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+2cos(2πx)+2cos(2πy)|<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(−ε p) for every ε>0 and all sufficiently large primes p, uniformly in the exponents. That proof uses simultaneous compression of the exponents, discriminant estimates, and cyclic resultants.
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BibTeX
@misc{cheng2026small,
title = {{Small sums of roots of unity: polylogarithmic exceptional sets and conditional separation}},
author = {Qi Cheng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27231},
eprint = {2610.27231},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27231},
}
RIS
TY - UNPB
TI - Small sums of roots of unity: polylogarithmic exceptional sets and conditional separation
AU - Qi Cheng
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.27231
UR - https://arxiv.org/abs/2610.27231
AB - 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+2cos(2πx)+2cos(2πy)|<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(−ε p) for every ε>0 and all sufficiently large primes p, uniformly in the exponents. That proof uses simultaneous compression of the exponents, discriminant estimates, and cyclic resultants.
ER -