{
  "id": "2109.03929",
  "version": "v1",
  "published": "2021-09-08T21:02:57.000Z",
  "updated": "2021-09-08T21:02:57.000Z",
  "title": "Independence inheritance and Diophantine approximation for systems of linear forms",
  "authors": [
    "Demi Allen",
    "Felipe A. Ramirez"
  ],
  "comment": "27 pages, 1 figure",
  "categories": [
    "math.NT"
  ],
  "abstract": "The classical Khintchine-Groshev theorem is a generalization of Khintchine's theorem on simultaneous Diophantine approximation, from approximation of points in $\\mathbb R^m$ to approximation of systems of linear forms in $\\mathbb R^{nm}$. In this paper, we present an inhomogeneous version of the Khintchine-Groshev theorem which does not carry a monotonicity assumption when $nm>2$. Our results bring the inhomogeneous theory almost in line with the homogeneous theory, where it is known by a result of Beresnevich and Velani (2010) that monotonicity is not required when $nm>1$. That result resolved a conjecture of Beresnevich, Bernik, Dodson, and Velani (2009), and our work resolves almost every case of the natural inhomogeneous generalization of that conjecture. Regarding the two cases where $nm=2$, we are able to remove monotonicity by assuming extra divergence of a measure sum, akin to a linear forms version of the Duffin-Schaeffer conjecture. When $nm=1$ it is known by work of Duffin and Schaeffer (1941) that the monotonicity assumption cannot be dropped. The key new result is an independence inheritance phenomenon; the underlying idea is that the sets involved in the $((n+k)\\times m)$-dimensional Khintchine-Groshev theorem ($k\\geq 0$) are always $k$-levels more probabilistically independent than the sets involved the $(n\\times m)$-dimensional theorem. Hence, it is shown that Khintchine's theorem itself underpins the Khintchine-Groshev theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-08T21:02:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "diophantine approximation",
      "khintchines theorem",
      "monotonicity assumption",
      "dimensional khintchine-groshev theorem",
      "independence inheritance phenomenon"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}