A localized uniformly Jarník set in continued fractions

Yuanhong Chen ¹   ; Yu Sun ²   ; Xiaojun Zhao ³

¹ School of Mathematics and Statistics Huazhong University of Science and Technology 430074 Wuhan, Hubei, P.R. China
² Faculty of Science Jiangsu University 212013 Zhenjiang, Jiangsu, P.R. China
³ School of Economics Peking University 100871 Beijing, P.R. China

Acta Arithmetica, Tome 167 (2015) no. 3, pp. 267-280

Résumé

For any $x\in [0,1]$, let $[a_1(x), a_2(x),\dots]$ be its continued fraction expansion and $\{q_n(x)\}_{n\ge 1}$ be the sequence of the denominators of its convergents. For any $\tau>0$, we call $$ U(\tau)=\bigg\{x \in [0,1): \bigg|x-\frac{p_n(x)}{q_n(x)}\bigg| \bigg(\frac{1}{q_n(x)}\bigg)^{{\tau+2}} \ {\text{for}}\ n\in \mathbb{N} \ {\text{ultimately}} \big\} $$ a uniformly Jarník set, a collection of points which can be uniformly well approximated by its convergents eventually. In this paper, instead of a constant function of $\tau$, we consider a localized version of the above set, namely $$ U_{\text{loc}}(\tau)=\bigg\{x \in [0,1): \bigg|x-\frac{p_n(x)}{q_n(x)}\bigg| \bigg(\frac{1}{q_n(x)}\bigg)^{{\tau(x)+2}} \ {\text{for}}\ n\in \mathbb N \ {\text{ultimately}}\biggr\}, $$ where $\tau:[0,1]\to \mathbb R^+$ is a continuous function. We call $U_{\text{loc}}(\tau)$ a localized uniformly Jarník set, and determine its Hausdorff dimension.

Détail

DOI : 10.4064/aa167-3-5

Keywords: dots its continued fraction expansion sequence denominators its convergents tau call tau bigg bigg x frac bigg bigg frac bigg tau text mathbb text ultimately uniformly jarn set collection points which uniformly approximated its convergents eventually paper instead constant function tau consider localized version above set namely text loc tau bigg bigg x frac bigg bigg frac bigg tau text mathbb text ultimately biggr where tau mathbb continuous function call text loc tau localized uniformly jarn set determine its hausdorff dimension