{
  "abstract": "The Kahane--Salem--Zygmund inequality provides unimodular $m$-linear forms with small supremum norm. We consider its dimension-free symmetric constant $C_m^{\\mathrm{sym}}$ over the real scalar field, defined by the requirement that, for every $n$, some real symmetric unimodular $m$-linear form $A:(\\ell_\\infty^n)^m\\to\\mathbb R$ satisfy \\[ \\|A\\|\\le C_m^{\\mathrm{sym}}n^{(m+1)/2}. \\] For complex scalars, Boas obtained under permutation symmetry an upper bound of order at most $\\sqrt{m\\log m}\\,\\sqrt{m!}$; over the real scalar field, an elementary argument gives the sharper order $\\sqrt m\\,\\sqrt{m!}$. We prove \\[ C_m^{\\mathrm{sym}}\\ge \\left(\\sqrt{\\frac2e}+o(1)\\right)\\frac{\\sqrt{m!}}m, \\] using the square-free Walsh spectrum of the diagonal polynomial. In the opposite direction, we establish \\[ C_m^{\\mathrm{sym}}\\le C_0\\sqrt{m!}, \\] where $C_0$ is absolute, removing the factor $\\sqrt m$ from the real upper bound. This estimate follows from a geometric argument in which rigidity for Gram permanents reduces the relevant configurations to sets controlled by Gaussian width. For unrestricted unimodular forms, we also obtain a quantitative rectangular estimate from truncated Hadamard matrices; in equal dimension, the normalized minimum is at most $1+o(1)$ whenever $m=o(n^{5/6})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30663",
  "author_total": 3,
  "authors": [
    {
      "name": "Daniel M. Pellegrino"
    },
    {
      "name": "Anselmo B. Raposo Jr."
    },
    {
      "name": "Eduardo V. Teixeira"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/265zecfbv6kph.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30663",
  "fulltext_url": null,
  "id": "265zecfbv6kph",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30663",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7214637439",
  "pdf_url": "https://arxiv.org/pdf/2609.30663",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/265zecfbv6kph.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30663",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30663"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Symmetric Kahane--Salem--Zygmund Inequalities and the Supremum Norm",
  "title_text": "Symmetric Kahane--Salem--Zygmund Inequalities and the Supremum Norm",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/265zecfbv6kph",
  "venue": "arXiv",
  "versions": []
}