{
  "id": "1811.07454",
  "version": "v2",
  "published": "2018-11-19T02:03:07.000Z",
  "updated": "2018-11-25T19:19:18.000Z",
  "title": "Product of sumsets over arbitrary finite fields",
  "authors": [
    "Mozhgan Mirzaei",
    "Thang Pham"
  ],
  "comment": "The paper has been withdrawn. Thanks to D. Koh and G. Petridis for pointing out the mistake in the main lemma",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\mathbb{F}_q$ be an arbitrary finite field of order $q$. For $A, B, C, D\\subset \\mathbb{F}_q$, we show that if $|A||B||C||D|\\gg q^{5/2},$ then the set $(A-B)(C-D)$ covers a positive proportion of all elements in $\\mathbb{F}_q.$ Our result significantly improves the recent exponent $\\frac{2}{3}-\\frac{1}{13542}$ given by Murphy and Petridis.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2018-11-25T19:19:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arbitrary finite field",
      "positive proportion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}