{
  "id": "1310.4348",
  "version": "v1",
  "published": "2013-10-16T12:30:36.000Z",
  "updated": "2013-10-16T12:30:36.000Z",
  "title": "On the Union of Arithmetic Progressions",
  "authors": [
    "Shoni Gilboa",
    "Rom Pinchasi"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We show that for every $\\varepsilon>0$ there is an absolute constant $c(\\varepsilon)>0$ such that the following is true. The union of any $n$ arithmetic progressions, each of length $n$, with pairwise distinct differences must consist of at least $c(\\varepsilon)n^{2-\\varepsilon}$ elements. We observe, by construction, that one can find $n$ arithmetic progressions, each of length $n$, with pairwise distinct differences such that the cardinality of their union is $o(n^2)$. We refer also to the non-symmetric case of $n$ arithmetic progressions, each of length $\\ell$, for various regimes of $n$ and $\\ell$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-10-16T12:30:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arithmetic progressions",
      "pairwise distinct differences",
      "non-symmetric case",
      "absolute constant",
      "cardinality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.4348G"
    }
  }
}