Fourier bounds and pseudorandom generators for product tests

We study the Fourier spectrum of functions f: {0, 1}mk → { -1, 0, 1} which can be written as a product of k Boolean functions fi on disjoint m-bit inputs. We prove that for every positive integer d, [EQUATION] Our upper bounds are tight up to a constant factor in the O(·). Our proof uses Schur-convexity, and builds on a new "level-d inequality" that bounds above [EQUATION] for any [0, 1]-valued function f in terms of its expectation, which may be of independent interest. As a result, we construct pseudorandom generators for such functions with seed length Õ(m + log(k/ε)), which is optimal up to polynomial factors in log m, log log k and log log(1/ε). Our generator in particular works for the well-studied class of combinatorial rectangles, where in addition we allow the bits to be read in any order. Even for this special case, previous generators have an extra Õ(log(1/ε)) factor in their seed lengths. We also extend our results to functions fi whose range is [-1, 1].