Inverse Conjecture for the Gowers norm is false

Shachar Lovett; Roy Meshulam; Alex Samorodnitsky

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arXiv · 0711.3388

Inverse Conjecture for the Gowers norm is false

Shachar Lovett·Roy Meshulam·Alex Samorodnitsky

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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 $d = p^2$.

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Shachar Lovett, Roy Meshulam, Alex Samorodnitsky. 2008-10-20. Inverse Conjecture for the Gowers norm is false. https://arxiv.org/abs/0711.3388

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