 Solution

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 ${\mathbb{F}}_p^{n+N}$.

Solved by gpt-5.6-sol high.