{
  "id": "1706.04170",
  "version": "v1",
  "published": "2017-06-13T17:13:58.000Z",
  "updated": "2017-06-13T17:13:58.000Z",
  "title": "Triangles capturing many lattice points",
  "authors": [
    "Nicholas F. Marshall",
    "Stefan Steinerberger"
  ],
  "comment": "12 pages, 8 figures",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "We study a combinatorial problem that recently arose in the context of shape optimization: among all triangles with vertices in $(0,0), (x,0), (0,y)$ and fixed area, which one encloses the most lattice points $\\mathbb{N}_{>0}^2$? Moreover, does its shape necessarily converge to the isosceles triangle $(x=y)$ as the area becomes large? Laugesen & Liu suggested that, in contrast to similar problems, there might not be a limiting shape. We prove this statement and show that there exists an infinite set of slopes $y/x$ such that any associated triangle captures more lattice points than any other fixed triangle for infinitely many (and arbitrarily large) areas; this set of slopes is a fractal subset of $[1/3, 3]$ and has Minkowski dimension at most $3/4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-13T17:13:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lattice points",
      "triangles capturing",
      "combinatorial problem",
      "associated triangle captures",
      "shape optimization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}