Mutual Annihilation of Tiles

TL;DR

This work investigates a finite-group analogue of the Fuglede conjecture by examining when a tiling complement of a subset A in $Z_n$ × $Z_n$ also serves as a spectral set, using the symplectic Fourier transform to study the geometry of the zero and difference sets. It establishes that the mutual annihilation between the difference set and the transform’s zero set persists in three structured regimes: (i) A is an order-$n$ subgroup, (ii) $n$ is prime with A complementing an order-$n$ subgroup, and (iii) $n=p^2$ with A complementing the non-cyclic order-$n$ subgroup; in each regime one component exhibits universality with respect to tiling complements or spectra. The analysis relies on a critical disjointness lemma and a counting lemma, all framed within symplectic time-frequency duality on finite groups. The results contribute precise criteria for when tiling and spectral properties align in finite abelian groups and offer tools for verifying universality in tiling–spectral configurations.

Abstract

Given $A\subset\mathbb Z_n^2$, the purpose of this article is to investigate when is the difference set $ΔA$ disjoint with the zero set of the Fourier transform of $A$. In the study of tiles in $\mathbb Z_n^2$, the author observed an interesting phenomenon that if $(A,B)$ is a tiling pair with $|A|=|B|$, then sometimes $(A,B)$ is also a spectral pair and vice versa. Moreover, in such cases actually one of the components would have universality (i.e., it is the universal spectrum/tiling complement for its tiling complements/spectra). It turns out that the disjointness is the critical property here, and shall be analyzed using the symplectic Fourier transform. Under such configuration it is shown that the phenomenon persists either (1) if $A$ is an order $n$ subgroup, or (2) if $n$ is a prime number and $A$ complements an order $n$ subgroup, or (3) if $n=p^2$, and $A$ complements the non-cyclic order $n$ subgroup. A side result binding number of elements in different subgroups is also given.

Figures (1)

• Figure 1: Left: Euclidean zero sets; $\quad$ Right: Symplectic zero sets

Theorems & Definitions (14)

• Example 1
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4: counting
• proof
• Lemma 5
• proof
• Lemma 6
• proof