Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Chi, Zi-Chao, Lu, Jian-Feng, Wang, He-Xiang
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G4105740819, G7884333606, G8827021649
OpenAlex ID: W7211880573