 Solution

Define the normalized progression averageWe first prove the needed four-term progression bound with torsion factors, valid for the arbitrary cyclic modulus in the question:For the factor at slope zero, write $y=x+3d$, $t=d$, and apply Cauchy-Schwarz to discard $f_3(y)$:For concise bookkeeping put $D_{h}f(x)=f(x)\overline{f(x+h)}$, the conjugate of the preceding derivative convention, and $F_{i,h}=D_{-(3-i)h}{f}_i$. Expanding the square with $t,t+h$ and setting $z=y-t$ gives ${\mathbb{E}}_{z,t,h}{F}_{0,h}(z-2t)F_{1,h}(z-t)F_{2,h}(z)$. Another Cauchy-Schwarz, now in $(z,h)$, discards $F_{2,h}$ and yieldsExpand with $t,t+k$, set $w=z-t$, and apply Cauchy-Schwarz a third time to discard $D_{-k}{F}_{1,h}(w)$. The remaining $t$-average is a full average over $x=w-t$, soNegating $h,k$ does not change the average. The nonnegative integrand depends only on $2k$ and $3h$. Averaging over the subgroups $2{\mathbb{Z}}_N$ and $3{\mathbb{Z}}_N$ costs at most their indices $t_2=\gcd (2,N)$ and $t_3=\gcd (3,N)$ compared with unrestricted increments. The unrestricted expression is exactly $\Vert f_0{\Vert }_{U^3}^8$, obtained by expanding the last squared average. Thus the bound is $(t_2{t}_3{)}^{1/8}\Vert f_0{\Vert }_{U^3}\le 6^{1/8}\Vert f_0{\Vert }_{U^3}$.

Reversing the progression handles slope three. To isolate slope one, eliminate slopes three, zero and two successively in the same calculation; the surviving increments have coefficients $-2,1,-1$, so only the index $t_2$ is needed. Reversal handles slope two. This proves the minimum-over-factors bound, without assuming that $2$ or $3$ is invertible modulo $N$.

If $\delta =0$, the requested lower bound is zero and is immediate. Otherwise let $g=1_A-\delta$. Telescoping the product of four copies of $1_A=\delta +g$ against the constant product ${\delta }^4$ gives four error terms, each with one factor $g$ and three factors bounded by one. HenceIn particular, choosing $\alpha \le {\delta }^{32}/(6\cdot 8^8)$ givesThese are ordered modular progression parameters $(x,d)$, as requested; small-order differences and $d=0$ are included. A claim of distinct nontrivial progressions would need a separate subtraction and size condition.