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Hausdorff dimension of Dirichlet non-improvable set versus well-approximable set
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 43 / Issue 8 / August 2023
- Published online by Cambridge University Press:
- 04 August 2022, pp. 2707-2731
- Print publication:
- August 2023
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- Article
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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
$$ \begin{align*} \mathcal{K}(\Phi):=\{x:a_{n+1}(x)\ge\Phi(q_{n}(x))\ \textrm{for infinitely many }n\in \mathbb{N}\}; \end{align*} $$
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
$$ \begin{align*} \mathcal{G}(\Phi):=\{x:a_{n}(x)a_{n+1}(x)\ge\Phi(q_{n}(x))\ \textrm{for infinitely many }n\in \mathbb{N}\}. \end{align*} $$
$\Phi _{1},\Phi _{2}:[1,+\infty )\rightarrow \mathbb {R}^{+}$
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.
