Let $f:\mathbb{Z}^+\to \mathbb{C}$ be bounded. Say we are interested in studying how $f$ behaves in short three-term arithmetic progressions. It is very well-known that we can bound
$$\sum\_{h\leq H} \sum\_{n\leq N} f(n) f(n+h) f(n+2 h)$$
by $o(H N)$ (to set ourselves a low bar...) provided that we can bound the *Gowers $U^2$ norm* $|f|\_{U^2}$ by $o(1)$, where $|f|\_{U^2}$ is defined by
$$|f|\_{U^2}^4=\frac{1}{H^2 N} \sum\_{n\leq N} \sum\_{h\_1,h\_2\leq H} f(n) \overline{f(n+h\_1) f(n+h\_2)} f(n+h\_1+h\_2).$$
The reduction is simple: change variables letting $n'=n+h$, put the sum on $n'$ on the outside and apply Cauchy-Schwarz.


What happens if what we want to show is rather that
$$\sum\_{h\leq H} \left|\sum\_{n\leq N} f(n) f(n+h) f(n+2 h)\right|^2 = o(H N^2)?$$Is the $U^2$ norm enough? Are higher $U^k$ norms enough? Is, say, the $U^3$ norm needed and enough?



Gowers norms and three-term arithmetic progressions in the mean
If you expand square as a double sum, you get a sum over copies of some configuration of 6 points. By the usual Cauchy--Schwarz argument, this sum is going to be controlled by some $U^d$ norm. The smallest $d$ such that this is controlled by $U^{d+1}$ is called ``true complexity'' of the system. For your case of 6 linear forms in 3 variables, the most relevant paper is probably <https://arxiv.org/abs/1705.06801> (see also references therein).


