## Arkiv för Matematik

### Noise correlation bounds for uniform low degree functions

#### 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
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.afm/1485907195

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

Mathematical Reviews number (MathSciNet)
MR3029335

Zentralblatt MATH identifier
1296.68102

Rights

#### 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

#### References

• Austrin, P. and Mossel, E., Approximation resistant predicates from pairwise independence, Comput. Complexity 18 (2009), 249–271.
• Benjamini, I., Kalai, G. and Schramm, O., Noise sensitivity of boolean functions and applications to percolation, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 5–43.
• Furstenberg, H. and Weiss, B., A mean ergodic theorem for $(1/N)\sum^{N}_{n=1}f(T^{n}x){}\cdot\allowbreak g(T^{n^{2}}x)$, in Convergence in Ergodic Theory and Probability (Columbus, OH, 1993 ), Ohio State Univ. Math. Res. Inst. Publ. 5, pp. 193–227, de Gruyter, Berlin, 1996.
• Gowers, W. T., A new proof of Szemerédi’s theorem for arithmetic progressions of length four, Geom. Funct. Anal. 8 (1998), 529–551.
• Gowers, W. T., A new proof of Szemerédi’s theorem, Geom. Funct. Anal. 11 (2001), 465–588.
• Gowers, W. T. and Wolf, J., The true complexity of a system of linear equations, Proc. Lond. Math. Soc. 100 (2010), 155–176.
• Green, B. and Tao, T., The primes contain arbitrarily long arithmetic progressions, Ann. of Math. 167 (2008), 481–547.
• Khot, S., On the power of unique 2-prover 1-round games, in Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing (Montreal, QC, 2002 ), pp. 767–775, ACM, New York, 2002.
• Khot, S., Kindler, G., Mossel, E. and O’Donnell, R., Optimal inapproximability results for max-cut and other 2-variable CSPs?, SIAM J. Comput. 37 (2007), 319–357.
• Lindeberg, J. W., Eine neue Herleitung des exponential-Gesetzes in der Wahrscheinlichkeitsrechnung, Math. Z. 15 (1922), 211–235.
• Mossel, E., Gaussian bounds for noise correlation of functions, Geom. Funct. Anal. 19 (2010), 1713–1756.
• Mossel, E., O’Donnell, R. and Oleszkiewicz, K., Noise stability of functions with low influences: invariance and optimality, Ann. of Math. 171 (2010), 295–341.
• O’Donnell, R., Computational Applications of Noise Sensitivity, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, 2003.
• Raghavendra, P., Optimal algorithms and inapproximability results for every CSP, in Proceedings of the 40th Annual ACM Symposium on Theory of Computing STOC’08 (Victoria, BC, 2008 ), pp. 245–254, ACM, New York, 2008.
• Rotar, V. I., Limit theorems for polylinear forms, J. Multivariate Anal. 9 (1979), 511–530.
• Roth, K. F., On certain sets of integers, J. Lond. Math. Soc. 28 (1953), 245–252.
• Samorodnitsky, A. and Trevisan, L., Gowers uniformity, influence of variables, and PCPs, in Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing (Seattle, WA, 2006 ), pp. 11–20, ACM, New York, 2006.
• Szemerédi, E., On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 299–345.