{
  "abstract": "For $p\\in[1,2]$, we prove uniform moment estimates for weighted sums \\[ \\sum_{j=1}^mζ_jw_j|\\langle{\\bf v}_j,y\\rangle|^p \\] of independent subgaussian random variables $ζ_j$, where the unit vectors $\\{{\\bf v}_j\\}_{j=1}^m$ and nonnegative weights $\\{w_j\\}_{j=1}^m$ form an isotropic frame. This allows us to obtain optimal sampling discretization of $L_p$ norms for each fixed $p\\in[1,2]$. Namely, for an $N$-dimensional space $X_N$ of bounded functions satisfying $\\|f\\|_\\infty\\le\\sqrt {KN}\\,\\|f\\|_{L_2}$ for every $f\\in X_N$, we show that $m\\ge C\\varepsilon^{-2}KN\\log(2N)$ independent samples $ξ^1,\\ldots,ξ^m$ provide the discretization \\[ (1-\\varepsilon)\\|f\\|_{L_p}^p \\le\\frac1m\\sum_{j=1}^m|f(ξ^j)|^p \\le(1+\\varepsilon)\\|f\\|_{L_p}^p \\quad\\forall f\\in X_N \\] with probability at least $1-\\exp(-c\\frac{m\\varepsilon^2}{KN})$. This implies a corresponding weighted discretization result that, for $1<p<2$, improves Talagrand's bound on the dimension $m$ needed to embed $N$-dimensional subspaces of $L_p$ into $\\ell_p^m$ with distortion at most $1+\\varepsilon$, giving $m\\le C\\varepsilon^{-2}N\\log(2N)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05930",
  "author_total": 2,
  "authors": [
    {
      "name": "Feng Dai"
    },
    {
      "name": "Egor Kosov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9qknc3sqgs07k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05930",
  "fulltext_url": null,
  "id": "9qknc3sqgs07k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05930",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05930",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9qknc3sqgs07k.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.05930",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05930"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Empirical approximation of $L^p$ norms for $p\\in (1, 2)$",
  "title_text": "Empirical approximation of L^p norms for p∈ (1, 2)",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/9qknc3sqgs07k",
  "venue": "arXiv",
  "versions": []
}