{
  "id": "2209.08509",
  "version": "v1",
  "published": "2022-09-18T08:56:24.000Z",
  "updated": "2022-09-18T08:56:24.000Z",
  "title": "Additive completition of thin sets",
  "authors": [
    "Jin-Hui Fang",
    "Csaba Sándor"
  ],
  "comment": "7 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Two sets $A,B$ of positive integers are called \\emph{exact additive complements}, if $A+B$ contains all sufficiently large integers and $A(x)B(x)/x\\rightarrow1$. Let $A=\\{a_1<a_2<\\cdots\\}$ be a set of positive integers. Denote $A(x)$ by the counting function of $A$ and $a^*(x)$ by the largest element in $A\\bigcap [1,x]$. Following the work of Ruzsa and Chen-Fang, we prove that, for exact additive complements $A,B$ with $\\frac{a_{n+1}}{na_n}\\rightarrow\\infty$, we have $A(x)B(x)-x\\ge \\frac{a^*(x)}{A(x)}+o\\left(\\frac{a^*(x)}{A(x)^2}\\right)$ as $x\\rightarrow +\\infty$. On the other hand, we also construct exact additive complements $A,B$ with $\\frac{a_{n+1}}{na_n}\\rightarrow\\infty$ such that $A(x)B(x)-x\\le \\frac{a^*(x)}{A(x)}+(1+o(1))\\left(\\frac{a^*(x)}{A(x)^2}\\right)$ holds for infinitely many positive integers $x$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-18T08:56:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B13",
      "11B34"
    ],
    "keywords": [
      "thin sets",
      "additive completition",
      "positive integers",
      "construct exact additive complements",
      "largest element"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}