Let $[a_1(x),a_2(x),\cdots,a_n(x),\cdots]$ be the continued fraction expansion of $x\in[0,1)$. In this paper, we study the increasing rate of the weighted product $a^{ t_0}_n(x)a^{ t_1}_{ n+1}(x)\cdots a^{ t_m}_{ n+m}(x)$ ,where $t_i\in \mathbb{ R}_+\ (0\leq i \leq m)$ are weights. More precisely, let $\varphi:\mathbb{ N}\to\mathbb{ R}_+$ be a function with $\varphi(n)/n\to \infty$ as $n\to \infty$. For any $(t_0,\cdots,t_m)\in \mathbb{ R}^{ m+1}_+$ with $t_i\geq 0$ and at least one $t_i\neq0 \ (0\leq i\leq m)$, the Hausdorff dimension of the set $$\underline{ E}(\{ t_i\}_{ i=0}^m,\varphi)=\left\{ x\in[0,1):\liminf\limits_{ n\to \infty}\dfrac{ \log \left(a^{ t_0}_n(x)a^{ t_1}_{ n+1}(x)\cdots a^{ t_m}_{ n+m}(x)\right)}{ \varphi(n)}=1\right\}$$ is obtained. Under the condition that $(t_0,\cdots,t_m)\in \mathbb{ R}^{ m+1}_+$ with $0<t_0\leq t_1\leq \cdots \leq t_m$, we also obtain the Hausdorff dimension of the set \begin{ equation*} \overline{ E}(\{ t_i\}_{ i=0}^m,\varphi)=\left\{ x\in[0,1):\limsup\limits_{ n\to \infty}\dfrac{ \log \left(a^{ t_0}_n(x)a^{ t_1}_{ n+1}(x)\cdots a^{ t_m}_{ n+m}(x)\right)}{ \varphi(n)}=1\right\}.\end{ equation*}
Categories: math.NT