On the fourier tails of bounded functions over the discrete cube

Irit Dinur*, Ehud Friedgut, Guy Kindler, Ryan O'Donnell

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

21 Scopus citations

Abstract

A theorem of Bourgain [4] on Fourier tails states that if f : {-1, 1} n → {-1, 1} is a boolean-valued function on the discrete cube such that for any k > 0, ∑|s|>k f̂(S)2 < k-1/2+o(1), then essentially, f depends on only 2O(k) coordinates. This and related theorems such as Friedgut's Theorem [12], KKL [16] the FKN Theorem [14], and the Majority Is Stablest Theorem [27] have proven useful for numerous results in theoretical computer science [3, 5, 9, 6, 7, 10, 11, 18, 19, 20, 24, 17, 25, 23, 22, 28, 29, 31]. In this paper we prove an analogue to Bourgain's Theorem for bounded functions on the discrete cube, f : {-1, 1}n → [-1, 1]; such functions arise naturally in hardness-of-approximation problems, as averages of boolean functions. Specifically, we show that for every k > 0, if ∑|s|>k f̂(S)2 < exp(-O(k2 log k)) then essentially, f depends on only 2O(k) coordinates. We also show, perhaps surprisingly, that this result is sharp up to the log k factor in the exponent. Our proof uses Fourier analysis, as well as some extremal properties of the Chebyshev polynomials.

Original languageAmerican English
Title of host publicationSTOC'06
Subtitle of host publicationProceedings of the 38th Annual ACM Symposium on Theory of Computing
PublisherAssociation for Computing Machinery (ACM)
Pages437-446
Number of pages10
ISBN (Print)1595931341, 9781595931344
DOIs
StatePublished - 2006
Externally publishedYes
Event38th Annual ACM Symposium on Theory of Computing, STOC'06 - Seattle, WA, United States
Duration: 21 May 200623 May 2006

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
Volume2006
ISSN (Print)0737-8017

Conference

Conference38th Annual ACM Symposium on Theory of Computing, STOC'06
Country/TerritoryUnited States
CitySeattle, WA
Period21/05/0623/05/06

Keywords

  • Boolean functions
  • Fourier analysis
  • Symmetry-breaking

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