With normalized averages on the finite abelian group $G$, the <Gowers U2 norm> is defined by
$$
\|f\|_{U^2}^4
=\mathbb E_{x,a,b}f(x)\overline{f(x+a)}\,\overline{f(x+b)}f(x+a+b).
$$
The <Gowers U3 norm> is defined by
$$
\|f\|_{U^3}^8
=\mathbb E_{x,a,b,c}
\prod_{\epsilon\in\{0,1\}^3}
\mathcal C^{|\epsilon|}f(x+\epsilon_1a+\epsilon_2b+\epsilon_3c),
$$
where $\mathcal C$ denotes complex conjugation.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 129 / 4 / i / Solution

{c}
{title2=$\|f\|_{U^2}$}

For a function on a finite abelian group,
$$
\|f\|_{U^2}^4=\mathbb E_{x,a,b}f(x)\overline{f(x+a)}\,\overline{f(x+b)}f(x+a+b)
=\sum_{\gamma\in\widehat G}|\widehat f(\gamma)|^4.
$$


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