{
  "id": "1911.13275",
  "version": "v1",
  "published": "2019-11-29T18:22:14.000Z",
  "updated": "2019-11-29T18:22:14.000Z",
  "title": "On strong infinite Sidon and $B_h$ sets and random sets of integers",
  "authors": [
    "David Fabian",
    "Juanjo Rué",
    "Christoph Spiegel"
  ],
  "comment": "15 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "A set of integers $S \\subset \\mathbb{N}$ is an $\\alpha$-strong Sidon set if the pairwise sums of its elements are far apart by a certain measure depending on $\\alpha$, more specifically if $| (x+w) - (y+z) | \\geq \\max \\{ x^{\\alpha},y^{\\alpha},z^{\\alpha},w^\\alpha \\}$ for every $x,y,z,w \\in S$ satisfying $\\{x,w\\} \\neq \\{y,z\\}$. We obtain a new lower bound for the growth of $\\alpha$-strong infinite Sidon sets when $0 \\leq \\alpha < 1$. We also further extend that notion in a natural way by obtaining the first non-trivial bound for $\\alpha$-strong infinite $B_h$ sets. In both cases, we study the implications of these bounds for the density of, respectively, the largest Sidon or $B_h$ set contained in a random infinite subset of $\\mathbb{N}$. Our theorems improve on previous results by Kohayakawa, Lee, Moreira and R\\\"odl.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-29T18:22:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random sets",
      "strong infinite sidon sets",
      "first non-trivial bound",
      "random infinite subset",
      "strong sidon set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}