The Spectra of Cantor-Type Measures with Consecutive Digits

• Published in: Bulletin of the Malaysian Mathematical Sciences Society Vol. 46; no. 4
• Main Authors: Zeng, Sai-Nan, Ai, Wen-Hui, Chen, Jia-Long
• Format: Journal Article
• Language: English
• Published: Singapore Springer Nature Singapore 01.07.2023; Springer Nature B.V
• Subjects: Applications of Mathematics; Integers; Mathematics; Mathematics and Statistics; Number theory; Prime numbers; Prime decomposition; Spectral measure; 11A51; Extreme cycle; 42C30; 28A80; Scaling spectra; Cantor-type set
• ISSN: 0126-6705; 2180-4206
• DOI: 10.1007/s40840-023-01518-x

For integers p , b ≥ 2 , suppose that D = 0 , 1 , . . . , b - 1 is a consecutive digit set. It’s noted that the Cantor measure μ p b , D is spectral with a spectrum Λ p b , p D = ∑ j = 0 finite p b j d j : d j ∈ p D . By building the connection with number theory, we aim to explore the conditions of the integer τ under which the scaling set τ Λ p b , p D is also the spectrum of μ p b , D . If so, we call τ complete. In particular, for prime numbers τ , τ 1 , τ 2 , . . . , τ m and τ i > p b - 1 , we investigate the sufficient conditions that the power of τ coprime to pb is complete and the power of τ 1 τ 2 · · · τ m is complete. Furthermore, when an integer τ coprime to b is incomplete while every proper divisor of it is complete, we call τ primitive. So we obtain some properties and a criteria for the primitive number.