{"paper":{"title":"Intersections of sumsets in additive number theory","license":"http://creativecommons.org/licenses/by/4.0/","headline":"In an additive abelian semigroup, the h-fold sumset of the intersection of a strictly decreasing sequence of sets equals the intersection of the h-fold sumsets only under certain conditions on the sequence.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Melvyn B. Nathanson","submitted_at":"2025-12-29T16:14:09Z","abstract_excerpt":"Let $A$ be a subset of an additive abelian semigroup $S$ and let $hA$ be the $h$-fold sumset of $A$. The following question is considered: Let $(A_q)_{q=1}^{\\infty}$ be a strictly decreasing sequence of sets in $S$ and let $A = \\bigcap_{q=1}^{\\infty} A_q$. When does one have \\[ hA = \\bigcap_{q=1}^{\\infty} hA_q \\] for some or all $h \\geq 2$?"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The question is considered: for a strictly decreasing sequence (A_q) with A = intersection A_q, when does hA = intersection hA_q hold for some or all h ≥ 2 in an additive abelian semigroup S.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The sequence (A_q) is strictly decreasing and S is an additive abelian semigroup, with no further restrictions stated on the sets or the semigroup operation.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Conditions are examined for equality hA = intersection hA_q where A is the intersection of a strictly decreasing sequence of sets A_q in an additive abelian semigroup.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"In an additive abelian semigroup, the h-fold sumset of the intersection of a strictly decreasing sequence of sets equals the intersection of the h-fold sumsets only under certain conditions on the sequence.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"5f1587c3f2516baba4dc8eecc2e6c1c34ae3505d1bfcabe2c1ff2d3e53954f25"},"source":{"id":"2512.23574","kind":"arxiv","version":3},"verdict":{"id":"f93b6086-a69e-4d83-a20b-e61ebf9aa632","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T19:13:22.689247Z","strongest_claim":"The question is considered: for a strictly decreasing sequence (A_q) with A = intersection A_q, when does hA = intersection hA_q hold for some or all h ≥ 2 in an additive abelian semigroup S.","one_line_summary":"Conditions are examined for equality hA = intersection hA_q where A is the intersection of a strictly decreasing sequence of sets A_q in an additive abelian semigroup.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The sequence (A_q) is strictly decreasing and S is an additive abelian semigroup, with no further restrictions stated on the sets or the semigroup operation.","pith_extraction_headline":"In an additive abelian semigroup, the h-fold sumset of the intersection of a strictly decreasing sequence of sets equals the intersection of the h-fold sumsets only under certain conditions on the sequence."},"references":{"count":13,"sample":[{"doi":"","year":1975,"title":"Erd˝ os and M","work_id":"63e23077-b392-4e9d-a9c5-b428ec55d88a","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"J. Fox, N. Kravitz, and S. Zhang, Finer control on relative sizes of iterated sumsets, arXiv: 2506.05691","work_id":"904e2521-4aa5-4a87-9c80-4f4d0e032283","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"N. Kravitz, Relative sizes of iterated sumsets, J. Number Theory 272 (2025), 113–128,","work_id":"f07a4a27-0fd1-486b-a316-2c8c38ec8dc9","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1974,"title":"M. B. Nathanson, Minimal bases and maximal nonbases in additive number theory, J. Number Theory 6 (1974), 324–333","work_id":"28a51239-40df-4666-afcc-c718a08accab","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"M. B. Nathanson, Inverse problems for sumset sizes of finite sets of integers, Fibonacci Quar- terly (2025), to appear. arXiv:2411.02365","work_id":"ae2a000b-a702-43b1-91ef-4a941217d794","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":13,"snapshot_sha256":"183070136e560246b9794d7064a06dc5243b9a57f7a6d0574364e82f28bcb13e","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"d063940538fadf24b000e24421d1c22c6415a1714b693c979bd508a9d0fa32d8"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}