{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:T2BPFDHK2MY6OVN3FYP2F75ZKM","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":"96897bfb5cc0c9ff043744314f156fae3198d65fdbc60119aca504358d07e0c1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-06-11T23:24:03Z","title_canon_sha256":"a83e367d5a77aa66bea6a5030aa4205c63c90de3f08eda52185970d19cd88959"},"schema_version":"1.0","source":{"id":"1106.2267","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1106.2267","created_at":"2026-05-18T03:58:48Z"},{"alias_kind":"arxiv_version","alias_value":"1106.2267v2","created_at":"2026-05-18T03:58:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.2267","created_at":"2026-05-18T03:58:48Z"},{"alias_kind":"pith_short_12","alias_value":"T2BPFDHK2MY6","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"T2BPFDHK2MY6OVN3","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"T2BPFDHK","created_at":"2026-05-18T12:26:41Z"}],"graph_snapshots":[{"event_id":"sha256:7c59bbb59029b0b70e5f012c25108d405e85c47af5f9caa7b2cc61296b9067ff","target":"graph","created_at":"2026-05-18T03:58:48Z","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":"A corollary of Kneser's theorem, one sees that any finite non-empty subset $A$ of an abelian group $G = (G,+)$ with $|A + A| \\leq (2-\\eps) |A|$ can be covered by at most $\\frac{2}{\\eps}-1$ translates of a finite group $H$ of cardinality at most $(2-\\eps)|A|$. Using some arguments of Hamidoune, we establish an analogue in the noncommutative setting. Namely, if $A$ is a finite non-empty subset of a nonabelian group $G = (G,\\cdot)$ such that $|A \\cdot A| \\leq (2-\\eps) |A|$, then $A$ is either contained in a right-coset of a finite group $H$ of cardinality at most $\\frac{2}{\\eps}|A|$, or can be co","authors_text":"Terence Tao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-06-11T23:24:03Z","title":"Noncommutative sets of small doubling"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.2267","kind":"arxiv","version":2},"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":"sha256:93a83e6880f654e263a7f2aabddd7609819f665caa1d4b4bf558ac1a0426f888","target":"record","created_at":"2026-05-18T03:58:48Z","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":"96897bfb5cc0c9ff043744314f156fae3198d65fdbc60119aca504358d07e0c1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-06-11T23:24:03Z","title_canon_sha256":"a83e367d5a77aa66bea6a5030aa4205c63c90de3f08eda52185970d19cd88959"},"schema_version":"1.0","source":{"id":"1106.2267","kind":"arxiv","version":2}},"canonical_sha256":"9e82f28cead331e755bb2e1fa2ffb9530c6ceb842037b95dc00d7c4935f0281b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9e82f28cead331e755bb2e1fa2ffb9530c6ceb842037b95dc00d7c4935f0281b","first_computed_at":"2026-05-18T03:58:48.735063Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:58:48.735063Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZwDYDJwaf4nWtMV67zhwmIrQFAKSiqEsegi3CjVmnuuN74iCefD2wvkKz++H7uTxn9+p9FE0VnTL7kg5ETiXBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:58:48.735578Z","signed_message":"canonical_sha256_bytes"},"source_id":"1106.2267","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:93a83e6880f654e263a7f2aabddd7609819f665caa1d4b4bf558ac1a0426f888","sha256:7c59bbb59029b0b70e5f012c25108d405e85c47af5f9caa7b2cc61296b9067ff"],"state_sha256":"6c1c3b8ca272b3749a31f038621945c06f5162c773daec733a34cc58e56837dd"}