Let $\mu$ be the probability measure induced by $S=\sum_{i=1}^{\infty}3^{-i}X_i$, 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. $\underline{\alpha}(s,a)$, $\overline{\alpha}(s,a)$) denote the local dimension (resp. lower, upper local dimension) of $s\in\mathrm{supp}\mu$, and let

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

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

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

$$\overline{\alpha}(4)=1, \underline{\alpha}(4)=11-\dfrac{\log(1+\sqrt{5})-\log 2}{\log 3} \text{ and } E=[\underline{\alpha}(4),\overline{\alpha}(4)].$$