TY - GEN

T1 - On the fourier tails of bounded functions over the discrete cube

AU - Dinur, Irit

AU - Friedgut, Ehud

AU - Kindler, Guy

AU - O'Donnell, Ryan

PY - 2006

Y1 - 2006

N2 - 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.

AB - 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.

KW - Boolean functions

KW - Fourier analysis

KW - Symmetry-breaking

UR - http://www.scopus.com/inward/record.url?scp=33748109882&partnerID=8YFLogxK

U2 - 10.1145/1132516.1132580

DO - 10.1145/1132516.1132580

M3 - Conference contribution

AN - SCOPUS:33748109882

SN - 1595931341

SN - 9781595931344

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 437

EP - 446

BT - STOC'06

PB - Association for Computing Machinery (ACM)

T2 - 38th Annual ACM Symposium on Theory of Computing, STOC'06

Y2 - 21 May 2006 through 23 May 2006

ER -