{"paper":{"title":"The Kakeya set and maximal conjectures for algebraic varieties over finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.CO","authors_text":"Jordan Ellenberg, Richard Oberlin, Terence Tao","submitted_at":"2009-03-10T22:35:22Z","abstract_excerpt":"Using the polynomial method of Dvir \\cite{dvir}, we establish optimal estimates for Kakeya sets and Kakeya maximal functions associated to algebraic varieties $W$ over finite fields $F$. For instance, given an $n-1$-dimensional projective variety $W \\subset \\P^n(F)$, we establish the Kakeya maximal estimate $$ \\| \\sup_{\\gamma \\ni w} \\sum_{v \\in \\gamma(F)} |f(v)| \\|_{\\ell^n(W)} \\leq C_{n,W,d} |F|^{(n-1)/n} \\|f\\|_{\\ell^n(F^n)}$$ for all functions $f: F^n \\to \\R$ and $d \\geq 1$, where for each $w \\in W$, the supremum is over all irreducible algebraic curves in $F^n$ of degree at most $d$ that pas"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.1879","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}