{
  "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 $|1+2\\cos(2πx)+2\\cos(2πy)|<\\exp(-C(\\log(2N))^{13})$. 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 $\\exp(-\\varepsilon p)$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27231",
  "author_total": 1,
  "authors": [
    {
      "name": "Qi Cheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0k22qc9wx8c7s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27231",
  "fulltext_url": null,
  "id": "0k22qc9wx8c7s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27231",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7223094522",
  "pdf_url": "https://arxiv.org/pdf/2610.27231",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0k22qc9wx8c7s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27231",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27231"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Small sums of roots of unity: polylogarithmic exceptional sets and conditional separation",
  "title_text": "Small sums of roots of unity: polylogarithmic exceptional sets and conditional separation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/0k22qc9wx8c7s",
  "venue": "arXiv",
  "versions": []
}