{
  "id": "2012.12017",
  "version": "v1",
  "published": "2020-12-22T14:04:50.000Z",
  "updated": "2020-12-22T14:04:50.000Z",
  "title": "On the structure of the $h$-fold sumsets",
  "authors": [
    "Jun-Yu Zhou",
    "Quan-Hui Yang"
  ],
  "comment": "8 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let~$A$ be a set of nonnegative integers. Let~$(h A)^{(t)}$ be the set of all integers in the sumset~$hA$ that have at least~$t$ representations as a sum of~$h$ elements of~$A$. In this paper, we prove that, if~$k \\geq 2$, and~$A=\\left\\{a_{0}, a_{1}, \\ldots, a_{k}\\right\\}$ is a finite set of integers such that~$0=a_{0}<a_{1}<\\cdots<a_{k}$ and $\\gcd\\left(a_{1}, a_2,\\ldots, a_{k}\\right)=1,$ then there exist integers ~$c_{t},d_{t}$ and sets~$C_{t}\\subseteq[0, c_{t}-2]$, $D_{t} \\subseteq[0, d_{t}-2]$ such that $$(h A)^{(t)}=C_{t} \\cup\\left[c_{t}, h a_{k}-d_{t}\\right] \\cup\\left(h a_{k-1}-D_{t}\\right) $$ for all~$h \\geq\\sum_{i=2}^{k}(ta_{i}-1)-1.$ This improves a recent result of Nathanson with the bound $h \\geq (k-1)\\left(t a_{k}-1\\right) a_{k}+1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-22T14:04:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B13",
      "G.2.1"
    ],
    "keywords": [
      "fold sumsets",
      "finite set",
      "representations",
      "nonnegative integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}