Fourier bounds and pseudorandom generators for product tests

TL;DR

This paper establishes tight Fourier bounds for product functions on disjoint inputs and constructs nearly optimal pseudorandom generators for these functions, advancing derandomization techniques for combinatorial rectangles.

Contribution

It introduces a new Fourier level-$d$ inequality and develops pseudorandom generators with near-optimal seed length for product functions and combinatorial rectangles.

Findings

Proved tight bounds on Fourier spectrum sums for product functions.

Constructed pseudorandom generators with seed length nearly optimal.

Extended results to functions with range [-1,1] using Schur-convexity.

Abstract

We study the Fourier spectrum of functions $f\colon \{0,1\}^{mk} \to \{-1,0,1\}$ which can be written as a product of $k$ Boolean functions $f_i$ on disjoint $m$-bit inputs. We prove that for every positive integer $d$, \[ \sum_{S \subseteq [mk]: |S|=d} |\hat{f_S}| = O(m)^d . \] Our upper bound is tight up to a constant factor in the $O(\cdot)$. Our proof builds on a new `level-$d$ inequality' that bounds above $\sum_{|S|=d} \hat{f_S}^2$ 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 $\tilde O(m + \log(k/\varepsilon))$, which is optimal up to polynomial factors in $\log m$, $\log\log k$ and $\log\log(1/\varepsilon)$. 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 $\tilde O(\log(1/\varepsilon))$ factor in their seed lengths. Using Schur-convexity, we also extend our results to functions $f_i$ whose range is $[-1,1]$.