{
  "abstract": "We construct arbitrarily large finite sets $A$ of real algebraic integers such that both $|A+A|$ and $|AA|$ are at most $|A|^{1.95835}$. The construction combines truncated ideal-smooth $S$-unit fibres with a coprime additive factor. A tensor-product rank argument, using a two-dimensional local feature at each selected prime ideal, controls the loss in the additive factor, while a direct estimate for an outer parallel body improves the sumset packing bound. The arithmetic input is an unramified pro-$2$ tower over a known degree-ten field, with simultaneous Frobenius cuts controlling the small prime ideals. We use unconditional Tsfasman--Vlăduţ inequalities for the joint class-number--regulator cost. All finite numerical comparisons entering the exponent are certified by outward rational interval arithmetic. No unproved hypothesis is used.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05725",
  "author_total": 3,
  "authors": [
    {
      "name": "Kaiqiang Zhang"
    },
    {
      "name": "Yuanguo Zeng"
    },
    {
      "name": "Yuyu Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2n0wgqg0n2tcj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05725",
  "fulltext_url": null,
  "id": "2n0wgqg0n2tcj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05725",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7220659839",
  "pdf_url": "https://arxiv.org/pdf/2610.05725",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2n0wgqg0n2tcj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05725",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05725"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Ideal-Smooth Sets and an Explicit Upper Bound for the Real Sum-Product Exponent",
  "title_text": "Ideal-Smooth Sets and an Explicit Upper Bound for the Real Sum-Product Exponent",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/2n0wgqg0n2tcj",
  "venue": "arXiv",
  "versions": []
}