{
  "id": "2106.15213",
  "version": "v1",
  "published": "2021-06-29T09:54:44.000Z",
  "updated": "2021-06-29T09:54:44.000Z",
  "title": "Values of Inhomogeneous Forms at S-integral points",
  "authors": [
    "Anish Ghosh",
    "Jiyoung Han"
  ],
  "comment": "27 pages",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "We prove effective versions of Oppenheim's conjecture for generic inhomogeneous forms in the S-arithmetic setting. We prove an effective result for fixed rational shifts and generic forms and we also prove a result where both the quadratic form and the shift are allowed to vary. In order to do so, we prove analogues of Rogers' moment formulae for $S$-arithmetic congruence quotients as well as for the space of affine lattices. We believe the latter results to be of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-29T09:54:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "s-integral points",
      "arithmetic congruence quotients",
      "quadratic form",
      "independent interest",
      "fixed rational shifts"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}