{
  "id": "1804.07560",
  "version": "v1",
  "published": "2018-04-20T11:44:14.000Z",
  "updated": "2018-04-20T11:44:14.000Z",
  "title": "Generalizations of some results about the regularity properties of an additive representation function",
  "authors": [
    "Sándor Z. Kiss",
    "Csaba Sándor"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $A = \\{a_{1},a_{2},\\dots{}\\}$ $(a_{1} < a_{2} < \\dots{})$ be an infinite sequence of nonnegative integers, and let $R_{A,2}(n)$ denote the number of solutions of $a_{x}+a_{y}=n$ $(a_{x},a_{y}\\in A)$. P. Erd\\H{o}s, A. S\\'ark\\\"ozy and V. T. S\\'os proved that if $\\lim_{N\\to\\infty}\\frac{B(A,N)}{\\sqrt{N}}=+\\infty$ then $|\\Delta_{1}(R_{A,2}(n))|$ cannot be bounded, where $B(A,N)$ denotes the number of blocks formed by consecutive integers in $A$ up to $N$ and $\\Delta_{l}$ denotes the $l$-th difference. Their result was extended to $\\Delta_{l}(R_{A,2}(n))$ for any fixed $l\\ge2$. In this paper we give further generalizations of this problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-20T11:44:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "additive representation function",
      "regularity properties",
      "generalizations",
      "infinite sequence",
      "th difference"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}