= Additive energy controls three-term progression mixing
{title2=$|T-\alpha\beta\gamma N^2|\leq\sqrt{\gcd(2,N)\beta\gamma}\,\epsilon^{1/4}N^2$}
If a subset $A$ of a cyclic group has normalized <additive energy> at most $\alpha^4+\epsilon$, every nonconstant <Fourier coefficient on a finite abelian group> of its indicator has modulus at most $\epsilon^{1/4}$. Expanding the count of $a+c=2b$ gives the frequency product $\widehat{1_A}(r)\widehat{1_C}(r)\widehat{1_B}(-2r)$. The <Cauchy-Schwarz inequality> and <Parseval identity on a finite group> bound its nonconstant part by the displayed quantity. Character doubling has multiplicity $\gcd(2,N)$, so no assumption that the modulus is odd is needed. The error is a normalized density error multiplied by $N^2$.
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