A two-dimensional continued fraction algorithm with Lagrange and Dirichlet properties

Drouin, Christian

Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 307-346.

On démontre dans cet article un Théorème de Lagrange, pour un certain algorithme de fraction continue en dimension 2, dont la définition géométrique est très naturelle. Des propriétés type Dirichlet sont aussi obtenues pour la convergence de cet algorithme. Ces propriétés proviennent de caractéristiques géométriques de l’algorithme. Les relations entre ces différentes propriétés sont étudiées. En lien avec l’algorithme présenté, sont rapidement évoqués les travaux de divers auteurs dans le domaine des fractions continues multidimensionnelles.

A Lagrange Theorem in dimension 2 is proved in this paper, for a particular two dimensional continued fraction algorithm, with a very natural geometrical definition. Dirichlet type properties for the convergence of this algorithm are also proved. These properties proceed from a geometrical quality of the algorithm. The links between all these properties are studied. In relation with this algorithm, some references are given to the works of various authors, in the domain of multidimensional continued fractions algorithms.

Classification : 11J70, 11J13, 11H06

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