A note on the inverse problem for the lattice points

TL;DR

This paper proves using algebraic topology that for certain compact sets covering the plane with integer translations, the difference set's lattice points are not confined to the axes, answering a question on relatively prime lattice points.

Contribution

It establishes a topological result about the distribution of lattice points in difference sets, providing a negative answer to a previously open question.

Findings

Difference set lattice points are not on the coordinate axes.

The result applies to sets covering the plane via integer translations.

It uses algebraic topology to prove the main theorem.

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.