The inverse problem for the lattice points

TL;DR

This paper uses algebraic topology to show that for certain compact sets covering the entire space with integer translations, the difference set's integer points are not confined to coordinate axes, addressing a question on lattice points.

Contribution

It proves a novel topological result about the distribution of lattice points in difference sets of specific compact sets, answering a question on relatively prime lattice points.

Findings

Integer points of the difference set are not on coordinate axes.

Provides a negative answer to a question by Hegarty and Nathanson.

Uses algebraic topology to analyze lattice point distributions.

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.