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 of {0, 1,.., N-n- 1} for n = 1, 2,..., except an extreme case, then the associated Cantor Moran measure mu = mu(N-n, B-n) =delta 1/N1B1 * delta 1/N1N2B2 * delta 1/N(1)N(22)N(3)B3 * ... =mu(n) * mu> n 0 with support inside [0, 1] always admits an exponential orthonormal basis E(Lambda) = {e(2 pi i lambda x) : lambda is an element of Lambda} : A E Al for L-2(mu), where Lambda is obtained from suitably modifying L-n. Here, mu(n) is the convolution of the first 71 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 v>n(.) = mu.n ((N-1 ... N-n)(-1) (.)).Such equi-positivity can be analyzed by the integral periodic zero set of the weak limit of {v>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 <= 4. Using the Bourgain's example that a sum of sine can be asymptotically small, we show 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 v > n uniformly converges on some unbounded set. Published by Elsevier Inc.