Dirichlet’s theorem, including the uniform setting and asymptotic setting, is one of the most fundamental results in Diophantine approximation. The improvement of the asymptotic setting leads to the well-approximable set (in words of continued fractions)

the improvement of the uniform setting leads to the Dirichlet non-improvable set

\begin{align*} \mathcal{G}(\Phi):=\{x:a_{n}(x)a_{n+1}(x)\ge\Phi(q_{n}(x))\ \textrm{for infinitely many }n\in \mathbb{N}\}. \end{align*} $$\begin{align*} \mathcal{G}(\Phi):=\{x:a_{n}(x)a_{n+1}(x)\ge\Phi(q_{n}(x))\ \textrm{for infinitely many }n\in \mathbb{N}\}. \end{align*}$$

Surprisingly, as a proper subset of Dirichlet non-improvable set, the well-approximable set has the same s-Hausdorff measure as the Dirichlet non-improvable set. Nevertheless, one can imagine that these two sets should be very different from each other. Therefore, this paper is aimed at a detailed analysis on how the growth speed of the product of two-termed partial quotients affects the Hausdorff dimension compared with that of single-termed partial quotients. More precisely, let \Phi _{1},\Phi _{2}:[1,+\infty )\rightarrow \mathbb {R}^{+}$\Phi _{1},\Phi _{2}:[1,+\infty )\rightarrow \mathbb {R}^{+}$ be two non-decreasing positive functions. We focus on the Hausdorff dimension of the set \mathcal {G}(\Phi _{1})\!\setminus\! \mathcal {K}(\Phi _{2})$\mathcal {G}(\Phi _{1})\!\setminus\! \mathcal {K}(\Phi _{2})$ . It is known that the dimensions of \mathcal {G}(\Phi )$\mathcal {G}(\Phi )$ and \mathcal {K}(\Phi )$\mathcal {K}(\Phi )$ depend only on the growth exponent of \Phi $\Phi$ . However, rather different from the current knowledge, it will be seen in some cases that the dimension of \mathcal {G}(\Phi _{1})\!\setminus\! \mathcal {K}(\Phi _{2})$\mathcal {G}(\Phi _{1})\!\setminus\! \mathcal {K}(\Phi _{2})$ will change greatly even slightly modifying \Phi _1$\Phi _1$ by a constant.