Abstract
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 \(||{f^{ > k}}||_2^2 \geqslant \Omega (Var[f]).\frac{1}{{\sqrt k }}for:{R^n} \to \{ - 1,1\} \). 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 \(\frac{1}{{\sqrt k {{\log }^{1.5}}k}}\) .
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Supported by the Koshland fellowship and by the Binational Science Foundation (BSF) grant no. 2008477.
Supported by NSF grants CCF-0747250 and CCF-0915893, and by a Sloan fellowship.
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Kindler, G., Kirshner, N. & O’Donnell, R. Gaussian noise sensitivity and Fourier tails. Isr. J. Math. 225, 71–109 (2018). https://doi.org/10.1007/s11856-018-1646-8
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DOI: https://doi.org/10.1007/s11856-018-1646-8