a1 Department of Mathematics, University of Crete, 714 09 Iraklion, Crete, Greece. e-mail: firstname.lastname@example.org
a2 Department of Mathematics, 253-37 Caltech, Pasadena, CA 91125, USA. e-mail: email@example.com
Several results are proved related to a question of Steinhaus: is there a set E2 such that the image of E under each rigid motion of IR2 contains exactly one lattice point? Assuming measurability, the analogous question in higher dimensions is answered in the negative, and on the known partial results in the two dimensional case are improved on. Also considered is a related problem involving finite sets of rotations.
(Received November 03 1997)