Spectrality of infinite convolutions with three-element digit sets

Let (0< rho <1) and let ({a_n, b_n}_{n=1}^infty ) be a sequence of integers with bounded from upper and lower. Associated with them there exists a unique Borel probability measure (mu _{rho , {0, a_n, b_n}}) generated by the following infinite convolution product

$$begin{aligned} mu _{rho , {0, a_n, b_n}}=delta _{rho {0, a_1, b_1}} *delta _{rho ^2 {0, a_2, b_2}} *delta _{rho ^3 {0, a_3, b_3}} *cdots end{aligned}$$

in the weak convergence, where (delta _E=frac{1}{# E}sum _{e in E} delta _e) and (hbox {gcd}(a_n, b_n)=1) for all (n in {{mathbb {N}}}). In this paper, we show that (L^2(mu _{rho , {0, a_n, b_n}})) admits an exponential orthonormal basis if and only if (rho ^{-1} in 3{{mathbb {N}}}) and  ({a_n, b_n} equiv {1, 2} (mathrm {mod} 3)) for all (n in {{mathbb {N}}}).