{
  "abstract": "We solve four questions raised by Bayart concerning coefficient summability for multilinear forms with prescribed supports. For every $m\\ge 2$ and $d\\in[1,m]$, we determine the product-summability exponent and the multilinear summability invariant: \\[ \\mathrm{prod}(m,d) =\\min\\left\\{\\frac{m}{d},\\,m-\\lceil d\\rceil+1\\right\\}, \\qquad γ_{\\mathrm{mult}}(m,d) =\\min\\left\\{m-\\lceil d\\rceil+1,\\frac{2m}{d+1}\\right\\}. \\] In particular, $γ_{\\mathrm{mult}}(4,2)=8/3$, showing that the multilinear invariant need not be an integer. We also prove that, for every $d\\in[1,m]$, there is a single infinite support of exact combinatorial dimension $d$ on which the dimensional Hardy--Littlewood bound is attained over both scalar fields for every anisotropic parameter $\\mathbf p=(p_1,\\ldots,p_m)$ with $\\sum_j 1/p_j<1$, simultaneously across the two regimes separated by $\\sum_j 1/p_j=1/2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06847",
  "author_total": 4,
  "authors": [
    {
      "name": "Anderson Barbosa"
    },
    {
      "name": "Daniel Pellegrino"
    },
    {
      "name": "Anselmo Raposo Jr"
    },
    {
      "name": "Eduardo Teixeira"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/20tq6yenmajqh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06847",
  "fulltext_url": null,
  "id": "20tq6yenmajqh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06847",
  "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.FA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7212042160",
  "pdf_url": "https://arxiv.org/pdf/2609.06847",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/20tq6yenmajqh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.06847",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06847"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp Summability on Supports of Prescribed Combinatorial Dimension",
  "title_text": "Sharp Summability on Supports of Prescribed Combinatorial Dimension",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/20tq6yenmajqh",
  "venue": "arXiv",
  "versions": []
}