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SIMULTANEOUS DYNAMICAL DIOPHANTINE APPROXIMATION IN BETA EXPANSIONS

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Classical measure theory

Published online by Cambridge University Press:  08 January 2020

WEILIANG WANG Open the ORCID record for WEILIANG WANG [Opens in a new window]  and
LU LI Open the ORCID record for LU LI [Opens in a new window]
WEILIANG WANG*
Affiliation:
School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, China Department of Mathematics, West Anhui University, Luan, Anhui 237012, China email [email protected]
LU LI
Affiliation:
Department of Mathematics, West Anhui University, Luan, Anhui 237012, China email [email protected]
*
✉[email protected]
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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 $T_{\unicode[STIX]{x1D6FD}}:x\mapsto \unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$. Let $f:[0,1]\rightarrow [0,1]$ and $g:[0,1]\rightarrow [0,1]$ be two Lipschitz functions. The main result of the paper is the determination of the Hausdorff dimension of the set

$$\begin{eqnarray}W(f,g,\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2})=\big\{(x,y)\in [0,1]^{2}:|T_{\unicode[STIX]{x1D6FD}}^{n}x-f(x)|<\unicode[STIX]{x1D6FD}^{-n\unicode[STIX]{x1D70F}_{1}(x)},|T_{\unicode[STIX]{x1D6FD}}^{n}y-g(y)|<\unicode[STIX]{x1D6FD}^{-n\unicode[STIX]{x1D70F}_{2}(y)}~\text{for infinitely many}~n\in \mathbb{N}\big\},\end{eqnarray}$$
where $\unicode[STIX]{x1D70F}_{1}$, $\unicode[STIX]{x1D70F}_{2}$ are two positive continuous functions with $\unicode[STIX]{x1D70F}_{1}(x)\leq \unicode[STIX]{x1D70F}_{2}(y)$ for all $x,y\in [0,1]$.

Keywords

beta expansionsDiophantine approximationHausdorff dimension

MSC classification

Primary: 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Secondary: 28A80: Fractals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 2 , October 2020 , pp. 186 - 195
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

The work is supported by the Provincial Natural Science Research Project of Anhui Colleges (grant KJ2019A0672).

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