Mutual Annihilation of Tiles

We prove that if the zero set of the Fourier transform of $A\subseteq {\mathbb{Z}}_n\times {\mathbb{Z}}_n$ contains an element of prime power order, then there is an equi-distribution relation in subsets of $A$ with respect to certain hyperplanes. With this we further show that if $A$ is a tiling complement of the subgroup generated by $(p,0)$ and $(0,p)$ in ${\mathbb{Z}}_{p^m}\times {\mathbb{Z}}_{p^m}$, then the zero set of its Fourier transform is disjoint with the orthogonal rotation of $A$. These results are motivated by a casual observation in ${\mathbb{Z}}_{p^2}\times {\mathbb{Z}}_{p^2}$.