{
  "id": "2205.02770",
  "version": "v1",
  "published": "2022-05-05T16:45:48.000Z",
  "updated": "2022-05-05T16:45:48.000Z",
  "title": "Additive properties of fractal sets on the parabola",
  "authors": [
    "Tuomas Orponen"
  ],
  "comment": "26 pages, 2 figures",
  "categories": [
    "math.CA",
    "math.CO"
  ],
  "abstract": "Let $0 \\leq s \\leq 1$, and let $\\mathbb{P} := \\{(t,t^{2}) \\in \\mathbb{R}^{2} : t \\in [-1,1]\\}$. If $K \\subset \\mathbb{P}$ is a closed set with $\\dim_{\\mathrm{H}} K = s$, it is not hard to see that $\\dim_{\\mathrm{H}} (K + K) \\geq 2s$. The main corollary of the paper states that if $0 < s < 1$, then adding $K$ once more makes the sum slightly larger: $$\\dim_{\\mathrm{H}} (K + K + K) \\geq 2s + \\epsilon, $$ where $\\epsilon = \\epsilon(s) > 0$. This information is deduced from an $L^{6}$ bound for the Fourier transforms of Frostman measures on $\\mathbb{P}$. If $0 < s < 1$, and $\\mu$ is a Borel measure on $\\mathbb{P}$ satisfying $\\mu(B(x,r)) \\leq r^{s}$ for all $x \\in \\mathbb{P}$ and $r > 0$, then there exists $\\epsilon = \\epsilon(s) > 0$ such that $$ \\|\\hat{\\mu}\\|_{L^{6}(B(R))}^{6} \\leq R^{2 - (2s + \\epsilon)} $$ for all sufficiently large $R \\geq 1$. The proof is based on a reduction to a $\\delta$-discretised point-circle incidence problem, and eventually to the $(s,2s)$-Furstenberg set problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-05T16:45:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "11B30"
    ],
    "keywords": [
      "fractal sets",
      "additive properties",
      "furstenberg set problem",
      "discretised point-circle incidence problem",
      "main corollary"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}