{
  "abstract": "We study fixed-field analogues of the lattice product construction recently used by Bloom, Sawin, Schildkraut, and Zhelezov to disprove the sum-product conjecture over the real numbers. These sets are of the form $A_n = P_nG_n \\subset F$, where $F$ is a fixed number field, $P_n$ can be rescaled to a set of algebraic integers whose conjugates are polynomially bounded in $|P_n \\setminus \\{0\\}|$, and $G_n$ occupies a positive proportion of a bounded box of scale $Y_n$ in a fixed positive-rank multiplicative lattice. We prove that these fixed-field constructions satisfy the sum-product conjecture, $\\max\\{|A_n+A_n|, |A_nA_n|\\} \\ge |A_n|^{2-o(1)}$ as $|A_n| \\to \\infty$, with no restriction on the relative growth of the two factors. More strongly, any fixed power saving in the product set forces an asymptotically maximal sumset: there exists $C > 0$ such that, for each fixed $δ\\in (0,1]$, if $|A_nA_n| \\le |A_n|^{2-δ}$ for all sufficiently large $n$, then $Y_n \\to \\infty$ and $|A_n+A_n|/|A_n|^2 = 1/2 + O_δ((\\log Y_n)^C/Y_n)$. In particular, the additive-box times unit-box construction of Bloom, Sawin, Schildkraut, and Zhelezov cannot produce sum-product counterexamples in any fixed totally real number field, at any relative scales. The proof combines a bound for factorizations of algebraic integers with a more general maximal-sumset theorem for coefficient sets whose nonzero-element ratios have small Weil height. The latter controls multiplication fibers and combines a lattice strip estimate with logarithmic separation derived from the theorems of Yu and Matveev.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26737",
  "author_total": 2,
  "authors": [
    {
      "name": "Harrison Lemley"
    },
    {
      "name": "David Chew"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/46466gnaa95fb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26737",
  "fulltext_url": null,
  "id": "46466gnaa95fb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26737",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7223059603",
  "pdf_url": "https://arxiv.org/pdf/2610.26737",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/46466gnaa95fb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.26737",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26737"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The sum-product conjecture holds for fixed-field lattice product constructions",
  "title_text": "The sum-product conjecture holds for fixed-field lattice product constructions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/46466gnaa95fb",
  "venue": "arXiv",
  "versions": []
}