Comptes Rendus
Number theory
A remark on Liao and Rams' result on the distribution of the leading partial quotient with growing speed en1/2 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 en1/2]
Comptes Rendus. Mathématique, Volume 355 (2017) no. 7, pp. 734-737.

Étant donné un réel x(0,1)Q, soit x=[a1(x),a2(x),] son développement en fraction continue. Soit Tn(x):=max{ak(x):1kn} le plus grand quotient partiel jusqu'à n. Pour tout α(0,),γ(0,), soit F(γ,α):={x(0,1)Q:limnTn(x)enγ=α}. Pour un ensemble E(0,1)Q, soit dimHE sa dimension de Hausdorff. Récemment, Lingmin Liao et Michal Rams ont montré que dimHF(γ,α)={1siγ(0,1/2)1/2siγ(1/2,) pour tout α(0,). Dans cet article, nous montrons que dimHF(1/2,α)=1/2 pour tout α(0,) en suivant la méthode de Liao et Rams, ce qui complète leur résultat.

For a real x(0,1)Q, let x=[a1(x),a2(x),] be its continued fraction expansion. Denote by Tn(x):=max{ak(x):1kn} the maximum partial quotient up to n. For any real α(0,),γ(0,), let F(γ,α):={x(0,1)Q:limnTn(x)enγ=α}. For a set E(0,1)Q, let dimHE be its Hausdorff dimension. Recently, Lingmin Liao and Michal Rams showed that dimHF(γ,α)={1ifγ(0,1/2)1/2ifγ(1/2,) for any α(0,). In this paper, we show that dimHF(1/2,α)=1/2 for any α(0,) following Liao and Rams' method, which supplements their result.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2017.05.012

Liangang Ma 1

1 Dept. of Mathematical Sciences, Binzhou University, Huanghe 5th road No. 391, City of Binzhou 256600, Shandong Province, PR China
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Liangang Ma. A remark on Liao and Rams' result on the distribution of the leading partial quotient with growing speed $ {\mathrm{e}}^{{n}^{1/2}}$ in continued fractions. Comptes Rendus. Mathématique, Volume 355 (2017) no. 7, pp. 734-737. doi : 10.1016/j.crma.2017.05.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.05.012/

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