{
  "id": "1509.01128",
  "version": "v1",
  "published": "2015-09-03T15:38:03.000Z",
  "updated": "2015-09-03T15:38:03.000Z",
  "title": "The Assouad dimensions of projections of planar sets",
  "authors": [
    "Jonathan M. Fraser",
    "Tuomas Orponen"
  ],
  "comment": "21 pages, 2 figures",
  "categories": [
    "math.CA",
    "math.DS",
    "math.MG"
  ],
  "abstract": "We consider the Assouad dimensions of orthogonal projections of planar sets onto lines. Our investigation covers both general and self-similar sets. For general sets, the main result is the following: if a set in the plane has Assouad dimension $s \\in [0,2]$, then the projections have Assouad dimension at least $\\min\\{1,s\\}$ almost surely. Compared to the famous analogue for Hausdorff dimension -- namely \\emph{Marstrand's Projection Theorem} -- a striking difference is that the words `at least' cannot be dispensed with: in fact, for many planar self-similar sets of dimension $s < 1$, we prove that the Assouad dimension of projections can attain both values $s$ and $1$ for a set of directions of positive measure. For self-similar sets, our investigation splits naturally into two cases: when the group of rotations is discrete, and when it is dense. In the `discrete rotations' case we prove the following dichotomy for any given projection: either the Hausdorff measure is positive in the Hausdorff dimension, in which case the Hausdorff and Assouad dimensions coincide; or the Hausdorff measure is zero in the Hausdorff dimension, in which case the Assouad dimension is equal to 1. In the `dense rotations' case we prove that every projection has Assouad dimension equal to one, assuming that the planar set is not a singleton. As another application of our results, we show that there is no \\emph{Falconer's Theorem} for Assouad dimension. More precisely, the Assouad dimension of a self-similar (or self-affine) set is not in general almost surely constant when one randomises the translation vectors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-03T15:38:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "28A78"
    ],
    "keywords": [
      "planar set",
      "projection",
      "hausdorff dimension",
      "hausdorff measure",
      "planar self-similar sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150901128F"
    }
  }
}