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

Abstract

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 reduced conductor: outside polylogarithmically many real embeddings, every such sum is at least quasipolynomially large. The proof is unconditional and combines a polynomial-vector-field multiplicity estimate, arithmetic jet determinants, analytic propagation, and interpolation determinants for nearby grid points. As a companion result, assuming a uniform number-field $abc$ conjecture, we prove that every sum of a fixed number of distinct prime-order roots of unity has modulus at least $\exp(-\varepsilon p)$ for every $\varepsilon>0$ and all sufficiently large primes $p$, uniformly in the exponents. That proof uses simultaneous compression of the exponents, discriminant estimates, and cyclic resultants.