Abstract:

It is well known that the self-similar measure $\mu_{k, a, b}$ defined by $\mu_{k, a, b}(\cdot)=\frac{1}{3}\sum\limits_{i=0}^{2}\mu_{k, a, b}(3k(\cdot)-ki)$ is a spectral measure with a spectrum $\Lambda(3k, C)=\left\{\sum\limits_{j=0}^{{\rm finite}}(3k)^jc_j:c_j\in C=\{0, 1, 2\}\right\}.$ In this paper, by applying the properties of congruences and the order of elements in the finite group, we obtain some conditions on the integer $p$ such that the set $p\Lambda(3k, C)$ is also a spectrum for $\mu_{k, a, b}$. Moreover, an example is given to explain our theory.

Key words: Self-similar measure, Spectral measures, Spectral eigenvalue, Multiple spectrum