configuration in question. It then proceeds to show that there is a substructure $S \subset \{1, \dots, N\}$ on which $A$ has increased density $\delta + \Omega_\delta(1)$. One then hopes to iterate the argument with $A \cap S$ in place of $A$ and $S$ in place of $\{1, \dots, N\}$.

One avenue to obtaining polylogarithmic bounds in a Szemerédi-type theorem is to obtain a constant proportion density increment $\delta + \Omega(\delta)$ on a substructure $S$ of polynomial size $|S| \approx N^{\Omega(1)}$. This was accomplished for three-term arithmetic progressions by Heath–Brown [@HeathBrownInteger] and Szemerédi [@SzemerediInteger] (in fact, they were able to handle a smaller lower bound on $|S|$).

An alternative strategy for obtaining polylogarithmic bounds is to obtain the weaker polynomial increment $\delta + \Omega(\delta^{O(1)})$, yet on a *dense* or *global* substructure $S$, that is, a substructure of size $|S| {\geqslant}\exp(-O(\delta^{-O(1)})) N$. This was accomplished by Sárközy [@SarkozyDifferenceI] for the configuration $x, x+y^2$ and for three-term arithmetic progressions by Bourgain [@BourgainTriples].

Both of these strategies are achievable for the nonlinear Roth configuration. The global structure strategy is perhaps the most natural, and may be accomplished by utilising a generalisation of Theorem \[global control\]. In this note we do not pursue this, 