Increasing rate of weighted product of partial quotients in continued fractions

Abstract

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*}