Anti-concentration of polynomials: Dimension-free covariance bounds and decay of Fourier coefficients

Abstract

We study random variables of the form f(X), when f is a degree d polynomial, and X is a random vector on Rn, motivated towards a deeper understanding of the covariance structure of X⊗d. For applications, the main interest is to bound Var(f(X)) from below, assuming a suitable normalization on the coefficients of f. Our first result applies when X has independent coordinates, and we establish dimension-free bounds. We also show that the assumption of independence can be relaxed and that our bounds carry over to uniform measures on isotropic Lp balls. Moreover, in the case of the Euclidean ball, we provide an orthogonal decomposition of Cov(X⊗d). Finally, we utilize the connection between anti-concentration and decay of Fourier coefficients to prove a high-dimensional analogue of the van der Corput lemma, thus partially answering a question posed by Carbery and Wright.