{
  "id": "2309.14229",
  "version": "v1",
  "published": "2023-09-25T15:41:28.000Z",
  "updated": "2023-09-25T15:41:28.000Z",
  "title": "On the expressive power of mod-$p$ linear forms on the Boolean cube",
  "authors": [
    "Thomas Karam"
  ],
  "comment": "26 pages",
  "categories": [
    "math.CO",
    "cs.IT",
    "math.IT",
    "math.NT"
  ],
  "abstract": "Let $(\\mathcal{A}_i)_{i \\in [s]}$ be a sequence of dense subsets of the Boolean cube $\\{0,1\\}^n$ and let $p$ be a prime. We show that if $s$ is assumed to be superpolynomial in $n$ then we can find distinct $i,j$ such that the two distributions of every mod-$p$ linear form on $\\mathcal{A}_i$ and $\\mathcal{A}_j$ are almost positively correlated. We also prove that if $s$ is merely assumed to be sufficiently large independently of $n$ then we may require the two distributions to have overlap bounded below by a positive quantity depending on $p$ only.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-25T15:41:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "boolean cube",
      "linear form",
      "expressive power",
      "dense subsets",
      "distributions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}