{
  "id": "2307.15822",
  "version": "v1",
  "published": "2023-07-28T21:45:46.000Z",
  "updated": "2023-07-28T21:45:46.000Z",
  "title": "Bounds for Periodic Energy and the Optimality of Two Periodic Point Configurations in the Plane",
  "authors": [
    "Doug Hardin",
    "Nathaniel Tenpas"
  ],
  "comment": "68 pages, 13 figures",
  "categories": [
    "math.CA",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We develop lower bounds for the energy of configurations in $\\mathbb{R}^d$ periodic with respect to a lattice. In certain cases, the construction of sharp bounds can be formulated as a finite dimensional, multivariate polynomial interpolation problem. We use this framework to show a scaling of the equitriangular lattice $A_2$ is universally optimal among all configurations of the form $\\omega_4+ A_2$ where $\\omega_4$ is a 4-point configuration in $\\mathbb{R}^2$. Likewise, we show a scaling and rotation of $A_2$ is universally optimal among all configurations of the form $\\omega_6+L$ where $\\omega_6$ is a 6-point configuration in $\\mathbb{R}^2$ and $L=\\mathbb{Z} \\times \\sqrt{3} \\mathbb{Z}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-28T21:45:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C35",
      "74G65",
      "52C05"
    ],
    "keywords": [
      "periodic point configurations",
      "periodic energy",
      "optimality",
      "multivariate polynomial interpolation problem",
      "universally optimal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 68,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}