Fourth Fourier moment bound for an indicator function (${\sum }_{\chi }|\widehat{1_A}(\chi ){|}^4\le {\alpha }^3$)

For a subset $A$ of a finite abelian group of density of a finite subset $\alpha$, take normalized Fourier coefficients on a finite abelian group and an unnormalized sum over frequencies. The bound follows from $|\widehat{1_A}(\chi )|\le \alpha$ and the Parseval identity on a finite group ${\sum }_{\chi }|\widehat{1_A}(\chi ){|}^2=\alpha$. It bounds the energy of the normalized convolution on a finite group $1_A\ast 1_A$.