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.
KeywordsLagrange’s theorem on continued fractions Klein polyhedron simplicial cone sail hyperbolic operator eigenbasis eigencone integer lattice semiperiodic boundary
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