{
  "id": "2010.00119",
  "version": "v1",
  "published": "2020-09-30T21:52:17.000Z",
  "updated": "2020-09-30T21:52:17.000Z",
  "title": "On the size of $A+λA$ for algebraic $λ$",
  "authors": [
    "Dmitry Krachun",
    "Fedor Petrov"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "For a finite set $A\\subset \\mathbb{R}$ and real $\\lambda$, let $A+\\lambda A:=\\{a+\\lambda b :\\, a,b\\in A\\}$. Combining a structural theorem of Freiman on sets with small doubling constants together with a discrete analogue of Pr\\'ekopa--Leindler inequality we prove a lower bound $|A+\\sqrt{2} A|\\geq (1+\\sqrt{2})^2|A|-O({|A|}^{1-\\varepsilon})$ which is essentially tight. We also formulate a conjecture about the value of $\\liminf |A+\\lambda A|/|A|$ for an arbitrary algebraic $\\lambda$. Finally, we prove a tight lower bound on the Lebesgue measure of $K+\\mathcal{T} K$ for a given linear operator $\\mathcal{T}\\in \\operatorname{End}(\\mathbb{R}^d)$ and a compact set $K\\subset \\mathbb{R}^d$ with fixed measure. This continuous result supports the conjecture and yields an upper bound in it.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-30T21:52:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tight lower bound",
      "compact set",
      "structural theorem",
      "linear operator",
      "lebesgue measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}