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

TL;DR

This paper explores the Herzog-Schönheim conjecture by connecting it to vanishing sums of roots of unity and convex polygons, providing new insights and partial results on the longstanding mathematical problem.

Contribution

It reformulates the Herzog-Schönheim conjecture as a problem involving roots of unity and convex polygons, offering novel approaches and partial proofs.

Findings

Reformulation of the conjecture using roots of unity

Establishment of connections with convex polygons

Partial results supporting the conjecture

Abstract

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.