{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:434FULTRYQAXPL6AKDJGMCTH2H","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":"6bc5e88fc6ecff242be98634b9c28245d16bf549de35bcfbca31bfd946047346","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-31T21:39:17Z","title_canon_sha256":"a0232d57a5600b567ba690a46312adcd82ff1efa3f0569d488c77a8e9922852e"},"schema_version":"1.0","source":{"id":"1301.0019","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1301.0019","created_at":"2026-05-18T03:37:28Z"},{"alias_kind":"arxiv_version","alias_value":"1301.0019v1","created_at":"2026-05-18T03:37:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.0019","created_at":"2026-05-18T03:37:28Z"},{"alias_kind":"pith_short_12","alias_value":"434FULTRYQAX","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"434FULTRYQAXPL6A","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"434FULTR","created_at":"2026-05-18T12:26:53Z"}],"graph_snapshots":[{"event_id":"sha256:8e6d35788644af34f7e3fe60825e368679db27e89da27b887afbcab2ed4effb8","target":"graph","created_at":"2026-05-18T03:37:28Z","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":"Let $\\xi$ be a real random variable with mean zero and variance one and $A={a_1,...,a_n}$ be a multi-set in $\\R^d$. The random sum\n  $$S_A := a_1 \\xi_1 + ... + a_n \\xi_n $$\nwhere $\\xi_i$ are iid copies of $\\xi$ is of fundamental importance in probability and its applications.\n  We discuss the small ball problem, the aim of which is to estimate the maximum probability that $S_A$ belongs to a ball with given small radius, following the discovery made by Littlewood-Offord and Erdos almost 70 years ago. We will mainly focus on recent developments that characterize the structure of those sets $A$ w","authors_text":"Hoi H. Nguyen, Van H. Vu","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-31T21:39:17Z","title":"Small ball probability, Inverse theorems, and applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.0019","kind":"arxiv","version":1},"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":"sha256:02ad91d88ef730a87218fef3709aa230dbb60fb07e3ddde5fb208137038459cc","target":"record","created_at":"2026-05-18T03:37:28Z","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":"6bc5e88fc6ecff242be98634b9c28245d16bf549de35bcfbca31bfd946047346","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-31T21:39:17Z","title_canon_sha256":"a0232d57a5600b567ba690a46312adcd82ff1efa3f0569d488c77a8e9922852e"},"schema_version":"1.0","source":{"id":"1301.0019","kind":"arxiv","version":1}},"canonical_sha256":"e6f85a2e71c40177afc050d2660a67d1db48a536b597acb34b13a80af167ff1a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e6f85a2e71c40177afc050d2660a67d1db48a536b597acb34b13a80af167ff1a","first_computed_at":"2026-05-18T03:37:28.567329Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:37:28.567329Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0kQATF7/Hef4tudL/35PAGAeI/bl+YTj1UtOlcuYcpdsK0Pey+U1mC2IS9hWpQrTyh/Wb3rOwRHGSBPolifhDg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:37:28.568157Z","signed_message":"canonical_sha256_bytes"},"source_id":"1301.0019","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:02ad91d88ef730a87218fef3709aa230dbb60fb07e3ddde5fb208137038459cc","sha256:8e6d35788644af34f7e3fe60825e368679db27e89da27b887afbcab2ed4effb8"],"state_sha256":"85e26c2c369579fb761428402dd9e7d47d9a9a40cf87bf1c3cd15b60a76f551f"}