{
  "id": "2211.15847",
  "version": "v1",
  "published": "2022-11-29T00:48:46.000Z",
  "updated": "2022-11-29T00:48:46.000Z",
  "title": "A counterexample to the periodic tiling conjecture",
  "authors": [
    "Rachel Greenfeld",
    "Terence Tao"
  ],
  "comment": "13 figures",
  "categories": [
    "math.CO",
    "math.DS"
  ],
  "abstract": "The periodic tiling conjecture asserts that any finite subset of a lattice $\\mathbb{Z}^d$ which tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture for sufficiently large $d$, which also implies a disproof of the corresponding conjecture for Euclidean spaces $\\mathbb{R}^d$. In fact, we also obtain a counterexample in a group of the form $\\mathbb{Z}^2 \\times G_0$ for some finite abelian $2$-group $G_0$. Our methods rely on encoding a \"Sudoku puzzle\" whose rows and other non-horizontal lines are constrained to lie in a certain class of \"$2$-adically structured functions\", in terms of certain functional equations that can be encoded in turn as a single tiling equation, and then demonstrating that solutions to this Sudoku puzzle exist but are all non-periodic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-29T00:48:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C22",
      "52C23"
    ],
    "keywords": [
      "counterexample",
      "sudoku puzzle",
      "periodic tiling conjecture asserts",
      "finite subset",
      "fact tiles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}