{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:3D6P5IA53S4OPD7CG5VTPMYNYT","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":"9536e68629d386c0a189c198100793d42d48b2fc65c7614a5b23152501632296","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2017-11-29T19:09:34Z","title_canon_sha256":"00055f2036ee78fc3614049495a85696d501b5d3deed8c20a815ff3ab10e5923"},"schema_version":"1.0","source":{"id":"1711.11060","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.11060","created_at":"2026-07-05T00:44:35Z"},{"alias_kind":"arxiv_version","alias_value":"1711.11060v3","created_at":"2026-07-05T00:44:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.11060","created_at":"2026-07-05T00:44:35Z"},{"alias_kind":"pith_short_12","alias_value":"3D6P5IA53S4O","created_at":"2026-07-05T00:44:35Z"},{"alias_kind":"pith_short_16","alias_value":"3D6P5IA53S4OPD7C","created_at":"2026-07-05T00:44:35Z"},{"alias_kind":"pith_short_8","alias_value":"3D6P5IA5","created_at":"2026-07-05T00:44:35Z"}],"graph_snapshots":[{"event_id":"sha256:3e78003f3c9a057430089d93e3df27506fb6f2dbf07cd066130518f049c834c3","target":"graph","created_at":"2026-07-05T00:44:35Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1711.11060/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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.","authors_text":"Max Wenqiang Xu, Xuancheng Shao","cross_cats":["math.CO"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2017-11-29T19:09:34Z","title":"A robust version of Freiman's $3k-4$ Theorem and applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.11060","kind":"arxiv","version":3},"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:b6606e3422174b928173b675d66366154b9497b9816fbad92c00e3b95d522285","target":"record","created_at":"2026-07-05T00:44:35Z","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":"9536e68629d386c0a189c198100793d42d48b2fc65c7614a5b23152501632296","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2017-11-29T19:09:34Z","title_canon_sha256":"00055f2036ee78fc3614049495a85696d501b5d3deed8c20a815ff3ab10e5923"},"schema_version":"1.0","source":{"id":"1711.11060","kind":"arxiv","version":3}},"canonical_sha256":"d8fcfea01ddcb8e78fe2376b37b30dc4f5019defa88bd2114b3e7faf55058f34","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d8fcfea01ddcb8e78fe2376b37b30dc4f5019defa88bd2114b3e7faf55058f34","first_computed_at":"2026-07-05T00:44:35.554817Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:44:35.554817Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7JlJvEyIIm2/C37uPsjgHo0iE17V/wsiQDuZL1j0SYzOKHixp+gz9I/Cu0MZDqgnbEz3M5JfsXmqQkCsEUZRAw==","signature_status":"signed_v1","signed_at":"2026-07-05T00:44:35.555169Z","signed_message":"canonical_sha256_bytes"},"source_id":"1711.11060","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b6606e3422174b928173b675d66366154b9497b9816fbad92c00e3b95d522285","sha256:3e78003f3c9a057430089d93e3df27506fb6f2dbf07cd066130518f049c834c3"],"state_sha256":"3d1fb20d493822bb6d5d648619f0b352a8cafbe9d0f1a4a4744e3d6a7499d06b"}