On the fourier tails of bounded functions over the discrete cube

  • Authors:

  • Irit Dinur;Ehud Friedgut;Guy Kindler;Ryan O'Donnell

  • Affiliations:

  • Hebrew University, Jerusalem, Israel;Hebrew University, Jerusalem, Israel;Microsoft Research;Microsoft Research

  • Venue:
  • Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
  • Year:
  • 2006

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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 -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 : (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 2 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.