A structure theorem for poorly anticoncentrated polynomials of Gaussians 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, p0, which can be decomposed as some function of polynomials q1,.,qm with qi normalized and m=Od(1), so that if X is a Gaussian random variable, the probability distribution on (q1(X),.,qm(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.