On the structure of spectral and tiling subsets of cyclic groups

The purpose of this paper is to investigate the properties spectral and tiling subsets cyclic groups, with an eye towards set conjecture in one dimension, which states that a bounded measurable subset $\mathbb{R}$ accepts orthogonal basis exponentials if only it tiles by translations. This strongly connected its discrete counterpart, namely every finite group, tile. tools presented herein are refinements recent ones used setting groups; structure vanishing sums roots unity prevalent notion throughout text, as well integers. We manage prove for groups order $p^mq^n$, when exponents $\leq6$ or $p^{m-2}<q^4$, also group $p_1^mp_2\dotsm p_n$ spectral.