October 5, 2025Open Access

Spectrality of Prime Size Tiles

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A tile with prime size in $ ext{Z}^d$ is shown to be spectral, highlighting unique mathematical properties.
The proof employs contradiction to demonstrate that tiling complements cannot annihilate $p$-subgroups effectively.
A transformation reveals that $p$ points in general linear positions in $ ext{Z}^d$ must be both tiling and spectral when $d$ meets specific conditions.
This research has implications for understanding how prime number characteristics influence tiling dynamics in higher dimensions.

Implication

The analysis shows tiles with prime size exhibit spectral properties, indicating the behavior of tiling complements in subgroup structures.

Abstract

We prove that if a tile in Zᵈ has prime size p, then it must be spectral. The proof is by contradiction, it is simply shown that the tiling complement of such a tile can not annihilate all p-subgroups. In addition, with a simple transformation we prove that any p points in general linear positions in Zᵈ must be both tiling and spectral if d≥ p-1.

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Cite This Study

Weiqi Zhou (2025) studied this question.

synapsesocial.com/papers/68e25382d6d66a53c247478e https://doi.org/10.48550/arxiv.2509.23752

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Implication

The analysis shows tiles with prime size exhibit spectral properties, indicating the behavior of tiling complements in subgroup structures.

Spectrality of Prime Size Tiles

Key Points

Implication

Abstract

Discussion

Cite This Study

Spectrality of Prime Size Tiles

Key Points

Implication

Abstract

Discussion

Cite This Study

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