{
  "id": "2307.06874",
  "version": "v1",
  "published": "2023-07-13T16:22:21.000Z",
  "updated": "2023-07-13T16:22:21.000Z",
  "title": "The sum-product problem for small sets",
  "authors": [
    "Ginny Ray Clevenger",
    "Haley Havard",
    "Patch Heard",
    "Andrew Lott",
    "Alex Rice",
    "Brittany Wilson"
  ],
  "comment": "10 pages, 1 table, 4 figures",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "For $A\\subseteq \\mathbb{R}$, let $A+A=\\{a+b: a,b\\in A\\}$ and $AA=\\{ab: a,b\\in A\\}$. For $k\\in \\mathbb{N}$, let $SP(k)$ denote the minimum value of $\\max\\{|A+A|, |AA|\\}$ over all $A\\subseteq \\mathbb{N}$ with $|A|=k$. Here we establish $SP(k)=3k-3$ for $2\\leq k \\leq 7$, the $k=7$ case achieved for example by $\\{1,2,3,4,6,8,12\\}$, while $SP(k)=3k-2$ for $k=8,9$, the $k=9$ case achieved for example by $\\{1,2,3,4,6,8,9,12,16\\}$. For $4\\leq k \\leq 7$, we provide two proofs using different applications of Freiman's $3k-4$ theorem; one of the proofs includes extensive case analysis on the product sets of $k$-element subsets of $(2k-3)$-term arithmetic progressions. For $k=8,9$, we apply Freiman's $3k-3$ theorem for product sets, and investigate the sumset of the union of two geometric progressions with the same common ratio $r>1$, with separate treatments of the overlapping cases $r\\neq 2$ and $r\\geq 2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-13T16:22:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "small sets",
      "sum-product problem",
      "product sets",
      "term arithmetic progressions",
      "element subsets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}