{
  "id": "1909.08688",
  "version": "v1",
  "published": "2019-09-18T20:24:17.000Z",
  "updated": "2019-09-18T20:24:17.000Z",
  "title": "Random gap processes and asymptotically complete sequences",
  "authors": [
    "Erin Crossen Brown",
    "Sevak Mkrtchyan",
    "Jonathan Pakianathan"
  ],
  "categories": [
    "math.PR",
    "math.CO",
    "math.NT"
  ],
  "abstract": "We study a process of generating random positive integer weight sequences $\\{ W_n \\}$ where the gaps between the weights $\\{ X_n = W_n - W_{n-1} \\}$ are i.i.d. positive integer-valued random variables. We show that as long as the gap distribution has finite $\\frac{1}{2}$-moment, almost surely, the resulting weight sequence is asymptotically complete, i.e., all large enough multiples of the gcd of the possible gap values can be written as a sum of distinct weights. We then show a much stronger result that if the gap distribution has a moment generating function with large enough radius of convergence, then every large enough multiple of the gcd of gap values can be written as a sum of $m$ distinct weights for any fixed $m \\geq 2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-18T20:24:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "60C05",
      "11P70",
      "11P81"
    ],
    "keywords": [
      "random gap processes",
      "asymptotically complete sequences",
      "distinct weights",
      "random positive integer weight sequences",
      "gap distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}