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

Itay Glazer, Dan Mikulincer

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.

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

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 -dimensional isotropic balls. We show that for a degree- homogeneous polynomial , with , the only obstruction to a dimension-free lower bound on its variance occurs when is an even integer and the coefficients of are close to those of . 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.
Paper Structure (32 sections, 38 theorems, 198 equations)

This paper contains 32 sections, 38 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Different notions of anti-concentration
  3. Main results: Fourier decay of log-concave measures
  4. Discussion and further questions
  5. Strategy of the proof of Theorem \ref{['thm:lpball']}
  6. Further related works
  7. Small-ball estimates in high-dimensional convex geometry
  8. Decay of Fourier coefficients
  9. Anti-concentration and singularities of polynomial maps
  10. Anti-concentration over arbitrary local fields
  11. Applications to algebraic combinatorics
  12. Small-ball estimates and asymptotic group theory
  13. Fourier coefficients of $L^{p}$-balls
  14. Some integrals on $L^{p}$-balls and $p$-Gaussian measures
  15. On the variance spectrum and a reduction to $H_{n}$-symmetric polynomials
  16. ...and 17 more sections

Key Result

Proposition 1.1

Let $X\sim\mu$ be a log-concave measure on $\mathbb{\mathbb{R}}^{n}$ and let $f:\mathbb{\mathbb{R}}^{n}\to\mathbb{\mathbb{R}}$ be a polynomial of degree $d\geq 2$. Then the following conditions are equivalent: where the constants $C_1,C_2,C_3$ depend on each other and on $d$ only.

Theorems & Definitions (90)

  • Proposition 1.1
  • proof
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Conjecture 1.7
  • Remark 1.8
  • Definition 1.9: Sch85
  • Theorem 2.1
  • ...and 80 more