Three examples of three-dimensional continued fractions in the sense of Klein

Oleg Karpenkov

doi:10.1016/j.crma.2006.04.023

Number Theory/Geometry

Three examples of three-dimensional continued fractions in the sense of Klein
[Trois exemples des fractions continues trois-dimensional en sens de Klein]

Présenté par : Vladimir Arnol'd

Oleg Karpenkov ¹

¹ CEREMADE – UMR 7534, université Paris-Dauphine France, 75775 Paris cedex 16, France

Comptes Rendus. Mathématique, Volume 343 (2006) no. 1, pp. 5-7.

Résumé
Abstract

Le problème de l'étude les plus simple fractions continues n-dimensional en sens de Klein pour $n ⩾ 2$ a été poser de V. Arnold. Le solution pour la case de $n = 2$ a presenté dans les articles de E. Korkina (1995) et G. Lachaud (1995). Dans la Note présente, on étude la case de $n = 3$ .

The problem of the investigation of the simplest n-dimensional continued fraction in the sense of Klein for $n ⩾ 2$ was posed by V. Arnold. The answer for the case $n = 2$ can be found in the works of E. Korkina (1995) and G. Lachaud (1995). In present Note we study the case $n = 3$ .

Reçu le : 2006-01-20
Accepté le : 2006-04-24
Publié le : 2006-06-09

DOI : 10.1016/j.crma.2006.04.023

Affiliations des auteurs :

Oleg Karpenkov ¹

¹ CEREMADE – UMR 7534, université Paris-Dauphine France, 75775 Paris cedex 16, France

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     title = {Three examples of three-dimensional continued fractions in the sense of {Klein}},
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Oleg Karpenkov. Three examples of three-dimensional continued fractions in the sense of Klein. Comptes Rendus. Mathématique, Volume 343 (2006) no. 1, pp. 5-7. doi : 10.1016/j.crma.2006.04.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.04.023/

Bibliographie
Cité par

[1] V.I. Arnold Continued Fractions, Moscow Center of Continuous Mathematical Education, Moscow, 2002

[2] O.N. Karpenkov On tori decompositions associated with two-dimensional continued fractions of cubic irrationalities, Funct. Anal. Appl., Volume 38 (2004) no. 2, pp. 28-37

[3] E.I. Korkina Two-dimensional continued fractions. The simplest examples, Proc. Steklov Inst. Math., Volume 209 (1995), pp. 143-166

[4] G. Lachaud, Voiles et Polyèdres de Klein, preprint no. 95-22, Laboratoire de Mathématiques Discrètes du C.N.R.S., Luminy, 1995

[5] J.-O. Moussafir, Voiles et Polyédres de Klein: Geometrie, Algorithmes et Statistiques, docteur en sciences thése, Université Paris IX-Dauphine, 2000

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