{
  "id": "1411.6868",
  "version": "v1",
  "published": "2014-11-25T13:51:36.000Z",
  "updated": "2014-11-25T13:51:36.000Z",
  "title": "Bisector energy and few distinct distances",
  "authors": [
    "Ben Lund",
    "Adam Sheffer",
    "Frank de Zeeuw"
  ],
  "comment": "18 pages, 2 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "We introduce the bisector energy of an $n$-point set $P$ in $\\mathbb{R}^2$, defined as the number of quadruples $(a,b,c,d)$ from $P$ such that $a$ and $b$ determine the same perpendicular bisector as $c$ and $d$. If no line or circle contains $M(n)$ points of $P$, then we prove that the bisector energy is $O(M(n)^{\\frac{2}{5}}n^{\\frac{12}{5}+\\epsilon} + M(n)n^2).$. We also prove the lower bound $\\Omega(M(n)n^2)$, which matches our upper bound when $M(n)$ is large. We use our upper bound on the bisector energy to obtain two rather different results: (i) If $P$ determines $O(n/\\sqrt{\\log n})$ distinct distances, then for any $0<\\alpha\\le 1/4$, either there exists a line or circle that contains $n^\\alpha$ points of $P$, or there exist $\\Omega(n^{8/5-12\\alpha/5-\\epsilon})$ distinct lines that contain $\\Omega(\\sqrt{\\log n})$ points of $P$. This result provides new information on a conjecture of Erd\\H{o}s regarding the structure of point sets with few distinct distances. (ii) If no line or circle contains $M(n)$ points of $P$, then the number of distinct perpendicular bisectors determined by $P$ is $\\Omega(\\min\\{M(n)^{-2/5}n^{8/5-\\epsilon}, M(n)^{-1} n^2\\})$. This appears to be the first higher-dimensional example in a framework for studying the expansion properties of polynomials and rational functions over $\\mathbb{R}$, initiated by Elekes and R\\'onyai.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-25T13:51:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bisector energy",
      "distinct distances",
      "point set",
      "upper bound",
      "circle contains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.6868L"
    }
  }
}