A localized uniformly Jarník set in continued fractions
Tom 167 / 2015
Streszczenie
For any , 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.
