Hausdorff dimension of a set in the theory of continued fractions

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 $\mathrm{\Phi }:\mathbb{N}\to (1,\mathrm{\infty })$ 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.