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We prove a structural result for degree-$d$ polynomials. In particular, weshow that any degree-$d$ polynomial, $p$ can be approximated by anotherpolynomial, $p_0$, which can be decomposed as some function of polynomials$q_1,...,q_m$ with $q_i$ normalized and $m=O_d(1)$, so that if $X$ is aGaussian 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 aboutpolynomial threshold functions, including producing better pseudorandomgenerators, obtaining a better invariance principle, and proving improvedbounds on noise sensitivity.

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