Periodicity of tiles in finite Abelian groups
Abstract
In this paper, we introduce the concept of periodic tiling (PT) property for finite abelian groups. A group has the PT property if any non-periodic set that tiles the group by translation has a periodic tiling complement. This property extends the scope beyond groups with the Hajós property. We classify all cyclic groups having the PT property. Additionally, we construct groups that possess the PT property but without the Hajós property. As byproduct, we identify new groups for which the implication ``Tile $\Longrightarrow$ Spectral" holds. For elementary $p$-groups having the PT property, we show that a tile must be a complete set of representatives of the cosets of some subgroup, by analyzing the structure of tiles.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2024
- DOI:
- 10.48550/arXiv.2408.12901
- arXiv:
- arXiv:2408.12901
- Bibcode:
- 2024arXiv240812901F
- Keywords:
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- Mathematics - Group Theory;
- Mathematics - Classical Analysis and ODEs;
- Mathematics - Combinatorics
- E-Print:
- 31 pages