Hausdorff dimension for the set of points connected with the generalized Jarn\'ik-Besicovitch set

Abstract

In this article we aim to investigate the Hausdorff dimension of the set of points $x \in [0,1)$ such that for any $r\in\mathbb{N},$ \begin{align*} a_{n+1}(x)a_{n+2}(x)\cdots a_{n+r}(x)\geq e^{\tau(x)(h(x)+\cdots+h(T^{n-1}(x)))} {align*} holds for infinitely many $n\in\mathbb{N},$ where $h$ and $\tau$ are positive continuous functions, $T$ is the Gauss map and $a_n(x)$ denote the $n$th partial quotient of $x$ in its continued fraction expansion. By appropriate choices of $r$, $\tau(x)$ snd $h(x)$ we obtain the classical Jarn\'{i}k-Besicovitch Theorem as well as more recent results by Wang-Wu-Xu, Wang-Wu, Huang-Wu-Xu and Hussain-Kleinbock-Wadleigh-Wang.