Gaussian noise sensitivity and Fourier tails

We observe a subadditivity property for the noise sensitivity of subsets of Gaussian space. For subsets of volume 1/2, this leads to an almost trivial proof of Borell’s Isoperimetric Inequality for ρ = cos(π/2ℓ), ℓ ∈ N. Rotational sensitivity also easily gives the Gaussian Isoperimetric Inequality for volume-1/2 sets and a.8787-factor UG-hardness for Max-Cut (within 10⁻⁴ of the optimum). As another corollary we show the Hermite tail bound ||f>k||22≥Ω(Var[f]).1kfor:Rn→{−1,1}. Combining this with the Invariance Principle shows the same Fourier tail bound for any Boolean f: {−1, 1}ⁿ → {−1, 1} with all its noisy-influences small, or more strongly, that a Boolean function with tail weight smaller than this bound must be close to a junta. This improves on a result of Bourgain, where the bound on the tail weight was only 1k1/2+ο(1). We also show a simplification of Bourgain’s proof that does not use Invariance and obtains the bound 1klog1.5k.