* 1_{-[H]} )(h)}{{\left\lfloor H \right\rfloor}^2}.$$ This is a probability measure on ${\mathbb{Z}}$ with support in the interval $(-H, H)$.

Iterating the density increment {#increment sec}
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In this section we prove Theorem \[main\] using the following lemma, which we will devote §§\[inverse theorem section\]–\[increment proof sec\] to proving.

\[increment lemma\] Let $q {\leqslant}N$ be positive integers and $\delta > 0$. Suppose that $A \subset [N]$ satisfies $|A| {\geqslant}\delta N$ and lacks the configuration $$\label{q config}
x,\ x+y, \ x+qy^2 \qquad (y \neq 0).$$ Then either $N \ll (q/\delta)^{O(1)}$ or there exists $q' {\leqslant}\exp{\left( O{\left( \delta^{-O(1)} \right)} \right)}$ and $N' {\geqslant}q^{-O(1)}\exp{\left( -O{\left( \delta^{-O(1)} \right)} \right)}N^{1/2} $ such that $$\label{increment}
 |A \cap (a + qq'\cdot[N'])| {\geqslant}(1+\Omega(1))\delta N'.$$

This is the same as the proof of [@PelusePrendivilleQuantitative Theorem 1.1], but using the improved density increment lemma above in place of the density increment lemma of [@PelusePrendivilleQuantitative]. Note first that if $A$ lacks the configuration , then the set $$\{x :a+qq'x\in A\},$$ lacks configurations of the form $$x,\ x+y,\ x+q^2q'y^2 \qquad (y \neq 0).$$

Let $A \subset [N]$ have size $\delta N$, and suppose that it has no non-linear Roth configurations . Setting $A_0 := A$, $N_0 := N$ and $q_0 = 1$, let 