On Intersections of Cantor Sets: Self-Similarity

TL;DR

This paper investigates the self-similarity properties of intersections of Cantor sets under translation, revealing that certain self-similar structures occur only when the translation is rational, even for classical Cantor sets.

Contribution

It characterizes when intersections of Cantor sets exhibit self-similarity, linking this property to rational translations within a specific class of Cantor sets.

Findings

Self-similar intersections occur only for rational translations.

Initial segments of intersections have contraction ratios as powers of the original.

Results apply even to the classical middle thirds Cantor set.

Abstract

Let C be a Cantor set. For a real number t let C+t be the translate of C by t, We say two real numbers s,t are equivalent if the intersection of C and C+s is a translate of the intersection of C and C+t. We consider a class of Cantor sets determined by similarities with one fixed positive contraction ratio. For this class of Cantor set, we show that an "initial segment" of the intersection of C and C+t is a self-similar set with contraction ratios that are powers of the contraction ratio used to describe C as a self- similar set if and only if t is equivalent to a rational number. Our results are new even for the middle thirds Cantor set.