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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1709.00080","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-08-31T20:56:48Z","cross_cats_sorted":[],"title_canon_sha256":"c5e3f47f5ab178f03dc727f45399a5385e7d5be81b11182a137b88d452982ba6","abstract_canon_sha256":"43779ac320d8969591e0a1be72d134d453e36e7ecea31c66815ab53fc5e55d3b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:58.526625Z","signature_b64":"iUS4yOpPwXBncu7NxSzHTWIw97f3ve7OtH7LIlfV4G00yucC5IrFEA4mvqKXKcdsJjmqSWX22ggSIuWCnkJiDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c7492afe01a49117d4d8934bef1b783211cf2e0d9177b3a17ebc66566dd93c1e","last_reissued_at":"2026-05-18T00:33:58.525992Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:58.525992Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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