{
  "id": "1812.03371",
  "version": "v1",
  "published": "2018-12-08T19:10:22.000Z",
  "updated": "2018-12-08T19:10:22.000Z",
  "title": "On Distinct Distances Between a Variety and a Point Set",
  "authors": [
    "Bryce McLaughlin",
    "Mohamed Omar"
  ],
  "comment": "8 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We consider the problem of Erd\\\"{o}s on determining the minimum number of distinct distances $f(n)$ that any set of $n$ points in the plane must determine. In a seminal paper, Erd\\\"{o}s showed that $f(n) = O(n/\\sqrt{\\log n})$, and conjectured $f(n)=\\Omega(n^c)$ for every $c<1$. This was resolved by Guth and Katz, who proved $f(n)=\\Omega(n / \\log n)$. The question remains where $f(n)$ falls asymptotically between $n/\\log n$ and $n/\\sqrt{\\log n}$. A recent result of Pohoata and Sheffer shows that if $\\Theta(n)$ of the $n$ points lie on a line, then Erd\\\"{o}s' upper bound is asymptotically tight, meaning $f(n)=\\Theta \\left( n/\\sqrt{\\log n} \\right)$. We generalize this result, proving that $f(n)=\\Theta \\left( n/\\sqrt{\\log n} \\right)$ if $\\Theta(n)$ of the points lie on an algebraic curve of a fixed degree.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-08T19:10:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "distinct distances",
      "point set",
      "points lie",
      "upper bound",
      "minimum number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}