of ${\mathbb{Z}}$ into intervals of length $M$ such that $\phi$ is constant on the intersection of every such interval with every congruence class mod $q$.

\[global control cor\] 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 1-bounded local function $\phi $ of resolution $\gg \delta^{O(1)} N$ and modulus $q \ll \delta^{-O(1)}$ such that $${\left| \sum_{x\in {\mathbb{Z}}} h(x)\phi(x) \right|} \gg \delta^{O(1)} N.$$

One cannot hope to prove that the functions $f$ and $g$ above also correlate globally with local functions, as the following example illustrates. For any positive integers $x_1, x_2 {\leqslant}N^{1/2}$, set $$f{\left( x_1 + (x_2-1){\left\lfloor N^{1/2} \right\rfloor} \right)} = \begin{cases} 
        1 & \text{ if } x_2 \equiv 0 \pmod 4,\\
        0 & \text{ if } x_2 \equiv 1 \pmod 4,\\ 
        -1 & \text{ if } x_2 \equiv 2 \pmod 4,\\
        0 & \text{ if } x_2 \equiv 3 \pmod 4; \end{cases}$$ and set $f(x) = 0$ everywhere else. Taking $g := f$ and $h:=1_{\{1,\dots, N\}}$, one can check that either $N \ll 1$ or $$\sum_{x\in {\mathbb{Z}}}\sum_{y\in {\mathbb{N}}} f(x)g(x+y) 