{
  "id": "1308.2919",
  "version": "v5",
  "published": "2013-08-13T17:11:30.000Z",
  "updated": "2015-01-18T03:31:56.000Z",
  "title": "Long progressions in sets of fractional dimension",
  "authors": [
    "Marc Carnovale"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We demonstrate $k+1$-term arithmetic progressions in certain subsets of the real line whose \"higher-order Fourier dimension\" is sufficiently close to 1. This Fourier dimension, introduced in previous work, is a higher-order (in the sense of Additive Combinatorics and uniformity norms) extension of the Fourier dimension of Geometric Measure Theory, and can be understood as asking that the uniformity norm of a measure, restricted to a given scale, decay as the scale increases. We further obtain quantitative information about the size and $L^p$ regularity of the set of common distances of the artihmetic progressions contained in the subsets of $\\mathbb{R}$ under consideration.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-08-16T13:57:45.000Z",
      "abstract": "We demonstrate $k+1$-term arithmetic progressions in certain subsets of the real line whose \"higher-order Fourier dimension\" is sufficiently close to 1. This Fourier dimension, introduced in previous work, is a higher-order (in the sense of Additive Combinatorics and uniformity norms) extension of the Fourier dimension of Geometric Measure Theory, and can be understood as asking that the uniformity norm of a measure, restricted to a given scale, decay as the scale increases. In fact, we obtain our results for measures supported in $\\R^d$, and for scaled and translated images of any collection of sufficiently \"distinct\" points $b_0,...,b_k$ for which a kind of multiple-recurrence is currently known; this includes, for instance, any $k+1$ points on the integer lattice $\\Z^d$ with pairwise distinct coordinates. We further obtain quantitative information about the size and $L^p$ regularity of the set of common distances of the artihmetic progressions contained in the subsets of $\\R$ under consideration, or in the case of sets in $\\R^d$, information on the size of the set of dilations which move a fixed $k+1$-point configuration inside of our set.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2015-01-18T03:31:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A78",
      "42A32",
      "42A38",
      "42A45",
      "11B25",
      "28A75"
    ],
    "keywords": [
      "fractional dimension",
      "long progressions",
      "uniformity norm",
      "point configuration inside",
      "term arithmetic progressions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.2919C"
    }
  }
}