{"paper":{"title":"A structure theorem for polynomial return-time sets in minimal systems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"In minimal systems, polynomial return-time sets coincide with those in the maximal infinite-step pronilfactor up to non-piecewise syndetic sets.","cross_cats":[],"primary_cat":"math.DS","authors_text":"Andreas Koutsogiannis, Anh N. Le, Daniel Glasscock, Donald Robertson, Florian K. Richter, Joel Moreira","submitted_at":"2025-11-03T21:40:30Z","abstract_excerpt":"We investigate the structure of return-time sets determined by orbits along polynomial tuples in minimal topological dynamical systems. Building on the topological characteristic factor theory of Glasner, Huang, Shao, Weiss, and Ye, we prove a structure theorem showing that, in a minimal system, return-time sets coincide -- up to a non-piecewise syndetic set -- with those in its maximal infinite-step pronilfactor. As applications, we establish three new multiple recurrence theorems concerning linear recurrence along dynamically defined syndetic sets and polynomial recurrence along arithmetic p"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove a structure theorem showing that, in a minimal system, return-time sets coincide -- up to a non-piecewise syndetic set -- with those in its maximal infinite-step pronilfactor.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The argument depends on the topological characteristic factor theory of Glasner, Huang, Shao, Weiss, and Ye to reduce the system to its pronilfactor.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"In minimal systems, polynomial return-time sets coincide up to non-piecewise syndetic sets with those in the maximal infinite-step pronilfactor, yielding new multiple recurrence theorems.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"In minimal systems, polynomial return-time sets coincide with those in the maximal infinite-step pronilfactor up to non-piecewise syndetic sets.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"08c322690d485472c998a0d64575c5e546bc45d07294303b4485e4e26f6772d2"},"source":{"id":"2511.02080","kind":"arxiv","version":2},"verdict":{"id":"49d33c35-0493-44cb-8d63-414e2e9df989","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-18T00:45:18.660303Z","strongest_claim":"We prove a structure theorem showing that, in a minimal system, return-time sets coincide -- up to a non-piecewise syndetic set -- with those in its maximal infinite-step pronilfactor.","one_line_summary":"In minimal systems, polynomial return-time sets coincide up to non-piecewise syndetic sets with those in the maximal infinite-step pronilfactor, yielding new multiple recurrence theorems.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The argument depends on the topological characteristic factor theory of Glasner, Huang, Shao, Weiss, and Ye to reduce the system to its pronilfactor.","pith_extraction_headline":"In minimal systems, polynomial return-time sets coincide with those in the maximal infinite-step pronilfactor up to non-piecewise syndetic sets."},"references":{"count":32,"sample":[{"doi":"","year":1988,"title":"J. Auslander. Minimal flows and their extensions , volume 153 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1988. Notas de Matem´ atica [Mathematical Notes], 122. 9, 1","work_id":"134b21d5-9f4e-4649-a983-e4017ce33172","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1971,"title":"K. Berg. Quasi-disjointness in ergodic theory. Trans. Amer. Math. Soc., 162:71–87, 1971. 15","work_id":"78caf364-a33e-45ca-9e80-406617b4b197","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2005,"title":"V. Bergelson, B. Host, and B. Kra. Multiple recurrence and nilsequences. Invent. Math. , 160(2):261–303, 2005. With an appendix by Imre Ruzsa. 29","work_id":"6dfb0817-f8cf-4719-b830-ed466b3c9901","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2008,"title":"V. Bergelson, A. Leibman, and E. Lesigne. Intersective polynomials and the polynomial Szemer´ edi theorem. Adv. Math., 219(1):369–388, 2008. 23","work_id":"e2b642b0-d1fe-4233-9418-f1b2bc198e2a","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2000,"title":"V. Bergelson and R. McCutcheon. An ergodic IP polynomial Szemer´ edi theorem. Mem. Amer. Math. Soc., 146(695):viii+106, 2000. 3, 4, 13","work_id":"173f7f55-125c-45e4-8b50-9f9615b6b832","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":32,"snapshot_sha256":"eea5f58e35b07a7fb0432bfe600a043e772cd66f1918e0488bef2124619b5da6","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"f16b68528335316e91b321610f3f73e7f1b4886eca9301c40c99a3de1d77c640"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}