{
  "id": "2410.19648",
  "version": "v1",
  "published": "2024-10-25T15:53:56.000Z",
  "updated": "2024-10-25T15:53:56.000Z",
  "title": "New results on embeddings of self-similar sets via renormalization",
  "authors": [
    "Amir Algom",
    "Michael Hochman",
    "Meng Wu"
  ],
  "comment": "28 pages",
  "categories": [
    "math.DS",
    "math.MG"
  ],
  "abstract": "For self-similar sets $X,Y\\subseteq \\mathbb{R}$, we obtain new results towards the affine embeddings conjecture of Feng-Huang-Rao (2014), and the equivalent weak intersections conjecture. We show that the conjecture holds when the defining maps of $X,Y$ have algebraic contraction ratios, and also for arbitrary $Y$ when the maps defining $X$ have algebraic-contraction ratios and there are sufficiently many of them relative to the number of maps defining $Y$. We also show that it holds for a.e. choice of the corresponding contraction ratios, and obtain bounds on the packing dimension of the exceptional parameters. Key ingredients in our argument include a new renormalization procedure in the space of embeddings, and Orponen's projection Theorem for Assouad dimension (2021).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-25T15:53:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "37C45"
    ],
    "keywords": [
      "self-similar sets",
      "equivalent weak intersections conjecture",
      "affine embeddings conjecture",
      "algebraic contraction ratios",
      "orponens projection theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}