{"paper":{"title":"Size of Sets with Small Sensitivity: a Generalization of Simon's Lemma","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Andris Ambainis, Jevg\\=enijs Vihrovs","submitted_at":"2014-05-31T12:31:08Z","abstract_excerpt":"We study the structure of sets $S\\subseteq\\{0, 1\\}^n$ with small sensitivity. The well-known Simon's lemma says that any $S\\subseteq\\{0, 1\\}^n$ of sensitivity $s$ must be of size at least $2^{n-s}$. This result has been useful for proving lower bounds on sensitivity of Boolean functions, with applications to the theory of parallel computing and the \"sensitivity vs. block sensitivity\" conjecture.\n  In this paper, we take a deeper look at the size of such sets and their structure. We show an unexpected \"gap theorem\": if $S\\subseteq\\{0, 1\\}^n$ has sensitivity $s$, then we either have $|S|=2^{n-s}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.0073","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":""},"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"}