= Four-term progression bound with torsion factors
{title2=$|\Lambda_4|\leq(t_2t_3)^{1/8}\min_i\|f_i\|_{U^3}$}
For a <finite abelian group> $G$, put $t_j=|\{x\in G:jx=0\}|$. If $|f_i|\leq1$, then $|\mathbb E_{x,d}\prod_{i=0}^3f_i(x+id)|\leq(t_2t_3)^{1/8}\min_i\|f_i\|_{U^3}$. Three applications of <Cauchy-Schwarz>, eliminating the factors at slopes $3,2,1$, bound the eighth power by $\mathbb E_{h,k}|\mathbb E_x\Delta_{-3h}\Delta_{-2k}f_0(x)|^2$, where $\Delta_hf(x)=f(x)\overline{f(x+h)}$. The integrand is nonnegative; averaging over $3G$ and $2G$ costs at most their indices $t_3$ and $t_2$ compared with the unrestricted average defining $\|f_0\|_{U^3}^8$. Reversal treats slope three. For slope one, elimination at slopes $3,0,2$ produces increments $-2h,k,-r$, costing only $t_2$; reversal treats slope two. In a <cyclic group>, $t_2\leq2,t_3\leq3$, so the constant is at most $6^{1/8}$ independently of group size.
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