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To what extent can $Q(\\xi_{1},\\dots,\\xi_{n})$ concentrate on a single value? This quadratic version of the classical Littlewood--Offord problem was popularised by Costello, Tao and Vu in their study of symmetric random matrices. In this paper, we obtain an essentially optimal bound for this problem, as conjectured by Nguyen and Vu.\n  Specifically, if $Q(\\xi_{1},\\dots,\\xi_{n})$ \"robustly depends on at least $m$ of the $\\xi_{i}$\" in the sense that there is no way to pin ","authors_text":"Lisa Sauermann, Matthew Kwan","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-12-21T13:17:50Z","title":"Resolution of the quadratic Littlewood--Offord problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2312.13826","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":"sha256:8b83eaf235f7433a6a5bf6ae07a2f1c7b8bce912858d4b0317f1e5b09148c9ee","target":"record","created_at":"2026-07-05T07:26:49Z","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":"0229d3cf389303626a7a8e7a69a7ae64bdb31b40ccfb81223c8d079665918886","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-12-21T13:17:50Z","title_canon_sha256":"f9b556e2c0f137be7b7ba6a66a0cd2cf1288a50295d28844603f259f7332dd99"},"schema_version":"1.0","source":{"id":"2312.13826","kind":"arxiv","version":1}},"canonical_sha256":"dda3b93446f3f867314474e77242a5a599f96fd5ef12114648e0f72c9841b84f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dda3b93446f3f867314474e77242a5a599f96fd5ef12114648e0f72c9841b84f","first_computed_at":"2026-07-05T07:26:49.969846Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:26:49.969846Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4bkJbj2zBUbypsqV1VvSeS3ny+oeto/SRgIydEh/52tcx98oNEfFza0wOW81Jb/VBQLMV1wJPaNIFFHxp+qHBg==","signature_status":"signed_v1","signed_at":"2026-07-05T07:26:49.970374Z","signed_message":"canonical_sha256_bytes"},"source_id":"2312.13826","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8b83eaf235f7433a6a5bf6ae07a2f1c7b8bce912858d4b0317f1e5b09148c9ee","sha256:9a1a22e3aebf1d151b4caefb2239a06c6e9a9d90179cfde670712ef1e7279efc"],"state_sha256":"ad4f08cfafc64656d603e0afdbcc1af4b59ca9ec3922b41e683011350cb0e837"}