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Given an integer vector $\\boldsymbol{a} = (a_1, \\dotsc, a_n)$, its concentration probability is the quantity $\\rho(\\boldsymbol{a}):=\\sup_{x\\in \\mathbb{Z}}\\Pr(\\epsilon_1 a_1+\\dots+\\epsilon_n a_n = x)$. The Littlewood-Offord problem asks for bounds on $\\rho(\\boldsymbol{a})$ under various hypotheses on $\\boldsymbol{a}$, whereas the inverse Littlewood-Offord problem, posed by Tao and Vu, asks for a characterization of all vectors $\\boldsymbol{a}$ for which $\\rho(\\boldsymbol"},"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":"1904.10425","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-04-23T17:05:13Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"cfa16b0e0cb1a1a83f20d6b8e856ed870da2e110e88b9cd2aad80123d5cc4973","abstract_canon_sha256":"ee69640e073508243213e0513507fca87fc1ea0787a8aa70b00d2b1c1b82a410"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:47:54.205263Z","signature_b64":"bSH8yHx95ll0/aob7tHrAUo/mrK7ftBCauBnaGeeglUwygcpHq3xWi+7ps3uktRvRFGp2l5auykDhUpme3bQCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7b09897d28d6602be8a736830fbca0e765f77218a5375fc04e39c2a18338cb58","last_reissued_at":"2026-05-17T23:47:54.204633Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:47:54.204633Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the counting problem in inverse Littlewood--Offord theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Asaf Ferber, Kyle Luh, Vishesh Jain, Wojciech Samotij","submitted_at":"2019-04-23T17:05:13Z","abstract_excerpt":"Let $\\epsilon_1, \\dotsc, \\epsilon_n$ be i.i.d. 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