{
  "abstract": "In a companion article \\cite{RJnodyn} the present authors proved a form of the Rudolph--Johnson theorem which is effective at finite resolution, bounding the low-frequency Fourier coefficients of a $\\times 2 \\times 3$-invariant measure by its normalized entropy. Here we read that bound in the other direction, as a constraint on eigenfunction behavior in potential counterexamples to Furstenberg's $\\times 2 \\times 3$ conjecture. We show that a near return of the coordinate function $x \\mapsto e^{2πix}$ to itself under the semigroup generated by $2$ and $3$ forces the resolution entropy to be small, and that adherence of the coordinate function to eigenfunctions of $x \\mapsto 2x$ forces it to decay. These rates are explicit in the rate of adherence and in the arithmetic of the eigenvalues.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04745",
  "author_total": 2,
  "authors": [
    {
      "name": "Peter Burton"
    },
    {
      "name": "Kate Juschenko"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3r0jfgc9fgs9f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04745",
  "fulltext_url": null,
  "id": "3r0jfgc9fgs9f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04745",
  "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.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7220677986",
  "pdf_url": "https://arxiv.org/pdf/2610.04745",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3r0jfgc9fgs9f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04745",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04745"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Entropic rates from eigenfunction structure for $\\times 2 \\times 3$",
  "title_text": "Entropic rates from eigenfunction structure for × 2 × 3",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/3r0jfgc9fgs9f",
  "venue": "arXiv",
  "versions": []
}