{
  "id": "2112.01802",
  "version": "v1",
  "published": "2021-12-03T09:22:43.000Z",
  "updated": "2021-12-03T09:22:43.000Z",
  "title": "Optimal and typical $L^2$ discrepancy of 2-dimensional lattices",
  "authors": [
    "Bence Borda"
  ],
  "comment": "24 pages",
  "journal": "Ann. Mat. Pura Appl. 203 (2024), 2157-2184",
  "doi": "10.1007/s10231-024-01440-4",
  "categories": [
    "math.NT"
  ],
  "abstract": "We undertake a detailed study of the $L^2$ discrepancy of rational and irrational 2-dimensional lattices either with or without symmetrization. We give a full characterization of lattices with optimal $L^2$ discrepancy in terms of the continued fraction partial quotients, and compute the precise asymptotics whenever the continued fraction expansion is explicitly known, such as for quadratic irrationals or Euler's number $e$. In the metric theory, we find the asymptotics of the $L^2$ discrepancy for almost every irrational, and the limit distribution for randomly chosen rational and irrational lattices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-12-03T09:22:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K38",
      "11J83"
    ],
    "keywords": [
      "discrepancy",
      "continued fraction partial quotients",
      "irrational lattices",
      "metric theory",
      "eulers number"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}