How likely can a point be in different Cantor sets

TL;DR

This paper studies the likelihood of a fixed point being in various Cantor sets parameterized by rom a class, revealing complex fractal structures and intersections with full Hausdorff dimension.

Contribution

It characterizes the parameter sets where a fixed point lies in Cantor sets, showing they are Cantor sets with full Hausdorff dimension and zero Lebesgue measure, and analyzes their intersections.

Findings

or each fixed point, the parameter set is a Cantor set with full Hausdorff dimension.

or any two points, the intersection of their parameter sets also has full Hausdorff dimension.

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 $x, y\in(0,1)$.