{
  "id": "1511.06010",
  "version": "v1",
  "published": "2015-11-18T22:42:01.000Z",
  "updated": "2015-11-18T22:42:01.000Z",
  "title": "A Roth type theorem for dense subsets of $\\mathbb{R}^d$",
  "authors": [
    "Brian Cook",
    "Ákos Magyar",
    "Malabika Pramanik"
  ],
  "categories": [
    "math.CO",
    "math.CA",
    "math.NT"
  ],
  "abstract": "Let $A\\subset\\mathbb{R}^d$ with $d>8$ be a measurable set of positive upper density. We prove that there exists a $\\lambda_0=\\lambda_0(A)$ such for all $\\lambda\\geq\\lambda_0$ there are $x,y\\in\\mathbb{R}^d$ such that $\\{x,x+y,x+2y\\}\\subset A$ and $|y|_4=\\lambda$, where $|y|_4=(\\sum_i y_i^4)^{1/4}$ is the $l^4$-norm of a point $y=(y_1,\\ldots,y_d)\\in\\mathbb{R}^d$. This means that dense subsets of $\\mathbb{R}^d$ contain 3-term progressions of arbitrary large gaps when the gap size is measured by the $l^4$-metric. This is known to be false in the ordinary $l^2$-metric and one of the goals of this note is to understand this phenomenon.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-18T22:42:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D10",
      "42A45",
      "11D72"
    ],
    "keywords": [
      "roth type theorem",
      "dense subsets",
      "arbitrary large gaps",
      "positive upper density",
      "progressions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151106010C"
    }
  }
}