{
  "abstract": "The celebrated (offline) Beck--Fiala conjecture asserts that every matrix $A\\in\\{0,1\\}^{n\\times T}$ with at most $d$ nonzero entries in each column has discrepancy $O(\\sqrt d)$. In this note, we consider the {\\it online} version of the conjecture. For every fixed $η>0$, we give a randomized online algorithm that achieves $O_η(\\sqrt d)$ prefix discrepancy for every fixed sequence of $d$-sparse vectors $a_t\\in[-1,1]^n$, except with probability at most \\[ C_ηT\\exp\\left[-\\frac{c_ηd}{\\log^{2+η}(ed)}\\right]. \\] In particular, this gives an optimal $O(\\sqrt d)$ discrepancy upper bound for $d\\geq C_η(\\log T)(\\log\\log T)^{2+η}$. This range of sparsity is optimal up to $\\log\\log T$ factors. The result also resolves in strong form the {\\it online Spencer problem}, corresponding to the case $d = n$. The algorithm is based on a compactly supported Metropolis walk, constructed by combining ideas from several recent works on online vector balancing. The proof was generated in conversation with ChatGPT 5.6 Pro; the authors provided high-level guidance in several rounds of prompting, followed by manual checking and rewriting of the proof.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.14238",
  "author_total": 2,
  "authors": [
    {
      "name": "Dylan J. Altschuler"
    },
    {
      "name": "Konstantin Tikhomirov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fdr6exqhx3k1m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.14238",
  "fulltext_url": null,
  "id": "fdr6exqhx3k1m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.14238",
  "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",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2612|Mathematical Approximation and Integration"
    ]
  ],
  "openalex": "W7169518119",
  "pdf_url": "https://arxiv.org/pdf/2607.14238",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fdr6exqhx3k1m.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:28Z",
      "native_id": "2607.14238",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.14238"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Online Beck--Fiala Down to Logarithmic Sparsity",
  "title_text": "Online Beck--Fiala Down to Logarithmic Sparsity",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/fdr6exqhx3k1m",
  "venue": "arXiv",
  "versions": []
}