Let $ [a_1(x),a_2(x),\ldots,a_n(x),\ldots] $ be the continued fraction expansion of $ x\in [0,1) $ . The study of real numbers with their partial quotients obeying various forms leads to the following set $ \begin{equation*} \mathcal{E}_1(\Phi): = \left\{ x\in[0,1)\colon\ a_{n+1}(x)\geq\Phi(n)\; \text{for infinitely many}\ n\in\mathbb{N} \right\}. \end{equation*} $ The second largest product of consecutive partial quotients will be the obstacle to obtain the limit theorems of the sum of products of consecutive partial quotients in continued fractions, which leads to the following set $ \begin{...
