The sets of Dirichlet non-improvable numbers vs well-approximable numbers

Abstract

Let $\Psi :[1,\infty )\rightarrow \mathbb{R}_{+}$ be a non-decreasing function, $a_{n}(x)$ the $n$'{th} partial quotient of $x$ and $q_{n}(x)$ the denominator of the $n$'{th} convergent. The set of $\Psi $-Dirichlet non-improvable numbers \begin{equation*} G(\Psi ):=\Big\{x\in \lbrack 0,1):a_{n}(x)a_{n+1}(x)\,>\,\Psi \big(q_{n}(x) \big)\ \mathrm{for\ infinitely\ many}\ n\in \mathbb{N}\Big\}, \end{equation*} is related with the classical set of $1/q^{2}\Psi (q)$-approximable numbers $ \mathcal{K}(\Psi )$ in the sense that $\mathcal{K}(3\Psi )\subset G(\Psi )$. Both of these sets enjoy the same $s$-dimensional Hausdorff measure criterion for $s\in (0,1)$. We prove that the set $G(\Psi )\setminus \mathcal{K}(3\Psi )$ is uncountable by proving that its Hausdorff dimension is the same as that for the sets $\mathcal{K}(\Psi )$ and $G(\Psi)$. This gives an affirmative answer to a question raised by Hussain-Kleinbock-Wadleigh-Wang (2018).