Dimension of homogeneous iterated function systems with algebraic translations

Abstract

Let $\hat{I}¼$ be the self-similar measure associated with a homogeneous iterated function system $\hat{I}¦=\{\hat{I}»x+t_j{\}}_{j=1}^m$ on $\mathbb{R}$ and a probability vector $(p_j{)}_{j=1}^m$, where $0\ne \hat{I}»\in (-1,1)$ and $t_j\in \mathbb{R}$. Recently by modifying the arguments of Varjú (2019), Rapaport and Varjú (2024) showed that if $t_1,\dots ,t_m$ are rational numbers and $0<\hat{I}»<1$, then $$\dim \hat{I}¼=\min \{1,\frac{{\sum }_{j=1}^m{p}_j\log p_j}{\log |\hat{I}»|}\}$$ unless $\hat{I}¦$ has exact overlaps. In this paper, we further show that the above equality holds in the case when $t_1,\dots ,t_m$ are algebraic numbers and $0<|\hat{I}»|<1$. This is done by adapting and extending the ideas employed in the recent papers of Breuillard, Rapaport and Varjú.