Limit theorems for sums of products of consecutive partial quotients of continued fractions

Abstract

Let $[a_1(x),a_2(x),\dots ,a_n(x),\dots ]$ be the continued fraction expansion of an irrational number $x\in (0,1)$. The study of the growth rate of the product of consecutive partial quotients $a_n(x)a_{n+1}(x)$ is associated with the improvements to Dirichlet's theorem (1842). We establish both the weak and strong laws of large numbers for the partial sums $S_n(x)={\sum }_{i=1}^n{a}_i(x)a_{i+1}(x)$ as well as, from a multifractal analysis point of view, investigate its increasing rate. Specifically, we prove the following results: \medskip \begin{itemize} \item For any $\epsilon >0$, the Lebesgue measure of the set $$\left\{x\in (0,1):\left|\frac{S_n(x)}{n{\log }^{2}n}-\frac{1}{2\log 2}\right|\ge \epsilon \right\}$$tends to zero as $n$ to infinity. \item For Lebesgue almost all $x\in (0,1)$, $$\underset{n\to \infty }{\lim }\frac{S_n(x)-\underset{1\le i\le n}{\max }{a}_i(x)a_{i+1}(x)}{n{\log }^{2}n}=\frac{1}{2\log 2}.$$ \item The Hausdorff dimension of the set $$E(\varphi ):=\left\{x\in (0,1):\underset{n\to \infty }{\lim }\frac{S_n(x)}{\varphi (n)}=1\right\}$$ is determined for a range of increasing functions $\varphi :\mathbb{N}\to {\mathbb{R}}^{+}$. \end{itemize}

27 pages