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?