Existence of biased functions matching KKL

Determine whether there exists a Boolean function f:{0,1}^n -> {0,1} such that Pr_{x uniform}[f(x)=0] = 1/n and the influence of each input bit (under the uniform distribution) equals c · log n / n^2 for some constant c; equivalently, investigate the existence of biased functions that match the Kahn–Kalai–Linial (KKL) theorem.

Background

Ajtai and Linial’s classical construction achieves resilience to coalitions of size on the order of n/log^2 n, while the KKL theorem suggests that the theoretical limit is closer to n/log n. The paper highlights that pushing resilience beyond the Ajtai–Linial barrier may require developing biased functions that achieve KKL-level influences.

Balanced functions such as TRIBES match KKL in the balanced regime, but when tuned so that Pr[TRIBES=0]=1/n, each bit’s influence becomes c·log^2 n/n^2, which is too large by a log n factor. The authors propose a concrete question aimed at constructing biased functions that meet the KKL bound on influence, as a potential first step toward improving resilience past the cn/log^2 n threshold.