Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 147 / 1 / ii

Write $1_A$ for the indicator function and use the normalized Fourier analysis on a finite abelian groupThere are $N^5$ sextuples satisfying the equation, since any five coordinates determine the sixth. The desired probability is consequentlyBy orthogonality of complex exponentials, the indicator of the equation isSubstitution makes all six sums independent and gives the sixth Fourier moment as a three-sum collision count:Because $1_A$ is real, $\hat{A}(-r)=\overline{\hat{A}(r)}$. Hence the probability is