{
  "id": "1907.12466",
  "version": "v1",
  "published": "2019-07-29T14:57:30.000Z",
  "updated": "2019-07-29T14:57:30.000Z",
  "title": "Equiangular lines with a fixed angle",
  "authors": [
    "Zilin Jiang",
    "Jonathan Tidor",
    "Yuan Yao",
    "Shengtong Zhang",
    "Yufei Zhao"
  ],
  "comment": "10 pages",
  "categories": [
    "math.CO",
    "math.MG"
  ],
  "abstract": "Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle. Fix $0 < \\alpha < 1$. Let $N_\\alpha(d)$ denote the maximum number of lines in $\\mathbb{R}^d$ with pairwise common angle $\\arccos \\alpha$. Let $k$ denote the minimum number (if it exists) of vertices of a graph whose adjacency matrix has spectral radius exactly $(1-\\alpha)/(2\\alpha)$. If $k < \\infty$, then $N_\\alpha(d) = \\lfloor k(d-1)/(k-1) \\rfloor$ for all sufficiently large $d$, and otherwise $N_\\alpha(d) = d + o(d)$. In particular, $N_{1/(2k-1)}(d) = \\lfloor k(d-1)/(k-1) \\rfloor$ for every integer $k\\geq 2$ and all sufficiently large $d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-29T14:57:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "equiangular lines",
      "fixed angle",
      "maximum number",
      "sufficiently large dimensions",
      "pairwise common angle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}