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KIAM Preprint № 11, Moscow, 2010
Authors: Bruno A. D.
The structure of the multidimensional Diophantine approximations
Abstract:
Let l linear forms and k quadratic forms (n = l + 2k) be given in the n-dimensional real space R. Absolute values of the forms define a map of the space R into the positive ortant S+ of the m-dimensional real space S, where m = l + k. Here the integer lattice in R is mapped into a set Z ⊂ S+. The closure of the convex hull G of the set Z ∖0 is a polyhedral set. Integer points from R, which are mapped in the boundary ∂G of the polyhedron G, give the best Diophantine approximations to root subspaces of all given forms. In the algebraic case, when the given forms are connected with roots of a polynomial of degree n, we prove that the polyhedron G has m− 1 independent periods. It is a generalization of the Lagrange Theorem, that continued fractions of a square irrationality is periodic.
Publication language: russian,  pages: 8
Research direction:
Mathematical problems and theory of numerical methods
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About authors:
  • Bruno Alexander Dmitrievich,  orcid.org/0000-0002-7465-1258KIAM RAS