Normalized Fourier analysis on a finite abelian group

For a finite abelian group $G$, use the normalized average ${\mathbb{E}}_x=|G{|}^{-1}{\sum }_x$, normalized convolutionand Fourier transform $\hat{f}(\gamma )={\mathbb{E}}_{x}f(x)\overline{\gamma (x)}$. Then Parseval's identity and the convolution identity take the forms

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.