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

Abstract

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 $$\begin{array}{rl}\mu =\mu (N_n,B_n)=& {\delta }_{\frac{1}{N_1}{B}_1}\ast {\delta }_{\frac{1}{N_1{N}_2}{B}_2}\ast {\delta }_{\frac{1}{N_1{N}_2{N}_3}{B}_3}\ast ...\\ =& {\mu }_n\ast {\mu }_{>n}\end{array}$$ 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.