The number of locally p-stable functions on Qn

Abstract

A Boolean function f:V(G)→{−1,1} on the vertex set of a graph G 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 [4] 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=Qn 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 Qn is eventually constant. The proofs use the Fourier decomposition of functions on the Boolean hypercube.