General inverse theorem for pairwise-connected k-wise correlations (Conjecture 3.13)

Establish Conjecture 3.13 by proving that for every integer k ≥ 1 there exists an integer k′ such that for all ε, α > 0 there exist d ∈ N and δ > 0 with the following property: for any pairwise-connected distribution μ over E^k in which each atom has mass at least α, and any 1-bounded functions f1, …, fk: E^n → C satisfying E_{(x1,…,xk)∼μ^{⊗n}}[f1(x1)⋯fk(xk)] ≥ ε, there exist a combinatorial degree-k′ function P: E^n → C with L2-norm 1 and a function L: E^n → C with L2-norm 1 and degree at most d such that |⟨f1, L · P⟩| ≥ δ.

Background

The paper studies inverse theorems for k-wise correlations under distributions over product spaces, with special attention to pairwise-connected distributions that arise in additive combinatorics and theoretical computer science. For k = 3, the authors prove a complete local and global inverse theory (Theorems 3.8–3.10), identifying product-function and low-degree structure after random restrictions.

Extending these results to k ≥ 4 remains open. Conjecture 3.13 proposes a general structural form—combinatorial low-degree functions together with low-degree polynomials—capturing functions that achieve noticeable k-wise correlation for any pairwise-connected distribution. Proving or refuting this conjecture would generalize the k = 3 theory and potentially enable new density-increment frameworks beyond Gowers uniformity norms.