Finding integral diagonal pairs in a two dimensional $\mathcal{N}$--set

TL;DR

This paper proves that in two-dimensional $ $-sets, there always exist distinct points whose difference is an integral vector not aligned with the axes, answering a question about the structure of such sets.

Contribution

The paper establishes that every two-dimensional $ $-set contains pairs of points with integral differences that are neither horizontal nor vertical, resolving a previously open question.

Findings

Existence of non-axis-aligned integral pairs in 2D $ $-sets

Answer to a question posed by Hegarty and Nathanson

Characterization of point differences in $ $-sets

Abstract

According to [1] an $n$-dimensional $\mathcal{N}$--set is a compact subset $A$ of $\mathbb{R}^n$ such that for every $x$ in $\mathbb{R}^n$ there is $y$ in $A$ with $y-x$ in $\mathbb{Z}^n$. We prove that every two dimensional $\mathcal{N}$--set $A$ must contain distinct points $x,y$ such that $x-y$ is in $\mathbb{Z}^2$ and $x-y$ is neither horizontal nor vertical. This answers a question of P. Hegarty and M. Nathanson.