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September 2010 New generalization of continued fraction, I
Alexander D. Bruno
Funct. Approx. Comment. Math. 43(1): 55-104 (September 2010). DOI: 10.7169/facm/1285679146


Let three homogeneous real linear forms be given in a three-dimensional real space. Their moduli give a mapping of the space into another space. In the second space, we consider the convex hull of images of all integer points of the first space except its origin. This convex hull is called the modular polyhedron. The best integer approximations to the root subspaces of these forms are given by the integer points whose images lie on the boundary of the modular polyhedron. Here we study the properties of the modular polyhedron and use them for the construction of an algorithm generalizing continued fraction. The algorithm gives the best approximations, and it is periodic for cubic irrationalities with positive discriminant. Attempts to generalize continued fraction were made by Euler, Jacobi, Dirichlet, Hermite, Poincare, Hurwitz, Klein, Minkowski, Voronoi, and by many others.


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Alexander D. Bruno. "New generalization of continued fraction, I." Funct. Approx. Comment. Math. 43 (1) 55 - 104, September 2010.


Published: September 2010
First available in Project Euclid: 28 September 2010

zbMATH: 1213.11145
MathSciNet: MR2683574
Digital Object Identifier: 10.7169/facm/1285679146

Primary: 11J70
Secondary: 11K60

Keywords: face , generalized continued fraction , lattice , modular polyhedron

Rights: Copyright © 2010 Adam Mickiewicz University


Vol.43 • No. 1 • September 2010
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