Measurable Steinhaus sets do not exist for finite sets or the integers in the plane

Mihail N. Kolountzakis, Michael Papadimitrakis

A Steinhaus set $S \subseteq \RR^d$ for a set $A \subseteq \RR^d$ is a set such that $S$ has exactly one point in common with $\tau A$, for every rigid motion $\tau$ of $\RR^d$. We show h…