{"paper":{"title":"VC-Dimension of Hyperplanes over Finite Fields","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alex Iosevich, Brian McDonald, Francisco Romero Acosta, Justin Cheigh, Livia Betti, Ruben Ascoli, Ryan Jeong, Santiago Velazquez Iannuzzelli, Steven J. Miller, Wyatt Milgrim, Xuyan Liu","submitted_at":"2023-07-19T19:24:01Z","abstract_excerpt":"Let $\\mathbb{F}_q^d$ be the $d$-dimensional vector space over the finite field with $q$ elements. For a subset $E\\subseteq \\mathbb{F}_q^d$ and a fixed nonzero $t\\in \\mathbb{F}_q$, let $\\mathcal{H}_t(E)=\\{h_y: y\\in E\\}$, where $h_y$ is the indicator function of the set $\\{x\\in E: x\\cdot y=t\\}$. Two of the authors, with Maxwell Sun, showed in the case $d=3$ that if $|E|\\geq Cq^{\\frac{11}{4}}$ and $q$ is sufficiently large, then the VC-dimension of $\\mathcal{H}_t(E)$ is 3. In this paper, we generalize the result to arbitrary dimension and improve the exponent in the case $d=3$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.10425","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2307.10425/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}