{
  "id": "1510.07596",
  "version": "v1",
  "published": "2015-10-26T19:05:38.000Z",
  "updated": "2015-10-26T19:05:38.000Z",
  "title": "Salem sets without arithmetic progressions",
  "authors": [
    "Pablo Shmerkin"
  ],
  "comment": "11 pages, no figures",
  "categories": [
    "math.CA",
    "math.NT"
  ],
  "abstract": "We construct Salem sets in $\\mathbb{R}/\\mathbb{Z}$ of any dimension (including $1$) which do not contain any arithmetic progressions of length $3$. Moreover, the sets can be taken to be Ahlfors regular if the dimension is less than $1$, and the measure witnessing the Fourier decay can be taken to be Frostman in the case of dimension $1$. This is in sharp contrast to the situation in the discrete setting (where Fourier uniformity is well known to imply existence of progressions), and helps clarify a result of {\\L}aba and Pramanik on pseudo-random subsets of the real line which do contain progressions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-26T19:05:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B25",
      "28A78",
      "42A38",
      "42A61"
    ],
    "keywords": [
      "arithmetic progressions",
      "construct salem sets",
      "contain progressions",
      "real line",
      "pseudo-random subsets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151007596S"
    }
  }
}