{
  "id": "1306.4941",
  "version": "v1",
  "published": "2013-06-20T17:40:26.000Z",
  "updated": "2013-06-20T17:40:26.000Z",
  "title": "Upper and lower bounds on $B_k^+$-sets",
  "authors": [
    "Craig Timmons"
  ],
  "comment": "26 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let $G$ be an abelian group. A set $A \\subset G$ is a \\emph{$B_k^+$-set} if whenever $a_1 + \\dots + a_k = b_1 + \\dots + b_k$ with $a_i, b_j \\in A$ there is an $i$ and a $j$ such that $a_i = b_j$. If $A$ is a $B_k$-set then it is also a $B_k^+$-set but the converse is not true in general. Determining the largest size of a $B_k$-set in the interval $\\{1, 2, \\dots, N \\} \\subset \\integers$ or in the cyclic group $\\integers_N$ is a well studied problem. In this paper we investigate the corresponding problem for $B_k^+$-sets. We prove non-trivial upper bounds on the maximum size of a $B_k^+$-set contained in the interval $\\{1, 2, \\dots, N \\}$. For odd $k \\geq 3$, we construct $B_k^+$-sets that have more elements than the $B_k$-sets constructed by Bose and Chowla. We prove a $B_3^+$-set $A \\subset \\integers_N$ has at most $(1 + o(1))(8N)^{1/3}$ elements. Finally we obtain new upper bounds on the maximum size of a $B_k^*$-set $A \\subset \\{1,2, \\dots, N \\}$, a problem first investigated by Ruzsa.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-06-20T17:40:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B75",
      "05D99"
    ],
    "keywords": [
      "lower bounds",
      "non-trivial upper bounds",
      "cyclic group",
      "abelian group",
      "maximum size"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.4941T"
    }
  }
}