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Comptes Rendus Mathématique
Volume 339, n° 10
pages 709-712 (novembre 2004)
Doi : 10.1016/j.crma.2004.09.026
Received : 5 June 2004 ;  accepted : 27 September 2004
Level sets of β -expansions
Ensembles de niveau des  -développements.
 

Aihua Fan a, b , Hao Zhu a
a Department of Mathematics, Wuhan University, 430072, Wuhan, China 
b LAMFA, UMR 6140 CNRS, université de Picardie, 33, rue Saint Leu, 80039 Amiens, France 

Abstract

Let   be the sequence of β -digits of a real number  , with the golden number   as basis. For any  , any   and any real number a , we consider the level set consisting of numbers x such that  . We prove that the Hausdorff dimension of this set is independent of a and , and that it is equal to   where  . To cite this article: A. Fan, H. Zhu, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

The full text of this article is available in PDF format.
Résumé

Soit   la suite de β -digits du nombre réel  , avec le nombre d'or   comme base. Pour tout  ,   et  , nous considérons l'ensemble de niveau qui est constitué des x tels que  . Nous prouvons que la dimension de Hausdorff de cet ensemble est independante de a et , et qu'elle est égale à   où  . Pour citer cet article : A. Fan, H. Zhu, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

The full text of this article is available in PDF format.


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