Let $\mu$ be the probability measure induced by $S=\sum_{i=0}^{\infty}q^{-i}X_i$, where $m \geqslant q \geqslant 2$ are integers and $X_0, X_1, \dots$ a sequence of independent identically distributed random variables each taking integer values $0, 1, \dots, m$ with equal probability $p = 1/(m + 1)$. Let $\alpha(s)$ (resp. $\alpha_*(s), \alpha^*(s)$) denote the local dimension (resp. lower, upper local dimension) of $s \in \mathrm{supp}\mu$, and let $$\alpha^*=\sup\{\alpha^*(s):s\in\mathrm{supp}\mu\}; \alpha_*=\inf\{\alpha_*(s):s\in \mathrm{supp}\mu\}.$$ We show that $$\alpha_*=\dfrac{\log(m+1)-\log(r+\sqrt{r^2+4(l+1)})+\log 2}{\log q}$$ for $rq\leqslant m < rq + r$; $r = 1,\dots, q − 1$ and $l = m − rq$.

The special case of our result, $m = rq$ $(l = 0)$, was obtained earlier in [6].