Solutions to the discrete Pompeiu problem and to the finite Steinhaus tiling problem

Let K be a nonempty finite subset of the Euclidean space Rk (k≥2). We prove that if a function f:Rk→C is such that the sum of f on every congruent copy of K is zero, then f vanishes everywhere. In fact, a stronger, weighted version is proved. As a corollary we find that every finite subset K of Rk having at least two elements is a Jackson set; that is, no subset of Rk intersects every congruent copy of K in exactly one point.