An inverse theorem for the Gowers U^3 norm
  The Gowers U^3 norm is one of a sequence of norms used in the study of arithmetic progressions. If G is an abelian group and A is a subset of G then the U^3(G) of the characteristic function 1_A is useful in the study of progressions of length 4 in A. We give a comprehensive study of the U^3(G) norm, obtaining a reasonably complete description of functions f : G -> C for which ||f||_{U^3} is large and providing links to recent results of Host, Kra and Ziegler in ergodic theory.   As an application we generalise a result of Gowers on Szemeredi's theorem. Writing r_4(G) for the size of the largest set A not containing four distinct elements in arithmetic progression, we show that r_4(G) << |G|(loglog|G|)^{-c} for some absolute constant c.   In future papers we will develop these ideas further, obtaining an asymptotic for the number of 4-term progressions p_1 < p_2 < p_3 < p_4 < N of primes as well as superior bounds for r_4(G).   Update, December 2023. Proposition 3.2 in the paper, which is stated without detailed proof, is incorrect. For a counterexample, see Candela, Gonzalez-Sanchez and Szegedy arXiv:2311.13899, Remark 4.3. Proposition 3.2 is invoked twice in the paper. First, it is used immediately after its statement to deduce the second part of Theorem 2.3. However, that theorem concerns only vector spaces over finite fields, and in this setting Proposition 3.2 is correct by standard linear algebra. The remark at the end of Section 3 that the argument works for arbitrary $G$ should, however, be deleted. The second application is in the proof of Lemma 10.6. It may well be possible to salvage this lemma, particularly if $P$ is assumed proper, but in any case it is only applied once, in the proof of Proposition 10.8. There, $P$ is proper and, more importantly, $H = \{0\}$ is trivial; in this setting Lemma 10.6 and its proof remain valid. 
Ben Green, Terence Tao