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- W3197594612 abstract "We study the Fourier spectrum of functions $fcolon {0,1}^{mk} to {-1,0,1}$ 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 $log m$, $loglog k$ and $loglog(1/varepsilon)$. 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 $[-1,1]$." @default.
- W3197594612 created "2021-09-13" @default.
- W3197594612 creator A5052748957 @default.
- W3197594612 date "2019-01-01" @default.
- W3197594612 modified "2023-09-23" @default.
- W3197594612 title "Fourier bounds and pseudorandom generators for product tests." @default.
- W3197594612 hasPublicationYear "2019" @default.
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