= Fourth Fourier moment bound for an indicator function
{title2=$\sum_\chi|\widehat{\mathbf1_A}(\chi)|^4\leq\alpha^3$}
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 $|\widehat{\mathbf1_A}(\chi)|\leq\alpha$ and the <Parseval identity on a finite group> $\sum_\chi|\widehat{\mathbf1_A}(\chi)|^2=\alpha$. It bounds the energy of the <normalized convolution on a finite group> $\mathbf1_A*\mathbf1_A$.
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