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A is Freiman isomorphic to a set in either the form of {0,2,...,2k} union B union {n} for some non-negative integer k at most n/2 -2 or the form of {0} union C union D union {n}, where n=2|A|-2, B is left dense in [2k,n-1], C is right dense in [1,u] for some u in [4,n-6], D i"},"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":"1308.0741","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.NT","submitted_at":"2013-08-03T20:51:23Z","cross_cats_sorted":[],"title_canon_sha256":"f8f83345e4ebe3dcb7a12cd95c3d5e9636cdb766231bb81364e7e7c1147df051","abstract_canon_sha256":"1ac9f74314b91da6ec2c11e9a9297dc5b8b2655e8c95e77b7b2554726d41a5db"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:16:48.337823Z","signature_b64":"ZzOSJH0wIjv/UoVP66H6hrqn7SV502qK0qpJFtCg2sAiDBvXAoGKHS/YuTsc444DMW6k8AXbUiOd4H106PK1DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"56be992ce0490bb4aeb0b16bf5002b57354e13f9c00d6f46799eeaae3b57e109","last_reissued_at":"2026-05-18T03:16:48.337349Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:16:48.337349Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Detailed Structure for Freiman's 3k-3 Theorem","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Renling Jin","submitted_at":"2013-08-03T20:51:23Z","abstract_excerpt":"Let A be a finite set of integers. We prove that if |A| is at least 2 and |A+A| is 3|A|-3, then one of the following is true:\n  1. A is a bi-arithmetic progression;\n  2. A+A contains an arithmetic progression of length 2|A|-1;\n  3. |A| is 6 and A is Freiman isomorphic to the set {(0,0),(0,1),(0,2),(1,0),(1,1),(2,0)};\n  4. A is Freiman isomorphic to a set in either the form of {0,2,...,2k} union B union {n} for some non-negative integer k at most n/2 -2 or the form of {0} union C union D union {n}, where n=2|A|-2, B is left dense in [2k,n-1], C is right dense in [1,u] for some u in [4,n-6], D i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.0741","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":"1308.0741","created_at":"2026-05-18T03:16:48.337417+00:00"},{"alias_kind":"arxiv_version","alias_value":"1308.0741v1","created_at":"2026-05-18T03:16:48.337417+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.0741","created_at":"2026-05-18T03:16:48.337417+00:00"},{"alias_kind":"pith_short_12","alias_value":"K27JSLHAJEF3","created_at":"2026-05-18T12:27:49.015174+00:00"},{"alias_kind":"pith_short_16","alias_value":"K27JSLHAJEF3JLVQ","created_at":"2026-05-18T12:27:49.015174+00:00"},{"alias_kind":"pith_short_8","alias_value":"K27JSLHA","created_at":"2026-05-18T12:27:49.015174+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/K27JSLHAJEF3JLVQWFV7KABLK4","json":"https://pith.science/pith/K27JSLHAJEF3JLVQWFV7KABLK4.json","graph_json":"https://pith.science/api/pith-number/K27JSLHAJEF3JLVQWFV7KABLK4/graph.json","events_json":"https://pith.science/api/pith-number/K27JSLHAJEF3JLVQWFV7KABLK4/events.json","paper":"https://pith.science/paper/K27JSLHA"},"agent_actions":{"view_html":"https://pith.science/pith/K27JSLHAJEF3JLVQWFV7KABLK4","download_json":"https://pith.science/pith/K27JSLHAJEF3JLVQWFV7KABLK4.json","view_paper":"https://pith.science/paper/K27JSLHA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1308.0741&json=true","fetch_graph":"https://pith.science/api/pith-number/K27JSLHAJEF3JLVQWFV7KABLK4/graph.json","fetch_events":"https://pith.science/api/pith-number/K27JSLHAJEF3JLVQWFV7KABLK4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/K27JSLHAJEF3JLVQWFV7KABLK4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/K27JSLHAJEF3JLVQWFV7KABLK4/action/storage_attestation","attest_author":"https://pith.science/pith/K27JSLHAJEF3JLVQWFV7KABLK4/action/author_attestation","sign_citation":"https://pith.science/pith/K27JSLHAJEF3JLVQWFV7KABLK4/action/citation_signature","submit_replication":"https://pith.science/pith/K27JSLHAJEF3JLVQWFV7KABLK4/action/replication_record"}},"created_at":"2026-05-18T03:16:48.337417+00:00","updated_at":"2026-05-18T03:16:48.337417+00:00"}