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There is a normalization omission in the printed conclusion: the discrepancy of a count of pairs must be bounded by a small function of the energy excess times $N^2$. We prove this normalized result, then give a counterexample to the literal bound without $N^2$. Write $ϵ$ for the energy excess parameter to avoid confusing it with an element of the third set.

For Fourier analysis on a finite abelian group, use $e_N(t)=e^{2\pi it/N}$ andCharacter orthogonality, ${\mathbb{E}}_x{e}_N(rx)=1$ for $r=0$ and zero otherwise, gives Fourier inversion and the Parseval identity on a finite group:Put $u=1_A$, $v=1_B$, $w=1_C$. Expanding the constraint $a_1+a_2=a_3+a_4$ by the same character orthogonality gives the additive energy identityThus ${\sum }_{r\ne 0}|\hat{u}(r){|}^4\le ϵ$, and every nonconstant Fourier coefficient on a finite abelian group of $u$ has modulus at most ${ϵ}^{1/4}$. Expanding the constraint in the triple count $T$ givesThe Cauchy-Schwarz inequality yieldsThe first squared norm is $\gamma$ by the Parseval identity on a finite group. For the second, multiplication by $2$ on ${\mathbb{Z}}_N$ has kernel size $h=\gcd (2,N)$ and every element of its image has exactly $h$ preimages. HenceThis proves that additive energy controls three-term progression mixing, including for even moduli:Thus the intended density-error function can be $F(ϵ)=\sqrt{2}{ϵ}^{1/4}$, which tends to zero. For odd moduli the factor $\sqrt{2}$ can be replaced by one.

To see why the literal unnormalized conclusion is false, take $N=q^2$ and let all three sets be the subgroup of multiples of $q$. Their densities are $1/q$. Every choice of three entries of that subgroup determines the fourth entry of an additive quadruple, so its energy is $q^3$. The excess parameter isEvery pair $(a,b)$ in the subgroup determines $c=2b-a$ there, giving $T=q^2$, whereas $\alpha \beta \gamma N^2=q$. Their difference is $q^2-q\to \infty$, which cannot be bounded by any function of $ϵ$ tending to zero. The missing $N^2$ is therefore a genuine source defect, not an alternative Fourier transform normalization.