{
  "id": "2105.10817",
  "version": "v1",
  "published": "2021-05-22T22:04:53.000Z",
  "updated": "2021-05-22T22:04:53.000Z",
  "title": "Asymptotics of the minimum values of Riesz potentials generated by greedy energy sequences on the unit circle",
  "authors": [
    "Abey López-García",
    "Ryan E. McCleary"
  ],
  "comment": "32 pages, several figures",
  "categories": [
    "math.CA"
  ],
  "abstract": "In this paper we investigate greedy energy sequences on the unit circle for the logarithmic and Riesz potentials. By definition, if $(a_n)_{n=0}^{\\infty}$ is a greedy $s$-energy sequence on the unit circle, the Riesz potential $U_{N,s}(x):=\\sum_{k=0}^{N-1}|a_k-x|^{-s}$, $s>0$, generated by the first $N$ points of the sequence attains its minimum value on the circle at the point $a_{N}$. In this work, we analyze the asymptotic properties of these extremal values $U_{N,s}(a_N)$, treating separately the cases $0<s<1$, $s=1$, and $s>1$. We present new second-order asymptotic formulas for $U_{N,s}(a_N)$ in the cases $0<s<1$ and $s=1$. A first-order result for $s>1$ is proved, and it is shown that the first-order normalized sequence $(U_{N,s}(a_N)/N^s)_{N=1}^{\\infty}$ is divergent in this case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-22T22:04:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31A15",
      "31C20",
      "11M06"
    ],
    "keywords": [
      "greedy energy sequences",
      "unit circle",
      "riesz potential",
      "minimum value",
      "second-order asymptotic formulas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}