{
  "id": "2301.11918",
  "version": "v1",
  "published": "2023-01-27T18:59:07.000Z",
  "updated": "2023-01-27T18:59:07.000Z",
  "title": "Regularity of almost-surely injective projections in Euclidean spaces",
  "authors": [
    "Krzysztof Barański",
    "Yonatan Gutman",
    "Adam Śpiewak"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "It is known that if a finite Borel measure $\\mu$ in a Euclidean space has Hausdorff dimension smaller than a positive integer $k$, then the orthogonal projection onto almost every $k$-dimensional linear subspace is injective on a set of full $\\mu$-measure. We study the regularity of the inverses of such projections. We prove that if $\\mu$ has a compact support $X$ and (respectively) the Hausdorff, upper box-counting or Assouad dimension of $X$ is smaller than $k$, then the inverse is (respectively) continuous, pointwise H\\\"older for some $\\alpha \\in (0,1)$ or pointwise H\\\"older for every $\\alpha \\in (0,1)$. The result generalizes to the case of typical linear perturbations of Lipschitz maps. Additionally, we construct a non-trivial measure on the plane which admits almost-surely injective projections in every direction, and show that no homogeneous self-similar measure has this property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-27T18:59:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "28A78",
      "28A80"
    ],
    "keywords": [
      "almost-surely injective projections",
      "euclidean space",
      "regularity",
      "hausdorff dimension smaller",
      "finite borel measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}