Spectral Tile Direction in the Group $\mathbb{Z}_{p^2} \times \mathbb{Z}_{q^2} \times \mathbb{Z}_r$

Thomas Fallon; Gergely Kiss; Azita Mayeli; Gábor Somlai

Spectral Tile Direction in the Group $\mathbb{Z}_{p^2} \times \mathbb{Z}_{q^2} \times \mathbb{Z}_r$

Classical Analysis and ODEs 2024-12-03 v3 Group Theory Number Theory

Authors: Thomas Fallon , Gergely Kiss , Azita Mayeli , Gábor Somlai

Abstract

In this paper, we investigate Fuglede's conjecture for $\mathbb{Z}_{p^2q^2r}$ and provide a proof under the condition $p^2q^2 \leq r$ . We develop a new technique by analyzing the divisibility of the mask polynomial of a given set by a system of cyclotomic polynomials. Combined with the so-called mod- $p$ -method, this technique serves as a powerful tool for studying Fuglede's conjecture in this and related cases.

Keywords

fuglede's conjecture spectral tile mask polynomial cyclotomic polynomial mod-p method finite abelian group

Cite

@article{arxiv.2308.16277,
  title  = {Spectral Tile Direction in the Group $\mathbb{Z}_{p^2} \times \mathbb{Z}_{q^2} \times \mathbb{Z}_r$},
  author = {Thomas Fallon and Gergely Kiss and Azita Mayeli and Gábor Somlai},
  journal= {arXiv preprint arXiv:2308.16277},
  year   = {2024}
}

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Spectral Tile Direction in the Group $\mathbb{Z}_{p^2} \times \mathbb{Z}_{q^2} \times \mathbb{Z}_r$

Abstract

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