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

TL;DR

This work proves that tiling and spectral sets coincide in the finite group $\mathbb Z_{p^2}\times\mathbb Z_{p^2}$ by recasting the problem in terms of symplectic spectral pairs and analyzing the corresponding zero sets. The authors develop a detailed toolbox based on the symplectic form, related transforms, and a refined subgroup structure of $\mathbb Z_{p^2}\times\mathbb Z_{p^2}$, including the families $K_k$, $H_{j,k}$, and $E_{j,k}$, to control tilings and spectra. They establish auxiliary results for sizes $|A|=p$ and $|A|=p^{2m-1}$, and prove that nontrivial spectral sets of intermediate sizes must tile, ultimately yielding the equivalence for all subsets. A weak periodic tiling property is also obtained, showing that in certain tiling configurations one component can be replaced by a periodic set without breaking the tiling relation. These results contribute a discrete analogue to Fuglede-type questions and offer insights into time-frequency structures in finite p-groups.

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. Symplectic structures appear naturally in time-frequency analysis and provides a perspective to reveal patterns that may not be so evident in the Euclidean setting. The main approach here is however still to count the size of the zero set and analyze its contents. 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.

Figures (2)

• Figure 1: Lagrangians in $\mathbb Z_4\times\mathbb Z_4$
• Figure 2: $K_k$ and $E_{j,k}$ in $\mathbb Z_4\times\mathbb Z_4$

Theorems & Definitions (31)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5: counting
• Lemma 6
• proof
• Lemma 7
• proof
• Lemma 8