{
  "id": "0902.3506",
  "version": "v2",
  "published": "2009-02-20T03:18:49.000Z",
  "updated": "2010-09-13T17:54:19.000Z",
  "title": "Sums and Products of Distinct Sets and Distinct Elements in $\\mathbb{C}$",
  "authors": [
    "Karsten Chipeniuk"
  ],
  "comment": "27 pages, Revised with corrections. Accepted by Integers: Electronic Journal of Combinatorial Number Theory",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let $A$ and $B$ be finite subsets of $\\mathbb{C}$ such that $|B|=C|A|$. We show the following variant of the sum product phenomenon: If $|AB|<\\alpha|A|$ and $\\alpha \\ll \\log |A|$, then $|kA+lB|\\gg |A|^k|B|^l$. This is an application of a result of Evertse, Schlickewei, and Schmidt on linear equations with variables taking values in multiplicative groups of finite rank, in combination with an earlier theorem of Ruzsa about sumsets in $\\mathbb{R}^d$. As an application of the case $A=B$ we give a lower bound on $|A^+|+|A^\\times|$, where $A^+$ is the set of sums of distinct elements of $A$ and $A^\\times$ is the set of products of distinct elements of $A$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-09-13T17:54:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A99",
      "11P99"
    ],
    "keywords": [
      "distinct elements",
      "distinct sets",
      "sum product phenomenon",
      "application",
      "lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0902.3506C"
    }
  }
}