h(x+y^2) \gg N^{3/2}.$$ However, for any arithmetic progression $P\subset\{1, \dots, N\}$, we have $${\left| \sum_{x \in P} f(x)\right|} \ll N^{1/2}.$$ Hence, for any local function $\phi$ of resolution ${\geqslant}\delta N$ and modulus ${\leqslant}\delta^{-1}$, the triangle inequality gives the discorrelation $${\left| \sum_{x \in {\mathbb{Z}}} f(x) \phi(x)\right|} \ll \delta^{-2} N^{1/2}.$$

This example is a local obstruction coming from the real numbers: the nature of our counting operator means that we cannot disentangle possible correlations between the $f$ and $g$ functions on subintervals of length $N^{1/2}$. We can, however, show that these are the only other possible obstructions to uniformity.

\[local control\] Let $\delta > 0$ and $f_1, f_2, f_3 : {\mathbb{Z}}\to {\mathbb{C}}$ be 1-bounded functions with support in ${\left\{1, \dots, N\right\}}$. Suppose that $${\left| \sum_{x\in{\mathbb{Z}}}\sum_{ y\in {\mathbb{N}}} f_1(x) f_2(x+y)f_3(x+y^2) \right|}{\geqslant}\delta N^{3/2}.$$ Then either $N \ll \delta^{-O(1)}$, or for each $i=1,2,3$ there is a 1-bounded local function $\phi_i$ of resolution $\gg \delta^{O(1)} N^{1/2}$ and modulus $q_i \ll \delta^{-O(1)}$ such that $${\left| \sum_{x\in {\mathbb{Z}}} f_i(x)\phi_i(x) \right|} \gg \delta^{O(1)} N.$$

Longer polynomial progressions {#longer intro}
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In analogy with the first author’s generalisation [@PeluseBounds] of [@PelusePrendivilleQuantitative], it is natural to ask whether the methods of this paper yield polylogarithmic bounds for sets of integers lacking 