Hausdorff dimension of a set in the theory of continued fractions

Abstract

In this article we calculate the Hausdorff dimension of the set \begin{equation*} \mathcal{F}(Φ)=\left\{ x\in \lbrack 0,1):\begin{aligned}a_{n+1}(x)a_n(x) \geq Φ(n) \ {\rm for \ infinitely \ many \ } n\in \mathbb N \ {\rm and } \\ a_{n+1}(x)< Φ(n) \ {\rm for \ all \ sufficiently \ large \ } n\in \mathbb N \end{aligned}\right\} \end{equation*} where $\Phi :\mathbb{N}\to (1,\infty )$ 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.

21 pages, preliminary version, any comments for improvements are appreciated