{
  "id": "2112.04144",
  "version": "v2",
  "published": "2021-12-08T06:50:24.000Z",
  "updated": "2022-01-27T22:27:08.000Z",
  "title": "On Hausdorff dimension in inhomogeneous Diophantine approximation over global function fields",
  "authors": [
    "Taehyeong Kim",
    "Seonhee Lim",
    "Frédéric Paulin"
  ],
  "comment": "53 pages",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "In this paper, we study inhomogeneous Diophantine approximation over the completion $K_v$ of a global function field $K$ (over a finite field) for a discrete valuation $v$, with affine algebra $R_v$. We obtain an effective upper bound for the Hausdorff dimension of the set \\[ \\mathbf{Bad}_A(\\epsilon)=\\left\\{\\boldsymbol{\\theta}\\in K_v^{\\,m} : \\liminf_{(\\mathbf{p},\\mathbf{q})\\in R_v^{\\,m} \\times R_v^{\\,n}, \\|\\mathbf{q}\\|\\to \\infty} \\|\\mathbf{q}\\|^n \\|A\\mathbf{q}-\\boldsymbol{\\theta}-\\mathbf{p}\\|^m \\geq \\epsilon \\right\\}, \\] of $\\epsilon$-badly approximable targets $\\boldsymbol{\\theta}\\in K_v^{\\,m}$ for a fixed matrix $A\\in\\mathscr{M}_{m,n}(K_v)$, using an effective version of entropy rigidity in homogeneous dynamics for an appropriate diagonal action on the space of $R_v$-grids. We further characterize matrices $A$ for which $\\mathbf{Bad}_A(\\epsilon)$ has full Hausdorff dimension for some $\\epsilon>0$ by a Diophantine condition of singularity on average. Our methods also work for the approximation using weighted ultrametric distances.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-01-27T22:27:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "global function field",
      "study inhomogeneous diophantine approximation",
      "appropriate diagonal action",
      "full hausdorff dimension",
      "affine algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}