{
  "id": "2203.09827",
  "version": "v1",
  "published": "2022-03-18T10:05:10.000Z",
  "updated": "2022-03-18T10:05:10.000Z",
  "title": "Sums of linear transformations",
  "authors": [
    "David Conlon",
    "Jeck Lim"
  ],
  "comment": "24 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "We show that if $\\mathcal{L}_1$ and $\\mathcal{L}_2$ are linear transformations from $\\mathbb{Z}^d$ to $\\mathbb{Z}^d$ satisfying certain mild conditions, then, for any finite subset $A$ of $\\mathbb{Z}^d$, $$|\\mathcal{L}_1 A+\\mathcal{L}_2 A|\\geq \\left(|\\det(\\mathcal{L}_1)|^{1/d}+|\\det(\\mathcal{L}_2)|^{1/d}\\right)^d|A|- o(|A|).$$ This result corrects and confirms the two-summand case of a conjecture of Bukh and is best possible up to the lower-order term for many choices of $\\mathcal{L}_1$ and $\\mathcal{L}_2$. As an application, we prove a lower bound for $|A + \\lambda \\cdot A|$ when $A$ is a finite set of real numbers and $\\lambda$ is an algebraic number. In particular, when $\\lambda$ is of the form $(p/q)^{1/d}$ for some $p, q, d \\in \\mathbb{N}$, each taken as small as possible for such a representation, we show that $$|A + \\lambda \\cdot A| \\geq (p^{1/d} + q^{1/d})^d |A| - o(|A|).$$ This is again best possible up to the lower-order term and extends a recent result of Krachun and Petrov which treated the case $\\lambda = \\sqrt{2}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-18T10:05:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D99",
      "11B13",
      "11B75",
      "11B30"
    ],
    "keywords": [
      "linear transformations",
      "lower-order term",
      "mild conditions",
      "finite subset",
      "result corrects"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}