Let $\mu$ be the probability measure induced by $S=\sum _{i=1}^{\mathrm{\infty }}{3}^{-i}{X}_i$, where $X_1,X_2,\dots$ are independent identically distributed random variables each taking integer values $0,1,a$ with equal probability $1/3$, where $a$ is a parameter. Let $\alpha (s,a)$ (resp. $\underset{―}{\alpha }(s,a)$, $\bar{\alpha }(s,a)$) denote the local dimension (resp. lower, upper local dimension) of $s\in \mathrm{supp}\mu$, and let

$$E(a)=\{\alpha :\alpha (s,a)=\alpha \text{ for some }s\in \mathrm{supp}\mu \},$$

$$\bar{\alpha }=sup\{\bar{\alpha }(s,a):s\in \mathrm{supp}\mu \};\underset{―}{\alpha }(a)=inf\{\underset{―}{\alpha }(s,a):s\in \mathrm{supp}\mu \}.$$

 In this paper, we prove that for $a=4$ we have 

$$\bar{\alpha }(4)=1,\underset{―}{\alpha }(4)=11-{\displaystyle \frac{\log (1+\sqrt{5})-\log 2}{\log 3}}\text{ and }E=[\underset{―}{\alpha }(4),\bar{\alpha }(4)].$$