{
  "id": "2012.11458",
  "version": "v1",
  "published": "2020-12-21T16:23:33.000Z",
  "updated": "2020-12-21T16:23:33.000Z",
  "title": "Decoupling for fractal subsets of the parabola",
  "authors": [
    "Alan Chang",
    "Jaume de Dios Pont",
    "Rachel Greenfeld",
    "Asgar Jamneshan",
    "Zane Kun Li",
    "José Madrid"
  ],
  "comment": "28 pages",
  "categories": [
    "math.CA",
    "math.NT"
  ],
  "abstract": "We consider decoupling for a fractal subset of the parabola. We reduce studying $l^{2}L^{p}$ decoupling for a fractal subset on the parabola $\\{(t, t^2) : 0 \\leq t \\leq 1\\}$ to studying $l^{2}L^{p/3}$ decoupling for the projection of this subset to the interval $[0, 1]$. This generalizes the decoupling theorem of Bourgain-Demeter in the case of the parabola. Due to the sparsity and fractal like structure, this allows us to improve upon Bourgain-Demeter's decoupling theorem for the parabola. In the case when $p/3$ is an even integer we derive theoretical and computational tools to explicitly compute the associated decoupling constant for this projection to $[0, 1]$. Our ideas are inspired by the recent work on ellipsephic sets by Biggs using nested efficient congruencing.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-21T16:23:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractal subset",
      "projection",
      "computational tools",
      "bourgain-demeters decoupling theorem",
      "ellipsephic sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}