{
  "id": "2308.08867",
  "version": "v1",
  "published": "2023-08-17T08:56:22.000Z",
  "updated": "2023-08-17T08:56:22.000Z",
  "title": "Sum-product phenomenon in quotients of rings of algebraic integers",
  "authors": [
    "Jincheng Tang",
    "Xin Zhang"
  ],
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "We obtain a bounded generation theorem over $\\mathcal O/\\mathfrak a$, where $\\mathcal O$ is the ring of integers of a number field and $\\mathfrak a$ a general ideal of $\\mathcal O$. This addresses a conjecture of Salehi-Golsefidy. Along the way, we obtain nontrivial bounds for additive character sums over $\\mathcal O/\\mathcal P^n$ for a prime ideal $\\mathcal P$ with the aid of certain sum-product estimates.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-17T08:56:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "algebraic integers",
      "sum-product phenomenon",
      "bounded generation theorem",
      "prime ideal",
      "additive character sums"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}