Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 129 / 2 / i / Solution

Take $x_1,x_2,x_3,x_4\in {\varphi }^{-1}(W)$ with $x_1+x_2=x_3+x_4$. Write $x_3=x_1+a$ and $x_4=x_1+b$; then $x_2=x_1+a+b$. Hencebelongs to $X$. All four values of $\varphi$ lie in the same affine subspace $W=V+w$, and their coefficients in $D$ sum to zero, so $D\in V$. The hypothesis $V\cap X=\{0\}$ gives $D=0$, precisely the additive-quadruple identity required of a Freiman homomorphism. This is the second-difference obstruction to a Freiman homomorphism.