Quantitative multiple recurrence for two and three transformations

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/2?), ? ∈ 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 \(||{f^{ > k}}||_2^2 \geqslant \Omega (Varf]).\frac{1}{{\sqrt k }}for:{R^n} \to \{ - 1,1\} \). 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 \(\frac{1}{{{k^{1/2 + o(1)}}}}\). We also show a simplification of Bourgain’s proof that does not use Invariance and obtains the bound \(\frac{1}{{\sqrt k {{\log }^{1.5}}k}}\).