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arXiv · 2412.03834

Tiling the field $\mathbb{Q}_p$ of $p$-adic numbers by a function

Shilei Fan
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Abstract

This study explores the properties of the function which can tile the field $\mathbb{Q}_p$ of $p$-adic numbers by translation. It is established that functions capable of tiling $\mathbb{Q}_p$ is by translation uniformly locally constancy. As an application, in the field $\mathbb{Q}_p$, we addressed the question posed by H. Leptin and D. M\"uller, providing the necessary and sufficient conditions for a discrete set to correspond to a uniform partition of unity. The study also connects these tiling properties to the Fuglede conjecture, which states that a measurable set is a tile if and only if it is spectral. The paper concludes by characterizing the structure of tiles in \(\mathbb{Q}_p \times \mathbb{Z}/2\mathbb{Z}\), proving that they are spectral sets.

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Shilei Fan. 2024-12-05. Tiling the field $\mathbb{Q}_p$ of $p$-adic numbers by a function. https://arxiv.org/abs/2412.03834

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