{"paper":{"title":"On polynomial progressions via transference","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Daniel Altman, Mehtaab Sawhney","submitted_at":"2025-06-16T00:38:54Z","abstract_excerpt":"We prove new cases of reasonable bounds for the polynomial Szemer\\'{e}di theorem both over $\\mathbb{Z}/N\\mathbb{Z}$ with $N$ prime and over the integers. In particular, we prove reasonable bounds for Szemer\\'edi's theorem in the integers with fixed polynomial common difference. That is, we prove for any polynomial $P(y)\\in \\mathbb{Z}[y]$ with $P(0) = 0$, that the largest subset $A\\subseteq [N]$ avoiding the pattern \\[x, x+P(y),\\ldots, x+ kP(y)\\] has size bounded by $\\ll_{P,k}N(\\log\\log\\log N)^{-\\Omega_{P,k}(1)}.$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.13010","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/2506.13010/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"}