{
  "abstract": "We study the complete number problem for a class of self-similar spectral measures on the real line. We say that a real number $t$ is complete for a spectral pair $(μ,Λ)$ if $(μ,tΛ)$ is also a spectral pair. In this paper we consider the partial homogeneous Cantor measure $μ_{N^r,\\mathcal D}$ and its canonical spectrum. Our main result fully characterizes all prime powers $p^n$ that are complete for $p>N$. In particular, we obtain a necessary and sufficient condition for an arbitrary prime $p$ to be complete. {We further establish a connection between the problem and certain Diophantine equations}, including the Nagell--Ljunggren equation, and obtain complete number results for a class of related models. As applications, known solutions of the Nagell--Ljunggren equation provide complete and incomplete prime-power examples.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04682",
  "author_total": 3,
  "authors": [
    {
      "name": "Zi-Chao Chi"
    },
    {
      "name": "Jian-Feng Lu"
    },
    {
      "name": "He-Xiang Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bevw9z2a4w6qm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04682",
  "fulltext_url": null,
  "id": "bevw9z2a4w6qm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04682",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7211880573",
  "pdf_url": "https://arxiv.org/pdf/2609.04682",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bevw9z2a4w6qm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2609.04682",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04682"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Primes are Complete for a Class of Cantor Spectral Pair",
  "title_text": "Primes are Complete for a Class of Cantor Spectral Pair",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/bevw9z2a4w6qm",
  "venue": "arXiv",
  "versions": []
}