{
  "id": "2408.06200",
  "version": "v1",
  "published": "2024-08-12T14:49:47.000Z",
  "updated": "2024-08-12T14:49:47.000Z",
  "title": "Dirichlet improvability in $L_p$-norms",
  "authors": [
    "Nikolay Moshchevitin",
    "Nikita Shulga"
  ],
  "comment": "31 pages, any comments are appreciated",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "For a norm $F$ on $\\mathbb{R}^2$, we consider the set of $F$-Dirichlet improvable numbers $\\mathbf{DI}_F$. In the most important case of $F$ being an $L_p$-norm with $p=\\infty$, which is a supremum norm, it is well-known that $\\mathbf{DI}_F = \\mathbf{BA}\\cup \\mathbb{Q}$, where $\\mathbf{BA}$ is a set of badly approximable numbers. It is also known that $\\mathbf{BA}$ and each $\\mathbf{DI}_F$ are of measure zero and of full Hausdorff dimension. Using classification of critical lattices for unit balls in $L_p$, we provide a complete and effective characterization of $\\mathbf{DI}_p:=\\mathbf{DI}_{F^{[p]}}$ in terms of the occurrence of patterns in regular continued fraction expansions, where $F^{[p]}$ is an $L_p$-norm with $p\\in[1,\\infty)$. This yields several corollaries. In particular, we resolve two open questions by Kleinbock and Rao by showing that the set $\\mathbf{DI}_{p}\\setminus \\mathbf{BA}$ is of full Hausdorff dimension, as well as proving some results about the size of the difference $\\mathbf{DI}_{p_1}\\setminus \\mathbf{DI}_{p_2}$. To be precise, we show that the set difference of Dirichlet improvable numbers in Euclidean norm ($p=2$) minus Dirichlet improvable numbers in taxicab norm ($p=1$) and vice versa, that is $\\mathbf{DI}_{2}\\setminus \\mathbf{DI}_{1}$ and $\\mathbf{DI}_{1}\\setminus \\mathbf{DI}_{2}$, are of full Hausdorff dimension. We also find all values of $p$, for which the set $\\mathbf{DI}_p^c\\cap\\mathbf{BA}$ has full Hausdorff dimension. Finally, our characterization result implies that the number $e$ satisfies $e\\in \\mathbf{DI}_p$ if and only if $p\\in(1,2)\\cup(p_0,\\infty)$ for some special constant $p_0\\approx2.57$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-12T14:49:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J83",
      "11H06",
      "11J70"
    ],
    "keywords": [
      "full hausdorff dimension",
      "dirichlet improvability",
      "characterization result implies",
      "regular continued fraction expansions",
      "minus dirichlet improvable numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}