Multiple spectra of self-similar measures with three digits on $$\mathbb{R}$$

Let $$q\ge3$$ be a positive integer, and let $$D=\{0,a,b\}$$ with $$\gcd(a,b)=1$$ . For the self-similar measure $$\mu_{q,D}$$ which is generated by the iterated function system $$\{\tau_d(x)=\frac{x+d}{q}\}_{d\in D}$$ , it is well known that if $$q$$ is divisible by 3 and $$\{a,b\}\equiv\{1,2\}\pmod 3$$ , $$\mu_{q,D}$$ is a spectral measure with a spectrum $$\Lambda(q,C)=\Big\{\sum_{j=0}^{n}q^ja_j:a_j\in C=\{0,q/3,2q/3\} {\rm and} n \in \mathbb{N} \Big\}.$$ In this paper, we study some positive integers $$p$$ such that $$p\Lambda(q,C)$$ is also a spectrum of $$\mu_{q,D}$$ .