{
  "id": "2310.18624",
  "version": "v1",
  "published": "2023-10-28T07:34:20.000Z",
  "updated": "2023-10-28T07:34:20.000Z",
  "title": "Block partitions in higher dimensions",
  "authors": [
    "Endre Csóka"
  ],
  "categories": [
    "math.CO",
    "math.GT"
  ],
  "abstract": "Consider a set $X\\subseteq \\mathbb{R}^d$ which is 1-dense, namely, it intersects every unit ball. We show that we can get from any point to any other point in $\\mathbb{R}^d$ in $n$ steps so that the intermediate points are in $X$, and the discrepancy of the step vectors is at most $2\\sqrt{2}$, or formally, $$\\sup\\limits_{\\substack{n\\in \\mathbb{Z}^+,\\ t\\in \\mathbb{R}^d\\\\ X\\text{ is 1-dense}}}\\,\\, \\inf\\limits_{\\substack{p_1,\\ldots, p_{n-1}\\in X\\\\ p_0=\\underline{0},\\ p_n=t}}\\,\\, \\max\\limits_{0\\leq i<j<n} \\big\\|(p_{i+1}-p_i)-(p_{j+1}-p_j)\\big\\|\\leq 2\\sqrt{2}.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-28T07:34:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "higher dimensions",
      "block partitions",
      "intermediate points",
      "unit ball"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}