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 : FN p → F is non-negligible, that is larger than a constant independent of N, then f can be non-trivially approximated by 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 any polynomial of degree 3 is exponentially small. Essentially the same result (with different correlation bounds) was independently obtained by Green and Tao [5]. 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 b...

• Type: Journal
• Year: 2007
• Where: ECCC
• Authors: Shachar Lovett, Roy Meshulam, Alex Samorodnitsky
• Keywords: Conjecture, ECCC 2007, Inverse Conjecture, fixed Prime Number