Measurable Steinhaus sets do not exist for finite sets or the integers in the plane
Mihail N. Kolountzakis, Michael Papadimitrakis · Bulletin of the London Mathematical Society · 2017
A Steinhaus set S ⊆ R d for a set A ⊆ R d is a set such that S has exactly one point in common with τ A , for every rigid motion τ of R d . We show here that if A is a finite set of at least two points then there is no such set S which is Lebesgue measurable. An old result of Komjáth says that there exists a Steinhaus set for A = Z × 0 in R 2 . We also show here that such a set cannot be Lebesgue measurable.