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

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 $|X|\le C^5$.