Residue Designs Modulo a Given Number on the Circle

The distribution of the total length of chords of the unit circle joining the vertices of a regular \(q\) -gon with numbers \(k\) and \(ak^{2} \ (\operatorname{mod} q)\) , \(k=1,2,\dots, q\) , is studied in the case where \(q\) is a prime and \(a\) ranges over the complete residual system modulo \(q\) .