{
  "id": "1808.06665",
  "version": "v1",
  "published": "2018-08-20T19:30:08.000Z",
  "updated": "2018-08-20T19:30:08.000Z",
  "title": "Cayley Digraphs Associated to Arithmetic Groups",
  "authors": [
    "David Covert",
    "Yeşim Demiroğlu Karabulut",
    "Jonathan Pakianathan"
  ],
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "We explore a paradigm which ties together seemingly disparate areas in number theory, additive combinatorics, and geometric combinatorics including the classical Waring problem, the Furstenberg-S\\'{a}rk\\\"{o}zy theorem on squares in sets of integers with positive density, and the study of triangles (also called $2$-simplices) in finite fields. Among other results we show that if $\\mathbb{F}_q$ is the finite field of odd order $q$, then every matrix in $Mat_d(\\mathbb{F}_q), d \\geq 2$ is the sum of a certain (finite) number of orthogonal matrices, this number depending only on $d$, the size of the matrix, and on whether $q$ is congruent to $1$ or $3$ (mod $4$), but independent of $q$ otherwise.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-20T19:30:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arithmetic groups",
      "cayley digraphs",
      "finite field",
      "geometric combinatorics",
      "orthogonal matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}