Classification of spectral self-similar measures with four-digit elements

Abstract: Let $\mu$ be a self-similar measure generated by iterated function system of four maps of equal contraction ratio $0<\rho<1$. We study when $\mu$ is a spectral measure which means that it admits an exponential orthonormal basis ${e^{2\pi i \lambda x}}_{\lambda\in\Lambda}$ in $L^2(\mu)$. By combining previous results of many authors and a careful study of some new cases, we completely classify all spectral self-similar measures with four maps. Moreover, the case allows us to propose a modified {\L}aba-Wang conjecture concerning when the self-similar measures are spectral in general cases.