Sixth Fourier moment as a three-sum collision count

For $A$ in a finite abelian group and normalized Fourier transform $\widehat{1_A}$, the normalized number of sextuples satisfyingisThis follows by inserting orthogonality of complex exponentials for the displayed equation: the three variables on one side contribute $\widehat{1_A}(\gamma {)}^3$ and those on the other side contribute its complex conjugate.