{
  "abstract": "We determine the sharp order of discrepancy in Tusnády's problem in every fixed dimension $d\\ge 2$. The largest combinatorial discrepancy of $N$ points in $\\mathbb{R}^d$ with respect to axis-parallel boxes is $Θ_d((\\log N)^{d-1/2})$, extending Wei's planar result and matching Nikolov's upper bound. More strongly, with high probability, this lower bound holds simultaneously for every subset obtained by deleting at most a sufficiently small fixed fraction of a uniform $N$-point sample. We apply this geometric estimate to online thinning of two independent samples of size $n$ from an unknown common distribution, where observations must be kept or discarded upon arrival. For every fixed dimension $k\\ge 1$ and expected deletion budget $ρn$, with $0<ρ<2$, we prove that the minimax expected prefix discrepancy over lower orthants is $Θ_{k,ρ}((\\log n)^{k+1/2})$. In dimension one, the optimal terminal discrepancy is instead $Θ_ρ((\\log\\log n)^{3/2})$, giving a sharp separation between the prefix and terminal objectives. For $k\\ge 2$, we obtain terminal bounds differing by a factor of $\\log n$. The lower bounds are based on Gaussian small-ball estimates for box-incidence matrices, while the upper bounds use stationary Metropolis thinning and, in one dimension, a terminal cancellation procedure.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08455",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuhao Lei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7vwgkd7vdy4ar.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08455",
  "fulltext_url": null,
  "id": "7vwgkd7vdy4ar",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08455",
  "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",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08455",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7vwgkd7vdy4ar.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.08455",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08455"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Tusnády's problem and online two-sample thinning",
  "title_text": "Tusnády's problem and online two-sample thinning",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:49Z",
  "url": "https://research.draconitic.com/paper/7vwgkd7vdy4ar",
  "venue": "arXiv",
  "versions": []
}