Inverse Conjecture for the Gowers norm is false

TL;DR

This paper disproves the inverse conjecture for the Gowers norm at p=2, d=4 by constructing a function with high Gowers norm but negligible correlation with degree 3 polynomials, challenging previous assumptions.

Contribution

It provides the first explicit counterexample to the inverse conjecture for the Gowers norm at specific parameters, showing the conjecture does not hold universally.

Findings

Counterexample with high Gowers norm and low polynomial correlation

Disproves the inverse conjecture for p=2, d=4

Implication that the conjecture fails for any prime p at d=p^2

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 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 \cite{gt07}. Their analysis uses a modification of a Ramsey-type argument of Alon and Beigel \cite{ab} 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 $d = p^2$.