Gowers uniformity norm ($U^k$ norm)

The Gowers uniformity norm measures the average multiplicative derivative of a function around affine cubes. The $U^k$ norm detects polynomial phases of degree below $k$.

For functions indexed by $ϵ\in \{0,1{\}}^k$, the Gowers inner product iswhere $\mathcal{C}$ denotes complex conjugation.

Repeated Cauchy-Schwarz inequality gives

A quadratic phase on ${\mathbb{F}}_p^n$ is a function $x\mapsto e_p(q(x))$, where $q$ is a quadratic form and $e_p(t)=e^{2\pi it/p}$. Its third multiplicative derivative is one, so its $U^3$ norm is one.

For fixed prime $p$, a one-bounded function on ${\mathbb{F}}_p^n$ with $U^3$ norm at least $\delta$ has correlation bounded below in terms of $p$ and $\delta$ with a quadratic phase, with the usual nonclassical formulation in small characteristic.