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

WriteApply the Cauchy-Schwarz inequality first in the variable carrying $f_1$ and then in the variable carrying $f_2$. After the invertible linear changes of variables permitted by $2,3\nmid |G|$, apply the assumed three-function $U^2$ estimate to the resulting multiplicative derivatives. The standard calculation givesBy the Derivative identity for the Gowers U3 norm, the last two factors are $\Vert f_3{\Vert }_{U^3}^8$ and $\Vert f_4{\Vert }_{U^3}^8$. Taking eighth roots proves

If $A\subseteq G$ has density $\alpha$, write $1_A=\alpha +f$. Expanding ${\Lambda }_4(1_A,1_A,1_A,1_A)$, the constant term is ${\alpha }^4$, and the displayed inequality bounds every nonconstant term after translation of one factor by a $U^3$ norm of the balanced function $f$. Thus sufficiently small $\Vert 1_A-\alpha {\Vert }_{U^3}$ makes the normalized number of four-term arithmetic progressions close to ${\alpha }^4$.