{
  "id": "1211.6567",
  "version": "v1",
  "published": "2012-11-28T10:22:49.000Z",
  "updated": "2012-11-28T10:22:49.000Z",
  "title": "Solving $a\\pm b=2c$ in the elements of finite sets",
  "authors": [
    "Vsevolod F. Lev",
    "Rom Pinchasi"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We show that if $A$ and $B$ are finite sets of real numbers, then the number of triples $(a,b,c)\\in A\\times B\\times (A\\cup B)$ with $a+b=2c$ is at most $(0.15+o(1))(|A|+|B|)^2$ as $|A|+|B|\\to\\infty$. As a corollary, if $A$ is antisymmetric (that is, $A\\cap(-A)=\\est$), then there are at most $(0.3+o(1))|A|^2$ triples $(a,b,c)$ with $a,b,c\\in A$ and $a-b=2c$. In the general case where $A$ is not necessarily antisymmetric, we show that the number of triples $(a,b,c)$ with $a,b,c\\in A$ and $a-b=2c$ is at most $(0.5+o(1))|A|^2$. These estimates are sharp.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-11-28T10:22:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite sets",
      "real numbers",
      "general case",
      "necessarily antisymmetric"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.6567L"
    }
  }
}