{
  "id": "2409.10412",
  "version": "v1",
  "published": "2024-09-16T15:46:16.000Z",
  "updated": "2024-09-16T15:46:16.000Z",
  "title": "The central limit theorem for entries of random matrices with specific rank over finite fields",
  "authors": [
    "Chin Hei Chan",
    "Maosheng Xiong"
  ],
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $\\mathbb{F}_q$ be the finite field of order $q$, and $\\mathcal{A}$ a non-empty proper subset of $\\mathbb{F}_q$. Let $\\mathbf{M}$ be a random $m \\times n$ matrix of rank $r$ over $\\mathbb{F}_q$ taken with uniform distribution. It was proved recently by Sanna that as $m,n \\to \\infty$ and $r,q,\\mathcal{A}$ are fixed, the number of entries of $\\mathbf{M}$ in $\\mathcal{A}$ approaches a normal distribution. The question was raised as to whether or not one can still obtain a central limit theorem of some sort when $r$ goes to infinity in a way controlled by $m$ and $n$. In this paper we answer this question affirmatively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-16T15:46:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B52",
      "11T99",
      "05C50",
      "60F05"
    ],
    "keywords": [
      "central limit theorem",
      "finite field",
      "random matrices",
      "specific rank",
      "non-empty proper subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}