Primes are Complete for a Class of Cantor Spectral Pair

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,𝒟 and its canonical spectrum. Our main result fully characterizes all prime powers pⁿ 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.

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BibTeX RIS

BibTeX

@misc{chi2026primes,
  title = {{Primes are Complete for a Class of Cantor Spectral Pair}},
  author = {Zi-Chao Chi and Jian-Feng Lu and He-Xiang Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.04682},
  eprint = {2609.04682},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04682},
}

RIS

TY  - UNPB
TI  - Primes are Complete for a Class of Cantor Spectral Pair
AU  - Zi-Chao Chi
AU  - Jian-Feng Lu
AU  - He-Xiang Wang
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.04682
UR  - https://arxiv.org/abs/2609.04682
AB  - 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,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.
ER  -

Abstract

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