{
  "id": "2401.08849",
  "version": "v1",
  "published": "2024-01-16T21:45:39.000Z",
  "updated": "2024-01-16T21:45:39.000Z",
  "title": "Inhomogeneous Diophantine Approximation on $M_0$-sets",
  "authors": [
    "Volodymyr Pavlenkov",
    "Evgeniy Zorin"
  ],
  "comment": "31 page",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "We prove new quantitative Schmidt-type theorem for Diophantine approximations with restraint denominators on fractals (more precisely, on $M_0$-sets). Our theorems introduce a sharp balance condition between the growth rate of the sequence of denominators and the decay rate of the Fourier transform of a Rajchman measure. Among the other things, this allows applications to sequences of denominators of polynomial growth. In particular, we infer new inhomogeneous Khintchine-J\\\"arnik type theorems with restraint denominators for a broad family of denominator sequences. Furthermore, our results provide non-trivial lower bounds for Hausdorff dimensions of intersections of two sets of inhomogeneously well-approximable numbers with restraint denominators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-16T21:45:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J83",
      "42A61"
    ],
    "keywords": [
      "inhomogeneous diophantine approximation",
      "restraint denominators",
      "non-trivial lower bounds",
      "sharp balance condition",
      "schmidt-type theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}