有限阿贝尔群中瓦片的周期性

Shilei Fan, Tao Zhang
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引用次数: 0

摘要

本文介绍了无穷无边群的周期性平铺(PT)属性概念。如果通过平移使群平铺的任何非周期性集合都有周期性平铺补集,那么这个群就具有 PT 特性。这一性质扩展了具有 Haj\'os 性质的群的范围。我们对具有 PT 性质的所有循环群进行了分类。此外,我们还构造了具有 PT 属性但不具有 Haj\'os 属性的群。作为副产品,我们识别出了蕴涵 "瓦片/长直箭/光谱 "成立的新群。对于具有 PT 特性的基本 $p$ 群,我们通过分析瓦片的结构,证明瓦片必须是某个子群余集代表的完整集合。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Periodicity of tiles in finite Abelian groups
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 Haj\'os property. We classify all cyclic groups having the PT property. Additionally, we construct groups that possess the PT property but without the Haj\'os 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.
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