{
  "id": "1706.09851",
  "version": "v1",
  "published": "2017-06-29T17:00:18.000Z",
  "updated": "2017-06-29T17:00:18.000Z",
  "title": "Pinned distance problem, slicing measures and local smoothing estimates",
  "authors": [
    "Alex Iosevich",
    "Bochen Liu"
  ],
  "categories": [
    "math.CA",
    "math.MG"
  ],
  "abstract": "We improve the Peres-Schlag result on pinned distances in sets of a given Hausdorff dimension. In particular, for Euclidean distances, with $$\\Delta^y(E) = \\{|x-y|:x\\in E\\},$$ we prove that for any $E, F\\subset{\\Bbb R}^d$, there exists a probability measure $\\mu_F$ on $F$ such that for $\\mu_F$-a.e. $y\\in F$, (1) $\\dim_{{\\mathcal H}}(\\Delta^y(E))\\geq\\beta$ if $\\dim_{{\\mathcal H}}(E) + \\frac{d-1}{d+1}\\dim_{{\\mathcal H}}(F) > d - 1 + \\beta$; (2) $\\Delta^y(E)$ has positive Lebesgue measure if $\\dim_{{\\mathcal H}}(E)+\\frac{d-1}{d+1}\\dim_{{\\mathcal H}}(F) > d$; (3) $\\Delta^y(E)$ has non-empty interior if $\\dim_{{\\mathcal H}}(E)+\\frac{d-1}{d+1}\\dim_{{\\mathcal H}}(F) > d+1$. We also show that in the case when $\\dim_{{\\mathcal H}}(E)+\\frac{d-1}{d+1}\\dim_{{\\mathcal H}}(F)>d$, for $\\mu_F$-a.e. $y\\in F$, $$ \\left\\{t\\in{\\Bbb R} : \\dim_{{\\mathcal H}}(\\{x\\in E:|x-y|=t\\}) \\geq \\dim_{{\\mathcal H}}(E)+\\frac{d+1}{d-1}\\dim_{{\\mathcal H}}(F)-d \\right\\} $$ has positive Lebesgue measure. This describes dimensions of slicing subsets of $E$, sliced by spheres centered at $y$. In our proof, local smoothing estimates of Fourier integral operators (FIO) plays a crucial role. In turn, we obtain results on sharpness of local smoothing estimates by constructing geometric counterexamples.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-29T17:00:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "35S30"
    ],
    "keywords": [
      "local smoothing estimates",
      "pinned distance problem",
      "slicing measures",
      "positive lebesgue measure",
      "fourier integral operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}