Herzog-Schonheim conjecture, vanishing sums of roots of unity and convex polygons

Let $G$ be a group and $H_1$,\ldots,$H_s$ be subgroups of $G$ of indices $d_1,\ldots,d_s$ respectively. In 1974, M. Herzog and J. Sch\"onheim conjectured that if $\{H_i\alpha_i\}_{i=1}^{i=s}$, $\alpha_i\in G$, is a coset partition of $G$, then $d_1,\ldots,d_s$ cannot be distinct. In this paper, we present the conjecture as a problem on vanishing sum of roots of unity and convex polygons and prove some results using this approach.Comment: arXiv admin note: substantial text overlap with arXiv:2001.03882, arXiv:1901.0989