Definition 4.8 ($U^3$ norm). Given $f:G\to ℂ$, define its $U^3$ norm by

$$\begin{array}{llll}\hfill \parallel f{\parallel }_{U^3(G)}^8& :={𝔼}_{{}_{\in }x,a,b,c}f(x)\overline{f(x+a)f(x+b)f(x+c)}& \hfill & \\ \hfill & f(x+a+b)f(x+b+c)f(x+a+c)\overline{f(x+a+b+c)}& \hfill & \\ \hfill & ={𝔼}_{x,h_1,h_2,h_3\in G}\prod _{𝜀\in {\{0,1\}}^3}{C}^{|𝜀|}f(x+𝜀\cdot h)& \hfill & \end{array}$$

where $Cg(x)=\overline{g(x)}$ and $|𝜀|$ denotes the number of ones in $𝜀$.