A note on the inverse problem for the lattice points

Abstract

Let $K\subseteq {\mathbb{R}}^2$ be a compact set such that $K+{\mathbb{Z}}^2={\mathbb{R}}^2$. We prove, via Algebraic Topology, that the integer points of the difference set of $K$, $(K-K)\cap {\mathbb{Z}}^2$, is not contained on the coordinate axes, $\mathbb{Z}\times \{0\}\cup \mathbb{Z}\times \{0\}$. This result gives a negative answer to a question posed by P. Hegarty and M. Nathanson on relatively prime lattice points.

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