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Comptes Rendus Mathématique
Volume 355, n° 7
pages 734-737 (juillet 2017)
Doi : 10.1016/j.crma.2017.05.012
Received : 10 August 2016 ;  accepted : 30 May 2017
A remark on Liao and Rams' result on the distribution of the leading partial quotient with growing speed   in continued fractions
Une remarque sur le résultat de Liao et Rams concernant la distribution des fractions continues dont le plus grand quotient partiel croît en  
 

Liangang Ma 1
 Dept. of Mathematical Sciences, Binzhou University, Huanghe 5th road No. 391, City of Binzhou 256600, Shandong Province, PR China 

Abstract

For a real  , let   be its continued fraction expansion. Denote by   the maximum partial quotient up to n . For any real  , let  . For a set  , let   be its Hausdorff dimension. Recently, Lingmin Liao and Michal Rams showed that   for any  . In this paper, we show that   for any   following Liao and Rams' method, which supplements their result.

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Résumé

Étant donné un réel  , soit   son développement en fraction continue. Soit   le plus grand quotient partiel jusqu'à n . Pour tout  , soit  . Pour un ensemble  , soit   sa dimension de Hausdorff. Récemment, Lingmin Liao et Michal Rams ont montré que   pour tout  . Dans cet article, nous montrons que   pour tout   en suivant la méthode de Liao et Rams, ce qui complète leur résultat.

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1  The author appreciates Dr Liao and Prof Rams' explanation of their results and helpful comments on the manuscript, as well as Dr Alexandre DeZotti's linguistic help in French greatly. Thanks is also given to the anonymous reviewer, who kindly helps to enhance the precision and readability of the work.


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