On Approximability of Satisfiable k-CSPs: IV
Amey Bhangale, Subhash Khot, Dor Minzer
TL;DR
This paper develops a comprehensive stability theory for 3-wise correlations with respect to a distribution μ on Σ×Γ×Φ, under mild connectivity assumptions and excluding Abelian embeddings into (ℤ,+). The core strategy introduces a master Abelian embedding that captures all embeddings, then uses the path trick to saturate and regularize these embeddings, pairing this with restriction-inverse and direct-product theorems to reveal a structured, low-complexity form for any highly correlated triple (f,g,h). The result shows that significant 3-wise correlation forces f,g,h to align with a product of an embedding-type function and a low-degree component in a finite Abelian group H, with explicit quantitative bounds, and that Horn-SAT obstructions can be overcome via a relaxed base-case framework. The approach integrates Fourier-analytic decomposition, Markov-operator noise stability, random restrictions, and a sequence of reductions to homogeneous settings, yielding potential invariance-principle-type consequences and applications to PCPs, additive combinatorics, and parallel repetition; it also opens avenues toward higher-arity predicates and stronger connections to Gowers norms. Overall, the work provides a robust structural understanding of when 3-wise correlations arise from Abelian-embedded, low-degree, or hybrid mechanisms, with tangible implications for hardness-of-approximation and discrete mathematics.
Abstract
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.