Arkiv för Matematik

Noise correlation bounds for uniform low degree functions

Per Austrin and Elchanan Mossel

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Abstract

We study correlation bounds under pairwise independent distributions for functions with no large Fourier coefficients. Functions in which all Fourier coefficients are bounded by δ are called δ-uniform. The search for such bounds is motivated by their potential applicability to hardness of approximation, derandomization, and additive combinatorics.

In our main result we show that $\operatorname{\mathbb {E}}[f_{1}(X_{1}^{1},\ldots,X_{1}^{n}) \ldots f_{k}(X_{k}^{1},\ldots,X_{k}^{n})]$ is close to 0 under the following assumptions:

  • the vectors $\{ (X_{1}^{j},\ldots,X_{k}^{j}) : 1 \leq j \leq n\}$ are independent identically distributed, and for each j the vector $(X_{1}^{j},\ldots,X_{k}^{j})$ has a pairwise independent distribution;
  • the functions fi are uniform;
  • the functions fi are of low degree.

We compare our result with recent results by the second author for low influence functions and to recent results in additive combinatorics using the Gowers norm. Our proofs extend some techniques from the theory of hypercontractivity to a multilinear setup.

Note

Work done while the first author was at the Royal Institute of Technology, funded by ERC Advanced investigator grant 226203 and a grant from the Mittag-Leffler Institute. Second author supported by BSF grant 2004105, NSF CAREER award DMS 0548249, DOD ONR grant N0014-07-1-05-06 and ISF grant 1300/08.

Article information

Source
Ark. Mat. Volume 51, Number 1 (2013), 29-52.

Dates
Received: 19 August 2010
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485907195

Digital Object Identifier
doi:10.1007/s11512-011-0145-5

Zentralblatt MATH identifier
1296.68102

Rights
2011 © Institut Mittag-Leffler

Citation

Austrin, Per; Mossel, Elchanan. Noise correlation bounds for uniform low degree functions. Ark. Mat. 51 (2013), no. 1, 29--52. doi:10.1007/s11512-011-0145-5. https://projecteuclid.org/euclid.afm/1485907195.


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