Infinite orthogonal exponentials for a class of Moran measures

For a sequence {M-n}(infinity)(n=1) of expanding matrices with M-n is an element of M-d(Z) and a sequence {D-n}(infinity)(n=1) of finite digit sets with D-n subset of Z(d), the Moran measure mu({Mn},{Dn}) is defined by the infinite convolution mu({Mn},{Dn}) = delta(M-11D1) * delta(M-11M-12D2) * delta(M-11M-12M-13D3) * ... and the convergence is in the weak sense. Under some additional assumptions, we show that L-2(mu({Mn},{Dn})) contains an infinite orthogonal set of exponential functions if and only if there exists an infinite subsequence {n(k)}(infinity)(k=1) of {n}(infinity)(n=1) such that (M*Mnk+1*(nk+2) ... M*(nk+1) Z(nk+1)) boolean AND Z(d) not equal theta for any k is an element of N+, where Z(nk+1) := {x : Sigma(alpha is an element of Dnk+1) e(2 pi i <alpha,x >) = 0} boolean AND [0, 1)(d). This extends the results of [J. L. Li, A necessary and sufficient condition for the finite mu(M,D)-orthogonality, Sci. China Math. 58 (2015) 2541-2548].