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

For functions $(f_{ϵ}{)}_{ϵ\in \{0,1{\}}^3}$ on ${\mathbb{F}}_p^n$, the Gowers inner product iswhere $\mathcal{C}$ is the complex conjugate operator. The Gowers uniformity norm isThe Gowers-Cauchy-Schwarz inequality states

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Repeated Cauchy-Schwarz inequality shows that the $U^3$ norm dominates the absolute mean of a function:Taking $f=1_A$ and using $\mathbb{E}{1}_A=\alpha$ gives

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For $h(x)=e_p(q(x))$, the product in the $U^3$ cube average isThe expression in parentheses is the third additive derivative of the quadratic form $q$, so it vanishes. Every cube contributes one and therefore the quadratic phase satisfies

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Apply the Cauchy-Schwarz inequality successively in the three shift variables to the correlation with the quadratic phase $h$. At the third step the third additive derivative of its phase vanishes, leavingThus $|⟨f,h⟩|\ge \delta$ implies

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Expand the $U^3$ cube product forWhenever all eight $x$-vertices of the cube lie in $S$, the Freiman homomorphism property makes every second additive derivative of $\varphi$ vanish. Consequently the coefficients of $y$ and of each of the three $y$-direction increments in the phase cancel, so the phase product around the cube is one. It follows thatPart b(i), applied to $S$, now givesSince $|g|\le 1$, one can apply the inverse theorem for the Gowers U3 norm over a finite field: $g$ has nontrivial correlation, quantitatively in $p$ and $\sigma$, with a quadratic phase on ${\mathbb{F}}_p^{n+N}$.

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