{
  "id": "1811.11000",
  "version": "v1",
  "published": "2018-11-27T14:11:36.000Z",
  "updated": "2018-11-27T14:11:36.000Z",
  "title": "Arithmetic compactifications of natural numbers and its applications",
  "authors": [
    "Fei Wei"
  ],
  "comment": "27 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "To study the arithmetic structure of natural numbers, we introduce a notion of entropy of arithmetic functions. This entropy possesses properties common both to Shannon's and Kolmogorov's entropies. We show that all zero-entropy arithmetic functions form a C*-algebra. We introduce the arithmetic compactification of natural numbers as the maximal ideal space of this C*-algebra. The arithmetic compactification is totally disconnected, and not extremely disconnected. We also compute the $K_{0}$-group and the $K_{1}$-group of the space of all continuous functions on the arithmetic compactification. As an application, we prove the following result: suppose that $X$ is a compact finite dimensional manifold with metric $d$, and $T$ a continuous map on $X$ with zero topological entropy. Then for any given $x\\in X$ and $\\varepsilon>0$, there is a map $f$ from natural numbers to $\\{T^{n_{1}}x,T^{n_{2}}x,\\ldots, T^{n_{k}}x\\}$, namely a finite set of points in the orbit of $x$, such that for each $i=1,\\ldots,k$, the characteristic function defined on $f^{-1}(\\{T^{n_{i}}x\\})$ has zero entropy and $\\sup_{n}d(T^{n}x, f(n))< \\varepsilon$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-27T14:11:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A55",
      "11N37"
    ],
    "keywords": [
      "natural numbers",
      "arithmetic compactification",
      "application",
      "entropy possesses properties common",
      "zero-entropy arithmetic functions form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}