Fuglede's conjecture on cyclic groups of order $p^n q$

TL;DR

This paper proves Fuglede's spectral set conjecture for cyclic groups of order p^n q, establishing the equivalence between tiling and spectral properties in this specific algebraic setting.

Contribution

It extends the class of cyclic groups for which Fuglede's conjecture is verified, specifically to groups of order p^n q, using properties of roots of unity.

Findings

Fuglede's conjecture holds for cyclic groups of order p^n q.

Tiling and spectral properties are equivalent in these groups.

The proof relies on the structure of roots of unity with at most two prime divisors.

Abstract

We show that the spectral set conjecture by Fuglede holds in the setting of cyclic groups of order $p^n q$, where $p$, $q$ are distinct primes and $n\geq1$. This means that a subset $E$ of such a group $G$ tiles the group by translation ($G$ can be partitioned into translates of $E$) if and only if there exists an orthogonal basis of $L^2(E)$ consisting of group characters. The main ingredient of the present proof is the structure of vanishing sums of roots of unity of order $N$, where $N$ has at most two prime divisors; the extension of this proof to the case of cyclic groups of order $p^n q^m$ seems therefore feasible. The only previously known infinite family of cyclic groups, for which Fuglede's conjecture is verified in both directions, is that of cyclic $p$-groups, i.e. $\mathbb{Z}_{p^n}$.