{
  "id": "2007.06109",
  "version": "v1",
  "published": "2020-07-12T22:05:52.000Z",
  "updated": "2020-07-12T22:05:52.000Z",
  "title": "Asymptotics of greedy energy sequences on the unit circle and the sphere",
  "authors": [
    "Abey López-García",
    "Ryan E. McCleary"
  ],
  "comment": "35 pages, 6 figures",
  "categories": [
    "math.CA",
    "math-ph",
    "math.MP"
  ],
  "abstract": "For a parameter $\\lambda>0$, we investigate greedy $\\lambda$-energy sequences $(a_{n})_{n=0}^{\\infty}$ on the unit sphere $S^{d}\\subset\\mathbb{R}^{d+1}$, $d\\geq 1$, satisfying the defining property that each $a_{n}$, $n\\geq 1$, is a point where the potential $\\sum_{k=0}^{n-1}|x-a_{k}|^{\\lambda}$ attains its maximum value on $S^{d}$. We show that these sequences satisfy the symmetry condition $a_{2k+1}=-a_{2k}$ for every $k\\geq 0$. The asymptotic distribution of the sequence undergoes a sharp transition at the value $\\lambda=2$, from uniform distribution ($\\lambda<2$) to concentration on two antipodal points ($\\lambda>2$). We investigate first-order and second-order asymptotics of the $\\lambda$-energy of the first $N$ points of the sequence, as well as the asymptotic behavior of the extremal values $\\sum_{k=0}^{n-1}|a_{n}-a_{k}|^{\\lambda}$ as $n$ tends to infinity. The second-order asymptotics is analyzed on the unit circle. It is shown that this asymptotic behavior differs from that of $N$ equally spaced points and a transition in the behavior takes place at $\\lambda=1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-12T22:05:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "greedy energy sequences",
      "unit circle",
      "second-order asymptotics",
      "asymptotic behavior differs",
      "sequence undergoes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}