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

Shilei Fan

TL;DR

This work analyzes tiling of the $p$-adic field $\mathbb{Q}_p$ by translations and establishes that any tiling function must be uniformly locally constant. It connects tiling by a set and by a function to uniform partitions of unity and to the Fuglede conjecture, proving a precise density relation for discrete tiling measures and providing a complete characterization of tiling structures in $\mathbb{Q}_p\times\mathbb{Z}/2\mathbb Z$ with spectrality results. The paper develops a robust $p$-adic harmonic-analytic framework using Bruhat-Schwartz distributions, Fourier analysis, and finite-group tiling theory to obtain both local and global tiling descriptions, including $p$-homogeneous tile structures and the spectral properties of tiles. Overall, it extends Fuglede-type tiling results to the infinite abelian group $\mathbb{Q}_p$ and an associated product, and it delivers detailed tile classifications across finite and $p$-adic settings that inform spectrality and partition structures in these groups.

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. Müller, 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.

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

TL;DR

This work analyzes tiling of the -adic field by translations and establishes that any tiling function must be uniformly locally constant. It connects tiling by a set and by a function to uniform partitions of unity and to the Fuglede conjecture, proving a precise density relation for discrete tiling measures and providing a complete characterization of tiling structures in with spectrality results. The paper develops a robust -adic harmonic-analytic framework using Bruhat-Schwartz distributions, Fourier analysis, and finite-group tiling theory to obtain both local and global tiling descriptions, including -homogeneous tile structures and the spectral properties of tiles. Overall, it extends Fuglede-type tiling results to the infinite abelian group and an associated product, and it delivers detailed tile classifications across finite and -adic settings that inform spectrality and partition structures in these groups.

Abstract

This study explores the properties of the function which can tile the field of -adic numbers by translation. It is established that functions capable of tiling is by translation uniformly locally constancy. As an application, in the field , we addressed the question posed by H. Leptin and D. Müller, 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 , proving that they are spectral sets.

Paper Structure

This paper contains 22 sections, 26 theorems, 111 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The field $\mathbb Q_p$ of $p$-adic numbers
  4. Fourier Transform
  5. $\mathbb{Z}$-module generated by $p^n$-th roots of unity
  6. Zeros of the Fourier Transform of a finite discrete signed measure
  7. Bruhat-Schwartz distributions in $\mathbb Q_p$
  8. Convolution and multiplication of distributions
  9. Proof of Theorem \ref{['thm-tilingperiodic']}
  10. Zeros of the Fourier transform of the measure $\nu$
  11. Proof of Theorem \ref{['thm-tilingperiodic']}
  12. Density of $\nu$ and the proof of Theorem \ref{['thm:card']}
  13. Structure of $\mathcal{Z}_{\widehat{\nu}}$
  14. Distribution of $\nu$
  15. Equality $D(\nu)=w/ \int_{\mathbb Q_p} f d\mathfrak{m}$
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $f \in L^{1}(\mathbb{Q}_p)$ and $V \subset \mathbb{Z} \setminus \{0\}$ be a finite set of non-zero integers. Suppose that $T$ is a locally finite subset in $\mathbb{Q}_p$, and $v_t \in V$ for $t \in T$ are such that for some $w \in \mathbb{R}$.Then $f$ is uniformly locally constancy, i.e. there exists $n\in \mathbb{Z}$ such that

Figures (2)

  • Figure 1: The set ${\mathbb Z}/3^4{\mathbb Z}=\{0,1,2,\cdots,80\}$ is considered as a tree ${\mathcal{T}}^{\left(4\right)}$.
  • Figure 2: For $p=3$, a ${{\mathcal{T}}_{I,J }}$-form tree with $n=5$, $I=\{0,2,4\} ,J=\{1,3\}$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Proposition 2.1: FFLS
  • Lemma 2.2: Sch64
  • Lemma 2.3: FFS
  • Corollary 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 24 more