{
  "abstract": "For every quartic Salem number $β\\in(1,2)$, we prove that the Bernoulli convolution $ν_{β^{-1}}$ with equal weights has correlation dimension strictly less than one. In particular, neither of the two corresponding measures has an $L^2$ density. We relate correlation dimension to exact collision probabilities and express these probabilities as positive integrals over a four-dimensional torus associated with the defining polynomial. The unit-circle conjugates produce a persistent oscillatory term; using carefully chosen return times of the corresponding rotation, we take differences that make this term small while controlling the length and cost of the resulting configurations. Many well-separated placements of these configurations then give a collision probability above the critical scale, which yields the strict dimension drop. The qualitative argument is independent of the computer-assisted estimates. Separate computer-assisted arguments give \\[ 0.99999<D_2(ν_{β_1^{-1}})<1-10^{-65}, \\qquad 0.99999<D_2(ν_{β_2^{-1}})<1-10^{-75}, \\] where $β_1<β_2$ are the two quartic Salem numbers in $(1,2)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09902",
  "author_total": 4,
  "authors": [
    {
      "name": "Guozheng Cheng"
    },
    {
      "name": "Xiang Fang"
    },
    {
      "name": "Xueqing Ma"
    },
    {
      "name": "Hongli Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/66was2z8x8qxh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09902",
  "fulltext_url": null,
  "id": "66was2z8x8qxh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09902",
  "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": "W7220999863",
  "pdf_url": "https://arxiv.org/pdf/2610.09902",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/66was2z8x8qxh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09902",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09902"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Strict correlation-dimension drop for quartic Salem Bernoulli convolutions",
  "title_text": "Strict correlation-dimension drop for quartic Salem Bernoulli convolutions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/66was2z8x8qxh",
  "venue": "arXiv",
  "versions": []
}