{
  "id": "2203.11247",
  "version": "v1",
  "published": "2022-03-21T18:15:11.000Z",
  "updated": "2022-03-21T18:15:11.000Z",
  "title": "The Assouad dimension of self-affine measures on sponges",
  "authors": [
    "Jonathan M. Fraser",
    "István Kolossváry"
  ],
  "comment": "21 pages, 1 figure",
  "categories": [
    "math.DS",
    "math.CA"
  ],
  "abstract": "We derive upper and lower bounds for the Assouad and lower dimensions of self-affine measures in $\\mathbb{R}^d$ generated by diagonal matrices and satisfying suitable separation conditions. The upper and lower bounds always coincide for $d=2,3$ yielding precise explicit formulae for the dimensions. Moreover, there are easy to check conditions guaranteeing that the bounds coincide for $d \\geq 4$. An interesting consequence of our results is that there can be a `dimension gap' for such self-affine constructions, even in the plane. That is, we show that for some self-affine carpets of `Bara\\'nski type' the Assouad dimension of all associated self-affine measures strictly exceeds the Assouad dimension of the carpet by some fixed $\\delta>0$ depending only on the carpet. We also provide examples of self-affine carpets of `Bara\\'nski type' where there is no dimension gap and in fact the Assouad dimension of the carpet is equal to the Assouad dimension of a carefully chosen self-affine measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-21T18:15:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "37D20",
      "37C45"
    ],
    "keywords": [
      "assouad dimension",
      "baranski type",
      "dimension gap",
      "self-affine carpets",
      "lower bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}