  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.

Classical Analysis and ODEs;Functional Analysis;Probability