{
  "abstract": "We identify the sharp iterated-logarithmic thresholds at the critical exponent for quantitative measure equivalence and orbit equivalence between integer lattices. To be more precise, let $n>m$ be two positive integers, $α$ be a positive number and $( β_j)_{j \\geqslant 1}$ be a finitely supported sequence of non-negative numbers. We show that there is a quantitatively $t^α\\cdot \\prod_{j \\geqslant 1} ( \\log^{(j)}{t} )^{-β_j} $-integrable measure equivalence from $\\mathbb{Z}^n$ to $\\mathbb{Z}^m$ if and only if either $α<m/n$ or $α=m/n$ and there is $j_* \\geqslant 1$ such that $β_j= 1$ for all $1 \\leqslant j < j_*$ and $β_{j_*} > 1$. Here $\\log^{(j)}$ is the $j$-fold iterated logarithm. The same characterization holds for quantitative orbit equivalence. This characterization greatly strengthens the previous best-known result, due to the work of Delabie, Koivisto, Le Maître and Tessera (2022) and the work of Correia (2025), which asserts that there is a $t^α$-integrable ($α>0$) measure equivalence (or orbit equivalence) from $\\mathbb{Z}^n$ to $\\mathbb{Z}^m$ if and only if $α<m/n$. In particular, our result solves, in much stronger forms, two open problems posed respectively by Delabie, Koivisto, Le Maître and Tessera, and by Naryshkin and Petrakos.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15816",
  "author_total": 1,
  "authors": [
    {
      "name": "Changhua Jiao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4321ezxad8n10.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15816",
  "fulltext_url": null,
  "id": "4321ezxad8n10",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15816",
  "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"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7213277322",
  "pdf_url": "https://arxiv.org/pdf/2609.15816",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4321ezxad8n10.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15816",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15816"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Sharp iterated-Logarithmic thresholds for quantitative measure and orbit equivalence between integer lattices",
  "title_text": "Sharp iterated-Logarithmic thresholds for quantitative measure and orbit equivalence between integer lattices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/4321ezxad8n10",
  "venue": "arXiv",
  "versions": []
}