{
  "id": "1010.0749",
  "version": "v3",
  "published": "2010-10-05T02:03:57.000Z",
  "updated": "2015-07-30T18:18:38.000Z",
  "title": "On directions determined by subsets of vector spaces over finite fields",
  "authors": [
    "Alex Iosevich",
    "Hannah Morgan",
    "Jonathan Pakianathan"
  ],
  "categories": [
    "math.CA",
    "math.CO",
    "math.NT"
  ],
  "abstract": "We prove that if a subset of a $d$-dimensional vector space over a finite field with $q$ elements has more than $q^{d-1}$ elements, then it determines all the possible directions. If a set has more than $q^k$ elements, it determines a $k$-dimensional set of directions. We prove stronger results for sets that are sufficiently random. This result is best possible as the example of a $k$-dimensional hyperplane shows. We can view this question as an Erd\\H os type problem where a sufficiently large subset of a vector space determines a large number of configurations of a given type. For discrete subsets of ${\\Bbb R}^d$, this question has been previously studied by Pach, Pinchasi and Sharir.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-05-13T21:12:07.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-07-30T18:18:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B05",
      "52C10"
    ],
    "keywords": [
      "finite field",
      "directions",
      "dimensional vector space",
      "vector space determines",
      "os type problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.0749I"
    }
  }
}