On spectral Cantor-Moran measures and a variant of Bourgain's sum of sine problem

In this paper, we show that if we have a sequence of Hadamard triples $\{(N_n,B_n,L_n)\}$ with $B_n\subset \{0,1,..,N_n-1\}$ for $n=1,2,...$, except an extreme case, then the associated Cantor-Moran measure $$ \begin{aligned} \mu = \mu(N_n,B_n) =& \delta_{\frac{1}{N_1}B_1}\ast\delta_{\frac{1}{N_1N_2}B_2}\ast \delta_{\frac{1}{N_1N_2N_3}B_3}\ast...\\ =& \mu_n\ast\mu_{>n} \end{aligned} $$ with support inside $[0,1]$ always admits an exponential orthonormal basis $E(\Lambda) = \{e^{2\pi i \lambda x}:\lambda\in\Lambda\}$ for $L^2(\mu)$, where $\Lambda$ is obtained from suitably modifying $L_n$. Here, $\mu_n$ is the convolution of the first $n$ Dirac measures and $\mu_{>n}$ denotes the tail-term. We show that the completeness of $E(\Lambda)$ in general depends on the ``equi-positivity" of the sequence of the pull-backed tail of the Cantor-Moran measure $\nu_{>n}(\cdot) = \mu_{>n}((N_1...N_n)^{-1}(\cdot))$. Such equi-positivity can be analyzed by the integral periodic zero set of the weak limit of $\{\nu_{>n}\}$. This result offers a new conceptual understanding of the completeness of exponential functions and it improves significantly many partial results studied by recent research, whose focus has been specifically on $\#B_n\le 4$. Using the Bourgain's example that a sum of sine can be asymptotically small, we shows that, in the extreme case, there exists some Cantor-Moran measure such that the equi-positive condition fails and the Fourier transform of the associated $\nu_{>n}$ uniformly converges on some unbounded set.