{
  "abstract": "We study the Assouad dimension, lower dimension, and Assouad spectrum of graphs of the functions \\(f(x)=\\sum_{n=1}^{\\infty}a_nϕ(b_nx+θ_n)\\), where \\(ϕ\\) is a nonconstant \\(1\\)-periodic \\(C^2\\) function, \\(a_n,b_n>0\\), \\((θ_n)\\) is an arbitrary real sequence, \\(\\sum_{n=1}^{\\infty}a_n<\\infty\\), and \\(b_{n+1}/b_n\\to\\infty\\). We give sufficient conditions for the graph over every nondegenerate compact interval to have Assouad dimension two or lower dimension one. For \\(a_n=b_n^{-α}\\), \\(0<α<1\\), an explicit frequency-gap condition ensures that the Assouad spectrum equals two for \\(α\\leq\\vartheta<1\\). Writing \\(β_*=\\limsup_{n\\to\\infty}\\log b_{n+1}/\\log b_n\\), we show that, if \\(β_*>(2-α)/α\\), the spectrum attains the general upper bound for \\(α\\)-Hölder graphs whenever \\((2-α)/β_*\\leq\\vartheta<α\\) and \\(\\vartheta>0\\). When \\(β_*=\\infty\\), this determines the complete spectrum. Together with Barański's formulas, our results yield graphs whose lower, Hausdorff, upper box, and Assouad dimensions are four distinct numbers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37388",
  "author_total": 2,
  "authors": [
    {
      "name": "Dong-Hong Xiong"
    },
    {
      "name": "Jun Jason Luo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bgh4727bhrpw6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37388",
  "fulltext_url": null,
  "id": "bgh4727bhrpw6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37388",
  "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.CA"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7214998897",
  "pdf_url": "https://arxiv.org/pdf/2609.37388",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bgh4727bhrpw6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2609.37388",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37388"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Assouad Dimensions and Spectra of Graphs of Weierstrass-Type Functions with Rapidly Growing Frequencies",
  "title_text": "Assouad Dimensions and Spectra of Graphs of Weierstrass-Type Functions with Rapidly Growing Frequencies",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/bgh4727bhrpw6",
  "venue": "arXiv",
  "versions": []
}