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