Approximation of biased Boolean functions of small total influence by DNFs

The influence of the kth coordinate on a Boolean function f : {0, 1}ⁿ → {0, 1} is the probability (with respect to the uniform measure on {0, 1}ⁿ) that flipping x_k changes the value f(x). The total influence I(f) is the sum of influences of the coordinates. The well-known ‘Junta theorem’ of Friedgut [Combinatorica 18 (1998) 27–35] asserts that if I(f) ≤ M,, then f can be ε-approximated by a function that depends on 2O(M/ε) coordinates. Friedgut's theorem has a wide variety of applications in mathematics and theoretical computer science. For a function Pr[f(x) = 1] = t ≤ 1/2,, the edge isoperimetric inequality on the cube implies that I(f) ≥ 2t log₂(1/t). Kahn and Kalai (2006) asked, in the spirit of the Junta theorem, whether any f such that I(f) is within a constant factor of the minimum, can be (εt) -approximated by a disjunctive normal form (DNF) of a ‘small’ size (that is, a union of a small number of subcubes). We answer the question by proving the following structure theorem: If I(f) ≤ 2t(log₂(1/t) +M),, then f can be (εt) -approximated by a DNF of size 2^2O(M/ε). The dependence on M is sharp up to the constant factor in the double exponent.