{
  "abstract": "A square in a matrix M = (a i, j ) is a 2 × 2 submatrix S with row and column indices {i, i + s} and { j, j + s}, respectively. A square is zero-sum if the sum of its entries equals 0. In 2021, Arévalo, Montejano and Roldán-Pensado proved that all large binary matrices M over {1, -1} of order n with absolute discrepancy | ∑ a i, j | ≤ n contain a zero-sum square, unless they are split. In 2023, Johnston showed that the same conclusion follows from the much weaker hypothesis | ∑ a i, j | ≤ n 2 /4. He further conjectured that the same conclusion would still follow from the weakest possible hypothesis, namely\n\n-2 if n is odd.\n\nWe prove this conjecture here. We also address a related problem raised by Johnston in the rectangular case.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Shalom Eliahou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c6nhfxh5gcgpn.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "c6nhfxh5gcgpn",
  "issns": [],
  "landing_url": "https://hal.science/hal-05753799v1",
  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05753799/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c6nhfxh5gcgpn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:36Z",
      "native_id": "hal-05753799",
      "source": "hal",
      "url": "https://hal.science/hal-05753799v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "Proof of a conjecture of Johnston on binary matrices with low discrepancy",
  "title_text": "Proof of a conjecture of Johnston on binary matrices with low discrepancy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:36Z",
  "url": "https://research.draconitic.com/paper/c6nhfxh5gcgpn",
  "venue": "HAL",
  "versions": []
}