TITLE: Extremal Obstructions to Gowers Uniformity
QUESTION [10 upvotes]: Recall the definition of the Gowers uniformity norm $\|f\|_{U^{k}(G)}$,
$$
 \|f\|_{U^{k}(G)} := \left( \mathbb{E}_{x,h_1,\ldots,h_k \in G} \Delta_{h_1} \ldots \Delta_{h_{k}} f(x) \right)^{2^{-k}} \,
$$
where the operator $\Delta_h$ is a multiplicative analog of a derivative given by
$$
 \Delta_h f(x) := f(x+h) \overline{f(x)} \,,
$$
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
$$
 \|f\|_2^2 = \sum_{h \in \mathbb{Z}_d} f(h) \overline{f(h)} = 1\,.
$$
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.

REPLY [5 votes]: I'm not quite sure where your lower bound comes from, but something close comes from functions such as
$$ f(x)=e(x^3/d)$$
This (after rescaling by $d^{-1/2}$ to match your definition) has $L^2$ norm 1, and $U^3$ norm
$$ \|f\|_{U^3}^8\leq\frac{2}{d^5}$$
In general, these `phase functions' of degree $n$ will give functions with small $U^n$ norm (because 'differentiating' such a phase function $n-1$ times gives a sum over linear phase functions and hence a lot of a cancellation), and I suspect the extremal example will be of this sort.