{
  "abstract": "Let $d \\geq 1$ be fixed and let \\[ D(n,d) := \\sum_{j=0}^{d} \\binom{n-1}{j}. \\] We show that if $x^{(1)}, \\dots, x^{(m)}$ are independent uniform points of $\\{\\pm 1\\}^n$ then uniformly for $m \\leq D(n,d)$, there exists a constant $C_d > 0$ such that \\[ \\mathbb{P}((x^{(1)})^{\\otimes d}, \\dots, (x^{(m)})^{\\otimes d} \\text{ are linearly independent}) = 1 - O_d\\left(\\frac{\\log^{C_d} n}{n^{1/2}} \\right). \\] This achieves the exact dimensional threshold and answers a question asked by Baldi and Vershynin. We discuss applications of the result to the semidefinite relaxation of the cut-polytope and to matrix factorization.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23858",
  "author_total": 1,
  "authors": [
    {
      "name": "Kyle Luh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a1szdawjyk28d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23858",
  "fulltext_url": null,
  "id": "a1szdawjyk28d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23858",
  "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.PR"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7214045230",
  "pdf_url": "https://arxiv.org/pdf/2609.23858",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a1szdawjyk28d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.23858",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23858"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Linear Independence of Random Boolean Tensor Powers at the Dimension Threshold",
  "title_text": "Linear Independence of Random Boolean Tensor Powers at the Dimension Threshold",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/a1szdawjyk28d",
  "venue": "arXiv",
  "versions": []
}