{"paper":{"title":"Resilience for the Littlewood-Offord Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Afonso S. Bandeira, Asaf Ferber, Matthew Kwan","submitted_at":"2016-09-26T19:46:27Z","abstract_excerpt":"Consider the sum $X(\\xi)=\\sum_{i=1}^n a_i\\xi_i$, where $a=(a_i)_{i=1}^n$ is a sequence of non-zero reals and $\\xi=(\\xi_i)_{i=1}^n$ is a sequence of i.i.d. Rademacher random variables (that is, $\\Pr[\\xi_i=1]=\\Pr[\\xi_i=-1]=1/2$). The classical Littlewood-Offord problem asks for the best possible upper bound on the concentration probabilities $\\Pr[X=x]$. In this paper we study a resilience version of the Littlewood-Offord problem: how many of the $\\xi_i$ is an adversary typically allowed to change without being able to force concentration on a particular value? We solve this problem asymptoticall"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.08136","kind":"arxiv","version":4},"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"}