{
  "id": "2408.07634",
  "version": "v1",
  "published": "2024-08-14T16:00:20.000Z",
  "updated": "2024-08-14T16:00:20.000Z",
  "title": "Sufficient conditions for the existence of packing asymptotics on linear sets of Lebesgue measure zero",
  "authors": [
    "Austin Anderson",
    "Steven Damelin"
  ],
  "comment": "37 pages, 2 figures",
  "categories": [
    "math.CA",
    "math.MG"
  ],
  "abstract": "We use a characterization of Minkowski measurability to study the asymptotics of best packing on cut-out subsets of the real line with Minkowski dimension $d\\in(0,1)$. Our main result is a proof that Minkowski measurability is a sufficient condition for the existence of best packing asymptotics on monotone rearrangements of these sets. For each such set, the main result provides an explicit constant of proportionality $p_d,$ depending only on the Minkowski dimension $d,$ that relates its packing limit and Minkowski content. We later use the Digamma function to study the limiting value of $p_d$ as $d\\to 1^-.$ For sharpness, we use renewal theory to prove that the packing constant of the $(1/2,1/3)$ Cantor set is less than the product of its Minkowski content and $p_d$. We also show that the measurability hypothesis of the main theorem is necessary by demonstrating that a monotone rearrangement of the complementary intervals of the 1/3 Cantor set has Minkowski dimension $d=\\log2/\\log3\\in(0,1),$ is not Minkowski measurable, and does not have convergent first-order packing asymptotics. The aforementioned characterization of Minkowski measurability further motivates the asymptotic study of an infinite multiple subset sum problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-14T16:00:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lebesgue measure zero",
      "packing asymptotics",
      "sufficient condition",
      "linear sets",
      "minkowski measurability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}