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

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.

Let $G_0={\mathbb{F}}_p^n$ and letFor each fixed first coordinate $d$, the second coordinates occurring in $\Gamma -\Gamma$ are values of $\varphi (x+d)-\varphi (x)$, of which there are at most $C$. ThereforeThe difference-set form of the Plünnecke-Ruzsa inequality now gives

Every $u\in X$ has the formso $\{0\}\times X\subseteq 2\Gamma -2\Gamma$. ConsequentlyThe set on the left has exactly $|X||\Gamma |$ elements, since its fiber over each $x$ is $\varphi (x)+X$. It follows thatand hence $|X|\le C^5$.

Choose $k$ so thatand take a uniformly random codimension-$k$ subspace $V\le {\mathbb{F}}_p^n$. Each fixed nonzero vector lies in $V$ with probability at most $p^{-k}$. The union bound givesThus some $V$ satisfies $V\cap X=\{0\}$.

For a uniformly random coset $W$ of this $V$, every value $\varphi (x)$ lies in $W$ with probability $p^{-k}$. Thereforeso some coset has inverse image $A$ of density at leastPart i says that $\varphi {|}_A$ is a Freiman homomorphism, proving the large Freiman-homomorphic restriction from bounded derivative images.