{
  "abstract": "Galatolo and Kim proved that the dynamical Borel-Cantelli property for decreasing sequences of balls is tightly connected with the waiting time problem. In systems where all such sequences are Borel-Cantelli, the time needed to enter a small ball $B$ for the first time scales as $μ(B)^{-1}$, and conversely, waiting time estimates yield Borel-Cantelli results for sequences of balls whose radii decrease in a controlled way. We extend this correspondence to the exponent-$s$ setting introduced by Tseng. For $s\\ge 1$, the $s$-exponent monotone shrinking target property ($s$MSTP) requires the Borel-Cantelli conclusion only for decreasing sequences of centered balls satisfying the stronger divergence condition $\\sum_nμ(B_n)^s=\\infty$. We prove that $s$MSTP forces the lower waiting time exponent, measured on the scale of $-\\logμ(B(y,r))$, to lie in the interval $[1,s]$ almost everywhere. That a quantitative ($s$-strong) form of the property bounds the upper exponent by $s$ and that, conversely, an exponent-$s$ waiting time estimate implies the Borel-Cantelli property for decreasing sequences of centered balls whose radii obey the calibrated decay condition matching the critical divergence exponent $s$. We also obtain the corresponding quantitative orbit approximation statement $\\liminf_n n^β\\,d(T^nx,y)=0$ for $β<1/(s\\,\\underline{d}_μ(y))$, show that the universal lower bound with exponent $1$ pins the theory to $s\\ge 1$, and discuss sharpness on circle rotations, where by results of Kurzweil, Kim-Seo and Tseng the picture is governed by the Diophantine type of the rotation number.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: Houston Journal of Mathematics 2026",
  "arxiv": "2607.11180",
  "author_total": 2,
  "authors": [
    {
      "name": "Dušan Bajović"
    },
    {
      "name": "Boris Petković"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/69yh4kenx93at.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.11180",
  "fulltext_url": null,
  "id": "69yh4kenx93at",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.11180",
  "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",
      "math.DS"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7168277254",
  "pdf_url": "https://arxiv.org/pdf/2607.11180",
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/69yh4kenx93at.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2607.11180",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.11180"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "An exponent-$s$ dynamical Borel-Cantelli lemma and the waiting time problem",
  "title_text": "An exponent-s dynamical Borel-Cantelli lemma and the waiting time problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/69yh4kenx93at",
  "venue": "arXiv",
  "versions": []
}