We prove that, for any unit in a real number field of degree , there exits only a finite number of n-tuples in which have a purely periodic expansion by the Jacobi-Perron algorithm. This generalizes the case of continued fractions for . For we give an explicit algorithm to compute all these pairs.
For any positive integer let denote the set of numbers with all partial quotients (except possibly the first) not exceeding . In this paper we characterize most products and quotients of sets of the form .
We give a very brief, but gentle, sketch of an introduction both to the Rosen continued fractions and to a geometric setting to which they are related, given in terms of Veech groups. We have kept the informal approach of the talk at the Numerations conference, aimed at an audience assumed to have heard of neither of the topics of the title.The Rosen continued fractions are a family of continued fraction algorithms, each gives expansions of real numbers in terms of elements of a corresponding algebraic...
Le thème de ce travail est la conversion entre le développement en fraction continuée d'un nombre réel et son développement en série de Engel. Chacun d'eux peut se traduire en terme de produits matriciels, produits qui sont à l'origine d'algorithmes, exprimés sous la forme de transducteurs, permettant de calculer un des développements à partir de l'autre. Cette méthode fournit des résultats nouveaux sur les nombres de Lucas, les nombres de Fredholm et sur toute une variété de nombres transcendants,...
The technique of singularization was developped by C. Kraaikamp during the nineties, in connection with his work on dynamical systems related to continued fraction algorithms and their diophantine approximation properties. We generalize this technique from one into two dimensions. We apply the method to the the two dimensional Brun’s algorithm. We discuss, how this technique, and related ones, can be used to transfer certain metrical and diophantine properties from one algorithm to the others. In...