 Additive energy controls three-term progression mixing

If a subset $A$ of a cyclic group has normalized additive energy at most ${\alpha }^4+ϵ$, every nonconstant Fourier coefficient on a finite abelian group of its indicator has modulus at most ${ϵ}^{1/4}$. 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 $N^2$.