{"paper":{"title":"Simultaneous popular polynomial differences over finite fields","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"David Conlon, Dingding Dong, Guo-Dong Hong","submitted_at":"2026-07-11T00:21:07Z","abstract_excerpt":"Green's popular difference theorem says that for every \\(\\varepsilon>0\\), all sufficiently large primes \\(p\\), and every set \\(A\\subseteq\\mathbb F_p\\) of density \\(\\alpha\\), there exists a nonzero \\(d\\in\\mathbb F_p\\) such that \\[\n  \\mathbb E_{x\\in\\mathbb F_p}\n  1_A(x)1_A(x+d)1_A(x+2d)\n  \\geq\n  \\alpha^3-\\varepsilon. \\] We show that a stronger simultaneous popular difference phenomenon holds for polynomial configurations. Namely, if $\\mathcal P=\\{P_1,\\dots,P_k\\} \\subset \\mathbb Z[t]$ is a fixed collection of linearly independent polynomials with zero constant terms, we show that for every \\(\\var"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.10051","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/2607.10051/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"}