Spectral eigenvalue problem of Cantor measures and Artin's primitive root conjecture

The eigenvalue problem for a probability measure $\mu$ with compact support in $\mathbb{R}$ is whether there exist a countable set $\mathrm{\Lambda }$ and a nonzero real $t\mathrm{\ne }1$ such that both $\mathrm{\Lambda }$ and $t\mathrm{\Lambda }$ are spectra of $\mu$, that is, the family $$E_{a\mathrm{\Lambda }}=\{e^{-2\pi ia\lambda x}:\lambda \in \mathrm{\Lambda }\}$$ is an orthonormal base for $L^2(\mu )$ for $a=1,t$. The eigenvalue problem was discovered independently by Strichartz \cite{Str00}, Łaba and Wang \cite{LW02} for the Cantor measures ${\mu }_{4,\{0,1\}}$ and ${\mu }_{6,\{0,1,2\}}$, respectively. In this paper, we investigate the spectral eigenvalue problem for the general spectral Cantor measure ${\mu }_{b,\mathcal{D}}$. This topic is naturally related to elementary number theory. Unexpectedly, however, our main results depend on the theory of integers, especially Artin’s primitive root conjecture. To some extent, our results suggest that Artin’s primitive root conjecture may hold and confirms some viewpoints implied by Minkowski in \cite{Min57}.