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On a class of self-similar sets which contain finitely many common points

Part of: Classical measure theory Elementary number theory Topological dynamics

Published online by Cambridge University Press:  30 May 2024

Kan Jiang ,
Derong Kong Open the ORCID record for Derong Kong [Opens in a new window] ,
Wenxia Li Open the ORCID record for Wenxia Li [Opens in a new window]  and
Zhiqiang Wang Open the ORCID record for Zhiqiang Wang [Opens in a new window]
Kan Jiang
Affiliation:
School of Mathematics and Statistics, Ningbo University, Ningbo 315211, People's Republic of China ([email protected])
Derong Kong
Affiliation:
College of Mathematics and Statistics, Center of Mathematics, Chongqing University, Chongqing 401331, People's Republic of China ([email protected])
Wenxia Li
Affiliation:
School of Mathematical Sciences, Key Laboratory of MEA (Ministry of Education) & Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, People's Republic of China ([email protected]; [email protected])
Zhiqiang Wang*
Affiliation:
School of Mathematical Sciences, Key Laboratory of MEA (Ministry of Education) & Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, People's Republic of China ([email protected]; [email protected])
*
*Corresponding author.
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Abstract

For $\lambda \in (0,\,1/2]$ let $K_\lambda \subset \mathbb {R}$ be a self-similar set generated by the iterated function system $\{\lambda x,\, \lambda x+1-\lambda \}$. Given $x\in (0,\,1/2)$, let $\Lambda (x)$ be the set of $\lambda \in (0,\,1/2]$ such that $x\in K_\lambda$. In this paper we show that $\Lambda (x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that for any $y_1,\,\ldots,\, y_p\in (0,\,1/2)$ there exists a full Hausdorff dimensional set of $\lambda \in (0,\,1/2]$ such that $y_1,\,\ldots,\, y_p \in K_\lambda$.

Keywords

Hausdorff dimensionthicknessself-similar setCantor setintersection

MSC classification

Primary: 28A78: Hausdorff and packing measures
Secondary: 28A80: Fractals 37B10: Symbolic dynamics 11A63: Radix representation; digital problems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 22
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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