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

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.