{
  "abstract": "In this paper, we investigate the Takagi-van der Waerden function, $$ T_r(x) = \\sum_{n=0}^{\\infty} \\frac{ϕ(r^n x)}{r^n} ,\\quad x\\in [0,1], \\quad r \\geq 2, $$ where $ϕ(x)={\\rm dist}(x,\\mathbb{Z})$ represents the distance from $x$ to the nearest integer. Lagarias and Maddock[13] introduced the notion of local level sets for the classical Takagi function $T_2$. They proved that if $y$ is uniformly distributed over the range of $T_2$, then the expected number of local level sets contained in the level set at height $y$ equals $3/2$. We define the corresponding local level sets for every integer $r\\geq2$ and prove that, if $y$ is uniformly distributed over the range of $T_r$, then the expected number of local level sets contained in the level set at height $y$ equals $(r+1)/r$ if $r$ is even, and $(r+1)/(r-1)$ if $r$ is odd.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08572",
  "author_total": 3,
  "authors": [
    {
      "name": "Lai Jiang"
    },
    {
      "name": "Ting-Ting Ying"
    },
    {
      "name": "Yi-Yang Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4h6jqzpx9tcdj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08572",
  "fulltext_url": null,
  "id": "4h6jqzpx9tcdj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08572",
  "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": "W7220966163",
  "pdf_url": "https://arxiv.org/pdf/2610.08572",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4h6jqzpx9tcdj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.08572",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08572"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Local level sets of the Takagi-van der Waerden function",
  "title_text": "Local level sets of the Takagi-van der Waerden function",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/4h6jqzpx9tcdj",
  "venue": "arXiv",
  "versions": []
}