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→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 wd(cB) = ∑cI⊆cB, abscI = d−1 (∏B∈cI√(abs B)) √(∑C∈cB∖cIabs C). We prove the all-center small-ball estimate supu∈R Prob{absP(X)−u≤ρ} ≤ Φd( Cdwd(cB)ρ/osc(P) ), where osc(P) is the length of the range of P and Φd(t) = t∑j = 0d−1logj(1/t)/j! on (0,1], capped at one. Every multi-affine polynomial admits the singleton partition, which yields the universal scale nd−1/2; an admissible partition into at most q blocks yields the improved scale q(d−1)/2nd/2. If cB has exactly d blocks, then wd(cB) = d∏B∈cB√(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 fP satisfying normfPLp(R) ≤ Cdpd−1 ( wd(cB)/osc(P) )1−1/p, 1<p<∞, with matching p- and block-scale growth for p≥2 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.
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BibTeX
@misc{abakumov2026productprofile,
title = {{Product-profile anti-concentration for block-structured multi-affine polynomials}},
author = {Evgeny Abakumov and Omer Friedland and Yosef Yomdin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19473},
eprint = {2609.19473},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19473},
}
RIS
TY - UNPB
TI - Product-profile anti-concentration for block-structured multi-affine polynomials
AU - Evgeny Abakumov
AU - Omer Friedland
AU - Yosef Yomdin
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.19473
UR - https://arxiv.org/abs/2609.19473
AB - 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→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) = ∑_cI⊆cB, abscI = d−1 (∏_B∈cI√(abs B)) √(∑_C∈cB∖cIabs C). We prove the all-center small-ball estimate sup_u∈R Prob{absP(X)−u≤ρ} ≤ Φ_d( C_dw_d(cB)ρ/osc(P) ), where osc(P) is the length of the range of P and Φ_d(t) = t∑_j = 0^d−1log^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)/2n^d/2. If cB has exactly d blocks, then w_d(cB) = d∏_B∈cB√(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 normf_P_L^p(R) ≤ C_dp^d−1 ( w_d(cB)/osc(P) )^1−1/p, 1<p<∞, with matching p- and block-scale growth for p≥2 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.
ER -