Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 129 / 4 / i / Solution

The Finite-field Szemerédi theorem for four-term arithmetic progressions says that for every $\delta >0$ there is $N=N(\delta )$ such that, for every prime $p\ge 5$, whenever $|{\mathbb{F}}_p^n|\ge N$ and $A\subseteq {\mathbb{F}}_p^n$ has density at least $\delta$, there exist $x,d\in {\mathbb{F}}_p^n$ with $d\ne 0$ and