{
  "id": "2502.04272",
  "version": "v1",
  "published": "2025-02-06T18:14:43.000Z",
  "updated": "2025-02-06T18:14:43.000Z",
  "title": "Strong Borel--Cantelli Lemmas for Recurrence",
  "authors": [
    "Tomas Persson",
    "Alejandro Rodriguez Sponheimer"
  ],
  "comment": "22 pages, 3 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $(X,T,\\mu,d)$ be a metric measure-preserving system for which $3$-fold correlations decay exponentially for Lipschitz continuous observables. Suppose that $(M_k)$ is a sequence satisfying some weak decay conditions and suppose there exist open balls $B_k(x)$ around $x$ such that $\\mu(B_k(x)) = M_k$. Under a short return time assumption, we prove a strong Borel--Cantelli lemma, including an error term, for recurrence, i.e., for $\\mu$-a.e. $x \\in X$, \\[ \\sum_{k=1}^{n} \\mathbf{1}_{B_k(x)} (T^k x) = \\Phi(n) + O \\bigl( \\Phi(n)^{1/2} (\\log \\Phi(n))^{3/2 + \\varepsilon} \\bigr), \\] where $\\Phi(n) = \\sum_{k=1}^{n} \\mu(B_k(x))$. Applications to systems include some non-linear piecewise expanding interval maps and hyperbolic automorphisms of $\\mathbf{T}^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-06T18:14:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B20",
      "37D20",
      "37A05"
    ],
    "keywords": [
      "strong borel-cantelli lemma",
      "recurrence",
      "short return time assumption",
      "weak decay conditions",
      "fold correlations decay"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}