Periodicity of tiles in finite Abelian groups

TL;DR

This paper introduces the periodic tiling (PT) property for finite abelian groups, classifies cyclic groups with this property, and explores its implications for tiling and spectral sets.

Contribution

It provides a complete classification of cyclic groups with the PT property and identifies non-cyclic groups with this property, extending understanding beyond the Hajós property.

Findings

Cyclic groups with the PT property are fully classified.

Certain non-cyclic groups have the PT property but not the Hajós property.

For elementary p-groups with PT, tiles are coset representatives.

Abstract

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 Haj\'os 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 Haj\'os 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.