{
  "abstract": "We prove new small-ball inequalities for boolean matrix-series. The leading example is $\\mathbb E_s[{\\text{det}(I-S^2)^β\\,\\mathbf 1_{\\{\\|S\\|<1\\}}}]\\ge e^{-O(βτ)}$, which holds for boolean matrix-series $S=\\sum_i s_iA_i$ formed using symmetric matrices $A_1,\\dots,A_n$ and uniformly random signs $s\\in\\{\\pm1\\}^n$. Specifically, this inequality holds for all $β\\ge1$ with $τ=\\sum_i\\text{Tr} A_i^2$, as soon as the maximum of $(\\text{Tr} A_i^2)_{i=1}^n$ and a certain variance term are bounded above by universal constants. The proof combines the Gaussian reciprocal estimate of (Akbas and Sra 2026), the directional-variation signing theorem of (Guo, Fang, and Lu 2026), and a replica argument that turns existence into a Gibbs law on good signings. Most notably, boolean small-ball delivers a new, interlacing-free proof of Kadison-Singer (most general case); it also recovers Matrix Spencer and Komlós as quick corollaries, while yielding more than six almost immediate proofs of an assortment of discrepancy theoretic problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20785",
  "author_total": 2,
  "authors": [
    {
      "name": "Emrullah Akbas"
    },
    {
      "name": "Suvrit Sra"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/29egzrbn34kch.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20785",
  "fulltext_url": null,
  "id": "29egzrbn34kch",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20785",
  "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.CO"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2612|Mathematical Approximation and Integration"
    ]
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  "openalex": "W7213610095",
  "pdf_url": "https://arxiv.org/pdf/2609.20785",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/29egzrbn34kch.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20785",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20785"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Boolean Small-Ball Inequalities for Discrepancy Theory",
  "title_text": "Boolean Small-Ball Inequalities for Discrepancy Theory",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/29egzrbn34kch",
  "venue": "arXiv",
  "versions": []
}