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

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.

The hypothesis says that the additive energy satisfies $E(A)\ge c|A{|}^3$. The Balog-Szemerédi-Gowers theorem supplies $A^{\prime }\subseteq A$ withBy the Freiman-Ruzsa theorem over a finite field, $A^{\prime }$ lies in a subspace $H$ withThus $A^{\prime }$ has density at least $C_2^{-1}$ in $H$. Applying the Finite-field Bogolyubov lemma inside $H$ gives a subspaceof codimension bounded in terms of $c,p$ alone. Thereforewhich is the additive energy produces a large subspace in a fourfold difference set result.

The Ruzsa triangle inequality applied to $A-A$ also bounds $|A+A|$ in terms of $C|A|$, so $A$ has bounded doubling. The Freiman-Ruzsa theorem places $A$ inside a proper coset progression $P$ whose rank and ratio $|P|/|A|$ are bounded only in terms of $C$. Hence $A$ has positive density bounded in terms of $C$ inside a bounded-rank progression.

For sufficiently large $|A|$, the Szemerédi theorem in a bounded-rank coset progression gives a nontrivial three-term arithmetic progression in $A$. This proves the small difference set forces a three-term arithmetic progression assertion.