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 $f:\F_p^N \to \F_p$ is nonnegligible, that is larger than a constant independent of $N$, then $f$ can be nontrivially approximated by a degree $d1$ 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 4th Gowers norm is nonnegligible, 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 \cite{gt07}. Their analysis uses a modification of a Ramseytype argument of Alon and Beigel \cite{ab} to show inapproximability of certain functions by lowdegree 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 $d = p^2$.
 Publication:

arXiv eprints
 Pub Date:
 November 2007
 DOI:
 10.48550/arXiv.0711.3388
 arXiv:
 arXiv:0711.3388
 Bibcode:
 2007arXiv0711.3388L
 Keywords:

 Mathematics  Combinatorics;
 11T06
 EPrint:
 20 pages