{
  "id": "2409.08061",
  "version": "v1",
  "published": "2024-09-12T14:09:48.000Z",
  "updated": "2024-09-12T14:09:48.000Z",
  "title": "Khintchine dichotomy for self-similar measures",
  "authors": [
    "Timothée Bénard",
    "Weikun He",
    "Han Zhang"
  ],
  "comment": "29 pages",
  "categories": [
    "math.DS",
    "math.NT",
    "math.PR"
  ],
  "abstract": "We establish the analogue of Khintchine's theorem for all self-similar probability measures on the real line. When specified to the case of the Hausdorff measure on the middle-thirds Cantor set, the result is already new and provides an answer to an old question of Mahler. The proof consists in showing effective equidistribution in law of expanding upper-triangular random walks on $\\text{SL}_{2}(\\mathbb{R})/\\text{SL}_{2}(\\mathbb{Z})$, a result of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-12T14:09:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "self-similar measures",
      "khintchine dichotomy",
      "self-similar probability measures",
      "expanding upper-triangular random walks",
      "middle-thirds cantor set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}