Small sums of five roots of unity

Abstract

AbstractMotivated by questions in number theory, Myerson asked how small the sum of five complex th roots of unity can be. We obtain a uniform bound of by perturbing the vertices of a regular pentagon, improving to infinitely often. The corresponding configurations were suggested by examining exact minimum values computed for . 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.