NON-SPECTRAL PROBLEM FOR CANTOR MEASURES

Abstract

The spectral and non-spectral problems of measures have been considered in recent years. For the Cantor measure [Formula: see text], Hu and Lau [Spectral property of the [Formula: see text]ernoulli convolutions, Adv. Math. 219(2) (2008) 554–567] showed that [Formula: see text] contains infinite orthogonal exponentials if and only if [Formula: see text] becomes some type of binomial number. In this paper, we classify the spectral number of the Cantor measure [Formula: see text] except the contraction ratio [Formula: see text] being some algebraic numbers called odd-trinomial number. When [Formula: see text] 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 [Formula: see text]. 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.