## 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 is non-trivially correlated to 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 d = 4, by presenting an explicit function whose 4-th Gowers norm is non-negligible, but whose correlation to any polynomial of degree 3 is exponentially small. Essentially the same result (with different correlation bounds) was independently obtained by Green and Tao (2009).

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 is non-trivially correlated to 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 d = 4, by presenting an explicit function whose 4-th Gowers norm is non-negligible, but whose correlation to any polynomial of degree 3 is exponentially small. Essentially the same result (with different correlation bounds) was independently obtained by Green and Tao (2009).

Original language | English |
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Article number | 1 |

Pages (from-to) | 131-145 |

Number of pages | 15 |

Journal | Theory of Computing |

Volume | 7 |

Issue number | 1 |

DOIs | |

State | Published - 12 Jul 2011 |

## Keywords

- Inverse Gowers conjecture
- Additive combinatorics
- Gowers norm