{
  "id": "2207.13155",
  "version": "v1",
  "published": "2022-07-26T19:13:07.000Z",
  "updated": "2022-07-26T19:13:07.000Z",
  "title": "Dimension drop for diagonalizable flows on homogeneous spaces",
  "authors": [
    "Dmitry Kleinbock",
    "Shahriar Mirzadeh"
  ],
  "comment": "33 pages",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "Let $X = G/\\Gamma$, where $G$ is a Lie group and $\\Gamma$ is a lattice in $G$, let $O$ be an open subset of $X$, and let $F = \\{g_t: t\\ge 0\\}$ be a one-parameter subsemigroup of $G$. Consider the set of points in $X$ whose $F$-orbit misses $O$; it has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of $X$. This conjecture is proved when $X$ is compact or when $G$ is a simple Lie group of real rank $1$, or, most recently, for certain special flows on the space of lattices. In this paper we prove this conjecture for arbitrary $\\operatorname{Ad}$-diagonalizable flows on irreducible quotients of semisimple Lie groups. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on $G/\\Gamma$. We also derive an application to jointly Dirichlet-Improvable systems of linear forms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-26T19:13:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A17",
      "37A25",
      "11J13"
    ],
    "keywords": [
      "diagonalizable flows",
      "dimension drop",
      "homogeneous spaces",
      "semisimple lie groups",
      "hausdorff dimension strictly smaller"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}