Size of Sets with Small Sensitivity: A Generalization of Simon’s Lemma

Abstract

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 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 $S\subseteq \{0, 1\}^n$ has sensitivity $s$, then we either have $|S|=2^{n-s}$ or $|S|\ge \frac{3}{2} 2^{n-s}$.

This provides new insights into the structure of low sensitivity subsets of the Boolean hypercube $\{0, 1\}^n$.

The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007–2013) under projects QALGO (Grant Agreement No. 600700) and RAQUEL (Grant Agreement No. 323970), ERC Advanced Grant MQC and Latvian State Research programme NexIT project No.1. Part of this work was done while Andris Ambainis was visiting Institute for Advanced Study, Princeton, supported by National Science Foundation under agreement No. DMS-1128155. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.