{
  "id": "1110.6792",
  "version": "v1",
  "published": "2011-10-31T13:41:54.000Z",
  "updated": "2011-10-31T13:41:54.000Z",
  "title": "On angles determined by fractal subsets of the Euclidean space via Sobolev bounds for bi-linear operators",
  "authors": [
    "Alex Iosevich",
    "Mihalis Mourgoglou",
    "Eyvindur Palsson"
  ],
  "categories": [
    "math.CA",
    "math.CO"
  ],
  "abstract": "We prove that if the Hausdorff dimension of a compact subset of ${\\mathbb R}^d$ is greater than $\\frac{d+1}{2}$, then the set of angles determined by triples of points from this set has positive Lebesgue measure. Sobolev bounds for bi-linear analogs of generalized Radon transforms and the method of stationary phase play a key role. These results complement those of V. Harangi, T. Keleti, G. Kiss, P. Maga, P. Mattila and B. Stenner in (\\cite{HKKMMS10}). We also obtain new upper bounds for the number of times an angle can occur among $N$ points in ${\\mathbb R}^d$, $d \\ge 4$, motivated by the results of Apfelbaum and Sharir (\\cite{AS05}) and Pach and Sharir (\\cite{PS92}). We then use this result to establish sharpness results in the continuous setting. Another sharpness result relies on the distribution of lattice points on large spheres in higher dimensions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-10-31T13:41:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "42B20",
      "52C10"
    ],
    "keywords": [
      "sobolev bounds",
      "fractal subsets",
      "euclidean space",
      "bi-linear operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.6792I"
    }
  }
}