We prove a stability result for general $3$-wise correlations over distributions satisfying mild connectivity properties. More concretely, we show that if $\Sigma,\Gamma$ and $\Phi$ are alphabets of constant size, and $\mu$ is a pairwise connected distribution over $\Sigma\times\Gamma\times\Phi$ with no $(\mathbb{Z},+)$ embeddings in which the probability of each atom is $\Omega(1)$, then the following holds. Any triplets of $1$-bounded functions $f\colon \Sigma^n\to\mathbb{C}$, $g\colon \Gamma^n\to\mathbb{C}$, $h\colon \Phi^n\to\mathbb{C}$ satisfying \[ \left|\mathbb{E}_{(x,y,z)\sim \mu^{\otimes n}}\big[f(x)g(y)h(z)\big]\right|\geq \varepsilon \] must arise from an Abelian group associated with the distribution $\mu$. More specifically, we show that there is an Abelian group $(H,+)$ of constant size such that for any such $f,g$ and $h$, the function $f$ (and similarly $g$ and $h$) is correlated with a function of the form $\tilde{f}(x) = \chi(\sigma(x_1),\ldots,\sigma(x_n)) L (x)$, where $\sigma\colon \Sigma \to H$ is some map, $\chi\in \hat{H}^{\otimes n}$ is a character, and $L\colon \Sigma^n\to\mathbb{C}$ is a low-degree function with bounded $2$-norm. En route we prove a few additional results that may be of independent interest, such as an improved direct product theorem, as well as a result we refer to as a ``restriction inverse theorem'' about the structure of functions that, under random restrictions, with noticeable probability have significant correlation with a product function. In companion papers, we show applications of our results to the fields of Probabilistically Checkable Proofs, as well as various areas in discrete mathematics such as extremal combinatorics and additive combinatorics.

翻译：我们证明了在满足温和连通性条件的分布上，一般三元相关性的稳定性结果。具体而言，我们证明：若Σ、Γ和Φ为常数大小的字母表，μ是Σ×Γ×Φ上的两两连通分布，且不具有任何(Z,+)嵌入，其中每个原子的概率为Ω(1)，则以下结论成立。任意三组1-有界函数f: Σⁿ→ℂ、g: Γⁿ→ℂ、h: Φⁿ→ℂ满足 \[ \left|\mathbb{E}_{(x,y,z)\sim \mu^{\otimes n}}\big[f(x)g(y)h(z)\big]\right|\geq \varepsilon \] 时，这些函数必然源于与该分布μ相关的阿贝尔群。更具体地说，存在一个常数大小的阿贝尔群(H,+)，使得对任意这样的f、g和h，函数f（g和h类似）与形如̃f(x) = χ(σ(x₁),…,σ(x_n)) L(x)的函数相关，其中σ: Σ→H为某个映射，χ∈Ĥ^{⊗n}为特征标，L: Σⁿ→ℂ为具有有界2-范数的低次函数。在此过程中，我们独立证明了若干可能具有独立意义的附加结果，包括改进的直积定理，以及我们称之为“限制逆定理”的关于函数结构的结果——该结构表明，在随机限制下，以显著概率与乘积函数具有高度相关性的函数具有特定形式。在配套论文中，我们将展示这些结果在概率可检验证明以及离散数学各领域（如极值组合学与加性组合学）中的应用。