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 | f^ (α) | = 1 / 2 k 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 / 2 k, - 1 / 2 k, 0, 1 / 2 k, 2 / 2 k} are indicator functions of two disjoint affine subspaces of dimension n- k or four disjoint affine subspace 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.