characterisation, an energy increment procedure closely following [@GreenTaoNewII] allows us to approximate an arbitrary set of integers by these level sets, up to an error that does not contribute substantially to the count of nonlinear Roth configurations. A combinatorial argument then allows us to deduce that our set must have a substantial density increment on one of these level sets, of the form $\delta + \Omega(\delta)$. As a result, our density increment procedure requires only $\log(\delta^{-1})+O(1)$ iterations, compared with the $O(\delta^{-O(1)})$ required in [@PelusePrendivilleQuantitative], and this yields the polylogarithmic improvement over our previous density increment iteration.

The remainder of this paper is organized as follows. We derive Theorem \[main\] in §\[increment sec\] via a density increment iteration. Our deduction uses a density increment lemma that is established in §§\[inverse theorem section\]–\[increment proof sec\]. We prove Theorem \[global control\] and Corollary \[global control cor\] in §\[global sec\]. Theorem \[local control\] will follow from combining Lemma \[partial inverse theorem\] with Corollary \[global control cor\].

Notation
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### Standard conventions

We use ${\mathbb{N}}$ to denote the positive integers. For a real number $X {\geqslant}1$, write $[X] = \{ 1,2, \ldots, {\left\lfloor X \right\rfloor}\}$. A complex-valued function is said to be *1-bounded* if the 