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.