The spectral eigenvalue set and Beurling dimension on self-similar measures

In this work, we study harmonic analysis in self-similar measures. A set $A$ is called a spectral eigenvalue set of μ if there exists $\Lambda \subset R$ such that the family $\{a\Lambda :a\in A\}$ are spectra for μ. Given a Hadamard triple $(q,D,L)$, Łaba and Wang [33] proved that the associated self-similar measure ${\mu }_{q,D}$ is spectral. We establish that the set$T=\{t\in Z:(q,D,tL\left)\text{ forms a Hadamard triple}\right\}\supseteq \{p\in Z:\gcd (p,q)=1\}$ constitutes a spectral eigenvalue set for ${\mu }_{q,D}$. Furthermore, we demonstrate that for any prescribed Beurling dimension $s\in [0,\frac{\log \#D}{\log q}]$, the corresponding spectra have the cardinality of the continuum. This result provides a complete answer to the question posed by Kong, Li and Wang [30]. As an application, we characterize the eigenvalue sets for N-Bernoulli convolutions, proving that $A$ is an eigenvalue set if and only if $A\subseteq \frac{1}{T}T$ for some $T\in T$.