{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19473",
  "author_total": 3,
  "authors": [
    {
      "name": "Evgeny Abakumov"
    },
    {
      "name": "Omer Friedland"
    },
    {
      "name": "Yosef Yomdin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6s7ygq96s5pcr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19473",
  "fulltext_url": null,
  "id": "6s7ygq96s5pcr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19473",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2612|Mathematical Approximation and Integration"
    ]
  ],
  "openalex": "W7213608897",
  "pdf_url": "https://arxiv.org/pdf/2609.19473",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6s7ygq96s5pcr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19473",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19473"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Product-profile anti-concentration for block-structured multi-affine polynomials",
  "title_text": "Product-profile anti-concentration for block-structured multi-affine polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/6s7ygq96s5pcr",
  "venue": "arXiv",
  "versions": []
}