= Cyclic 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 $|\widehat{1_D}|\geq\alpha^{3/2}/\sqrt8$. <Parseval identity on a finite group> bounds their number and the discarded fourth moment. Their nearly constant phases make $1_D*1_D*1_{-D}*1_{-D}$ positive on the <Bohr set>.
Back to article page
