{
  "abstract": "We present a counterexample, produced by OpenAI, to Littlewood's 1961 theorem on $L^α$-flat polynomials. The theorem asserts that no $L^α$-flat polynomials exist within the class of trigonometric polynomials of the form$$\\sum_{m=1}^{n} a_m \\cos(mθ+α_m), \\qquad n \\in \\mathbb{N}^*,$$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": "2610.30797",
  "author_total": 1,
  "authors": [
    {
      "name": "el Houcein el Abdalaoui"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/51nk24rjbvgsw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30797",
  "fulltext_url": null,
  "id": "51nk24rjbvgsw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30797",
  "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",
      "2613|Approximation Theory and Sequence Spaces"
    ]
  ],
  "openalex": "W7223315899",
  "pdf_url": "https://arxiv.org/pdf/2610.30797",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/51nk24rjbvgsw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.30797",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30797"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "OpenAI partially disproves Littlewood's 1961 $L^α$-flat theorem",
  "title_text": "OpenAI partially disproves Littlewood's 1961 L^α-flat theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/51nk24rjbvgsw",
  "venue": "arXiv",
  "versions": []
}