For functions $(f_\epsilon)_{\epsilon\in\{0,1\}^3}$ on $\mathbb F_p^n$, the <Gowers inner product> is
$$
\left\langle(f_\epsilon)\right\rangle_{U^3}
=\mathbb E_{x,h_1,h_2,h_3}
\prod_{\epsilon\in\{0,1\}^3}
\mathcal C^{|\epsilon|}f_\epsilon(x+\epsilon_1h_1+\epsilon_2h_2+\epsilon_3h_3),
$$
where $\mathcal C$ is the <complex conjugate> operator. The <Gowers uniformity norm> is
$$
\|f\|_{U^3}
=\left\langle(f)_{\epsilon\in\{0,1\}^3}\right\rangle_{U^3}^{1/8}.
$$
The <Gowers-Cauchy-Schwarz inequality> states
$$
\left|\left\langle(f_\epsilon)\right\rangle_{U^3}\right|
\leq\prod_{\epsilon\in\{0,1\}^3}\|f_\epsilon\|_{U^3}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 129 / 4 / a / Solution

{c}
{title2=$U^k$ norm}
{wiki}

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 $k$.


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 Incoming links: Gowers uniformity norm