Dimension of homogeneous iterated function systems with algebraic translations

Let μ be the self-similar measure associated with a homogeneous iterated function system Φ = {λ x + t_j }_j=1^m on R and a probability vector (p_j)_j=1^m, where 0≠λ∈ (-1,1) and t_j∈ R. Recently by modifying the arguments of Varjú (2019), Rapaport and Varjú (2024) showed that if t_1,…, t_m are rational numbers and 0<λ<1, then μ =min{ 1, ∑_j=1^m p_jlog p_j/log |λ| } unless Φ has exact overlaps. In this paper, we further show that the above equality holds in the case when t_1,…, t_m are algebraic numbers and 0<|λ|<1. This is done by adapting and extending the ideas employed in the recent papers of Breuillard, Rapaport and Varjú.