  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 $\{0, 1\}^n$.
