{
  "abstract": "In 2006 Isaac reduced the simple normality to base $2$ of $\\sqrt s$, for $s$ not a perfect square, to a property of dyadic expansions that he called Condition~(TU): the upper frequency of the digit~$1$ in $\\sqrtν$, along any subsequence, depends only on the tail of the expansion of $ν$. We show that Condition~(TU) is false, already for the full sequence. We give two proofs. The first is elementary and turns Isaac's own reduction against the condition. The second uses the theorem of Kaufman and Mosquera--Shmerkin on the Fourier decay of smooth images of self-similar measures, and gives counterexamples for almost every point of a Cantor measure. All results are machine-verified in Lean~4; the second proof uses the two cited theorems as hypotheses.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21899",
  "author_total": 1,
  "authors": [
    {
      "name": "Ralf Stephan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2442dh01r6dzt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21899",
  "fulltext_url": null,
  "id": "2442dh01r6dzt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21899",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7222683823",
  "pdf_url": "https://arxiv.org/pdf/2610.21899",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2442dh01r6dzt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21899",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21899"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Isaac's Condition (TU) is false",
  "title_text": "Isaac's Condition (TU) is false",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/2442dh01r6dzt",
  "venue": "arXiv",
  "versions": []
}