{"paper":{"title":"Biasing Boolean Functions and Collective Coin-Flipping Protocols over Arbitrary Product Distributions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"cs.DM","authors_text":"David Zuckerman, Hamed Hatami, Lianna Hambardzumyan, Pooya Hatami, Yuval Filmus","submitted_at":"2019-02-20T06:34:09Z","abstract_excerpt":"The seminal result of Kahn, Kalai and Linial shows that a coalition of $O(\\frac{n}{\\log n})$ players can bias the outcome of any Boolean function $\\{0,1\\}^n \\to \\{0,1\\}$ with respect to the uniform measure. We extend their result to arbitrary product measures on $\\{0,1\\}^n$, by combining their argument with a completely different argument that handles very biased coordinates.\n  We view this result as a step towards proving a conjecture of Friedgut, which states that Boolean functions on the continuous cube $[0,1]^n$ (or, equivalently, on $\\{1,\\dots,n\\}^n$) can be biased using coalitions of $o("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.07426","kind":"arxiv","version":1},"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"}