Dimension of homogeneous iterated function systems with algebraic translations

Let $\mu$ be the self-similar measure associated with a homogeneous iterated function system $\Phi = \{ \lambda x + t_j \}_{j=1}^m$ on ${\Bbb R}$ and a probability vector $(p_{j})_{j=1}^m$, where $0\neq \lambda\in (-1,1)$ and $t_j\in {\Bbb R}$. Recently by modifying the arguments of Varj\'u (2019), Rapaport and Varj\'u (2024) showed that if $t_1,\ldots, t_m$ are rational numbers and $0<\lambda<1$, then $$\dim \mu =\min\Big \{ 1, \; \frac{\sum_{j=1}^m p_{j}\log p_{j}}{ \log |\lambda| }\Big\}$$ unless $\Phi$ has exact overlaps. In this paper, we further show that the above equality holds in the case when $t_1,\ldots, t_m$ are algebraic numbers and $0<|\lambda|<1$. This is done by adapting and extending the ideas employed in the recent papers of Breuillard, Rapaport and Varj\'u.