{
  "id": "1906.01151",
  "version": "v1",
  "published": "2019-06-04T01:34:26.000Z",
  "updated": "2019-06-04T01:34:26.000Z",
  "title": "Inhomogeneous Diophantine Approximation on $M_0$-sets with restricted denominators",
  "authors": [
    "Andrew D. Pollington",
    "Sanju Velani",
    "Agamemnon Zafeiropoulos",
    "Evgeniy Zorin"
  ],
  "comment": "58 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $F \\subseteq [0,1]$ be a set that supports a probability measure $\\mu$ with the property that $ |\\widehat{\\mu}(t)| \\ll (\\log |t|)^{-A}$ for some constant $ A > 0 $. Let $\\mathcal{A}= (q_n)_{n\\in \\mathbb{N}} $ be a sequence of natural numbers. If $\\mathcal{A}$ is lacunary and $A >2$, we establish a quantitative inhomogeneous Khintchine-type theorem in which (i) the points of interest are restricted to $F$ and (ii) the denominators of the `shifted' rationals are restricted to $\\mathcal{A}$. The theorem can be viewed as a natural strengthening of the fact that the sequence $(q_nx {\\rm \\ mod \\, } 1)_{n\\in \\mathbb{N}} $ is uniformly distributed for $\\mu$ almost all $x \\in F$. Beyond lacunary, our main theorem implies the analogous quantitative result for sequences $\\mathcal{A}$ for which the prime divisors are restricted to a finite set of $k$ primes and $A > 2k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-04T01:34:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J83"
    ],
    "keywords": [
      "inhomogeneous diophantine approximation",
      "restricted denominators",
      "main theorem implies",
      "finite set",
      "prime divisors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 58,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}