{"paper":{"title":"A robust version of Freiman's $3k-4$ Theorem and applications","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Max Wenqiang Xu, Xuancheng Shao","submitted_at":"2017-11-29T19:09:34Z","abstract_excerpt":"We prove a robust version of Freiman's $3k - 4$ theorem on the restricted sumset $A+_{\\Gamma}B$, which applies when the doubling constant is at most $\\tfrac{3+\\sqrt{5}}{2}$ in general and at most $3$ in the special case when $A = -B$. As applications, we derive robust results with other types of assumptions on popular sums, and structure theorems for sets satisfying almost equalities in discrete and continuous versions of the Riesz-Sobolev inequality."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.11060","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1711.11060/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}