On Approximability of Satisfiable $k$-CSPs: VI

Abstract

We prove local and global inverse theorems for general $3$-wise correlations over pairwise-connected distributions. Let $μ$ be a distribution over $Σ\times Γ\times Φ$ such that the supports of $μ_{xy}$, $μ_{xz}$, and $μ_{yz}$ are all connected, and let $f: Σ^n \to \mathbb{C}$, $g: Γ^n \to \mathbb{C}$, $h: Φ^n \to \mathbb{C}$ be $1$-bounded functions satisfying \[ \left|\mathbb{E}_{(x,y,z) \sim μ^{\otimes n}}[f(x)g(y)h(z)]\right| \geq \varepsilon. \] In this setting, our local inverse theorem asserts that there is $δ:=\textsf{exp}(-\varepsilon^{-O_μ(1)})$ such that with probability at least $δ$, a random restriction of $f$ down to $δn$ coordinates $δ$-correlates to a product function. To get a global inverse theorem, we prove a restriction inverse theorem for general product functions, stating that if a random restriction of $f$ down to $δn$ coordinates is $δ$-correlated with a product function with probability at least $δ$, then $f$ is $2^{-\textsf{poly}(\log(1/δ))}$-correlated with a function of the form $L\cdot P$, where $L$ is a function of degree $\textsf{poly}(1/δ)$, $\|L\|_2\leq 1$, and $P$ is a product function. We show applications to property testing and to additive combinatorics. In particular, we show the following result via a density increment argument. Let $Σ$ be a finite set and $S \subseteq Σ\times Σ\times Σ$ such that: (1) $(x, x, x) \in S$ for all $x \in S$, and (2) the supports of $S_{xy}$, $S_{xz}$, and $S_{yz}$ are all connected. Then, any set $A \subseteq Σ^n$ with $|Σ|^{-n}|A| \geq Ω((\log \log \log n)^{-c})$ contains $x, y, z \in A$, not all equal, such that $(x_i,y_i,z_i) \in S$ for all $i$. This gives the first reasonable bounds for the restricted 3-AP problem over finite fields.