Qi Cheng

Small sums of roots of unity: polylogarithmic exceptional sets and conditional separation

We prove a polylogarithmic exceptional-set estimate for symmetric five-term sums of roots of unity of arbitrary order. More precisely, at most $O((\log(2N))^{11})$ points of the $N$-grid satisfy $|1+2\cos(2\pi x)+2\cos(2\pi y)|<\exp(-C(\log(2N))^{13})$. This implies the corresponding bound for small conjugates at the r...