Gaussian Bounds for Noise Correlation of Resilient Functions

Gaussian bounds on noise correlation of functions play an important role in hardness of approximation, in quantitative social choice theory and in testing. The author (2008) obtained sharp gaussian bounds for the expected correlation of $\ell$ low influence functions $f^{(1)},\ldots, f^{(\ell)} : \Omega^n \to [0,1]$, where the inputs to the functions are correlated via the $n$-fold tensor of distribution $\mathcal{P}$ on $\Omega^{\ell}$. It is natural to ask if the condition of low influences can be relaxed to the condition that the function has vanishing Fourier coefficients. Here we answer this question affirmatively. For the case of two functions $f$ and $g$, we further show that if $f,g$ have a noisy inner product that exceeds the gaussian bound, then the Fourier supports of their large coefficients intersect.