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Fuglede's spectral set conjecture states that a subset $\Omega$ of a locallycompact abelian group $G$ tiles the group by translation if and only if thereexists a subset of continuous group characters which is an orthogonal basis of$L^2(\Omega)$. We prove that Fuglede's conjecture holds on cyclic groups$\mathbb{Z}_{pqr}$ with $p,q,r$ distinct primes.

Fuglede's conjecture holds on cyclic groups $\mathbb{Z}_{pqr}$