  The 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 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 is a prevalent notion throughout the text,
as well as the structure of tiling subsets of integers. We manage to prove the
conjecture for cyclic groups of order $p^mq^n$, when one of the exponents is
$\leq6$ or when $p^{m-2}<q^4$, and also prove that a tiling subset of a cyclic
group of order $p_1^mp_2\dotsm p_n$ is spectral.
