that the exponent $c = 2^{-150}$ is permissible, where 150 represents the combined number of times we utilise the Cauchy–Schwarz inequality in [@PelusePrendivilleQuantitative] and this paper

We also show, in a quantitatively effective manner, that the major arc Fourier coefficients of a set determine how many nonlinear Roth configurations the set contains.

\[global control\] Let $\delta > 0$ and $f, g, h : {\mathbb{Z}}\to {\mathbb{C}}$ be 1-bounded functions with support in ${\left\{1, \dots, N\right\}}$. Suppose that $${\left| \sum_{x\in{\mathbb{Z}}}\sum_{ y\in {\mathbb{N}}} f(x) g(x+y)h(x+y^2) \right|}{\geqslant}\delta N^{3/2}.$$ Then either $N \ll \delta^{-O(1)}$, or there is a frequency $\alpha \in {\mathbb{R}}$ and a positive integer $q \ll \delta^{-O(1)}$ such that[^1] ${\left\| q\alpha\right\|} \ll \delta^{-O(1)}/N$ and $${\left| \sum_{x\in {\mathbb{Z}}} h(x)e(\alpha x) \right|} \gg \delta^{O(1)} N.$$

In the nomenclature of [@TaoObstructions], the major arc linear phases are the only obstructions to uniformity for the nonlinear Roth configuration.

The major arc Fourier coefficients of a subset of $\{1,\dots,N\}$ essentially measure its distribution in arithmetic progressions of common difference $\ll 1$ and length $\gg N$. To illustrate this, the following definition is useful.

\[local function def\] We call a function $\phi : {\mathbb{Z}}\to {\mathbb{C}}$ a *local function of resolution $M$ and modulus $q$* if there exists a partition 