and instead give details for a constant-proportion density increment, as our argument is somewhat cleaner in this form.

More specifically, we show that if $A \subset {\left\{1,\dots, N\right\}}$ has density $\delta$ and lacks nontrivial configurations of the form $x, x+y, x+y^2$, then there exists an arithmetic progression $P$ of length $|P| \gg \delta^{O(1)} N^{1/2}$ and common difference $q \ll \delta^{-O(1)}$ such that we have the density increment $$\label{constant proportion}
\frac{|A\cap P|}{|P|} {\geqslant}(1+\Omega(1)) \frac{|A|}{N}.$$ As outlined in [@PelusePrendivilleQuantitative], the ‘almost bounded’ size of $q$ allows us to iterate this procedure. (In [@PelusePrendivilleQuantitative], we obtain the weaker density increment $(1+\Omega(\delta^{O(1)}))|A|/N$, which leads to the extra logarithm appearing in the bound there.)

We obtain the constant-proportion increment by combining the local $U^1$-control result of [@PelusePrendivilleQuantitative] with a strategy of Heath–Brown [@HeathBrownInteger] and Szemerédi [@SzemerediInteger], which has a very robust formulation due to Green and Tao [@GreenTaoNewII]. To accomplish this, we first give a structural characterisation of sets lacking the nonlinear Roth configuration (Theorem \[local control\]). These sets resemble the level sets of the product of a function that is constant on intervals of length $N^{1/2}$ and a function that is constant on congruence classes modulo a bounded $q$.

Having obtained such a structural 