Boolean Functions with Small Approximate Spectral Norm

The sum of the absolute values of the Fourier coefficients of a function f : F-n(2) -> R is called the spectral norm off. Green and Sanders' quantitative version of Cohen's idempotent theorem states that if the spectral norm off : F-n(2) -> {0,1} is at most M, then the support off belongs to the ring of sets generated by at most l(M) cosets, where l(M) is a constant that only depends on M. We prove that the above statement can be generalized to approximate spectral norms if and only if the support off and its complement satisfy a certain arithmetic connectivity condition. In particular, our theorem provides a new proof of the quantitative Cohen's theorem for F-n(2).