  We prove new results on the polarizing random walk framework introduced in
recent works of Chattopadhyay {et al.} [CHHL19,CHLT19] that exploit $L_1$
Fourier tail bounds for classes of Boolean functions to construct pseudorandom
generators (PRGs). We show that given a bound on the $k$-th level of the
Fourier spectrum, one can construct a PRG with a seed length whose quality
scales with $k$. This interpolates previous works, which either require Fourier
bounds on all levels [CHHL19], or have polynomial dependence on the error
parameter in the seed length [CHLT10], and thus answers an open question in
[CHLT19]. As an example, we show that for polynomial error, Fourier bounds on
the first $O(\log n)$ 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-$k$ Lagrange remainder term using
multilinearity and random restrictions. Interestingly, our analysis relies only
on the \emph{level-k unsigned Fourier sum}, which is potentially a much smaller
quantity than the $L_1$ notion in previous works. By generalizing a connection
established in [CHH+20], we give a new reduction from constructing PRGs to
proving correlation bounds. Finally, using these improvements we show how to
obtain a PRG for $\mathbb{F}_2$ polynomials with seed length close to the
state-of-the-art construction due to Viola [Vio09], which was not known to be
possible using this framework.
