{
  "id": "1609.07802",
  "version": "v1",
  "published": "2016-09-25T21:16:54.000Z",
  "updated": "2016-09-25T21:16:54.000Z",
  "title": "On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions",
  "authors": [
    "Pablo Shmerkin"
  ],
  "comment": "65 pages",
  "categories": [
    "math.DS",
    "math.CA",
    "math.CO",
    "math.NT"
  ],
  "abstract": "We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the $L^q$-dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg's long-standing conjecture on the dimension of the intersections of $\\times p$ and $\\times q$-invariant sets. Among several other applications, we also show that Bernoulli convolutions have an $L^q$ density for all finite $q$, outside of a zero-dimensional set of exceptions. The proof of the main result is inspired by M. Hochman's approach to the dimensions of self-similar measures and his inverse theorem for entropy. Our method can be seen as an extension of Hochman's theory from entropy to $L^q$ norms, and likewise relies on an inverse theorem for the decay of $L^q$ norms of discrete measures under convolution. This central piece of our approach may be of independent interest, and is an application of well-known methods and results in additive combinatorics: the asymmetric version of the Balog-Szemer\\'{e}di-Gowers Theorem due to Tao-Vu, and some constructions of Bourgain.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-25T21:16:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K55",
      "28A80",
      "37C45",
      "28A78",
      "28D05",
      "37A45"
    ],
    "keywords": [
      "furstenbergs intersection conjecture",
      "dynamically driven self-similar measures",
      "inverse theorem",
      "contain proper self-similar measures",
      "main result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 65,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}