{
  "id": "1310.5275",
  "version": "v4",
  "published": "2013-10-19T22:25:17.000Z",
  "updated": "2015-07-09T04:15:56.000Z",
  "title": "Polynomials and Primes in Generalized Arithmetic Progressions (Revised Version)",
  "authors": [
    "Ernie Croot",
    "Neil Lyall",
    "Alex Rice"
  ],
  "comment": "14 pages, typos corrected, numerology improved, properness hypotheses eliminated",
  "categories": [
    "math.NT",
    "math.CA"
  ],
  "abstract": "We provide upper bounds on the density of a symmetric generalized arithmetic progression lacking nonzero elements of the form h(n) for natural numbers n, or h(p) with p prime, for appropriate polynomials h with integer coefficients. The prime variant can be interpreted as a multi-dimensional, polynomial extension of Linnik's Theorem. This version is a revision of the published version. Most notably, the properness hypotheses have been removed from Theorems 2 and 3, and the numerology in Theorem 2 has been improved.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-07-10T18:54:33.000Z",
      "title": "Polynomials and Primes in Generalized Arithmetic Progressions",
      "abstract": "We provide upper bounds on the density of a symmetric generalized arithmetic progression lacking nonzero elements of the form h(n) for natural numbers n, or h(p) with p prime, for appropriate polynomials h with integer coefficients. The prime variant can be interpreted as a multi-dimensional, polynomial extension of Linnik's Theorem.",
      "comment": "14 pages, typos and references corrected, funding info added, final version",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-07-09T04:15:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "polynomial",
      "arithmetic progression lacking nonzero elements",
      "generalized arithmetic progression lacking nonzero",
      "upper bounds",
      "symmetric generalized arithmetic progression lacking"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.5275C"
    }
  }
}