{"paper":{"title":"Improved estimates for polynomial Roth type theorems in finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dong Dong, Will Sawin, Xiaochun Li","submitted_at":"2017-08-31T20:56:48Z","abstract_excerpt":"We prove that, under certain conditions on the function pair $\\varphi_1$ and $\\varphi_2$, bilinear average $p^{-1}\\sum_{y\\in \\mathbb{F}_p}f_1(x+\\varphi_1(y)) f_2(x+\\varphi_2(y))$ along curve $(\\varphi_1, \\varphi_2)$ satisfies certain decay estimate. As a consequence, Roth type theorems hold in the setting of finite fields. In particular, if $\\varphi_1,\\varphi_2\\in \\mathbb{F}_p[X]$ with $\\varphi_1(0)=\\varphi_2(0)=0$ are linearly independent polynomials, then for any $A\\subset \\mathbb{F}_p, |A|=\\delta p$ with $\\delta>c p^{-\\frac{1}{12}}$, there are $\\gtrsim \\delta^3p^2$ triplets $x,x+\\varphi_1(y"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.00080","kind":"arxiv","version":3},"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"}