### Spectrality of infinite convolutions with three-element digit sets

Authors and Corporations: Spectrality of infinite convolutions with three-element digit sets Monatshefte für Mathematik, 183, 2017, 3, p. 465-485 Springer Vienna 465-485 0026-9255 1436-5081 1436-5081 Let $$0< \rho <1$$ 0<ρ<1[/itex] and let $$\{a_n, b_n\}_{n=1}^\infty$$ {an,bn}n=1[/itex] be a sequence of integers with bounded from upper and lower. Associated with them there exists a unique Borel probability measure $$\mu _{\rho , \{0, a_n, b_n\}}$$ μρ,{0,an,bn}[/itex] 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} μρ,{0,an,bn}=δρ{0,a1,b1}δρ2{0,a2,b2}δρ3{0,a3,b3}[/itex] in the weak convergence, where $$\delta _E=\frac{1}{\# E}\sum _{e \in E} \delta _e$$ δE=1#EeEδe[/itex] and $$\hbox {gcd}(a_n, b_n)=1$$ gcd(an,bn)=1[/itex] for all $$n \in {{\mathbb {N}}}$$ nN[/itex] . In this paper, we show that $$L^2(\mu _{\rho , \{0, a_n, b_n\}})$$ L2(μρ,{0,an,bn})[/itex] admits an exponential orthonormal basis if and only if $$\rho ^{-1} \in 3{{\mathbb {N}}}$$ ρ-13N[/itex] and  $$\{a_n, b_n\} \equiv \{1, 2\} \ (\mathrm {mod} \ 3)$$ {an,bn}{1,2}(mod3)[/itex] for all $$n \in {{\mathbb {N}}}$$ nN[/itex] . E-Article Springer Journals English