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On a class of self-similar sets which contain finitely many common points
Published online by Cambridge University Press: 30 May 2024
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 $y_1,\,\ldots,\, y_p \in K_\lambda$
.
MSC classification
Information
- Type
- Research Article
- Information
- Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 156 , Issue 2 , April 2026 , pp. 543 - 564
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh
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