Fuglede's conjecture for cyclic groups of square-free order

Abstract

We prove Fuglede's conjecture for finite cyclic groups of square-free order. Specifically, if N is square-free and A is a nonempty subset of Z_{N} with |A|=m, then A is spectral if and only if it tiles Z_{N}; moreover, these conditions hold precisely when m divides N and A is a complete set of representatives modulo m. This gives a complete structural description of spectral and tiling sets in Z_{N}.