{
  "id": "math/0501314",
  "version": "v4",
  "published": "2005-01-20T05:00:43.000Z",
  "updated": "2011-12-31T18:13:25.000Z",
  "title": "The Gaussian primes contain arbitrarily shaped constellations",
  "authors": [
    "Terence Tao"
  ],
  "comment": "58 pages, no figures. An issue (pointed out by Lilian Matthiesen) regarding the need to ensure the linear forms are not commensurate to their conjugate has been addressed",
  "journal": "J. d.Analyse Mathematique 99 (2006), 109-176",
  "categories": [
    "math.CO",
    "math.NT",
    "math.PR"
  ],
  "abstract": "We show that the Gaussian primes $P[i] \\subseteq \\Z[i]$ contain infinitely constellations of any prescribed shape and orientation. More precisely, given any distinct Gaussian integers $v_0,...,v_{k-1}$, we show that there are infinitely many sets $\\{a+rv_0,...,a+rv_{k-1}\\}$, with $a \\in \\Z[i]$ and $r \\in \\Z \\backslash \\{0\\}$, all of whose elements are Gaussian primes. The proof is modeled on a recent paper by Green and Tao and requires three ingredients. The first is a hypergraph removal lemma of Gowers and R\\\"odl-Skokan; this hypergraph removal lemma can be thought of as a generalization of the Szemer\\'edi-Furstenberg-Katznelson theorem concerning multidimensional arithmetic progressions. The second ingredient is the transference argument of Green and Tao, which allows one to extend this hypergraph removal lemma to a relative version, weighted by a pseudorandom measure. The third ingredient is a Goldston-Yildirim type analysis for the Gaussian integers, which yields a pseudorandom measure which is concentrated on Gaussian \"almost primes\".",
  "revisions": [
    {
      "version": "v4",
      "updated": "2011-12-31T18:13:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C65",
      "11P32"
    ],
    "keywords": [
      "primes contain arbitrarily shaped constellations",
      "gaussian primes contain arbitrarily",
      "hypergraph removal lemma",
      "theorem concerning multidimensional arithmetic",
      "concerning multidimensional arithmetic progressions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 58,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......1314T"
    }
  }
}