{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:7OUJXZZQFCQHJPGAS2MNRASN4Z","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":"e0aa3d9c50c6731a7dd0652ed392fd28638e7309de292c2915eeb14bfe07298e","cross_cats_sorted":["math.CO","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-06-05T16:16:12Z","title_canon_sha256":"dcf0f64215954e8eecd49f66ed94547f83b57de52a20d38aed2785413f393e31"},"schema_version":"1.0","source":{"id":"1506.01958","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.01958","created_at":"2026-05-18T01:35:43Z"},{"alias_kind":"arxiv_version","alias_value":"1506.01958v2","created_at":"2026-05-18T01:35:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.01958","created_at":"2026-05-18T01:35:43Z"},{"alias_kind":"pith_short_12","alias_value":"7OUJXZZQFCQH","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_16","alias_value":"7OUJXZZQFCQHJPGA","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_8","alias_value":"7OUJXZZQ","created_at":"2026-05-18T12:29:10Z"}],"graph_snapshots":[{"event_id":"sha256:e8dcf57933a9bd1babfadce0db5cb858af3dd789f19f77a00dd367f3c534afec","target":"graph","created_at":"2026-05-18T01:35:43Z","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":"In 1943, Littlewood and Offord proved the first anti-concentration result for sums of independent random variables. Their result has since then been strengthened and generalized by generations of researchers, with applications in several areas of mathematics.\n  In this paper, we present the first non-abelian analogue of Littlewood-Offord result, a sharp anti-concentration inequality for products of independent random variables.","authors_text":"Pham H. Tiep, Van H. Vu","cross_cats":["math.CO","math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-06-05T16:16:12Z","title":"Non-abelian Littlewood-Offord inequalities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.01958","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":"sha256:94116b95caa07197ab83db61456ddd530b474bb3569db8be1bbf1986712d62c4","target":"record","created_at":"2026-05-18T01:35:43Z","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":"e0aa3d9c50c6731a7dd0652ed392fd28638e7309de292c2915eeb14bfe07298e","cross_cats_sorted":["math.CO","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-06-05T16:16:12Z","title_canon_sha256":"dcf0f64215954e8eecd49f66ed94547f83b57de52a20d38aed2785413f393e31"},"schema_version":"1.0","source":{"id":"1506.01958","kind":"arxiv","version":2}},"canonical_sha256":"fba89be73028a074bcc09698d8824de6415dfc7a77d50d487d32169ce23d2313","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fba89be73028a074bcc09698d8824de6415dfc7a77d50d487d32169ce23d2313","first_computed_at":"2026-05-18T01:35:43.287490Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:35:43.287490Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ESBbFbFidDX2xBoza+BIhE22cWGVorfDzqLL1xrXxgvlnysXUhRmYYrZY0WOsgeScZduoV6Pa6/4Ee0KzrReBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:35:43.287927Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.01958","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:94116b95caa07197ab83db61456ddd530b474bb3569db8be1bbf1986712d62c4","sha256:e8dcf57933a9bd1babfadce0db5cb858af3dd789f19f77a00dd367f3c534afec"],"state_sha256":"8f1663ec5724716ef376a643f4acd3f991ff0b2f93845065512fbb1db7b60de1"}