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 | | 2 2 ≥ Ω ( V a r [ f ] ) . 1 k f o r : R n → { − 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 1 k 1 / 2 + ο ( 1 ) . We also show a simplification of Bourgain’s proof that does not use Invariance and obtains the bound 1 k log 1.5 k .