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 ⊂ { 0 , 1 , . . , N n − 1 } for n = 1 , 2 , . . . , except an extreme case, then the associated Cantor-Moran measure μ = μ ( N n , B n ) = δ 1 N 1 B 1 ⁎ δ 1 N 1 N 2 B 2 ⁎ δ 1 N 1 N 2 N 3 B 3 ⁎ . . . = μ n ⁎ μ > n with support inside [ 0 , 1 ] always admits an exponential orthonormal basis E ( Λ ) = { e 2 π i λ x : λ ∈ Λ } for L 2 ( μ ) , where Λ is obtained from suitably modifying L n . Here, μ n is the convolution of the first n Dirac measures and μ > n denotes the tail-term. We show that the completeness of E ( Λ ) in general depends on the “equi-positivity” of the sequence of the pull-backed tail of the Cantor-Moran measure ν > n ( ⋅ ) = μ > n ( ( N 1 . . . N n ) − 1 ( ⋅ ) ) . Such equi-positivity can be analyzed by the integral periodic zero set of the weak limit of { ν > 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 ν > n uniformly converges on some unbounded set.