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

Abstract

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