{
  "id": "1602.01634",
  "version": "v1",
  "published": "2016-02-04T11:14:39.000Z",
  "updated": "2016-02-04T11:14:39.000Z",
  "title": "Salem sets, equidistribution and arithmetic progressions",
  "authors": [
    "Paul Potgieter"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "Arithmetic progressions of length $3$ may be found in compact subsets of the reals that satisfy certain Fourier-dimensional as well as Hausdorff-dimensional requirements. It has been shown that a very similar result holds in the integers under analogous conditions, with Fourier dimension being replaced by the decay of a discrete Fourier transform. By using a construction of Salem's, we show that this correspondence can be made more precise. Specifically, we show that a subset of the integers can be mapped to a compact subset of the continuum in a way which preserves equidistribution properties as well as arithmetic progressions of arbitrary length, and vice versa. We use the method to characterise Salem sets in $\\mathbb{R}$ through discrete, equidistributed approximations. Finally, we discuss how this method sheds light on the generation of Salem sets by stationary stochastic processes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-04T11:14:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B05",
      "11B25",
      "28A78",
      "42A38",
      "43A46",
      "60G17"
    ],
    "keywords": [
      "arithmetic progressions",
      "compact subset",
      "discrete fourier transform",
      "similar result holds",
      "preserves equidistribution properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160201634P"
    }
  }
}