{"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/exponentially-larger-affine-and-projective","title":"Exponentially Larger Affine and Projective Caps","arxiv_id":"2211.09772","date":"2022-11-17","proceeding":null,"authors":["Christian Elsholtz","Gabriel F. Lipnik"],"abstract":"In spite of a recent breakthrough on upper bounds of the size of cap sets (by Croot, Lev and Pach (2017) and Ellenberg and Gijswijt (2017)), the classical cap set constructions had not been affected. In this work, we introduce a very different method of construction for caps in all affine spaces with odd prime modulus $p$. Moreover, we show that for all primes $p \\equiv 5 \\bmod 6$ with $p \\leq 41$, the new construction leads to an exponentially larger growth of the affine and projective caps in $\\mathrm{AG}(n,p)$ and $\\mathrm{PG}(n,p)$. For example, when $p=23$, the existence of caps with growth $(8.0875\\ldots)^n$ follows from a three-dimensional example of Bose (1947), and the only improvement had been to $(8.0901\\ldots)^n$ by Edel (2004), based on a six-dimensional example. We improve this lower bound to $(9-o(1))^n$.","url_abs":"https://arxiv.org/abs/2211.09772v1","url_pdf":"https://arxiv.org/pdf/2211.09772v1.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":"exponentially-larger-affine-and-projective","repo_url":"https://gitlab.com/galipnik/large-caps","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"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}