Bogolyubov lemma

If $A$ has positive density in a finite abelian group, then $A+A-A-A$ contains a structured neighbourhood of zero. In a finite-dimensional vector space this neighbourhood can be taken to be a large vector subspace; in a cyclic group it can be taken to be a Bohr set.

For a set of density $\alpha$ in a finite cyclic group, select frequencies whose normalized Fourier coefficients on a finite abelian group have magnitude at least ${\alpha }^{3/2}/2$. Parseval identity bounds their number by $4{\alpha }^{-2}$. Fourier inversion of the fourfold convolution shows it is positive on the Bohr set in phase-distance convention of radius $1/6$. This is an explicit form of the Bogolyubov lemma.

For $D\subseteq \mathbb{Z}/q\mathbb{Z}$ of subset density $\alpha$, $2D-2D$ contains a Bohr set of rank at most $8{\alpha }^{-2}$ and a fixed positive width, including for composite $q$. With normalized Fourier coefficients on a finite abelian group, retain the frequencies where $|\widehat{1_D}|\ge {\alpha }^{3/2}/\sqrt{8}$. Parseval identity on a finite group bounds their number and the discarded fourth moment. Their nearly constant phases make $1_D\ast 1_D\ast 1_{-D}\ast 1_{-D}$ positive on the Bohr set.

If $A\subseteq {\mathbb{F}}_p^n$ has density $\alpha$, then $2A-2A$ contains a vector subspace of codimension at most $2{\alpha }^{-2}$. It is the annihilator of the Fourier spectrum on which $|\widehat{1_A}|$ is at least ${\alpha }^{3/2}/\sqrt{2}$.

For fixed $c>0$ and prime $p$, if $A\subseteq {\mathbb{F}}_p^N$ has at least $c|A{|}^3$ additive quadruples, then $2A-2A$ contains a vector subspace of size at least $c^{\prime }(c,p)|A|$. Combine the Balog-Szemerédi-Gowers theorem, the Freiman-Ruzsa theorem over a finite field, and the Finite-field Bogolyubov lemma.

For every $\alpha >0$, if $A$ has density at least $\alpha$ in a cyclic group of prime order, then $2A-2A$ contains a proper generalized arithmetic progression of rank $O_{\alpha }(1)$ and size ${\Omega }_{\alpha }(|G|)$.