A note on an inverse problem for lattice points

Let K⊆R2 be a compact set such that K+Z2=R2 and let K−K={a−b|a,b∈K}. We prove, via Algebraic Topology, that the integer points of the difference set of K, (K−K)∩Z2, is not contained on the coordinate axes, Z×{0}∪{0}×Z. This result implies the negative answer to the inverse problem posed by M.B. Nathanson (2010) [5].