{"paper":{"title":"Extensions of the Furstenberg-S\\'ark\\\"ozy theorem via the arithmetic level-$d$ inequality","license":"http://creativecommons.org/licenses/by/4.0/","headline":"For any intersective polynomial h the largest subset of {1,...,X} without nonzero h(n) differences has a quasipolynomial upper bound on size.","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Andrew Lott, Carlo Francisco E. Adajar, Chian Yeong Chuah, Krishnamohan Nandakumar, Mukul Rai Choudhuri, Nagendar Reddy Ponagandla, Rishika Agrawal, Steve Fan, Swaroop Hegde","submitted_at":"2026-05-15T17:29:24Z","abstract_excerpt":"Very recently, Green and Sawhney obtained a quasipolynomial bound in the Furstenberg--S\\'ark\\\"ozy theorem for square differences by proving an ''arithmetic level-$d$'' inequality, thereby yielding a greatly improved density increment scheme. We adapt their method to general intersective polynomials $h\\in\\mathbb{Z}[x]$ and obtain an analogous quasipolynomial upper bound for the largest subset of $\\{1,2,\\dots,X\\}$ whose difference set contains no nonzero element of the form $h(n)$ with $n\\in \\mathbb{Z}$. This is the best quantitative upper bound presently known for sets lacking intersective poly"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We adapt their method to general intersective polynomials h∈Z[x] and obtain an analogous quasipolynomial upper bound for the largest subset of {1,2,…,X} whose difference set contains no nonzero element of the form h(n) with n∈Z.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The arithmetic level-d inequality remains effective uniformly across all auxiliary polynomials arising in the iteration.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Extends Furstenberg-Sárközy to general intersective polynomials h via uniform arithmetic level-d inequality, yielding the best known quasipolynomial density bound.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For any intersective polynomial h the largest subset of {1,...,X} without nonzero h(n) differences has a quasipolynomial upper bound on size.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"85c3d1ddae52dc8e435031c0485bbd71a0ce4fd06833a6b04b933ece387c3bea"},"source":{"id":"2605.16216","kind":"arxiv","version":1},"verdict":{"id":"9809a34e-c3d3-4d1c-8a65-ee134b8e1583","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T18:27:12.380056Z","strongest_claim":"We adapt their method to general intersective polynomials h∈Z[x] and obtain an analogous quasipolynomial upper bound for the largest subset of {1,2,…,X} whose difference set contains no nonzero element of the form h(n) with n∈Z.","one_line_summary":"Extends Furstenberg-Sárközy to general intersective polynomials h via uniform arithmetic level-d inequality, yielding the best known quasipolynomial density bound.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The arithmetic level-d inequality remains effective uniformly across all auxiliary polynomials arising in the iteration.","pith_extraction_headline":"For any intersective polynomial h the largest subset of {1,...,X} without nonzero h(n) differences has a quasipolynomial upper bound on size."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16216/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T19:01:18.879111Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T18:40:53.147500Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"shingle_duplication","ran_at":"2026-05-19T17:49:44.669110Z","status":"skipped","version":"0.1.0","findings_count":0},{"name":"citation_quote_validity","ran_at":"2026-05-19T17:49:44.158729Z","status":"skipped","version":"0.1.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T17:33:24.782416Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"external_links","ran_at":"2026-05-19T17:31:33.196841Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"cited_work_retraction","ran_at":"2026-05-19T17:22:06.226459Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T16:41:55.388848Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"5f543e25500c8321b86f5fac8c79c1cd3ccac193f9078e2094aa3670b84a68a7"},"references":{"count":25,"sample":[{"doi":"","year":2024,"title":"Arala,A maximal extension of the Bloom–Maynard bound for sets without square differences, Funct","work_id":"a70de516-ef89-4f8d-9e05-3ec07304e681","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1994,"title":"A. 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