{
  "abstract": "Let $K$ be a fixed number field. We prove a quantitative expansion theorem for mixed sumsets $A+B$, assuming only that $A$ is covered by relatively few cosets of a multiplicative group of small finite valuation rank. The constants are independent of the generators, their heights, and their prime ideal supports. As a consequence, if $A\\subset K^\\times$, $|A|=N\\ge 3$, and each element has nonzero valuation at at most $k$ finite places, then every $B\\subset K$ with $|B|=N$ satisfies $$ \\max\\{|A+B|,|AA|\\} \\ge N^2\\exp\\left[-C_K\\left((k+1)\\log(\\log N+2)+\\sqrt{(k+1)\\log N}\\right)\\right]. $$ This is $N^{2-o(1)}$ for $k=o(\\log N/\\log\\log N)$. We also obtain a trade-off between the additive and multiplicative deficits, estimates for iterated linear sumsets, and a transfer theorem for product constructions $A=PG$ in terms of the multiplicative deficit of $P$. The proof recovers local scales from an exact sum and bounds the remaining conditional entropy by the overlap of entropy measures on fixed trees.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36930",
  "author_total": 3,
  "authors": [
    {
      "name": "Kaiqiang Zhang"
    },
    {
      "name": "Yuanguo Zeng"
    },
    {
      "name": "Yuyu Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/12mnd443xa8n2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36930",
  "fulltext_url": null,
  "id": "12mnd443xa8n2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36930",
  "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",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7224103307",
  "pdf_url": "https://arxiv.org/pdf/2610.36930",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/12mnd443xa8n2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.36930",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36930"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Near-quadratic sum-product estimates in fixed number fields",
  "title_text": "Near-quadratic sum-product estimates in fixed number fields",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/12mnd443xa8n2",
  "venue": "arXiv",
  "versions": []
}