How likely can a point be in different Cantor sets

Abstract

Let $m\in {\mathbb{N}}_{\ge 2}$, and let $\mathcal{K}=\{K_{\lambda }:\lambda \in (0,1/m]\}$ be a class of Cantor sets, where $K_{\lambda }=\{{\sum }_{i=1}^{\infty }{d}_i{\lambda }^i:d_i\in \{0,1,\dots ,m-1\},i\ge 1\}$. We investigate in this paper the likelyhood of a fixed point in the Cantor sets of $\mathcal{K}$. More precisely, for a fixed point $x\in (0,1)$ we consider the parameter set $\Lambda (x)=\{\lambda \in (0,1/m]:x\in K_{\lambda }\}$, and show that $\Lambda (x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, by constructing a sequence of Cantor subsets with large thickness in $\Lambda (x)$ we prove that the intersection $\Lambda (x)\cap \Lambda (y)$ also has full Hausdorff dimension for any $x,y\in (0,1)$.

29 pages, 5 figures. We added some statements in the abstract and the introduction

Cited by in corpus (1)

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