In this article we aim to investigate the Hausdorff dimension of the set of points $x \in [0,1)$ such that for any $r\in \mathbb {N}$ , $$ \begin{align*} a_{n+1}(x)a_{n+2}(x)\cdots a_{n+r}(x)\geq e^{\tau(x)(h(x)+\cdots+h(T^{n-1}(x)))} \end{align*} $$ holds for infinitely many $n\in \mathbb {N}$ , where h and $\tau $ are positive continuous functions, T is the Gauss map and $a_{n}(x)$ denotes the nth partial quotient of x in its continued fraction expansion. By appropriate choices of $r,\tau (x)$ and $h(x)$ we obtain various classical results including the famous Jarník–Besicovitch theorem.

中文翻译：

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与广义 JARNÍK-BESICOVITCH 集有关的点集的 HAUSDORFF 维数

在本文中，我们旨在研究点集的 Hausdorff 维数$x \in [0,1)$这样对于任何$r\in \mathbb {N}$,$$ \begin{align*} a_{n+1}(x)a_{n+2}(x)\cdots a_{n+r}(x)\geq e^{\tau(x)(h( x)+\cdots+h(T^{n-1}(x)))} \end{对齐*} $$适用于无限多$n\in \mathbb {N}$， 在哪里H和$\头$是正连续函数，吨是高斯图和$a_{n}(x)$表示n的部分商X在它的连续分数扩展中。通过适当的选择$r,\tau (x)$和$h(x)$我们获得了各种经典结果，包括著名的 Jarník-Besicovitch 定理。