Open status of structural characterization for k-wise correlations when k ≥ 4

Determine, for k ≥ 4, the structural characterization of 1-bounded functions f1, …, fk: E^n → C that achieve noticeable k-wise correlation under a pairwise-connected distribution μ over E^k, i.e., given E_{(x1,…,xk)∼μ^{⊗n}}[f1(x1)⋯fk(xk)] ≥ ε, ascertain the precise structure of the functions fi as posed in Question 2.6.

Background

Question 2.6 asks for the structure of 1-bounded functions achieving nontrivial k-wise correlation with respect to a pairwise-connected distribution. For k = 3, the authors resolve this via product-function and low-degree structure after random restrictions (Theorems 3.8–3.10).

For k ≥ 4, the problem remains open and is believed to be challenging, with potential connections to higher-order additive combinatorics and inverse Gowers-norm theorems.