@inproceedings{07c766499b834e9ea32259fbcb9fbfcf,

title = "Inverse conjecture for the gowers norm is false",

abstract = "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.",

keywords = "Gowers norm, Low degree tests, Multivariate polynomials",

author = "Shachar Lovett and Roy Meshulam and Alex Samorodnitsky",

year = "2008",

doi = "10.1145/1374376.1374454",

language = "American English",

isbn = "9781605580470",

series = "Proceedings of the Annual ACM Symposium on Theory of Computing",

publisher = "Association for Computing Machinery (ACM)",

pages = "547--556",

booktitle = "STOC'08",

address = "United States",

note = "40th Annual ACM Symposium on Theory of Computing, STOC 2008 ; Conference date: 17-05-2008 Through 20-05-2008",

}