{"paper":{"title":"The Reverse Littlewood--Offord problem of Erd\\H{o}s","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Bhargav Narayanan, Sam Spiro, Tomas Juskevicius, Xiaoyu He","submitted_at":"2024-08-20T17:41:37Z","abstract_excerpt":"Let $\\epsilon_{1},\\ldots,\\epsilon_{n}$ be a sequence of independent Rademacher random variables. We prove that there is a constant $c>0$ such that for any unit vectors $v_1,\\ldots,v_n\\in \\mathbb{R}^2$,\n  $$\\Pr\\left[||\\epsilon_1 v_1+\\ldots+\\epsilon_n v_n||_2 \\leq \\sqrt{2}\\right]\\geq \\frac{c}{n}.$$\n  This resolves the only remaining conjecture from the seminal paper of Erd\\H{o}s on the Littlewood--Offord problem, and it is sharp both in the sense that the constant $\\sqrt{2}$ cannot be reduced and that the magnitude $n^{-1}$ is best possible. We also prove polynomial bounds for the analogous prob"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.11034","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2408.11034/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}