{"paper":{"title":"Beating Product Constructions for Linear Equations Over Finite Fields","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Fred Tyrrell, Paul Hametner","submitted_at":"2026-06-10T15:16:54Z","abstract_excerpt":"We show that for any $A\\subseteq \\mathbb{F}_q^n$ lacking non-trivial solutions to a translation-invariant linear equation of genus one, meaning that no nonempty proper subset of the coefficients sums to $0$, there is a set $B\\subseteq \\mathbb{F}_q^m$ in some higher dimension which also lacks non-trivial solutions, such that \\[|B|^{1/m}>|A|^{1/n}.\\] In particular, this implies that no fixed cap set in $\\mathbb{F}_3^n$ gives an asymptotically optimal lower bound by direct products alone."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.12194","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/2606.12194/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"}