Periodicity of tiles in finite Abelian groups
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914047647023104 |
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| author | Fan, Shilei Zhang, Tao |
| author_facet | Fan, Shilei Zhang, Tao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12901 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Periodicity of tiles in finite Abelian groups Fan, Shilei Zhang, Tao Group Theory Classical Analysis and ODEs Combinatorics 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. |
| title | Periodicity of tiles in finite Abelian groups |
| topic | Group Theory Classical Analysis and ODEs Combinatorics |
| url | https://arxiv.org/abs/2408.12901 |