A localized uniformly Jarník set in continued fractions
Volume 167 / 2015
Abstract
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.
