AN INVERSE THEOREM FOR THE GOWERS U4-NORM

@article{Green2010ANIT,
  title={AN INVERSE THEOREM FOR THE GOWERS U4-NORM},
  author={Ben Green and Terence Tao and Tamar D. Ziegler},
  journal={Glasgow Mathematical Journal},
  year={2010},
  volume={53},
  pages={1 - 50}
}
Abstract We prove the so-called inverse conjecture for the Gowers Us+1-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we show that if f : [N] → ℂ is a function with |f(n)| ≤ 1 for all n and ‖f‖U4 ≥ δ then there is a bounded complexity 3-step nilsequence F(g(n)Γ) which correlates with f. The approach seems to generalise so as to prove the inverse conjecture for s ≥ 4 as well, and a longer paper will follow concerning this. By combining the main result… 
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