{
  "id": "2506.04187",
  "version": "v2",
  "published": "2025-06-04T17:32:03.000Z",
  "updated": "2025-06-19T09:01:00.000Z",
  "title": "Toward Khintchine's theorem with a moving target: extra divergence or finitely centered target",
  "authors": [
    "Gilbert Michaud",
    "Felipe A. Ramírez"
  ],
  "comment": "22 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Sz{\\\"u}sz's inhomogeneous version (1958) of Khintchine's theorem (1924) gives conditions on $\\psi:\\mathbb{N}\\to\\mathbb{R}_{\\geq 0}$ under which for almost every real number $\\alpha$ there exist infinitely many rationals $p/q$ such that \\begin{equation*} \\lvert\\alpha - \\frac{p+\\gamma}{q}\\rvert < \\frac{\\psi(q)}{q}, \\end{equation*} where $\\gamma\\in\\mathbb{R}$ is some fixed inhomogeneous parameter. It is often interpreted as a statement about visits of $q\\alpha\\,(\\bmod 1)$ to a shrinking target centered around $\\gamma\\,(\\bmod 1)$, viewed in $\\mathbb{R}/\\mathbb{Z}$. Hauke and the second author have conjectured that Sz{\\\"u}sz's result continues to hold if the target is allowed to move as well as shrink, that is, if the inhomogeneous parameter $\\gamma$ is allowed to depend on the denominator $q$ of the approximating rational. We show that the conjecture holds under an ``extra divergence'' assumption on $\\psi$. We also show that it holds when the inhomogeneous parameter's movement is constrained to a finite set. As a byproduct, we obtain a finite-colorings version of the inhomogeneous Khintchine theorem, giving rational approximations with monochromatic denominators.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-06-19T09:01:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J83",
      "11K60"
    ],
    "keywords": [
      "khintchines theorem",
      "finitely centered target",
      "extra divergence",
      "moving target",
      "real number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}