A note on an inverse problem for lattice points

Let K ⊆ R 2 be a compact set such that K + Z 2 = R 2 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 ) ∩ Z 2 , 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].