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, $\mathbb{Z}\times\{0\}\times\ldots\times\{0\}\cup\{0\}\times\mathbb{Z}\times\ldots\times\{0\}\cup\ldots\cup\{0\}\times\{0\}\times\ldots\times\mathbb{Z}$. This result gives a negative answer to a question posed by P. Hegarty and M. Nathanson on relatively prime lattice points.