Small sums of roots of unity

In my research project I am looking at a lower bound for Kloosterman sums, which are sums of roots of unity. The best known lower bound for a sum of $k$ $N$th roots of unity is $k^{-N}$, which comes from a simple algebraic number theory argument. In a 1986 paper, "How Small Can a Sum of Roots of Unity Be?", Gerry (Gerald) Myerson makes the comment that when $k$ is approximately N, the bound becomes $c_kN^{-1}$ for some constant $c_k$ but gives no reference for this idea. Specifially, he says that choosing roots near the vertices of a $k$-gon gives this estimate. Where does this come from?