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arXiv · 2412.16865

Mutual Annihilation of Tiles

Weiqi Zhou
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Abstract

We prove that if the zero set of the Fourier transform of $A\subseteq\mathbb Z_n\times\mathbb Z_n$ contains an element of prime power order, then there is an equi-distribution relation in subsets of $A$ with respect to certain hyperplanes. With this we further show that if $A$ is a tiling complement of the subgroup generated by $(p,0)$ and $(0,p)$ in $\mathbb Z_{p^m}\times\mathbb Z_{p^m}$, then the zero set of its Fourier transform is disjoint with the orthogonal rotation of $A$. These results are motivated by a casual observation in $\mathbb Z_{p^2}\times\mathbb Z_{p^2}$.

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Weiqi Zhou. 2024-12-22. Mutual Annihilation of Tiles. https://arxiv.org/abs/2412.16865

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