A Generalization of a Theorem of Rothschild and van Lint

TL;DR

This paper extends Rothschild and van Lint's classical theorem by characterizing Boolean functions with Fourier coefficients in a specific set, linking their structure to indicator functions of affine subspaces, using additive combinatorics tools.

Contribution

It generalizes the classical theorem to functions with Fourier coefficients in a broader set, revealing their structure as indicator functions of multiple affine subspaces.

Findings

Boolean functions with Fourier coefficients in {-2/2^k, -1/2^k, 0, 1/2^k, 2/2^k} are indicator functions of affine subspaces.

Such functions correspond to unions of two or four disjoint affine subspaces.

The structure is characterized using bounds from additive combinatorics.

Abstract

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.