{
  "id": "1504.02410",
  "version": "v1",
  "published": "2015-04-09T18:19:01.000Z",
  "updated": "2015-04-09T18:19:01.000Z",
  "title": "Sets of recurrence as bases for the positive integers",
  "authors": [
    "Jakub Konieczny"
  ],
  "comment": "33 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "We study sets of the form $A = \\big\\{ n \\in \\mathbb N \\big| \\lVert p(n) \\rVert_{\\mathbb R / \\mathbb Z} \\leq \\varepsilon(n) \\big\\}$ for various real valued polynomials $p$ and decay rates $\\varepsilon$. In particular, we ask when such sets are bases of finite order for the positive integers. We show that generically, $A$ is a basis of order $2$ when $\\operatorname{deg} p \\geq 3$, but not when $\\operatorname{deg} p = 2$, although then $A + A$ still has asymptotic density $1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-09T18:19:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "positive integers",
      "recurrence",
      "real valued polynomials",
      "decay rates",
      "finite order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150402410K"
    }
  }
}