{
  "abstract": "In this follow-up article of a multi-part series on (strictly) preperiodic point-counting, we inspect an astonishing relationship between the set of (strictly) $1_{m}$-preperiodic points of a polynomial map $φ_{d, c}$ defined by $φ_{d, c}(z) = z^d + c$ for all $c, z \\in \\mathcal{O}_{K}$ and the coefficient $c$, where $K$ is any number field of degree $n\\geq 1$, $d>2$ is an integer and $m\\in \\mathbb{Z}_{\\geq 1}$ is any fixed (eventual period). As before, we study counting problems that are inspired by torsion point-counting in arithmetic statistics and (strictly) preperiodic point-counting in arithmetic dynamics. In doing so, we then first prove that for any prime $p\\geq 3$ and for any fixed $\\ell \\in \\mathbb{Z}_{\\geq 1}$ and fixed (eventual period) $m\\in \\mathbb{Z}_{\\geq 1}$, the average number of distinct $1_{m}$-preperiodic integral points of any $φ_{p^{\\ell}, c}$ modulo prime $p\\mathcal{O}_{K}$ is unbounded or zero as $c\\to \\infty$. Inspired further by work of Doyle-Poonen along with conjectural work of Hutz and $\\textit{abc}(\\textit{d})$-conditional work of Panraksa on $K$-rational preperiodic points of any $φ_{(p-1)^{\\ell}, c}$ for any prime $p\\geq 5$ in arithmetic dynamics, we then also prove that for any fixed (eventual period) $m \\in \\mathbb{Z}_{ \\geq 1}$, the average number of distinct $1_{m}$-preperiodic integral points of any $φ_{(p-1)^{\\ell}, c}$ modulo prime $p\\mathcal{O}_{K}$ is unbounded or zero as $c\\to \\infty$. Finally, we then apply counting and distribution results from arithmetic statistics, and then get counting and statistical results on arithmetic objects arising naturally in our polynomial discrete dynamical settings. Together with the previous articles in the series, it might then be fair to view this article as an attempt to contribute to \"arithmetic statistical dynamics\".",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.14468",
  "author_total": 1,
  "authors": [
    {
      "name": "Brian Kintu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0kae1ja7w5sav.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.14468",
  "fulltext_url": null,
  "id": "0kae1ja7w5sav",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.14468",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7164851628",
  "pdf_url": "https://arxiv.org/pdf/2606.14468",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0kae1ja7w5sav.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2606.14468",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.14468"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Counting the number of $1_{m}$-preperiodic $\\mathcal{O}_{K}$-points of a discrete dynamical system with applications from arithmetic statistics, VII",
  "title_text": "Counting the number of 1_m-preperiodic 𝒪_K-points of a discrete dynamical system with applications from arithmetic statistics, VII",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/0kae1ja7w5sav",
  "venue": "arXiv",
  "versions": []
}