{
  "abstract": "In the setting of abelian groups, Haussler's packing lemma implies that any subset of bounded $\\operatorname{VC}$-dimension can be approximated by one of a small number of representative translates. While this exact statement is provably false under the weaker hypothesis that $A$ has bounded $\\operatorname{VC}_2$-dimension, we show that in $\\mathbb{F}_p^n$ a similar statement does hold after localization to cosets of subspaces depending on the individual translates. More precisely, given $ε>0$ and $A\\subseteq \\mathbb{F}_p^n$ with $\\operatorname{VC}_2(A)<k$, we show that there exist $r\\leq ε^{-O_{k,p}(1)}$ and $g_1,\\ldots, g_r\\in \\mathbb{F}_p^n$ with the property that for every $x\\in \\mathbb{F}_p^n$, there is a partition $\\mathcal{C}_x$ of $\\mathbb{F}_p^n$ into cosets of (possibly distinct) subgroups, each of index at most $\\exp(ε^{-O_{k,p}(1)})$, and a subcollection $Ω_x\\subseteq \\mathcal{C}_x$ whose union has size at least $(1-ε)|\\mathbb{F}_p^n|$, such that on each $C\\in Ω_x$, the translate $A-x$ looks like one of $A-g_1,\\ldots, A-g_r$, in the sense that $|((A-x)Δ(A-g_{i_{x,C}}))\\cap C|<ε|C|$ for some $1\\leq i_{x,C}\\leq r$. We also prove a second version of this result in which each partition $\\mathcal{C}_x$ is the set of cosets of a single subgroup $H_x$ of quasipolynomial index but where the bound on $r$ is a quadruple exponential.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01101",
  "author_total": 2,
  "authors": [
    {
      "name": "C. Terry"
    },
    {
      "name": "J. Wolf"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/513qapb3t9hqs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01101",
  "fulltext_url": null,
  "id": "513qapb3t9hqs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01101",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.LO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01101",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/513qapb3t9hqs.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01101",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01101"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "A localized packing lemma for sets of bounded $\\operatorname{VC}_2$-dimension",
  "title_text": "A localized packing lemma for sets of bounded VC_2-dimension",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/513qapb3t9hqs",
  "venue": "arXiv",
  "versions": []
}