{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:5256D2ITYEZ5YJAZD7K6Y3UQKA","short_pith_number":"pith:5256D2IT","schema_version":"1.0","canonical_sha256":"eebbe1e913c133dc24191fd5ec6e905008739b09e2fbd85421f7fa71bc3f2ec8","source":{"kind":"arxiv","id":"1105.5153","version":1},"attestation_state":"computed","paper":{"title":"Inverse Additive Problems for Minkowski Sumsets I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"D. Grynkiewicz, G. A. Freiman, O. Serra, Y. V. Stanchescu","submitted_at":"2011-05-25T21:13:58Z","abstract_excerpt":"We give the structure of discrete two-dimensional finite sets $A,\\,B\\subseteq \\R^2$ which are extremal for the recently obtained inequality $|A+B|\\ge (\\frac{|A|}{m}+\\frac{|B|}{n}-1)(m+n-1)$, where $m$ and $n$ are the minimum number of parallel lines covering $A$ and $B$ respectively. Via compression techniques, the above bound also holds when $m$ is the maximal number of points of $A$ contained in one of the parallel lines covering $A$ and $n$ is the maximal number of points of $B$ contained in one of the parallel lines covering $B$. When $m,\\,n\\geq 2$, we are able to characterize the case of "},"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":"1105.5153","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-05-25T21:13:58Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"847f3bd877126ed3645af1bebc86db0a753a32dea743e4b94c91716e289ef861","abstract_canon_sha256":"7f24a1271acf87f950ef20741d42f99f7fe9b555df096474fd75d67f5709a2a4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:02.394597Z","signature_b64":"OIoGHVe6cOy7wkDOo1DdtctZFXbMyewEVJ2NzsC+zQxwfS1X1t1S2JLtyKA3ga3FUWWGIcG9XYS1Tp3cWBCgCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eebbe1e913c133dc24191fd5ec6e905008739b09e2fbd85421f7fa71bc3f2ec8","last_reissued_at":"2026-05-18T03:07:02.394085Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:02.394085Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Inverse Additive Problems for Minkowski Sumsets I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"D. Grynkiewicz, G. A. Freiman, O. Serra, Y. V. Stanchescu","submitted_at":"2011-05-25T21:13:58Z","abstract_excerpt":"We give the structure of discrete two-dimensional finite sets $A,\\,B\\subseteq \\R^2$ which are extremal for the recently obtained inequality $|A+B|\\ge (\\frac{|A|}{m}+\\frac{|B|}{n}-1)(m+n-1)$, where $m$ and $n$ are the minimum number of parallel lines covering $A$ and $B$ respectively. Via compression techniques, the above bound also holds when $m$ is the maximal number of points of $A$ contained in one of the parallel lines covering $A$ and $n$ is the maximal number of points of $B$ contained in one of the parallel lines covering $B$. When $m,\\,n\\geq 2$, we are able to characterize the case of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.5153","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":""},"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":"1105.5153","created_at":"2026-05-18T03:07:02.394167+00:00"},{"alias_kind":"arxiv_version","alias_value":"1105.5153v1","created_at":"2026-05-18T03:07:02.394167+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.5153","created_at":"2026-05-18T03:07:02.394167+00:00"},{"alias_kind":"pith_short_12","alias_value":"5256D2ITYEZ5","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_16","alias_value":"5256D2ITYEZ5YJAZ","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_8","alias_value":"5256D2IT","created_at":"2026-05-18T12:26:20.644004+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/5256D2ITYEZ5YJAZD7K6Y3UQKA","json":"https://pith.science/pith/5256D2ITYEZ5YJAZD7K6Y3UQKA.json","graph_json":"https://pith.science/api/pith-number/5256D2ITYEZ5YJAZD7K6Y3UQKA/graph.json","events_json":"https://pith.science/api/pith-number/5256D2ITYEZ5YJAZD7K6Y3UQKA/events.json","paper":"https://pith.science/paper/5256D2IT"},"agent_actions":{"view_html":"https://pith.science/pith/5256D2ITYEZ5YJAZD7K6Y3UQKA","download_json":"https://pith.science/pith/5256D2ITYEZ5YJAZD7K6Y3UQKA.json","view_paper":"https://pith.science/paper/5256D2IT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1105.5153&json=true","fetch_graph":"https://pith.science/api/pith-number/5256D2ITYEZ5YJAZD7K6Y3UQKA/graph.json","fetch_events":"https://pith.science/api/pith-number/5256D2ITYEZ5YJAZD7K6Y3UQKA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5256D2ITYEZ5YJAZD7K6Y3UQKA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5256D2ITYEZ5YJAZD7K6Y3UQKA/action/storage_attestation","attest_author":"https://pith.science/pith/5256D2ITYEZ5YJAZD7K6Y3UQKA/action/author_attestation","sign_citation":"https://pith.science/pith/5256D2ITYEZ5YJAZD7K6Y3UQKA/action/citation_signature","submit_replication":"https://pith.science/pith/5256D2ITYEZ5YJAZD7K6Y3UQKA/action/replication_record"}},"created_at":"2026-05-18T03:07:02.394167+00:00","updated_at":"2026-05-18T03:07:02.394167+00:00"}