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

Abstract:

According to Nathanson, 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.