  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.
