Anti-concentration of polynomials: $L^{p}$ balls and symmetric measures

TL;DR

This paper establishes variance bounds for polynomials under log-concave measures, leading to new anti-concentration results, especially for uniform measures on $L^{p}$ balls and symmetric log-concave measures, extending previous work.

Contribution

It provides the first variance bounds for polynomials on isotropic $L^{p}$ balls and symmetric log-concave measures, advancing understanding of anti-concentration phenomena.

Findings

Variance bounds depend on the degree and measure symmetry.

Dimension-free small-ball and Fourier decay estimates are derived.

Results extend Carbery and Wright's questions on anti-concentration.

Abstract

We begin with the observation, based on previous results, that dimension-free lower bounds on the variance of a polynomial under a log-concave measure yield dimension-free small-ball and Fourier decay estimates. Motivated by this, we establish variance bounds for polynomials on log-concave random vectors beyond the classical setting of product measures. First, we consider the family of uniform measures on the $n$-dimensional isotropic $L^{p}$ balls. We show that for a degree-$d$ homogeneous polynomial $f=\sum_{I}a_{I}x^{I}$, with $\sum_{I}a_{I}^{2}=1$, the only obstruction to a dimension-free lower bound on its variance occurs when $p=d$ is an even integer and the coefficients of $f$ are close to those of $\frac{1}{\sqrt{n}}\left\Vert x\right\Vert _{p}^{p}$. Second, we consider general isotropic log-concave measures that are invariant under coordinate permutations and reflections, and determine the minimal variance for quadratic and cubic polynomials. These variance bounds lead to new dimension-free anti-concentration results in both settings, addressing a natural extension of a question posed by Carbery and Wright.