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 $|\widehat{1_D}|\ge {\alpha }^{3/2}/\sqrt{8}$. 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.