{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:S5EZZJNXBJF632X2ORUFTCTLS4","short_pith_number":"pith:S5EZZJNX","schema_version":"1.0","canonical_sha256":"97499ca5b70a4bedeafa7468598a6b9713d71ac13e0ffd531f0bce8948e4b18f","source":{"kind":"arxiv","id":"1305.2431","version":1},"attestation_state":"computed","paper":{"title":"Large restricted sumsets in general abelian group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alain Plagne, Susana C. Lopez, Yahya Ould Hamidoune","submitted_at":"2013-05-10T20:19:34Z","abstract_excerpt":"Let A, B and S be three subsets of a finite Abelian group G. The restricted sumset of A and B with respect to S is defined as A\\wedge^{S} B= {a+b: a in A, b in B and a-b not in S}. Let L_S=max_{z in G}| {(x,y): x,y in G, x+y=z and x-y in S}|. A simple application of the pigeonhole principle shows that |A|+|B|>|G|+L_S implies A\\wedge^S B=G. We then prove that if |A|+|B|=|G|+L_S then |A\\wedge^S B|>= |G|-2|S|. We also characterize the triples of sets (A,B,S) such that |A|+|B|=|G|+L_S and |A\\wedge^S B|= |G|-2|S|. Moreover, in this case, we also provide the structure of the set G\\setminus (A\\wedge^"},"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":"1305.2431","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-05-10T20:19:34Z","cross_cats_sorted":[],"title_canon_sha256":"d9e9eb21773a1633ccb87af8a6ce8f849bcc0ce9f3bc48709144ac364908b4db","abstract_canon_sha256":"462ea18613335b3c73bd8d94f7188600474c2c029c73974506e40d3d57aec734"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:25:57.882808Z","signature_b64":"QVXzWvk3qJvigoSGJBfNJdjxBYN5L4UBNo/ZimT19bmLte3inQrQUeZPdVBbV6AmlbWSrLhwmfCrfkHT7lOKBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"97499ca5b70a4bedeafa7468598a6b9713d71ac13e0ffd531f0bce8948e4b18f","last_reissued_at":"2026-05-18T03:25:57.882166Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:25:57.882166Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Large restricted sumsets in general abelian group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alain Plagne, Susana C. Lopez, Yahya Ould Hamidoune","submitted_at":"2013-05-10T20:19:34Z","abstract_excerpt":"Let A, B and S be three subsets of a finite Abelian group G. The restricted sumset of A and B with respect to S is defined as A\\wedge^{S} B= {a+b: a in A, b in B and a-b not in S}. Let L_S=max_{z in G}| {(x,y): x,y in G, x+y=z and x-y in S}|. A simple application of the pigeonhole principle shows that |A|+|B|>|G|+L_S implies A\\wedge^S B=G. We then prove that if |A|+|B|=|G|+L_S then |A\\wedge^S B|>= |G|-2|S|. We also characterize the triples of sets (A,B,S) such that |A|+|B|=|G|+L_S and |A\\wedge^S B|= |G|-2|S|. 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