{
  "abstract": "We prove Fuglede's conjecture for finite cyclic groups of square-free order. Specifically, if \\(N\\) is square-free and \\(A\\subseteq \\mathbb Z_N\\) is nonempty with \\(|A|=m\\), then \\(A\\) is spectral if and only if it tiles \\(\\mathbb Z_N\\); moreover, these conditions hold precisely when \\(m\\mid N\\) and \\(A\\) is a complete set of representatives modulo \\(m\\). This gives a complete structural description of spectral and tiling sets in \\(\\mathbb Z_N\\). The key new ingredient is a prime-coordinate elimination theorem: if \\(P\\nmid nm\\) is an odd prime, then every \\(m\\)-point spectral pair in \\(\\mathbb Z_n\\times\\mathbb Z_P\\) projects injectively to a spectral pair in \\(\\mathbb Z_n\\). Its proof combines character-matrix orthogonality, cyclotomic coefficient comparison, and a \\(P\\)-adic divisibility argument for associated Gram matrices. Iterating this reduction yields the result for arbitrary square-free \\(N\\), with no restriction on the number or relative sizes of its prime factors.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02218",
  "author_total": 2,
  "authors": [
    {
      "name": "Hu Tan"
    },
    {
      "name": "Ying Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/86dd30a0n40fd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02218",
  "fulltext_url": null,
  "id": "86dd30a0n40fd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02218",
  "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.CO"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7220469715",
  "pdf_url": "https://arxiv.org/pdf/2610.02218",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/86dd30a0n40fd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02218",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02218"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Fuglede's Conjecture for Cyclic Groups of Square-free Order",
  "title_text": "Fuglede's Conjecture for Cyclic Groups of Square-free Order",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/86dd30a0n40fd",
  "venue": "arXiv",
  "versions": []
}