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THE SPECTRAL EIGENVALUES OF A CLASS OF PRODUCT-FORM SELF-SIMILAR SPECTRAL MEASURE
- Part of
-
- Journal:
- Journal of the Australian Mathematical Society / Volume 119 / Issue 2 / October 2025
- Published online by Cambridge University Press:
- 21 July 2025, pp. 246-263
- Print publication:
- October 2025
-
- Article
- Export citation
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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 
$1\leq p_1<p_2<\cdots <p_s<q$. 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 
$\mu _{M,D}$.
