Recall the definition of the Gowers uniformity norm $\|f\|_{U^{k}(G)}$,
\begin{align}
\|f\|_{U^{k}(G)} := \left( \mathbb{E}_{x,h_1,\ldots,h_k \in G} \Delta_{h_1} \ldots \Delta_{h_{k}} f(x) \right)^{2^{-k}} \,
\end{align}
where the operator $\Delta_h$ is a multiplicative analog of a derivative given by
\begin{align}
\Delta_h f(x) := f(x+h) \overline{f(x)} \,,
\end{align}
and $G$ is a finite abelian group. I'm specifically interested in the case $G=\mathbb{Z}_d$ of integers modulo $d$, and $k=3$. Therefore, I'll just use the shorthand notation
$\|f\|_{U^{k}(\mathbb{Z}_d)} = \|f\|_{U^{k}}$.
I'm interested in functions $f:\mathbb{Z}_d \to \mathbb{C}$ which have some fixed value of $\|f\|_2$, say 1, meaning that
\begin{align}
\|f\|_2^2 = \sum_{h \in \mathbb{Z}_d} f(h) \overline{f(h)} = 1\,.
\end{align}
Then my question is,
What are the functions having unit 2-norm which minimize $\|f\|_{U^3}$?
I can prove a lower bound of $\|f\|^8_{U^3} \ge \frac{2}{d^{4} (d+1)}$, so such functions cannot have arbitrarily small Gowers norm. This bound seems to be tight for all values of $d$ (via numerics) but there is no obvious function which provably saturates the bound for all $d$.
From what I can tell, it appears that such obstructions to Gowers uniformity, like the 2-norm constraint above, have been studied before. But I cannot tell if such extremal problems have been studied, or even if they are thought to be tractable.
