{"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"}