The maximal angular gap among rectangular grid points (Q1307425)

Take \(mn\) points in the plane of the form \((a_i, b_j)_{i=1}^m{}_{j=1}^n\). The rays connecting these points to the origin divide the plane into \(mn\) sectors, and the maximal one must be \(\geq 2\pi/(mn)\). This bound is improved when at least one of \(m, n\) is odd, and cases of equality are described. It is proved that for \(m=n\) a configuration exists with maximal angle \( <220/n^2\).