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The structure of tiles in $\mathbb{Z}_{p^n}\times \mathbb{Z}_q$ and $\mathbb{Z}_{p^n}\times \mathbb{Z}_p$

Shilei Fan, Mamateli Kadir, Peishan Li

Abstract

In this paper, we provide a geometric characterization of tiles in the finite abelian groups \( \mathbb{Z}_{p^n} \times \mathbb{Z}_q \) and \( \mathbb{Z}_{p^n} \times \mathbb{Z}_p \) using the concept of a \( p \)-homogeneous tree, which provides an intuitively visualizable criterion.

The structure of tiles in $\mathbb{Z}_{p^n}\times \mathbb{Z}_q$ and $\mathbb{Z}_{p^n}\times \mathbb{Z}_p$

Abstract

In this paper, we provide a geometric characterization of tiles in the finite abelian groups and using the concept of a -homogeneous tree, which provides an intuitively visualizable criterion.

Paper Structure

This paper contains 14 sections, 25 theorems, 66 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The structure of tiles in ${\mathbb Z}_{p^n}\times{\mathbb Z}_q$
  3. The structure of tiles in ${\mathbb Z}_{p^n}\times{\mathbb Z}_p$
  4. Preliminaries
  5. Fourier transform on the finite groups.
  6. Basic properties of tiles.
  7. ${\mathbb Z}$-module generated by $p^n$-th roots of unity.
  8. $p$-homogeneity of tiles in ${\mathbb Z}_{p^n}$
  9. The structure of tiles on ${\mathbb Z}_{p^nq}$
  10. CM condition for tiles in finite group ${\mathbb Z}_N$.
  11. The structure of tiles in ${\mathbb Z}_{p^nq}$
  12. The structure of tiles in ${\mathbb Z}_{p^n}\times{\mathbb Z}_p$
  13. Equidistribution property
  14. Proof of the theorem \ref{['main2']}

Key Result

Theorem 1.1

Let $\Omega=\bigsqcup\limits_{j=0}^{q-1}( \Omega_{j} \times \left \{ j \right \})$ be a tile in ${\mathbb Z}_{p^n}\times{\mathbb Z}_q$ with $\Omega_j\subset {\mathbb Z}_{p^n}$. Then we have the following two cases.

Figures (7)

  • Figure 1: Consider the set ${\mathbb Z}_{3^4}=\{0, 1, 2, \cdots, 80\}$ as a tree $\mathcal{T}^{\left(4\right)}$.
  • Figure 2: The subtree $\mathcal{T}_{C}$ of $\mathcal{T}^{\left(3\right)}$ with $C=\{0,2,3,4,\cdots, 24,26\}$.
  • Figure 3: For $p=3$, a ${{\mathcal{T}}_I}$-form tree with $n=5$, $I=\{0,2,4\}$.
  • Figure 4: For $p=2$, a $p$- homogeneous tree.
  • Figure 5: Consider set $\{0, 4, 8, 9, 13, 17, 18, 22, 26\}$ as a $p$-homogeneous tree.
  • ...and 2 more figures

Theorems & Definitions (38)

  • Conjecture : Fuglede 1974
  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2: Z2022
  • Lemma 2.3
  • proof
  • Lemma 2.4: S20
  • Lemma 2.5: SS09book
  • ...and 28 more