A problem of Steinhaus: Can all placements of a planar set contain exactly one lattice point?

Steinhaus asked whether there exists a subset of the plane which, no matter how translated and rotated, always contains exactly one point with integer coordinates. The best result on this problem so far has been that of Beck who showed, using harmonic analysis, that no such bounded measurable set exists. By simplifying Beck's method and by pushing it to its limits we improve his result. We show that if a set is measurable and there exists a direction such that the set is very small outside large strips parallel to the direction, then the set cannot have the property of Steinhaus.