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 $$q$$ .