A generalization of a theorem of Rothschild and van Lint

Abstract

A classical result of Rothschild and van Lint asserts that if every non-zero Fourier coefficient of a Boolean function f over F2n has the same absolute value, namely |fˆ(α)|=1/2k for every α in the Fourier support of f, then f must be the indicator function of some affine subspace of dimension n−k. In this paper we slightly generalize their result. Our main result shows that, roughly speaking, Boolean functions whose Fourier coefficients take values in the set {−2/2k,−1/2k,0,1/2k,2/2k} are indicator functions of two disjoint affine subspaces of dimension n−k or four disjoint affine subspaces of dimension n−k−1. Our main technical tools are results from additive combinatorics which offer tight bounds on the affine span size of a subset of F2n when the doubling constant of the subset is small.