Abstract

The Fuglede conjecture establishes the relationship between spectral and tilings. Spectral is one of the core concepts in harmonic analysis. Since tilings are relatively easy to verify while spectral are more difficult, researchers are particularly interested in what kind of tiling sets are spectral sets (T-S for short). We mainly focus on the structure of tiling sets in finite abelian groups. Many researchers have considered tiling sets with special structures in finite abelian groups, such as good groups, groups with Redei property, and quasi-periodic groups. However, there are only a few good groups and groups with Redei property. Additionally, there are groups with quasi-periodic properties that do not have T-S, and there are groups with T-S that do not have quasi-periodic properties. Therefore, it is necessary to propose a more suitable property that is easy to verify, has a good structure, and satisfies T-S.