{
  "id": "1607.00670",
  "version": "v1",
  "published": "2016-07-03T19:01:26.000Z",
  "updated": "2016-07-03T19:01:26.000Z",
  "title": "Generalizations of Furstenberg's Diophantine result",
  "authors": [
    "Asaf Katz"
  ],
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "We prove two generalizations of Furstenberg's Diophantine result regarding density of an orbit of an irrational point in the one-torus under the action of multiplication by a non-lacunary multiplicative semi-group of $\\mathbb{N}$. We show that for any sequences $\\{a_{n} \\},\\{b_{n} \\}\\subset\\mathbb{Z}$ for which the quotients of successive elements tend to $1$ as $n$ goes to infinity, and any infinite sequence $\\{c_{n} \\}$, the set $\\{a_{n}b_{m}c_{k}x : n,m,k\\in\\mathbb{N} \\}$ is dense modulo $1$ for every irrational $x$. Moreover, by ergodic-theoretical methods, we prove that if $\\{a_{n} \\},\\{b_{n} \\}$ are sequence having smooth $p$-adic interpolation for some prime number $p$, then for every irrational $x$, the sequence $\\{p^{n}a_{m}b_{k}x : n,m,k\\in\\mathbb{N} \\}$ is dense modulo 1.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-03T19:01:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "generalizations",
      "furstenbergs diophantine result regarding density",
      "dense modulo",
      "irrational point",
      "non-lacunary multiplicative semi-group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}