{
  "id": "1311.4788",
  "version": "v1",
  "published": "2013-11-19T15:57:17.000Z",
  "updated": "2013-11-19T15:57:17.000Z",
  "title": "Group actions and geometric combinatorics in ${\\mathbb F}_q^d$",
  "authors": [
    "M. Bennett",
    "D. Hart",
    "A. Iosevich",
    "J. Pakianathan",
    "M. Rudnev"
  ],
  "comment": "26 pages",
  "categories": [
    "math.CO",
    "math.CA",
    "math.NT"
  ],
  "abstract": "In this paper we apply a group action approach to the study of Erd\\H os-Falconer type problems in vector spaces over finite fields and use it to obtain non-trivial exponents for the distribution of simplices. We prove that there exists $s_0(d)<d$ such that if $E \\subset {\\mathbb F}_q^d$, $d \\ge 2$, with $|E| \\ge Cq^{s_0}$, then $|T^d_d(E)| \\ge C'q^{d+1 \\choose 2}$, where $T^d_k(E)$ denotes the set of congruence classes of $k$-dimensional simplices determined by $k+1$-tuples of points from $E$. Non-trivial exponents were previously obtained by Chapman, Erdogan, Hart, Iosevich and Koh (\\cite{CEHIK12}) for $T^d_k(E)$ with $2 \\leq k \\leq d-1$. A non-trivial result for $T^2_2(E)$ in the plane was obtained by Bennett, Iosevich and Pakianathan (\\cite{BIP12}). These results are significantly generalized and improved in this paper. In particular, we establish the Wolff exponent $\\frac{4}{3}$, previously established in \\cite{CEHIK12} for the $q\\equiv3\\mbox{ mod }4$ case to the case $q\\equiv1\\mbox{ mod }4$, and this results in a new sum-product type inequality. We also obtain non-trivial results for subsets of the sphere in ${\\mathbb F}_q^d$, where previous methods have yielded nothing. The key to our approach is a group action perspective which quickly leads to natural and effective formulae in the style of the classical Mattila integral from geometric measure theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-19T15:57:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C10",
      "42B20",
      "11L05"
    ],
    "keywords": [
      "geometric combinatorics",
      "non-trivial result",
      "non-trivial exponents",
      "os-falconer type problems",
      "group action approach"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.4788B"
    }
  }
}