Spectral Eigen-subspace and Tree Structure for a Cantor Measure

In this work we investigate the question of constructions of the possible Fourier bases $E(\mathrm{\Lambda })=\{e^{2\pi i\lambda x}:\lambda \in \mathrm{\Lambda }\}$ for the Hilbert space $L^2({\mu }_4)$, where ${\mu }_4$ is the standard middle-fourth Cantor measure and $\mathrm{\Lambda }$ is a countable discrete set. We show that the set \mathop \bigcap_{p\in 2\Z+1}\left\{Λ\subset \R: \text{$E(Λ)$ and $E(pΛ)$ are Fourier bases for $L^2(μ_4)$}\right\} has the cardinality of the continuum. We also give other characterizations on the orthonormal set of exponential functions being a basis for the space $L^2({\mu }_4)$ from the viewpoint of measure and dimension. Moreover, we provide a method of constructing explicit discrete set $\mathrm{\Lambda }$ such that $E(\mathrm{\Lambda })$ and its all odd scaling sets $E(\mathrm{\Lambda }),p\in 2\mathbb{Z}+1,$ are still Fourier bases for $L^2({\mu }_4)$.