THE SPECTRAL EIGENVALUES OF A CLASS OF PRODUCT-FORM SELF-SIMILAR SPECTRAL MEASURE

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 $1\leq p_1<p_2<\cdots <p_s<q$ . 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 $\mu _{M,D}$ .