{
  "abstract": "Spencer's 1985 ``six standard deviations suffice'' theorem shows that every $A \\in [-1,1]^{n \\times n}$ has a sign vector $x \\in \\{-1,1\\}^n$ with $\\|Ax\\|_\\infty \\le 6\\sqrt{n}$. We show the upper bound $\\sqrt{3\\operatorname{arsinh}(10)}\\sqrt{n}+4 < 2.9992 \\sqrt{n} + 4$ by directly rounding the minimizer of a potential function to a vertex of the cube. We also show that, for every power of two $n \\ge 2^{50}$, there exists a matrix $A \\in \\{-1,1\\}^{n \\times n}$ such that $\\|Ax\\|_\\infty >1.0000002\\sqrt{n}$ for every choice of signs $x \\in \\{-1,1\\}^n$. The construction simply replaces a $2^{-22}$ fraction of the columns of a Hadamard matrix with independent random sign vectors. This is the first improvement over the $\\sqrt{n}$ lower bound of Olson and Spencer (1978), which uses a Hadamard matrix.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34471",
  "author_total": 2,
  "authors": [
    {
      "name": "Victor Reis"
    },
    {
      "name": "Zhao Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/abncz9md7wzzs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34471",
  "fulltext_url": null,
  "id": "abncz9md7wzzs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34471",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214734665",
  "pdf_url": "https://arxiv.org/pdf/2609.34471",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/abncz9md7wzzs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.34471",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34471"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Three Standard Deviations Suffice While One Does Not",
  "title_text": "Three Standard Deviations Suffice While One Does Not",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/abncz9md7wzzs",
  "venue": "arXiv",
  "versions": []
}