Size of Sets with Small Sensitivity: a Generalization of Simon's Lemma

TL;DR

This paper investigates the structure and size of sets with small sensitivity in the Boolean hypercube, revealing a gap theorem that classifies such sets into two distinct size categories and deepening understanding of their structure.

Contribution

It introduces a gap theorem for the size of sensitivity-bounded sets, classifies these sets into constructible and irreducible types, and provides new structural insights.

Findings

Sets with sensitivity s are either of size 2^{n-s} or at least 1.5 times that size.

Classifies sensitivity sets into constructible and irreducible categories.

Provides a lower bound on the size of irreducible sensitivity sets.

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 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 $|S|\geq \frac{3}{2} 2^{n-s}$. This is shown via classifying such sets into sets that can be constructed from low-sensitivity subsets of $\{0, 1\}^{n'}$ for $n'<n$ and irreducible sets which cannot be constructed in such a way and then proving a lower bound on the size of irreducible sets. This provides new insights into the structure of low sensitivity subsets of the Boolean hypercube $\{0, 1\}^n$.