Inverse conjecture for the gowers norm is false

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 sign_p^N → double strok F sign_p 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.