Large spectrum (${\mathrm{Spec}}_{\rho }(f)$)

For a set $A$ of density $\alpha$ in a finite abelian group,

Chang's theorem says that ${\mathrm{Spec}}_{\rho }(1_A)$ is contained in the span of a dissociated set of size $O({\rho }^{-2}\log ({\alpha }^{-1}))$.

A subset $\Lambda$ of an abelian group is dissociated when a relationforces every coefficient to vanish.