Fractional Pseudorandom Generators from Any Fourier Level

We prove new results on the polarizing random walk framework introduced in recent works of Chattopadhyay et al.[CHHL19, CHLT19] that exploit Fourier tail bounds for classes of Boolean functions to construct pseudorandom generators (PRGs). We show that given a bound on the -th level of the Fourier spectrum, one can construct a PRG with a seed length whose quality scales with . This interpolates previous works, which either require Fourier bounds on all levels [CHHL19], or has polynomial dependence on the error parameter in the seed length [CHLT19], and thus answers an open question in [CHLT19]. As an example, we show that for polynomial error, Fourier bounds on the first levels is sufficient to recover the seed length in [CHHL19], which requires bounds on the entire tail. We obtain our results by an alternate analysis of fractional PRGs using Taylor's theorem and bounding the degree- Lagrange …