Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $f:\F_p^N \to \F_p$ 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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The categories of this paper are: plasm-ph
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The categories of this paper are: supr-con
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The categories of this paper are: acc-phys
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The categories of this paper are: cs
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-th
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The categories of this paper are: q-alg
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The categories of this paper are: stat
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The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: math-ph
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The categories of this paper are: solv-int
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The categories of this paper are: nlin
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The categories of this paper are: econ
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The categories of this paper are: q-bio
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The categories of this paper are: hep-ex