{
  "id": "2112.09352",
  "version": "v2",
  "published": "2021-12-17T07:05:29.000Z",
  "updated": "2023-06-20T17:22:39.000Z",
  "title": "Additive energies on discrete cubes",
  "authors": [
    "Jaume de Dios Pont",
    "Rachel Greenfeld",
    "Paata Ivanisvili",
    "José Madrid"
  ],
  "comment": "16 pages, 3 figures",
  "categories": [
    "math.CO",
    "math.CA"
  ],
  "abstract": "We prove that for $d\\geq 0$ and $k\\geq 2$, for any subset $A$ of a discrete cube $\\{0,1\\}^d$, the $k-$higher energy of $A$ (the number of $2k-$tuples $(a_1,a_2,\\dots,a_{2k})$ in $A^{2k}$ with $a_1-a_2=a_3-a_4=\\dots=a_{2k-1}-a_{2k}$) is at most $|A|^{\\log_{2}(2^k+2)}$, and $\\log_{2}(2^k+2)$ is the best possible exponent. We also show that if $d\\geq 0$ and $2\\leq k\\leq 10$, for any subset $A$ of a discrete cube $\\{0,1\\}^d$, the $k-$additive energy of $A$ (the number of $2k-$tuples $(a_1,a_2,\\dots,a_{2k})$ in $A^{2k}$ with $a_1+a_2+\\dots+a_k=a_{k+1}+a_{k+2}+\\dots+a_{2k}$) is at most $|A|^{\\log_2{ \\binom{2k}{k}}}$, and $\\log_2{ \\binom{2k}{k}}$ is the best possible exponent. We discuss the analogous problems for the sets $\\{0,1,\\dots,n\\}^d$ for $n\\geq 2$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-06-20T17:22:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B30"
    ],
    "keywords": [
      "discrete cube",
      "additive energy",
      "higher energy",
      "analogous problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}