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

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. $\mathbb{Z}_{p^n}$.