{
  "annotation_id": "96c2d9c2-89ef-4129-ba15-c7c901440c7e",
  "date_created": "2026-02-17T05:53:19.841000Z",
  "date_modified": "2026-02-17T05:53:19.841000Z",
  "file_hash": "0254caca6c117101ab8fb4e9bcee34de8d6171573960368050d552da59251857",
  "private": false,
  "record": {
    "abstract": "In 1997, Erd\\H{o}s asked whether for arbitrarily large $n$ there exists a set of $n$ points in $\\mathbb{R}^2$ that determines $O(\\frac{n}{\\sqrt{\\log n}})$ distinct distances while satisfying the local constraint that every 4-point subset determines at least 3 distinct pairwise distances. We construct $n$-point sets from an $m\\times m$ box of the lattice $L = \\{(x,\\sqrt{2}y):x,y \\in \\mathbb{Z}\\} \\subset \\mathbb{R}^2.$ The distinct distance bound follows from applying Bernays\u0027 theorem to the number of integers represented by the binary quadratic form $u^2 + 2v^2$. The local 4-point constraint is verified through Perucca\u0027s similarity classification of the six similarity types determining exactly two distances.",
    "arxiv_id": "2601.09102",
    "authors": [
      "Benjamin Grayzel"
    ],
    "categories": [
      "math.CO"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "title": "Solution to a Problem of Erd\\H{o}s Concerning Distances and Points",
    "url": "https://arxiv.org/abs/2601.09102",
    "version": "v2"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "91da311e-8158-4b99-b8d1-446aa1fc1c2a",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}