Gowers Norms

Gowers Norms

Let $Z$ to be a finite abelian group, let $k\ge 2$ be an integer, $f:\mathbb{Z}\to \mathbb{C}$ a function, $\Vert f{\Vert }_{U^k(Z)}^{2^k}:={\mathbb{E}}_{x,h_1,\dots ,h_k}{\prod }_{\omega \in \{0,1{\}}^k}{\mathcal{C}}^{|\omega |}f(x+\omega \cdot \vec{h})$, with $\mathcal{C}$ is the complex conjugation, $\omega =({\omega }_1,{\omega }_2,\dots ,{\omega }_k)$, ${\omega }_i\in \{0,1\}$, $|\omega |={\sum }_{i=1}^k{\omega }_i$. Define $\Vert f{\Vert }_{U^k(Z)}$ the $k$ th Gowers norm to be the unique real and non-negative $2^k$-th root.

Example: Take $k=2$, we get $\Vert f{\Vert }_{U^2(Z)}^4={\mathbb{E}}_{x,h_1,h_2}\left[f(x)\cdot \overline{f(x+h_1)}\cdot \overline{f(x+h_2)}\cdot f(x+h_1+h_2)\right]$

Definition

For $f:[N]\to \mathbb{C}$ a function, embedding $[N]\to \mathbb{Z}/N^{\prime }\mathbb{Z}$ with $N^{\prime }>10kN$ a large prime. Define $\Vert f{\Vert }_{U^k[N]}=\frac{1}{\Vert 1_N{\Vert }_{U^k(\mathbb{Z}/N^{\prime }\mathbb{Z})}}\Vert f{\Vert }_{U^k(\mathbb{Z}/N^{\prime }\mathbb{Z})}$