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

Let $[a_1(x),a_2(x),\ldots, a_n(x), \ldots]$ be the continued fraction expansion of an irrational number $x\in (0, 1)$. The study growth rate product consecutive partial quotients $a_n(x)a_{n+1}(x)$ is associated with improvements to Dirichlet's theorem (1842). We establish both weak and strong laws large numbers for sums $S_n(x)= \sum_{i=1}^n a_i(x)a_{i+1}(x)$ as well as, from a multifractal analysis point view, investigate its increasing rate. Specifically, we prove following results: \medskip \begin{itemize} \item For any $\epsilon>0$, Lebesgue measure set $$\left\{x\in(0, 1): \left|\frac{ S_n(x)}{n\log^2 n}-\frac1{2\log2}\right|\geq \epsilon\right\}$$tends zero $n$ infinity. almost all (0,1)$, $$\lim\limits_{n\rightarrow \infty} \frac{S_n(x)-\max\limits_{1\leq i \leq n}a_i(x)a_{i+1}(x)}{n\log^2n}=\frac{1}{2\log2}.$$ Hausdorff dimension $$E(\phi):=\left\{x\in(0,1):\lim\limits_{n\rightarrow \infty}\frac{S_n(x)}{\phi(n)}=1\right\}$$ determined range functions $\phi: \mathbb N\to R^+$. \end{itemize}