{
  "id": "2103.11418",
  "version": "v1",
  "published": "2021-03-21T15:12:29.000Z",
  "updated": "2021-03-21T15:12:29.000Z",
  "title": "On the Mattila-Sjölin distance theorem for product sets",
  "authors": [
    "Doowon Koh",
    "Thang Pham",
    "Chun-Yen Shen"
  ],
  "comment": "8 pages",
  "categories": [
    "math.CA",
    "math.CO"
  ],
  "abstract": "Let $A$ be a compact set in $\\mathbb{R}$, and $E=A^d\\subset \\mathbb{R}^d$. We know from the Mattila-Sj\\\"olin's theorem if $\\dim_H(A)>\\frac{d+1}{2d}$, then the distance set $\\Delta(E)$ has non-empty interior. In this paper, we show that the threshold $\\frac{d+1}{2d}$ can be improved whenever $d\\ge 5$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-21T15:12:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "mattila-sjölin distance theorem",
      "product sets",
      "compact set",
      "distance set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}