{
  "id": "2102.13046",
  "version": "v1",
  "published": "2021-02-25T17:55:24.000Z",
  "updated": "2021-02-25T17:55:24.000Z",
  "title": "Divergence of separated nets with respect to displacement equivalence",
  "authors": [
    "Michael Dymond",
    "Vojtěch Kaluža"
  ],
  "comment": "Contains a small amount of modified content from arXiv:1903.05923",
  "categories": [
    "math.MG",
    "math.FA"
  ],
  "abstract": "We introduce a hierachy of equivalence relations on the set of separated nets of a given Euclidean space, indexed by concave increasing functions $\\phi\\colon (0,\\infty)\\to(0,\\infty)$. Two separated nets are called $\\phi$-displacement equivalent if, roughly speaking, there is a bijection between them which, for large radii $R$, displaces points of norm at most $R$ by something of order at most $\\phi(R)$. We show that the spectrum of $\\phi$-displacement equivalence spans from the established notion of bounded displacement equivalence, which corresponds to bounded $\\phi$, to the indiscrete equivalence relation, coresponding to $\\phi(R)\\in \\Omega(R)$, in which all separated nets are equivalent. In between the two ends of this spectrum, the notions of $\\phi$-displacement equivalence are shown to be pairwise distinct with respect to the asymptotic classes of $\\phi(R)$ for $R\\to\\infty$. We further undertake a comparison of our notion of $\\phi$-displacement equivalence with previously studied relations on separated nets. Particular attention is given to the interaction of the notions of $\\phi$-displacement equivalence with that of bilipschitz equivalence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-25T17:55:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51F99",
      "51M05",
      "52C99",
      "26B35",
      "26B10"
    ],
    "keywords": [
      "separated nets",
      "divergence",
      "indiscrete equivalence relation",
      "displacement equivalence spans",
      "concave increasing functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}