{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-vc-dimension-of-quadratic-residues-in","title":"The VC-dimension of quadratic residues in finite fields","arxiv_id":"2210.03789","date":"2022-10-07","proceeding":null,"authors":["Brian McDonald","Anurag Sahay","Emmett L. Wyman"],"abstract":"We study the Vapnik-Chervonenkis (VC) dimension of the set of quadratic residues (i.e. squares) in finite fields, $\\mathbb F_q$, when considered as a subset of the additive group. We conjecture that as $q \\to \\infty$, the squares have the maximum possible VC-dimension, viz. $(1+o(1))\\log_2 q$. We prove, using the Weil bound for multiplicative character sums, that the VC-dimension is $\\geq (\\frac{1}{2} + o(1))\\log_2 q$. We also provide numerical evidence for our conjectures. The results generalize to multiplicative subgroups $\\Gamma \\subseteq \\mathbb F_q^\\times$ of bounded index.","url_abs":"https://arxiv.org/abs/2210.03789v2","url_pdf":"https://arxiv.org/pdf/2210.03789v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-vc-dimension-of-quadratic-residues-in","repo_url":"https://github.com/anuragsahay/vcdimsquares","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}