{"paper":{"title":"Anti-concentration in most directions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Amir Yehudayoff, Anup Rao","submitted_at":"2018-11-15T18:11:54Z","abstract_excerpt":"We prove anti-concentration bounds for the inner product of two independent random vectors. For example, we show that if $A,B$ are subsets of the cube $\\{\\pm 1\\}^n$ with $|A| \\cdot |B| \\geq 2^{1.01 n}$, and $X \\in A$ and $Y \\in B$ are sampled independently and uniformly, then the inner product $\\langle X, Y \\rangle$ takes on any fixed value with probability at most $O(\\tfrac{1}{\\sqrt{n}})$. Extending Hal\\'asz work, we prove stronger bounds when the choices for $x$ are unstructured. We also describe applications to communication complexity, randomness extraction and additive combinatorics."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.06510","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"}