{
  "id": "1405.0241",
  "version": "v2",
  "published": "2014-05-01T18:11:30.000Z",
  "updated": "2014-12-03T01:23:08.000Z",
  "title": "A structure theorem for multiplicative functions over the Gaussian integers and applications",
  "authors": [
    "Wenbo Sun"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1303.4329 by other authors",
  "categories": [
    "math.NT",
    "math.CO",
    "math.DS"
  ],
  "abstract": "We prove a structure theorem for multiplicative functions on the Gaussian integers, showing that every bounded multiplicative function on the Gaussian integers can be decomposed into a term which is approximately periodic and another which has a small U^{3}-Gowers uniformity norm. We apply this to prove partition regularity results over the Gaussian integers for certain equations involving quadratic forms in three variables. For example, we show that for any finite coloring of the Gaussian integers, there exist distinct nonzero elements x and y of the same color such that x^{2}-y^{2}=n^{2} for some Gaussian integer n. The analog of this statement over Z remains open.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-01T18:11:30.000Z",
      "abstract": "We prove a structure theorem for multiplicative functions on the Gaussian integers, showing that every bounded multiplicative function on the Gaussian integers can be decomposed into a term which is approximately periodic and another which has a small U^3-Gowers uniformity norm. We apply this to prove partition regularity results over the Gaussian integers for certain equations involving quadratic forms in three variables. For example, we show that for any finite coloring of the Gaussian integers, there exists dfferent nonzero elements x and y of the same color such that x^2-y^2=n^2 for some Gaussian integer n, the analog of which over Z remains open.",
      "comment": "42 pages. arXiv admin note: text overlap with arXiv:1303.4329 by other authors",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-03T01:23:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D10",
      "11B30",
      "11D09",
      "11N37",
      "11N60",
      "37A45"
    ],
    "keywords": [
      "gaussian integer",
      "multiplicative function",
      "structure theorem",
      "applications",
      "dfferent nonzero elements"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.0241S"
    }
  }
}