Product-profile anti-concentration for block-structured multi-affine polynomials

Abstract

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.