{
  "id": "2001.02551",
  "version": "v1",
  "published": "2020-01-08T14:46:34.000Z",
  "updated": "2020-01-08T14:46:34.000Z",
  "title": "Intersection between pencils of tubes, discretized sum-product, and radial projections",
  "authors": [
    "Bochen Liu",
    "Chun-Yen Shen"
  ],
  "categories": [
    "math.CA",
    "math.CO",
    "math.MG"
  ],
  "abstract": "In this paper we prove the following results in the plane. They are related to each other, while each of them has its own interest. First we obtain an $\\epsilon_0$-increment on intersection between pencils of $\\delta$-tubes, under non-concentration conditions. In fact we show it is equivalent to the discretized sum-product problem, thus the $\\epsilon_0$ follows from Bourgain's celebrated result. Then we prove a couple of new results on radial projections. We also discussion about the dependence of $\\epsilon_0$ and make a new conjecture. A tube condition on Frostman measures, after careful refinement, is also given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-08T14:46:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "radial projections",
      "intersection",
      "tube condition",
      "frostman measures",
      "bourgains celebrated result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}