Gowers inner product
= Gowers inner product
{c}
{title2=$\langle(f_\epsilon)\rangle_{U^k}$}
For functions indexed by $\epsilon\in\{0,1\}^k$, the Gowers inner product is
$$
\left\langle(f_\epsilon)\right\rangle_{U^k}
=\mathbb E_{x,h_1,\ldots,h_k}
\prod_{\epsilon\in\{0,1\}^k}
\mathcal C^{|\epsilon|}f_\epsilon(x+\epsilon\mathbin\cdot h),
$$
where $\mathcal C$ denotes complex conjugation.