{
  "id": "2410.19640",
  "version": "v1",
  "published": "2024-10-25T15:47:04.000Z",
  "updated": "2024-10-25T15:47:04.000Z",
  "title": "On the dimension of $αβ$-sets",
  "authors": [
    "Michael Hochman"
  ],
  "comment": "22 pages",
  "categories": [
    "math.DS",
    "math.MG"
  ],
  "abstract": "We show that the Feng-Xiong lower bound of $1/2$ for the box dimension of $\\alpha\\beta$-sets is tight. We also study how much of an $\\alpha\\beta$-orbit ``carries the dimension'': deleting an arbitararily small positive density set of times can cause the box dimension to drop to zero, but the Assouad dimension cannot drop below $1/4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-25T15:47:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "37C45",
      "11J13",
      "11J83"
    ],
    "keywords": [
      "box dimension",
      "arbitararily small positive density set",
      "feng-xiong lower bound",
      "assouad dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}