Inverse Conjecture for the Gowers Norm is False

Shachar Lovett, Roy Meshulam, Alex Samorodnitsky

Research output: Contribution to journalArticlepeer-review


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).
Original languageEnglish
Article number1
Pages (from-to)131-145
Number of pages15
JournalTheory of Computing
Issue number1
StatePublished - 12 Jul 2011


  • Inverse Gowers conjecture
  • Additive combinatorics
  • Gowers norm


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