Mutual Annihilation of Tiles

TL;DR

This paper investigates the Fourier transform properties of subsets in finite abelian groups, revealing conditions for equidistribution and disjointness of zero sets, especially in the context of tiling complements in groups of prime power order.

Contribution

It establishes new Fourier-analytic conditions for tiling complements in groups of prime power order, linking zero sets to equidistribution and disjointness properties.

Findings

Zero set of Fourier transform contains prime power order elements implies equidistribution.

Tiling complements in groups of prime power order have disjoint zero sets and orthogonal rotations.

Results are motivated by observations in $Z_{p^2} imesZ_{p^2}$.

Abstract

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 $\mathbb Z_{p^2}\times\mathbb Z_{p^2}$.