On Tiling and Spectral Sets in $\mathbb Z_{p^2}\times\mathbb Z_{p^2}$

TL;DR

This paper proves that in the group rac{Z}{p^2} imesrac{Z}{p^2}, tiling and spectral sets are equivalent by using symplectic Fourier analysis, extending understanding of these sets in finite abelian groups.

Contribution

It introduces the use of symplectic Fourier transform to analyze tiling and spectral sets, establishing their equivalence in rac{Z}{p^2} imesrac{Z}{p^2} and providing auxiliary results for related group sizes.

Findings

Tiling and spectral sets coincide in rac{Z}{p^2} imesrac{Z}{p^2}.

Symplectic Fourier transform offers more flexibility in analysis.

Auxiliary results for sets of sizes p and p^{2m-1} are provided.

Abstract

Let $p$ be a prime number, it is shown that tiling and spectral sets coincide in $\mathbb Z_{p^2}\times\mathbb Z_{p^2}$ by considering equivalently symplectic spectral pairs. The main approach is still to analyze the zero set of the Fourier transform. The zero set of the symplectic Fourier transform differs from the zero set of the usual Fourier transform by an orthogonal rotation, but using the symplectic Fourier transform allows more freedom when applying change of basis. Some auxiliary results concerning tiling sets and spectral sets of sizes $p$ and $p^{2m-1}$ in $\mathbb Z_{p^m}\times\mathbb Z_{p^m}$ are also presented.