{
  "abstract": "Cyclotomic divisibility imposes strong prime-power structure on integer tiles. We study unsupported cyclotomic divisors: mixed-order divisors for which none of the prime-power components of the order divides the mask, although every prime in the order divides the tile cardinality. Kiss, Łaba, Marshall and Somlai asked whether such a phenomenon can occur in the three-prime setting. We prove that unsupported cyclotomic divisors already occur for periods with three distinct prime factors. For primes \\(p<q<r\\), we characterize the square-period case: an unsupported factor \\(Φ_{pqr}\\) occurs in a tiling of \\(\\ZZ_{(pqr)^2}\\) if and only if \\(r\\in\\langle p,q\\rangle\\), and every such tile lies in a single residue class modulo \\(r\\). Among cyclic tilings with the unsupported order dividing the specified modulus, the smallest modulus is \\(180\\); if the order has three distinct prime factors, it is \\(900\\). An Apéry-set construction gives examples for every triple at period \\(p^2q^2r^3\\).The proof of our results combines Fourier rigidity, a three-cylinder decomposition, and an integer mass obstruction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06677",
  "author_total": 2,
  "authors": [
    {
      "name": "Hu Tan"
    },
    {
      "name": "Ying Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cyf5x92eb6p1m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06677",
  "fulltext_url": null,
  "id": "cyf5x92eb6p1m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06677",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7212084708",
  "pdf_url": "https://arxiv.org/pdf/2609.06677",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cyf5x92eb6p1m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06677",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06677"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Unsupported Cyclotomic Divisors in Three-Prime Integer Tilings",
  "title_text": "Unsupported Cyclotomic Divisors in Three-Prime Integer Tilings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/cyf5x92eb6p1m",
  "venue": "arXiv",
  "versions": []
}