{
  "id": "1507.01605",
  "version": "v1",
  "published": "2015-07-06T20:03:28.000Z",
  "updated": "2015-07-06T20:03:28.000Z",
  "title": "Leading Digit Laws on Linear Lie Groups",
  "authors": [
    "Corey Manack",
    "Steven J. Miller"
  ],
  "comment": "Version 1.0, 17 pages, 1 figure",
  "categories": [
    "math.NT",
    "math.MG",
    "math.PR"
  ],
  "abstract": "We determine the leading digit laws for the matrix components of a linear Lie group $G$. These laws generalize the observations that the normalized Haar measure of the Lie group $\\mathbb{R}^+$ is $dx/x$ and that the scale invariance of $dx/x$ implies the distribution of the digits follow Benford's law, which is the probability of observing a significand base $B$ of at most $s$ is $\\log_B(s)$; thus the first digit is $d$ with probability $\\log_B(1 + 1/d)$). Viewing this scale invariance as left invariance of Haar measure, we determine the power laws in significands from one matrix component of various such $G$. We also determine the leading digit distribution of a fixed number of components of a unit sphere, and find periodic behavior when the dimension of the sphere tends to infinity in a certain progression.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-06T20:03:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K06",
      "60F99",
      "28C10",
      "15B52",
      "15B99"
    ],
    "keywords": [
      "linear lie group",
      "leading digit laws",
      "haar measure",
      "scale invariance",
      "matrix component"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150701605M"
    }
  }
}