Year: 2025
Type: article
Abstract: Let $\mu _{M,D}$ be the self-similar measure generated by $M=RN^q$ and the product-form digit set $D=\{0,1,\ldots ,N-1\}\oplus N^{p_1}\{0,1,\ldots ,N-1\}\oplus \cdots \oplus N^{p_s}\{0,1,\ldots ,N-1\}$ , where $R\geq 2$ , $N\geq 2$ , q , $p_i(1\leq i\leq s)$ are integers with $\gcd (R,N)=1$ and $1\leq p_1 . In this paper, we first show that $\mu _{M,D}$ is a spectral measure with a model spectrum $\Lambda $ . Then, we completely settle two types of spectral eigenvalue problems for $\mu _{M,D}$ . In the first case, for a real t , we give a necessary and sufficient condition under which $t\Lambda $ is also a spectrum of $\mu _{M,D}$ . In the second case, we characterize all possible real numbers t such that $\Lambda '\subset \mathbb {R}$ and $t\Lambda '$ are both spectra of $\mu _{M,D}$ .
Authors: XIAO-YU YAN, WEN-HUI AI
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G3409788273, G8928941908
OpenAlex ID: W4412505791