The proper generalization of the continued fraction
by
Alexander Bruno
Keldysh Institute of Applied Mathematics of RAS, Miusskaja sq. 4, Moscow 125047, Russia

(i) it is simple;

(ii) it gives the best rational approximations to the number;

(iii) it is periodic for square irrational numbers.

In 18-20 centuries a lot of mathematicians attempted to generalize the algorithm for vectors (Euler, Hermit, Jacobi, Dirichlet, Poincare, Hurwitz, Minkowski, Klein, Arnold etc.). But their algorithms had not properties (i) and (ii) together with the property

(iii') periodicity for cubic irrational numbers.

We propose a new concept of the usual continued fraction with the following generalization which has properties (i), (ii), (iii').

In a three-dimensional space we consider three gomogeneous linear forms. In another three-dimensional space, where coordinates are absolute values of these forms, we consider the convex hull of points corresponding to all integer points of the first space, except the origin. The proposed generalization of the continued fraction is a motion along the surface of the convex hull [1, 3]. In [2] there are results of computation of the surfaces for eleven cubic forms being products of three linear forms. They show periodic structures and confirm the correctness of the proposed generalized algorithm.

References

1. A.D. Bruno. The correct generalization of the continued fraction. Preprint no. 86 of Keldysh Institute of Applied Mathematics. Moscow, 2003. 17 p. (in Russian)

2. A.D. Bruno and V.I. Parusnikov. Polyhedra of absolute values for triples of linear forms. Preprint no. 93 of Keldysh Institute of Applied Mathematics. Moscow, 2003. 20 p. (in Russian)

3. A.D. Bruno. On generalization of the continued fraction. Preprint no. 10 of Keldysh Institute of Applied Mathematics. Moscow, 2004. 31 p. (in Russian)

Date received: April 6, 2004