Kahn–Kalai conjecture on DNF approximation of low-influence biased Boolean functions

Let $f\colon\left\{0,1\right\}^{n}\to\left\{0,1\right\}$ be a monotone Boolean function with expectation $\mathbb{E}[f]=\mu$, and let $I[f]$ denote its total influence. A DNF of width $w$ is a union of sub-cubes of co-dimension at most $w$. Kahn–Kalai's conjecture. For any $C,\epsilon,\mu>0$, there exists $w=O_{C,\epsilon}\left(\log\left(1/\mu\right)\right)$ such that, whenever

$$I[f]\le C\mu\log(1/\mu),$$

$f$ can be $\epsilon\mu$-approximated by a DNF of width at most $w$. This conjecture concerns the structure of biased Boolean functions whose influence is within a constant factor of the minimum possible; the paper presents it as an open problem, while the stated approximation theorem is the target assertion.