Boolean Functions With Low Average Sensitivity Depend On Few Coordinates

Consider a function f : f0;1g n ! f0;1g. The sensitivity of a point v 2 f0;1g n is jfv 0 : f (v 0) 6 = f (v); dist(v; v 0) = 1gj, i.e. the number of neighbors of the point in the discrete cube on which the value of f diiers. The average sensitivity of f is the average of the sensitivity of all points in f0;1g n. (This can also be interpreted as the sum of the innuences of the n variables on f , or as a measure of the edge boundary of the set which f is the characteristic function of.) We show here that if the average sensitivity of f is k then f can be approximated by a function depending on c k coordinates where c is a constant depending only on the accuracy of the approximation but not on n. We also present a more general version of this theorem, where the sensitivity is measured with respect to a product measure which is not the uniform measure on the cube.