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

Liangang Ma

doi:10.1016/j.crma.2017.05.012

Number theory

A remark on Liao and Rams' result on the distribution of the leading partial quotient with growing speed

e^{n^{1 / 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

e^{n^{1 / 2}}

]

Présenté par : the Editorial Board

Liangang Ma ¹

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

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

Résumé
Abstract

Étant donné un réel $x \in (0, 1) ∖ Q$ , soit $x = [a_{1} (x), a_{2} (x), \dots]$ son développement en fraction continue. Soit $T_{n} (x) : = \max {a_{k} (x) : 1 \leq k \leq n}$ le plus grand quotient partiel jusqu'à n. Pour tout $α \in (0, \infty), γ \in (0, \infty)$ , soit $F (γ, α) : = {x \in (0, 1) ∖ Q : \lim_{n \to \infty} \frac{T_{n} (x)}{e^{n^{γ}}} = α}$ . Pour un ensemble $E \subset (0, 1) ∖ Q$ , soit $\dim_{H} E$ sa dimension de Hausdorff. Récemment, Lingmin Liao et Michal Rams ont montré que $\dim_{H} F (γ, α) = {\begin{matrix} 1 & s i γ \in (0, 1 / 2) \\ 1 / 2 & s i γ \in (1 / 2, \infty) \end{matrix}$ pour tout $α \in (0, \infty)$ . Dans cet article, nous montrons que $\dim_{H} F (1 / 2, α) = 1 / 2$ pour tout $α \in (0, \infty)$ en suivant la méthode de Liao et Rams, ce qui complète leur résultat.

For a real $x \in (0, 1) ∖ Q$ , let $x = [a_{1} (x), a_{2} (x), \dots]$ be its continued fraction expansion. Denote by $T_{n} (x) : = \max {a_{k} (x) : 1 \leq k \leq n}$ the maximum partial quotient up to n. For any real $α \in (0, \infty), γ \in (0, \infty)$ , let $F (γ, α) : = {x \in (0, 1) ∖ Q : \lim_{n \to \infty} \frac{T_{n} (x)}{e^{n^{γ}}} = α}$ . For a set $E \subset (0, 1) ∖ Q$ , let $\dim_{H} E$ be its Hausdorff dimension. Recently, Lingmin Liao and Michal Rams showed that $\dim_{H} F (γ, α) = {\begin{matrix} 1 & i f γ \in (0, 1 / 2) \\ 1 / 2 & i f γ \in (1 / 2, \infty) \end{matrix}$ for any $α \in (0, \infty)$ . In this paper, we show that $\dim_{H} F (1 / 2, α) = 1 / 2$ for any $α \in (0, \infty)$ following Liao and Rams' method, which supplements their result.

Reçu le : 2016-08-10
Accepté le : 2017-05-30
Publié le : 2017-06-28

DOI : 10.1016/j.crma.2017.05.012

Affiliations des auteurs :

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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     title = {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},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {734--737},
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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/

Bibliographie
Cité par

[1] E. Cesaratto; B. Vallée Hausdorff dimension of real numbers with bounded digit averages, Acta Arith., Volume 125 (2006), pp. 115-162

[2] A. Fan; L. Liao; B. Wang; J. Wu On Kintchine exponents and Lyapunov exponents of continued fractions, Ergod. Theory Dyn. Syst., Volume 29 (2009), pp. 73-109

[3] L. Fang; K. Song A remark on the extreme value theory for continued fractions, 2016 | arXiv

[4] G. Iommi; T. Jordan Multifractal analysis of Birkhoff averages for countable Markov maps, Ergod. Theory Dyn. Syst., Volume 35 (2015), pp. 2559-2586

[5] L. Liao; M. Rams Subexponentially increasing sums of partial quotients in continued fraction expansions, Math. Proc. Camb. Philos. Soc., Volume 160 (2016) no. 3, pp. 401-412

[6] T. Okano Explicit continued fractions with expected partial quotient growth, Proc. Amer. Math. Soc., Volume 130 (2002) no. 3, pp. 1603-1605

[7] W. Philipp A conjecture of Erdös on continued fractions, Acta Arith., Volume 28 (1975/76) no. 4, pp. 379-386

[8] J. Wu; J. Xu The distribution of the largest digit in continued fraction expansions, Math. Proc. Camb. Philos. Soc., Volume 146 (2009) no. 1, pp. 207-212

[9] J. Wu; J. Xu On the distribution for sums of partial quotients in continued fraction expansions, Nonlinearity, Volume 24 (2011) no. 4, pp. 1177-1187

[10] J. Xu On sums of partial quotients in continued fraction expansions, Nonlinearity, Volume 21 (2008) no. 9, pp. 2113-2120

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