The inverse problem for the lattice points

Abstract

Fix an positive integer $n$. Let $K\subseteq {\mathbb{R}}^n$ be a compact set such that $K+{\mathbb{Z}}^n={\mathbb{R}}^n$. We prove, via Algebraic Topology, that the integer points of the difference set of $K$, $(K-K)\cap {\mathbb{Z}}^n$, is not contained on the coordinate axes, $\mathbb{Z}\times \{0\}\times \dots \times \{0\}\cup \{0\}\times \mathbb{Z}\times \dots \times \{0\}\cup \dots \cup \{0\}\times \{0\}\times \dots \times \mathbb{Z}$. 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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• Achievable Sets in Z^n