块结构多重仿射多项式的乘积轮廓反集中不等式
Product-profile anti-concentration for block-structured multi-affine polynomials
- Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS, LAMA UMR 8050（巴黎东部克雷泰伊大学）
- Institut de Mathématiques de Jussieu-Paris Rive Gauche (IMJ-PRG), Sorbonne Université（索邦大学）
- Weizmann Institute of Science（魏茨曼科学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对立方体上的多重仿射多项式，建立了乘积轮廓反集中、密度和Remez估计，并证明了所得尺度在块结构下最优，方法结合加权块选择与递推。
AI中文摘要：
我们在立方体上建立了多重仿射多项式的乘积轮廓反集中、密度和Remez估计。设$X$在$[0,1]^n$上均匀分布，$P:[0,1]^n\to\R$是精确度为$d$的非恒定多重仿射多项式，且所有变量都是活跃的。对于变量的一个划分$\cB$，使得没有单项式包含来自同一块的两个变量，定义$$ w_d(\cB) = \sum_{\cI\subseteq\cB, \abs{\cI} = d-1} \left(\prod_{B\in\cI}\sqrt{\abs B}\right) \sqrt{\sum_{C\in\cB\setminus\cI}\abs C}. $$ 我们证明了全中心小球估计$$ \sup_{u\in\R} \Prob\{\abs{P(X)-u}\le\rho\} \le \Phi_d\left( C_dw_d(\cB)\frac{\rho}{\osc(P)} \right), $$ 其中$\osc(P)$是$P$的值域长度，$\Phi_d(t) = t\sum_{j = 0}^{d-1}\log^j(1/t)/j!$在$(0,1]$上，上限为1。每个多重仿射多项式都允许单例划分，这产生了通用尺度$n^{d-1/2}$；一个至多包含$q$个块的可容许划分产生了改进的尺度$q^{(d-1)/2}n^{d/2}$。如果$\cB$恰好有$d$个块，那么$$ w_d(\cB) = d\prod_{B\in\cB}\sqrt{\abs B}, $$ 并且中心化块平均的乘积表明，小球的轮廓和对于完整块大小向量的依赖都是最优的，直到仅依赖于$d$的常数。我们还证明了$P(X)$具有密度$f_P$，满足$$ \norm{f_P}_{L^p(\R)} \le C_dp^{d-1} \left( \frac{w_d(\cB)}{\osc(P)} \right)^{1-1/p}, \quad 1<p<\infty, $$ 对于$p\ge2$的块乘积模型，具有匹配的$p$和块尺度增长。作为一个应用，我们推导了具有相同结构尺度的平移不变商Remez不等式。证明结合了加权块选择、仿射立方体切片、降阶收缩以及乘积轮廓背后的精确递推。
英文摘要：
We establish product-profile anti-concentration, density, and Remez estimates for multi-affine polynomials on the cube. Let $X$ be uniformly distributed on $[0,1]^n$, and let $P:[0,1]^n\to\R$ be a nonconstant multi-affine polynomial of exact degree $d$, with all variables active. For a partition $\cB$ of the variables such that no monomial contains two variables from the same block, define $$ w_d(\cB) = \sum_{\cI\subseteq\cB, \abs{\cI} = d-1} \left(\prod_{B\in\cI}\sqrt{\abs B}\right) \sqrt{\sum_{C\in\cB\setminus\cI}\abs C}. $$ We prove the all-center small-ball estimate $$ \sup_{u\in\R} \Prob\{\abs{P(X)-u}\leρ\} \le Φ_d\left( C_dw_d(\cB)\fracρ{\osc(P)} \right), $$ where $\osc(P)$ is the length of the range of $P$ and $Φ_d(t) = t\sum_{j = 0}^{d-1}\log^j(1/t)/j!$ on $(0,1]$, capped at one. Every multi-affine polynomial admits the singleton partition, which yields the universal scale $n^{d-1/2}$; an admissible partition into at most $q$ blocks yields the improved scale $q^{(d-1)/2}n^{d/2}$. If $\cB$ has exactly $d$ blocks, then $$ w_d(\cB) = d\prod_{B\in\cB}\sqrt{\abs B}, $$ and products of centered block averages show that both the small-ball profile and the dependence on the full block-size vector are optimal, up to constants depending only on $d$. We also prove that $P(X)$ has a density $f_P$ satisfying $$ \norm{f_P}_{L^p(\R)} \le C_dp^{d-1} \left( \frac{w_d(\cB)}{\osc(P)} \right)^{1-1/p}, \quad 1<p<\infty, $$ with matching $p$- and block-scale growth for $p\ge2$ on the block-product models. As an application, we derive translation-invariant quotient Remez inequalities with the same structural scale. The proof combines weighted block selection, affine cube slicing, degree-lowering contractions, and the exact recursion underlying the product profile.
