{
  "abstract": "Let $A\\subseteq\\mathbb{R}$ support a probability measure whose Fourier--Stieltjes transform tends to zero at infinity. We prove that, for every $\\varepsilon\\in(0,1)$, there is a closed, $1$-periodic, nowhere dense set $E\\subseteq\\mathbb{R}$ such that \\[ m(E\\cap I)\\ge1-\\varepsilon \\] for every interval $I$ of length $1$, while $E$ contains no affine copy of $A$. Thus every set supporting a Rajchman measure satisfies the Erdős similarity conjecture in a uniform large-set form. The proof combines equidistribution modulo one for large dilates of the measure with a multiscale family of low-density periodic blockers; no quantitative rate of Fourier decay is used. We also refine a classical theorem of Ivašev-Musatov, showing that for every Hausdorff gauge $h$ there is an $h$-null compact Rajchman support $K$ satisfying \\[ \\overline{\\dim}_{\\mathrm B}^{\\log} K=\\dim_{\\mathrm P}^{\\log} K=1. \\] The value $1$ is sharp for both dimensions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04456",
  "author_total": 5,
  "authors": [
    {
      "name": "A. Iosevich"
    },
    {
      "name": "N. Kulkarni"
    },
    {
      "name": "N. Mora Cuéllar"
    },
    {
      "name": "I. Rojas Aravena"
    },
    {
      "name": "A. Yavicoli"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b89ewd6s0d79m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04456",
  "fulltext_url": null,
  "id": "b89ewd6s0d79m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04456",
  "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.AP"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7211867743",
  "pdf_url": "https://arxiv.org/pdf/2609.04456",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b89ewd6s0d79m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.04456",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04456"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The Erdős similarity conjecture and Rajchman measures",
  "title_text": "The Erdős similarity conjecture and Rajchman measures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/b89ewd6s0d79m",
  "venue": "arXiv",
  "versions": []
}