NON-SPECTRAL PROBLEM FOR CANTOR MEASURES.

The spectral and non-spectral problems of measures have been considered in recent years. For the Cantor measure μ ρ , Hu and Lau [Spectral property of the B ernoulli convolutions, Adv. Math.219(2) (2008) 554–567] showed that L 2 (μ ρ) contains infinite orthogonal exponentials if and only if ρ becomes some type of binomial number. In this paper, we classify the spectral number of the Cantor measure μ ρ except the contraction ratio ρ being some algebraic numbers called odd-trinomial number. When ρ is an odd-trinomial number, we provide an exponential and polynomial estimations of the upper bound of the spectral number related to the algebraic degree of ρ. Some examples on odd-trinomial number via generalized Fibonacci numbers are provided such that the spectral number of them can be determined. Our study involves techniques from polynomial theory, especially the decomposition theory on trinomial. [ABSTRACT FROM AUTHOR]