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 $(\mu ,\Lambda )$ if $(\mu ,t\Lambda )$ is also a spectral pair. In this paper we consider the partial homogeneous Cantor measure ${\mu }_{N^r,\mathcal{D}}$ 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.