{
  "id": "2309.04387",
  "version": "v1",
  "published": "2023-09-08T15:37:37.000Z",
  "updated": "2023-09-08T15:37:37.000Z",
  "title": "Asymptotics of the optimal values of potentials generated by greedy energy sequences on the unit circle",
  "authors": [
    "Abey López-García",
    "Erwin Miña-Díaz"
  ],
  "comment": "20 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "For the Riesz and logarithmic potentials, we consider greedy energy sequences $(a_n)_{n=0}^\\infty$ on the unit circle $S^1$, constructed in such a way that for every $n\\geq 1$, the discrete potential generated by the first $n$ points $a_0,\\ldots,a_{n-1}$ of the sequence attains its minimum (say $U_n$) at $a_n$. We obtain asymptotic formulae that describe the behavior of $U_n$ as $n\\to\\infty$, in terms of certain bounded arithmetic functions with a doubling periodicity property. As previously shown in \\cite{LopMc2}, after properly translating and scaling $U_n$, one obtains a new sequence $(F_n)$ that is bounded and divergent. We find the exact value of $\\liminf F_n$ (the value of $\\limsup F_n$ was already given in \\cite{LopMc2}), and show that the interval $[\\liminf F_n,\\limsup F_n]$ comprises all the limit points of the sequence $(F_n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-08T15:37:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31C20",
      "31A15",
      "11M06"
    ],
    "keywords": [
      "greedy energy sequences",
      "unit circle",
      "optimal values",
      "bounded arithmetic functions",
      "limit points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}