On the arithmetic difference of middle Cantor sets

TL;DR

This paper characterizes the arithmetic differences of middle Cantor sets, explores conditions for these differences to contain intervals or have zero measure, and introduces a new stability concept related to their intersections.

Contribution

It provides a comprehensive characterization of differences of middle Cantor sets, introduces a new stability notion, and analyzes the structure of these differences under various transformations.

Findings

Characterization of triples where $C_eta - rac{1}{eta} C_eta$ equals an interval

Identification of dense subsets where differences contain intervals or have zero measure

Existence of a new stability in the intersection of regular Cantor sets

Abstract

Suppose that $\mathcal{C}$ is the space of all middle Cantor sets. We characterize all triples $(\alpha,~\beta,~\lambda)\in \mathcal{C}\times\mathcal{C}\times \mathbb{R}^*$ that satisfy $C_\alpha- \lambda C_\beta=[-\lambda,~1]. $ Also all triples (that are dense in $\mathcal{C}\times\mathcal{C}\times \mathbb{R}^*$) has been determined such that $C_\alpha- \lambda C_\beta$ forms the attractor of an iterated function system. Then we found a new family of the pair of middle Cantor sets $\mathcal{P}$ in a way that for each $(C_\alpha,~ C_\beta)\in\mathcal{P}$, there exists a dense subfield $F\subset \mathbb{R}$ such that for each $\mu \in F$, the set $C_\alpha- \mu C_\beta$ contains an interval or has zero Lebesgue measure. In sequel, conditions on the functions $f, ~g$ and the pair $(C_\alpha,~C_\beta)$ is provided which $f(C_{\alpha})- g(C_{\beta})$ contains an interval. This leads us to denote another type of stability in the intersection of two Cantor sets. We prove the existence of this stability for regular Cantor sets that have stable intersection and its absence for those which the sum of their Hausdorff dimension is less than one. At the end, special middle Cantor sets $C_\alpha$ and $C_\beta$ are introduced. Then the iterated function system corresponding to the attractor $C_{\alpha}-\frac{2\alpha}{\beta}C_\beta$ is characterized. Some specifications of the attractor has been presented that keep our example as an exception. We also show that $\sqrt{C_{\alpha}}$ - $\sqrt{C_{\beta}}$ contains at least one interval.