Boolean functions whose Fourier transform is concentrated on the first two levels

Abstract

In this note we describe Boolean functions f(x1,x2,…,xn) whose Fourier coefficients are concentrated on the lowest two levels. We show that such a function is close to a constant function or to a function of the form f=xk or f=1−xk. This result implies a “stability” version of a classical discrete isoperimetric result and has an application in the study of neutral social choice functions. The proofs touch on interesting harmonic analysis issues.