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Product Space Models of Correlation: Between Noise Stability and Additive Combinatorics

Author(s)
Hązła, Jan; Holenstein, Thomas; Mossel, Elchanan
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Abstract
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.
Date issued
2018
URI
https://hdl.handle.net/1721.1/137506
Department
Massachusetts Institute of Technology. Institute for Data, Systems, and Society; Massachusetts Institute of Technology. Department of Mathematics
Journal
Discrete Analysis
Publisher
Alliance of Diamond Open Access Journals
Citation
Hązła, Jan, Holenstein, Thomas and Mossel, Elchanan. 2018. "Product Space Models of Correlation: Between Noise Stability and Additive Combinatorics." Discrete Analysis, 20.
Version: Final published version

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