On Intersections of Cantor Sets: Self-Similarity

Abstract

Let $C$ be a Cantor subset of the real line. For a real number $t$, let $C+t$ be the translate of $C$ by $t$. We say two real numbers $s,t$ are translation 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 selfsimilar set if and only if $t$ is translation equivalent to a rational number. Many of our results are new even for the middle thirds Cantor set.