Dirichlet non-improvable sets and partial quotient growth in continued fractions

Abstract

Dirichlet’s theorem is a fundamental result in metric Diophantine approximation. The refinement of its asymptotic setting yields the $\psi$-well-approximable set (via continued fractions)

and the refinement of its uniform setting leads to the $\psi$-Dirichlet non-improvable set

where $a_n(x)$ denotes the $n$-th partial quotient of $x$ and $q_n(x)$ is the denominator of the $n$-th convergent. Surprisingly, as a proper subset of $\mathcal{G}(\psi ),$ the set $\mathcal{K}(\psi )$ shares the same $s$-dimensional Hausdorff measure as $\mathcal{G}(\psi ).$ Furthermore, the difference set $\mathcal{G}(\psi )\setminus \mathcal{K}(\psi )$ has the same Hausdorff dimension as $\mathcal{K}(\psi )$. Therefore, to gain deeper insights into these results, this paper investigates how the growth rate of the product of partial quotients $a_n(x)a_{n+1}(x)$ affects the Hausdorff dimension relative to that of individual partial quotients $a_n(x)$. More precisely, for any $\alpha ,\beta ,\gamma \in [0,\infty )$, we investigate the Hausdorff dimension of the intersection of

and