{
  "id": "math/0309215",
  "version": "v3",
  "published": "2003-09-12T15:04:15.000Z",
  "updated": "2004-02-09T16:56:14.000Z",
  "title": "A lower bound for periods of matrices",
  "authors": [
    "Pietro Corvaja",
    "Zeev Rudnick",
    "Umberto Zannier"
  ],
  "comment": "Added references and corrected a few misprints. Added condition that A be ergodic for a remark in the introduction",
  "doi": "10.1007/s00220-004-1184-6",
  "categories": [
    "math.NT",
    "math-ph",
    "math.MP",
    "nlin.CD"
  ],
  "abstract": "For a nonsingular integer matrix A, we study the growth of the order of A modulo N. We say that a matrix is exceptional if it is diagonalizable, and a power of the matrix has all eigenvalues equal to powers of a single rational integer, or all eigenvalues are powers of a single unit in a real quadratic field. For exceptional matrices, it is easily seen that there are arbitrarily large values of N for which the order of A modulo N is logarithmically small. In contrast, we show that if the matrix is not exceptional, then the order of A modulo N goes to infinity faster than any constant multiple of log N.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2004-02-09T16:56:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "11Jxx"
    ],
    "keywords": [
      "lower bound",
      "nonsingular integer matrix",
      "single rational integer",
      "real quadratic field",
      "eigenvalues equal"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer",
      "journal": "Communications in Mathematical Physics",
      "year": 2004,
      "month": "Dec",
      "volume": 252,
      "number": "1-3",
      "pages": 535
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004CMaPh.252..535C"
    }
  }
}