= Large Fourier coefficient from a deficit of three-term progressions
{title2=$\max_{r\ne0}|\widehat{1_A-\alpha}(r)|>\alpha^2/2$}
In a <cyclic group> of odd prime order, the normalized ordered three-term <arithmetic progression> count is $\sum_r\widehat{1_A}(r)^2\widehat{1_A}(-2r)$. The zero frequency contributes $\alpha^3$. If the count is less than $\alpha^3/2$, its nonzero-frequency contribution has magnitude greater than $\alpha^3/2$. The <Parseval identity on a finite group> bounds that magnitude by $\alpha\max_{r\ne0}|\widehat{1_A-\alpha}(r)|$, proving the displayed estimate. The count includes progressions of zero common difference.
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