{
  "id": "1103.6000",
  "version": "v2",
  "published": "2011-03-30T17:35:20.000Z",
  "updated": "2013-02-25T14:11:52.000Z",
  "title": "Arithmetic progressions in sumsets and L^p-almost-periodicity",
  "authors": [
    "Ernie Croot",
    "Izabella Laba",
    "Olof Sisask"
  ],
  "comment": "15 pages; to appear in Combinatorics, Probability and Computing",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "We prove results about the L^p-almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in L^p, and gives a very short proof of a theorem of Green that if A and B are subsets of {1,...,N} of sizes alpha N and beta N then A+B contains an arithmetic progression of length at least about exp(c (alpha beta log N)^{1/2}). Another almost-periodicity result improves this bound for densities decreasing with N: we show that under the above hypotheses the sumset A+B contains an arithmetic progression of length at least about exp(c (alpha log N/(log(beta^{-1}))^3)^{1/2}).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-02-25T14:11:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B30"
    ],
    "keywords": [
      "arithmetic progression",
      "alpha beta log",
      "alpha log",
      "short proof",
      "general lemma"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.6000C"
    }
  }
}