{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:75EYMRD7RNZTNLUAN2DH22Q63W","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"a7d7b2f2d471c438738505e0e806663f12479e8bd287a287fb5cf9d8d2ffcb14","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2008-05-31T16:35:34Z","title_canon_sha256":"c3673c053da349224e00a101920fb3430487c6ffae1524aca581c7c34d43577b"},"schema_version":"1.0","source":{"id":"0806.0092","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0806.0092","created_at":"2026-05-18T02:15:52Z"},{"alias_kind":"arxiv_version","alias_value":"0806.0092v1","created_at":"2026-05-18T02:15:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0806.0092","created_at":"2026-05-18T02:15:52Z"},{"alias_kind":"pith_short_12","alias_value":"75EYMRD7RNZT","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"75EYMRD7RNZTNLUA","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"75EYMRD7","created_at":"2026-05-18T12:25:56Z"}],"graph_snapshots":[{"event_id":"sha256:6de5a944c6d65d409b36046c89e8f8601702ec3be09d084afde49e1f2fde7ec1","target":"graph","created_at":"2026-05-18T02:15:52Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"For a subset A of a finite abelian group G we define Sigma(A)={sum_{a\\in B}a:B\\subset A}. In the case that Sigma(A) has trivial stabiliser, one may deduce that the size of Sigma(A) is at least quadratic in |A|; the bound |Sigma(A)|>= |A|^{2}/64 has recently been obtained by De Vos, Goddyn, Mohar and Samal. We improve this bound to the asymptotically best possible result |Sigma(A)|>= (1/4-o(1))|A|^{2}.\n  We also study a related problem in which A is any subset of Z_{n} with all elements of A coprime to n; it has recently been shown, by Vu, that if such a set A has the property Sigma(A) is not Z","authors_text":"Simon Griffiths","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2008-05-31T16:35:34Z","title":"Asymptotically tight bounds on subset sums"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0806.0092","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a474f7aea260f4d71a8cb80b7f066b6865667bc6be3d04fed3d428c35b62ee3e","target":"record","created_at":"2026-05-18T02:15:52Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"a7d7b2f2d471c438738505e0e806663f12479e8bd287a287fb5cf9d8d2ffcb14","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2008-05-31T16:35:34Z","title_canon_sha256":"c3673c053da349224e00a101920fb3430487c6ffae1524aca581c7c34d43577b"},"schema_version":"1.0","source":{"id":"0806.0092","kind":"arxiv","version":1}},"canonical_sha256":"ff4986447f8b7336ae806e867d6a1edd990c4d99c34c73e0474277ca2c40b2d9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ff4986447f8b7336ae806e867d6a1edd990c4d99c34c73e0474277ca2c40b2d9","first_computed_at":"2026-05-18T02:15:52.075483Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:15:52.075483Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QIKoPTsdd6rtbkgtUd2wvJSUOzcfl4Vg55e/tnPuJNE2UYJFa8VSvbOAReyhpvAws01dVcZgBGAcWVOA+br7BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:15:52.075852Z","signed_message":"canonical_sha256_bytes"},"source_id":"0806.0092","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a474f7aea260f4d71a8cb80b7f066b6865667bc6be3d04fed3d428c35b62ee3e","sha256:6de5a944c6d65d409b36046c89e8f8601702ec3be09d084afde49e1f2fde7ec1"],"state_sha256":"0c5a506a5d8c4a18828aa38ab2e4a9c34e8ed1002c30ffea74c187c3d38e67b1"}