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 $a_n^{t_0}(x)a_{n+1}^{t_1}(x)\cdots a_{n+m}^{t_m}(x)$ ,where $t_i\in {\mathbb{R}}_{+}(0\le i\le m)$ are weights. More precisely, let $\phi :\mathbb{N}\to {\mathbb{R}}_{+}$ be a function with $\phi (n)/n\to \infty$ as $n\to \infty$. For any $(t_0,\cdots ,t_m)\in {\mathbb{R}}_{+}^{m+1}$ with $t_i\ge 0$ and at least one $t_i\ne 0(0\le i\le m)$, the Hausdorff dimension of the set $$\underline{E}(\{t_i{\}}_{i=0}^m,\phi )=\left\{x\in [0,1):\underset{n\to \infty }{\mathrm{liminf}}\frac{\log \left(a_n^{t_0}(x)a_{n+1}^{t_1}(x)\cdots a_{n+m}^{t_m}(x)\right)}{\phi (n)}=1\right\}$$ is obtained. Under the condition that $(t_0,\cdots ,t_m)\in {\mathbb{R}}_{+}^{m+1}$ with $0<t_0\le t_1\le \cdots \le t_m$, we also obtain the Hausdorff dimension of the set \begin{equation*} \overline{E}(\{t_i\}_{i=0}^m,φ)=\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)}{φ(n)}=1\right\}.\end{equation*}

References in corpus (1)

• Metrical properties for the weighted products of multiple partial quotients in continued fractions