2024/08/23 by Shilei Fan, Tao Zhang, Fan, Shilei +1 · 1 citation
Computer Science · Materials Science · Mathematics · #Cellular Automata and Applications #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Quasicrystal Structures and Properties #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2408.12901
openalex publication_date 2024/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we introduce the periodic tiling (PT) property for finite abelian groups. A finite abelian group is said to have the PT property if every non-periodic set that tiles the group by translation admits a periodic tiling complement. This notion extends the scope beyond groups with the Hajós property. We give a complete classification of cyclic groups possessing the PT property and identify certain non-cyclic groups that enjoy the PT property but fail to satisfy the Hajós property.. As a byproduct, we obtain new families of groups for which the implication ``Tile \Longrightarrow Spectral" holds. Furthermore, for elementary p-groups with the PT property, by analyzing the structure of tiles, we prove that every tile is a complete set of representatives of the cosets of some subgroup.