In this article we calculate the Hausdorff dimension of the set \begin{ equation*} \mathcal{ F}(\Phi )=\left\{ x\in \lbrack 0,1):\begin{ aligned}a_{ n+1}(x)a_n(x) \geq \Phi(n) \ { \rm for \ infinitely \ many \ } n\in \mathbb N \ { \rm and } \\ a_{ n+1}(x)< \Phi(n) \ { \rm for \ all \ sufficiently \ large \ } n\in \mathbb N \end{ aligned}\right\} \end{ equation*} where $\Phi :\mathbb{ N}\rightarrow (1,\infty)$ is any function with $\lim_{ n\to \infty} \Phi(n)=\infty.$ This in turn contributes to the metrical theory of continued fractions as well as gives insights about the set of Dirichlet non-improvable numbers.
Categories: math.DS math.NT