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
Comptes Rendus. Mathématique, Volume 355 (2017) no. 7, pp. 734-737.

For a real $x∈(0,1)∖Q$, let $x=[a1(x),a2(x),⋯]$ be its continued fraction expansion. Denote by $Tn(x):=max{ak(x):1≤k≤n}$ the maximum partial quotient up to n. For any real $α∈(0,∞),γ∈(0,∞)$, let $F(γ,α):={x∈(0,1)∖Q:limn→∞⁡Tn(x)enγ=α}$. 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.

Étant donné un réel $x∈(0,1)∖Q$, soit $x=[a1(x),a2(x),⋯]$ son développement en fraction continue. Soit $Tn(x):=max{ak(x):1≤k≤n}$ le plus grand quotient partiel jusqu'à n. Pour tout $α∈(0,∞),γ∈(0,∞)$, soit $F(γ,α):={x∈(0,1)∖Q:limn→∞⁡Tn(x)enγ=α}$. 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.

Accepted:
Published online:
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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