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

Let $μ$ be a self-similar measure generated by iterated function system of four maps of equal contraction ratio $0<ρ<1$. We study when $μ$ is a spectral measure which means that it admits an exponential orthonormal basis $\{e^{2πi λx}\}_{λ\inΛ}$ in $L^2(μ)$. 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 Łaba-Wang conjecture concerning when the self-similar measures are spectral in general cases.

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