{
  "id": "2311.09399",
  "version": "v1",
  "published": "2023-11-15T21:52:15.000Z",
  "updated": "2023-11-15T21:52:15.000Z",
  "title": "Tight lower bound on $|A+λA|$ for algebraic integer $λ$",
  "authors": [
    "D. Krachun",
    "F. Petrov"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove an asymptotically tight lower bound on $|A+\\lambda A|$ for $A\\subset \\mathbb{C}$ and algebraic integer $\\lambda$. The proof combines strong version of Freiman's theorem, structural theorem on dense subsets of a hypercubic lattice and a generalisation of the continuous result on tight bound for the measure of $K+\\tau K$ for a compact subset $K\\subset \\mathbb{R}^d$ of unit Lebesgue measure and a fixed linear operator $\\tau\\colon\\mathbb{R}^d\\to \\mathbb{R}^d$, obtained in our previous work.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-15T21:52:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B13"
    ],
    "keywords": [
      "algebraic integer",
      "asymptotically tight lower bound",
      "unit lebesgue measure",
      "dense subsets",
      "fixed linear operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}