The spectral and non-spectral problems of measures have been considered in recent years. For the Cantor measure μρ, Hu and Lau [Spectral property of the Bernoulli convolutions, Adv. Math. 219(2) (2008) 554–567] showed that L2(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 μρ 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.

中文翻译：

---

CANTOR 测量的非谱问题

近年来已经考虑了测量的光谱和非光谱问题。对于康托尔测量μρ, Hu 和 Lau [光谱性质的乙埃努利卷积，进阶。数学。 219(2) (2008) 554–567] 表明大号2(μ ρ)包含无限正交指数当且仅当ρ变成某种类型的二项式数。在本文中，我们对康托测度的谱数进行分类μρ除了收缩率ρ是一些称为奇三项数的代数数。什么时候ρ是奇三项式数，我们提供了与代数度相关的谱数上限的指数和多项式估计ρ. 提供了一些关于通过广义斐波那契数的奇三项式数的示例，以便可以确定它们的谱数。我们的研究涉及多项式理论的技术，尤其是关于三项式的分解理论。