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A DICHOTOMY LAW FOR THE DIOPHANTINE PROPERTIES IN $\unicode[STIX]{x1D6FD}$-DYNAMICAL SYSTEMS

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

Published online by Cambridge University Press:  16 May 2016

Michael Coons ,
Mumtaz Hussain  and
Bao-Wei Wang
Michael Coons
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia email michael.coons@newcastle.edu.au
Mumtaz Hussain
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia email mumtaz.hussain@newcastle.edu.au, drhussainmumtaz@gmail.com
Bao-Wei Wang
Affiliation:
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China email bwei_wang@hust.edu.cn
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Abstract

Let $\unicode[STIX]{x1D6FD}>1$ be a real number and define the $\unicode[STIX]{x1D6FD}$-transformation on $[0,1]$ by $T_{\unicode[STIX]{x1D6FD}}:x\mapsto \unicode[STIX]{x1D6FD}x\,\text{mod}\,1$. Further, define

$$\begin{eqnarray}W_{y}(T_{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6F9}):=\{x\in [0,1]:|T_{\unicode[STIX]{x1D6FD}}^{n}x-y|<\unicode[STIX]{x1D6F9}(n)\text{ for infinitely many }n\}\end{eqnarray}$$
and
$$\begin{eqnarray}W(T_{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6F9}):=\{(x,y)\in [0,1]^{2}:|T_{\unicode[STIX]{x1D6FD}}^{n}x-y|<\unicode[STIX]{x1D6F9}(n)\text{ for infinitely many }n\},\end{eqnarray}$$
 where $\unicode[STIX]{x1D6F9}:\mathbb{N}\rightarrow \mathbb{R}_{{>}0}$ is a positive function such that $\unicode[STIX]{x1D6F9}(n)\rightarrow 0$ as $n\rightarrow \infty$. In this paper, we show that each of the above sets obeys a Jarník-type dichotomy, that is, the generalized Hausdorff measure is either zero or full depending upon the convergence or divergence of a certain series. This work completes the metrical theory of these sets.

MSC classification

Primary: 11J83: Metric theory
Secondary: 11K60: Diophantine approximation 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
Type
Research Article
Information
Mathematika , Volume 62 , Issue 3 , 2016 , pp. 884 - 897
Copyright
Copyright © University College London 2016 

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