A Generalization of a Theorem of Rothschild and van Lint

A classical result of Rothschild and van Lint asserts that if every non-zero Fourier coefficient of a Boolean function f over \(\mathbb {F}_2^{n}\) has the same absolute value, namely \(|\hat{f}(\alpha )|=1/2^k\) for every \(\alpha \) 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 \(\mathbb {F}_2^{n}\) when the doubling constant of the subset is small.