Anti-concentration of polynomials: $L^{p}$ balls and symmetric measures
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911540804845568 |
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| author | Glazer, Itay Mikulincer, Dan |
| author_facet | Glazer, Itay Mikulincer, Dan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22664 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Anti-concentration of polynomials: $L^{p}$ balls and symmetric measures Glazer, Itay Mikulincer, Dan Probability Classical Analysis and ODEs Functional Analysis 52A23 (Primary) 46B07, 42B10, 42B20, 20C30 (Secondary) 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. |
| title | Anti-concentration of polynomials: $L^{p}$ balls and symmetric measures |
| topic | Probability Classical Analysis and ODEs Functional Analysis 52A23 (Primary) 46B07, 42B10, 42B20, 20C30 (Secondary) |
| url | https://arxiv.org/abs/2603.22664 |