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 $K_{\lambda}=\{\sum_{i=1}^\infty d_i\lambda^i: d_i\in\{0,1,\ldots, m-1\}, i\ge 1\}$. 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 $x, y\in(0,1)$.