{
  "id": "0711.1800",
  "version": "v2",
  "published": "2007-11-12T15:29:00.000Z",
  "updated": "2007-11-13T14:57:39.000Z",
  "title": "Arithmetic and Geometric Progressions in Productsets over Finite Fields",
  "authors": [
    "Igor E. Shparlinski"
  ],
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Given two sets $\\cA, \\cB \\subseteq \\F_q$ of elements of the finite field $\\F_q$ of $q$ elements, we show that the productset $$ \\cA\\cB = \\{ab | a \\in \\cA, b \\in\\cB\\} $$ contains an arithmetic progression of length $k \\ge 3$ provided that $k<p$, where $p$ is the characteristic of $\\F_q$, and $# \\cA # \\cB \\ge 3q^{2d-2/k}$. We also consider geometric progressions in a shifted productset $\\cA\\cB +h$, for $f \\in \\F_q$, and obtain a similar result.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-11-13T14:57:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B83",
      "11T23",
      "11T30"
    ],
    "keywords": [
      "finite field",
      "geometric progressions",
      "arithmetic progression",
      "similar result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0711.1800S"
    }
  }
}