{
  "abstract": "We disprove the sum-product conjecture for real numbers by constructing arbitrarily large $A\\subset \\mathbb{R}$ (whose elements are algebraic integers in a number field of degree $\\asymp \\log\\lvert A\\rvert$) such that \\[\\max(\\lvert A+A\\rvert ,\\lvert AA\\rvert)\\leq \\lvert A\\rvert^{2-c}\\] where $c>0$ is an absolute constant. We also disprove the many sums and products conjecture by constructing, for any $k\\geq 3$, arbitrarily large $A\\subset \\mathbb{R}$ such that \\[\\max(\\lvert kA\\rvert,\\lvert A^{(k)}\\rvert)\\leq \\lvert A\\rvert^{C\\frac{\\log k}{\\log\\log k}}\\] for some constant $C>0$. We obtain similar constructions for $p$-adics, finite fields, and function fields in positive characteristic, and also obtain new lower bounds for the number of solutions to linear equations in a multiplicative group and the number of solutions to the unit equation in sufficiently many variables.",
  "abstract_shown": true,
  "affiliations": [
    "Department of Mathematics , Stanford University , Stanford CA",
    "Department of Mathematics , Princeton University , Princeton , NJ 08540",
    "Department of Mathematics , University of Manchester , Manchester , M13 9PL"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 4,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Thomas F. Bloom",
      "orcid": "0000-0002-2454-4008"
    },
    {
      "affs": [
        0
      ],
      "name": "Will Sawin",
      "orcid": "0000-0003-0344-5438"
    },
    {
      "affs": [
        1,
        2
      ],
      "name": "Carl Schildkraut",
      "orcid": "0009-0005-2574-0309"
    },
    {
      "affs": [
        1
      ],
      "name": "Dmitrii Zhelezov",
      "orcid": "0000-0002-8092-577X"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9jjxgkzezbvt3.bib",
  "cited_by": 29,
  "doi": "10.48550/arxiv.2605.28781",
  "fulltext_url": null,
  "id": "9jjxgkzezbvt3",
  "issns": [
    "2331-8422"
  ],
  "landing_url": "https://arxiv.org/abs/2605.28781",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7162658663",
  "pdf_url": "https://arxiv.org/pdf/2605.28781",
  "pmcid": null,
  "pmid": null,
  "published": "2026-05-27",
  "publisher": "Cornell University",
  "ris": "https://research.draconitic.com/paper/9jjxgkzezbvt3.ris",
  "sources": [
    {
      "first_seen": "2026-11-07T13:20:41Z",
      "native_id": "W7162658663",
      "source": "openalex",
      "url": "https://arxiv.org/abs/2605.28781"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2607|Limits and Structures in Graph Theory",
      "subfield": "combinatorics"
    }
  ],
  "title": "The sum-product conjecture is false for real numbers",
  "title_text": "The sum-product conjecture is false for real numbers",
  "type": "preprint",
  "updated_at": "2026-11-09T13:20:59Z",
  "url": "https://research.draconitic.com/paper/9jjxgkzezbvt3",
  "venue": "arXiv (Cornell University)",
  "versions": []
}