Gaussian noise sensitivity and Fourier tails

Guy Kindler*, Naomi Kirshner, Ryan O’Donnell

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


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.

Original languageAmerican English
Pages (from-to)71-109
Number of pages39
JournalIsrael Journal of Mathematics
Issue number1
StatePublished - 1 Apr 2018

Bibliographical note

Publisher Copyright:
© 2018, Hebrew University of Jerusalem.


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