NDLI: On the fourier tails of bounded functions over the discrete cube

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On the fourier tails of bounded functions over the discrete cube

Content Provider	ACM Digital Library
Author	Dinur, Irit Friedgut, Ehud O'Donnell, Ryan Kindler, Guy
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, [Σ\|S\| > k $f(S)^{2}$ < k-1/2 + o(1), ] then essentially, f depends on only $2^{O(k)}$ 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 : $(^{n}$ ⋺ [-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 [Σ\|S\| > k $f(S)^{2}$ < $exp(-O(k^{2}$ log k))] then essentially, f depends on only $2^{O(k)}$ 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.
Starting Page	437
Ending Page	446
Page Count	10
File Format	PDF
ISBN	1595931341
DOI	10.1145/1132516.1132580
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2006-05-21
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Boolean functions Symmetry-breaking Fourier analysis
Content Type	Text
Resource Type	Article

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