{
  "id": "2407.06944",
  "version": "v1",
  "published": "2024-07-09T15:19:56.000Z",
  "updated": "2024-07-09T15:19:56.000Z",
  "title": "Additive energies of subsets of discrete cubes",
  "authors": [
    "Xuancheng Shao"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "For a positive integer $n \\geq 2$, define $t_n$ to be the smallest number such that the additive energy $E(A)$ of any subset $A \\subset \\{0,1,\\cdots,n-1\\}^d$ and any $d$ is at most $|A|^{t_n}$. Trivially we have $t_n \\leq 3$ and $$ t_n \\geq 3 - \\log_n\\frac{3n^3}{2n^3+n} $$ by considering $A = \\{0,1,\\cdots,n-1\\}^d$. In this note, we investigate the behavior of $t_n$ for large $n$ and obtain the following non-trivial bounds: $$ 3 - (1+o_{n\\rightarrow\\infty}(1)) \\log_n \\frac{3\\sqrt{3}}{4} \\leq t_n \\leq 3 - \\log_n(1+c), $$ where $c>0$ is an absolute constant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-09T15:19:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B30"
    ],
    "keywords": [
      "additive energy",
      "discrete cubes",
      "smallest number",
      "non-trivial bounds",
      "absolute constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}