{
  "id": "1407.1654",
  "version": "v1",
  "published": "2014-07-07T10:33:19.000Z",
  "updated": "2014-07-07T10:33:19.000Z",
  "title": "Sum-ratio estimates over arbitrary finite fields",
  "authors": [
    "Oliver Roche-Newton"
  ],
  "comment": "12 pages. This note is not intended for journal publication, since the main result and proof are too similar to the work of arXiv:1106.1148. However, the subtle differences between the proof of the main result here and that of arXiv:1106.1148 mean that it is necessary to record the result carefully, particularly with future applications in mind",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "The aim of this note is to record a proof that the estimate $$\\max{\\{|A+A|,|A:A|\\}}\\gg{|A|^{12/11}}$$ holds for any set $A\\subset{\\mathbb{F}_q}$, provided that $A$ satisfies certain conditions which state that it is not too close to being a subfield. An analogous result was established in \\cite{LiORN}, with the product set $A\\cdot{A}$ in the place of the ratio set $A:A$. The sum-ratio estimate here beats the sum-product estimate in \\cite{LiORN} by a logarithmic factor, with slightly improved conditions for the set $A$, and the proof is arguably a little more intuitive. The sum-ratio estimate was mentioned in \\cite{LiORN}, but a proof was not given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-07T10:33:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "68R05",
      "11B75"
    ],
    "keywords": [
      "arbitrary finite fields",
      "sum-ratio estimate",
      "conditions",
      "product set",
      "ratio set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.1654R"
    }
  }
}