{
  "id": "1810.11553",
  "version": "v1",
  "published": "2018-10-26T23:33:25.000Z",
  "updated": "2018-10-26T23:33:25.000Z",
  "title": "On the Lebesgue measure, Hausdorff dimension, and Fourier dimension of sums and products of subsets of Euclidean space",
  "authors": [
    "Kyle Hambrook",
    "Krystal Taylor"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We investigate the Lebesgue measure, Hausdorff dimension, and Fourier dimension of sets of the form $RY +Z, $ where $R \\subseteq (0,\\infty)$ and $Y, Z \\subset \\mathbb{R}^d$. Most notable, for each $\\alpha \\in [0,1]$ and for each non-empty set $Y \\subseteq \\mathbb{R}$, we prove the existence of a compact set $R \\subseteq (0,\\infty)$ such that $\\dim_H(R) = \\dim_F(R) = \\alpha$ and $\\dim_F(RY) \\geq \\min\\{ 1, \\dim_F(R) + \\dim_F(Y)\\}$. This work is a contribution to the set of problems which study the measure and dimension of images of subsets of Euclidean space under Lipschitz maps.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-26T23:33:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B10",
      "28A75",
      "28A80"
    ],
    "keywords": [
      "hausdorff dimension",
      "fourier dimension",
      "lebesgue measure",
      "euclidean space",
      "non-empty set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}