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

Let $K$ be a nonempty finite subset of the Euclidean space ${\mathbb{R}}^k$ $(k\ge 2)$. We prove that if a function $f\text{\hspace{0.17em}⁣}:{\mathbb{R}}^k\to \mathbb{C}$ 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 ${\mathbb{R}}^k$ having at least two elements is a Jackson set; that is, no subset of ${\mathbb{R}}^k$ intersects every congruent copy of $K$ in exactly one point.