{
  "id": "2105.13872",
  "version": "v1",
  "published": "2021-05-28T14:28:29.000Z",
  "updated": "2021-05-28T14:28:29.000Z",
  "title": "Khintchine's theorem and Diophantine approximation on manifolds",
  "authors": [
    "Victor Beresnevich",
    "Lei Yang"
  ],
  "comment": "19 pages",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "In this paper we initiate a new approach to studying approximations by rational points to smooth submanifolds of $\\mathbb{R}^n$. Our main result is a convergence Khintchine type theorem for arbitrary nondegenerate submanifolds of $\\mathbb{R}^n$, which resolves a longstanding problem in the theory of Diophantine approximation. Furthermore, we refine this result using Hausdorff $s$-measures and consequently obtain the exact value of the Hausdorff dimension of $\\tau$-well approximable points lying on any nondegenerate submanifold for a range of Diophantine exponents $\\tau$ close to $1/n$. Our approach uses geometric and dynamical ideas together with a new technique of `major and minor arcs'. In particular, we establish sharp upper bounds for the number of rational points of bounded height lying near `major arcs' and give explicit exponentially small bounds for the measure of `minor arcs'. The latter uses a result of Bernik, Kleinbock and Margulis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-28T14:28:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J83",
      "11J13",
      "11K60",
      "11K55"
    ],
    "keywords": [
      "diophantine approximation",
      "khintchines theorem",
      "convergence khintchine type theorem",
      "minor arcs",
      "rational points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}