On the intersections of localized Jarník sets and localized uniformly Jarník sets in continued fractions

Abstract

For $x\in [0,1),$ let $[a_{1}(x),a_{2}(x),\ldots ]$ be its continued fraction expansion and $\big \{\frac{p_{n}(x)}{q_{n}(x)}, n\ge 1\big \}$ be the sequence of rational convergents of x. In this note, we consider the intersections of localized Jarník sets and localized uniformly Jarník sets in continued fractions and obtain the Hausdorff dimension of the exceptional set

$$\begin{aligned}&{\mathcal {J}}_{\text {loc}}(\tau _{1},\tau _{2})=\Big \{x\in [0,1):\liminf _{n\rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_{n}(x)}=\tau _{1}(x),\\&\quad \limsup _{n\rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_{n}(x)}=\tau _{2}(x)\Big \}, \end{aligned}$$

where $\tau _{1}(x)$ and $\tau _{2}(x)$ are non-negative continuous functions defined on [0,1].