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

TL;DR

This paper investigates functions that can tile the p-adic field dic numbers through translation, establishing their local constancy, and connects these tiling properties to spectral sets and the Fuglede conjecture.

Contribution

It characterizes tiling functions in dics, provides conditions for partitions of unity, and links tiling to spectral properties in p-adic spaces.

Findings

Functions tiling dics are uniformly locally constant.

Necessary and sufficient conditions for discrete sets to form partitions of unity.

Tiles in dics dics dics are spectral sets.

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.