NON-SPECTRAL PROBLEM FOR CANTOR MEASURES

Xin-Rong Dai () and Meng Zhu
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Xin-Rong Dai: School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P. R. China
Meng Zhu: School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P. R. China

FRACTALS (fractals), 2021, vol. 29, issue 06, 1-12

Abstract: The spectral and non-spectral problems of measures have been considered in recent years. For the Cantor measure Î¼Ï , Hu and Lau [Spectral property of the Bernoulli convolutions, Adv. Math. 219(2) (2008) 554–567] showed that L2(μ Ï ) contains infinite orthogonal exponentials if and only if Ï becomes some type of binomial number. In this paper, we classify the spectral number of the Cantor measure Î¼Ï except the contraction ratio Ï being some algebraic numbers called odd-trinomial number. When Ï 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 Ï . 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.

Keywords: Spectral Number; Cantor Measures; Odd-Trinomial Number (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S0218348X21501577

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