{
  "abstract": "Let $(ϕ_i)_{i=1}^n$ be an orthonormal system on a probability space with $\\|ϕ_i\\|_\\infty\\leq L$. We prove that, for every $2\\leq k<n$, there is a set $I$ of at least $n-k$ indices such that $$ \\|f\\|_2\\leq CM_k\\sqrt{\\log(e+M_k)}\\|f\\|_1 $$ for every $f$ in the span of $(ϕ_i)_{i\\in I}$, where $M_k=L\\sqrt{\\frac{n}{k}\\log(2k)}$. This replaces the logarithmic exponent $\\frac{5}{2}$ in the corresponding theorem of Guédon, Mendelson, Pajor, and Tomczak-Jaegermann by $\\frac{1}{2}$. We also obtain a partition into two sets of cardinality $\\frac{n}{2}+O(\\sqrt n)$ for which both spans satisfy the resulting norm comparison, and for uniformly bounded systems the proportional estimate is $C\\sqrt{\\log n\\log\\log n}$ for sufficiently large $n$. This is the third paper in a series systematically using gradient descent as a proof mechanism in harmonic analysis, following the $Λ(p)$ and Bernoulli restricted isometry papers of Burstein, Iosevich, and Krause, and the descent here is entropic mirror descent. We represent each function in an $L_1$ ball by a probability measure of controlled relative entropy, and this entropy pays for the corrections to the predictions of its coefficients. Processing the thresholds from large to small, we obtain a single weighted bound for all the corrections, which we turn into a finite description of approximate coefficient squares and then into a conditional Rademacher estimate, and symmetrization completes the argument directly on the $L_1$ ball. We also record consequences for concentration and for recovery from coefficient measurements.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39025",
  "author_total": 3,
  "authors": [
    {
      "name": "Will Burstein"
    },
    {
      "name": "Alex Iosevich"
    },
    {
      "name": "Ben Krause"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c1j3rqs62hya4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39025",
  "fulltext_url": null,
  "id": "c1j3rqs62hya4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39025",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39025",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c1j3rqs62hya4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39025",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39025"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A Hierarchical Online-Compression Approach to Proportional Subsets of Bounded Orthonormal Systems",
  "title_text": "A Hierarchical Online-Compression Approach to Proportional Subsets of Bounded Orthonormal Systems",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/c1j3rqs62hya4",
  "venue": "arXiv",
  "versions": []
}