Hausdorff dimension of Dirichlet non-improvable set versus well-approximable set

Abstract

AbstractDirichlet’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) $$ \begin{align*} \mathcal{K}(\Phi):=\{x:a_{n+1}(x)\ge\Phi(q_{n}(x))\ \textrm{for infinitely many }n\in \mathbb{N}\}; \end{align*} $$ the improvement of the uniform setting leads to the Dirichlet non-improvable set $$ \begin{align*} \mathcal{G}(\Phi):=\{x:a_{n}(x)a_{n+1}(x)\ge\Phi(q_{n}(x))\ \textrm{for infinitely many }n\in \mathbb{N}\}. \end{align*} $$ 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 $\Phi _{1},\Phi _{2}:[1,+\infty )\rightarrow \mathbb {R}^{+}$ be two non-decreasing positive functions. We focus on the Hausdorff dimension of the set $\mathcal {G}(\Phi _{1})\!\setminus\! \mathcal {K}(\Phi _{2})$ . It is known that the dimensions of $\mathcal {G}(\Phi )$ and $\mathcal {K}(\Phi )$ depend only on the growth exponent of $\Phi $ . However, rather different from the current knowledge, it will be seen in some cases that the dimension of $\mathcal {G}(\Phi _{1})\!\setminus\! \mathcal {K}(\Phi _{2})$ will change greatly even slightly modifying $\Phi _1$ by a constant.