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

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  27 June 2019

AYREENA BAKHTAWAR ,
PHILIP BOS  and
MUMTAZ HUSSAIN
AYREENA BAKHTAWAR
Affiliation:
Department of Mathematics and Statistics, La Trobe University, PO Box 199, Bendigo3552, Australia email [email protected], [email protected], [email protected]
PHILIP BOS
Affiliation:
Department of Mathematics and Statistics, La Trobe University, PO Box 199, Bendigo3552, Australia email [email protected], [email protected], [email protected]
MUMTAZ HUSSAIN
Affiliation:
Department of Mathematics and Statistics, La Trobe University, PO Box 199, Bendigo3552, Australia email [email protected], [email protected], [email protected]
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Abstract

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,

$$\begin{eqnarray}G(\unicode[STIX]{x1D6F9}):=\{x\in [0,1):a_{n}(x)a_{n+1}(x)>\unicode[STIX]{x1D6F9}(q_{n}(x))\text{ for infinitely many }n\in \mathbb{N}\},\end{eqnarray}$$
is related with the classical set of $1/q^{2}\unicode[STIX]{x1D6F9}(q)$-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].

Keywords

uniform Diophantine approximationDirichlet’s theoremJarnik theoremHausdorff dimension

MSC classification

Primary: 11K50: Metric theory of continued fractions 11K60: Diophantine approximation 11J70: Continued fractions and generalizations 11J83: Metric theory
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 12 , December 2020 , pp. 3217 - 3235
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Copyright
© Cambridge University Press, 2019

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