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The sets of Dirichlet non-improvable numbers versus well-approximable numbers
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 40 / Issue 12 / December 2020
- Published online by Cambridge University Press:
- 27 June 2019, pp. 3217-3235
- Print publication:
- December 2020
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- Article
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Let
$\unicode[STIX]{x1D6F9}:[1,\infty )\rightarrow \mathbb{R}_{+}$ 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 
$\unicode[STIX]{x1D6F9}$-Dirichlet non-improvable numbers, is related with the classical set of
$$\begin{eqnarray}G(\unicode[STIX]{x1D6F9}):=\{x\in [0,1):a_{n}(x)a_{n+1}(x)>\unicode[STIX]{x1D6F9}(q_{n}(x))\text{ for infinitely many }n\in \mathbb{N}\},\end{eqnarray}$$
$1/q^{2}\unicode[STIX]{x1D6F9}(q)$-approximable numbers 
${\mathcal{K}}(\unicode[STIX]{x1D6F9})$ in the sense that 
${\mathcal{K}}(3\unicode[STIX]{x1D6F9})\subset G(\unicode[STIX]{x1D6F9})$. Both of these sets enjoy the same 
$s$-dimensional Hausdorff measure criterion for 
$s\in (0,1)$. We prove that the set 
$G(\unicode[STIX]{x1D6F9})\setminus {\mathcal{K}}(3\unicode[STIX]{x1D6F9})$ is uncountable by proving that its Hausdorff dimension is the same as that for the sets 
${\mathcal{K}}(\unicode[STIX]{x1D6F9})$ and 
$G(\unicode[STIX]{x1D6F9})$. This gives an affirmative answer to a question raised by Hussain et al [Hausdorff measure of sets of Dirichlet non-improvable numbers. Mathematika 64(2) (2018), 502–518].
