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

Overview & Synopsis

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 $|1+2\cos(2πx)+2\cos(2πy)|0$ and all sufficiently large primes $p$, uniformly in the exponents. That proof uses simultaneous compression of the exponents, discriminant estimates, and cyclic resultants.