{
  "id": "1511.05874",
  "version": "v1",
  "published": "2015-11-18T16:55:03.000Z",
  "updated": "2015-11-18T16:55:03.000Z",
  "title": "On polynomial configurations in fractal sets",
  "authors": [
    "Kevin Henriot",
    "Izabella Laba",
    "Malabika Pramanik"
  ],
  "comment": "38 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "We show that subsets of $\\mathbb{R}^n$ of large enough Hausdorff and Fourier dimension contain polynomial patterns of the form \\begin{align*} ( x ,\\, x + A_1 y ,\\, \\dots,\\, x + A_{k-1} y ,\\, x + A_k y + Q(y) e_n ), \\quad x \\in \\mathbb{R}^n,\\ y \\in \\mathbb{R}^m, \\end{align*} where $A_i$ are real $n \\times m$ matrices, $Q$ is a real polynomial in $m$ variables and $e_n = (0,\\dots,0,1)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-18T16:55:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractal sets",
      "polynomial configurations",
      "fourier dimension contain polynomial patterns",
      "real polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151105874H"
    }
  }
}