{
  "abstract": "We give a self-contained proof of Bourgain's finite $Λ(p)$ selection theorem with an explicit probability bound. For every $p>2$ and $N\\ge2$, given $N$ pairwise orthogonal functions bounded by one on a probability space, a uniformly chosen subsystem of cardinality $\\lceil N^{\\frac{2}{p}}\\rceil$ satisfies the $Λ(p)$ inequality with probability at least $1-\\frac{1}{N}$, with a constant depending only on $p$. The inequality holds simultaneously for all complex coefficient vectors. More generally, for every fixed $A>0$, the success probability is at least $1-\\frac{1}{N^A}$ with a constant depending only on $p$ and $A$, and independent of $N$. The proof uses a greedy approximation in $L^r$, with $r>2$, whose potential decreases throughout all approximation scales. A single weighted bound on the cumulative number of updates controls a union bound over discrete update histories, and increasing the weight assigned to each history gives the prescribed failure exponent without changing the cardinality exponent.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12566",
  "author_total": 3,
  "authors": [
    {
      "name": "Will Burstein"
    },
    {
      "name": "Alex Iosevich"
    },
    {
      "name": "Ben Krause"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ap4rh912mqhvp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12566",
  "fulltext_url": null,
  "id": "ap4rh912mqhvp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12566",
  "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",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7213036547",
  "pdf_url": "https://arxiv.org/pdf/2609.12566",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ap4rh912mqhvp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12566",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12566"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Bourgain's Lambda(p) selection theorem: a greedy proof with polynomial failure bounds",
  "title_text": "Bourgain's Lambda(p) selection theorem: a greedy proof with polynomial failure bounds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/ap4rh912mqhvp",
  "venue": "arXiv",
  "versions": []
}