Fuglede's Conjecture for Cyclic Groups of Square-free Order

We prove Fuglede's conjecture for finite cyclic groups of square-free order. Specifically, if \(N\) is square-free and \(A\subseteq \mathbb Z_N\) is nonempty with \(|A|=m\), then \(A\) is spectral if and only if it tiles \(\mathbb Z_N\); moreover, these conditions hold precisely when \(m\mid N\) and