Gowers–Wolf conjecture on the true complexity of linear systems

Let $G$ be a finite Abelian group, and let $\mathcal{L}=(L_1,\dots,L_m)$ be a system of distinct linear forms in $d$ variables. The true complexity of $\mathcal{L}$ is the smallest $k$ such that, for every $\epsilon>0$, there exists $\delta>0$ with the property that every function $f:G\to\mathbb{C}$ satisfying $\|f\|_\infty\leq 1$ and $\|f\|_{U^{k+1}}\leq\delta$ also satisfies

$$\left|\mathbb{E}_{x_1,\dots,x_d\in G}\prod_{i=1}^m f\bigl(L_i(x_1,\dots,x_d)\bigr)\right|\leq\epsilon.$$

Gowers–Wolf conjecture. The true complexity of $\mathcal{L}$ is equal to the smallest $k$ such that the functions $L_1^{k+1},\dots,L_m^{k+1}$ are linearly independent. This conjecture identifies the precise degree of uniformity controlling averages associated with a system of linear forms. The paper presents the linear independence condition as the conjectural converse to the known obstruction from linearly dependent powers, while the general assertion is not resolved in the supplied source.