{
  "abstract": "We present a counterexample, produced by OpenAI, to Littlewood's 1961 theorem on $L^\\alpha$-flat polynomials. The theorem asserts that no $L^\\alpha$-flat polynomials exist within the class of trigonometric polynomials of the form$$\\sum_{m=1}^{n} a_m \\cos(m\\theta+\\alpha_m), \\qquad n \\in \\mathbb{N}^*,$$satisfying$$k\\, n^2 \\sum_{m=1}^{n} a_m^2 \\;\\leq\\; \\sum_{m=1}^{n} m^2 a_m^2,$$where $k$ is an absolute constant (alternatively expressed on the right-hand side as $K = 1/k$). We show that OpenAI's counterexample is contained within the class of hyper-lacunary sequences. Consequently, it offers only a limited refutation, as such sequences fail to satisfy an implicit assumption of Littlewood's theorem. We also discuss broader questions surrounding this counterexample and provide a brief historical overview.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "El Houcein El Abdalaoui"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7cvg1rd51hs38.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "7cvg1rd51hs38",
  "issns": [],
  "landing_url": "https://hal.science/hal-05830397v1",
  "language": "en",
  "licence": "CC-BY-NC-4.0",
  "licence_label": "CC BY-NC 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05830397/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7cvg1rd51hs38.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:00Z",
      "native_id": "hal-05830397",
      "source": "hal",
      "url": "https://hal.science/hal-05830397v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "OpenAI PARTIALLY DISPROVES LITTLEWOOD'S 1961 Theorem",
  "title_text": "OpenAI PARTIALLY DISPROVES LITTLEWOOD'S 1961 Theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:00Z",
  "url": "https://research.draconitic.com/paper/7cvg1rd51hs38",
  "venue": "HAL",
  "versions": []
}