{
  "abstract": "We construct explicit translational tiles without spectra in three finite abelian $p$-groups. The first is a $64$-point subset of $\\Z_4^4\\times\\Z_2^2$. The other two are a $512$-point subset of $\\F_2^{13}$ and a $2187$-point subset of $\\F_3^9$. Consequently, the tile-to-spectral implication fails for finite abelian $p$-groups, and it already fails within the class of elementary abelian groups for both $p=2$ and $p=3$. Two elementary mechanisms organize the examples. A two-layer obstruction turns a spectral non-tile with two suitable tiling complements into a tile without a spectrum. A fiber--clique obstruction converts a family of tiling complements with controlled common Fourier zeros into an elementary abelian counterexample. All coordinate data are included. The finite claims are certified by three short, self-contained programs using exact integer arithmetic and exhaustive searches; the accompanying source files recompute every assertion used in the proofs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00087",
  "author_total": 2,
  "authors": [
    {
      "name": "Shilei Fan"
    },
    {
      "name": "Mamateli Kadir"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/36wfvrcaqn7hy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00087",
  "fulltext_url": null,
  "id": "36wfvrcaqn7hy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00087",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7206176960",
  "pdf_url": "https://arxiv.org/pdf/2609.00087",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/36wfvrcaqn7hy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.00087",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00087"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Translational tiles without spectra in finite abelian p-groups",
  "title_text": "Translational tiles without spectra in finite abelian p-groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/36wfvrcaqn7hy",
  "venue": "arXiv",
  "versions": []
}