Spectrality of Prime Size Tiles
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912953162268672 |
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| author | Zhou, Weiqi |
| author_facet | Zhou, Weiqi |
| contents | We prove that if a tile in $\mathbb Z^d$ 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 $\mathbb Z^d (d\ge p-1)$ must be both tiling and spectral. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_23752 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectrality of Prime Size Tiles Zhou, Weiqi Classical Analysis and ODEs Combinatorics 42A99, 05B45 We prove that if a tile in $\mathbb Z^d$ 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 $\mathbb Z^d (d\ge p-1)$ must be both tiling and spectral. |
| title | Spectrality of Prime Size Tiles |
| topic | Classical Analysis and ODEs Combinatorics 42A99, 05B45 |
| url | https://arxiv.org/abs/2509.23752 |