Size of sets with small sensitivity: A generalization of simon’s Lemma

Abstract

We study the structure of sets S ⊆ {0, 1}n with small sensitivity. The well-known Simon’s lemma says that any S ⊆ {0, 1}n of sensitivity s must be of size at least 2n−s. This result has been useful for proving lower bounds on the sensitivity of Boolean functions, with applications to the theory of parallel computing and the “sensitivity vs. block sensitivity” conjecture. 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 ⊆ {0, 1}n has sensitivity s, then we either have |S| = 2n−s or |S| ≥ 3 2 2n−s. This provides new insights into the structure of low sensitivity subsets of the Boolean hypercube {0, 1}n.