Gowers U3 norm
= Gowers U3 norm
{c}
{title2=$\|f\|_{U^3}$}
The third Gowers norm is the eighth root of the conjugated product around a three-dimensional affine cube:
$$
\|f\|_{U^3}^8=\mathbb E_{x,a,b,c}
\prod_{\epsilon\in\{0,1\}^3}\mathcal C^{|\epsilon|}f(x+\epsilon_1a+\epsilon_2b+\epsilon_3c).
$$