A group ring approach to Fuglede's conjecture in cyclic groups

TL;DR

This paper introduces a new algebraic approach using group rings to analyze Fuglede's conjecture in cyclic groups, proving its validity in specific cases like rac{p^{n}qr}{ ext{cyclic groups}}.

Contribution

The paper develops a novel group ring method to study spectral sets in cyclic groups and proves Fuglede's conjecture for rac{p^{n}qr}{ ext{cyclic groups}}.

Findings

Fuglede's conjecture holds in rac{p^{n}qr}{ ext{cyclic groups}}.

Introduces a new algebraic tool for spectral set analysis.

Advances understanding of spectral sets in cyclic groups.

Abstract

Fuglede's conjecture states that a subset $\Omega\subseteq\mathbb{R}^{n}$ of positive and finite Lebesgue measure is a spectral set if and only if it tiles $\mathbb{R}^{n}$ by translation. The conjecture does not hold in both directions for $\mathbb{R}^n$, $n\ge3$. However, this conjecture remains open in $\mathbb{R}$ and $\mathbb{R}^2$. Cyclic groups play important roles in the study of Fuglede's conjecture in $\mathbb{R}$. In this paper, we introduce a new tool to study the spectral sets in cyclic groups. In particular, we prove that Fuglede's conjecture holds in $\mathbb{Z}_{p^{n}qr}$.