Small sums of five roots of unity

TL;DR

This paper investigates the minimal sums of five complex roots of unity, providing bounds that improve upon previous results by analyzing geometric configurations and rational approximations.

Contribution

It introduces new bounds on the minimal sum of five roots of unity, achieved through geometric perturbations and analysis of rational approximations, advancing understanding in number theory.

Findings

Established a uniform bound of O(n^{-4/3}) for the sum of five roots of unity.

Improved the bound to O(n^{-7/3}) infinitely often.

Identified configurations related to rational approximations that explain minimal sums.

Abstract

Motivated by questions in number theory, Myerson asked how small the sum of 5 complex nth roots of unity can be. We obtain a uniform bound of O(n^{-4/3}) by perturbing the vertices of a regular pentagon, improving to O(n^{-7/3}) infinitely often. The corresponding configurations were suggested by examining exact minimum values computed for n <= 221000. These minima can be explained at least in part by selection of the best example from multiple families of competing configurations related to close rational approximations.