Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 129 / 3 / i / Solution

The finite-field Bogolyubov lemma states that if $A\subseteq G={\mathbb{F}}_p^n$ has density $\alpha$, then $2A-2A$ contains a subspace of codimension at most $2{\alpha }^{-2}$.

Use normalized Fourier analysis on a finite abelian group and put $f=1_A$. DefineBy Parseval identity,so $|S|\le 2{\alpha }^{-2}$. LetThen $V$ is a subspace of codimension at most $|S|$.

The normalized representation function of $2A-2A$ iswhere $\tilde{f}(x)=f(-x)$. For $x\in V$, all terms indexed by $S$ are nonnegative real numbers, whileThe trivial character alone contributes ${\alpha }^4$, so $r(x)>0$. Hence $x\in 2A-2A$, proving the Finite-field Bogolyubov lemma.