{
  "id": "1901.08734",
  "version": "v1",
  "published": "2019-01-25T04:27:32.000Z",
  "updated": "2019-01-25T04:27:32.000Z",
  "title": "Fuglede's conjecture fails in $\\mathbb{Z}_{p}^{4}$ for odd primes",
  "authors": [
    "Samuel Ferguson",
    "Nat Sothanaphan"
  ],
  "comment": "5 pages",
  "categories": [
    "math.NT",
    "math.CA"
  ],
  "abstract": "We disprove Fuglede's conjecture in $\\mathbb{Z}_p^4$ for all odd primes $p$, extending the result of Aten et al. that it fails in $\\mathbb{Z}_4^p$ for primes $p \\equiv 3$ (mod $4$) and in $\\mathbb{Z}_p^5$ for all odd primes $p$. We also prove the conjecture in $\\mathbb{Z}_2^4$, resolving all cases of a vector space of dimension four over prime fields. We show that the method of proof fails for $\\mathbb{Z}_{2}^{5}$; nevertheless, the authors have verified that the conjecture holds for $\\mathbb{Z}_{2}^{5}$ and $\\mathbb{Z}_{2}^{6}$ using a computer program. Finally, we slightly modify Terry Tao's counterexample to show that the conjecture fails in $\\mathbb{Z}_2^{10}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-25T04:27:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A15",
      "43A40",
      "43A70",
      "43A75"
    ],
    "keywords": [
      "fugledes conjecture fails",
      "odd primes",
      "slightly modify terry taos counterexample",
      "disprove fugledes conjecture",
      "prime fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}