{
  "id": "1706.05646",
  "version": "v1",
  "published": "2017-06-18T13:28:32.000Z",
  "updated": "2017-06-18T13:28:32.000Z",
  "title": "Balanced words in higher dimensions",
  "authors": [
    "Siddhartha Bhattacharya"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "For $d\\ge 1$, a word $w\\in \\{ 0,1\\}^{\\Z^d}$ is called balanced if there exists $M > 0$ such that for any two rectangles $R, R^{'}\\subset\\Z^d$ that are translates of each other, the number of occurrences of the symbol $1$ in $R$ and $R^{'}$ differ by at most $M$. It is known that for every balanced word $w$, the asymptotic frequency of the symbol $1$ ( called the density of $w$ ) exists. In this paper we show that there exist two dimensional balanced words with irrational densities, answering a question raised by Berth\\'e and Tijdeman.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-18T13:28:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "higher dimensions",
      "asymptotic frequency",
      "dimensional balanced words",
      "irrational densities",
      "rectangles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}