TY - GEN

T1 - Inverse conjecture for the gowers norm is false

AU - Lovett, Shachar

AU - Meshulam, Roy

AU - Samorodnitsky, Alex

PY - 2008

Y1 - 2008

N2 - Let p be a fixed prime number and N be a large integer. The "Inverse Conjecture for the Gowers Norm" states that if the "d-th Gowers norm" of a function f : double strok F signpN → double strok F signp is non-negligible, that is larger than a constant independent of N, then f has a non-trivial correlation with a degree d - 1 polynomial. The conjecture is known to hold for d = 2, 3 and for any prime p. In this paper we show the conjecture to be false for p = 2 and for d = 4, by presenting an explicit function whose 4-th Gowers norm is non-negligible, but whose correlation with any polynomial of degree 3 is exponentially small. Essentially the same result, with different bounds for correlation, was independently obtained by Green and Tao [8]. Their analysis uses a modification of a Ramsey-type argument of Alon and Beigel [1] to show inapproximability of certain functions by low-degree polynomials. We observe that a combination of our results with the argument of Alon and Beigel implies the inverse conjecture to be false for any prime p, for d = p2.

AB - Let p be a fixed prime number and N be a large integer. The "Inverse Conjecture for the Gowers Norm" states that if the "d-th Gowers norm" of a function f : double strok F signpN → double strok F signp is non-negligible, that is larger than a constant independent of N, then f has a non-trivial correlation with a degree d - 1 polynomial. The conjecture is known to hold for d = 2, 3 and for any prime p. In this paper we show the conjecture to be false for p = 2 and for d = 4, by presenting an explicit function whose 4-th Gowers norm is non-negligible, but whose correlation with any polynomial of degree 3 is exponentially small. Essentially the same result, with different bounds for correlation, was independently obtained by Green and Tao [8]. Their analysis uses a modification of a Ramsey-type argument of Alon and Beigel [1] to show inapproximability of certain functions by low-degree polynomials. We observe that a combination of our results with the argument of Alon and Beigel implies the inverse conjecture to be false for any prime p, for d = p2.

KW - Gowers norm

KW - Low degree tests

KW - Multivariate polynomials

UR - http://www.scopus.com/inward/record.url?scp=57049124369&partnerID=8YFLogxK

U2 - 10.1145/1374376.1374454

DO - 10.1145/1374376.1374454

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AN - SCOPUS:57049124369

SN - 9781605580470

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

SP - 547

EP - 556

BT - STOC'08

PB - Association for Computing Machinery (ACM)

T2 - 40th Annual ACM Symposium on Theory of Computing, STOC 2008

Y2 - 17 May 2008 through 20 May 2008

ER -