{
  "abstract": "Let $q$ be a prime power, $F=\\mathbb F_q(\\!(t^{-1})\\!)$, $G=\\mathrm{SL}_2(F)$, $Γ=\\mathrm{SL}_2(\\mathbb F_q[t])$, and $K=\\mathrm{SL}_2(\\mathbb F_q[\\![t^{-1}]\\!])$, and let $U<G$ be the upper unipotent subgroup. We study right $K$-spherical averages along $U$ on $X=Γ\\backslash G$. Expanding translates of compact $U$-orbits and compact-open F$\\unicode{x00F8}$lner-ball averages become terminal layers of rooted descendant shadows in the Bruhat--Tits tree. In the even sector, we compute the Haar height law and signed finite-scale discrepancy exactly. This yields $K$-spherical equidistribution for compact-orbit translates and, for irrational boundary endpoints, for F$\\unicode{x00F8}$lner-ball averages. For a depth-$N$ shadow rooted at height $k$, with cutoff $M=N-k\\ge0$, bounded-profile errors are $O_q(q^{-M})$ in the backward state and $O_q(q^{-M/2})$ uniformly for moving roots, while the shadow law eventually agrees exactly with the Haar law on every fixed finite height window. For $|Φ(2m)|\\le Cq^{αm}$, $α<2$, three rate regimes arise, with a linear-in-scale factor at $α=1$ and explicit moving-root dependence. Artin continued-fraction digits eventually encode the cutoff and these rates excursion by excursion through individual digit degrees.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.08704",
  "author_total": 1,
  "authors": [
    {
      "name": "Sanghoon Kwon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/07x0pdsfqa0h3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.08704",
  "fulltext_url": null,
  "id": "07x0pdsfqa0h3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.08704",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7167900056",
  "pdf_url": "https://arxiv.org/pdf/2607.08704",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/07x0pdsfqa0h3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2607.08704",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.08704"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "$K$-spherical horospherical averages on the Nagao quotient: tree combinatorics and exact discrepancy",
  "title_text": "K-spherical horospherical averages on the Nagao quotient: tree combinatorics and exact discrepancy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/07x0pdsfqa0h3",
  "venue": "arXiv",
  "versions": []
}