{
  "id": "2312.01986",
  "version": "v1",
  "published": "2023-12-04T16:04:03.000Z",
  "updated": "2023-12-04T16:04:03.000Z",
  "title": "Quantitative inhomogeneous Diophantine approximation for systems of linear forms",
  "authors": [
    "Manuel Hauke"
  ],
  "comment": "13 pages, comments are highly appreciated!",
  "categories": [
    "math.NT",
    "math.PR"
  ],
  "abstract": "The inhomogeneous Khintchine-Groshev Theorem is a classical generalization of Khintchine's Theorem in Diophantine approximation, by approximating points in $\\mathbb{R}^m$ by systems of linear forms in $n$ variables. Analogous to the question considered by Duffin and Schaeffer for Khintchine's Theorem (which is the case $m = n = 1$), the question arises for which $m,n$ the monotonicity can be safely removed. If $m = n = 1$, it is known that monotonicity is needed. Recently, Allen and Ramirez showed that for $mn \\geq 3$, the monotonicity assumption is unnecessary, conjecturing this to also hold when $mn = 2$. In this article, we confirm this conjecture for the case $(m,n)=(1,2)$ whenever the inhomogeneous parameter is a non-Liouville irrational number. Furthermore, under mild assumptions on the approximation function, we show an asymptotic formula (with almost square-root cancellation), which is not even known for homogeneous approximation. The proof makes use of refined overlap estimates in the 1-dimensional setting, which may have other applications including the inhomogeneous Duffin-Schaeffer conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-04T16:04:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J20",
      "11K60",
      "11J83",
      "11J71"
    ],
    "keywords": [
      "quantitative inhomogeneous diophantine approximation",
      "linear forms",
      "khintchines theorem",
      "non-liouville irrational number",
      "monotonicity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}