Spectrality of certain Moran measures with three-element digit sets

Let Dn={0,an,bn}={0,1,2}(mod3),pn∈3Z+, n≥1, satisfy supn≥1⁡max⁡{|an|,|bn|}pn<∞. It is well-known that there exists a unique Borel probability measure μ{pn},{Dn} generated by the following infinite convolution productμ{pn},{Dn}=δp1−1D1⁎δ(p1p2)−1D2⁎⋯ in the weak convergence. In this paper, we give some conditions to ensure that there exists a discrete set Λ such that the exponential function system {e2πiλx}λ∈Λ forms an orthonormal basis for L2(μ{pn},{Dn}).