{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2007:2QAV4AZPQWBBHHA2DBCVEQL45F","short_pith_number":"pith:2QAV4AZP","schema_version":"1.0","canonical_sha256":"d4015e032f8582139c1a184552417ce9697aa08952c3ec74e031e5289b025c2e","source":{"kind":"arxiv","id":"math/0703503","version":2},"attestation_state":"computed","paper":{"title":"The Littlewood-Offord Problem and invertibility of random matrices","license":"","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Mark Rudelson, Roman Vershynin","submitted_at":"2007-03-16T21:58:20Z","abstract_excerpt":"We prove two basic conjectures on the distribution of the smallest singular value of random n times n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n^{-1/2}, which is optimal for Gaussian matrices. Moreover, we give a optimal estimate on the tail probability. This comes as a consequence of a new and essentially sharp estimate in the Littlewood-Offord problem: for i.i.d. random variables X_k and real numbers a_k, determine the probability P that the sum of a_k X_k lies near some number v. For arbitrary coefficients a_k "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/0703503","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2007-03-16T21:58:20Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"1c4b8b9d92f90f60ab175ec898bbd537032e11da37273e825df58f08fc448764","abstract_canon_sha256":"b81c4124eb0a53abf8f831f93cf92e3702547ed613cad189aa387817be11aa58"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:54:09.240894Z","signature_b64":"GOrtbHyumoFJRa0F5h9/u13DzObDKR5+NHhCmPsuR3sw3e4aaq2XZEz2s2qlHQFIdNt8c+6PpEYLkEQTy7PLAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d4015e032f8582139c1a184552417ce9697aa08952c3ec74e031e5289b025c2e","last_reissued_at":"2026-05-18T00:54:09.240329Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:54:09.240329Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Littlewood-Offord Problem and invertibility of random matrices","license":"","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Mark Rudelson, Roman Vershynin","submitted_at":"2007-03-16T21:58:20Z","abstract_excerpt":"We prove two basic conjectures on the distribution of the smallest singular value of random n times n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n^{-1/2}, which is optimal for Gaussian matrices. Moreover, we give a optimal estimate on the tail probability. This comes as a consequence of a new and essentially sharp estimate in the Littlewood-Offord problem: for i.i.d. random variables X_k and real numbers a_k, determine the probability P that the sum of a_k X_k lies near some number v. For arbitrary coefficients a_k "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0703503","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0703503","created_at":"2026-05-18T00:54:09.240419+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0703503v2","created_at":"2026-05-18T00:54:09.240419+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0703503","created_at":"2026-05-18T00:54:09.240419+00:00"},{"alias_kind":"pith_short_12","alias_value":"2QAV4AZPQWBB","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_16","alias_value":"2QAV4AZPQWBBHHA2","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_8","alias_value":"2QAV4AZP","created_at":"2026-05-18T12:25:54.717736+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/2QAV4AZPQWBBHHA2DBCVEQL45F","json":"https://pith.science/pith/2QAV4AZPQWBBHHA2DBCVEQL45F.json","graph_json":"https://pith.science/api/pith-number/2QAV4AZPQWBBHHA2DBCVEQL45F/graph.json","events_json":"https://pith.science/api/pith-number/2QAV4AZPQWBBHHA2DBCVEQL45F/events.json","paper":"https://pith.science/paper/2QAV4AZP"},"agent_actions":{"view_html":"https://pith.science/pith/2QAV4AZPQWBBHHA2DBCVEQL45F","download_json":"https://pith.science/pith/2QAV4AZPQWBBHHA2DBCVEQL45F.json","view_paper":"https://pith.science/paper/2QAV4AZP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0703503&json=true","fetch_graph":"https://pith.science/api/pith-number/2QAV4AZPQWBBHHA2DBCVEQL45F/graph.json","fetch_events":"https://pith.science/api/pith-number/2QAV4AZPQWBBHHA2DBCVEQL45F/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2QAV4AZPQWBBHHA2DBCVEQL45F/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2QAV4AZPQWBBHHA2DBCVEQL45F/action/storage_attestation","attest_author":"https://pith.science/pith/2QAV4AZPQWBBHHA2DBCVEQL45F/action/author_attestation","sign_citation":"https://pith.science/pith/2QAV4AZPQWBBHHA2DBCVEQL45F/action/citation_signature","submit_replication":"https://pith.science/pith/2QAV4AZPQWBBHHA2DBCVEQL45F/action/replication_record"}},"created_at":"2026-05-18T00:54:09.240419+00:00","updated_at":"2026-05-18T00:54:09.240419+00:00"}