Intersections of middle-$\alpha$ Cantor sets with a fixed translation

TL;DR

This paper investigates the intersection properties of middle-$eta$ Cantor sets with a fixed translation, revealing a complex fractal structure of parameters where intersections occur and their dimensional characteristics.

Contribution

It characterizes the set of parameters where Cantor sets intersect under translation, computes local dimensions, and describes the structure of level sets with specific intersection dimensions.

Findings

$\Lambda(t)$ is a topological Cantor set with zero Lebesgue measure and full Hausdorff dimension.

The local dimension of $\Lambda(t)$ is explicitly calculated, demonstrating a dimensional variation principle.

The dimension of level sets $\Lambda_eta(t)$ is explicitly determined, and the set where Hausdorff and packing dimensions differ has full Hausdorff dimension.

Abstract

For $\lambda\in(0,1/3]$ let $C_\lambda$ be the middle-$(1-2\lambda)$ Cantor set in $\mathbb R$. Given $t\in[-1,1]$, excluding the trivial case we show that \[ \Lambda(t):=\left\{\lambda\in(0,1/3]: C_\lambda\cap(C_\lambda+t)\ne\emptyset\right\} \] is a topological Cantor set with zero Lebesgue measure and full Hausdorff dimension. In particular, we calculate the local dimension of $\Lambda(t)$, which reveals a dimensional variation principle. Furthermore, for any $\beta\in[0,1]$ we show that the level set \[ \Lambda_\beta(t):=\left\{\lambda\in\Lambda(t): \dim_H(C_\lambda\cap(C_\lambda+t))=\dim_P(C_\lambda\cap(C_\lambda+t))=\beta\frac{\log 2}{-\log \lambda}\right\} \] has equal Hausdorff and packing dimension $(-\beta\log\beta-(1-\beta)\log\frac{1-\beta}{2})/\log 3$. We also show that the set of $\lambda\in\Lambda(t)$ for which $\dim_H(C_\lambda\cap(C_\lambda+t))\ne\dim_P(C_\lambda\cap(C_\lambda+t))$ has full Hausdorff dimension.