The Number of Locally $p$-stable Functions on $Q_n$

A Boolean function $f:V \to \{-1,1\}$ on the vertex set of a graph $G=(V,E)$ is locally $p$-stable if for every vertex $v$ the proportion of neighbours $w$ of $v$ with $f(v)=f(w)$ is exactly $p$. This notion was introduced by Gross and Grupel in [1] while studying the scenery reconstruction problem. They give an exponential type lower bound for the number of isomorphism classes of locally $p$-stable functions when $G=Q_n$ is the $n$-dimensional Boolean hypercube and ask for more precise estimates. In this paper we provide such estimates by improving the lower bound to a double exponential type lower bound and finding a matching upper bound. We also show that for a fixed $k$ and increasing $n$, the number of isomorphism classes of locally $(1-k/n)$-stable functions on $Q_n$ is eventually constant. The proofs use the Fourier decomposition of functions on the Boolean hypercube.