= Gowers uniformity norm
{c}
{title2=$U^k$ norm}
{wiki}
For a <positive integer> $k$ and a <function> $f:G\to\mathbb C$ on a finite <abelian group>, with uniform <expectations>, define
$$
\|f\|_{U^k}^{2^k}=\mathbb E_{x,h_1,\ldots,h_k}\prod_{\epsilon\in\{0,1\}^k}\mathcal C^{|\epsilon|}f(x+\epsilon_1h_1+\cdots+\epsilon_kh_k),
$$
where $\mathcal C$ is <complex conjugation>. The alternating conjugations measure agreement around additive cubes. The expression is nonnegative; for $k\geq2$ its $2^k$th root is a <norm>. At $k=1$ it is only the <seminorm> $|\mathbb E f|$.
With the <multiplicative derivative> $\partial_hf(x)=f(x+h)\overline{f(x)}$,
$$
\|f\|_{U^{k+1}}^{2^{k+1}}=\mathbb E_h\|\partial_hf\|_{U^k}^{2^k}.
$$
Repeated <multiplicative derivatives> remove low-degree <polynomial phases>: a unit-modulus phase of degree less than $k$ has $U^k$ <norm> one. The <Gowers U2 norm> is the fourth moment of <Fourier coefficients on a finite abelian group> and detects linear correlations; the <Gowers U3 norm> controls four-term <arithmetic progression> counts through the <generalized von Neumann inequality for four-term progressions>.<Decompositions, approximate structure, transference, and the Hahn-Banach theorem by Timothy Gowers>[Sections 2.3–2.6]{ref}
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