{
  "abstract": "Consider the task of online vector balancing for stochastic arrivals $X_1,\\ldots,{X_T}$, where the $X_i$ are independent uniformly random $d$--sparse binary vectors in $\\{0,1\\}^n$. This is a random analogue of the online Beck--Fiala problem. We show that uniformly for $2\\le d\\le n/2$ and $T = Θ(n)$, the optimal online prefix discrepancy $\\max\\limits_{t\\leq T}\\left\\|\\sum_{i=1}^tσ_i X_i\\right\\|_\\infty$ is of order \\[ Θ\\big(\\max\\{\\sqrt d,\\log\\log n\\}\\big). \\] The upper bound is achieved by an efficient online algorithm. Thus, for $d\\le(\\log\\log n)^2$, the optimal discrepancy is $Θ(\\log\\log n)$ and is independent of the sparsity up to constant factors, whereas above this scale it is $Θ(\\sqrt d)$, matching the order of the offline discrepancy. This identifies the threshold at which sparsity begins to govern the online discrepancy of the random Beck--Fiala model.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14975",
  "author_total": 2,
  "authors": [
    {
      "name": "Dylan J. Altschuler"
    },
    {
      "name": "Konstantin Tikhomirov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5js4gmds4c6yq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14975",
  "fulltext_url": null,
  "id": "5js4gmds4c6yq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14975",
  "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.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2612|Mathematical Approximation and Integration"
    ]
  ],
  "openalex": "W7213368609",
  "pdf_url": "https://arxiv.org/pdf/2609.14975",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5js4gmds4c6yq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14975",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14975"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The threshold for online balancing of i.i.d. binary vectors",
  "title_text": "The threshold for online balancing of i.i.d. binary vectors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/5js4gmds4c6yq",
  "venue": "arXiv",
  "versions": []
}