A Structure Theorem for Poorly Anticoncentrated Gaussian Chaoses and Applications to the Study of Polynomial Threshold Functions

We prove a structural result for degree-d polynomials. In particular, we show that any degree-d polynomial, p can be approximated by another polynomial, p
₀
, which can be decomposed as some function of polynomials q1,· · ·, q
_m
with q
_i
normalized and m=O
_d
(1), so that if X is a Gaussian random variable, the probability distribution on (q
₁
(X), · · · , q
_m
(X)) does not have too much mass in any small box. Using this result, we prove improved versions of a number of results about polynomial threshold functions, including producing better pseudorandom generators, obtaining a better invariance principle, and proving improved bounds on noise sensitivity.