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

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.