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