Increasing rate of weighted product of partial quotients in continued fractions

TL;DR

This paper investigates the growth rate of weighted products of partial quotients in continued fractions, determining the Hausdorff dimension of sets characterized by specific liminf and limsup growth conditions under certain weight and function constraints.

Contribution

It provides explicit formulas for the Hausdorff dimension of sets where weighted products of partial quotients grow at a specified rate, extending understanding of metric properties of continued fractions.

Findings

Hausdorff dimension of sets with specified liminf growth rate derived

Hausdorff dimension of sets with specified limsup growth rate derived

Results depend on weights and growth functions satisfying certain conditions

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*}