  There is a common theme to some research questions in additive combinatorics
and noise stability. Both study the following basic question: Let $\mathcal{P}$
be a probability distribution over a space $\Omega^\ell$ with all $\ell$
marginals equal. Let $\underline{X}^{(1)}, \ldots, \underline{X}^{(\ell)}$
where $\underline{X}^{(j)} = (X_1^{(j)}, \ldots, X_n^{(j)})$ be random vectors
such that for every coordinate $i \in [n]$ the tuples $(X_i^{(1)}, \ldots,
X_i^{(\ell)})$ are i.i.d. according to $\mathcal{P}$.
  A central question that is addressed in both areas is:
  - Does there exist a function $c_{\mathcal{P}}()$ independent of $n$ such
that for every $f: \Omega^n \to [0, 1]$ with $\mathrm{E}[f(X^{(1)})] = \mu >
0$: \begin{align*} \mathrm{E} \left[ \prod_{j=1}^\ell f(X^{(j)}) \right]
  \ge c(\mu) > 0 \, ? \end{align*}
  Instances of this question include the finite field model version of Roth's
and Szemer\'edi's theorems as well as Borell's result about the optimality of
noise stability of half-spaces.
  Our goal in this paper is to interpolate between the noise stability theory
and the finite field additive combinatorics theory and address the question
above in further generality than considered before. In particular, we settle
the question for $\ell = 2$ and when $\ell > 2$ and $\mathcal{P}$ has bounded
correlation $\rho(\mathcal{P}) < 1$. Under the same conditions we also
characterize the _obstructions_ for similar lower bounds in the case of $\ell$
different functions. Part of the novelty in our proof is the combination of
analytic arguments from the theories of influences and hyper-contraction with
arguments from additive combinatorics.
