{
  "id": "1904.11537",
  "version": "v1",
  "published": "2019-04-25T18:55:11.000Z",
  "updated": "2019-04-25T18:55:11.000Z",
  "title": "A counterexample to Fuglede's conjecture in $(\\mathbb{Z}/p\\mathbb{Z})^4$ for all odd primes",
  "authors": [
    "Sam Mattheus"
  ],
  "comment": "This result was found simultaneously and independently with [arXiv:1901.08734]. However, it contained an error which was fixed afterwards (again independently). So although not suitable for publication, we record it here as it gives a more geometrical approach to more or less the same construction",
  "categories": [
    "math.CO",
    "math.CA"
  ],
  "abstract": "In this short note we construct a spectral, non-tiling set of size $2p$ in $(\\mathbb{Z}/p\\mathbb{Z})^4$, $p$ odd prime. This example complements a previous counterexample in [arXiv:1509.01090], which existed only for $p \\equiv 3 \\pmod{4}$. On the contrary we show that the conjecture does hold in $(\\mathbb{Z}/2\\mathbb{Z})^4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-25T18:55:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "odd prime",
      "fugledes conjecture",
      "counterexample",
      "short note"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}