{
  "id": "1203.0017",
  "version": "v1",
  "published": "2012-02-29T21:09:46.000Z",
  "updated": "2012-02-29T21:09:46.000Z",
  "title": "On Congruences with Products of Variables from Short Intervals and Applications",
  "authors": [
    "Jean Bourgain",
    "Moubariz Garaev",
    "Sergei Konyagin",
    "Igor Shparlinski"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We obtain upper bounds on the number of solutions to congruences of the type $$ (x_1+s)...(x_{\\nu}+s)\\equiv (y_1+s)...(y_{\\nu}+s)\\not\\equiv0 \\pmod p $$ modulo a prime $p$ with variables from some short intervals. We give some applications of our results and in particular improve several recent estimates of J. Cilleruelo and M. Z. Garaev on exponential congruences and on cardinalities of products of short intervals, some double character sum estimates of J. B. Friedlander and H. Iwaniec and some results of M.-C. Chang and A. A. Karatsuba on character sums twisted with the divisor function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-02-29T21:09:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D79",
      "11L40"
    ],
    "keywords": [
      "short intervals",
      "applications",
      "double character sum estimates",
      "upper bounds",
      "exponential congruences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1203.0017B"
    }
  }
}