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We prove that if $\\langle Y \\rangle$ is commutative, $Y$ is non-empty, and $X+2Y \\neq X + Y + y$ for some $y \\in Y$, then $$ |X+Y| \\ge |X|+\\min(\\gamma(Y), |Y| - 1). $$ Actually, this is obtained from a more general result, which improves on previous work of the author on sumsets in cancellativ"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1604.02136","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-07T19:46:19Z","cross_cats_sorted":["math.GR","math.NT"],"title_canon_sha256":"50817ee96973ae314ca6b9794c6e62d594837f7925d02140eda95e66b4fe6153","abstract_canon_sha256":"40c1b197091a35eea84699b4335ccd5054c867289e8c8fa88b0645f3b4319a6c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:38.123371Z","signature_b64":"JjC7TdC6KEyUGR5iFWsCzveucdUU/oSb73oZzX7ukcaXo4NgqYIMGM8ZHHhXPM+QSHCqNY28wDWA6bpqMllsCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3aac2a8a2fe0f27032c68515f9e86c6b157bb67de844af2703fc6fe1e2bc926b","last_reissued_at":"2026-05-18T01:15:38.122719Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:38.122719Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cauchy-Davenport type inequalities, I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.NT"],"primary_cat":"math.CO","authors_text":"Salvatore Tringali","submitted_at":"2016-04-07T19:46:19Z","abstract_excerpt":"Let $\\mathbb G = (G, +)$ be a group (either abelian or not). Given $X, Y \\subseteq G$, we denote by $\\langle Y \\rangle$ the subsemigroup of $\\mathbb G$ generated by $Y$, and we set $$\\gamma(Y) := \\sup_{y_0 \\in Y} \\inf_{y_0 \\ne y \\in Y} {\\rm ord}(y - y_0)$$ if $|Y| \\ge 2$ and $\\gamma(Y) := |Y|$ otherwise. We prove that if $\\langle Y \\rangle$ is commutative, $Y$ is non-empty, and $X+2Y \\neq X + Y + y$ for some $y \\in Y$, then $$ |X+Y| \\ge |X|+\\min(\\gamma(Y), |Y| - 1). $$ Actually, this is obtained from a more general result, which improves on previous work of the author on sumsets in cancellativ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.02136","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1604.02136","created_at":"2026-05-18T01:15:38.122816+00:00"},{"alias_kind":"arxiv_version","alias_value":"1604.02136v2","created_at":"2026-05-18T01:15:38.122816+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.02136","created_at":"2026-05-18T01:15:38.122816+00:00"},{"alias_kind":"pith_short_12","alias_value":"HKWCVCRP4DZH","created_at":"2026-05-18T12:30:19.053100+00:00"},{"alias_kind":"pith_short_16","alias_value":"HKWCVCRP4DZHAMWG","created_at":"2026-05-18T12:30:19.053100+00:00"},{"alias_kind":"pith_short_8","alias_value":"HKWCVCRP","created_at":"2026-05-18T12:30:19.053100+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HKWCVCRP4DZHAMWGQUK7T2DMNM","json":"https://pith.science/pith/HKWCVCRP4DZHAMWGQUK7T2DMNM.json","graph_json":"https://pith.science/api/pith-number/HKWCVCRP4DZHAMWGQUK7T2DMNM/graph.json","events_json":"https://pith.science/api/pith-number/HKWCVCRP4DZHAMWGQUK7T2DMNM/events.json","paper":"https://pith.science/paper/HKWCVCRP"},"agent_actions":{"view_html":"https://pith.science/pith/HKWCVCRP4DZHAMWGQUK7T2DMNM","download_json":"https://pith.science/pith/HKWCVCRP4DZHAMWGQUK7T2DMNM.json","view_paper":"https://pith.science/paper/HKWCVCRP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1604.02136&json=true","fetch_graph":"https://pith.science/api/pith-number/HKWCVCRP4DZHAMWGQUK7T2DMNM/graph.json","fetch_events":"https://pith.science/api/pith-number/HKWCVCRP4DZHAMWGQUK7T2DMNM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HKWCVCRP4DZHAMWGQUK7T2DMNM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HKWCVCRP4DZHAMWGQUK7T2DMNM/action/storage_attestation","attest_author":"https://pith.science/pith/HKWCVCRP4DZHAMWGQUK7T2DMNM/action/author_attestation","sign_citation":"https://pith.science/pith/HKWCVCRP4DZHAMWGQUK7T2DMNM/action/citation_signature","submit_replication":"https://pith.science/pith/HKWCVCRP4DZHAMWGQUK7T2DMNM/action/replication_record"}},"created_at":"2026-05-18T01:15:38.122816+00:00","updated_at":"2026-05-18T01:15:38.122816+00:00"}