Stable intersection of middle-$\alpha$ Cantor sets

TL;DR

This paper introduces a pair of middle-$\alpha$ Cantor sets with stable intersection despite their thickness product being less than one, and shows their arithmetic difference contains intervals for all nonzero scalars.

Contribution

It demonstrates the existence of middle-$\alpha$ Cantor sets with stable intersection under conditions previously thought to prevent it, and analyzes their arithmetic differences.

Findings

Stable intersection occurs even when the product of thicknesses is less than one.

The difference set $C_\alpha - \lambda C_\beta$ contains an interval for all nonzero $\lambda$.

Provides new insights into the structure of Cantor sets and their intersections.

Abstract

In the present paper, We introduce a pair of middle Cantor sets namely $(C_\alpha, C_\beta)$ having stable intersection, while the product of their thickness is smaller than one. Furthermore, the arithmetic difference $C_\alpha- \lambda C_\beta$ contains at least one interval for each nonzero number $\lambda$.