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

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 L_p 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.