On the structure of spectral and tiling subsets of cyclic groups

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.