关于可满足k-CSP的近似性：IV
On Approximability of Satisfiable k-CSPs: IV
AI总结：
本文证明了满足温和连通性条件下一般三元相关性的稳定性定理，刻画了高相关函数必来源于常数大小阿贝尔群的结构，并给出改进的直积定理和限制逆定理，应用于PCP与组合数学。
摘要：
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。更具体地说，我们证明：如果Σ、Γ和Φ是常数大小的字母表，μ是Σ×Γ×Φ上的成对连通分布，且不存在(ℤ,+)嵌入使得每个原子的概率为Ω(1)，则以下结论成立。任何满足\\[\left|\mathbb{E}_{(x,y,z)\sim μ^{\otimes n}}\big[f(x)g(y)h(z)\big]\right|\geq \varepsilon\\]的1-有界函数三元组$f\colon Σ^n\to\mathbb{C}$，$g\colon Γ^n\to\mathbb{C}$，$h\colon Φ^n\to\mathbb{C}$必定来源于与分布μ相关联的阿贝尔群。更具体地说，我们证明存在一个常数大小的阿贝尔群$(H,+)$，使得对于任何这样的$f,g$和$h$，函数$f$（类似地$g$和$h$）与形如$\tilde{f}(x) = χ(σ(x_1),\ldots,σ(x_n)) L (x)$的函数相关，其中$σ\colon Σ\to H$是某个映射，$χ\in \hat{H}^{\otimes n}$是一个特征，$L\colon Σ^n\to\mathbb{C}$是一个具有有界2-范数的低次函数。在此过程中，我们证明了一些可能具有独立意义的额外结果，例如改进的直积定理，以及一个我们称之为“限制逆定理”的结果，该结果涉及在随机限制下以显著概率与乘积函数具有显著相关性的函数的结构。在伴随论文中，我们展示了我们的结果在概率可检查证明领域以及离散数学的各个领域（如极值组合学和加性组合学）中的应用。
英文摘要：
We prove a stability result for general $3$-wise correlations over distributions satisfying mild connectivity properties. More concretely, we show that if $Σ,Γ$ and $Φ$ are alphabets of constant size, and $μ$ is a pairwise connected distribution over $Σ\timesΓ\timesΦ$ with no $(\mathbb{Z},+)$ embeddings in which the probability of each atom is $Ω(1)$, then the following holds. Any triplets of $1$-bounded functions $f\colon Σ^n\to\mathbb{C}$, $g\colon Γ^n\to\mathbb{C}$, $h\colon Φ^n\to\mathbb{C}$ satisfying \[ \left|\mathbb{E}_{(x,y,z)\sim μ^{\otimes n}}\big[f(x)g(y)h(z)\big]\right|\geq \varepsilon \] must arise from an Abelian group associated with the distribution $μ$. 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 $\tilde{f}(x) = χ(σ(x_1),\ldots,σ(x_n)) L (x)$, where $σ\colon Σ\to H$ is some map, $χ\in \hat{H}^{\otimes n}$ is a character, and $L\colon Σ^n\to\mathbb{C}$ 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.
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