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Comptes Rendus Mathématique
Volume 343, n° 7
pages 441-445 (octobre 2006)
Doi : 10.1016/j.crma.2006.09.007
Received : 10 Mars 2006 ;  accepted : 5 September 2006
Sur lʼensemble exceptionnel de Weyl
About Weylʼs exceptional set
 

Bakir Farhi
Département de mathématiques, université du Maine, avenue Olivier-Messiaen, 72085 Le Mans cedex 9, France 

Résumé

Il est connu depuis H. Weyl quʼétant donné un réel  , lʼensemble   des réels   pour lesquels la suite   nʼest pas équirépartie modulo 1, est de mesure de Lebesgue nulle. Dans cette Note, sans utiliser la notion de la dimension de Hausdorff, on démontre entre autres que ces ensembles   sont infinis non dénombrables. Pour citer cet article : B. Farhi, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

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

It is well known since H. Weylʼs work that for any given real number  , the set   consisting of positive real numbers for which the sequence   is not uniformly distributed modulo 1, has Lebesgue measure zero. In this Note, without use the concept of Hausdorff dimension, one shows among other things that these sets   are uncountable. To cite this article: B. Farhi, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

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


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