Boolean Functions With Low Average Sensitivity Depend On Few Coordinates
. The sensitivity of a point \(\) is \(\) dist\(\), i.e. the number of neighbors of the point in the discrete cube on which the value of \(\) differs. The average sensitivity of \(\) is the average of the sensitivity of all points in \(\). (This can also be interpreted as the sum of the influences of the \(\) variables on \(\), or as a measure of the edge boundary of the set which \(\) is the characteristic function of.) We show here that if the average sensitivity of \(\) is \(\) then \(\) can be approximated by a function depending on \(\) coordinates where \(\) is a constant depending only on the accuracy of the approximation but not on \(\). 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.
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