On multidimensional generalization of the Lagrange theorem on continued fractions

摘要

The Lagrange theorem on continued fractions states that a number is aquadratic surd if and only if its continued fraction expansion is eventuallyperiodic. The current paper is devoted to a multidimensional generalization ofthis fact. As a multidimensional analog of continued fractions so called Kleinpolyhedra are considered: given an irrational lattice in an n-dimensionalEuclidean space, its Klein polyhedron is defined as the convex hull of nonzerolattice points with nonnegative coordinates.