AbstractThe purpose of this paper is to investigate the properties of spectral and tiling subsets of cyclic groups, with an eye towards the spectral set conjecture [9] in one dimension, which states that a bounded measurable subset of$\mathbb {R}$accepts an orthogonal basis of exponentials if and only if it tiles$\mathbb {R}$by translations. This conjecture is strongly connected to its discrete counterpart, namely that, in every finite cyclic group, a subset is spectral if and only if it is a tile. The tools presented herein are refinements of recent ones used in the setting of cyclic groups; the structure of vanishing sums of roots of unity [20] is a prevalent notion throughout the text, as well as the structure of tiling subsets of integers [1]. We manage to prove the conjecture for cyclic groups of order$p^{m}q^{n}$, when one of the exponents is$\leq 6$or when$p^{m-2}<q^{4}$, and also prove that a tiling subset of a cyclic group of order$p_{1}^{m}p_{2}\dotsm p_{n}$is spectral.