  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.
