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

Abstract

Let $p$, $q$, and $r$ be distinct primes such that $p^2{q}^2<r$. We prove that every spectral set in the cyclic group ${\mathbb{Z}}_{p^2{q}^{2}r}$ is a tile. Since the reverse direction is already known, this shows that ${\mathbb{Z}}_{p^2{q}^{2}r}$ is a Fuglede group under this condition. The proof is based on divisibility properties of mask polynomials and on the structure of spectral sets in finite cyclic groups.

18 pages. A gap in one of the main steps has been found and repaired with a new, complete argument